research paper on indices

What Is a Journal Index, and Why is Indexation Important?

• Research Process
• Peer Review

A journal index, or a list of journals organized by discipline, subject, region and other factors, can be used by other researchers to search for studies and data on certain topics. As an author, publishing your research in an indexed journal increases the credibility and visibility of your work. Here we help you to understand journal indexing better - as well as benefit from it.

Updated on May 13, 2022

A journal index, also called a ‘bibliographic index' or ‘bibliographic database', is a list of journals organized by discipline, subject, region or other factors.

Journal indexes can be used to search for studies and data on certain topics. Both scholars and the general public can search journal indexes.

Journals in indexes have been reviewed to ensure they meet certain criteria. These criteria may include:

• Ethics and peer review policies
• Assessment criteria for submitted articles
• Editorial board transparency

What is a journal index?

Indexed journals are important, because they are often considered to be of higher scientific quality than non-indexed journals. You should aim for publication in an indexed journal for this reason. AJE's Journal Guide journal selection tool can help you find one.

Journal indexes are created by different organizations, such as:

• Public bodies- For example, PubMed is maintained by the United States National Library of Medicine. PubMed is the largest index for biomedical publications.
• Analytic companies- For example: the Web of Science Core Collection is maintained by Clarivate Analytics. The WOS Core Collection includes journals indexed in the following sub-indexes: (1) Science Citation Index Expanded (SCIE); (2) Social Sciences Citation Index (SSCI); (3) Arts & Humanities Citation Index (AHCI); (4) Emerging Sources Citation Index.
• Publishers- For example, Scopus is owned by Elsevier and maintained by the Scopus Content Selection and Advisory Board . Scopus includes journals in all disciplines, but the majority are science and technology journals.

Key types of journal indexes

You can choose from a range of journal indexes. Some are broad and are considered “general indexes”. Others are specific to certain fields and are considered “specialized indexes”.

For example:

• The Science Citation Index Expanded includes mostly science and technology journals
• The Arts & Humanities Citation Index includes mostly arts and humanities journals
• PubMed includes mostly biomedical journals
• The Emerging Sources Citation Index includes journals in all disciplines

Which index you choose will depend on your research subject area.

Some indexes, such as Web of Science , include journals from many countries. Others, such as the Chinese Academy of Science indexing system , are specific to certain countries or regions.

Choosing the type of index may depend on factors such as university or grant requirements.

Some indexes are open to the public, while others require a subscription. Many people searching for research papers will start with free search engines, such as Google Scholar , or free journal indexes, such as the Web of Science Master Journal List . Publishing in a journal in one or more free indexes increases the chance of your article being seen.

Journals in subscription-based indexes are generally considered high-quality journals. If the status of the journal is important, choose a journal in one or more subscription-based indexes.

Most journals belong to more than one index. To improve the visibility and impact of your article, choose a journal featured in multiple indexes.

How does journal indexing work?

All journals are checked for certain criteria before being added to an index. Each index has its own set of rules, but basic publishing standards include the following:

• An International Standard Serial Number (ISSN). ISSNs are unique to each journal and indicate that the journal publishes issues on a recurring basis.
• An established publishing schedule.
• Digital Object Identifiers (DOIs) . DOIs are unique letter/number codes assigned to digital objects. The benefit of a DOI is that it will never change, unlike a website link.
• Copyright requirements. A copyright policy helps protect your work and outlines the rules for the use or sharing of your work, whether it's copyrighted or has some form of creative commons licensing .
• Other requirements can include conflict of interest statements, ethical approval statements, an editorial board listed on the website, and published peer review policies.

To be included in an index, a journal must submit an application and undergo an audit by the indexation board. Index board members (called auditors) will confirm certain information, such as the full listing of the editorial board on the website, the inclusion of ethics statements in published articles, established appeal and retraction processes, and more.

Why is journal indexing important?

As an author, publishing your research in an indexed journal increases the credibility and visibility of your work. Indexed journals are generally considered to be of higher scientific quality than non-indexed journals.

With the growth of fully open access journals and online-only journals, recognizing “predatory” journals and their publishers has become difficult. Indexing a journal in one or more well-known databases is a good sign the journal is credible.

Moreover, more and more institutions are requiring publication in an indexed journal as a requirement for graduation, promotion, or grant funding.

As an author, it is important to ensure that your research is seen by as many eyes as possible. Index databases are often the first places scholars and the public will search for specific information. Publishing a paper in a non-indexed journal could be harmful in this context.

However, there are some exceptions, such as medical case reports.

Many journals don't accept medical case reports because they don't have high citation rates. However, several primary and secondary journals have been created specifically for case reports. Examples include the primary journal, BMC Medical Case Reports, and the secondary journal, European Heart Journal - Case Reports.

While many of these journals are indexed, they may not be indexed in the major indexes, though they are still highly acceptable journals.

Open access and indexation

With the recent increase in open access publishing, many journals have started offering an open access option. Other journals are completely open access, meaning they do not offer a traditional subscription service.

Open access journals have many benefits, such as:

• High visibility. Anyone can access and read your paper.
• Publication speed. It is generally quicker to post an article online than to publish it in a traditional journal format.

Identifying credible open access journals

Open access has made it easier for predatory journal publishers to attract unsuspecting or new authors. These predatory journal publishers often publish any article for a fee without peer review and with questionable ethical and copyright policies. Here we show you eight ways to spot predatory open access journals .

One way to identify credible open access journals is their index status. However, be aware that some predatory journals will falsely list indexes or display logos on their website. It is good practice to make sure the journal is indexed on the index's website before submitting your article to that journal.

Major journal indexing services

There are several journal indexes out there. Some of the most popular indexes are as follows:

Life Sciences and Hard Sciences

• Science Citation Index Expanded (SCIE) Master Journal List
• Engineering Index
• Web of Science (now published by Clarivate Analytics, formerly by ISI and Thomson Reuters)
• Chinese Academy of Sciences (CAS)

Humanities and Social Sciences

• Arts & Humanities Citation Index (AHCI) Master Journal List
• Social Sciences Citation Index (SSCI) Master Journal List

Indexation and impact factors

It is easy to assume that indexed journals will have higher impact factors, but indexation and impact factor are unrelated.

Many credible journals don't have impact factors, but they are indexed in several well-known indexes. Therefore, the lack of an impact factor may not accurately represent the credibility of a journal.

Of course, impact factors may be important for other reasons, such as institutional requirements or grant funding. Read this authoritative piece on the uses, importance, and limitations of impact factors .

Final Thoughts

Selecting an indexed journal is an important part of the publication journey. Indexation can tell you a lot about a journal. Publishing in an indexed journal can increase the visibility and credibility of your research. If you're having trouble selecting a journal for publication, consider learning more about AJE's journal recommendation service .

Catherine Zettel Nalen, MS

Academic Editor, Specialist, and Journal Recommendation Team Lead

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On the Methodological Framework of Composite Indices: A Review of the Issues of Weighting, Aggregation, and Robustness

• Open access
• Published: 17 January 2018
• Volume 141 , pages 61–94, ( 2019 )

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• Salvatore Greco 1 , 2 ,
• Alessio Ishizaka 2 ,
• Menelaos Tasiou   ORCID: orcid.org/0000-0001-7700-450X 3 &
• Gianpiero Torrisi   ORCID: orcid.org/0000-0003-4497-2365 1 , 3  

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In recent times, composite indicators have gained astounding popularity in a wide variety of research areas. Their adoption by global institutions has further captured the attention of the media and policymakers around the globe, and their number of applications has surged ever since. This increase in their popularity has solicited a plethora of methodological contributions in response to the substantial criticism surrounding their underlying framework. In this paper, we put composite indicators under the spotlight, examining the wide variety of methodological approaches in existence. In this way, we offer a more recent outlook on the advances made in this field over the past years. Despite the large sequence of steps required in the construction of composite indicators, we focus particularly on two of them, namely weighting and aggregation. We find that these are where the paramount criticism appears and where a promising future lies. Finally, we review the last step of the robustness analysis that follows their construction, to which less attention has been paid despite its importance. Overall, this study aims to provide both academics and practitioners in the field of composite indices with a synopsis of the choices available alongside their recent advances.

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1 Introduction

In the past decades, we have witnessed an enormous upsurge in available information, the extent and use of which are characterised by the founder of the World Economic Forum as the ‘Fourth Industrial Revolution’ (Schwab 2016 , para. 2). While Schwab focuses on the use and future impact of these data—ranging from policy and business analysis to artificial intelligence—one of the key underlying points is that this enormous and exponential increase in available information hides another issue: the need for its interpretation and consolidation. Indeed, an ever-increasing variety of information, broadly speaking in the form of indicators, increases the difficulty involved in interpreting a complex system. To illustrate this, consider for example a phenomenon like well-being. In principle, it is a very complex concept that is particularly difficult to capture with only a single indicator (Decancq and Lugo 2013 ; Decancq and Schokkaert 2016 ; Patrizii et al. 2017 ). Hence, one should enlarge the range of indicators to encompass all the necessary information on a matter that is generally multidimensional in nature (Greco et al. 2016 ). However, in such a case, it would be very difficult for the public to understand ‘well-being’ by, say, identifying common trends among several individual indices. They would understand a complex concept more easily in the form of a sole number that encompasses this plethora of indicators (Saltelli 2007 ). Reasonably, this argument may raise more questions than it might answer. For instance, how would this number be produced? Which aspects of a concept would it encompass? How would they be aggregated into the form of a simple interpretation for the public and so on? This issue, and the questions that it raises, introduce the concept of ‘composite indicators’.

Defining ‘composite’ (sometimes also encountered as ‘synthetic’) indicators should be a straightforward task given their widespread use nowadays. Even though it appears that there is no single official definition to explain this concept, the literature provides a wide variety of definitions. According to the European Commission’s first state-of-the-art report (Saisana and Tarantola 2002 , p. 5), composite indicators are ‘[…] based on sub-indicators that have no common meaningful unit of measurement and there is no obvious way of weighting these sub-indicators’. Freudenberg ( 2003 , p. 5) identifies composite indicators as ‘synthetic indices of multiple individual indicators’. Another potential definition provided by the OECD’s first handbook for constructing composite indicators (Nardo et al. 2005 , p. 8) is that a composite indicator ‘[…] is formed when individual indicators are compiled into a single index, on the basis of an underlying model of the multi-dimensional concept that is being measured’. This list of definitions could continue indefinitely. By pooling them together, a common pattern emerges and relates to the central idea of the landmark work of Rosen ( 1991 ). Essentially, a composite indicator might reflect a ‘complex system’ that consists of numerous ‘components’, making it easier to understand in full rather than reducing it back to its ‘spare parts’. Although this ‘complexity’, from a biologist’s viewpoint, refers to the causal impact that organisations exert on the system as a whole, the intended meaning here is astonishingly appropriate for the aim of composite indicators. After all, Rosen asserts that this ‘complexity’ is a universal and interdisciplinary feature.

Despite their vague definition, composite indicators have gained astounding popularity in all areas of research. From social aspects to governance and the environment, the number of their applications is constantly growing at a rapid pace (Bandura 2005 , 2008 , 2011 ). For instance, Bandura ( 2011 ) identifies over 400 official composite indices that rank or assess a country according to some economic, political, social, or environmental measures. In a complementary report by the United Nations’ Development Programme, Yang ( 2014 ) documents over 100 composite measures of human progress. While these inventories are far from being exhaustive—compared with the actual number of applications in existence—they give us a good understanding of the popularity of composite indicators. Moreover, a search for ‘composite indicators’ in SCOPUS, conducted in January 2017, shows this trend (see Fig.  1 ). The increase over the past 20 years is exponential, and the number of yearly publications shows no sign of a decline. Moreover, their widespread adoption by global institutions (e.g. the OECD, World Bank, EU, etc.) has gradually captured the attention of the media and policymakers around the globe (Saltelli 2007 ), while their simplicity has further strengthened the case for their adoption in several practices.

Results for ‘composite indicators’ on SCOPUS for the period 1997–2016

Nevertheless, composite indicators have not always been so popular, and there was a time when considerable criticism surrounded their use (Sharpe 2004 ). In fact, according to the author, their very existence was responsible for the creation of two camps in the literature: aggregators versus non - aggregators . In brief, Footnote 1 the first group supports the construction of synthetic indices to describe an overall complex phenomenon, while the latter opposes it, claiming that the final product is statistically meaningless. While it seems idealistic to assume that this debate will ever be resolved (Saisana et al. 2005 ), it quickly drew the attention of policymakers and the public. Sharpe ( 2004 ) describes the example of the Human Development Index (HDI), which has received a vast amount of criticism since its creation due to the arbitrariness of its methodological framework (Ray 2008 ). However, it is the most well-known composite index to date. Moreover, it led the 1998 Nobel Prize-winning economist A. K. Sen, once one of the main critics of aggregators, to change his position due to the attention that the HDI attracted and the debate that it fostered afterwards (Saltelli 2007 ). He characterised it as a ‘success’ that would not have happened in the case of non-aggregation (Sharpe 2004 , p. 11). Seemingly, this might be considered as the first win for the camp of aggregators. Nevertheless, the truth is that we are still far from settling the disputes and the criticism concerning the stages of the construction process (Saltelli 2007 ).

This is natural, as there are many stages in the construction process of a composite index and criticism could grow simultaneously regarding each of them (Booysen 2002 ). Moreover, if the procedure followed is not clear and reasonably justified to everyone, there is considerable room for manipulation of the outcome (Grupp and Mogee 2004 ; Grupp and Schubert 2010 ). Working towards a solution to this problem, the OECD ( 2008 , p. 15) identifies a ten-step process, namely a ‘checklist’. Its aim is to establish a common guideline as a basis for the development of composite indices and to enhance the transparency and the soundness of the process. Undeniably, this checklist aids the developer in gaining a better understanding of the benefits and drawbacks of each choice and overall in achieving the kind of coherency required in the steps of constructing a composite index. In practice, though, this hardly reduces the criticism that an index might receive. This is because, even if one does indeed achieve perfect coherency (from choosing the theoretical framework to developing the final composite index), there might still be certain drawbacks in the methodological framework itself.

The purpose of this study is to review the literature with respect to the methodological framework used to construct a composite index. While the existing literature contains a number of reviews of composite indicators, the vast majority particularly focuses on covering the applications for a specific discipline. To be more precise, several reviews of composite indicators’ applications exist in the fields of sustainability (Bohringer and Jochem 2007 ; Singh et al. 2009 , 2012 ; Pissourios 2013 ; Huang et al. 2015 ), the environment (Juwana et al. 2012 ; Wirehn et al. 2015 ), innovation (Grupp and Mogee 2004 ; Grupp and Schubert 2010 ), and tourism (Mendola and Volo 2017 ). However, the concept of composite indicators is interdisciplinary in nature, and it is applied to practically every area of research (Saisana and Tarantola 2002 ). Since the latest reviews on the methodological framework of composite indices were published a decade ago (Booysen 2002 ; Saisana and Tarantola 2002 ; Freudenberg 2003 ; Sharpe 2004 ; Nardo et al. 2005 ; OECD 2008 ) and a great number of new publications have appeared since then (see Fig.  1 ), we re-examine the literature focusing on the methodological framework of composite indicators and more specifically on the weighting, aggregation, and robustness steps. These steps are the focus of the paramount criticism as well as the recent development. In the following, Sect.  2 describes the weighting schemes found in the literature and Sect.  3 covers the step of aggregation. Section  4 provides an overview of the methods used for robustness checks following the construction of an index, and Sect.  5 contains a discussion and concluding remarks.

2 On the Weighting of Composite Indicators

The meaning of weighting in the construction of composite indicators is twofold (OECD 2008 , pp. 31–33). First, it refers to the ‘explicit importance’ that is attributed to every criterion in a composite index. More specifically, a weight may be considered as a kind of coefficient that is attached to a criterion, exhibiting its importance relative to the rest of the criteria. Second, it relates to the implicit importance of the attributes, as this is shown by the ‘trade-off’ between the pairs of criteria in an aggregation process. A more detailed description of the latter and the difference between these two meanings is presented in Sect.  3 , in which we describe the stage of aggregation and explain the distinction between ‘compensatory’ and ‘non-compensatory’ approaches.

Undeniably, the selection of weights might have a significant effect on the units ranked. For instance, Saisana et al. ( 2005 ) show that, in the case of the Technology Achievement Index, changing the weights of certain indicators seems to affect several of the units evaluated, especially those that are ranked in middle positions. Footnote 2 Grupp and Mogee ( 2004 ) and Grupp and Schubert ( 2010 , p. 69) present two further cases of science and technology indicators, for which the country rankings could significantly change or otherwise be ‘manipulated’ in the case of different weighting schemes. This is a huge challenge in the construction of a composite indicator, often referred to as the ‘index problem’ (Rawls 1971 ). Basically, even if we reach an agreement about the indicators that are to be used, the question that follows—and the most ‘pernicious’ one (Freudenberg 2003 )—is how a weighting scheme might be achieved. Although far from reaching a consensus (Cox et al. 1992 ), the literature tries to solve this puzzle in several ways. Before we venture further to analyse the weighting approaches in existence, we should first note that no weighting system is above criticism. Each approach has its benefits and drawbacks, and there is no ultimate case of a clear winner or a kind of ‘one-size-fits-all’ solution. On the contrary, it is up to the index developer to choose a weighting system that is best fitted to the purpose of the construction, as disclosed in the theoretical framework (see OECD 2008 , p. 22).

2.1 No or Equal Weights

As simple as it sounds, the first option is not to distribute any weights to the indicators, otherwise called an ‘attributes-based weighting system’ (see e.g. Slottje 1991 , pp. 686–688). This system may have two consequences. First, the overall score (index) could simply be the non-weighted arithmetic average of the normalised indicators (Booysen 2002 ; Singh et al. 2009 ; Karagiannis 2017 ). A common problem that appears here, though, is that of ‘double counting’ Footnote 3 (Freudenberg 2003 ; OECD 2008 ). Of course, this issue might partially be moderated by averaging the collinear indicators as well prior to their aggregation into a composite (Kao et al. 2008 ). The second alternative in the absence of weights is that the composite index is equal to the sum of the individual rankings that each unit obtains in each of the sub-indicators (e.g. see the Information and Communication Technologies Index in Saisana and Tarantola 2002 , p. 9). By relying solely on aggregating rankings, this approach fails to achieve the purpose of vastly improving the statistical information, as it does not benefit from the absolute level of information of the indicators (Saisana and Tarantola 2002 ).

Equal weighting is the most common scheme appearing in the development of composite indicators (Bandura 2008 ; OECD 2008 ). It is important to note here that the difference between distributing equal weights and not distributing weights at all (e.g. the ‘non-weighted arithmetic average’ discussed above) is that equal weighting schemes could be applied hierarchically. More specifically, if the indicators are grouped into a higher order (e.g. a dimension) and the weighting is distributed equally dimension-wise, then it does not necessarily mean that the individual indicators will have equal weights (OECD 2008 ). For instance, ISTAT ( 2015 ) provides the ‘BES’, a broad data set of 134 socio-economic indicators for the 20 Italian regions. These are unevenly grouped into 12 dimensions. If equal weights are applied to the highest hierarchy level (e.g. dimensions) a priori, then the sub-indicators are not weighted equally due to the different number of indices in each dimension. In general, there are various justifications for most applications choosing equal weights a priori. These include: (1) simplicity of construction, (2) a lack of theoretical structure to justify a differential weighting scheme, (3) no agreement between decision makers, (4) inadequate statistical and/or empirical knowledge, and, finally, (5) alleged objectivity (see Freudenberg 2003 ; OECD 2008 ; Maggino and Ruviglioni 2009 ; Decancq and Lugo 2013 ). Nevertheless, it is often found that equal weighting is not adequately justified (Greco et al. 2017 ). For instance, choosing equal weights due to the ‘simplicity of the construction’, Footnote 4 instead of an alternative scheme that is based on a proper theoretical and methodological framework, bears a huge oversimplification cost, especially in certain aggregation schemes (Paruolo et al. 2013 ). Furthermore, we could argue that, conceptually, equal weights miss the point of differentiating between essential and less important indicators by treating them all equally. In any case, the co-operation of experts and the public in an open debate might resolve the majority of the aforementioned justifications (Freudenberg 2003 ). Finally, considering equal weights as an ‘objective’ technique (relative to the ‘subjective’ exercise of a developer who sets the weights arbitrarily) is far from being undisputable. Quoting Chowdhury and Squire ( 2006 , p. 762), setting weights to be equal ‘[seems] obviously convenient, but also universally considered to be wrong’. Ray ( 2008 , p. 5) and Mikulić et al. ( 2015 ) claim that equal weighting is not only wrong—as it does not convey the realistic image—but also an equally ‘subjective judgement’ to other arbitrary weighting schemes in existence. This last argument prepares the scene for the consideration of a plurality of weighting systems, mainly related to the representation of the preferences of a ‘plurality of individuals’ (see e.g. Greco et al. 2017 ).

2.2 Plurality of Weighting Systems

Understandably, the decision maker could choose from a range of weighting schemes, depending on the structure and quality of the data or her beliefs. More specifically, in the first case, higher weighting could be assigned to indicators with broader coverage (as opposed to those with multiple cases of treated missing data) or those taken from more trustworthy sources, as a way to account for the quality of the indicators (Freudenberg 2003 ). However, an issue here is that this could result in a ‘biased selection’ in favour of proxies that are not able to identify and capture properly the information desired to measure (see e.g. Custance and Hillier 1998 , pp. 284–285; OECD 2008 , p. 32). Moreover, indicators should be chosen carefully a priori and according to a conceptual and quality framework (OECD 2008 ). Otherwise, a ‘garbage in–garbage out’ outcome may be produced (Funtowicz and Ravetz 1990 ), which in this case is that of a composite indicator reflecting ‘insincere’ dimensions in relation to those desired (Munda 2005a ). When the weighting scheme is chosen by the developer of an index, naturally this means that it is conceived as ‘subjective’, since it relies purely on the developer’s perceptions (Booysen 2002 ). There are several participatory approaches in the literature to make this subjective exercise as transparent as possible. These involve a single or several stakeholders deciding on the weighting scheme to be chosen. Stakeholders could be expert analysts, policymakers, or even citizens to whom policies are addressed. From a social viewpoint, the combination of all of them in an open debate could be an ideal approach theoretically (Munda 2005b , 2007 ), Footnote 5 but it is only viable if a well-defined framework for a national policy exists (OECD 2008 ). Indeed, if one could imagine a framework on which policies will be based, enlarging the set of decision makers to include all the participants’ preferences is probably the desired outcome (Munda 2005a ). However, if the objective is not well defined or the number of indicators is very large and it is probably impossible to reach a consensus about their importance, this procedure could result in an endless debate and disagreement between the participants (Saisana and Tarantola 2002 ). Moreover, if the objective involves an international comparison, in which no doubt the problem is significantly enlarged, common ground is even harder to achieve or simply ‘inconsistent outcomes’ may be produced (OECD 2008 , p. 32). For instance, one country’s most important objective could be different from another country’s (e.g. economy vs environment). In general, participatory methods are seen as a conventional way for transparent and subjective judgements, and they could be effective and of great use when they fulfil the aforementioned requirements. However, since these techniques may yield alternative weighting schemes (Saisana et al. 2005 ), Footnote 6 one should carefully choose the most suitable according to their properties, of which we provide a brief overview in the following subsections.

2.2.1 Budget Allocation Process

In the budget allocation process (BAP), a set of chosen decision makers (e.g. a panel of experts) is given ‘ n ’ points to distribute to the indicators, or groups of indicators (e.g. dimensions), and then an average of the experts’ choices is used (Jesinghaus 1997 ). Footnote 7 Two prerequisites are the careful selection of the group of experts and the total number of indicators that will be evaluated. A rule of thumb is to have fewer than 10 indicators so that the approach is optimally executed cognitively. Otherwise, problems of inconsistency could be introduced (Saisana and Tarantola 2002 ). The BAP is used for estimating the weights in one of the Economic Freedom Indices (Gwartney et al. 1996 ) and by the European Commission (JRC) for the creation of the ‘e-Business Readiness Index’ (Pennoni et al. 2005 ) and the ‘Internal Market Index’ (Tarantola et al. 2004 ). Moreover, several studies in the literature use this method; for the most recent see, for example, Hermans et al. ( 2008 ), Couralet et al. ( 2011 ), Zhou et al. ( 2012 ), and Dur and Yigitcanlar ( 2015 ). A specific issue with the BAP arises during the process of indicator comparison. Decision makers might be led to ‘circular thinking’ (see e.g. Saisana et al. 2005 , p. 314), the probability of which increases with the number of indicators to be evaluated. Circular thinking is both moderated and verifiable in the analytic hierarchy process (AHP), which is discussed in the following subsection.

2.2.2 Analytic Hierarchy Process

Originally introduced by Saaty in the 1970s (Saaty 1977 , 1980 ), the AHP translates a complex problem into a hierarchy consisting of three levels: the ultimate goal, the criteria, and the alternatives (Ishizaka and Nemery 2013 , pp. 13–14). Experts have to assign the importance of each criterion relative to the others. More specifically, pairwise comparisons among criteria are carried out by the decision makers. These are expressed on an ordinal scale with nine levels, ranging from ‘equally important’ to ‘much more important’, representing how many times more important one criterion is than another one. Footnote 8 The weights elicited with the AHP are less prone to errors of judgement, as discussed in the previous subsection. This happens because, in addition to setting the weights relatively, a consistency measure is introduced (namely the ‘inconsistency ratio’), assessing the cognitive intuition of decision makers in the pairwise comparison setting (OECD 2008 ). Despite its popularity as a technique to elicit weights (Singh et al. 2007 ; Hermans et al. 2008 ), it still suffers from the same problem as the BAP (Saisana and Tarantola 2002 ). That is, on the occasion that the number of indicators is very large, it exerts cognitive stress on decision makers, which in the AHP is amplified due to the pairwise comparisons required (Ishizaka 2012 ).

2.2.3 Conjoint Analysis

Conjoint analysis (CA) is commonly encountered in consumer research and marketing (Green et al. 2001 ; OECD 2008 ; Wind and Green 2013 ), but applications in the field of composite indices follow suit, mainly in the case of quality-of-life indicators (Ülengin et al. 2001 , 2002 ; Malkina-Pykh and Pykh 2008 ). CA is a disaggregation method. It could be seen as the exact opposite of the AHP, as it moves from the overall priority to determining the weight of the criteria. More specifically, the model first seeks the preferences of individuals (e.g. experts or the public) regarding a set of alternatives (e.g. countries, firms, or products) and then decomposes them according to the individual indicators. Theoretically, the indicators’ weights are obtained via the calculation of the marginal rates of substitution of the overall probability function. Footnote 9 In practice we can derive the importance of a criterion by dividing the range of importance of that criterion in the respondent’s opinion by the total sum of ranges of all the criteria (Maggino and Ruviglioni 2009 ). While it might seem easier to obtain a preference estimation of the ultimate objective first and then search for the importance of its determinants (in contrast to the AHP), CA carries alternative limitations. Its major drawbacks are its overall complexity, the requirement of a large sample, and an overall pre-specified utility function, which is very difficult to estimate (OECD 2008 ; Wind and Green 2013 ).

What one might derive from the above section is that participatory techniques are helpful tools overall. They make the subjectivity behind the process of weighting the indicators controllable and, most importantly, transparent. In fact, this whole act of gathering a panel consisting of experts, policymakers, or even citizens, who will mutually decide on the importance of the factors at stake, is a natural and desired behaviour in a society (Munda 2005a ). Nevertheless, it is rather difficult to apply in contexts in which the phenomena to be measured are not well defined and/or the number of underlying indicators is very large. These approaches then stop being consistent, and they ultimately become unmanageable and ineffective. What is more, in the case in which the participatory audience does not clearly understand a framework (e.g. to evaluate the importance of an indicator/phenomenon or what it actually represents), these methods would lead to biased results (OECD 2008 ).

2.3 Data-Driven Weights

In the aftermath of participatory approaches, this ‘subjectivity’ behind the arbitrariness in decision makers’ weight selection is dismissed by other statistical methods that claim to be more ‘objective’. Footnote 10 This property is increasingly claimed to be desirable in the choice of weights (Ray 2008 ), thus stirring up interest in approaches like correlation analysis or regression analysis, principal component analysis (PCA) or factor analysis (FA), and data envelopment analysis (DEA) models and their variations. These so-called ‘data-driven techniques’ (Decancq and Lugo 2013 , p. 19), as their name suggests, emerge from the data themselves under a specific mathematical function. Therefore, it is often argued that they potentially do not suffer from the aforementioned problems of ‘manipulation’ of the results (Grupp and Mogee 2004 , p. 1382) and the subjective, direct weighting exercise of various decision makers (Ray 2008 , p. 9). However, these approaches bear a different kind of criticism, deeply rooted in the core of their philosophy. More specifically, Decancq and Lugo ( 2013 , p. 9) distinguish these techniques from the aforementioned ones based on the ‘is–ought’ distinction that is found in the work of a notable philosopher of the eighteenth century, David Hume. In the authors’ words: ‘it is impossible to derive a statement about values from a statement about facts’ (p. 9). In other words, they claim that one should be very cautious in deriving the importance of a concept (e.g. indicator/dimension) based on what the data ‘consider’ to be a fact, as this appears to be the ‘is’ that we observe but not the ‘ought’ that we are seeking. After all, statistical relationships between indicators—for example in the form of correlation—do not always represent the actual influence between them (Saisana and Tarantola 2002 ). This appears to be one side of the criticism that these approaches receive, and it is related to the philosophical aspect underlying their use. Further criticism appearing in the literature is focused on their specific properties, which we will examine individually in the following subsections.

2.3.1 Correlation Analysis

Correlation analysis is mostly used in the first steps of the construction process to examine the structure and the dynamics of the indicators in the data set (Booysen 2002 ; OECD 2008 ). For instance, it might determine a very strong correlation between two sub-indicators within a dimension, which, depending on the school of thought (e.g. see Saisana et al. 2005 , p. 314), may then be moderated by accounting for it in the weighting step (OECD 2008 ; Maggino and Ruviglioni 2009 ). Nevertheless, this approach might still serve as a tool to obtain objective weights (Ray 2008 ). According to the author, there are two ways in which weights might be elicited using correlation analysis. The first is based on a simple correlation matrix, with the indicator weights being proportional to the sum of the absolute values of that row or column, respectively. In the second method, known as ‘capacity of information’ (Hellwig 1969 ; Ray 1989 ), first the developer chooses a distinctive variable in the data set that, according to the author, plays the role of an endogenous criterion. Then the developer computes the correlation of each indicator with that distinctive variable. These correlation coefficients are used to determine the weights of the indicators, with those having the highest correlation accordingly gaining the highest weights. More specifically, an indicator’s weight is given by the ratio of the squared correlation coefficient of that indicator with the distinctive variable to the sum of the squared correlation coefficients of the rest of the indicators with that variable (Ray 2008 ). One issue with both the aforementioned uses is that the correlation could be statistically insignificant. Moreover, even if statistical significance applies, it does not imply causality but rather shows a similar or opposite co-movement between indicators (Freudenberg 2003 ; OECD 2008 ).

2.3.2 Multiple Linear Regression Analysis

Multiple linear regression analysis is another approach through which weights can be elicited. By moving beyond simple statistical correlation, the decision maker is able to explore the causal link between the sub-indicators and a chosen output indicator. However, this raises two concerns that the developer must bear in mind. First, these models assume strict linearity, which is hardly the norm with composite indices (Saisana et al. 2005 ). Second, if there was an objective and effective output measure for the sub-indicators to be regressed on, there would not be a need for a composite index in the first place (Saisana and Tarantola 2002 ). With respect to the latter, according to the authors, an indicator that is generally assumed to capture the wider phenomenon to be studied might be used. For instance, in the National Innovative Capacity Index (Porter and Stern 2001 ), the dependent variable used in the regression analysis is the log of patents. The authors argue that this is a broadly accepted variable in the literature, as it sufficiently captures the levels of innovation in a country. In the absence of such a specific indicator, the gross national product per capita could serve as a more generalised variable (Ray 2008 ), as it is often linked to most socio-economic aspects that a composite index might be aiming to measure. However, that would dismiss the whole momentum that composite indicators have gained by refraining from following the common approach of solely economic output (Costanza et al. 2009 ; Stiglitz et al. 2009 ; Decancq and Schokkaert 2016 ; Patrizii et al. 2017 ). Finally, in the case in which a developer has multiple such output variables, canonical correlation analysis could be used, which is a generalisation of the previous case (see e.g. Saisana and Tarantola 2002 , p. 53).

2.3.3 Principal Component Analysis and Factor Analysis

Principal component analysis (PCA; Pearson 1901 ) and factor analysis (FA; Spearman 1904 ) are statistical approaches with the aim of reductionism. More specifically, the core of their philosophy is to capture the highest variance possible in the original variables (standardised for this purpose) with as few components as possible (Ram 1982 ). In PCA the original data may be described by a series of equations, as many as the number of indicators. These equations essentially represent linear transformations of the original data, constructed in such a way that the maximum variance of the original variables is explained with the first equation, the second-highest variance (which is not explained by the first equation) is explained by the second equation, and so on. In FA the outcome is rather similar, but the idea is somewhat different. Here the original data supposedly depend on underlying common and specific factors, which can possibly explain the variance in the original data set. FA is slightly more complex than PCA in the sense that it involves an additional step, in which a choice has to be made by the developer (e.g. the choice of an extraction method). Finally, for both PCA and FA, certain choices must be made by the decision maker; hence, subjectivity is introduced to a certain degree. These choices involve the number of components/factors to be retained or the rotation method to be used. Nonetheless, several criteria or rules of thumb exist in the literature for each of the two approaches to facilitate the proper choice (e.g. see OECD 2008 , pp. 66–67 and p. 70).

In general, there are several applications using FA or PCA to elicit the weights for the indicators, especially in the context of well-being and poverty. Footnote 11 One of the first applications is that of Ram ( 1982 ), using PCA in the case of a physical quality-of-life indicator, followed by Noorbakhsh ( 1996 ), who uses PCA to weigh the components of HDI. Naturally, further applications follow suit both in the literature (Klasen 2000 ; McGillivray 2005 ; Dreher 2006 ) and in official indicators provided by large organisations (e.g. the Internal Market Index, Science and Technology Indicator, and Business Climate Indicator, see Saisana and Tarantola 2002 ; the Environmental Degradation Index, see Bandura 2008 ). The standard procedure in using PCA as a weight elicitation technique is to use the factor loadings of the first component to serve as weights for the indicators (Greyling and Tregenna 2016 ). However, sometimes the first component alone is not adequate to explain a large portion of the variance of the indicators; thus, more components are needed. Nicoletti et al. ( 2000 ) develop indicators of product market regulation, illustrating how these can be accomplished using FA. The authors use PCA as the extraction method and rotate the components with the varimax technique, in this way minimising the number of indicators with high loadings on each component. By considering the factor loadings of all the retained factors (see Nicoletti et al. 2000 , pp. 19–22), this allows the preservation of the largest proportion of the variation in the original data set.

This method is frequently used in composite indicators produced by large organisations (e.g. the Business Climate Indicator, Relative Intensity of Regional Problems in the Community, and General Indicator of Science and Technology, see Saisana and Tarantola 2002 ) and can be found in several studies in the literature (Mariano and Murasawa 2003 ; Gupta 2008 ; Hermans et al. 2008 ; Ediger and Berk 2011 ; Salvati and Carlucci 2014 ; Riedler et al. 2015 ; Li et al. 2016 ; Tapia et al. 2017 ). However, according to Saisana and Tarantola ( 2002 ), the use of these approaches is not feasible in certain cases, due to either negative weights assigned (e.g. the Environmental Sustainability Index) or a very low correlation among the indicators (e.g. synthetic environmental indices). Finally, PCA can be used for cases in which the elicitation of weights is not the main goal. For instance, Ogwang and Abdou ( 2003 ) review the use of these models in selecting the ‘principal variables’. More specifically, PCA/FA could be used to select a single or a subset of variables to include in the construction of a composite index that can explain the variation of the overall data set adequately. Thus, they could serve as an aiding tool, enabling the developer to gain a better understanding of the dimensionality in the considered phenomenon or the structure of the indicators accordingly.

Understandably, these approaches might seem popular (e.g. with respect to their use in the literature) and convenient (e.g. with respect to the objectivity and transparency in their process). Nevertheless, it is important to note a few issues relating to their use at this point. First, property-wise, the use of PCA/FA involves the assumptions of having continuous indicators and a linear relationship among them. In the case in which these assumptions do not hold, the use of non-linear PCA (or otherwise categorical PCA; CATPCA) is suggested (see e.g. Greyling and Tregenna 2016 , p. 893). Second, the nature and philosophy of these approaches rely on the statistical properties of the data, which can be seen as both an advantage and a drawback. For instance, this reductionism could be proven to be very useful in some cases in which problems of ‘double counting’ exist. On the other hand, if there is no correlation between the indicators or the variation of a variable is very small, these techniques might even fail to work. Footnote 12 Furthermore, the weights that are assigned endogenously by PCA/FA do not necessarily correspond to the actual linkages among the indicators, particularly statistical ones (Saisana and Tarantola 2002 ). Therefore, one should be cautious about how to interpret these weights and especially about the extent to which one might use these methods, as the truth is that they do not necessarily reflect a sound theoretical framework (De Muro et al. 2011 ). Additionally, a general issue with both these approaches is that they are sensitive to the construction of the data. More specifically, if, in an evaluation exercise using PCA/FA, several units are added or subtracted afterwards (especially outliers), this may significantly change the weights that are used to construct the overall index (Nicoletti et al. 2000 ). However, this issue is addressed with robust variations of PCA (e.g. see Ruymgaart 1981 ; Li and Chen 1985 ; Hubert et al. 2005 ). Finally, with the obtained weights being inconsistent over time and space, the comparison might eventually prove to be very difficult (De Muro et al. 2011 , p. 6).

2.3.4 Data Envelopment Analysis (DEA)

Originally developed by Charnes et al. ( 1978 ), DEA uses mathematical programming to measure the relative performance of several units (e.g. businesses, institutions, countries, etc.), and hence to evaluate them, based on a so-called ‘efficiency’ score (see Cooper et al. 2000 ). This score is obtained by a ratio (the weighted sum of outputs to the weighted sum of inputs) that is computed for every unit under a minimisation/maximisation function set by the developer. From this linear programming formulation, a set of weights (one for each unit) is endogenously determined in such a way as to maximise their ‘efficiency’ under some given constraints (Hermans et al. 2008 ). According to Mahlberg and Obersteiner ( 2001 , in Despotis 2005a , p. 970), the first authors to propose the use of DEA in the HDI context, this approach constitutes a more realistic application, because each country is ‘benchmarked against best practice countries’. In the context of composite indicators, the classic DEA formulation is adjusted, as usually all the indicators are treated as outputs, thereby considering no inputs (see Hermans et al. 2008 ). Therefore, the denominator of the abovementioned ratio—that is, the weighted inputs of the units—comprises a dummy variable equal to one, whereas the nominator—that is, the weighted outputs—comprises a weighted sum of the indicators that forms the overall composite index (Yang et al. 2017 ). In this field, this model is mostly referred to as the classic ‘benefit-of-the-doubt’ approach (Cherchye 2001 ; Cherchye et al. 2004 , 2007 ), originally introduced by Melyn and Moesen ( 1991 ) in a context of macroeconomic evaluation.

Due to the desirable properties of the endogenously calculated differential weighting, applications in the literature follow suit (e.g. Takamura and Tone 2003 ; Despotis 2005a ; Murias et al. 2006 ; Zhou et al. 2007 ; Cherchye et al. 2008 ; Hermans et al. 2008 ; Antonio and Martin 2012 ; Gaaloul and Khalfallah 2014 ; Martin et al. 2017 ). Indeed, the differential weighting scheme between units (e.g. countries) is potentially a desirable property for policymakers, because each unit chooses its own weights in such a way as to maximise its performance. Footnote 13 Thus, any potential conflicts, for example the chosen weights not favouring any unit, are in fact dismissed (Yang et al. 2017 ). This is a key reason for the huge success of this approach (Cherchye et al. 2007 , 2008 ). To understand this argument better, one may consider the following example of two countries. Let us imagine that these countries have different policy goals for different areas (e.g. economy vs environment); thus, each spends its resources accordingly. Potentially, they could perform better in different areas precisely for that particular reason. Therefore, in a weighting exercise, each country would choose to weigh significantly higher those exact dimensions on which it performs better to reflect that effect. However, this argument is criticised for the following reasons. First, on a theoretical basis, this approach dismisses one of the three basic requirements in social choice theory, which acts as a response to Arrow’s theorem (Arrow 1963 ): ‘neutrality’. In brief, neutrality states that ‘all alternatives (e.g. countries) must be treated equally’ (OECD 2008 , p. 105). Footnote 14 Second, if we indeed accept that each unit could declare its own preferences in the weighting process, for example according to the different policies that they follow (Cherchye et al. 2007 , 2008 )—thus entirely dismissing the ‘neutrality’ principle—another problem that arises in the process is related to the calculation of these weights. More specifically, consider an example of a DEA approach, in which the desired output is the maximisation of the value of the composite index from each unit’s perspective. Executing this technique with the basic constraints (e.g. see Despotis 2005a , or Cherchye et al. 2007 ) will probably result in all the weighting capacity being assigned to the indicator with the highest value (e.g. see Hermans et al. 2008 , pp. 1340–1341). Furthermore, since these DEA models are output-maximised, holding the unitary input constant, it often occurs that, in the absence of further constraints, after the maximisation/minimisation process, a multiplicity of equilibria is introduced (Fusco 2015 , p. 622). Meanwhile, the majority of the units evaluated will be deemed to be efficient (e.g. they are assigned a value equal to ‘1’) Footnote 15 (Zhou et al. 2007 ; Decancq and Lugo 2013 ; Yang et al. 2017 ).

A simple solution to this problem is for more constraints to be placed by the decision maker, controlling, for instance, the lower and upper bounds of the weights of each indicator or group of indicators (e.g. dimensions). Footnote 16 For instance, Hermans et al. ( 2008 ) ask a panel of experts to assign weights to several indicators, using their opinions as binding constraints on the weights to be chosen by the DEA model. In the absence of information on such restrictions, the classic BoD model could be transformed into a ‘pessimistic’ one (Zhou et al. 2007 ; Rogge 2012 ). More specifically, while the classic BoD model finds the most favourable weights for each unit, the ‘pessimistic’ BoD model finds the least favourable weights. They are afterwards combined (either by a weighted or by a non-weighted average) to form a single, final index score (Zhou et al. 2007 ). There are several other methods in the literature Footnote 17 that deal with the issue of adjusting the discrimination in BoD models, the most popular being the super-efficiency (Andersen and Petersen 1993 ), cross-efficiency (Sexton et al. 1986 ; Doyle and Green 1994 ; Green et al. 1996 ), PCA-DEA (Adler and Yazhemsky 2010 ), and DEA entropy (Nissi and Sarra 2016 ) models.

Another issue with most BoD models regards the differential weighting inherent in the process. The beneficial weights obtained by the model prove to be a challenge when comparability among the units is at stake. More specifically, each unit has a different set of weights, making it difficult to compare them by simply looking at the overall score. For this reason, a number of techniques exist in the literature that arrive at a common weighting scheme (e.g. see, among others, Despotis 2005a ; Hatefi and Torabi 2010 ; Kao 2010 ; Morais and Camanho 2011 ; Sun et al. 2013 ). Of course, this rather decreases the desirability of this method—that of favourable weights in the eyes of policymakers—based on which this approach gained such momentum in the first place (Decancq and Lugo 2013 ).

Finally, we will discuss some recent developments in this area regarding the function or type of aggregation. More specifically, with respect to the aggregation function, while the classic BoD model is often specified as a weighted sum, recent studies present multiplicative forms merely to account for the issue of complete compensation, as it is introduced in the basic model of the weighted sum (e.g. see Blancas et al. 2013 ; Giambona and Vassallo 2014 ; Tofallis 2014 ; van Puyenbroeck and Rogge 2017 ). With respect to the type of aggregation, Rogge ( 2017 ), based on an earlier work of Färe and Zelenyuk ( 2003 ), Footnote 18 puts forward the idea of aggregating individual composite indicators into groups of composite indices. According to the author, one could be interested in analysing the performance of a cluster of individual units (e.g. groups of countries) rather than simply examining the units themselves. After the individual units’ performance is determined through classic BoD, a second aggregation takes place, again through BoD, but this time the indicators are the scores of countries, obtained in the previous step, and the weights reflect the shares of units in the aggregate form.

3 On the Aggregation of Composite Indicators

Weighting the indicators naturally leads to the final step in forming a composite index: ‘aggregation’. According to the latest handbook on constructing composite indices, aggregation methods may be divided into three distinctive categories: linear , geometric , and multi - criteria (see OECD 2008 , p. 31, Table 4). However, this division might send a somewhat misleading message, since all these methods are included in the multi-criteria decision analysis framework. Footnote 19 Another distinctive categorisation of the aggregation methods in the literature would be that of choosing between ‘compensatory’ and ‘non-compensatory’ approaches (Munda 2005b ). As we highlighted at the beginning of the previous section, the interpretation of the weights could be twofold: ‘trade-offs’ or ‘importance coefficients’. Footnote 20 The choice of the proper annotation, though, essentially boils down to the choice of the proper aggregation method (Munda 2005a , p. 118; OECD 2008 , p. 33). Quoting the latter: ‘To ensure that weights remain a measure of importance, other aggregation methods should be used, in particular methods that do not allow compensability’. In other words, ‘compensability’ is inseparably connected with the term ‘trade-off’ (and vice versa), and, as a result, its very definition is presented as such (Bouyssou 1986 ). According to the author (p. 151): ‘A preference relation is non-compensatory if no trade-offs occur and is compensatory otherwise. The definition of compensation therefore boils down to that of a trade-off’. Consequently, according to the latter categorisation of aggregation approaches (i.e. that of ‘compensatory’ and ‘non-compensatory’), the linear Footnote 21 and geometric Footnote 22 aggregation schemes lie within the ‘compensatory’ aggregation scheme, while the ‘non-compensatory’ aggregation scheme contains other multi-criteria approaches, considering preferential relationships from the pairwise comparisons of the indicators (e.g. see OECD 2008 , pp. 112–113). Similar to the issue of a non-existent perfect weighting scheme, there is no such thing as a ‘perfect aggregation’ scheme (Arrow 1963 ; Arrow and Raynaud 1986 ). Each approach is mostly fit for a different purpose and involves some benefits and drawbacks accordingly. In the following two subsections, we provide a brief overview of this situation by analysing the two aggregation settings and their properties, respectively.

3.1 Compensatory Aggregation

Among the compensatory aggregation approaches, the linear one is the most commonly used in composite indicators (Saisana and Tarantola 2002 ; Freudenberg 2003 ; OECD 2008 ; Bandura 2008 , 2011 ). Two general issues must be considered in this additive utility-based approach. The first is that it assumes ‘preferential independence’ among indicators (OECD 2008 , p. 103; Fusco 2015 , p. 621), something that is conceptually considered as a very strong assumption to make (Ting 1971 ). Second, there is a chasm between the two perceptions of weights, translated into importance measures and trade-offs. More specifically, if one sets the weights by considering them as importance measures for the indicators, one will soon find that this is far from actually happening in this aggregation setting, and this situation is the norm rather than the exception (Anderson and Zalinski 1988 ; Munda and Nardo 2005 ; Billaut et al. 2010 ; Rowley et al. 2012 ; Paruolo et al. 2013 ). Quoting the latter (p. 611): ‘This gives rise to a paradox, of weights being perceived by users as reflecting the importance of a variable, where this perception can be grossly off the mark’. This happens because the weights in this setting should be perceived as trade-offs between pairs of indicators and therefore assigned as such from the very beginning. Decancq and Lugo ( 2013 ) stress this point by showing how weights in this setting express the marginal rates of substitution among pairs of indicators. Understandably, this trade-off implies constant compensability between indicators and dimensions; thus, a unit could compensate for the loss in one dimension with a gain in another (OECD 2008 ; Munda and Nardo 2009 ). This, however, is far from desirable in certain cases. For instance, Munda ( 2012 , p. 338) considers an example of a hypothetical sustainability index, in which economic growth could compensate for a loss in the environmental dimension in the case of a compensatory approach. Of course, this argument could easily be extended to applications in other socio-economic areas, Footnote 23 albeit with the following point: constant compensation is always assumed in linear aggregation at the rate of substitution among pairs of indicators (e.g. w a / w b ) (Decancq and Lugo 2013 , p. 17). That is something that should be taken into consideration at the very beginning of the construction stage, the theoretical framework (OECD 2008 ).

One partial solution to that issue could be to use geometric aggregation instead. This approach is adopted when the developer of an index prefers only ‘some’ degree of compensability (OECD 2008 , p. 32). While linear aggregation assumes constant trade-offs for all cases, geometric aggregation offers inferior compensability for indices with lower values (diminishing returns) (van Puyenbroeck and Rogge 2017 ). This makes it far more appealing in a benchmarking exercise in which, for instance, regions with lower scores in a given dimension will not be able to compensate fully in other dimensions (Greco et al. 2017 ). Moreover, the same regions could be even more motivated to increase their lower scores, as the marginal increase in these indicators will be much higher in contrast to regions that already achieve high scores (Munda and Nardo 2005 ). Therefore, under these circumstances, a switch from linear to geometric aggregation could even be considered both appealing and more realistic. One such case is that of probably the most well-known composite index to date, the Human Development Index (HDI). Having received paramount criticism (Desai 1991 ; Sagar and Najam 1998 ; Chowdhury and Squire 2006 ; Ray 2008 ; Davies 2009 ), the developers of the HDI switched the aggregation function from linear to geometric in 2010, addressing one of their main methodological criticisms. More specifically, in their yearly report (UNDP 2010 , p. 216), they state the following: ‘It thus addresses one of the most serious criticisms of the linear aggregation formula, which allowed for perfect substitution across dimensions’. There is no doubt that, compared with the linear type of aggregation, geometric is the solid first step towards a solution to the issue of an index’s compensability. In fact, it is argued that, under such circumstances, it provides more meaningful results (see e.g. Ebert and Welsch 2004 ). However, this still appears to be only a partial solution or a ‘trade-off’ between compensatory and non-compensatory techniques (Zhou et al. 2010 , p. 171). Therefore, if complete ‘inelasticity’ of compensation, or the meaning of weights to be interpreted solely as ‘importance coefficients’, is the actual objective of a composite index, a non-compensatory approach is ideal and strongly suggested to be reconsidered (Paruolo et al. 2013 , p. 632).

3.2 Non-compensatory Aggregation

Non-compensatory aggregation techniques (Vansnick 1990 ; Vincke 1992 ; Roy 1996 ) are mainly based on ELECTRE methods (see e.g. Figueira et al. 2013 , 2016 ) and PROMETHEE methods (Brans and Vincke 1985 ; Brans and De Smet 2016 ). Given the weights for each criterion (interpreted as ‘importance coefficients’ in this exercise) and some other preference parameters (e.g. indifference, preference, and veto thresholds), the mathematical aggregation is divided into the following steps: (1) ‘pair-wise comparison of units according to the whole set of indicators’ and (2) ‘ranking of units in a partial, or complete pre-order’ (Munda and Nardo 2009 , p. 1516). The first step creates the ‘outranking matrix’ Footnote 24 (Roy and Vincke 1984 ), which essentially discloses the pairwise comparisons of the alternatives (e.g. countries) for each criterion (Munda and Nardo 2009 ). Moving to the second step (i.e. the exploitation procedure of the outranking matrix), an approach must be selected regarding the proper aggregation. The exploitation procedures can mainly be divided into the Condorcet- and the Borda-type approach (Munda and Nardo 2003 ). These two are radically different Footnote 25 and as such yield different results (Fishburn 1973 ). Moulin ( 1988 ) argues that the Borda-type approach is ideal when just one alternative should be chosen. Otherwise, the Condorcet-type approach is the most ‘consistent’ and thus the most preferable for ranking the considered alternatives (Munda and Nardo 2003 , p. 10). A big issue with the Condorcet approach, though, is that of the presence of cycles, Footnote 26 the probability of which increases with both the number of criteria and the number of alternatives to be evaluated (Fishburn 1973 ). A large amount of work has been carried out with the aim of providing solutions to this issue (Kemeny 1959 ; Young and Levenglick 1978 ; Young 1988 ). A ‘satisfying’ one is for the ranking of alternatives to be obtained according to the maximum likelihood principle, Footnote 27 which essentially chooses as the final ranking the one with the ‘maximum pair-wise support’ (Munda 2012 , p. 345). While this approach enjoys ‘remarkable properties’ (Saari and Merlin 2000 , p. 404), one drawback is that it is computationally costly, making it unmanageable when the number of alternatives increases considerably (Munda 2012 ). Nevertheless, the C–K–Y–L approach is of great use for the concept of a non-compensatory aggregation scheme, and it could be used as a solid alternative solution to the common practice of linear aggregation schemes. Munda ( 2012 ) applies this approach to the case of the Environmental Sustainability Index (ESI), produced by Yale University and Columbia University in collaboration with the World Economic Forum and the European Commission (Joint Research Centre). According to the author, there are noticeable differences in the rankings between the two approaches (linear and non-compensatory), mostly apparent in the countries ranked among the middle positions and less apparent among those ranked first or last.

Despite its desirable properties, judging from the number of applications existing in this literature, the non-compensatory multi-criteria approach (NCMCA) is not met hugely popular. This could be attributed to the simplicity of construction of other methods (e.g. linear or geometric aggregation) or the issue of being computationally costly to calculate. Furthermore, NCMCA approaches are so far used to provide the developer with a ranking of the units evaluated; thus, one can only follow the rankings through time (Saltelli et al. 2005 , p. 364), swapping the absolute level of information in possession with an ordinal scale. Despite these drawbacks, Paruolo et al. ( 2013 , p. 631) urge developers to reflect on the cost of oversimplification that other techniques bear (e.g. linear), and, whenever possible, to use NCMC approaches, in which the weights exhibit the actual importance of the criteria. Otherwise, the authors suggest that the developers of an index should inform the audience to which the index is targeted that, in the other settings (e.g. linear or geometric aggregation), weights express the relative importance of the indicators (trade-offs) and not the nominal ones that were originally assigned.

3.3 Mixed Strategies

Owing to the unresolved issues of choosing a weighting and an aggregation approach, several methodologies appear in the literature, dealing with these steps in different manners. These methodologies are hybrid in the sense that they do not particularly fit into one category or the other both weighting- and aggregation-wise. This is because they use a combination of different approaches to solving the aforementioned issues. These are discussed further below.

3.3.1 Mazziotta–Pareto Index (MPI)

The Mazziotta–Pareto Index (MPI), originally introduced in 2007 (Mazziotta and Pareto 2007 ), aims to produce a composite index that penalises substitutability among the indicators, as this is introduced in the case of linear aggregation. More specifically, in linear aggregation a unit that performs very well in one indicator can offset a poor performance in another, proportionally to the ratio of their weights. In the MPI this is addressed by adding (subtracting) a component to (from) a non-weighted arithmetic mean (depending on the direction of the index), designed in such a way as to penalise this unbalance between the indicators (De Muro et al. 2011 ). This component, usually referred to as a ‘penalty’, is equal to a multiplication term of the unit’s standard deviation and the coefficient of variation among its indicators. Essentially, what the authors aim for is a simplistic methodology calculation-wise that favours not only a high-performing unit on average (as in the linear aggregation) but also a consistent one throughout all the indicators. Due to the desirability of simplicity, the MPI’s use of the arithmetic mean still bears the cost of compensability regarding aggregation. Nevertheless, one could argue that it is fairly adjusted to account for the unbalance among the indicators with its ‘penalty’ component. A newer variant of the index allows for the ‘absolute assessment’ of the units over time (Mazziotta and Pareto 2016 , p. 989). To achieve this, the authors change the normalisation method from a modified z-score to a rescaling of the original variables according to two policy ‘goalposts’. These are a minimum and a maximum value that accordingly represent the potential range to be covered by each indicator in a certain period. In this way the normalised indicators exhibit absolute changes over time instead of the relative changes that are captured by the standardisation approach used in their previous model. As an illustrative application, the authors measure the well-being of the OECD countries in 2011 and 2014.

3.3.2 Penalty for a Bottleneck

Working towards the creation of the Global Entrepreneurship and Development Index, Ács et al. ( 2014 ) present a novel methodology in the field of composite indices, known as the ‘penalty for a bottleneck’. Although different from the MPI methodologically, their approach is conceptually in line with penalising the unbalances when producing the overall index. This penalisation is achieved by ‘correcting’ the sub-indicators prior to the aggregation stage. More specifically, a component of an exponential function adjusts all the sub-indicators according to the overall weakest-performing indicator (minimum value) of that unit (otherwise described as a ‘bottleneck’). After the unbalance-adjusted indicators have been computed, a non-weighted arithmetic mean is used to construct the final index. In this way the complete compensability, as introduced in the linear aggregation setting, is significantly reduced. However, an issue raised here by the authors is that the amount of the ‘penalty’ adjustment is in fact unknown, as it depends on each data set and on the presence or otherwise of any outliers in an indicator’s value. This is something that, as they state, also implies that the solution is not always optimal. Despite the original development of this approach towards the measurement of national innovation and entrepreneurship at the country level, the authors claim that this methodology can be extended to the evaluation of any unit and for any discipline beyond innovation.

3.3.3 Mean–Min Function

The mean–min function, developed by Tarabusi and Guarini ( 2013 ), is another approach working towards the penalisation of the unbalances in the construction of a composite index. What the authors aim to achieve is an intermediate but controllable case between the zero penalisation of the arithmetic mean and the maximum penalisation of the min function. Footnote 28 To achieve this, they start with the non-weighted arithmetic average—as in the case of the MPI—from which they subtract a penalty component. This comprises the difference between the arithmetic average and the min function, interacted with two variables, 0 ≤ α ≤ 1 and β ≥ 0, to control the amount of penalisation intended by the developer. For α = 0, the equation is reduced back to the arithmetic average, while, for α = 1 (and β = 0), it is reduced back to the min function. Therefore, ‘ β ’ can be seen as a coefficient that determines the compensability between the arithmetic mean and the min function. One issue that is potentially encountered here, though, is that of the subjectivity, or even ignorance, behind the control of penalisation. In other words, what should the values of ‘ α ’ and ‘ β ’ be to determine the proper penalisation intended? The authors suggest that, in the case of standardised variables, a reasonable value could be that of α = β = 1, as this introduces progressive compensability.

3.3.4 ZD Model

The ZD model is developed by Yang et al. ( 2017 ) for an ongoing project of the Taiwan Institute of Economic Research. The core idea behind it is inspired by the well-known Z-score, in which the mean stands as a reference point, with values lower (higher) than it exhibiting worse (better) performance. Similarly, a virtual unit (e.g. country, region, or firm) is constructed in such a way as to perform equally to the average of each indicator to be used as such a reference point. The evaluation of the units is attained by a DEA-like model and thus presented in the form of an ‘efficiency’ score. More specifically, this score is obtained by minimising the sum of the differences between the units that are above average and those that are below average. In this way a common set of weights is achieved for all the units, which exhibits the smallest total difference between the relative performance of the unit evaluated and that of the average. The limitations of this approach are the same as those appearing in the rest of the DEA-like models in the literature, as described in Sect.  2.3.4 .

3.3.5 Directional Benefit-of-the-Doubt (BoD)

Directional BoD, introduced by Fusco ( 2015 ), is another approach using a DEA-like model for the construction of composite indicators. According to the author, one of the main drawbacks of the classic BoD model (see Sect.  2.3.4 ) is that it still assumes complete compensability among the indicators. This is attributed to the nature of the linear aggregation setting. To overcome this issue, Fusco ( 2015 ) suggests including a ‘directional penalty’ in the classic BoD model by using the directional distance function introduced by Chambers et al. ( 1998 ). To obtain the direction, ‘ g ’, the slope of the first principal component is used. The output’s (viz. the overall index) distance to the frontier is then evaluated, and the directional BoD estimator is obtained by solving a simple linear problem. According to the author, there is one limitation to this approach regarding the methodological framework. The overall index scores obtained with this approach are sensitive to outliers, as both the DEA and the PCA approach that are used suffer from this drawback. To moderate this issue, robust frontier and PCA techniques could be used instead (see Fusco 2015 , p. 629).

4 On the Robustness of Composite Indicators

Composite indicators involve a long sequence of steps that need to be followed meticulously. There is no doubt that ‘incompatible’ or ‘naive’ choices (i.e. without knowing the actual consequences) in the steps of weighting and aggregation may result in a ‘meaningless’ synthetic measure. However, in such a case, the developer is inevitably compelled to draw wrong conclusions from it. This is one of the indicators’ main drawbacks and needs extreme caution (Saisana and Tarantola 2002 ), especially when indices are used in policy practices (Saltelli 2007 ). One example of such a case is presented by Billaut et al. ( 2010 ). The authors examine the ‘Shanghai Ranking’, a composite index used to rank the best 500 universities in the world. They claim that, despite the paramount criticism that this index receives in the literature (regarding both its theoretical and its methodological framework), it attracts such interest in the academic and policymaking communities that policies are designed on behalf of the latter, heavily influenced by the ranking of the index. However, if the construction of an index fully neglects the aggregation techniques’ properties, it ‘vitiates’ the whole purpose of evaluation and eventually shows a distorted picture of reality (Billaut et al. 2010 , p. 260). Indeed, a misspecified aggregate measure may radically alter the results, and drawing conclusions from it is inadvisable in policy practices (Saltelli 2007 ; OECD 2008 ).

Regardless of the composite’s objective (e.g. serving as a tool for policymakers or otherwise), these aggregate measures ought to be tested for their robustness as a whole (OECD 2008 ). This will act as a ‘quality assurance’ tool that illustrates how sensitive the index is to changes in the steps followed to construct it and will highly reduce the possibilities to convey a misleading message (Saisana et al. 2005 ). Despite its importance, robustness analysis is often found to be completely missing for the vast majority of the composite indices (OECD 2008 ), while some only partially use it (Freudenberg 2003 ; Dobbie and Dail 2013 ). To understand its importance better, we will analyse this concept further in the subsequent sections, covering all its potential forms.

4.1 Traditional Techniques: Uncertainty and Sensitivity Analyses

Robustness analysis is usually accomplished through ‘uncertainty analysis’, ‘sensitivity analysis’, or their ‘synergistic use’ (Saisana et al. 2005 , p. 308). These are characterised as the ‘traditional techniques’ (Permanyer 2011 , p. 308). Putting it simply, uncertainty analysis (UA) refers to the changes that are observed in the final outcome (viz. the composite index value) from a potentially different choice made in the ‘inputs’ (viz. the stages to construct the composite index). On the other hand, sensitivity analysis (SA) measures how much variance of the overall output is attributed to those uncertainties (Saisana et al. 2005 ). It is often seen that these two are treated separately, with UA being the most frequent kind of robustness used (Freudenberg 2003 ; Dobbie and Dail 2013 ). However, both are needed to give the developer, and the audience to which the index is referred, a better understanding. Footnote 29 By solely applying uncertainty analysis, the developer may observe how the performance of a unit (e.g. ranking) deviates with changes in the steps of the construction phase. This is usually illustrated in a scatter plot, with the vertical axis exhibiting the country performance (e.g. ranking) and the horizontal axis exhibiting the input source of uncertainty being tested for (e.g. alternative weighting or aggregation scheme) (OECD 2008 ). To gain a better understanding, however, it is also important to identify the portion of this variation in the rankings that is attributed to that particular change. For instance, is it the weighting scheme that mainly changes the rankings, is it the aggregation scheme that affects them, or is it a combination of these changes in the inputs (interactions) that has a greater effect on the final output? These questions are answered via the use of sensitivity analysis, and they are generally expressed in terms of sensitivity measures for each input tested. More specifically, they show by how much the variance would decrease in the index if that uncertainty input were removed (OECD 2008 ). Understandably, with the use of both, a composite index might convey a more robust picture (Saltelli et al. 2005 ), and it can even be proven useful in dissolving some of the criticisms surrounding composite indicators (e.g. see Saisana et al. 2005 , for an example using the Environmental Sustainability Index). Having discussed the concept of robustness analysis through the use of uncertainty and sensitivity analyses, we will now briefly discuss how these are applied after the construction of a composite index. Footnote 30

The first step in uncertainty analysis is to choose which input factors will be tested (Saisana et al. 2005 ). These are essentially the choices made in each step (e.g. selection of the indicators, imputation of missing data, normalisation, and weighting and aggregation schemes) where applicable. Ideally, one should address all sources of uncertainty (OECD 2008 ). These inputs are translated into scalar factors, which, in a Monte Carlo simulation environment, are randomly chosen in each iteration. Then the following outputs are captured and monitored accordingly: (1) the overall index value; (2) the difference in the values of the composite index between two units of interest (e.g. countries or regions); and (3) the average shift in the rank of each unit.

Unlike UA, sensitivity is applied to only two of the above-mentioned outputs, which are relevant to the evaluation of the quality of the composite. These are (2) and (3) as mentioned in the previous paragraph (Saisana et al. 2005 ). According to the authors, variance-based techniques are more appropriate due to the non-linear nature of composite indices. For each input factor being tested, a sensitivity index is computed, showing the proportion of the overall variance of the composite that is explained, ceteris paribus, by changes in this output. These sensitivity indices are calculated for all the input factors via a decomposition formula (see Saisana et al. 2005 , p. 311). To obtain an even better understanding, it is also important to identify the interactions between the considered inputs (e.g. how a change in an input factor interacts with a change in another). For this exercise, total sensitivity indices are produced. According to the authors, the most commonly used method is the one by Sobol ( 1993 ), in a computationally improved form given by Saltelli ( 2002 ).

4.2 Stochastic Multi-criteria Acceptability Analysis

Stochastic multi-attribute acceptability analysis (SMAA; Lahdelma et al. 1998 ; Lahdelma and Salminen 2001 ) has become popular in multiple criteria decision analysis for dealing with the issue of uncertainty in the data or the preferences required by the decision maker during the evaluation process (e.g. see Tervonen and Figueira 2008 ). SMAA has recently been introduced in the field of composite indicators as a technique to deal with uncertainties in the construction process. More specifically, Doumpos et al. ( 2016 ) use this approach to create a composite index that evaluates the overall financial strength of 1200 cross-country banks in different weighting scenarios. Footnote 31 SMAA can prove to be a great tool in the hands of indices’ developers, and it can extend beyond its use as an uncertainty tool. For instance, Greco et al. ( 2017 ) propose SMAA to deal with the issue of weighting in composite indicators by taking into consideration the whole set of potential weight vectors. In this way it is possible to consider a population in which preferences (represented by each vector of weights) are distributed according to a considered probability. In a complementary interpretation, the plurality of weight vectors can be imagined as a representative of the preferences of a plurality of selves, of which each individual can be imagined to be composed (see e.g. Elster 1987 ). On the basis of these premises, SMAA is applied to the ‘whole space’ of weight vectors for the considered dimensions, obtaining a probabilistic ranking. Essentially, this output illustrates the probability that each considered entity (a country, a region, a city, etc.) attains the first, the second and so on position, as well as the probability that each entity is preferred to another one. Moreover, Greco et al. ( 2017 ) introduce a specific SMAA-based class of multidimensional concentration and polarisation indices (the latter extending the EGR index) (Esteban and Ray 1994 ; Esteban et al. 2007 ), measuring the concentration and the polarisation of the probability of a given entity being ranked in a given position or better/worse (e.g. the concentration and the polarisation to be ranked in the third or a better/worse position).

The use of SMAA as a tool that extends beyond its standard practice (e.g. dealing with uncertainty) is a significant first step towards a conceptual issue in the construction of composite indicators: representative weights. More specifically, constructing a composite index using a single set of weights automatically implies that they are representative of the whole population (Greco et al. 2017 ). Quoting the authors (p. 3): ‘[…] the usual approach considering a single vector of weights levels out all the individuals, collapsing them to an abstract and unrealistic set of ‘representative agents”’. Now one can imagine a cross-country comparison using a single set of weights that act as a representative set for all the countries involved. Understandably, it is a rather difficult assumption to make, given Arrow’s theorem (Arrow 1950 ). Decancq and Lugo ( 2013 , p. 10) describe this fundamental problem with a simple example of a theoretical well-being index. According to the authors, the literature is well documented with respect to the variation of personal opinions on what a ‘good life’ is. Therefore, following the same reasoning, how can a developer assume that a set of weights acts as a representative of all this variation? Quoting the authors (p. 10): ‘Whose value judgements on the “good life” are reflected in the weights?’ This is a classic example of a conflictual situation in public policy, arising due to the existence of a plurality of social actors (see e.g. Munda 2016 ). This issue of the representative agent (see e.g. Hartley and Hartley 2002 ) has long been criticised in the economics literature, one of the most well-known criticisms being made by Kirman ( 1992 ). According to Decancq et al. ( 2013 ), inevitably there are many individuals who are ‘worse off’ when a policymaker chooses a single set of weights. On the one hand, SMAA extends above and beyond the issue of representativeness by providing the developer of an index with the option to include all possible viewpoints. However, for every viewpoint taken into account, a different ranking is produced; thus, a choice has to be made afterwards regarding how to deal with these outcomes. Usually, the mode ranking is chosen, obtained by the ranking acceptability indices (see e.g. Greco et al. 2017 ). Moreover, in its current form, SMAA can only provide the developer with a ranking of the units evaluated. Thus, it still suffers from the same issue as other non-compensatory techniques: swapping the available information in possession with an ordinal scale in the form of a ranking.

4.3 Other Approaches

Several other approaches appear in the literature, with which the robustness of composite indices may be evaluated or which may simply provide more robust rankings. An example of the latter is given by Cherchye et al. ( 2008 ), presenting a new approach according to which several units may be ranked ‘robustly’ (i.e. rankings are not reversed for a wide set of weighting vectors or aggregation schemes). To achieve this, they propose a generalised version of the Lorenz dominance criterion, which leaves to the user the choice of how ‘weak’ or ‘strong’ the dominance relationship will be for the ranking to be considered robust. This approach can be implemented via linear programming, an illustrative application of which is given with the well-known HDI. In regard to the robustness evaluation, Foster et al. ( 2012 ) present another approach, Footnote 32 in which several other weight vectors are considered to monitor the existence of rank reversals. In essence, by changing the weights among the indicators, this approach measures how well the units’ rankings are preserved (e.g. in terms of percentage). In an illustrative application, the authors examine three well-known composite indices, namely the HDI, the Index of Economic Freedom, and the Environmental Performance Index. Similar to Foster et al. ( 2012 ), Permanyer ( 2011 ) suggests considering the whole space of weight vectors, though the objective is slightly different this time. The author proposes to find three sets of weights according to which: (1) a unit, say ‘ α ’, is not ranked below another unit, say ‘ β ’; (2) units ‘ α ’ and ‘ β ’ are equally ranked; and (3) ‘ β ’ dominates ‘ α ’. Essentially, the original intended weight vector set by the developer can fairly be considered to be ‘robust’ the further it is from the second subset (viz. the set of weights according to which ‘ α ’ is equal to unit ‘ β ’), because the closer to it that it is, the more possible it is for a rank reversal to happen. This intuitive approach is further extended to multiple examples and specifications, details of which can be found in Permanyer ( 2011 , pp. 312–316). An illustrative example is provided using the well-known HDI, the Gender-related Development Index, and the Human Poverty Index.

While still considering the robustness evaluation, Paruolo et al. ( 2013 ) propose another approach, which is mainly concerned with the perception of weights and the actual effect that they have on the final output. More specifically, the authors stress how far off the mark a weighting scheme might be when it is assigned in comparison with the actual effect that it has on the overall index (what they call the ‘main effect’, p. 610). This effect is notably apparent in the case of linear aggregation. They propose to measure this effect via Karl Pearson’s correlation ratio, often applied in sensitivity analysis as a first-order measure. According to the authors, this measure can potentially fill a gap in the criticism regarding the difference between the stated importance (given by the weighting) and the actual importance achieved (after the aggregation has taken part) in the case of compensatory aggregation. In a recent study, Becker et al. ( 2017 ) extend this area of research by introducing three tools to aid the developers of composite indices in gaining a better insight into the effect that weights have on the final synthetic measure. The first tool is based on Paruolo et al. ( 2013 ), estimating the main effect of weights using either Gaussian processes or penalised splines, depending on the size of the considered data set and thus the computational cost. The second tool relates to the isolation of indicators’ correlation in the main effect measured by Karl Pearson’s correlation ratio. Using a regression-based approach, the correlation effect can be isolated from this first-order measure so that the developer has an insight into the pure effect of the weights on the composite index, regardless of the correlation among indicators. Finally, the authors propose a third tool allowing stated weights to be aligned perfectly with their actual importance in the final index.

Undeniably, robustness analysis in any form, ‘traditional’ or otherwise, may act as a quality assurance tool. This exhibits the strength of an index by delineating all its potential forms in the case of different choices made in the inputs. However, one of the first points stressed in the OECD’s Handbook is that one cannot interpret an assessment of robustness as the validation of a ‘sensible’ index (OECD 2008 , p. 35). Rather, it is the creation of a sound theoretical framework that determines whether the index is actually sensible. Robustness might only help the developer to answer the questions related to the fit of the model and the meaning of its concept (OECD 2008 ). Unfortunately, but no doubt reasonably, the Handbook cannot provide any form of aid to the developer regarding which theoretical framework fits best. Quoting the authors (OECD 2008 , p. 17): ‘[…] our opinion is that the peer community is ultimately the legitimate forum to judge the soundness of the framework and fitness for purpose of the derived composite’. However, they do urge developers to bear in mind that, whichever framework is used, transparency is of the utmost importance. Making an effort to reduce this uncertainty stemming from the creation of the theoretical framework, Burgass et al. ( 2017 ) suggest the following actions: first, the use of systems modelling (either quantitative or qualitative) to aid the developers of indices to make the proper choices; second, the promotion of open discussions among modellers, experts, and stakeholders to construct a sound theoretical framework that works for all.

5 Conclusions

In this paper we have put composite indicators under the spotlight, examining a wide variety of the methodological approaches in existence. We particularly focused on the issues of weighting and aggregation, the reason being that we find that these are the focus of the paramount criticism in the literature and interesting developments. Additionally, we considered the robustness section of composite indicators that follows their construction. We find that it is an area that attracts increased attention for two main reasons. First, it illustrates how ‘sound’ an index is, when changes occur in the steps leading to its construction, while at the same time further enhancing its overall transparency. This is of the utmost importance given the uncertainties introduced in the previous stages of the construction. Second, uncertainty techniques like SMAA stimulate interest in considering the preferences of different classes of individuals, as they are represented by different weighting vectors. This allows the measurement of the uncertainty (e.g. through probabilistic rankings) but most importantly overcomes the issue of the representative agent that is inherent in the single, allegedly representative, weight vector.

As previously outlined, the purpose of this review was mainly to compensate for the absence of a recent similar study. More specifically, the most recent review studies that focus on the methodological framework, irrespective of the research discipline, are now over a decade old. With the number of applications constantly growing, we took the opportunity to re-examine this topic and offer a more recent outlook. There was by no means any intention to replace any previous studies, like the Handbook on Constructing Composite Indicators . In fact, we find it to be a remarkable and indispensable manuscript for both newcomers in this field and developers who would like to base their work on it. On the contrary, this study offers some recent developments on a heated topic that continues to attract the interest of the public and remains at the forefront of upcoming developments. In the following, we offer some concluding remarks to summarise this study and our thoughts about future development.

In an era of ever-increasing availability of information, composite indicators meet the need for consolidation, aggregating a plethora of indicators into a sole number that encompasses and summarises all this information. Their success and widespread use by global organisations, academics, the media, and policymakers around the world can be attributed to this irresistible characteristic. However successful, they should be interpreted with extreme caution, especially when important conclusions are to be drawn on the basis of these measures (e.g. by policymakers, media, or even the public). This is because their validity is intrinsically linked to their construction, and, as highlighted in this paper, there is no element in their construction that is above criticism. Each approach in every single step has both its benefits and its drawbacks. More specifically, in the weighting stage, developers encounter a wide variety of approaches along a subjective to objective spectrum. Approaches falling at the former end could assign a more meaningful set of weights, according to a theoretical framework or an expert’s opinion. However, with the norm being the lack of a theoretical framework and the existence of ‘biasedness’ in each developer’s opinion, they may result in inconsistencies and broad criticism. At the other end of this spectrum (i.e. ‘objective’ approaches), these kinds of inconsistencies or subjectivity are claimed to be missing. Nonetheless, their criticism involves accusations of assigning conceptually meaningless weights that are driven by the data, while they are often considered unrealistic. What is more, irrespective of their classification (e.g. as ‘objective’ or ‘subjective’), all these methods assume that the weights are representative of the whole population associated with the evaluation. This is something that should be taken into account by the developer when interpreting the results, as it is argued that it is a rather strong hypothesis to make. With respect to the step of aggregation, developers’ choices are still burdensome. More specifically, they suffer from a trade-off between compensability and complexity or a loss of information. That is because, moving from ample compensation (e.g. linear aggregation) to a complete lack of it (e.g. NCMA), the developer soon finds that the complexity and the computational cost increase dramatically.

Understandably, each choice made for the construction of a composite index appears to be ‘between the devil and the deep blue sea’. The developer is compelled to make compromises in each stage, valiantly bearing their drawbacks at the end. Despite often being omitted, robustness analysis should follow the construction of an index. It is an excellent quality assurance tool in the hands of the developer that further enhances the overall transparency. However, it should not be misinterpreted as a guarantee of the sensibility of the composite index. This mainly lies in the evaluation of the theoretical framework, which for this reason should be completely transparent. In fact, robustness could be guaranteed when each choice concisely links back to the aim of construction. As suggested in the literature, a great way to achieve this is to hold an open discussion between the modeller and the implicated stakeholders (e.g. experts, policymakers, or even the public).

Moving forwards, we see a promising trajectory towards eliminating the main criticism surrounding composite indicators. More specifically, it is apparent from the latest publications that, after a vast amount of suggestions in the literature, there is a shift towards the spectrum of non-compensatory approaches. The newly presented methodologies act in favour of adjusting the compensability inherent in the linear aggregation setting, thereby considering one of the main key criticisms in the literature, that of aggregation. Moreover, some recent tools appearing in the sensitivity literature deal with this issue in a different manner, by trying to match the stated and the actual importance of indicators in the final index, compensation and correlation aside. Furthermore, much work has been carried out in the DEA literature to address significant issues, such as improving the discriminatory power, dealing with compensability, or classifying units’ performances into groups. Last, but not least, another interesting development in the literature is the introduction of SMAA, a tool that extends above and beyond the concept of the representative agent by considering the viewpoints of the whole population associated with the evaluation process. From the above, it is apparent that the recent literature has followed a long and interesting route, providing solutions on all fronts. Undeniably, there is still great room for improvement and a long road ahead to reach a pleasing state. However, after all, the interest in composite indicators is currently growing at an ever-increasing pace, and their future is seemingly somewhat promising.

For a more detailed analysis of the debate between the two groups, see Sharpe ( 2004 , pp. 9–11).

Freudenberg ( 2003 ) presents a similar case during the construction of an index of innovation performance.

In brief, ‘double counting’ refers to the issue of implicitly weighting an indicator higher than the desired level. This happens when two collinear indicators are included in the aggregation process without moderating their weighting for this effect.

This is often justified by referring to ‘Occam’s razor’ (see Cherchye et al. 2007 , p. 759).

Quoting the author (Munda 2005a , p. 132): ‘When science is used for policy making, an appropriate management of decisions implies including the multiplicity of participants and perspectives’.

The authors interview 20 experts to set the weights for the 8 sub-indicators of the Technology Achievement Index (TAI) according to the budget allocation process (BAP) and analytic hierarchy process (AHP) techniques. They observe that, in the majority of the cases, the interviewees’ responses were in disagreement when the method changed, revealing how human judgement alters according to the way in which the same question is formulated (e.g. in the BAP versus in the AHP).

For an illustrative example of this approach, see Hermans et al. ( 2008 , pp. 1339–1340).

For example, a value of ‘1’ represents equal importance, while a value of ‘5’ represents five times higher importance, and so on. For a more detailed analysis of this approach and a comprehensive example, see Ishizaka and Nemery ( 2013 , pp. 13–20).

For a more detailed analysis of this approach, see Hair et al. ( 1995 , in OECD 2008 , p. 98) or Green and DeSarbo ( 1978 ).

The literature considers these techniques to be more ‘objective’, as they are not based on any subjective valuation of a decision maker (e.g. see Booysen 2002 , p. 127; Zhou et al. 2007 , p. 293; Decancq and Lugo 2013 , p. 9).

For a review of these, see Krishnakumar and Nagar ( 2008 ), and for an illustrative example and analysis of the steps, see Greyling and Tregenna ( 2016 ).

Nardo et al. ( 2005 , p. 64) mention two such examples of failed uses of PCA/FA; namely the Economic Sentiment Indicator and the development of an index of environmental sustainability.

The reason behind it is given by Lovell et al. ( 1995 , p. 508, in Cherchye et al. 2007 , p. 117): ‘Equality across components is unnecessarily restrictive, and equality across nations and through time is undesirably restrictive. Both penalize a country for a successful pursuit of an objective, at the acknowledged expense of another conflicting objective. What is needed is a weighting scheme which allows weights to vary across objectives, over countries and through time’.

A similar argument is found in Adler et al. ( 2002 ), with the authors claiming that it is amiss to rank several units (e.g. countries) based on a differential set of weights.

In fact, Zhou et al. ( 2007 ) show that, if a unit is dominating all the rest on a specific indicator, then this unit will always be efficient, as it will assign all the weight capacity to that particular indicator.

An extensive review of such constraints can be found in Allen et al. ( 1997 ) and Allen and Thanassoulis ( 2004 ) and three practical applications in Despotis ( 2005a , b ) and Hermans et al. ( 2008 ).

For a comprehensive review see for example Adler et al. ( 2002 ), Angulo-Meza and Lins ( 2002 ), or Podinovski and Thanassoulis ( 2007 ).

A complementary version of the idea, or ‘postscript’ as the authors characterise it, is presented by Färe and Karagiannis ( 2014 ).

Linear and geometric aggregation methods (otherwise called ‘simple additive weighting’ and ‘weighted product’) are also part of the MCDA domain (e.g. see Zhou and Ang 2009 , p. 85).

Often also referred to as “symmetrical importance” (see Podinovskii 1994 , p. 241).

The composite index is formed through an additive utility function, in which the composite equals the sum of the products of weights and indicators.

The composite index is formed through a Cobb–Douglas type function (multiplicative function), in which the composite equals the product of the indicators, each raised to the power of the weight assigned.

For example, see Desai ( 1991 ) and Ravallion ( 1997 ) for a critique on the additive model and the implied trade-offs of the Human Development Index (HDI).

A detailed explanation of the calculation process can be found in Saltelli et al. ( 2005 ) and Munda ( 2012 ). Put simply, each country is pairwise compared with the rest of the countries in each indicator. Each time a country ‘outranks’ another on an indicator, it is given the weight of that indicator as a score, while each time it ranks equally, half of the weight is given to each indicator.

For a brief overview of these, see Greco et al. ( 2017 , pp. 4–5), and for an illustrated example see Munda ( 2012 , p. 342).

For instance, given that we have three objects, say a, b, and c, a cycle occurs when a is preferred to b and b is preferred to c but c is also preferred to a. This is a common problem in the Condorcet-type approaches; see e.g. Fishburn ( 1973 ) and Moulin ( 1988 ).

Mostly known as ‘Kemeny’s rule’ but often referred as ‘C–K–Y–L’ from the initials of Condorcet, Kemeny, Young, and Levenglick, named like this after Munda ( 2012 , p. 345).

With the ‘min function’, the overall index value is equal to the value of the worst-performing indicator, implying the maximum potential penalisation.

An illustrative example can be found in OECD ( 2008 , pp. 117–131), examining the case of the Technology Achievement Index (TAI).

For a more detailed analysis of the procedure, the reader is referred to Saisana et al. ( 2005 , pp. 309–321).

For a similar application, see also Doumpos et al. ( 2017 ).

Conceptually introduced by Foster et al. ( 2010 ).

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Greco, S., Ishizaka, A., Tasiou, M. et al. On the Methodological Framework of Composite Indices: A Review of the Issues of Weighting, Aggregation, and Robustness. Soc Indic Res 141 , 61–94 (2019). https://doi.org/10.1007/s11205-017-1832-9

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An empirical examination of investor sentiment and stock market volatility: evidence from India

• Haritha P H 1 &
• Abdul Rishad 2  

Financial Innovation volume  6 , Article number:  34 ( 2020 ) Cite this article

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Understanding the irrational sentiments of the market participants is necessary for making good investment decisions. Despite the recent academic effort to examine the role of investors’ sentiments in market dynamics, there is a lack of consensus in delineating the structural aspect of market sentiments. This research is an attempt to address this gap. The study explores the role of irrational investors’ sentiments in determining stock market volatility. By employing monthly data on market-related implicit indices, we constructed an irrational sentiment index using principal component analysis. This sentiment index was modelled in the GARCH and Granger causality framework to analyse its contribution to volatility. The results showed that irrational sentiment significantly causes excess market volatility. Moreover, the study indicates that the asymmetrical aspects of an inefficient market contribute to excess volatility and returns. The findings are crucial for retail investors as well as portfolio managers seeking to make an optimum portfolio to maximise profits.

Introduction

There has been growing academic attention in the past decade on investors’ sentiments and their potential impact on market performance. Investor sentiment is the expectation of market participants about the future cash flows (returns) and investment risk (De Long et al. 1990 ). Because traditional stock market theories comprehended market dynamics under the theoretical framework of the efficient market hypothesis (EMH) and random walk theory, they did not consider investor sentiment as an important aspect. However, they failed to explain the heterogeneous behaviour of investors in the capital market. Investors’ sentiment is a vital aspect of the capital market, as it contributes to frequent fluctuations in the stock price and thus creates uncertainty about future returns on investments. In the past few decades, there have been radical changes in the Indian financial environment, especially in the basic structure—for example, shifting from a savings-oriented economy to an investment-oriented economy. These changes have increased heterogeneity in the composition of participants and impacted investors’ risk-taking behaviour.

As per the EMH in classical financial theory, market participants exhibit rational risk aversion. Moreover, the information efficiency of the market does not allow participants to outperform the market (Fama 1965 ). The classical theory fails to explain the presence of systematic mispricing in the capital markets resulting from sentimental factors. Behavioural financial theories claim that irrational behaviour of noise traders and arbitrators causes a disparity in asset prices from their intrinsic (fundamental) values. Recent theoretical advances in behavioural finance and empirical evidence both have rejected the hypotheses of classical financial theory because of its assumption of rationality of agents in capital markets. In the previous decade, rational participants did not seem to have played a leading role in bringing the value of assets up to the then current value of anticipated cash flows (Baker and Wurgler 2007 ). Behavioural finance offers an alternative model that claims that economic phenomena can be better understood if the investors are accepted to not be entirely rational. In this context, the asset pricing not only includes the risk-related anticipated rates but also the impact of investor expectations on the returns. Behavioural finance explains the relationship between investment and the investor’s psychology. Investor behaviour is reflected in the stock prices, and market fluctuations, which ultimately shape the market, are themselves shaped by the psychology of the investors. Baker and Wurgler ( 2006 ) argued that market sentiment creates a tendency for investors to be optimistic or pessimistic while speculating prices instead of deciding on fundamental factors.

Previous studies sought to detect the predictability of sentiments as a systematic risk factor valued as per certain conditions in the market. Studies from developed economies like the USA are far ahead in understanding the sentiment-related market dynamics (Barberis et al. 1998 ; Lee et al. 2002 ; Neal and Wheatley 1998 ). Academic study of investor sentiment in developing economies with rapidly growing capital markets is still in infancy. Previous research has mainly focused on the influence of investors’ sentiment on investment returns, whereas the effect of sentiment on the conditional volatility structure of the market is less explored. Also, even among those studies that consider sentiment as a critical factor influencing the time-varying stock return, volatility and potential profitability relating to noise traders were the main aspects of focus. During the periods of high sentiment and low sentiment, noise traders act differently to keep their positions secure. During the high sentiment episodes, their participation and trading is more aggressive compared to that during a low sentiment episode. This is caused by naive and unaware noise traders’ misjudgement of potential risk. Past academic studies about emerging economies have not explored such factors in-depth. The present study is an attempt to address the above-mentioned issues by using a market-oriented sentiment index. We developed an investors’ sentiment index by using multiple sentiment yardsticks mentioned by Baker and Wurgler ( 2006 ). Considering the investors’ sentiments’ contribution to volatility in emerging markets, the current study aimed to establish new empirical evidence to add a more comparative dimension to the existing literature. The findings can help market participants to understand the role of investor sentiment in the determination of conditional volatility of the market and to take decisions to optimise the portfolio.

This study developed the aggregate sentiment index (ASI) from market-oriented sentimental factors such as trading volume, put-call ratio, advance-decline ratio, market turnover, share turnover and number of initial public offers (IPOs) in the period. The use of a constructed sentiment index under the GARCH framework to estimate the association between stock market volatility and investor sentiment makes this study different from existing studies. The findings indicate the persistence of volatility in market indices. Such persistent connection between the sentiment index and stock volatility suggests that investor sentiment is one of the most crucial determinants of Indian stock market volatility.

Theoretical background

According to the conventional theory of ‘market noise’ proposed by the Black ( 1986 ), noise traders operate on noisy signals in financial markets and balance both the systematic and non-systematic risk of an asset. According to this theory, noise makes markets inefficient to some extent and prevents investors from benefitting from inefficiencies. The significance of sentimental factors in asset pricing theories is substantiated by empirical literature from developed economies. The question of how irrational beliefs held by investors affect the market through asset pricing and expected returns is explained in behavioural finance theories. The theoretical model developed by De Long et al. ( 1990 ) explained this phenomenon as ‘Some investors, denominated noise traders, were subject to sentiment – a belief about future cash flows and risks of securities not supported by economic fundamentals of the underlying asset(s) – while other investors were rational arbitrageurs, free of sentiment. The irrational beliefs were caused by noise, interpreted by the irrational traders as information, thus the term noise traders.’

Theoretically, noise is part of irrational behaviour; the irrational traders consider noise as information. Interestingly, proponents of an efficient market claimed that noise traders were exploited by rational arbitrageurs who drove prices towards fundamental equilibrium values. Thus, noise was a reaction of noise traders to the activities of rational arbitrageurs that caused overpricing or underpricing of stocks during periods of high and low sentiment (Lemmon and Portniaguina 2006 ; Baker and Wurgler 2006 ). Researchers have been unable to satisfactorily explain the interaction between rational and irrational investors. The continuing debate on this issue significantly contributes to the literature but concentrates mainly on the role of noise traders in anticipated asset yields and volatility of return. It is not understood how the market reacts to noise, which is caused by a large number of small events. This behaviour can be observed among investors from advanced economies because they believe that systematic risk and return anomaly is associated with irrational investment behaviour (Brown and Cliff 2004 ; Qiu and Welch 2006 ; Lemmon and Portniaguina 2006 ). With this theoretical background, our study examines the role of irrational feelings of investors and their impact on the volatility of the Indian capital market.

Literature review

The development of behavioural finance theories triggered a discussion on the impact of investor sentiment on asset returns in the integrated stock market. According to theoretical and empirical research, investor sentiment strongly influences stock prices with inevitable consequences on portfolio selection and asset management, as psychological differences of heterogeneous investors have implications on the pricing of assets in the market. The influence of investor’s sentiment in asset price volatility is widely described as a combination of investors’ reaction to the current market situation and unjustified expectation of the future cash flows (Baker and Wurgler 2006 , 2007 ).

As a psychological factor, it is not easy to estimate investors’ sentiment because of their subjective and qualitative nature. However, different proxies have been used to measure sentiment. These indicators of the sentiment index are classified as indirect and direct measures. In direct measures, researchers measure the individual investor sentiment via surveys and polling techniques. They are highly sampling-dependent, and the chances of sampling errors are high. Moreover, they may not be able to give a broad picture of the prevailing sentiment. Indirect measures use market-determined sentiment proxies, such as trading volume, turnover volatility ratio, put-call ratio, advance-decline ratio, market turnover and share turnover for measuring the same. They posit that investors’ sentiment are reflected in the structure and breadth of the market and understanding these dynamics helps to capture the irrational aspects of the market. The consistent and theoretically comprehensible nature of the sentiment index has led to its wide adoption (Baker and Wurgler 2006 ; Brown and Cliff 2004 ; Chen et al. 1993 ; Clarke and Statman 1998 ; DeBondt and Thaler 1985 ; Elton et al. 1998 ; Fisher and Statman 2000 ; Lee et al. 2002 ; Neal and Wheatley 1998 ; Sias et al. 2001 ). According to Zhou ( 2018 ), investor sentiment indicates the distance of the asset’s value from its economic bases. This can be measured from different sources, such as official documents, media reports and market surveys. Mushinada and Veluri ( 2018 ) used trading volume and return volatility for understanding the relationship between sentiments and returns. Their findings showed that post-investment analysis was essential to correct errors in previous behavioural estimations. Market participants’ behaviour is heterogeneous because of the risk-return expectation, and it creates noise in the market. These findings contradict with the premises of the efficient market hypothesis that postulate that markets turn information efficient when investors behave rationally.

In the past few decades, empirical studies across the globe have investigated the connection between investors’ sentiment and stock returns for understanding and substantiating theories of market inefficiency (Brown and Cliff 2004 ; Fisher and Statman 2000 ). Chi et al. ( 2012 ) examined the impact of investor sentiment on stock returns and volatility by using mutual fund flows as an investor sentiment proxy in the Chinese stock market. They found that investor sentiment has a great impact on stock returns. The relationship between stock market volatility and investor sentiment has also been reported as statistically significant. Supporting these findings, Zhou and Yang ( 2019 ) stated that the construction of a theoretical model of stochastic investor sentiment influences investor crowdedness and also affects asset prices. Their result indicated that optimistic (pessimistic) expectations of investors can move asset prices above (below) the basic value. By examining the long-term association between investor sentiment in the stock and bond market, Fang et al. ( 2018 ) showed that the index of investor sentiment is positively associated with market volatility. Contradicting the fundamental tenets of the efficient market hypothesis, Shiller ( 1981 ) argued that investors are not completely rational, which could affect market prices aside from fundamental variables. Wang et al. ( 2006 ) noted that the sensitivity of investor sentiments to the information flow affected both market return and volatility. Chiu et al. ( 2018 ) found a positive relationship between investor sentiment, market volatility and macroeconomic variables. Jiang et al. ( 2019 ) constructed the fund manager’s sentiment index as a predictor of aggregate stock market returns. They found that when managers had a high level of sentiment, it caused a reduction in overall income surprises from total investment. Li ( 2014 ) pointed out that the sentiment index has strong predictive power for Chinese stock market returns. Retail investors’ attention will help to mitigate the crash risk, as the retail investors’ attention will not allow any irrational or noise traders to overrun the rational market participants (Wen et al. 2019 ).

Verma and Verma ( 2007 ) studied the role of retail and noise traders in price volatility to yield similar results. Verma and Soydemir’s ( 2009 ) empirical examination of the rational and irrational investors’ impact on market prices also supported the previous finding. They discovered that individual and institutional investors’ feelings influenced the market. Further evidence shows that the response of the market to volatility is not homogeneous; it is heterogeneous depending on the variations in shareholder sentiment. These findings are validated by Gupta ( 2019 ) who found that sentiments of fund managers are a stronger predictor than the returns, when it comes to forecasting volatility. Yang and Copeland ( 2014 ) found that the investor sentiment index has a long-term and short-term asymmetrical impact on volatility. They concluded that bearish sentiment is associated with lower returns than bullish sentiment, which accelerates market return. This shows that the bullish feeling has positive effects on short-term volatility, whereas in the long-term, it has a negative effect on volatility. These findings agree with the findings discussed by Qiang and Shue-e ( 2009 ), namely that positive and negative sentiment create different impacts on stock price variation. Baker et al. ( 2012 ) constructed investor sentiment indicators of six nations but because of the disintegration of different markets owing to the heterogeneous behaviour of investors across the globe, the indicators were not viable. Other studies have reported that investors’ sentiments are driven by overall funding patterns irrespective of the investor being individual or institutional (Baker and Wurgler 2000 ; Henderson et al. 2006 ). Investors’ sentiment has a mutual relationship with the expected return of public bonds and the expected return from the stock market (Bekaert et al. 2010 ).

In the context of the Indian stock market, Sehgal et al. ( 2009 ) discussed the fundamental aspects of investor sentiment and its relationship with market performance. They identified several factors that might act (individually and together) as indicators of market behaviour and investor sentiments’ influence on market behaviour. The authors used macroeconomic factors such as real GDP, corporate profits, inflation, interest rates and liquidity in the economy and market-based factors such as the put-call ratio, advance-decline ratio, earnings surprises, the price to earnings ratio and price to book value as potential factors to explain the underlying investor sentiment at the aggregate market level. They also suggested the development of a sentiment index based on these macroeconomic and market indicators. Using some of these indicators, Dash and Mahakud ( 2013 ) examined the explanatory power of an index of investor sentiment on aggregate returns. They found a significant relationship between the investor sentiment index and stock returns across industries in the Indian stock market. Rahman and Shamsuddin ( 2019 ) studied the excess price ratio and its influence on investor sentiment and found that the price to earnings ratio increased with a rise in investors’ sentiment. Kumari and Mahakud ( 2016 ) and Chandra and Thenmozhi ( 2013 ) studied the impact of investor sentiment in the Indian capital market. They found a positive relationship between investor sentiment and market volatility. Verma and Verma ( 2007 ) showed that investor sentiment has a positive impact on asset return, but it makes an adverse impact on individual and institutional investors owing to market volatility. Aggarwal and Mohanty ( 2018 ) studied the impact of the investor sentiment index on the Indian stock market and found that there is a positive relationship between stock returns and investor sentiments. However, most of these studies focused on the general effect of investors’ sentiment on stock returns. Such an approach restricts our understanding of the phenomenon of investors’ sentiment and its influence on market dynamics to a single dimension. In the present study, we explored the role of investor sentiment in determining excess market returns and volatility.

Data and variables

Being a qualitative factor, it is not easy to quantify the market behaviour of investors. Past studies have used multiple ways to measure investors’ sentiment. Some studies have relied on media reports, events and other publicly available documents to collect information on investor behaviour, and other studies have conducted surveys among investors for the same. Some other researchers have used market-based indicators such as price movements and trading activities for constructing sentiment indexes. A few researchers have used single variables as an indicator of investor sentiment. For instance, Mushinada and Veluri ( 2018 ) used trading volume as an indicator of investor sentiment. Using a single variable may not be sufficient to explain market sentiments because there are multiple factors that cause variation in these single variable proxies. Latest studies have constructed the sentiment index by using multiple market-based indicators that directly reflect the participants’ behaviour. Following Baker et al. ( 2012 ), this study employed multiple market-based indicators for constructing the sentiment index for the period from January 2000 to December 2016. We used the monthly average closing price of the NIFTY 50 (Nifty) stock index to measure market volatility and return. The diversified market representation of the Nifty index over the other benchmark BSE SENSEX (Sensex) motivated us to select the former. The study used monthly data because of the scarcity of high-frequency data on market-related indicators. The data was collected from the official websites and various reports of the National Stock Exchange, Reserve Bank of India and Securities and Exchange Board of India.

The study employed Bollerslev ( 1986 ) generalized autoregressive conditional heteroskedastic model (GARCH) to measure volatility using the conditional variance equation and to capture the dynamics of volatility clustering. This helped us to examine how the investors’ sentiment reacts to market volatility. This model helps verify whether the investors’ shocks are persistent or not. For the serial correlation, this study used the autoregressive conditional heteroskedasticity-Lagrange multiplier (ARCH-LM) of Engle and Ng ( 1993 ), autoregressive conditional heteroskedasticity (ARCH) test of Engle ( 1982 ) and Mcleod and Ll ( 1983 ) tests for the estimation of models. We also employed the Granger causality test to check the direction of causality between sentiment and market volatility.

Construction of investors’ sentiment index

The present study adopted the framework developed by Baker et al. ( 2012 ) to construct the investors’ sentiment index. It considered six variables: trading volume, put-call ratio, advance-decline ratio, market turnover, share turnover and the number of IPOs. The number of IPOs was defined as the total number of IPOs during the period. Baker et al. ( 2012 ) argued that firms try to procure more capital when the market value of the firm is high and repurchase their shares when the market value is low. The intention is to take advantage of the market sentiment until it reaches the fundamental value. In a bullish market, new issue of shares will transfer wealth from new shareholders to the company or to the existing shareholders. This market timing hypothesis suggests that higher (lower) value or number of IPOs means that the market sentiment is bullish (bearish) (Baker and Wurgler 2006 ). The number of IPOs reflects the market pulse; hence, they can be considered as an important component of the sentiment index.

The share turnover ratio is one of the conventional yardsticks for measuring the liquidity position, which reflects the active participation of traders and investors in the market. It is the ratio of the total value of shares traded during the period to the average market capitalization for the period. Turnover is vital in gauging investors’ sentiment in the market. Irrational investors actively participate in the market when they are optimistic and accelerate the volume of turnover (Baker and Stein 2004 ). Theoretically, the relationship between market returns and turnover is expected to be negative (Jones 2001 ). The presence of high turnover ensures liquidity and reduces the chances of abnormal returns.

Market turnover (MT) is the ratio of trading volume to the number of shares listed on the stock exchange. Market sentiment can be sensed from the turnover of the market because turnover will be low in bearish markets and high in bullish markets (Karpoff 1987 ). Small turnover is usually preceded by a price decline, whereas high turnover is associated with an increase in price (Ying 1966 ). Thus, the turnover information is a significant component of measuring the sentiment of market participants.

The advances and declines ratio (ADR) is a market-breadth indicator that analyses the proportion between the number of advancing shares and declining shares. The increasing (decreasing) trends in the ADR confirm the upward (downward) trend in the market (Brown and Cliff 2004 ). Generally, the ADR ratio is expected to be positive because investors’ sentiment makes the market active. Thus, the ADR ratio helps to recognise the recent trend and can be used as an indicator of market performance.

The put-call ratio (PCR) is another indicator to measure the dynamics of the secondary market. This sentiment indicator is measured as the ratio between transactions on all the put options and the call options on Nifty. A higher (lower) ratio indicates a bullish (bearish) sentiment in the market. Incorporating PCR to measure the aggregate sentiment index yields accurate results because it reflects the expectations of market participants. When market participants expect a bearish trend, they try to shield their positions. When trade volumes of put options are higher relative to the trade quantity of call options, the ratio will go up (Brown and Cliff 2004 ; Finter and Ruenzi 2012 ; Wang et al. 2006 ). This derivative market proxy is considered as an indicator of a bullish trend because the bearish market PCR will be small (Brown and Cliff 2004 ).

The trading volume (TV) is a key variable for constructing the sentiment index. It is measured as the monthly average of the Nifty daily trade volume. Frequent trades in an active market increase the volume and create liquidity in the market. Therefore, researchers have used market turnover as a proxy for investor sentiment (Qiang and Shue-e 2009 ; Zhu 2012 ; Li 2014 ; Chuang et al. 2010 ). The present study considered TV as one of the indicators of market sentiment.

Macroeconomic factors that are often flashed in the media tend to influence investor sentiment quite significantly. Factors like the levels of inflation, corporate debt, economic growth rate and foreign exchange rate and reserves tend to affect the behaviour of market participants to a certain extent. Therefore, this study used variables such as the exchange rate, Wholesale Price Index (WPI), Index of Industrial Production (IIP), Net Foreign Institutional Investment (FII) and Term Spread (TS) to measure the intensity of aggregate investor sentiment on market volatility.

Unit root tests

Ensuring the stationarity of the variables is necessary for consistent estimators. This study used the augmented Dickey-Fuller test (ADF) (Dickey and Fuller 1981 ) to analyse the presence of unit roots in the time series properties of each variable. Table  1 shows the results of unit root analysis using the ADF test. Unit root tests were run with the linear trend and at levels and intercept. The result shows that all variables expect MT are stationary at level. MT was converted to stationarity by taking the first difference.

• Principal component analysis

Principal component analysis (PCA) is a multivariate method in which several interconnected quantitative dependent variables describing the observations are analysed. PCA aims to find and extract the most significant information from the data by compressing the size and simplifying the data without losing the important information (Abdi and Williams 2010 ). It consists of several steps for conducting the linear transformations of a large number of correlated variables to obtain a comparatively few unrelated elements. In this way, information is clustered together into narrow sets and multicollinearity is eliminated. The principal goal of PCA is to summarize the indicator data through a common set of variables as efficiently as possible.

First, the six orthogonal sentiment proxies and their first lags were used as factor loadings to calculate the raw sentiment index. The study started with estimating the initial principal component of the six indicators and their lags, which gave a first-stage index with 12 loading factors, namely the six proxies and their lags. Then, we calculated the correlation between the initial index and the current and lagged values of the indicators. Finally, we estimated the sentiment as the first principal component of the correlation matrix of six variables, which were the respective proxy’s lead or lag. We chose whichever had a higher correlation with the first-stage index to rescale the coefficients so that the index had unit variance (Table 2 ). This process yielded a parsimonious index.

Investors’ sentiment and stock market volatility

Following the theoretical and empirical models proposed by Baker and Wurgler ( 2007 ), Brown and Cliff ( 2004 ) and Baker et al. ( 2012 ), this study used market-related indicators for the construction of the investor sentiment index in the initial stage. This study used six indirect proxies to create the sentiment index by considering the first principal component and the lagged components of the variable. The first principal component explains the sample’s variance. Researchers have argued that certain proxies take longer periods to reflect the investors’ sentiment. Therefore, the present study followed the approach of Baker and Wurgler ( 2006 ) and Ding et al. ( 2017 ) to reflect the investors’ sentiment accurately and to assess the PCA with levels as well as their lags to find the main factors.

The GARCH (1,1) model was used to estimate the impact of sentiment on market volatility and stock returns. The GARCH model helps analyse the volatility characteristics of the datasets, especially for financial data, as it has the unique characteristics of heteroscedasticity and volatility clustering (Fig.  1 ). The specific character of financial time series data limits the use of conventional econometrics models to estimate the parameters. The GARCH model helps to capture volatility clustering and to manage issues of heteroskedasticity.

The rational aggregate sentiment index

Stock market volatility can be estimated in two ways: with the help of market-determined option prices or by time series modelling. Non-availability of option prices led us to choose the time-series method. There are multiple indices available to measure the dynamics of the Indian capital market. Among them, Nifty, which consists of 50 companies from different sectors, and Sensex, which covers 30 companies, are prominent. Inclusion of diversified sectors and wider market coverage (market capitalisation) motivated us to select Nifty as the indicator for measuring market volatility. The indicator of market returns and the aggregate sentiment index showed volatility clustering (Fig.  2 ), and the heteroskedastic behaviour was confirmed through the ARCH-LM test. This satisfied the prerequisites for estimating the GARCH model.

Investors’ sentiment and stock index return

The GARCH ( p, q ) model, introduced by Engle ( 1982 ) and Bollerslev ( 1986 ), can be expressed as follows:

where r t is the log Nifty return (the positive value of r t indicates a bullish trend in the market, and the negative value shows a bearish trend in the market). It is calculated by.

\( {r}_t=\frac{P_1-{P}_0}{P_0} \) , where P 0 and P 1 represent the price at time t-1 and t. γ is the coefficient of the lagged value of the Nifty return ( r t  − 1 ).  c 0 is the constant of the mean equation; ω is the constant in the variance equations; and  ε t is the error term.  I t  − 1 represents the information available to the market participants. \( {\varepsilon}_{t-1}^2 \) is the ARCH term and \( {\sigma}_{t-1}^2 \) is the GARCH term that explains the instantaneous variance at time t − 1. α +β > 1β ≥ 0, which shows the persistence of volatility. A value close to 1 indicates the persistence of volatility and indicates a low level of mean reversion in the system. By increasing the number of the ARCH and GARCH terms, the model can be generalized to a GARCH (p,q) model. For a well-specified GARCH model, ω > 0, α > 0 and  β  ≥ 0 should be satisfied.

We modified the basic GARCH model by incorporating a sentiment variable in the equation,

where δ represents the coefficient of the sentiment index.

Empirical results

The estimated result of the GARCH (1,1) model is presented in Table  3 . The coefficients of the ARCH (α) and GARCH terms (β) are statistically significant and different from zero. In addition, the sum of α + β is close to unity. This indicates the high persistence of volatility, that is, the mean reversal process is very slow because of the persistent shocks. The result of the ARCH-LM test indicates the absence of further ARCH effects, which means the model captures the ARCH effects. The statistically significant coefficient of Q and Q 2 at the 20th lag indicates the absence of further autocorrelation in the model.

Sentiment is a crucial element that directly influences market behaviour. The conventional capital asset pricing model theory states that investors should be rewarded according to their risk-taking behaviour. However, the impact of sentiment on market volatility may cause market uncertainty and lead to less returns. If the market participants fail to earn a market risk premium for their expected volatility, they will move away from the market, which further causes volatility in the market. This vicious circle may cause a bearish trend and languid growth and development of the market. The conditional volatility graph shows that the impact of negative sentiment is higher than that of positive sentiment. This indicates that when sentiments are positive, investors actively participate in the market with the expectation of higher returns. However, this causes more speculative activities in the markets and may cause overvaluation of scrips. In contrast, during the dominance of negative sentiments, investors move away from the market because of the negative expectation of market returns. Therefore, it can be theorised that during positive sentiment, companies explore the opportunity to enter the market through IPOs. Similarly, dividend declaration, bonus issue and a rights issue also trigger positive sentiments.

Conditional volatility

The conditional variance graph from the GARCH (1,1) model shows the dynamics of market volatility of the Nifty returns (Fig.  3 ). Up to May 2008, volatility was high, though it can be deemed as moderate when compared to that during the subprime crisis period. During this period, volatility increased exponentially, and this trend continued up to February 2010. Later, the volatility reduced substantially.

• Granger causality test

The Granger causality test examines the direction of cause among different series (Granger 1969 ). A time series x t Granger-causes another time series y t if series y t can be predicted with better accuracy by using the past values of x t rather than by not doing so. This study examined the causal relationship between the sentiment index (S ent ) and stock market return. Tests between the aggregate sentiment index and stock returns were modelled for understanding the leading and the lagging variables. We found that investors’ sentiment leads to volatility of the market returns. However, volatility in the returns does not cause sentiment (Table  4 ).

De Long et al. ( 1990 ) pointed out that noise traders’ pressure in a market with a strong bullish sentiment on the price to move beyond the fundamental value causes a drop in the expected return. However, if bullish noise traders dominate the market, it causes a rapid upward movement in the market prices because of the upsurge in demand for the high-risk scrips. The expected level of market risk will be higher, creating a ‘hold more’ effect because of the expectation of higher returns. The intensity of sentiments on stock returns closely depends on the effect that dominates the market expectation. The unidirectional causality of sentiments to volatility indicates that the price-pressure effect (noise traders’ pressure on prices reduces the expected return) dominates the market and that noise traders benefit during episodes of a high sentiment index. This way, sentiment leads to volatility. However, once the noise traders start making profit, their expectation on return and risk will increase. Thus, it may not create a reverse causality in a developing market because of information inefficiency. In another way, it can be explained that when investors’ irrational sentiment is positive, their expectation on return is also positive. This may lead to speculative activities on their part to exploit the situation, exciting them to invest more. This leads to volatility in the market. On the other hand, market uncertainty causes withdrawal of market makers and encourages investors to stay inactive because of the uncertain expectation on the return in a risky market. Moreover, in such a situation, investors are always concerned about fundamentally induced equilibrium prices that give the fair value of assets. Following the arguments of Wen et al. ( 2019 ), retail investors should be more attentive in collecting information to minimise their information asymmetry for managing their potential risk.

This research provides a comprehensive examination of the impact of investor sentiment on stock market volatility. The study constructed a sentiment index by using a linear combination of different-market oriented proxies weighted using principal component analysis. The study found an asymmetrical relationship when the sentiment index was decomposed into positive and negative sentiment. The positive sentiment index has a positive effect on excess market return, but the intensity of negative sentiment is less on negative returns. These results imply that when investors are more optimistic about the market generating excess returns, their extreme optimism leads to more speculative activities that tempt them to invest even more. The study also found persistency of market volatility and the sentiment index, which shows the contemporaneous impact on sentiment and excess market returns. The findings reveal that investors consider the market as weak-efficient. This shows that the efficient market hypothesis may not be sufficient in explaining the market behaviour of emerging markets like India. The results indicate the scope for arbitration in the Indian market and thus invalidate the explanation of efficient market volatility in India. This further indicates a deviation from a random walk, but it is difficult to predict the volatility of the market sufficiently to produce excess returns.

The results help to understand the role of non-fundamental factors in driving the Indian equity market away from a fundamentally oriented equilibrium and in influencing the risk-return perception. They also show that sentiment is relatively correlated with unexpected stock returns, and the correlation differs significantly over time. This contradicts with the traditional capital market theories and supports the behavioural theories on capital markets. Proper examination of the market sentiment helps investors and fund managers decide their entry and exit points for investment. By taking the investor sentiment into account as a significant determinant of stock market volatility in asset price models, investors can enhance their portfolio performance. The results can also help policymakers’ efforts to stabilize stock market volatility and uncertainty in order to protect investors’ wealth and attract more investors. Therefore, future research should aim to develop investors’ sentiments from available high-frequency data by incorporating additional comprehensive investor sentiment factors to reflect real-time information.

Availability of data and materials

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Abbreviations

Indian Aggregate Sentiment Index

Gross domestic product

Price to earnings ratio

Generalized autoregressive conditional heteroskedastic model

Autoregressive conditional heteroscedasticity-Lagrange multiplier

Put-call ratio

Initial public offer

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Haritha P H

Central University of Himachal Pradesh, Dharamsala, Himachal Pradesh, India

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The authors analyze the role of irrational investor sentiment in determining Indian stock market volatility using monthly information from India’s National Stock Exchange between June 2000 and December 2016. Sentiment index with the assistance of principal component analysis was developed using market-related implicit indices. Further, this sentiment index was modelled in the GARCH and Granger Causality framework to analyses its contribution to volatility. The results show that irrational sentiment significantly causes excess market volatility. Moreover, the study reveals that the asymmetrical aspects of an inefficient market contribute to excess volatility and returns. The findings reveal that investors consider the market as weak-efficient. This shows that the efficient market hypothesis may not be sufficient in explaining the market behaviour of emerging markets like India. The author(s) read and approved the final manuscript.

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P H, H., Rishad, A. An empirical examination of investor sentiment and stock market volatility: evidence from India. Financ Innov 6 , 34 (2020). https://doi.org/10.1186/s40854-020-00198-x

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Short-term stock market price trend prediction using a comprehensive deep learning system

• Jingyi Shen 1 &
• M. Omair Shafiq   ORCID: orcid.org/0000-0002-1859-8296 1  

Journal of Big Data volume  7 , Article number:  66 ( 2020 ) Cite this article

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In the era of big data, deep learning for predicting stock market prices and trends has become even more popular than before. We collected 2 years of data from Chinese stock market and proposed a comprehensive customization of feature engineering and deep learning-based model for predicting price trend of stock markets. The proposed solution is comprehensive as it includes pre-processing of the stock market dataset, utilization of multiple feature engineering techniques, combined with a customized deep learning based system for stock market price trend prediction. We conducted comprehensive evaluations on frequently used machine learning models and conclude that our proposed solution outperforms due to the comprehensive feature engineering that we built. The system achieves overall high accuracy for stock market trend prediction. With the detailed design and evaluation of prediction term lengths, feature engineering, and data pre-processing methods, this work contributes to the stock analysis research community both in the financial and technical domains.

Introduction

Stock market is one of the major fields that investors are dedicated to, thus stock market price trend prediction is always a hot topic for researchers from both financial and technical domains. In this research, our objective is to build a state-of-art prediction model for price trend prediction, which focuses on short-term price trend prediction.

As concluded by Fama in [ 26 ], financial time series prediction is known to be a notoriously difficult task due to the generally accepted, semi-strong form of market efficiency and the high level of noise. Back in 2003, Wang et al. in [ 44 ] already applied artificial neural networks on stock market price prediction and focused on volume, as a specific feature of stock market. One of the key findings by them was that the volume was not found to be effective in improving the forecasting performance on the datasets they used, which was S&P 500 and DJI. Ince and Trafalis in [ 15 ] targeted short-term forecasting and applied support vector machine (SVM) model on the stock price prediction. Their main contribution is performing a comparison between multi-layer perceptron (MLP) and SVM then found that most of the scenarios SVM outperformed MLP, while the result was also affected by different trading strategies. In the meantime, researchers from financial domains were applying conventional statistical methods and signal processing techniques on analyzing stock market data.

The optimization techniques, such as principal component analysis (PCA) were also applied in short-term stock price prediction [ 22 ]. During the years, researchers are not only focused on stock price-related analysis but also tried to analyze stock market transactions such as volume burst risks, which expands the stock market analysis research domain broader and indicates this research domain still has high potential [ 39 ]. As the artificial intelligence techniques evolved in recent years, many proposed solutions attempted to combine machine learning and deep learning techniques based on previous approaches, and then proposed new metrics that serve as training features such as Liu and Wang [ 23 ]. This type of previous works belongs to the feature engineering domain and can be considered as the inspiration of feature extension ideas in our research. Liu et al. in [ 24 ] proposed a convolutional neural network (CNN) as well as a long short-term memory (LSTM) neural network based model to analyze different quantitative strategies in stock markets. The CNN serves for the stock selection strategy, automatically extracts features based on quantitative data, then follows an LSTM to preserve the time-series features for improving profits.

The latest work also proposes a similar hybrid neural network architecture, integrating a convolutional neural network with a bidirectional long short-term memory to predict the stock market index [ 4 ]. While the researchers frequently proposed different neural network solution architectures, it brought further discussions about the topic if the high cost of training such models is worth the result or not.

There are three key contributions of our work (1) a new dataset extracted and cleansed (2) a comprehensive feature engineering, and (3) a customized long short-term memory (LSTM) based deep learning model.

We have built the dataset by ourselves from the data source as an open-sourced data API called Tushare [ 43 ]. The novelty of our proposed solution is that we proposed a feature engineering along with a fine-tuned system instead of just an LSTM model only. We observe from the previous works and find the gaps and proposed a solution architecture with a comprehensive feature engineering procedure before training the prediction model. With the success of feature extension method collaborating with recursive feature elimination algorithms, it opens doors for many other machine learning algorithms to achieve high accuracy scores for short-term price trend prediction. It proved the effectiveness of our proposed feature extension as feature engineering. We further introduced our customized LSTM model and further improved the prediction scores in all the evaluation metrics. The proposed solution outperformed the machine learning and deep learning-based models in similar previous works.

The remainder of this paper is organized as follows. “ Survey of related works ” section describes the survey of related works. “ The dataset ” section provides details on the data that we extracted from the public data sources and the dataset prepared. “ Methods ” section presents the research problems, methods, and design of the proposed solution. Detailed technical design with algorithms and how the model implemented are also included in this section. “ Results ” section presents comprehensive results and evaluation of our proposed model, and by comparing it with the models used in most of the related works. “ Discussion ” section provides a discussion and comparison of the results. “ Conclusion ” section presents the conclusion. This research paper has been built based on Shen [ 36 ].

Survey of related works

In this section, we discuss related works. We reviewed the related work in two different domains: technical and financial, respectively.

Kim and Han in [ 19 ] built a model as a combination of artificial neural networks (ANN) and genetic algorithms (GAs) with discretization of features for predicting stock price index. The data used in their study include the technical indicators as well as the direction of change in the daily Korea stock price index (KOSPI). They used the data containing samples of 2928 trading days, ranging from January 1989 to December 1998, and give their selected features and formulas. They also applied optimization of feature discretization, as a technique that is similar to dimensionality reduction. The strengths of their work are that they introduced GA to optimize the ANN. First, the amount of input features and processing elements in the hidden layer are 12 and not adjustable. Another limitation is in the learning process of ANN, and the authors only focused on two factors in optimization. While they still believed that GA has great potential for feature discretization optimization. Our initialized feature pool refers to the selected features. Qiu and Song in [ 34 ] also presented a solution to predict the direction of the Japanese stock market based on an optimized artificial neural network model. In this work, authors utilize genetic algorithms together with artificial neural network based models, and name it as a hybrid GA-ANN model.

Piramuthu in [ 33 ] conducted a thorough evaluation of different feature selection methods for data mining applications. He used for datasets, which were credit approval data, loan defaults data, web traffic data, tam, and kiang data, and compared how different feature selection methods optimized decision tree performance. The feature selection methods he compared included probabilistic distance measure: the Bhattacharyya measure, the Matusita measure, the divergence measure, the Mahalanobis distance measure, and the Patrick-Fisher measure. For inter-class distance measures: the Minkowski distance measure, city block distance measure, Euclidean distance measure, the Chebychev distance measure, and the nonlinear (Parzen and hyper-spherical kernel) distance measure. The strength of this paper is that the author evaluated both probabilistic distance-based and several inter-class feature selection methods. Besides, the author performed the evaluation based on different datasets, which reinforced the strength of this paper. However, the evaluation algorithm was a decision tree only. We cannot conclude if the feature selection methods will still perform the same on a larger dataset or a more complex model.

Hassan and Nath in [ 9 ] applied the Hidden Markov Model (HMM) on the stock market forecasting on stock prices of four different Airlines. They reduce states of the model into four states: the opening price, closing price, the highest price, and the lowest price. The strong point of this paper is that the approach does not need expert knowledge to build a prediction model. While this work is limited within the industry of Airlines and evaluated on a very small dataset, it may not lead to a prediction model with generality. One of the approaches in stock market prediction related works could be exploited to do the comparison work. The authors selected a maximum 2 years as the date range of training and testing dataset, which provided us a date range reference for our evaluation part.

Lei in [ 21 ] exploited Wavelet Neural Network (WNN) to predict stock price trends. The author also applied Rough Set (RS) for attribute reduction as an optimization. Rough Set was utilized to reduce the stock price trend feature dimensions. It was also used to determine the structure of the Wavelet Neural Network. The dataset of this work consists of five well-known stock market indices, i.e., (1) SSE Composite Index (China), (2) CSI 300 Index (China), (3) All Ordinaries Index (Australian), (4) Nikkei 225 Index (Japan), and (5) Dow Jones Index (USA). Evaluation of the model was based on different stock market indices, and the result was convincing with generality. By using Rough Set for optimizing the feature dimension before processing reduces the computational complexity. However, the author only stressed the parameter adjustment in the discussion part but did not specify the weakness of the model itself. Meanwhile, we also found that the evaluations were performed on indices, the same model may not have the same performance if applied on a specific stock.

Lee in [ 20 ] used the support vector machine (SVM) along with a hybrid feature selection method to carry out prediction of stock trends. The dataset in this research is a sub dataset of NASDAQ Index in Taiwan Economic Journal Database (TEJD) in 2008. The feature selection part was using a hybrid method, supported sequential forward search (SSFS) played the role of the wrapper. Another advantage of this work is that they designed a detailed procedure of parameter adjustment with performance under different parameter values. The clear structure of the feature selection model is also heuristic to the primary stage of model structuring. One of the limitations was that the performance of SVM was compared to back-propagation neural network (BPNN) only and did not compare to the other machine learning algorithms.

Sirignano and Cont leveraged a deep learning solution trained on a universal feature set of financial markets in [ 40 ]. The dataset used included buy and sell records of all transactions, and cancellations of orders for approximately 1000 NASDAQ stocks through the order book of the stock exchange. The NN consists of three layers with LSTM units and a feed-forward layer with rectified linear units (ReLUs) at last, with stochastic gradient descent (SGD) algorithm as an optimization. Their universal model was able to generalize and cover the stocks other than the ones in the training data. Though they mentioned the advantages of a universal model, the training cost was still expensive. Meanwhile, due to the inexplicit programming of the deep learning algorithm, it is unclear that if there are useless features contaminated when feeding the data into the model. Authors found out that it would have been better if they performed feature selection part before training the model and found it as an effective way to reduce the computational complexity.

Ni et al. in [ 30 ] predicted stock price trends by exploiting SVM and performed fractal feature selection for optimization. The dataset they used is the Shanghai Stock Exchange Composite Index (SSECI), with 19 technical indicators as features. Before processing the data, they optimized the input data by performing feature selection. When finding the best parameter combination, they also used a grid search method, which is k cross-validation. Besides, the evaluation of different feature selection methods is also comprehensive. As the authors mentioned in their conclusion part, they only considered the technical indicators but not macro and micro factors in the financial domain. The source of datasets that the authors used was similar to our dataset, which makes their evaluation results useful to our research. They also mentioned a method called k cross-validation when testing hyper-parameter combinations.

McNally et al. in [ 27 ] leveraged RNN and LSTM on predicting the price of Bitcoin, optimized by using the Boruta algorithm for feature engineering part, and it works similarly to the random forest classifier. Besides feature selection, they also used Bayesian optimization to select LSTM parameters. The Bitcoin dataset ranged from the 19th of August 2013 to 19th of July 2016. Used multiple optimization methods to improve the performance of deep learning methods. The primary problem of their work is overfitting. The research problem of predicting Bitcoin price trend has some similarities with stock market price prediction. Hidden features and noises embedded in the price data are threats of this work. The authors treated the research question as a time sequence problem. The best part of this paper is the feature engineering and optimization part; we could replicate the methods they exploited in our data pre-processing.

Weng et al. in [ 45 ] focused on short-term stock price prediction by using ensemble methods of four well-known machine learning models. The dataset for this research is five sets of data. They obtained these datasets from three open-sourced APIs and an R package named TTR. The machine learning models they used are (1) neural network regression ensemble (NNRE), (2) a Random Forest with unpruned regression trees as base learners (RFR), (3) AdaBoost with unpruned regression trees as base learners (BRT) and (4) a support vector regression ensemble (SVRE). A thorough study of ensemble methods specified for short-term stock price prediction. With background knowledge, the authors selected eight technical indicators in this study then performed a thoughtful evaluation of five datasets. The primary contribution of this paper is that they developed a platform for investors using R, which does not need users to input their own data but call API to fetch the data from online source straightforward. From the research perspective, they only evaluated the prediction of the price for 1 up to 10 days ahead but did not evaluate longer terms than two trading weeks or a shorter term than 1 day. The primary limitation of their research was that they only analyzed 20 U.S.-based stocks, the model might not be generalized to other stock market or need further revalidation to see if it suffered from overfitting problems.

Kara et al. in [ 17 ] also exploited ANN and SVM in predicting the movement of stock price index. The data set they used covers a time period from January 2, 1997, to December 31, 2007, of the Istanbul Stock Exchange. The primary strength of this work is its detailed record of parameter adjustment procedures. While the weaknesses of this work are that neither the technical indicator nor the model structure has novelty, and the authors did not explain how their model performed better than other models in previous works. Thus, more validation works on other datasets would help. They explained how ANN and SVM work with stock market features, also recorded the parameter adjustment. The implementation part of our research could benefit from this previous work.

Jeon et al. in [ 16 ] performed research on millisecond interval-based big dataset by using pattern graph tracking to complete stock price prediction tasks. The dataset they used is a millisecond interval-based big dataset of historical stock data from KOSCOM, from August 2014 to October 2014, 10G–15G capacity. The author applied Euclidean distance, Dynamic Time Warping (DTW) for pattern recognition. For feature selection, they used stepwise regression. The authors completed the prediction task by ANN and Hadoop and RHive for big data processing. The “ Results ” section is based on the result processed by a combination of SAX and Jaro–Winkler distance. Before processing the data, they generated aggregated data at 5-min intervals from discrete data. The primary strength of this work is the explicit structure of the whole implementation procedure. While they exploited a relatively old model, another weakness is the overall time span of the training dataset is extremely short. It is difficult to access the millisecond interval-based data in real life, so the model is not as practical as a daily based data model.

Huang et al. in [ 12 ] applied a fuzzy-GA model to complete the stock selection task. They used the key stocks of the 200 largest market capitalization listed as the investment universe in the Taiwan Stock Exchange. Besides, the yearly financial statement data and the stock returns were taken from the Taiwan Economic Journal (TEJ) database at www.tej.com.tw/ for the time period from year 1995 to year 2009. They conducted the fuzzy membership function with model parameters optimized with GA and extracted features for optimizing stock scoring. The authors proposed an optimized model for selection and scoring of stocks. Different from the prediction model, the authors more focused on stock rankings, selection, and performance evaluation. Their structure is more practical among investors. But in the model validation part, they did not compare the model with existed algorithms but the statistics of the benchmark, which made it challenging to identify if GA would outperform other algorithms.

Fischer and Krauss in [ 5 ] applied long short-term memory (LSTM) on financial market prediction. The dataset they used is S&P 500 index constituents from Thomson Reuters. They obtained all month-end constituent lists for the S&P 500 from Dec 1989 to Sep 2015, then consolidated the lists into a binary matrix to eliminate survivor bias. The authors also used RMSprop as an optimizer, which is a mini-batch version of rprop. The primary strength of this work is that the authors used the latest deep learning technique to perform predictions. They relied on the LSTM technique, lack of background knowledge in the financial domain. Although the LSTM outperformed the standard DNN and logistic regression algorithms, while the author did not mention the effort to train an LSTM with long-time dependencies.

Tsai and Hsiao in [ 42 ] proposed a solution as a combination of different feature selection methods for prediction of stocks. They used Taiwan Economic Journal (TEJ) database as data source. The data used in their analysis was from year 2000 to 2007. In their work, they used a sliding window method and combined it with multi layer perceptron (MLP) based artificial neural networks with back propagation, as their prediction model. In their work, they also applied principal component analysis (PCA) for dimensionality reduction, genetic algorithms (GA) and the classification and regression trees (CART) to select important features. They did not just rely on technical indices only. Instead, they also included both fundamental and macroeconomic indices in their analysis. The authors also reported a comparison on feature selection methods. The validation part was done by combining the model performance stats with statistical analysis.

Pimenta et al. in [ 32 ] leveraged an automated investing method by using multi-objective genetic programming and applied it in the stock market. The dataset was obtained from Brazilian stock exchange market (BOVESPA), and the primary techniques they exploited were a combination of multi-objective optimization, genetic programming, and technical trading rules. For optimization, they leveraged genetic programming (GP) to optimize decision rules. The novelty of this paper was in the evaluation part. They included a historical period, which was a critical moment of Brazilian politics and economics when performing validation. This approach reinforced the generalization strength of their proposed model. When selecting the sub-dataset for evaluation, they also set criteria to ensure more asset liquidity. While the baseline of the comparison was too basic and fundamental, and the authors did not perform any comparison with other existing models.

Huang and Tsai in [ 13 ] conducted a filter-based feature selection assembled with a hybrid self-organizing feature map (SOFM) support vector regression (SVR) model to forecast Taiwan index futures (FITX) trend. They divided the training samples into clusters to marginally improve the training efficiency. The authors proposed a comprehensive model, which was a combination of two novel machine learning techniques in stock market analysis. Besides, the optimizer of feature selection was also applied before the data processing to improve the prediction accuracy and reduce the computational complexity of processing daily stock index data. Though they optimized the feature selection part and split the sample data into small clusters, it was already strenuous to train daily stock index data of this model. It would be difficult for this model to predict trading activities in shorter time intervals since the data volume would be increased drastically. Moreover, the evaluation is not strong enough since they set a single SVR model as a baseline, but did not compare the performance with other previous works, which caused difficulty for future researchers to identify the advantages of SOFM-SVR model why it outperforms other algorithms.

Thakur and Kumar in [ 41 ] also developed a hybrid financial trading support system by exploiting multi-category classifiers and random forest (RAF). They conducted their research on stock indices from NASDAQ, DOW JONES, S&P 500, NIFTY 50, and NIFTY BANK. The authors proposed a hybrid model combined random forest (RF) algorithms with a weighted multicategory generalized eigenvalue support vector machine (WMGEPSVM) to generate “Buy/Hold/Sell” signals. Before processing the data, they used Random Forest (RF) for feature pruning. The authors proposed a practical model designed for real-life investment activities, which could generate three basic signals for investors to refer to. They also performed a thorough comparison of related algorithms. While they did not mention the time and computational complexity of their works. Meanwhile, the unignorable issue of their work was the lack of financial domain knowledge background. The investors regard the indices data as one of the attributes but could not take the signal from indices to operate a specific stock straightforward.

Hsu in [ 11 ] assembled feature selection with a back propagation neural network (BNN) combined with genetic programming to predict the stock/futures price. The dataset in this research was obtained from Taiwan Stock Exchange Corporation (TWSE). The authors have introduced the description of the background knowledge in detail. While the weakness of their work is that it is a lack of data set description. This is a combination of the model proposed by other previous works. Though we did not see the novelty of this work, we can still conclude that the genetic programming (GP) algorithm is admitted in stock market research domain. To reinforce the validation strengths, it would be good to consider adding GP models into evaluation if the model is predicting a specific price.

Hafezi et al. in [ 7 ] built a bat-neural network multi-agent system (BN-NMAS) to predict stock price. The dataset was obtained from the Deutsche bundes-bank. They also applied the Bat algorithm (BA) for optimizing neural network weights. The authors illustrated their overall structure and logic of system design in clear flowcharts. While there were very few previous works that had performed on DAX data, it would be difficult to recognize if the model they proposed still has the generality if migrated on other datasets. The system design and feature selection logic are fascinating, which worth referring to. Their findings in optimization algorithms are also valuable for the research in the stock market price prediction research domain. It is worth trying the Bat algorithm (BA) when constructing neural network models.

Long et al. in [ 25 ] conducted a deep learning approach to predict the stock price movement. The dataset they used is the Chinese stock market index CSI 300. For predicting the stock price movement, they constructed a multi-filter neural network (MFNN) with stochastic gradient descent (SGD) and back propagation optimizer for learning NN parameters. The strength of this paper is that the authors exploited a novel model with a hybrid model constructed by different kinds of neural networks, it provides an inspiration for constructing hybrid neural network structures.

Atsalakis and Valavanis in [ 1 ] proposed a solution of a neuro-fuzzy system, which is composed of controller named as Adaptive Neuro Fuzzy Inference System (ANFIS), to achieve short-term stock price trend prediction. The noticeable strength of this work is the evaluation part. Not only did they compare their proposed system with the popular data models, but also compared with investment strategies. While the weakness that we found from their proposed solution is that their solution architecture is lack of optimization part, which might limit their model performance. Since our proposed solution is also focusing on short-term stock price trend prediction, this work is heuristic for our system design. Meanwhile, by comparing with the popular trading strategies from investors, their work inspired us to compare the strategies used by investors with techniques used by researchers.

Nekoeiqachkanloo et al. in [ 29 ] proposed a system with two different approaches for stock investment. The strengths of their proposed solution are obvious. First, it is a comprehensive system that consists of data pre-processing and two different algorithms to suggest the best investment portions. Second, the system also embedded with a forecasting component, which also retains the features of the time series. Last but not least, their input features are a mix of fundamental features and technical indices that aim to fill in the gap between the financial domain and technical domain. However, their work has a weakness in the evaluation part. Instead of evaluating the proposed system on a large dataset, they chose 25 well-known stocks. There is a high possibility that the well-known stocks might potentially share some common hidden features.

As another related latest work, Idrees et al. [ 14 ] published a time series-based prediction approach for the volatility of the stock market. ARIMA is not a new approach in the time series prediction research domain. Their work is more focusing on the feature engineering side. Before feeding the features into ARIMA models, they designed three steps for feature engineering: Analyze the time series, identify if the time series is stationary or not, perform estimation by plot ACF and PACF charts and look for parameters. The only weakness of their proposed solution is that the authors did not perform any customization on the existing ARIMA model, which might limit the system performance to be improved.

One of the main weaknesses found in the related works is limited data-preprocessing mechanisms built and used. Technical works mostly tend to focus on building prediction models. When they select the features, they list all the features mentioned in previous works and go through the feature selection algorithm then select the best-voted features. Related works in the investment domain have shown more interest in behavior analysis, such as how herding behaviors affect the stock performance, or how the percentage of inside directors hold the firm’s common stock affects the performance of a certain stock. These behaviors often need a pre-processing procedure of standard technical indices and investment experience to recognize.

In the related works, often a thorough statistical analysis is performed based on a special dataset and conclude new features rather than performing feature selections. Some data, such as the percentage of a certain index fluctuation has been proven to be effective on stock performance. We believe that by extracting new features from data, then combining such features with existed common technical indices will significantly benefit the existing and well-tested prediction models.

The dataset

This section details the data that was extracted from the public data sources, and the final dataset that was prepared. Stock market-related data are diverse, so we first compared the related works from the survey of financial research works in stock market data analysis to specify the data collection directions. After collecting the data, we defined a data structure of the dataset. Given below, we describe the dataset in detail, including the data structure, and data tables in each category of data with the segment definitions.

Description of our dataset

In this section, we will describe the dataset in detail. This dataset consists of 3558 stocks from the Chinese stock market. Besides the daily price data, daily fundamental data of each stock ID, we also collected the suspending and resuming history, top 10 shareholders, etc. We list two reasons that we choose 2 years as the time span of this dataset: (1) most of the investors perform stock market price trend analysis using the data within the latest 2 years, (2) using more recent data would benefit the analysis result. We collected data through the open-sourced API, namely Tushare [ 43 ], mean-while we also leveraged a web-scraping technique to collect data from Sina Finance web pages, SWS Research website.

Data structure

Figure  1 illustrates all the data tables in the dataset. We collected four categories of data in this dataset: (1) basic data, (2) trading data, (3) finance data, and (4) other reference data. All the data tables can be linked to each other by a common field called “Stock ID” It is a unique stock identifier registered in the Chinese Stock market. Table  1 shows an overview of the dataset.

Data structure for the extracted dataset

The Table  1 lists the field information of each data table as well as which category the data table belongs to.

In this section, we present the proposed methods and the design of the proposed solution. Moreover, we also introduce the architecture design as well as algorithmic and implementation details.

Problem statement

We analyzed the best possible approach for predicting short-term price trends from different aspects: feature engineering, financial domain knowledge, and prediction algorithm. Then we addressed three research questions in each aspect, respectively: How can feature engineering benefit model prediction accuracy? How do findings from the financial domain benefit prediction model design? And what is the best algorithm for predicting short-term price trends?

The first research question is about feature engineering. We would like to know how the feature selection method benefits the performance of prediction models. From the abundance of the previous works, we can conclude that stock price data embedded with a high level of noise, and there are also correlations between features, which makes the price prediction notoriously difficult. That is also the primary reason for most of the previous works introduced the feature engineering part as an optimization module.

The second research question is evaluating the effectiveness of findings we extracted from the financial domain. Different from the previous works, besides the common evaluation of data models such as the training costs and scores, our evaluation will emphasize the effectiveness of newly added features that we extracted from the financial domain. We introduce some features from the financial domain. While we only obtained some specific findings from previous works, and the related raw data needs to be processed into usable features. After extracting related features from the financial domain, we combine the features with other common technical indices for voting out the features with a higher impact. There are numerous features said to be effective from the financial domain, and it would be impossible for us to cover all of them. Thus, how to appropriately convert the findings from the financial domain to a data processing module of our system design is a hidden research question that we attempt to answer.

The third research question is that which algorithms are we going to model our data? From the previous works, researchers have been putting efforts into the exact price prediction. We decompose the problem into predicting the trend and then the exact number. This paper focuses on the first step. Hence, the objective has been converted to resolve a binary classification problem, meanwhile, finding an effective way to eliminate the negative effect brought by the high level of noise. Our approach is to decompose the complex problem into sub-problems which have fewer dependencies and resolve them one by one, and then compile the resolutions into an ensemble model as an aiding system for investing behavior reference.

In the previous works, researchers have been using a variety of models for predicting stock price trends. While most of the best-performed models are based on machine learning techniques, in this work, we will compare our approach with the outperformed machine learning models in the evaluation part and find the solution for this research question.

Proposed solution

The high-level architecture of our proposed solution could be separated into three parts. First is the feature selection part, to guarantee the selected features are highly effective. Second, we look into the data and perform the dimensionality reduction. And the last part, which is the main contribution of our work is to build a prediction model of target stocks. Figure  2 depicts a high-level architecture of the proposed solution.

High-level architecture of the proposed solution

There are ways to classify different categories of stocks. Some investors prefer long-term investments, while others show more interest in short-term investments. It is common to see the stock-related reports showing an average performance, while the stock price is increasing drastically; this is one of the phenomena that indicate the stock price prediction has no fixed rules, thus finding effective features before training a model on data is necessary.

In this research, we focus on the short-term price trend prediction. Currently, we only have the raw data with no labels. So, the very first step is to label the data. We mark the price trend by comparing the current closing price with the closing price of n trading days ago, the range of n is from 1 to 10 since our research is focusing on the short-term. If the price trend goes up, we mark it as 1 or mark as 0 in the opposite case. To be more specified, we use the indices from the indices of n  −  1 th day to predict the price trend of the n th day.

According to the previous works, some researchers who applied both financial domain knowledge and technical methods on stock data were using rules to filter the high-quality stocks. We referred to their works and exploited their rules to contribute to our feature extension design.

However, to ensure the best performance of the prediction model, we will look into the data first. There are a large number of features in the raw data; if we involve all the features into our consideration, it will not only drastically increase the computational complexity but will also cause side effects if we would like to perform unsupervised learning in further research. So, we leverage the recursive feature elimination (RFE) to ensure all the selected features are effective.

We found most of the previous works in the technical domain were analyzing all the stocks, while in the financial domain, researchers prefer to analyze the specific scenario of investment, to fill the gap between the two domains, we decide to apply a feature extension based on the findings we gathered from the financial domain before we start the RFE procedure.

Since we plan to model the data into time series, the number of the features, the more complex the training procedure will be. So, we will leverage the dimensionality reduction by using randomized PCA at the beginning of our proposed solution architecture.

Detailed technical design elaboration

This section provides an elaboration of the detailed technical design as being a comprehensive solution based on utilizing, combining, and customizing several existing data preprocessing, feature engineering, and deep learning techniques. Figure  3 provides the detailed technical design from data processing to prediction, including the data exploration. We split the content by main procedures, and each procedure contains algorithmic steps. Algorithmic details are elaborated in the next section. The contents of this section will focus on illustrating the data workflow.

Detailed technical design of the proposed solution

Based on the literature review, we select the most commonly used technical indices and then feed them into the feature extension procedure to get the expanded feature set. We will select the most effective i features from the expanded feature set. Then we will feed the data with i selected features into the PCA algorithm to reduce the dimension into j features. After we get the best combination of i and j , we process the data into finalized the feature set and feed them into the LSTM [ 10 ] model to get the price trend prediction result.

The novelty of our proposed solution is that we will not only apply the technical method on raw data but also carry out the feature extensions that are used among stock market investors. Details on feature extension are given in the next subsection. Experiences gained from applying and optimizing deep learning based solutions in [ 37 , 38 ] were taken into account while designing and customizing feature engineering and deep learning solution in this work.

Applying feature extension

The first main procedure in Fig.  3 is the feature extension. In this block, the input data is the most commonly used technical indices concluded from related works. The three feature extension methods are max–min scaling, polarizing, and calculating fluctuation percentage. Not all the technical indices are applicable for all three of the feature extension methods; this procedure only applies the meaningful extension methods on technical indices. We choose meaningful extension methods while looking at how the indices are calculated. The technical indices and the corresponding feature extension methods are illustrated in Table  2 .

After the feature extension procedure, the expanded features will be combined with the most commonly used technical indices, i.e., input data with output data, and feed into RFE block as input data in the next step.

Applying recursive feature elimination

After the feature extension above, we explore the most effective i features by using the Recursive Feature Elimination (RFE) algorithm [ 6 ]. We estimate all the features by two attributes, coefficient, and feature importance. We also limit the features that remove from the pool by one, which means we will remove one feature at each step and retain all the relevant features. Then the output of the RFE block will be the input of the next step, which refers to PCA.

Applying principal component analysis (PCA)

The very first step before leveraging PCA is feature pre-processing. Because some of the features after RFE are percentage data, while others are very large numbers, i.e., the output from RFE are in different units. It will affect the principal component extraction result. Thus, before feeding the data into the PCA algorithm [ 8 ], a feature pre-processing is necessary. We also illustrate the effectiveness and methods comparison in “ Results ” section.

After performing feature pre-processing, the next step is to feed the processed data with selected i features into the PCA algorithm to reduce the feature matrix scale into j features. This step is to retain as many effective features as possible and meanwhile eliminate the computational complexity of training the model. This research work also evaluates the best combination of i and j, which has relatively better prediction accuracy, meanwhile, cuts the computational consumption. The result can be found in the “ Results ” section, as well. After the PCA step, the system will get a reshaped matrix with j columns.

Fitting long short-term memory (LSTM) model

PCA reduced the dimensions of the input data, while the data pre-processing is mandatory before feeding the data into the LSTM layer. The reason for adding the data pre-processing step before the LSTM model is that the input matrix formed by principal components has no time steps. While one of the most important parameters of training an LSTM is the number of time steps. Hence, we have to model the matrix into corresponding time steps for both training and testing dataset.

After performing the data pre-processing part, the last step is to feed the training data into LSTM and evaluate the performance using testing data. As a variant neural network of RNN, even with one LSTM layer, the NN structure is still a deep neural network since it can process sequential data and memorizes its hidden states through time. An LSTM layer is composed of one or more LSTM units, and an LSTM unit consists of cells and gates to perform classification and prediction based on time series data.

The LSTM structure is formed by two layers. The input dimension is determined by j after the PCA algorithm. The first layer is the input LSTM layer, and the second layer is the output layer. The final output will be 0 or 1 indicates if the stock price trend prediction result is going down or going up, as a supporting suggestion for the investors to perform the next investment decision.

Design discussion

Feature extension is one of the novelties of our proposed price trend predicting system. In the feature extension procedure, we use technical indices to collaborate with the heuristic processing methods learned from investors, which fills the gap between the financial research area and technical research area.

Since we proposed a system of price trend prediction, feature engineering is extremely important to the final prediction result. Not only the feature extension method is helpful to guarantee we do not miss the potentially correlated feature, but also feature selection method is necessary for pooling the effective features. The more irrelevant features are fed into the model, the more noise would be introduced. Each main procedure is carefully considered contributing to the whole system design.

Besides the feature engineering part, we also leverage LSTM, the state-of-the-art deep learning method for time-series prediction, which guarantees the prediction model can capture both complex hidden pattern and the time-series related pattern.

It is known that the training cost of deep learning models is expansive in both time and hardware aspects; another advantage of our system design is the optimization procedure—PCA. It can retain the principal components of the features while reducing the scale of the feature matrix, thus help the system to save the training cost of processing the large time-series feature matrix.

Algorithm elaboration

This section provides comprehensive details on the algorithms we built while utilizing and customizing different existing techniques. Details about the terminologies, parameters, as well as optimizers. From the legend on the right side of Fig.  3 , we note the algorithm steps as octagons, all of them can be found in this “ Algorithm elaboration ” section.

Before dive deep into the algorithm steps, here is the brief introduction of data pre-processing: since we will go through the supervised learning algorithms, we also need to program the ground truth. The ground truth of this research is programmed by comparing the closing price of the current trading date with the closing price of the previous trading date the users want to compare with. Label the price increase as 1, else the ground truth will be labeled as 0. Because this research work is not only focused on predicting the price trend of a specific period of time but short-term in general, the ground truth processing is according to a range of trading days. While the algorithms will not change with the prediction term length, we can regard the term length as a parameter.

The algorithmic detail is elaborated, respectively, the first algorithm is the hybrid feature engineering part for preparing high-quality training and testing data. It corresponds to the Feature extension, RFE, and PCA blocks in Fig.  3 . The second algorithm is the LSTM procedure block, including time-series data pre-processing, NN constructing, training, and testing.

Algorithm 1: Short-term stock market price trend prediction—applying feature engineering using FE + RFE + PCA

The function FE is corresponding to the feature extension block. For the feature extension procedure, we apply three different processing methods to translate the findings from the financial domain to a technical module in our system design. While not all the indices are applicable for expanding, we only choose the proper method(s) for certain features to perform the feature extension (FE), according to Table  2 .

Normalize method preserves the relative frequencies of the terms, and transform the technical indices into the range of [0, 1]. Polarize is a well-known method often used by real-world investors, sometimes they prefer to consider if the technical index value is above or below zero, we program some of the features using polarize method and prepare for RFE. Max-min (or min-max) [ 35 ] scaling is a transformation method often used as an alternative to zero mean and unit variance scaling. Another well-known method used is fluctuation percentage, and we transform the technical indices fluctuation percentage into the range of [− 1, 1].

The function RFE () in the first algorithm refers to recursive feature elimination. Before we perform the training data scale reduction, we will have to make sure that the features we selected are effective. Ineffective features will not only drag down the classification precision but also add more computational complexity. For the feature selection part, we choose recursive feature elimination (RFE). As [ 45 ] explained, the process of recursive feature elimination can be split into the ranking algorithm, resampling, and external validation.

For the ranking algorithm, it fits the model to the features and ranks by the importance to the model. We set the parameter to retain i numbers of features, and at each iteration of feature selection retains Si top-ranked features, then refit the model and assess the performance again to begin another iteration. The ranking algorithm will eventually determine the top Si features.

The RFE algorithm is known to have suffered from the over-fitting problem. To eliminate the over-fitting issue, we will run the RFE algorithm multiple times on randomly selected stocks as the training set and ensure all the features we select are high-weighted. This procedure is called data resampling. Resampling can be built as an optimization step as an outer layer of the RFE algorithm.

The last part of our hybrid feature engineering algorithm is for optimization purposes. For the training data matrix scale reduction, we apply Randomized principal component analysis (PCA) [ 31 ], before we decide the features of the classification model.

Financial ratios of a listed company are used to present the growth ability, earning ability, solvency ability, etc. Each financial ratio consists of a set of technical indices, each time we add a technical index (or feature) will add another column of data into the data matrix and will result in low training efficiency and redundancy. If non-relevant or less relevant features are included in training data, it will also decrease the precision of classification.

The above equation represents the explanation power of principal components extracted by PCA method for original data. If an ACR is below 85%, the PCA method would be unsuitable due to a loss of original information. Because the covariance matrix is sensitive to the order of magnitudes of data, there should be a data standardize procedure before performing the PCA. The commonly used standardized methods are mean-standardization and normal-standardization and are noted as given below:

Mean-standardization: \(X_{ij}^{*} = X_{ij} /\overline{{X_{j} }}\) , which \(\overline{{X_{j} }}\) represents the mean value.

Normal-standardization: \(X_{ij}^{*} = (X_{ij} - \overline{{X_{j} }} )/s_{j}\) , which \(\overline{{X_{j} }}\) represents the mean value, and \(s_{j}\) is the standard deviation.

The array fe_array is defined according to Table  2 , row number maps to the features, columns 0, 1, 2, 3 note for the extension methods of normalize, polarize, max–min scale, and fluctuation percentage, respectively. Then we fill in the values for the array by the rule where 0 stands for no necessity to expand and 1 for features need to apply the corresponding extension methods. The final algorithm of data preprocessing using RFE and PCA can be illustrated as Algorithm 1.

Algorithm 2: Price trend prediction model using LSTM

After the principal component extraction, we will get the scale-reduced matrix, which means i most effective features are converted into j principal components for training the prediction model. We utilized an LSTM model and added a conversion procedure for our stock price dataset. The detailed algorithm design is illustrated in Alg 2. The function TimeSeriesConversion () converts the principal components matrix into time series by shifting the input data frame according to the number of time steps [ 3 ], i.e., term length in this research. The processed dataset consists of the input sequence and forecast sequence. In this research, the parameter of LAG is 1, because the model is detecting the pattern of features fluctuation on a daily basis. Meanwhile, the N_TIME_STEPS is varied from 1 trading day to 10 trading days. The functions DataPartition (), FitModel (), EvaluateModel () are regular steps without customization. The NN structure design, optimizer decision, and other parameters are illustrated in function ModelCompile () .

Some procedures impact the efficiency but do not affect the accuracy or precision and vice versa, while other procedures may affect both efficiency and prediction result. To fully evaluate our algorithm design, we structure the evaluation part by main procedures and evaluate how each procedure affects the algorithm performance. First, we evaluated our solution on a machine with 2.2 GHz i7 processor, with 16 GB of RAM. Furthermore, we also evaluated our solution on Amazon EC2 instance, 3.1 GHz Processor with 16 vCPUs, and 64 GB RAM.

In the implementation part, we expanded 20 features into 54 features, while we retain 30 features that are the most effective. In this section, we discuss the evaluation of feature selection. The dataset was divided into two different subsets, i.e., training and testing datasets. Test procedure included two parts, one testing dataset is for feature selection, and another one is for model testing. We note the feature selection dataset and model testing dataset as DS_test_f and DS_test_m, respectively.

We randomly selected two-thirds of the stock data by stock ID for RFE training and note the dataset as DS_train_f; all the data consist of full technical indices and expanded features throughout 2018. The estimator of the RFE algorithm is SVR with linear kernels. We rank the 54 features by voting and get 30 effective features then process them using the PCA algorithm to perform dimension reduction and reduce the features into 20 principal components. The rest of the stock data forms the testing dataset DS_test_f to validate the effectiveness of principal components we extracted from selected features. We reformed all the data from 2018 as the training dataset of the data model and noted as DS_train_m. The model testing dataset DS_test_m consists of the first 3 months of data in 2019, which has no overlap with the dataset we utilized in the previous steps. This approach is to prevent the hidden problem caused by overfitting.

Term length

To build an efficient prediction model, instead of the approach of modeling the data to time series, we determined to use 1 day ahead indices data to predict the price trend of the next day. We tested the RFE algorithm on a range of short-term from 1 day to 2 weeks (ten trading days), to evaluate how the commonly used technical indices correlated to price trends. For evaluating the prediction term length, we fully expanded the features as Table  2 , and feed them to RFE. During the test, we found that different length of the term has a different level of sensitive-ness to the same indices set.

We get the close price of the first trading date and compare it with the close price of the n _ th trading date. Since we are predicting the price trend, we do not consider the term lengths if the cross-validation score is below 0.5. And after the test, as we can see from Fig.  4 , there are three-term lengths that are most sensitive to the indices we selected from the related works. They are n  = {2, 5, 10}, which indicates that price trend prediction of every other day, 1 week, and 2 weeks using the indices set are likely to be more reliable.

How do term lengths affect the cross-validation score of RFE

While these curves have different patterns, for the length of 2 weeks, the cross-validation score increases with the number of features selected. If the prediction term length is 1 week, the cross-validation score will decrease if selected over 8 features. For every other day price trend prediction, the best cross-validation score is achieved by selecting 48 features. Biweekly prediction requires 29 features to achieve the best score. In Table  3 , we listed the top 15 effective features for these three-period lengths. If we predict the price trend of every other day, the cross-validation score merely fluctuates with the number of features selected. So, in the next step, we will evaluate the RFE result for these three-term lengths, as shown in Fig.  4 .

We compare the output feature set of RFE with the all-original feature set as a baseline, the all-original feature set consists of n features and we choose n most effective features from RFE output features to evaluate the result using linear SVR. We used two different approaches to evaluate feature effectiveness. The first method is to combine all the data into one large matrix and evaluate them by running the RFE algorithm once. Another method is to run RFE for each individual stock and calculate the most effective features by voting.

Feature extension and RFE

From the result of the previous subsection, we can see that when predicting the price trend for every other day or biweekly, the best result is achieved by selecting a large number of features. Within the selected features, some features processed from extension methods have better ranks than original features, which proves that the feature extension method is useful for optimizing the model. The feature extension affects both precision and efficiency, while in this part, we only discuss the precision aspect and leave efficiency part in the next step since PCA is the most effective method for training efficiency optimization in our design. We involved an evaluation of how feature extension affects RFE and use the test result to measure the improvement of involving feature extension.

We further test the effectiveness of feature extension, i.e., if polarize, max–min scale, and calculate fluctuation percentage works better than original technical indices. The best case to leverage this test is the weekly prediction since it has the least effective feature selected. From the result we got from the last section, we know the best cross-validation score appears when selecting 8 features. The test consists of two steps, and the first step is to test the feature set formed by original features only, in this case, only SLOWK, SLOWD, and RSI_5 are included. The next step is to test the feature set of all 8 features we selected in the previous subsection. We leveraged the test by defining the simplest DNN model with three layers.

The normalized confusion matrix of testing the two feature sets are illustrated in Fig.  5 . The left one is the confusion matrix of the feature set with expanded features, and the right one besides is the test result of using original features only. Both precisions of true positive and true negative have been improved by 7% and 10%, respectively, which proves that our feature extension method design is reasonably effective.

Confusion matrix of validating feature extension effectiveness

Feature reduction using principal component analysis

PCA will affect the algorithm performance on both prediction accuracy and training efficiency, while this part should be evaluated with the NN model, so we also defined the simplest DNN model with three layers as we used in the previous step to perform the evaluation. This part introduces the evaluation method and result of the optimization part of the model from computational efficiency and accuracy impact perspectives.

In this section, we will choose bi-weekly prediction to perform a use case analysis, since it has a smoothly increasing cross-validation score curve, moreover, unlike every other day prediction, it has excluded more than 20 ineffective features already. In the first step, we select all 29 effective features and train the NN model without performing PCA. It creates a baseline of the accuracy and training time for comparison. To evaluate the accuracy and efficiency, we keep the number of the principal component as 5, 10, 15, 20, 25. Table  4 recorded how the number of features affects the model training efficiency, then uses the stack bar chart in Fig.  6 to illustrate how PCA affects training efficiency. Table  6 shows accuracy and efficiency analysis on different procedures for the pre-processing of features. The times taken shown in Tables  4 , 6 are based on experiments conducted in a standard user machine to show the viability of our solution with limited or average resource availability.

Relationship between feature number and training time

We also listed the confusion matrix of each test in Fig.  7 . The stack bar chart shows that the overall time spends on training the model is decreasing by the number of selected features, while the PCA method is significantly effective in optimizing training dataset preparation. For the time spent on the training stage, PCA is not as effective as the data preparation stage. While there is the possibility that the optimization effect of PCA is not drastic enough because of the simple structure of the NN model.

How does the number of principal components affect evaluation results

Table  5 indicates that the overall prediction accuracy is not drastically affected by reducing the dimension. However, the accuracy could not fully support if the PCA has no side effect to model prediction, so we looked into the confusion matrices of test results.

From Fig.  7 we can conclude that PCA does not have a severe negative impact on prediction precision. The true positive rate and false positive rate are barely be affected, while the false negative and true negative rates are influenced by 2% to 4%. Besides evaluating how the number of selected features affects the training efficiency and model performance, we also leveraged a test upon how data pre-processing procedures affect the training procedure and predicting result. Normalizing and max–min scaling is the most commonly seen data pre-procedure performed before PCA, since the measure units of features are varied, and it is said that it could increase the training efficiency afterward.

We leveraged another test on adding pre-procedures before extracting 20 principal components from the original dataset and make the comparison in the aspects of time elapse of training stage and prediction precision. However, the test results lead to different conclusions. In Table  6 we can conclude that feature pre-processing does not have a significant impact on training efficiency, but it does influence the model prediction accuracy. Moreover, the first confusion matrix in Fig.  8 indicates that without any feature pre-processing procedure, the false-negative rate and true negative rate are severely affected, while the true positive rate and false positive rate are not affected. If it performs the normalization before PCA, both true positive rate and true negative rate are decreasing by approximately 10%. This test also proved that the best feature pre-processing method for our feature set is exploiting the max–min scale.

Confusion matrices of different feature pre-processing methods

In this section, we discuss and compare the results of our proposed model, other approaches, and the most related works.

Comparison with related works

From the previous works, we found the most commonly exploited models for short-term stock market price trend prediction are support vector machine (SVM), multilayer perceptron artificial neural network (MLP), Naive Bayes classifier (NB), random forest classifier (RAF) and logistic regression classifier (LR). The test case of comparison is also bi-weekly price trend prediction, to evaluate the best result of all models, we keep all 29 features selected by the RFE algorithm. For MLP evaluation, to test if the number of hidden layers would affect the metric scores, we noted layer number as n and tested n  = {1, 3, 5}, 150 training epochs for all the tests, found slight differences in the model performance, which indicates that the variable of MLP layer number hardly affects the metric scores.

From the confusion matrices in Fig.  9 , we can see all the machine learning models perform well when training with the full feature set we selected by RFE. From the perspective of training time, training the NB model got the best efficiency. LR algorithm cost less training time than other algorithms while it can achieve a similar prediction result with other costly models such as SVM and MLP. RAF algorithm achieved a relatively high true-positive rate while the poor performance in predicting negative labels. For our proposed LSTM model, it achieves a binary accuracy of 93.25%, which is a significantly high precision of predicting the bi-weekly price trend. We also pre-processed data through PCA and got five principal components, then trained for 150 epochs. The learning curve of our proposed solution, based on feature engineering and the LSTM model, is illustrated in Fig.  10 . The confusion matrix is the figure on the right in Fig.  11 , and detailed metrics scores can be found in Table  9 .

Model prediction comparison—confusion matrices

Learning curve of proposed solution

Proposed model prediction precision comparison—confusion matrices

The detailed evaluate results are recorded in Table  7 . We will also initiate a discussion upon the evaluation result in the next section.

Because the resulting structure of our proposed solution is different from most of the related works, it would be difficult to make naïve comparison with previous works. For example, it is hard to find the exact accuracy number of price trend prediction in most of the related works since the authors prefer to show the gain rate of simulated investment. Gain rate is a processed number based on simulated investment tests, sometimes one correct investment decision with a large trading volume can achieve a high gain rate regardless of the price trend prediction accuracy. Besides, it is also a unique and heuristic innovation in our proposed solution, we transform the problem of predicting an exact price straight forward to two sequential problems, i.e., predicting the price trend first, focus on building an accurate binary classification model, construct a solid foundation for predicting the exact price change in future works. Besides the different result structure, the datasets that previous works researched on are also different from our work. Some of the previous works involve news data to perform sentiment analysis and exploit the SE part as another system component to support their prediction model.

The latest related work that can compare is Zubair et al. [ 47 ], the authors take multiple r-square for model accuracy measurement. Multiple r-square is also called the coefficient of determination, and it shows the strength of predictor variables explaining the variation in stock return [ 28 ]. They used three datasets (KSE 100 Index, Lucky Cement Stock, Engro Fertilizer Limited) to evaluate the proposed multiple regression model and achieved 95%, 89%, and 97%, respectively. Except for the KSE 100 Index, the dataset choice in this related work is individual stocks; thus, we choose the evaluation result of the first dataset of their proposed model.

We listed the leading stock price trend prediction model performance in Table  8 , from the comparable metrics, the metric scores of our proposed solution are generally better than other related works. Instead of concluding arbitrarily that our proposed model outperformed other models in related works, we first look into the dataset column of Table  8 . By looking into the dataset used by each work [ 18 ], only trained and tested their proposed solution on three individual stocks, which is difficult to prove the generalization of their proposed model. Ayo [ 2 ] leveraged analysis on the stock data from the New York Stock Exchange (NYSE), while the weakness is they only performed analysis on closing price, which is a feature embedded with high noise. Zubair et al. [ 47 ] trained their proposed model on both individual stocks and index price, but as we have mentioned in the previous section, index price only consists of the limited number of features and stock IDs, which will further affect the model training quality. For our proposed solution, we collected sufficient data from the Chinese stock market, and applied FE + RFE algorithm on the original indices to get more effective features, the comprehensive evaluation result of 3558 stock IDs can reasonably explain the generalization and effectiveness of our proposed solution in Chinese stock market. However, the authors of Khaidem and Dey [ 18 ] and Ayo [ 2 ] chose to analyze the stock market in the United States, Zubair et al. [ 47 ] performed analysis on Pakistani stock market price, and we obtained the dataset from Chinese stock market, the policies of different countries might impact the model performance, which needs further research to validate.

Proposed model evaluation—PCA effectiveness

Besides comparing the performance across popular machine learning models, we also evaluated how the PCA algorithm optimizes the training procedure of the proposed LSTM model. We recorded the confusion matrices comparison between training the model by 29 features and by five principal components in Fig.  11 . The model training using the full 29 features takes 28.5 s per epoch on average. While it only takes 18 s on average per epoch training on the feature set of five principal components. PCA has significantly improved the training efficiency of the LSTM model by 36.8%. The detailed metrics data are listed in Table  9 . We will leverage a discussion in the next section about complexity analysis.

Complexity analysis of proposed solution

This section analyzes the complexity of our proposed solution. The Long Short-term Memory is different from other NNs, and it is a variant of standard RNN, which also has time steps with memory and gate architecture. In the previous work [ 46 ], the author performed an analysis of the RNN architecture complexity. They introduced a method to regard RNN as a directed acyclic graph and proposed a concept of recurrent depth, which helps perform the analysis on the intricacy of RNN.

The recurrent depth is a positive rational number, and we denote it as \(d_{rc}\) . As the growth of \(n\) \(d_{rc}\) measures, the nonlinear transformation average maximum number of each time step. We then unfold the directed acyclic graph of RNN and denote the processed graph as \(g_{c}\) , meanwhile, denote \(C(g_{c} )\) as the set of directed cycles in this graph. For the vertex \(v\) , we note \(\sigma_{s} (v)\) as the sum of edge weights and \(l(v)\) as the length. The equation below is proved under a mild assumption, which could be found in [ 46 ].

They also found that another crucial factor that impacts the performance of LSTM, which is the recurrent skip coefficients. We note \(s_{rc}\) as the reciprocal of the recurrent skip coefficient. Please be aware that \(s_{rc}\) is also a positive rational number.

According to the above definition, our proposed model is a 2-layers stacked LSTM, which \(d_{rc} = 2\) and \(s_{rc} = 1\) . From the experiments performed in previous work, the authors also found that when facing the problems of long-term dependency, LSTMs may benefit from decreasing the reciprocal of recurrent skip coefficients and from increasing recurrent depth. The empirical findings above mentioned are useful to enhance the performance of our proposed model further.

This work consists of three parts: data extraction and pre-processing of the Chinese stock market dataset, carrying out feature engineering, and stock price trend prediction model based on the long short-term memory (LSTM). We collected, cleaned-up, and structured 2 years of Chinese stock market data. We reviewed different techniques often used by real-world investors, developed a new algorithm component, and named it as feature extension, which is proved to be effective. We applied the feature expansion (FE) approaches with recursive feature elimination (RFE), followed by principal component analysis (PCA), to build a feature engineering procedure that is both effective and efficient. The system is customized by assembling the feature engineering procedure with an LSTM prediction model, achieved high prediction accuracy that outperforms the leading models in most related works. We also carried out a comprehensive evaluation of this work. By comparing the most frequently used machine learning models with our proposed LSTM model under the feature engineering part of our proposed system, we conclude many heuristic findings that could be future research questions in both technical and financial research domains.

Our proposed solution is a unique customization as compared to the previous works because rather than just proposing yet another state-of-the-art LSTM model, we proposed a fine-tuned and customized deep learning prediction system along with utilization of comprehensive feature engineering and combined it with LSTM to perform prediction. By researching into the observations from previous works, we fill in the gaps between investors and researchers by proposing a feature extension algorithm before recursive feature elimination and get a noticeable improvement in the model performance.

Though we have achieved a decent outcome from our proposed solution, this research has more potential towards research in future. During the evaluation procedure, we also found that the RFE algorithm is not sensitive to the term lengths other than 2-day, weekly, biweekly. Getting more in-depth research into what technical indices would influence the irregular term lengths would be a possible future research direction. Moreover, by combining latest sentiment analysis techniques with feature engineering and deep learning model, there is also a high potential to develop a more comprehensive prediction system which is trained by diverse types of information such as tweets, news, and other text-based data.

Abbreviations

Long short term memory

Principal component analysis

Recurrent neural networks

Artificial neural network

Deep neural network

Dynamic Time Warping

Recursive feature elimination

Support vector machine

Convolutional neural network

Stochastic gradient descent

Rectified linear unit

Multi layer perceptron

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This research is supported by Carleton University, in Ottawa, ON, Canada. This research paper has been built based on the thesis [ 36 ] of Jingyi Shen, supervised by M. Omair Shafiq at Carleton University, Canada, available at https://curve.carleton.ca/52e9187a-7f71-48ce-bdfe-e3f6a420e31a .

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How to Construct an Index for Research

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An index is a composite measure of variables, or a way of measuring a construct--like religiosity or racism--using more than one data item. An index is an accumulation of scores from a variety of individual items. To create one, you must select possible items, examine their empirical relationships, score the index, and validate it.

Item Selection

The first step in creating an index is selecting the items you wish to include in the index to measure the variable of interest. There are several things to consider when selecting the items. First, you should select items that have face validity. That is, the item should measure what it is intended to measure. If you are constructing an index of religiosity, items such as church attendance and frequency of prayer would have face validity because they appear to offer some indication of religiosity.

A second criterion for choosing which items to include in your index is unidimensionality. That is, each item should represent only one dimension of the concept you are measuring. For example, items reflecting depression should not be included in items measuring anxiety, even though the two might be related to one another.

Third, you need to decide how general or specific your variable will be. For example, if you only wish to measure a specific aspect of religiosity, such as ritual participation, then you would only want to include items that measure ritual participation, such as church attendance, confession, communion, etc. If you are measuring religiosity in a more general way, however, you would want to also include a more balanced set of items that touch on other areas of religion (such as beliefs, knowledge, etc.).

Lastly, when choosing which items to include in your index, you should pay attention to the amount of variance that each item provides. For example, if an item is intended to measure religious conservatism, you need to pay attention to what proportion of respondents would be identified as religiously conservative by that measure. If the item identifies nobody as religiously conservative or everyone as a religiously conservative, then the item has no variance and it is not a useful item for your index.

Examining Empirical Relationships

The second step in index construction is to examine the empirical relationships among the items you wish to include in the index. An empirical relationship is when respondents’ answers to one question help us predict how they will answer other questions. If two items are empirically related to each other, we can argue that both items reflect the same concept and we can, therefore, include them in the same index. To determine if your items are empirically related, crosstabulations, correlation coefficients , or both may be used.

Index Scoring

The third step in index construction is scoring the index. After you have finalized the items you are including in your index, you then assign scores for particular responses, thereby making a composite variable out of your several items. For example, let’s say you are measuring religious ritual participation among Catholics and the items included in your index are church attendance, confession, communion, and daily prayer, each with a response choice of "yes, I regularly participate" or "no, I do not regularly participate." You might assign a 0 for "does not participate" and a 1 for "participates." Therefore, a respondent could receive a final composite score of 0, 1, 2, 3, or 4 with 0 being the least engaged in Catholic rituals and 4 being the most engaged.

Index Validation

The final step in constructing an index is validating it. Just like you need to validate each item that goes into the index, you also need to validate the index itself to make sure that it measures what it is intended to measure. There are several methods for doing this. One is called item analysis in which you examine the extent to which the index is related to the individual items that are included in it. Another important indicator of an index’s validity is how well it accurately predicts related measures. For example, if you are measuring political conservatism, those who score the most conservative in your index should also score conservative in other questions included in the survey.

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Research Impact

• Author Identification
• Citation Indexes

Citation Indexes Overview

Other citation index numbers.

• Journal Impact
• More Resources

What are index numbers?

Citation index numbers provide a way to measure impact beyond raw citation counts. Index numbers can be calculated for individual articles, a group/list of publications, or even all the articles published in a journal or field (see our Journal Impact page).

What is the "best" index number?

Generally, the "best" measurement depends on what matters to you. The h-index is the most widely known index measurement. Some alternative measurements, like the g-index, address specific issues with the h-index. Other measurements target recent publications and citations, such as the the contemporary h-index. 

Alternatives to the h-index include:

• g-index :  Gives more weight to highly cited publications. The original h-index is insensitive to high "outliers" -- a few papers that have very high citation counts will not sway the h-index score (much). The g-index allows highly cited papers to play a larger role in the index, and tends to emphasize visibility and "lifetime achievement."
• hc-index (contemporary h-index) :  Gives more weight to recent publications. The original h-index favors senior researchers with extensive publication records, even if they have ceased publishing. The hc-index attempts to correct this and favors researchers currently publishing.
• i10-index: Measures the number of papers that have at least 10 citations. Introduced (and used) by Google Scholar.
• m-quotient: Divides the h-index by the number of years since the researcher's first published paper. m-quotient was proposed as a way to help younger researchers who may not have long publication lists.

For more index measurements, we suggest " Reflections on the  h- index ," by Prof. Anne-Wil Harzing, University of Melbourne.

What is the h-index?

The h-index attempts to correlate a researcher's total publications and total citations. It was proposed by Jorge E. Hirsch in 2005 (" An index to quantify an individual's scientific research output ," PNAS November 15, 2005 vol. 102 no. 46 16569-16572). For more information, see the Wikipedia article .

Graph of the h-index, from Wikipedia.

How do I calculate my h-index?

• Web of Science or Google Scholar will automatically calculate the h-index for the list of publications in your profile. 
• Publish or Perish will calculate h-index (and many other index numbers) for an author's publications. 
• If you want to calculate an h-index manually, Hirsch defines the h-index as follows: "A scientist has index  h  if  h  of his or her  Np  papers have at least  h  citations each and the other ( Np – h ) papers have ≤h  citations each."
• << Previous: Citation Metrics
• Next: Journal Impact >>
• Last Updated: Jan 26, 2024 11:12 AM
• URL: https://guides.library.georgetown.edu/researchimpact

topological indices Recently Published Documents

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Topological indices based on 2- or 3-eccentricity to predict anti-HIV activity

On degree based topological indices of tio2 crystal via m-polynomial.

Topological indices (TI) (descriptors) of a molecular graph are very much useful to study various physiochemical properties. It is also used to develop the quantitative structure-activity relationship (QSAR), quantitative structure-property relationship (QSPR) of the corresponding chemical compound. Various techniques have been developed to calculate the TI of a graph. Recently a technique of calculating degree-based TI from M-polynomial has been introduced. We have evaluated various topological descriptors for 3-dimensional TiO2 crystals using M-polynomial. These descriptors are constructed such that it contains 3 variables (m, n and t) each corresponding to a particular direction. These 3 variables facilitate us to deeply understand the growth of TiO2 in 1 dimension (1D), 2 dimensions (2D), and 3 dimensions (3D) respectively. HIGHLIGHTS Calculated degree based Topological indices of a 3D crystal from M-polynomial A relation among various Topological indices is established geometrically Variations of Topological Indices along three dimensions (directions) are shown geometrically Harmonic index approximates the degree variation of oxygen atom

Computing Reverse Degree Based Topological Indices of Vanadium Carbide

Computation of vertex degree-based molecular descriptors of hydrocarbon structure.

Topological indices are such numbers or set of numbers that describe topology of structures. Nearly 400 topological indices are calculated so far. The prognostication of physical, chemical, and biological attributes of organic compounds is an important and still unsolved problem of computational chemistry. Topological index is the tool to predict the physicochemical properties such as boiling point, melting point, density, viscosity, and polarity of organic compounds. In this study, some degree-based molecular descriptors of hydrocarbon structure are calculated.

On Reverse Valency Based Topological Indices of Metal–Organic Framework

Computing some degree-based topological indices of honeycomb networks.

A topological index is a numeric quantity related with the chemical composition claiming to correlate the chemical structure with different chemical properties. Topological indices serve to predict physicochemical properties of chemical substance. Among different topological indices, degree-based topological indices would be helpful in investigating the anti-inflammatory activities of certain chemical networks. In the current study, we determine the neighborhood second Zagreb index and the first extended first-order connectivity index for oxide network O X n , silicate network S L n , chain silicate network C S n , and hexagonal network H X n . Also, we determine the neighborhood second Zagreb index and the first extended first-order connectivity index for honeycomb network H C n .

On ev and ve-Degree Based Topological Indices of Silicon Carbides

Statistical analysis on the topological indices of clustered graphs, on analysis of banhatti indices for hyaluronic acid curcumin and hydroxychloroquine.

Topological indices are numerical numbers assigned to the graph/structure and are useful to predict certain physical/chemical properties. In this paper, we give explicit expressions of novel Banhatti indices, namely, first K Banhatti index B 1 G , second K Banhatti index B 2 G , first K hyper-Banhatti index HB 1 G , second K hyper-Banhatti index HB 2 G , and K Banhatti harmonic index H b G for hyaluronic acid curcumin and hydroxychloroquine. The multiplicative version of these indices is also computed for these structures.

On the Sum of Degree-Based Topological Indices of Rhombus-Type Silicate and Oxide Structures

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Indexed journal: What does it mean?

Yatan pal singh balhara.

Department of Psychiatry and De-addiction, Lady Hardinge Medical College and Smt Sucheta Kriplani Hospital, New Delhi, India E-mail: moc.liamg@arahlabspy

Indexation of a journal is considered a reflection of its quality. Indexed journals are considered to be of higher scientific quality as compared to non-indexed journals. Indexation of medical journals has become a debatable issue. For a long-time Index Medicus has been the most comprehensive index of medical scientific journal articles. It is being publication since 1879. Over the years, many other popular indexation services have developed. These include MedLine, PubMed, EMBASE, SCOPUS, EBSCO Publishing's Electronic Databases, SCIRUS among others. There are various regional and national versions of Index Medicus such as African Index Medicus.

A related and equally controversial issue is that of impact factor (IF).[ 1 ] IF is used as a proxy for the relative importance of a journal within its field. IF is awarded to the journals indexed in Thomson Reuters Journal Citation Reports. IF has been criticised for manipulation and incorrect application.[ 2 ] There are multiple factors that could bias the calculation of the IF.[ 3 ] These include coverage and language preference of the database, procedures used to collect citations, algorithm used to calculate the IF, citation distribution of journals, online availability of publications, negative citations, preference of journal publishers for articles of a certain type, publication lag, citing behaviour across subjects, and possibility of exertion of influence from journal editors.[ 4 ] Interestingly, IF is not available for all indexed journals. In fact, not all journals indexed even in Index Medicus/MedLine/PubMed are indexed in the Thomson Reuters Journal Citation Reports. Similarly, not all journals indexed in Thomson Reuters Journal Citation Reports and consequently have an IF are listed in Index Medicus/PubMed/MedLine.

This brings us to the question which indexation is best and most valid? How to compare the quality of articles published in journals indexed with different indexation services? These questions are of particular relevance for two main reasons. First, importance of publications is being increasingly recognised by the academic institutions. MCI guidelines also recommend indexed publications for teaching faculty in medical colleges. Consequently many more authors would be publishing than ever before.[ 4 ] Selection of high quality journal becomes a difficult decision for the authors as there is no clarity on the issue. Should one aim at only the journals indexed in Index Medicus/MedLine/PubMed? Is it appropriate to make submissions to journals having a high impact factor although they are not indexed with Index Medicus/MedLine/PubMed?

Second, recently many more indexation services have come up. These include Caspur, DOAJ, Expanded Academic ASAP, Genamics Journal Seek, Hinari, Index Copernicus, Open J Gate, Primo Central, Pro Quest, SCOLOAR, SIIC databases, Summon by Serial Solutions, Ulrich's International Periodical Directory. Are these indexations services equally relevant? Would a journal indexed with any of these databases be considered “indexed”?

These are some questions that warrant discussion. Associations of editors of medical journals such as International Committee of Medical Journal Editors could play a pivotal role in such discussion.

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Producer price index by commodity: pulp, paper, and allied products: wood pulp (wpu0911).

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Units:   Index 1982=100 , Not Seasonally Adjusted

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U.S. Bureau of Labor Statistics, Producer Price Index by Commodity: Pulp, Paper, and Allied Products: Wood Pulp [WPU0911], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/WPU0911, April 26, 2024.

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