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Augmented Dickey Fuller Test (ADF Test) – Must Read Guide

  • November 2, 2019
  • Selva Prabhakaran

Augmented Dickey Fuller test (ADF Test) is a common statistical test used to test whether a given Time series is stationary or not. It is one of the most commonly used statistical test when it comes to analyzing the stationary of a series.

1. Introduction

In ARIMA time series forecasting , the first step is to determine the number of differencing required to make the series stationary.

Since testing the stationarity of a time series is a frequently performed activity in autoregressive models, the ADF test along with KPSS test is something that you need to be fluent in when performing time series analysis .

Another point to remember is the ADF test is fundamentally a statistical significance test. That means, there is a hypothesis testing involved with a null and alternate hypothesis and as a result a test statistic is computed and p-values get reported.

It is from the test statistic and the p-value, you can make an inference as to whether a given series is stationary or not.

So, how exactly does the ADF test work? let’s see the mathematical intuition behind the test with clear examples.

Let’s begin.

2. What is a Unit Root Test?

The ADF test belongs to a category of tests called ‘Unit Root Test’, which is the proper method for testing the stationarity of a time series.

So what does a ‘Unit Root’ mean?

Unit root is a characteristic of a time series that makes it non-stationary. Technically speaking, a unit root is said to exist in a time series of the value of alpha = 1 in the below equation.

null and alternative hypothesis of adf test

where, Y t is the value of the time series at time ‘t’ and X e is an exogenous variable (a separate explanatory variable, which is also a time series).

What does this mean to us?

The presence of a unit root means the time series is non-stationary. Besides, the number of unit roots contained in the series corresponds to the number of differencing operations required to make the series stationary.

Alright, let’s come back to topic.

null and alternative hypothesis of adf test

3. Dickey-Fuller Test

Before going into ADF test, let’s first understand what is the Dickey-Fuller test.

A Dickey-Fuller test is a unit root test that tests the null hypothesis that α=1 in the following model equation. alpha is the coefficient of the first lag on Y.

Null Hypothesis (H0): alpha=1

null and alternative hypothesis of adf test

  • y(t-1) = lag 1 of time series
  • delta Y(t-1) = first difference of the series at time (t-1)

Fundamentally, it has a similar null hypothesis as the unit root test. That is, the coefficient of Y(t-1) is 1, implying the presence of a unit root. If not rejected, the series is taken to be non-stationary.

The Augmented Dickey-Fuller test evolved based on the above equation and is one of the most common form of Unit Root test.

4. How does Augmented Dickey Fuller (ADF) Test work?

As the name suggest, the ADF test is an ‘augmented’ version of the Dickey Fuller test.

The ADF test expands the Dickey-Fuller test equation to include high order regressive process in the model.

null and alternative hypothesis of adf test

If you notice, we have only added more differencing terms, while the rest of the equation remains the same. This adds more thoroughness to the test.

The null hypothesis however is still the same as the Dickey Fuller test.

A key point to remember here is: Since the null hypothesis assumes the presence of unit root, that is α=1, the p-value obtained should be less than the significance level (say 0.05) in order to reject the null hypothesis. Thereby, inferring that the series is stationary.

However, this is a very common mistake analysts commit with this test. That is, if the p-value is less than significance level, people mistakenly take the series to be non-stationary.

5. ADF Test in Python

So, how to perform a Augmented Dickey-Fuller test in Python?

The statsmodel package provides a reliable implementation of the ADF test via the adfuller() function in statsmodels.tsa.stattools . It returns the following outputs:

  • The p-value
  • The value of the test statistic
  • Number of lags considered for the test
  • The critical value cutoffs.

When the test statistic is lower than the critical value shown, you reject the null hypothesis and infer that the time series is stationary.

Alright, let’s run the ADF test on the a10 dataset from the fpp package from R. This dataset counts the total monthly scripts for pharmaceutical products falling under ATC code A10. The original source of this dataset is the Australian Health Insurance Commission.

As see earlier, the null hypothesis of the test is the presence of unit root, that is, the series is non-stationary.

null and alternative hypothesis of adf test

The packages and the data is loaded, we have everything needed to perform the test using adfuller() .

An optional argument the adfuller() accepts is the number of lags you want to consider while performing the OLS regression.

By default, this value is 12*(nobs/100)^{1/4} , where nobs is the number of observations in the series. But, optionally you can specify either the maximum number of lags with maxlags parameter or let the algorithm compute the optimal number iteratively.

This can be done by setting the autolag='AIC' . By doing so, the adfuller will choose a the number of lags that yields the lowest AIC. This is usually a good option to follow.

The p-value is obtained is greater than significance level of 0.05 and the ADF statistic is higher than any of the critical values.

Clearly, there is no reason to reject the null hypothesis. So, the time series is in fact non-stationary.

6. ADF Test on stationary series

Now, let’s see another example of performing the test on a series of random numbers which is usually considered as stationary.

Let’s use np.random.randn() to generate a randomized series.

The p-value is very less than the significance level of 0.05 and hence we can reject the null hypothesis and take that the series is stationary.

Let’s visualise the series as well to confirm.

null and alternative hypothesis of adf test

7. Conclusion

We saw how the Augmented Dickey Fuller Test works and how to perform it using statsmodels . Now given any time series, you should be in a position to perform the ADF Test and make a fair inference on whether the series is stationary or not.

In the next one we’ll see how to perform the KPSS test .

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  • Last updated February 13, 2024
  • In AI Mysteries

Augmented Dickey-Fuller (ADF) Test In Time-Series Analysis

  • by Yugesh Verma

null and alternative hypothesis of adf test

When we make a model for forecasting purposes in time series analysis, we require a stationary time series for better prediction. So the first step to work on modeling is to make a time series stationary. Testing for stationarity is a frequently used activity in autoregressive modeling. We can perform various tests like the KPSS, Phillips–Perron, and Augmented Dickey-Fuller. This article is more focused on the Dickey-Fuller test. The article will see the mathematics behind the test and how we can implement it in a time series.

ADF (Augmented Dickey-Fuller) test is a statistical significance test which means the test will give results in hypothesis tests with null and alternative hypotheses. As a result, we will have a p-value from which we will need to make inferences about the time series, whether it is stationary or not.

Before going into the ADF test, we must know about the unit root test because the ADF test belongs to the unit root test.

Unit Root Test

A unit root test tests whether a time series is not stationary and consists of a unit root in time series analysis. The presence of a unit root in time series defines the null hypothesis, and the alternative hypothesis defines time series as stationary.

Mathematically the unit root test can be represented as  

null and alternative hypothesis of adf test

  • Dt is the deterministic component.
  • z t is the stochastic component.
  • ɛ t is the stationary error process.

The unit root test’s basic concept is to determine whether the z t (stochastic component ) consists of a unit root or not.

There are various tests which include unit root tests.

  • Augmented Dickey-Fuller test.
  • Phillips-perron test.
  • ADF-GLS test
  • Breusch-godfrey test.
  • Ljung-Box test.
  • Durbin-watson test.

 Let’s move into our motive, which is the Dickey-Fuller test.

Explanation of the Dickey-Fuller test.

A simple AR model can be represented as:

null and alternative hypothesis of adf test

  • y t is variable of interest at the time t
  • ρ is a coefficient that defines the unit root
  • u t  is noise or can be considered as an error term.

If ρ = 1, the unit root is present in a time series, and the time series is non-stationary.

If a regression model can be represented as 

null and alternative hypothesis of adf test

Where 

  • Δ is a difference operator.

So here, if ρ = 1, which means we will get the differencing as the error term and if the coefficient has some values smaller than one or bigger than one, we will see the changes according to the past observation.

There can be three versions of the test.

  • test for a unit root
  • test for a unit root with constant
  • test for a unit root with the constant and deterministic trends with time

So if a time series is non-stationary, it will tend to return an error term or a deterministic trend with the time values. If the series is stationary, then it will tend to return only an error term or deterministic trend. In a stationary time series, a large value tends to be followed by a small value, and a small value tends to be followed by a large value. And in a non-stationary time series the large and the small value will accrue with probabilities that do not depend on the current value of the time series.

The augmented dickey- fuller test is an extension of the dickey-fuller test, which removes autocorrelation from the series and then tests similar to the procedure of the dickey-fuller test. 

The augmented dickey fuller test works on the statistic, which gives a negative number and rejection of the hypothesis depends on that negative number; the more negative magnitude of the number represents the confidence of presence of unit root at some level in the time series.

 We apply ADF on a model, and it can be represented mathematically as

null and alternative hypothesis of adf test

  • ɑ is a constant 
  • ???? is the coefficient at time. 
  • p is the lag order of the autoregressive process.

Here in the mathematical representation of ADF, we have added the differencing terms that make changes between ADF and the Dickey-Fuller test.

The unit root test is then carried out under the null hypothesis ???? = 0 against the alternative hypothesis of ???? < 0. Once a value for the test statistic.

null and alternative hypothesis of adf test

it can be compared to the relevant critical value for the Dickey-Fuller test. The test has a specific distribution simply known as the Dickey–Fuller table for critical values.

A key point to remember here is: Since the null hypothesis assumes the presence of a unit root, the p-value obtained by the test should be less than the significance level (say 0.05) to reject the null hypothesis. Thereby, inferring that the series is stationary.

Implementation of ADF Test

To perform the ADF test in any time series package, statsmodel provides the implementation function adfuller().

Function adfuller() provides the following information.

  • Value of the test statistic
  • Number of lags for testing consideration
  • The critical values

Next in the article, we will perform the ADF test with airline passengers data that is non-stationary, and temperature data that is stationary.

Importing the libraries:

Reading the airline-passengers data 

Checking for some values of the data.

data.head()

null and alternative hypothesis of adf test

Plotting the data.

data.plot(figsize=(14,8), title='alcohol data series')

null and alternative hypothesis of adf test

Here we can see that the data we are using is non-stationary because the number of passengers is integrated positively with time.

Now that we have all the things we require, we can perform our test on the time series.

Taking out the passengers number as a series.

null and alternative hypothesis of adf test

Performing the ADF test on the series:

Extracting the values from the results:

null and alternative hypothesis of adf test

Here in the results, we can see that the p-value for time series is greater than 0.05, and we can say we fail to reject the null hypothesis and the time series is non-stationary. 

Now, let’s check the test for stationary data.

Loading the data.

Checking for some head values of the data:

null and alternative hypothesis of adf test

Here we can see that the data has the average temperature values for every day.

data.plot(figsize=(14,8), title='temperature data series')

null and alternative hypothesis of adf test

Here we can see that in the data, the larger value follows the next smaller value throughout the time series, so we can say the time series is stationary and check it with the ADF test.

Extracting temperature in a series.

null and alternative hypothesis of adf test

Performing ADF test.

result = adfuller(series, autolag='AIC')

Checking the results:

null and alternative hypothesis of adf test

In the results, we can see that the p-value obtained from the test is less than 0.05 so we are going to reject the null hypothesis “Time series is stationary”, that means the time series is non-stationary.

In the article, we have seen why we need to perform the ADF test and the algorithms that the ADF and dickey-fuller test follow to make inferences about any time series. Statsmodel is one of the packages which allows us to perform many kinds of tests and analysis regarding time series analysis.

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null and alternative hypothesis of adf test

Methods and formulas for Augmented Dickey-Fuller Test

In this topic, regression models, test statistic, mackinnon's approximate p-values, determination of lag order.

null and alternative hypothesis of adf test

Each Augmented Dickey-Fuller test uses the following hypotheses:

null and alternative hypothesis of adf test

The null hypothesis says that a unit root is in the time series sample, which means that the mean of the data is not stationary. Rejecting the null hypothesis indicates that the mean of the data is stationary or trend stationary, depending on the model for the test.

null and alternative hypothesis of adf test

Under the null hypothesis, the asymptotic distribution of the test statistic does not follow a standard distribution. Fuller (1976) 1 provides a table with common percentiles of the asymptotic distribution. MacKinnon (1994 2 , 2010 3 ) applies response surface approximations to simulated data to provide an approximate p-value for any value of the ADF test statistic.

If the specifications for the analysis use 0.01, 0.05, or 0.1 as the significance level, then the evaluation of the null hypothesis compares the test statistic to the critical value for that significance level. If the test statistic is less than or equal to the critical value, reject the null hypothesis.

If the specifications for the analysis give a different significance level, then the evaluation of the null hypothesis compares the approximate p-value to the significance level. If the p-value is less than the significance level, reject the null hypothesis.

Critical values for the significance levels 0.01, 0.05, and 0.1

null and alternative hypothesis of adf test

Approximate p-values

The calculation of the approximate p-value comes from Mackinnon (1994). Compare the p-value to the significance level to make a decision. If the p-value is less than or equal to the significance level, reject the null hypothesis.

null and alternative hypothesis of adf test

The selection of the lag order depends on the criterion in the specifications of the analysis. If the specifications for the analysis do not include a criterion, then the regression model for the test is the maximum order of p .

In the calculations to determine the lag order, the number of observations depends on the maximum lag order such that m = n – p – 1.

The calculation of each criteria follows:

Akaike Information Criterion (AIC)

The analysis evaluates a regression model for each lag order in the specifications of the analysis. The lag order for the test is the regression model with the minimum value of the AIC.

null and alternative hypothesis of adf test

Bayesian Information Criterion (BIC)

The analysis evaluates a regression model for each lag order in the specifications of the analysis. The lag order for the test is the regression model with the minimum value of the BIC.

null and alternative hypothesis of adf test

t-statistic

null and alternative hypothesis of adf test

where i = 1, …, p

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Stationarity and detrending (ADF/KPSS) ¶

Stationarity means that the statistical properties of a time series i.e. mean, variance and covariance do not change over time. Many statistical models require the series to be stationary to make effective and precise predictions.

Two statistical tests would be used to check the stationarity of a time series – Augmented Dickey Fuller (“ADF”) test and Kwiatkowski-Phillips-Schmidt-Shin (“KPSS”) test. A method to convert a non-stationary time series into stationary series shall also be used.

This first cell imports standard packages and sets plots to appear inline.

Sunspots dataset is used. It contains yearly (1700-2008) data on sunspots from the National Geophysical Data Center.

Some preprocessing is carried out on the data. The “YEAR” column is used in creating index.

The data is plotted now.

../../../_images/examples_notebooks_generated_stationarity_detrending_adf_kpss_8_1.png

ADF test is used to determine the presence of unit root in the series, and hence helps in understand if the series is stationary or not. The null and alternate hypothesis of this test are:

Null Hypothesis: The series has a unit root.

Alternate Hypothesis: The series has no unit root.

If the null hypothesis in failed to be rejected, this test may provide evidence that the series is non-stationary.

A function is created to carry out the ADF test on a time series.

KPSS test ¶

KPSS is another test for checking the stationarity of a time series. The null and alternate hypothesis for the KPSS test are opposite that of the ADF test.

Null Hypothesis: The process is trend stationary.

Alternate Hypothesis: The series has a unit root (series is not stationary).

A function is created to carry out the KPSS test on a time series.

The ADF tests gives the following results – test statistic, p value and the critical value at 1%, 5% , and 10% confidence intervals.

ADF test is now applied on the data.

Based upon the significance level of 0.05 and the p-value of ADF test, the null hypothesis can not be rejected. Hence, the series is non-stationary.

The KPSS tests gives the following results – test statistic, p value and the critical value at 1%, 5% , and 10% confidence intervals.

KPSS test is now applied on the data.

Based upon the significance level of 0.05 and the p-value of KPSS test, there is evidence for rejecting the null hypothesis in favor of the alternative. Hence, the series is non-stationary as per the KPSS test.

It is always better to apply both the tests, so that it can be ensured that the series is truly stationary. Possible outcomes of applying these stationary tests are as follows:

Here, due to the difference in the results from ADF test and KPSS test, it can be inferred that the series is trend stationary and not strict stationary. The series can be detrended by differencing or by model fitting.

Detrending by Differencing ¶

It is one of the simplest methods for detrending a time series. A new series is constructed where the value at the current time step is calculated as the difference between the original observation and the observation at the previous time step.

Differencing is applied on the data and the result is plotted.

../../../_images/examples_notebooks_generated_stationarity_detrending_adf_kpss_20_1.png

ADF test is now applied on these detrended values and stationarity is checked.

Based upon the p-value of ADF test, there is evidence for rejecting the null hypothesis in favor of the alternative. Hence, the series is strict stationary now.

KPSS test is now applied on these detrended values and stationarity is checked.

Based upon the p-value of KPSS test, the null hypothesis can not be rejected. Hence, the series is stationary.

Conclusion ¶

Two tests for checking the stationarity of a time series are used, namely ADF test and KPSS test. Detrending is carried out by using differencing. Trend stationary time series is converted into strict stationary time series. Requisite forecasting model can now be applied on a stationary time series data.

Statology

Statistics Made Easy

Augmented Dickey-Fuller Test in R (With Example)

A time series is said to be “stationary” if it has no trend, exhibits constant variance over time, and has a constant autocorrelation structure over time.

One way to test whether a time series is stationary is to perform an augmented Dickey-Fuller test , which uses the following null and alternative hypotheses:

H 0 : The time series is non-stationary. In other words, it has some time-dependent structure and does not have constant variance over time.

H A : The time series is stationary.

If the p-value from the test is less than some significance level (e.g. α = .05), then we can reject the null hypothesis and conclude that the time series is stationary.

The following step-by-step example shows how to perform an augmented Dickey-Fuller test in R for a given time series.

Example: Augmented Dickey-Fuller Test in R

Suppose we have the following time series data in R:

Before we perform an augmented Dickey-Fuller test on the data, we can create a quick plot to visualize the data:

null and alternative hypothesis of adf test

To perform an augmented Dickey-Fuller test, we can use the adf.test() function from the tseries library. 

The following code shows how to use this function:

Here’s how to interpret the most important values in the output:

  • Test statistic:  -2.2048
  • P-value:  0.4943

Since the p-value is not less than .05, we fail to reject the null hypothesis.

This means the time series is non-stationary. In other words, it has some time-dependent structure and does not have constant variance over time.

Additional Resources

The following tutorials explain how to perform other common tasks in R:

How to Perform a Mann-Kendall Trend Test in R How to Plot a Time Series in R How to Detrend Data

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This topic contains the following sections:

  • ADF Test Specification
  • Restrictions and Recommendations
  • ADF Test Implementation and Usage
  • Code Sample

null and alternative hypothesis of adf test

t is the time index,

α is an intercept constant called a drift ,

β is the coefficient on a time trend,

γ is the coefficient presenting process root, i.e. the focus of testing,

p is the lag order of the first-differences autoregressive process,

e t is an independent identically distributes residual term.

ADF Hypotheses

The ADF test ensures that the null hypothesis is accepted unless there is strong evidence against it to reject in favor of the alternate stationarity hypothesis.

ADF testing technique involves Ordinary Least Squares (OLS) method to find the coefficients of the model chosen. To estimate the significance of the coefficients in focus, the modified T (Student)-statistic (known as Dickey-Fuller statistic ) is computed and compared with the relevant critical value: if the test statistic is less than the critical value then the null hypothesis is rejected. Each version of the test has its own critical value which depends on the size of the sample.

The ADF Test has low statistical power in distinguishing between true unit root processes ( γ = 0 ) and near unit root processes ( γ is close to zero). Also, several authors have shown that the ADF test tends to reject the non-stationarity hypothesis far too often, when the series have large (long-run) moving average processes. If there is evidence or doubt about such cases then it is recommended to additionally apply other stationarity tests (for instance, the KPSS Test ) to ensure the results.

Uncertainty about lags order, p , which has to be determined when applying the test. As it is shown by Perron (1989) , including extra regressors of lagged first-differences does not affect the size of the test (probability of Type I Error ) but decreases test power. On the contrary, too few lags may affect the size of the test. Perron suggested a procedure to define the adequate lag order. An alternative approach is to examine information criteria (such as the Bayesian , Hannan-Quinn , etc.) computed for different lags.

Uncertainty about what test version to use, i.e. about including the intercept and time trend terms. Inappropriate exclusion or inclusion of these terms substantially affects test reliability. Using of prior knowledge (for instance, as result of visual inspection of a given time series) about whether the intercept and time trend should be included is the mostly recommended (but not always possible) informal way to overcome the difficulty mentioned. The more formal (algorithmic) approach is known as testing strategy and requires several additional experiments (series of ordered tests) to decide what model is adequate for a given observed process. There are several testing strategies developed, the most known of them are those suggested by Dolado et al. (1990) , by Elder and Kennedy (2001) and by Enders (2004) .

The ADFTest class implements ADF Test procedure according to the specification above. This class inherits from the OneSampleTest class.

The following constructor creates an instance of the ADFTest class.

The following public properties provide the details about the model and decision made:

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Augmented Dickey-Fuller test

Description

h = adftest( y ) returns the rejection decision h from conducting an augmented Dickey-Fuller test for a unit root in a univariate time series y .

[ h , pValue , stat , cValue ] = adftest( y ) also returns the p -value pValue , test statistic stat , and critical value cValue of the test.

StatTbl = adftest( Tbl ) returns the table StatTbl containing variables for the test results, statistics, and settings from conducting an augmented Dickey-Fuller test for a unit root in the last variable of the input table or timetable Tbl . To select a different variable in Tbl to test, use the DataVariable name-value argument.

[ ___ ] = adftest( ___ , Name=Value ) specifies options using one or more name-value arguments in addition to any of the input argument combinations in previous syntaxes. adftest returns the output argument combination for the corresponding input arguments.

Some options control the number of tests to conduct. The following conditions apply when adftest conducts multiple tests:

adftest treats each test as separate from all other tests.

If you specify y , all outputs are vectors.

If you specify Tbl , each row of StatTbl contains the results of the corresponding test.

For example, adftest(Tbl,DataVariable="GDP",Alpha=0.025,Lags=[0 1]) conducts two tests, at a level of significance of 0.025, for the presence of a unit root in the variable GDP of the table Tbl . The first test includes 0 lagged difference terms in the AR model, and the second test includes 1 lagged difference term in the AR model.

[ ___ , reg ] = adftest( ___ ) additionally returns a structure of regression statistics for the hypothesis test reg .

collapse all

Conduct Dickey-Fuller Test Without Augmentation on Vector of Data

Test a time series for a unit root using the default autoregressive model without augmented difference terms. Input the time series data as a numeric vector.

Load the Canadian inflation rate data and extract the CPI-based inflation rate INF_C .

Test the time series for a unit root.

The result h = 0 indicates that this test fails to reject the null hypothesis of a unit root against the autoregressive alternative.

Return Test p -Value and Decision Statistics

Load Canadian inflation rate data and extract the CPI-based inflation rate INF_C .

Test the time series for a unit root. Return the test decision, p -value, test statistic, and critical value.

Conduct Dickey-Fuller Test Without Augmentation on Table Variable

Test a time series, which is one variable in a table, for a unit root using the default autoregressive model without augmented difference terms.

Load Canadian inflation rate data, which contains yearly measurements on five time series variables in the table DataTable .

Test the long-term bond rate series INT_L , the last variable in the table, for a unit root.

adftest returns test results and settings in the table StatTbl , where variables correspond to test results ( h , pValue , stat , and cValue ) and settings ( Lags , Alpha , Model , and Test ), and rows correspond to individual tests (in this case, adftest conducts one test).

By default, adftest tests the last variable in the table. To select a variable from an input table to test, set the DataVariable option.

Conduct Augmented Dickey-Fuller Test Against Trend-Stationary Alternative

Test a time series for a unit root against a trend-stationary alternative augmented with lagged difference terms.

Load a GDP data set. Compute the log of the series.

Test for a unit root against a trend-stationary alternative, augmenting the model with 0, 1, and 2 lagged difference terms.

adftest treats the three lag choices as three separate tests, and returns a vector with rejection decisions for each test. The values h = 0 indicate that all three tests fail to reject the null hypothesis of a unit root against the trend-stationary alternative.

Choose Number of Lags for Test by Inspecting OLS Statistics

Test a time series for a unit root against trend-stationary alternatives augmented with different numbers of lagged difference terms. Look at the regression statistics corresponding to each of the alternative models to choose how many lagged difference terms to include in the augmented model.

Load a US macroeconomic data set Data_USEconModel.mat . Compute the log of the GDP and include the result as a new variable called LogGDP in the data set.

Test for a unit root in the logged GDP series using three different choices for the number of lagged difference terms. Return the regression statistics for each alternative model.

adftest treats each of the three lag choices as separate tests, and returns results and settings for each test along the rows of the table StatTbl . reg is a 3-by-1 structure array containing regression statistics corresponding to each of the three alternative models.

Display the names of the coefficients, their t -statistics and corresponding p -values, and the BIC resulting from the regression of the three alternative models.

The first model has no added difference terms, the second model has one difference term ( b1 ), and the third model has two difference terms ( b1 and b2 ). These results indicate that the coefficient of the first difference term is significantly different from zero in both the second and third models, but the coefficient of the second term in the third model is not at a 0.05 significance level. This result suggests augmenting the model with one lagged difference term is adequate.

Compare the BIC for each of the three alternatives.

Of the three alternative models, the model augmented with one lagged difference term is the best because it yields the lowest BIC.

Input Arguments

Y — univariate time series data numeric vector.

Univariate time series data, specified as a numeric vector. Each element of y represents an observation.

Data Types: double

Tbl — Time series data table | timetable

Time series data, specified as a table or timetable. Each row of Tbl is an observation.

Specify a single series (variable) to test by using the DataVariable argument. The selected variable must be numeric.

adftest removes missing observations, represented by NaN values, from the input series.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN , where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: adftest(Tbl,DataVariable="GDP",Alpha=0.025,Lags=[0 1]) conducts two separate tests at a level of significance of 0.025 on the variable G GDP of the input table Tbl . The first test includes 0 lagged difference terms in the AR model, and the second test includes 1 lagged difference term in the AR model.

Lags — Number of lagged difference terms 0 (default) | nonnegative integer | vector of nonnegative integers

Number p of lagged difference terms to include in the AR model, specified as a nonnegative integer or vector of nonnegative integers.

adftest conducts a separate test for each element in Lags .

Example: Lags=[0 1] includes no lags in the AR model for the first test, and then includes Δ y t – 1 in the AR model for the second test.

Model — Model variant "AR" (default) | "ARD" | "TS" | character vector | string vector | cell vector of character vectors

Model variant, specified as a model variant name, or a string vector or cell vector of model names. This table contains the supported model variant names.

adftest conducts a separate test for each model variant name in Model .

Example: Model=["AR" "ARD"] uses the stationary AR model as the alternative hypothesis for the first test, and then uses the stationary AR model with drift as the alternative hypothesis for the second test.

Data Types: char | cell | string

Test — Test statistic "t1" (default) | "t2" | "F" | character vector | string vector | cell vector of character vectors

Test statistic, specified as a test name, or a string vector or cell vector of test names. This table contains the supported test names.

adftest conducts a separate test for each test name in Test .

Example: Test="F" computes the F test statistic for all tests.

Alpha — Nominal significance level 0.05 (default) | numeric scalar | numeric vector

Nominal significance level for the hypothesis test, specified as a numeric scalar between 0.001 and 0.999 or a numeric vector of such values.

adftest conducts a separate test for each value in Alpha .

Example: Alpha=[0.01 0.05] uses a level of significance of 0.01 for the first test, and then uses a level of significance of 0.05 for the second test.

DataVariable — Variable in Tbl to test last variable (default) | string scalar | character vector | integer | logical vector

Variable in Tbl to test, specified as a string scalar or character vector containing a variable name in Tbl.Properties.VariableNames , or an integer or logical vector representing the index of a name. The selected variable must be numeric.

Example: DataVariable="GDP"

Example: DataVariable=[false true false false] or DataVariable=2 tests the second table variable.

Data Types: double | logical | char | string

When adftest conducts multiple tests, the function applies all single settings (scalars or character vectors) to each test.

All vector-valued specifications that control the number of tests must have equal length.

If you specify the vector y and any value is a row vector, all outputs are row vectors.

A lagged and differenced time series has a reduced sample size. Absent presample values, if the test series y t is defined for t = 1,…, T , the lagged series y t – k is defined for t = k +1,…, T . The first difference applied to the lagged series y t – k further reduces the time base to k +2,…, T . With p lagged differences, the common time base is p +2,…, T and the effective sample size is T –( p +1).

Output Arguments

H — test rejection decisions logical scalar | logical vector.

Test rejection decisions, returned as a logical scalar or vector with length equal to the number of tests. adftest returns h when you supply the input y .

Values of 1 indicate rejection of the unit-root null model in favor of the alternative model.

Values of 0 indicate failure to reject the unit-root null model.

pValue — Test statistic p -values numeric scalar | numeric vector

Test statistic p -values, returned as a numeric scalar or vector with length equal to the number of tests. adftest returns pValue when you supply the input y .

The p -value of test statistic ( Test ) "t1" or "t2" is a left-tail probability.

The p -value of test statistic "F" is a right-tail probability.

When test statistics are outside tabulated critical values, adftest returns maximum ( 0.999 ) or minimum ( 0.001 ) p -values.

stat — Test statistics numeric scalar | numeric vector

Test statistics, returned as a numeric scalar or vector with length equal to the number of tests. adftest returns stat when you supply the input y .

adftest computes test statistics using ordinary least squares (OLS) estimates of the coefficients in the alternative model.

cValue — Critical values numeric scalar | numeric vector

Critical values, returned as a numeric scalar or vector with length equal to the number of tests. adftest returns cValue when you supply the input y .

The critical value of test statistic ( Test ) "t1" or "t2" is for a left-tail probability.

The critical value of test statistic "F" is for a right-tail probability.

StatTbl — Test summary table

Test summary, returned as a table with variables for the outputs h , pValue , stat , and cValue , and with a row for each test. adftest returns StatTbl when you supply the input Tbl .

StatTbl contains variables for the test settings specified by Lags , Alpha , Model , and Test .

reg — Regression statistics structure array

Regression statistics from the OLS estimation of coefficients in the alternative model, returned as a structure array with number of records equal to the number of tests.

Each element of reg has the fields in this table. You can access a field using dot notation, for example, reg(1).coeff contains the coefficient estimates of the first test.

Augmented Dickey-Fuller Test for Unit Root

The augmented Dickey-Fuller test for a unit root assesses the null hypothesis of a unit root in the time series y t , where

y t = c + δ t + ϕ y t − 1 + β 1 Δ y t − 1 + … + β p Δ y t − p + ε t ,

Δ is the differencing operator such that Δ y t = y t − y t − 1 .

p is the number of lagged difference terms (see Lags ).

c is the drift coefficient (see Model ).

δ is the deterministic trend coefficient (see Model ).

ε t is a mean zero innovation process.

The null hypothesis of a unit root is

H 0 : ϕ = 1.

Under the alternative hypothesis, ϕ < 1.

Variants of the model allow for different growth characteristics (see Model ). The model with δ = 0 has no trend component, and the model with c = 0 and δ = 0 has no drift or trend.

A test that fails to reject the null hypothesis, fails to reject the possibility of a unit root.

To draw valid inferences from the test, determine a suitable value for Lags .

One method is to begin with a maximum lag, such as the one recommended in [7] , and then test down by assessing the significance of β ^ p , the coefficient of the largest lagged change in y t . The usual t statistic is appropriate, as returned in the reg output structure.

Another method is to combine a measure of fit, such as the SSR, with information criteria, such as AIC, BIC, and HQC. These statistics are also returned in the reg output structure. For more details, see [6] .

With a specific testing strategy in mind, determine the value of Model by the growth characteristics of y t . If you include too many regressors (see Lags ), the test loses power; if you include too few regressors, the test is biased towards favoring the null model [4] . In general, if a series grows, the "TS" model (see Model ) provides a reasonable trend-stationary alternative to a unit-root process with drift. If a series is does not grow, the "AR" and "ARD" models provide reasonable stationary alternatives to a unit-root process without drift. The "ARD" alternative model has a mean of c /(1 – a ); the "AR" alternative model has mean 0.

Dickey-Fuller statistics follow nonstandard distributions under the null hypothesis (even asymptotically). adftest uses tabulated critical values, generated by Monte Carlo simulations, for a range of sample sizes and significance levels of the null model with Gaussian innovations and five million replications per sample size. adftest interpolates critical values cValue and p -values pValue from the tables. Tables for tests of Test types "t1" and "t2" are identical to those for pptest . For small samples, tabulated values are valid only for Gaussian innovations. For large samples, values are also valid for non-Gaussian innovations.

[1] Davidson, R., and J. G. MacKinnon. Econometric Theory and Methods . Oxford, UK: Oxford University Press, 2004.

[2] Dickey, D. A., and W. A. Fuller. "Distribution of the Estimators for Autoregressive Time Series with a Unit Root." Journal of the American Statistical Association . Vol. 74, 1979, pp. 427–431.

[3] Dickey, D. A., and W. A. Fuller. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root." Econometrica . Vol. 49, 1981, pp. 1057–1072.

[4] Elder, J., and P. E. Kennedy. "Testing for Unit Roots: What Should Students Be Taught?" Journal of Economic Education . Vol. 32, 2001, pp. 137–146.

[5] Hamilton, James D. Time Series Analysis . Princeton, NJ: Princeton University Press, 1994.

[6] Ng, S., and P. Perron. "Unit Root Tests in ARMA Models with Data-Dependent Methods for the Selection of the Truncation Lag." Journal of the American Statistical Association . Vol. 90, 1995, pp. 268–281.

[7] Schwert, W. "Tests for Unit Roots: A Monte Carlo Investigation." Journal of Business and Economic Statistics . Vol. 7, 1989, pp. 147–159.

Version History

Introduced in R2009b

kpsstest | lmctest | pptest | vratiotest | i10test

  • Unit Root Tests
  • Unit Root Nonstationarity

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Quantitative Finance & Algo Trading Blog by QuantInsti

Augmented Dickey Fuller (ADF) Test for a Pairs Trading Strategy

By Chainika Thakar

Augmented Dickey Fuller Test, denoted by "ADF", serves the purpose of finding out which stocks can be paired together in the pairs trading strategy. For the strategy to work effectively, it is important to find the pair of stocks that are co-integrated.

The statistical calculation for testing the co-integration is done with Augmented Dickey Fuller Test. In this blog, we will be discussing the ADF test for pairs trading strategy.

This blog covers:

What is a pairs trading strategy?

Stationary time series, non-stationary time series, what is adf testing, what does the adf test check in pairs trading, difference between df test and adf test.

  • Working of the Augmented Dickey Fuller test

Formula of ADF test

Calculation of adf test, adf test using excel, adf test using python, advantages of adf testing, disadvantages of adf testing.

The origin of the pairs trading strategy was back in the early 1980s. The strategy was initially developed by stock market technical analysts and researchers who were employees of the renowned investment banking and financial company – Morgan Stanley.

In order to implement the pairs trading strategy, one needs technical and statistical analysis skills. The pairs trading strategy basically requires two stocks or securities to co-integrate so that they can be considered as a pair for trading.

Hence, a positive correlation is needed.

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After the securities show co-integration, from the pair of stocks, you enter a long position in one stock and a short position in the other. The positions are based on the current market price of both the stocks and their standard deviation.

Here is a short, educational video that explains the pairs trading method in about 3 minutes. Find out how to combine securities and optimizing your investments using a market-neutral method.

Essential terms used in the Augmented Dickey Fuller Test

Stationarity means that the statistical properties of a time series i.e. mean, variance and covariance do not change over time which implies that the time series has no unit root.

In the case of stationarity, trading signals are generated assuming that the prices of both stocks will revert to the mean eventually. Hence, we can take the advantage of the prices that deviate from the mean for a short period of time.

A time series with a unit root is non-stationary and will have changes in its mean, variance and covariance over time. Due to the non-stationarity of time series, the trading signals cannot be generated.

Below, you can see a plot of an asset which is a non-stationary time series with a deterministic trend represented by the “blue” curve and its detrended stationary time series represented by the “red” curve.

Stationary and non stationary time series

​​The unit root is a characteristic of a time series that makes it non-stationary. Technically speaking, a unit root is said to exist in a time series of the value of alpha = 1 in the below equation.

where, Yt is the value of the time series at the time ‘t’ and Xe is an exogenous variable (a separate explanatory variable, which is also a time series).

The Augmented Dickey Fuller test (ADF) is a modification of the Dickey-Fuller (DF) unit root.

Dickey-Fuller used a combination of T-statistics and F-statistics to detect the presence of a unit root in time series .

ADF test in pairs trading is done to check the co-integration between two stocks (presence of unit root).

If there is a unit root present in the time series, it implies that the time series is non-stationary and the stocks are not co-integrated. Hence, stocks cannot be traded together.

Alternatively, if the null hypothesis gets rejected and the stocks show co-integration; it implies that the time series is stationary and the stocks can be traded.

Statistical Arbitrage: from A to Z

The main point of conducting an ADF test in pairs trading strategy is to ensure if the pair of stocks considered in the strategy are stationary or not.

Hence, for the pair of stocks to be traded in the pairs trading strategy, it is required that the time series is stationary. A stationary time series makes effective and precise predictions.

Also, a stationary time series means that the pair of stocks is co-integrated and can be traded together by generating trading signals.

The following are the differences between the Dickey-Fuller test and the Augmented Dickey Fuller test (ADF test).

Dickey-Fuller Test

The Dickey-Fuller Test is a statistical test that is used to determine if there is a unit root in the data i.e., whether the time series is stationary or non-stationary. The test was developed by Robert Dickey and Thomas Fuller in 1979.

Augmented Dickey-Fuller Test

The augmented Dickey-Fuller test is an extension of the standard Dickey-Fuller test, which also checks for both stationarity and non-stationarity in the time series.

The main difference from the Dickey Fuller Test is that the Augmented Dickey Fuller test can also be applied on a large sized set of time series models. The large sized time series models can be more complicated and hence, the DF test was modified into the ADF Test. Also, the ADF Test works on the data with missing values.

  • The primary difference between the two tests is that the ADF is utilised for a larger sized set of time series models which can be more complicated.
  • The ADF test is an alternative to DF because it can be used even when there are missing values in the data.

Working of the Augmented Dickey-Fuller test

Now, let us find the working of an Augmented Dickey-Fuller test. We will begin with the hypothesis and advance to the calculation and its working in both Excel as well as Python.

The Augmented Dickey-Fuller test is based on two hypothesis:

  • The null hypothesis states that there exists a unit root in the time series and is non-stationary.
  • The alternative hypothesis states that there exists no unit root in the time series and is stationary or trend stationary.

Let us first see the formula for Dickey Fuller Test (which is the origin of Augmented Dicket Fuller Test), and that is:

where, yt = value in the time series at time t or lag of 1 time series delta yt = first difference of the series at time (t-1)

Now, we will see the formula for Augmented Dickey Fuller test, and it goes as follows:

You can see that the formula for ADF is the same equation as the DF with the only difference being the addition of differencing terms representing a larger time series.

The test is most easily performed by rewriting the model:

The hypothesis (1) = 0 again corresponds to

H0 : π = 0    HA : π < 0

The t−test for H0 is denoted as the augmented Dickey-Fuller (ADF) test.

Let us see the step-by-step working of the ADF test in the excel sheet.

Step 1: Get the data for two stocks that you want to perform an ADF test on

In this example, I have taken the two stocks Zymergen Inc (NASDAQ: ZY) and Zynerba Pharmaceuticals Inc (NASDAQ: ZYNE) both belong to the Pharmaceutical industry and are listed on American Stock Exchange “NASDAQ”.

Also, the number of observations that I'm using here is 60.

Take a look at the image below, to understand how the data with close price of each stock is represented side by side in the excel sheet.

Data for two stocks

Step 2: Perform a linear regression on the two stocks using a set number of observations

Make sure you also request the residuals to be outputted.

*Note*: When running this in a pairs trading strategy you will have to run the ADF test every day to make sure that the Null hypothesis is rejected (Null hypothesis = there is a unit root).

Here’s how it will be done in the excel sheet.

Linear regression on two stocks

The X Variable Coefficient of -3.98 is what we will be using for the hedge ratio.

Step 3: Calculate the difference between the residuals in the column “Delta”

Difference between residual and delta

Step 4: Calculate the t-1 residual in the next column

Calculation of t-1 residual

Step 5: Perform a linear regression on the Delta and t-1 column

Linear regression

Step 6: Compare the t-test statistic and the critical value

In order to reject the null hypothesis that there is a unit root present, the t-statistic, in this case, must be within the critical value range. The critical values for an ADF test have their own distribution and here’s a snapshot of some of the critical values:

Critical values

  • We will be using the Critical value of -2.89 since we have less than 100 observations
  • The t-stat is -1.109.
  • Therefore the null hypothesis is rejected and we can say that the data of the pair is co-integrated.

Now, let us find out how the ADF Test can be done using Python with the same pair of stocks as used above.

Now, we will check the cointegration by running the Augmented Dickey Fuller test. Using the statsmodels.api library, we compute the Ordinary Least Squared regression on the closing price of the commodity pair and store the result of the regression in the variable named ‘result’.

Using the statsmodels.tsa.stattools library, we run the adfuller test by passing the residual of the regression as the input and store the result of this computation in the array “c_t”.

This array contains values like the t-statistic, p-value, and critical value parameters. Here, we consider a significance level of 0.1 (90% confidence level). “c_t[0]” contains the t-statistic, “c_t[1]” contains the p-value and “c_t[4]” stores a dictionary containing critical value parameters for different confidence levels.

For co-integration we consider two conditions:

  • First, we check whether the t-stat is less than or equal to the critical value parameter (c_t[0] <= c_t[4]['10%'])
  • Second, we find out whether the p-value is lesser than the significance level (c_t[1] <= 0.1).

If both these conditions are true, we print that the “Pair of securities is co-integrated”, else print that the “Pair of securities is not co-integrated”.

The output above shows that the pair of securities is co-integrated and hence, can be traded together.

Let us now look at the advantages of using the ADF test.

  • The first and foremost advantage of implementing the pairs trading strategy is that it can help you to maximise your returns.
  • ADF test also helps in mitigating the potential risk at the same time since it hedges the portfolio of two stocks.

ADF testing comes with its own set of cons as well. Let us take a look at these cons.

  • The biggest con is that the ADF test relies on a high statistical correlation between the securities which might be difficult for the ones who are not great at statistical analysis.
  • Also, historical trends may be mostly accurate, but they do not guarantee an indication of future trends. Pairs traders usually look for a correlation of 0.80 which can also lower the chances of expected returns.

Augmented Dickey Fuller Test is a statistical test that can be performed in both Excel as well as Python and is relevant for those who employ a pairs trading strategy. ADF test is an effective test for finding out which stocks can be paired together to maximise the returns on the strategy.

If you also wish to find out more about this statistical test, you can enrol in our course on Statistical Arbitrage Trading and find out all about the ADF test. Learning about the ADF testing can help you find more trading opportunities with different pairs of stocks. Additionally, you will learn how to create trading models using Excel and Python as well as how to backtest them.

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Note: The original post has been revamped on 25th July 2022 for accuracy, and recentness.

Disclaimer: All investments and trading in the stock market involve risk. Any decision to place trades in the financial markets, including trading in stock or options or other financial instruments is a personal decision that should only be made after thorough research, including a personal risk and financial assessment and the engagement of professional assistance to the extent you believe necessary. The trading strategies or related information mentioned in this article is for informational purposes only.

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  • Knowledge Base
  • Null and Alternative Hypotheses | Definitions & Examples

Null & Alternative Hypotheses | Definitions, Templates & Examples

Published on May 6, 2022 by Shaun Turney . Revised on June 22, 2023.

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :

  • Null hypothesis ( H 0 ): There’s no effect in the population .
  • Alternative hypothesis ( H a or H 1 ) : There’s an effect in the population.

Table of contents

Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, similarities and differences between null and alternative hypotheses, how to write null and alternative hypotheses, other interesting articles, frequently asked questions.

The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”:

  • The null hypothesis ( H 0 ) answers “No, there’s no effect in the population.”
  • The alternative hypothesis ( H a ) answers “Yes, there is an effect in the population.”

The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample. It’s critical for your research to write strong hypotheses .

You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.

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null and alternative hypothesis of adf test

The null hypothesis is the claim that there’s no effect in the population.

If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.

Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept . Be careful not to say you “prove” or “accept” the null hypothesis.

Null hypotheses often include phrases such as “no effect,” “no difference,” or “no relationship.” When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).

You can never know with complete certainty whether there is an effect in the population. Some percentage of the time, your inference about the population will be incorrect. When you incorrectly reject the null hypothesis, it’s called a type I error . When you incorrectly fail to reject it, it’s a type II error.

Examples of null hypotheses

The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.

*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .

The alternative hypothesis ( H a ) is the other answer to your research question . It claims that there’s an effect in the population.

Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.

The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

Alternative hypotheses often include phrases such as “an effect,” “a difference,” or “a relationship.” When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes < or >). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.

Examples of alternative hypotheses

The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.

Null and alternative hypotheses are similar in some ways:

  • They’re both answers to the research question.
  • They both make claims about the population.
  • They’re both evaluated by statistical tests.

However, there are important differences between the two types of hypotheses, summarized in the following table.

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To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.

General template sentences

The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:

Does independent variable affect dependent variable ?

  • Null hypothesis ( H 0 ): Independent variable does not affect dependent variable.
  • Alternative hypothesis ( H a ): Independent variable affects dependent variable.

Test-specific template sentences

Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.

Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

  • Normal distribution
  • Descriptive statistics
  • Measures of central tendency
  • Correlation coefficient

Methodology

  • Cluster sampling
  • Stratified sampling
  • Types of interviews
  • Cohort study
  • Thematic analysis

Research bias

  • Implicit bias
  • Cognitive bias
  • Survivorship bias
  • Availability heuristic
  • Nonresponse bias
  • Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (“ x affects y because …”).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses . In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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aTSA Alternative Time Series Analysis

  • Package overview
  • accurate: Accurate Computation
  • adf.test: Augmented Dickey-Fuller Test
  • arch.test: ARCH Engle's Test for Residual Heteroscedasticity
  • aTSA-package: Alternative Time Series Analysis
  • coint.test: Cointegration Test
  • ecm: Error Correction Model
  • estimate: Estimate an ARIMA Model
  • expsmooth: Simple Exponential Smoothing
  • forecast: Forecast From ARIMA Fits
  • Holt: Holt's Two-parameter Exponential Smoothing
  • identify: Identify a Time Series Model
  • kpss.test: Kwiatkowski-Phillips-Schmidt-Shin Test
  • MA: Moving Average Filter
  • pp.test: Phillips-Perron Test
  • stationary.test: Stationary Test for Univariate Time Series
  • stepar: Stepwise Autoregressive Model
  • trend.test: Trend Test
  • ts.diag: Diagnostics for ARIMA fits
  • Winters: Winters Three-parameter Smoothing
  • Browse all...

adf.test : Augmented Dickey-Fuller Test In aTSA: Alternative Time Series Analysis

Description Usage Arguments Details Value Note Author(s) References See Also Examples

Description

Performs the Augmented Dickey-Fuller test for the null hypothesis of a unit root of a univarate time series x (equivalently, x is a non-stationary time series).

The Augmented Dickey-Fuller test incorporates three types of linear regression models. The first type ( type1 ) is a linear model with no drift and linear trend with respect to time:

dx[t] = ρ*x[t-1] + β[1]*dx[t-1] + ... + β[nlag - 1]*dx[t - nlag + 1] +e[t],

where d is an operator of first order difference, i.e., dx[t] = x[t] - x[t-1] , and e[t] is an error term.

The second type ( type2 ) is a linear model with drift but no linear trend:

dx[t] = μ + ρ*x[t-1] + β[1]*dx[t-1] + ... + β[nlag - 1]*dx[t - nlag + 1] +e[t].

The third type ( type3 ) is a linear model with both drift and linear trend:

dx[t] = μ + β*t + ρ*x[t-1] + β[1]*dx[t-1] + ... + β[nlag - 1]*dx[t - nlag + 1] +e[t].

We use the default nlag = floor(4*(length(x)/100)^(2/9)) to calcuate the test statistic. The Augmented Dickey-Fuller test statistic is defined as

ADF = ρ.hat/S.E(ρ.hat),

where ρ.hat is the coefficient estimation and S.E(ρ.hat) is its corresponding estimation of standard error for each type of linear model. The p.value is calculated by interpolating the test statistics from the corresponding critical values tables (see Table 10.A.2 in Fuller (1996)) for each type of linear models with given sample size n = length( x ). The Dickey-Fuller test is a special case of Augmented Dickey-Fuller test when nlag = 2.

A list containing the following components:

Missing values are removed.

Fuller, W. A. (1996). Introduction to Statistical Time Series, second ed., New York: John Wiley and Sons.

pp.test , kpss.test , stationary.test

Example output

Related to adf.test in atsa ..., r package documentation, browse r packages, we want your feedback.

null and alternative hypothesis of adf test

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  • 9.1 Null and Alternative Hypotheses
  • Introduction
  • 1.1 Definitions of Statistics, Probability, and Key Terms
  • 1.2 Data, Sampling, and Variation in Data and Sampling
  • 1.3 Frequency, Frequency Tables, and Levels of Measurement
  • 1.4 Experimental Design and Ethics
  • 1.5 Data Collection Experiment
  • 1.6 Sampling Experiment
  • Chapter Review
  • Bringing It Together: Homework
  • 2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs
  • 2.2 Histograms, Frequency Polygons, and Time Series Graphs
  • 2.3 Measures of the Location of the Data
  • 2.4 Box Plots
  • 2.5 Measures of the Center of the Data
  • 2.6 Skewness and the Mean, Median, and Mode
  • 2.7 Measures of the Spread of the Data
  • 2.8 Descriptive Statistics
  • Formula Review
  • 3.1 Terminology
  • 3.2 Independent and Mutually Exclusive Events
  • 3.3 Two Basic Rules of Probability
  • 3.4 Contingency Tables
  • 3.5 Tree and Venn Diagrams
  • 3.6 Probability Topics
  • Bringing It Together: Practice
  • 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable
  • 4.2 Mean or Expected Value and Standard Deviation
  • 4.3 Binomial Distribution (Optional)
  • 4.4 Geometric Distribution (Optional)
  • 4.5 Hypergeometric Distribution (Optional)
  • 4.6 Poisson Distribution (Optional)
  • 4.7 Discrete Distribution (Playing Card Experiment)
  • 4.8 Discrete Distribution (Lucky Dice Experiment)
  • 5.1 Continuous Probability Functions
  • 5.2 The Uniform Distribution
  • 5.3 The Exponential Distribution (Optional)
  • 5.4 Continuous Distribution
  • 6.1 The Standard Normal Distribution
  • 6.2 Using the Normal Distribution
  • 6.3 Normal Distribution—Lap Times
  • 6.4 Normal Distribution—Pinkie Length
  • 7.1 The Central Limit Theorem for Sample Means (Averages)
  • 7.2 The Central Limit Theorem for Sums (Optional)
  • 7.3 Using the Central Limit Theorem
  • 7.4 Central Limit Theorem (Pocket Change)
  • 7.5 Central Limit Theorem (Cookie Recipes)
  • 8.1 A Single Population Mean Using the Normal Distribution
  • 8.2 A Single Population Mean Using the Student's t-Distribution
  • 8.3 A Population Proportion
  • 8.4 Confidence Interval (Home Costs)
  • 8.5 Confidence Interval (Place of Birth)
  • 8.6 Confidence Interval (Women's Heights)
  • 9.2 Outcomes and the Type I and Type II Errors
  • 9.3 Distribution Needed for Hypothesis Testing
  • 9.4 Rare Events, the Sample, and the Decision and Conclusion
  • 9.5 Additional Information and Full Hypothesis Test Examples
  • 9.6 Hypothesis Testing of a Single Mean and Single Proportion
  • 10.1 Two Population Means with Unknown Standard Deviations
  • 10.2 Two Population Means with Known Standard Deviations
  • 10.3 Comparing Two Independent Population Proportions
  • 10.4 Matched or Paired Samples (Optional)
  • 10.5 Hypothesis Testing for Two Means and Two Proportions
  • 11.1 Facts About the Chi-Square Distribution
  • 11.2 Goodness-of-Fit Test
  • 11.3 Test of Independence
  • 11.4 Test for Homogeneity
  • 11.5 Comparison of the Chi-Square Tests
  • 11.6 Test of a Single Variance
  • 11.7 Lab 1: Chi-Square Goodness-of-Fit
  • 11.8 Lab 2: Chi-Square Test of Independence
  • 12.1 Linear Equations
  • 12.2 The Regression Equation
  • 12.3 Testing the Significance of the Correlation Coefficient (Optional)
  • 12.4 Prediction (Optional)
  • 12.5 Outliers
  • 12.6 Regression (Distance from School) (Optional)
  • 12.7 Regression (Textbook Cost) (Optional)
  • 12.8 Regression (Fuel Efficiency) (Optional)
  • 13.1 One-Way ANOVA
  • 13.2 The F Distribution and the F Ratio
  • 13.3 Facts About the F Distribution
  • 13.4 Test of Two Variances
  • 13.5 Lab: One-Way ANOVA
  • A | Appendix A Review Exercises (Ch 3–13)
  • B | Appendix B Practice Tests (1–4) and Final Exams
  • C | Data Sets
  • D | Group and Partner Projects
  • E | Solution Sheets
  • F | Mathematical Phrases, Symbols, and Formulas
  • G | Notes for the TI-83, 83+, 84, 84+ Calculators

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 , the — null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are reject H 0 if the sample information favors the alternative hypothesis or do not reject H 0 or decline to reject H 0 if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 9.1

H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.

Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following: H 0 : μ = 2.0 H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

  • H 0 : μ __ 66
  • H a : μ __ 66

Example 9.3

We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: H 0 : μ ≥ 5 H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

  • H 0 : μ __ 45
  • H a : μ __ 45

Example 9.4

An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

  • H 0 : p __ 0.40
  • H a : p __ 0.40

Collaborative Exercise

Bring to class a newspaper, some news magazines, and some internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

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9.1: Null and Alternative Hypotheses

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The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

\(H_0\): The null hypothesis: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

\(H_a\): The alternative hypothesis: It is a claim about the population that is contradictory to \(H_0\) and what we conclude when we reject \(H_0\). This is usually what the researcher is trying to prove.

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are "reject \(H_0\)" if the sample information favors the alternative hypothesis or "do not reject \(H_0\)" or "decline to reject \(H_0\)" if the sample information is insufficient to reject the null hypothesis.

\(H_{0}\) always has a symbol with an equal in it. \(H_{a}\) never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example \(\PageIndex{1}\)

  • \(H_{0}\): No more than 30% of the registered voters in Santa Clara County voted in the primary election. \(p \leq 30\)
  • \(H_{a}\): More than 30% of the registered voters in Santa Clara County voted in the primary election. \(p > 30\)

Exercise \(\PageIndex{1}\)

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

  • \(H_{0}\): The drug reduces cholesterol by 25%. \(p = 0.25\)
  • \(H_{a}\): The drug does not reduce cholesterol by 25%. \(p \neq 0.25\)

Example \(\PageIndex{2}\)

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are:

  • \(H_{0}: \mu = 2.0\)
  • \(H_{a}: \mu \neq 2.0\)

Exercise \(\PageIndex{2}\)

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol \((=, \neq, \geq, <, \leq, >)\) for the null and alternative hypotheses.

  • \(H_{0}: \mu \_ 66\)
  • \(H_{a}: \mu \_ 66\)
  • \(H_{0}: \mu = 66\)
  • \(H_{a}: \mu \neq 66\)

Example \(\PageIndex{3}\)

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:

  • \(H_{0}: \mu \geq 5\)
  • \(H_{a}: \mu < 5\)

Exercise \(\PageIndex{3}\)

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

  • \(H_{0}: \mu \_ 45\)
  • \(H_{a}: \mu \_ 45\)
  • \(H_{0}: \mu \geq 45\)
  • \(H_{a}: \mu < 45\)

Example \(\PageIndex{4}\)

In an issue of U. S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses.

  • \(H_{0}: p \leq 0.066\)
  • \(H_{a}: p > 0.066\)

Exercise \(\PageIndex{4}\)

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (\(=, \neq, \geq, <, \leq, >\)) for the null and alternative hypotheses.

  • \(H_{0}: p \_ 0.40\)
  • \(H_{a}: p \_ 0.40\)
  • \(H_{0}: p = 0.40\)
  • \(H_{a}: p > 0.40\)

COLLABORATIVE EXERCISE

Bring to class a newspaper, some news magazines, and some Internet articles . In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

In a hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we:

  • Evaluate the null hypothesis , typically denoted with \(H_{0}\). The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality \((=, \leq \text{or} \geq)\)
  • Always write the alternative hypothesis , typically denoted with \(H_{a}\) or \(H_{1}\), using less than, greater than, or not equals symbols, i.e., \((\neq, >, \text{or} <)\).
  • If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis.
  • Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.

Formula Review

\(H_{0}\) and \(H_{a}\) are contradictory.

  • If \(\alpha \leq p\)-value, then do not reject \(H_{0}\).
  • If\(\alpha > p\)-value, then reject \(H_{0}\).

\(\alpha\) is preconceived. Its value is set before the hypothesis test starts. The \(p\)-value is calculated from the data.References

Data from the National Institute of Mental Health. Available online at http://www.nimh.nih.gov/publicat/depression.cfm .

Null and Alternative Hypothesis Assignment (PREJAN)

IMAGES

  1. Augmented Dickey-Fuller (ADF) Test

    null and alternative hypothesis of adf test

  2. Difference between Null and Alternative Hypothesis

    null and alternative hypothesis of adf test

  3. Augmented Dickey Fuller Test (ADF) Unit Root Test Null Hypothesis: the

    null and alternative hypothesis of adf test

  4. Hypothesis Testing

    null and alternative hypothesis of adf test

  5. Research Hypothesis Generator

    null and alternative hypothesis of adf test

  6. Results of the ADF unit root test Null hypothesis: the variable has a

    null and alternative hypothesis of adf test

VIDEO

  1. Hypothesis (complete)

  2. Testing of Hypothesis,Null, alternative hypothesis, type-I & -II Error etc @VATAMBEDUSRAVANKUMAR

  3. Null vs Alternative Hypothesis

  4. Null Hypothesis vs Alternate Hypothesis

  5. Paired T-Test (Dependent T-Test)- Hypothesis Testing- One Sided- Statistics Made Easy (20)

  6. Hypothesis Testing|| 8948156741

COMMENTS

  1. Augmented Dickey Fuller Test (ADF Test)

    Fundamentally, it has a similar null hypothesis as the unit root test. That is, the coefficient of Y (t-1) is 1, implying the presence of a unit root. If not rejected, the series is taken to be non-stationary. The Augmented Dickey-Fuller test evolved based on the above equation and is one of the most common form of Unit Root test. 4.

  2. Augmented Dickey-Fuller test

    In statistics, an augmented Dickey-Fuller test ( ADF) tests the null hypothesis that a unit root is present in a time series sample. The alternative hypothesis is different depending on which version of the test is used, but is usually stationarity or trend-stationarity.

  3. Augmented Dickey-Fuller (ADF) Test In Time-Series Analysis

    ADF (Augmented Dickey-Fuller) test is a statistical significance test which means the test will give results in hypothesis tests with null and alternative hypotheses. As a result, we will have a p-value from which we will need to make inferences about the time series, whether it is stationary or not.

  4. Methods and formulas for Augmented Dickey-Fuller Test

    The null hypothesis says that a unit root is in the time series sample, which means that the mean of the data is not stationary. Rejecting the null hypothesis indicates that the mean of the data is stationary or trend stationary, depending on the model for the test. Test statistic The test statistic for the ADF has the following form: where

  5. Dickey-Fuller Test

    The null hypothesis of DF test is that there is a unit root in an AR model, which implies that the data series is not stationary. The alternative hypothesis is generally stationarity or trend stationarity but can be different depending on the version of the test is being used. Consider the following example to understand DF test. (8.1)

  6. PDF Augmented Dickey-Fuller Unit Root Tests

    The null hypothesis of the Augmented Dickey-Fuller t-test is H0 θ=: 0 (i.e. the data needs to be differenced to make it stationary) versus the alternative hypothesis of H1 θ<: 0 (i.e. the data is stationary and doesn't need to be differenced) c. When the time series has a trend in it (either up or down) and is potentially slow-turning around a trend line you would draw through the

  7. Stationarity and detrending (ADF/KPSS)

    The null and alternate hypothesis for the KPSS test are opposite that of the ADF test. Null Hypothesis: The process is trend stationary. Alternate Hypothesis: The series has a unit root (series is not stationary). A function is created to carry out the KPSS test on a time series. [6]:

  8. Augmented Dickey-Fuller Test in R (With Example)

    One way to test whether a time series is stationary is to perform an augmented Dickey-Fuller test, which uses the following null and alternative hypotheses: H0: The time series is non-stationary. In other words, it has some time-dependent structure and does not have constant variance over time. HA: The time series is stationary.

  9. Augmented Dickey-Fuller (ADF) Test

    The ADF test ensures that the null hypothesis is accepted unless there is strong evidence against it to reject in favor of the alternate stationarity hypothesis. ADF testing technique involves Ordinary Least Squares (OLS) method to find the coefficients of the model chosen.

  10. PDF The Asymptotic Size and Power of the Augmented Dickey-Fuller Test for a

    tribution of the ADF test tn under the null hypothesis of a unit root. The asymptotic behavior of tn under the alternative of stationarity is also derived in Section 2, and its consequences for the power properties of the ADF test are discussed. Section 3 show cases a real data example where the reduced power of the ADF test is manifested in an

  11. Augmented Dickey-Fuller test

    Under the alternative hypothesis, ϕ < 1. Variants of the model allow for different growth characteristics (see Model). The model with δ = 0 has no trend component, and the model with c = 0 and δ = 0 has no drift or trend. A test that fails to reject the null hypothesis, fails to reject the possibility of a unit root.

  12. Augmented Dickey Fuller (ADF) Test for a Pairs Trading Strategy

    Alternatively, if the null hypothesis gets rejected and the stocks show co-integration; it implies that the time series is stationary and the stocks can be traded. ... The ADF test is an alternative to DF because it can be used even when there are missing values in the data. Working of the Augmented Dickey-Fuller test. Now, let us find the ...

  13. An Introduction to the Augmented Dickey-Fuller Unit Root Test

    The likelihood of making a type I error, or rejecting the null hypothesis when it is true, is the ADF test's significance level. The researcher usually chooses the significance threshold, which is often set at 0.05 or 0.01. The test is more rigorous and less likely to reject the null hypothesis the lower the significance threshold.

  14. Dickey-Fuller Test

    The Dickey-Fuller test is a way to determine whether the above process has a unit root. The approach used is quite straightforward. First calculate the first difference, i.e. i.e. If we use the delta operator, defined by Δyi = yi - yi-1 and set β = φ - 1, then the equation becomes the linear regression equation.

  15. Dickey-Fuller test

    In statistics, the Dickey-Fuller test tests the null hypothesis that a unit root is present in an autoregressive (AR) time series model. The alternative hypothesis is different depending on which version of the test is used, but is usually stationarity or trend-stationarity.

  16. r

    ADF: Reject or keep null hypothesis (difference p-value & test statistic) Ask Question Asked 8 years, 7 months ago Modified 8 years, 7 months ago Viewed 4k times 2 I'm working at my thesis and wanted to test my time series for Stationarity by using the Augmented Dickey Fuller Test with the program R.

  17. Null & Alternative Hypotheses

    The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test: Null hypothesis (H0): There's no effect in the population. Alternative hypothesis (Ha or H1): There's an effect in the population.

  18. Alternative Hypothesis for ADF Test

    1 Answer Sorted by: 2 Your test statistic involves the quantity (ρ − 1) ( ρ − 1). If the value of the statistic is negative, it indicates that ρ < 1 ρ < 1, and so you are rejecting the null of a unit root with statistical evidence that ρ < 1 ρ < 1 and so that that process is stationary.

  19. adf.test: Augmented Dickey-Fuller Test in aTSA: Alternative Time Series

    Performs the Augmented Dickey-Fuller test for the null hypothesis of a unit root of a univarate time series x (equivalently, x is a non-stationary time series). Usage Arguments Details The Augmented Dickey-Fuller test incorporates three types of linear regression models.

  20. 9.1 Null and Alternative Hypotheses

    The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. H 0, the —null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

  21. 9.1: Null and Alternative Hypotheses

    The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. \(H_0\): The null hypothesis: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

  22. Unit root tests, stationarity, and the null hypothesis

    The null hypothesis is "the differences, yt+1 −yt y t + 1 − y t, are stationary". You're suggesting switching it to the opposite, but one won't be able to carry out such a test, as very-close-to-stationary will look just like stationary. But what you are really saying is that one should only take differences and act as if they're stationary ...

  23. Null and Alternative Hypothesis Assignment (PREJAN)

    Answer the following questions: 1. A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses. H 0 : P = 0.25 H a : P ≠ 0.25 2. We want to test whether the mean height of eighth graders is 66 inches. Fill in the correct symbol ( =,≠,≥,<,≤,>) for the null and ...