null and alternative hypothesis of adf test

Applied Time Series Analysis for Fisheries and Environmental Sciences

5.3 dickey-fuller and augmented dickey-fuller tests, 5.3.1 dickey-fuller test.

The Dickey-Fuller test is testing if \(\phi=0\) in this model of the data: \[y_t = \alpha + \beta t + \phi y_{t-1} + e_t\] which is written as \[\Delta y_t = y_t-y_{t-1}= \alpha + \beta t + \gamma y_{t-1} + e_t\] where \(y_t\) is your data. It is written this way so we can do a linear regression of \(\Delta y_t\) against \(t\) and \(y_{t-1}\) and test if \(\gamma\) is different from 0. If \(\gamma=0\) , then we have a random walk process. If not and \(-1<1+\gamma<1\) , then we have a stationary process.

5.3.2 Augmented Dickey-Fuller test

The Augmented Dickey-Fuller test allows for higher-order autoregressive processes by including \(\Delta y_{t-p}\) in the model. But our test is still if \(\gamma = 0\) . \[\Delta y_t = \alpha + \beta t + \gamma y_{t-1} + \delta_1 \Delta y_{t-1} + \delta_2 \Delta y_{t-2} + \dots\]

The null hypothesis for both tests is that the data are non-stationary. We want to REJECT the null hypothesis for this test, so we want a p-value of less that 0.05 (or smaller).

5.3.3 ADF test using adf.test()

The adf.test() from the tseries package will do a Augmented Dickey-Fuller test (Dickey-Fuller if we set lags equal to 0) with a trend and an intercept. Use ?adf.test to read about this function. The function is

x are your data. alternative="stationary" means that \(-2<\gamma<0\) ( \(-1<\phi<1\) ) and alternative="explosive" means that is outside these bounds. k is the number of \(\delta\) lags. For a Dickey-Fuller test, so only up to AR(1) time dependency in our stationary process, we set k=0 so we have no \(\delta\) ’s in our test. Being able to control the lags in our test, allows us to avoid a stationarity test that is too complex to be supported by our data.

5.3.3.1 Test on white noise

Let’s start by doing the test on data that we know are stationary, white noise. We will use an Augmented Dickey-Fuller test where we use the default number of lags (amount of time-dependency) in our test. For a time-series of 100, this is 4.

The null hypothesis is rejected.

Try a Dickey-Fuller test. This is testing with a null hypothesis of AR(1) stationarity versus a null hypothesis with AR(4) stationarity when we used the default k .

Notice that the test-statistic is smaller. This is a more restrictive test and we can reject the null with a higher significance level.

5.3.3.2 Test on white noise with trend

Try the test on white noise with a trend and intercept.

The null hypothesis is still rejected. adf.test() uses a model that allows an intercept and trend.

5.3.3.3 Test on random walk

Let’s try the test on a random walk (nonstationary).

The null hypothesis is NOT rejected as the p-value is greater than 0.05.

Try a Dickey-Fuller test.

Notice that the test-statistic is larger.

5.3.3.4 Test the anchovy data

The p-value is greater than 0.05. We cannot reject the null hypothesis. The null hypothesis is that the data are non-stationary.

5.3.4 ADF test using ur.df()

The ur.df() Augmented Dickey-Fuller test in the urca package gives us a bit more information on and control over the test.

The ur.df() function allows us to specify whether to test stationarity around a zero-mean with no trend, around a non-zero mean with no trend, or around a trend with an intercept. This can be useful when we know that our data have no trend, for example if you have removed the trend already. ur.df() allows us to specify the lags or select them using model selection.

5.3.4.1 Test on white noise

Let’s first do the test on data we know is stationary, white noise. We have to choose the type and lags . If you have no particular reason to not include an intercept and trend, then use type="trend" . This allows both intercept and trend. When you might you have a particular reason not to use "trend" ? When you have removed the trend and/or intercept.

Next you need to chose the lags . We will use lags=0 to do the Dickey-Fuller test. Note the number of lags you can test will depend on the amount of data that you have. adf.test() used a default of trunc((length(x)-1)^(1/3)) for the lags, but ur.df() requires that you pass in a value or use a fixed default of 1.

lags=0 is fitting the following model to the data:

z.diff = gamma * z.lag.1 + intercept + trend * tt

z.diff means \(\Delta y_t\) and z.lag.1 is \(y_{t-1}\) . You are testing if the effect for z.lag.1 is 0.

When you use summary() for the output from ur.df() , you will see the estimated values for \(\gamma\) (denoted z.lag.1 ), intercept and trend. If you see *** or ** on the coefficients list for z.lag.1 , it suggest that the effect of z.lag.1 is significantly different than 0 and this supports the assumption of stationarity. However, the test level shown is for independent data not time series data. The correct test levels (critical values) are shown at the bottom of the summary output.

Note urca:: in front of summary() is needed if you have not loaded the urca package with library(urca) .

We need to look at information at the bottom of the summary output for the test statistics and critical values. The part that looks like this

The first test statistic number is for \(\gamma=0\) and will be labeled tau , tau2 or tau3 .

In our example with white noise, notice that the test statistic is LESS than the critical value for tau3 at 5 percent. This means the null hypothesis is rejected at \(\alpha=0.05\) , a standard level for significance testing.

5.3.4.2 When you might want to use ur.df()

If you remove the trend (and/or level) from your data, the ur.df() test allows you to increase the power of the test by removing the trend and/or level from the model.

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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.

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.

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.

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:

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

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.

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 (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.

​​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.

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.

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”

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

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

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:

• 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 any one of these conditions is 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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