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?

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.

## Essential terms used in the Augmented Dickey Fuller Test

### Stationary time series

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.

### Non-stationary time series

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.

### Unit root

​​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.

$$Y_{t} =\alpha y_{t-1} + \beta X_{e} + \epsilon$$

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.

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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.

## Difference between DF test and ADF test

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.

### Key points

• 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.

### Hypothesis

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

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

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

$$y_{t} = c + \beta_{t} + \alpha Y_{t-1} + \phi\Delta Y_{t-1} + e_{t}$$

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:

$$y_{t} = c + \beta_{t} + \alpha Y_{t-1} + \phi\Delta Y_{t-1} + \phi_{2}\Delta Y_{t-2} .. + \phi_{p}\Delta Y_{t-p}$$

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:

$$Y_{t} - Y_{t-1} = (\phi1 - 1)Y_{t-1} + \phi2Y_{t-2} + \phi3Y_{t-3} + \varepsilon t$$ $$Y_{t} - Y_{t-1} = (\phi1 - 1)Y_{t-1} + (\phi2 + \phi3)Y_{t-2} + \phi3(Y_{t-3} - Y_{t-2}) + \varepsilon t$$ $$Y_{t} - Y_{t-1} = (\phi1 + \phi2 + \phi3-1) Y_{t-1} + (\phi2 + \phi3)(Y_{t-2} - Y_{t-1}) + \phi 3(Y_{t-3} - Y_{t-2}) + \varepsilon t$$ $$Y_{t} = \pi Y_{t-1} + c_{1} Y_{t-1} + c_{2} Y_{t-2} + \varepsilon t$$

where,

$$\pi = \phi1 + \phi2 + \phi3 - 1 = -\phi(1)$$ $$c_{1} = -(\phi2 + \phi3)$$ $$c_{2} = - \phi3$$

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.

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### 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”

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### 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.

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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” contains the t-statistic, “c_t” contains the p-value and “c_t” stores a dictionary containing critical value parameters for different confidence levels.

For co-integration we consider two conditions:

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

Statistical Arbitrage: from A to Z

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

• 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.

### Conclusion

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.