# Dickey-Fuller stationarity test¶

The Dickey-Fuller test checks the stationarity of a scalar time series using one time series. It assumes that the process with , discretized on the time grid writes:

(1)

where and where or or both can be assumed to be equal to 0.

The Dickey-Fuller test checks whether the random perturbation at time vanishes with time.

When and , the model (1) is said to have a drift. When and , the model (1) is said to have a linear trend.

In the model (1), the only way to have stochastic non stationarity is to have (if , then the process diverges with time which is readily seen in the data). In the general case, the Dickey-Fuller test is a unit root test to detect whether against :

The test statistics and its limit distribution depend on the a priori knowledge we have on and . In case of absence of a priori knowledge on the structure of the model, several authors have proposed a global strategy to cover all the sub-cases of the model (1), depending on the possible values on and .

The strategy implemented is recommended by Enders (Applied Econometric Times Series, Enders, W., second edition, John Wiley & sons editions, 2004.).

We note the data, by the Wiener process, and , .

1. We assume the model (2):

(2)

The coefficients are estimated by using ordinary least-squares fitting, which leads to:

(3)

We first test:

(4)

thanks to the Student statistics:

where is the least square estimate of the standard deviation of , given by:

which converges in distribution to the Dickey-Fuller distribution associated to the model with drift and trend:

The null hypothesis from (4) is accepted when where is the test threshold of level .

The quantiles of the Dickey-Fuller statistics for the model with drift and linear trend are:

1.1. Case 1: The null hypothesis from (4) is rejected

We test whether :

(5)

where the statistics converges in distribution to the Student distribution Student with , where is the least square estimate of the standard deviation of , given by:

The decision to be taken is:
• If from (5) is rejected, then the model 1 (2) is confirmed. And the test (4) proved that the unit root is rejected : . We then conclude that the final model is : with which is a trend stationary model.

• If from (5) is accepted, then the model 1 (2) is not confirmed, since the trend presence is rejected and the test (4) is not conclusive (since based on a wrong model). We then have to test the second model (7).

1.2. Case 2: The null hypothesis from (4) is accepted

We test whether :

(6)

with the Fisher statistics:

where is the sum of the square errors of the model 1 (2) assuming from (6) and is the same sum when we make no assumption on and .

The statistics converges in distribution to the Fisher-Snedecor distribution FisherSnedecor with . The null hypothesis from (4) is accepted when where is the test threshold of level .

The decision to be taken is:
• If from (6) is rejected, then the model 1 (2) is confirmed since the presence of linear trend is confirmed. And the test (4) proved that the unit root is accepted: . We then conclude that the model is: which is a non stationary model.

• If from (6) is accepted, then the model 1 (2) is not confirmed, since the presence of the linear trend is rejected and the test (4) is not conclusive (since based on a wrong model). We then have to test the second model (7).

2. We assume the model (7):

(7)

The coefficients are estimated as follows:

(8)

We first test:

(9)

thanks to the Student statistics:

where is the least square estimate of the standard deviation of , given by:

which converges in distribution to the Dickey-Fuller distribution associated to the model with drift and no linear trend:

The null hypothesis from (9) is accepted when where is the test threshold of level .

The quantiles of the Dickey-Fuller statistics for the model with drift are:

2.1. Case 1: The null hypothesis from (9) is rejected

We test whether :

(10)

where the statistics converges in distribution to the Student distribution Student with , where is the least square estimate of the standard deviation of , given by:

The decision to be taken is:
• If from (10) is rejected, then the model 2 (7) is confirmed. And the test (9) proved that the unit root is rejected: . We then conclude that the final model is: with which is a stationary model.

• If from (10) is accepted, then the model 2 (7) is not confirmed, since the drift presence is rejected and the test (4) is not conclusive (since based on a wrong model). We then have to test the third model (12).

2.2. Case 2: The null hypothesis from (9) is accepted

We test whether :

(11)

with a Fisher test. The statistics is:

where is the sum of the square errors of the model 2 (7) assuming from (11) and is the same sum when we make no assumption on and .

The statistics converges in distribution to the Fisher-Snedecor distribution FisherSnedecor with . The null hypothesis from (4) is accepted if when where is the test threshold of level .

The decision to be taken is:
• If from (11) is rejected, then the model 2 (7) is confirmed since the presence of the drift is confirmed. And the test (9) proved that the unit root is accepted: . We then conclude that the model is: which is a non stationary model.

• If from (11) is accepted, then the model 2 (7) is not confirmed, since the drift presence is rejected and the test (9) is not conclusive (since based on a wrong model). We then have to test the third model (12).

3. We assume the model (12):

(12)

The coefficients are estimated as follows:

(13)

We first test:

(14)

thanks to the Student statistics:

where is the least square estimate of the standard deviation of , given by:

which converges in distribution to the Dickey-Fuller distribution associated to the random walk model:

The null hypothesis from (14) is accepted when where is the test threshold of level .

The quantiles of the Dickey-Fuller statistics for the random walk model are:

The decision to be taken is:
• If from (14) is rejected, we then conclude that the model is : where which is a stationary model.

• If from (14) is accepted, we then conclude that the model is: which is a non stationary model.

API: