# 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 .