Dickey Fuller Test

Dickey fuller test was developed by statisticians David Dickey and Wayne Fuller in 1979. In statistics, the Dickey-Fuller Test tests the null hypothesis of whether a unit root is present in an autoregressive model. The alternative hypothesis tests stationarity in the model.

 

Suppose for an AR(1) model:

y_t = θ y_t-1+ ε_t           Where,  t= time index , θ= coefficient ,  ε_t = error term

 

The unit root null hypothesis against the stationary alternative corresponds to

H₀ : θ = 1           H_A : θ < 1

 

Alternatively, the model can be formulated as

?y_t = (θ − 1)y_t−1 + t = πy_t-1 + ε_t        Where, π = θ − 1 = θ(1).

 

The unit root hypothesis translates into

H₀ : π = 0              H_A : π < 0

The Dickey-fuller test is simply the t-test for H₀ and the t-statistic has a specific distribution known as the Dickey–Fuller table.

 

There are three main versions of the test:

 

1. Test for a unit root:

?y_t =  πy_t-1 + ε_t   

 

2. Test for a unit root with drift:

?y_t =  a₀ + πy_t-1 + ε_t  

 

3. Test for a unit root with drift and deterministic time trend:

?y_t =  a₀ + a₁t + πy_t-1 + ε_t