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When would an Augmented Dickey Fuller test be needed instead of Dickey fuller test?
Recall that the asymptotic distribution of Dickey Fuller test were derived for the null hypothesis for an autoregressive process AR(1), i.e., xt=αxt-1+ut H0 : α=1 H1 : α<1
However if the data generating process was actually an AR(p) process, then the asymptotic distributions will be different. Hence a different decomposition is needed in which Δx becomes the dependent variable
Δxt= Φxt-1 + Σj=1p-1 αjΔxt-j + ut
In this model, the process is transformed to AR(p-1) instead of the AR(p)
