# Rationale of Augmented Dickey Fuller Test on lag difference

In the wiki page of ADF test, its testing procedure is applied to the model $$\Delta y_t = \alpha+\beta t+\gamma y_{t-1} + \delta_1 \Delta y_{t-1} + \dots + \delta_{p-1} \Delta y_{t-p+1} + \epsilon_t$$ where $$\alpha$$ is a constant, $$\beta$$ is the coefficient on a time trend and $$p$$ the lag order of the AR process.

Question: It seems that equation above can be derived from AR's equation. If this is the case, why do we transform from original equation of AR process to equation above?

Hi: The equation could be written with $$y_t$$ on the LHS but that would get messy because the null in the ADF test is that there is a unit root. ( i.e: $$\gamma = 0$$ results in $$y_t = y_{t-1}$$ if you split the difference on the LHS into its components ). So, by considering the difference as the LHS, the null corresponds to $$\gamma = 0$$ which is the usual convention ( think of it a complicated regression model ) for testing a null hypotheses.
Then, because, the LHS is the difference, you then need the lagged differences to capture any serial correlation in the regression. So, that's why the rest of the terms on the RHS ( except for the one with $$\gamma$$ as the coefficient ) are lagged differences. These lagged differences are not really involved in the hypothesis test. They are only there to capture serial correlation in the response ( i.e: the LHS ).