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 ).
It doesn't matter whether you formulate the Dickey-Fuller regression with $y_t$ or $\Delta y_t$ on the left hand side. The $t$-statistics are exactly the same. It's a matter of preference. (The second formulation seems more common.)
First Regression
The Dickey-Fuller regression $$ y_t = \gamma y_{t-1} + \epsilon_t, $$ is not robust with respect to serial correlation in the error term $\{ \epsilon_t \}$.
The augmentation of the regression by lagged terms comes from an attempt to control for such serial correlation---specifically, in the case where the error term follows an ARMA process.
Suppose $\epsilon_t$ follows an ARMA(1,1) specification: $$ y_t = \gamma y_{t-1} + \epsilon_t, $$ where $$ (1 - \phi L) \epsilon_t = (1 - \theta L) \nu_t. $$ $L$ is the lag operator and $\{ \nu_t\}$ is white noise.
The inverted MA representation of $\epsilon_t$ $$ \nu_t = (1 - \theta L)^{-1} (1 - \phi L) \epsilon_t = \sum_{h = 0} \psi_h \epsilon_{t-h} $$ implies $$ \epsilon_t = \psi_1 \epsilon_{t-1} + \psi_2 \epsilon_{t-2} + \cdots + \nu_t. $$ Substituting back into the model, $$ y_t = \gamma y_{t-1} + \psi_1 \epsilon_{t-1} + \psi_2 \epsilon_{t-2} + \cdots + \nu_t. $$ Under the unit root null $H_0: \gamma = 1$, $\epsilon_t = \Delta y_t$. So the regression model becomes (with linear trend) $$ y_t = \alpha + \beta t + \gamma_1 y_{t-1} + \psi_1 \Delta y_{t-1} + \psi_2 \Delta y_{t-2} + \cdots + \nu_t. $$ The Dickey-Fuller statistic is the $t$ statistic for testing $H_0: \gamma_1 = 1$.
Second Regression
Subtracting $y_{t-1}$ from both sides, $$ \Delta y_t = \alpha + \beta t + \gamma_2 y_{t-1} + \psi_1 \Delta y_{t-1} + \psi_2 \Delta y_{t-2} + \cdots + \nu_t. $$ The Dickey-Fuller statistic is the $t$ statistic for testing $H_0: \gamma_2 = 0$.
Simple regression algebra tells you that $\hat{\gamma}_2 = \hat{\gamma}_1 -1$. Moreover the standard errors are the same---these two regressions have the same residuals and regressors. This implies the standard errors of $\hat{\gamma}_2$ and $\hat{\gamma}_1$ are the same. So the $t$-statistics are the same.
