# Deriving the mathematical model for Augmented Dickey Fuller test

This is a question on the mathematical derivation of the model used in the augmented Dickey Fuller test and hence, I choose not to raise this question in cross-validated.

If a process $\{X_t\}$ follows the AR(p) model with mean $\mu$ given by $$X_t - \mu = \phi_1(X_t-\mu)+ ... + \phi_p(X_{t-p}-\mu) + Z_t$$, where $\{Z_t\} \sim WN(0,\sigma^2)$

Question: How can I show that the above model can be rewritten as  $$\nabla X_t = X_t - X_{t-1} = \phi^*_0 + \phi^*_1 X_{t-1} + \phi^*_2\nabla X_{t-1} + ... + \phi^*_p \nabla X_{t-p+1} + Z_t$$  , where $\phi^*_0=\mu(1- \phi_1 - ... - \phi_p), \phi^*_1 = \sum^p_{i=1}\phi_i-1, \phi^*_j = -\sum^p_{i=j} \phi_i, j=2,...p$