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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 $\phi^*_0=\mu(1- \phi_1 - ... - \phi_p), \phi^*_1 = \sum^p_{i=1}\phi_i-1, \phi^*_j = -\sum^p_{i=j} \phi_i, j=2,...p$

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I'm doing some casual reading on this. Try playing around with the indices and expanding every mention of X sub t-1 in the solution with X sub t - (X sub t - X sub t-1) to define Xsub t-1. It also appears one series can be can be subtracted from the other which may provide for deltaXsubt-1 in the solution, and that mu, because it is known to be zero, is not a factor. I could be wrong, however, if the value of mu is not equivalent to the value defined in the WNormal (mu, sigma), where mu = 0.

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