A non-stationary $AR(1)$ process, which is a random walk, is constant in mean, but not constant in variance. How about the other $AR(p)$ processes with the order $p>1$? Are they constant in mean?

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#### Best Answer

If we define non-stationary AR(p) processes as the ones having single unit root, they all can be written as

$$X_t=X_{t-1}+Z_t$$

where $Z_t$ is a stationary linear process. Then the answer is yes, they are constant in mean.

In general situation is more complicated even for the AR(1) process, since in general we can define it as

$$X_t=mu+rho X_{t-1}+Z_t$$

Now if $rho=1$ the process is not constant in mean.