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?


1 Answer 1


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


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.

  • 3
    $\begingroup$ This answer is lacking precision by not differentiating the conditional and unconditional statistics. A lot of text on this topic has similar problems. It is spreading like a plague. $\endgroup$ Nov 21, 2013 at 9:01
  • $\begingroup$ Weak stationarity is defined using non-conditional means. Any stationary ARMA process has non-constant unconditional mean, so such diferentiation is meaningless. And conditional on what? For any random variable I can find such sigma algebra that it its conditional mean is not a constant. Finally practically every time series textbook or for that matter text on ARMA processes does not talk about conditional mean of the process. So if it is a plague it has already spread. $\endgroup$
    – mpiktas
    Dec 24, 2013 at 10:11

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