The Wold representation says that any stationary process is equivalent to a MA($\infty$) process which, in turn, is the limit of a MA(q) process. Does that mean that anything that can be proved for an MA(q) immediately generalizes to any stationary process? I suspect this is not the case. but can't really come up with a counter-example.

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    $\begingroup$ For the process to be stationary there are conditions on the moving average parameters. $\endgroup$ Commented Feb 6, 2017 at 22:26

1 Answer 1


Proposition. Let $y_t$ be a series from $MA(q)$. Then for any $q$, there is $s$ such that $Cov(y_t,y_{t-s})=0$.

Obviously, the above proposition is true. But there is no such $s$ for $AR(1)$ process with non-zero AR coefficient.

  • $\begingroup$ Nice example, but you did not make any conclusion. Since it was not self-study, it makes sense to spell it out fully. $\endgroup$ Commented Feb 7, 2017 at 6:36
  • $\begingroup$ What conclusion? I provided an counter-example to the OP's question, which means that his hypothesis is false. $\endgroup$
    – Julius
    Commented Feb 7, 2017 at 6:46
  • $\begingroup$ That's what I meant. This sentence was missing :) $\endgroup$ Commented Feb 7, 2017 at 7:02

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