2
$\begingroup$

E.g. $ y_t= A_1y_{t−1} + u_t, u_t ∼ (0, \Sigma_u)$

Would it be possible to let the time series to have zero-mean but is not stationary?

$\endgroup$
2
  • $\begingroup$ E.g. mean stays constant, but variance changes over time. $\endgroup$
    – Tim
    Jun 1 '18 at 21:17
  • $\begingroup$ It seems that in some text book (e.g. afriheritage.org/TTT/…), "zero-mean" seems to mean that the intercept is 0. I want to be sure about that. $\endgroup$
    – Aqqqq
    Jun 2 '18 at 4:35
1
$\begingroup$

Certainly. The most famous example is the Random Walk model,

$$y_t = y_{t-1} + u_t, \;\;\; u_t \sim WN(\sigma^2)$$

Since then $y_t = \sum_{i=1}^t u_i$ it follows that

$$E(y_t)=0, \text{Var}(y_t) = t\sigma^2$$

Then $y_t$ is first-order stationary (constant mean), but not second-order (or weakly) stationary.

$\endgroup$
2
  • $\begingroup$ Just to make sure, does "zero mean" really mean that the mean is 0 or does it refer to the intercept? $\endgroup$
    – Aqqqq
    Jun 3 '18 at 6:01
  • 1
    $\begingroup$ Zero-mean means expected value equals zero. And in a time series model it implies that the intercept is zero. $\endgroup$ Jun 3 '18 at 8:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.