# Why theoretical ACF in the not stationary case is not exist?

My professor told us that ACF will not exist if the model is not stationary but leave us to figure out why.

• Please put a self-study tag and read the policy related to homework questions. A hint for the above question is that each $X_t$ in a time series is a different random variable. So correlation between $X_t, X_{t+\tau}$ should be a function of $t, \tau$. The way ACF is defined, this correlation should be independent of $t$. Under what conditions can that be true? Commented Oct 19, 2020 at 7:16
• Your professor must mean some form of weak stationarity. At stats.stackexchange.com/a/282875/919 I provided a counterexample to the statement you quote: namely, an example of a time series process that has an ACF but is not stationary.
– whuber
Commented Oct 19, 2020 at 13:05

You can define auto-correlation functions for non-stationary time series.

• An auto-correlation like this

$$\sum_{t=0}^n X_t \overline{X_{t+\tau}}$$

can always be computed for a sample.

• Also a correlation function in terms of the expectation can always be expressed (except if the covariance or 2nd moment is not defined which may occur with distributions that have infinite or undefined higher moments)

$$\varphi(t_1,t_2) = E[X_{t_1} \overline{X_{t_2}}]$$

or expressed in terms of a time difference.

$$\varphi(t,\tau) = E[X_t \overline{X_{t+\tau}}]$$

For a non-stationary time series this type of correlation function in terms of the expectation will not only depend on the time difference $$\tau$$ but also on the time $$t$$. It is gonna be dependent on both changes in the mean and/or variance and so this expectation is not constant for any point in time and varies with $$t$$,

However,

• The typical correlation function is only a function of the time difference

$$\varphi(\tau) = E[X_t \overline{X_{t+\tau}}]$$

For this to make sense and be well-defined, this requires $$\varphi(t,\tau)$$ to be dependent on $$\tau$$ only. That is, it needs to be independent of time

$$\forall t: \varphi(t,\tau) = \varphi(\tau)$$

For a non-stationary time series you do not have that : $$\varphi(t,\tau)$$ is constant in time $$t$$. Tondescribe the auto-correlation as a function of $$\tau$$ only would be ambiguous. Which time point are you gonna choose to define the ACF of the series?

But there is a middleway:

• If you standardise the auto-correlation function

$$\frac{E[(X_t -\mu_t) \overline{(X_{t+\tau} -\mu_{t+\tau})}]}{\sigma_t \sigma_{t+\tau}}$$

then for some non-stationary time series this type of auto-correlation function may still work. (And actually, also for the non-standardised auto-correlation function you can have that for some non-stationary sequences the auto-correlation is defined. As Whuber shower here by generating two distributions that are different, but still have equal covariance)

• I think there is a failure of logic in this answer. You correctly establish that the "ACF" must be a function of $(t,\tau)$ rather than $\tau$ alone: but that does not demonstrate the claim in the question. Indeed, it looks perfectly possible that for some $\tau$ this expectation will not vary with $t,$ controverting your statements in the second bullet. What is needed is to analyze that possibility in the context of a clear definition of what you mean by "stationary."
– whuber
Commented Oct 19, 2020 at 13:01