# Can a Covariance-Stationary series be Heteroskedastic?

Can a time series that satisfies the covariance-stationary properties (mean of x(t) and covariance between x(t) and x(t+m) are both constant relative to t) be heteroskedastic (variance of x(t) changes with respect to t)?

• No but the conditional variance can be time-varying. – Taylor Dec 5 '17 at 1:39

No.

You're asking for $\text{cov}(X_t,X_{t+m})$ to not depend on $t$, only on $m$, for all integers $m$. In particular, for $m=0$:

$$\text{cov}(X_t,X_{t+0}) = \text{cov}(X_t,X_t) = \text{var}(X_t)$$

So, the variance can't depend on $t$ either, that's included in the restrictions on the autocovariance function.

Edit: I suppose I didn't anticipate the confusion behind the question as the others did, but they are correct: the variance of the transition distribution (or conditional distribution) can depend on $t$ and still be stationary, as in a (stationary) GARCH model.

That distinction applies to the mean as well, and is maybe simpler to understand there. Take $X_t = \phi X_{t-1} + \varepsilon_t$, with $|\phi|<1$, which is stationary. The mean $E(X_t)=0$ doesn't depend on $t$, but the transition distribution is $X_t|X_{t-1} \sim \mathcal{N}(\phi X_{t-1},\sigma^2)$, so does depend on $t$.

Both answers that I have seen so far are correct: The unconditional variance of a (covariance-) stationary process does not depend on the point in time. However, the conditional variance (to which you might be refering to) may depend on the past of the process.

You might want to check out Conditonally Heteroskedastic Models. Autoregressive Moving Average models are used to model the conditional mean of a time series while GARCH models are used to describe the conditional variance of a time series.