Stationarity of time series with product of white noise time series

Is the time series $$\{Yt\}$$ given by $$Y_{t} = Z_{t} - \frac{1}{2}Z_{t-1}Z_{t-2}$$

With $$Z_{t} \sim{N}(0\,1)$$, weakly stationary?

I do not know how to check if the above stated formula is stationary? I am struggling how to interpret the product of the same white noise time series with different lag?

Assume $$k> 0$$ without loss of generality, and assuming white $$Z_t$$: \begin{align}\operatorname{cov}(Y_t,Y_{t-k})&=\operatorname{cov}(Z_t-\frac{1}{2}Z_{t-1}Z_{t-2},Z_{t-k}-\frac{1}{2}Z_{t-k-1}Z_{t-k-2})\\&=-\frac{1}{2}\operatorname{cov}(Z_{t-1}Z_{t-2},Z_{t-k})+\frac{1}{4}\operatorname{cov}(Z_{t-1}Z_{t-2},Z_{t-1-k}Z_{t-2-k})\end{align}
Check this expression for $$k=1,2$$ and greater values. You'll see that the covariance is always $$0$$. Also, the mean process will be constant.
• Yes, I shortened that a bit because covariance of $Z_t$ with any of the past values will be $0$ Feb 15, 2021 at 11:26