$\newcommand{\Cov}{\operatorname{Cov}}$Suppose $\{W_t\}$ and $\{Y_t\}$ are two independent normal white noise series with $\operatorname{Var}(W_t)=2\operatorname{Var}(Y_t)=4$. Let $X_t = W_t-0.5W_{t-1}$ and $Z_t=Y_t+0.4Y_{t-1}-0.4Y_{t-2}$. Put $V_t=X_t-Z_t$. Find the $\Cov(V_t,V_{t-1})$, $k=0,1,2,3,\ldots$
So I tried doing this:
$$\Cov(V_t,V_{t-1}) = E[(W_t-0.5W_{t-1}-Y_t-0.4Y_{t-1}+0.4Y_{t-2})(W_{t-1}-0.5W_{t-2}-Y_{t-1}-0.4Y_{t-2}+0.4Y_{t-3})]$$
For $k=0$, $\Cov(V_t,V_{t-1})=1$
For $k=1$, $\Cov(V_t,V_{t-1})=-4.8$
For $k=2$, $\Cov(V_t,V_{t-1})=-0.8$
For $k>2$, $\Cov(V_t,V_{t-1})=0$
Is this correct? Any help/contribution will be greatly appreciated. Thank you.