Time Series (White Noise)

#1
Suppose \(W_t\) and \(Y_t\) are two independent normal white noise series with \(Var(W_t)=2Var(Y_t)=4\). 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 \(Cov(V_t,V_{t-1})\), \(k=0,1,2,3,..\)



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.