Time Series (White Noise)

Asano Kyouya

New Member
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.