1. ## Time series question

Suppose I have two series:

$\displaystyle x_t=0.3x_{t-1}+w_t$

$\displaystyle y_t=x_t+0.5w_t$

where $\displaystyle w_t\sim WN(0,1)$.

I try to find the autocovariance function of $\displaystyle y_t$ as followed:

$\displaystyle \gamma(h)=Cov(y_{t-h},y_t)=Cov(x_{t-h}+0.5w_{t-h},x_t+0.5w_t)$

$\displaystyle =Cov(x_{t-h},x_t)+0.5Cov(w_{t-h},x_t)+0.5Cov(w_t,x_{t-h})+0.5^2Cov(w_{t-h},w_t)$

Then $\displaystyle \gamma(0)=1/(1-0.3^2)+0.5^2$

$\displaystyle \gamma(1)=0.3/(1-0.3^2)$

$\displaystyle \gamma(2)=0.3^2/(1-0.3^2)$ and so on..

My question is: am I right to assume both $\displaystyle 0.5Cov(w_{t-h},x_t)$ and $\displaystyle 0.5Cov(w_t,x_{t-h})$ are 0 at any lag h?

By the way, is the series $\displaystyle y_t$ weakly stationary such that $\displaystyle \gamma(h)$ is independent of t?

2. ## Re: Time series question

My question is: am I right to assume both $0.5Cov(w_{t-h},x_t)$ and $0.5Cov(w_t,x_{t-h})$ are 0 at any lag h?
try evaluating both of the covariances at lag 1 and you will almost certainly find that one of them is 0 and one of them isn't. Try to understand why.

Spoiler:

$0.5Cov(w_t,x_{t-h})$: x_{t-h} is not correlated to future values of the white noise, because by definition future white noise is independent of the current x value.

$0.5Cov(w_{t-h},x_t)$: : x is correlated to past values of white noise, because it is correlated to its own history. **Unless there is a tremendous fluke of algebra, i haven't actually worked it out for your particular series.

Looking at that series i expect it is stationary but i dont know how to prove it without doing the whole question for you. Have a go if you like and post if you get stuck.

3. ## Re: Time series question

Ok it seems that

$\displaystyle \gamma(0)=1/(1-0.3^2)+0.5^2+1$

$\displaystyle \gamma(1)=0.3/(1-0.3^2)+0.5(0.3)$

$\displaystyle \gamma(2)=0.3^2/(1-0.3^2)+0.5(0.3)(0.3)$

...

It seems to me that $\displaystyle E[y_t]$ and $\displaystyle \gamma(h)$ are not governed by t..