Autocovariance Function

Hey there!

I have a question relating to Applied Time Series that I'm having a bit of trouble with.

For all parts of this question, let \( Z_t; t \geq 0 \) be a sequence of independent normal variables with mean 0 and variance \( \sigma^2 \), and let a, b and c be constants.

For the series given by \( aZ_t+bZ_{t-1}+cZ_{t-4}+c\) find its mean and autocovariance functions.

I was okay finding the mean
\( \mu_x (t)=E[X_t ]=E[aZ_t+bZ_{t-1}+cZ_{t-4}+c] \)
\( =E[aZ_t]+E[bZ_{t-1}]+E[cZ_{t-4}]+E[c] \)
\( =c \)

but I'm getting really confused with the covariance.

At the moment this is what I think I need to do:

\( \gamma_x(t,t+h) = Cov(X_t, X_{t+h}) \)
\( = E[X_tX_{t+h}]-\mu^2_x \)

so then after I have expanded it all out and whatnot I end up with 0? Just want to check whether this is correct or if I've managed to royally stuff that up!

Thanks in advance :D

P.S. if my math is in fact correct that makes this a stationary time series as the mean is constant and the covariance does not depend on t?


Ambassador to the humans
It shouldn't come out to be 0 for all values of h. You can figure it out for an arbitrary value of h up to a point but then at some point you need to start plugging in values for h.
Okay right so does that mean that:

\( \gamma_x(h) = \gamma_x(t,t+h) = \)
\(a^2\sigma^2 \) if h=0,
\(b^2\sigma^2 \) if |h|=1,
\(c^2\sigma^2 \) if |h|=4, and
0 for all other h

or is that not right?


TS Contributor
You need to count more carefully. E.g. at \( |h| = 0 \), (which you are calculating the variance)

\( \gamma(0) = (a^2 + b^2 + c^2)\sigma^2 \).

And yes you just need to count up to \( |h| = 4 \). But do not miss the \( |h| = 3 \) case.

Make sure you double check your solutions again.