Autocovariance & Autocorrelation

Calculate the autocovariance function funtion (γk) and autocorrelation function (ρk) of the following first order moving process:

Xt = u + Et + BEt-1

I know γk/γ0 = ρk

But to calculate γk?

γk = Cov(Xt,Xt-k) = Cov(u + Et + BEt-1,Xt-k) = ??

Where do you go from here to get γk?

γ0 = Bσ^2 (Am I correct here??)

 

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John's revised post below

For MA processes, write it out in full for each lag...

Code:

γ0 = Cov(Xt,Xt) = Cov(Et + BEt-1,
                                Et + BEt-1)

Et is a series of INDEPENDENT identically distributed random variables.
Covariances are just like multiplying out brackets but keep the covariance outside.
So, notice how I've written it so that only when the time period on the Et is the same, is there a non-zero covariance because the Et are INDEPENDENT of each other.

Code:

γ0 = Cov(Xt,Xt) = Cov(Et + BEt-1,
                                Et + BEt-1)

= σ^2 + B^2σ^2 

Similarly

Code:

γ1 = Cov(Xt,Xt-1) = Cov(Et + BEt-1,
                                           Et-1 + BEt-2)

= Bσ^2 

And

Code:

γ2 = Cov(Xt,Xt-2) = Cov(Et + BEt-1,
                                                   Et-2 + BEt-3)

= 0

You can see that there is no covariance for lag 2 and beyond.
More generally, for an MA(q), γ cuts out for lags > q

John

 

Thanks guys! Great help. Much appreciated