Autocovariance & Autocorrelation

[CorkActuary]

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??)

 

[John Potter]

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

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.

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

= σ^2 + B^2σ^2

Similarly

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

= Bσ^2

And

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

 

[John Potter]

PS Sorry that's annoying - whatever software this thing uses has taken away my spaces so that you can't notice how I've written it out :-(

I want the Et to appear directly underneath the Et
the Et-1 to appear directly underneath the Et-1

I've got a slide that demonstrates this nicely so just give me a shout johnpotter@bpp.com if still not clear.

 

[didster]

Like this?? You can use the CODE tags (#button in this site's editor) to keep characters of a single fixed width (I uses same width as W rather than what it actually needs to occupy) maintain spaces etc (or at least that's how it works in general elsewhere). Useful for primitive alignments using spaces (and old school ASCII art).


John's revised post below

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

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.

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

= σ^2 + B^2σ^2

Similarly

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

= Bσ^2

And

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

 

[John Potter]

Thank You Didster!
Really appreciated

 

[CorkActuary]

Thanks guys! Great help. Much appreciated