time series exam questions

I may be wrong, but I feel as though the time series questions from past exams are not treated consistently in the solutions.

In particular, the questions asking to calculate the autocorrelation and autocovariance functions. Sometimes the solution includes specific treatment of covariance of the Xt series with the Zt series and sometimes not. Please can someone clarify when we must explicitly break down the covariance between the Xt function and the white noise process and when not?

March 2006 paper question 4: the solution for autocovariance with lag k gives a to the power of k and then includes the solution for Y(0) - I cannot understand how this has been put together. Please could someone explain this as well?

Thanks very much.

 

Another example - April 2005 exam, question 4: when the autocovariance of lag 0 is calculated (cov(Xt,Xt)), Xt is expanded on the one side and not on the other (cov(Xt,0.8Xt + Zt....)), whilst in other tine series questions Xt is expanded on both sides, so it would be cov(0.8Xt + Zt....,0.8Xt + Zt...) for Y(0). The different forms of cov give difference answers and I don't understand when each is supposed to be used.

 

I really don't know what your getting at, i have never seen a covariance expanded in the way you describe. Doing that would be sure to turn into a complete mess. Try it yourself on the example you gave and you will see it goes absolutley nowhere. Please elaborate.

 

Sorry - I should have described it better.

Question 4 of the April 2005 exam calculates the autocovariance in the following way:

Yt = 0.8Yt-1 + Zt + 0.2Zt-1
(a) cov(Yt,Zt) = var(Z)
(b) cov(Yt,Yt) = cov(Yt, 0.8Yt-1 + Zt + 0.2Zt-1) = 0.8y(1) + var(Z) + 0.2var(Z)

I calculated (a) cov(Yt,Zt) as cov(0.8Yt-1 + Zt + 0.2Zt-1,Zt) which is 1var(Z) + 0.2var(Z) = 1.2var(Z)

I am also confused about (b) as Question 4 of April 2006 calculated covariance y(0) as:

cov(Xt,Xt) = cov(aXt-1 + et, aXt-1 + et)

instead of cov(Xt, aXt-1 + et), which is the way it is done in (b). This is what I mean by sometimes it is expanded and sometimes not, and the two ways of doing it give different answers. So in doing the past exams I always "expand" it, like in April 2006 example, but it then always looks different to the solutions, which are not always expanded.

So sorry for this long question! Thanks very very much for any help!

 

Never do it in the way that the april 2006 describes. It might be all right if you have 2 terms but trying to workout the auto covariance for an ARIMA(2,2) process for instance will be horrendous. The best way to get the yule-walker equations is to set up a relation of the form Yk = Cov(Xt,Xt-k) and replace the Xt and leave the lagged X as it is then expand and repeat for as many lags as you need. As for your other question about the white noise terms, you went wrong in your working there because Cov(Zt-1,Zt) = 0, white noise is a sequence of independent random variables so it can only be non zero when the lag is 0 (deleting your last term gives the required answer). Always be very very careful with white noise, especially for high order autoregressive and moving average models.

 

Thanks very much for your help, Devon. I thought that the white noise process covariance could be equal to the variance up to lag of 1. Feel a bit better about this now, thanks again!