CS2 - September 2019 - Q8.(v) - Yule-Walker Equations

[Stephan Iliffe]

Is there an error in the suggested solution. Shouldn’t gamma 1 also have a 1/n.

if gamma 1 is just the covariance function

https://www.actuaries.org.uk/docume...d-survival-analysis-exam-paper-september-2019

 

[Dave Johnson]

Hi Stephan

You are essentially correct - however the missing factor is actually 1/(n-1) since the summation runs from t=2 to n (since we are looking at pairs of observations with lag 1).

Well done getting so far this question. It is tough even before you take into account the typos in the question and the examiners report.

Dave

 

[Stephan Iliffe]

Thanks Dave

this makes sense. I suppose then even for alpha there would’ve been 1/(n-1) at the top and 1/n at the bottom leading to an n/(n-1).

the question did take some time to work out but you get alpha by using p 1 = gamma 1 / gamma 0

and sigma squared by using gamma 0 = cov(yt,yt) =alpha * gamma 1 + sigma squared.

have you seen any other similar questions like this anywhere else. I’m feeling a little bit unsure of myself now that my answers don’t match the memo which is usually correct..

 

[Dave Johnson]

Hi Stephan

Something has been pointed out to me that I missed in my original answer. The usual formula for estimating sample autocovariance is given on p16 of chapter 14 using a factor of 1/n for all lags. There is some further explanation in the notes about why this is used rather than the more natural factor 1/(n-k) that I quoted in my earlier response.

One thing not mentioned in the notes is that using a factor of 1/n for all lags means these factors will cancel in the estimate for rho at all lags, simplifying the formula (which you alluded to in your comment).

Dave

 

[Dave Johnson]

In answer to your second question, I'm not totally sure what you mean by "similar questions". If you mean questions where there are errors in the examiner's report then it does happen occasionally, and so it's good practice to be inquisitive, as you have been here.