Sept 2021 Question 7(iii) -1st,2nd and 3rd Yule Walker Equations for Time Series

[Bill SD]

Hi,
Sept 2021 Question 7(iii) provides a 'first Yule Walker' equation and asks students to provide the 'second and third Yule Walker' equations.

Are these covered anywhere in the Time Series core reading/notes or were students just expected to guess the format and solution (which is implied by the Examiners Report commentary (copied below*) but seems exceedingly difficult)?

The Examiners Report simply provides the answers without any explanation (so perhaps I'm missing something obvious :
2nd equation: rho_1 - (alpha + beta) + alpha * beta * rho_1 = 0
3rd equation: rho_2 - (alpha + beta) * rho_1 + alpha * beta = 0

* The Examiners Report commentary at end of the question adds that:
The most common errors were Misunderstanding what was meant by the “second” and “third” Yule-Walker equations. The statement of the first Yule-Walker equation in the question was intended to make this clear.]

Thanks in advance

 

[Dave Johnson]

Hi

There are two things going on here, which I will cover in turn.

The first is that there is a formula in the Core Reading for the Yule-Walker equations of an \( AR(2) \) process (see section 3.4 of chapter 13 of the ActEd notes). It is given by:

\( \gamma_k = \alpha_1 \gamma_{k-1} + \alpha_2 \gamma_{k-2} + \sigma^2 1_{\{k=0\}} \) for \( 0 \le k \le 2 \)

For \( \gamma_0 \) this becomes:

\( \gamma_0 = \alpha_1 \gamma_1 + \alpha_2 \gamma_2 + \sigma^2 \)

The second thing is that we can rewrite this as:

\( 1 - \alpha_1 \rho_1 - \alpha_2 \rho_2 = \frac{\sigma^2}{\gamma_0} \)

Which is the form given in the question. Note here that \( \alpha_1 \) and \( \alpha_2 \) represent the autoregressive coefficients and shouldn't be confused with \( \alpha \) and \( \beta \) given in the question.

Knowing this makes it straightforward to write the second and third Yule-Walker equations in the same format:

\( \rho_1 - \alpha_1 - \alpha_2 \rho_1 = 0 \)
\( \rho_2 - \alpha_1 \rho_1 - \alpha_2 = 0 \)

Since this formula is given in the notes, this is a legitimate "write down" question, ie you get no marks for showing your working.

This is a tough question but it's well worth understanding it. The Examiners' Report noted that this part was poorly answered, which makes it the sort of question that allows the strongest candidates to stand out. This question is similar to questions from previous exams, and time series are examined in every CS2 sitting.

Hope that helps.

Dave

 

[Priyanka Malhotra]

Hi Dave, hope you are doing well! Thank you for explaining this part, although I do have one question from the examiners' report, could you please help me understand that why the highlighted part in equation below has + and not a minus sign when I compare it with the explanation given, thanks in advance!

rho_1 - (alpha + beta) + alpha * beta * rho_1 = 0
rho_2 - (alpha + beta) * rho_1 + alpha * beta = 0

 

[Dave Johnson]

Hi Priyanka,

I think the issue may be that you need to rearrange the equation for the time series to get the right signs for \( \alpha \) and \( \beta \). We can write the time series as:

\( X_t = (\alpha + \beta)X_{t-1} - \alpha \beta X_{t-2} \)

The we can see that the coefficients are \( \alpha + \beta \) and \( - \alpha \beta \).

This should get you the correct signs.

Hope that helps.

Dave