Sept 2012 Q9

[Rieks]

question 9 on September 2012 exam. In part ii they ask you to find alpha and beta, and the hint they give refers to using the Yule walker method. Looking through the notes from chapters 12 and 13 I don't seem to be able to solve that question. Can you maybe give me a detailed answer or indicate where I can find notes regarding the Yule Walker? (There are notes about finding the co variance and correlation in a simpler model, but I don't know how to apply YW to situations as encountered in the question mentioned above)...

 

[Katherine Young]

We have an AR(2) process:

\[Y_t=(\alpha+\beta)Y_{t-1}-\alpha\beta Y_{t-2}+e_t\]


The Yule-Walker equations are:

\[\gamma_1=(\alpha+\beta) \gamma_0 - \alpha\beta \gamma_1\]
\[\gamma_2=(\alpha+\beta) \gamma_1 - \alpha\beta \gamma_0\]

Rearranging the first Yule-Walker equation gives:

\[ \gamma_1= \frac{(\alpha+\beta)}{1+\alpha\beta } \gamma_0 \]

Substituting this into the second Yule-Walker equation gives:

\[\gamma_2=(\alpha+\beta) \frac{(\alpha+\beta)}{1+\alpha\beta } \gamma_0-\alpha\beta \gamma_0\]

Now divide through by \[ \gamma_0 \] to give:

\[ \rho_1= \frac{(\alpha+\beta)}{1+\alpha\beta } \]

and

\[\rho_2=(\alpha+\beta) \frac{(\alpha+\beta)}{1+\alpha\beta } -\alpha\beta =(\alpha+\beta) \rho_1 - \alpha\beta \]

We are given that \[\rho_1=0.2\] and \[\rho_2=0.7\] so now you can simply use the method of moments to solve for alpha and beta.

I hope that helps.

 

[Rieks]

Thank you very much, this makes it a lot simpler! It's the first step I didn't get...

 

[Learner56]

Hi Katherine young, sorry how did you construct the intial equation before determining the Yule-walker equations
i.e I am a bit confused about how alpha + beta (and also alpha*beta) gets to be the coefficients there, respectively.

 

[John Lee]

It's given in the question:

\((1-(\alpha + \beta)B + \alpha\beta B^2)X_t = e_t\)