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Question 40.2 ASM
I thought the partial autocorrelations were the solutions to the phi's in the Yule-Walker equations for an AR(2) process. If this is the case a1=phi1=0.5 and a2=phi2= -0.5 (given in problem). What is rho3 (autocorrelation @ lag 3).
Yule Walker equations: rho1 = phi1/(1-phi2) rho2 = (phi1)^2/(1-phi2) + phi2
Plugging in phi1 and phi2, I get rho1=1/3 and rho2 = -1/3
Then plugging into the equation for rho3 = (phi1)(rho2) + (phi2)(rho1) = (0.5)(-1/3)+(-0.5)(1/3) = -1/3
I'm not sure what ASM is doing, but they come up with -0.34375.
Can someone enlighten me? Am I missing something very obvious or is this a mistake in the ASM manual?
You are missing the idea that the ak values are the solution to phik in a k-equation linear system. It's better to solve this from the viewpoint of finding rho1, rho2, and then rho3 than assuming you know phi1 and phi2. I don't see any error in the solution manual.
The recursive linear equations are:
Rho1 = (phi1) + (phi2)(rho1)
Rho2 = (phi1)(rho1) + (phi2)
If you simplify these 2 equations, you get the formulas from my previous post. I am not assuming that I know phi1 and phi2, they are given in the problem. I think that I know where the confusion is coming from. What I am assuming is that phi1 = a1(ASM notation) = partial autocorr coeff at lag1 and phi2 = a2(ASM notation) = partial autocorr coeff at lag2. I¡¦m not sure why the solution says that rho1 = a1= partial autocorr coeff at lag1. Isn¡¦t this only true of an AR(1) process, not an AR(2). Do you agree or do you still disagree?? Why is the solution solving for phi1? I thought that phi1 = the partial autocorr coeff at lag 1 which is given as a1.
03-26-2002, 02:29 PM
I'll take a stab at this....
I think the subtle concept is that the solutions to the phi's could be different, depending on how many equations you consider. That is, if you only look at the first equation, the solution to phi_1 could be (always is?) different than the solution to phi_1 if you consider the first 2 equations together.
So knowing a_1 = .5 doesn't tell you anything about phi_1 in the system of 2 equations involving phi_2.
Hope that helps.
I think your first sentence in your first post isn't correct--a_k is the solution to phi_k in the Y-W equations for an AR(k) process. So a_1 is the solution if you assume it's an AR(1) process, etc.
<font size=-1>[ This Message was edited by: phdmom on 2002-03-26 14:34 ]</font>
<font size=-1>[ This Message was edited by: phdmom on 2002-03-26 14:37 ]</font>
I think phdmom's right. This was messing with me when I studied that section.
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