On this question, I can see that for Model X, we can derive an estimate for a1 by calculating: Autocov(0)=Cov(a0+a1yt-1+et,yt) = a1*autocov(1) + sigma^2 Autocov(1)=Cov(a0+a1yt-l+et,yt-1) = a1*autocov(0), thus a1 = autocov(1)/autocov(0) = r1 However, why can we not solve by: Autocov(0)=Cov(a0+a1yt-1+et,a0+a1yt-1+et) = a1^2*autocov(0) + sigma^2 Autocov(1)=Cov(a0+a1yt-1+et,a0+a1yt-2+et-1) = a1^2*autocov(1)... here we get a1^2 = autocov(1)/autocov(1) which is not the same as above?
Hi Edward Your method does give the same answer eventually, but I think it's a bit less efficient. Using the Yule-Walker equations I can set up the simultaneous equations: g0 = a1 * g1 + s^2 g1 = a1 * g0 From which I can get: g0 = s^2 / (1 - a1^2) g1 = a1 * s^2 / (1 - a1^2) Using your alternative method for g1 I get (noting that cov(yt, et) = s^2): g1 = cov(a0 + a1 * yt-1 + et, a0 + a1 * yt-2 + et-1) = a1^2 * g1 + a1 * s^2 From which: g1 = a1 * s^2 / (1 - a1^2) So they do boil down to the same thing eventually, but I think it is harder work to do it the second way. Hope that helps, Dave
