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Autoregressive Models
Consider first the AR(1) case. For this process, we have the simple relationship ρ1 = φ.
In the method of moments, ρ1 is equated to r1, the lag 1 sample autocorrelation. Thus we can estimate φ by
Now consider the AR(2) case. The relationships between the parameters φ1 and φ2 and various moments are given by the Yule-Walker equations (4.3.13)
Setting k = 1 and using ρ0 = 1 and ρ−1 = ρ1, we get
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Sample moments If we have a sample of i.i.d random variables, a natural way to estimate a population mean is to take the sample mean, the same is true for other moments of a
