The method of moments estimator for AR processes can be had with the Yule-Walker equations. But how is it derived?

The equation for AR(1):

$$Y_t =aY_{t-1}+\epsilon_t$$

Where $\epsilon $ ~ $N(0,\sigma^2 )$.

So the moment conditions are $E(\epsilon)=0, E(\epsilon^2)=\sigma^2$

But the "standard" solution can't be used as $E(\epsilon)=E(Y_t-aY_{t-1})=0$ is true for any $a$.


I figured the answer:

Multiply the equation by $Y_{t-1}$






And for $\sigma^2$:









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