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My question regards validating the legitimacy of representing an AR(1) as a MA($\infty $) process. In my notes this is done by showing convergence in mean square, saying that:

$\mathbb{E}[(\sum_{i=0}^{n-1}\phi^i\epsilon_{t-i}-Y_t)^2]=\mathbb{E}[\phi^{2n}Y_{t-n}^2]$

And then noting that:

$\phi^{2n}\gamma_0 \rightarrow 0$ as $n \rightarrow \infty$

The second step is fine. Could anyone show how the first equality is made? I get the feeling this shouldn't be too hard, but I just don't see it.

[For context, see 3.5.1 of these notes, which are similar]

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up vote 2 down vote accepted

Are you sure the RHS of your equation is correct?

As for where it comes from, after you correct it, I believe it follows directly from the equation at the bottom of page 21 of the document you point to, squaring both sides and taking expectations.

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Thanks Glen, common problem of trying too hard and thinking it must be complicated... :), and no the Y on the RHS was missing its square. –  conjectures Feb 20 '13 at 20:19
    
@conjectures I know how you feel, having been in similar situations many times. But these days I usually tend to go with the sentiment of the Ancient Simian Proverb, and assume most such things associated with techniques in standard practice are likely to be reasonably straightforward. It's usually right. The trick is figuring out what I've missed or done wrong that makes it seem harder. –  Glen_b Feb 20 '13 at 22:22
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