## The Gaussian AR(1) Case without Intercept: Part 1

Consider the AR(1) model without intercept, rewritten as3

Ayt = a0yt-1 + ut, where ut is iid N(0, a2), (29.2)

and y t is observed for t = 1, 2,…, n. For convenience I will assume that

yt = 0 for t < 0. (29.3)

This assumption is, of course, quite unrealistic, but is made for the sake of transparency of the argument, and will appear to be innocent.

The OLS estimator of a 0 is:

7 о = = a о + І=П

If -2 < a 0 < 0, so that yt is stationary, then it is a standard exercise to verify that л/n (a0 – a0) ^ N(0, 1 – (1 + a0)2) in distribution. On the other hand, if a0 = 0, so that yt is a unit root process, this result reads: a0 ^ N(0, 0) in distribution,

hence plimn^^/n a 0 = 0...

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