Section A

You must answer one question from this section.

1. Assume that the stochastic process {(xt, yt) : t = 1, 2, . . . , n} follows the linear
model (with no intercept) yt = βxt + ut where ut is a stationary moving average
of order two, that is, ut = εt + θ1εt−1 + θ2εt−2 and {εt : t = 1, . . . , n} is an
independent and identically distributed sequence of random variables with mean
zero and variance one.

For simplicity, assume that xt is independent of us, for any t and s.

(a) Compute the population autocovariance function of ut, that is, compute Cov(ut, ut+k)
as a function of k = 0, 1, 2, . . . .

(b) Let βb denote the ordinary least square estimator of β. Compute the variance
of βb (conditional on the regressors).

(c) Assume that θ1, θ2 > 0 and {xt : t = 1, . . . , n} is positively autocorrelated.
What would be the effect of ignoring the autocorrelation of ut on the t-test
for the null hypothesis β = 0 against β ̸= 0?

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