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Problems 177
that the mean is known to be zero in writing down the latter equations, so that
the sample autocovariances are ˆγ (h) 1
n
n−h
t1 Xt+hXt for h ≥ 0.)
5.11. Given two observations x1 and x2 from the causal AR(1) process satisfying
Xt φXt−1 + Zt, {Zt} ∼WN 0,σ 2 ,
and assuming that |x1| |x2|, find the maximum likelihood estimates of φ
and σ 2 .
5.12. Derive a cubic equation for the maximum likelihood estimate of the coefficient
φ of a causal AR(1) process based on the observations X1,...,Xn.
5.13. Use the result of Problem A.7 and the approximate large-sample normal distribution
of the maximum likelihood estimator ˆφp to establish the approximation
(5.5.1).