Suppose you observe yt = xt + t where:

xt follows a stationary AR(1) process with AR parameter and inno vation variance v, i.e., xt = xt 1 + t with independent innovations N(0v);

The t are independent measurement errors with t N(0w);

The t and t series are mutually independent.

It easily follows that q = V (yt) = s +w where s = V(xt) = v (1 2)

(a) Show that yt = yt 1 + t where t = t+ t t 1

(b) Show that the lag 1 correlation in the t sequence is given by the expression w(w(1 + 2)+v)

(c) Find an expression for the lag k autocorrelation of the yt process in terms of k , and the signal to noise ratio s q Comment on this result.

(d) Is yt an AR(1) process? Is it Markov? Discuss and provide theoretical rationalization.