Question 1

(a) Let Xt be a zero-mean, unit-variance stationary process with autocorrelation k. Suppose that t is a non-constant function and that t is a positive-valued non-constant function. The observed series is formed as Yt=t+tXt.

(i) Find the mean and covariance function for the Yt process

(ii) Is the process Yt stationary?

(b) Consider the model Yt = 5 + e - 0.7et-1 + 0.5et-2, where et "IID N(0, o? ). (i) Find the autocovariance function for the Yt process. (ii) Using b(i) or otherwise, determine on, and $22- (c) Suppose that {Y} is generated according to Yt = e+ + cet- + cet-2 + cet-3 + ... + ceo fort > 0, e " IID N(0, ) and {VY} is the first difference of {Y{} . (i) Determine {VY+} (ii) Is {VY } stationary? Justify your answer. (ii) Identify {VY } as a specific ARIMA process specifying p, d, q, o and e