- Question 2 29 Marks A Let X Be A Stationary Time Series And Define Odd Y X 3 For Reven I Show That C 1 (64.65 KiB) Viewed 16 times
Question 2 [29 marks] (a) Let (X,) be a stationary time series, and define odd, Y₁ = | X₁ + 3 for reven. (i) Show that Cov (Y,, Y-k) is independent of r for all lags k. (3 marks) (ii) Is (Y,) stationary? Explain your answer. (2 marks) (b) Consider the process X, = Bo +21+, where is a model parameter, and €1, €2,... are independent and identically distributed random variables with mean 0 and variance 2. (i) Find the autocorrelation function of Y,= (1-B)X,. (6 marks) (2 marks) (ii) Is Y,= (1-B)X, covariance stationary? Explain your answer. (c) Suppose that X₁ = En-1n (t = 1, 2, ...) where 1, 2,... are independent and identically distributed random variables with mean 0 and variance 2. (i) Find the mean function for (X,). (ii) Find the autocorrelation function for (X,). (1 marks) (6 marks) (2 marks) (iii) Is (X,) stationary? Why or why not? (d) Consider the process X₁ = ₁X₁-1 +6₁ + 0₁6₁-1+0₂61-2 where ₁, 01 and 2 are model parameters; and €1,2,... are independent and identically distributed random variables with mean 0 and variance a (i) Identify the process/model. (2 marks) (ii) Show that the process is stationary if |1| < 1. (3 marks) (iii) Under what conditions on 0₁ and 02 is the process invertible? (2 marks)