STAT 443 assignment2

# STAT 443 assignment2 - STAT 443 Assignment 2 This...

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Unformatted text preview: STAT 443: Assignment 2 This assignment is to be handed in at the start of your lecture of Friday 2nd March . Please make sure that you write your name, student number and section number on your script. This material covers the slides on stationary processes and regression. 1 Stationary processes 1. Show the process { X t } defined by X t = Z t + C ( Z t- 1 + Z t- 2 + ··· ) where C is a constant, Z t ∼ WN (0 ,σ 2 ), is non-stationary. Also show that the series of first differences Y t = X t- X t- 1 is stationary and compute its acf. 2. Assume that { X t } is a stationary time series, with mean μ and ACVF γ ( h ). Let P n X n + h be the optimal (in terms of MSE) linear prediction of X n + h based on X 1 , ··· ,X n . (a) Show explicitly how to construct P n X n + h in the case n = 2 and h = 2 (b) In the case n = 2 and h = 2, show that P n X n + h is an unbiased estimator i.e. E ( X n + h- P n X n + h ) = 0 (c) In the case n = 2 and h = 2, show that P n X n + h is uncorrelated with X j i.e. E [( X n + h- P n X n + h )( X j- μ )] = 0 j = 1 , 2. 3. Consider the stationary solution to X t- φX t- 1 = Z t + θZ t- 1 where Z t ∼ WN (0 ,σ 2 ) (a) show by direct substitution that X t = Z t + ( θ + φ ) ∞ X j =1 φ j- 1 Z t- j is a solution. 1 (b) Show that when | φ | < 1 lim N →∞ N X j =1 | φ j- 1 | < ∞ . (c) Compute Cov ( X n ,X n +2 ). 4. (a) Find the ACVF of the time series Y t = Z t + 0 . 3 Z t- 1- . 4 Z t- 2 where Z t ∼ WN (0 , 1) (b) For this time series find P 1 Y 2 and its mean square error 5. Suppose that Z t is an iid sequence with Z t ∼ N (0 , 1). Consider the process defined by...
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STAT 443 assignment2 - STAT 443 Assignment 2 This...

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