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STAT 443 assignment3

STAT 443 assignment3 - STAT 443 Assignment 3 This...

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STAT 443: Assignment 3 This assignment is to be handed in at the start of your lecture of Monday 2 April . Please make sure that you write your name, student number and section number on your script. 1 ARIMA processes 1. Determine which of the following ARMA processes are stationary or causal, where Z t WN (0 , σ 2 ). (a) X t + 0 . 3 X t - 1 - 0 . 5 X t - 2 = Z t (b) X t + X t - 2 = Z t + 0 . 2 Z t - 1 + 0 . 7 Z t - 2 (c) X t + 1 . 8 X t - 1 + 0 . 81 X t - 2 + 0 . 1 X t - 3 = 0 . 5 Z t + 0 . 5 Z t - 1 (d) X t - 0 . 5 X t - 1 + 0 . 5 X t - 2 = Z t - 2 Z t - 1 2. For the processes in Question 1 which are causal compute the first six coefficients ψ i , i = 0 , · · · , 5 in the MA ( ) representation X t = j =0 ψ j Z t - j . [You can use MAPLE output if the results are explained clearly] 3. Let W = ( W 1 , · · · , W n ) T , U and V be random vectors with finite second moments and denote the best linear predictor of U using W by P ( U | W ) in a mean squared error sense. Prove the following results (a) E [ U - P ( U | W )] = 0 (b) E [( U - P ( U | W )) W ] = 0 (c) If α i and β are constants then P ( α 1 U + α 2 V + β | W ) = α 1 P ( U | W )+ α 2 P ( V | W )+ β (d) P ( n i =1 α i W i + β | W ) = n i =1 α i W i + β 4. For an AR (1) process { X t } find the best linear predictor of X 2 given X 1 and X 3 .
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