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Unformatted text preview: 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 . 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 . 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...
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This note was uploaded on 03/12/2012 for the course STAT 443 taught by Professor Yuliagel during the Winter '09 term at Waterloo.
 Winter '09
 YuliaGel
 Forecasting

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