12_07ans - STAT 420 Examples for 12/007/2010 Fall 2010 1....

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Unformatted text preview: STAT 420 Examples for 12/007/2010 Fall 2010 1. Consider the AR(2) process for which it is known that = 0, Y t = 1 Y t 1 + 2 Y t 2 + e t Based on a series of length N = 100, we observe y 99 = 2, y 100 = 3. a) Suppose r 1 = 0.50, r 2 = 0.55. Use the method of moments to estimate 1 and 2 . Yule-Walker equations for an AR(2) process: 1 = 1 + 2 1 2 = 1 1 + 2 0.50 = 1 0.50 2 2 1 = 2 1 2 0.55 = 0.50 1 + 2 0.55 = 0.50 1 + 2 0.45 = 1.5 1 1 = 0.30 1 = 2 ( 0.30 ) 2 1 = 0.60 2 2 = 0.40 a ) If 1 and 2 are equal to your answers to part (a), is this process stationary? ( B ) = 1 + 0.3 B 0.4 B 2 = ( 1 0.5 B ) ( 1 + 0.8 B ) The roots of ( z ) = 0 are z 1 = 2 and z 2 = 1.25. Both are outside the unit circle. That is, | z 1 | > 1, | z 2 | > 1. The process is stationary. OR An AR(2) model is stationary if 1 < 2 < 1, 2 + 1 < 1, 2 1 < 1. 1 < 0.4 < 1, 0.4 0.3 < 1, 0.4 + 0.3 < 1. This process is stationary. b) Use your answers to part (a) to forecast y 101 , y 102 , and y 103 . Y N + 1 = + 1 ( Y N ) + 2 ( Y N 1 ) + e N + 1 1 + N y = ( ) 1 N y = E N ( Y N + 1 ) = + 1 ( y N ) + 2 ( y N 1 ) Y N + 2 = + 1 ( Y N + 1...
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This note was uploaded on 12/17/2010 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois, Urbana Champaign.

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12_07ans - STAT 420 Examples for 12/007/2010 Fall 2010 1....

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