12_06ans - STAT 420 Examples for Fall 2007 1 Consider the...

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STAT 420 Examples for 12/06/2007 Fall 2007 1. Consider the AR(2) process for which it is known that μ = 0, Y t = φ 1 Y t – 1 + φ 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 φ 1 and φ 2 . Yule-Walker equations for an AR(2) process: ρ 1 = φ 1 + φ 2 ρ 1 ρ 2 = φ 1 ρ 1 + φ 2 0.50 = φ 1 – 0.50 φ 2 × 2 1 = 2 φ 1 φ 2 0.55 = – 0.50 φ 1 + φ 2 0.55 = – 0.50 φ 1 + φ 2 ° 0.45 = 1.5 φ 1 ° 1 ˆ φ = 0.30 ° 1 = 2 ( 0.30 ) φ 2 1 = – 0.60 – φ 2 ° 2 ˆ φ = 0.40 b) Use your answers to part (a) to forecast y 101 , y 102 , and y 103 . Y N + 1 = μ + φ 1 ( Y N μ ) + φ 2 ( Y N – 1 μ ) + e N + 1 1 ˆ + N y = ( ) 1 ˆ N y = E N ( Y N + 1 ) = μ + φ 1 ( y N μ ) + φ 2 ( y N – 1 μ ) Y N + 2 = μ + φ 1 ( Y N + 1 μ ) + φ 2 ( Y N μ ) + e N + 2 2 ˆ + N y = ( ) 2 ˆ N y = E N ( Y N + 2 ) = μ + φ 1 ( 1 ˆ + N y μ ) + φ 2 ( y N μ ) Y N + 3 = μ + φ 1 ( Y N + 2 μ ) + φ 2 ( Y N + 1 μ ) + e N + 3 3 ˆ + N y = ( ) 3 ˆ N y = E N ( Y N + 3 ) = μ + φ 1 ( 2 ˆ + N y μ ) + φ 2 ( 1 ˆ + N y μ )

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Y N + k = μ + φ 1 ( Y N + k – 1 μ ) + φ 2 ( Y N + k
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