420Hw13ans - STAT 420(10 points(due Friday December 7 by...

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STAT 420 Fall 2007 Homework #13 (10 points) (due Friday, December 7, by 3:00 p.m.) 1. Given the time series 64 55 41 59 48 71 35 57 40 58 calculate r 1 and r 2 . ( Note: In practice reliable autocorrelation estimates are only obtained from series consisting of approximately 50 observations or more. ) 10 58 40 57 35 71 48 59 41 55 64 + + + + + + + + + = y = 52.8 . t y y y t - ( ) 2 y y t - ( ) ( ) 1 y y y y t t - - + ( ) ( ) 2 y y y y t t - - + 64 11.2 125.44 24.64 –132.16 55 2.2 4.84 –25.96 13.64 41 –11.8 139.24 –73.16 56.64 59 6.2 38.44 –29.76 112.84 48 –4.8 23.04 –87.36 85.44 71 18.2 331.24 –323.96 76.44 35 –17.8 316.84 –74.76 227.84 57 4.2 17.64 –53.76 21.84 40 –12.8 163.84 –66.56 58 5.2 27.04 1187.60 –710.64 462.52 ( ) ( ) ( ) 60 . 1187 64 . 710 1 2 1 1 1 1 - = - - - = ° ° = - = + N N t t t t t y y y y y y r 0.5984 . ( ) ( ) ( ) 60 . 1187 52 . 462 1 2 2 1 2 2 = - - - = ° ° = - = + N N t t t t t y y y y y y r 0.3895 .
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2. The model ( Y t μ ) = φ ( Y t – 1 μ ) + e t has been fitted to a time series giving φ ˆ = 0.6 , ° ˆ = 5.0 , and σ ˆ e 2 = 2.25 . The last five values of the series are 5.5 , 4.8 , 4.5 , 5.6 , 4.7 . Using the time of the last observation as the forecast origin, calculate the forecasts and approximate 95% probability limits for the next three observations. φ ˆ = 0.6 , ° ˆ = 5.0 , σ ˆ e 2 = 2.25 , y N = 4.7 . 1 ˆ + N y = ° ˆ + φ ˆ ( y N ° ˆ ) = 5.0 + 0.6 ( 4.7 – 5.0 ) = 4.82 .
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