Slides30-2010

# Slides30-2010 - Restricted Least Squares and Hypothesis...

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Restricted Least Squares and Hypothesis Testing: Lecture XXX Charles B. Moss November 19, 2010 Charles B. Moss () Restricted Least Squares November 19, 2010 1 / 16

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1 Resticted Least Squares 2 Testing Linear Restrictions Charles B. Moss () Restricted Least Squares November 19, 2010 2 / 16
Resticted Least Squares Consider ftting the linear model y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + β 4 x 4 + ± (1) to the data presented in Table 1. Charles B. Moss () Restricted Least Squares November 19, 2010 3 / 16

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Table: Regression Data for Restricted Least Squares Observation yx 1 x 2 x 3 x 4 1 75.72173 4.93638 9.76352 4.39735 2.27485 2 45.11874 6.95106 3.11080 -1.96920 3.59838 3 51.61298 4.69639 4.17138 3.84384 2.73787 4 92.53986 10.22038 8.93246 1.73695 5.36207 5 118.74310 12.05240 12.22066 6.40735 4.92600 6 80.78596 10.42798 5.58383 1.61742 9.30154 7 43.79312 2.94557 5.16446 1.21681 4.75092 8 47.84554 3.54233 5.58659 2.18433 3.65499 9 63.02817 4.56528 6.52987 4.40254 5.36942 10 88.83397 11.47854 8.82219 0.70927 2.94652 11 104.06740 11.87840 8.53466 5.21573 8.91658 12 57.40342 7.99115 7.42219 -3.62246 -2.19067 13 76.62745 7.14806 7.39096 5.19569 3.00548 14 109.96540 10.34953 9.82083 7.82591 7.09768 15 72.66822 7.74594 4.79418 5.39538 6.29685 16 68.22719 4.10721 8.51792 4.00252 3.88681 17 122.50920 12.77741 11.57631 6.85352 7.63219 18 70.71453 9.69691 6.54209 0.53160 0.79405 19 70.00971 6.46460 6.62652 4.31049 5.03634 20 75.82481 6.31186 8.49487 3.38461 5.53753 21 38.82780 3.04641 2.99413 2.69198 6.26460 22 79.15832 8.85780 7.29142 3.33994 2.86917 23 62.29580 5.82182 6.16096 4.18066 1.73678 24 80.63698 4.97058 9.83663 6.71842 3.47608 25 77.32687 5.90209 8.56241 5.42130 4.70082 26 23.34500 1.57363 2.82311 0.95729 0.69178 27 81.54044 9.25334 6.43342 5.02273 3.84773 28 67.16680 10.77622 5.21271 -0.87349 -1.17348 29 47.92786 6.96800 2.39798 -0.56746 6.08363 30 48.58950 7.06326 3.24990 -0.77682 3.09636 Charles B. Moss () Restricted Least Squares November 19, 2010 4 / 16
Solving for the least squares estimates ˆ β = ( X 0 X ) 1 ( X 0 y ) = 4 . 7238 4 . 0727 3 . 9631 2 . 0185 0 . 9071 (2) Estimating the variance matrix ˆ s 2 = y 0 y ( y 0 X )( X 0 X ) 1 ( X 0 y ) 30 5 =1 . 2858 V ± ˆ β s 2 ( X 0 X ) 1 (3) Charles B. Moss () Restricted Least Squares November 19, 2010 5 / 16

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V ± ˆ β = 0 . 5037 0 . 0111 0 . 0460 0 . 0252 0 . 0285 0 . 0111 0 . 0079 0 . 0068 0 . 0044 0 . 0033 0 . 0460 0 . 0068 0 . 0164 0 . 0104 0 . 0047 0 . 0252 0 . 0044 0 . 0104 0 . 0141 0 . 0070 0 . 0285 0 . 0033 0 . 0047 0 . 0070 0 . 0104 (4) Charles B. Moss () Restricted Least Squares November 19, 2010 6 / 16
Next, consider the hypothesis that β 1 = β 2 (which seems plausible given the results above).

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## This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.

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Slides30-2010 - Restricted Least Squares and Hypothesis...

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