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Lecture30-2010 - Restricted Least Squares and Hypothesis...

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Restricted Least Squares and Hypothesis Testing: Lecture XXX Charles B. Moss November 29, 2010 I. Resticted Least Squares A. Consider fitting 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. Solving for the least squares estimates ˆ β = ( X X ) 1 ( X y ) = 4 . 7238 4 . 0727 3 . 9631 2 . 0185 0 . 9071 (2) Estimating the variance matrix ˆ s 2 = y y ( y X ) ( X X ) 1 ( X y ) 30 5 = 1 . 2858 V ˆ β = ˆ s 2 ( X X ) 1 = 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 (3) 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XXX Fall 2010 Table 1: Regressin Data for Restricted Least Squares Observation y x 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 2
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XXX Fall 2010 B. Next, consider the hypothesis that β 1 = β 2
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