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Homework2 Solutions

Homework2 Solutions - Mehmet Soytas Wednesday Applied...

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Mehmet Soytas Applied Econometrics I, Summer, 2009 Wednesday, May 21, 2009 (due: Wednesday, May 27, 2009) Homework Assignment 2 1. Show that the OLS formula for the coe¢ cient ° 1 in the multiple regression model Y i = ° 0 + ° 1 X 1 i + ° 2 X 2 i + u i reduces to the OLS formula of ° 1 in the one regressor case if the correlation between X 1 i and X 2 i is zero. The formula for ^ ° 1 in the multiple regression model is ( x 1 i = X 1 i ° ° X 1 , x 2 i = X 2 i ° ° X 2 , y i = Y i ° ° Y ): ^ ° 1 = P n i =1 y i x 1 i P n i =1 x 2 2 i ° P n i =1 y i x 2 i P n i =1 x 1 i x 2 i P n i =1 x 2 1 i P n i =1 x 2 2 i ° ( P n i =1 x 1 i x 2 i ) 2 Correlation between X 1 i and X 2 i is de°ned as : ± X 1 ;X 2 = cov ( X 1 ;X 2 ) p var ( X 1 i ) p var ( X 2 i ) 1 n ° 1 P n i =1 x 1 i x 2 i p 1 n ° 1 P n i =1 x 2 1 i p 1 n ° 1 P n i =1 x 2 2 i If the correlation is equal to zero, ± X 1 ;X 2 = 1 n ° 1 P n i =1 x 1 i x 2 i p 1 n ° 1 P n i =1 x 2 1 i p 1 n ° 1 P n i =1 x 2 2 i = 0 ) 1 n ° 1 P n i =1 x 1 i x 2 i = 0 ) P n i =1 x 1 i x 2 i = 0 If we replace P n i =1 x 1 i x 2 i with 0 in the formula of ^ ° 1 . ^ ° 1 = P n i =1 y i x 1 i P n i =1 x 2 2 i ° P n i =1 y i x 2 i (0) P n i =1 x 2 1 i P n i =1 x 2 2 i ° (0) 2 = P n i =1 y i x 1 i P n i =1 x 2 2 i P n i =1 x 2 1 i P n i =1 x 2 2 i = P n i =1 y i x 1 i P n i =1 x 2 1 i = The formula for ^ ° 1 in the one variable regression model 2. For this question, use the California Testscore Data Set and your regression package (STATA). a. First perform a multiple regression of testscores on a constant, the student-teacher ratio, and the percent of English learners. Record the coe¢ cients. Run reg testscr str el_pct,r Next, do the following three step procedure instead: b. First, regress the testscore on a constant and the percent of English learners. Calculate the residuals and store them under the name resYX2. Run reg testscr el_pct,r generate resYX2=testscr-664.7394-(-0.6711562)el_pct c. Second, regress the student-teacher ratio on a constant and the percent of English learners. Calculate the residuals from this regression and store these under the name resX1X2. Run reg str el_pct,r generate resX1X2=str-19.33432-0.019413el_pct d. Finally regress resYX2 on resX1X2. Explain intuitively why the simple regression coe¢ cient in the last regression is identical to the regression coe¢ cient on the student-teacher ratio in the multiple regression Run
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