Homework1 Solution

# Homework1 Solution - Mehmet Soytas Wednesday Applied...

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Applied Econometrics I, Summer, 2009 Wednesday, May 13, 2009 (due: Monday, May 18, 2009) Homework Assignment 1 Solutions E ( u i j X i ) = 0 , implies that E ( Y i j X i ) = & 0 + 1 X i . Population Regression Function : Y i = & 0 + 1 X i + u i : Take expectation of both sides with respect to X . E ( Y i j X i ) = E ( & 0 + 1 X i + u i j X i ) E ( Y i j X i ) = E ( & 0 j X i ) + E ( 1 X i j X i ) + E ( u i j X i ) E ( Y i j X i ) = & 0 + 1 E ( X i j X i ) + E ( u i j X i ) E ( Y i j X i ) = & 0 + 1 X i + E ( u i j X i ) E ( Y i j X i ) = & 0 + 1 X i Step 2 follows from the fact that expectation is a linear operator (That basically means you can take expectation of individual terms in a summation). Step 3 uses the fact that expectation of a constant is itself. Step 4 is result of the fact that expectation of a random variable wrt itself is equal to that random variable . Last step uses the OLS assumption given in the question E ( u i j X i ) = 0 . 2. Write down the expressions for the R 2 for the regression of Y on X and then X on Y . Compare them. Write the R 2 of the regression of Y on X R 2 = ESS TSS = P n i =1 ( ^ Y i Y ) 2 P n i =1 ( Y i Y ) 2 observe that ^ Y i = ^ & 0 + ^ 1 X i = P n i =1 ( ^ & 0 + ^ 1 X i Y i ) 2 P n i =1 ( Y i Y ) 2 use ^ & 0 = Y ^ 1 X = P n i =1 ( Y ^ 1 X + ^ 1 X i Y ) 2 P n i =1 ( Y i Y ) 2 = P n i =1 ( ^ 1 ( X i X )) 2 P n i =1 ( Y i Y ) 2 = ^ 2 1 P n i =1 ( X i X ) 2 P n i =1 ( Y i Y ) 2 use ^ 1 = P n i =1 ( X i X )( Y i Y ) P n i =1 ( X i X ) 2 = P n i =1 ( X i X )( Y i Y ) P n i =1 ( X i X ) 2 ± 2 P n i =1 ( X i X ) 2 P n i =1 ( Y i Y ) 2 = ( P n i =1 ( X i X )( Y i Y ) ) 2 P n i =1 ( X i X ) 2 P n i =1 ( Y i Y ) 2 = ( 1 n 1 P n i =1 ( X i X )( Y i Y ) ) 2 1 n 1 P n i =1 ( X i X ) 2 1 n 1 P n i =1 ( Y i Y ) 2 = ² 1 n 1 P n i =1 ( X i X )( Y i Y ) p 1 n 1 P n i =1 ( X i X ) 2 p 1 n 1 P n i =1 ( Y i Y ) 2 ³ 2 = ² cov ( X;Y ) p var ( X ) p var ( Y ) ³ 2 = ´ ± X;Y µ 2 Symmetrically R 2 of the regression of X on Y will be equal to ´ ± Y;X µ 2 . Since from the ± X;Y = ± Y;X : Therefore R 2 from both regressions will be equal. 3. Go to the web address http://wps.aw.com/aw_stock_ie_2/50/13016/3332253.cw/index.html. Down- load the data "California Test Score Data Used in Chapters 4-9". ± Replicate the regression result we saw in the lecture. Basically run a regression of testscr on str

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Homework1 Solution - Mehmet Soytas Wednesday Applied...

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