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Homework4Key - Economics 4261 Introduction to Econometrics...

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Unformatted text preview: Economics 4261 Introduction to Econometrics Fall 2010 HOMEWORK 4 : REGRESSION INFERENCE KEY Question 1 Consider the regression model y X , with K predictors (excluding intercept). Using our standard assumptions, (a) Show SSR e 1 e 1 M X . (b) Let s 2 SSR n K 1 Using Assumptions A1-A4, show that E r s 2 | X s 2 . This will require you to use the trace operator mentioned in HW1. You should proceed in two steps: (1) Notice that SSR 1 M X is a quadratic form. Write this out using sum notation and show that E r 1 M X | X s 2 trace r M s . (2) Show that trace r M s n K 1. Note that since V ar p | X q 2 p X 1 X q 1 , regression packages printout y V ar p | X q s 2 p X 1 X q 1 .................................................................................................................. (For this proof, I let M M X and P P X . ) Using the hint, we proceed in two steps. The first step tells us that SSR 2 trace p X q . Then under the second part, we establish the form of trace p X q . Step 1 First note that E r 1 X | X s n i 1 n j 1 m ij E r i j | X s , where the above holds for all i,j , and since M is just a function of X , we can take it outside the expectation. Next, by spherical errors, E r i j | X s 0 for all i j . For i j , we have E r i j | X s 2 . Putting these two facts together yields E r 1 X | X s n i 1 m ii E r 2 i | X s 2 n i 1 m ii 2 trace p M q . Step 2 Next we need to obtain the expression for trace p M q . We note that trace p M q trace p I n q trace p P q . The trace of the identity matrix is n , hence trace p M q n trace p P q . Recall that P X p X 1 X q 1 X 1 so kwilliams@umn.edu http://www.econ.umn.edu/ ~ will3324 Page 1 of 9 Economics 4261 Introduction to Econometrics Fall 2010 trace p P q trace r X p X 1 X q 1 X 1 s trace rp X 1 X q 1 p X 1 X qs trace r I #regression coefficients s . The above work holds from the first homework assignment. The total number of regressors is K . We also have a coefficient for the intercept. Thus, trace r I #regression coefficients s K 1, which implies trace p M q n K 1 . Lets check our work: E r s 2 | X s SSR n K 1 SSR 1 n K 1 E r 1 M X | X s 1 n K 1 2 trace r M s 1 n K 1 2 p n K 1 q 1 n K 1 2 . kwilliams@umn.edu http://www.econ.umn.edu/ ~ will3324 Page 2 of 9 Economics 4261 Introduction to Econometrics Fall 2010 Question 2 Consider the linear model y i 1 x 1 i 2 x 2 i i , such that i P 1 ,..., 10. Suppose X 1 X 10 50 5 50 304 34 5 34 5 and SSE 10 i 1 p y i y q 2 . 4987 and MSE . 1752 Perform calculations at the 95% confidence level....
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This note was uploaded on 02/24/2011 for the course ECON 4261 taught by Professor Staff during the Spring '08 term at Minnesota.

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Homework4Key - Economics 4261 Introduction to Econometrics...

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