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Unformatted text preview: Problem Set 4. Answer Key Introduction to Econometrics  Fall 2006 Due Date: Tuesday, November 7th, BEFORE CLASS. Each student is responsible for handing one handwritten solution. If working in groups, indicate the members of the group (to facilitate grading). PLEASE SHOW YOUR WORK . 1. Consider the following multiple regression model: Y i = & + & 1 X 1 i + & 2 X 2 i + u i You collect a random sample of 1000 observations and you estimate following regression: b Y i = 2 : 52 (1 : 12) + 1 : 25 (0 : 95) X 1 i & : 89 ( : 64) X 2 i R 2 = 0 : 22 Assume that all the lest squares assumptions hold. (a) Estimate the expected change in Y i when X 1 i is increased by 1 unit and X 2 i is constant. Estimate the expected change in Y i when X 1 i is constant and X 2 i is decreased by 2 units. Estimate the expected change in Y i when X 1 i is decreased by 2 unit and X 2 i is increased by 1 units. Solution: The expected change in Y i given X 1 i and X 2 i can be estimated as: & b Y i = 1 : 25& X 1 i & : 89& X 2 i ( i ) & X 1 i = 1 and & X 2 i = 0 , so & b Y i = 1 : 25 ( ii ) & X 1 i = 0 and & X 2 i = & 2 , so & b Y i = ( & 2) ( & : 89) = 1 : 78 ( iii ) & X 1 i = & 2 and & X 2 i = 1 , so & b Y i = ( & 2) (1 : 25) + (1) ( & : 89) = & 3 : 39 (b) Compute the R 2 of this regression. Solution: R 2 = 1 & n & 1 n & k & 1 SSR TSS , but we also know that: R 2 = 1 & SSR TSS , so: R 2 = 1 & n & 1 n & k & 1 & 1 & R 2 In this case, n = 1000 and k = 2 , so: R 2 = 1 & 1000 & 1 1000 & 2 & 1 (1 & : 22) = 0 : 2184 2. Consider the following multiple regression model: Y i = & + & 1 X 1 i + & 2 X 2 i + u i where E ( u i j X 1 i ;X 2 i ) = 0 : (a) Assume X 2 i = 2 + X 1 i : Can you compute the OLS coe cients? Explain. Solution: X 2 i = 2 + X 1 i = 2 X i + X 1 i ; so the regressor X 2 i can be written as a perfect linear function of the other regressors. It follows that the regressor are perfectly multicollinear and the OLS coe cients cannot be computed. (b) Assume again that X 2 i = 2 + X 1 i : Can you write a single variable model: Y i = + 1 X 1 i + u i , equivalent to the multiple regression model above? Can you compute the OLS coe cients of this single variable model? What is the intuition here?...
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 Fall '08
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 Econometrics

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