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Unformatted text preview: Problem Set 4 Solutions Econ 120C / Winter 2008 Due date: March 5, 2008 1 Exercises Suppose the true regression model is Y i = + 1 X 1 i + 2 X 2 i + u i , (1) where X 1 i and X 2 i are exogenous explanatory variables. However, the re searcher can only directly obtain data on X 1 i ; consequently, he runs the regression Y i = + 1 X 1 i + v i (2) using ordinary least squares (OLS) regression. Use the notation Y j = Cov ( Y, X j ) for j = 1 , 2 and 12 = Cov ( X 1 , X 2 ) as needed in your answers. 1. Express the error term v i in terms of quantities in the true regression model. First solve for v i in terms of other quantities in the estimated regression equation; then use the true regression model to express the answer in terms of u i , 2 and X 2 i : v i = Y i ( + 1 X 1 i ) (3) = 2 X 2 i + u i . 2. What is Cov ( X 1 i , v i )? 1 Use Formula (2.33) in Stock and Watson to write: Cov ( X 1 i , v i ) = Cov ( X 1 i , 2 X 2 i + u i ) = 2 Cov ( X 1 i , X 2 i ) + Cov ( X 1 i , u i ) = 2 12 + 0 = 2 12 . 3. Under what condition on 12 = Cov ( X 1 i , X 2 i ) is X 1 i still exogenous in the estimated regression model? Explain. 2 12 = 0, which either implies that X 2 does not enter the regression equation ( 2 = 0 ), X 1 and X 2 are uncorrelated( 12 = 0 ) or both. 4. What is the formula for the OLS slope coefficient estimator, 1 , in the estimated regression model? The formula for the OLS slope coefficient estimator may be derived 1 based on the condition that Cov ( X 1 i , v i ) = 0 (which will not hold if X 1 is endogenous). Using this condition with Stock and Watsons Formula (2.33), we can write Cov ( X 1 i , v i ) = Cov ( X 1 i , Y i ( + 1 X 1 i )) = Cov ( X 1 i , Y i ) + Cov ( X 1 i , ( + 1 X 1 i )) (4) = Cov ( X 1 i , Y i ) 1 Cov ( X 1 i , X 1 i ) = Y 1 1 2 X 1 = 0 . Solving for 1 , we get 1 = Y 1 / 2 X 1 . (5) The estimator for 1 is then obtained by replacing the population para meters with their sample estimators 2 : 1 = S Y 1 /S 2 1 , (6) where S Y 1 = n i =1 ( Y i Y )( X 1 i X 1 ) n 1 , (7) 1 It is also possible to use calculus to derive the OLS slope coefficient estimator. 2 This procedure defines the OLS slope coefficient estimator as a method of moments estimator; calculus may be used to show this is the estimator which follows from solving for the linear regression function parameter values that minimize the sum of squared residuals. 2 and S 1 = n i =1 ( X 1 i X 1 ) 2 n 1 , (8) 5. Show that 1 is (generally) not a consistent estimator of 1 . The Law of Large Numbers implies that sample covariance and variance estimators are consistent; for instance: plim ( S 1 ) = plim ( n i =1 ( X 1 i X 1 ) 2 n 1 ) = plim ( n n 1 n i =1 ( X 1 i X 1 ) 2 n ) (9) = plim ( n n 1 ) plim ( n i =1 ( X 1 i X 1 ) 2 n ) = 1 E [( X 1 X 1 ) 2 ] = 2 X 1 , where Slutskys theorem and the continuous mapping theorem are at...
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This note was uploaded on 04/08/2008 for the course ECON 120C taught by Professor Stohs during the Winter '08 term at UCSD.
 Winter '08
 Stohs
 Econometrics

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