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Problem Set 4

# Problem Set 4 - Problem Set 4 Solutions Econ 120C Winter...

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Problem Set 4 Solutions Econ 120C / Winter 2008 Due date: March 5, 2008 1 Exercises Suppose the true regression model is Y i = β 0 + β 1 X 1 i + β 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 = β 0 + β 1 X 1 i + v i (2) using ordinary least squares (OLS) regression. Use the notation σ Y j = Cov ( Y, X j ) for j = 1 , 2 and σ 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 , β 2 and X 2 i : v i = Y i - ( β 0 + β 1 X 1 i ) (3) = β 2 X 2 i + u i . 2. What is Cov ( X 1 i , v i )? 1

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Use Formula (2.33) in Stock and Watson to write: Cov ( X 1 i , v i ) = Cov ( X 1 i , β 2 X 2 i + u i ) = β 2 Cov ( X 1 i , X 2 i ) + Cov ( X 1 i , u i ) = β 2 σ 12 + 0 = β 2 σ 12 . 3. Under what condition on σ 12 = Cov ( X 1 i , X 2 i ) is X 1 i still exogenous in the estimated regression model? Explain. β 2 σ 12 = 0, which either implies that X 2 does not enter the regression equation ( β 2 = 0 ), X 1 and X 2 are uncorrelated( σ 12 = 0 ) or both. 4. What is the formula for the OLS slope coefficient estimator, ˆ β 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 Watson’s Formula (2.33), we can write Cov ( X 1 i , v i ) = Cov ( X 1 i , Y i - ( β 0 + β 1 X 1 i )) = Cov ( X 1 i , Y i ) + Cov ( X 1 i , - ( β 0 + β 1 X 1 i )) (4) = Cov ( X 1 i , Y i ) - β 1 Cov ( X 1 i , X 1 i ) = σ Y 1 - β 1 σ 2 X 1 = 0 . Solving for β 1 , we get β 1 = σ Y 1 2 X 1 . (5) The estimator for β 1 is then obtained by replacing the population para- meters with their sample estimators 2 : ˆ β 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 ˆ β 1 is (generally) not a consistent estimator of β 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 - μ X 1 ) 2 ] = σ 2 X 1 , where Slutsky’s theorem and the continuous mapping theorem are at the last two steps along with the Law of Large Numbers. A similar argument can be used to show plim ( S Y 1 ) = σ Y 1 . (10) Using the ratio version of Slutsky’s theorem, we get plim ( ˆ β 1 ) = plim ( S Y 1 /S 2 1 ) = σ Y 1 2 X 1 . (11) But substitution from the population regression model shows that σ Y 1 = Cov ( Y i , X 1 i ) = Cov ( β 0 + β 1 X 1 i + β 2 X 2 i + u i , X 1 i )) = Cov (

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Problem Set 4 - Problem Set 4 Solutions Econ 120C Winter...

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