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Lecture 10 Multiple Linear Regression Model

# Lecture 10 Multiple Linear Regression Model - Economics 326...

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Economics 326 Methods of Empirical Research in Economics Lecture 10: Multiple regression model Vadim Marmer University of British Columbia March 3, 2011

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Why we need a multiple regression model I There are many factors a/ecting the outcome variable Y . I If we want to estimate the marginal e/ect of one of the factors (regressors), we need to control for other factors. I Suppose that we are interested in the e/ect of X 1 on Y , but Y is a/ected by both X 1 and X 2 : Y i = β 0 + β 1 X 1 , i + β 2 X 2 , i + U i . I Suppose we regress Y only against X 1 : ˆ β 1 = n i = 1 ( X 1 , i ° ¯ X 1 ) Y i n i = 1 ( X 1 , i ° ¯ X 1 ) 2 . 1/17
Omitted variable bias Since Y depends on X 2 : Y i = β 0 + β 1 X 1 , i + β 2 X 2 , i + U i , I We have: ˆ β 1 = n i = 1 ( X 1 , i ° ¯ X 1 ) ( β 0 + β 1 X 1 , i + β 2 X 2 , i + U i ) n i = 1 ( X 1 , i ° ¯ X 1 ) 2 = β 1 + β 2 n i = 1 ( X 1 , i ° ¯ X 1 ) X 2 , i n i = 1 ( X 1 , i ° ¯ X 1 ) 2 + n i = 1 ( X 1 , i ° ¯ X 1 ) U i n i = 1 ( X 1 , i ° ¯ X 1 ) 2 . I Assume that E ( U i j X 1 , i , X 2 , i ) = 0 . Now, conditional on X °s: E ° ˆ β 1 ± = β 1 + β 2 n i = 1 ( X 1 , i ° ¯ X 1 ) X 2 , i n i = 1 ( X 1 , i ° ¯ X 1 ) 2 6 = β 1 . The exception is when n i = 1 ( X 1 , i ° ¯ X 1 ) X 2 , i = 0 . 2/17

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Omitted variable bias I When the true model is Y i = β 0 + β 1 X 1 , i + β 2 X 2 , i + U i , but we regress only on X 1 , Y i = β 0 + β 1 X 1 , i + V i , where V i is the new error term: V i = β 2 X 2 , i + U i . I If X 1 and X 2 are related, we can no longer say that E ( V i j X 1 , i ) = 0 . I When X 1 changes, X 2 changes as well, which contaminates estimation of the e/ect of X 1 on Y . I As a result, ˆ β 1 from the regression of Y on X 1 alone is biased. 3/17
Multiple linear regression model I The econometrician observes the data: f ( Y i , X 1 , i , X 2 , i , . . . , X k , i ) : i = 1 , . . . , n g . I The model: Y i = β 0 + β 1 X 1 , i + β 2 X 2 , i + . . . + β k X k , i + U i , E ( U i j X 1 , i , X 2 , i , . . . , X k , i ) = 0 . I We also assume no multicollinearity: None of the regressors are constant and there are no exact linear relationships among the regressors. 4/17

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Interpretation of the coe¢ cients Y i = β 0 + β 1 X 1 , i + β 2 X 2 , i + . . . + β k X k , i + U i .
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