Lecture_5 - Lecture 5 Stock and Watson, Chapter 6 Multiple...

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1 Lecture 5 Stock and Watson, Chapter 6 Multiple regression: using omitted variable bias as our motivation, we will now consider data when we record the values of more than 1 regressor for each unit in the sample
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2 Omitted variable bias The bias in the OLS estimator that occurs as a result of an omitted factor is called omitted variable bias. For omitted variable bias to occur, the omitted factor “ Z ” must be: 1. A determinant of Y (i.e. Z is part of u ); and 2. Correlated with the regressor X ( i.e. corr( Z , X ) 0) Sign of bias = sign (effect of Z on Y) *sign(corr(Z, X)) Bias is positive : implies regression with omitted variable is an overestimate of the true effect of X on Y
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3 Omitted variable bias, ctd. In the test score example, consider Z = fraction of learners for whom English is a second language, Y = outcomes in a standardized test and X = class size in district: 1. Z is a determinant of Y : Sign? 2. Z is correlated with X : Why? Sign? Accordingly, 1 ˆ is biased. What is the direction of this bias?
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4 Districts with fewer English Learners have higher test scores Districts with lower percent EL ( PctEL ) have smaller classes Among districts with comparable PctEL , the effect of class size i small (recall overall “test score gap” = 7.4)
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5 The Population Multiple Regression Model Consider the case of two regressors: Y i = 0 + 1 X 1 i + 2 X 2 i + u i , i = 1,…, n Y is the dependent variable X 1 , X 2 are the two independent variables ( regressors ) ( Y i , X 1 i , X 2 i ) denote the i th observation on Y , X 1 , and X 2 . 0 = unknown population intercept 1 = effect on Y of a change in X 1 , holding X 2 constant 2 = effect on Y of a change in X 2 , holding X 1 constant u i = the regression error (omitted factors)
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6 Interpretation of coefficients in multiple regression Y i = 0 + 1 X 1 i + 2 X 2 i + u i , i = 1,…, n Consider changing X 1 by X 1 while holding X 2 constant: Population regression line before the change: Y = 0 + 1 X 1
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Lecture_5 - Lecture 5 Stock and Watson, Chapter 6 Multiple...

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