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6-1 Introduction to Multiple Regression (SW Chapter 6) Outline 1. Omitted variable bias 2. Causality and regression analysis 3. Multiple regression and OLS 4. Measures of fit 5. Sampling distribution of the OLS estimator

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6-2 Omitted Variable Bias (SW Section 6.1) The error u arises because of factors that influence Y but are not included in the regression function; so, there are always omitted variables. Sometimes, the omission of those variables can lead to bias in the OLS estimator.
6-3 Omitted variable bias, ctd. 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) Both conditions must hold for the omission of Z to result in omitted variable bias .

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6-4 Omitted variable bias, ctd. In the test score example: 1. English language ability (whether the student has English as a second language) plausibly affects standardized test scores: Z is a determinant of Y . 2. Immigrant communities tend to be less affluent and thus have smaller school budgets – and higher STR : Z is correlated with X . Accordingly, 1 ˆ β is biased. What is the direction of this bias? What does common sense suggest? If common sense fails you, there is a formula…
6-5 Omitted variable bias, ctd. A formula for omitted variable bias: recall the equation, 1 ˆ β β 1 = 1 2 1 ( ) ( ) n i i i n i i X X u X X = = - - = 1 2 1 1 n i i X v n n s n = - where v i = ( X i X ) u i ( X i μ X ) u i . Under Least Squares Assumption 1, E [( X i μ X ) u i ] = cov( X i , u i ) = 0. But what if E [( X i μ X ) u i ] = cov( X i , u i ) = σ Xu 0?

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6-6 Omitted variable bias, ctd. In general (that is, even if Assumption #1 is not true), 1 ˆ β β 1 = 1 2 1 1 ( ) 1 ( ) n i i i n i i X X u n X X n = = - - p 2 Xu X σ σ = u Xu X X u σ σ σ σ σ × = u Xu X σ ρ σ , where ρ Xu = corr( X , u ). If assumption #1 is valid, then ρ Xu = 0, but if not we have….
6-7 The omitted variable bias formula : 1 ˆ β p β 1 + u Xu X σ ρ σ If an omitted factor Z is both : (1) a determinant of Y (that is, it is contained in u ); and (2) correlated with X , then ρ Xu 0 and the OLS estimator 1 ˆ β is biased (and is not consistent). The math makes precise the idea that districts with few ESL students (1) do better on standardized tests and (2) have smaller classes (bigger budgets), so ignoring the ESL factor results in overstating the class size effect. Is this is actually going on in the CA data ?

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6-8 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 is small (recall overall “test score gap” = 7.4)
6-9 Digression on causality and regression analysis What do we want to estimate?

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