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# ch5 - Multiple Regression(SW Chapter 5 OLS estimate of the...

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5-1 Multiple Regression (SW Chapter 5) OLS estimate of the Test Score / STR relation: ± TestScore = 698.9 – 2.28 × STR , R 2 = .05, SER = 18.6 (10.4) (0.52) Is this a credible estimate of the causal effect on test scores of a change in the student-teacher ratio? No : there are omitted confounding factors (family income; whether the students are native English speakers) that bias the OLS estimator: STR could be “picking up” the effect of these confounding factors.

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5-2 Omitted Variable Bias (SW Section 5.1) 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 ; and 2. correlated with the regressor X . Both conditions must hold for the omission of Z to result in omitted variable bias .
5-3 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…

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5-4 A formula for omitted variable bias: recall the equation, 1 ˆ β 1 = 1 2 1 () n ii i n i i X Xu 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 X ) u i ] = cov( X i , u i ) = 0. But what if E [( X i X ) u i ] = cov( X i , u i ) = σ Xu 0?
5-5 Then 1 ˆ β 1 = 1 2 1 () n ii i n i i X Xu X X = = = 1 2 1 1 n i i X v n n s n = ⎛⎞ ⎜⎟ ⎝⎠ so E ( 1 ˆ ) – 1 = 1 2 1 n i n i i X E XX = = 2 Xu X σ = uX u X σσ × where holds with equality when n is large; specifically, 1 ˆ p 1 + u Xu X ρ , where Xu = corr( X , u )

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5-6 Omitted variable bias formula : 1 ˆ β p 1 + u Xu X σ ρ ⎛⎞ ⎜⎟ ⎝⎠ . 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 Xu 0 and the OLS estimator 1 ˆ is biased. 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 ?
5-7 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)

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5-8 Three ways to overcome omitted variable bias 1. Run a randomized controlled experiment in which treatment ( STR ) is randomly assigned: then PctEL is still a determinant of TestScore , but PctEL is uncorrelated with STR . ( But this is unrealistic in practice. ) 2. Adopt the “cross tabulation” approach, with finer
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ch5 - Multiple Regression(SW Chapter 5 OLS estimate of the...

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