# Chapter 6 - Multiple Regression(SW Chapters 6 OLS estimate...

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6-1 Multiple Regression (SW Chapters 6) OLS estimate of the Test Score / STR relation: n 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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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 1 = 1 2 1 () n ii i n i i XX u = = = 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?

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6-6 Omitted variable bias, ctd. In general (that is, even if Assumption #1 is not true), 1 ˆ β 1 = 1 2 1 1 () 1 n ii i n i i XX u n n = = p 2 Xu X σ = uX u u σσ ⎛⎞ × ⎜⎟ ⎝⎠ = u Xu X ρ , where Xu = corr( X , u ). If assumption #1 is valid, then Xu = 0, but if not ?
6-7 The 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 (and is not consistent).

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6-8 Omitted variable bias, ctd. Y i = β 0 + 1 X 1 i + u i , i = 1,…, n But we have omitted X 2 so that u i = 2 X 2 i + ε i with E( ε i | X 1 i , X 2 i ) = 0 cov( X 1i , u i ) = E [( X 1i μ X ) u i ] = E [( X 1i X ) ( 2 X 2 i + ε i ) ] = 2 E [( X 1i X ) X 2 i ] = 2 cov( X 1i , X 2 i ) so that Bias is approximately 1 1 2 Xu X σ = 12 1 2 2 XX X βσ 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.
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Chapter 6 - Multiple Regression(SW Chapters 6 OLS estimate...

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