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Lecture 6 - Introduction to Multiple Regression(SW Chapter...

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1 ˆ β Introduction to Multiple Regression (SW Chapter 6) 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. 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 . 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 . [omission leads to over estimation] 2. Immigrant communities tend to be less affluent and thus have smaller school budgets – and higher STR : Z is correlated with X . Accordingly, is biased. What is the direction of this bias? What does common sense suggest? If common sense fails you, there is a formula… A formula for omitted variable bias: recall the equation,
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1 ˆ β 1 2 1 ( ) ( ) n i i i n i i X X u X X = = - - 1 ˆ β β 1 = = where v i = ( X i – ) 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 ˆ 0? In general (that is, even if Assumption #1 is not true), β 1 = = = , where ρ Xu = corr( X , u ). If assumption #1 is valid, then ρ Xu = 0, but if not we have…. The omitted variable bias formula : 2
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1 ˆ β 1 ˆ β β 1 + 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 ˆ 0 and the OLS estimator 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 ? 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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Omitted Variable Bias: 4
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Example: A researcher plans to study the causal effect of police on crime using data from a random sample of U.S. counties. He plans to regress the country’s crime rate on the (per capita) size of the country’s police force. a. Explain why this regression is likely to suffer from omitted variable bias. Which variables would you add to the regression to control for important omitted variables? b. Determine whether the regression will likely to over- or under-estimate the effect of police on the crime rate.
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