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Unformatted text preview: 61Introduction to Multiple Regression (SW Chapter 6)Outline1.Omitted variable bias 2.Causality and regression analysis 3.Multiple regression and OLS 4.Measures of fit 5.Sampling distribution of the OLS estimator 62Omitted Variable Bias (SW Section 6.1) The error uarises 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. 63Omitted variable bias, ctd. The bias in the OLS estimator that occurs as a result of an omitted factor is called omitted variablebias. For omitted variable bias to occur, the omitted factor “Z” must be: 1.A determinant of Y(i.e. Zis part of u); and2.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. 64Omitted 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: Zis a determinant of Y. 2.Immigrant communities tend to be less affluent and thus have smaller school budgets – and higher STR: Zis correlated with X. Accordingly, 1ˆβis biased. What is the direction of this bias? •What does common sense suggest? •If common sense fails you, there is a formula… 65Omitted variable bias, ctd. A formula for omitted variable bias: recall the equation, 1ˆβ– β1= 121()()niiiniiXXuXX==∑∑= 1211niiXvnnsn=∑where vi= (Xi– X)ui≈(Xi– μX)ui. Under Least Squares Assumption 1, E[(Xi– μX)ui] = cov(Xi,ui) = 0. But what if E[(Xi– μX)ui] = cov(Xi,ui) = σXu≠0? 66Omitted variable bias, ctd. In general (that is, even if Assumption #1 is not true), 1ˆβ– β1= 1211()1()niiiniiXXunXXn==∑∑p→2XuXσσ= uXuXXuσσσσσ×= uXuXσρσ, where ρXu= corr(X,u). If assumption #1 is valid, then ρXu= 0, but if not we have…. 67The omitted variable bias formula: 1ˆβp→β1+ uXuXσρσIf an omitted factor Zis both: (1) a determinant of Y(that is, it is contained in u); and(2) correlated with X, then ρXu≠0 and the OLS estimator 1ˆβis 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? 68•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) 69Digression on causality and regression analysis What do we want to estimate?•What is, precisely, a causal effect? •The commonsense definition of causality isn’t precise enough for our purposes. enough for our purposes....
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This note was uploaded on 05/25/2011 for the course ECON 2007 taught by Professor J during the Spring '11 term at UCL.
 Spring '11
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