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# lecture12_short - ECON 103 Lecture 12 Internal Validity of...

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Unformatted text preview: ECON 103, Lecture 12: Internal Validity of Multiple Regression Analysis Maria Casanova February 23rd (version 0) Maria Casanova Lecture 12 Requirements for this lecture: Chapter 9 of Stock and Watson Maria Casanova Lecture 12 0. Introduction Studies based on regression analysis are internally valid if the estimated regression coefficients: are unbiased are consistent have associated standard errors that yield confidence intervals with the desired confidence level. Maria Casanova Lecture 12 0. Introduction Why do we care about potential bias of the OLS coefficients? The answer is related to the difference between 1 Describing the data and 2 Empirically modeling the social or economic processes involved. The second task is more ambitious, and will usually be our objective when we run a regression. Maria Casanova Lecture 12 0. Introduction In this lecture: We will see 5 reasons why the OLS estimator of the multiple regression coefficients might be biased. The 5 sources of bias arise because the explanatory variable(s) is (are) correlated with the error term in the population regression, violating the first Least Squares Assumption: Ass1: E ( ε i | X i ) = 0 = ⇒ Cov ( ε i , X i ) = 0 In none of the 5 cases will the bias go to 0 as we increase the sample size, so the OLS estimator will also be inconsistent. We will investigate the circumstances that lead to inconsistent standard errors Maria Casanova Lecture 12 0. Introduction The potential sources of bias are: 1 Omitted variable bias 2 Misspecification of the functional form of the regression function 3 Imprecise measurement of the independent variable (”errors in variables”) 4 Sample selection 5 Simultaneous causality Maria Casanova Lecture 12 1. Omitted variable bias Omitted variable bias arises when a variable that is omitted from the regression both... a) Determines Y b) Is correlated with one or more of the included regressors. Let the correlation between X i and ε i be Corr ( X i , ε i ) = ρ x ε . Then the OLS estimator has the limit: ˆ β 1 p-→ β 1 + ρ x ε σ ε σ x The bias in ˆ β 1 is given by the term ρ x ε ( σ ε /σ x ). Maria Casanova Lecture 12 1. Omitted variable bias Bias( ˆ β 1 ) = ρ x ε σ ε σ x Notice that: The size of the bias does not depend on the sample size. In the presence of omitted variable bias ˆ β 1 is both biased and inconsistent . The size of the bias depends on the correlation between the regressor and the error term ( ρ x ε ). The direction of the bias depends on whether X and ε are positively or negatively correlated. Maria Casanova Lecture 12 1. Omitted variable bias Proof that both a) and b) are necessary conditions [ > advanced topic < ] Imagine that the following is the population regression function for wages: Wage i = β + β 1 fitness i + β 2 age i + u i , where E ( u i | fitness i , age i ) = 0 (i.e. L.S. Ass1 is satisfied) We estimate the following regression: Wage i = β + β 1 fitness i + ε i Then: ε i = β 2 age i + u i Maria Casanova Lecture 12 1. Omitted variable bias1....
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lecture12_short - ECON 103 Lecture 12 Internal Validity of...

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