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Unformatted text preview: Introduction to Multiple Regression (SW Chapter 6&7) Outline 1. Omitted variable bias 2. Causality and regression analysis 3. Multiple regression and OLS 4. Measures of fit 5. Sampling distribution of the OLS estimator 6. Hypothesis testing 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. 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 . 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… Omitted Variable Bias: 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 underestimate the effect of police on the crime rate. Example: Regress log(wage rate) on a constant, experience, education to estimate return to schooling. Does this regression suffer omitted variable bias? Does the regression under or overestimate the return to schooling? More generally, 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 gradations of STR and PctEL – within each group, all classes have the same PctEL , so we control for PctEL ( But soon we will run out of data, and what about other determinants like family income and parental education ?) 3. Use a regression in which the omitted variable ( PctEL ) is no longer omitted: include PctEL as an additional regressor in a multiple regression. The Population Multiple Regression Model (SW Section 6.2) Consider the case of two regressors: Y i = β + β 1 X 1 i + β...
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This note was uploaded on 11/23/2009 for the course FIN 5290 taught by Professor Li during the Fall '09 term at Temple.
 Fall '09
 Li

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