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Unformatted text preview: Introduction Assumption 1 Revisited Omitted Variable Bias Econ 103, UCLA, Fall 2010 Introduction to Econometrics Lecture 6: Multiple Regression Sarolta Laczó October 12, 2010 Introduction Assumption 1 Revisited Omitted Variable Bias Introduction Simple Regression: Linear regression model with one regressor. Y i = β + β 1 X i + u i Multiple Regression: Linear regression model with more than one regressors. Y i = β + β 1 X 1 i + β 2 X 2 i + u i Why do we need to add additional regressors? Introduction Assumption 1 Revisited Omitted Variable Bias Introduction/2 Outline: Revisiting the OLS assumptions Assumption 1: E ( u i  X i ) = 0 . Omitted Variable Bias (OVB) When does it occur? Why is it a problem? We examine these questions both in our example (test scores and STR) and formally. Introduction Assumption 1 Revisited Omitted Variable Bias Assumption 1 Revisited/1 Recall our OLS Assumption 1: E ( u i  X i ) = 0 . In words, the error term u i has a conditional mean of zero given X i . We can also write it as E ( u i  X i = x i ) = 0 . E ( u i  X i = x i ) is the expected value of u i given that X i equals any number x i . E.g. E ( u i  STR i = 20) is the expected value of the residual given that STR i = 20 . So Assumption 1 says that the expected value of u i is 0 regardless of what X i is, or, more loosely, that u i and X i are unrelated. This also means that the expected value of omitted variables is 0 for any X i . Introduction Assumption 1 Revisited Omitted Variable Bias Assumption 1 Revisited/2 Under Assumption 1, the predicted value of the dependent variable, ˆ Y i , is also the expected value of Y i given X i . Why? The model is Y i = β + β 1 X i + u i Take conditional expectations of both sides: E ( Y i  X i = x i ) = E ( β + β 1 X i + u i  X i = x i ) = E ( β  X i = x i ) + E ( β 1 X i  X i = x i ) + E ( u i  X i = x i ) = β + β 1 X i So our predicted value, ˆ Y i = ˆ β + ˆ β 1 X i is also an estimate of the expected value E ( Y i  X i = x i ) . Introduction Assumption 1 Revisited Omitted Variable Bias Assumption 1 Revisited/3 Examples: 1 \ Test Scores i = 689 . 9 (9 . 47) 2 . 28 (0 . 48) STR i The expected value of test scores given e.g. STR i = 21 is 698 . 9 2 . 28 × 21 = 615 . 02 . 2 [ Sales i = 59 . 1 (10 . 2) + 5 . 67 (3 . 87) AdCampaign i The expected number of sales in a region given that advertising campaign A is used is 59 . 1 + 5 . 67 × 1 = 64 . 77 units. In a region where advertising campaign B is used, expected sales are 59 . 1 units. Introduction Assumption 1 Revisited...
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This note was uploaded on 10/23/2010 for the course STAT 10 taught by Professor Davis during the Spring '10 term at UCLA.
 Spring '10
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