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Unformatted text preview: ECON 103, Lecture 8: Multiple Regression I Maria Casanova May 5 (version 0) Maria Casanova Lecture 8 1. Introduction Simple Regression: Linear regression model with one regressor. Y i = + 1 X i + u i Multiple Regression: Linear regression model with more than one regressors. Y i = + 1 X 1 i + 2 X 2 i + u i Why do we need to add additional regressors? Maria Casanova Lecture 8 1. Introduction Outline: Revisiting the first OLS assumption 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 through examples and formally. Maria Casanova Lecture 8 2. Assumption 1 Revisited 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 error term 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 . Maria Casanova Lecture 8 2. Assumption 1 Revisited What does Assumption 1 mean? Under Assumption 1, the predicted value of the dependent variable, Y i , is an estimate of the expected value of Y i given X i . Why? The model is Y i = + 1 X i + u i Take conditional expectations of both sides: E ( Y i  X i = x i ) = E ( + 1 X i + u i  X i = x i ) = E (  X i = x i ) + E ( 1 X i  X i = x i ) + E ( u i  X i = x i ) = + 1 X i So Y i = + 1 X i is an estimate of E ( Y i  X i = x i ). Maria Casanova Lecture 8 2. Assumption 1 Revisited 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....
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 Spring '07
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