Econ103fall10lec7

Econ103fall10lec7 - Introduction OLS Assumptions for...

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Introduction OLS Assumptions for Multiple Regression Estimation Goodness of Fit Econ 103, UCLA, Fall 2010 Introduction to Econometrics Lecture 7: Multiple Regression Sarolta Laczó October 14, 2010
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Introduction OLS Assumptions for Multiple Regression Estimation Goodness of Fit Introduction/1 General Multiple Regression Model: Y i = β 0 + β 1 X 1 i + β 2 X 2 i + ... + β k X ki + u i We have k regressors: X 1 i , X 2 i , ..., X ki k slope coefficients (parameters): β 1 , β 2 , ..., β k Each slope coefficient β j measures the effect of a one unit change in the corresponding regressor X ji , holding all else (e.g. the other regressors) constant. β 0 is the intercept, as before the residual u i : still omitted variables (but hopefully there are less in here since we are including more regressors)
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Introduction OLS Assumptions for Multiple Regression Estimation Goodness of Fit Introduction/2 Outline: OLS assumptions for multiple regression New assumption: no perfect multicollinearity between regressors Estimation Formally Adding % Still Learning English to our regression of Test Scores on STR Dummy variables in multiple regression Measures of goodness of fit
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Introduction OLS Assumptions for Multiple Regression Estimation Goodness of Fit OLS Assumptions for Multiple Regression/1 As in the simple regression model, we need to make some assumptions in order to estimate the coefficients β 0 , β 1 , ..., β k . The first 3 are very similar to our previous set of assumptions. 1 E ( u i | X 1 i = x 1 i , X 2 i = x 2 i , ..., X ki = x ki ) . In words, the expectation of u i is zero regardless of the values of the k regressors. 2 ( X 1 i , X 2 i , ..., X ki , Y i ) are independently and identically distributed ( i.i.d. ). This is true with random sampling. 3 ( X 1 i , X 2 i , ..., X ki , Y i ) have finite fourth moments. That is, large outliers are unlikely (this is generally true in economic data).
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Introduction OLS Assumptions for Multiple Regression Estimation Goodness of Fit OLS Assumptions for Multiple Regression/2 We also need a fourth assumption in the multiple regression model. This fourth assumption addresses how the various X ji ’s are related to each other. 4 The regressors ( X 1 i , X 2 i , ..., X ki ) are not perfectly multicollinear . This means that none of the regressors can be written as a perfect linear function of only the other regressors. Assumption 4 is rarely violated in practice, and when it is, it is typically by accident. However, if the correlation between any too regressors is ‘high’, that will also be problematic. (We will discuss this later.)
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Introduction OLS Assumptions for Multiple Regression Estimation Goodness of Fit OLS Assumptions for Multiple Regression/3 Example for perfect multicollinearity and why it is a problem: We have a sample of grades ( Y i ), and we interview the students in order to measure:
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