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lecture10 - ECON 103 Lecture 10 Inference with OLS(contd...

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ECON 103, Lecture 10: Inference with OLS (contd.) Maria Casanova February 11th (version 0) Maria Casanova Lecture 10
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Requirements for this lecture: Chapter 7 of Stock and Watson Maria Casanova Lecture 10
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1. Inference with the OLS estimators 1. HYPOTHESIS TESTING We can: Test a single hypothesis about a single coefficient β j Test a single hypothesis about a linear combination of coefficients (e.g. β j - β k ) Test multiple hypotheses about linear combinations of the β j ’s Today we will cover cases 2 and 3. Maria Casanova Lecture 10
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1.1 Joint hypotheses The t-test procedure is valuable for testing statistical significance of an individual regression coefficient (or a linear combination of coefficients). HOWEVER, the t-test procedure IS NOT valid for testing joint hypotheses. Maria Casanova Lecture 10
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1.1 Joint hypotheses Example: We estimate the following regression by OLS for a sample of manual workers: Wage = β 0 + β 1 Edu + β 2 Exp + β 3 Region + ε A joint hypothesis would be that neither education nor experience have an effect on a manual worker’s wage. Formally: H 0 : β 1 = 0 and β 2 = 0 vs H 1 : β 1 6 = 0 and/or β 2 6 = 0 In general, a joint hypothesis is a hypothesis that imposes two or more restrictions on the regression coefficients. Maria Casanova Lecture 10
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1.1 Joint hypotheses This is the intuition behind the procedure to test joint hypotheses: When we impose restrictions of the type H 0 : β 1 = 0 and β 2 = 0, the residual sum of squares (or RSS ) increases for two reasons: 1 Because the OLS estimators are chosen to minimize the sum of squared residuals, the RSS always increases when variables are dropped from the model = This is an algebraic fact. 2 If X 1 and X 2 are contributing to explaining Y (i.e. β 1 and/or β 2 6 = 0), the RSS (unexplained variation in Y ) also increases when we drop them from the model.
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