Lecture 9

Lecture 9 - LECTURE 9: HYPOTHESIS TESTING JOINT HYPOTHESIS...

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LECTURE 9: HYPOTHESIS TESTING JOINT HYPOTHESIS T-Test procedure is valuable for testing the statistical significance of an individual regression coefficient but is NOT valid for testing joint hypotheses. Conceptually: want to test how much the RESIDUAL SUM OF SQUARES increases when we impose the restrictions. If it increases by too much, then we will reject the Null Hypothesis. Because the OLS estimates 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. The question is, how much? Y i = β 0 + β 1 X 1i + β 2 X 2i + ε i H o : β 1 = β 2 = 0 H 1 : not H 0 General Case : 1. Write down the linear restrictions: β 1 = 0 β 2 = 0 2. Estimate the UNCONSTRAINED MODEL (full model, no restrictions): Y i = β 0 + β 1 X 1i + β 2 X 2i + ε i 3. Get the RSS for the unconstrained model (RSS U ) 4. Plug the LINEAR RESTRICTIONS IN: Y i = β 0 + ε i 5. Estimate the CONSTRAINED/RESTRICED MODEL.
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NOTE: The Restricted Model always has fewer parameters than the unrestricted model. 6. Get the RSS for the constrained or restricted model (RSS R ) NOTE: RSS R is always larger than the RSS U (since we know that OLS minimizes the RSS) 7. Calculate the F-statistic k n , m U U R F ~ k n RSS m ) RSS RSS ( F Where: m = number of linear restrictions k = number of parameters in the unrestricted regression NOTE: F is always non-negative F measures the relative increase in RSS when moving from the unrestricted model to the restricted model: It can be shown that: k n ) R 1 ( m R R F 2 U 2 R 2 U STATA EXAMPLE: F-TEST EXAMPLE . reg lincome education Source |SS df MS Number of obs = 65241
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-------------+------------------------------ F( 1, 65239) =12031.82 Model |13276.8994 1 13276.8994 Prob > F = 0.0000 Residual |71990.0625 65239 1.10348201 R-squared = 0.1557
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Lecture 9 - LECTURE 9: HYPOTHESIS TESTING JOINT HYPOTHESIS...

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