lect15_2010

# lect15_2010 - 1 21 Introduction to Econometrics Econ 322...

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Unformatted text preview: 1 / 21 Introduction to Econometrics Econ 322 Fall, 2010 Lecture 15: Joint Hypothesis Tests October 25, 2010 Topics Covered triangleright Topics Covered Joint tests of hypotheses Testing Joint Linear Hypotheses The F-test with homoskedastic errors Confidence Regions Regression Statistics for GLRM Adjusted R 2 Overall F-stat Regressions using “time series” 2 / 21 1. tests of multiple hypotheses 2. the “F-test” 3. multiple regression diagnostics 4. regression with “time series” Joint tests of hypotheses Topics Covered triangleright Joint tests of hypotheses Testing Joint Linear Hypotheses The F-test with homoskedastic errors Confidence Regions Regression Statistics for GLRM Adjusted R 2 Overall F-stat Regressions using “time series” 3 / 21 square Suppose we estimate the following regression equation: y i = β + β 1 X 1 i + β 2 X 2 i + epsilon1 i , using OLS with heteroscedasticity-consistent standard errors. square Suppose we would like to test the following hypotheses: H : β 1 = 0 and β 2 = 0 vrs H A : either β 1 negationslash = 0 or β 2 negationslash = 0 or both square In this case the test is testing the overall “importance” of the regression. If we cannot reject at least one of these two hypotheses then we don’t have a very good regression! Testing the two hypotheses the naive way Topics Covered triangleright Joint tests of hypotheses Testing Joint Linear Hypotheses The F-test with homoskedastic errors Confidence Regions Regression Statistics for GLRM Adjusted R 2 Overall F-stat Regressions using “time series” 4 / 21 square One way you might think about testing these two hypotheses is to perform two separate tests. That is, first test whether β 1 = 0 and then test whether β 2 = 0 . square This approach is not quite correct. Suppose we did the first test and chose to reject the first hypothesis at the 5% level. Then we did the second test and chose to reject the hypothesis at the 5% level as well. square The question is: is the size of this test equal to 5%? square The answer to this question is NO. Let’s see why: square The size of a test is the probability that we reject the null hypothesis when the null is true. That is, size = Prob ( reject H | H true ) . Testing Two Hypothesis (Cont) Topics Covered triangleright Joint tests of hypotheses Testing Joint Linear Hypotheses The F-test with homoskedastic errors Confidence Regions Regression Statistics for GLRM Adjusted R 2 Overall F-stat Regressions using “time series” 5 / 21 square Suppose we follow the testing strategy above. Then we would form the two test statistics t 1 = ˆ β 1- SE ( ˆ β 1 ) , and t 2 = ˆ β 2- SE ( ˆ β 2 ) . square Then we would reject the null if Testing Two Hypothesis (Cont) Topics Covered triangleright Joint tests of hypotheses Testing Joint Linear Hypotheses The F-test with homoskedastic errors Confidence Regions Regression Statistics for GLRM Adjusted R 2 Overall F-stat Regressions using “time series” 6 / 21 square So the probability of rejecting the NULL when it is true is given by square Thus, assuming t 1 and t 2 are independent we have Testing Two Hypothesis (Cont)...
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lect15_2010 - 1 21 Introduction to Econometrics Econ 322...

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