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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 Ftest with homoskedastic errors Confidence Regions Regression Statistics for GLRM Adjusted R 2 Overall Fstat Regressions using “time series” 2 / 21 1. tests of multiple hypotheses 2. the “Ftest” 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 Ftest with homoskedastic errors Confidence Regions Regression Statistics for GLRM Adjusted R 2 Overall Fstat 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 heteroscedasticityconsistent 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 Ftest with homoskedastic errors Confidence Regions Regression Statistics for GLRM Adjusted R 2 Overall Fstat 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 Ftest with homoskedastic errors Confidence Regions Regression Statistics for GLRM Adjusted R 2 Overall Fstat 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 Ftest with homoskedastic errors Confidence Regions Regression Statistics for GLRM Adjusted R 2 Overall Fstat 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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 Fall '11
 LANDONLANE
 Statistics, Econometrics, Regression Analysis, Null hypothesis, confidence regions, Joint Linear, Overall Fstat

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