Tests of Exclusion Restrictions on Regression Coefficients: Formulation and Interpretation

Transcription

1 ECONOMICS 5* -- NOTE 6 ECON 5* -- NOTE 6 Tests of Excluson Restrctons on Regresson Coeffcents: Formulaton and Interpretaton The populaton regresson equaton (PRE) for the general multple lnear regresson model takes the form: Y = β + β X + β X + L + β X + u () k k where u s an d (ndependently and dentcally dstrbuted) random error term. PRE () consttutes the unrestrcted model for tests of excluson restrctons. Excluson restrctons set one or more regresson coeffcents equal to zero. Common types of excluson restrctons:. One slope coeffcent equals zero. Tests of the ndvdual sgnfcance of a sngle slope coeffcent.. All slope coeffcents jontly equal zero. Tests of the jont sgnfcance of all slope coeffcents.. Some slope coeffcents equal zero. Tests of the jont sgnfcance of a subset of slope coeffcents. ECON 5* -- Note 6: Tests of Excluson Restrctons: Formulaton Page of 9 pages

2 ECONOMICS 5* -- NOTE 6. Tests of Excluson Restrctons: Formulaton TEST : One slope coeffcent equals zero. The null hypothess s: H : β j = ; the slope coeffcent of regressor X j equals zero. The alternatve hypothess s: H : β j ; the slope coeffcent of regressor X j s not equal to zero. Queston addressed by ths test : Is the regressor X j relevant n explanng the dependent varable Y (controllng for the effects on Y of all the other ncluded regressors)? Is the explanatory varable X j ndvdually relevant n explanng the dependent varable Y? Does the explanatory varable X j have an ndvdually sgnfcant margnal effect on the dependent varable Y? Does the true PRF for the dependent varable Y nclude X j? Names (labels) for ths test: a test of the ndvdual relevance of the explanatory varable X j. a test of the ndvdual sgnfcance of the slope coeffcent for X j. ECON 5* -- Note 6: Tests of Excluson Restrctons: Formulaton Page of 9 pages

3 ECONOMICS 5* -- NOTE 6 TEST : All slope coeffcents jontly equal zero. The null hypothess s: H : β j = j =,,..., k. β = and β = and... β k =. All K slope coeffcents equal zero;.e., the slope coeffcents are jontly equal to zero. The alternatve hypothess s: H : β j j =,,..., k. β and/or β and/or β... and/or β k. At least one of the slope coeffcents does not equal zero. Queston addressed by ths test : Is the regresson model gven by PRE () relevant n explanng the dependent varable Y? Are the k = K explanatory varables n PRE () jontly relevant n explanng the dependent varable Y? Names (labels) for ths test: a test of the overall sgnfcance of the regresson model a test of the jont sgnfcance of the slope coeffcents ECON 5* -- Note 6: Tests of Excluson Restrctons: Formulaton Page of 9 pages

4 ECONOMICS 5* -- NOTE 6 TEST : Some regresson coeffcents equal zero. Example : Suppose the unrestrcted model s gven by the PRE Y = β + β X + β X + β X + u ( =,..., N) () so that K = 4. The null hypothess s: H : β j = j =,. β = and β =. The two slope coeffcents β and β both equal zero;.e., the slope coeffcents β and β are jontly equal to zero. The alternatve hypothess s: H : β j j =,. β and/or β. At least one of the slope coeffcents β and β does not equal zero. Queston addressed by ths test : Are the explanatory varables X and X jontly relevant n explanng the dependent varable Y? Does the true PRF for the dependent varable Y nclude both the explanatory varables X and X? Names (labels) for ths test: a test of the jont relevance of the regressors X and X a test of the jont sgnfcance of the slope coeffcents for X and X ECON 5* -- Note 6: Tests of Excluson Restrctons: Formulaton Page 4 of 9 pages

5 ECONOMICS 5* -- NOTE 6 Example : A model of North Amercan car prces from Stata Tutorals Suppose the unrestrcted model s gven by the PRE prce = β + β weght + β weght + β mpg + u ( =,..., N) () The margnal effect of the varable weght on the dependent varable prce s obtaned by takng the partal dervatve of the regresson functon n () wth respect to weght : prce weght = β + β weght ( =,..., N). A suffcent condton for the margnal effect of weght to equal zero for all observatons s that the coeffcents β and β equal zero. The null hypothess s: H : β j = j =, ; β = and β = The two slope coeffcents β and β both equal zero;.e., the slope coeffcents β and β are jontly equal to zero. The margnal effect on prce of weght s zero. The alternatve hypothess s: H : β j j =, ; β and/or β At least one of the slope coeffcents β and β does not equal zero. The margnal effect on prce of weght s not equal zero. Queston addressed by ths test : Is the explanatory varable weght relevant n determnng prce? Does the true PRF for the dependent varable prce nclude both the regressors weght and weght? ECON 5* -- Note 6: Tests of Excluson Restrctons: Formulaton Page 5 of 9 pages

6 ECONOMICS 5* -- NOTE 6. Defntons. The Restrcted and Unrestrcted Models The unrestrcted model s the PRE that corresponds to, or s mpled by, the alternatve hypothess H. It s the PRE that s presumed to be true f the null hypothess H s false. The restrcted model s the PRE that corresponds to, or s mpled by, the null hypothess H. It s the PRE that s presumed to be true f the null hypothess H s true. It s obtaned by substtutng the coeffcent restrctons specfed by the null hypothess H nto the unrestrcted model. ECON 5* -- Note 6: Tests of Excluson Restrctons: Formulaton Page 6 of 9 pages

7 ECONOMICS 5* -- NOTE 6. Examples Suppose the unrestrcted model s gven by the PRE Y = β + β X + β X + β X + u ( =,..., N) () The number of free (unrestrcted) regressons coeffcents n () s K = 4. Test : a test of the ndvdual sgnfcance of one slope coeffcent. The null hypothess s: H : β = ; the slope coeffcent of regressor X equals zero. The alternatve hypothess s: H : β ; the slope coeffcent of regressor X s not equal to zero. The unrestrcted model correspondng to the alternatve hypothess H s smply PRE (): Y = β + β X + β X + β X + u ( =,..., N) () The number of free (unrestrcted) regresson coeffcents n model () s K = K = 4. The restrcted model correspondng to the null hypothess H s obtaned by settng β = n the unrestrcted model (): Y = β + β X + β X + u ( =,..., N) () The number of free (unrestrcted) regresson coeffcents n model () s K =. Number of coeffcent restrctons specfed by the null hypothess H s q = K K = K K = 4 =. ECON 5* -- Note 6: Tests of Excluson Restrctons: Formulaton Page 7 of 9 pages

8 ECONOMICS 5* -- NOTE 6 Test : a test of the jont sgnfcance of all slope coeffcents. Y = β + β X + β X + β X + u ( =,..., N) () The null hypothess s: H : β j = j =,, ; β = and β = and β =. All of the K = k = 4 = slope coeffcents equal zero;.e., the slope coeffcents are jontly equal to zero. The alternatve hypothess s: H : β j j =,, ; β and/or β and/or β. At least one of the slope coeffcents does not equal zero. The unrestrcted model correspondng to the alternatve hypothess H s smply PRE (): Y = β + β X + β X + β X + u ( =,..., N) () The number of free (unrestrcted) regresson coeffcents n the unrestrcted model () s K = K = 4. The restrcted model correspondng to the null hypothess H s obtaned by settng β = and β = and β = n the unrestrcted model (): Y = β + u ( =,..., N) (4) The number of free (unrestrcted) regresson coeffcents n the restrcted model (4) s K =. Number of coeffcent restrctons specfed by the null hypothess H s q = K K = K K = 4 =. ECON 5* -- Note 6: Tests of Excluson Restrctons: Formulaton Page 8 of 9 pages

9 ECONOMICS 5* -- NOTE 6 Test : a test of the jont sgnfcance of two of the slope coeffcents. Y = β + β X + β X + β X + u ( =,..., N) () The null hypothess s: H : β j = j =, ; β = and β =. The two slope coeffcents β and β both equal zero;.e., the slope coeffcents β and β are jontly equal to zero. The alternatve hypothess s: H : β j j =, ; β and/or β. At least one of the slope coeffcents β and β does not equal zero. The unrestrcted model correspondng to the alternatve hypothess H s smply PRE (): Y = β + β X + β X + β X + u ( =,..., N) () The number of free (unrestrcted) regresson coeffcents n the unrestrcted model () s K = K = 4. The restrcted model correspondng to the null hypothess H s obtaned by settng β = and β = n the unrestrcted model (): Y = β + β X + u ( =,..., N) (5) The number of free (unrestrcted) regresson coeffcents n the restrcted model (5) s K =. Number of coeffcent restrctons specfed by the null hypothess H s q = K K = K K = 4 =. ECON 5* -- Note 6: Tests of Excluson Restrctons: Formulaton Page 9 of 9 pages