Marginal Effects of Explanatory Variables: Constant or Variable? 1. Constant Marginal Effects of Explanatory Variables: A Starting Point

Transcription

1 CONOMICS * -- NOT CON * -- NOT Margnal ffects of xplanatory Varables: Constant or Varable?. Constant Margnal ffects of xplanatory Varables: A Startng Pont Nature: A contnuous explanatory varable has a constant margnal effect on the dependent varable f t enters the regressor set only lnearly and addtvely. Model : Y β + β + β + u () Model contans only two explanatory varables and and two regressors. The populaton regresson functon, or condtonal mean functon, f (, ) n Model takes the form (Y, ) β + β + β. The margnal effects on Y of the two explanatory varables and n equaton () are obtaned analytcally by partally dfferentatng Y, or the condtonal mean of Y gven and, wth respect to each of the explanatory varables and.. The margnal effect of n Model s: Y β a constant. The margnal effect of n Model s: Y β a constant CON * -- Note : Margnal ffects of xplanatory Varables Page of pages

2 CONOMICS * -- NOT xample of Model : prce β + β wgt + β mpg + u (*) where prce the prce of the -th car (n US dollars); wgt the weght of the -th car (n pounds); mpg the mles per gallon (fuel effcency) for the -th car (n mles per gallon).. * Model : constant margnal effects of wgt and mpg. regress prce wgt mpg Source SS df MS Number of obs F(, 7).7 Model Prob > F. Resdual R-squared Adj R-squared.7 Total Root MS prce Coef. Std. rr. t P> t [9% Conf. Interval] wgt mpg _cons CON * -- Note : Margnal ffects of xplanatory Varables Page of pages

3 CONOMICS * -- NOT. Varable Margnal ffects and Interacton Terms: Squares and Cross Products of Contnuous xplanatory Varables Nature: Interactons between two contnuous varables refer to products of pars of explanatory varables. If j and h are two contnuous explanatory varables, the nteracton term between them s the product j h. The nteracton of the varable j wth tself s smply the product. Incluson of these regressors n a lnear regresson model allows for varable or nonconstant margnal effects of the explanatory varables on the condtonal mean of the dependent varable Y. Usage: Interacton terms between contnuous varables allow the margnal effect of one explanatory varable to be a lnear functon of both tself and other explanatory varables. Model : j j j Y β + β + β + β + β + β + u () Model contans only two explanatory varables and but fve regressors. Formally, the populaton regresson functon (Y, ) f (, ) n PR () can be derved as a second-order Taylor seres approxmaton to the functon f (, ). A second-order Taylor seres approxmaton to the populaton regresson functon f (, ) takes the form (Y, ) β + β + β + β + β + β. CON * -- Note : Margnal ffects of xplanatory Varables Page of pages

4 CONOMICS * -- NOT Margnal ffects n Model : The margnal effects on Y of the two explanatory varables and n equaton () are obtaned analytcally by partally dfferentatng Y, or the condtonal mean of Y gven and, wth respect to each of the explanatory varables and. The populaton regresson functon for Model s: (Y, ) β + β + β + β + β + β. The margnal effect of n Model s: Y β + β + β a lnear functon of both and. The margnal effect of n Model s: Y β + β + β a lnear functon of both and Squares of Contnuous xplanatory Varables Purpose: Allow for ncreasng or decreasng margnal effects of an explanatory varable on the dependent varable -- sometmes called ncreasng or decreasng margnal returns. CON * -- Note : Margnal ffects of xplanatory Varables Page of pages

5 CONOMICS * -- NOT Determnng whether the margnal effect of s ncreasng or decreasng Whether the margnal effect of s ncreasng or decreasng.e., whether exhbts ncreasng or decreasng margnal returns s determned by the sgn of the regresson coeffcent β on the regressor n Model. Y β + β + β + β + β + β + u () We prevously saw that the margnal effect of n Model s gven by the frstorder partal dervatve of Y, or (Y, ), wth respect to Y : β + β + β To determne whether the margnal effect of n Model s ncreasng or decreasng n, we need to examne the second-order partal dervatve of Y, or (Y, ), wth respect to : Y β. The margnal effect of s ncreasng n meanng exhbts ncreasng margnal returns when Y β >.e., when β >. The margnal effect of s decreasng n meanng exhbts decreasng margnal returns when Y β <.e., when β < CON * -- Note : Margnal ffects of xplanatory Varables Page of pages

6 CONOMICS * -- NOT Determnng whether the margnal effect of s ncreasng or decreasng Whether the margnal effect of s ncreasng or decreasng.e., whether exhbts ncreasng or decreasng margnal returns s determned by the sgn of the regresson coeffcent β on the regressor n Model. Y β + β + β + β + β + β + u () We prevously saw that the margnal effect of n Model s gven by the frstorder partal dervatve of Y, or (Y, ), wth respect to Y : β + β + β To determne whether the margnal effect of n Model s ncreasng or decreasng n, we need to examne the second-order partal dervatve of Y, or (Y, ), wth respect to : Y β. The margnal effect of s ncreasng n meanng exhbts ncreasng margnal returns when Y β >.e., when β >. The margnal effect of s decreasng n meanng exhbts decreasng margnal returns when Y β <.e., when β < CON * -- Note : Margnal ffects of xplanatory Varables Page 6 of pages

7 CONOMICS * -- NOT Products of Two Contnuous xplanatory Varables Purpose: Allow for relatonshps of complementarty or substtutablty between and n determnng Y. Whether and are complementary or substtutable s determned by the sgn of the regresson coeffcent β on the nteracton term n Model. Y β + β + β + β + β + β + u () The margnal effects of and n Model are gven by the frst-order partal dervatves of Y, or (Y, ), wth respect to and : Y Y β + β + β β + β + β To determne whether the margnal effect of ( ) n Model s ncreasng or decreasng n ( ), we need to examne the second-order cross partal dervatve of Y, or (Y, ), wth respect to and : Y β. The margnal effect of s ncreasng n (or the margnal effect of s ncreasng n ) -- meanng and are complementary -- when Y β >. The margnal effect of s decreasng n (or the margnal effect of s decreasng n ) -- meanng and are substtutable -- when Y β < CON * -- Note : Margnal ffects of xplanatory Varables Page 7 of pages

8 CONOMICS * -- NOT xample of Model : prce β + β wgt + β mpg + β wgt + β mpg + β wgt mpg + u. (*) where prce the prce of the -th car (n US dollars); wgt the weght of the -th car (n pounds); wgt the square of wgt ; mpg the mles per gallon (fuel effcency) for the -th car (n mles per gallon); mpg the square of mpg ; wgt mpg the product of wgt and mpg for the -th car.. * Model : varable margnal effects of wgt and mpg. regress prce wgt mpg wgtsq mpgsq wgtmpg Source SS df MS Number of obs F(, 68).8 Model Prob > F. Resdual R-squared Adj R-squared.78 Total Root MS prce Coef. Std. rr. t P> t [9% Conf. Interval] wgt mpg wgtsq mpgsq wgtmpg _cons CON * -- Note : Margnal ffects of xplanatory Varables Page 8 of pages

9 CONOMICS * -- NOT Hypothess Tests on the Margnal ffects of wgt and mpg prce β + β wgt + β mpg + β wgt + β mpg + β wgt mpg + u (*) Test : The margnal effect of wgt on prce s zero for all cars. The margnal effect of wgt on prce n Model * s: prce wgt ( prce wgt, mpg ) wgt Null and Alternatve Hypotheses β + β wgt + β mpg H : β and β and β H : β and/or β and/or β specfes three coeffcent restrctons Unrestrcted Model Correspondng to H : regresson equaton (*) Restrcted Model Correspondng to H : set β and β and β n (*). prce β + β mpg + β mpg + u. * Test : Test hypothess that margnal effect of wgt equals zero for all cars. test wgt wgtsq wgtmpg ( ) wgt. ( ) wgtsq. ( ) wgtmpg. F(, 68) 6. Prob > F.7 CON * -- Note : Margnal ffects of xplanatory Varables Page 9 of pages

10 CONOMICS * -- NOT Test : The margnal effect of mpg on prce s zero for all cars. prce β + β wgt + β mpg + β wgt + β mpg + β wgt mpg + u (*) The margnal effect of mpg on prce n Model * s: prce mpg ( prce wgt, mpg ) mpg Null and Alternatve Hypotheses β + β mpg + β wgt H : β and β and β H : β and/or β and/or β specfes three coeffcent restrctons Unrestrcted Model Correspondng to H : regresson equaton (*) Restrcted Model Correspondng to H : set β and β and β n (*). prce β + β wgt + β wgt + u. * Test : Test hypothess that margnal effect of mpg equals zero for all cars. test mpg mpgsq wgtmpg ( ) mpg. ( ) mpgsq. ( ) wgtmpg. F(, 68). Prob > F.6 CON * -- Note : Margnal ffects of xplanatory Varables Page of pages

11 CONOMICS * -- NOT Test : The margnal effect of wgt on prce s constant. prce β + β wgt + β mpg + β wgt + β mpg + β wgt mpg + u (*) The margnal effect of wgt on prce n Model * s: prce wgt ( prce wgt, mpg ) wgt β + β wgt + β mpg β (a constant) f β and β Null and Alternatve Hypotheses H : β and β H : β and/or β specfes two coeffcent restrctons Unrestrcted Model Correspondng to H : regresson equaton (*) Restrcted Model Correspondng to H : set β and β n (*). prce β + β wgt + β mpg + β mpg + u. * Test : Test hypothess that margnal effect of wgt s constant. test wgtsq wgtmpg ( ) wgtsq. ( ) wgtmpg. F(, 68) 8.8 Prob > F. CON * -- Note : Margnal ffects of xplanatory Varables Page of pages

12 CONOMICS * -- NOT Test : The margnal effect of mpg on prce s constant. prce β + β wgt + β mpg + β wgt + β mpg + β wgt mpg + u (*) The margnal effect of mpg on prce n Model * s: prce mpg ( prce wgt, mpg ) mpg β + β mpg + β wgt β (a constant) f β and β Null and Alternatve Hypotheses H : β and β H : β and/or β specfes two coeffcent restrctons Unrestrcted Model Correspondng to H : regresson equaton (*) Restrcted Model Correspondng to H : set β and β n (*). prce β + β wgt + β mpg + β wgt + u. * Test : Test hypothess that margnal effect of mpg s constant. test mpgsq wgtmpg ( ) mpgsq. ( ) wgtmpg. F(, 68).7 Prob > F.7 CON * -- Note : Margnal ffects of xplanatory Varables Page of pages

13 CONOMICS * -- NOT Test : The margnal effects of both wgt and mpg on prce are constant. prce β + β wgt + β mpg + β wgt + β mpg + β wgt mpg + u (*) The margnal effect of wgt on prce n Model * s: prce wgt ( prce wgt, mpg ) wgt β + β wgt + β mpg β (a constant) f β and β The margnal effect of mpg on prce n Model * s: prce mpg ( prce wgt, mpg ) mpg β + β mpg + β wgt β (a constant) f β and β Null and Alternatve Hypotheses H : β and β and β H : β and/or β and/or β specfes three coeffcent restrctons Unrestrcted Model Correspondng to H : regresson equaton (*) Restrcted Model Correspondng to H s Model *: set β and β and β n (*). prce β + β wgt + β mpg + u (*). * Test : Test hypothess that margnal effects of wgt and mpg are constants. test wgtsq mpgsq wgtmpg ( ) wgtsq ( ) mpgsq ( ) wgtmpg F(, 68) 8.7 Prob > F. CON * -- Note : Margnal ffects of xplanatory Varables Page of pages