Extensions of the Linear ModelWalter Sosa-EscuderoEcon 471. Applied Econometrics. Spring 2009March 4, 2009Walter Sosa-Escudero Extensions of the Linear ModelDummy explanatory variablesDo men earn more than women just because of gender?Problem: men and women may differ in more dimensions thatjust gender (education, experience, etc).A simple modelWi= β1+ β2Xi+ β3Di+ uiWi= wage, Xi= education, Di= 1 if i is male, and 0 otherwise.Diis a dummy or binary explanatory variable.Problem: we cannot interpret β3as a derivative (we cannot moveDimarginally).Walter Sosa-Escudero Extensions of the Linear ModelCompute expected wages for males and females:E(Wi|Di= 1) = β1+ β2Xi+ β3E(Wi|Di= 0) = β1+ β2XiThen:E(Wi|Di= 1) − E(Wi|Di= 0) = β3β3as the difference in expected wages between males and femalesholding everything else constant.Walter Sosa-Escudero Extensions of the Linear ModelImplicitely, we are estimating a model for men and womenforcing both models to have the same slope but maybedifferent intercepts (see figure).Estimation is straightforward: just regress Wion Xiand Di.Note that this model fits all the classical assumptions so nonew estimation/inference strategies should be changed, justthe interpretation.A simple test that gender is not an issue is a t-test ofH0: β3= 0Walter Sosa-Escudero Extensions of the Linear ModelSome remarks:To distinguish between one category or the other we used onlyone dummy. What happens if we try to estimate this model:Wi= β1+ β2Xi+ β3D1i+ β4D1i+ uiD1i= 1 if male, 0 if not, and D2i= 1 if female and 0otherwise?Interpretation depends on how we define dummies. Considerthe following model:Wi= β1+ β2Xi+ β∗3D∗i+ uiD∗i= 1 if female and 0 otherwise. Note that β∗3= −β3in theoriginal case. It does not matter which category we assign to1 as long as we keep a consistent interpretation.Walter Sosa-Escudero Extensions of the Linear ModelDependent variable in logs Consider:wi= β1+ β2Xi+ β3Di+ uiwhere wi= ln Wi. Call wMithe log wage for males, and wFifor females, then:wMi− wFi= ln WMi− ln WFi= lnWMiWFi= β3so,WMi/WFi− 1 = eβ3− 1eβ3− 1 = how much more proportionally is the wage formales higher than that for a female.Walter Sosa-Escudero Extensions of the Linear ModelRemember from a basic calculus that for small β3eβ3− 1 ' β3then, when the explanatory variable is in logs and β3is small, wecan interpret β3directly as giving the proportional differencebetween categories.Walter Sosa-Escudero Extensions of the Linear ModelThe effect of unions on wages.Earnings function. 100 individuals. Source: Johnston andDiNardo (1994, p. 172.)Dependent Variable: log wagesExplanatory Variables : GRADE (years of education),POTEXP (experience), EXP2 (experience squared), UNION(1 if individual belongs to a union)Walter Sosa-Escudero Extensions of the Linear ModelVariable Coefficient Std. Error t-Statistic Prob.---------------------------------------------------C 0.595106 0.283485 2.099248 0.0384GRADE 0.083543 0.020093 4.157841 0.0001POTEXP 0.050274 0.014137 3.556214 0.0006EXP2 -0.000562 0.000288 -1.951412 0.0540UNION 0.165929 0.124454 1.333248 0.1856Union is not significant. What would be the interpretation if it were significant?Walter Sosa-Escudero Extensions of the Linear ModelOther uses of dummy variablesSlope dummy variables:Wi= β1+ β2Xi+ β3(DiXi) + uiwere Di= 1 for males and zero otherwise. Now, for menWi= β1+ (β2+ β3)Xi+ uiand for women:Wi= β0+ β1Xi+ uiWalter Sosa-Escudero Extensions of the Linear ModelTo give an interpretation for β2take the derivative of Wiwithrespect to Xiin both equations to get:∂Wi(men)∂Xi= β2+ β3∂Wi(women)∂Xi= β2then:β2=∂Wi(men)∂Xi−∂Wi(women)∂XiIn this model β3measures the differnce in slope between the modelof males and females.If β3is positive, this means that experience has a larger effect onincreasing wages for males than for females. As before, a simpletest that both slopes are the same can be performed by testingH0: β3= 0 in (3.6)Walter Sosa-Escudero Extensions of the Linear ModelExample: returns to education revisited.ln w = β1+ β2E + β3A + β4A2+ β5man + β6man ∗ E + uE = education, A = experience, A2= experience squared,man = 1 if male.We are letting education have a differential effect formales-females.Walter Sosa-Escudero Extensions of the Linear Modelregress logW aedu age age2 man man_aedu------------------------------------------------------------------------------logW | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------aedu | .0888402 .0017526 50.69 0.000 .0854049 .0922755age | .0622024 .0019397 32.07 0.000 .0584005 .0660044age2 | -.0005635 .0000232 -24.30 0.000 -.000609 -.000518man | .0346514 .026336 1.32 0.188 -.0169689 .0862716man_aedu | .0034475 .0023337 1.48 0.140 -.0011268 .0080217_cons | -1.623832 .0428326 -37.91 0.000 -1.707786 -1.539877The interactive effect is not significant: education affect males and females alike.What would be the interpretation if it were significant?Walter Sosa-Escudero Extensions of the Linear ModelIt is natural to propose the following model that allows for boththe intercept and the slope to differ between classes:Wi= β1+ β2Xi+ β3Di+ β4(DiXi) + uiIn this case, if β3= 0 and β4= 0 then the model for men andwomen coincides.Walter Sosa-Escudero Extensions of the Linear ModelMore than two categories:Suppose we observe experience and education, and that educationis observed as a category: less than complete high school, completehigh-school or incomplete college, and complete college degree.Obviously, individuals belong to only one of the three categories.Consider the following model:Wi= β1+ β2Xi+ β3D1i+ β4D2i+ uiwhereD1i=1 Maximum education of i is high school0 Maximum education of i is not high schoolD2i=1 Maximum education of i is college0 Maximum education of i is not collegeWalter Sosa-Escudero Extensions of the Linear ModelThis model provides all the information we need. Compute thefollowing expectaions for individuals in each category:E(Wi, Less than High school) = β1+ β2XiE(Wi, High school) = β1+ β3+ β2XiE(Wi, College) = β1+ β4+ β2Xiβ3measures the impact on expected wages of finishing high schoolcompared to not finishing it. β4measures the effect of completingcollege with respect having just unfinished high school.Walter Sosa-Escudero Extensions of the Linear ModelIn general, each of the coefficients of the dummy