Using Dummy Variables to Analyze the Difference in Average Wages Between Men and Women, Study notes of Economic Analysis

Download Using Dummy Variables to Analyze the Difference in Average Wages Between Men and Women and more Study notes Economic Analysis in PDF only on Docsity! Dummy Explanatory Variables Christopher Taber Department of Economics University of Wisconsin-Madison April 5, 2010 Categorical Variables Lets go back to something we thought about very early on in this course: the difference in average wages between men and women Suppose you want to test whether men make more money than women That is you have the following null hypothesis H0 : E (W | Male) = E (W | Female) where W is hourly earnings. How do you do this? Let E (Wi | mi) = β0 + β1mi . Why is this useful? Now notice that E (Wi | Male) = E (Wi | mi = 1) = β0 + β1 E (Wi | Female) = E (Wi | mi = 0) = β0 Solving out this means that β0 = E (Wi | Female) β1 = E (Wi | Male)− E (Wi | Female) But then testing H0 : E (W | Male) = E (W | female) is equivalent to testing H0 : β1 = 0 We already know how to do this. Lets see. Adding Conditioning Variables But that isn’t all. We might be worried that women have less labor market experience than men. An interesting null hypothesis might be H0 : E (W | Male,Experience) = E (W | Female,Experience) That is, comparing men and women with the same level of experience, do they earn the same amount of money? This is easy to do, we just write the model as E (Wi | mi) = β0 + β1mi + β2Expi + β3Exp2 i . Then testing whether β1 = 0 tests exactly what we want. One could think of just running separate regressions for men and for women E (W | Male,Experience) = βm 0 + βm 1 Expi E (W | Female,Experience) = βf 0 + βf 1Expi Lets see what this looks like But what if I want to test whether these things are the same? That is I might want to run a joint test of whether men and women face the same earnings profile The key here is an interaction. Think about the model E (Wi | Gender,Experience) = β0+β1mi+β2Expi+β3 (mi × Expi) Then E (W | Male,Experience) = βm 0 + βm 1 Expi = E (Wi | mi = 1,Expi) = β0 + β1 + β2Expi + β3Expi E (W | Female,Experience) = βf 0 + βf 1Expi = E (Wi | mi = 0,Expi) = β0 + β2Expi Thus β0 = βf 0 β1 = βm 0 − βf 0 β2 = βf 2 β3 = βm 2 − βf 2 Testing that the profiles are the same is equivalent to testing the joint null hypothesis: H0 : β1 = 0 β3 = 0 More than Two Categories So far we have dealt with categorical variables with only 2 categories, but this is clearly not the only interesting case For example think about race where we could think of (at least) 5 groups Race could be African American Asian Hispanic Native American All others We are still going to have the dummy variable trap, but in this case it means we must omit 1 category Let B,Ai ,Hi ,Ni be dummy variables for black, asian, hispanic, and native american respectively. That is for example Bi = { 1 Person i is African American 0 otherwise Then we can think of the regression E (Wi | Race) = β0 + β1Bi + β2Ai + β3Hi + β4Ni Note that we have 5 basic population equations (for the 5 races) and 5 parameters so we seemed to have solved the dummy variable trap problem. How do we interpret the parameters? E (Wi | African American) = β0 + β1 E (Wi | Asian) = β0 + β2 E (Wi | Hispanic) = β0 + β3 E (Wi | Native American) = β0 + β4 E (Wi | All Others) = β0 Thus solving out one can show that β0 = E (Wi | All Others) β1 = E (Wi | African American)− E (Wi | All Others) β2 = E (Wi | Asian)− E (Wi | All Others) β3 = E (Wi | Hispanic)− E (Wi | All Others) β4 = E (Wi | Native American)− E (Wi | All Others) Thus the left out group matters a lot in the interpretation of the parameters What would happen if we just thought about the model as: E (Wi | Race,Gender) = β0 + β1mi + β2Bi Note that in this model E (Wi |White Male) = β0 + β1 E (Wi |White Female) = β0 E (Wi | Black Male) = β0 + β1 + β2 E (Wi | Black Female) = β0 + β2 Now actually we have 4 equations and three parameters so we can’t solve out exactly. Note that E (Wi |White Male)− E (Wi |White Female) = β1 and E (Wi | Black Male)− E (Wi | Black Female) = β1 We have imposed this, but it may not be true. How do we relax this? An interaction between Bi and mi . E (Wi | Race,Gender) = β0 + β1mi + β2Bi + β3 (mi × Bi) then E (Wi |White Male) = β0 + β1 E (Wi |White Female) = β0 E (Wi | Black Male) = β0 + β1 + β2 + β3 E (Wi | Black Female) = β0 + β2 Now E (Wi |White Male)− E (Wi |White Female) = β1 and E (Wi | Black Male)− E (Wi | Black Female) = β1 + β3