Multiple Regression Analysis with Qualitative Information

1. Multiple Regression Analysis with Qualitative Information

2. Multiple Regression Analysis with Qualitative Information = the wage gain/loss if the person is a woman rather than a man (holding other things fixed) Dummy variable: =1 ifthepersonis a woman =0 ifthepersonis man • Qualitative Information • Examples: gender, race, industry, region, rating grade, … • A way to incorporate qualitative information is to use dummy variables • They may appear as the dependent or as independent variables • A single dummy independent variable

3. Multiple Regression Analysis with Qualitative Information Alternative interpretationofcoefficient: i.e. thedifference in mean wage betweenmenandwomenwiththe same levelofeducation. Interceptshift Graphical Illustration

4. Multiple Regression Analysis with Qualitative Information This model cannotbeestimated (perfectcollinearity) When using dummy variables, one category always has to be omitted: The basecategoryaremen The basecategoryarewomen Alternatively, one could omit the intercept: Disadvantages: 1) More difficulttotestfordiffe-rencesbetweentheparameters 2) R-squaredformulaonly valid ifregressioncontainsintercept Dummy variable trap

5. Multiple Regression Analysis with Qualitative Information Holding education, experience, and tenure fixed, women earn $1.81 less per hour than men • Estimated wage equation with intercept shift • Does that mean that women are discriminated against? • Not necessarily. Being female may be correlated with other produc-tivity characteristics that have not been controlled for.

6. Multiple Regression Analysis with Qualitative Information Not holding other factors constant, women earn $2.51per hour less than men, i.e. the difference between the mean wage of men and that of women is $2.51. • Comparing means of subpopulations described by dummies • Discussion • It can easily be tested whether difference in means is significant • The wage difference between men and women is larger if no other things are controlled for; i.e. part of the difference is due to differ-ences in education, experience, and tenure between men and women

7. Multiple Regression Analysis with Qualitative Information Hourstraining per employee Dummy variable indicating whether firm received a training grant • Further example: Effects of training grants on hours of training • This is an example of program evaluation • Treatment group (= grant receivers) vs. control group (= no grant) • Is the effect of treatment on the outcome of interest causal?

8. Multiple Regression Analysis with Qualitative Information Dummyindicatingwhetherhouseisofcolonial style As thedummyforcolonial style changesfrom 0 to 1, thehousepriceincreasesby 5.4 percentagepoints • Using dummy explanatory variables in equations for log(y)

9. Multiple Regression Analysis with Qualitative Information Holding otherthingsfixed, marriedwomenearn 19.8% lessthansinglemen (= thebasecategory) • Using dummy variables for multiple categories • 1) Define membership in each category by a dummy variable • 2) Leave out one category (which becomes the base category)

10. Multiple Regression Analysis with Qualitative Information Municipalbond rate Creditratingfrom 0-4 (0=worst, 4=best) Thisspecificationwouldprobably not beappropriateasthecreditratingonlycontainsordinalinformation. A betterwaytoincorporatethisinformationistodefinedummies: Dummiesindicatingwhethertheparticularratingapplies, e.g. CR1=1 if CR=1, and CR1=0 otherwise. All effectsaremeasured in comparisontotheworstrating (= basecategory). Incorporating ordinal information using dummy variables Example: City credit ratings and municipal bond interest rates

11. Multiple Regression Analysis with Qualitative Information Interaction term = intercept men = slope men = intercept women = slope women The returntoeducationisthe same formenandwomen The whole wage equationisthe same formenandwomen Interactions involving dummy variables Allowing for different slopes Interesting hypotheses

12. Multiple Regression Analysis with Qualitative Information Interactingboththeinterceptandtheslopewiththefemaledummyenablesoneto model completelyindependent wage equationsformenandwomen Graphical illustration

13. Multiple Regression Analysis with Qualitative Information Doesthismeanthatthereisnosignificantevidenceoflowerpayforwomenatthe same levelsofeduc, exper, andtenure? No: thisisonlytheeffectforeduc = 0. Toanswerthequestiononehastorecentertheinteractionterm, e.g. aroundeduc = 12.5 (= averageeducation). Noevidenceagainsthypothesisthatthereturntoeducationisthe same formenandwomen Estimated wage equation with interaction term

14. Multiple Regression Analysis with Qualitative Information College grade pointaverage Standardizedaptitudetest score High school rank percentile Total hoursspent in collegecourses Testing for differences in regression functions across groups Unrestricted model (contains full set of interactions) Restricted model (same regression for both groups)

15. Multiple Regression Analysis with Qualitative Information All interactioneffectsarezero, i.e. the same regressioncoefficientsapplytomenandwomen Testedindividually, thehypothesisthattheinteractioneffectsarezerocannotberejected Null hypothesis Estimation of the unrestricted model

16. Multiple Regression Analysis with Qualitative Information Null hypothesisisrejected • Joint test with F-statistic • Alternative way to compute F-statistic in the given case • Run separate regressions for men and for women; the unrestricted SSR is given by the sum of the SSR of these two regressions • Run regression for the restricted model and store SSR • If the test is computed in this way it is called the Chow-Test • Important: Test assumes a constant error variance accross groups

17. Multiple Regression Analysis with Qualitative Information Ifthedependent variable onlytakes on thevalues 1 and 0 Linear probability model (LPM) In the linear probability model, thecoefficientsdescribetheeffectoftheexplanatory variables on theprobabilitythat y=1 A Binary dependent variable: the linear probability model Linear regression when the dependent variable is binary

18. Multiple Regression Analysis with Qualitative Information =1 if in laborforce, =0 otherwise Non-wifeincome (in thousanddollars per year) Ifthenumberofkidsundersixyearsincreasesbyone, the pro- probabilitythatthewomanworks falls by 26.2% Does not looksignificant (but seebelow) Example: Labor force participation of married women

19. Multiple Regression Analysis with Qualitative Information Graph for nwifeinc=50, exper=5, age=30, kindslt6=1, and kidsge6=0 The maximumlevelofeducation in the sample iseduc=17. Forthegi-vencase, thisleadsto a predictedprobabilitytobe in thelaborforceofabout 50%. Negative predictedprobability but noproblembecausenowoman in the sample haseduc < 5. Example: Female labor participation of married women (cont.)

20. Multiple Regression Analysis with Qualitative Information VarianceofBer-noulli variable • Disadvantages of the linear probability model • Predicted probabilities may be larger than one or smaller than zero • Marginal probability effects sometimes logically impossible • The linear probability model is necessarily heteroskedastic • Heteroskedasticity consistent standard errors need to be computed • Advantanges of the linear probability model • Easy estimation and interpretation • Estimated effects and predictions are often reasonably good in practice

21. Multiple Regression Analysis with Qualitative Information The firm‘s scrap rate =1 if firm receivedtraininggrant, =0 otherwise Noapparenteffectofgrant on productivity Treatment group: grant receivers,Control group: firms that received no grant Grants weregiven on a first-come, first-servedbasis. Thisis not the same asgivingthem out randomly. Itmightbethecasethatfirmswithlessproductiveworkerssaw an opportunitytoimproveproductivityandappliedfirst. More on policy analysis and program evaluation Example: Effect of job training grants on worker productivity

22. Multiple Regression Analysis with Qualitative Information The dummyindicatingwhetheror not there was treatmentisunrelatedtootherfactorsaffectingtheoutcome. • Self-selection into treatment as a source for endogeneity • In the given and in related examples, the treatment status is probably related to other characteristics that also influence the outcome • The reason is that subjects self-select themselves into treatment depending on their individual characteristics and prospects • Experimental evaluation • In experiments, assignment to treatment is random • In this case, causal effects can be inferred using a simple regression

23. Multiple Regression Analysis with Qualitative Information Dummyindicatingwhetherloan was approved Racedummy Creditrating • Further example of an endogenous dummy regressor • Are nonwhite customers discriminated against? • It is important to control for other characteristics that may be important for loan approval (e.g. profession, unemployment) • Omitting important characteristics that are correlated with the non-white dummy will produce spurious evidence for discrimination