Copyright © 2006 The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Dummy Variable Regression Models chapter ten

10-2 Qualitative or “Dummy” Variables Variables in regression models do not have to be numerically quantitative, but may be qualitative in nature. Qualitative or dummy variables only take values of zero (0) and one (1) in order to represent categorical data. Also called indicator, binary, categorical or dichotomous variables.

10-3 Examples Sex or gender: female = 1, male = 0 Time period: before = 0, after = 1 States, regions, or countries One variable for each category: D T = 1 if Tennessee, D T =0 otherwise But one must be left out: 49 variables for 50 states Any other suitable category Union or non-union; urban or rural; age categories, etc.

10-4 Does expenditure on food vary by sex? See data tables 10-1 and Create a dummy variable for sex D = 1 if female, 0 if male Regress food expenditure on D A device to see if two mean values are different Not significantly different here

10-5 Table 10-1 Food expenditure in relation to after-tax income, sex, and age.

10-6 Table 10-2 Food expenditure in relation to after-tax income and sex.

10-7 Does the assignment of 0 or 1 matter? It doesn’t matter which category is assigned a “1” and which a “0” Let D = 1 for male, 0 for female The resulting regression is identical except for sign of coeff. of D → Men on average spend about $500/year more on food than women

10-8 The Dummy Variable Trap Suppose D 1 = 1 for female, 0 for male D 2 = 1 for male, 0 for female Then D 1 = 1 – D 2 and D 2 = 1 – D 1 Perfect collinearity! Don’t fall for the dummy variable trap. Number of dummy variables = number of categories minus 1

10-9 ANCOVA Models Regression on one quantitative and one qualitative variable Regress expenditure on food (Y) on income (X) and dummy for sex: D = 1 female, D = 0 male. Significant dummy indicates different intercepts (Fig. 10-2)

10-10 Figure 10-2 Food expenditure in relation to after-tax income.

10-11 Interaction terms and Slope Differentials In the food expenditure example, are the slope coefficients of after- tax income the same for male and female? Consider this regression model → B 4 is the differential slope coefficient or slope drifter D i enters in both additive and interactive (or mltiplicative) forms. Fig shows the possibilities and Table 10-6 shows results

10-12 Figure 10-4 Comparing two regressions.

10-13 Table 10-6 Results of regression (10.23).

10-14 Do Results Indicate Specification Error? Compare Table 10-6 to our earlier result → Neither the differential intercept or slope coefficients are significant Suggests specification error by including D i X i Generally, start with more comprehensive model and reduce it after diagnostic testing.

10-15 Dummy Variable as Dependent Variable Problems with OLS: P(Y = 1|X) u i follows binomial distribution (approaches normal as n gets large) The error term is heteroscedastic R 2 is not meaningful Estimated Y values may be 1 Assumes P(Y = 1) changes linearly with X For example the probability that a family owns a home increases by the same amount as income increases, regardless of income level. Alternatives: Probit and Logit models.