Discrete Choice Modeling William Greene Stern School of Business New York University Modeling Categorical Variables Theoretical foundations Econometric methodology

Models Statistical bases Econometric methods Applications Binary Outcome Multinomial Unordered Choice Ordered Outcome Self Reported Health Satisfaction Categorical Variables

Observed outcomes Inherently discrete: number of occurrences, e.g., family size Multinomial: The observed outcome indexes a set of unordered labeled choices. Implicitly continuous: The observed data are discrete by construction, e.g., revealed preferences; our main

subject Implications For model building For analysis and prediction of behavior Binary Choice Models Agenda A Basic Model for Binary Choice Specification

Maximum Likelihood Estimation Estimating Partial Effects A Random Utility Approach Underlying Preference Scale, U*(choices) Revelation of Preferences: U*(choices) < 0 Choice 0

U*(choices) > 0 Choice 1 Simple Binary Choice: Insurance Censored Health Satisfaction Scale 0 = Not Healthy 1 = Healthy Count Transformed to Indicator

Redefined Multinomial Choice Fly Ground A Model for Binary Choice Yes or No decision (Buy/NotBuy, Do/NotDo) Example, choose to visit physician or not

Model: Net utility of visit at least once Random Utility Uvisit = +1Age + 2Income + Sex + Choose to visit if net utility is positive Net utility = Uvisit Unot visit Data: X y = [1,age,income,sex]

= 1 if choose visit, Uvisit > 0, 0 if not. Choosing Between the Two Alternatives Modeling the Binary Choice Uvisit = + 1 Age + 2 Income + 3 Sex + Chooses to visit: Uvisit > 0 + 1 Age + 2 Income + 3 Sex + > 0 > -[ + 1 Age + 2 Income + 3 Sex ] Probability Model for Choice Between Two Alternatives Probability is governed by , the random

part of the utility function. > -[ + 1Age + 2Income + 3Sex ] What Can Be Learned from the Data? (A Sample of Consumers, i = 1,,N) Are the characteristics relevant? Predicting behavior - Individual E.g., will a person visit the physician? Will a person purchase the insurance? - Aggregate E.g., what proportion of the population will visit the physician? Buy the insurance?

Analyze changes in behavior when attributes change E.g., how will changes in education change the proportion who buy the insurance? Application: Health Care Usage German Health Care Usage Data (GSOEP), 7,293 Individuals, Varying Numbers of Periods Data downloaded from Journal of Applied Econometrics Archive. This is an unbalanced panel with 7,293 individuals. They can be used for regression, count models, binary choice, ordered choice, and bivariate binary choice. This is a large data set. There are altogether 27,326 observations. The number of observations ranges from 1 to 7. (Frequencies are: 1=1525, 2=2158, 3=825, 4=926, 5=1051, 6=1000, 7=987). Variables in the file are DOCTOR HOSPITAL HSAT DOCVIS

HOSPVIS PUBLIC ADDON HHNINC = = = = = = = = 1(Number of doctor visits > 0)

1(Number of hospital visits > 0) health satisfaction, coded 0 (low) - 10 (high) number of doctor visits in last three months number of hospital visits in last calendar year insured in public health insurance = 1; otherwise = 0 insured by add-on insurance = 1; otherwise = 0 household nominal monthly net income in German marks / 10000. (4 observations with income=0 were dropped) HHKIDS = children under age 16 in the household = 1; otherwise = 0 EDUC = years of schooling AGE = age in years FEMALE = 1 for female headed household, 0 for male

Application 27,326 Observations 1 to 7 years, panel 7,293 households observed We use the 1994 year, 3,337 household observations Descriptive Statistics ========================================================= Variable Mean

Std.Dev. Minimum Maximum --------+-----------------------------------------------DOCTOR| .657980 .474456 .000000 1.00000 AGE| 42.6266 11.5860 25.0000 64.0000 HHNINC| .444764 .216586 .340000E-01 3.00000 FEMALE| .463429

.498735 .000000 1.00000 Binary Choice Data An Econometric Model Choose to visit iff Uvisit > 0 Uvisit = + Age + Income + Sex + 1 2 3

Uvisit > 0 > -( + 1 Age + 2 Income + 3 Sex) < + 1 Age + 2 Income + 3 Sex Probability model: For any person observed by the analyst, Prob(visit) = Prob[ < + 1 Age + 2 Income + 3 Sex] Note the relationship between the unobserved and the outcome +1Age + 2 Income + 3

Sex Modeling Approaches Nonparametric relationship Semiparametric index function

Minimal Assumptions Minimal Conclusions Stronger assumptions Robust to model misspecification (heteroscedasticity) Still weak conclusions Parametric Probability function and index Strongest assumptions complete specification

Strongest conclusions Possibly less robust. (Not necessarily) Nonparametric Regressions P(Visit)=f(Age) P(Visit)=f(Income) Klein and Spady Semiparametric No specific distribution assumed Note necessary normalizations. Coefficients are relative to

FEMALE. Prob(yi = 1 | xi ) =G(xx) G is estimated by kernel methods Fully Parametric Index Function: U* = xx + Observation Mechanism: y = 1[U* > 0] Distribution: ~ f(); Normal, Logistic, Maximum Likelihood Estimation: Max() logL = i log Prob(Yi = yi|xi) Parametric: Logit Model

What do these mean? Parametric vs. Semiparametric .02365/.63825 = .04133 -.44198/.63825 = -.69249 Parametric Model Estimation How to estimate , 1, 2, 3? Its not regression

The technique of maximum likelihood L y 0 Prob[ y 0] y 1 Prob[ y 1] Prob[y=1] = Prob[ > -( + 1 Age + 2 Income + 3 Sex)] Prob[y=0] = 1 - Prob[y=1] Requires a model for the probability

Completing the Model: F() The distribution Normal: PROBIT, natural for behavior Logistic: LOGIT, allows thicker tails Gompertz: EXTREME VALUE, asymmetric,

Does it matter? Yes, large difference in estimates Not much, quantities of interest are more stable. Estimated Binary Choice Models LOGIT Variable Constant Age Income

Sex PROBIT Estimate EXTREME VALUE t-ratio Estimate t-ratio

Estimate t-ratio -0.42085 -2.662 -0.25179 -2.600 0.00960

0.078 0.02365 7.205 0.01445 7.257 0.01878 7.129 -0.44198

-2.610 -0.27128 -2.635 -0.32343 -2.536 0.63825 8.453

0.38685 8.472 0.52280 8.407 Log-L -2097.48 -2097.35 -2098.17

Log-L(0) -2169.27 -2169.27 -2169.27 Ignore the t ratios for now. Effect on Predicted Probability of an Increase in Age + 1 (Age+1) + 2 (Income) + 3 Sex

(1 is positive) Partial Effects in Probability Models Prob[Outcome] = some F(+1Income) Partial effect = F(+1Income) / x (derivative)

Partial effects are derivatives Result varies with model Logit: F(+1Income) /x = Prob * (1-Prob) Probit: F(+1Income)/x = Normal density

Extreme Value: F(+1Income)/x = Scaling usually erases model differences Prob * (-log Prob)

Estimated Partial Effects Partial Effect for a Dummy Variable Prob[yi = 1|xi,di] = F(xi+di) = conditional mean Partial effect of d Prob[yi = 1|xi,di=1]- Prob[yi = 1|xi,di=0]

Probit: x (di ) x Partial Effect Dummy Variable Computing Partial Effects

Compute at the data means? Simple Inference is well defined. Average the individual effects More appropriate?

Asymptotic standard errors are problematic. Average Partial Effects Probability = Pi F( ' xi ) Pi F( ' xi ) Partial Effect = f ( ' xi ) = di xi xi 1 n 1 n Average Partial Effect = i 1 di i 1 f ( ' xi ) n

n are estimates of =E[di ] under certain assumptions. Average Partial Effects vs. Partial Effects at Data Means ============================================= Variable Mean Std.Dev. S.E.Mean ============================================= --------+-----------------------------------ME_AGE| .00511838 .000611470 .0000106 ME_INCOM| -.0960923 .0114797 .0001987

ME_FEMAL| .137915 .0109264 .000189 A Nonlinear Effect P = F(age, age2, income, female) ---------------------------------------------------------------------Binomial Probit Model Dependent variable DOCTOR Log likelihood function -2086.94545 Restricted log likelihood -2169.26982 Chi squared [ 4 d.f.]

164.64874 Significance level .00000 --------+------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------|Index function for probability Constant| 1.30811*** .35673 3.667 .0002 AGE| -.06487*** .01757

-3.693 .0002 42.6266 AGESQ| .00091*** .00020 4.540 .0000 1951.22 INCOME| -.17362* .10537 -1.648 .0994 .44476

FEMALE| .39666*** .04583 8.655 .0000 .46343 --------+------------------------------------------------------------Note: ***, **, * = Significance at 1%, 5%, 10% level. ---------------------------------------------------------------------- Nonlinear Effects This is the probability implied by the model. Partial Effects? ---------------------------------------------------------------------Partial derivatives of E[y] = F[*] with

respect to the vector of characteristics They are computed at the means of the Xs Observations used for means are All Obs. --------+------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Elasticity --------+------------------------------------------------------------|Index function for probability AGE| -.02363*** .00639 -3.696 .0002 -1.51422 AGESQ| .00033*** .729872D-04

4.545 .0000 .97316 INCOME| -.06324* .03837 -1.648 .0993 -.04228 |Marginal effect for dummy variable is P|1 - P|0. FEMALE| .14282*** .01620 8.819 .0000

.09950 --------+------------------------------------------------------------- Separate partial effects for Age and Age2 make no sense. They are not varying partially. Practicalities of Nonlinearities PROBIT ; Lhs=doctor ; Rhs=one,age,agesq,income,female ; Partial effects $ PROBIT PARTIALS ; Lhs=doctor ; Rhs=one,age,age*age,income,female $

; Effects : age $ Partial Effect for Nonlinear Terms Prob [ 1Age 2 Age 2 3 Income 4 Female] Prob [ 1Age 2 Age 2 3 Income 4 Female] (1 2 2 Age) Age (1.30811 .06487 Age .0091Age 2 .17362 Income .39666 Female) [( .06487 2(.0091) Age] Must be computed at specific values of Age, Income and Female Average Partial Effect: Averaged over Sample Incomes and Genders for Specific Values of Age

Interaction Effects Prob = ( + 1Age 2 Income 3Age*Income ...) Prob ( + 1Age 2 Income 3 Age*Income ...)( 2 3Age) Income The "interaction effect" 2 Prob x (x)(1 3 Income)(2 3 Age) (x)3 IncomeAge = (x)(x)12 if 3 0. Note, nonzero even if 3 0. Partial Effects? The software does not know that Age_Inc = Age*Income. ---------------------------------------------------------------------Partial derivatives of E[y] = F[*] with respect to the vector of characteristics

They are computed at the means of the Xs Observations used for means are All Obs. --------+------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Elasticity --------+------------------------------------------------------------|Index function for probability Constant| -.18002** .07421 -2.426 .0153 AGE| .00732*** .00168 4.365 .0000

.46983 INCOME| .11681 .16362 .714 .4753 .07825 AGE_INC| -.00497 .00367 -1.355 .1753 -.14250 |Marginal effect for dummy variable is P|1 - P|0. FEMALE|

.13902*** .01619 8.586 .0000 .09703 --------+------------------------------------------------------------- Models for Visit Doctor Direct Effect of Age Income Effect Income Effect on Health for Different Ages

Interaction Effect The "interaction effect" 2 Prob x (x)(1 3 Income)( 2 3 Age) (x)3 IncomeAge = (x)(x)12 if 3 0. Note, nonzero even if 3 0. Interaction effect if x = 0 is (0)3 It's not possible to trace this effect for nonzero x. Nonmonotonic in x and 3 . Answer: Don't rely on the numerical values of parameters to inform about interaction effects. Examine the model implications and the data more closely. Gender Age Interaction Effects

Interaction Effects Margins and Odds Ratios .8617 .9144 Overall take up rate of public insurance is greater for females than males. What does the binary choice model say about the difference? Binary Choice Models Average Partial Effects Other things equal, the take up rate is about .02 higher in female headed households.

The gross rates do not account for the facts that female headed households are a little older and a bit less educated, and both effects would push the take up rate up. Odds Ratio Probit and Logit Models Examine a probability model with one continuous X and one dummy D Prob(Takeup) F(+X+D)X+D)D) 1-Prob(Takeup) 1 F(+X+D)X+D)D) Symmetric Probability Distributions Odds ratio = F(+X+D)X+D)D) Odds ratio =

F(--X+D)X-D)D) Ratio of Odds Ratios Probit and Logit Models F(+X+D)X+D)D) Ratio of Odds Ratios comparing D=1 to D=0 is F(--X+D)X-D)D) F(+X+D)X) F(--X+D)X) For the probit model, this does not simplify. For the logit model, the ratio is exp(+X+D)X+D)D)/[1+exp(+X+D)X+D)D)]

1/[1+exp(+X+D)X+D)D)] e exp(+X+D)X)/[1+exp(+X+D)X)] 1/[1+exp(+X+D)X)] Odds Ratios for Insurance Takeup Model Logit vs. Probit

Reporting Odds Ratios Margins are about units of measurement Partial Effect Takeup rate for female headed households is about 91.7% Other things equal, female headed households are about .02 (about 2.1%) more likely

to take up the public insurance Odds Ratio The odds that a female headed household takes up the insurance is about 14. The odds go up by about 26% for a female headed household compared to a

male headed household.