Discrete Choice Modeling

Transcription

1 [Part 4] 1/43 Discrete Choice Modeling 0 Introduction 1 Summary 2 Binary Choice 3 Panel Data 4 Bivariate Probit 5 Ordered Choice 6 Count Data 7 Multinomial Choice 8 Nested Logit 9 Heterogeneity 10 Latent Class 11 Mixed Logit 12 Stated Preference 13 Hybrid Choice William Greene Stern School of Business New York University

2 [Part 4] 2/43 Multivariate Binary Choice Models Bivariate Probit Models Analysis of bivariate choices Marginal effects Prediction Simultaneous Equations and Recursive Models A Sample Selection Bivariate Probit Model The Multivariate Probit Model Specification Simulation based estimation Inference Partial effects and analysis The panel probit model

3 [Part 4] 3/43 Application: Health Care Usage German Health Care Usage Data, 7,293 Individuals, Varying Numbers of Periods Variables in the file are 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=1079, 3=825, 4=926, 5=1051, 6=1000, 7=887). Note, the variable NUMOBS below tells how many observations there are for each person. This variable is repeated in each row of the data for the person. DOCTOR = 1(Number of doctor visits > 0) HOSPITAL = 1(Number of hospital visits > 0) HSAT = health satisfaction, coded 0 (low) - 10 (high) DOCVIS = number of doctor visits in last three months HOSPVIS = number of hospital visits in last calendar year PUBLIC = insured in public health insurance = 1; otherwise = 0 ADDON = insured by add-on insurance = 1; otherswise = 0 HHNINC = household nominal monthly net income in German marks / (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 MARRIED = marital status EDUC = years of education

4 [Part 4] 4/43 Gross Relation Between Two Binary Variables Cross Tabulation Suggests Presence or Absence of a Bivariate Relationship

5 [Part 4] 5/43 Tetrachoric Correlation A correlation measure for two binary variables Can be defined implicitly y * = μ + ε, y =1(y * > 0) y * = μ + ε,y =1(y * > 0) ε1 0 1 ρ ~ N, ε2 0 ρ 1 ρ is the tetrachoric correlation between y 1 and y2

6 [Part 4] 6/43 Log Likelihood Function n logl = logφ (2y -1)μ,(2y -1)μ,(2y -1)(2y -1)ρ 2 i=1 n i=1 2 i1 1 i2 2 i1 i2 = logφ q μ,q μ,q q ρ 2 i1 1 i2 2 i1 i2 Note : q = (2y -1) = -1 if y = 0 and +1 if y = 1. i1 i1 i1 i1 Φ = Bivariate normal CDF - must be computed using quadrature Maximized with respect to μ,μ and ρ. 1 2

7 [Part 4] 7/43 Estimation FIML Estimates of Bivariate Probit Model Maximum Likelihood Estimates Dependent variable DOCHOS Weighting variable None Number of observations Log likelihood function Number of parameters Variable Coefficient Standard Error b/st.er. P[ Z >z] Index equation for DOCTOR Constant Index equation for HOSPITAL Constant Tetrachoric Correlation between DOCTOR and HOSPITAL RHO(1,2)

8 [Part 4] 8/43 A Bivariate Probit Model Two Equation Probit Model (More than two equations comes later) No bivariate logit there is no reasonable bivariate counterpart Why fit the two equation model? Analogy to SUR model: Efficient Make tetrachoric correlation conditional on covariates i.e., residual correlation

9 [Part 4] 9/43 Bivariate Probit Model y * = βx + ε, y =1(y * > 0) y * = βx + ε,y =1(y * > 0) ε1 0 1 ρ ~ N, ε2 0 ρ 1 The variables in x and x may be the same or 2 2 different. There is no need for each equation to have its 'own variable.' ρ is the conditional tetrachoric correlation between y and y (The equations can be fit one at a time. Use FIML for (1) efficiency and (2) to get the estimate of ρ.) 1 2.

10 [Part 4] 10/43 ML Estimation of the Bivariate Probit Model (2y i1-1) βx 1 i1, n logl = logφ 2 (2y i=1 i2-1) βx 2 i2, (2y i1-1)(2yi2-1)ρ 2 = logφ q βx,q βx,q q ρ n i=1 2 i1 1 i1 i2 2 i2 i1 i2 Note : q = (2y -1) = -1 if y = 0 and +1 if y = 1. i1 i1 i1 i1 Φ = Bivariate normal CDF - must be computed using quadrature Maximized with respect to β, β and ρ. 1 2

11 [Part 4] 11/43 Application to Health Care Data x1=one,age,female,educ,married,working x2=one,age,female,hhninc,hhkids BivariateProbit ;lhs=doctor,hospital ;rh1=x1 ;rh2=x2;marginal effects $

12 [Part 4] 12/43 Parameter Estimates FIML Estimates of Bivariate Probit Model Dependent variable DOCHOS Log likelihood function Estimation based on N = 27326, K = Variable Coefficient Standard Error b/st.er. P[ Z >z] Mean of X Index equation for DOCTOR Constant *** AGE.01402*** FEMALE.32453*** EDUC *** MARRIED WORKING *** Index equation for HOSPITAL Constant *** AGE.00509*** FEMALE.12143*** HHNINC HHKIDS Disturbance correlation RHO(1,2).29611***

13 [Part 4] 13/43 Marginal Effects What are the marginal effects Effect of what on what? Two equation model, what is the conditional mean? Possible margins? Derivatives of joint probability = Φ 2 (β 1 x i1, β 2 x i2,ρ) Partials of E[y ij x ij ] =Φ(β j x ij ) (Univariate probability) Partials of E[y i1 x i1,x i2,y i2 =1] = P(y i1,y i2 =1)/Prob[y i2 =1] Note marginal effects involve both sets of regressors. If there are common variables, there are two effects in the derivative that are added.

14 [Part 4] 14/43 Bivariate Probit Conditional Means Prob[y =1,y =1] = Φ ( βx, βx,ρ) i1 i2 2 1 i1 2 i2 This is not a conditional mean. For a generic Prob[y i1 =1,y i2 =1] = g β + g x i β i1 1 i2 2 β 2xi2 -ρβ1x i1 β 1xi1 -ρβ2xi2 g i1 =φ( βx 1 i1)φ,g 2 i2 = φ( βx 2 i2 )Φ 1-ρ 2 1-ρ The term in β is 0 if x does not appear in x and likewise for β. 1 i i1 2 Φ 2( β1x i1, β2xi2,ρ) E[y i1 xi1, xi2,y i2 =1] = Prob[y i1 =1 xi1, xi2,y i2 =1] = Φ( βx ) x that might appear in either index function, E[y i1 xi1, x 1 i 2,y i2 =1] Φ 2( β1x i1, β2xi2,ρ)φ( β2xi2) = g β + g β - β 2 x Φ( βx ) [Φ( βx )] 2 i2 i1 1 i2 2 2 i 2 i2 2 i2 g i1 gi2 Φ 2( β1x i1, β2x i2,ρ)φ( β2x i2) = β β Φ( β2xi2 ) Φ( β2xi2 ) [Φ( β2xi2 )] 1 2

15 [Part 4] 15/43 Marginal Effects: Decomposition Marginal Effects for Ey1 y2= Variable Efct x1 Efct x2 Efct z1 Efct z AGE FEMALE EDUC MARRIED WORKING HHNINC HHKIDS

16 [Part 4] 16/43 Direct Effects Derivatives of E[y 1 x 1,x 2,y 2 =1] wrt x Partial derivatives of E[y1 y2=1] with respect to the vector of characteristics. They are computed at the means of the Xs. Effect shown is total of 4 parts above. Estimate of E[y1 y2=1] = Observations used for means are All Obs. These are the direct marginal effects Variable Coefficient Standard Error b/st.er. P[ Z >z] Mean of X AGE FEMALE EDUC MARRIED WORKING HHNINC (Fixed Parameter) HHKIDS (Fixed Parameter)

17 [Part 4] 17/43 Indirect Effects Derivatives of E[y 1 x 1,x 2,y 2 =1] wrt x Partial derivatives of E[y1 y2=1] with respect to the vector of characteristics. They are computed at the means of the Xs. Effect shown is total of 4 parts above. Estimate of E[y1 y2=1] = Observations used for means are All Obs. These are the indirect marginal effects Variable Coefficient Standard Error b/st.er. P[ Z >z] Mean of X AGE D FEMALE EDUC (Fixed Parameter) MARRIED (Fixed Parameter) WORKING (Fixed Parameter) HHNINC HHKIDS

18 Marginal Effects: Total Effects Sum of Two Derivative Vectors Discrete Choice Modeling [Part 4] 18/ Partial derivatives of E[y1 y2=1] with respect to the vector of characteristics. They are computed at the means of the Xs. Effect shown is total of 4 parts above. Estimate of E[y1 y2=1] = Observations used for means are All Obs. Total effects reported = direct+indirect Variable Coefficient Standard Error b/st.er. P[ Z >z] Mean of X AGE FEMALE EDUC MARRIED WORKING HHNINC HHKIDS

19 [Part 4] 19/43 Marginal Effects: Dummy Variables Using Differences of Probabilities Analysis of dummy variables in the model. The effects are computed using E[y1 y2=1,d=1] - E[y1 y2=1,d=0] where d is the variable. Variances use the delta method. The effect accounts for all appearances of the variable in the model Variable Effect Standard error t ratio (deriv) FEMALE ( ) MARRIED ( ) WORKING ( ) HHKIDS ( )

20 [Part 4] 20/43 Average Partial Effects

21 [Part 4] 21/43 Model Simulation

22 Model Simulation Discrete Choice Modeling [Part 4] 22/43

23 [Part 4] 23/43 A Simultaneous Equations Model Simultaneous Equations Model y * = βx + θ y + ε, y = 1(y * > 0) y * = βx + θ y + ε,y = 1(y * > 0) ε1 0 1 ρ ~N, ε2 0 ρ 1 This model is not identified. Incoherent. (Not estimable. The computer can compute 'estimates' but they have no meaning.)

24 Fully Simultaneous Model FIML Estimates of Bivariate Probit Model Dependent variable DOCHOS Log likelihood function Variable Coefficient Standard Error b/st.er. P[ Z >z] Mean of X Index equation for DOCTOR Constant *** AGE.01124*** FEMALE.27070*** EDUC MARRIED WORKING HOSPITAL *** Index equation for HOSPITAL Constant *** AGE *** FEMALE *** HHNINC HHKIDS DOCTOR *** Disturbance correlation RHO(1,2) *** ******** Discrete Choice Modeling [Part 4] 24/43

25 A Recursive Simultaneous Equations Model Recursive Simultaneous Equations Model Discrete Choice Modeling [Part 4] 25/43 y * = βx + ε, y = 1(y * > 0) y * = βx + θ y + ε,y = 1(y * > 0) ε1 0 1 ρ ~ N, ε2 0 ρ 1 This model is identified. It can be consistently and efficiently estimated by full information maximum likelihood. Treated as a bivariate probit model, ignoring the simultaneity. Bivariate ; Lhs = y1,y2 ; Rh1=,y2 ; Rh2 = $

26 [Part 4] 26/43 Application: Gender Economics at Liberal Arts Colleges Journal of Economic Education, fall, 1998.

27 Estimated Recursive Model Discrete Choice Modeling [Part 4] 27/43

28 [Part 4] 28/43 Estimated Effects: Decomposition

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31 [Part 4] 31/43 Causal Inference? Causal Inference? There is no partial (marginal) effect for PIP. PIP cannot change partially (marginally). It changes because something else changes. (X or I or u 2.) The calculation of ME PIP does not make sense.

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37 [Part 4] 37/43 A Sample Selection Model Sample Selection Model y * = βx + ε, y =1(y * > 0) y * = βx + ε,y =1(y * > 0) ε1 0 1 ρ ~ N, ε2 0 ρ 1 y is only observed when y = f(y,y ) = Prob[y =1 y =1]*Prob[y =1] (y =1,y =1) = Prob[y = 0 y =1]*Prob[y =1] (y = 0,y =1) = Prob[y = 0] (y = 0) 2 2

38 [Part 4] 38/43 Sample Selection Model: Estimation f(y,y ) = Prob[y = 1 y =1] *Prob[y =1] (y =1,y =1) = Prob[y = 0 y =1]*Prob[y =1] (y = 0,y =1) = Prob[y = 0] (y = 0) Terms in the log likelihood: 2 2 (y =1,y =1) Φ ( βx, βx,ρ) (Bivariate normal) i1 2 i2 (y = 0,y =1) Φ (- βx, βx,-ρ) (Bivariate normal) i1 2 i2 (y = 0) Φ(- βx ) (Univariate normal) 2 2 i2 Estimation is by full information maximum likelihood. There is no "lambda" variable.

39 [Part 4] 39/43 Application: Credit Scoring American Express: 1992 N = 13,444 Applications Observed application data Observed acceptance/rejection of application N 1 = 10,499 Cardholders Observed demographics and economic data Observed default or not in first 12 months Full Sample is in AmEx.lpj; description shows when imported.

40 [Part 4] 40/43 The Multivariate Probit Model Multiple Equations Analog to SUR Model for M Binary Variables y * = βx + ε, y =1(y * > 0) y * = βx + ε, y =1(y * > 0) y * = β x + ε, y =1(y * > 0) M M M M M M ε 0 1 ρ... ρ ε2 0 ρ1 ~ N M, 1... ρ εm 0 ρ1m ρ 2M M logl = logφ [q βx,q β x,...,q β x Σ*] Σ mn* N i=1 2 2M M i1 1 i1 i2 2 i2 im M im 1 if m = n or q q ρ if not. im in mn

41 [Part 4] 41/43 MLE: Simulation Estimation of the multivariate probit model requires evaluation of M-order Integrals The general case is usually handled with the GHK simulator. Much current research focuses on efficiency (speed) gains in this computation. The Panel Probit Model is a special case. (Bertschek-Lechner, JE, 1999) Construct a GMM estimator using only first order integrals of the univariate normal CDF (Greene, Emp.Econ, 2003) Estimate the integrals with simulation (GHK) anyway.

42 [Part 4] 42/ Multivariate Probit Model: 3 equations. Dependent variable MVProbit Log likelihood function Variable Coefficient Standard Error b/st.er. P[ Z >z] Mean of X Index function for DOCTOR Constant ** [ ] AGE.01664*** [ ] FEMALE.30931*** [ ] EDUC [ ] MARRIED [ ] WORKING *** [ ] Index function for HOSPITAL Constant *** [ ] AGE.00717** [ ] FEMALE [ ] HHNINC [ ] HHKIDS [ ] Index function for PUBLIC Constant *** [ ] AGE.00661** [ ] HSAT *** [ ] MARRIED [ ] Correlation coefficients R(01,02).28381*** [ was ] R(01,03) R(02,03)

43 [Part 4] 43/43 Marginal Effects There are M equations: Effect of what on what? NLOGIT computes E[y1 all other ys, all xs] Marginal effects are derivatives of this with respect to all xs. (EXTREMELY MESSY) Standard errors are estimated with bootstrapping.