Discrete Choice Modeling

Transcription

1 [Part 6] 1/55 0 Introduction 1 Summary 2 Binary Choice 3 Panel Data 4 Bivariate Probit 5 Ordered Choice 6 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 6] 2/55 Application: Major Derogatory Reports AmEx Credit Card Holders N = 1310 (of 13,777) Number of major derogatory reports in 1 year Issues: Nonrandom selection Excess zeros

3 [Part 6] 3/55 Histogram for Credit Data Histogram for MAJORDRG NOBS= 1310, Too low: 0, Too high: 0 Bin Lower limit Upper limit Frequency Cumulative Frequency ======================================================================== (.8038) 1053(.8038) (.1038) 1189(.9076) (.0382) 1239(.9458) (.0183) 1263(.9641) (.0130) 1280(.9771) (.0076) 1290(.9847) (.0038) 1295(.9885) (.0046) 1301(.9931) (.0000) 1301(.9931) (.0015) 1303(.9947) (.0008) 1304(.9954) (.0031) 1308(.9985) (.0008) 1309(.9992) (.0000) 1309(.9992) (.0008) 1310(1.0000)

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5 [Part 6] 5/55 Doctor Visits

6 [Part 6] 6/55 Basic Modeling for Counts of Events E.g., Visits to site, number of purchases, number of doctor visits Regression approach Quantitative outcome measured Discrete variable, model probabilities Poisson probabilities loglinear model Prob[Y = j x ] = λ = exp( β'x i ) = E[y i i i i exp(-λ i)λ j! x ] i j i

7 Poisson Model for Doctor Visits [Part 6] 7/ Poisson Regression Dependent variable DOCVIS Log likelihood function Restricted log likelihood Chi squared [ 6 d.f.] Significance level McFadden Pseudo R-squared Estimation based on N = 27326, K = 7 Information Criteria: Normalization=1/N Normalized Unnormalized AIC Chi- squared = RsqP=.0818 G - squared = RsqD=.0601 Overdispersion tests: g=mu(i) : Overdispersion tests: g=mu(i)^2: Variable Coefficient Standard Error b/st.er. P[ Z >z] Mean of X Constant.77267*** AGE.01763*** EDUC *** FEMALE.29287*** MARRIED HHNINC *** HHKIDS ***

8 [Part 6] 8/55 Partial Effects Partial derivatives of expected val. with respect to the vector of characteristics. Effects are averaged over individuals. Observations used for means are All Obs. Conditional Mean at Sample Point Scale Factor for Marginal Effects E[y x x Variable Coefficient Standard Error b/st.er. P[ Z >z] Mean of X AGE.05613*** EDUC *** FEMALE.93237*** MARRIED HHNINC *** HHKIDS *** i i i ] = λ β i

9 [Part 6] 9/55 Poisson Model Specification Issues Equi-Dispersion: Var[y i x i ] = E[y i x i ]. Overdispersion: If i = exp[ x i + ε i ], E[y i x i ] = γexp[ x i ] Var[y i ] > E[y i ] (overdispersed) ε i ~ log-gamma Negative binomial model ε i ~ Normal[0, 2 ] Normal-mixture model ε i is viewed as unobserved heterogeneity ( frailty ). Normal model may be more natural. Estimation is a bit more complicated.

10 [Part 6] 10/55 exp(-λ i)λ Prob[Y i = j xi] = j! λ = exp( β'x ) = E[y i i i Moment Equations : x ] Estimation: Nonlinear Least Squares: i 1 i 1 Inefficient but robust if nonpoisson Maximum Likelihood: Moment Equations : i N j i Min Max N i 1 y N y x i i i i i 1 i i 2 log log( y!) Efficient, also robust to some kinds of NonPoissonness y N i i i x y i i i i

11 Poisson Model for Doctor Visits [Part 6] 11/ Poisson Regression Dependent variable DOCVIS Log likelihood function Restricted log likelihood Chi squared [ 6 d.f.] Significance level McFadden Pseudo R-squared Estimation based on N = 27326, K = 7 Information Criteria: Normalization=1/N Normalized Unnormalized AIC Chi- squared = RsqP=.0818 G - squared = RsqD=.0601 Overdispersion tests: g=mu(i) : Overdispersion tests: g=mu(i)^2: Variable Coefficient Standard Error b/st.er. P[ Z >z] Mean of X Constant.77267*** AGE.01763*** EDUC *** FEMALE.29287*** MARRIED HHNINC *** HHKIDS ***

12 Variable Coefficient Standard Error b/st.er. P[ Z >z] Mean of X Standard Negative Inverse of Second Derivatives Constant.77267*** AGE.01763*** EDUC *** FEMALE.29287*** MARRIED HHNINC *** HHKIDS *** Robust Sandwich Constant.77267*** AGE.01763*** EDUC *** FEMALE.29287*** MARRIED HHNINC *** HHKIDS *** Cluster Correction Constant.77267*** AGE.01763*** EDUC *** FEMALE.29287*** MARRIED HHNINC *** HHKIDS *** [Part 6] 12/55

13 [Part 6] 13/55 Negative Binomial Specification Prob(Y i =j x i ) has greater mass to the right and left of the mean Conditional mean function is the same as the Poisson: E[y i x i ] = λ i =Exp( x i ), so marginal effects have the same form. Variance is Var[y i x i ] = λ i (1 + α λ i ), α is the overdispersion parameter; α = 0 reverts to the Poisson. Poisson is consistent when NegBin is appropriate. Therefore, this is a case for the ROBUST covariance matrix estimator. (Neglected heterogeneity that is uncorrelated with x i.)

14 NegBin Model for Doctor Visits [Part 6] 14/ Negative Binomial Regression Dependent variable DOCVIS Log likelihood function NegBin LogL Restricted log likelihood Poisson LogL Chi squared [ 1 d.f.] Reject Poisson model Significance level McFadden Pseudo R-squared Estimation based on N = 27326, K = 8 Information Criteria: Normalization=1/N Normalized Unnormalized AIC NegBin form 2; Psi(i) = theta Variable Coefficient Standard Error b/st.er. P[ Z >z] Mean of X Constant.80825*** AGE.01806*** EDUC *** FEMALE.32596*** MARRIED HHNINC *** HHKIDS *** Dispersion parameter for count data model Alpha ***

15 [Part 6] 15/55 Marginal Effects Scale Factor for Marginal Effects POISSON Variable Coefficient Standard Error b/st.er. P[ Z >z] Mean of X AGE.05613*** EDUC *** FEMALE.93237*** MARRIED HHNINC *** HHKIDS *** Scale Factor for Marginal Effects NEGATIVE BINOMIAL AGE.05767*** EDUC *** FEMALE *** MARRIED HHNINC *** HHKIDS ***

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18 Model Formulations [Part 6] 18/55 Poisson exp( i) i Prob[ Y yi xi], (1 y ) exp( x ), y 0,1,..., i 1,..., N i i i E[ y x ] Var[ y x ] i i i i y i E[y i x i ]=λ i

19 NegBin-P Model [Part 6] 19/ Negative Binomial (P) Model Dependent variable DOCVIS Log likelihood function Restricted log likelihood Chi squared [ 1 d.f.] Variable Coefficient Standard Error b/st.er. NB-2 NB-1 Poisson Constant.60840*** AGE.01710*** EDUC *** FEMALE.36386*** MARRIED.03670* HHNINC *** HHKIDS *** Dispersion parameter for count data model Alpha *** Negative Binomial. General form, NegBin P P ***

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21 [Part 6] 21/55 Zero Inflation ZIP Models Two regimes: (Recreation site visits) Zero (with probability 1). (Never visit site) Poisson with Pr(0) = exp[- x i ]. (Number of visits, including zero visits this season.) Unconditional: Pr[0] = P(regime 0) + P(regime 1)*Pr[0 regime 1] Pr[j j >0] = P(regime 1)*Pr[j regime 1] Two inflation Number of children These are latent class models

22 [Part 6] 22/55 Zero Inflation Models exp(-λ i)λi Prob(y i = j x i) =, λ i = exp( βx i) j! j Zero Inflation = ZIP Prob(0 regime) = F( γz ) i ZIP - tau = ZIP(τ) [Not generally used] Prob(0 regime) = F( βx ) i

23 [Part 6] 23/55 Notes on Zero Inflation Models Poisson is not nested in ZIP. γ = 0 in ZIP does not produce Poisson; it produces ZIP with P(regime 0) = ½. Standard tests are not appropriate Use Vuong statistic. ZIP model almost always wins. Zero Inflation models extend to NB models ZINB is a standard model Creates two sources of overdispersion Generally difficult to estimate

24 ZIP Model [Part 6] 24/ Zero Altered Poisson Regression Model Logistic distribution used for splitting model. ZAP term in probability is F[tau x Z(i) ] Comparison of estimated models Pr[0 means] Number of zeros Log-likelihood Poisson Act.= Prd.= Z.I.Poisson Act.= Prd.= Vuong statistic for testing ZIP vs. unaltered model is Distributed as standard normal. A value greater than favors the zero altered Z.I.Poisson model. A value less than rejects the ZIP model. Variable Coefficient Standard Error b/st.er. P[ Z >z] Mean of X Poisson/NB/Gamma regression model Constant *** AGE.01100*** EDUC *** FEMALE.10943*** MARRIED *** HHNINC *** HHKIDS *** Zero inflation model Constant *** FEMALE *** EDUC.04114***

25 [Part 6] 25/55 Marginal Effects for Different Models Scale Factor for Marginal Effects POISSON Variable Coefficient Standard Error b/st.er. P[ Z >z] Mean of X AGE.05613*** EDUC *** FEMALE.93237*** MARRIED HHNINC *** HHKIDS *** Scale Factor for Marginal Effects NEGATIVE BINOMIAL - 2 AGE.05767*** EDUC *** FEMALE *** MARRIED HHNINC *** HHKIDS *** Scale Factor for Marginal Effects ZERO INFLATED POISSON AGE.03427*** EDUC *** FEMALE.97958*** MARRIED *** HHNINC *** HHKIDS ***

26 [Part 6] 26/55 Zero Inflation Models Zero Inflation = ZIP exp(-λ i)λi Prob(y i = j x i) =, λ i = exp( βxi) j! Prob(0 regime) = F( γz ) i j

27 An Unidentified ZINB Model [Part 6] 27/55

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30 [Part 6] 30/55 Partial Effects for Different Models

31 [Part 6] 31/55 The Vuong Statistic for Nonnested Models Model 0: logl = logf (y x, ) = m i,0 0 i i 0 i,0 Model 0 is the Zero Inflation Model Model 1: logl = logf (y x, ) = m i,1 1 i i 1 i,1 Model 1 is the Poisson model (Not nested. =0 implies the splitting probability is 1/2, not 1) 0 i i 0 Define ai mi,0 mi,1 log f 1 (y i x i, 1 ) [a] V s / n a f (y x, ) 1 n f 0(y i x i, 0) n i 1 log n f 1(y i x i, 1) 1 n f 0(y i x i, 0) f 0(y i x i, 0) i 1 log log n 1 f 1(y i x i, 1) f 1(y i x i, 1) Limiting distribution is standard normal. Large + favors model 0, large - favors model 1, < V < 1.96 is inconclusive. 2

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33 [Part 6] 33/55 Two part model: A Hurdle Model Model 1: Probability model for more than zero occurrences Model 2: Model for number of occurrences given that the number is greater than zero. Applications common in health economics Usage of health care facilities Use of drugs, alcohol, etc.

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35 [Part 6] 35/55 Hurdle Model Two Part Model Prob[y > 0] = F( γ'x) Prob[y=j] Prob[y=j] Prob[y = j y > 0] = = Prob[y>0] 1 Pr ob[y 0 x] A Poisson Hurdle Model with Logit Hurdle exp( γ'x) Prob[y>0]= 1+exp( γ'x ) j exp(- ) Prob[y=j y>0,x]=, =exp( β'x) j![1 exp(- )] F( γ'x)exp( β'x) E[y x] =0 Prob[y=0]+Prob[y>0] E[y y>0] = 1-exp[-exp( β'x )] Marginal effects involve both parts of the model.

36 [Part 6] 36/55 Hurdle Model for Doctor Visits

37 [Part 6] 37/55 Partial Effects

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41 [Part 6] 41/55 Application of Several of the Models Discussed in this Section

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43 [Part 6] 43/55 See also: van Ophem H Modeling selectivity in count data models. Journal of Business and Economic Statistics 18: Winkelmann finds that there is no correlation between the decisions A significant correlation is expected [T]he correlation comes from the way the relation between the decisions is modeled.

44 [Part 6] 44/55 Probit Participation Equation Poisson-Normal Intensity Equation

45 [Part 6] 45/55 Bivariate-Normal Heterogeneity in Participation and Intensity Equations Gaussian Copula for Participation and Intensity Equations

46 [Part 6] 46/55 Correlation between Heterogeneity Terms Correlation between Counts

47 [Part 6] 47/55 Panel Data Models Heterogeneity; λ it = exp(β x it + c i ) Fixed Effects Poisson: Standard, no incidental parameters issue Negative Binomial Hausman, Hall, Griliches (1984) put FE in variance, not the mean Use brute force to get a conventional FE model Random Effects Poisson Log-gamma heterogeneity becomes an NB model Contemporary treatments are using normal heterogeneity with simulation or quadrature based estimators NB with random effects is equivalent to two effects one time varying one time invariant. The model is probably overspecified Random Parameters: Mixed models, latent class models, hiererchical all extended to Poisson and NB

48 [Part 6] 48/55 Poisson (log)normal Mixture

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50 [Part 6] 50/55 A Peculiarity of the FENB Model True FE model has λ i =exp(α i +x it β). Cannot be fit if there are time invariant variables. Hausman, Hall and Griliches (Econometrica, 1984) has α i appearing in θ. Produces different results Implies that the FEM can contain time invariant variables.

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52 [Part 6] 52/55 See: Allison and Waterman (2002), Guimaraes (2007) Greene, Econometric Analysis (2012)

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54 [Part 6] 54/55 Bivariate Random Effects

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