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

[Part 5] 2/43 Discrete Choice Modeling Ordered Choice Models Ordered Discrete Outcomes  E.g.: Taste test, credit rating, course grade, preference scale  Underlying random preferences: Existence of an underlying continuous preference scale Mapping to observed choices  Strength of preferences is reflected in the discrete outcome  Censoring and discrete measurement  The nature of ordered data

[Part 5] 3/43 Discrete Choice Modeling Ordered Choice Models Ordered Preferences at IMDB.com

[Part 5] 4/43 Discrete Choice Modeling Ordered Choice Models Health Satisfaction (HSAT) Self administered survey: Health Care Satisfaction? (0 – 10) Continuous Preference Scale

[Part 5] 5/43 Discrete Choice Modeling Ordered Choice Models Modeling Ordered Choices  Random Utility (allowing a panel data setting) U it =  +  ’x it +  it = a it +  it  Observe outcome j if utility is in region j  Probability of outcome = probability of cell Pr[Y it =j] = F(  j – a it ) - F(  j-1 – a it )

[Part 5] 6/43 Discrete Choice Modeling Ordered Choice Models Ordered Probability Model

[Part 5] 7/43 Discrete Choice Modeling Ordered Choice Models Combined Outcomes for Health Satisfaction

[Part 5] 8/43 Discrete Choice Modeling Ordered Choice Models Ordered Probabilities

[Part 5] 9/43 Discrete Choice Modeling Ordered Choice Models

[Part 5] 10/43 Discrete Choice Modeling Ordered Choice Models Coefficients

[Part 5] 11/43 Discrete Choice Modeling Ordered Choice Models Partial Effects in the Ordered Choice Model Assume the β k is positive. Assume that x k increases. β’x increases. μ j - β’x shifts to the left for all 5 cells. Prob[y=0] decreases Prob[y=1] decreases – the mass shifted out is larger than the mass shifted in. Prob[y=3] increases – same reason in reverse. Prob[y=4] must increase. When β k > 0, increase in x k decreases Prob[y=0] and increases Prob[y=J]. Intermediate cells are ambiguous, but there is only one sign change in the marginal effects from 0 to 1 to … to J

[Part 5] 12/43 Discrete Choice Modeling Ordered Choice Models Partial Effects of 8 Years of Education

[Part 5] 13/43 Discrete Choice Modeling Ordered Choice Models An Ordered Probability Model for Health Satisfaction | Ordered Probability Model | | Dependent variable HSAT | | Number of observations | | Underlying probabilities based on Normal | | Cell frequencies for outcomes | | Y Count Freq Y Count Freq Y Count Freq | | | | | | | | | |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| Index function for probability Constant FEMALE EDUC AGE HHNINC HHKIDS Threshold parameters for index Mu(1) Mu(2) Mu(3) Mu(4) Mu(5) Mu(6) Mu(7) Mu(8) Mu(9)

[Part 5] 14/43 Discrete Choice Modeling Ordered Choice Models Ordered Probability Partial Effects | Marginal effects for ordered probability model | | M.E.s for dummy variables are Pr[y|x=1]-Pr[y|x=0] | | Names for dummy variables are marked by *. | |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| These are the effects on Prob[Y=00] at means. *FEMALE EDUC D AGE D HHNINC *HHKIDS These are the effects on Prob[Y=01] at means. *FEMALE EDUC D AGE D HHNINC *HHKIDS repeated for all 11 outcomes These are the effects on Prob[Y=10] at means. *FEMALE EDUC AGE HHNINC *HHKIDS

[Part 5] 15/43 Discrete Choice Modeling Ordered Choice Models Ordered Probit Marginal Effects

[Part 5] 16/43 Discrete Choice Modeling Ordered Choice Models Analysis of Model Implications  Partial Effects  Fit Measures  Predicted Probabilities Averaged: They match sample proportions. By observation Segments of the sample Related to particular variables

[Part 5] 17/43 Discrete Choice Modeling Ordered Choice Models Predictions from the Model Related to Age

[Part 5] 18/43 Discrete Choice Modeling Ordered Choice Models Fit Measures  There is no single “dependent variable” to explain.  There is no sum of squares or other measure of “variation” to explain.  Predictions of the model relate to a set of J+1 probabilities, not a single variable.  How to explain fit? Based on the underlying regression Based on the likelihood function Based on prediction of the outcome variable

[Part 5] 19/43 Discrete Choice Modeling Ordered Choice Models Log Likelihood Based Fit Measures

[Part 5] 20/43 Discrete Choice Modeling Ordered Choice Models

[Part 5] 21/43 Discrete Choice Modeling Ordered Choice Models A Somewhat Better Fit

[Part 5] 22/43 Discrete Choice Modeling Ordered Choice Models Different Normalizations  NLOGIT Y = 0,1,…,J, U* = α + β’x + ε One overall constant term, α J-1 “cutpoints;” μ -1 = -∞, μ 0 = 0, μ 1,… μ J-1, μ J = + ∞  Stata Y = 1,…,J+1, U* = β’x + ε No overall constant, α=0 J “cutpoints;” μ 0 = -∞, μ 1,… μ J, μ J+1 = + ∞

[Part 5] 23/43 Discrete Choice Modeling Ordered Choice Models

[Part 5] 24/43 Discrete Choice Modeling Ordered Choice Models

[Part 5] 25/43 Discrete Choice Modeling Ordered Choice Models Generalizing the Ordered Probit with Heterogeneous Thresholds

[Part 5] 26/43 Discrete Choice Modeling Ordered Choice Models Hierarchical Ordered Probit

[Part 5] 27/43 Discrete Choice Modeling Ordered Choice Models Ordered Choice Model

[Part 5] 28/43 Discrete Choice Modeling Ordered Choice Models HOPit Model

[Part 5] 29/43 Discrete Choice Modeling Ordered Choice Models Differential Item Functioning

[Part 5] 30/43 Discrete Choice Modeling Ordered Choice Models A Vignette Random Effects Model

[Part 5] 31/43 Discrete Choice Modeling Ordered Choice Models Vignettes

[Part 5] 32/43 Discrete Choice Modeling Ordered Choice Models

[Part 5] 33/43 Discrete Choice Modeling Ordered Choice Models

[Part 5] 34/43 Discrete Choice Modeling Ordered Choice Models

[Part 5] 35/43 Discrete Choice Modeling Ordered Choice Models Panel Data  Fixed Effects The usual incidental parameters problem Practically feasible but methodologically ambiguous Partitioning Prob(y it > j|x it ) produces estimable binomial logit models. (Find a way to combine multiple estimates of the same β.  Random Effects Standard application Extension to random parameters – see above

[Part 5] 36/43 Discrete Choice Modeling Ordered Choice Models Incidental Parameters Problem Table 9.1 Monte Carlo Analysis of the Bias of the MLE in Fixed Effects Discrete Choice Models (Means of empirical sampling distributions, N = 1,000 individuals, R = 200 replications)

[Part 5] 37/43 Discrete Choice Modeling Ordered Choice Models A Dynamic Ordered Probit Model

[Part 5] 38/43 Discrete Choice Modeling Ordered Choice Models Model for Self Assessed Health  British Household Panel Survey (BHPS) Waves 1-8, Self assessed health on 0,1,2,3,4 scale Sociological and demographic covariates Dynamics – inertia in reporting of top scale  Dynamic ordered probit model Balanced panel – analyze dynamics Unbalanced panel – examine attrition

[Part 5] 39/43 Discrete Choice Modeling Ordered Choice Models Dynamic Ordered Probit Model It would not be appropriate to include h i,t-1 itself in the model as this is a label, not a measure

[Part 5] 40/43 Discrete Choice Modeling Ordered Choice Models Testing for Attrition Bias Three dummy variables added to full model with unbalanced panel suggest presence of attrition effects.

[Part 5] 41/43 Discrete Choice Modeling Ordered Choice Models Attrition Model with IP Weights Assumes (1) Prob(attrition|all data) = Prob(attrition|selected variables) (ignorability) (2) Attrition is an ‘absorbing state.’ No reentry. Obviously not true for the GSOEP data above. Can deal with point (2) by isolating a subsample of those present at wave 1 and the monotonically shrinking subsample as the waves progress.

[Part 5] 42/43 Discrete Choice Modeling Ordered Choice Models Inverse Probability Weighting

[Part 5] 43/43 Discrete Choice Modeling Ordered Choice Models Estimated Partial Effects by Model

[Part 5] 44/43 Discrete Choice Modeling Ordered Choice Models Partial Effect for a Category These are 4 dummy variables for state in the previous period. Using first differences, the estimated for SAHEX means transition from EXCELLENT in the previous period to GOOD in the previous period, where GOOD is the omitted category. Likewise for the other 3 previous state variables. The margin from ‘POOR’ to ‘GOOD’ was not interesting in the paper. The better margin would have been from EXCELLENT to POOR, which would have (EX,POOR) change from (1,0) to (0,1).

[Part 5] 45/43 Discrete Choice Modeling Ordered Choice Models Model Extensions  Multivariate Bivariate Multivariate  Inflation and Two Part Zero inflation Sample Selection Endogenous Latent Class

[Part 5] 46/43 Discrete Choice Modeling Ordered Choice Models A Sample Selection Model