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1. William Greene Department of Economics Stern School of Business Topics in MicroeconometricsUniversity of QueenslandBrisbane, QLDJuly 7-9, 2010

2. Ordered Choices and Count Data

3. Ordered Preferences at IMDB.com

4. Translating Movie Preferences Into a Discrete Outcome

5. Health Satisfaction (HSAT) Self administered survey: Health Care Satisfaction? (0 – 10) Continuous Preference Scale

6. Modeling Ordered Choices Random Utility (allowing a panel data setting) Uit = +’xit+ it =ait+ it Observe outcome j if utility is in region j Probability of outcome = probability of cell Pr[Yit=j] = F(j – ait) - F(j-1 – ait)

7. Ordered Probability Model

8. Combined Outcomes for Health Satisfaction

9. Ordered Probabilities

10. Probabilities for Ordered Choices μ1 =1.1479 μ2=2.5478 μ3=3.0564

11. 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

12. Coefficients and Partial Effects

13. Partial Effects in the Ordered Probability Model Assume the βk is positive. Assume that xk increases. β’x increases. μj- 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 βk > 0, increase in xk 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

14. Partial Effects of 8 Years of Education

15. An Ordered Probability Model for Health Satisfaction +---------------------------------------------+ | Ordered Probability Model | | Dependent variable HSAT | | Number of observations 27326 | | Underlying probabilities based on Normal | | Cell frequencies for outcomes | | Y Count Freq Y Count Freq Y Count Freq | | 0 447 .016 1 255 .009 2 642 .023 | | 3 1173 .042 4 1390 .050 5 4233 .154 | | 6 2530 .092 7 4231 .154 8 6172 .225 | | 9 3061 .112 10 3192 .116 | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Index function for probability Constant 2.61335825 .04658496 56.099 .0000 FEMALE -.05840486 .01259442 -4.637 .0000 .47877479 EDUC .03390552 .00284332 11.925 .0000 11.3206310 AGE -.01997327 .00059487 -33.576 .0000 43.5256898 HHNINC .25914964 .03631951 7.135 .0000 .35208362 HHKIDS .06314906 .01350176 4.677 .0000 .40273000 Threshold parameters for index Mu(1) .19352076 .01002714 19.300 .0000 Mu(2) .49955053 .01087525 45.935 .0000 Mu(3) .83593441 .00990420 84.402 .0000 Mu(4) 1.10524187 .00908506 121.655 .0000 Mu(5) 1.66256620 .00801113 207.532 .0000 Mu(6) 1.92729096 .00774122 248.965 .0000 Mu(7) 2.33879408 .00777041 300.987 .0000 Mu(8) 2.99432165 .00851090 351.822 .0000 Mu(9) 3.45366015 .01017554 339.408 .0000

16. Ordered Probability 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 .00200414 .00043473 4.610 .0000 .47877479 EDUC -.00115962 .986135D-04 -11.759 .0000 11.3206310 AGE .00068311 .224205D-04 30.468 .0000 43.5256898 HHNINC -.00886328 .00124869 -7.098 .0000 .35208362 *HHKIDS -.00213193 .00045119 -4.725 .0000 .40273000 These are the effects on Prob[Y=01] at means. *FEMALE .00101533 .00021973 4.621 .0000 .47877479 EDUC -.00058810 .496973D-04 -11.834 .0000 11.3206310 AGE .00034644 .108937D-04 31.802 .0000 43.5256898 HHNINC -.00449505 .00063180 -7.115 .0000 .35208362 *HHKIDS -.00108460 .00022994 -4.717 .0000 .40273000 ... repeated for all 11 outcomes These are the effects on Prob[Y=10] at means. *FEMALE -.01082419 .00233746 -4.631 .0000 .47877479 EDUC .00629289 .00053706 11.717 .0000 11.3206310 AGE -.00370705 .00012547 -29.545 .0000 43.5256898 HHNINC .04809836 .00678434 7.090 .0000 .35208362 *HHKIDS .01181070 .00255177 4.628 .0000 .40273000

17. The Single Crossing Effect The marginal effect for EDUC is negative for Prob(0),…,Prob(7), then positive for Prob(8)…Prob(10). One “crossing.”

18. Nonlinearities A nonlinear index function could generalize the relationship between the covariates and the probabilities: U* = … + β1x + β2x2 + β3x3 + … ε ∂Prob(Y=j|X)/∂x = Scale * (β1+ 2β2x + 3β3x2) The partial effect of income is more directly dependent on the value of income. (It also enters the scale part of the partial effect.)

19. Nonlinearity

20. Predictions of the Model:Kids +----------------------------------------------+ |Variable Mean Std.Dev. Minimum Maximum | +----------------------------------------------+ |Stratum is KIDS = 0.000. Nobs.= 2782.000 | +--------+-------------------------------------+ |P0 | .059586 .028182 .009561 .125545 | |P1 | .268398 .063415 .106526 .374712 | |P2 | .489603 .024370 .419003 .515906 | |P3 | .101163 .030157 .052589 .181065 | |P4 | .081250 .041250 .028152 .237842 | +----------------------------------------------+ |Stratum is KIDS = 1.000. Nobs.= 1701.000 | +--------+-------------------------------------+ |P0 | .036392 .013926 .010954 .105794 | |P1 | .217619 .039662 .115439 .354036 | |P2 | .509830 .009048 .443130 .515906 | |P3 | .125049 .019454 .061673 .176725 | |P4 | .111111 .030413 .035368 .222307 | +----------------------------------------------+

21. Predictions from the Model Related to Age

22. 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

23. Log Likelihood Based Fit Measures

24. A Somewhat Better Fit

25. An Aggregate Prediction Measure

26. Heterogeneity in OC Models • Heterogeneity in Thresholds • Pudney and Shields – Nursing • Harris, Greene, et al. – Obesity and BMI • Scale Heterogeneity: Heteroscedasticity • Standard Models of Heterogeneity in Discrete Choice Models • Panel Data: Fixed and Random Effects • Latent Class Models • Random Parameters

27. Count Data: 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

28. Doctor Visits

29. 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”

30. Poisson Model for Doctor Visits ---------------------------------------------------------------------- Poisson Regression Dependent variable DOCVIS Log likelihood function -103727.29625 Restricted log likelihood -108662.13583 Chi squared [ 6 d.f.] 9869.67916 Significance level .00000 McFadden Pseudo R-squared .0454145 Estimation based on N = 27326, K = 7 Information Criteria: Normalization=1/N Normalized Unnormalized AIC 7.59235 207468.59251 Chi- squared =255127.59573 RsqP= .0818 G - squared =154416.01169 RsqD= .0601 Overdispersion tests: g=mu(i) : 20.974 Overdispersion tests: g=mu(i)^2: 20.943 --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- Constant| .77267*** .02814 27.463 .0000 AGE| .01763*** .00035 50.894 .0000 43.5257 EDUC| -.02981*** .00175 -17.075 .0000 11.3206 FEMALE| .29287*** .00702 41.731 .0000 .47877 MARRIED| .00964 .00874 1.103 .2702 .75862 HHNINC| -.52229*** .02259 -23.121 .0000 .35208 HHKIDS| -.16032*** .00840 -19.081 .0000 .40273 --------+-------------------------------------------------------------

31. 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 3.1835 Scale Factor for Marginal Effects 3.1835 --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- AGE| .05613*** .00131 42.991 .0000 43.5257 EDUC| -.09490*** .00596 -15.923 .0000 11.3206 FEMALE| .93237*** .02555 36.491 .0000 .47877 MARRIED| .03069 .02945 1.042 .2973 .75862 HHNINC| -1.66271*** .07803 -21.308 .0000 .35208 HHKIDS| -.51037*** .02879 -17.730 .0000 .40273 --------+-------------------------------------------------------------

32. Poisson Model Specification Issues Equi-Dispersion: Var[yi|xi] = E[yi|xi]. Overdispersion: If i = exp[’xi + εi], E[yi|xi] = γexp[’xi] Var[yi] > E[yi] (overdispersed) εi ~ log-Gamma  Negative binomial model εi ~ Normal[0,2]  Normal-mixture model εi is viewed as unobserved heterogeneity (“frailty”). Normal model may be more natural. Estimation is a bit more complicated.

33. Negative Binomial Specification The Poisson estimator is consistent when there is unmeasured heterogeneity in the conditional mean.Therefore, this is a case for the ROBUST covariance matrix estimator. (Neglected heterogeneity that is uncorrelated with xi.)

34. Poisson Model for Doctor Visits ---------------------------------------------------------------------- Poisson Regression Dependent variable DOCVIS Log likelihood function -103727.29625 Restricted log likelihood -108662.13583 Chi squared [ 6 d.f.] 9869.67916 Significance level .00000 McFadden Pseudo R-squared .0454145 Estimation based on N = 27326, K = 7 Information Criteria: Normalization=1/N Normalized Unnormalized AIC 7.59235 207468.59251 Chi- squared =255127.59573 RsqP= .0818 G - squared =154416.01169 RsqD= .0601 Overdispersion tests: g=mu(i) : 20.974 Overdispersion tests: g=mu(i)^2: 20.943 --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- Constant| .77267*** .02814 27.463 .0000 AGE| .01763*** .00035 50.894 .0000 43.5257 EDUC| -.02981*** .00175 -17.075 .0000 11.3206 FEMALE| .29287*** .00702 41.731 .0000 .47877 MARRIED| .00964 .00874 1.103 .2702 .75862 HHNINC| -.52229*** .02259 -23.121 .0000 .35208 HHKIDS| -.16032*** .00840 -19.081 .0000 .40273 --------+-------------------------------------------------------------

35. Alternative Covariance Matrices --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- | Standard – Negative Inverse of Second Derivatives Constant| .77267*** .02814 27.463 .0000 AGE| .01763*** .00035 50.894 .0000 43.5257 EDUC| -.02981*** .00175 -17.075 .0000 11.3206 FEMALE| .29287*** .00702 41.731 .0000 .47877 MARRIED| .00964 .00874 1.103 .2702 .75862 HHNINC| -.52229*** .02259 -23.121 .0000 .35208 HHKIDS| -.16032*** .00840 -19.081 .0000 .40273 --------+------------------------------------------------------------- | Robust – Sandwich Constant| .77267*** .08529 9.059 .0000 AGE| .01763*** .00105 16.773 .0000 43.5257 EDUC| -.02981*** .00487 -6.123 .0000 11.3206 FEMALE| .29287*** .02250 13.015 .0000 .47877 MARRIED| .00964 .02906 .332 .7401 .75862 HHNINC| -.52229*** .06674 -7.825 .0000 .35208 HHKIDS| -.16032*** .02657 -6.034 .0000 .40273 --------+------------------------------------------------------------- | Cluster Correction Constant| .77267*** .11628 6.645 .0000 AGE| .01763*** .00142 12.440 .0000 43.5257 EDUC| -.02981*** .00685 -4.355 .0000 11.3206 FEMALE| .29287*** .03213 9.116 .0000 .47877 MARRIED| .00964 .03851 .250 .8023 .75862 HHNINC| -.52229*** .08295 -6.297 .0000 .35208 HHKIDS| -.16032*** .03455 -4.640 .0000 .40273

36. Negative Binomial Specification Prob(Yi=j|xi) has greater mass to the right and left of the mean Conditional mean function is the same as the Poisson: E[yi|xi] = λi=Exp(’xi), so marginal effects have the same form. Variance is Var[yi|xi] = λi(1 + α λi), 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 xi.)

37. NegBin Model for Doctor Visits ---------------------------------------------------------------------- Negative Binomial Regression Dependent variable DOCVIS Log likelihood function -60134.50735 NegBin LogL Restricted log likelihood -103727.29625 Poisson LogL Chi squared [ 1 d.f.] 87185.57782 Reject Poisson model Significance level .00000 McFadden Pseudo R-squared .4202634 Estimation based on N = 27326, K = 8 Information Criteria: Normalization=1/N Normalized Unnormalized AIC 4.40185 120285.01469 NegBin form 2; Psi(i) = theta --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- Constant| .80825*** .05955 13.572 .0000 AGE| .01806*** .00079 22.780 .0000 43.5257 EDUC| -.03717*** .00386 -9.622 .0000 11.3206 FEMALE| .32596*** .01586 20.556 .0000 .47877 MARRIED| -.00605 .01880 -.322 .7477 .75862 HHNINC| -.46768*** .04663 -10.029 .0000 .35208 HHKIDS| -.15274*** .01729 -8.832 .0000 .40273 |Dispersion parameter for count data model Alpha| 1.89679*** .01981 95.747 .0000 --------+-------------------------------------------------------------

38. Marginal Effects +--------------------------------------------------------------------- Scale Factor for Marginal Effects 3.1835 POISSON --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- AGE| .05613*** .00131 42.991 .0000 43.5257 EDUC| -.09490*** .00596 -15.923 .0000 11.3206 FEMALE| .93237*** .02555 36.491 .0000 .47877 MARRIED| .03069 .02945 1.042 .2973 .75862 HHNINC| -1.66271*** .07803 -21.308 .0000 .35208 HHKIDS| -.51037*** .02879 -17.730 .0000 .40273 --------+------------------------------------------------------------- Scale Factor for Marginal Effects 3.1924 NEGATIVE BINOMIAL --------+------------------------------------------------------------- AGE| .05767*** .00317 18.202 .0000 43.5257 EDUC| -.11867*** .01348 -8.804 .0000 11.3206 FEMALE| 1.04058*** .06212 16.751 .0000 .47877 MARRIED| -.01931 .06382 -.302 .7623 .75862 HHNINC| -1.49301*** .16272 -9.176 .0000 .35208 HHKIDS| -.48759*** .06022 -8.097 .0000 .40273 --------+-------------------------------------------------------------

39. Model Formulations E[yi |xi ]=λi

40. NegBin-1 Model ---------------------------------------------------------------------- Negative Binomial Regression Dependent variable DOCVIS Log likelihood function -60025.78734 Restricted log likelihood -103727.29625 NegBin form 1; Psi(i) = theta*exp[bx(i)] --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- Constant| .62584*** .05816 10.761 .0000 AGE| .01428*** .00073 19.462 .0000 43.5257 EDUC| -.01549*** .00359 -4.314 .0000 11.3206 FEMALE| .33028*** .01479 22.328 .0000 .47877 MARRIED| .04324** .01852 2.335 .0196 .75862 HHNINC| -.24543*** .04540 -5.406 .0000 .35208 HHKIDS| -.14877*** .01745 -8.526 .0000 .40273 |Dispersion parameter for count data model Alpha| 6.09246*** .06694 91.018 .0000 --------+-------------------------------------------------------------

41. NegBin-P Model ---------------------------------------------------------------------- Negative Binomial (P) Model Dependent variable DOCVIS Log likelihood function -59992.32903 Restricted log likelihood -103727.29625 Chi squared [ 1 d.f.] 87469.93445 --------+----------------------------------------- Variable| Coefficient Standard Error b/St.Er. --------+----------------------------------------- Constant| .60840*** .06452 9.429 AGE| .01710*** .00082 20.782 EDUC| -.02313*** .00414 -5.581 FEMALE| .36386*** .01640 22.187 MARRIED| .03670* .02030 1.808 HHNINC| -.35093*** .05146 -6.819 HHKIDS| -.16902*** .01911 -8.843 |Dispersion parameter for count data model Alpha| 3.85713*** .14581 26.453 |Negative Binomial. General form, NegBin P P| 1.38693*** .03142 44.140 --------+------------------------------------------------------------- NB-2 NB-1 Poisson

42. Marginal Effects for Different Models Scale Factor for Marginal Effects 3.1835 POISSON Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- AGE| .05613*** .00131 42.991 .0000 43.5257 EDUC| -.09490*** .00596 -15.923 .0000 11.3206 FEMALE| .93237*** .02555 36.491 .0000 .47877 MARRIED| .03069 .02945 1.042 .2973 .75862 HHNINC| -1.66271*** .07803 -21.308 .0000 .35208 HHKIDS| -.51037*** .02879 -17.730 .0000 .40273 --------+------------------------------------------------------------- Scale Factor for Marginal Effects 3.1924 NEGATIVE BINOMIAL - 2 AGE| .05767*** .00317 18.202 .0000 43.5257 EDUC| -.11867*** .01348 -8.804 .0000 11.3206 FEMALE| 1.04058*** .06212 16.751 .0000 .47877 MARRIED| -.01931 .06382 -.302 .7623 .75862 HHNINC| -1.49301*** .16272 -9.176 .0000 .35208 HHKIDS| -.48759*** .06022 -8.097 .0000 .40273 --------+------------------------------------------------------------- Scale Factor for Marginal Effects 3.1835 NEGATIVE BINOMIAL - 1 AGE| .04547*** .00263 17.285 .0000 43.5257 EDUC| -.04933*** .01196 -4.125 .0000 11.3206 FEMALE| 1.05145*** .05456 19.272 .0000 .47877 MARRIED| .13766** .06154 2.237 .0253 .75862 HHNINC| -.78134*** .15139 -5.161 .0000 .35208 HHKIDS| -.47361*** .05885 -8.048 .0000 .40273 --------+------------------------------------------------------------- Scale Factor for Marginal Effects 3.0077 NEGATIVE BINOMIAL - P AGE| .05143*** .00246 20.934 .0000 43.5257 EDUC| -.06957*** .01241 -5.605 .0000 11.3206 FEMALE| 1.09436*** .04968 22.027 .0000 .47877 MARRIED| .11038* .06109 1.807 .0708 .75862 HHNINC| -1.05547*** .15411 -6.849 .0000 .35208 HHKIDS| -.50835*** .05753 -8.836 .0000 .40273