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ECON 6002 Econometrics Memorial University of Newfoundland Qualitative and Limited Dependent Variable Models Adapted from Vera Tabakova’s notes 16.1 Models with Binary Dependent Variables 16.2 The Logit Model for Binary Choice 16.3 Multinomial Logit 16.4 Conditional Logit 16.5 Ordered Choice Models 16.6 Models for Count Data and extensions 16.7 Limited Dependent Variables Principles of Econometrics, 3rd Edition Slide 16-2 When the dependent variable in a regression model is a count of the number of occurrences of an event, the outcome variable is y = 0, 1, 2, 3, … These numbers are actual counts, and thus different from the ordinal numbers of the previous section. Examples include: The number of trips to a physician a person makes during a year. The number of fishing trips taken by a person during the previous year. The number of children in a household. The number of automobile accidents at a particular intersection during a month. The number of televisions in a household. The number of alcoholic drinks a college student takes in a week. Principles of Econometrics, 3rd Edition Slide16-3 If Y is a Poisson random variable, then its probability function is e y f y P Y y , y! y! y y 1 y 2 1 y 0,1, 2, “rate” E Y exp 1 2 x (16.27) Also equal To the variance (16.28) This choice defines the Poisson regression model for count data. Principles of Econometrics, 3rd Edition Slide16-4 If we observe 3 individuals: one faces no event, the other two two events each: L 1 , 2 P Y 0 P Y 2 P Y 2 ln L 1 , 2 ln P Y 0 ln P Y 2 ln P Y 2 e y ln P Y y ln y ln ln y ! y! exp 1 2 x y 1 2 x ln y ! ln L 1 , 2 exp 1 2 xi yi 1 2 xi ln yi ! N i 1 Principles of Econometrics, 3rd Edition Slide16-5 E y0 0 exp 1 2 x0 Pr Y y exp 0 0y y! , y 0,1,2, So now you can calculate the predicted probability of a certain number y of events Principles of Econometrics, 3rd Edition Slide16-6 E yi i 2 xi (16.29) You may prefer to express this marginal effect as a %: %E y E yi E yi 100 1002 % xi xi Principles of Econometrics, 3rd Edition Slide16-7 E yi i exp 1 2 xi Di E yi | Di 0 exp 1 2 xi If there is a dummy Involved, be careful, remember E yi | Di 1 exp 1 2 xi exp 1 2 xi exp 1 2 xi 100 % 100 e 1 % exp 1 2 xi Which would be identical to the effect of a dummy In the log-linear model we saw under OLS Principles of Econometrics, 3rd Edition Slide16-8 Extensions: overdispersion Under a plain Poisson the mean of the count is assumed to be equal to the average (equidispersion) This will often not hold Real life data are often overdispersed For example: • a few women will have many affairs and many women will have few • a few travelers will make many trips to a park and many will make few • etc. Principles of Econometrics, 3rd Edition Slide16-9 Extensions: overdispersion use "C:\bbbECONOMETRICS\Rober\GRAD\GROSMORNE.dta", clear . poisson visits Travelcost educat income Iteration 0: log likelihood = -1321.4696 1:persontrip log likelihood = -1321.4665 Travelcost Iteration 2: log likelihood = -1321.4665 Iteration . poisson educat income, nolog Poisson Poisson regression regression Number of obs LR chi2(3) Prob > chi2 Pseudo R2 Log likelihood = -1321.4665 Log likelihood = -2541.5165 visits Travelcost persontrip educat income Travelcost _cons educat income _cons Coef. Std. Err. z -.3299655 .0529402 Coef. Std.-6.23 Err. -.0307667 .026493 -1.16 -.0019933 .0007191 -2.77 -.9570718 .0435943 .8765791 .1125493 7.79 -.0206209 -.0014578 2.144476 Principles of Econometrics, 3rd Edition .0163568 .0004404 .0688666 P>|z| Number of = 919 obs = 56.61 LR chi2(3) = Prob >0.0000 chi2 = 0.0210 Pseudo R2 = = = = [95% Conf. Interval] 0.000 z -.4337264 P>|z| -.2262045 [95% Conf. 0.246 -.0826921 .0211587 0.006 -.0034027 -.0005839 -21.95 .6559865 0.000 1.097172 -1.042515 0.000 -1.26 -3.31 31.14 919 671.71 0.0000 0.1167 0.207 0.001 0.000 -.0526797 -.002321 2.0095 Interval] -.8716285 .0114379 -.0005946 2.279452 Slide16-10 Extensions: negative binomial Under a plain Poisson the mean of the count is assumed to be equal to the average (equidispersion) The Poisson will inflate your t-ratios in this case, making you think that your model works better than it actually does Or use a Negative Binomial model instead (nbreg) or even a Generalised Negative Binomial (gnbreg) , which will allow you to model the overdispersion parameter as a function of covariates of our choice You can also test for overdispersion, to test whether the problem is significant Principles of Econometrics, 3rd Edition Slide16-11 Extensions: negative binomial sum visits Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------visits | 966 1.416149 1.718147 1 26 Principles of Econometrics, 3rd Edition Slide16-12 Extensions: negative binomial . nbreg persontrip Travelcost educat income, nolog Negative binomial regression Number of obs LR chi2(3) Prob > chi2 Pseudo R2 Dispersion = mean Log likelihood = -2038.1155 persontrip Coef. Std. Err. Travelcost educat income _cons -.7135986 -.0218888 -.0014357 1.994577 .0489137 .0248201 .0006578 .1037 /lnalpha -1.190022 alpha .3042145 -14.59 -0.88 -2.18 19.23 P>|z| 0.000 0.378 0.029 0.000 919 236.04 0.0000 0.0547 [95% Conf. Interval] -.8094676 -.0705353 -.0027249 1.791329 -.6177295 .0267578 -.0001465 2.197826 .0724583 -1.332038 -1.048006 .0220429 .2639388 .3506361 Likelihood-ratio test of alpha=0: Principles of Econometrics, 3rd Edition z = = = = chibar2(01) = 1006.80 Prob>=chibar2 = 0.000 Slide16-13 Extensions: excess zeros Often the numbers of zeros in the sample cannot be accommodated properly by a Poisson or Negative Binomial model They would underpredict them too There is said to be an “excess zeros” problem You can then use hurdle models or zero inflated or zero augmented models to accommodate the extra zeros Principles of Econometrics, 3rd Edition Slide16-14 Extensions: excess zeros 0 .2 Proportion nbvargr Is a very useful command .4 They would underpredict them too .6 Often the numbers of zeros in the sample cannot be accommodated properly by a Poisson or Negative Binomial model 0 2 4 6 8 10 k mean = 3.296; overdispersion = 5.439 observed proportion poisson prob Principles of Econometrics, 3rd Edition neg binom prob Slide16-15 Extensions: excess zeros You can then use hurdle models or zero inflated or zero augmented models to accommodate the extra zeros They will also allow you to have a different process driving the value of the strictly positive count and whether the value is zero or strictly positive EXAMPLES: •Number of extramarital affairs versus gender •Number of children before marriage versus religiosity In the continuous case, we have similar models (e.g. Cragg’s Model) and an example is that of size of Insurance Claims from fires versus the age of the building Principles of Econometrics, 3rd Edition Slide16-16 Extensions: excess zeros You can then use hurdle models or zero inflated or zero augmented models to accommodate the extra zeros Hurdle Models A hurdle model is a modified count model in which there are two processes, one generating the zeros and one generating the positive values. The two models are not constrained to be the same. In the hurdle model a binomial probability model governs the binary outcome of whether a count variable has a zero or a positive value. If the value is positive, the "hurdle is crossed," and the conditional distribution of the positive values is governed by a zero-truncated count model. Example: smokers versus non-smokers, if you are a smoker you will smoke! Principles of Econometrics, 3rd Edition Slide16-17 Extensions: excess zeros Hurdle Models In Stata Joseph Hilbe’s downloadable ado hplogit will work, although it does not allow for two different sets of variables, just two different sets of coefficients See also hnblogit Example: smokers versus non-smokers, if you are a smoker you will smoke! Principles of Econometrics, 3rd Edition Slide16-18 Extensions: excess zeros You can then use hurdle models or zero inflated or zero augmented models to accommodate the extra zeros Zero-inflated models (initially suggested by D. Lambert) attempt to account for excess zeros in a subtly different way. In this model there are two kinds of zeros, "true zeros" and excess zeros. Zero-inflated models estimate also two equations, one for the count model and one for the excess zero's. The key difference is that the count model allows zeros now. It is not a truncated count model, but allows for “corner solutions” Example: meat eaters (who sometimes just did not eat meat that week) versus vegetarians who never ever do Principles of Econometrics, 3rd Edition Slide16-19 Extensions: excess zeros webuse fish We want to model how many fish are being caught by fishermen at a state park. Visitors are asked how long they stayed, how many people were in the group, were there children in the group and how many fish were caught. Some visitors do not fish at all, but there is no data on whether a person fished or not. Some visitors who did fish did not catch any fish (and admitted it ) so there are excess zeros in the data because of the people that did not fish. Principles of Econometrics, 3rd Edition Slide16-20 Extensions: excess zeros 150 . histogram count, discrete freq 0 50 Frequency 100 Lots of zeros! 0 50 100 150 count Principles of Econometrics, 3rd Edition Slide16-21 Extensions: excess zeros . zip naffairs age male relig , inflate( age male relig ) vuong nolog Zero-inflated Poisson regression Number of obs Nonzero obs Zero obs = = = 601 150 451 Inflation model = logit Log likelihood = -810.055 LR chi2(3) Prob > chi2 = = 29.67 0.0000 naffairs Coef. Std. Err. z P>|z| [95% Conf. Interval] naffairs age male relig _cons .015609 -.1598035 -.0971114 1.581638 .0038029 .0686006 .0292688 .1577305 4.10 -2.33 -3.32 10.03 0.000 0.020 0.001 0.000 .0081555 -.2942583 -.1544772 1.272492 .0230625 -.0253487 -.0397456 1.890784 age male relig _cons -.019041 -.1791471 .2884574 .9322364 .0104841 .1948003 .0841492 .3901503 -1.82 -0.92 3.43 2.39 0.069 0.358 0.001 0.017 -.0395895 -.5609488 .1235281 .1675558 .0015075 .2026546 .4533867 1.696917 inflate Vuong test of zip vs. standard Poisson: Principles of Econometrics, 3rd Edition z = Vuong test 11.66 Pr>z = 0.0000 Slide16-22 Extensions: excess zeros . zinb naffairs age male relig , inflate( age male relig ) vuong nolog Zero-inflated negative binomial regression Number of obs Nonzero obs Zero obs = = = 601 150 451 Inflation model = logit Log likelihood = -726.405 LR chi2(3) Prob > chi2 = = 8.92 0.0304 naffairs Coef. Std. Err. z P>|z| [95% Conf. Interval] naffairs age male relig _cons .0258188 -.2214886 -.1472717 1.273196 .0107692 .1660362 .0749567 .3874106 2.40 -1.33 -1.96 3.29 0.017 0.182 0.049 0.001 .0047115 -.5469135 -.2941842 .5138849 .046926 .1039364 -.0003593 2.032506 age male relig _cons -.014892 -.2309299 .274744 .6673066 .0113465 .2091759 .0904315 .433002 -1.31 -1.10 3.04 1.54 0.189 0.270 0.002 0.123 -.0371308 -.6409071 .0975014 -.1813618 .0073468 .1790474 .4519865 1.515975 /lnalpha -.2743069 .2532933 -1.08 0.279 -.7707527 .2221388 alpha .7600988 .1925279 .4626647 1.248745 inflate Vuong test of zinb vs. standard negative binomial: z = Principles of Econometrics, 3rd Edition Vuong test 2.82 Pr>z = 0.0024 Slide16-23 Extensions: truncation • Count data can be truncated too (usually at zero) • So ztp and ztnb can accommodate that • Example: you interview visitors at the recreational site, so they all made at least that one trip •In the continuous case we would have to use the truncreg command Principles of Econometrics, 3rd Edition Slide16-24 Extensions: truncation This model works much better and showcases the bias in the previous estimates: • . ztp persontrip Travelcost educat income, nolog Zero-truncated Poisson regression Number of obs LR chi2(3) Prob > chi2 Pseudo R2 Log likelihood = -2412.6552 persontrip Coef. Travelcost educat income _cons -1.380461 -.0170332 -.0013521 2.278878 Std. Err. .0571736 .0175026 .000473 .0728394 z -24.15 -0.97 -2.86 31.29 P>|z| 0.000 0.330 0.004 0.000 = = = = 919 885.68 0.0000 0.1551 [95% Conf. Interval] -1.492519 -.0513376 -.0022791 2.136116 -1.268403 .0172712 -.0004251 2.421641 Smaller now estimated Consumer Surplus Principles of Econometrics, 3rd Edition Slide16-25 Extensions: truncation This model works much better and showcases the bias in the previous estimates: • Now accounting for overdispersion . ztnb persontrip Travelcost educat income, nolog Zero-truncated negative binomial regression Number of obs LR chi2(3) Prob > chi2 Pseudo R2 Dispersion = mean Log likelihood = -1866.326 persontrip Coef. Travelcost educat income _cons -1.079011 -.0216377 -.0016369 2.015503 .068793 .0322941 .0008563 .1344308 /lnalpha -.6368613 alpha .52895 Std. Err. -15.68 -0.67 -1.91 14.99 P>|z| 0.000 0.503 0.056 0.000 919 263.89 0.0000 0.0660 [95% Conf. Interval] -1.213843 -.084933 -.0033152 1.752024 -.9441795 .0416576 .0000413 2.278983 .101849 -.8364818 -.4372409 .053873 .433232 .6458158 Likelihood-ratio test of alpha=0: Principles of Econometrics, 3rd Edition z = = = = chibar2(01) = 1092.66 Prob>=chibar2 = 0.000 Slide16-26 Extensions: truncation and endogenous stratification Example: you interview visitors at the recreational site, so they all made at least that one trip • You interview patients at the doctors’ office about how often they visit the doctor • You ask people in George St. how often the go to George St… • •Then you are oversampling “frequent visitors” and biasing your estimates, perhaps substantially Principles of Econometrics, 3rd Edition Slide16-27 Extensions: truncation and endogenous stratification •Then you are oversampling “frequent visitors” and biasing your estimates, perhaps substantially •It turns out to be supereasy to deal with a Truncated and Endogenously Stratified Poisson Model (as shown by Shaw, 1988): Simply run a plain Poisson on “Count-1” and that will work (In STATA: poisson on the corrected count) It is more complex if there is overdispersion though Principles of Econometrics, 3rd Edition Slide16-28 Extensions: truncation and endogenous stratification •Supereasy to deal with a Truncated and Endogenously Stratified Poisson Model . poisson persontripminusone Travelcost educat income, nolog Poisson regression Number of obs LR chi2(3) Prob > chi2 Pseudo R2 Log likelihood = -2474.3262 persontrip~e Coef. Travelcost educat income _cons -1.657986 -.0202144 -.0016285 2.191885 Std. Err. .0620722 .0191574 .0005184 .0792934 z -26.71 -1.06 -3.14 27.64 P>|z| 0.000 0.291 0.002 0.000 = = = = 919 1071.95 0.0000 0.1780 [95% Conf. Interval] -1.779646 -.0577622 -.0026446 2.036473 -1.536327 .0173333 -.0006124 2.347298 Much smaller now estimated Consumer Surplus Principles of Econometrics, 3rd Edition Slide16-29 Extensions: truncation and endogenous stratification •Endogenously Stratified Negative Binomial Model (as shown by Shaw, 1988; Englin and Shonkwiler, 1995): . nbstrat persontrip Travelcost educat income, nolog Negative Binomial with Endogenous Stratification Log likelihood = -1837.3183 Travelcost educat income _cons -1.152915 -.0229483 -.0017368 1.189429 .0695958 .0318753 .0008447 .1561017 -16.57 -0.72 -2.06 7.62 0.000 0.472 0.040 0.000 -1.289321 -.0854228 -.0033923 .8834757 -1.01651 .0395261 -.0000813 1.495383 /lnalpha .092944 .1482435 0.63 0.531 -.197608 .3834959 alpha 1.0974 .1626825 .8206915 1.467406 4.007 0.000 P>|z| 919 283.49 0.0000 Coef. = = z = = = persontrip AIC Statistic Deviance Std. Err. Number of obs Wald chi2(3) Prob > chi2 [95% Conf. Interval] BIC Statistic = Dispersion = -6243.307 0.000 Even after accounting for overdispersion, CS estimate is relatively low Principles of Econometrics, 3rd Edition Slide16-30 Extensions: truncation and endogenous stratification •How do we calculate the pseudo-R2 for this model??? . nbstrat persontrip Travelcost educat income, nolog Negative Binomial with Endogenous Stratification Log likelihood = -1837.3183 Travelcost educat income _cons -1.152915 -.0229483 -.0017368 1.189429 .0695958 .0318753 .0008447 .1561017 -16.57 -0.72 -2.06 7.62 0.000 0.472 0.040 0.000 -1.289321 -.0854228 -.0033923 .8834757 -1.01651 .0395261 -.0000813 1.495383 /lnalpha .092944 .1482435 0.63 0.531 -.197608 .3834959 alpha 1.0974 .1626825 .8206915 1.467406 Principles of Econometrics, 3rd Edition 4.007 0.000 P>|z| 919 283.49 0.0000 Coef. = = z = = = persontrip AIC Statistic Deviance Std. Err. Number of obs Wald chi2(3) Prob > chi2 [95% Conf. Interval] BIC Statistic = Dispersion = -6243.307 0.000 Slide16-31 Extensions: truncation and endogenous stratification •GNBSTRAT will also allow you to model the overdispersion parameter in this case, just as gnbreg did for the plain case •See also Hilbe’s CENPOIS: Stata module to estimate censored maximum likelihood Poisson regression models And in general take a good look at: Hilbe, J. (2011). Negative Binomial Regression, 2nd ed. Cambridge, UK: Cambridge University Press. Principles of Econometrics, 3rd Edition Slide16-32 Extensions: endogeneity • Sample selection models and endogenous switching (ssm and espoisson) •(See also movestay would work for a continuous dependent variable in a similar setting) •Endogenous treatment models •Mtreatnb allows for a multinomial treatment •(from Stata Help: mtreatnb fits a treatment-effects model that considers the effects of an endogenously chosen multinomial treatment on another endogenous count outcome, conditional on two sets of independent variables. The treatment variable is modeled via a multinomial logit and the outcome via a negative binomial regression. The model is fitted using maximum simulated likelihood. The simulator uses Halton sequences.) Principles of Econometrics, 3rd Edition Slide16-33 Extensions: multivariate models • bivariate poisson, and my personal favourite, at least for the name: the SUPREME model •King, G. A seemingly unrelated Poisson regression model Sociological Methods and Research, 1989, 17, 235–255 •bivariate NB (seemingly unrelated negative binomial) •Hausman et al. (1984) and a bit more flexible in: Winkelmann, R. Seemingly unrelated negative binomial regression Oxford Bulletin of Economics and Statistics, 2000, 62, 553-560 Principles of Econometrics, 3rd Edition Slide16-34 Extensions: mixed-effects Poisson • xtmepoisson (from STATA help) fits mixed-effects models for count responses. •Mixed models contain both fixed effects and random effects. The fixed effects are analogous to standard regression coefficients and are estimated directly. •The random effects are not directly estimated (although they may be obtained postestimation) but are summarized according to their estimated variances and covariances. •Random effects may take the form of either random intercepts or random coefficients, and the grouping structure of the data may consist of multiple levels of nested groups Principles of Econometrics, 3rd Edition Slide16-35 Extensions: mixed-effects Poisson • xtmepoisson (from STATA help) fits mixed-effects models for count responses. •The distribution of the random effects is assumed to be Gaussian. The conditional distribution of the response given the random effects is assumed to be Poisson •Because the log likelihood for this model has no closed form, it is approximated by adaptive Gaussian quadrature. Principles of Econometrics, 3rd Edition Slide16-36 Extensions: finite mixture models • •AKA Latent Class Models •Fmm •See examples in the works by Deb and Trivedi for medical care (see Cameron & Trivedi MMA and MUS) •And, again Hilbe (2011) Principles of Econometrics, 3rd Edition Slide16-37 Extensions: panels and pseudo panels • Xtpoisson, xtnb •Xtgee in general Principles of Econometrics, 3rd Edition Slide16-38 NOTE: what is the exposure • Count models often need to deal with the fact that the counts may be measured over different observation periods, which might be of different length (in terms of time or some other relevant dimension) For example, the number of accidents are recorded for 50 different intersections. However, the number of vehicles that pass through the intersections can vary greatly. Five accidents for 30,000 vehicles is very different from five accidents for 1,500 vehicles. Count models account for these differences by including the log of the exposure variable in model with coefficient constrained to be one. The use of exposure is often superior to analyzing rates as response variables as such, because it makes use of the correct probability distributions Principles of Econometrics, 3rd Edition Slide16-39 binary choice models censored data conditional logit count data models feasible generalized least squares Heckit identification problem independence of irrelevant alternatives (IIA) index models individual and alternative specific variables individual specific variables latent variables likelihood function limited dependent variables linear probability model Principles of Econometrics, 3rd Edition logistic random variable logit log-likelihood function marginal effect maximum likelihood estimation multinomial choice models multinomial logit odds ratio ordered choice models ordered probit ordinal variables Poisson random variable Poisson regression model probit selection bias tobit model truncated data Slide 16-40 Hoffmann, 2004 for all topics Long, S. and J. Freese for all topics, most of all for postestimation and reporting tricks Cameron and Trivedi’s book for count data Winkelmann’s 2008 book on count data is free as an ebook from the QEII Hilbe (2011) for NB related models and count data models in general Cameron&Trivedi’s MUS and MMA Greene’s Econometric Analysis Agresti, A. (2001) Categorical Data Analysis (2nd ed). New York: Wiley