20. Count Data - faculty.econ.ucdavis.edu
Transcript of 20. Count Data - faculty.econ.ucdavis.edu
20. Count DataA. Colin Cameron Pravin K. Trivedi Copyright 2006
These slides were prepared in 2002.They cover material similar to Chapter 20 of our subsequent bookMicroeconometrics: Methods and Applications, Cambridge Univer-sity Press, 2005.
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OUTLINE
1. Introduction and Example2. Poisson Regression- MLE and Quasi-MLE
3. Richer Fully Parametric Cross-section models- Negative binomial- Hurdle or two-part and with-zeros- Finite mixtures and latent class
4. Complications- Time Series, Multivariate- Panel (emphasized here)- Sample Selection, Endogeneity- Semiparametric, Bayesian
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1A. INTRODUCTION
� Count data models are for dependent variable y =0, 1, 2, ...
� Two leading examples:� y: Number of doctor visits (usually cross-section)x: health status, age, gender, ....
� y: Number of patent applications (usually panel)x: current and lagged R&D expenditure
� Here emphasize cross-section data and short panels.
� Many approaches and issues are general nonlinear model issues.
� Pecking order: Continuous, tobit, binary/multinomial, duration, counts.
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1B. HEALTH EXAMPLE
� Many surveys such as U.S. National Health Interview Survey (NHIS) measurehealth use as counts as people have better recall of counts than of dollars spent.
� Australian Health Survey 1977-78 has many such measures.e.g. Number of Doctor Visits in past 2 weeks n =5190
# Visits 0 1 2 3 4 5 6 7 8 9Freq 4141 782 174 30 24 9 12 12 5 1Rel Freq .798 .151 .033 .006 .005 .002 .002 .002 .000 .001
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1B. HEALTH EXAMPLE (continued)
� Interest is in role of health insurance on health service use.
� Regressors grouped into four categories:
� Socioeconomic: SEX, AGE, AGESQ, INCOME
� Health insurance status indicators:LEVYPLUS, FREEPOOR, FREEREPA, LEVY (omitted)
� Recent health status measures: ILLNESS, ACTDAYS
� Long-term health status measures:HSCORE, CHCOND1, CHCOND2.
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2. POISSON REGRESSION: SUMMARY
� Poisson regression is straightforward, many packages do poisson regression,and coef�cients are easily interpreted as semi-elasticities.
� Do Poisson rather than OLS with dependent variable y or ln y (with adjust-ment for ln 0) or variance-stabilizing transformations such aspy:
� Poisson MLE consistent provided only that E[yjx] = exp(x0�):But when do Poisson make sure standard errors etc. are robust toV[yjx] 6= E[yjx]:
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2A. POISSON MODEL
� From stochastic process theory, natural model for counts is y � Poisson(�):
� Density f (y) = e���y=y!
� Moments E[y] = � V[y] = �
� Regression model lets the Poisson rate parameter vary across individuals withx in way to ensure � > 0. Exponential function achieves this.
� = E[yjx] = exp(x0�):
� This common starting fully parametric model is too restrictive.
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2B. POISSON MLE
� MLE is straightforward given data independent over i.
� The ML f.o.c. are that the residual is orthogonal to the regressors.As a result consistency does not require Poisson distribution (see below).Xn
i=1(yi � exp(x0i�))xi = 0:
� Detailsf (y) = e���y=y! and � = exp(x0�)) ln f (y) = � exp(x0�)+yx0� � ln y!
) L(�) =Pn
i=1 f� exp(x0i�)+yix0i� � ln yi!g) @L=@� =
Pni=1 f� exp(x0i�)xi + yix0ig :
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2B. POISSON MLE: DOCTOR VISITS REGRESSION
Variable Coeff Robust Average dE[yjx]=dxjst.error* dE[yjx]=dxj at x=�x
ONE �2:224 :190 � �SEX :157 :056� :047 :035AGE (years=100) 1:056 1:001 � �AGESQ �:849 1:078 � �INCOME ($10; 000) �:205 :088� �:062 �:047ILLNESS :187 :018� :056 :043ACTDAYS :127 :005� :038 :029HSCORE :030 :010� :009 :007CHCOND1 (not limit) :114 :066 :026 :026CHCOND2 (not limit) :141 :083 :032 :032
* Note that the usual ML standard errors are not used as explained below.
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2B. POISSON MLE: DOCTOR VISITS REGRESSION (continued)
� Dependent variable is number of doctor visits.Regressors also include LEVYPLUS, FREEPOOR, FREEREPA
� Robust se is standard error assuming V[yjx] = �� E[yjx] (see below)
� Average effect over the sample of change in xj is1
n
Xn
i=1@E[yijxi]=@xij =
1
n
Xn
i=1exp(x0i�)� �j
� Effect of change in xj evaluated at x = �x is@E[yjx]=@xjjx=�x = exp(�x
0�)� �j
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2C. POISSON MODEL: COEFFICIENT INTERPRETATION
� Key result for E[yjx] = exp(x0�) is that:
@E[yjx]@xj
= exp(x0�)� �j = E[yjx]� �j
1. Conditional mean is strictly monotonic increasing (or decreasing) inxj according to the sign of �j.
2. Coef�cients are semi-elasticities:�j is proportionate change in conditional mean when xij changes by one unit.
3. Like all single-index models, if one coef�cient is double another, then effectof one-unit change of associated regressor is double that of other.
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2C. POISSON MODEL: COEFFICIENT INTERPRETATION (cont.)As an example of coef�cient interpretation consider the following.
� DVISITS = number of doctor visits.� ACTDAYS = number of days of reduced activity.
� Poisson regression of DVISITS on ACTDAYS yieldsE[DVISITSjACTDAYS] = exp(�1:529 + 0:158 � ACTDAYS)
� So one more days of reduced activity leads toa 15.8 percent increase in doctor visits (calculus method)or 100� [exp(0:158)� 1] = 17:1 percent increase (noncalculus).
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2D. POISSON QUASI-MLE
� What are properties of Poisson MLE if density is misspeci�ed?
� Poisson MLE is consistent provided only that E[yjx] = exp(x0�).Not a general ML result. Holds in just a few models.
� Still need to correct standard errors if overdispersion (variance > mean) orunderdispersion (variance < mean). Possible methods:� 1. MLE s.e. Assume Poisson, i.e. variance equals mean. Wrong.� 2. GLM Robust s.e. Assume variance = � times mean and calculate �.� 3. White robust. Assume no functional form for the variance.
� Data usually overdispersed, so 1. is wrong.Use 2. or 3. to get robust standard errors.
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2D. POISSON QUASI-MLE: CONSISTENCY
� The MLE f.o.c. are Xn
i=1(yi � exp(x0i�))xi = 0:
� So MLE is consistent ifE[yijxi] = exp(x0i�):
� Thus consistency requires �only� correct conditional mean!� Property shared by generalized linear models based on linear exponentialfamily: normal, binomial, bernoulli, gamma, exponential, Poisson.
� Generalized linear models is standard framework in statistics for nonlinearcross-section regression, including counts.
� Econometrics instead uses either ML/quasi-ML or GMM.
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2D. POISSON QUASI-MLE: ROBUST STANDARD ERRORS
� Correct (robust) standard errors for Poisson quasi-MLE.
� Let �i = exp(x0i�) and �2i = V[yijxi].� Then V[b�] = (Pi �ixix
0i)�1 �P
i �2ixix
0i
�(P
i �ixix0i)�1 :
� If �2i = �i get usual Poisson MLE variance (P
i �ixix0i)�1.
� If �2i = ��i then get � (P
i �ixix0i)�1. b� = (n� k)�1Pi(yi � b�i)2xi
Usually � > 1) Poisson MLE overstates t statistics.� If �2i unspeci�ed then use White robust with �2i replaced by (yi � b�)2.
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2E: POISSON: SUMMARY
� Poisson regression is straightforward, many packages do this and coef�cientsare easily interpreted as semi-elasticities.
� Do Poisson rather than OLS with dependent variable y or ln y (with adjust-ment for ln 0) or variance-stabilizing transformations such aspy:
� Poisson MLE consistent provided only that E[yjx] = exp(x0�):But when do Poisson make sure standard errors etc. are robust toV[yjx] 6= E[yjx]:
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OUTLINE
1. Introduction and Example2. Poisson Regression- MLE and Quasi-MLE
3. Richer Fully Parametric Cross-section models- Negative binomial- Hurdle or two-part and with-zeros- Finite mixtures and latent class
4. Complications- Time Series, Multivariate- Panel (emphasized here)- Sample Selection, Endogeneity- Semiparametric, Bayesian
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3. RICHER PARAMETRIC MODELS
� Data frequently exhibit �non-Poisson� features:� Overdispersion: conditional variance exceeds conditional mean, whereasPoisson imposes equality.
� Excess zeros: higher frequency of zeros than predicted by Poisson withgiven mean.
� Truncation from left: small counts excluded, e.g. 0.� Censoring from right: counts larger than some speci�ed integer aregrouped.
� This provides motivation for richer parametric models than basic Poisson.
� Some still have E[yjx] = exp(x0�). So only ef�ciency gains are issue.Others have different conditional mean in which case usual Poisson QMLE isinconsistent.
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3A. NEGATIVE BINOMIAL MODEL
� Negative binomial (Negbin 2) permits overdispersion.
f (yj�; �) = �(y + ��1)
�(y + 1)�(��1)
���1
��1 + �
���1��
��1 + �
�y:
� Same mean E[yjx] = � = exp(x0�):� Different variance
E[yjx] = � + ��2 = exp(x0�) + �(exp(x0�))2:� Estimate by ML.� In practice little ef�ciency gain over Poisson with robust standard errors.
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3A. NEGATIVE BINOMIAL MODEL: DOCTOR VISITS
Variable Poisson Negbin2 Poisson Negbin2/ Coeff / se Coeff Coeff st.error st.errorONE �2:224 �2:190 :190 :222SEX :157 :217 :056� :066�
AGE (years=100) 1:056 �:216 1:001 1:233AGESQ �:849 :609 1:078 1:380INCOME ($10; 000) �:205 �:142 :088� :098�
ILLNESS :187 :214 :018� :026�
ACTDAYS :127 :144 :005� :008�
HSCORE :030 :038 :010� :014�
CHCOND1 (not limit) :114 :099 :066 :077CHCOND2 (not limit) :141 :190 :083 :095� � 1:077 � :098�
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3B. MIXTURE MODELS
� Mixture motivation for the negative binomial model is to assumeyj� � Poisson (�)
where � = �� is the product of two components:� observed individual heterogeneity � = exp(x0�)� unobserved individual heterogeneity � � Gamma[1; �]
� Integrating out h(yj�) =Rf (yj�; �)g(�)d� gives
yj� � Negative Binomial [�; � + ��2]:
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3B. MIXTURE MODELS (continued)
� A wide range of models, called mixture models, can be generated by specify-ing different distributions of �:e.g. Poisson-Inverse Gaussian.
� Even if no closed form solution can estimate using� numerical integration e.g. Gaussian quadrature, or� monte carlo integration e.g. maximum simulated likelihood
h(yj�) =Zf (yj�; �)g(�)d� ' 1
S
SXs=1
f (yj�; �(s));
where �(s), s = 1; :::; S are S independent draws from g(�) and S !1:
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3C. LEFT-TRUNCATION AT ZERO
� Sampling rule is such that observe only positive counts.
� Untruncated density is f (yjx;�) e.g. Negbin2.
� Truncated density is
f (yjx;�;y � 0) = f (yjx;�)Pr[y � 0jx;�] =
f (yjx;�)[1� f (0jx;�)]:
� Estimate by MLE.� Inconsistent if any aspect of model misspeci�ed.
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3D. RIGHT-CENSORING AT c
� Sampling rule is that observe only 0, 1, 2, ..., c� 1, c or more.
� Uncensored density is f (yjx;�) and cdf is F (yjx;�) e.g. Negbin2.
� Censored density is �f (yjx;�) y � c� 11� F (cjx;�) y = c
� Estimate by MLE.� Inconsistent if any aspect of model misspeci�ed.
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3E. HURDLE MODEL or TWO-PART MODEL
� Suppose process for zeros differs from that for nonzeros.
� Density is
f (yjx1;x1;�1;�2) =
8<: f1(yjx1;�1) y = 01� f1(0jx1;�1)1� f2(0jx2;�2)
� f2(yjx2;�2) y � 1
� Estimate by MLE.� Inconsistent if any aspect of model misspeci�ed.� Hurdles negative binomial often works well.
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3F. WITH-ZEROS MODEL
� Suppose there is extra reason for zeros.
� Density isf (yjx1;x1;�1;�2)
=
�f1(0jx1;�1) + [1� f1(0jx1;�1)]� �f2(0jx2;�2) y = 0[1� f1(0jx1;�1)]� �f2(yjx2;�2) y � 1
� Estimate by MLE.� Inconsistent if any aspect of model misspeci�ed.� Not used much in econometrics.
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3F. FINITE MIXTURES MODEL
� Density is weighted sum of two (or more) densities.
� Density isf (yjx1;x1;�1;�2; �1) = �1f1(yjx1;�1) + (1� �1)f2(yjx2;�2):
� Estimate by MLE.� Inconsistent if any aspect of model misspeci�ed.
� Permits �exible models e.g. bimodal from Poissons.� Can be viewed as a �nite mixture model.
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3G. LATENT CLASS MODEL
� Observation is drawn from one of two (or more) densities, where we don'tknow which density drawn from.
� Let d1 = 1 if type 1 and d1 = 0 otherwiseand d2 = 1 if type 2 and d2 = 0 otherwise
� Density is
f (yjx1;x1;�1;�2; �1; �2) =2Yj=1
[�jfj(yjxj;�j)]dj:
� Estimate by ML using EM algorithm as dj not observed.� Nice interpretation e.g. �sick� type and �healthy� type and people haveprobability of being drawn from either type.
� Similar to unobserved heterogeneity in duration data models.
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3H. MODEL EVALUATION
� Formal tests for overdispersion or underdispersion exist.� Various R-squareds for count data models have been proposed.� For complete data on y the choice between fully parametric approachand moment-based estimators depends on whether want to predict countprobabilities rather than just the mean.
� For fully parametric models� Choice between nested models using likelihood ratio tests.� Choice between non-nested mixture models using Akaike's informationcriterion and extensions.
� Calculate a predicted frequency distribution as the average over observationsof the predicted probabilities for each count. Compare this to the observedfrequency distribution.
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OUTLINE
1. Introduction and Example2. Poisson Regression- MLE and Quasi-MLE
3. Richer Fully Parametric Cross-section models- Negative binomial- Hurdle or two-part and with-zeros- Finite mixtures and latent class
4. Complications- Time Series, Multivariate- Panel (emphasized here)- Sample Selection, Endogeneity- Semiparametric, Bayesian
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4A. TIME SERIES DATA
� Examples are number of strikes and number of trades of a given stock in aone-hour period.
� Many different approaches are possible� Integer valued ARMA: e.g. INAR(1) is yt = � � yt�1 + "twhere � � yt�1 is the number of successes in yt�1 trials, � is probability ofsuccess in one trial, "t is Poisson.
� Autoregressive: e.g. AR(1) is yt � Poisson(�yt�1)with adjustment if yt�1 = 0:
� Serially-correlated error models� State-space models: yt � Poisson(�t) and �t = g(�t�1)� Hidden-Markov models: Different models in different regimes with Markovtransition probabilities.
� Discrete ARMA models.
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4B. MULTIVARIATE DATA
� Example is number of doctor visits and number of hospital stays.� Multivariate Poisson and NB exist but are too restrictive.� GMM approach generalizes SUR to variance a multiple of mean.e.g. E[yjijxj] = exp(x0ji�j) for j = 1; 2,and V[yjijxj] = �j exp(x0ji�j)and Cov[y1i; y2ijxj] = � exp(x01i�1)
1=2 exp(x01i�2)1=2
� Parametric approach induces correlation through common latent variable.e.g. yjijxj �Poisson(exp(x0ji�j + �i)) where �i � g(�).Estimation is by simulated ML if there is no closed form solution.
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4C. PANEL DATA
� Number of patents applications by company in several years.
� Now have (yit;xit), i = 1; :::; n; t = 1; :::; T:
� Consider only short panel where T is small and n!1.
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4C. PANEL REVIEW: LINEAR MODEL FOR PANEL DATA
� Model with individual-speci�c effect is
yit = x0it�+�i + "it:
� Different people have different unobserved intercept �i.
� We want to consistently estimate slope parameters �.
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4C. PANEL REVIEW: LINEAR MODEL - RANDOM EFFECTS
� The approach in most applied statistics� �i is independent of regressors with mean 0 and variance �2�.� Then do feasible GLS to get ef�cient estimates.� Or even do OLS but make sure get correct standard errors that control forwithin-individual clustering.
� Can extend to richer random effects models.
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4C. PANEL REVIEW: LINEAR MODEL - FIXED EFFECTS
� The approach in econometrics.� �i may be correlated with regressors.� e.g. High �i means high unobserved propensity to see doctor.May also mean likely to have generous insurance.
� More fundamental problem: OLS and GLS inconsistent.� Solution is to difference out �i
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4C. PANEL REVIEW: LINEAR MODEL - FIXED EFFECTS (cont)
� Either look at deviations from individual meani.e. Deviation from doctor visits this year from individual's average
yit � �yi = (xit � �xi)0� + ("it � �"i)
� Or look at deviations from last year for individuali.e. Deviation from doctor visits this year from individual's average
yit � yi;t�1 = (xit � xi;t�1)0� + ("it � "i;t�1)
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4C. PANEL DATA: POISSON MODEL
� Poisson panel model isf (yitjxit;�; �i) � Poisson[�it = �i�it]
� Poisson[�it = �i exp(x0it�)]
� Poisson[�it = exp(ln�i + x0it�)]
where �i is unobserved and possibly correlated with xit.
� So the usual mean �it is rescaled by a time invariant multiple �i.
� The two key issues are� correct standard errors allowing for clustering via �i� consistent estimates of � if �i is correlated with xit.
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4C. PANEL DATA: POISSON MOMENT-BASED ESTIMATION
� Assume regressors xit are strictly exogenous, soE[yitjxi1; : : : ;xiT ; �i] = �i�it:
� Average over t for given iE[�yijxi1; : : : ;xiT ; �i] = �i��i
� SoE��yit � (�it=��i)�yi
�jxi1;:::;xiT
�= 0:
� ThusE
�xit
�yit �
�it��i�yi
��= 0:
� b�GMM solves the corresponding sample moment conditionsnXi=1
TXt=1
xit
�yit �
�it��i�yi
�= 0; where �it = exp(x0it�):
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4C. PANEL DATA: POISSON MOMENT BASED-ESTIMATION (contin-ued)
� Similar to linear model except instead of work with the difference (yit� �yi) weconsider the quasi-difference (yit � [�it=��i]�yi).
� Similar qualitative conclusions to linear model� Consistency of b�GMM requires only correct speci�c of the mean!� Consistent for � in either Fixed effects or random effects model.
� Robust inference is based on standard errors that do not require mean =variance.
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4C. PANEL DATA: POISSON FIXED EFFECTS
� Additionally assume the Poisson distribution and both � and �i are parame-ters to be estimated.
� Get Fixed effects MLE of � alone by concentrating out �i.Some math yields b�ML = b�GMM !
� Or do conditional MLE based on the conditional density f (yi1; :::; yitjyi). Thenb�CML = b�GMM !� Robust inference is based on standard errors that do not require mean =variance.
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4C. PANEL DATA: POISSON RANDOM EFFECTS
� Assume the Poisson distribution and �i are i.i.d. gamma distributed withmean 1 and variance 1=�.
� Obtain MLE of � and � yields f.o.c. for � ofnXi=1
TXt=1
xit
�yit � �it
�yi + �=T��i + �=T
�= 0; where �it = exp(x0it�):
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4C. PANEL DATA: OTHER COUNT MODELS
� Fixed and random effects for negative binomial also exist.But ef�ciency gains may not be great.
� For Fixed effects models use preceding moment-based estimator with robuststandard errors.
� For random effects this estimator is also consistent.Or can assume �exible distributions. Even if no closed form solution fordensity can use simulation methods.
� For dynamic models i.e. lagged dependent variable as regressor, instead usethe quasi-difference
E [(yit � (�it�1=�it)yit�1) jyit�1; :::; yi1;xit;:::;xi1] = 0;analogous to working with (yit � yit�1) in the linear model.
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4D. SAMPLE SELECTION
� Suppose process for zeros differs from that for nonzeros.e.g. visit doctor or not differs from process for further visits.
� Generalize the two-part model (or hurdle model) to permit correlation inunobservables across the two parts, similar to generalized tobit. Not donemuch.
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4E. ENDOGENEITY
� Problem isE[(yi � exp(x0i�))jxi] 6= 0:
� Assume existence of instruments zi such thatE[(yi � exp(x0i�))jzi] = 0:
� Then if dim[zi] = dim[�] estimate � by solvingXn
i=1(yi � exp(x0i�))zi = 0:
� And if dim[zi] > dim[�] then use GMM.
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4F. SEMIPARAMETERIC
� Focus on estimating conditional mean.
� Most generally E[yijxi] = g(xi) and estimate function g(�):
� Kernel regression works well in one dimension.� In higher dimensions need more structure.e.g. the single-index form E[yijxi] = g(x0i�):
� Flexible parametric may be an alternative method.e.g. series expansions.
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4G. BAYESIAN
� Poisson with gamma prior yields closed form solution.
� But can now use richer models, e.g. negative binomial and normal prior, andcompute using MCMC methods.
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5. SUMMARY OF COUNT REGRESSION
� For cross-section count data basic approaches are� Moment-based: Let E[yjx] = exp(x0�) and do Poisson QMLE with robusts.e.'s.
� Fully parametric: MLE of richer models than Poisson.
� For panel count data� Specify multiplicative individual speci�c effect.� Moment-based: Estimation based on quasi-differenceE��yit � (�it=��i)�yi
�jxi1;:::;xiT
�= 0 with robust s.e.'s.
� Fully parametric: MLE of richer models than Poisson-gamma.� Use E [(yit � (�it�1=�it)yit�1) jyit�1; :::; yi1;xit;:::;xi1] = 0 if model isdynamic.
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5. SUMMARY OF COUNT REGRESSION (continued)
� The cross-section and static panel count models can be estimated in STATA,LIMDEP and TSP.
� Count methods also exist (though no off-the-shelf programs) for the usualcomplications� Time Series data� Multivariate data� Measurement error� Sample selection� Endogenous regressors� Semiparametric approach� Bayesian approach.
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6. REFERENCES [Recent Examples plus some classics]1. BooksCameron, A.C., and P.K. Trivedi (1998), Regression Analysis of CountData,Econometric Society Monograph No.30, Cambridge University Press.Winkelmann, R. (2000), Econometric Analysis of Count Data, 3rd edition,Springer.2 and 3A. Cross-Section Poisson and Negative BinomialCameron, A.C., and P.K. Trivedi (1986), �Econometric Models Based on CountData: Comparisons and Applications of Some Estimators,� Journal of AppliedEconometrics, 1, 29-53.3E. Hurdle Model or Two-Part Model and With-ZeroesMullahy, J. (1986), �Speci�cation and Testing of Some Modi�ed Count DataModels,� Journal of Econometrics, 33, 341-365.3F. Finite Mixture ModelsDeb, P. and P.K.Trivedi (1997), �Demand for Medical Care by the Elderly: AFinite Mixture Approach,� Journal of Applied Econometrics, 12(3), 313-36.
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3G. Latent Class ModelsDeb, P. and P.K.Trivedi (2001), �The Structure of Demand for Health Care: LatentClass versus Two-part Models,� Journal of Health Economics, forthcoming.4A. Time Series DataBrannas, K. and J. Hellstrom (2001), �Generalized Integer-Valued Autoregres-sion,� Econometric Reviews, 20(4), 425-43.4B. Multivariate DataTrivedi, P.K. and Munkin, M.K. (1999), �Simulated Maximum LikelihoodEstimation of Multivariate Mixed-Poisson Regression Models, with Application�,Econometrics Journal, 2(1), 29-48.4C. Panel DataHausman, J.A., B.H. Hall and Z. Griliches (1984), �Econometric Modelsfor Count Data With an Application to the Patents-R and D Relationship,�Econometrica, 52, 909-938.Blundell, R., R. Grif�th and F. Windmeijer (2002), �Individual Effects andDynamics in Count Data,� Journal of Econometrics, 108, 113-131.
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Windmeijer, F. (2002), �EXPEND, A Gauss programme for non-linear GMMestimation of exponential models with endogenous regressors for cross sectionand panel (dynamic) count data models", cemmap working paper CWP14/02.4D. Sample SelectionWinkelmann, R. (1998), �Count Data Models with Selectivity,� EconometricReviews, 17(4), 339-59.4E. EndogeneityMullahy, J. (1997), �Instrumental Variable Estimation of Poisson RegressionModels: Application to Models of Cigarette Smoking Behavior,� Review ofEconomics and Statistics, 79, 586-593.Windmeijer, F. (2000), �Moment Conditions for Fixed Effects Count Data Modelswith Endogenous Regressors,� Economics Letters, 68(1), 21-24.4G. BayesianChib, S., E.Greenberg and R.Winkelmann (1998), �Posterior Simulation andBayes Factors in Panel Count Data,� Journal of Econometrics, 86(1), 33-54.
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