Day 1B Count Data Regression: Part 2

Transcription

1 Day 1B Count Data Regression: Part 2 A. Colin Cameron Univ. of Calif. - Davis... for Center of Labor Economics Norwegian School of Economics Advanced Microeconometrics Aug 28 - Sept 1, Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School of Economics Aug 28 - Sept Advanced 1, 2017Microeconomet 1 / 47

2 1. ntroduction 1. ntroduction Count data models are for dependent variable y = 0, 1, 2,... Previously considered basic cross-section Poisson, negative binomial, GLM s Richer models: hurdle, zero-in ated,.. Now consider mixture models endogenous regressors panel data. Many of these methods generalize to other nonlinear models. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School of Economics Aug 28 - Sept Advanced 1, 2017Microeconomet 2 / 47

3 1. ntroduction Outline 1 ntroduction 2 Finite Mixture Models 3 Endogenous Regressors 4 Short panel count regression A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School of Economics Aug 28 - Sept Advanced 1, 2017Microeconomet 3 / 47

4 2. Finite mixtures model Finite mixtures model 2. Finite mixtures model Density is weighted sum of two (or more) densities Permits exible models e.g. bimodal from Poissons. For an m-component model f (yjx, θ, π) = m j=1 π jf j (yjx, θ j ), 0 π j 1, m j=1 π j = 1. For a 2-component model f (yjx, θ 1, θ 2, π) = πf 1 (yjx, θ 1 ) + (1 π)f 2 (yjx, θ 2 ) MLE maximizes ln L(θ) = N i=1 ln(πf 1(yjx, θ 1 ) + (1 π)f 2 (yjx, θ 2 )). Can restrict some parameters to be the same. e.g. only intercept di ers EM algorithm often used rather than Newton-Raphson. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School of Economics Aug 28 - Sept Advanced 1, 2017Microeconomet 4 / 47

5 2. Finite mixtures model Finite mixtures model Determining the number of components is a nonstandard inference problem as testing at boundary of parameter space. Simple approach is to use BC or CAC. Or do appropriate bootstrap for the likelihood ratio test. An alternative to MLE is minimum Hellinger distance estimation. " # d(θ) = (p k=0 k ) 1/2 1 1/2 2 N N i=1 f (y i = kjx i, θ, π) where p k equals fraction of observations with y i = k. attraction is that it is less in uenced by outlying observations estimate using an iterative method (HELMX) A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School of Economics Aug 28 - Sept Advanced 1, 2017Microeconomet 5 / 47

6 2. Finite mixtures model Latent class model Latent class model Finite mixture model can be interpreted as a latent class model. There are two types of people (given observables x) e.g. sick type and healthy type there is a probability of being drawn from either type. Similar to unobserved heterogeneity in duration data models. Stata version 15 command fmm 2, vce(robust): poisson docvis $xlist Before version 15 there was user-written command fmm A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School of Economics Aug 28 - Sept Advanced 1, 2017Microeconomet 6 / 47

7 2. Finite mixtures model Latent class model Finite mixture model Number of obs = 3,677 Log pseudolikelihood = Class (base outcome) Robust Coef. Std. Err. z P> z [95% Conf. nterval] 2.Class _cons Class : 1 Response : docvis Model : poisson Robust Coef. Std. Err. z P> z [95% Conf. nterval] docvis private medicaid age age educyr actlim totchr _cons Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School of Economics Aug 28 - Sept Advanced 1, 2017Microeconomet 7 / 47

8 2. Finite mixtures model Latent class model Class : 2 Response : docvis Model : poisson Robust Coef. Std. Err. z P> z [95% Conf. nterval] docvis private medicaid age age educyr actlim totchr _cons Log-likelihood comparison across models: Poisson ; 2-component Poisson ; 2-component NB ; 2-component NB Last is almost exactly same as hurdle NB and ZNB (-10493).. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School of Economics Aug 28 - Sept Advanced 1, 2017Microeconomet 8 / 47

9 2. Finite mixtures model Latent class model Component 1 occurs with probability 0.65 and is low use. Component 2 occurs with probability 0.35 and is high use.. * Obtain latent class probabilities. estat lcprob Latent class marginal probabilities Number of obs = 3,677 Delta method Margin Std. Err. [95% Conf. nterval] Class * Obtain predicted mean for each individual by component. predict yfit* (option mu assumed). summarize yfit1 yfit2 Variable Obs Mean Std. Dev. Min Max yfit1 3, yfit2 3, A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School of Economics Aug 28 - Sept Advanced 1, 2017Microeconomet 9 / 47

10 2. Finite mixtures model Latent class model Density Predicted mean (# doctor visits in class 1.Class) Density Predicted mean (# doctor visits in class 2.Class). Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 10 / 47

11 3. Endogenous regressors Linear models 3. Endogenous regressor: linear model Begin with review of the linear regression model: y i = x 0 i β + u i. f regressors are correlated with error then OLS is inconsistent. Reason: OLS b β = (X 0 X) 1 X 0 y = β + (X 0 X) 1 X 0 u so plim b β = β + (plim N 1 X 0 X) 1 plim N 1 X 0 u 6= β if plim N 1 X 0 u 6= 0. Solution: Assume the existence of an instrument z where changes in z are associated with changes in x but changes in z do not led to change in y (aside from indirectly via x) z! x! y " % u Leads to instrumental variables (V) estimator and two-stage least squares (2SLS) estimator. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 11 / 47

12 3. Endogenous regressors Linear models Formally key assumption is: E[u i jz i ] = 0 Just-identi ed case (# instruments = # endogenous) bβ V = (Z 0 X) 1 Z 0 y. Over-identi ed case (# instruments > # endogenous) bβ 2SLS = [X 0 Z(Z 0 Z) 1 Z 0 X] 1 X 0 Z(Z 0 Z) 1 Z 0 y. Example: log-earnings (y) regressed on years of school (x) ability is an omitted regressor so part of error (u) and clearly correlated with x instrument z is correlated with years of school but not directly with earnings example of z may be distance from school or college. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 12 / 47

13 3. Endogenous regressors Linear models Several nterpretations of Linear V/2SLS 1. Method of Moments E[u i jz i ] = 0 ) E[z i u i ] = 0 ) E[z i (y i xi 0 β)] = 0. V solves corresponding sample moment condition N i =1 z i (y i xi 0 β) = 0 And if overidenti ed do generalized method of moments (GMM). 2. Control Function add predicted residual to control for endogeneity. OLS with additional regressor the residual from rst-stage OLS regression of y 2 on all exogenous regressors. 3. Two-Stage Least Squares OLS with endogenous regressor y 2 replaced by its predicted value by 2 from rst-stage OLS regression of y 2 on all exogenous regressors. Only methods 1 and 3 extend to nonlinear models such as Poisson. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 13 / 47

14 3. Endogenous regressors Nonlinear GMM Poisson endogenous method 1: nonlinear GMM Problem is E[(y i exp(x 0 i β))jx i ] 6= 0. Assume existence of instruments z i such that E[(y i exp(x 0 i β))jz i ] = 0 ) E[z i (y i exp(x 0 i β))] = 0 Just-identi ed case: b β MM solves n i=1 (y i exp(x 0 i β))z i = 0. Over-identi ed case b β GMM minimizes n i=1 (y i exp(x 0 i β))z i 0 W n i=1 (y i exp(x 0 i β))z i usually W = (Z 0 Z) 1 (called nonlinear 2SLS). A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 14 / 47

15 3. Endogenous regressors Nonlinear GMM Literature exists on weighting matrix W and whether to use di erent moment condition such as (yi exp(xi 0 E β) exp(xi 0β) z i = 0 Mullahy (1997), Windmeijer and Santos Silva (1997), Windmeijer (2008). We use the simpler n i=1(y i exp(x 0 i β))z i = 0. The following output was obtained using own code written using Mata A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 15 / 47

16 3. Endogenous regressors Nonlinear GMM NL2SLS: Example with private (private insurance) endogenous nstruments are income and ssiratio (soc sec income / total income) Estimate by nonlinear 2SLS:. ereturn display Coef. Std. Err. z P> z [95% Conf. nterval] private medicaid age age educyr actlim totchr cons private was (0.036) and is now (0.340) standard errors much larger with V Also medicaid changes a lot. Others change little. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 16 / 47

17 3. Endogenous regressors Control function approach Poisson endogenous method 2: control function Add error in Poisson model (allows for overdispersion and endogeneity) Structural eqn: y 1i Poisson[µ i = exp(β 1 y 2i + z1i 0 β 2 + u 1i )] Reduced-form eqn: y 2i = γ 1 z 2i + z1i 0 γ 2 + v 2i Error model: u 1i = αv 2i + ε i Then µ i jy 2i, z 1i, v 2i, ε i = exp(β 1 y 2i + z 0 1i β 2 + αv 2i + ε i ) = exp(ε i ) exp(β 1 y 2i + z 0 1i β 2 + αv 2i ) µ i jy 2i, z 1i, v 2i = E[exp(ε i )] exp(β 1 y 2i + z 0 1i β 2 + αv 2i ) = exp(β 1 y 2i + z 0 1i β 2 + αv 2i ) where if ε i is i.i.d. then E[exp(ε i )] is a constant that is absorbed in β 2. Control function approach 1. OLS of y 2 on z 2 and z 1 gives residual bv 2i = y 2i bγ 1 z 1i z2i 0 bγ Poisson of y 1i on y 2i, z 1i and bv 2i gives V estimate. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 17 / 47

18 3. Endogenous regressors Control function approach Control function approach for same example. First-stage: OLS for reduced form. global xlist2 medicaid age age2 educyr actlim totchr. regress private $xlist2 income ssiratio, vce(robust) Linear regression Number of obs = 3677 F( 8, 3668) = Prob > F = R squared = Root MSE = Robust private Coef. Std. Err. t P> t [95% Conf. nterval] medicaid age age educyr actlim totchr income ssiratio _cons A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 18 / 47

19 3. Endogenous regressors Control function approach Second stage: Poisson with rst-stage predicted residual as regressor. predict lpuhat, residual.. * Second stage Poisson with robust SEs. poisson docvis private $xlist2 lpuhat, vce(robust) nolog Poisson regression Number of obs = 3677 Wald chi2(8) = Prob > chi2 = Log pseudolikelihood = Pseudo R2 = Robust docvis Coef. Std. Err. z P> z [95% Conf. nterval] private medicaid age age educyr actlim totchr lpuhat _cons private is (0.245) compared to (0.340) for NL2SLS A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 19 / 47

20 Here little change in standard errors. A. Colin Cameron Univ. of Calif. - Davis... for Center of Labor Economics Norwegian School of Economics Advanced Microeconomet Count Regression: Part 2 Aug 28 - Sept 1, / Endogenous regressors Control function approach Should bootstrap to get correct s.e. s (lpuhat is a generated regressor). * Program and bootstrap for Poisson two step estimator. program endogtwostep, eclass 1. version tempname b 3. capture drop lpuhat2 4. regress private $xlist2 income ssiratio 5. predict lpuhat2, residual 6. poisson docvis private $xlist2 lpuhat2 7. matrix `b' = e(b) 8. ereturn post `b' 9. end. bootstrap _b, reps(400) seed(10101) nodots nowarn: endogtwostep Bootstrap results Number of obs = 3,677 Replications = 400 Observed Bootstrap Normal based Coef. Std. Err. z P> z [95% Conf. nterval] private medicaid age age educyr actlim totchr lpuhat _cons

21 3. Endogenous regressors Structural approach Poisson endogenous method 3: structural approach Example with binary endogenous regressor y 2i is Outcome eqn: y 1i Poisson[µ i = exp(β 1 y 2i + z1i 0 β 2 + δ 1u i )] Participation eqn: Pr[y 2i = 1] = Λ(z2i 0 β 2 + λ 1u i ) Error model: u i N [0, 1] Estimate using Stata 15 command gsem global xlist2 medicaid age age2 educyr actlim totchr gsem (docvis <- private $xlist2 L, poisson) /// (private <- $xlist2 income ssiratio L, logit), var(l@1) Can also extend the two-part (hurdle) model to incorporate selection This allows for correlation due to unobservables between process for y = 0 or not and process for positives. Terza (1998). A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 21 / 47

22 4. Panel data Overview 4. Panel data: linear model review Focus is on model with individual-speci c e ect y it = xit 0 β + α i + ε it, i = 1,..., N, t = 1,..., T. So di erent people have di erent unobserved intercept α i. Goal is to consistently estimate slope parameters β. Focus on short panel with N! and T small. f α i is a random e ect then pooled OLS with cluster-robust s.e. s is okay random e ects GLS/MLE may be more e cient f α i is a xed e ect, so Cov[α i, x it ] 6= 0 pooled OLS and random e ects GLS/MLE are inconsistent need xed e ects: regress (y it ȳ i ) on (x it x i ) or rst di erences: regress y it on x it. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 22 / 47

23 4. Panel data Overview Panel data: poisson model review These results carry over qualitatively to Poisson panel regression. The Poisson individual e ects model speci es conditional mean E[y it jx it, β, α i ] = exp(δ i + xit 0 β) = α i exp(xit 0 β) The e ect is multiplicative rather than additive. f α i is a random e ect then pooled Poisson with cluster-robust s.e. s is okay random e ects Poisson GLS/MLE may be more e cient f α i is a xed e ect, so Cov[α i, x it ] 6= 0 pooled Poisson and random e ects Poisson are inconsistent need xed e ects explained below or quasi rst di erences A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 23 / 47

24 4. Panel data Data example: Doctor visits (RAND) Data example: Doctor visits (RAND) Data from RAND health insurance experiment. y is number of doctor visits.. use mus18data.dta, clear. describe mdu lcoins ndisease female age lfam child id year storage display value variable name type format label variable label mdu float %9.0g number face to fact md visits lcoins float %9.0g log(coinsurance+1) ndisease float %9.0g count of chronic diseases ba female float %9.0g female age float %9.0g age that year lfam float %9.0g log of family size child float %9.0g child id float %9.0g person id, leading digit is sit year float %9.0g study year A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 24 / 47

25 4. Panel data Data example: Doctor visits (RAND) Dependent variable mdu is very overdispersed: bv[y] = ' 7 ȳ.. summarize mdu lcoins ndisease female age lfam child id year Variable Obs Mean Std. Dev. Min Max mdu lcoins ndisease female age lfam child id year A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 25 / 47

26 4. Panel data Data example: Doctor visits (RAND) Declare as panel data xtset id year Panel is unbalanced. Most are in for 3 years or 5 years.. xtdescribe id: , ,..., n = 5908 year: 1, 2,..., 5 T = 5 Delta(year) = 1 unit Span(year) = 5 periods (id*year uniquely identifies each observation) Distribution of T_i: min 5% 25% 50% 75% 95% max A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 26 / 47

27 4. Panel data Data example: Doctor visits (RAND) Time series plots for the rst 20 individuals. Much serial correlation. quietly xtline mdu if _n<=85, overlay legend(off) number face to fact md visits study year A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 27 / 47

28 4. Panel data Data example: Doctor visits (RAND) For mdu both within and between variation are important.. * Panel summary of dependent variable. xtsum mdu Variable Mean Std. Dev. Min Max Observations mdu overall N = between n = 5908 within T bar = Only time-varying regressors are age, lfam and child and these have mainly between variation this will make within or xed estimator very imprecise. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 28 / 47

29 4. Panel data Panel count estimators: Summary Panel Poisson Consider four panel Poisson estimators Pooled Poisson with cluster-robust errors Population-averaged Poisson (GEE) Poisson random e ects (gamma and normal) Poisson xed e ects Can additionally apply most of these to negative binomial. And can extend FE to dynamic panel Poisson where y i,t 1 is a regressor. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 29 / 47

30 4. Panel data Pooled Poisson Panel Poisson method 1: pooled Poisson Specify y it jx it, β Poisson[exp(xit 0 β)] Pooled Poisson of y it on intercept and x it gives consistent β. But get cluster-robust standard errors where cluster on the individual. These control for both overdispersion and correlation over t for given i. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 30 / 47

31 4. Panel data Pooled Poisson Pooled Poisson with cluster-robust standard errors. * Pooled Poisson estimator with cluster robust standard errors. poisson mdu lcoins ndisease female age lfam child, vce(cluster id) teration 0: log pseudolikelihood = teration 1: log pseudolikelihood = teration 2: log pseudolikelihood = Poisson regression Number of obs = Wald chi2(6) = Prob > chi2 = Log pseudolikelihood = Pseudo R2 = (Std. Err. adjusted for 5908 clusters in id) Robust mdu Coef. Std. Err. z P> z [95% Conf. nterval] lcoins ndisease female age lfam child _cons By comparison, the default (non cluster-robust) s.e. s are 1/4 as large. ) The default (non cluster-robust) t-statistics are 4 times as large!! A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 31 / 47

32 4. Panel data Population-averaged Poisson (PA or GEE) Panel Poisson method 2: population-averaged This method the standard method in statistics. Again assume there is no xed e ects problem. But want more e cient estimator than pooled Poisson. Assume that for the i th observation moments are like for GLM Poisson E[y it jx it ] = exp(x 0 it β) V[y it jx it ] = φ exp(x 0 it β). Assume constant correlation between y it and y is Cov[y it, y is jx it, x is ] = ρ. Estimate by the generalized estimating equations (GEE) estimator or population-averaged estimator (PA) of Liang and Zeger (1986). This is essentially nonlinear feasible GLS Get a cluster-robust estimate of the variance matrix that is correct even if Cov[y it, y is jx it, x is ] 6= ρ. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 32 / 47

33 4. Panel data Population-averaged Poisson (PA or GEE) Population-averaged Poisson with unstructured correlation xtpoisson mdu lcoins ndisease female age lfam child, /// pa corr(unstr) vce(robust) GEE population averaged model Number of obs = Group and time vars: id year Number of groups = 5908 Link: log Obs per group: min = 1 Family: Poisson avg = 3.4 Correlation: unstructured max = 5 Wald chi2(6) = Scale parameter: 1 Prob > chi2 = (Std. Err. adjusted for clustering on id) Semi robust mdu Coef. Std. Err. z P> z [95% Conf. nterval] lcoins ndisease female age lfam child _cons Generally s.e. s are within 10% of pooled Poisson cluster-robust s.e. s. The default (non cluster-robust) t-statistics are times larger, because do not control for overdispersion. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 33 / 47

34 4. Panel data Population-averaged Poisson (PA or GEE) The correlations Cor[y it, y is jx i ] for PA (unstructured) are not equal. But they are not declining as fast as AR(1).. matrix list e(r) symmetric e(r)[5,5] c1 c2 c3 c4 c5 r1 1 r r r r A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 34 / 47

35 4. Panel data Poisson random e ects Panel Poisson method 3: random e ects This method is used more in econometrics. No FE but want more e cient estimator than pooled Poisson. Poisson random e ects model is y it jx it, β, α i Poiss[α i exp(xit 0 β)] Poiss[exp(ln α i + xit 0 β)] where α i is unobserved but is not correlated with x it. RE estimator 1: Assume α i is Gamma[1, η] distributed closed-form solution exists (negative binomial) E[y it jx it, β] = exp(xit 0 β) RE estimator 2: Assume ln α i is N [0, σ 2 ε ] distributed closed-form solution does not exist (one-dimensional integral) can extend to slope coe cients (higher-dimensional integral) E[y it jx it, β] 6= exp(xit 0 β)!!! Now di erent conditional mean. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 35 / 47

36 4. Panel data Poisson random e ects Poisson random e ects (gamma) with panel bootstrap se s xtpoisson mdu lcoins ndisease female age lfam child, re vce(cluster id) Random effects Poisson regression Number of obs = 20,186 Group variable: id Number of groups = 5,908 Random effects u_i ~ Gamma Obs per group: min = 1 avg = 3.4 max = 5 Wald chi2(6) = Prob > chi2 = Log pseudolikelihood = (Std. Err. adjusted for 5,908 clusters in id) Robust Std. Err. z P> z [95% Conf. nterval] mdu Coef. lcoins ndisease female age lfam child _cons /lnalpha alpha of alpha=0: chibar2(01) = 3.9e+04 Prob >= chibar2 = LR test The default (non cluster-robust) t-statistics are 2.5 times larger because default do not control for overdispersion. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 36 / 47

37 4. Panel data Poisson xed e ects Panel Poisson method 4: xed e ects Poisson xed e ects model is y it jx it, β, α i Poiss[α i exp(x 0 it β)] Poiss[exp(ln α i + x 0 it β)] where α i is unobserved and is possibly correlated with x it. n theory need to estimate β and α 1,..., α N. potential incidental parameters problem N + K parameters and NT observations with N!. but no problem as can eliminate α i. Eliminate α i by quasi-di erencing as follows E[y it jx i1,..., x it, α i ] = α i λ it λ it = exp(x 0 it β) ) E[ȳ i jx i1,..., x it, α i ] = α i λ i λ i = T 1 t λ it ) E y it (λ it / λ i )ȳ i jxi1,..., x it = 0 The rst line assumes regressors x it are strictly exogenous. This is stronger than weakly exogenous. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 37 / 47

38 4. Panel data Poisson xed e ects The nal result implies E x it y it λ it λ i ȳ i = 0. Poisson xed e ects estimator solves the corresponding sample moment conditions N i=1 T t=1 x λ it it y it ȳ i = 0, where λ it = exp(xit λ 0 β). i Get cluster-robust standard errors Bootstrap xtpoisson, re or use add-on xtpqml. Consistency requires E[y it jx i1,..., x it, α i ] = α i exp(xit 0 β). A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 38 / 47

39 4. Panel data Poisson xed e ects Poisson xed e ects with panel bootstrap se s xtpoisson mdu lcoins ndisease female age lfam child, fe vce(robust) Conditional fixed effects Poisson regression Number of obs = 17,791 Group variable: id Number of groups = 4,977 Obs per group: min = 2 avg = 3.6 max = 5 Wald chi2(3) = 4.58 Log pseudolikelihood = Prob > chi2 = (Std. Err. adjusted for clustering on id) Robust mdu Coef. Std. Err. z P> z [95% Conf. nterval] age lfam child The default (non cluster-robust) t-statistics are 2 times larger. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 39 / 47

40 4. Panel data Poisson xed e ects Remarkably the Poisson FE estimator for β can also be obtained in the following ways under fully parametric assumption that y it jx it, β, α i Poiss[α i exp(x 0 it β)] 1. Obtain the MLE of β and α 1,..., α N. 2. Obtain the conditional MLE based on the conditional density f (y i1,..., y it jȳ i, x i1,..., x it, β, α i ) = T t=1 f (y it jx it, β, α i ) f (ȳ i jx i1,..., x it, β, α i ) But should then use cluster-robust standard errors and not default ML se s. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 40 / 47

41 4. Panel data Poisson xed e ects Strength of xed e ects versus random e ects Allows α i to be correlated with x it. So consistent estimates if regressors are correlated with the error provided regressors are correlated only with the time-invariant component of the error An alternative to V to get causal estimates. Limitations: Coe cients of time-invariant regressors are not identi ed For identi ed regressors standard errors can be much larger Marginal e ect in a nonlinear model depend on α i and α i is unknown. ME j = E[y it ]/ x it,j = α i exp(x 0 it β)β j A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 41 / 47

42 4. Panel data Estimator comparison Panel Poisson: estimator comparison Compare following estimators pooled Poisson with cluster-robust s.e. s pooled population averaged Poisson with unstructured correlations and cluster-robust s.e. s random e ects Poisson with gamma random e ect and cluster-robust s.e. s random e ects Poisson with normal random e ect and default s.e. s xed e ects Poisson and cluster-robust s.e. s Find that similar results for all but FE note that these data are not good to illustrate FE as regressors have little within variation. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 42 / 47

43 4. Panel data Estimator comparison * Comparison of Poisson panel estimators * using panel-robust standard errors global xlist lcoins ndisease female age lfam child quietly poisson mdu $xlist, vce(cluster id) estimates store POOLED quietly xtpoisson mdu $xlist, pa corr(unstr) vce(robust) estimates store POPAVE quietly xtpoisson mdu $xlist, re vce(cluster id) estimates store RE_GAMMA quietly xtpoisson mdu $xlist, re normal vce(cluster id) estimates store RE_NORMAL quietly xtpoisson mdu $xlist, fe vce(robust) estimates store FXED estimates table POOLED POPAVE RE_GAMMA RE_NORMAL FXED, /// equations(1) b(%8.4f) se stats(n ll) stfmt(%8.0f) A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 43 / 47

44 4. Panel data Estimator comparison Comparison of di erent Poisson panel estimators with panel-robust s.e. s Variable POOLED POPAVE RE_GAMMA RE_NOR~L FXED #1 lcoins ndisease female age lfam child _cons lnalpha _cons lnsig2u _cons Statistics N ll legend: b/se A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 44 / 47

45 4. Panel data Panel negative binomial Panel negative binomial Fixed and random e ects for negative binomial also exist. But e ciency gains may not be great simplest to work with Poisson but make sure get cluster-robust standard errors to control for overdispersion. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 45 / 47

46 4. Panel data Dynamic panel Dynamic panels Extend Arellano-Bond setup for linear model to nonlinear model. Sequential moment conditions E[y it jy it 1,..., y i1, x it,..., x i1, α i ] = α i λ it. Obvious model is α i λ it = α i exp(ρy i,t 1 + xit 0 β) but this is potentially explosive. May be better to use linear feedback: α i λ it = α i fρy i,t 1 + exp(xit 0 β)g Then E[y it (λ i,t 1 /λ it )y it 1 jy it 1,..., y i1, x it,..., x i1 ] = 0 So can do GMM based on moment condition E[z it fy it (λ i,t 1 /λ it )y it 1 g] = 0 where z it = (y it 1, x it ) for example. These can be coded up in Stata using the gmm command. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 46 / 47

47 4. Panel data Dynamic panel References n addition to those given in preceding set of slides. Finite Mixture Models Deb, P. and P.K. Trivedi (2022), The structure of demand for health care: latent class versus two-part models, Journal of health economics, Panel Data Cameron, A.C., and P.K. Trivedi (2015), Count Panel Data, in B. Baltagi ed., Oxford Handbook of Panel Data, Oxford: Oxford University Press, pp A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 2 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 47 / 47