In this note we study the conditions under which leading models for underreported counts are identified. In particular, we highlight a peculiar identification problem that afflicts two of the most popular models in this class

Papadopoulos, Georgios

Santos Silva, Joao M C

University of Essex Research Repository

ISSN 1755-5361                   Discussion Paper Series              Identification issues in models for underreported counts  Georgios Papadopoulos and J. M. C. Santos Silva     Note : The Discussion Papers in this series are prepared by members of the Department of Economics, University of Essex, for private circulation to interested readers. They often represent preliminary reports on work in progress and should therefore be neither quoted nor referred to in published work without the written consent of the author.                                 University of Essex           Department of Economics    No. 657 July 2008 Identification issues in models for underreportedcounts∗Georgios PapadopoulosDepartment of Economics, University of EssexJ. M. C. Santos SilvaDepartment of Economics, University of Essex and CEMAPREJuly 25, 2008AbstractIn this note we study the conditions under which leading models for underre-ported counts are identified. In particular, we highlight a peculiar identificationproblem that aﬄicts two of the most popular models in this class.JEL classification code: C13, C25.Key words: Poisson-logit; Restrictions, Stopped-sum distributions.∗We are grateful to Bill Greene, Pravin Trivedi and Rainer Winkelmann for helpful comments anddiscussions. The usual disclaimer applies. The data used in the illustration in Section 3 were kindlymade available by Rainer Winkelmann and are from the public use version of the German Socio-Economic Panel Study. We thank the Deutsches Institut für Wirtschaftsforschung for authorizingits use. Santos Silva gratefully acknowledges the partial financial support from Fundação para aCiência e Tecnologia, program POCTI, partially funded by FEDER.Address for correspondence: Department of Economics, University of Essex, Wivenhoe Park,Colchester CO4 3SQ, UK. E-mail: gpapad@essex.ac.uk (Papadopoulos), jmcss@essex.ac.uk (SantosSilva).11. INTRODUCTIONUnderreporting is likely to be pervasive in survey data. Therefore, models that ac-count for this type of measurement error are possibly useful in many applications. Forthe particular case of count data, models accounting for underreporting are describedin the monographs of Cameron and Trivedi (1998) and Winkelmann (2008), and areimplemented in popular statistical packages (e.g., Econometric Software, Inc., 2007).In this note we study the conditions under which leading models for underreportedcounts are identified, and highlight a peculiar identification problem that aﬄicts twoof the most popular specifications.The reminder of this note is organized as follows. Section 2 presents the identifica-tion results, Section 3 discusses the practical consequences of the main results and,finally, Section 4 contains brief concluding remarks.2. RESULTSModels for underreported counts are based on the assumption that the number ofoccurrences reported in a given period by individual i is given byyi =y∗iXj=1bij, (1)where y∗i is the total (unobserved) number of occurrences and bij is a Bernoulli randomvariable that takes the value 1 when the jth occurrence is reported. Throughout,we assume a regression framework in which the object of interest is the conditionaldistribution of y∗i , given the set of regressors xi. For convenience, xi is written asxi = (x1i, x2i), where x1i and x2i may be identical, overlapping or disjoint.Perhaps the most popular specification for underreported counts is the Poisson-logit model introduced by Winkelmann and Zimmermann (1993),1 which is obtained1See also Mukhopadhyay and Trivedi (1995).2as a special case of (1) when it is assumed that, conditionally on xi: y∗i followsa Poisson distribution with parameter λi = exp (x01iβ), Pr (bij = 1|xi) = Λi =exp (x02iγ) / (1 + exp (x02iγ)), and y∗i and bij are independent.2 Under these assump-tions, it is possible to show that, conditionally on xi, yi follows a Poisson law withparameter μi = λiΛi.3Although Poisson regression is generally well behaved, the Poisson-logit is a double-index model whose likelihod function may have multiple maxima. Moreover, identi-fication of θ = (β0, γ0)0 is problematic, even under the maintained strong parametricassumptions. For example, θ is not identified when x2i is a subset of x1i and Λi is con-stant (see, Cameron and Trivedi, 1998, or Winkelmann, 2008). Even in less extremesituations, identification of the Poisson-logit model is aﬄicted by a subtle problem.Indeed, it is trivial to show thatμi ≡ exp (x01iβ)exp (x02iγ)1 + exp (x02iγ)= exp (x01iβ + x02iγ)exp (−x02iγ)1 + exp (−x02iγ)≡ μai . (2)Since the likelihood function of a Poisson regression model depends on xi onlythrough the conditional mean, (2) implies that there are two Poisson-logit regressionmodels with conditional means μi ≡ λiΛi and μai ≡ λi exp (x02iγ) (1− Λi), which willlead exactly to the same value of the likelihood function. Therefore, unless appropriaterestrictions are imposed on θ, these two models with very diﬀerent specifications ofE [yi|xi] are observationally equivalent.42If y∗i and bij are conditionally independent, (1) implies that yi has a stopped-sum distribution(see Johnson, Kemp and Kotz, 2005, Ch. 9).3This result can be traced back to Catcheside (1948).4Zero inflated models of the type introduced by Mullahy (1986) and Lambert (1992) are oftenspecified with a conditional mean of the form μi ≡ λiΛi, where Λi represents the probability of zero-inflation. The likelihood function for zero-inflated models, however, depends separately on μi andΛi, and therefore the identification problem discussed here does not arise if the model is estimatedby maximum-likelihood.3To explore the consequences of (2), it is convenient to consider the case in whichxi = x1i = x2i, which leads toexp (x0iβ)exp (x0iγ)1 + exp (x0iγ)= exp (x0i (β + γ))exp (−x0iγ)1 + exp (−x0iγ). (3)Now, identification can be studied by analyzing the non-sample information neededto distinguish θ = (β0, γ0)0 from θa = (β0 + γ0,−γ0)0.Starting with the case in which the researcher has information on the logit part ofthe model, it is obvious that θ is identified when the sign of at least one (non-zero)element of γ is known a priori. Alternatively, when some elements of γ are known tobe zero, although there are still two observationally equivalent models, it is possibleto identify the elements of β corresponding to the zeros in γ because in this case therelevant elements of β and βa = β + γ are identical. Turning now to the possiblenon-sample information on β, identification of θ requires the knowledge of at leastone element of this vector, for example as a result of an exclusion restriction. Whenthat is the case, μi and μai can be distinguished because μai is not consistent with thenon-sample information. The practical consequences of these results will be exploredbelow.Although consistency of the Poisson-logit estimator only requires E [yi|xi] = μi, it ispossible to generalize this model to account for possible overdispersion. The standardway of doing this is to assume that, conditionally on xi and on an unobservable indi-vidual eﬀect εi, y∗i follows a Poisson distribution with parameter λi = exp (x01iβ + εi),Pr (bij = 1|xi, εi) = exp (x02iγ) / (1 + exp (x02iγ)), and y∗i and bij are independent. Now,conditionally on xi and εi, yi has a Poisson distribution with parameter λiΛi exp (εi),where λi and Λi are defined as before.Under the usual assumption that exp (εi) has a gamma distribution with unit meanand variance αi, the distribution of yi, conditionally on xi only, is negative-binomialwith mean μi = λiΛi and variance ωi = μi + αiμ2i .4The leading member of this family of models is the NegBin2-logit, for which αi isconstant. As in the Poisson-logit, the likelihood function of the NegBin2-logit dependson xi only through μi (see Cameron and Trivedi, 1998). Therefore identification of θrequires exactly the same conditions established for the Poisson-logit model.When αi is allowed to depend on the regressors, identification may be easier. Inparticular, identification is possible when αi is a function of λi. For example, assumingthat αi = δ/λi, we obtain a NegBin1-logit model with mean μi = λiΛi and varianceωi = μi + δλiΛ2. In this case, due to additional structure imposed on the variance,the likelihood function does not depend on xi only through μi and therefore the resultin (2) does not imply the existence of an identification problem, even when x1i = x2i.It should be noted, however, that in this case identification of the conditional meanof yi is achieved by assuming a parametric specification for the conditional variance.This has obvious consequences for the robustness of the estimator.An alternative way of accounting for unobserved heterogeneity in count data isto use of finite-mixture models, which in some cases have a natural and attractiveinterpretation (see, e.g., Deb and Trivedi, 1997 and 2002). Although we are not awareof any model based on finite-mixtures that also accounts for underreporting, it is easyto see that, if the probability of underreporting is allowed to vary across classes, theconditions for the identification of the mixture model will be exactly the same thatare required for the identification of the models for each class. Therefore, the resultspresented above can easily be extended to this class of models.Of course, strictly speaking, the identification problems of the Poisson-logit andNegBin2-logit do not extend to models where Pr (bij = 1|xi) is not of the logit form,like in a Poisson-probit model or in the specification suggested byWinkelmann (1998).In spite of this, the findings in the next section suggest that the identification resultspresented here are likely to be useful at least for some of these models.53. CONSEQUENCES FOR PRACTITIONERSAs noted before, the likelihood function of a Poisson-logit model may have multiplemaxima. Moreover, the results in Section 2 suggest that very diﬀerent sets of parame-ters may lead to very similar values of the likelihood function. This will certainly bethe case when identification hinges on an exclusion restriction that is “weak” in thesense that either the regressors excluded from x1 are highly colinear with remainingelements of this vector, or the elements of γ corresponding to the regressors excludedfrom x1 are small. Naturally, the same result is true for the NegBin2-logit model.If the practitioner is unaware of this, he may be puzzled to find that his estimatedmodel fits the data quite well, despite having estimated parameters with implausiblevalues or “wrong signs”. To exemplify this situation, we reconsider a well knownempirical illustration.Winkelmann (2008) uses a Poisson-logit to model the number of job oﬀers when onlydata on voluntary job changes is available. The partial observability resulting fromthe fact that only the accepted job oﬀers are counted makes the use of the Poisson-logit potentially appropriate.5 The data used in the illustration is a subsample of theGerman Socio-Economic Panel consisting of 1962 males workers aged between 25 and50 in 1974. Table 1 presents two sets of results from the estimation of a Poisson-logitmodel using the sample and specification considered in Winkelmann (2008).6 Thenames of regressors in Table 1 are self-explanatory, but a complete description of theregressors and further details on the data can be found in Winkelmann (2008).The first two columns in Table 1, labeled β˜ and γ˜, coincide with the estimatesgiven in Winkelmann (2008). The final two columns, labeled βˆ and γˆ, correspond to5It should be made clear that the results presented in Winkelmann (2008) are merely illustrativeand that the same data set is used to exemplify the estimation of many diﬀerent count data models.6Standard errors computed from the observed information matrix are given in parenthesis.6the estimates obtained when using as starting values β0 = β˜ + γ˜ and γ0 = −γ˜, andimposing the that the element of β0 corresponding to Single is zero.Table 1: Poisson-logit regressions for number of job changesβ˜ γ˜ βˆ γˆIntercept 0.81208 0 0.84395 0(0.21841) – (0.22583) –Education −0.32237 3.73211 3.25160 −3.58186(0.15870) (2.05257) (2.04117) (1.98077)Experience −0.66889 −6.04420 −6.40329 5.72807(0.13372) (3.89619) (3.70472) (3.67749)Experience2 0.07179 3.32178 3.19228 −3.12084(0.04796) (2.24479) (2.11346) (2.09315)Union −0.29149 0 −0.29352 0(0.06498) – (0.06496) –Single 0 0.37970 0 −0.08352– (1.37139) – (0.11589)German −0.39741 0 −0.39614 0(0.07614) – (0.07622) –Qualified white collar 0.06936 0 0.06771 0(0.13112) – (0.13117) –Ordinary white collar 0.17865 0 0.18113 0(0.14752) – (0.14770) –Qualified blue collar 0.13240 0 0.13270 0(0.08245) – (0.08252) –Log-likelihood −2039.3549 −2039.13127In this particular sample, the model is identified by the exclusion of Single fromx1i. However, Single has a relatively small, and statistically insignificant, eﬀect onthe logit part of the model. Therefore, although θ˜ =³β˜0, γ˜0´0and θ0 =¡β00, γ00¢0do not lead to the same value for the likelihood function, the results of the two lastcolumns reveal that there is a second maximum of the likelihood function for θ veryclose to θ0. Indeed, we note that γˆ ' −γ˜ and βˆ ' β˜ + γ˜. Moreover, we find thatθˆ =³βˆ0, γˆ0´0leads to a value of the log-likelihood function which is extremely closeto, and actually slightly better than, the one obtained with θ˜.These results are not limited to the Poisson-logit specification. For this particulardata, comparable results can be found for the Poisson-probit and for the NegBin2-logit models (these results are available from the authors upon request). Therefore,even in cases where identification is guaranteed by appropriate restrictions or by thestructure of the model, the results of the previous section may be helpful in thatthey can be used to guide the researcher in the search for the global maximum of thelikelihood function.4. CONCLUDING REMARKSThis note reviews the conditions needed for the identification of the Poisson-logitand other leading models for underreported counts. In general, identification is easierwhen at least one of the regressors aﬀects the probability of reporting without aﬀectingthe underlying count process. In case such regressor is not available, identificationmay still be achieved by using additional information. This information can take theform of restrictions on the signs of some coeﬃcients in the probability of reporting,or it can be used to specify a skedastic function that disentangles the eﬀect of theregressors on the mean of the count process and on the probability of reporting.Even when the model is identified, the likelihood function is likely to have multiplemaxima. Therefore, when estimating this sort of model, it is important not to accept8a maximum of the likelihood function as global without having performed a thoroughsearch for other maxima. In particular, having obtained one set of estimates, ourresults can be used to obtain starting values that are often close to a set of parametervalues that leads to an alternative maximum of the likelihood function.REFERENCESCameron, A.C. and P.K. Trivedi (1998). Regression Analysis of Count Data, Cam-bridge, MA: Cambridge University Press.Catcheside, D.G. (1948). “Genetic Eﬀect of Radiation,” Advances in Genetics, 2,271-358.Deb, P. and P.K. Trivedi (1997). “Demand for Medical Care by the Elderly: AFinite Mixture Approach,” Journal of Applied Econometrics, 12, 313-336.Deb, P. and P.K. Trivedi (2002). “The Structure of Demand for Health Care: LatentClass Versus Two-Part Models,” Journal of Health Economics, 21, 601-625.Econometric Software, Inc. (2007). LIMDEP 9.0, New York: Plainview.Johnson, N.L.; A.W. Kemp, and S. Kotz, (2005). Univariate Discrete Distributions,3rd ed., New York: John Wiley & Sons.Lambert, D. (1992). “Zero-Inflated Poisson Regression, With an Application toDefects in Manufacturing,” Technometrics, 34, 1-14.Mukhopadhyay, K. and P.K. Trivedi (1995). Regression Models for Under-recordedCount Data, Paper presented at the Econometric Society 7th World Congress,Tokyo.Mullahy, J. (1986). “Specification and Testing in Some Modified Count Data Mod-els,” Journal of Econometrics, 33, 341-365.Winkelmann, R. (1998). “Count Data Models with Selectivity,” Econometric Re-views, 17, 339-359.9Winkelmann, R. (2008). Econometric Analysis of Count Data, 5th ed., Berlin:Springer-Verlag.Winkelmann, R and K.F. Zimmermann (1993). Poisson Logistic Regression, De-partment of Economics, University of Munich, Working Paper No 93-18.10

Identification issues in models for underreported counts

Abstract

Similar works

Full text

Available Versions

University of Essex Research Repository

University of East Anglia digital repository