Specification and testing of some modified count data models

This paper explores the specification and testing of some modified count data models. These alternatives permit more flexible specification of the data-generating process (dgp) than do familiar count data models (e.g., the Poisson), and provide a natural means for modeling data that are over- or underdispersed by the standards of the basic models. In the cases considered, the familiar forms of the distributions result as parameter-restricted versions of the proposed modified distributions. Accordingly, score tests of the restrictions that use only the easily-computed ML estimates of the standard models are proposed. The tests proposed by Hausman (1978) and White (1982) are also considered. The tests are then applied to count data models estimated using survey microdata on beverage consumption.     

SEMI-PARAMETRIC ESTIMATION OF HURDLE REGRESSION MODELS WITH AN APPLICATION TO MEDICAID UTILIZATION

This paper develops a semi-parametric estimation method for hurdle (two-part) count regression models. The approach in each stage is based on Laguerre series expansion for the unknown density of the unobserved heterogeneity. The semi-parametric hurdle model nests Poisson and negative binomial hurdle models, which have been used in recent applied literature. The empirical part of the paper evaluates the impact of managed care programmes for Medicaid eligibles on utilization of health-care services using a key utilization variable, the number of doctor and health centre visits. Health status measures and age seem to be more important in determining health-care utilization than other socio-economic and enrollment variables. The semi-parametric approach is particularly useful for the analysis of overdispersed individual level data characterized by a large proportion of non-users, and highly skewed distribution of counts for users. © 1997 John Wiley & Sons, Ltd.     

The Relative Performance of Poisson and Negative Binomial Regression Estimators

Negative binomial estimators are commonly used in estimating models with count‐data dependent variables. In this paper, sampling experiments are used to evaluate the performance of these estimators relative to the simpler Poisson estimator in finite‐sample situations. The results do not suggest a clear preference for negative binomial estimators in situations in which the underlying dependent variables are overdispersed, unless the researcher is comfortable in assumptions about the precise form of the overdispersion.     

Individual effects and dynamics in count data models

In this paper we examine the panel data estimation of dynamic models for count data that include correlated fixed effects and predetermined variables. Use of a linear feedback model is proposed. A quasi-differenced GMM estimator is consistent for the parameters in the dynamic model, but when series are highly persistent, there is a problem of weak instrument bias. An estimator is proposed that utilises pre-sample information of the dependent count variable, which is shown in Monte Carlo simulations to possess desirable small sample properties. The models and estimators are applied to data on US patents and R&D expenditure.     

Multiple imputation of incomplete zero‐inflated count data

Empirical count data are often zero‐inflated and overdispersed. Currently, there is no software package that allows adequate imputation of these data. We present multiple‐imputation routines for these kinds of count data based on a Bayesian regression approach or alternatively based on a bootstrap approach that work as add‐ons for the popular multiple imputation by chained equations (mice ) software in R (van Buuren and Groothuis‐Oudshoorn , Journal of Statistical Software, vol. 45, 2011, p. 1). We demonstrate in a Monte Carlo simulation that our procedures are superior to currently available count data procedures. It is emphasized that thorough modeling is essential to obtain plausible imputations and that model mis‐specifications can bias parameter estimates and standard errors quite noticeably. Finally, the strengths and limitations of our procedures are discussed, and fruitful avenues for future theory and software development are outlined.     

Modelling Hospital Admission and Length of Stay by Means of Generalised Count Data Models

For a large heterogeneous group of patients, we analyse probabilities of hospital admission and distributional properties of lengths of hospital stay conditional on individual determinants. Bayesian structured additive regression models for zero‐inflated and overdispersed count data are employed. In addition, the framework is extended towards hurdle specifications, providing an alternative approach to cover particularly large frequencies of zero quotes in count data. As a specific merit, the model class considered embeds linear and nonlinear effects of covariates on all distribution parameters. Linear effects indicate that the quantity and severity of prior illness are positively correlated with the risk of hospital admission, while medical prevention (in the form of general practice visits) and rehabilitation reduce the expected length of future hospital stays. Flexible nonlinear response patterns are diagnosed for age and an indicator of a patients' socioeconomic status. We find that social deprivation exhibits a positive impact on the risk of admission and a negative effect on the expected length of future hospital stays of admitted patients. Copyright © 2015 John Wiley & Sons, Ltd.     

A simple approach for the analysis of generalizea linear mixed models

A broad class of generalized linear mixed models, e.g. variance components models for binary data, percentages or count data, will be introduced by incorporating additional random effects into the linear predictor of a generalized linear model structure. Parameters are estimated by a combination of quasi-likelihood and iterated MINQUE (minimum norm quadratic unbiased estimation), the latter being numerically equivalent to REML (restricted, or residual, maximum likelihood). First, conditional upon the additional random effects, observations on a working variable and weights are derived by quasi-likelihood, using iteratively re-weighted least squares. Second, a linear mixed model is fitted to the working variable, employing the weights for the residual error terms, by iterated MINQUE. The latter may be regarded as a least squares procedure applied to squared and product terms of error contrasts derived from the working variable. No full distributional assumptions are needed for estimation. The model may be fitted with standardly available software for weighted regression and REML.     

Distribution free estimation of heteroskedastic binary response models using Probit/Logit criterion functions

In this paper estimators for distribution free heteroskedastic binary response models are proposed. The estimation procedures are based on relationships between distribution free models with a conditional median restriction and parametric models (such as Probit/Logit) exhibiting (multiplicative) heteroskedasticity. The first proposed estimator is based on the observational equivalence between the two models, and is a semiparametric sieve estimator (see, e.g. Gallant and Nychka (1987), Ai and Chen (2003) and Chen et al. (2005)) for the regression coefficients, based on maximizing standard Logit/Probit criterion functions, such as NLLS and MLE. This procedure has the advantage that choice probabilities and regression coefficients are estimated simultaneously. The second proposed procedure is based on the equivalence between existing semiparametric estimators for the conditional median model (,  and ) and the standard parametric (Probit/Logit) NLLS estimator. This estimator has the advantage of being implementable with standard software packages such as Stata. Distribution theory is developed for both estimators and a Monte Carlo study indicates they both perform well in finite samples.     

HETEROGENEITY,EXCESS ZEROS,AND THE STRUCTURE OF COUNT DATA MODELS

This paper demonstrates that the unobserved heterogeneity commonly assumed to be the source of overdispersion in count data models has predictable implications for the probability structure of such mixture models. In particular, the common observation of excess zeros is a strict implication of unobserved heterogeneity. This result has important implications for using count model estimates for predicting certain interesting parameters. Test statistics to detect such heterogeneity-related departures from the null model are proposed and applied in a health-care utilization example, suggesting that a null Poisson model should be rejected in favour of a mixed alternative. © 1997 John Wiley & Sons, Ltd.     

Modelling Time Series of Counts in Epidemiology

We review generalized dynamic models for time series of count data. Usually temporal counts are modelled as following a Poisson distribution, and a transformation of the mean depends on parameters which evolve smoothly with time. We generalize the usual dynamic Poisson model by considering continuous mixtures of the Poisson distribution. We consider Poisson‐gamma and Poisson‐log‐normal mixture models. These models have a parameter for each time t which captures possible extra‐variation present in the data. If the time interval between observations is short, many observed zeros might result. We also propose zero inflated versions of the models mentioned above. In epidemiology, when a count is equal to zero, one does not know if the disease is present or not. Our model has a parameter which provides the probability of presence of the disease given no cases were observed. We rely on the Bayesian paradigm to obtain estimates of the parameters of interest, and discuss numerical methods to obtain samples from the resultant posterior distribution. We fit the proposed models to artificial data sets and also to a weekly time series of registered number of cases of dengue fever in a district of the city of Rio de Janeiro, Brazil, during 2001 and 2002.     

A latent class poisson regression model for heterogeneous count data

In this paper an approach is developed that accommodates heterogeneity in Poisson regression models for count data. The model developed assumes that heterogeneity arises from a distribution of both the intercept and the coefficients of the explanatory variables. We assume that the mixing distribution is discrete, resulting in a finite mixture model formulation. An EM algorithm for estimation is described, and the algorithm is applied to data on customer purchases of books offered through direct mail. Our model is compared empirically to a number of other approaches that deal with heterogeneity in Poisson regression models.     

Statistical inference on seemingly unrelated varying coefficient partially linear models

This paper is concerned with the statistical inference on seemingly unrelated varying coefficient partially linear models. By combining the local polynomial and profile least squares techniques, and estimating the contemporaneous correlation, we propose a class of weighted profile least squares estimators (WPLSEs) for the parametric components. It is shown that the WPLSEs achieve the semiparametric efficiency bound and are asymptotically normal. For the non‐parametric components, by applying the undersmoothing technique, and taking the contemporaneous correlation into account, we propose an efficient local polynomial estimation. The resulting estimators are shown to have mean‐squared errors smaller than those estimators that neglect the contemporaneous correlation. In addition, a class of variable selection procedures is developed for simultaneously selecting significant variables and estimating unknown parameters, based on the non‐concave penalized and weighted profile least squares techniques. With a proper choice of regularization parameters and penalty functions, the proposed variable selection procedures perform as efficiently as if one knew the true submodels. The proposed methods are evaluated using wide simulation studies and applied to a set of real data.     

Semiparametric time series models driven by latent factor

We introduce a class of semiparametric time series models (SemiParTS) driven by a latent factor process. The proposed SemiParTS class is flexible because, given the latent process, only the conditional mean and variance of the time series are specified. These are the primary features of SemiParTS: (i) no parametric form is assumed for the conditional distribution of the time series given the latent process; (ii) it is suitable for a wide range of data: non-negative, count, bounded, binary, and real-valued time series; (iii) it does not constrain the dispersion parameter to be known. The quasi-likelihood inference is employed in order to estimate the parameters in the mean function. Here, we derive explicit expressions for the marginal moments and for the autocorrelation function of the time series process so that a method of moments can be employed to estimate the dispersion parameter and also the parameters related to the latent process. Simulated results that aim to check the proposed estimation procedure are presented. Forecasting procedures are proposed and evaluated in simulated and real data. Analyses of the number of admissions in a hospital due to asthma and a total insolation time series illustrate the potential for practical situations that involve the proposed models.     

The second-order bias and mean squared error of estimators in time-series models

We develop analytical results on the second-order bias and mean squared error of estimators in time-series models. These results provide a unified approach to developing the properties of a large class of estimators in linear and nonlinear time-series models and they are valid for both normal and nonnormal samples of observations, and where the regressors are stochastic. The estimators included are the generalized method of moments, maximum likelihood, least squares, and other extremum estimators. Our general results are applied to four time-series models. We investigate the effects of nonnormality on the second-order bias results for two of these models, while for all four models, the second-order bias and mean squared error results are given under normality. Numerical results for some of these models are also presented.     

Efficiency and productivity of the US banking industry, 1998–2005: evidence from the Fourier cost function satisfying global regularity conditions

This paper provides estimates of bank efficiency and productivity in the United States, over the period from 1998 to 2005, using (for the first time) the globally flexible Fourier cost functional form, as originally proposed by Gallant ( 1982 ), and estimated subject to global theoretical regularity conditions, using procedures suggested by Gallant and Golub ( 1984 ). We find that failure to incorporate monotonicity and curvature into the estimation results in mismeasured magnitudes of cost efficiency and misleading rankings of individual banks in terms of cost efficiency. We also find that the largest two subgroups (with assets greater than 1 billion in 1998 dollars) are less efficient than the other subgroups and that the largest four bank subgroups (with assets greater than $ 400 million) experienced significant productivity gains and the smallest eight subgroups experienced insignificant productivity gains or even productivity losses. Copyright © 2008 John Wiley & Sons, Ltd.     

Repeat tourism in Uruguay: modelling truncated distributions of count data

This paper studies the determinants of repeat visiting in Uruguay, where loyal visitors are a relevant part of the total. From a statistical point of view, the number of times a visitor has been to a place constitutes count data. In this regard available information on Uruguay presents relevant limitations. Count data is in fact reported only for those who visited the country up to five times, whereas records about the most frequent visitors are collapsed into one residual category. This implies that the classic models for count data such as Poisson or negative binomial cannot be put into consideration. The paper suggests instead modelling the available part of the empirical distribution through quantile count data regression. It is a model based on measures of location rather than mean values, which allows estimating tourists’ behaviour as the number of visits increases. A set of explanatory variables related to budgetary constraints, socioeconomic, trip-related and psychographic characteristics are taken as regressors to the considered count data.     

Computation and analysis of multiple structural change models

In a recent paper, Bai and Perron ( 1998 ) considered theoretical issues related to the limiting distribution of estimators and test statistics in the linear model with multiple structural changes. In this companion paper, we consider practical issues for the empirical applications of the procedures. We first address the problem of estimation of the break dates and present an efficient algorithm to obtain global minimizers of the sum of squared residuals. This algorithm is based on the principle of dynamic programming and requires at most least‐squares operations of order O(T²) for any number of breaks. Our method can be applied to both pure and partial structural change models. Second, we consider the problem of forming confidence intervals for the break dates under various hypotheses about the structure of the data and the errors across segments. Third, we address the issue of testing for structural changes under very general conditions on the data and the errors. Fourth, we address the issue of estimating the number of breaks. Finally, a few empirical applications are presented to illustrate the usefulness of the procedures. All methods discussed are implemented in a GAUSS program. Copyright © 2002 John Wiley & Sons, Ltd.     

Least squares model averaging by Mallows criterion

This paper is in response to a recent paper by Hansen (2007) who proposed an optimal model average estimator with weights selected by minimizing a Mallows criterion. The main contribution of Hansen’s paper is a demonstration that the Mallows criterion is asymptotically equivalent to the squared error, so the model average estimator that minimizes the Mallows criterion also minimizes the squared error in large samples. We are concerned with two assumptions that accompany Hansen’s approach. The first is the assumption that the approximating models are strictly nested in a way that depends on the ordering of regressors. Often there is no clear basis for the ordering and the approach does not permit non-nested models which are more realistic from a practical viewpoint. Second, for the optimality result to hold the model weights are required to lie within a special discrete set. In fact, Hansen noted both difficulties and called for extensions of the proof techniques. We provide an alternative proof which shows that the result on the optimality of the Mallows criterion in fact holds for continuous model weights and under a non-nested set-up that allows any linear combination of regressors in the approximating models that make up the model average estimator. These results provide a stronger theoretical basis for the use of the Mallows criterion in model averaging by strengthening existing findings.     

An empirical application of stochastic volatility models

This paper studies the empirical performance of stochastic volatility models for twenty years of weekly exchange rate data for four major currencies. We concentrate on the effects of the distribution of the exchange rate innovations for both parameter estimates and for estimates of the latent volatility series. The density of the log of squared exchange rate innovations is modelled as a flexible mixture of normals. We use three different estimation techniques: quasi-maximum likelihood, simulated EM, and a Bayesian procedure. The estimated models are applied for pricing currency options. The major findings of the paper are that: (1) explicitly incorporating fat-tailed innovations increases the estimates of the persistence of volatility dynamics; (2) the estimation error of the volatility time series is very large; (3) this in turn causes standard errors on calculated option prices to be so large that these prices are rarely significantly different from a model with constant volatility. © 1998 John Wiley & Sons, Ltd.