Incidental Parameters Problem Research Articles

Robust standard errors in transformed likelihood estimation of dynamic panel data models with cross-sectional heteroskedasticity

This paper extends the transformed maximum likelihood approach for estimation of dynamic panel data models by Hsiao et al. (2002) to the case where the errors are cross-sectionally heteroskedastic. This extension is not trivial due to the incidental parameters problem and its implications for estimation and inference. We approach the problem by working with a mis-specified homoskedastic model, and then show that the transformed maximum likelihood estimator continues to be consistent even in the presence of cross-sectional heteroskedasticity. We also obtain standard errors that are robust to cross-sectional heteroskedasticity of unknown form. By means of Monte Carlo simulations, we investigate the finite sample behavior of the transformed maximum likelihood estimator and compare it with various GMM estimators proposed in the literature. Simulation results reveal that, in terms of median absolute errors and accuracy of inference, the transformed likelihood estimator outperforms the GMM estimators in almost all cases.

Comment on ‘IV estimation of panels with factor residuals’ by D. Robertson and V. Sarafidis

This comment discusses two issues related to the paper by Robertson and Sarafidis (2014, hereafter RS). The first issue is the potential incidental parameters problem that may arise when data are cross-sectionally heteroskedastic. The second issue is an identification problem in the estimation of panel models with weakly exogenous instrumental variables.

Panel data partially linear model with fixed effects, spatial autoregressive error components and unspecified intertemporal correlation

This paper considers the estimating problem of a panel data partially linear model with spatial autoregressive errors and fixed effects. In addition, we allow the idiosyncratic errors to be intertemporally correlated. By combining the polynomial spline series approximation, the semiparametric least squares method and the difference based technique, a new generalized moment estimator for the autoregressive parameter of the spatial model is constructed. Its consistency and asymptotic normality are established. In order to avoid the incidental parameter problem, a difference based intertemporal covariance matrix estimator is proposed. Based on the estimated spatially and time-wise correlated error structure, we further construct a weighted difference based semiparametric least squares estimator (WDSLSE) and a weighted difference based polynomial spline series estimator (WDPSSE) for the parametric and nonparametric components of the mean model, respectively. We develop an asymptotic theory for these two estimators, including the asymptotic normality, asymptotic efficiency and convergence rate. In particular, we show that the parametric component estimator has the same asymptotic distribution as that based on completely known spatial autoregressive parameter and intertemporal covariance matrix. Simulation studies demonstrate that our asymptotic theory is applicable for finite samples, and the analysis of a real data set illustrates the usefulness of our developed methodology.

Consistent estimation of the fixed effects stochastic frontier model

In this paper we consider a fixed-effects stochastic frontier model. That is, we have panel data, fixed individual (firm) effects, and the usual stochastic frontier analysis (SFA) composed error.Maximum likelihood estimation (MLE) of this model has been considered by Greene (2005a,b). It is subject to the “incidental parameters problem”, that is, to possible inconsistency due to the number of parameters growing with the number of firms. In the linear regression model with normal errors, it is known that the MLE of the regression coefficients is consistent, and the inconsistency due to the incidental parameters problem applies only to the error variance. Greene’s simulations suggest that the same is true in the fixed effects SFA model.In this paper we take a somewhat different approach. We consider MLE based only on the joint density of the deviations from means. In the linear regression model with normal errors, this estimator is the same as the full MLE for the regression coefficients, but it yields a consistent estimator of the error variance. For the SFA model, the MLE based on the deviations from means is not the same as the full MLE, and it has the advantage of not being subject to the incidental parameters problem.The derivation of the joint density of the deviations from means is made possible by results in the statistical literature on the closed skew normal family of distributions. These results may be of independent interest to researchers in this area.Simulations indicate that our within MLE estimator performs quite well in finite samples.We also present an empirical example.

Bayesian Inference for Censored Panel Regression Model

It was recognized by some researchers that the disturbance variance in a censored regression model is frequently underestimated by the maximum likelihood method. This underestimation has implications for the estimation of marginal effects and asymptotic standard errors. For instance, the actual coverage probability of the confidence interval based on a maximum likelihood estimate can be significantly smaller than the nominal confidence level; consequently, a Bayesian estimation is considered to overcome this difficulty. The behaviors of the maximum likelihood and Bayesian estimators of disturbance variance are examined in a fixed effects panel regression model with a limited dependent variable, which is known to have the incidental parameter problem. Behavior under random effect assumption is also investigated.

A Test for Slope Heterogeneity in Fixed Effects Models

Typical panel data models make use of the assumption that the regression parameters are the same for each individual cross-sectional unit. We propose tests for slope heterogeneity in panel data models. Our tests are based on the conditional Gaussian likelihood function in order to avoid the incidental parameters problem induced by the inclusion of individual fixed effects for each cross-sectional unit. We derive the Conditional Lagrange Multiplier test that is valid in cases where N → ∞ and T is fixed. The test applies to both balanced and unbalanced panels. We expand the test to account for general heteroskedasticity where each cross-sectional unit has its own form of heteroskedasticity. The modification is possible if T is large enough to estimate regression coefficients for each cross-sectional unit by using the MINQUE unbiased estimator for regression variances under heteroskedasticity. All versions of the test have a standard Normal distribution under general assumptions on the error distribution as N → ∞. A Monte Carlo experiment shows that the test has very good size properties under all specifications considered, including heteroskedastic errors. In addition, power of our test is very good relative to existing tests, particularly when T is not large.

Spatial Panel Data Models with Common Shocks

Spatial effects and common-shocks effects are of increasing empirical importance. Each type of effects has been analyzed separately in a growing literature. This paper considers a joint modeling of both types of effects. Joint modeling allows one to evaluate which type is present or more important. A large number of incidental parameters exist under the joint modeling. Heteroscedasticity is also allowed. The quasi maximum likelihood method (MLE) is proposed to estimate the model. This paper demonstrates that the quasi-MLE can effectively deal with the incidental parameters problem. An inferential theory including consistency, rate of convergence and limiting distributions is developed. The quasi-MLE can be easily implemented via the EM algorithm, as confirmed by the Monte Carlo simulations. The simulation further reveals the excellent finite sample properties of the quasi-MLE. Some potential extensions are discussed.

Panel data models with nonadditive unobserved heterogeneity: Estimation and inference

This paper considers fixed effects estimation and inference in linear and nonlinear panel data models with random coefficients and endogenous regressors. The quantities of interest—means, variances, and other moments of the random coefficients—are estimated by cross sectional sample moments of generalized method of moments (GMM) estimators applied separately to the time series of each individual. To deal with the incidental parameter problem introduced by the noise of the within-individual estimators in short panels, we develop bias corrections. These corrections are based on higher-order asymptotic expansions of the GMM estimators and produce improved point and interval estimates in moderately long panels. Under asymptotic sequences where the cross sectional and time series dimensions of the panel pass to infinity at the same rate, the uncorrected estimators have asymptotic biases of the same order as their asymptotic standard deviations. The bias corrections remove the bias without increasing variance. An empirical example on cigarette demand based on Becker, Grossman, and Murphy (1994) shows significant heterogeneity in the price effect across U.S. states.

Grouped effects estimators in fixed effects models

We consider estimation of nonlinear panel data models with common and individual specific parameters. Fixed effects estimators are known to suffer from the incidental parameters problem, which can lead to large biases in estimates of common parameters. Pooled estimators, which ignore heterogeneity across individuals, are also generally inconsistent. We assume that individuals in the data are grouped on multiple levels where groups are defined by some observable external classification. We consider “group effects” estimators, where individual specific parameters are assumed common across groups at some level. We provide conditions under which group effects estimates of common parameters are asymptotically unbiased and normal. The conditions suggest a tradeoff between two sources of bias, one due to incidental parameters and the other due to misspecification of unobserved heterogeneity.

Testing for heteroskedasticity in fixed effects models

We derive tests for heteroskedasticity after fixed effects estimation of linear panel models. The asymptotic results are based on a ‘large N–fixed T’ framework, where the incidental parameters problem is bypassed by utilizing a (pseudo) likelihood function conditional on the sufficient statistic for these parameters. A simple ‘studentization’ produces distribution free tests that can easily be implemented using an artificial regression based on residuals after fixed effects estimation. A Monte Carlo exploration suggests that the tests perform well in small samples such as those encountered in practice.

K-Step Bootstrap Bias Correction for Fixed Effects Estimators in Nonlinear Panel Models

Fixed effects estimators in nonlinear panel models with fixed and short time series length T usually suffer from inconsistency because of the incidental parameters problem first noted by Neyman and Scott (1948). Moreover, even if T grows but at a rate not faster than the cross sectional sample size n, they are asymptotically biased, and therefore the associated confidence intervals have a large coverage error. This paper analyzes the properties of the parametric bootstrap bias corrected maximum likelihood (ML) estimators of nonlinear panel models with fixed effects. We assume that each time series follows a finite order Markov process. We show that the bootstrap bias corrected estimators are asymptotically normal and centered at the true parameter. In particular, we propose using the k-step parametric bootstrap procedure to alleviate the computational cost of implementing the standard bootstrap. We also apply the standard and k-step bootstrap bias correction to average marginal effect estimation and to the double bootstrap for confidence interval construction. Our Monte Carlo simulations show that the k-step bootstrap bias corrected estimator reduces the bias remarkably well in finite samples.

Estimation in the presence of many nuisance parameters: Composite likelihood and plug-in likelihood

We consider the incidental parameters problem in this paper, i.e. the estimation for a small number of parameters of interest in the presence of a large number of nuisance parameters. By assuming that the observations are taken from a multiple strictly stationary process, the two estimation methods, namely the maximum composite quasi-likelihood estimation (MCQLE) and the maximum plug-in quasi-likelihood estimation (MPQLE) are considered. For the MCQLE, we profile out nuisance parameters based on lower-dimensional marginal likelihoods, while the MPQLE is based on some initial estimators for nuisance parameters. The asymptotic normality for both the MCQLE and the MPQLE is established under the assumption that the number of nuisance parameters and the number of observations go to infinity together, and both the estimators for the parameters of interest enjoy the standard root-n convergence rate. Simulation with a spatial–temporal model illustrates the finite sample properties of the two estimation methods.

Finite Mixture for Panels with Fixed Effects

This paper develops finite mixture models with fixed effects for two families of distributions for which the incidental parameter problem has a solution. Analytical results are provided for mixtures of Normals and mixtures of Poisson. We provide algorithms based on the expectations-maximization (EM) approach as well as computationally simpler equivalent estimators that can be used in the case of the mixtures of normals. We design and implement a Monte Carlo study that examines the finite sample performance of the proposed estimator and also compares it with other estimators such the Mundlak-Chamberlain conditionally correlated random effects estimator. The results of Monte Carlo experiments suggest that our proposed estimators of such models have excellent finite sample properties, even in the case of relatively small T and moderately sized N dimensions. The methods are applied to models of healthcare expenditures and counts of utilization using data from the Health and Retirement Study.

Robust Standard Errors in Transformed Likelihood Estimation of Dynamic Panel Data Models

This paper extends the transformed maximum likelihood approach for estimation of dynamic panel data models by Hsiao, Pesaran and Tahmiscioglu (2002) to the case where the errors are cross-sectionally heteroskedastic. This extension is not trivial due to the incidental parameters problem that arises, and its implications for estimation and inference. We approach the problem by working with a mis-specified homoskedastic model. It is shown that the transformed maximum likelihood estimator continues to be consistent even in the presence of cross-sectional heteroskedasticity. We also obtain standard errors that are robust to cross-sectional heteroskedasticity of unknown form. By means of Monte Carlo simulation, we investigate the finite sample behavior of the transformed maximum likelihood estimator and compare it with various GMM estimators proposed in the literature. Simulation results reveal that, in terms of median absolute errors and accuracy of inference, the transformed likelihood estimator outperforms the GMM estimators in almost all cases.

Measurement of labour quality growth caused by unobservable characteristics

The standard economy-wide indices of labour quality (or human capital) largely ignore the role of unobservable worker characteristics. In this article, we develop a methodology for identifying the contributions of both observable and unobservable worker characteristics in the presence of the incidental parameter problem. Based on data for Switzerland over the period 1991 to 2006, we find that a large part of growth in labour quality is caused by shifts in the distribution of unobservable worker characteristics. The overall index differs little from the standard indices, but contributions to growth attributed to education and age are corrected downwards.

Consistent Estimation of the 'True' Fixed-Effects Stochastic Frontier Model

The classic stochastic frontier panel data models provide no mechanism to disentangle individual time in- variant unobserved heterogeneity from ine ffi ciency. Greene (2005a,b) proposed a fixed-e ffects model spec- ification that distinguishes these two latent components and allows a time varying ine ffi ciency distribution. However, the maximum likelihood estimator proposed by Greene leads to biased ine ffi ciency estimates due to the incidental parameters problem. In this paper, we propose two alternative estimation procedures that, by relying on a first di fference data transformation, achieve consistency for n!1 with fixed T. Evidence from Monte Carlo simulations shows good finite sample performances of both approaches even in presence of small samples.

Robust Standard Errors in Transformed Likelihood Estimation of Dynamic Panel Data Models

This paper extends the transformed maximum likelihood approach for estimation of dynamic panel data models by Hsiao, Pesaran, and Tahmiscioglu (2002) to the case where the errors are crosssectionally heteroskedastic. This extension is not trivial due to the incidental parameters problem that arises, and its implications for estimation and inference. We approach the problem by working with a mis-specified homoskedastic model. It is shown that the transformed maximum likelihood estimator continues to be consistent even in the presence of cross-sectional heteroskedasticity. We also obtain standard errors that are robust to cross-sectional heteroskedasticity of unknown form. By means of Monte Carlo simulation, we investigate the finite sample behavior of the transformed maximum likelihood estimator and compare it with various GMM estimators proposed in the literature. Simulation results reveal that, in terms of median absolute errors and accuracy of inference, the transformed likelihood estimator outperforms the GMM estimators in almost all cases.

Estimating Nonlinear Models with Multiple Fixed Effects: A Computational Note*

AbstractIn this paper we consider estimation of nonlinear panel data models that include multiple individual fixed effects. Estimation of these models is complicated both by the difficulty of estimating models with possibly thousands of coefficients and also by the incidental parameters problem; that is, noisy estimates of the fixed effects when the time dimension is short contaminate the estimates of the common parameters due to the nonlinearity of the problem. We propose a simple variation of existing bias‐corrected estimators, which can exploit the additivity of the effects for numerical optimization. We exhibit the performance of the estimators in simulations.

Remedying the Neyman–Scott phenomenon in model discrimination

The objective of this paper is to investigate through simulation the possible presence of the incidental parameters problem when performing frequentist model discrimination with stratified data. In this context, model discrimination amounts to considering a structural parameter taking values in a finite space, with k points, k≥2. This setting seems to have not yet been considered in the literature about the Neyman–Scott phenomenon. Here we provide Monte Carlo evidence of the severity of the incidental parameters problem also in the model discrimination setting and propose a remedy for a special class of models. In particular, we focus on models that are scale families in each stratum. We consider traditional model selection procedures, such as the Akaike and Takeuchi information criteria, together with the best frequentist selection procedure based on maximization of the marginal likelihood induced by the maximal invariant, or of its Laplace approximation. Results of two Monte Carlo experiments indicate that when the sample size in each stratum is fixed and the number of strata increases, correct selection probabilities for traditional model selection criteria may approach zero, unlike what happens for model discrimination based on exact or approximate marginal likelihoods. Finally, two examples with real data sets are given.

Bias corrections for two-step fixed effects panel data estimators

This paper introduces large- T bias-corrected estimators for nonlinear panel data models with both time invariant and time varying heterogeneity. These models include systems of equations with limited dependent variables and unobserved individual effects, and sample selection models with unobserved individual effects. Our two-step approach first estimates the reduced form by fixed effects procedures to obtain estimates of the time varying heterogeneity underlying the endogeneity/selection bias. We then estimate the primary equation by fixed effects including an appropriately constructed control variable from the reduced form estimates as an additional explanatory variable. The fixed effects approach in this second step captures the time invariant heterogeneity while the control variable accounts for the time varying heterogeneity. Since either or both steps might employ nonlinear fixed effects procedures it is necessary to bias adjust the estimates due to the incidental parameters problem. This problem is exacerbated by the two-step nature of the procedure. As these two-step approaches are not covered in the existing literature we derive the appropriate correction thereby extending the use of large- T bias adjustments to an important class of models. Simulation evidence indicates our approach works well in finite samples and an empirical example illustrates the applicability of our estimator.