Nonlinear Panel Data Models Research Articles

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.

Geographically and Temporally Weighted Likelihood Regression: Exploring the Spatiotemporal Determinants of Land Use Change

Urban areas possess complex spatial configurations. These patterns are produced by cumulative changes in land use and land cover as human and natural environments are influenced by markets forces, policy, and changes in the natural landscape. To understand the mechanisms underlying these complex patterns, it is important to develop models that can capture the complexity of the underlying economic process. This includes spatiotemporal variation in the variables as well as spatiotemporal heterogeneity or non-stationarity in the model. The objective of this paper is to build on previous work in spatial nonparametric modeling and propose a spatiotemporal technique for nonlinear panel data models. Using a series of Monte Carlo experiments, we demonstrate how extending a geographically weighted likelihood regression (GWLR) model to account for temporal heterogeneity can improve the performance of the model when heterogeneity exists in the spatial and temporal dimension. We also show how the technique can be used in modeling real world land use changes by applying our proposed technique to a panel of historical subdivision development from an urbanizing county in the Baltimore/Towson Metropolitan Statistical Area (MSA). Our results demonstrate that the method provides better performance than a standard parametric model. We also demonstrate how the spatiotemporal marginal effects from the model can be used to conduct policy analysis at multiple spatial and temporal scales, which is not possible using the standard global parameter estimates. Our proposed technique is simple to execute and can be implemented using any statistical software package.

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.

Estimation of a nonlinear panel data model with semiparametric individual effects

This paper investigates identification and estimation of a class of nonlinear panel data, single-index models. The model allows for unknown time-specific link functions, and semiparametric specification of the individual-specific effects. We develop an estimator for the parameters of interest, and propose a powerful new kernel-based modified backfitting algorithm to compute the estimator. We derive uniform rates of convergence results for the estimators of the link functions, and show the estimators of the finite-dimensional parameters are root-N consistent with a Gaussian limiting distribution. We study the small sample properties of the estimator via Monte Carlo techniques.

Identification and [formula omitted]-consistent estimation of a nonlinear panel data model with correlated unobserved effects

This paper investigates identification and root-n-consistent estimation of a class of single-index panel data models in which the link function is unknown, the unobserved individual effects may be correlated with all the explanatory variables, and all the explanatory variables may be predetermined. We propose two sets of sufficient conditions, one in which link function is assumed to be strictly increasing, and the other in which it is not. We propose simple kernel-based estimators for the models, and derive consistency and asymptotic normality results for the proposed estimators. Finally, we present results of two Monte Carlo studies of the estimators.

Model Selection in the Presence of Incidental Parameters

This paper considers model selection in nonlinear panel data models where incidental parameters or large-dimensional nuisance parameters are present. Primary interest typically centres on selecting a model that best approximates the underlying structure involving parameters that are common within the panel after concentrating out the incidental parameters. It is well known that conventional model selection procedures are often inconsistent in panel models and this can be so even without nuisance parameters (Han et al, 2012). Modifications are then needed to achieve consistency. New model selection information criteria are developed here that use either the Kullback-Leibler information criterion based on the profile likelihood or the Bayes factor based on the integrated likelihood with the robust prior of Arellano and Bonhomme (2009). These model selection criteria impose heavier penalties than those associated with standard information criteria such as AIC and BIC. The additional penalty, which is data-dependent, properly reflects the model complexity arising from the presence of incidental parameters. A particular example is studied in detail involving lag order selection in dynamic panel models with fixed individual effects. The new criteria are shown to control for over/under-selection probabilities in these models and lead to consistent order selection criteria.

Spillovers and absorptive capacity in the decision to innovate of Spanish firms: the role of human capital

This paper investigates whether the existence of knowledge spillovers and the capacity of firms to assimilate them, which we relate with R&D intensity and some human resource management practices, are associated with the decision to innovate of Spanish firms. In order to do this, we employ data from the ‘Central de Balances’ database, which covers both manufacturing and services firms during the period 2003–2007, and use an estimator proposed by Wooldridge [2005. Simple solutions to the initial conditions problem in dynamic nonlinear panel data models with unobserved heterogeneity. Journal of Applied Econometrics 20, no. 1: 39–54] for dynamic random effects discrete choice models. The empirical exercise provides evidence on the positive link between spillovers and the innovative behaviour of companies, not just for the knowledge generated in the same industry, but also for that generated in the same region or by the public sector. Moreover, this link is stronger for those firms with a higher capacity to absorb those spillovers. This ability not only works through firms’ R&D capabilities, but also through factors such as the quality of the labour force, the share of temporary employment and the amount of resources spent in training. In addition to these factors, we find that innovation performance exhibits a high degree of inertia. Further, some other observed firm characteristics, such as size, sales growth, export behaviour, sector capital intensity or financial structure variables, are also found to be relevant determinants of the likelihood of innovation.

Lumpy Price Adjustments: A Microeconometric Analysis

Based on a reduced-form state-dependent pricing model with random thresholds, we specify and estimate a nonlinear panel data model with an unobserved factor representing the common cost or demand components of the unobserved optimal price. Using this model, we are able to assess the relative importance of common and idiosyncratic shocks in explaining the frequency and magnitude of price changes in the case of a wide variety of consumer products in Belgium and France. We find that the mean level and variability of the random thresholds are key for explaining differences across products in the frequency of price changes. We also find that the idiosyncratic shocks represent the most important driver of the magnitude of price changes. Supplementary materials for this article are available online.

Nonlinear Panel Data Analysis

Nonlinear panel data models arise naturally in economic applications, yet their analysis is challenging. Here we provide a progress report on some recent advances in the area. We start by reviewing the properties of random-effects likelihood approaches. We emphasize a link with Bayesian computation and Markov chain Monte Carlo, which provides a convenient approach to estimation and inference. The relaxation of parametric assumptions on the distribution of individual effects raises serious identification problems. In discrete choice models, common parameters and average marginal effects are generally set identified. The availability of continuous outcomes, however, provides opportunities for point identification. We end by reviewing recent progress on non-fixed-T approaches. In panel applications in which the time dimension is not negligible relative to the size of the cross section, it makes sense to view the estimation problem as a time-series finite-sample bias. Several perspectives to bias reduction are now available. We review their properties, with a special emphasis on random-effects methods.

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.

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.

Estimating Non-Linear Models with Multiple Fixed Effects: A Computational Note

In 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.

PANEL DATA MODELS WITH FINITE NUMBER OF MULTIPLE EQUILIBRIA

We study a nonlinear panel data model in which the fixed effects are assumed to have finite support. The fixed effects estimator is known to have the incidental parameters problem. We contribute to the literature by making a qualitative observation that the incidental parameters problem in this model may not be not as severe as in the conventional case. Because fixed effects have finite support, the probability of correctly identifying the fixed effect converges to one even when the cross sectional dimension grows as fast as some exponential function of the time dimension. As a consequence, the finite sample bias of the fixed effects estimator is expected to be small.

Simplified Implementation of the Heckman Estimator of the Dynamic Probit Model and a Comparison with Alternative Estimators*

AbstractThis paper presents a convenient shortcut method for implementing the Heckman estimator of the dynamic random effects probit model and other dynamic nonlinear panel data models using standard software. It then compares the estimators proposed by Heckman, Orme and Wooldridge, based on three alternative approximations, first in an empirical model for the probability of unemployment and then in a set of simulation experiments. The results indicate that none of the three estimators dominates the other two in all cases. In most cases, all three estimators display satisfactory performance, except when the number of time periods is very small.

A Penalty Function Approach to Bias Reduction in Nonlinear Panel Models with Fixed Effects

We consider estimation of nonlinear panel data models with individual specific fixed effects. Estimation of these models is complicated since estimation of the fixed effects when the time dimension is short generally results in inconsistent estimates of all model parameters. We present a penalized objective function that reduces the bias in the resulting point estimates. The penalty function is simple to construct and requires no modification for models with multiple individual specific parameters. We illustrate the approach through a series of simulations that suggest the approach is effective in reducing bias and in an empirical study of insider trading activity.

Discrete choice modeling with nonstationary panels applied to exchange rate regime choice

Abstract This paper develops a regression limit theory for discrete choice nonstationary panels with large cross section ( N ) and time series ( T ) dimensions. Some results emerging from this theory are directly applicable in the wider context of M-estimation. This includes an extension of work by Wooldridge [Wooldridge, J.M., 1994. Estimation and Inference for Dependent Processes. In: Engle, R.F., McFadden, D.L. (Eds.). Handbook of Econometrics, vol. 4, North-Holland, Amsterdam] on the limit theory of local extremum estimators to multi-indexed processes in nonlinear nonstationary panel data models. It is shown that the maximum likelihood (ML) estimator is consistent without an incidental parameters problem and has a limit theory with a fast rate of convergence N 1 / 2 T 3 / 4 (in the stationary case, the rate is N 1 / 2 T 1 / 2 ) for the regression coefficients and thresholds, and a normal limit distribution. In contrast, the limit distribution is known to be mixed normal in time series modeling, as shown in [Park, J.Y., Phillips, P.C.B., 2000, Nonstationary binary choice. Econometrica, 68, 1249–1280] (hereafter PP), and [Phillips, P.C.B., Jin, S., Hu, L., 2007. Nonstationary discrete choice: A corrigendum and addendum. Journal of Econometrics 141(2), 1115–1130] (hereafter, PJH). The approach is applied to exchange rate regime choice by monetary authorities, and we provide an analysis of the empirical phenomenon known as “fear of floating”.

Persistence in health limitations: A European comparative analysis

This paper investigates the persistence in health limitations for individuals within the member states of the European Union. We use the full eight waves of data available in the European Community Household Panel (ECHP) to explore the relative contributions of state dependence, unobserved heterogeneity and socioeconomic characteristics, in particular income, education and activity status, and how these vary across countries. We focus on binary measures of health limitations, constructed from the answers to the question: “Are you hampered in your daily activities by any physical or mental health problem, illness or disability?”. Dynamic non-linear panel data models are specified and estimated using both pooled and random effects probit and logit models together with complementary log–log models. The random effects probit specifications are preferred. Results reveal high state dependence of health limitations, which remains after controlling for measures of socioeconomic status. There is heterogeneity in the socioeconomic gradient across countries.

Panel data methods for fractional response variables with an application to test pass rates

We revisit the effects of spending on student performance using data from the state of Michigan. In addition to exploiting a dramatic change in funding in the mid-1990s and subsequent nonsmooth changes, we propose nonlinear panel data models that recognize the bounded nature of the pass rate. Importantly, we show how to estimate average partial effects, which can be compared across many different models (linear and nonlinear) under different assumptions and estimated using many different methods. We find that spending has nontrivial and statistically significant effects, although the diminishing effect is not especially pronounced.

Inference in panel data models under attrition caused by unobservables

This paper concerns identification and estimation of a finite-dimensional parameter in a panel data-model under nonignorable sample attrition. Attrition can depend on second period variables which are unobserved for the attritors but an independent refreshment sample from the marginal distribution of the second period values is available. This paper shows that under a quasi-separability assumption, the model implies a set of conditional moment restrictions where the moments contain the attrition function as an unknown parameter. This formulation leads to (i) a simple proof of identification under strictly weaker conditions than those in the existing literature and, more importantly, (ii) a sieve-based root-n consistent estimate of the finite-dimensional parameter of interest. These methods are applicable to both linear and nonlinear panel data models with endogenous attrition and analogous methods are applicable to situations of endogenously missing data in a single cross-section. The theory is illustrated with a simulation exercise, using Current Population Survey data where a panel structure is introduced by the rotation group feature of the sampling process.

Sequential numerical integration in nonlinear state space models for microeconometric panel data

AbstractThis paper discusses the estimation of a class of nonlinear state space models including nonlinear panel data models with autoregressive error components. A health economics example illustrates the usefulness of such models. For the approximation of the likelihood function, nonlinear filtering algorithms developed in the time‐series literature are considered. Because of the relatively simple structure of these models, a straightforward algorithm based on sequential Gaussian quadrature is suggested. It performs very well both in the empirical application and a Monte Carlo study for ordered logit and binary probit models with an AR(1) error component. Copyright © 2008 John Wiley & Sons, Ltd.