Heteroskedasticity and spatiotemporal dependence robust inference for linear panel models with fixed effects

This paper studies robust inference for linear panel models with fixed effects in the presence of heteroskedasticity and spatiotemporal dependence of unknown forms. We propose a bivariate kernel covariance estimator that nests existing estimators as special cases. Our estimator improves upon existing estimators in terms of robustness, efficiency, and adaptiveness. For distributional approximations, we considered two types of asymptotics: the increasing-smoothing asymptotics and the fixed-smoothing asymptotics. Under the former asymptotics, the Wald statistic based on our covariance estimator converges to a chi-square distribution. Under the latter asymptotics, the Wald statistic is asymptotically equivalent to a distribution that can be well approximated by an F distribution. Simulation results show that our proposed testing procedure works well in finite samples.     

Estimation for spatial dynamic panel data with fixed effects: The case of spatial cointegration

Yu et al. (2008) establish asymptotic properties of quasi-maximum likelihood estimators for a stable spatial dynamic panel model with fixed effects when both the number of individuals n and the number of time periods T are large. This paper investigates unstable cases where there are unit roots generated by temporal and spatial correlations. We focus on the spatial cointegration model where some eigenvalues of the data generating process are equal to 1 and the outcomes of spatial units are cointegrated as in a vector autoregressive system. The asymptotics of the QML estimators are developed by reparameterization, and bias correction for the estimators is proposed. We also consider the 2SLS and GMM estimations when T could be small.     

Nonparametric fixed effects model for panel data with locally stationary regressors

We develop methods for inference in nonparametric time-varying fixed effects panel data models that allow for locally stationary regressors and for the time series length $T$ and cross-section size $N$ both being large. We first develop a pooled nonparametric profile least squares dummy variable approach to estimate the nonparametric function, and establish the optimal convergence rate and asymptotic normality of the resultant estimator. We then propose a test statistic to check whether the bivariate nonparametric function is time-varying or the time effect is separable, and derive the asymptotic distribution of the proposed test statistic. We present several simulated examples and two real data analyses to illustrate the finite sample performance of the proposed methods.     

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$N$ consistent with a Gaussian limiting distribution. We study the small sample properties of the estimator via Monte Carlo techniques.     

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.     

Asymptotics for panel quantile regression models with individual effects

This paper studies panel quantile regression models with individual fixed effects. We formally establish sufficient conditions for consistency and asymptotic normality of the quantile regression estimator when the number of individuals, n$n$, and the number of time periods, T$T$, jointly go to infinity. The estimator is shown to be consistent under similar conditions to those found in the nonlinear panel data literature. Nevertheless, due to the non-smoothness of the objective function, we had to impose a more restrictive condition on T$T$ to prove asymptotic normality than that usually found in the literature. The finite sample performance of the estimator is evaluated by Monte Carlo simulations.     

Semiparametric efficient estimation of dynamic panel data models

This paper extends the semiparametric efficient treatment of panel data models pursued by Park and Simar [Park, B.U., Simar, L., 1994. Efficient semiparametric estimation in stochastic frontier models. Journal of the American Statistical Association 89, 929–936] and Park et al. [Park, B.U., Sickles, R.C., Simar, L., 1998. Stochastic frontiers: a semiparametric approach. Journal of Econometrics 84, 273–301; Park, B.U., Sickles, R.C., Simar, L., 2003. Semiparametric efficient estimation of AR(1) panel data models. Journal of Econometrics 117, 279–309] to a dynamic panel setting. We develop a semiparametric efficient estimator under minimal assumptions when the panel model contains a lagged dependent variable. We apply this new estimator to analyze the structure of demand between city pairs for selected U.S. airlines during the period 1979 I–1992 IV.     

Quantile treatment effects in the regression discontinuity design

We introduce a nonparametric estimator for local quantile treatment effects in the regression discontinuity (RD) design. The procedure uses local distribution regression to estimate the marginal distributions of the potential outcomes. We illustrate the procedure through Monte Carlo simulations and an application on the distributional effects of a universal pre-K program in Oklahoma. We find that participation in a pre-K program significantly raises the lower end and the middle of the distribution of test scores.     

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$n$ 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.     

Nonresponse in dynamic panel data models

To verify whether data are missing at random (MAR) we need to observe the missing data. There are only two exceptions: when the relationship between the probability of responding and the missing variables is either imposed by introducing untestable assumptions or recovered using additional data sources. In this paper, we briefly review the estimation and test procedures for selectivity in panel data. Furthermore, by extending the MAR definition from a static setting to the case of dynamic panel data models, we prove that some tests for selectivity are not verifying the MAR condition.     

The effects of dynamic feedbacks on LS and MM estimator accuracy in panel data models

The finite sample behavior is analyzed of particular least squares (LS) and a range of (generalized) method of moments (MM) estimators in panel data models with individual effects and both a lagged dependent variable regressor and another explanatory variable. The latter may be affected by lagged feedbacks from the dependent variable too. Asymptotic expansions indicate how the order of magnitude of bias of MM estimators tends to increase with the number of moment conditions exploited. They also provide analytic evidence on how the bias of the various estimators depends on the feedbacks and on other model characteristics such as prominence of individual effects and correlation between observed and unobserved heterogeneity. Simulation results corroborate the theoretical findings and reveal that in small samples of models with dynamic feedbacks none of the techniques examined dominates regarding bias and mean squared error over all parametrizations examined.     

On the effect of mean-nonstationarity in dynamic panel data models

In this paper, we investigate the effect of mean-nonstationarity on the first-difference generalized method of moments (FD-GMM) estimator in dynamic panel data models. We find that when data is mean-nonstationary and the variance of individual effects is significantly larger than that of disturbances, the FD-GMM estimator performs quite well. We demonstrate that this is because the correlation between the lagged dependent variable and instruments gets larger owing to the unremoved individual effects, i.e., instruments become strong. This implies that, under mean-nonstationarity, the FD-GMM estimator does not always suffer from the weak instruments problem even when data is persistent.     

The optimal choice of moments in dynamic panel data models

This paper derives an approximation of the mean square error (MSE) of the GMM estimator in dynamic panel data models. The approximation is based on higher-order asymptotic theory under double asymptotics. While first-order theory under double asymptotics provides information about the bias, it does not provide enough information about the variance of the estimator. Higher-order theory enables us to obtain information about the variance. From this result, a procedure for choosing the number of instruments is proposed. The simulations confirm that the proposed procedure improves the precision of the estimator.     

Pseudo conditional maximum likelihood estimation of the dynamic logit model for binary panel data

We show how the dynamic logit model for binary panel data may be approximated by a quadratic exponential model. Under the approximating model, simple sufficient statistics exist for the subject-specific parameters introduced to capture the unobserved heterogeneity between subjects. The latter must be distinguished from the state dependence which is accounted for by including the lagged response variable among the regressors. By conditioning on the sufficient statistics, we derive a pseudo conditional likelihood estimator of the structural parameters of the dynamic logit model, which is simple to compute. Asymptotic properties of this estimator are studied in detail. Simulation results show that the estimator is competitive in terms of efficiency with estimators recently proposed in the econometric literature.     

Nonparametric trending regression with cross-sectional dependence

Panel data, whose series length T$T$ is large but whose cross-section size N$N$ need not be, are assumed to have common time trend, of unknown form. The model includes additive, unknown, individual-specific components and allows for spatial or other cross-sectional dependence and/or heteroscedasticity. A simple smoothed nonparametric trend estimate is shown to be dominated by an estimate which exploits availability of cross-sectional data. Asymptotically optimal bandwidth choices are justified for both estimates. Feasible optimal bandwidths, and feasible optimal trend estimates, are asymptotically justified, finite sample performance of the latter being examined in a Monte Carlo study. Potential extensions are discussed.     

Testing for serial correlation,spatial autocorrelation and random effects using panel data

This paper considers a spatial panel data regression model with serial correlation on each spatial unit over time as well as spatial dependence between the spatial units at each point in time. In addition, the model allows for heterogeneity across the spatial units using random effects. The paper then derives several Lagrange multiplier tests for this panel data regression model including a joint test for serial correlation, spatial autocorrelation and random effects. These tests draw upon two strands of earlier work. The first is the LM tests for the spatial error correlation model discussed in Anselin and Bera [1998. Spatial dependence in linear regression models with an introduction to spatial econometrics. In: Ullah, A., Giles, D.E.A. (Eds.), Handbook of Applied Economic Statistics. Marcel Dekker, New York] and in the panel data context by Baltagi et al. [2003. Testing panel data regression models with spatial error correlation. Journal of Econometrics 117, 123–150]. The second is the LM tests for the error component panel data model with serial correlation derived by Baltagi and Li [1995. Testing AR(1) against MA(1) disturbances in an error component model. Journal of Econometrics 68, 133–151]. Hence, the joint LM test derived in this paper encompasses those derived in both strands of earlier works. In fact, in the context of our general model, the earlier LM tests become marginal LM tests that ignore either serial correlation over time or spatial error correlation. The paper then derives conditional LM and LR tests that do not ignore these correlations and contrast them with their marginal LM and LR counterparts. The small sample performance of these tests is investigated using Monte Carlo experiments. As expected, ignoring any correlation when it is significant can lead to misleading inference.     

Estimating dynamic panel data discrete choice models with fixed effects

This paper considers the estimation of dynamic binary choice panel data models with fixed effects. It is shown that the modified maximum likelihood estimator (MMLE) used in this paper reduces the order of the bias in the maximum likelihood estimator from O(T^-1) to O(T^-2), without increasing the asymptotic variance. No orthogonal reparametrization is needed. Monte Carlo simulations are used to evaluate its performance in finite samples where T is not large. In probit and logit models containing lags of the endogenous variable and exogenous variables, the estimator is found to have a small bias in a panel with eight periods. A distinctive advantage of the MMLE is its general applicability. Estimation and relevance of different policy parameters of interest in this kind of models are also addressed.     

Nonparametric dynamic panel data models: Kernel estimation and specification testing

Motivated by the first-differencing method for linear panel data models, we propose a class of iterative local polynomial estimators for nonparametric dynamic panel data models with or without exogenous regressors. The estimators utilize the additive structure of the first-differenced model—the fact that the two additive components have the same functional form, and the unknown function of interest is implicitly defined as a solution of a Fredholm integral equation of the second kind. We establish the uniform consistency and asymptotic normality of the estimators. We also propose a consistent test for the correct specification of linearity in typical dynamic panel data models based on the _L2$L_2$ distance of our nonparametric estimates and the parametric estimates under the linear restriction. We derive the asymptotic distributions of the test statistic under the null hypothesis and a sequence of Pitman local alternatives, and prove its consistency against global alternatives. Simulations suggest that the proposed estimators and tests perform well for finite samples. We apply our new method to study the relationships among economic growth, the initial economic condition and capital accumulation, and find a significant nonlinear relation between economic growth and the initial economic condition.     

Asymptotic theory for nonparametric regression with spatial data

Nonparametric regression with spatial, or spatio-temporal, data is considered. The conditional mean of a dependent variable, given explanatory ones, is a nonparametric function, while the conditional covariance reflects spatial correlation. Conditional heteroscedasticity is also allowed, as well as non-identically distributed observations. Instead of mixing conditions, a (possibly non-stationary) linear process is assumed for disturbances, allowing for long range, as well as short-range, dependence, while decay in dependence in explanatory variables is described using a measure based on the departure of the joint density from the product of marginal densities. A basic triangular array setting is employed, with the aim of covering various patterns of spatial observation. Sufficient conditions are established for consistency and asymptotic normality of kernel regression estimates. When the cross-sectional dependence is sufficiently mild, the asymptotic variance in the central limit theorem is the same as when observations are independent; otherwise, the rate of convergence is slower. We discuss the application of our conditions to spatial autoregressive models, and models defined on a regular lattice.