Sieve estimation of panel data models with cross section dependence

In this paper we consider the problem of estimating semiparametric panel data models with cross section dependence, where the individual-specific regressors enter the model nonparametrically whereas the common factors enter the model linearly. We consider both heterogeneous and homogeneous regression relationships when both the time and cross-section dimensions are large. We propose sieve estimators for the nonparametric regression functions by extending Pesaran’s (2006) common correlated effect (CCE) estimator to our semiparametric framework. Asymptotic normal distributions for the proposed estimators are derived and asymptotic variance estimators are provided. Monte Carlo simulations indicate that our estimators perform well in finite samples.     

A test of cross section dependence for a linear dynamic panel model with regressors

This paper proposes a new testing procedure for detecting error cross section dependence after estimating a linear dynamic panel data model with regressors using the generalised method of moments (GMM). The test is valid when the cross-sectional dimension of the panel is large relative to the time series dimension. Importantly, our approach allows one to examine whether any error cross section dependence remains after including time dummies (or after transforming the data in terms of deviations from time-specific averages), which will be the case under heterogeneous error cross section dependence. Finite sample simulation-based results suggest that our tests perform well, particularly the version based on the [Blundell, R., Bond, S., 1998. Initial conditions and moment restrictions in dynamic panel data models. Journal of Econometrics 87, 115–143] system GMM estimator. In addition, it is shown that the system GMM estimator, based only on partial instruments consisting of the regressors, can be a reliable alternative to the standard GMM estimators under heterogeneous error cross section dependence. The proposed tests are applied to employment equations using UK firm data and the results show little evidence of heterogeneous error cross section dependence.     

A Lagrange Multiplier test for cross-sectional dependence in a fixed effects panel data model

It is well known that the standard Breusch and Pagan (1980) LM test for cross-equation correlation in a SUR model is not appropriate for testing cross-sectional dependence in panel data models when the number of cross-sectional units (n)$\left(n\right)$ is large and the number of time periods (T)$\left(T\right)$ is small. In fact, a scaled version of this LM test was proposed by Pesaran (2004) and its finite sample bias was corrected by Pesaran et al. (2008). This was done in the context of a heterogeneous panel data model. This paper derives the asymptotic bias of this scaled version of the LM test in the context of a fixed effects homogeneous panel data model. This asymptotic bias is found to be a constant related to n$n$ and T$T$, which suggests a simple bias corrected LM test for the null hypothesis. Additionally, the paper carries out some Monte Carlo experiments to compare the finite sample properties of this proposed test with existing tests for cross-sectional dependence.     

Bias in dynamic panel models under time series misspecification

We consider within-group estimation of higher-order autoregressive panel models with exogenous regressors and fixed effects, where the lag order is possibly misspecified. Even when disregarding the misspecification bias, the fixed-effect bias formula is quite different from the correctly specified case though its asymptotic order remains the same under the stationarity. We suggest bias reduction methods under the possible time series misspecification.     

Quantile regression for dynamic panel data with fixed effects

This paper studies a quantile regression dynamic panel model with fixed effects. Panel data fixed effects estimators are typically biased in the presence of lagged dependent variables as regressors. To reduce the dynamic bias, we suggest the use of the instrumental variables quantile regression method of Chernozhukov and Hansen (2006) along with lagged regressors as instruments. In addition, we describe how to employ the estimated models for prediction. Monte Carlo simulations show evidence that the instrumental variables approach sharply reduces the dynamic bias, and the empirical levels for prediction intervals are very close to nominal levels. Finally, we illustrate the procedures with an application to forecasting output growth rates for 18 OECD countries.     

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.     

Fixed effects instrumental variables estimation in correlated random coefficient panel data models

We provide a set of conditions sufficient for consistency of a general class of fixed effects instrumental variables (FE-IV) estimators in the context of a correlated random coefficient panel data model, where one ignores the presence of individual-specific slopes. We discuss cases where the assumptions are met and violated. Monte Carlo simulations verify that the FE-IV estimator of the population averaged effect performs notably better than other standard estimators, provided a full set of period dummies is included. We also propose a simple test of selection bias in unbalanced panels when we suspect the slopes may vary by individual.     

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.     

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.     

Estimating a common deterministic time trend break in large panels with cross sectional dependence

This paper develops an estimation procedure for a common deterministic time trend break in large panels. The dependent variable in each equation consists of a deterministic trend and an error term. The deterministic trend is subject to a change in the intercept, slope or both, and the break date is common for all equations. The estimation method is simply minimizing the sum of squared residuals for all possible break dates. Both serial and cross sectional correlations are important factors that decide the rate of convergence and the limiting distribution of the break date estimate. The rate of convergence is faster when the errors are stationary than when they have a unit root. When there is no cross sectional dependence among the errors, the rate of convergence depends on the number of equations and thus is faster than the univariate case. When the errors have a common factor structure with factor loadings correlated with the intercept and slope change parameters, the rate of convergence does not depend on the number of equations and thus reduces to the univariate case. The limiting distribution of the break date estimate is also provided. Some Monte Carlo experiments are performed to assess the adequacy of the asymptotic results. A brief empirical example using the US GDP price index is offered.     

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.     

A generalized nonlinear IV unit root test for panel data with cross-sectional dependence

This paper proposes a unit root test for panel data with cross-sectional dependence. The test generalizes the nonlinear IV unit root test of Chang (2002) to the case where there exist some common factors in panels. The main idea is to eliminate the cross-sectional dependence through the method of principal components as in Bai and Ng (2004) and then apply Chang’s test to the treated data. Under certain conditions, the proposed test is consistent and has a standard normal limiting distribution under the null hypothesis. Simulation results show that the proposed test compares favorably to other alternative tests.     

Long difference instrumental variables estimation for dynamic panel models with fixed effects

This paper proposes a new instrumental variables estimator for a dynamic panel model with fixed effects with good bias and mean squared error properties even when identification of the model becomes weak near the unit circle. We adopt a weak instrument asymptotic approximation to study the behavior of various estimators near the unit circle. We show that an estimator based on long differencing the model is much less biased than conventional implementations of the GMM estimator for the dynamic panel model. We also show that under the weak instrument approximation conventional GMM estimators are dominated in terms of mean squared error by an estimator with far less moment conditions. The long difference (LD) estimator mimics the infeasible optimal procedure through its reliance on a small set of moment conditions.     

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.     

Residual‐based IV estimation of dynamic panel data models with fixed effects

In dynamic panel regression, when the variance ratio of individual effects to disturbance is large, the system‐GMM estimator will have large asymptotic variance and poor finite sample performance. To deal with this variance ratio problem, we propose a residual‐based instrumental variables (RIV) estimator, which uses the residual from regressing Δy_i,t?1 on as the instrument for the level equation. The RIV estimator proposed is consistent and asymptotically normal under general assumptions. More importantly, its asymptotic variance is almost unaffected by the variance ratio of individual effects to disturbance. Monte Carlo simulations show that the RIV estimator has better finite sample performance compared to alternative estimators. The RIV estimator generates less finite sample bias than difference‐GMM, system‐GMM, collapsing‐GMM and Level‐IV estimators in most cases. Under RIV estimation, the variance ratio problem is well controlled, and the empirical distribution of its t‐statistic is similar to the standard normal distribution for moderate sample sizes.     

First difference maximum likelihood and dynamic panel estimation

First difference maximum likelihood (FDML) seems an attractive estimation methodology in dynamic panel data modeling because differencing eliminates fixed effects and, in the case of a unit root, differencing transforms the data to stationarity, thereby addressing both incidental parameter problems and the possible effects of nonstationarity. This paper draws attention to certain pathologies that arise in the use of FDML that have gone unnoticed in the literature and that affect both finite sample performance and asymptotics. FDML uses the Gaussian likelihood function for first differenced data and parameter estimation is based on the whole domain over which the log-likelihood is defined. However, extending the domain of the likelihood beyond the stationary region has certain consequences that have a major effect on finite sample and asymptotic performance. First, the extended likelihood is not the true likelihood even in the Gaussian case and it has a finite upper bound of definition. Second, it is often bimodal, and one of its peaks can be so peculiar that numerical maximization of the extended likelihood frequently fails to locate the global maximum. As a result of these pathologies, the FDML estimator is a restricted estimator, numerical implementation is not straightforward and asymptotics are hard to derive in cases where the peculiarity occurs with non-negligible probabilities. The peculiarities in the likelihood are found to be particularly marked in time series with a unit root. In this case, the asymptotic distribution of the FDMLE has bounded support and its density is infinite at the upper bound when the time series sample size T→∞$T\to \infty$. As the panel width n→∞$n\to \infty$ the pathology is removed and the limit theory is normal. This result applies even for T$T$ fixed and we present an expression for the asymptotic distribution which does not depend on the time dimension. We also show how this limit theory depends on the form of the extended likelihood.     

Some properties of the LIML estimator in a dynamic panel structural equation

We investigate the finite sample and asymptotic properties of the within-groups (WG), the random-effects quasi-maximum likelihood (RQML), the generalized method of moment (GMM) and the limited information maximum likelihood (LIML) estimators for a panel autoregressive structural equation model with random effects when both T (time-dimension) and N (cross-section dimension) are large. When we use the forward-filtering due to Alvarez and Arellano (2003), the WG, the RQML and GMM estimators are significantly biased when both T and N are large while T/N is different from zero. The LIML estimator gives desirable asymptotic properties when T/N converges to a constant.     

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 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.