panel data models

This paper investigates efficient estimation of heterogeneous coefficients in panel data models with common shocks, which have been a particular focus of recent theo- retical and empirical literature. We propose a new two-step method to estimate the heterogeneous coefficients. In the first step, the maximum likelihood (ML) method is first conducted to estimate the loadings and idiosyncratic variances. The second step estimates the heterogeneous coefficients by using the structural relations implied by the model and replacing the unknown parameters with their ML estimates. We establish the asymptotic theory of our estimator, including consistency, asymptotic representa- tion, and limiting distribution. The two-step estimator is asymptotically efficient in the sense that it has the same limiting distribution as the infeasible generalized least squares (GLS) estimator. Intensive Monte Carlo simulations show that the proposed estimator performs robustly in a variety of data setups.

Most of the existing work in the literature on nonstationary nonlinear panel data models requires a large number of time periods- see e.g. Moon and Phillips (2000). One exception is Chen and Khan (2008), who assumed correlated random effects. Here, we look for assump- tions motivated from the previous literature, that aim at relaxing stationarity. The issue is that standard mean and median independence assumptions on the marginals of ǫ’s do not allow us to provide any restrictions on β, i.e., the sharp set is the trivial set- i.e. the original parameter space. The intuition is that the marginal median independence assumption places no restriction on the conditional median of (ǫ i1 − ǫ i2 ). Also, mean independence assumptions

The rest of the paper is organized as follows. The next section presents the model, describes our methodology, and gives asymptotic properties of our estimators. We first consider panel data models with only individual effects, then we extend our methodology to treat two-way effects models. Section 3 presents some Monte Carlo evidence on how our estimator behaves in the finite sample setting. All mathematical proofs are provided in the appendix.

portant respects. First, our benchmark experiment explicitly controls for the loading factor and the signal-to-noise ratio, two fundamental control parameters in dynamic panel data models (Kiviet, 1995). Under a more rigorous simulation design, we show that the IV/GMM estima- tors are sensitive to the control parameters and perform less well than in FH’s simulations. Second, while FH only examine a bias-corrected estimator based on an analytical approach (LSDVC), we further consider two recently developed bias correction methods based on itera- tive sampling algorithms, namely BC and II. Compared to LSDVC, these methods rely on less restrictive assumptions, and are computationally less demanding, especially in the case of BC. We find that both perform well with BC being least affected by autocorrelation. Third, we com- pare the IV/GMM and bias-corrected estimators with DPF in an important simulation exercise allowing for the fractional nature of the dependent variable (Loudermilk, 2007; Elsas and Flo- rysiak, 2014). Inconsistent with FH’s conjecture, we show the non-negligible effect of severe censoring and the relevance of DPF in this case. Fourth, while FH conclude that LSDVC and SYS-GMM are the most appropriate methods, we observe poor performance from SYS-GMM and the related IV/GMM estimators (especially in the presence of autocorrelation). We, thus, caution against the use of these methods and recommend the more robust bias-corrected esti- mators. Finally, we are able to corroborate our simulation results using empirical applications to two highly relevant areas of corporate finance, capital structure and cash holdings. In sum, our study provides new simulation and empirical results with general and relevant implications for future empirical corporate research.

PANEL DATA MODELS WITH INTERACTIVE EFFECTS 3 Similarly, we allow common regressors, which do not vary across individuals, such as prices and policy variables. The corresponding regression coefficients are individual-dependent so that individuals respond differently to policy or price changes. In our view, this is a sensible way to incorporate time-invariant and common regressors. For example, wages associated with education and with gender are more likely to change over time rather than remain constant. In our analysis, time invariant regressors are treated as the components of λ i that are observable, and common regressors as the components of f t that

Early development of spatial models has been summarized by a number of books, including Cliff and Ord (1973), Anselin (1988), and Cressie (1993). GMM estimation of spatial models are studied by Kelijian and Prucha (1998, 1999, 2010), and Kapoor et al. (2007), among others. The maximum likelihood method is considered by Ord (1975), Anselin (1988), Lee (2004a), Yu et al. (2008) and Lee and Yu (2010), and so on. For panel data models with multiple common shocks, Ahn et al. (2013) consider the fixed- T GMM estimation. Pesaran (2006) proposes the correlated random effects method by including additional regressors, which are the cross-sectional averages of dependent and the explanatory variables. The principal components method is studied by Bai (2009) and Moon and Weidner (2009). Bai and Li (2014) consider the maximum likelihood method.

To deal with these problems, we propose a new simple test that accommodates cross-sectional dependence by using the results of Bai (2009), Song (2013), and Ando and Bai (2014). These studies considered panel data models with interactive fixed effects. Our proposed test statistic, denoted by ˆ Γ, is a modified version of Swamy’s (1970) test statistic, similar to that of Pesaran and Yamagata (2008). An advantage of our testing procedure is that it provides a robust test under cross-sectionally correlated errors with heteroskedasticity. Furthermore, the proposed test works even when the set of predictors and the unobservable errors that contain the factor structure are correlated. We investigate the asymptotic distribution of our test statistic, and show that the test has a standard normal distribution as N, T → ∞ such that √ T /N → 0. Monte Carlo experiments show that the proposed test tends to have the correct size and satisfactory power as N, T → ∞ .

The paper develops a general Bayesian framework for robust linear static panel data models using ε-contamination. A two-step approach is employed to derive the conditional type-II maximum likelihood (ML-II) posterior distribution of the coefficients and individual effects. The ML-II posterior densities are weighted averages of the Bayes estimator under a base prior and the data-dependent empirical Bayes estimator. Two-stage and three stage hier- archy estimators are developed and their finite sample performance is investigated through a series of Monte Carlo experiments. These include standard random effects as well as Mundlak-type, Chamberlain-type and Hausman-Taylor-type models. The simulation results underscore the relatively good performance of the three-stage hierarchy estimator. Within a single theoretical framework, our Bayesian approach encompasses a variety of specifications while conventional methods require separate estimators for each case. We illustrate the per- formance of our estimator relative to classic panel estimators using data on earnings and crime.

Spatial panel data models are widely used in empirical studies. The existing theories of spatial models so far have largely confine the analysis under the assump- tion of parameters stabilities. This is unduely restrictive, since a large number of studies have well documented the presence of structural changes in the relationship of economic variables. This paper proposes and studies spatial panel data models with structural change. We consider using the quasi maximum likelihood method to estimate the model. Static and dynamic models are both considered. Large-T and fixed-T setups are both considered. We provide a relatively complete asymptotic theory for the maximum likelihood estimators, including consistency, convergence rates and limiting distributions of the regression coefficients, the timing of structural change and variance of errors. We study the hypothesis testing for the presence of structural change. The three super-type statistics are proposed. The Monte Carlo simulation results are consistent with our theoretical results and show that the max- imum likelihood estimators have good finite sample performance.

This paper is concerned with the use of the Durbin-Wu-Hausman test for correlated effects with panel data. The assumptions underlying the construc- tion of the statistic are too strong in many empirical cases. The consequences of deviations from the basic assumptions are investigated. The size distortion is assessed. In the case of measurement error, the Hausman test is found to be a test of the difference in asymptotic biases of between and within group estima- tors. However, its ‘size’ is sensitive to the relative magnitude of the intra-group and inter-group variations of the covariates, and can be so large as to preclude the use of the statistic in this case. We show to what extent some assumptions can be relaxed in a panel data context and we discuss an alternative robust formulation of the test. Power considerations are presented.

First, estimating heterogeneous panels by EM-REML yields an unbiased estimation of the variance components. This is important as the unbiased estimator of the variance-covariance matrix of the random coefficients proposed by Swamy (1970) is often negative definite. In such cases, the author suggests eliminating a term to obtain a non-negative definite matrix. Although not unbiased, this alternative estimator is consistent when T tends to infinity. Lee and Griffiths (1979) derive a recursive system of equations as a solution to the maximization of the likelihood function of the data which incorporates the prior likelihood of the random coefficients. However, I demonstrate that their estimate of the coefficients residuals’ variance- covariance matrix does not satisfy the law of total variance. Differently from the latter, we consider the joint likelihood of the observed data and the random coefficients as an incomplete data problem (in a sense which will be more clear later on). We show that maximizing the expected value of the joint likelihood function with respect to the conditional distribution of the random coefficients residuals given the observed data is necessary to obtain an unbiased estimator of the random coefficients covariance matrix. As a result, our approach should be preferred when T is relatively small. Another interesting feature of the EM (compared to the papers mentioned in this paragraph) is that it allows us to make inference on the random coefficients population. Indeed, in general, it gives a probability distribution over the missing data.

However, as in the error-component model, the Swamy estimator of the random coefficient covariance matrix is not necessarily nonnegative definite. Our aim is to investigate the consequences of this drawback in finite samples, in particular when testing hypotheses. At the same time, we propose a solution to the above mentioned problem by applying the EM algorithm. In particular, following the seminal papers of Dempster et al. (1977), and Patterson and Thompson (1971), we propose to estimate heterogeneous panels by applying the EM algorithm to obtain tractable closed form solutions of restricted maximum likelihood (REML) estimates of both fixed and random components of the regression coefficients as well as the variance parameters. The proposed estimation procedure is quite general, as we consider a broad framework which incorporates various panel data models as special case. Our approach yields an estimator of the average effects which is asymptotically related to both the GLS and the Mean Group estimator, and which performs relatively well in finite sample as shown in our limited Monte Carlo analysis. We also review some of the existing sampling and Bayesian methods commonly used to estimate heterogeneous panel data, to highlight similarities and differences with the EM-REML approach.

In contrast, the coefficient for the EU dummy is always identified, but its estimates vary considerably across models. In models (1), (2) and (6) it is identified mostly (in the case of FE, only) from the time dimension, i.e., from the change in trade of type-B countries. In the other three models identification is based more on the cross-sectional dimension, i.e., comparing EU pairs to non-EU pairs before 2004. Apart from that, in this case, the RE parameter estimates happen to be quite close to the FE estimates (except for model (5)).

This paper considers the estimation methods for dynamic panel data (DPD) models with fixed effects which suggested in econometric literature, such as least squares (LS) and generalized method of moments (GMM). These methods obtain biased estimators for DPD models. The LS estimator is inconsistent when the time dimension ( ) is short regardless of the cross sectional dimension ( ). Although consistent estimates can be obtained by GMM procedures, the inconsistent LS estimator has a relatively low variance and hence can lead to an estimator with lower root mean square error after the bias is removed. Therefore, we discuss in this paper the different methods to correct the bias of LS and GMM estimations. The analytical expressions for the asymptotic biases of the LS and GMM estimators have been presented for large N and finite T. Finally, we display new estimators that presented by Youssef and Abonazel (2015) as more efficient estimators than the conventional estimators.

predictive perspective. Specifically, the panel data model is evaluated from a predictive point of view, and we propose an estimator of the expected mean squared error (MSE). The criterion is developed by correcting the asymptotic bias in the MSE as an estimate of the expected MSE. To prove the consistency of the selection of the number of group- specific pervasive factors, we extend the analysis of Bai (2009). There exist several references concerning model selection of panel data models with factor structures. Ando and Tsay (2013) investigated the model selection problem for large panel data models with the interactive fixed effects of Bai (2009), where the slope coefficients are common to each unit. Ando and Bai (2013) studied the panel data model selection problem under heterogeneous slopes and hierarchical factor error structures. These results are for panel data models where group membership is known. Therefore, our problem is different, as we need to further develop the criterion for selecting the number of groups. Panel data models with homogeneous regression coefficients between the groups involve parsimonious specifications that may be suitable for some applications. How- ever, there is evidence that homogeneity of the parameters is rejected (see for example Hsiao and Tahmiscioglu (1997), Lin and Ng (2012)). To deal with the presence of unobserved heterogeneity, we therefore extend the proposed model to the flexible yet parsimonious approach. This approach delivers estimates of group-specific regression parameters, together with interpretable estimates of unit-specific time patterns and group membership. After we describe the model estimation procedure, the consistency and asymptotic distribution of the proposed estimator are established. To determine the number of group-specific pervasive factors, the magnitude of the regularization parameter and the number of groups, we again develop a new C p -type criterion for

12 2. Relationship between employment protection regulations and long-term unemployment rate: We observe no causal relationship between LU and REGLAB or TEMPOLAB. In conformity with this result, our panel data models show that neither REGLAB nor TEMPOLAB has a short-run relationship with LU. Examining only the statistically significant long-run relationships, we observe that the PMG model shows a negative relationship between REGLAB and LU while the DFE model shows a similar relationship between TEMPOLAB and LU (Tables 1 and 2 Parts I.B and II.B). So the contention that strictness of employment protection aggravates long-term unemployment problem cannot be supported by our causality test and panel data modelling.

In the literature, model specifications and estimation strategies, including the maximum like- lihood (ML), GMM and Bayesian methods, receive considerably more attention than specification testing and other forms of hypothesis tests for the SDPD model. For some recent surveys on spatial panel data models, see Anselin et al. (2008), Elhorst (2010a,b, 2014) and Lee and Yu (2010b, 2015b). Yu et al. (2008, 2012), Yu and Lee (2010) and Lee and Yu (2010a, 2011, 2012b, 2016) consider the ML approach for dynamic spatial panel data models when both the number of individuals and the number of time periods are large under various scenarios. The (quasi) ML estimator (QMLE) suggested in these studies may have asymptotic bias contingent on the asymptotic setup assumed for the growth rate of the number of individuals and the number of time periods. For example, the QMLE in Yu et al. (2008) has asymptotic bias when the number of individuals are asymptotically proportional to number of time periods, i.e., when n/T → k < ∞ . The limiting distributions of bias corrected versions suggested in Yu et al. (2008) are only properly centered when T grows relatively fast compare to n 1/3 . For the asymptotic scenario where the cross-sectional dimension is large and the time dimension is fixed, Elhorst (2005) and Su and Yang (2015) consider the ML approach for the dynamic panel data models that have spatial autoregressive processes in the disturbance term. Lee and Yu (2015a) suggest a QMLE to a static two-way panel data model that has disturbances with dynamic and spatial correlations which might be spatially stable or unstable. The result- ing QMLE is consistent and has a properly centered asymptotic normal distribution regardless of whether T is large or not, and whether the process defined for the disturbance term is stable or not.

In these contexts, strict exogeneity of covariates other than the lagged dependent variable, conditional on unobserved heterogeneity, is required for consistent estimation of the regression and state dependence parameters, when the estimation relies on correlated random effects or on fixed effects which are eliminated when conditioning on suitable sufficient statistics for the individual unobserved heterogeneity. However, the assumption of strict exogeneity is likely to be violated in practice because there may be feedback effects from the past of the outcome variable on the present values of the covariates, namely the model covariates may be Granger- caused by the response variable Granger (1969). While in linear models the mainstream approach to overcome this problem is to consider instrumental variables (Anderson and Hsiao, 1981; Arellano and Bond, 1991; Arellano and Bover, 1995; Blundell and Bond, 1998), considerably fewer results are available for nonlinear binary panel data models with predetermined covariates. This is particularly true with short binary panel data and no general solution is yet available, despite the relevance of binary these type of data in microeconomic applications.

As these multidimensional panel data models are frequently used to deal with flow types of data like trade, capital movements (FDI), etc., it is important to have a closer look at the case when, by nature, we do not observe self flow. This means that from the (ijt) indexes we do not have observations for the dependent variable of the model when i = j for any t. This is the first step to relax our initial assumption that N 1 = N 2 = N and that the observation sets i and j are equivalent.

We apply our proposed formulation to the problem of estimating the labor supply of married women. The same empirical application is considered by Fern´andez-Val (2009) and Dhaene and Jochmans (2015). The sample is drawn from the Panel Study of Income Dynamics (PSID), that consists of n = 1, 908 married women between 19 and 59 years of age in 1980, followed for T = 7 time occasions, from 1979 to 1985. We specify a static logit model for the probability of being employed at time t, conditional on the number of children of a certain age in the family, namely the number of kids between 0 and 2 years old, between 3 and 5, and between 6 and 17, on the husband’s income, and on the woman’s age and age squared. We also specify a dynamic logit model, that is we include lagged participation in the set of model covariates.