Asymptotics for Panel Models with Common Shocks

This article develops a novel asymptotic theory for panel models with common shocks. We assume that contemporaneous correlation can be generated by both the presence of common regressors among units and weak spatial dependence among the error terms. Several characteristics of the panel are considered: cross-sectional and time-series dimensions can either be fixed or large; factors can either be observable or unobservable; the factor model can describe either a cointegration relationship or a spurious regression, and we also consider the stationary case. We derive the rate of convergence and the limit distributions for the ordinary least square (OLS) estimates of the model parameters under all the aforementioned cases.     

The accuracy of normal approximation in a heterogeneous panel data unit root test

Tests for unit roots in panel data have become very popular. Two attractive features of panel data unit root tests are the increased power compared to time-series tests, and the often well-behaved limiting distributions of the tests. In this paper we apply Monte Carlo simulations to investigate how well the normal approximation works for a heterogeneous panel data unit root test when there are only a few cross sections in the sample. We find that the normal approximation, which should be valid for large numbers of cross-sectional units, works well, at conventional significance levels, even when the number of cross sections is as small as two. This finding is valuable for the applied researcher since critical values will be easy to obtain and p-values will be readily available.     

考虑截面相关条件下的异质性面板数据协整回归模型的估计

 面板数据的非平稳分析是近年来迅速发展的方向,其中考虑截面相关情形下面板数据的协整分析的发展备受关注。Bai &; Kao（2006）得出了截面相关条件下面板协整估计的因子模型,但该模型只考虑了被解释变量截面相关情形,未考虑解释变量的截面相关,且假定各截面间长期协方差矩阵相同。本文在Bai（2006）考虑截面相关条件下面板数据协整回归模型估计的基础上将其结论推广至被解释变量和解释变量均截面相关及截面长期协方差矩阵不相同即异质性时的情形,并试图通过Monte Carlo 模拟讨论其小样本性质。并且由于截面间长期协方差矩阵异质性的存在,本文还针对两变量的协整系统提出了系数检验的组间均值t统计量。     

Stationary bootstrapping for panel cointegration tests under cross-sectional dependence

Stationary bootstrapping is applied to panel cointegration tests which are based on the ordinary least-squares estimator and the seemingly unrelated regression (SUR) estimator of the residual unit root. Large sample validity of stationary bootstrapping is established. A finite sample experiment reveals that size performances of the bootstrap tests are much less sensitive to cross-sectional correlation than those of existing tests and a test based on the SUR estimator has substantially better power than existing tests.     

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.     

Combining p-Values in Non-Stationary Panels

This article summarizes and discusses the existing p-value pooling approaches and compares their performances in the context of panel unit root tests. When the data are free of contemporaneous correlation, most tests achieve very high power. However, in the presence of contemporaneous correlation, most tests suffer from moderate to severe size distortions. When the panel contains both stationary and nonstationary series, the power of tests increases as the cross-sectional units grows. Among all the tests under study, the mean-of-Z test yields the highest power for the benchmark model, while the Fisher test is most robust for complicated model structures.     

Cross-Sectional Dependence in Panel Data Analysis

This article provides an overview of the existing literature on panel data models with error cross-sectional dependence (CSD). We distinguish between weak and strong CSD and link these concepts to the spatial and factor structure approaches. We consider estimation under strong and weak exogeneity of the regressors for both T fixed and T large cases. Available tests for CSD and methods for determining the number of factors are discussed in detail. The finite-sample properties of some estimators and statistics are investigated using Monte Carlo experiments.     

Double unit root tests for cross-sectionally dependent panel data

This paper proposes various double unit root tests for cross-sectionally dependent panel data. The cross-sectional correlation is handled by the projection method [P.C.B. Phillips and D. Sul, Dynamic panel estimation and homogeneity testing under cross section dependence, Econom. J. 6 (2003), pp. 217–259; H.R. Moon and B. Perron, Testing for a unit root in panels with dynamic factors, J. Econom. 122 (2004), pp. 81–126] or the subtraction method [J. Bai and S. Ng, A PANIC attack on unit roots and cointegration, Econometrica 72 (2004), pp. 1127–1177]. Pooling or averaging is applied to combine results from different panel units. Also, to estimate autoregressive parameters the ordinary least squares estimation [D.P. Hasza and W.A. Fuller, Estimation for autoregressive processes with unit roots, Ann. Stat. 7 (1979), pp. 1106–1120] or the symmetric estimation [D.L. Sen and D.A. Dickey, Symmetric test for second differencing in univariate time series, J. Bus. Econ. Stat. 5 (1987), pp. 463–473] are used, and to adjust mean functions the ordinary mean adjustment or the recursive mean adjustment are used. Combinations of different methods in defactoring to eliminate the cross-sectional dependency, integrating results from panel units, estimating the parameters, and adjusting mean functions yields various available tests for double unit roots in panel data. Simple asymptotic distributions of the proposed test statistics are derived, which can be used to find critical values of the test statistics.

We perform a Monte Carlo experiment to compare the performance of these tests and to suggest optimal tests for a given panel data. Application of the proposed tests to a real data, the yearly export panel data sets of several Latin–American countries for the past 50 years, illustrates the usefulness of the proposed tests for panel data, in that they reveal stronger evidence of double unit roots than the componentwise double unit root tests of Hasza and Fuller [Estimation for autoregressive processes with unit roots, Ann. Stat. 7 (1979), pp. 1106–1120] or Sen and Dickey [Symmetric test for second differencing in univariate time series, J. Bus. Econ. Stat. 5 (1987), pp. 463–473].     

Testing for panel cointegration in an error-correction framework with an application to the Fisher hypothesis

In this article, three innovative panel error-correction model (PECM) tests are proposed. These tests are based on the multivariate versions of the Wald (W), likelihood ratio (LR), and Lagrange multiplier (LM) tests. Using Monte Carlo simulations, the size and power of the tests are investigated when the error terms exhibit both cross-sectional dependence and independence. We find that the LM test is the best option when the error terms follow independent white-noise processes. However, in the more empirically relevant case of cross-sectional dependence, we conclude that the W test is the optimal choice. In contrast to previous studies, our method is general and does not rely on the strict assumption that a common factor causes the cross-sectional dependency. In an empirical application, our method is also demonstrated in terms of the Fisher effect—a hypothesis about the existence of which there is still no clear consensus. Based on our sample of the five Nordic countries we utilize our powerful test and discover evidence which, in contrast to most previous research, confirms the Fisher effect.     

Common Correlated Effects Estimation of Dynamic Panels with Cross-Sectional Dependence

We derive inconsistency expressions for dynamic panel data estimators under error cross-sectional dependence generated by an unobserved common factor in both the fixed effect and the incidental trends case. We show that for a temporally dependent factor, the standard within groups (WG) estimator is inconsistent even as both N and T tend to infinity. Next we investigate the properties of the common correlated effects pooled (CCEP) estimator of Pesaran (2006) which eliminates the error cross-sectional dependence using cross-sectional averages of the data. In contrast to the static case, the CCEP estimator is only consistent when next to N also T tends to infinity. It is shown that for the most relevant parameter settings, the inconsistency of the CCEP estimator is larger than that of the infeasible WG estimator, which includes the common factors as regressors. Restricting the CCEP estimator results in a somewhat smaller inconsistency. The small sample properties of the various estimators are analyzed using Monte Carlo experiments. The simulation results suggest that the CCEP estimator can be used to estimate dynamic panel data models provided T is not too small. The size of N is of less importance.     

A Bayesian method of distinguishing unit root from stationary processes based on panel data models with cross-sectional dependence

In this paper we develop a Bayesian approach to detecting unit roots in autoregressive panel data models. Our method is based on the comparison of stationary autoregressive models with and without individual deterministic trends, to their counterpart models with a unit autoregressive root. This is done under cross-sectional dependence among the error terms of the panel units. Simulation experiments are conducted with the aim to assess the performance of the suggested inferential procedure, as well as to investigate if the Bayesian model comparison approach can distinguish unit root models from stationary autoregressive models under cross-sectional dependence. The approach is applied to real exchange rate series for a panel of the G7 countries and to a panel of US nominal interest rates data.     

Panel Seasonal Unit Root Test: Further Simulation Results and An Application to Unemployment Data

Summary: In this paper the seasonal unit root test of Hylleberg et al. (1990) is generalized to cover a heterogenous panel. The procedure follows the work of Im, Pesaran and Shin (2002) and is independently proposed by Otero et al. (2004). Test statistics are given and critical values are obtained by simulation. Moreover, the properties of the tests are analyzed for different deterministic and dynamic specifications. Evidence is presented that for a small time series dimension the power is low even for increasing cross section dimension. Therefore, it seems necessary to have a higher time series dimension than cross section dimension. The test is applied to unemployment data in industrialized countries. In some cases seasonal unit roots are detected. However, the null hypotheses of panel seasonal unit roots are rejected. The null hypothesis of a unit root at the zero frequency is not rejected, thereby supporting the presence of hysteresis effects. * The research of this paper was supported by the Deutsche Forschungsgemeinschaft. The paper was presented at the workshop “Unit roots and cointegration in panel data” in Frankfurt, October 2004 and in the poster-session at the EC2 meeting in Marseille, December 2004. We are grateful to the participants of the workshops and an anonymous referee for their helpful comments.     

Unit Root Testing in Heteroscedastic Panels Using the Cauchy Estimator

The Cauchy estimator of an autoregressive root uses the sign of the first lag as instrumental variable. The resulting IV t-type statistic follows a standard normal limiting distribution under a unit root case even under unconditional heteroscedasticity, if the series to be tested has no deterministic trends. The standard normality of the Cauchy test is exploited to obtain a standard normal panel unit root test under cross-sectional dependence and time-varying volatility with an orthogonalization procedure. The article’s analysis of the joint N, T asymptotics of the test suggests that (1) N should be smaller than T and (2) its local power is competitive with other popular tests. To render the test applicable when N is comparable with, or larger than, T, shrinkage estimators of the involved covariance matrix are used. The finite-sample performance of the discussed procedures is found to be satisfactory.     

中国人口出生率下降与经济发展

中国人口出生率下降以及人口老龄化引起了人们对计划生育政策调整的激烈讨论。作者根据全国27个地区1980~2011年的面板数据,在横截面存在相关性的情形下,运用面板单位根和协整方法,定量考察了人口出生率、人口死亡率、城镇职工工资水平、农村居民收入水平之间的关系。研究发现：1）尽管我国实行了计划生育政策,但人口出生率与经济发展变量之间仍然存在着长期稳定的关系;2）城市和农村的经济发展对于人口出生率的影响存在显著差异。前者会刺激人口出生率的提高,但后者会降低人口出生率,且影响程度比前者大。3）人口出生率与经济发展变量之间的关系存在明显的地区差异。根据以上结论,作者认为放松计划生育政策不会引起我国人口数量猛增,反而会优化人口结构,提高人口整体素质。     

Robust block bootstrap panel predictability tests*

This article develops two block bootstrap-based panel predictability test procedures that are valid under very general conditions. Some of the allowable features include cross-sectional dependence, heterogeneous predictive slopes, persistent predictors, and complex error dynamics, including cross-unit endogeneity. While the first test procedure tests if there is any predictability at all, the second procedure determines the units for which predictability holds in case of a rejection by the first. A weak unit root framework is adopted to allow persistent predictors, and a novel theory is developed to establish asymptotic validity of the proposed bootstrap. Simulations are used to evaluate the performance of our tests in small samples, and their implementation is illustrated through an empirical application to stock returns.     

The Estimation and Inference of a Panel Cointegration Model with a Time Trend

This article investigates the asymptotic properties of coefficient estimators in the panel cointegration model with a time trend. We find that the bias of OLS estimator for the slope coefficient in the panel cointegration model with a time trend is distinct from that in the panel cointegration model without a time trend. Meanwhile, the variance of the limiting distribution for the slope coefficient is larger in the panel cointegration model with a time trend than without a time trend.     

Comparisons of tests for multivariate cointegration

This paper compares the small sample properties of different tests for multivariate cointegration like Johansen's trace test, stock &; Watson's common trend test, Phillips &; Ouliaris' principal component test, as well as cointegration rank decisions based on order selection criteria. Under the null hypothesis of non-cointegration we find a slow convergence rate of the test statistics. In bivariate models the Phillips &; Ouliaris test is extremely dependent on the specification and is outperformed by the other procedures. For trivariate processes we find dependence of the power results on the dynamic specification. The lag order is successfully estimated by order selection criteria.     

Testing for stationarity in large panels with cross-dependence,and US evidence on unit labor cost

A new stationarity test for heterogeneous panel data with large cross-sectional dimension is developed and used to examine a panel with growth rates of unit labor cost in the USA. The test allows for strong cross-unit dependence in the form of unbounded long-run correlation matrices, for which a simple parameterization is proposed. A KPSS-type distribution results asymptotically if letting T→∞ be followed by N→∞. Some evidence against stationarity (short memory) is found for the examined series.     

Local power of panel unit root tests allowing for structural breaks

The asymptotic local power of least squares–based fixed-T panel unit root tests allowing for a structural break in their individual effects and/or incidental trends of the AR(1) panel data model is studied. Limiting distributions of these tests are derived under a sequence of local alternatives, and analytic expressions show how their means and variances are functions of the break date and the time dimension of the panel. The considered tests have nontrivial local power in a N^?1/2 neighborhood of unity when the panel data model includes individual intercepts. For panel data models with incidental trends, the power of the tests becomes trivial in this neighborhood. However, this problem does not always appear if the tests allow for serial correlation in the error term and completely vanishes in the presence of cross-section correlation. These results show that fixed-T tests have very different theoretical properties than their large-T counterparts. Monte Carlo experiments demonstrate the usefulness of the asymptotic theory in small samples.     

A Meta Analytic Approach to Testing for Panel Cointegration

We propose new tests for panel cointegration by extending the panel unit root tests of Choi (2001 Choi , I. ( 2001 ). Unit root tests for panel data . Journal of International Money and Finance 20 ( 2 ): 249 – 272 .[Crossref], [Web of Science ®] , [Google Scholar]) and Maddala and Wu (1999 Maddala , G. , Wu , S. ( 1999 ). A comparative study of unit root tests with panel data and a new simple test . Oxford Bulletin of Economics and Statistics 61 ( S1 ): 631 – 652 .[Crossref] , [Google Scholar]) to the panel cointegration case. The tests are flexible, intuitively appealing, and relatively easy to compute. We investigate the finite sample behavior in a simulation study. Several variants of the tests compare favorably in terms of both size and power with other widely used panel cointegration tests.