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.     

Testing panel data regression models with spatial error correlation

This paper derives several lagrange multiplier (LM) tests for the panel data regression model with spatial error correlation. These tests draw upon two strands of earlier work. The first is the LM tests for the spatial error correlation model discussed in Anselin (Spatial Econometrics: Methods and Models, Kluwer Academic Publishers, Dordrecht; Rao's score test in spatial econometrics, J. Statist. Plann. Inference 97 (2001) 113) and Anselin et al. (Regional Sci. Urban Econom. 26 (1996) 77), and the second is the LM tests for the error component panel data model discussed in Breusch and Pagan (Rev. Econom. Stud. 47(1980) 239) and Baltagi et al. (J. Econometrics 54 (1992) 95). The idea is to allow for both spatial error correlation as well as random region effects in the panel data regression model and to test for their joint significance. Additionally, this paper derives conditional LM tests, which test for random regional effects given the presence of spatial error correlation. Also, spatial error correlation given the presence of random regional effects. These conditional LM tests are an alternative to the one-directional LM tests that test for random regional effects ignoring the presence of spatial error correlation or the one-directional LM tests for spatial error correlation ignoring the presence of random regional effects. We argue that these joint and conditional LM tests guard against possible misspecification. Extensive Monte Carlo experiments are conducted to study the performance of these LM tests as well as the corresponding likelihood ratio tests.     

Testing for heteroskedasticity and serial correlation in a random effects panel data model

This paper considers a panel data regression model with heteroskedastic as well as serially correlated disturbances, and derives a joint LM test for homoskedasticity and no first order serial correlation. The restricted model is the standard random individual error component model. It also derives a conditional LM test for homoskedasticity given serial correlation, as well as, a conditional LM test for no first order serial correlation given heteroskedasticity, all in the context of a random effects panel data model. Monte Carlo results show that these tests along with their likelihood ratio alternatives have good size and power under various forms of heteroskedasticity including exponential and quadratic functional forms.     

A REVIEW AND COMPARISON OF TESTS OF CROSS-SECTION INDEPENDENCE IN PANELS

Abstract.  In this paper we review and compare diagnostic tests of cross-section independence in the disturbances of panel regression models. We examine tests based on the sample pairwise correlation coefficient or on its transformations, and tests based on the theory of spacings. The ultimate goal is to shed some light on the appropriate use of existing diagnostic tests for cross-equation error correlation. Our discussion is supported by means of a set of Monte Carlo experiments and a small empirical study on health. Results show that tests based on the average of pairwise correlation coefficients work well when the alternative hypothesis is a factor model with non-zero mean loadings. Tests based on spacings are powerful in identifying various forms of strong cross-section dependence, but have low power when they are used to capture spatial correlation.     

Robust Small Area Estimation under Spatial Non‐stationarity

The effective use of spatial information in a regression‐based approach to small area estimation is an important practical issue. One approach to account for geographic information is by extending the linear mixed model to allow for spatially correlated random area effects. An alternative is to include the spatial information by a non‐parametric mixed models. Another option is geographic weighted regression where the model coefficients vary spatially across the geography of interest. Although these approaches are useful for estimating small area means efficiently under strict parametric assumptions, they can be sensitive to outliers. In this paper, we propose robust extensions of the geographically weighted empirical best linear unbiased predictor. In particular, we introduce robust projective and predictive estimators under spatial non‐stationarity. Mean squared error estimation is performed by two analytic approaches that account for the spatial structure in the data. Model‐based simulations show that the methodology proposed often leads to more efficient estimators. Furthermore, the analytic mean squared error estimators introduced have appealing properties in terms of stability and bias. Finally, we demonstrate in the application that the new methodology is a good choice for producing estimates for average rent prices of apartments in urban planning areas in Berlin.     

Goodness-of-fit tests in linear EV regression with replications

This paper proposes a goodness-of-fit test for checking the adequacy of parametric forms of the regression error density functions in linear errors-in-variables regression models. Instead of assuming the distribution of the measurement error to be known, we assume that replications of the surrogates of the latent variables are available. The test statistic is based upon a weighted integrated squared distance between a nonparametric estimator and a semi-parametric estimator of the density functions of certain residuals. Under the null hypothesis, the test statistic is shown to be asymptotically normal. Consistency and local power results of the proposed test under fixed alternatives and local alternatives are also established. Finite sample performance of the proposed test is evaluated via simulation studies. A real data example is also included to demonstrate an application of the proposed test.     

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.     

Misspecified models with dependent observations

General results are given in this paper which allow the development of a theory of estimation and inference for situations in which the model of a data-generating process has been misspecified. Observations may come from time-series, cross-section, panel, or experimental data. The nonlinear regression model is examined in some detail. Conditions are provided which ensure the consistency and asymptotic normality of the least-squares estimator with respect to the parameter vector of a weighted least-squares approximation to the underlying data-generating process. A specification-robust estimator of the asymptotic covariance matrix is given, allowing a proper treatment of inference in potentially misspecified models. The properties of the approximation and the covariance estimator are exploited to yield new tests for model specification.     

Small‐sample testing inference in symmetric and log‐symmetric linear regression models

This paper deals with the issue of testing hypotheses in symmetric and log‐symmetric linear regression models in small and moderate‐sized samples. We focus on four tests, namely, the Wald, likelihood ratio, score, and gradient tests. These tests rely on asymptotic results and are unreliable when the sample size is not large enough to guarantee a good agreement between the exact distribution of the test statistic and the corresponding chi‐squared asymptotic distribution. Bartlett and Bartlett‐type corrections typically attenuate the size distortion of the tests. These corrections are available in the literature for the likelihood ratio and score tests in symmetric linear regression models. Here, we derive a Bartlett‐type correction for the gradient test. We show that the corrections are also valid for the log‐symmetric linear regression models. We numerically compare the various tests and bootstrapped tests, through simulations. Our results suggest that the corrected and bootstrapped tests exhibit type I probability error closer to the chosen nominal level with virtually no power loss. The analytically corrected tests as well as the bootstrapped tests, including the Bartlett‐corrected gradient test derived in this paper, perform with the advantage of not requiring computationally intensive calculations. We present a real data application to illustrate the usefulness of the modified tests.     

A simple approach for the analysis of generalizea linear mixed models

A broad class of generalized linear mixed models, e.g. variance components models for binary data, percentages or count data, will be introduced by incorporating additional random effects into the linear predictor of a generalized linear model structure. Parameters are estimated by a combination of quasi-likelihood and iterated MINQUE (minimum norm quadratic unbiased estimation), the latter being numerically equivalent to REML (restricted, or residual, maximum likelihood). First, conditional upon the additional random effects, observations on a working variable and weights are derived by quasi-likelihood, using iteratively re-weighted least squares. Second, a linear mixed model is fitted to the working variable, employing the weights for the residual error terms, by iterated MINQUE. The latter may be regarded as a least squares procedure applied to squared and product terms of error contrasts derived from the working variable. No full distributional assumptions are needed for estimation. The model may be fitted with standardly available software for weighted regression and REML.     

Testing for Error Correction in Panel Data*

This paper proposes new error correction‐based cointegration tests for panel data. The limiting distributions of the tests are derived and critical values provided. Our simulation results suggest that the tests have good small‐sample properties with small size distortions and high power relative to other popular residual‐based panel cointegration tests. In our empirical application, we present evidence suggesting that international healthcare expenditures and GDP are cointegrated once the possibility of an invalid common factor restriction has been accounted for.     

Testing a linear dynamic panel data model against nonlinear alternatives

The most popular econometric models in the panel data literature are the class of linear panel data models with unobserved individual- and/or time-specific effects. The consistency of parameter estimators and the validity of their economic interpretations as marginal effects depend crucially on the correct functional form specification of the linear panel data model. In this paper, a new class of residual-based tests is proposed for checking the validity of dynamic panel data models with both large cross-sectional units and time series dimensions. The individual and time effects can be fixed or random, and panel data can be balanced or unbalanced. The tests can detect a wide range of model misspecifications in the conditional mean of a dynamic panel data model, including functional form and lag misspecification. They check a large number of lags so that they can capture misspecification at any lag order asymptotically. No common alternative is assumed, thus allowing for heterogeneity in the degrees and directions of functional form misspecification across individuals. Thanks to the use of panel data with large $N$ and $T$, the proposed nonparametric tests have an asymptotic normal distribution under the null hypothesis without requiring the smoothing parameters to grow with the sample sizes. This suggests better nonparametric asymptotic approximation for the panel data than for time series or cross sectional data. This is confirmed in a simulation study. We apply the new tests to test linear specification of cross-country growth equations and found significant nonlinearities in mean for OECD countries’ growth equation for annual and quintannual panel data.     

Optimal critical values of pre-tests when estimating the regression error variance: analytical findings under a general loss structure

This paper re-visits the problem of estimating the regression error variance in a linear multiple regression model after preliminary hypothesis tests for either linear restrictions on the coefficients or homogeneity of variances. There is an extensive literature that discusses these problems, particularly in terms of the sampling properties of the pre-test estimators using various loss functions as the basis for risk analysis. In this paper, a unified framework for analysing the risk properties of these estimators is developed under a general class of loss structures that incorporates virtually all first-order differentiable losses. Particular consideration is given to the choice of critical values for the pre-tests. Analytical results indicate that an α-level substantially higher than those normally used may be appropriate for optimal risk properties under a wide range of loss functions. The paper also generalizes some known analytical results in the pre-test literature and proves other results only previously shown numerically.     

Efficient estimation for error component seemingly unrelated nonparametric regression models

Multivariate panel data provides a unique opportunity in studying the joint evolution of multiple response variables over time. In this paper, we propose an error component seemingly unrelated nonparametric regression model to fit the multivariate panel data, which is more flexible than the traditional error component seemingly unrelated parametric regression. By applying the undersmoothing technique and taking both of the correlations within and among responses into account, we propose an efficient two-stage local polynomial estimation for the unknown functions. It is shown that the resulting estimators are asymptotically normal, and have the same biases as the standard local polynomial estimators, which are only based on the individual response, and smaller asymptotic variances. The performance of the proposed procedure is evaluated through a simulation study and a real data set.     

Comparison of intraday probabilistic forecasting of solar irradiance using only endogenous data

Accurate solar forecasts are necessary to improve the integration of solar renewables into the energy grid. In recent years, numerous methods have been developed for predicting the solar irradiance or the output of solar renewables. By definition, a forecast is uncertain. Thus, the models developed predict the mean and the associated uncertainty. Comparisons are therefore necessary and useful for assessing the skill and accuracy of these new methods in the field of solar energy.The aim of this paper is to present a comparison of various models that provide probabilistic forecasts of the solar irradiance within a very strict framework. Indeed, we consider focusing on intraday forecasts, with lead times ranging from 1 to 6 h. The models selected use only endogenous inputs for generating the forecasts. In other words, the only inputs of the models are the past solar irradiance data. In this context, the most common way of generating the forecasts is to combine point forecasting methods with probabilistic approaches in order to provide prediction intervals for the solar irradiance forecasts. For this task, we selected from the literature three point forecasting models (recursive autoregressive and moving average (ARMA), coupled autoregressive and dynamical system (CARDS), and neural network (NN)), and seven methods for assessing the distribution of their error (linear model in quantile regression (LMQR), weighted quantile regression (WQR), quantile regression neural network (QRNN), recursive generalized autoregressive conditional heteroskedasticity (GARCHrls), sieve bootstrap (SB), quantile regression forest (QRF), and gradient boosting decision trees (GBDT)), leading to a comparison of 20 combinations of models.None of the model combinations clearly outperform the others; nevertheless, some trends emerge from the comparison. First, the use of the clear sky index ensures the accuracy of the forecasts. This derived parameter permits time series to be deseasonalized with missing data, and is also a good explanatory variable of the distribution of the forecasting errors. Second, regardless of the point forecasting method used, linear models in quantile regression, weighted quantile regression and gradient boosting decision trees are able to forecast the prediction intervals accurately.     

Large panels with common factors and spatial correlation

This paper considers methods for estimating the slope coefficients in large panel data models that are robust to the presence of various forms of error cross-section dependence. It introduces a general framework where error cross-section dependence may arise because of unobserved common effects and/or error spill-over effects due to spatial or other forms of local dependencies. Initially, this paper focuses on a panel regression model where the idiosyncratic errors are spatially dependent and possibly serially correlated, and derives the asymptotic distributions of the mean group and pooled estimators under heterogeneous and homogeneous slope coefficients, and for these estimators proposes non-parametric variance matrix estimators. The paper then considers the more general case of a panel data model with a multifactor error structure and spatial error correlations. Under this framework, the Common Correlated Effects (CCE) estimator, recently advanced by Pesaran (2006), continues to yield estimates of the slope coefficients that are consistent and asymptotically normal. Small sample properties of the estimators under various patterns of cross-section dependence, including spatial forms, are investigated by Monte Carlo experiments. Results show that the CCE approach works well in the presence of weak and/or strong cross-sectionally correlated errors.     

Good neighbours matter: economic geography and the diffusion of human rights

Using multi-country panel data, this paper investigates the geopolitical and economic aspects of human rights performance. Human rights performance depends on the relative levels of economic development and spatial proximity to ‘good’ and ‘bad’ neighbours. The paper tests for basic effects of income, and applies spatial weighting models, to analyse the neighbours’ impact on human rights levels, treating this impact as partly endogenous. It takes into account size and distance when comparing each country’s human rights performance with what would be predicted from a weighted average of its neighbours’ performance. There are (1) geographical clusters and (2) size and proximity effects for human rights performance.     

Exact inference for the linear model with groupwise heteroscedastic spherical disturbances

Exact nonparametric inference on a single coefficient in a linear regression model, as considered by Bekker (Working Paper, Department of Economics, University of Groningen, 1997), is elaborated for the case of spherically distributed heteroscedastic disturbances. Instead of approximate inference based on feasible weighted least squares, exact inference is formulated based on partial rotational invariance of the distribution of the vector of disturbances. Thus, classical exact inference based on t-statistics is generalized to exact inference that remains valid in a groupwise heteroscedastic context. The approach is applied to a basic two-sample problem, and to the random- and fixed-effects models for panel data.     

Error Correction Testing in Panels with Common Stochastic Trends

This paper develops panel data tests for the null hypothesis of no error correction in a model with common stochastic trends. The asymptotic distributions of the new test statistics are derived and simulation results are provided to suggest that they perform well in small samples. Copyright © 2015 John Wiley & Sons, Ltd.     

Asymptotically optimal tests for some growth curve models under non-normal error structure

This paper considers growth curve models consisting of two stages. In the first stage we have a regression model with random parameter. The second stage is given by a linear model for the mean of the random parameter of the first stage. In the present paper we prove asymptotic optimality of tests for linear functions of the mean of the random parameter under non-normal error. Received: September 1999