Consistency,asymptotic unbiasedness and bounds on the bias of s2 in the linear regression model with error component disturbances

The OLS estimator of the disturbance variance in the linear regression model with error component disturbances is shown to be weakly consistent and asymptotically unbiased without any restrictions on the regressor matrix. Also, simple exact bounds on the expected value of s² are given for both the one-way and two-way error component models.     

Testing for random individual effects using recursive residuals

This paper derives a simple ANOVA-F-statistic which tests for random individual effects in a one-way error component model, using recursive residuals. Power comparisons are performed for this F-test when it is computed using true disturbances and recursive residuals from a panel data regression. Under the null, both statistics have an exact F distribution. The standardized version of the Breusch and Pagan (1980) Lagrange Multiplier test (SLM) as well as a fixed effects F-statistic (FE) recommended by Moulton and Randolph (1989), are also included in this comparison. The exact power function can be computed in all cases using Imhof's (1961) procedure. Our results suggest that the F-test based on recursive residuals is inferior to the popular SLM and FE tests based on computational simplicity, power comparisons and its sensitivity to the K observations starting the recursion.     

A Parametric Bootstrap Test for Two-Way ANOVA Model Without Interaction Under Heteroscedasticity

In this article we consider the two-way ANOVA model without interaction under heteroscedasticity. For the problem of testing equal effects of factors, we propose a parametric bootstrap (PB) approach and compare it with existing the generalized F (GF) test. The Type I error rates and powers of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test performs better than the GF test. The PB test performs very satisfactorily even for small samples while the GF test exhibits poor Type I error properties when the number of factorial combinations or treatments goes up. It is also noted that the same tests can be used to test the significance of random effect variance component in a two-way mixed-effects model under unequal error variances.     

A Joint Score Test for Heteroscedasticity in the Two Way Error Components Model

This article extends the work by Holly and Gardiol (2000) (A score test for individual heteroscedasticity in a one-way error component model. In: Krishnakumar, J., Ronchetti, E., Eds. Panel Data Econometrics: Future Directions. Elsevier, North-Holland, Amsterdam, pp. 199–211, Ch. 10) to the two-way error components model. It deals exclusively with a joint heteroscedasticity test by first deriving Rao's efficient score statistics. Then, based on appropriate set of assumptions, we deduce the asymptotic distribution of the score under contiguous alternatives. Finally, we provide the expression for the score test statistic in the presence of heteroscedasticity and discuss its asymptotic local power.     

Estimating partially linear panel data models with one-way error components

We consider the problem of estimating a partially linear panel data model whenthe error follows an one-way error components structure. We propose a feasiblesemiparametric generalized least squares (GLS) type estimator for estimating the coefficient of the linear component and show that it is asymptotically more efficient than a semiparametric ordinary least squares (OLS) type estimator. We also discussed the case when the regressor of the parametric component is correlated with the error, and propose an instrumental variable GLS-type semiparametric estimator.     

A DISTRIBUTION-FREE TEST FOR TWO-WAY LAYOUT1

Many nonparametric tests have been proposed for the hypothesis of no row (treatment) effect in a one-way layout design. Examples of such tests are Kruskal-Wallis H-test, Bhapkar's (1961) V-test and Deshpande's (1965) L-test. However not many tests are available for testing the same hypothesis in a two-way layout design without interaction. Perhaps the only “established” test is the one due to Friedman (1937). However, it applies to the case of one observation per cell only. In this paper, a new distribution-free test is proposed for the hypothesis of row effect in a two-way layout design. It applies to the case of several observations per cell, not necessarily equal. The asymptotic efficiency of the proposed test relative to other tests is studied.     

Saddlepoint <Emphasis Type="Italic">p</Emphasis>-values and confidence intervals for a class of two sample permutation tests for current status and panel count data

The current status and panel count data frequently arise from cancer and tumorigenicity studies when events currently occur. A common and widely used class of two sample tests, for current status and panel count data, is the permutation class. We manipulate the double saddlepoint method to calculate the exact mid-p-values of the underlying permutation distributions of this class of tests. Permutation simulations are replaced by analytical saddlepoint computations which provide extremely accurate mid-p-values that are exact for most practical purposes and almost always more accurate than normal approximations. The method is illustrated using two real tumorigenicity panel count data. To compare the saddlepoint approximation with the normal asymptotic approximation, a simulation study is conducted. The speed and accuracy of the saddlepoint method facilitate the calculation of the confidence interval for the treatment effect. The inversion of the mid-p-values to calculate the confidence interval for the mean rate of development of the recurrent event is discussed.     

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 Monte Carlo simulation study of two-sided tolerance intervals in balanced one-way random effects model for non-normal errors

Recently, Ong and Mukerjee [Probability matching priors for two-sided tolerance intervals in balanced one-way and two-way nested random effects models. Statistics. 2011;45:403–411] developed two-sided Bayesian tolerance intervals, with approximate frequentist validity, for a future observation in balanced one-way and two-way nested random effects models. These were obtained using probability matching priors (PMP). On the other hand, Krishnamoorthy and Lian [Closed-form approximate tolerance intervals for some general linear models and comparison studies. J Stat Comput Simul. 2012;82:547–563] studied closed-form approximate tolerance intervals by the modified large-sample (MLS) approach. We compare the performances of these two approaches for normal as well as non-normal error distributions. Monte Carlo simulation methods are used to evaluate the resulting tolerance intervals with regard to achieved confidence levels and expected widths. It turns out that PMP tolerance intervals are less conservative for data with large number of classes and small number of observations per class and the MLS procedure is preferable for smaller sample sizes.     

Partially linear models and polynomial spline approximations for the analysis of unbalanced panel data

In this paper, we study the estimation of the unbalanced panel data partially linear models with a one-way error components structure. A weighted semiparametric least squares estimator (WSLSE) is developed using polynomial spline approximation and least squares. We show that the WSLSE is asymptotically more efficient than the corresponding unweighted estimator for both parametric and nonparametric components of the model. This is a significant improvement over previous results in the literature which showed that the simply weighting technique can only improve the estimation of the parametric component. The asymptotic normalities of the proposed WSLSE are also established.     

An LM-type test for idiosyncratic unit roots in the exact factor model with integrated factors

We consider an exact factor model with integrated factors and propose an LM-type test for unit roots in the idiosyncratic component. We show that, for a fixed number of panel individuals (N) and when the number of time points (T) tends to infinity, the limiting distribution of the LM-type statistic is a weighted sum of independent Chi-square variables with one degree of freedom, and when T tends to infinity followed by N tending to infinity, the limiting distribution is standard normal. The results should contribute to the challenging task of deriving likelihood-based unit-root tests in dynamic factor models.     

A Note on Two-Way ECM Estimation of SUR Systems on Unbalanced Panel Data

This article considers the two-way error components model (ECM) estimation of seemingly unrelated regressions (SUR) on unbalanced panel by generalized least squares (GLS). As suggested by Biørn (2004 Biørn , E. ( 2004 ). Regression systems for unbalanced panel data: a stepwise maximum likelihood procedure . Journal of Econometrics 122 : 281 – 291 . [Google Scholar]) for the one-way case, in order to use the standard results for the balanced case the individuals are arranged in groups according to the number of times they are observed. Thus, the GLS estimator can be interpreted as a matrix weighted average of the group specific GLS estimators with weights equal to the inverse of their respective covariance matrices.     

An adaptive test for the one-way layout

An adaptive test is proposed for the one-way layout. This test procedure uses the order statistics of the combined data to obtain estimates of percentiles, which are used to select an appropriate set of rank scores for the one-way test statistic. This test is designed to have reasonably high power over a range of distributions. The adaptive procedure proposed for a one-way layout is a generalization of an existing two-sample adaptive test procedure. In this Monte Carlo study, the power and significance level of the F-test, the Kruskal-Wallis test, the normal scores test, and the adaptive test were evaluated for the one-way layout. All tests maintained their significance level for data sets having at least 24 observations. The simulation results show that the adaptive test is more powerful than the other tests for skewed distributions if the total number of observations equals or exceeds 24. For data sets having at least 60 observations the adaptive test is also more powerful than the F-test for some symmetric distributions.     

The equivalence between likelihood ratio test and F-test for testing variance component in a balanced one-way random effects model

In mixed linear models, it is frequently of interest to test hypotheses on the variance components. F-test and likelihood ratio test (LRT) are commonly used for such purposes. Current LRTs available in literature are based on limiting distribution theory. With the development of finite sample distribution theory, it becomes possible to derive the exact test for likelihood ratio statistic. In this paper, we consider the problem of testing null hypotheses on the variance component in a one-way balanced random effects model. We use the exact test for the likelihood ratio statistic and compare the performance of F-test and LRT. Simulations provide strong support of the equivalence between these two tests. Furthermore, we prove the equivalence between these two tests mathematically.     

Two-way fixed effects analysis of variance when the error variances may be unequal

The method in Krutchkoff (1988b) for one-way ANOVA is extended to the two-way situation. The K- ANOVA application program is discussed.     

Conditional Score Tests for Heteroscedasticity in the Two-Way Error Components Model

This paper extends the one-way heteroskedasticity score test of Holly and Gardiol (2000, In: Krishnakumar, J, Ronchetti, E (Eds.), Panel Data Econometrics: Future Directions, North-Holland, Amsterdam, pp. 199–211) to two conditional Lagrange Multiplier (LM) tests of heteroskedasticity under contiguous alternatives within the two-way error components model framework. In each case, the derivation of Rao's efficient score statistics for testing heteroskedasticity is first obtained. Then, based on a specific set of assumptions, the asymptotic distribution of the score under contiguous alternatives is established. Finally, the expression for the score test statistic in the presence of heteroskedasticity and related asymptotic local powers of these score test statistics are derived and discussed.     

Testing for error cross-sectional uncorrelatedness in a two-way error components panel data model

This paper proposes a new test for the error cross-sectional uncorrelatedness in a two-way error components panel data model based on large panel data sets. By virtue of an existing statistic under the raw data circumstance, an analogous test statistic using the within residuals of the model is constructed. We show that the resulting statistic needs bias correction to make valid inference, and then propose a method to implement feasible correction. Simulation shows that the test based on the feasible bias-corrected statistic performs well. Additionally, we employ a real data set to illustrate the use of the new test.     

Parametric bootstrap inferences for unbalanced panel data models

This article presents parametric bootstrap (PB) approaches for hypothesis testing and interval estimation for the regression coefficients of panel data regression models with incomplete panels. Some simulation results are presented to compare the performance of the PB approaches with the approximate inferences. Our studies show that the PB approaches perform satisfactorily for various sample sizes and parameter configurations, and the performance of PB approaches is mostly better than the approximate methods with respect to the coverage probabilities and the Type I error rates. The PB inferences have almost exact coverage probabilities and Type I error rates. Furthermore, the PB procedure can be simply carried out by a few simulation steps, and the derivation is easier to understand and to be extended to the multi-way error component regression models with unbalanced panels. Finally, the proposed approaches are illustrated by using a real data example.     

The limiting distribution of the F-statistic from nonnormal universes

We consider a linear regression model with an unbalanced 1-fold nested error structure, where group effect and error are from nonnormal universes. The limiting distribution of the F-statistic in this model is derived, as the sample size is large and group sizes take values from a finite set of distinct integers. The result is used to approximate the F-distribution quantile and to test the significance of the random effect variance component. Results are also applicable to the F-statistic in the one-way random-effects model. The effects of departure from normality on the F-statistic distribution are given.     

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.