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Tests for the existence of group effects and interactions for two-way models with dependent errors
Abstract
In this paper, we propose tests for the existence of random effects and interactions for two-way models with dependent errors. We prove that the proposed tests are asymptotically distribution-free which have asymptotically size and are consistent. We elucidate the nontrivial power under the local alternative when a sample size tends to infinity and the number of groups is fixed. A simulation study is performed to investigate the finite-sample performance of the proposed tests. In the real data analysis, we apply our tests to the daily log-returns of 24 stock prices from six countries and four sectors. We find that there is no strong evidence to support the existence of substantial differences in the log-return across countries, nor to the existence of interactions between countries and sectors. However, there exists random effect differences in the daily log-return series across different sectors.
... Environmental and meteorology studies are other application areas where periodic models show great promise, see Ghysels (1994); Franses and Paap (1994). In some real data, the periodic models may be suited better than random models which proposed by many authors such as, Beran et al. (1996); Akharif et al. (2020);Ou Larbi et al. (2021); Goto et al. (2023); Regui et al. (2024). The present paper focuses on obtaining, using the ULAN property satisfied by a periodic regression model, an adaptive estimator of the parameters of a periodic regression model in short panel data. ...
This paper proposes the use of the adaptive estimation method for estimating the periodic regression parameters in short panel data. This will go through three phases. The first phase aims to show that the periodic regression model verifies the Uniform Local Asymptotic Normality (ULAN), the second phase focuses on constructing the local asymptotically Minimax (LAM) estimators, and the last phase deals with constructing the Adaptive Estimators (AE) of the periodic regression model using the results of phase one and phase two. The results obtained in the simulation show that the Adaptive Estimator is always better than the Least Squares Estimator. The AE is more efficient in the case of an asymmetric score function. Real data are used to compare the two methods and show that the periodic coefficient regression model outperforms both traditional regression and random regression models.
This chapter extends the one-way models with time-dependent errors and correlated groups dealt with in Chapter 4 to the two-way models with time-dependent errors and correlated groups. We propose a test for the existence of random effects in Section 5.1 and a test for the existence of random interactions in Section 5.2. We show that the tests have asymptotically size and are consistent. The nontrivial power under the local alternative is elucidated. These tests can be applied to the fixed and mixed effect models. This chapter is mainly based on Goto et al. (2023).
In this paper we consider estimation of common structural breaks in panel data models with interactive fixed effects which are unobservable. We introduce a penalized principal component (PPC) estimation procedure with an adaptive group fused LASSO to detect the multiple structural breaks in the models. Under some mild conditions, we show that with probability approaching one the proposed method can correctly determine the unknown number of breaks and consistently estimate the common break dates. Furthermore, we estimate the regression coefficients through the post-LASSO method and establish the asymptotic distribution theory for the resulting estimators. The developed methodology and theory are applicable to the case of dynamic panel data models. The Monte Carlo simulation results demonstrate that the proposed method works well in finite samples with low false detection probability when there is no structural break and high probability of correctly estimating the break numbers when the structural breaks exist. We finally apply our method to study the environmental Kuznets curve for 74 countries over 40 years and detect two breaks in the data. * The authors would like to thank the editor, an associate editor and two referees for the helpful and insightful comments which greatly improve an earlier version of the paper.
1 Continuity of the Moore{Penrose inverse The paper gives an elementary proof of the theorem on the continuity of the Moore{ Penrose inverse in a C⁄-algebra that does not require the concept of the conorm, but uses instead a C⁄ modiflcation of Izumino's inequality kbyk • 4kayk valid when a; b have the Moore{Penrose inverse and satisfy the inequalities kb ¡ ak < 1 2ka yk¡1 and kbby ¡ aayk < 1. The paper then studies the conditions for the difierentiability of the Moore{Penrose inverse in a C⁄-algebra and gives an explicit formula for the derivative. The Moore{Penrose inverse of an element a of a unital C⁄-algebra A with the unit e is the unique element ay of A satisfying the equations
We consider the problem of testing for the existence of fixed effects and random effects in one-way models, where the groups are correlated and the disturbances are dependent. The classical F-statistic in the analysis of variance is not asymptotically distribution-free in this setting. To overcome this problem, we propose a new test statistic for this problem without any distributional assumptions, so that the test statistic is asymptotically distribution-free. The proposed test statistic takes the form of a natural extension of the classical F-statistic in the sense of distribution-freeness. The new tests are shown to be asymptotically size α and consistent. The nontrivial power under local alternatives is also elucidated. The theoretical results are justified by numerical simulations for the model with disturbances from linear time series with innovations of symmetric random variables, heavy-tailed variables, and skewed variables, and furthermore from GARCH models. The proposed test is applied to log-returns for stock prices and uncovers random effects in sectors.
Supplementary information:
The online version contains supplementary material available at 10.1007/s11749-022-00828-9.
Extending rank-based inference to a multivariate setting such as multiple-output regression or MANOVA with unspecified d-dimensional error density has remained an open problem for more than half a century. None of the many solutions proposed so far is enjoying the combination of distribution-freeness and efficiency that makes rank-based inference a successful tool in the univariate setting. A concept of center-outward multivariate ranks and signs based on measure transportation ideas has been introduced recently. Center-outward ranks and signs are not only distribution-free but achieve in dimension d > 1 the (essential) maximal ancillarity property of traditional univariate ranks. In the present case, we show that fully distribution-free testing procedures based on center-outward ranks can achieve parametric efficiency. We establish the Hájek representation and asymptotic normality results required in the construction of such tests in multiple-output regression and MANOVA models. Simulations and an empirical study demonstrate the excellent performance of the proposed procedures.
Factorial experiments are well-understood when the given observations are outcomes of random variables. However, when we observe spatial point patterns in each combination of factors cells, the methodology is much less developed. Motivated by a real problem of locations of bubbles in a mineral flotation experiment where the interest is analysing if the spatial distribution might be affected by frother concentrations and volumetric airflow rates, we develop an approach for statistical testing of two-way factorial experiments for spatial point patterns. We describe the point patterns through the K-function, a second-order summary statistic, and develop a set of new Fisher-based statistics using weighted means. For inference by Monte Carlo, we use random permutations of weighted residuals depending on the null hypothesis. We conduct simulation experiments to demonstrate the performance of the new test statistics and present the results of the real problem.
This textbook offers a comprehensive introduction to panel data econometrics, an area that has enjoyed considerable growth over the last two decades. Micro and Macro panels are becoming increasingly available, and methods for dealing with these types of data are in high demand among practitioners. Software programs have fostered this growth, including freely available programs in R and numerous user-written programs in both Stata and EViews.
Written by one of the world’s leading researchers and authors in the field, Econometric Analysis of Panel Data has established itself as the leading textbook for graduate and postgraduate courses on panel data. It provides up-to-date coverage of basic panel data techniques, illustrated with real economic applications and datasets, which are available at the book’s website on springer.com.
This new sixth edition has been fully revised and updated, and includes new material on dynamic panels, limited dependent variables and nonstationary panels, as well as spatial panel data. The author also provides empirical illustrations and examples using Stata and EViews.
“This is a definitive book written by one of the architects of modern, panel data econometrics. It provides both a practical introduction to the subject matter, as well as a thorough discussion of the underlying statistical principles without taxing the reader too greatly."
Professor Kajal Lahiri, State University of New York, Albany, USA.
"This book is the most comprehensive work available on panel data. It is written by one of the leading contributors to the field, and is notable for its encyclopaedic coverage and its clarity of exposition. It is useful to theorists and to people doing applied work using panel data. It is valuable as a text for a course in panel data, as a supplementary text for more general courses in econometrics, and as a reference."
Professor Peter Schmidt, Michigan State University, USA.
“Panel data econometrics is in its ascendancy, combining the power of cross section averaging with all the subtleties of temporal and spatial dependence. Badi Baltagi provides a remarkable roadmap of this fascinating interface of econometric method, enticing the novitiate with technical gentleness, the expert with comprehensive coverage and the practitioner with many empirical applications.”
Professor Peter C. B. Phillips, Cowles Foundation, Yale University, USA.
Random coefficient regression models are the regression counterparts of the classical random effects models in Analysis of Variance and panel data analysis. While several heuristic methods have been proposed for the detection of such random regression coefficients, little is known on their optimality properties. Based on a nonstandard ULAN property, we are proposing locally asymptotically optimal (in the Hájek-Le Cam sense) parametric, pseudo-Gaussian, and rank-based procedures for this problem. The asymptotic relative efficiencies (with respect to the pseudo-Gaussian procedure) of rank-based tests turn out to be quite high under heavy-tailed and skewed densities, demonstrating the importance of a careful choice of scores. Simulations reveal the excellent finite-sample performances of a class of rank-based procedures based on data-driven scores.
Spectral analysis of multivariate time series has been an active field of methodological and applied statistics for the past 50 years. Since the success of the fast Fourier transform algorithm, the analysis of serial auto- and cross-correlation in the frequency domain has helped us to understand the dynamics in many serially correlated data without necessarily needing to develop complex parametric models. In this work, we give a nonexhaustive review of the mostly recent nonparametric methods of spectral analysis of multivariate time series, with an emphasis on model-based approaches. We try to give insights into a variety of complimentary approaches for standard and less standard situations (such as nonstationary, replicated, or high-dimensional time series), discuss estimation aspects (such as smoothing over frequency), and include some examples stemming from life science applications (such as brain data).
Expected final online publication date for the Annual Review of Statistics and Its Application, Volume 7 is March 7, 2020. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
This study establishes a new approach for the analysis of variance (ANOVA) of time series. ANOVA has been sufficiently tailored for cases with independent observations, but there has recently been substantial demand across many fields for ANOVA in cases with dependent observations. For example, ANOVA for dependent observations is important to analyze differences among industry averages within financial data. Despite this demand, the study of ANOVA for dependent observations is more nascent than that of ANOVA for independent observations, and, thus, in this analysis, we study ANOVA for dependent observations. Specifically, we show the asymptotics of classical tests proposed for independent observations and give a sufficient condition for the observations to be asymptotically distributed. If this sufficient condition is not satisfied, we suggest a likelihood ratio test based on the Whittle likelihood and derive an asymptotic distribution of our test. Finally, we provide some numerical examples using simulated and real financial data as applications of these results.
Panel data analysis is an important topic in statistics and econometrics. In such analysis, it is very common to assume the impact of a covariate on the response variable remains constant across all individuals. While the modelling based on this assumption is reasonable when only the global effect is of interest, in general, it may overlook some individual/subgroup attributes of the true covariate impact. In this paper, we propose a data driven approach to identify the groups in panel data with interactive effects induced by latent variables. It is assumed that the impact of a covariate is the same within each group, but different between the groups. An EM based algorithm is proposed to estimate the unknown parameters, and a binary segmentation based algorithm is proposed to detect the grouping. We then establish asymptotic theories to justify the proposed estimation, grouping method, and the modelling idea. Simulation studies are also conducted to compare the proposed method with the existing approaches, and the results obtained favour our method. Finally, the proposed method is applied to analyse a data set about income dynamics, which leads to some interesting findings.
Time series data obtained from neurophysiological signals is often high-dimensional and the length of the time series is often short relative to the number of dimensions. Thus, it is difficult or sometimes impossible to compute statistics that are based on the spectral density matrix because estimates of these matrices are often numerically unstable. In this work, we discuss the importance of regularization for spectral analysis of high dimensional time series and propose shrinkage estimation for estimating high dimensional spectral density matrices. We use and develop the multivariate Time-frequency Toggle (TFT) bootstrap procedure for multivariate time series to estimate the shrinkage parameters, and show that the multivariate TFT bootstrap is theoretically valid. We show via simulations and an fMRI data set that failure to regularize the estimates of the spectral density matrix can yield unstable statistics, and that this can be alleviated by shrinkage estimation. © 2014, Institute of Mathematical Statistics. All rights reserved.
The partially linear additive model arises in many scientific endeavors. We look at inference given panel data and a serially correlated error component structure. By combining polynomial spline series approximation with least squares and the estimation of correlations, we propose a weighted semiparametric least squares estimator (WSLSE) for the parametric components, and a weighted polynomial spline series estimator (WPSSE) for the nonparametric components. The WSLSE is shown to be asymptotically normal and more efficient than the unweighted one. In addition, based on the WSLSE and WPSSE, a two-stage local polynomial estimator (TSLLE) of the nonparametric components is proposed that takes both contemporaneous correlations and additive structures into account. The TSLLE has several advantages, including higher asymptotic efficiency and an oracle property that achieves the same asymptotic distribution of each additive component as if the parametric and other nonparametric components were known with certainty. Some simulation studies were conducted to illustrate the finite sample performance of the proposed procedure. An example of applications to a set of panel data from a wage study is illustrated.
When analysing periodic data, there are not one but two time intervals to be considered. Often these intervals correspond to seasons and years. Specifically, one expects relationships to exist (a) between observations for successive seasons in a particular year, and (b) between the observations for the same season in successive years. Periodic data of this type may be tabulated in s rows due to s seasons and n columns due to n years, where s is the periodicity of the data. The usual two-way analysis of variance for examining the seasonal and annual effects may be highly misleading because of the correlation among observations within rows as well as within columns. In the present paper, we propose modified F-tests for testing the presence of row and column effects in a two-way layout assuming that observations follow a seasonal ARIMA process. A Gaussian approximation to the distribution of a linear combination of quadratic forms is proposed to obtain percentile values of the test statistics. The test is illustrated with the airline series. /// Notre étude porte sur l'analyse de données périodiques pour lesquelles deux intervalles de temps plutôt qu'un doivent être pris enconsideration. Il arrive souvent que ces intervalles soient des saisons et des années. De façon plus spécifique, on s'attend à ce qu'il y ait des relations (a) entre les observations portant sur les saisons successives au cours d'une même années et (b) entre les observations portant sur les mêmes saisons au cours d'années successives. Des données périodiques de ce caractère peuvent être représentées par un tableau comportant s rangées pour les saisons et n colonnes pour les année. La technique habituelle de l'analyse de la variance à deux modalités peut mener à des conclusions trompeuses lorsque l'on veut determiner la présence d'effets annuels et saisonniers, ceci étant dû à la corrélation des données d'une même rangée et d'une même colonne. Nous proposons dans cet article des tests de Fisher modifiés qui permettent de déterminer la présence d'effets de rangées et de colonnes dans le contexte d'une analyse à deux modalités et ce, en supposant que les observations soient issues d'un processus saisonnier du type 'ARIMA'. Nous proposons une approximation normale de la distribution d'une combinaison linéaire de formes quadratiques afin d'obtenir des valeurs critiques pour la statistique obtenue. En guise d'application, on analyse des données provenant de compagnies aériennes.
An insightful approach to the analysis of variance in the study of linear models. Linear Models explores the theory of linear models and the dynamic relationships that these models have with Analysis of Variance (ANOVA), experimental design, and random and mixed-model effects. This one-of-a-kind book emphasizes an approach that clearly explains the distribution theory of linear models and experimental design starting from basic mathematical concepts in linear algebra. The author begins with a presentation of the classic fixed-effects linear model and goes on to illustrate eight common linear models, along with the value of their use in statistics. From this foundation, subsequent chapters introduce concepts pertaining to the linear model, starting with vector space theory and the theory of least-squares estimation. An outline of the Helmert matrix is also presented, along with a thorough explanation of how the ANOVA is created in both typical two-way and higher layout designs, ultimately revealing the distribution theory. Other important topics covered include: Vector space theory The theory of least squares estimation Gauss-Markov theorem Kronecker products Diagnostic and robust methods for linear models Likelihood approaches to estimation A discussion of Bayesian theory is also included for purposes of comparison and contrast, and numerous illustrative exercises assist the reader with uncovering the nature of the models, using both classic and new data sets. Requiring only a working knowledge of basic probability and statistical inference, Linear Models is a valuable book for courses on linear models at the upper-undergraduate and graduate levels. It is also an excellent reference for practitioners who use linear models to conduct research in the fields of econometrics, psychology, sociology, biology, and agriculture.
This paper considers the maximum likelihood estimation of panel data models
with interactive effects. Motivated by applications in economics and other
social sciences, a notable feature of the model is that the explanatory
variables are correlated with the unobserved effects. The usual within-group
estimator is inconsistent. Existing methods for consistent estimation are
either designed for panel data with short time periods or are less efficient.
The maximum likelihood estimator has desirable properties and is easy to
implement, as illustrated by the Monte Carlo simulations. This paper develops
the inferential theory for the maximum likelihood estimator, including
consistency, rate of convergence and the limiting distributions. We further
extend the model to include time-invariant regressors and common regressors
(cross-section invariant). The regression coefficients for the time-invariant
regressors are time-varying, and the coefficients for the common regressors are
cross-sectionally varying.
We study asymptotic results for F tests in analysis of variance models as the number of factor levels goes to ∞ but the number of observations for each factor combination is fixed. Asymptotic derivations of the type discussed in this article would be relevant whenever both the numerator and denominator degrees of freedom go to ∞ (at the same rate). We consider null and alternative distributions of F, the usual F statistic, for fixed-effects and random-effects, balanced and unbalanced, one-way and two-way, and normal and nonnormal analysis of variance (ANOVA) models. The results may be most relevant for random-effects and mixed models. For example, we may have an agricultural experiment in which the number of cows is quite large but the number of measurements on each cow is small. The results would also be relevant for fixed-effects models in which there are many factor levels but not many observations for each factor level.
In this paper we propose a bootstrap method for panel data linear regression models with individual fixed effects. The method consists of applying the standard moving blocks bootstrap of Künsch (1989, Annals of Statistics 17, 1217–1241) and Liu and Singh (1992, in R. LePage & L. Billiard (eds.), Exploring the Limits of the Bootstrap) to the vector containing all the individual observations at each point in time. We show that this bootstrap is robust to serial and cross-sectional dependence of unknown form under the assumption that n (the cross-sectional dimension) is an arbitrary nondecreasing function of T (the time series dimension), where T → ∞, thus allowing for the possibility that both n and T diverge to infinity. The time series dependence is assumed to be weak (of the mixing type), but we allow the cross-sectional dependence to be either strong or weak (including the case where it is absent). Under appropriate conditions, we show that the fixed effects estimator (and also its bootstrap analogue) has a convergence rate that depends on the degree of cross-section dependence in the panel. Despite this, the same studentized test statistics can be computed without reference to the degree of cross-section dependence. Our simulation results show that the moving blocks bootstrap percentile-t intervals have very good coverage properties even when the degree of serial and cross-sectional correlation is large, provided the block size is appropriately chosen.
We propose a general bootstrap procedure to approximate the null distribution of non-parametric frequency domain tests about the spectral density matrix of a multivariate time series. Under a set of easy-to-verify conditions, we establish asymptotic validity of the bootstrap procedure proposed. We apply a version of this procedure together with a new statistic to test the hypothesis that the spectral densities of not necessarily independent time series are equal. The test statistic proposed is based on an "L"2-distance between the non-parametrically estimated individual spectral densities and an overall, 'pooled' spectral density, the latter being obtained by using the whole set of "m" time series considered. The effects of the dependence between the time series on the power behaviour of the test are investigated. Some simulations are presented and a real life data example is discussed. Copyright (c) 2009 Royal Statistical Society.
A central limit theorem is proved for the sample covariances of a linear process. The sufficient conditions for the theorem are described by more natural ones than usual. We apply this theorem to the parameter estimation of a fitted spectral model, which does not necessarily include the true spectral density of the linear process. We also deal with estimation problems for an autoregressive signal plus white noise. A general result is given for efficiency of Newton-Raphson iterations of the likelihood equation.
The author considers frequency domain time series analysis, where smoothing in nonparametric spectrum estimation is data-dependent. Uniform convergence of spectrum estimates is established and applied to a semiparametric model, parameterized over possibly only a subset of the frequencies, in which disturbances have nonparametric autocorrelation. Optimal instruments depend on the disturbance spectrum and frequency response function, which is nonparametric in incomplete systems. The author justifies feasible, optimal parameter estimates. The degree of smoothing is allowed to depend on the data in a general way. The author proves consistency of a cross-validation method of automatic smoothing and applies it to a semiparametric model. Copyright 1991 by The Econometric Society.
This paper proposes an econometric methodology to deal with life cycle earnings and mobility among discrete earnings classes. First, we use panel data on male log earnings to estimate an earnings function with permanent and serially correlated transitory components due to both measured and unmeasured variables. Assuming that the error components are normally distributed, we develop statements for the probability that an individual's earnings will fall into a particular but arbitrary time sequence of poverty states. Using these statements, we illustrate the implications of our earnings model for poverty dynamics and compare our approach to Markov chain models of income mobility.