We propose a new kernel estimator for nonparametric regression with unknown error distribution. We show that the proposed estimator is adaptive in the sense that it is asymptotically equivalent to the infeasible local likelihood estimator (Staniswalis, 1989, Journal of the American Statistical Association 84, 276–283; Fan, Farmen, and Gijbels, 1998, Journal of the Royal Statistical Society, Series B 60, 591–608; and Fan and Chen, 1999, Journal of the Royal Statistical Society, Series B 61, 927–943), which requires knowledge of the error distribution. Hence, our estimator improves on standard nonparametric kernel estimators when the error distribution is not normal. A Monte Carlo experiment is conducted to investigate the finite-sample performance of our procedure.We thank Yuichi Kitamura, Yanqin Fan, Joel Horowitz, Roger Koenker, Jens Perch Nielsen, Peter Phillips, Peter Robinson, Tom Rothenberg, and two referees for helpful comments. Financial support from the NSF and the ESRC (UK) is gratefully acknowledged.

In this paper, we address two important issues in semiparametric survival model selection for censored data generated by the Archimedean copula family: method of estimating the parametric copulas and data reuse. We demonstrate that for selection among candidate copula models that might all be misspecified, estimators of the parametric copulas based on minimizing the selection criterion function may be preferred to other estimators. To handle the issue of data reuse, we put model selection in the context of hypothesis testing and propose a simple test for model selection from a finite number of parametric copulas. Results from a simulation study and two empirical illustrations confirm our theoretical findings.We thank the editor Peter Phillips, three anonymous referees, and Hal White for their comments, which greatly improved the paper. An earlier version of this paper was presented at the 2005 World Congress Meetings of the Econometric Society, the 2005 Joint Statistical Meetings, the University of Waterloo, and the University of Western Ontario. Chen acknowledges support from the National Science Foundation and the C.V. Starr Center at NYU. Fan acknowledges support from the National Science Foundation. We thank Demian Pouzo for excellent research assistance on the numerical work in this paper and Weijing Wang for providing us with the Fortran code for computing the bivariate Kaplan–Meier estimator of Dabrowska (1988).

In this paper we deal with the identification of an autoregressive model for an observed time series and the detection of a unit root in its characteristic polynomial. This is a big issue concerned with distinguishing stationary time series from time series for which differencing is required to induce stationarity. We consider a Bayesian approach, and particular attention is devoted to the problem of the sensitivity of the standard Bayesian analysis with respect to the choice of the prior distribution for the autoregressive coefficients.We thank three anonymous referees for their useful comments, which have improved the final version of the paper.

This paper deals with issues of structure and parametrization of VECH models proposed in Bollerslev, Engle, and Wooldridge (1988) and Baba, Engle, Kraft, and Kroner (BEKK) models. Both general models and also restricted versions such as the widely used diagonal VECH (DVECH) and factor generalized autoregressive conditional heteroskedastic (F-GARCH) models are discussed. A simple algorithm is presented that checks whether a given VECH model may be cast as a BEKK model. It is shown that in the bivariate case BEKK models are as general as VECH models. In higher dimensional cases however, VECH models allow for more flexibility. In addition, a parametrization for a generic, i.e., open and dense, class of BEKK models is given, and the frequently cited parametrization by Engle and Kroner (1995, Econometric Theory 11, 122–150) is analyzed. Two shortcomings of the latter are pointed out. Finally, parametrizations for BEKK(p,q,K) models with K ≤ n, including DVECH, F-GARCH, and a generalization of the latter, are discussed.This research was supported by the project P17065 “Identification of multivariate dynamic systems with a focus on dimension reduction” of the Austrian Science Foundation (FWF).

Generalized autoregressive conditional heteroskedasticity (GARCH) models with Markov-switching regimes are often used for volatility analysis of financial time series. Such models imply less persistence in the conditional variance than the standard GARCH model and potentially provide a significant improvement in volatility forecast. Nevertheless, conditions for asymptotic wide-sense stationarity have been derived only for some degenerated models. In this paper, we introduce a comprehensive approach for stationarity analysis of Markov-switching GARCH models, which manipulates a backward recursion of the model's second-order moment. A recursive formulation of the state-dependent conditional variances is developed, and the corresponding conditions for stationarity are obtained. In particular, we derive necessary and sufficient conditions for the asymptotic wide-sense stationarity of two different variants of Markov-switching GARCH processes and obtain expressions for their asymptotic variances in the general case of m-state Markov chains and (p,q)-order GARCH processes.The authors thank Professor Rami Atar for helpful discussions. The authors thank the co-editor Bruce Hansen and the three anonymous referees for their helpful comments and suggestions and in particular the referee who proposed a generalization of the proof in Appendix B. This research was supported by the Israel Science Foundation (grant 1085/05).

This paper focuses on the analysis of efficiency, peakedness, and majorization properties of linear estimators under heavy-tailedness assumptions. We demonstrate that peakedness and majorization properties of log-concavely distributed random samples continue to hold for convolutions of α-symmetric distributions with α > 1. However, these properties are reversed in the case of convolutions of α-symmetric distributions with α < 1.

We show that the sample mean is the best linear unbiased estimator of the population mean for not extremely heavy-tailed populations in the sense of its peakedness. In such a case, the sample mean exhibits monotone consistency, and an increase in the sample size always improves its performance. However, efficiency of the sample mean in the sense of peakedness decreases with the sample size if it is used to estimate the location parameter under extreme heavy-tailedness. We also present applications of the results in the study of concentration inequalities for linear estimators.The results in this paper constitute a part of the author's dissertation “New Majorization Theory in Economics and Martingale Convergence Results in Econometrics” presented to the faculty of the Graduate School of Yale University in candidacy for the degree of Doctor of Philosophy in Economics in March 2005. Some of the results were originally contained in the work circulated in 2003–2005 under the titles “Shifting Paradigms: On the Robustness of Economic Models to Heavy-Tailedness Assumptions” and “On the Robustness of Economic Models to Heavy-Tailedness Assumptions.” I am indebted to my advisers, Donald Andrews, Peter Phillips, and Herbert Scarf, for all their support and guidance in all stages of the current project. I also thank the associate editor, two anonymous referees, Donald Brown, Aydin Cecen, Gary Chamberlain, Brian Dineen, Darrell Duffie, Xavier Gabaix, Tilmann Gneiting, Philip Haile, Wolfgang Härdle, Boyan Jovanovic, Samuel Karlin, Benoît Mandelbrot, Alex Maynard, Marcelo Morreira, Ingram Olkin, Ben Polak, Gustavo Soares, Kevin Song, and the participants at seminars at the Departments of Economics at Yale University, University of British Columbia, the University of California at San Diego, Harvard University, the London School of Economics and Political Science, Massachusetts Institute of Technology, the Université de Montréal, McGill University, and New York University, the Division of the Humanities and Social Sciences at California Institute of Technology, Nuffield College, University of Oxford, and the Department of Statistics at Columbia University, and the participants at the 18th New England Statistics Symposium at Harvard University, April 2004, the International Conference on Stochastic Finance, Lisbon, Portugal, September 2004, and the Conference “Heavy Tails and Stable Paretian Distributions in Finance and Macroeconomics” in celebration of the 80th birthday of Benoît Mandelbrot, Deutsche Bundesbank, Eltville, Germany, November 2005, for many helpful comments and discussions.

This paper considers generalized method of moments (GMM) estimation of the inclusive panel AR(1) model that contains the covariance stationary panel AR(1) model and the panel AR(1) model with a unit root as special cases. The paper presents a two-step optimal linear GMM (OLGMM) estimator for the inclusive model that is asymptotically equivalent to the optimal nonlinear GMM estimator of Ahn and Schmidt (1997, Journal of Econometrics 76, 309–321) when the data are covariance stationary. Next the paper derives the asymptotic distribution of the OLGMM estimator when the model has a unit root under a variety of assumptions about the initial observations and the initial estimator. It is shown that in most cases the OLGMM estimator is superconsistent. In addition it is shown that the iterated OLGMM estimator is superefficient when the variance of the initial observations is finite and fixed, i.e., small compared to the cross-sectional dimension of the panel. The paper also conducts a Monte Carlo study in which the finite-sample properties of various GMM estimators for the inclusive panel AR(1) model are compared.I thank Steve Bond and Frank Windmeijer for kindly making one of their computer programs available to me. I also thank two anonymous referees and a co-editor for very helpful comments. This research was funded by the ESRC under grant R000239139.

In what follows all processes referred to are weakly stationary. Let us call the real part of a complex ARMA(p,q) process a Re CARMA(p,q) process. Every real ARMA(p,q) process can trivially be written as a Re CARMA(p,q) process. Provided the moment properties of complex linear processes are appropriately specified, the following inverse result is available: every Re CARMA(p,q) process is spectrally equivalent to a real ARMA(2p,p + q) process or some simpler process. Thus the ARMA and Re CARMA classes are spectrally equivalent. The question of whether an ARMA or a Re CARMA parametrization is better in a given context then arises. If cyclicality is present, and especially if we wish to treat cycles, growth, and decay together, in a model whose parameters are easy to interpret, then a Re CARMA approach may be helpful.The author thanks Paolo Paruolo, A.M. Robert Taylor, and an anonymous referee for helpful suggestions.

We show that the order of integration of a vector autoregressive process is equal to the difference between the multiplicity of the unit root in the characteristic equation and the multiplicity of the unit root in the adjoint matrix polynomial. The equivalence with the standard I(1) and I(2) conditions (Johansen, 1996, Likelihood-Based Inference in Cointegrated Vector Auto-Regressive Models) is proved.I am very grateful to Søren Johansen for his precious insights and his continuous help throughout the development of the paper. I also thank Paolo Paruolo (the coeditor) and the referees for valuable feedback. This paper was written while the author was a Post Doc at the Department of Economics, University of Copenhagen, and the hospitality of this institution is gratefully acknowledged.

I am pleased to announce that Guido Kuersteiner and Robert Taylor have joined the Board of Co-Editors of Econometric Theory for five-year terms beginning on 1 January 2007.