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).

This paper considers the information available to invariant unit root tests at and near the unit root. Because all invariant tests will be functions of the maximal invariant, the Fisher information in this statistic will be the available information. The main finding of the paper is that the available information for all tests invariant to a linear trend is zero at the unit root. This result applies for any sample size, over a variety of distributions and correlation structures, and is robust to the inclusion of any other deterministic component. In addition, an explicit upper bound upon the power of all invariant unit root tests is shown to depend solely upon the information. This bound is illustrated via a brief simulation study that also examines the impact that different invariance requirements have on power.Thanks are due to Francesco Bravo, Giovanni Forchini, Les Godfrey, Robert Taylor, participants at seminars at the Universities of Birmingham and York and at the ESRC Econometric study group conference, Bristol, 2004, and also to Bruce Hansen, Joel Horowitz, and five anonymous referees. Revisions of this paper have greatly benefited from comments and suggestions from Grant Hillier and Peter Phillips.

Andrews (1999, Econometrica 67, 1341–1383) derived the first-order asymptotic theory for a very general class of estimators when a parameter is on a boundary. We derive the second-order asymptotic theory in this setting in some special cases. We focus on the behavior of the quasi maximum likelihood estimator (QMLE) in stationary and nonstationary generalized autoregressive conditionally heteroskedastic (GARCH) models when constraints are imposed in the maximization procedure. We show how in this case both a first- and a second-order bias appear in the estimator and how the bias can be quite large. We provide two types of bias correction mechanisms for the researcher to choose in practice: either to bias correct only for a first-order bias or for a first- and second-order bias. We show that when some constraints are imposed, it is advisable to bias correct not only for the first-order bias but also for the second-order bias.We thank Bruce Hansen and two referees for helpful comments. The first author gratefully acknowledges financial support from the MSU Intramural Research Grants Program. The second author gratefully acknowledges financial support from the ESRC.

In nonstationary heterogeneous panels where the number of units is finite and where each unit cointegrates, a large number of conditions needs to be satisfied for cointegration to be preserved in the aggregate relationship. In reality, the conditions most likely will not hold. This paper takes a closer look at what happens when the conditions are violated. In this case, the question of whether an aggregate relationship is observationally equivalent to a cointegrating equation is of particular interest. We derive a measure of the degree of noncointegration of the aggregate estimates, and we explore its asymptotic properties.We thank Stephen Hall for inspiring discussions and comments during the initial stage of this research. We are grateful to participants in the 9th Conference on Panel Data (Geneva, June 22–23, 2000) and the 2000 Meeting of the Society for Economic Dynamics (San José, Costa Rica, June 29–July 2, 2000) and to colleagues at the University of Bergamo (Italy), London Business School (UK), and Cass Business School (UK) for useful comments. We have enormously benefited from comments and suggestions from two anonymous referees, the editor, Peter C.B. Phillips, and the co-editor, Bruce Hansen. The usual disclaimer applies. G. Urga acknowledges ESRC (Project R022251032) for financial support. Lorenzo Trapani acknowledges financial support from Cass Business School (RAE Development Funds Scheme) and ESRC Postdoctoral Fellowship Scheme (A.N. PTA-026-27-1107).

This paper provides a root-n consistent, asymptotically normal weighted least squares estimator of the coefficients in a truncated regression model. The distribution of the errors is unknown and permits general forms of unknown heteroskedasticity. Also provided is an instrumental variables based two-stage least squares estimator for this model, which can be used when some regressors are endogenous, mismeasured, or otherwise correlated with the errors. A simulation study indicates that the new estimators perform well in finite samples. Our limiting distribution theory includes a new asymptotic trimming result addressing the boundary bias in first-stage density estimation without knowledge of the support boundary.This research was supported in part by the National Science Foundation through grant SBR-9514977 to A. Lewbel. The authors thank Thierry Magnac, Dan McFadden, Jim Powell, Richard Blundell, Bo Honoré, Jim Heckman, Xiaohong Chen, and Songnian Chen for helpful comments. Any errors are our own.

The asymptotic distributions of augmented Dickey–Fuller (ADF) unit root tests for autoregressive processes with a unit or near-unit root are discussed in the presence of multiple stochastic level shifts of large size occurring independently in time. The distributions depend on a Brownian motion and a Poisson-type jump process. Due to the latter, tests based on standard critical values experience power losses increasing rapidly with the number and the magnitude of the shifts. A new approach to unit root testing is suggested which requires no knowledge of either the location or the number of level shifts, and which dispenses with the assumption of independent shift occurrence. It is proposed to remove possible shifts from a time series by weighting its increments according to how likely it is, with respect to an ad hoc postulated distribution, a shift to have occurred in each period. If the number of level shifts is bounded in probability, the limiting distributions of the proposed test statistics coincide with those of ADF statistics under standard conditions. A Monte Carlo experiment shows that, despite their generality, the new tests perform well in finite samples.We are grateful to Pentti Saikkonen (the co-editor) and two referees for their helpful and constructive comments on earlier versions of this paper. We also thank Søren Johansen, Helmut Lütkepohl, Anders Rahbek, Robert Taylor, and seminar participants at the European University Institute, the University of Copenhagen, and the URCT Conference held in Faro, Portugal, September 29 to October 1, 2005, for useful comments. Both authors are grateful to the Danish Social Sciences Research Council, project 2114-04-001, for financial support.

We study asymptotic properties of the local Whittle estimator of the long memory parameter for a wide class of fractionally integrated nonlinear time series models. In particular, we solve the conjecture posed by Phillips and Shimotsu (2004, Annals of Statistics 32, 656–692) for Type I processes under our framework, which requires a global smoothness condition on the spectral density of the short memory component. The formulation allows the widely used fractional autoregressive integrated moving average (FARIMA) models with generalized autoregressive conditionally heteroskedastic (GARCH) innovations of various forms, and our asymptotic results provide a theoretical justification of the findings in simulations that the local Whittle estimator is robust to conditional heteroskedasticity. Additionally, our conditions are easily verifiable and are satisfied for many nonlinear time series models.We thank Liudas Giraitis for providing the manuscript by Dalla, Giraitis, and Hidalgo (2006). We are grateful to the two referees and the editor for their detailed comments, which led to substantial improvements. We also thank Michael Stein for helpful comments on an earlier version. The work is supported in part by NSF grant DMS-0478704.

Manfred Deistler was born in St. Pölten, Austria, on September 23, 1941, as the younger of two brothers. After attending elementary school and high school (Gymnasium) in St. Pölten, he studied electrical engineering at the University of Technology Vienna, then known as the Technische Hochschule Wien. He graduated with the degree of “Diplom-Ingenieur” (which is roughly equivalent to a master's degree) in 1964. After working in industry for a year and a half, he attended the Institute of Advanced Studies Vienna and earned a diploma in economics from this institution in 1968. During this period he started work on a Ph.D. thesis in applied mathematics at the Technischen Hochschule Wien, obtaining the Ph.D. degree in 1970. After academic appointments at the University of Regensburg as an assistant professor and at the University of Bonn as an associate professor, he was appointed full professor of econometrics at the University of Technology Vienna in 1978, where he has been ever since. He is married to Dr. Gerda Deistler-Russ, and they have a daughter, Gerda.

Manfred Deistler has worked extensively in econometrics and systems identification, which has resulted in numerous papers and an important book with Ted Hannan [1]. He is also a Fellow of the Econometric Society, a Fellow of the Journal of Econometrics, and a Fellow of IEEE (The Institute of Electrical and Electronic Engineers). He is a corresponding member of the Austrian Academy of Sciences. He is or has been a member of the editorial boards of numerous scientific journals, such as Econometrica, Econometric Reviews, Econometric Theory, Empirical Economics, Journal of Econometrics, Journal of Forecasting, Journal of Mathematical Systems and Control, Journal of Multivariate Analysis, Journal of Time Series Analysis, SIAM Journal on Control and Optimization, SIAM Journal on Matrix Analysis and Applications, and Surveys on Mathematics for Industry. He also is a member of the advisory board of Econometric Theory. Manfred Deistler has also been a most important force in establishing econometrics in Austria.

The interview was conducted on February 17, 2006, in Manfred Deistler's office at the University of Technology Vienna in our common native language, German, and later was transcribed into English. The help of Ingmar Prucha in preparing the interview and comments by Hannes Ledolter, Hannes Leeb, and Richard Nickl on a first draft are gratefully acknowledged.

Dynamic economic theories usually have implications on and only on the conditional mean dynamics of economic processes. Using a generalized spectral derivative approach, Hong and Lee (2005, Review of Economic Studies 72, 499–541) recently proposed a new class of omnibus nonparametric specification tests for linear and nonlinear time series conditional mean models, where the dimension of the conditioning information set may be infinite. The tests can detect a wide range of model misspecifications in mean while being robust to conditional heteroskedasticity and time-varying higher order moments of unknown form. They enjoy an asymptotic “nuisance parameter–free” property in the sense that parameter estimation uncertainty has no impact on the asymptotic N(0,1) distribution of the test statistics. As a result, only the estimated residuals from the null parametric model are needed to implement the tests, and no specific estimation is required.

Although parameter estimation uncertainty has no impact on the asymptotic distribution of the tests, it may have significant impact on the finite-sample distribution, and such an impact may become more substantial as the number of estimated parameters increases. In this paper, we adopt the Wooldridge (1990, Econometric Theory 6, 17–43) device for parametric m-tests to the Hong and Lee (2005) nonparametric tests to reduce the impact of parameter estimation uncertainty. Asymptotic size and power properties of the modified tests are investigated, and simulation studies show that the modified tests generally have better sizes in finite samples and are robust to parameter estimation uncertainty. In the meantime, the size improvement does not cause loss of power against a wide range of alternatives when using the empirical critical values for the tests. These results suggest that the modified generalized spectral derivative tests can be a useful tool in time series conditional mean modeling.We thank Bruce Hansen (the co-editor) and three referees for insightful and helpful comments and suggestions. We also are grateful for financial support from the National Science Foundation Grant SES–0111769 and the Cheung Kong Visiting Scholarship by the Chinese Ministry of Education and Xiamen University, China.

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).

Much of the credit for the outpouring of research on nonlinear models in the 1970s—particularly limited dependent variable models—should go to Takeshi Amemiya. His classic 1973 Econometrica article on estimation of the parameters of the Tobit model [11] set a new standard for mathematical rigor in theoretical econometrics. During his career, which spans nearly four decades, Takeshi's research contributions to econometrics have touched on most of the central models for empirical economics, including linear and nonlinear simultaneous equations models, distributed lag models, heteroskedastic and random coefficient models, qualitative response models, censored and truncated regression and selection models, transformed regression models, choice-based sampling models, and duration models. His 1985 text, Advanced Econometrics [44], became the standard reference for second-year graduate microeconometrics courses at the leading graduate programs in economics.

Takeshi's research accomplishments have been accompanied by numerous professional honors. He is a Fellow of the Econometric Society, the American Statistical Association, and the American Academy of Arts and Sciences and was awarded Guggenheim and Humboldt Fellowships. Takeshi also has a distinguished record of professional service; he served briefly as a co-editor of Econometrica and has been a co-editor of the Journal of Econometrics since 1982.

This interview was conducted in Takeshi Amemiya's office at Stanford University in March 2004. In recent years his interests have turned to comparative study of Greek and Japanese culture and mythology—his scholarship includes a Japanese translation of an English research monograph on Aristotle's Ethics [58]—and the discussion starts with this topic before turning to his unusual education, career as an econometric theorist, and variety of outside interests. His remarks reveal both his careful attention to detail and the dry humor that his colleagues and former students (like me) savor.

We consider quadratic forms of martingale differences and establish a central limit theorem under mild and easily verifiable conditions. By approximating Fourier transforms of stationary processes by martingales, our central limit theorem is applied to the smoothed periodogram estimate of spectral density functions. Our results go beyond earlier ones by allowing a variety of nonlinear time series and by avoiding strong mixing and/or summability conditions on joint cumulants.We thank the two reviewers for their detailed comments, which led to substantial improvements. The work is supported in part by NSF grant DMS-0478704.

In this paper we consider estimating the rank of an unknown symmetric matrix based on a symmetric, asymptotically normal estimator of the matrix. The related positive definite limit covariance matrix is assumed to be estimated consistently and to have either a Kronecker product or an arbitrary structure. These assumptions are standard although they exclude the case when the matrix estimator is positive or negative semidefinite. We adapt and reexamine here some available rank tests and introduce a new rank test based on the sum of eigenvalues of the matrix estimator. We discuss two applications where rank estimation in symmetric matrices is of interest, and we also provide a small simulation study.The first author acknowledges the support of an Alfred P. Sloan Foundation Research Fellowship and NSF Grant SES-0196372. We thank the co-editor and the two referees for useful comments and suggestions. CEMPRE—Centro de Estudos Macroeconómicos e Previsão—is supported by the Fundação para a Ciência e a Tecnologia, Portugal, through the Programa Operacional Ciência, Tecnologia e Inovação (POCTI) of the Quadro Comunitário de Apoio III, which is financed by FEDER and Portuguese funds.

In this paper, we provide a Doob-style consistency theorem for stationary models. Many applications involving Bayesian inference deal with non independent and identically distributed data, in particular, with stationary data. However, for such models, there is still a theoretical gap to be filled regarding the asymptotic properties of Bayesian procedures. The primary goal to be achieved is establishing consistency of the sequence of posterior distributions. Here we provide an answer to the problem. Bayesian methods have recently gained growing popularity in economic modeling, thus implying the timeliness of the present paper. Indeed, we secure Bayesian procedures against possible inconsistencies. No results of such a generality are known up to now.The authors are grateful for the comments and suggestions of two referees. Antonio Lijoi and Igor Prünster were supported by the Italian Ministry of University and Research, grants 2006134525 and 2006133449, respectively. The research of Stephen G. Walker was funded by an EPSRC Advanced Research Fellowship.

Most nonparametric identification results for the mixed proportional hazards model for single spell duration data depend crucially on the proportional hazards assumption. Here, it is shown that variation in covariates over time, combined with variation across observations, is sufficient to ensure identification without the proportional hazards assumption. The required variation over time is minimal, and the mixed hazards model is identified without the proportional hazards assumption in the presence of standard time-varying covariates.Thanks to Kåre Bævre, Zhiyang Jia, Tore Schweder, Rolf Aaberge, and John K. Dagsvik for useful comments.

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 updates Baltagi's (2003, Econometric Theory 19, 165–224) rankings of academic institutions by publication activity in econometrics over the period 1989–1999 to the period 1989–2005. This ranking is based on 16 leading international journals that publish econometrics articles. It is compared with the prior rankings by Hall (1987, Econometric Theory 3, 171–194; 1990, Econometric Theory 6, 1–16) for the period 1980–1988. In addition, a list of the top 150 individual producers of econometrics in these 16 journals over this 17-year period is provided. This is done for theoretical econometrics and also for all contributions in econometrics. Sensitivity analysis is provided using (i) alternative weighting factors given to the 16 journals taking into account impact citations, excluding self-citations, size and age of the journal, (ii) alternative time intervals, namely, 2000–2005, 1995–2005, and 1989–2005, (iii) alternative ranking using the number of articles published in these journals, (iv) separate rankings for both institutions and individuals by journal, (v) rankings for institutions and individuals based on publications in three core econometrics journals.I thank Peter C.B. Phillips for entrusting me with this study and P.M. Kisor for her research assistance. Also, I thank three referees for their helpful comments and suggestions.

For use in asymptotic analysis of nonlinear time series models, we show that with (Xt,t ≥ 0) a (geometrically) ergodic Markov chain, the general version of the strong law of large numbers applies. That is, the average converges almost surely to the expectation of φ(Xt,Xt+1,…) irrespective of the choice of initial distribution of, or value of, X0. In the existing literature, the less general form has been studied.We thank Paolo Paruolo (the co-editor) and the referee for valuable comments. Also we thank the Danish Social Sciences Research Council (grant 2114-04-0001) for financial support.

An asymptotic expansion is given for the autocovariance matrix of a vector of stationary long-memory processes with memory parameters d ∈ [0,½). The theory is then applied to deliver formulas for the long-run covariance matrices of multivariate time series with long memory.Phillips acknowledges partial support from a Kelly Fellowship and from the NSF under grant SES 04-142254.This may be proved directly using a Fourier integral asymptotic expansion when the spectrum of the short-memory component is analytic.

This note shows that the orthogonal shocks obtained by imposing a recursive structure on the contemporaneous impacts of the errors in a vector-error correction model (VECM) are related to the orthogonal permanent-transitory shocks in a Gonzalo and Ng (2001, Journal of Economic Dynamics and Control 25, 1527–1546) decomposition provided some of the variables in the system are weakly exogenous. Specifically, when there are as many weakly exogenous variables as common trends and the weakly exogenous variables are ordered first, the orthogonal shocks obtained directly from the VECM constitute a permanent-transitory decomposition, and the permanent shocks are identical to those in the Gonzalo and Ng decomposition.We are grateful to the co-editor Paolo Paruolo and an anonymous referee for useful comments, which have helped to improve the content and exposition of this note. Financial support from the Australian Research Council is gratefully acknowledged.