Under minimal assumptions on the distribution of income, we demonstrate that Chakravarty's empirical (1988, International Economic Review 29, 147–156) E-Gini index is consistent and asymptotically normal. We also derive an explicit formula for the asymptotic variance of the index and then construct a consistent and computationally straightforward estimator for it.Sincere thanks are due to the co-editor Oliver B. Linton and two anonymous referees whose constructive criticism, advice, and queries helped me in reshaping the paper considerably. As advised by a referee, I had the great pleasure of communicating with Garry F. Barrett and Stephen G. Donald and learning about their interesting and closely related results. The starting point of my work on the project was correspondence with Joseph L. Gastwirth in the spring of 2000 that resulted in our joint work on the S-Gini index. I am grateful to Joseph L. Gastwirth for his time, his advice, and his numerous suggestions that followed. The help, in addition to interest in the project, by Ying Zhang of the Statistical Laboratory at the University of Western Ontario is greatly appreciated; the analysis of the dependence of σF,α2 on parameters presented in Table 1 is due to her. This research was partially supported by an NSERC of Canada individual research grant at the University of Western Ontario.

We present a method of estimating density-related functionals, without prior knowledge of the density's functional form. The approach revolves around the specification of an explicit formula for a new class of distributions that encompasses many of the known cases in statistics, including the normal, gamma, inverse gamma, and mixtures thereof. The functionals are based on a couple of hypergeometric functions. Their parameters can be estimated, and the estimates then reveal both the functional form of the density and the parameters that determine centering, scaling, etc. The function to be estimated always leads to a valid density, by design, namely, one that is nonnegative everywhere and integrates to 1. Unlike fully nonparametric methods, our approach can be applied to small datasets. To illustrate our methodology, we apply it to finding risk-neutral densities associated with different types of financial options. We show how our approach fits the data uniformly very well. We also find that our estimated densities' functional forms vary over the dataset, so that existing parametric methods will not do uniformly well.We thank Hans-Jürg Büttler, Aleš Černý, Tony Culyer, Les Godfrey, David Hendry, Sam Kotz, Steve Lawford, Peter Phillips, Bas Werker, and three anonymous referees for their comments. We also thank for their feedback the participants at the seminars and conferences where this paper has been invited, in particular the 1998 CEPR Finance Network Workshop, the 1998 METU conference, the 1998 FORC (Warwick) conference “Options: Recent Advances,” Money Macro & Finance Group, the Swiss National Bank, Imperial College, Tilburg University, Université Libre de Bruxelles, the University of Oxford, Southampton University, and UMIST. The first author acknowledges support from the ESRC (UK) grant R000239538. The second author acknowledges help from the HEC Foundation and the European Community TMR grant “Financial Market Efficiency and Economic Efficiency.” This paper was written when the second author was affiliated with HEC-Paris.

It is a challenging task to review the research carried out by Professor Sargan which has covered such a broad range of areas and which offers general and unifying approaches to tackling problems and to obtaining results.I can only apologize for my audacity in accepting to deliver this laudatio. In order to remedy the limitation of my resources, I have benefited from Manuel Arellano's help and I have used the papers by David Hendry and Ken Wallis (1984), Peter Phillips (1985), and Esfandiar Maasoumi (1988a, 1988b). These papers review and summarize the works of the common master, who is considered to be one of the major contributors to Econometrics; in the words of Professor T.W. Anderson: “one of the most distinguished econometricians of our generation.” I am grateful to Peter Phillips and Grayham Mizon for their comments in preparing the English version of this laudatio.

This paper investigates the asymptotic theory for a vector autoregressive moving average–generalized autoregressive conditional heteroskedasticity (ARMA-GARCH) model. The conditions for the strict stationarity, the ergodicity, and the higher order moments of the model are established. Consistency of the quasi-maximum-likelihood estimator (QMLE) is proved under only the second-order moment condition. This consistency result is new, even for the univariate autoregressive conditional heteroskedasticity (ARCH) and GARCH models. Moreover, the asymptotic normality of the QMLE for the vector ARCH model is obtained under only the second-order moment of the unconditional errors and the finite fourth-order moment of the conditional errors. Under additional moment conditions, the asymptotic normality of the QMLE is also obtained for the vector ARMA-ARCH and ARMA-GARCH models and also a consistent estimator of the asymptotic covariance.The authors thank the co-Editor, Bruce Hansen, and two referees for very helpful comments and suggestions and acknowledge the financial support of the Australian Research Council.

In Pötscher (1991, Econometric Theory 7, 163–185) the asymptotic distribution of a post-model-selection estimator, both unconditional and conditional on selecting a correct model (minimal or not), has been derived. Limitations of these results are (i) that they do not provide information on the distribution of the post-model-selection estimator conditional on selecting an incorrect model and (ii) that the quality of this asymptotic approximation to the finite-sample distribution is not uniform with respect to the underlying parameters. In the present paper we first obtain the unconditional and also the conditional finite-sample distribution of the post-model-selection estimator, which turn out to be complicated and difficult to interpret. Second, we obtain approximations to the finite-sample distributions that are as simple and easy to interpret as the asymptotic distributions obtained in Pötscher (1991) but at the same time are close to the finite-sample distributions uniformly with respect to the underlying parameters. As a by-product, we also obtain the asymptotic distribution conditional on selecting an incorrect model.We thank the co-editor Richard Smith and the two referees for helpful comments on a previous version of this paper. Hannes Leeb's research was supported by the Austrian Science Foundation (FWF), project P13868-MAT.

Let g(λ) be the spectral density of a stationary process and let fθ(λ), θ ∈ Θ, be a fitted spectral model for g(λ). A minimum contrast estimator of θ is defined that minimizes a distance between , where is a nonparametric spectral density estimator based on n observations. It is known that is asymptotically Gaussian efficient if g(λ) = fθ(λ). Because there are infinitely many candidates for the distance function , this paper discusses higher order asymptotic theory for in relation to the choice of D. First, the second-order Edgeworth expansion for is derived. Then it is shown that the bias-adjusted version of is not second-order asymptotically efficient in general. This is in sharp contrast with regular parametric estimation, where it is known that if an estimator is first-order asymptotically efficient, then it is automatically second-order asymptotically efficient after a suitable bias adjustment (e.g., Ghosh, 1994, Higher Order Asymptotics, p. 57). The paper establishes therefore that for semiparametric estimation it does not hold in general that “first-order efficiency implies second-order efficiency.” The paper develops verifiable conditions on D that imply second-order efficiency.This paper was written while the first author was visiting the University of Bristol as a Benjamin Meaker Professor. The second author was previously at Bristol and is now supported by a fellowship of the Royal Netherlands Academy of Arts and Sciences. We are grateful to the co-editor Pentti Saikkonen and two anonymous referees for their valuable comments, which significantly improved the paper.

We present the form of the optimal instrument in a system of multiperiod conditional moment restrictions in the presence of conditional heteroskedasticity. Using Hansen's (1985, Journal of Econometrics 30, 203–228) and Hansen, Heaton, and Ogaki's (1988, Journal of the American Statistical Association 83, 863–871) work on efficiency bounds for generalized method of moments estimators, we show that this form is an autoregressive recurrence parametrized by a few auxiliary processes that are defined through a system of nonlinear stochastic restrictions, with a stability condition among them. In general, the system does not allow inversion and obtaining an explicit solution for the auxiliary parameters.This is a part of my Ph.D. dissertation (Anatolyev, 2000). I am grateful to Kenneth West for introducing me to the subject and providing advice and encouragement. I also thank Bruce Hansen, Yuichi Kitamura, seminar participants at Indiana University, Pennsylvania State University, Rutgers University, University of British Columbia, University of Virginia, and University of Wisconsin, the co-editor Richard Smith, and two referees for helpful comments and suggestions. Research for this paper was supported in part by a fellowship from the Economics Education and Research Consortium, with funds provided by the government of Sweden through the Eurasia Foundation. The opinions expressed in this paper are those of the author and do not necessarily reflect the views of the government of Sweden, the Eurasia Foundation, or any other member of the Economics Education and Research Consortium.

We examine the identifiability of a nonlinear state space system under general assumptions. The discrete time evolution of the state is generated by a recurrent Elman network. For a large set of Elman networks we determine the class of observationally equivalent minimal systems, i.e., minimal systems that exhibit the same input-output behavior.The authors are grateful for discussions with M. Deistler and D. Bauer. A.A. Al-Falou was funded by the ERNSI network within the European Union program Training and Mobility of Researchers (TMR). D. Trummer was funded by the FWF (Austrian Science Fund). We thank an anonymous referee for helpful comments and suggestions.

This paper analyzes the large sample behavior of the seasonal unit root tests of Dickey, Hasza, and Fuller (1984, Journal of the American Statistical Association 79, 355–367) when applied to a series that admits a unit root at the zero but not seasonal spectral frequencies. We show that in such cases the Dickey et al. statistics have nondegenerate limiting distributions. Consequently, there is a nonzero probability that, taken in isolation, they will lead the applied researcher to accept the seasonal unit root null hypothesis and hence, incorrectly, take seasonal differences of the series, even asymptotically. The same conclusion holds if the process displays unit root behavior at any of the zero and/or seasonal frequencies. Our results therefore prove a conjecture made on the basis of Monte Carlo simulation evidence, in Ghysels, Lee, and Noh (1994, Journal of Econometrics 62, 415–442) that the tests of Dickey et al., unlike those of Hylleberg, Engle, Granger, and Yoo (1990, Journal of Econometrics 44, 215–238), are unable to separate between unit roots at the zero and seasonal frequencies.I thank Peter Burridge, Bruce Hansen, and three anonymous referees for helpful comments on earlier drafts of this paper.

In this paper we introduce a general method for estimating semiparametrically the different components in separable models. The family of separable models is quite popular in economic research because this structure offers clear interpretation, has straightforward economic consequences, and is often justified by theory. This family is also of statistical interest because it allows us to estimate high-dimensional complexity semiparametrically without running into the curse of dimensionality. We consider even the case when multiple indices appear in the objective function; thus we can estimate models that are typical in economic analysis, such as those that contain limited dependent variables. The idea of the new method is mainly based on a generalized profile likelihood approach. Although this requires some hypotheses on the conditional error distribution, it yields a quite general usable method with low computational costs but high accuracy even for small samples. We give estimation procedures and provide some asymptotic theory. Implementation is discussed; simulations and an application demonstrate its feasibility and good finite-sample behavior.This research was financially supported by Dirección General de Investigación del Ministerio de Ciencia y Tecnología under research grants BEC2001-1121 and BEC2001-1270; Dirección General de Enseñanza Superior del Ministerio de Educación y Ciencia under Subprograma de estancias de investigadores españoles en centros de investigación españoles y extranjeros, ref. PR2000-0096; and by the Danish Social Science Research fund. We also thank M. Delgado, O. Linton, two anonymous referees, and all participants of the working group STAPH on functional statistics in Toulouse, the activities of which are available on line at http://www.lsp.upstlse.fr/Fp/Ferraty/staph.html.

In this paper we provide some weak convergence results for sample statistics of the product of a variable with its kth-order lag. We assume the variable is a stationary vector that can be represented by linear process, and the lag length k is allowed to be a function of the sample size. Employing the Beveridge–Nelson decomposition, we derive a new functional central limit theorem for this situation and establish related stochastic integral convergence results. We then consider the behavior of associated long-run variance estimators and also extend our analysis to the case where the sample statistics are based on regression residuals. We illustrate the potential range of application of these techniques in the context of (i) testing for I(0) versus I(1) behavior and (ii) estimation and testing in a heteroskedastically cointegrated regression model.We thank the co-editor and the referees for helpful comments on earlier drafts.

This paper proposes a test of the null of underidentification in the nonlinear-in-parameters generalized method of moments model. It can be thought of as a nonlinear analog of the usual linear instrumental variables first-stage F-test. It can be used as a diagnostic to warn a researcher when conventional asymptotic theory is unlikely to work well.I am grateful to Don Andrews, Jon Faust, John Fernald, Jim Stock, and two anonymous referees for their helpful comments on earlier drafts of this manuscript. I am also grateful to George Tauchen for providing me with the code for generating artificial asset price data. All errors are my sole responsibility.

In this paper, functional limit theorems for general fractional processes are established under quite weak conditions. The results are then used to derive weak convergence of general nonstationary fractionally integrated processes and to characterize unit root distribution in a model with error being a fractional autoregressive moving average process or a nonstationary fractionally integrated process.The authors thank three referees and an associate editor for their detailed reading of this paper and valuable comments, which have led to this much improved version of the paper.

I review the part played by John Denis Sargan in the formation of the “LSE approach” to dynamic econometric modeling. Despite his unassuming demeanor and his location at LSE—which had earlier dismissed a substantive role for econometric evidence—Sargan nevertheless radically altered the econometric approach of a generation, establishing a powerful approach to empirical modeling of economic time series. His main contributions to econometric methodology, and the subsequent research, are discussed as a complement to the other papers in this memorial volume.Financial support from the U.K. Economic and Social Research Council under grant L138251009 is gratefully acknowledged. I am indebted to Alok Bhargava, Julia Campos, Meghnad Desai, Neil Ericsson, Toni Espasa, Grayham Mizon, Peter Phillips, Timo Teräsvirta, and Ken Wallis for helpful comments and to Peter Phillips for the invitation to contribute this paper to Econometric Theory.

We modify the procedure for constructing exact bounds confidence regions introduced by Dufour (1990, Econometrica 58, 475–494) so as to drop the requirement that the confidence regions for the nuisance parameters are marginal with respect to the parameters of interest, i.e., that they are the same for all values of the parameters of interest. We illustrate this modified procedure with an application to a dependent variable heteroskedasticity model and, using a Monte Carlo study, compare the confidence regions constructed by this procedure with asymptotically justified regions.I thank the editor and three referees for useful comments. As always, I retain responsibility for any remaining errors.

A cointegration relation is often interpreted as a long-run equilibrium relation whereas short-run deviations are allowed to a certain extent. In practice, however, real data often fail to confirm cointegration for a well understood economic relation. This paper proposes that in many cases failure of confirming cointegration is due to nonstationary deviations in a relatively small portion of the data period (“short-run”) whereas a cointegration relation prevails in the other periods. We call such a situation segmented cointegration. We study in this paper how to detect segmented cointegration and how to identify the period of short-run deviations.I thank Pentti Saikkonen and three anonymous referees for helpful comments. I am deeply indebted to Peter Phillips for encouragement, discussion, and helpful comments on this work. The research of this paper is supported by the Research Grants Council of Hong Kong (grant HKUST6178/98H) and FRAP B of SUNY-Albany (account 320-9709W).

It was June 1940. The Second World War was already raging in full swing in its devastation. A young man not yet 20 set out on a 500-mile train journey from the coastal city of Vishakapatnam, India, to Calcutta, the second largest city of the British Empire, after obtaining a first-class first degree in mathematics and with a glimmer of hope of finding a job in the military. The young man was not so lucky; he was deemed too young for the job. However, while in Calcutta, through a chance encounter, he visited the Indian Statistical Institute (ISI) founded in 1931 by Professor Prasanta Chandra Mahalanobis, a Cambridge-trained physicist. As a last resort he applied for a one-year training program in statistics there. Very promptly he received a positive reply from Professor Mahalanobis admitting him to the Training Section of the ISI from January 1, 1941. This young man was none other than perhaps the most well known statistician in the world today, Calyampudi Radhakrishna Rao (or simply Dr. Rao, as he was universally known to his students and colleagues at the ISI. The “professor” title was reserved for Mahalanobis). Such was the beginning of his spectacular career in statistics, a field chosen as a last resort, and he never looked back.

This paper considers estimation of a sample selection model subject to conditional heteroskedasticity in both the selection and outcome equations. The form of heteroskedasticity allowed for in each equation is multiplicative, and each of the two scale functions is left unspecified. A three-step estimator for the parameters of interest in the outcome equation is proposed. The first two stages involve nonparametric estimation of the “propensity score” and the conditional interquartile range of the outcome equation, respectively. The third stage reweights the data so that the conditional expectation of the reweighted dependent variable is of a partially linear form, and the parameters of interest are estimated by an approach analogous to that adopted in Ahn and Powell (1993, Journal of Econometrics 58, 3–29). Under standard regularity conditions the proposed estimator is shown to be -consistent and asymptotically normal, and the form of its limiting covariance matrix is derived.We are grateful to B. Honoré, R. Klein, E. Kyriazidou, L.-F. Lee, J. Powell, two anonymous referees, and the co-editor D. Andrews and also to seminar participants at Princeton, Queens, UCLA, and the University of Toronto for helpful comments. Chen's research was supported by RGC grant HKUST 6070/01H from the Research Grants Council of Hong Kong.

A common test in econometrics is the Dickey–Fuller test, which is based on the test statistic . We investigate the behavior of the test statistic if the data yt are given by an exponential random walk exp(Zt) where Zt = Zt−1 + σεt and the εt are independent and identically distributed random variables. The test statistic DF(T) is a nonlinear transformation of the partial sums of εt process. Under certain moment conditions on the εt we show that tends to one as λ → 0. For the particular case that the εt define a simple random walk it is shown that plimT→∞DF(T)/T exists and the limit is evaluated. The theoretical results are illustrated by some simulation experiments.We gratefully acknowledge the help of an anonymous referee whose comments on the first two versions of this paper enabled us to reduce the number of mistakes and to increase the clarity of presentation. The authors' research was supported in part by Sonderforschungsbereich 475, University of Dortmund.