This paper is concerned with estimating a conditional quantile function that is assumed to be partially linear. The paper develops a simple estimator of the parametric component of the conditional quantile. The semiparametric efficiency bound for the parametric component is derived, and two types of efficient estimators are considered. Asymptotic properties of the proposed estimators are established under regularity conditions. Some Monte Carlo experiments indicate that the proposed estimators perform well in small samples.This paper is a part of my Ph.D. dissertation submitted to the University of Iowa. I am grateful to my adviser, Joel Horowitz, for his insightful comments, suggestions, guidance, and support. I also thank John Geweke, Gene Savin, two anonymous referees, the co-editor Oliver Linton, and participants at the 2001 Midwest Econometrics Group Annual Meeting in Kansas City for many helpful comments and suggestions. Of course, the responsibility for any errors is mine.

We develop an asymptotic theory for quadratic forms of the autocorrelations of squared residuals from a GARCH(p,q) model. Denoting by , k ≥ 1, these autocorrelations computed from a realization of length n, we show that the statistic is a matrix computed from the data, converges to the chi-square distribution with K degrees of freedom for any 1 ≤ i1 < ··· < iK. Our results are valid under weak assumptions on the innovations and model coefficients that admit that arbitrary low-order moments of the observations can be infinite. The matrix and its asymptotic limit D depend on the distribution of the innovations. A small simulation study illustrates the theory and shows, in particular, that using the matrix D computed under the assumption of normal innovations may lead to incorrect conclusions if the innovations have a different distribution.We thank the two referees for their comments and Professor Bruce E. Hansen, the co-editor in charge, for his sound advice on how to improve the paper. The work of István Berkes was supported by the Hungarian National Foundation for Scientific Research, grant T 29621. The work of Lajos Horváth and Piotr Kokoszka was supported by NATO grant PST.CLG.977607.

This issue of Econometric Theory is a Memorial Issue to commemorate John Denis Sargan. It brings together two of Denis Sargan's essays on econometrics, a laudation by Antoni Espasa, and three separate memorial essays written by David F. Hendry, Peter M. Robinson, and myself. Also included are some photographs of Denis Sargan and his family that were most kindly given to us by Mary Sargan for publication in this issue.

An unobserved components and a conventional first-order autoregressive (AR) model with constant are analyzed with the adjusted profile likelihood of Cox and Reid (1987, Journal of the Royal Statistical Society, Series B 49, 1–39; 1993, Journal of the Royal Statistical Society, Series B 55, 467–71). Both the unobserved components model and the Cox–Reid adjustment can provide more accurate estimates of an AR coefficient of unity. The unobserved components model yields more powerful unit-root tests. In general the most powerful test utilizes the adjustment. Under the unobserved components model the adjusted statistics follow the asymptotic distributions better than the unadjusted.The basis of this research is my D. Phil. thesis (1997) for the University of Oxford. I am grateful to David Hendry for supervision and to Pentti Saikkonen for most helpful advice. I appreciate also the CBRT Young Economist Award with which I was honored after presenting a previous version at the 1998 erc/METU International Conference on Economics. I remain responsible for any errors. Financial support by the Yrjö Jahnsson Foundation and Academy of Finland is gratefully acknowledged.

The variance-ratio (VR) test statistic, which is based on k-period differences of the data, is commonly used in empirical finance and economics to test the random walk hypothesis. We obtain the asymptotic power function of the VR test statistic when the differencing period k is increasing with the sample size n such that k/n → δ > 0. We show that the test is inconsistent against a variety of mean-reverting alternatives, confirm the result in simulations, and then characterize the functional form of the asymptotic power in terms of δ and these alternatives.

The possibility that a structural equation may not be identified casts doubt on measures of estimator precision that are usually used. Using the Fieller–Creasy problem for illustration, we argue that an observed identifiability test statistic is directly relevant to the precision with which the structural parameters can be estimated, and hence we argue that inference in such models should be conditioned on the observed value of that statistic (or statistics).

We examine in detail the effects of such conditioning on the properties of the ordinary least squares (OLS) and two-stage least squares (TSLS) estimators for the coefficients of the endogenous variables in a single structural equation. We show that (a) conditioning has very little impact on the properties of the OLS estimator but a substantial impact on those of the TSLS estimator; (b) the conditional variance of the TSLS estimator can be very much larger than its unconditional variance (when the identifiability statistic is small) or very much smaller (when the identifiability statistic is large); and (c) conditional mean-square-error comparisons of the two estimators favor the OLS estimator when the sample evidence only weakly supports the identifiability hypothesis but favor TSLS when that evidence moderately supports identifiability.

Finally, we note that another consequence of our argument is that the statistic upon which Anderson–Rubin confidence sets are based is in fact nonpivotal.We are grateful for the constructive comments offered by Peter Phillips and three anonymous referees that greatly improved the paper. Giovanni Forchini acknowledges support from ESRC grant NR00429424115.

Forty-five years ago when I started to study econometrics seriously I was full of confidence that the application of mathematics and statistical methodology would soon lead to a better understanding of the economic system, and, ultimately, to better public and private policies. Now, with the world economy poised on the edge of a world slump and Eastern Europe and the undeveloped world in turmoil my general hope seems optimistic. Even within econometrics, experience proves that the more we study econometrics the more there is to discover. I will concentrate on a brief survey of the problems in successfully modelling an economy both for understanding the way that it works and for the purpose of deciding on successful government policies.

This paper studies power functions and envelopes for covariate augmented unit root tests. The power functions are calculated by integrating the characteristic function, allowing accurate evaluation of the power envelope and the power functions. Using the power functions, we study the selection among point optimal invariant unit root tests. An “optimal” point optimal test is proposed based on minimizing the integrated power difference. We find that when there are covariate effects, optimal tests use a local alternative where the power envelope has an approximate value of 0.75.We thank Pentti Saikkonen and two referees for helpful comments.

We consider the K-statistic, Kleibergen's (2002, Econometrica 70, 1781–1803) adaptation of the Anderson–Rubin (AR) statistic in instrumental variables regression. Whereas Kleibergen (2002) especially analyzes the asymptotic behavior of the statistic, we focus on finite-sample properties in a Gaussian framework. The AR statistic then has an F-distribution. The finite-sample distribution of the K-statistic is, however, affected by nuisance parameters. We consider two extreme cases for the nuisance parameters, which provide tight bounds for the exact distribution. The first case amounts to perfect identification—which is similar to the asymptotic case—where the statistic has an F-distribution. In the other extreme case there is total underidentification. For the latter case we show how to compute the exact distribution. We thus provide tight bounds for exact confidence sets based on the K-statistic. Asymptotically the two bounds converge, except when there is a large number of redundant instruments.The authors' research documented in this paper has been funded by the NWO Vernieuwingsimpuls research grant “Empirical Comparison of Economic Models.”

In this note we consider the asymptotic power functions of some bootstrap unit root tests under local alternatives and show that they are in fact the same as for ordinary unit root tests. This is regardless of whether the differences of the observations, i.e., the so-called restricted residuals, or the ordinary least squares residuals are used to construct the resampled observations. We also consider models containing a constant and a linear trend and the DF-GLS tests proposed by Elliott, Rothenberg, and Stock (1996, Econometrica 64, 813–836). A small Monte Carlo experiment is included.I thank the associate editor, Bruce E. Hansen, and three anonymous referees for very constructive comments on the previous versions of the manuscript.

This paper investigates the so-called one-step local quasi–maximum likelihood estimator for the unit root process with GARCH(1,1) errors. When the scaled conditional errors (the ratio of the disturbance to the conditional standard deviation) follow a symmetric distribution, the asymptotic distribution of the estimated unit root is derived only under the second-order moment condition. It is shown that this distribution is a functional of a bivariate Brownian motion as in Ling and Li (1998, Annals of Statistics 26, 84–125) and can be used to construct the unit root test.The authors thank the co-editor, Bruce Hansen, and two referees for very helpful comments and suggestions. W.K. Li's research is partially supported by the Hong Kong Research Grants Council. Ling's research is supported by RGC Competitive Earmarked Research grant HKUST6113/02P.

This paper proposes model selection criteria (MSC) for unconditional moment models using generalized empirical likelihood (GEL) statistics. The use of GEL-statistics in lieu of J-statistics (in the spirit of Andrews, 1999, Econometrica 67, 543–564; and Andrews and Lu, 2001, Journal of Econometrics 101, 123–164) leads to an alternative interpretation of the MSCs that emphasizes the common information-theoretic rationale underlying model selection procedures for both parametric and semiparametric models. The result of this paper also provides a GEL-based model selection alternative to the information criteria–based nonnested tests for generalized method of moments models considered in Kitamura (2000, University of Wisconsin). The results of a Monte Carlo experiment are reported to illustrate the finite-sample performance of the selection criteria and their impact on parameter estimation.The authors gratefully acknowledge support from the NSF (Hong: SES-0079495, Shum: SES-0003352) and the Fellowship of Woodrow Wilson Scholars (Preston). We thank the co-editor Don Andrews, Xiaohong Chen, John Geweke, Bo Honore, Yuichi Kitamura, Serena Ng, Harry Paarsch, Gautam Tripathi, and two anonymous referees for insightful suggestions and helpful comments.

I was very pleased to be invited to the Universidad Carlos III as visiting professor, to meet all my old Spanish friends again, and to have the opportunity to give this public lecture. My address will have two themes: a brief review of the history and current status of the London School of Economics, and then a discussion of the strong tradition which has developed there of the study of mathematical methods in the social sciences, particularly in economics, by building mathematical models of economic behaviour, and then estimating these models statistically. That briefly is how I define econometrics.

This paper examines the effects of temporal aggregation on the asymptotic variances of estimators in cointegrated systems. Two important findings are obtained. First, estimators based on flow data alone are more efficient than when the data are all stocks or a mixture of stocks and flows. Second, estimators based on flow data are as efficient as when the data are recorded continuously. A method of improving efficiency with stock variables is also proposed, and an empirical illustration of the method is provided in the context of long-run money demand regressions.I thank Roy Bailey, Rex Bergstrom, Roderick McCrorie, a co-editor, and two anonymous referees for helpful comments. I also thank Katsumi Shimotsu for help with some data issues. None of these individuals are implicated, however, in any possible shortcomings of this paper. The financial support provided by the ESRC under grant R000221818 is gratefully acknowledged.

It is well known that the unrestricted bootstrap estimator of the slope parameter in the random walk model without drift converges to a random distribution. This bootstrap failure is commonly attributed to the discontinuity of the limit distribution of the least-squares estimator in the parameter of interest. We demonstrate by counterexample that this type of continuity is not essential for the validity of the bootstrap nor is it essential that the rate of convergence of the estimator remain constant over the whole parameter space.We thank Don Andrews, Shinichi Sakata, Jonathan Wright, and two anonymous referees for very helpful comments. The views expressed in this paper do not necessarily reflect those of the European Central Bank or its members.

We propose an estimator for the maximal moment exponent of a GARCH(1,1) sequence. We establish its consistency asymptotic normality with rate n−1/2. Finite sample properties are investigated by means of a small simulation study.The research for this paper was partially supported by NSF grant INT-0223262. István Berkes and Lajos Horváth were supported by the Hungarian National Foundation for Scientific Research, grant T 29621. Piotr Kokoszka and Lajos Horváth were supported by NATO grant PST.CLG.977607.

In this paper, we quantify the asymptotic bias of the Florens-Zmirou (1993, Journal of Applied Probability 30, 790–804) and Jiang and Knight (1997, Econometric Theory 13, 615–645) estimator for the diffusion coefficient when the step of discretization is fixed, and then we propose a bias adjustment that partially compensates for the distortion. Also, we show that our estimators have all the asymptotic properties of the Florens-Zmirou and Jiang and Knight estimator when the step of discretization goes to zero. We provide some examples.I thank the editor Peter C.B. Phillips and the two referees for comments and suggestions that led to considerable improvement of the paper. I am also grateful to Carlos Braumann and Tom Kundert for helpful comments. This research was supported by the Fundação para a Ciência e a Tecnologia (FCT) and by POCTI.

In this paper, two competing types of multistep predictors, i.e., plug-in and direct predictors, are considered in autoregressive (AR) processes. When a working model AR(k) is used for the h-step prediction with h > 1, the plug-in predictor is obtained from repeatedly using the fitted (by least squares) AR(k) model with an unknown future value replaced by their own forecasts, and the direct predictor is obtained by estimating the h-step prediction model's coefficients directly by linear least squares. Under rather mild conditions, asymptotic expressions for the mean-squared prediction errors (MSPEs) of these two predictors are obtained in stationary cases. In addition, we also extend these results to models with deterministic time trends. Based on these expressions, performances of the plug-in and direct predictors are compared. Finally, two examples are given to illustrate that some stationary case results on these MSPEs can not be generalized to the nonstationary case.The author is deeply grateful to the co-editor Pentti Saikkonen and two referees for their helpful suggestions and comments on a previous version of this paper.

It is possible to view econometric history as the retrospective assessment of intellectual resource allocations. This paper uses such a perspective to examine the history of extraneous estimation. Independently proposed by Marschak (1939, Econometrica 7, 332–335) and Wold (1940, Efterfrågan på jordbruksprodukter och dess känslighet för prisoch inkomstförändringar (The demand for agricultural products and its sensitivity to price and income changes)), extraneous estimation was widely used as a collinearity solution in demand analysis from 1950 to 1980. We give a detailed account of the origins of this procedure and the theoretical work that clarified the statistical structure that underpinned it. The subsequent literature is then assessed in terms of the optimal use of available econometric tools and the interaction between theory and applications. We find that optimal econometric procedures were rarely used and that theoretical refinements continued to be published well after extraneous estimation had ceased to be a useful tool for demand analysis.An earlier version of this paper titled “To Pool or Not to Pool? Why Is That Still a Question?” was presented at the Thirteenth Annual Meeting of the Canadian Econometrics Study Group, University of Waterloo, September 21–22, 1996. I thank Robin Carter for his comments at that presentation. Another version titled “The Optimum Amount of Econometrics” was presented on October 31, 1998, at Concordia University on the occasion of Gordon Fisher's seventieth birthday. I am indebted to Lady Giovanna Stone and Peter Jones of King's College Library for arranging access to Sir Richard Stone's papers and to the staff of the UCLA Research Library for access to the Jacob Marschak Archive. James Tobin was kind enough to share some of his recollections of the “extraneous estimation” period. George Szava-Kovats and Chris Minns provided able research assistance, and Charlene Hill typed the manuscript with her customary efficiency. Colin and Sally Ash, Reinhold Bergstrom, Jim Hamilton, Balder von Hohenbalken, Joe Ostroy, and Martin Weale provided help of the sort that does not lend itself to detailed elaboration. An earlier version of the paper had the benefit of a close and perceptive reading by John Aldrich. The referees' and editor's comments have also helped shape the paper into its present form, and I thank them for their contributions.

We consider the limiting behavior of the overidentifying restrictions test in the presence of neglected structural instability at a single “break point.” It is shown that the test need not be consistent against this type of misspecification. If it is consistent then it emerges that the limiting behavior of this test statistic depends on the covariance matrix estimator employed. In this paper we consider the case in which a heteroskedasticity autocorrelation covariance (HAC) is used. It is shown that (i) if the HAC estimator is based on uncentered autocovariances then the overidentifying restrictions test diverges at rate T/bT where T is the sample size and bT is the bandwidth; (ii) if the HAC estimator is based on centered autocovariances then the rate of increase of the overidentifying restrictions test is either T/bT or T depending on the form of the instability. These results are used to provide conditions for the consistency of the method of moment selection of Andrews (1999, Econometrica 67, 543–564) when certain elements of the candidate set of moments are misspecified as a result of neglected structural instability.This work was begun while Hall was a Senior Research Fellow and Peixe was a graduate student at the Department of Economics, University of Birmingham, UK, and this support is gratefully acknowledged. Peixe also gratefully acknowledges financial support from FCT under Grant PRAXIS XXI/BD/13453/97. We are very grateful for the very useful comments of Don Andrews and two anonymous referees.