In applied work economists often seek to relate a given response variable y to some causal parameter μ* associated with it. This parameter usually represents a summarization based on some explanatory variables of the distribution of y, such as a regression function, and treating it as a conditional expectation is central to its identification and estimation. However, the interpretation of μ* as a conditional expectation breaks down if some or all of the explanatory variables are endogenous. This is not a problem when μ* is modeled as a parametric function of explanatory variables because it is well known how instrumental variables techniques can be used to identify and estimate μ*. In contrast, handling endogenous regressors in nonparametric models, where μ* is regarded as fully unknown, presents difficult theoretical and practical challenges. In this paper we consider an endogenous nonparametric model based on a conditional moment restriction. We investigate identification-related properties of this model when the unknown function μ* belongs to a linear space. We also investigate underidentification of μ* along with the identification of its linear functionals. Several examples are provided to develop intuition about identification and estimation for endogenous nonparametric regression and related models.We thank Jeff Wooldridge and two anonymous referees for comments that greatly improved this paper.

In this paper, some structural properties of a family of generalized autoregressive conditionally heteroskedastic (GARCH) processes are considered. First, a sufficient and necessary condition for the strict stationarity of this family of GARCH processes is given. Second, some simple conditions for the existence of the moments of the family of GARCH processes are also derived. Finally, we describe the tail of the marginal distribution of the family of GARCH processes.I am grateful to Bruce E. Hansen and an anonymous referee for helpful comments on the initial version.

We define a three-step procedure for more efficient estimation of the nonparametric regression mean with nonparametric autocorrelated errors. The procedure is based upon a nonparametric prewhitening transformation of the dependent variable that has to be estimated from the data by a local polynomial technique. We establish the asymptotic distribution of our estimator under weak dependence conditions and show that it is more efficient than the conventional local polynomial estimator. Furthermore, we consider criterion functions based on the linear exponential family, which include the local polynomial least squares criterion as a special case. Simulation evidence suggests that significant gains can be achieved in finite samples with our approach.The authors thank Oliver Linton for his many constructive and helpful suggestions. The very insightful comments from the referees are also gratefully acknowledged. The second author gratefully acknowledges financial support from the Academic Senate, UCR.

A new one-sided test for serial correlation in multivariate time series models is proposed. The test is based on a comparison between a multivariate spectral density estimator and the spectral density under the null hypothesis of no serial correlation. Duchesne and Roy (2004, Journal of Multivariate Analysis 89, 148–180) considered a multivariate kernel-based spectral density estimator. However, when the spectral density exhibits irregular features (because of strong autocorrelation or seasonality, among other factors), it is expected that a multivariate wavelet-based spectral density estimator will capture more effectively the local behavior of the spectral density. We consider a test based on a wavelet spectral density estimator, which represents a generalization of a test proposed by Lee and Hong (2001, Econometric Theory 17, 386–423). The asymptotic distribution of the new test is established under the null hypothesis, which is N(0,1). We propose and justify a suitable data-driven method to choose the smoothing parameter of the wavelet estimator (called the finest scale in that context). The new test should be powerful when the spectral density contains peaks or bumps. This is confirmed in a simulation study, where kernel-based and wavelet-based estimators are compared.The author thanks the co-editor Pentti Saikkonen and two referees for their constructive remarks and suggestions. Many thoughtful comments of the referees led to significant improvements of the paper. This work was supported by grants from the National Science and Engineering Research Council of Canada and the Fonds québécois de la recherche sur la nature et les technologies du Québec (Canada).

This paper proposes a new testing procedure to detect the presence of a cointegrating relationship that follows a globally stationary smooth transition process. In the context of nonlinear smooth transition error correction models (ECMs) we provide two simple operational versions of the tests. First, we obtain the associated nonlinear ECM-based tests. Second, we derive the nonlinear analogue of the residual-based test for cointegration in linear models. We derive the asymptotic distributions of the proposed tests. Monte Carlo simulation exercises confirm that our proposed tests have much better power than the linear counterparts against the alternative of a globally stationary nonlinear cointegrating process. In an application to the price-dividend relationship, our test is able to find cointegration, whereas the linear-based tests fail to do so.We are grateful to an associate editor, two anonymous referees, Richard Baillie, In Choi, Atanas Christev, Hira Koul, Richard Harris, Cheng Hsiao, Changjin Kim, Joon Park, Peter Schmidt, Yoonjae Whang, Jeff Wooldridge, and seminar participants at University of Edinburgh, Heriot-Watt University, Korea University, University of Leeds, Michigan State University, University of Newcastle, and Sungkunkwan University for their helpful comments. Partial financial support from the ESRC (grant R000223399) is gratefully acknowledged. The usual disclaimer applies.

Testing goodness (or lack) of fit for distributions of observable and nonobservable random variables is one of the main topics in statistics. When they exist, the corresponding density functions and their shapes allow researchers to easily recognize the underlying distribution functions. The present paper is concerned with the densities of (unobservable) generalized autoregressive conditional heteroskedasticity (GARCH) innovations and also with developing goodness-of-fit tests for the densities. Specifically, we construct and investigate large-sample properties of a kernel-type density estimator for GARCH innovations based on (observable) residuals.The authors sincerely thank the Co-Editor Oliver Linton and three anonymous referees for constructive criticism and suggestions that helped us to prepare a much revised version of the original manuscript. The feedback by participants of the Conference on Statistical Models for Financial Data at the University of Graz in May 2004 is also greatly appreciated. The research of the first author was partially supported by NSF grant INT-0223262 and NATO grant PST.EAP.CLG 980599. The research of the second author was partially supported by a Discovery Research Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada.

In this paper we provide an asymptotic analysis of generalized bipower measures of the variation of price processes in financial economics. These measures encompass the usual quadratic variation, power variation, and bipower variations that have been highlighted in recent years in financial econometrics. The analysis is carried out under some rather general Brownian semimartingale assumptions, which allow for standard leverage effects.Ole E. Barndorff-Nielsen's work is supported by the Centre for Analytical Finance (CAF), which is funded by the Danish Social Science Research Council. Neil Shephard's research is supported by the UK's ESRC through the grant “High frequency financial econometrics based upon power variation.” We thank the editor, Peter Phillips, and the referees for their stimulating comments on an earlier version.

The solution of differential equations lies at the heart of many problems in structural economics. In econometrics the general nonparametric analysis of consumer welfare is historically the most obvious application, but there are also many applications in finance and other fields. This work considers the general nonparametric form for these problems and identification conditions. It derives a kernel-based estimator and shows consistency and asymptotic normality.

In particular, the link with inverse problems allows us to define it in terms of a well-posed inverse problem and to stress the regularity properties of the estimated solution.I thank Jean-Pierre Florens, Richard Blundell, Eric Renault, and Christine Thomas-Agnan for stimulating conversations, suggestions, and advice. I am very grateful to Oliver Linton and two anonymous referees for most helpful comments.

The single-index model is one of the most popular semiparametric models in applied quantitative sciences. Two new estimation methods have been proposed recently by Hristache, Juditski, and Spokoiny (2001, Annals of Statistics 29, 595–623) and Xia, Tong, Li, and Zhu (2002, Journal of the Royal Statistical Society, Series B 64, 363–410), respectively. However, their asymptotic distributions have not been investigated yet. In this paper, alternative versions for the methods are investigated. Asymptotic distributions of the estimators are derived. Efficiency comparisons between the estimation methods are made.The author is most grateful to Professor O. Linton and Professor W. Härdle for helpful discussions. Valuable comments from two anonymous reviewers have improved the presentation of the paper substantially. The research has been partially supported by NUS research grant R-155-000-048-112, National University of Singapore, Singapore, and the Alexander von Humboldt Foundation, Germany.

The bias-reduced log-periodogram estimator of Andrews and Guggenberger (2003, Econometrica 71, 675–712) for the long-memory parameter d in a stationary long-memory time series reduces the asymptotic bias of the original log-periodogram estimator of Geweke and Porter-Hudak (1983) by an order of magnitude but inflates the asymptotic variance by a multiplicative constant cr, for example, c1 = 2.25 and c2 = 3.52. In this paper, we introduce a new, computationally attractive estimator by taking a weighted average of estimators over different bandwidths. We show that, for each fixed r ≥ 0, the new estimator can be designed to have the same asymptotic bias properties as but its asymptotic variance is changed by a constant cr* that can be chosen to be as small as desired, in particular smaller than cr. The same idea is also applied to the local-polynomial Whittle estimator in Andrews and Sun (2004, Econometrica 72, 569–614) leading to the weighted estimator . We establish the asymptotic bias, variance, and mean-squared error of the weighted estimators and show their asymptotic normality. Furthermore, we introduce a data-dependent adaptive procedure for selecting r and the bandwidth m and show that up to a logarithmic factor, the resulting adaptive weighted estimator achieves the optimal rate of convergence. A Monte Carlo study shows that the adaptive weighted estimator compares very favorably to several other adaptive estimators.We thank a co-editor and three referees for very helpful suggestions. We are grateful for the constructive comments offered by Marc Henry, Javier Hidalgo, and especially Katsumi Shimotsu.

This paper extends current theory on the identification and estimation of vector time series models to nonstationary processes. It examines the structure of dynamic simultaneous equations systems or ARMAX processes that start from a given set of initial conditions and evolve over a given, possibly infinite, future time horizon. The analysis proceeds by deriving the echelon canonical form for such processes. The results are obtained by amalgamating ideas from the theory of stochastic difference equations with adaptations of the Kronecker index theory of dynamic systems. An extension of these results to the analysis of unit-root, partially nonstationary (cointegrated) time series models is also presented, leading to straightforward identification conditions for the error correction, echelon canonical form. An innovations algorithm for the evaluation of the exact Gaussian likelihood is given. The asymptotic properties of the approximate Gaussian estimator and the exact maximum likelihood estimator based upon the algorithm are derived for the cointegrated case. Examples illustrating the theory are discussed, and some experimental evidence is also presented.I thank two referees for insightful comments and helpful suggestions on the content and presentation of this paper. I am particularly grateful for the correction of errors in earlier drafts and reference to the work of B. Hanzon. Financial support under ARC grant DP0343811 is gratefully acknowledged.

This paper is concerned with the problem of combining a data set that identifies the conditional distribution P(y|x) with one that identifies the conditional distribution P(z|x) to identify the regressions E(y|x,·) ≡ [E(y|x,z = j),j ∈ Z] when the conditional distribution P(y|x,z) is unknown. Cross and Manski (2002, Econometrica 70, 357–368) studied this problem and showed that the identification region of E(y|x,·) can be precisely calculated when y has finite support. Here we generalize the result of Cross and Manski, showing that the identification region can be precisely calculated also in the case in which y has infinite support.We are grateful to the co-editor Paolo Paruolo, an anonymous referee, Maria Goltsman, Nick Kiefer, Tymon Tatur, and Tim Vogelsang for useful comments. Any remaining errors are our own responsibility.Financial support from Northwestern University's Dissertation Year Fellowship is gratefully acknowledged.

We revisit the exact properties of two-stage least squares and limited information maximum likelihood estimators in a structural equation/instrumental variables regression under Gaussian assumptions. Simple derivations based on conditioning serve both to demystify the apparently complicated formulas, and to isolate the key quantities that determine the properties of the estimators. Some recent results obtained under weak-instrument asymptotics are sharpened and clarified by the exact analysis.Thanks to Peter Phillips and several anonymous referees for helpful comments that improved the paper considerably.

This paper considers the estimation of the parameters of general systems of stochastic differential-difference equations in which the lag parameters themselves are treated as unknown and are not restricted to be integers and therefore form part of the parameter vector to be estimated. The asymptotic properties of an infeasible frequency domain maximum likelihood estimator are established in addition to those of a feasible version based on truncating an infinite series that arises in the computation of the spectral density function of the observed discrete time series. Precise conditions that the truncation parameter must satisfy for the asymptotic results to hold are provided.We are grateful to Gordon Kemp, Andrew Harvey, the Editor, and two anonymous referees for helpful comments on an earlier version of this paper. Any remaining errors are the sole responsibility of the authors. The first author thanks the Economic and Social Research Council for financial support under grant number R00429434216, and the second author thanks the Leverhulme Trust for financial support in the form of a Philip Leverhulme Prize.

This paper studies a semiparametric nonstationary binary choice model. Imposing a spherical normalization constraint on the parameter for identification purposes, we find that the maximum score estimator and smoothed maximum score estimator are at least [square root of n]-consistent. Comparing this rate to the convergence rate of the parametric maximum likelihood estimator (MLE), we show that when a normalization restriction is imposed on the parameter, the Park and Phillips (2000, Econometrica 68, 1249–1280) parametric MLE converges at a rate of n3/4 and its limiting distribution is a mixed normal. Finally, we show briefly how to apply our estimation method to a nonstationary single-index model.The first draft of the paper was written while Guerre was visiting the economics department of the University of Southern California. We thank Peter C.B. Phillips, a co-editor, and three anonymous referees for helpful comments and John Dolfin for proofreading. Guerre thanks the economics department of the University of Southern California for its hospitality during his visit. Moon appreciates financial support of the University of Southern California faculty development award.

We investigate the convergence in distribution of integrals of stochastic processes satisfying a functional limit theorem. We allow a large class of continuous Gaussian processes in the limit. Depending on the continuity properties of the underlying process, local Lebesgue or Riemann integrability is required.We are grateful to the referees and Benedikt Pötscher for their helpful and constructive comments. The research of the first author was partially supported by OTKA grants T37668 and T43037 and NSF-OTKA grant INT-0223262. The research of the second author was partially supported by NATO grant PST.EAP.CLG 980599 and NSF-OTKA grant INT-0223262.

The stationarity conditions for an autoregressive (AR) process in general are reduced to a remarkably simple inequality if the lag coefficients are restricted to be identical. The condition is not only analytically elegant but also applicable in checking the validity of the stationarity conditions for such a restricted AR process of any order.We are deeply indebted to Professor Paolo Paruolo, NP co-editor of Econometric Theory, and anonymous referees for constructive comments and suggestions that led to significant improvements. Errors, if any, are solely ours.

We propose a closed-form estimator for the linear GARCH(1,1) model. The estimator has the advantage over the often used quasi-maximum likelihood estimator (QMLE) that it can be easily implemented and does not require the use of any numerical optimization procedures or the choice of initial values of the conditional variance process. We derive the asymptotic properties of the estimator, showing T(κ−1)/κ-consistency for some κ ∈ (1,2) when the fourth moment exists and -asymptotic normality when the eighth moment exists. We demonstrate that a finite number of Newton–Raphson iterations using our estimator as starting point will yield asymptotically the same distribution as the QMLE when the fourth moment exists. A simulation study confirms our theoretical results.The first author's research was supported by the Shoemaker Foundation. The second author's research was supported by the Economic and Social Science Research Council of the United Kingdom.

This note uses fixed bandwidth (fixed-b) asymptotic theory to suggest a new approach to testing cointegration parameters in a single-equation cointegration environment. It is shown that the standard tests still have asymptotic distributions that are free of serial correlation nuisance parameters regardless of the bandwidth or kernel used, even if the regressors in the cointegration relationship are endogenous.I thank Tim Vogelsang for his guidance, Pentti Saikkonen for the promptness with which he provided me with an old working paper of his, Wayne Fuller for several discussions, and an anonymous referee for many helpful comments. The usual disclaimer applies.

In Wooldridge (2005), I posed a problem whose solution involved showing that the so-called ignorability of treatment assumption, commonly used in the treatment effects literature, was necessarily violated. In particular, in a setting with multiple treatment effects and without distributional or functional form assumptions, I assumed that treatment was randomized with respect to the counterfactual outcomes but not with respect to the covariates. I applied the law of iterated expectations to provide a simple proof that ignorability (given the covariates) necessarily fails.