This paper develops semiparametric kernel-based estimators of risk-specific hazard functions for competing risks data. Both discrete and continuous failure times are considered. The maintained assumption is that the hazard function depends on explanatory variables only through an index. In contrast to existing semiparametric estimators, proportional hazards is not assumed. The new estimators are asymptotically normally distributed. The estimator of index coefficients is root-n consistent. The estimator of hazard functions achieves the one-dimensional rate of convergence. Thus the index assumption eliminates the “curse of dimensionality.” The estimators perform well in Monte Carlo experiments.I thank Denise Doiron for stimulating my interest in this research project and Catherine de Fontenay, Hans Christian Kongsted, Lars Muus, seminar participants, and two anonymous referees for comments on an earlier version of the paper. I gratefully acknowledge the hospitality of the University of Aarhus and the University of Copenhagen, where part of this research was undertaken.

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.

This paper is concerned with deriving formulae for higher order derivatives of exogenous variables for use in estimating the parameters of an open secondorder continuous time model with mixed stock and flow data and first and second order derivatives of exogenous variables which are not observable. This should provide the basis for the future estimation of continuous time models in a range of applied areas using the new Gaussian estimation computer program developed by Nowman [4].

Most work on the asymptotic properties of least absolute deviations (LAD) estimators makes use of the assumption that the common distribution of the disturbances has a density that is both positive and finite at zero. We consider the implications of weakening this assumption in a number of regression settings, primarily with a time series orientation. These models include ones with deterministic and stochastic trends, and we pay particular attention to the case of a simple unit root model. The way in which the conventional assumption on the error distribution is modified is motivated in part by N.V. Smirnov's work on domains of attraction in the asymptotic theory of sample quantiles. The approach adopted usually allows for simple characterizations (often featuring a single parameter, γ), of both the shapes of the limiting distributions of the LAD estimators and their convergence rates. The present paper complements the closely related recent work of K. Knight.

The issue of identification of covariance structures, which arises in a number of different contexts, has been so far linked to conditions on the true parameters to be estimated. In this paper, this limitation is removed.

As done by Johansen (1995, Journal of Econometrics 69, 112–132) in the context of linear models, the present paper provides necessary and sufficient conditions for the identification of a covariance structure that depends only on the constraints and can therefore be checked independently of estimated parameters.

A structure condition is developed, which only depends on the structure of the constraints. It is shown that this condition, if coupled with the familiar order condition, provides a sufficient condition for identification. In practice, because the structure condition holds if and only if a certain matrix, constructed from the constraint matrices, is invertible, automatic software checking for identification is feasible even for large-scale systems.

Most of the paper focuses on structural vector autoregressions, but extensions to other statistical models are also briefly discussed.I thank all the participants at the meeting held in Pavia on June 11, 2004, in honor of Carlo Giannini for their comments; it goes without saying that Carlo himself provided not only acute observations on the day but also the main inspiration for this piece of work. Sadly, Carlo passed away on September 11, 2004, and this paper is dedicated to his memory. Pär Österholm spotted several mistakes in an earlier version and helped me clarify some implementation details. Thanks are also due to Gianni Amisano, Bruce Hansen, Giulio Palomba, Paolo Paruolo, and two anonymous referees. The usual disclaimer obviously applies.

This note suggests a simple modification to the Kwiatkowski, Phillips, Schmidt, and Shin (1992, Journal of Econometrics, 54, 159–178) test (KPSS test) so that it is applicable to testing the null hypothesis of near integration against a unit root alternative. The modified KPSS test is shown not to suffer from the asymptotic size distortion problems of the original KPSS test that are described by Müller (2005, Journal of Econometrics 128, 195–213). The test also has good asymptotic and finite-sample properties relative to the point optimal tests of Müller (2005) and Elliott and Müller (2006, Journal of Econometrics 135, 285–310).

Most work in the area of nonlinear econometric modeling is based on a single equation and assumes exogeneity of the explanatory variables. Recently, work by Caner and Hansen (2004) and Psaradakis, Sola, and Spagnolo (2005) has considered the possibility of estimating nonlinear models by methods that take into account endogeneity but provide no tests for exogeneity. This paper examines the problem of testing for exogeneity in nonlinear threshold models. We suggest new Hausman-type tests and discuss the use of the bootstrap to improve the properties of asymptotic tests. The theoretical properties of the tests are discussed and an extensive Monte Carlo study is undertaken.

In the past thirty-five years, time-series econometrics developed from infancy to relative maturity. A large part of that development is due to Robert F. Engle, whose work is distinguished by exceptional creativity in the empirical modeling of dynamic economic and financial phenomena. Engle's footsteps range widely, from early work on band-spectral regression, testing, and exogeneity through more recent work on cointegration, autoregressive conditional heteroskedasticity (ARCH) models, and ultra-high-frequency financial asset return dynamics. The booming field of financial econometrics, which did not exist twenty-five years ago, is built in large part on the volatility models pioneered by Engle, and their many variations and extensions, which have found widespread application in financial risk management, asset pricing, and asset allocation.

We began the interview in fall 1998 at Spruce in Chicago, the night before the annual NBER/NSF Time Series Seminar; continued in fall 2000 at Tabla in New York; continued again in summer 2001 at the Conference on Market Microstructure and High-Frequency Data in Finance, Sandbjerg Estate, Denmark; and wrapped up by telephone in January 2003.For their cheerful and effective assistance in transcribing this interview, I thank (without implicating) Sean Campbell, Michele Souli, and Clara Vega.

This paper presents a set of questions prepared by Clive Granger with responses by David Hendry on the use of PcGets (see Hendry and Krolzig, 2001) in data modeling and as a new research tool. PcGets is an Ox package (see Doornik, 2001) implementing automatic general-to-specific (Gets) modeling for linear regression models based on the theory of reduction, as in Hendry (1995, Ch. 9).

This paper studies the consistency of the Gaussian maximum likelihood estimator in a cointegrated vector autoregressive model with nonlinear time trends in cointegrating relations. The results are proved in a coordinate free framework that readily allows for general nonlinear parameter restrictions and makes it possible to show the consistency of reduced form parameter estimators without assuming identifiability of underlying structural parameters. Various consistency results for structural parameter estimators can then be obtained by imposing suitable identification conditions for the parameters of interest but not necessarily for nuisance parameters. Orders of consistency are also obtained because they are needed to develop a related asymptotic theory of statistical inference.

This paper proposes a new Bayesian approach for estimating, nonparametrically, functional parameters in econometric models that are characterized as the solution of a linear inverse problem. By using a Gaussian process prior we propose the posterior mean as an estimator and prove frequentist consistency of the posterior distribution. The latter provides the frequentist validation of our Bayesian procedure. We show that the minimax rate of contraction of the posterior distribution can be obtained provided that either the regularity of the prior matches the regularity of the true parameter or the prior is scaled at an appropriate rate. The scaling parameter of the prior distribution plays the role of a regularization parameter. We propose a new data-driven method for optimally selecting in practice this regularization parameter. We also provide sufficient conditions such that the posterior mean, in a conjugate-Gaussian setting, is equal to a Tikhonov-type estimator in a frequentist setting. Under these conditions our data-driven method is valid for selecting the regularization parameter of the Tikhonov estimator as well. Finally, we apply our general methodology to two leading examples in econometrics: instrumental regression and functional regression estimation.

This paper generalizes the univariate results of Chan and Tran (1989, Econometric Theory 5, 354–362) and Phillips (1990, Econometric Theory 6, 44–62) to multivariate time series. We develop the limit theory for the least-squares estimate of a VAR(l) for a random walk with independent and identically distributed errors and for I(1) processes with weakly dependent errors whose distributions are in the domain of attraction of a stable law. The limit laws are represented by functional of a stable process. A semiparametric correction is used in order to asymptotically eliminate the “bias” term in the limit law. These results are also an extension of the multivariate limit theory for square-integrable disturbances derived by Phillips and Durlauf (1986, Review of Economic Studies 53, 473–495). Potential applications include tests for multivariate unit roots and cointegration.

Two issues have come up in the specification of a prior in the Bayesian analysis of time series with possible unit roots. The first issue deals with the singularity that is due to the local identification problem of the unconditional mean of an AR(1) process in the limit of a random walk. This singularity problem is related to the difference between a structural parameterization and the linear reduced form in a standard regression model with fixed regressors. The second is related to the time series nature of the regressor in an AR(1) model. In this paper we will concentrate on the parameterization issue. First, it is shown that the posterior of the autoregressive parameter can be very sensitive to the degree of prior dependence between the unconditional mean and the autocorrelation parameter. Second, the time series nature of the problem suggests a particular form of this dependence.

This paper proposes a version of the integrated conditional moment (ICM) test that is optimal for a class of composite alternatives. The ICM test is built on the fact that a random function based on a correctly specified model should have zero mean, whereas any misspecification in the conditional mean implies a divergent mean for the random function. We derive test statistics that are optimal for each basis element of an orthonormal decomposition of the function space for which the random function is an element. We then use a weighted summation of these test statistics to compose the single test statistic that is optimal for any pair of alternatives that are symmetric about zero. This test is equivalent to using a particular measure in the ICM test of Bierens and Ploberger (1997, Econometrica 65, 1129–1152).

We study the convergence in distribution of M-estimators over a convex kernel. Under convexity, the limit distribution of M-estimators can be obtained under minimal assumptions. We consider the case when the limit is arbitrary, not necessarily normal. If some Taylor expansions hold, the limit distribution is stable. As an application, we examine the limit distribution of M-estimators for the multivariate linear regression model. We obtain the distributional convergence of M-estimators for the multivariate linear regression model for a wide range of sequences of regressors and different types of conditions on the sequence of errors.

We consider the least-squares estimator in a strictly stationary first-order autoregression without an estimated intercept. We study its continuous time asymptotic distribution based on an asymptotic framework where the sampling interval converges to zero as the sample size increases. We derive a momentgenerating function which permits the calculation of percentage points and moments of this asymptotic distribution and assess the adequacy of the approximation to the finite sample distribution. In general, the approximation is excellent for values of the autoregressive parameter near one. We also consider the behavior of the power function of tests based on the normalized leastsquares estimator. Interesting nonmonotonic properties are uncovered. This analysis extends the study of Perron [15] and helps to provide explanations for the finite sample results established by Nankervis and Savin [13].

This note adds to the recent research project on treatment choice under ambiguity. I generalize the Manski (Journal of Econometrics, in press) analysis of minimax regret treatment choice by considering a more general setting and, more importantly, by solving for the treatment rule given finitely many (as opposed to two) treatments. The most interesting finding is that with three or more undominated treatments, the minimax regret treatment rule may assign the same treatment to all subjects; thus, the most salient feature of the two-treatment case does not generalize.I thank Chuck Manski, the co-editor, and especially an anonymous referee for helpful comments. Financial support through the Robert Eisner Memorial Fellowship, Department of Economics, Northwestern University, in addition to a Dissertation Year Fellowship, The Graduate School, Northwestern University, is gratefully acknowledged.

It is well known that a one-step scoring estimator that starts from any N1/2-consistent estimator has the same first-order asymptotic efficiency as the maximum likelihood estimator. This paper extends this result to k-step estimators and test statistics for k ≥ 1, higher order asymptotic efficiency, and general extremum estimators and test statistics.

The paper shows that a k-step estimator has the same higher order asymptotic efficiency, to any given order, as the extremum estimator toward which it is stepping, provided (i) k is sufficiently large, (ii) some smoothness and moment conditions hold, and (iii) a condition on the initial estimator holds.

For example, for the Newton–Raphson k-step estimator based on an initial estimator in a wide class, we obtain asymptotic equivalence to integer order s provided 2k ≥ s + 1. Thus, for k = 1, 2, and 3, one obtains asymptotic equivalence to first, third, and seventh orders, respectively. This means that the maximum differences between the probabilities that the (N1/2-normalized) k-step and extremum estimators lie in any convex set are o(1), o(N−3/2), and o(N−3), respectively.

Shibata (1981, Biometrika 68, 45–54) considers data-generating mechanisms belonging to a certain class of linear regressions with errors that are independent and identically normally distributed. He compares the variable selection criteria AIC (Akaike information criterion) and BIC (Bayesian information criterion) using the following type of comparison. For each fixed possible data–generating mechanism, these criteria are compared as the data length increases. The results of this comparison have been interpreted as meaning that, in the context of the data-generating mechanisms considered by Shibata, AIC is better than BIC for large data lengths. Shibata's comparison is pointwise in the space of data–generating mechanisms (as the data length increases). Such comparisons are potentially misleading. We consider a simple class of data-generating mechanisms satisfying Shibata's assumptions and carry out a different type of comparison. For each fixed data length (possibly large) we compare the variable selection criteria for every possible data-generating mechanism in this class. According to this comparison, for this class of data-generating mechanisms no matter how large the data length AIC is not better than BIC.