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.

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.

A central limit theorem is proved for dependent stochastic processes. Global heterogeneity of the distribution of the terms is permitted, including asymptotically unbounded moments. The approach is to adapt a CLT for martingale differences due to McLeish and show that suitably defined Bernstein blocks satisfy the required conditions.

This paper studies the asymptotic validity of the Anderson–Rubin (AR) test and the J test for overidentifying restrictions in linear models with many instruments. When the number of instruments increases at the same rate as the sample size, we establish that the conventional AR and J tests are asymptotically incorrect. Some versions of these tests, which are developed for situations with moderately many instruments, are also shown to be asymptotically invalid in this framework. We propose modifications of the AR and J tests that deliver asymptotically correct sizes. Importantly, the corrected tests are robust to the numerosity of the moment conditions in the sense that they are valid for both few and many instruments. The simulation results illustrate the excellent properties of the proposed tests.

Tests for Granger-causality have been performed in numerous empirical studies. These tests are usually based on finite order vector autoregressive (VAR) processes, and the assumption is made that the model fitted to the available data corresponds to the true data generating mechanism. In the present study, the more general assumption is made that a finite order VAR model is fitted to a potentially infinite order process. The order is assumed to increase with the sample size. Asymptotic properties of tests for Granger-causality as well as other types of causality concepts are derived. Some limited small sample results are obtained using simulation methods.

Value-at-risk (VaR) and expected shortfall (ES) are now both widely used risk measures. However, users have not paid much attention to the estimation risk issues, especially in the case of heteroskedastic financial time series. The key challenge arises from the fact that the estimated generalized autoregressive conditional heteroskedasticity (GARCH) innovations are not the true independent innovations. The purpose of this work is to provide an analytical method to assess the precision of conditional VaR and ES in the GARCH model estimated by the filtered historical simulation (FHS) method based on the asymptotic behavior of the residual empirical distribution function in GARCH processes. The proposed method is evaluated by simulation and proved valid.

A vector autoregressive model allowing for unit roots as well as an explosive characteristic root is developed. The Granger-Johansen representation shows that this results in processes with two common features: a random walk and an explosively growing process. Cointegrating and coexplosive vectors can be found that eliminate these common factors. The likelihood ratio test for a simple hypothesis on the coexplosive vectors is analyzed. The method is illustrated using data from the extreme Yugoslavian hyperinflation of the 1990s.

The distributions of products and ratios of random variables are of interest in many areas of the sciences. In this note, the exact distributions of the product XY and the ratio X/Y are derived when X and Y are gamma and Weibull random variables distributed independently of each other.The authors thank the referee and the editor for carefully reading the paper and for their help in improving the paper.

In this paper we propose an estimation procedure for a censored regression model where the latent regression function has a partially linear form. Based on a conditional quantile restriction, we estimate the model by a two stage procedure. The first stage nonparametrically estimates the conditional quantile function at in-sample and appropriate out-of-sample points, and the second stage involves a simple weighted least squares procedure. The proposed procedure is shown to have desirable asymptotic properties under regularity conditions that are standard in the literature. A small scale simulation study indicates that the estimator performs well in moderately sized samples.

This paper presents the likelihood ratio (LR) test for the number of cointegrating relations in the I(2) vector autoregressive model. It is shown that the asymptotic distribution of the LR test for the cointegration ranks is identical to the asymptotic distribution of the much applied test statistic based on the two-step estimation procedure in Johansen (1995, Econometric Theory 11, 25–59), Paruolo (1996, Journal of Econometrics 72, 313–356), and Rahbek, Kongsted, and Jørgensen (1999, Journal of Econometrics 90, 265–289). By construction the LR test statistic is smaller than the non-LR test statistic from the two-step procedure, and application of the LR test may change rank selection in empirical work. Based on a study of existing empirical applications and related Monte Carlo simulations we conclude that the LR test has much better size properties when compared to the two-step-based test. Overall, we propose use of the LR test for rank determination in I(2) analysis.Discussions with Søren Johansen, Hans Christian Kongsted, and Bent Nielsen are gratefully acknowledged. We also thank the editor and three anonymous referees for very constructive comments that have led to a much improved version of the paper. This research was supported by Danish Social Sciences Research Council grant 2114-04-0001.

The paper traces Trygve Haavelmo's training and early career as an econometrician from graduation in economics at the University of Oslo in 1933 until his departure for the United States in 1939. The overwhelming influence on Haavelmo in this period was Ragnar Frisch, whose econometric laboratory at the University of Oslo was Haavelmo's workplace and training ground. In the latter part of the period Haavelmo traveled in Europe, mostly within the network of econometricians Frisch had been instrumental in establishing. Haavelmo's work with Frisch, his interaction with other econometricians and statisticians, and his own scholarly work are set out in some detail, allowing assessment to be made of the development of Haavelmo's econometric ideas. Of particular interest is how far his ideas had evolved by 1939. This paper deals with Frisch and his research program in the early 1930s. Haavelmo's activities are narrated by and large chronologically. A sequel to this paper will deal with Haavelmo's scientific activities while in the United States from 1939 to 1944.I have benefited greatly from extensive, generous, and very insightful advice—and prodding—from three anonymous referees on earlier drafts. I am also most grateful for advice and encouragement from the editor. To one of the anonymous referees I also owe the title. I absolve the referees and take full responsibility for all remaining errors and shortcomings. The paper draws on correspondence and documents from the Rockefeller Archive Center, Tarrytown, New York, and the Frisch Correspondence Files at the National Library of Norway and in addition from the Frisch and Haavelmo archives, currently at the Department of Economics, University of Oslo. I am most grateful to professor emeritus Tore S. Thonstad, who has done a great job of organizing the Frisch and Haavelmo archives. I have also benefited from the comprehensive Frisch bibliography prepared by professor emeritus Kåre N. Edvardsen. I thank J.J. Polak for reminiscences, Yngve Willassen for information, E.S. Jansen and J. Kiviet for a push to submit, and Hilde Bojer and Inger Bjerkodden for encouragement.

We study estimation and inference of the expected shortfall for time series with infinite variance. Both the smoothed and nonsmoothed estimators are investigated. The rate of convergence is determined by the tail thickness parameter, and the limiting distribution is in the stable class with parameters depending on the tail thickness parameter of the time series and on the dependence structure, which makes inference complicated. A subsampling procedure is proposed to carry out statistical inference. We also analyze a nonparametric estimator of the conditional expected shortfall. A Monte Carlo experiment is conducted to evaluate the finite sample performance of the proposed inference procedure, and an empirical application to emerging market exchange rates (from October 1997 to October 2008) is conducted to highlight the proposed study.

Recently, Arcones and Giné (1992, pp. 13–47, in R. LePage & L. Billard [eds.], Exploring the Limits of Bootstrap, New York: Wiley) established that the bootstrap distribution of the M-estimator converges weakly to the limit distribution of the estimator in probability. In contrast, Brown and Newey (1992, Bootstrapping for GMM, Seminar note) discovered that the bootstrap distribution of the GMM overidentification test statistic does not converge weakly to the x2 distribution. In this paper, it is shown that the bootstrap distribution of the GMM estimator converges weakly to the limit distribution of the estimator in probability. Asymptotic coverage probabilities of the confidence intervals based on the bootstrap percentile method are thus equal to their nominal coverage probability.

Problems with the asymptotic theory of nonlinear maximum likelihood estimation in integrated and cointegrated systems are discussed in this paper. One problem is that standard proofs of consistency generally do not apply; another one is that, even if the consistency has been established, it can be difficult to deduce the limiting distribution of a maximum likelihood estimator from a conventional Taylor series expansion of the score vector. It is argued in this paper that the latter difficulty can generally be resolved if, in addition to consistency, an appropriate result of the order of consistency of the long-run parameter estimator of the model is available and the standardized sample information matrix satisfies a suitable extension of previous stochastic equicontinuity conditions. To make this idea applicable in particular cases, extensions of the author's recent stochastic equicontinuity results, relevant for many integrated and cointegrated systems with nonlinearities in parameters, are provided. As an illustration, a simple regression model with integrated and stationary regressors and nonlinearities in parameters is considered. In this model, the consistency and order of consistency of the long-run parameter estimator are obtained by employing extensions of well-known sufficient conditions for consistency. These conditions are applicable quite generally, and their verification in the special case of this paper suggests how to proceed in more complex models.

In this paper, we study new definitions of noncausality, set in a continuous time framework, illustrated by the intuitive example of stochastic volatility models. Then, we define CIMA processes (i.e., processes admitting a continuous time invertible moving average representation), for which canonical representations and sufficient conditions of invertibility are given. We can provide for those CIMA processes parametric characterizations of noncausality relations as well as properties of interest for structural interpretations. In particular, we examine the example of processes solutions of stochastic differential equations, for which we study the links between continuous and discrete time definitions, find conditions to solve the possible problem of aliasing, and set the question of testing continuous time noncausality on a discrete sample of observations. Finally, we illustrate a possible generalization of definitions and characterizations that can be applied to continuous time fractional ARMA processes.

We present a goodness-of-fit test for time series models based on the discrete spectral average estimator. Unlike current tests of goodness of fit, the asymptotic distribution of our test statistic allows the null hypothesis to be either a short- or long-range dependence model. Our test is in the frequency domain, is easy to compute, and does not require the calculation of residuals from the fitted model. This is especially advantageous when the fitted model is not a finite-order autoregressive model. The test statistic is a frequency domain analogue of the test by Hong (1996, Econometrica 64, 837–864), which is a generalization of the Box and Pierce (1970, Journal of the American Statistical Association 65, 1509–1526) test statistic. A simulation study shows that our test has power comparable to that of Hong's test and superior to that of another frequency domain test by Milhoj (1981, Biometrika 68, 177–187).

The strong consistency of regression quantile statistics (Koenker and Bassett [4]) in linear models with iid errors is established. Mild regularity conditions on the regression design sequence and the error distribution are required. Strong consistency of the associated empirical quantile process (introduced in Bassett and Koenker [1]) is also established under analogous conditions. However, for the proposed estimate of the conditional distribution function of Y, no regularity conditions on the error distribution are required for uniform strong convergence, thus establishing a Glivenko-Cantelli-type theorem for this estimator.

Dealing with noninvertible, infinite-order moving average (MA) models, we study the asymptotic properties of an estimator of the noninvertible coefficient. The estimator is constructed acting as if the data were generated from a Gaussian MA process. Allowing for two cases on the initial values of the error process, we first discuss the condition for the existence of a consistent estimator. We then compute the probability of the estimator occurring at the boundary of the invertibility region. Some approximations are also suggested to the limiting distribution of the normalized estimator.

We consider the stochastic process , t = 2, …, n, where sw(xt, xi) is a similarity function between the tth and the ith observations and {εt} is a random disturbance term. This process was originally axiomatized by Gilboa, Lieberman, and Schmeidler (2006, Review of Economics and Statistics 88, 433–444) as a way by which agents, or even nature, reason. In the present paper, consistency and the asymptotic distribution of the quasi-maximum likelihood estimator of the parameters of the model are established. Connections to other models and techniques are drawn. In its general form, the model does not fall within any class of nonstationary econometric models for which asymptotic theory is available. For this reason, the developments in this paper are new and nonstandard.

This paper develops a test of the unit root null hypothesis against a stationary threshold process. This testing problem is nonstandard and complicated because a parameter is unidentified and the process is nonstationary under the null hypothesis. We derive an asymptotic distribution for the test, which is not pivotal without simplifying assumptions. A residual-based block bootstrap is proposed to calculate the asymptotic p-values. The asymptotic validity of the bootstrap is established, and a set of Monte Carlo simulations demonstrates its finite-sample performance. In particular, the test exhibits considerable power gains over the augmented Dickey–Fuller (ADF) test, which neglects threshold effects.