This paper considers an empirical likelihood method to estimate the parameters of the quantile regression (QR) models and to construct confidence regions that are accurate in finite samples. To achieve the higher order refinements, we smooth the estimating equations for the empirical likelihood. We show that the smoothed empirical likelihood estimator is first-order asymptotically equivalent to the standard QR estimator and establish that confidence regions based on the smoothed empirical likelihood ratio have coverage errors of order n−1 and may be Bartlett corrected to produce regions with errors of order n−2, where n denotes the sample size. Our result is an extension of the previous result of Chen and Hall (1993, Annals of Statistics 21, 1166–1181) to the regression context. Monte Carlo experiments suggest that the smoothed empirical likelihood confidence regions may be more accurate in small samples than the confidence regions that can be constructed from the smoothed bootstrap method recently suggested by Horowitz (1998, Econometrica 66, 1327–1351).I thank the co-editor Bruce Hansen and anonymous referees for valuable suggestions and comments. I also thank Song X. Chen, Joel Horowitz, Yuichi Kitamura, Oliver Linton, Whitney Newey, Peter Phillips, and Richard Smith for helpful comments. Parts of this paper were written while I was visiting the Cowles Foundation at Yale University, whose hospitality is gratefully acknowledged. Young-Hyun Cho has provided excellent research assistance. This work was supported by the Korea Research Foundation grant KRF 2003-041-B00072.

In this paper we analyze the asymptotic properties of the popular distribution tail index estimator by Hill (1975) for dependent, heterogeneous processes. We develop new extremal dependence measures that characterize a massive array of linear, nonlinear, and conditional volatility processes with long or short memory. We prove that the Hill estimator is weakly and uniformly weakly consistent for processes with extremes that form mixingale sequences and asymptotically normal for processes with extremes that are near epoch dependent (NED) on some arbitrary mixing functional. The extremal persistence assumptions in this paper are known to hold for mixing, Lp-NED, and some non-Lp-NED processes, including ARFIMA, FIGARCH, explosive GARCH, nonlinear ARMA-GARCH, and bilinear processes, and nonlinear distributed lags like random coefficient and regime-switching autoregressions.

Finally, we deliver a simple nonparametric estimator of the asymptotic variance of the Hill estimator and prove consistency for processes with NED extremes.

This paper introduces the structural threshold regression (STR) model that allows for an endogenous threshold variable as well as for endogenous regressors. This model provides a parsimonious way of modeling nonlinearities and has many potential applications in economics and finance. Our framework can be viewed as a generalization of the simple threshold regression framework of Hansen (2000, Econometrica 68, 575–603) and Caner and Hansen (2004, Econometric Theory 20, 813–843) to allow for the endogeneity of the threshold variable and regime-specific heteroskedasticity. Our estimation of the threshold parameter is based on a two-stage concentrated least squares method that involves an inverse Mills ratio bias correction term in each regime. We derive its asymptotic distribution and propose a method to construct confidence intervals. We also provide inference for the slope parameters based on a generalized method of moments. Finally, we investigate the performance of the asymptotic approximations using a Monte Carlo simulation, which shows the applicability of the method in finite samples.

A method is presented for computing maximum likelihood, or Gaussian, estimators of the structural parameters in a continuous time system of higherorder stochastic differential equations. It is argued that it is computationally efficient in the standard case of exact observations made at equally spaced intervals. Furthermore it can be applied in situations where the observations are at unequally spaced intervals, some observations are missing and/or the endogenous variables are subject to measurement error. The method is based on a state space representation and the use of the Kalman–Bucy filter. It is shown how the Kalman-Bucy filter can be modified to deal with flows as well as stocks.

We consider nonparametric estimation of spectral densities of stationary processes, a fundamental problem in spectral analysis of time series. Under natural and easily verifiable conditions, we obtain consistency and asymptotic normality of spectral density estimates. Asymptotic distribution of maximum deviations of the spectral density estimates is also derived. The latter result sheds new light on the classical problem of tests of white noises.

In this paper we consider the moment structure of a class of first-order exponential generalized autoregressive conditional heteroskedasticity (GARCH) models. This class contains as special cases both the standard exponential GARCH model and the symmetric and asymmetric logarithmic GARCH model. Conditions for the existence of any arbitrary moment are given. Furthermore, the expressions for the kurtosis and the autocorrelations of positive powers of absolute-valued observations are derived. The properties of the autocorrelation structure are discussed and compared to those of the standard first-order GARCH process. In particular, it is seen that, contrary to the standard GARCH case, the decay rate of the autocorrelations of squared errors is not constant and that the rate can be quite rapid in the beginning, depending on the parameters of the model.

This paper presents central limit theorems for triangular arrays of mixingale and near-epoch-dependent random variables. The central limit theorem for near-epoch-dependent random variables improves results from the literature in various respects. The approach is to define a suitable Bernstein blocking scheme and apply a martingale difference central limit theorem, which in combination with weak dependence conditions renders the result. The most important application of this central limit theorem is the improvement of the conditions that have to be imposed for asymptotic normality of minimization estimators.

An asymptotic theory is developed for multivariate regression in cointegrated systems whose variables are moderately integrated or moderately explosive in the sense that they have autoregressive roots of the form ρni = 1 + ci/nα, involving moderate deviations from unity when α ∈ (0, 1) and ci ∈ ℝ are constant parameters. When the data are moderately integrated in the stationary direction (with ci < 0), it is shown that least squares regression is consistent and asymptotically normal but suffers from significant bias, related to simultaneous equations bias. In the moderately explosive case (where ci > 0) the limit theory is mixed normal with Cauchy-type tail behavior, and the rate of convergence is explosive, as in the case of a moderately explosive scalar autoregression (Phillips and Magdalinos, 2007, Journal of Econometrics 136, 115–130). Moreover, the limit theory applies without any distributional assumptions and for weakly dependent errors under conventional moment conditions, so an invariance principle holds, unlike the well-known case of an explosive autoregression. This theory validates inference in cointegrating regression with mildly explosive regressors. The special case in which the regressors themselves have a common explosive component is also considered.

We propose a semiparametric varying-coefficient estimator that admits both qualitative and quantitative covariates along with a test for correct specification of parametric varying-coefficient models. The proposed estimator is exceedingly flexible and has a wide range of potential applications including hierarchical (mixed) settings, small area estimation, etc. A data-driven cross-validatory bandwidth selection method is proposed that can handle both the qualitative and quantitative covariates and that can also handle the presence of potentially irrelevant covariates, each of which can result in finite-sample efficiency gains relative to the conventional frequency (sample-splitting) estimator that is often found in such settings. Theoretical underpinnings including rates of convergence and asymptotic normality are provided. Monte Carlo simulations are undertaken to assess the proposed estimator’s finite-sample performance relative to the conventional semiparametric frequency estimator and to assess the finite-sample performance of the proposed test for correct parametric specification.

Time series data are often well modeled by using the device of an autoregressive root that is local to unity. Unfortunately, the localizing parameter (c) is not consistently estimable using existing time series econometric techniques and the lack of a consistent estimator complicates inference. This paper develops procedures for the estimation of a common localizing parameter using panel data. Pooling information across individuals in a panel aids the identification and estimation of the localizing parameter and leads to consistent estimation in simple panel models. However, in the important case of models with concomitant deterministic trends, it is shown that pooled panel estimators of the localizing parameter are asymptotically biased. Some techniques are developed to overcome this difficulty, and consistent estimators of c in the region c < 0 are developed for panel models with deterministic and stochastic trends. A limit distribution theory is also established, and test statistics are constructed for exploring interesting hypotheses, such as the equivalence of local to unity parameters across subgroups of the population. The methods are applied to the empirically important problem of the efficient extraction of deterministic trends. They are also shown to deliver consistent estimates of distancing parameters in nonstationary panel models where the initial conditions are in the distant past. In the development of the asymptotic theory this paper makes use of both sequential and joint limit approaches. An important limitation in the operation of the joint asymptotics that is sometimes needed in our development is the rate condition n/T → 0. So the results in the paper are likely to be most relevant in panels where T is large and n is moderately large.

Presently, conditions ensuring the validity of bootstrap methods for the sample mean of (possibly heterogeneous) near epoch dependent (NED) functions of mixing processes are unknown. Here we establish the validity of the bootstrap in this context, extending the applicability of bootstrap methods to a class of processes broadly relevant for applications in economics and finance. Our results apply to two block bootstrap methods: the moving blocks bootstrap of Künsch (1989, Annals of Statistics 17, 1217–1241) and Liu and Singh (1992, in R. LePage & L. Billiard (eds.), Exploring the Limits of the Bootstrap, 224–248) and the stationary bootstrap of Politis and Romano (1994a, Journal of the American Statistical Association 89, 1303–1313). In particular, the consistency of the bootstrap variance estimator for the sample mean is shown to be robust against heteroskedasticity and dependence of unknown form. The first-order asymptotic validity of the bootstrap approximation to the actual distribution of the sample mean is also established in this heterogeneous NED context.

Semiparametric generalized additive models are a powerful tool in quantitative econometrics. With response Y, covariates X,T, the considered model is E(Y |X;T) = G{XTβ + α + m1(T1) + ··· + md(Td)}. Here, G is a known link, α and β are unknown parameters, and m1,…,md are unknown (smooth) functions of possibly higher dimensional covariates T1,…,Td. Estimates of m1,…,md, α, and β are presented, and asymptotic distributions are given for both the nonparametric and the parametric part. The main focus of the paper is application of bootstrap methods. It is shown how bootstrap can be used for bias correction, hypothesis testing (e.g., component-wise analysis), and the construction of uniform confidence bands. Further, bootstrap tests for model specification and parametrization are given, in particular for testing additivity and link function specification. The practical performance of the methods is illustrated in a simulation study.This research was supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 373 “Quantifikation und Simulation ökonomischer Prozesse,” Humboldt-Universität zu Berlin, DFG project MA 1026/6-2, the Spanish “Dirección General de Enseñanza Superior,” no. BEC2001-1270, and the grant “Nonparametric methods in finance and insurance” from the Danish Social Science Research Council. We thank Marlene Müller, Oliver Linton, and two anonymous referees for helpful discussion.

We develop a generally applicable framework for constructing efficient estimators of regression models via quantile regressions. The proposed method is based on optimally combining information over multiple quantiles and can be applied to a broad range of parametric and nonparametric settings. When combining information over a fixed number of quantiles, we derive an upper bound on the distance between the efficiency of the proposed estimator and the Fisher information. As the number of quantiles increases, this upper bound decreases and the asymptotic variance of the proposed estimator approaches the Cramér–Rao lower bound under appropriate conditions. In the case of nonregular statistical estimation, the proposed estimator leads to super-efficient estimation. We illustrate the proposed method for several widely used regression models. Both asymptotic theory and Monte Carlo experiments show the superior performance over existing methods.

Pötscher (1991, Econometric Theory7, 163–181) has recently considered the question of how the use of a model selection procedure affects the asymptotic distribution of parameter estimators and related statistics. An important potential application of such results is to the generation of confidence regions for the parameters of interest. It is demonstrated that a great deal of care must be exercised in any attempt at such an application. We also consider the effect of model selection on prediction regions. It is demonstrated that the use of asymptotic results for the construction of prediction regions requires the same sort of care as the use of such results for the construction of confidence regions.

The purpose of this paper is to describe the performance of generalized empirical likelihood (GEL) methods for time series instrumental variable models specified by nonlinear moment restrictions as in Stock and Wright (2000, Econometrica 68, 1055–1096) when identification may be weak. The paper makes two main contributions. First, we show that all GEL estimators are first-order equivalent under weak identification. The GEL estimator under weak identification is inconsistent and has a nonstandard asymptotic distribution. Second, the paper proposes new GEL test statistics, which have chi-square asymptotic null distributions independent of the strength or weakness of identification. Consequently, unlike those for Wald and likelihood ratio statistics, the size of tests formed from these statistics is not distorted by the strength or weakness of identification. Modified versions of the statistics are presented for tests of hypotheses on parameter subvectors when the parameters not under test are strongly identified. Monte Carlo results for the linear instrumental variable regression model suggest that tests based on these statistics have very good size properties even in the presence of conditional heteroskedasticity. The tests have competitive power properties, especially for thick-tailed or asymmetric error distributions.This paper is a revision of Guggenberger's job market paper “Generalized Empirical Likelihood Tests under Partial, Weak, and Strong Identification.” We are thankful to the editor, P.C.B. Phillips, and three referees for very helpful suggestions on an earlier version of this paper. Guggenberger gratefully acknowledges the continuous help and support of his adviser, Donald Andrews, who played a prominent role in the formulation of this paper. He thanks Peter Phillips and Joseph Altonji for their extremely valuable comments. We also thank Vadim Marner for help with the simulation section and John Chao, Guido Imbens, Michael Jansson, Frank Kleibergen, Marcelo Moreira, Jonathan Wright, and Motohiro Yogo for helpful comments. Aspects of this research have been presented at the 2003 Econometric Society European Meetings; York Econometrics Workshop 2004; Seminaire Malinvaud; CREST-INSEE; and seminars at Albany, Alicante, Austin (Texas), Brown, Chicago, Chicago GSB, Harvard/MIT, Irvine, ISEG/Universidade Tecnica de Lisboa, Konstanz, Laval, Madison (Wisconsin), Mannheim, Maryland, NYU, Penn, Penn State, Pittsburgh, Princeton, Rice, Riverside, Rochester, San Diego, Texas A&M, UCLA, USC, and Yale. We thank all the seminar participants. Guggenberger and Smith received financial support through a Carl Arvid Anderson Prize Fellowship and a 2002 Leverhulme Major Research Fellowship, respectively.

In this paper we study doubly robust estimators of various average and quantile treatment effects under unconfoundedness; we also consider an application to a setting with an instrumental variable. We unify and extend much of the recent literature by providing a very general identification result which covers binary and multi-valued treatments; unnormalized and normalized weighting; and both inverse-probability weighted (IPW) and doubly robust estimators. We also allow for subpopulation-specific average treatment effects where subpopulations can be based on covariate values in an arbitrary way. Similar to Wooldridge (2007), we then discuss estimation of the conditional mean using quasi-log likelihoods (QLL) from the linear exponential family.

The problem addressed in this paper is to test the null hypothesis that a time series process is uncorrelated up to lag K in the presence of statistical dependence. We propose an extension of the Box–Pierce Q-test that is asymptotically distributed as chi-square when the null is true for a very general class of dependent processes that includes non-martingale difference sequences. The test is based on a consistent estimator of the asymptotic covariance matrix of the sample autocorrelations under the null. The finite sample performance of this extension is investigated in a Monte Carlo study.

This paper derives a functional central limit theorem for the partial sums of fractionally integrated processes, otherwise known as I(d) processes for |d| < 1/2. Such processes have long memory, and the limit distribution is the so-called fractional Brownian motion, having correlated increments even asymptotically. The underlying shock variables may themselves exhibit quite general weak dependence by being near-epoch-dependent functions of mixing processes. Several weak convergence results for stochastic integrals having fractional integrands and weakly dependent integrators are also obtained. Taken together, these results permit I(p + d) integrands for any integer p ≥ 1.

We show the consistency of the log-periodogram regression estimate of the long memory parameter for long range dependent linear, not necessarily Gaussian, time series when we make a pooling of periodogram ordinates. Then, we study the asymptotic behavior of the tapered periodogram of long range dependent time series for frequencies near the origin, and we obtain the asymptotic distribution of the log-periodogram estimate for possibly non-Gaussian observation when the tapered periodogram is used. For these results we rely on higher order asymptotic properties of a vector of periodogram ordinates of the linear innovations. Finally, we assess the validity of the asymptotic results for finite samples via Monte Carlo simulation.

This paper considers dynamic time series binary choice models. It proves near epoch dependence and strong mixing for the dynamic binary choice model with correlated errors. Using this result, it shows in a time series setting the validity of the dynamic probit likelihood procedure when lags of the dependent binary variable are used as regressors, and it establishes the asymptotic validity of Horowitz’s smoothed maximum score estimation of dynamic binary choice models with lags of the dependent variable as regressors. For the semiparametric model, the latent error is explicitly allowed to be correlated. It turns out that no long-run variance estimator is needed for the validity of the smoothed maximum score procedure in the dynamic time series framework.