This paper considers a series estimator of E[α(Y)|λ(X) = λ̄], (α,λ) ∈ 𝛢 × Λ, indexed by function spaces, and establishes the estimator's uniform convergence rate over λ̄ ∈ R, α ∈ 𝛢, and λ ∈ Λ, when 𝛢 and Λ have a finite integral bracketing entropy. The rate of convergence depends on the bracketing entropies of 𝛢 and Λ in general. In particular, we demonstrate that when each λ ∈ Λ is locally uniformly ℒ2-continuous in a parameter from a space of polynomial discrimination and the basis function vector pK in the series estimator keeps the smallest eigenvalue of E[pK(λ(X))pK(λ(X))‼] above zero uniformly over λ ∈ Λ, we can obtain the same convergence rate as that established by de Jong (2002, Journal of Econometrics 111, 1–9).

High-breakdown-point regression estimators protect against large errors and data contamination. We generalize the concept of trimming used by many of these robust estimators, such as the least trimmed squares and maximum trimmed likelihood, and propose a general trimmed estimator, which renders robust estimators applicable far beyond the standard (non)linear regression models. We derive here the consistency and asymptotic distribution of the proposed general trimmed estimator under mild β-mixing conditions and demonstrate its applicability in nonlinear regression and limited dependent variable models.

This paper derives asymptotic normality of a class of M-estimators in the generalized autoregressive conditional heteroskedastic (GARCH) model. The class of estimators includes least absolute deviation and Huber's estimator in addition to the well-known quasi maximum likelihood estimator. For some estimators, the asymptotic normality results are obtained only under the existence of fractional unconditional moment assumption on the error distribution and some mild smoothness and moment assumptions on the score function.

This paper develops a generalized autoregressive conditional correlation (GARCC) model when the standardized residuals follow a random coefficient vector autoregressive process. As a multivariate generalization of the Tsay (1987, Journal of the American Statistical Association 82, 590–604) random coefficient autoregressive (RCA) model, the GARCC model provides a motivation for the conditional correlations to be time varying. GARCC is also more general than the Engle (2002, Journal of Business & Economic Statistics 20, 339–350) dynamic conditional correlation (DCC) and the Tse and Tsui (2002, Journal of Business & Economic Statistics 20, 351–362) varying conditional correlation (VCC) models and does not impose unduly restrictive conditions on the parameters of the DCC model. The structural properties of the GARCC model, specifically, the analytical forms of the regularity conditions, are derived, and the asymptotic theory is established. The Baba, Engle, Kraft, and Kroner (BEKK) model of Engle and Kroner (1995, Econometric Theory 11, 122–150) is demonstrated to be a special case of a multivariate RCA process. A likelihood ratio test is proposed for several special cases of GARCC. The empirical usefulness of GARCC and the practicality of the likelihood ratio test are demonstrated for the daily returns of the Standard and Poor's 500, Nikkei, and Hang Seng indexes.

We propose an easy to use derivative-based two-step estimation procedure for semiparametric index models, where the number of indexes is not known a priori. In the first step various functionals involving the derivatives of the unknown function are estimated using nonparametric kernel estimators, in particular the average outer product of the gradient (AOPG). By testing the rank of the AOPG we determine the required number of indexes. Subsequently, we estimate the index parameters in a method of moments framework, with moment conditions constructed using the estimated average derivative functionals. The estimator readily extends to multiple equation models and is shown to be root-N-consistent and asymptotically normal.

Aragon, Daouia, and Thomas-Agnan (2005, Econometric Theory 21, 358–389) introduced a new nonparametric frontier estimation. We prove the weak convergence of the empirical conditional quantile function. The distribution of the limit depends on the unknown conditional quantile density function. We provide a method to construct uniform confidence bands without estimating the conditional quantile density.

We consider a model Yt = σtηt in which (σt) is not independent of the noise process (ηt) but σt is independent of ηt for each t. We assume that (σt) is stationary, and we propose an adaptive estimator of the density of ln(σt2) based on the observations Yt. Under a new dependence structure, the τ-dependency defined by Dedecker and Prieur (2005, Probability Theory and Related Fields 132, 203–236), we prove that the rates of this nonparametric estimator coincide with the rates obtained in the independent and identically distributed (i.i.d.) case when (σt) and (ηt) are independent. The results apply to various linear and nonlinear general autoregressive conditionally heteroskedastic (ARCH) processes. They are illustrated by simulations applying the deconvolution algorithm of Comte, Rozenholc, and Taupin (2006, Canadian Journal of Statistics 34, 431–452) to a new noise density.

Nonparametric data envelopment analysis (DEA) estimators based on linear programming methods have been widely applied in analyses of productive efficiency. The distributions of these estimators remain unknown except in the simple case of one input and one output, and previous bootstrap methods proposed for inference have not been proved consistent, making inference doubtful. This paper derives the asymptotic distribution of DEA estimators under variable returns to scale. This result is used to prove consistency of two different bootstrap procedures (one based on subsampling, the other based on smoothing). The smooth bootstrap requires smoothing the irregularly bounded density of inputs and outputs and smoothing the DEA frontier estimate. Both bootstrap procedures allow for dependence of the inefficiency process on output levels and the mix of inputs in the case of input-oriented measures, or on input levels and the mix of outputs in the case of output-oriented measures.

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.

This note considers a puzzling phenomenon that is observed in some semiparametric estimation problems. In some cases, using estimated values of the nuisance parameters provides a more efficient estimator for the parameters of interest than does using the true values. This phenomenon takes place even in cases of semi-nonparametric models in which the nuisance parameters are infinite-dimensional and cannot be estimated at the parametric rate. We examine the structure and present the necessary and sufficient condition for the occurrence of this puzzle. We also provide a simple sufficient condition. It shows that the puzzle occurs when the term accounting for the effect of estimation of nuisance parameters is included in the tangent space. This condition is often satisfied when the estimating equation does not bring any restriction on the form of the nuisance parameters. Our simple sufficient condition can be applied to many important estimators.