Many data sets used by economists and other social scientists are collected by stratified sampling. The sampling scheme used to collect the data induces a probability distribution on the observed sample that differs from the target or underlying distribution for which inference is to be made. If this effect is not taken into account, subsequent statistical inference can be seriously biased. This paper shows how to do efficient semiparametric inference in moment restriction models when data from the target population are collected by three widely used sampling schemes: variable probability sampling, multinomial sampling, and standard stratified sampling.

This paper analyzes tests for a shift in the trend function of a time series at an unknown date based on ordinary least squares (OLS) estimates of the trend function. Inference about the trend parameters depends on the serial correlation structure of the data through the long-run variance (zero frequency spectral density) of the errors. Asymptotically pivotal tests can be obtained by the use of serial correlation robust standard errors that require an estimate of the long-run variance. The focus is on the class of nonparametric kernel estimators of the long-run variance. Tests based on these estimators present two problems for practitioners. The first is the choice of kernel and bandwidth. The second is the well-known overrejection problem caused by strong serial correlation (or a possible unit root) in the errors.We provide solutions to both problems by using the fixed-b asymptotic framework of Kiefer and Vogelsang (2005, Econometric Theory, 21, 1130–1164) in conjunction with the scaling factor approach of Vogelsang (1998, Econometrica 65, 123–148). Our results provide practitioners with a family of OLS-based trend function structural change tests that are size robust to the presence of strong serial correlation or a unit root. Specific recommendations are provided for the tuning parameters (kernel and bandwidth) in a way that maximizes asymptotic integrated power.

This paper proposes a nonparametric simultaneous test for parametric specification of the conditional mean and variance functions in a time series regression model. The test is based on an empirical likelihood (EL) statistic that measures the goodness of fit between the parametric estimates and the nonparametric kernel estimates of the mean and variance functions. A unique feature of the test is its ability to distribute natural weights automatically between the mean and the variance components of the goodness-of-fit measure. To reduce the dependence of the test on a single pair of smoothing bandwidths, we construct an adaptive test by maximizing a standardized version of the empirical likelihood test statistic over a set of smoothing bandwidths. The test procedure is based on a bootstrap calibration to the distribution of the empirical likelihood test statistic. We demonstrate that the empirical likelihood test is able to distinguish local alternatives that are different from the null hypothesis at an optimal rate.

In this paper we clarify the relations between the existing sets of regularity conditions for convergence rates of nonparametric indirect regression (NPIR) and nonparametric instrumental variables (NPIV) regression models. We establish minimax risk lower bounds in mean integrated squared error loss for the NPIR and NPIV models under two basic regularity conditions: the approximation number and the link condition. We show that both a simple projection estimator for the NPIR model and a sieve minimum distance estimator for the NPIV model can achieve the minimax risk lower bounds and are rate optimal uniformly over a large class of structure functions, allowing for mildly ill-posed and severely ill-posed cases.

This paper provides a (saddlepoint) tail probability approximation for the distribution of an optimal unit root test. Under restrictive assumptions, Gaussianity, and known covariance structure, the order of error of the approximation is given. More generally, when innovations are a linear process in martingale differences, the estimated saddlepoint is proved to yield valid asymptotic inference. Numerical evidence, considered over a range of models, demonstrates some finite-sample superiority over approximations for a directly comparable test based on simulation of its limiting stochastic representation.

Testing for white noise has been well studied in the literature of econometrics and statistics. For most of the proposed test statistics, such as the well-known Box–Pierce test statistic with fixed lag truncation number, the asymptotic null distributions are obtained under independent and identically distributed assumptions and may not be valid for dependent white noise. Because of recent popularity of conditional heteroskedastic models (e.g., generalized autoregressive conditional heteroskedastic [GARCH] models), which imply nonlinear dependence with zero autocorrelation, there is a need to understand the asymptotic properties of the existing test statistics under unknown dependence. In this paper, we show that the asymptotic null distribution of the Box–Pierce test statistic with general weights still holds under unknown weak dependence as long as the lag truncation number grows at an appropriate rate with increasing sample size. Further applications to diagnostic checking of the autoregressive moving average (ARMA) and fractional autoregressive integrated moving average (FARIMA) models with dependent white noise errors are also addressed. Our results go beyond earlier ones by allowing non-Gaussian and conditional heteroskedastic errors in the ARMA and FARIMA models and provide theoretical support for some empirical findings reported in the literature.

New notions of tail and nontail dependence are used to characterize separately extremal and nonextremal information, including tail log-exceedances and events, and tail-trimmed levels. We prove that near epoch dependence (McLeish, 1975; Gallant and White, 1988) and L0-approximability (Pötscher and Prucha, 1991) are equivalent for tail events and tail-trimmed levels, ensuring a Gaussian central limit theory for important extreme value and robust statistics under general conditions. We apply the theory to characterize the extremal and nonextremal memory properties of possibly very heavy-tailed GARCH processes and distributed lags. This in turn is used to verify Gaussian limits for tail index, tail dependence, and tail-trimmed sums of these data, allowing for Gaussian asymptotics for a new tail-trimmed least squares estimator for heavy-tailed processes.

In this paper we propose a new approximate factor model for large cross-section and time dimensions. Factor loadings are assumed to be smooth functions of time, which allows considering the model as locally stationary while permitting empirically observed time-varying second moments. Factor loadings are estimated by the eigenvectors of a nonparametrically estimated covariance matrix. As is well known in the stationary case, this principal components estimator is consistent in approximate factor models if the eigenvalues of the noise covariance matrix are bounded. To show that this carries over to our locally stationary factor model is the main objective of our paper. Under simultaneous asymptotics (cross-section and time dimension go to infinity simultaneously), we give conditions for consistency of our estimators. A simulation study illustrates the performance of these estimators.

This paper considers the first-order large sample properties of the generalized empirical likelihood (GEL) class of estimators for models specified by nonsmooth indicators. The GEL class includes a number of estimators recently introduced as alternatives to the efficient generalized method of moments (GMM) estimator that may suffer from substantial biases in finite samples. These include empirical likelihood (EL), exponential tilting (ET), and the continuous updating estimator (CUE). This paper also establishes the validity of tests suggested in the smooth moment indicators case for overidentifying restrictions and specification. In particular, a number of these tests avoid the necessity of providing an estimator for the Jacobian matrix that may be problematic for the sample sizes typically encountered in practice.

This paper studies the estimation of a nonparametric function ϕ from the inverse problem r = Tϕ given estimates of the function r and of the linear transform T. We show that rates of convergence of the estimator are driven by two types of assumptions expressed in a single Hilbert scale. The two assumptions quantify the prior regularity of ϕ and the prior link existing between T and the Hilbert scale. The approach provides a unified framework that allows us to compare various sets of structural assumptions found in the econometric literature. Moreover, general upper bounds are also derived for the risk of the estimator of the structural function ϕ as well as that of its derivatives. It is shown that the bounds cover and extend known results given in the literature. Two important applications are also studied. The first is the blind nonparametric deconvolution on the real line, and the second is the estimation of the derivatives of the nonparametric instrumental regression function via an iterative Tikhonov regularization scheme.

Using the power kernels of Phillips, Sun, and Jin (2006, 2007), we examine the large sample asymptotic properties of the t-test for different choices of power parameter (ρ). We show that the nonstandard fixed-ρ limit distributions of the t-statistic provide more accurate approximations to the finite sample distributions than the conventional large-ρ limit distribution. We prove that the second-order corrected critical value based on an asymptotic expansion of the nonstandard limit distribution is also second-order correct under the large-ρ asymptotics. As a further contribution, we propose a new practical procedure for selecting the test-optimal power parameter that addresses the central concern of hypothesis testing: The selected power parameter is test-optimal in the sense that it minimizes the type II error while controlling for the type I error. A plug-in procedure for implementing the test-optimal power parameter is suggested. Simulations indicate that the new test is as accurate in size as the nonstandard test of Kiefer and Vogelsang (2002a, 2002b), and yet it does not incur the power loss that often hurts the performance of the latter test. The results complement recent work by Sun, Phillips, and Jin (2008) on conventional and bT HAC testing.

A multivariate extension of the exponential continuous time GARCH (p, q) model (ECOGARCH) is introduced and studied. Stationarity and mixing properties of the new stochastic volatility model are investigated, and ways to model a component-wise leverage effect are presented.

In this paper we propose a bootstrap method for panel data linear regression models with individual fixed effects. The method consists of applying the standard 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) to the vector containing all the individual observations at each point in time. We show that this bootstrap is robust to serial and cross-sectional dependence of unknown form under the assumption that n (the cross-sectional dimension) is an arbitrary nondecreasing function of T (the time series dimension), where T → ∞, thus allowing for the possibility that both n and T diverge to infinity. The time series dependence is assumed to be weak (of the mixing type), but we allow the cross-sectional dependence to be either strong or weak (including the case where it is absent). Under appropriate conditions, we show that the fixed effects estimator (and also its bootstrap analogue) has a convergence rate that depends on the degree of cross-section dependence in the panel. Despite this, the same studentized test statistics can be computed without reference to the degree of cross-section dependence. Our simulation results show that the moving blocks bootstrap percentile-t intervals have very good coverage properties even when the degree of serial and cross-sectional correlation is large, provided the block size is appropriately chosen.

We propose a new estimator for the density of a random variable observed with an additive measurement error. This estimator is based on the spectral decomposition of the convolution operator, which is compact for an appropriate choice of reference spaces. The density is approximated by a sequence of orthonormal eigenfunctions of the convolution operator. The resulting estimator is shown to be consistent and asymptotically normal. While most estimation methods assume that the characteristic function (CF) of the error does not vanish, we relax this assumption and allow for isolated zeros. For instance, the CF of the uniform and symmetrically truncated normal distributions have isolated zeros. We show that, in the presence of zeros, the density is identified even though the convolution operator is not one-to-one. We propose two consistent estimators of the density. We apply our method to the estimation of the measurement error density of hourly income collected from survey data.

We propose nonnested tests for competing conditional moment restriction models using the method of conditional empirical likelihood, recently developed by Kitamura, Tripathi, and Ahn (2004) and Zhang and Gijbels (2003). To define the test statistics, we use the implied conditional probabilities from conditional empirical likelihood, which take into account the full implications of conditional moment restrictions. We propose three types of nonnested tests: the moment-encompassing, Cox-type, and efficient score-encompassing tests. We derive the asymptotic null distributions and investigate their power properties against a sequence of local alternatives and a fixed global alternative. Our tests have distinct global power properties from some of the existing tests based on finite-dimensional unconditional moment restrictions. Simulation experiments show that our tests have reasonable finite sample properties and dominate some of the existing nonnested tests in terms of size-corrected powers.

Fan and Yao (1998) proposed an efficient method to estimate the conditional variance of heteroskedastic regression models. Chen, Cheng, and Peng (2009) applied variance reduction techniques to the estimator of Fan and Yao (1998) and proposed a new estimator for conditional variance to account for the skewness of financial data. In this paper, we apply empirical likelihood methods to construct confidence intervals for the conditional variance based on the estimator of Fan and Yao (1998) and the reduced variance modification of Chen et al. (2009). Simulation studies and data analysis demonstrate the advantage of the empirical likelihood method over the normal approximation method.

In this comment we identify a lacuna in a proof in the paper by M. Caner published in 1997 in Econometric Theory concerning the weak limit behavior of various expressions involving heavy-tailed multivariate vectors and the convergence of stochastic integrals. In a later paper (Caner, 1998) the results for these limit relations are used to formulate tests for cointegration with infinite variance errors.

We show that any invariant test for spatial autocorrelation in a spatial error or spatial lag model with equal weights matrix has power equal to size. This result holds under the assumption of an elliptical distribution. Under Gaussianity, we also show that any test whose power is larger than its size for at least one point in the parameter space must be biased.