The convergence rates of the sums of α-mixing (or strongly mixing) triangular arrays of heterogeneous random variables are derived. We pay particular attention to the case where central limit theorems may fail to hold, due to relatively strong time-series dependence and/or the nonexistence of higher-order moments. Several previous studies have presented various versions of laws of large numbers for sequences/triangular arrays, but their convergence rates were not fully investigated. This study is the first to investigate the convergence rates of the sums of α-mixing triangular arrays whose mixing coefficients are permitted to decay arbitrarily slowly. We consider two kinds of asymptotic assumptions: one is that the time distance between adjacent observations is fixed for any sample size n; and the other, called the infill assumption, is that it shrinks to zero as n tends to infinity. Our convergence theorems indicate that an explicit trade-off exists between the rate of convergence and the degree of dependence. While the results under the infill assumption can be seen as a direct extension of those under the fixed-distance assumption, they are new and particularly useful for deriving sharper convergence rates of discretization biases in estimating continuous-time processes from discretely sampled observations. We also discuss some examples to which our results and techniques are useful and applicable: a moving-average process with long lasting past shocks, a continuous-time diffusion process with weak mean reversion, and a near-unit-root process.

This paper presents uniform convergence rates for kernel regression estimators, in the setting of a structural nonlinear cointegrating regression model. We generalise the existing literature in three ways. First, the domain to which these rates apply is much wider than the domains that have been considered in the existing literature, and can be chosen so as to contain as large a fraction of the sample as desired in the limit. Second, our results allow the regression disturbance to be serially correlated, and cross-correlated with the regressor; previous work on this problem (of obtaining uniform rates) having been confined entirely to the setting of an exogenous regressor. Third, we permit the bandwidth to be data-dependent, requiring it to satisfy only certain weak asymptotic shrinkage conditions. Our assumptions on the regressor process are consistent with a very broad range of departures from the standard unit root autoregressive model, allowing the regressor to be fractionally integrated, and to have an infinite variance (and even infinite lower-order moments).

The tail index as a measure of tail thickness provides information that is not captured by standard volatility measures. It may however change over time. Currently available procedures for detecting those changes for dependent data (e.g., Quintos et al., 2001) are all based on comparing Hill (1975) estimates from different subsamples. We derive tests for a wide class of other tail index estimators. The limiting distribution of the test statistics is shown not to depend on the particular choice of the estimator, while the assumptions on the dependence structure allow for sufficient generality in applications. A simulation study investigates empirical sizes and powers of the tests in finite samples.

We consider the problem of estimating the common time of a change in the mean parameters of panel data when dependence is allowed between the cross-sectional units in the form of a common factor. A CUSUM type estimator is proposed, and we establish first and second order asymptotics that can be used to derive consistent confidence intervals for the time of change. Our results improve upon existing theory in two primary directions. Firstly, the conditions we impose on the model errors only pertain to the order of their long run moments, and hence our results hold for nearly all stationary time series models of interest, including nonlinear time series like the ARCH and GARCH processes. Secondly, we study how the asymptotic distribution and norming sequences of the estimator depend on the magnitude of the changes in each cross-section and the common factor loadings. The performance of our results in finite samples is demonstrated with a Monte Carlo simulation study, and we consider applications to two real data sets: the exchange rates of 23 currencies with respect to the US dollar, and the GDP per capita in 113 countries.

This paper studies the paired nonseparable measurement error models, where two measurements, X and Y, are produced by mutually independent unobservables, U, V, and W, through the system, X = g(U,V) and Y = h(U,W). We propose restrictions to identify the marginal distribution of the common component U and the conditional distributions of X and Y given U. Applying this method to twin panel data, we find the following robust reporting patterns for years of education: (1) self reports are accurate only when the true years of education are 16 or 18, typically corresponding to advanced university degrees in the US education system; (2) sibling reports are accurate whenever the true years of education are 12, 14, 16, and 18, which are typical diploma years.

Many econometric models can be analyzed as finite mixtures. We focus on two-component mixtures, and we show that they are nonparametrically point identified by a combination of an exclusion restriction and tail restrictions. Our identification analysis suggests simple closed-form estimators of the component distributions and mixing proportions, as well as a specification test. We derive their asymptotic properties using results on tail empirical processes and we present a simulation study that documents their finite-sample performance.

The family of multiplicative error models is important for studying non-negative variables such as realized volatility, trading volume, and duration between consecutive financial transactions. Methods are developed for testing the parametric specification of a multiplicative error model, which consists of separate parametric models for the conditional mean and the error distribution. The same method can also be used for testing the specification of the error distribution provided the conditional mean is correctly specified. A bootstrap method is proposed for computing the p-values of the tests and is shown to be consistent. The proposed tests have nontrivial asymptotic power against a class of O(n−1/2)-local alternatives. The tests performed well in a simulation study, and they are illustrated using a data example on realized volatility.

This paper examines the identification power of assumptions that formalize the notion of complementarity in the context of a nonparametric bounds analysis of treatment response. I extend the literature on partial identification via shape restrictions by exploiting cross-dimensional restrictions on treatment response when treatments are multidimensional; the assumption of supermodularity can strengthen bounds on average treatment effects in studies of policy complementarity. This restriction can be combined with a statistical independence assumption to derive improved bounds on treatment effect distributions, aiding in the evaluation of complex randomized controlled trials. Complementarities arising from treatment effect heterogeneity can be incorporated through supermodular instrumental variables to strengthen identification in studies with one or multiple treatments. An application examining the long-run impact of zoning on the evolution of urban spatial structure illustrates the value of the proposed identification methods.

We study the properties of the integrated score estimator (ISE), which is the Laplace version of Manski’s maximum score estimator (MMSE). The ISE belongs to a class of estimators whose basic asymptotic properties were studied in Jun, Pinkse, and Wan (2015, Journal of Econometrics 187(1), 201–216). Here, we establish that the MMSE, or more precisely $$\root 3 \of n |\hat \theta _M - \theta _0 |$$, (locally first order) stochastically dominates the ISE under the conditions necessary for the MMSE to attain its $\root 3 \of n $ convergence rate and that the ISE has the same convergence rate as Horowitz’s smoothed maximum score estimator (SMSE) under somewhat weaker conditions. An implication of the stochastic dominance result is that the confidence intervals of the MMSE are for any given coverage rate wider than those of the ISE, provided that the input parameter αn is not chosen too large. Further, we introduce an inference procedure that is not only rate adaptive as established in Jun et al. (2015), but also uniform in the choice of αn. We propose three different first order bias elimination procedures and we discuss the choice of input parameters. We develop a computational algorithm for the ISE based on the Gibbs sampler and we examine implementational issues in detail. We argue in favor of normalizing the norm of the parameter vector as opposed to fixing one of the coefficients. Finally, we evaluate the computational efficiency of the ISE and the performance of the ISE and the proposed inference procedure in an extensive Monte Carlo study.

This paper examines the problem of identification and inference on a conditional moment condition model with missing data, with special focus on the case when the conditioning covariates are missing. We impose no assumption on the distribution of the missing data and we confront the missing data problem by using a worst case scenario approach.

We characterize the sharp identified set and argue that this set is usually too complex to compute or to use for inference. Given this difficulty, we consider the construction of outer identified sets (i.e. supersets of the identified set) that are easier to compute and can still characterize the parameter of interest. Two different outer identification strategies are proposed. Both of these strategies are shown to have nontrivial identifying power and are relatively easy to use and combine for inferential purposes.

We provide conditions for the existence and the uniqueness of strictly stationary solutions of the usual Dynamic Conditional Correlation GARCH models (DCC-GARCH). The proof is based on Tweedie’s (1988) criteria, after having rewritten DCC-GARCH models as nonlinear Markov chains. We also study the existence of their moments and discuss the tightness of our sufficient conditions.

We present a new framework for the identification of competing risks models, which also include Roy models. We show that by establishing a Hicksian-type decomposition, the direction of covariate effects on the marginal distributions of the competing risks model can be identified under weak restrictions. Our approach leaves the marginal distributions and their joint distribution completely unspecified, except that the associated copula is invariant in the covariates. Results from simulations and two data examples suggest that our method often outperforms existing comparable approaches in terms of the range of durations for which the direction of the covariate effect is identified, particularly for long duration.

We derive nonparametric efficiency bounds for regular estimators of integrated smooth transformations of instantaneous variances, in particular, integrated power variance. We find that realized variance attains the efficiency bound for integrated variance under both regular and irregular sampling schemes. For estimating higher powers such as integrated quarticity, the block-based procedures of Mykland and Zhang (2009) can get arbitrarily close to the nonparametric bounds, when observation times are equidistant. Moreover, the estimator in Jacod and Rosenbaum (2013), whose efficiency was documented for the submodel assuming constant volatility, is efficient also for nonconstant volatility paths. When the observation times are possibly random but predictable, we provide an estimator, similar to that of Kristensen (2010), which can get arbitrarily close to the nonparametric bound. Finally, parametric information about the functional form of volatility leads to a lower efficiency bound, unless the volatility process is piecewise constant.

This paper analyzes a generalized class of flat-top realized kernel estimators for the quadratic variation spectrum, that is, the decomposition of quadratic variation into integrated variance and jump variation. The underlying log-price process is contaminated by additive noise, which consists of two orthogonal components to accommodate α-mixing dependent exogenous noise and an asymptotically non-degenerate endogenous correlation structure. In the absence of jumps, the class of estimators is shown to be consistent, asymptotically unbiased, and mixed Gaussian at the optimal rate of convergence, n1/4. Exact bounds on lower-order terms are obtained, and these are used to propose a selection rule for the flat-top shrinkage. Bounds on the optimal bandwidth for noise models of varying complexity are also provided. In theoretical and numerical comparisons with alternative estimators, including the realized kernel, the two-scale realized kernel, and a bias-corrected pre-averaging estimator, the flat-top realized kernel enjoys a higher-order advantage in terms of bias reduction, in addition to good efficiency properties. The analysis is extended to jump-diffusions where the asymptotic properties of a flat-top realized kernel estimate of the total quadratic variation are established. Apart from a larger asymptotic variance, they are similar to the no-jump case. Finally, the estimators are used to design two classes of (medium) blocked realized kernels, which produce consistent, non-negative estimates of integrated variance. The blocked estimators are shown to have no loss either of asymptotic efficiency or in the rate of consistency relative to the flat-top realized kernels when jumps are absent. However, only the medium blocked realized kernels achieve the optimal rate of convergence under the jump alternative.

We consider a nonparametric Bayesian model for conditional densities. The model is a finite mixture of normal distributions with covariate dependent multinomial logit mixing probabilities. A prior for the number of mixture components is specified on positive integers. The marginal distribution of covariates is not modeled. We study asymptotic frequentist behavior of the posterior in this model. Specifically, we show that when the true conditional density has a certain smoothness level, then the posterior contraction rate around the truth is equal up to a log factor to the frequentist minimax rate of estimation. An extension to the case when the covariate space is unbounded is also established. As our result holds without a priori knowledge of the smoothness level of the true density, the established posterior contraction rates are adaptive. Moreover, we show that the rate is not affected by inclusion of irrelevant covariates in the model. In Monte Carlo simulations, a version of the model compares favorably to a cross-validated kernel conditional density estimator.

Censored quantile regressions have received a great deal of attention in the literature. In a linear setup, recent research has found that an estimator based on the idea of “redistribution-of-mass” in Efron (1967, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 4, pp. 831–853, University of California Press) has better numerical performance than other available methods. In this paper, this idea is combined with the local polynomial kernel smoothing for nonparametric quantile regression of censored data. We derive the uniform Bahadur representation for the estimator and, more importantly, give theoretical justification for its improved efficiency over existing estimation methods. We include an example to illustrate the usefulness of such a uniform representation in the context of sufficient dimension reduction in regression analysis. Finally, simulations are used to investigate the finite sample performance of the new estimator.

The exponential GARCH (EGARCH) model introduced by Nelson (1991) is a popular model for discrete time volatility since it allows for asymmetric effects and naturally ensures positivity even when including exogenous variables. Estimation and inference are usually done via maximum likelihood. Although some progress has been made recently, a complete distribution theory of MLE for EGARCH models is still missing. Furthermore, the estimation procedure itself may be highly sensitive to starting values, the choice of numerical optimization algorithm, etc. We present an alternative estimator that is available in a simple closed form and which could be used, for example, as starting values for MLE. The estimator of the dynamic parameter is independent of the innovation distribution. For the other parameters we assume that the innovation distribution belongs to the class of Generalized Error Distributions (GED), profiling out its parameter in the estimation procedure. We discuss the properties of the proposed estimator and illustrate its performance in a simulation study and an empirical example.

This paper studies the identification and estimation of weighted average derivatives of conditional location functionals including conditional mean and conditional quantiles in settings where either the outcome variable or a regressor is interval-valued. Building on Manski and Tamer (2002, Econometrica 70(2), 519–546) who study nonparametric bounds for mean regression with interval data, we characterize the identified set of weighted average derivatives of regression functions. Since the weighted average derivatives do not rely on parametric specifications for the regression functions, the identified set is well-defined without any functional-form assumptions. Under general conditions, the identified set is compact and convex and hence admits characterization by its support function. Using this characterization, we derive the semiparametric efficiency bound of the support function when the outcome variable is interval-valued. Using mean regression as an example, we further demonstrate that the support function can be estimated in a regular manner by a computationally simple estimator and that the efficiency bound can be achieved.

The method of moments procedure proposed by Carrasco and Florens (2000) permits full exploitation of the information contained in the characteristic function and yields an estimator which is asymptotically as efficient as the maximum likelihood estimator. However, this estimation procedure depends on a regularization or tuning parameter α that needs to be selected. The aim of the present paper is to provide a way to optimally choose α by minimizing the approximate mean square error (AMSE) of the estimator. Following an approach similar to that of Donald and Newey (2001), we derive a higher-order expansion of the estimator from which we characterize the finite sample dependence of the AMSE on α. We propose to select the regularization parameter by minimizing an estimate of the AMSE. We show that this procedure delivers a consistent estimator of α. Moreover, the data-driven selection of the regularization parameter preserves the consistency, asymptotic normality, and efficiency of the CGMM estimator. Simulation experiments based on a CIR model show the relevance of the proposed approach.