We investigate the properties of several bootstrap-based inference procedures for semiparametric density-weighted average derivatives. The key innovation in this paper is to employ an alternative asymptotic framework to assess the properties of these inference procedures. This theoretical approach is conceptually distinct from the traditional approach (based on asymptotic linearity of the estimator and Edgeworth expansions), and leads to different theoretical prescriptions for bootstrap-based semiparametric inference. First, we show that the conventional bootstrap-based approximations to the distribution of the estimator and its classical studentized version are both invalid in general. This result shows a fundamental lack of “robustness” of the associated, classical bootstrap-based inference procedures with respect to the bandwidth choice. Second, we present a new bootstrap-based inference procedure for density-weighted average derivatives that is more “robust” to perturbations of the bandwidth choice, and hence exhibits demonstrable superior theoretical statistical properties over the traditional bootstrap-based inference procedures. Finally, we also examine the validity and invalidity of related bootstrap-based inference procedures and discuss additional results that may be of independent interest. Some simulation evidence is also presented.

For a certain class of martingales, convergence to a mixture of normal distributions is established under convergence in distribution for the conditional variance. This is less restrictive in comparison with the classical martingale limit theorem, where one generally requires convergence in probability. The extension partially removes a barrier in the applications of the classical martingale limit theorem to nonparametric estimation and inference with nonstationarity and enhances the effectiveness of the classical martingale limit theorem as one of the main tools to investigate asymptotics in statistics, econometrics, and other fields. The main result is applied to investigate limit behavior of the conventional kernel estimator in a nonlinear cointegrating regression model, which improves existing works in the literature.

Economic and financial time series frequently exhibit time irreversible dynamics. For instance, there is considerable evidence of asymmetric fluctuations in many macroeconomic and financial variables, and certain game theoretic models of price determination predict asymmetric cycles in price series. In this paper, we make two primary contributions to the econometric literature on time reversibility. First, we propose a new test of time reversibility, applicable to stationary Markov chains. Compared to existing tests, our test has the advantage of being consistent against arbitrary violations of reversibility. Second, we explain how a circulation density function may be used to characterize the nature of time irreversibility when it is present. We propose a copula-based estimator of the circulation density and verify that it is well behaved asymptotically under suitable regularity conditions. We illustrate the use of our time reversibility test and circulation density estimator by applying them to five years of Canadian gasoline price markup data.

This paper determines the properties of standard generalized method of moments (GMM) estimators, tests, and confidence sets (CSs) in moment condition models in which some parameters are unidentified or weakly identified in part of the parameter space. The asymptotic distributions of GMM estimators are established under a full range of drifting sequences of true parameters and distributions. The asymptotic sizes (in a uniform sense) of standard GMM tests and CSs are established.

The paper also establishes the correct asymptotic sizes of “robust” GMM-based Wald, t, and quasi-likelihood ratio tests and CSs whose critical values are designed to yield robustness to identification problems.

The results of the paper are applied to a nonlinear regression model with endogeneity and a probit model with endogeneity and possibly weak instrumental variables.

In this paper, we introduce a family of contrasts for parametric inference in ARCH models the volatility of which exhibits some degeneracy. We focus specifically on ARCH processes with a linear volatility (called LARCH processes), for which the Gaussian quasi-likelihood estimator may be inconsistent. Our approach generalizes that of Beran and Schützner (2009) and gives an interesting alternative to the WLSE used by Francq and Zakoïan (2010) for an autoregressive process with LARCH errors. The family of contrasts is indexed by a single parameter that controls the smoothness of an approximated quasi-likelihood function. Under mild conditions, the resulting estimators are shown to be strongly consistent and asymptotically normal. The optimal asymptotic variance is also given. For LARCH processes, an atypical result is obtained: under assumptions, we show that the limiting distribution of the estimators can be arbitrarily close to a Gaussian distribution supported on a line. Extensions to multivariate processes are also discussed.

This paper considers a decision maker who prefers to make a point decision when the object of interest is interval-identified with regular bounds. When the bounds are just identified along with known interval length, the local asymptotic minimax decision with respect to a symmetric convex loss function takes an obvious form: an efficient lower bound estimator plus the half of the known interval length. However, when the interval length or any nontrivial upper bound for the length is not known, the minimax approach suffers from triviality because the maximal risk is associated with infinitely long identified intervals. In this case, this paper proposes a local asymptotic minimax regret approach and shows that the midpoint between semiparametrically efficient bound estimators is optimal.

On the evening of March 7, 2008, the New Zealand Econometric Study Group Meeting held its Conference Dinner. The venue was the Owen Glenn Building, the spectacular new home of the Auckland Business School and the Department of Economics at the University of Auckland. The meeting was organized by my colleagues, co-authors, and close companions Donggyu Sul and Chirok Han. Chirok did double duty by videotaping the evening, Donggyu coordinated festivities with consummate skill, and we settled in to a memorable evening.

Econometricians, old friends, former students, two of my former teachers, faculty, and senior administrators were gathered together to celebrate my 60th birthday. Many had traveled long distances from overseas and navigated busy schedules to come to this event. It was a singular honor. My wife and daughter were with me. Opening speeches from Bas Sharp and John McDermott broke the ice with endearing tales from the past and jokes about some mysterious hole in my vita. I stood at the front table, looked out, and felt a glow of fellowship envelop me. I was fortunate indeed. Life had bestowed many gifts. The warmth of family, friends, and collegiality were at the top of the list. My education and early training in New Zealand were a clear second.

What follows is a graduate student story. It draws on the first part of the speech I gave that evening at the NZESG conference dinner. It mixes personal reflections with recollections of the extraordinary New Zealanders who shaped my thinking as a graduate student and beginning researcher—people who have had an enduring impact on my work and career as an econometrician. The story traces out these human initial conditions and unit roots that figure in my early life of teaching and research.

This paper deals with asymptotically efficient estimation in exchangeable nonlinear dynamic panel models with common unobservable factors. These models are relevant for applications to large portfolios of credits, corporate bonds, or life insurance contracts. For instance, the Asymptotic Risk Factor (ARF) model is recommended in the current regulation in Finance (Basel II and Basel III) and Insurance (Solvency II) for risk prediction and computation of the required capital. The specification accounts for both micro- and macrodynamics, induced by the lagged individual observations and the common stochastic factors, respectively. For large cross-sectional and time dimensions n and T, we derive the efficiency bound and introduce computationally simple efficient estimators for both the micro- and macroparameters. The results are based on an asymptotic expansion of the log-likelihood function in powers of 1/n, and are linked to granularity theory. The results are illustrated with the stochastic migration model for credit risk analysis.

We consider pure-jump transaction-level models for asset prices in continuous time, driven by point processes. In a bivariate model that admits cointegration, we allow for time deformations to account for such effects as intraday seasonal patterns in volatility and nontrading periods that may be different for the two assets. We also allow for asymmetries (leverage effects). We obtain the asymptotic distribution of the log-price process. For the weak fractional cointegration case, we obtain the asymptotic distribution of the ordinary least squares estimator of the cointegrating parameter based on data sampled from an equally spaced discretization of calendar time, and we justify a feasible method of hypothesis testing for the cointegrating parameter based on the corresponding t-statistic. In the strong fractional cointegration case, we obtain the limiting distribution of a continuously averaged tapered estimator as well as other estimators of the cointegrating parameter, and we find that the rate of convergence can be affected by properties of intertrade durations. In particular, the persistence of durations (hence of volatility) can affect the degree of cointegration. We also obtain the rate of convergence of several estimators of the cointegrating parameter in the standard cointegration case. Finally, we consider the properties of the ordinary least squares estimator of the regression parameter in a spurious regression, i.e., in the absence of cointegration.

We provide a first in-depth look at robust estimation of integrated quarticity (IQ) based on high-frequency data. IQ is the key ingredient enabling inference about volatility and the presence of jumps in financial time series and is thus of considerable interest in applications. We document the significant empirical challenges for IQ estimation posed by commonly encountered data imperfections and set forth three complementary approaches for improving IQ-based inference. First, we show that many common deviations from the jump-diffusive null can be dealt with by a novel filtering scheme that generalizes truncation of individual returns to truncation of arbitrary functionals on return blocks. Second, we propose a new family of efficient robust neighborhood truncation (RNT) estimators for integrated power variation based on order statistics of a set of unbiased local power variation estimators on a block of returns. Third, we find that ratio-based inference, originally proposed in this context by Barndorff-Nielsen and Shephard (2002, Journal of Applied Econometrics 17, 457–477), has desirable robustness properties in the face of regularly occurring data imperfections and thus is well suited for empirical applications. We confirm that the proposed filtering scheme and the RNT estimators perform well in our extensive simulation designs and in an application to the individual Dow Jones 30 stocks.

Econometric analysis of continuous time models has drawn the attention of Peter Phillips for 40 years, resulting in many important publications by him. In these publications he has dealt with a wide range of continuous time models and the associated econometric problems. He has investigated problems from univariate equations to systems of equations, from asymptotic theory to finite sample issues, from parametric models to nonparametric models, from identification problems to estimation and inference problems, and from stationary models to nonstationary and nearly nonstationary models. This paper provides an overview of Peter Phillips’ contributions in the continuous time econometrics literature. We review the problems that have been tackled by him, outline the main techniques suggested by him, and discuss the main results obtained by him. Based on his early work, we compare the performance of three asymptotic distributions in a simple setup. Results indicate that the in-fill asymptotics significantly outperforms the long-span asymptotics and the double asymptotics.

This paper considers convolution equations that arise from problems such as measurement error and nonparametric regression with errors in variables with independence conditions. The equations are examined in spaces of generalized functions to account for possible singularities; this makes it possible to consider densities for arbitrary and not only absolutely continuous distributions, and to operate with Fourier transforms for polynomially growing regression functions. Results are derived for identification and well-posedness in the topology of generalized functions for the deconvolution problem and for some regression models. Conditions for consistency of plug-in estimation for these models are provided.

Testing for causality is of critical importance for many econometric applications. For bivariate AR(1) processes, the limit distributions of causality tests based on least squares estimation depend on the presence of nonstationary processes. When nonstationary processes are present, the limit distributions of such tests are usually very complicated, and the full-sample bootstrap method becomes inconsistent as pointed out in Choi (2005, Statistics and Probability Letters 75, 39–48). In this paper, a profile empirical likelihood method is proposed to test for causality. The proposed test statistic is robust against the presence of nonstationary processes in the sense that one does not have to determine the existence of nonstationary processes a priori. Simulation studies confirm that the proposed test statistic works well.

This paper considers sieve maximum likelihood estimation of seminonparametric (SNP) models with an unknown density function as non-Euclidean parameter, next to a finite-dimensional parameter vector. The density function involved is modeled via an infinite series expansion, so that the actual parameter space is infinite-dimensional. It will be shown that under low-level conditions the sieve estimators of these parameters are consistent, and the estimators of the Euclidean parameters are $\sqrt N$ asymptotically normal, given a random sample of size N. The latter result is derived in a different way than in the sieve estimation literature. It appears that this asymptotic normality result is in essence the same as for the finite dimensional case. This approach is motivated and illustrated by an SNP discrete choice model.

When estimating integrated volatilities based on high-frequency data, simplifying assumptions are usually imposed on the relationship between the observation times and the price process. In this paper, we establish a central limit theorem for the realized volatility in a general endogenous time setting. We also establish a central limit theorem for the tricity under the hypothesis that there is no endogeneity, based on which we propose a test and document that this endogeneity is present in financial data.

An economic time series can often be viewed as a noisy proxy for an underlying economic variable. Measurement errors will influence the dynamic properties of the observed process and may conceal the persistence of the underlying time series. In this paper we develop instrumental variable (IV) methods for extracting information about the latent process. Our framework can be used to estimate the autocorrelation function of the latent volatility process and a key persistence parameter. Our analysis is motivated by the recent literature on realized volatility measures that are imperfect estimates of actual volatility. In an empirical analysis using realized measures for the Dow Jones industrial average stocks, we find the underlying volatility to be near unit root in all cases. Although standard unit root tests are asymptotically justified, we find them to be misleading in our application despite the large sample. Unit root tests that are based on the IV estimator have better finite sample properties in this context.

The Gibbs posterior is a useful tool for risk minimization, which adopts a Bayesian framework and can incorporate convenient computational algorithms such as Markov chain Monte Carlo. We derive risk bounds for the Gibbs posterior using some general nonasymptotic inequalities, which can be used to derive nearly optimal convergence rates and select models to optimally balance the approximation errors and the stochastic errors. These inequalities are formulated in a very general way that does not require the empirical risk to be a usual sample average over independent observations. We apply this framework to study the convergence rate of the GMM (generalized method of moments) risk and derive an oracle inequality for the ranking risk, where models are selected based on the Gibbs posterior with a nonadditive empirical risk.