This paper focuses on the analysis of efficiency, peakedness, and majorization properties of linear estimators under heavy-tailedness assumptions. We demonstrate that peakedness and majorization properties of log-concavely distributed random samples continue to hold for convolutions of α-symmetric distributions with α > 1. However, these properties are reversed in the case of convolutions of α-symmetric distributions with α < 1.

We show that the sample mean is the best linear unbiased estimator of the population mean for not extremely heavy-tailed populations in the sense of its peakedness. In such a case, the sample mean exhibits monotone consistency, and an increase in the sample size always improves its performance. However, efficiency of the sample mean in the sense of peakedness decreases with the sample size if it is used to estimate the location parameter under extreme heavy-tailedness. We also present applications of the results in the study of concentration inequalities for linear estimators.The results in this paper constitute a part of the author's dissertation “New Majorization Theory in Economics and Martingale Convergence Results in Econometrics” presented to the faculty of the Graduate School of Yale University in candidacy for the degree of Doctor of Philosophy in Economics in March 2005. Some of the results were originally contained in the work circulated in 2003–2005 under the titles “Shifting Paradigms: On the Robustness of Economic Models to Heavy-Tailedness Assumptions” and “On the Robustness of Economic Models to Heavy-Tailedness Assumptions.” I am indebted to my advisers, Donald Andrews, Peter Phillips, and Herbert Scarf, for all their support and guidance in all stages of the current project. I also thank the associate editor, two anonymous referees, Donald Brown, Aydin Cecen, Gary Chamberlain, Brian Dineen, Darrell Duffie, Xavier Gabaix, Tilmann Gneiting, Philip Haile, Wolfgang Härdle, Boyan Jovanovic, Samuel Karlin, Benoît Mandelbrot, Alex Maynard, Marcelo Morreira, Ingram Olkin, Ben Polak, Gustavo Soares, Kevin Song, and the participants at seminars at the Departments of Economics at Yale University, University of British Columbia, the University of California at San Diego, Harvard University, the London School of Economics and Political Science, Massachusetts Institute of Technology, the Université de Montréal, McGill University, and New York University, the Division of the Humanities and Social Sciences at California Institute of Technology, Nuffield College, University of Oxford, and the Department of Statistics at Columbia University, and the participants at the 18th New England Statistics Symposium at Harvard University, April 2004, the International Conference on Stochastic Finance, Lisbon, Portugal, September 2004, and the Conference “Heavy Tails and Stable Paretian Distributions in Finance and Macroeconomics” in celebration of the 80th birthday of Benoît Mandelbrot, Deutsche Bundesbank, Eltville, Germany, November 2005, for many helpful comments and discussions.

This note discusses identification in nonparametric, continuous triangular systems. It provides conditions that are both necessary and sufficient for the existence of control functions satisfying conditional independence and support requirements. Confirming a commonly noticed pattern, these conditions restrict the admissible dimensionality of unobserved heterogeneity in the first-stage structural relation, or more generally the dimensionality of the family of conditional distributions of second-stage heterogeneity given explanatory variables and instruments. These conditions imply that no such control function exists without assumptions that seem hard to justify in most applications. In particular, none exists in the context of a generic random coefficient model.

We argue that many methodological confusions in time-series econometrics may be seen as arising out of ambivalence or confusion about the error terms. Relationships between macroeconomic time series are inexact, and, inevitably, the early econometricians found that any estimated relationship would only fit with errors. Slutsky interpreted these errors as shocks that constitute the motive force behind business cycles. Frisch tried to dissect the errors further into two parts: stimuli, which are analogous to shocks, and nuisance aberrations. However, he failed to provide a statistical framework to make this distinction operational. Haavelmo, and subsequent researchers at the Cowles Commission, saw errors in equations as providing the statistical foundations for econometric models and required that they conform to a priori distributional assumptions specified in structural models of the general equilibrium type, later known as simultaneous-equations models. Because theoretical models were at that time mostly static, the structural modeling strategy relegated the dynamics in time-series data frequently to nuisance, atheoretical complications. Revival of the shock interpretation in theoretical models came about through the rational expectations movement and development of the vector autoregression modeling approach. The so-called London School of Economics dynamic specification approach decomposes the dynamics of the modeled variable into three parts: short-run shocks, disequilibrium shocks, and innovative residuals, with only the first two of these sustaining an economic interpretation.

Comparisons of estimators are made on the basis of their mean squared errors and their concentrations of probability computed by means of asymptotic expansions of their distributions when the disturbance variance tends to zero and alternatively when the sample size increases indefinitely. The estimators include k-class estimators (limited information maximum likelihood, two-stage least squares, and ordinary least squares) and linear combinations of them as well as modifications of the limited information maximum likelihood estimator and several Bayes' estimators. Many inequalities between the asymptotic mean squared errors and concentrations of probability are given. Among medianunbiasedestimators, the limited information maximum likelihood estimator dominates the median-unbiased fixed k-class estimator.

The Markov property is a fundamental property in time series analysis and is often assumed in economic and financial modeling. We develop a new test for the Markov property using the conditional characteristic function embedded in a frequency domain approach, which checks the implication of the Markov property in every conditional moment (if it exists) and over many lags. The proposed test is applicable to both univariate and multivariate time series with discrete or continuous distributions. Simulation studies show that with the use of a smoothed nonparametric transition density-based bootstrap procedure, the proposed test has reasonable sizes and all-around power against several popular non-Markov alternatives in finite samples. We apply the test to a number of financial time series and find some evidence against the Markov property.

A general framework for asymptotic tests is proposed. The framework contains as particular cases tests based on various estimation techniques: maximum likelihood methods, pseudo-maximum likelihood (PML) methods and quasi-generalized PML methods, m-estimation methods, moments or generalized moments method, and asymptotic least squares. Moreover the null hypothesis has a general mixed form, including the usual implicit and explicit form.

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 study a family of nonparametric tests of density ratio ordering between two continuous probability distributions on the real line. Density ratio ordering is satisfied when the two distributions admit a nonincreasing density ratio. Equivalently, density ratio ordering is satisfied when the ordinal dominance curve associated with the two distributions is concave. To test this property, we consider statistics based on the Lp-distance between an empirical ordinal dominance curve and its least concave majorant. We derive the limit distribution of these statistics when density ratio ordering is satisfied. Further, we establish that, when 1 ≤ p ≤ 2, the limit distribution is stochastically largest when the two distributions are equal. When 2 < p ≤ ∞, this is not the case, and in fact the limit distribution diverges to infinity along a suitably chosen sequence of concave ordinal dominance curves. Our results serve to clarify, extend, and amend assertions appearing previously in the literature for the cases p = 1 and p = ∞. We provide numerical evidence confirming their relevance in finite samples.

This paper proposes (apparently) novel standard error formulas for the density-weighted average derivative estimator of Powell, Stock, and Stoker (Econometrica 57, 1989). Asymptotic validity of the standard errors developed in this paper does not require the use of higher-order kernels, and the standard errors are “robust” in the sense that they accommodate (but do not require) bandwidths that are smaller than those for which conventional standard errors are valid. Moreover, the results of a Monte Carlo experiment suggest that the finite sample coverage rates of confidence intervals constructed using the standard errors developed in this papercoincide (approximately) with the nominal coverage rates across a nontrivial range of bandwidths.

I am grateful to Professor Neuberg for taking the time to study my book Causality and for summarizing its main ideas in a language familiar to readers of Econometric Theory. I will comment on several issues that I believe warrant further discussion. First, I will summarize in my own style what economists can learn from Causality, and, second, I will correct several inaccuracies in the summary of Professor Neuberg.

In this paper we derive matrix formulae in closed form for higher order moments and give sufficient conditions for higher order stationarity of Markov switching VARMA models. We provide asymptotic theory for sample higher order moments which can be used for testing multivariate normality. As an application, we propose new definitions of multivariate skewness and kurtosis measures for such models, and relate them with the existing concepts in the literature. Our work completes the statistical analysis developed in the fundamental paper of Francq and Zakoïan (2001, Econometric Theory 18, 815–818) and relates with the concepts of multivariate skewness and kurtosis proposed by Mardia (1970, Biometrika 57, 519–530), Móri, Rohatgi, and Székely (1993, Theory of Probability and its Applications 38, 547–551), and Kollo (2008, Journal of Multivariate Analysis 99, 2328–2338). Under suitable assumptions, our results imply that the sample estimators of the skewness and kurtosis measures proposed by these authors are consistent and asymptotically normally distributed. Finally, we check our theory statements numerically via Monte Carlo simulations.

Fractionally integrated time series, which have become an important modeling tool over the last two decades, are obtained by applying the fractional filter to a weakly dependent (short memory) sequence. Weakly dependent sequences are characterized by absolutely summable impulse response coefficients of their Wold representation. The weights bn decay at the rate nd−1 and are not absolutely summable for long memory models (d > 0). It has been believed that this rate is inherited by the impulse responses of any long memory fractionally integrated model. We show that this conjecture does not hold in such generality, and we establish a simple necessary and sufficient condition for the rate nd−1 to be inherited by fractionally integrated processes.

In this paper, we calculate Jeffreys prior for an AR(1) process with and without a constant and a time trend when using the exact likelihood function. We show how this prior can be calculated for the explosive region, even though the unconditional variance of the process is infinite. The calculations lend additional support to the Schotman-van Dijk [6] procedure for restricting the location and the variance of the time trend coefficient. The results show that flat priors are reasonable for the nonexplosive region in an AR(1) without a constant and a time trend where the variance is known and the initial observation is zero, i.e., for the special case studied by Sims and Uhlig [7]. Differences to a flat prior analysis remain in particular for nonzero initial observations, however. For the explosive region, the unconditional prior diverges as the root diverges, supporting findings by Phillips [4]. This paper thus provides a useful perspective as well as some reconciliation for the different stands taken in the literature about priors and Bayesian inference for potentially nonstationary time series.

We propose a characteristic function based test for conditional independence, applicable to both cross-sectional and time series data. We also derive a class of derivative tests, which deliver model-free tests for such important hypotheses as omitted variables, Granger causality in various moments and conditional uncorrelatedness. The proposed tests have a convenient asymptotic null N (0, 1) distribution, and are asymptotically locally more powerful than a variety of related smoothed nonparametric tests in the literature. Unlike other smoothed nonparametric tests for conditional independence, we allow nonparametric estimators for both conditional joint and marginal characteristic functions to jointly determine the asymptotic distributions of the test statistics. Monte Carlo studies demonstrate excellent power of the tests against various alternatives. In an application to testing Granger causality, we document the existence of nonlinear relationships between money and output, which are missed by some existing tests.

This note provides a simple proof for the problem of estimating and testing for multiple breaks in a single equation framework with regressors that are endogenous. We show based on standard assumptions about the regressors, instruments, and errors that the second-stage regression of the instrumental variable procedure involves regressors and errors that satisfy all the assumptions in Perron and Qu (2006, Journal of Econometrics 134, 373–399) so that the results about consistency, rate of convergence and limit distributions of the estimates of the break dates, in addition to the limit distributions of the tests, are obtained as simple consequences. The results are obtained within a unified framework for various cases about the nature of the reduced form: stable, no structural changes but time variations in the parameters, structural changes at dates that are common to those of the structural form, and structural changes occurring at arbitrary dates.

We develop order T−1 asymptotic expansions for the quasi-maximum likelihood estimator (QMLE) and a two-step approximate QMLE in the GARCH(l,l) model. We calculate the approximate mean and skewness and, hence, the Edgeworth-B distribution function. We suggest several methods of bias reduction based on these approximations.

The least absolute shrinkage and selection operator (LASSO) is a widely used statistical methodology for simultaneous estimation and variable selection. It is a shrinkage estimation method that allows one to select parsimonious models. In other words, this method estimates the redundant parameters as zero in the large samples and reduces variance of estimates. In recent years, many authors analyzed this technique from a theoretical and applied point of view. We introduce and study the adaptive LASSO problem for discretely observed multivariate diffusion processes. We prove oracle properties and also derive the asymptotic distribution of the LASSO estimator. This is a nontrivial extension of previous results by Wang and Leng (2007, Journal of the American Statistical Association, 102(479), 1039–1048) on LASSO estimation because of different rates of convergence of the estimators in the drift and diffusion coefficients. We perform simulations and real data analysis to provide some evidence on the applicability of this method.

Standard risk measures, such as the value-at-risk (VaR), or the expected shortfall, have to be estimated, and their estimated counterparts are subject to estimation uncertainty. Replacing, in the theoretical formulas, the true parameter value by an estimator based on n observations of the profit and loss variable induces an asymptotic bias of order 1/n in the coverage probabilities. This paper shows how to correct for this bias by introducing a new estimator of the VaR, called estimation-adjusted VaR (EVaR). This adjustment allows for a joint treatment of theoretical and estimation risks, taking into account their possible dependence. The estimator is derived for a general parametric dynamic model and is particularized to stochastic drift and volatility models. The finite sample properties of the EVaR estimator are studied by simulation and an empirical study of the S&P index is proposed.

We revisit the exact properties of two-stage least squares and limited information maximum likelihood estimators in a structural equation/instrumental variables regression under Gaussian assumptions. Simple derivations based on conditioning serve both to demystify the apparently complicated formulas, and to isolate the key quantities that determine the properties of the estimators. Some recent results obtained under weak-instrument asymptotics are sharpened and clarified by the exact analysis.Thanks to Peter Phillips and several anonymous referees for helpful comments that improved the paper considerably.

This paper considers the unit root tests in models with structural change. Particular attention is given to their dependency on the limiting ratios of the subsample sizes between breaks. The dependency is analyzed in detail, and the invariant testing procedure based on a transformed model is developed. The required transformation is essentially identical to the generalized least-squares correction for heteroskedasticity. The limiting distributions of the new tests do not depend on the relative sizes of the subsamples and are shown to be simple mixtures of the limiting distributions of the corresponding tests from the independent unit root models without structural change.