This book seeks to integrate research on cause and effect inference from cognitive science, econometrics, epidemiology, philosophy, and statistics. It puts forward the work of its author, his collaborators, and others over the past two decades as a new account of cause and effect inference that can aid practical researchers in many fields, including econometrics. Pearl adheres to several propositions on cause and effect inference. Though cause and effect relations are fundamentally deterministic (he explicitly excludes quantum mechanical phenomena from his concept of cause and effect), cause and effect analysis involves probability language. Probability language helps to convey uncertainty about cause and effect relations but is insufficient to fully express those relations. In addition to conditional probabilities of events, cause and effect analysis requires graphs or diagrams and a language that distinguishes intervention or manipulation from observation. Cause and effect analysis also requires counterfactual reasoning and causal assumptions in addition to observations and statistical assumptions.

Data envelopment analysis (DEA) and free disposal hull (FDH) estimators are widely used to estimate efficiencies of production units. In applications, both efficiency scores for individual units as well as average efficiency scores are typically reported. While several bootstrap methods have been developed for making inference about the efficiencies of individual units, until now no methods have existed for making inference about mean efficiency levels. This paper shows that standard central limit theorems do not apply in the case of means of DEA or FDH efficiency scores due to the bias of the individual scores, which is of larger order than either the variance or covariances among individual scores. The main difficulty comes from the fact that such statistics depend on efficiency estimators evaluated at random points. Here, new central limit theorems are developed for means of DEA and FDH scores, and their efficacy for inference about mean efficiency levels is examined via Monte Carlo experiments.

We consider least absolute error estimation in a dynamic nonlinear model with neither independent nor identically distributed errors. The estimator is shown to be consistent and asymptotically normal, with asymptotic covariance matrix depending on the errors through the heights of their density functions at their medians (zero). A consistent estimator of the asymptotic covariance matrix of the estimator is given, and the Wald, Lagrange multiplier, and likelihood ratio tests for linear restrictions on the parameters are discussed. A Lagrange multiplier test for heteroscedasticity based upon the absolute residuals is analyzed. This will be useful whenever the heights of the density functions are related to the dispersions.

We consider asymmetric kernel density estimators and smoothed histograms when the unknown probability density function f is defined on [0,+∞). Uniform weak consistency on each compact set in [0,+∞) is proved for these estimators when f is continuous on its support. Weak convergence in L1 is also established. We further prove that the asymmetric kernel density estimator and the smoothed histogram converge in probability to infinity at x = 0 when the density is unbounded at x = 0. Monte Carlo results and an empirical study of the shape of a highly skewed income distribution based on a large microdata set are finally provided.We thank O. Linton and the three referees for constructive criticism and M.P. Feser and J. Litchfield for kindly providing the Brazilian data. We are grateful to I. Gijbels, J.M. Rolin, and I. Van Keilegom for their stimulating remarks and to participants at the workshop on statistical modeling (UCL 2002), LAMES (Sao Paulo 2002), L1 Norm conference (Neuchatel 2002), Geneva econometrics seminar, and KUL econometrics seminar for their comments. Part of this research was done when the second author was visiting THEMA and IRES. The first, resp. second, author gratefully acknowledges financial support from the “Projet d'Actions de Recherche Concertées” grant 98/03-217, and from the IAP research network grant P5/24 of the Belgian state, resp. the Swiss National Science Foundation through the National Centre of Competence in Research: Financial Valuation and Risk Management (NCCR-FINRISK).

The notion of completeness between two random elements has been considered recently to provide identification in nonparametric instrumental problems. This condition is quite abstract, however, and characterizations have been obtained only in special cases. This paper considers a nonparametric model between the two variables with an additive separability and a large support condition. In this framework, different versions of completeness are obtained, depending on which regularity conditions are imposed. This result allows one to establish identification in an instrumental nonparametric regression with limited endogenous regressor, a case where the control variate approach breaks down.

This paper proposes nonparametric tests of change in the distribution function of a time series. The limiting null distributions of the test statistics depend on a nuisance parameter, and critical values cannot be tabulated a priori. To circumvent this problem, a new simulation-based statistical method is developed. The validity of our simulation procedure is established in terms of size, local power, and test consistency. The finite-sample properties of the proposed tests are evaluated in a set of Monte Carlo experiments, and the distributional stability in financial markets is examined.

A new class of large-sample covariance and spectral density matrix estimators is proposed based on the notion of flat-top kernels. The new estimators are shown to be higher-order accurate when higher-order accuracy is possible. A discussion on kernel choice is presented as well as a supporting finite-sample simulation. The problem of spectral estimation under a potential lack of finite fourth moments is also addressed. The higher-order accuracy of flat-top kernel estimators typically comes at the sacrifice of the positive semidefinite property. Nevertheless, we show how a flat-top estimator can be modified to become positive semidefinite (even strictly positive definite) while maintaining its higher-order accuracy. In addition, an easy (and consistent) procedure for optimal bandwidth choice is given; this procedure estimates the optimal bandwidth associated with each individual element of the target matrix, automatically sensing (and adapting to) the underlying correlation structure.

Improvements in data acquisition and processing techniques have led to an almost continuous flow of information for financial data. High-resolution tick data are available and can be quite conveniently described by a continuous-time process. It is therefore natural to ask for possible extensions of financial time series models to a functional setup. In this paper we propose a functional version of the popular autoregressive conditional heteroskedasticity model. We will establish conditions for the existence of a strictly stationary solution, derive weak dependence and moment conditions, show consistency of the estimators, and perform a small empirical study demonstrating how our model matches with real data.

We use local polynomial fitting to estimate the nonparametric M-regression function for strongly mixing stationary processes {(Yi, Xi)}. We establish a strong uniform consistency rate for the Bahadur representation of estimators of the regression function and its derivatives. These results are fundamental for statistical inference and for applications that involve plugging such estimators into other functionals where some control over higher order terms is required. We apply our results to the estimation of an additive M-regression model.

This paper considers the consistency property of some test statistics based on a time series of data. While the usual consistency criterion is based on keeping the sampling interval fixed, we let the sampling interval take any equispaced path as the sample size increases to infinity. We consider tests of the null hypotheses of the random walk and randomness against positive autocorrelation (stationary or explosive). We show that tests of the unit root hypothesis based on the first-order correlation coefficient of the original data are consistent as long as the span of the data is increasing. Tests of the same hypothesis based on the first-order correlation coefficient of the first-differenced data are consistent against stationary alternatives only if the span is increasing at a rate greater than T½, where T is the sample size. On the other hand, tests of the randomness hypothesis based on the first-order correlation coefficient applied to the original data are consistent as long as the span is not increasing too fast. We provide Monte Carlo evidence on the power, in finite samples, of the tests Studied allowing various combinations of span and sampling frequencies. It is found that the consistency properties summarize well the behavior of the power in finite samples. The power of tests for a unit root is more influenced by the span than the number of observations while tests of randomness are more powerful when a small sampling frequency is available.

Although it is only during the last decade that continuous-time models have been extensively used in applied econometric work, the development of statistical methods applicable to such models commenced over 40 years ago. The first significant contribution to the problem of estimating the parameters of continuous-time stochastic models from discrete data was made by the British statistician Bartlett [1946] only three years after the pioneering contribution of Haavelmo [1943] on simultaneous equations models. Moreover, by this time the fundamental mathematical theory of continuous-time stochastic models was already well developed, major contributions having been made by some of the leading mathematicians of the twentieth century, including Einstein, Weiner, and Kolmogorov.

In this article, we investigate a bias in an asymptotic expansion of the simulated maximum likelihood estimator introduced by Lerman and Manski (pp. 305–319 in C. Manski and D. McFadden (eds.), Structural Analysis of Discrete Data with Econometric Applications, Cambridge: MIT Press, 1981) for the estimation of discrete choice models. This bias occurs due to the nonlinearity of the derivatives of the log likelihood function and the statistically independent simulation errors of the choice probabilities across observations. This bias can be the dominating bias in an asymptotic expansion of the simulated maximum likelihood estimator when the number of simulated random variables per observation does not increase at least as fast as the sample size. The properly normalized simulated maximum likelihood estimator even has an asymptotic bias in its limiting distribution if the number of simulated random variables increases only as fast as the square root of the sample size. A bias-adjustment is introduced that can reduce the bias. Some Monte Carlo experiments have demonstrated the usefulness of the bias-adjustment procedure.

The constant conditional correlation general autoregressive conditional heteroskedasticity (GARCH) model is among the most commonly applied multivariate GARCH models and serves as a benchmark against which other models can be compared. In this paper we consider an extension to this model and examine its fourth-moment structure. The extension, first defined by Jeantheau (1998, Econometric Theory 14, 70–86), is motivated by the result found and discussed in this paper that the squared observations from the extended model have a rich autocorrelation structure. This means that already the first-order model is capable of reproducing a whole variety of autocorrelation structures observed in financial return series. These autocorrelations are derived for the first- and the second-order constant conditional correlation GARCH model. The usefulness of the theoretical results of the paper is demonstrated by reconsidering an empirical example that appeared in the original paper on the constant conditional correlation GARCH model.This research has been supported by the Swedish Research Council of Humanities and Social Sciences and the Tore Browaldh's Foundation. A part of this work was carried out while the second author was visiting the School of Finance and Economics, University of Technology, Sydney, whose kind hospitality is gratefully acknowledged. The paper has been presented at the Econometric Society European Meeting, Venice, August 2002. We thank participants for comments and two anonymous referees for their remarks. Any errors and shortcomings in the paper remain our own responsibility.

This paper is concerned with estimating a conditional quantile function that is assumed to be partially linear. The paper develops a simple estimator of the parametric component of the conditional quantile. The semiparametric efficiency bound for the parametric component is derived, and two types of efficient estimators are considered. Asymptotic properties of the proposed estimators are established under regularity conditions. Some Monte Carlo experiments indicate that the proposed estimators perform well in small samples.This paper is a part of my Ph.D. dissertation submitted to the University of Iowa. I am grateful to my adviser, Joel Horowitz, for his insightful comments, suggestions, guidance, and support. I also thank John Geweke, Gene Savin, two anonymous referees, the co-editor Oliver Linton, and participants at the 2001 Midwest Econometrics Group Annual Meeting in Kansas City for many helpful comments and suggestions. Of course, the responsibility for any errors is mine.

For purposes of maximum likelihood estimation, we show how to compute the Gaussian likelihood function when the data are generated by a higher-order continuous-time vector ARMAX model and are observed as stocks and flows at different frequencies. Continuous-time ARMAX models are analogous to discrete-time autoregressive moving-average models with distributed-lag exogenous variables. Stocks are variables observed at points in time and flows are variables observed as integrals over sampling intervals. We derive the implied state-space model of the discrete-time data and show how to use it to compute the Gaussian likelihood function with Kalman-filtering, prediction-error, decomposition of the data.

This paper considers a general nonlinear econometric model framework that contains a large class of estimators defined as solutions to optimization problems. For this framework we derive several asymptotically equivalent forms of a test statistic for the local (in a way made precise in the paper) multivariate nonlinear inequality constraints test H: h(β) ≥ 0 versus K: β ∈ RK. We extend these results to consider local hypotheses tests of the form H: h1(β) ≥ 0 and h2(β) = 0 versus K: β ∈ RK. For each test we derive the asymptotic distribution for any size test as a weighted sum of χ2-distributions. We contrast local as opposed to global inequality constraints testing and give conditions on the model and constraints when each is possible. This paper also extends the well-known duality results in testing multivariate equality constraints to the case of nonlinear multivariate inequality constraints and combinations of nonlinear inequality and equality constraints.

This paper develops the asymptotic theory for least absolute deviation estimation of a shift in linear regressions. Rates of convergence and asymptotic distributions for the estimated regression parameters and the estimated shift point are derived. The asymptotic theory is developed both for fixed magnitude of shift and for shift with magnitude converging to zero as the sample size increases. Asymptotic distributions are also obtained for trending regressors and for dependent disturbances. The analysis is carried out in the framework of partial structural change, allowing some parameters not to be influenced by the shift. Efficiency relative to least-squares estimation is also discussed. Monte Carlo analysis is performed to assess how informative the asymptotic distributions are.

This paper introduces a new estimation method for arbitrary temporal heterogeneity in panel data models. The paper provides a semiparametric method for estimating general patterns of cross-sectional specific time trends. The methods proposed in the paper are related to principal component analysis and estimate the time-varying trend effects using a small number of common functions calculated from the data. An important application for the new estimator is in the estimation of time-varying technical efficiency considered in the stochastic frontier literature. Finite sample performance of the estimators is examined via Monte Carlo simulations. We apply our methods to the analysis of productivity trends in the U.S. banking industry.

We point out the close relationship between the integrated conditional moment tests in Bierens (1982, Journal of Econometrics 20, 105–134) and Bierens and Ploberger (1997, Econometrica 65, 1129–1152) with the complex-valued exponential weight function and the kernel-based tests in Härdle and Mammen (1993, Annals of Statistics 21, 1926–1947), Li and Wang (1998, Journal of Econometrics 87, 145–165), and Zheng (1996, Journal of Econometrics 75, 263–289). It is well established that the integrated conditional moment tests of Bierens (1982) and Bierens and Ploberger (1997) are more powerful than kernel-based nonparametric tests against Pitman local alternatives. In this paper we analyze the power properties of the kernel-based tests and the integrated conditional moment tests for a sequence of “singular” local alternatives, and show that the kernel-based tests can be more powerful than the integrated conditional moment tests for these “singular” local alternatives. These results suggest that integrated conditional moment tests and kernel-based tests should be viewed as complements to each other. Results from a simulation study are in agreement with the theoretical results.

This paper considers an empirical likelihood method to estimate the parameters of the quantile regression (QR) models and to construct confidence regions that are accurate in finite samples. To achieve the higher order refinements, we smooth the estimating equations for the empirical likelihood. We show that the smoothed empirical likelihood estimator is first-order asymptotically equivalent to the standard QR estimator and establish that confidence regions based on the smoothed empirical likelihood ratio have coverage errors of order n−1 and may be Bartlett corrected to produce regions with errors of order n−2, where n denotes the sample size. Our result is an extension of the previous result of Chen and Hall (1993, Annals of Statistics 21, 1166–1181) to the regression context. Monte Carlo experiments suggest that the smoothed empirical likelihood confidence regions may be more accurate in small samples than the confidence regions that can be constructed from the smoothed bootstrap method recently suggested by Horowitz (1998, Econometrica 66, 1327–1351).I thank the co-editor Bruce Hansen and anonymous referees for valuable suggestions and comments. I also thank Song X. Chen, Joel Horowitz, Yuichi Kitamura, Oliver Linton, Whitney Newey, Peter Phillips, and Richard Smith for helpful comments. Parts of this paper were written while I was visiting the Cowles Foundation at Yale University, whose hospitality is gratefully acknowledged. Young-Hyun Cho has provided excellent research assistance. This work was supported by the Korea Research Foundation grant KRF 2003-041-B00072.