We consider robust methods for estimation and unit root (UR) testing in autoregressions with infrequent outliers whose number, size, and location can be random and unknown. We show that in this setting standard inference based on ordinary least squares estimation of an augumented Dickey–Fuller (ADF) regression may not be reliable, because (a) clusters of outliers may lead to inconsistent estimation of the autoregressive parameters and (b) large outliers induce a jump component in the asymptotic distribution of UR test statistics. In the benchmark case of known outlier location, we discuss why the augmentation of the ADF regression with appropriate dummy variables not only ensures consistent parameter estimation but also gives rise to UR tests with significant power gains, growing with the number and the size of the outliers. In the case of unknown outlier location, the dummy-based approach is compared with a robust, mixed Gaussian, quasi maximum likelihood (QML) approach, novel in this context. It is proved that, when the ordinary innovations are Gaussian, the QML and the dummy-based approach are asymptotically equivalent, yielding UR tests with the same asymptotic size and power. Moreover, as a by-product of QML the outlier dates can be consistently estimated. When the innovations display tails fatter than Gaussian, the QML approach ensures further power gains over the dummy-based method. Simulations show that the QML ADF-type t-test, in conjunction with standard Dickey–Fuller critical values, yields the best combination of finite-sample size and power.

The solution of differential equations lies at the heart of many problems in structural economics. In econometrics the general nonparametric analysis of consumer welfare is historically the most obvious application, but there are also many applications in finance and other fields. This work considers the general nonparametric form for these problems and identification conditions. It derives a kernel-based estimator and shows consistency and asymptotic normality.

In particular, the link with inverse problems allows us to define it in terms of a well-posed inverse problem and to stress the regularity properties of the estimated solution.I thank Jean-Pierre Florens, Richard Blundell, Eric Renault, and Christine Thomas-Agnan for stimulating conversations, suggestions, and advice. I am very grateful to Oliver Linton and two anonymous referees for most helpful comments.

Time variation is a fundamental problem in statistical and econometric analysis of macroeconomic and financial data. Recently, there has been considerable focus on developing econometric modelling that enables stochastic structural change in model parameters and on model estimation by Bayesian or nonparametric kernel methods. In the context of the estimation of covariance matrices of large dimensional panels, such data requires taking into account time variation, possible dependence and heavy-tailed distributions. In this paper, we introduce a nonparametric version of regularization techniques for sparse large covariance matrices, developed by Bickel and Levina (2008) and others. We focus on the robustness of such a procedure to time variation, dependence and heavy-tailedness of distributions. The paper includes a set of results on Bernstein type inequalities for dependent unbounded variables which are expected to be applicable in econometric analysis beyond estimation of large covariance matrices. We discuss the utility of the robust thresholding method, comparing it with other estimators in simulations and an empirical application on the design of minimum variance portfolios.

This paper compares the asymptotic local power properties of some tests of a null model against a single nonnested alternative and against multiple nonnested alternatives, denoted hereafter as paired and joint tests, respectively. It is demonstrated that the ranking of tests on the basis of asymptotic local powers depends on the choice of local hypothesis. When a local null hypothesis is employed, it is not possible to rank the Wald and Cox-type paired or joint tests. However, when the local hypothesis is specified with reference to one of the alternative models under consideration, a ranking of different test procedures becomes possible. Under a local alternative hypothesis, it is shown that the paired Wald test will never have greater asymptotic local power than a paired Cox-type test.

This paper derives an exact discrete time representation corresponding to a triangular cointegrated continuous time system with mixed stock and flow variables and observable stochastic trends. The discrete time model inherits the triangular structure of the underlying continuous time system and does not suffer from the apparent excess differencing that has been found in some related work. It can therefore serve as a basis for the study of the asymptotic sampling properties of estimators of the model's parameters. Some further analytical and computational results that enable Gaussian estimation to be implemented are also provided.

Andrews (1999, Econometrica 67, 1341–1383) derived the first-order asymptotic theory for a very general class of estimators when a parameter is on a boundary. We derive the second-order asymptotic theory in this setting in some special cases. We focus on the behavior of the quasi maximum likelihood estimator (QMLE) in stationary and nonstationary generalized autoregressive conditionally heteroskedastic (GARCH) models when constraints are imposed in the maximization procedure. We show how in this case both a first- and a second-order bias appear in the estimator and how the bias can be quite large. We provide two types of bias correction mechanisms for the researcher to choose in practice: either to bias correct only for a first-order bias or for a first- and second-order bias. We show that when some constraints are imposed, it is advisable to bias correct not only for the first-order bias but also for the second-order bias.We thank Bruce Hansen and two referees for helpful comments. The first author gratefully acknowledges financial support from the MSU Intramural Research Grants Program. The second author gratefully acknowledges financial support from the ESRC.

The exponentially tilted family of densities is used to discuss the relationship between the encompassing principle and the M-test or conditional moment testing principle. It is shown that the two principles are capable of generating the same test statistics and in this sense equivalent. However, there are differences in motivation and emphasis underlying the principles that are important in econometric modeling. In addition, parsimonious encompassing interpretations for test statistics, including those of Cox (1961, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability I, pp. 105-123; 1962, Journal of the Royal Statistical Society, Series B 24, 406-424) and Atkinson (1970, Journal of the Royal Statistical Society, Series B 32, 323-344) are provided.

The paper introduces a new nonparametric specification test for dynamic regression models. The test combines chi-square statistics based on Fourier series regression. A data-driven choice of the regression order, which uses the square root of the number of Fourier coefficients, is proposed. The benefits of the new test are (1) the selection procedure produces explicit and chi-square critical values that give a finite-sample size close to the nominal size; (2) the test is adaptive rate-optimal and detects local alternatives converging to the null with a rate that can be made arbitrarily close to the parametric rate. Simulation experiments illustrate the practical relevance of the new test.The first author acknowledges financial support from the Fonds Québécois de la Recherche sur la Société et la Culture (FQRSC). The second author acknowledges financial support from LSTA.

This note considers a puzzling phenomenon that is observed in some semiparametric estimation problems. In some cases, using estimated values of the nuisance parameters provides a more efficient estimator for the parameters of interest than does using the true values. This phenomenon takes place even in cases of semi-nonparametric models in which the nuisance parameters are infinite-dimensional and cannot be estimated at the parametric rate. We examine the structure and present the necessary and sufficient condition for the occurrence of this puzzle. We also provide a simple sufficient condition. It shows that the puzzle occurs when the term accounting for the effect of estimation of nuisance parameters is included in the tangent space. This condition is often satisfied when the estimating equation does not bring any restriction on the form of the nuisance parameters. Our simple sufficient condition can be applied to many important estimators.

This paper proposes novel inference procedures for instrumental variable models in the presence of many, potentially weak instruments that are robust to the presence of heteroskedasticity. First, we provide an Anderson–Rubin-type test for the entire parameter vector that is valid under assumptions weaker than previously proposed Anderson–Rubin-type tests. Second, we consider the case of testing a subset of parameters under the assumption that a consistent estimator for the parameters not under test exists. We show that under the null, the proposed statistics have Gaussian limiting distributions and derive alternative chi-square approximations. An extensive simulation study shows the competitive finite sample properties in terms of size and power of our procedures. Finally, we provide an empirical application using college proximity instruments to estimate the returns to education.

Over the last three decades, G.S. Maddala (universally known as “G.S.”) has been a familiar name to all doing econometric work. His substantial contributions to the discipline through numerous books and articles single him out as one of the most distinguished econometricians of our time. Because of his extraordinary ability to synthesize and exposit complex methodological results in simple intuitive terms, he has influenced econometric research in other areas of social science also. According to the Social Science Citation Index, G.S. has been one of the top five most cited econometricians during each of the years 1988–1993. He writes econometrics in plain English with a characteristic sense of wit and humor. There cannot be too many empirical economists around the world who have not been influenced by G.S.'s writings in some way or other. Often he has taken a critical look at evolving econometric techniques, particularly those that have no practical applications, and has not hesitated to go against the tide of the profession.

Estimation of simultaneous equations with limited (or transformed) endogenous regressors has been difficult in the parametric literature for various reasons. In this paper, we propose a nonparametric two-stage method that is analogous to two-stage least-squares estimation. A simultaneous censored model is used to illustrate our approach, and then its generalization to other cases is developed. The technical highlight is in handling a nondifferentiable second-stage minimand with an infinite-dimensional first-stage nuisance parameter when the first-stage error is not orthogonal to the second.

Model selection procedures have become an integral part of almost any statistical or econometric analysis of data. Prominent examples include the selection of explanatory variables in (non)linear regression or lag-length selection in time series models. It is safe to say that these methods have become an integral part of the toolbox of modern data analysis. Moreover, research on model selection (or related procedures such as model averaging and shrinkage procedures) and its implications for statistical analysis has intensified in recent years. This is witnessed by the fact that it is difficult to pick up any recent issue of an econometrics or statistics journal and avoid encountering a paper where some form of model selection, or model averaging, or general shrinkage estimation is considered.

Econometrics as practiced by Arthur (Art) Goldberger demonstrates extraordinary sensitivity to issues of measurement and model specification, and unusual care and caution in interpretation of results, as well as a thorough and comprehensive mastery of econometric theory. His landmark 1964 book, Econometric Theory, set a new standard of rigor in econometrics, and at the same time treated the important problems posed by limited and qualitative dependent variables years before any other text. Art Goldberger's work ranges from early contributions to macro modeling through demand analysis, multivariate modeling with latent variables, and models for sample selectivity, to his highly regarded work on important social issues of heritability of IQ, effectiveness of public versus private schools, and measurement of salary discrimination. Goldberger's influential work, especially on modeling latent or unobservable variables, is widely known and applied in sociology and psychology as well as in economics. Art has been at Wisconsin for many years, and this association is an important reason for Wisconsin's continuing reputation as a leading center for quantitative social sciences.

The quality and influence of Art Goldberger's work has earned him many professional honors. He is a Fellow of the Econometric Society, the American Statistical Association, the American Academy of Arts and Sciences, and the American Association for the Advancement of Science, and has twice been a Guggenheim Fellow. He is a Distinguished Fellow of the American Economic Association and a member of the National Academy of Sciences. Art gave the Woytinsky Lecture at the University of Michigan in 1985.

This interview took place on May 5 in Art Goldberger's office overlooking Lake Mendota. Art's remarks cover a wide range of topics, and I hope they are of interest to social scientists generally as well as to econometricians.

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.

In this paper we are concerned with analyzing the behavior of a semiparametric estimator that corrects for endogeneity in a nonparametric regression by assuming mean independence of residuals from instruments only. Because it is common in many applications, we focus on the case where endogenous regressors and additional instruments are jointly normal, conditional on exogenous regressors. This leads to a severely ill-posed inverse problem. In this setup, we show first how to test for conditional normality. More importantly, we then establish how to exploit this knowledge when constructing an estimator, and we derive the large sample behavior of such an estimator. In addition, in a Monte Carlo experiment we analyze its finite sample behavior. Our application comes from consumer demand. We obtain new and interesting findings that highlight both the advantages and the difficulties of an approach that leads to ill-posed inverse problems. Finally, we discuss the somewhat problematic relationship between endogenous nonparametric regression models and the recently emphasized issue of unobserved heterogeneity in structural models.

A mixture copula is a linear combination of several individual copulas that can be used to generate dependence structures not belonging to existing copula families. Because different pairs of markets may exhibit quite different dependence structures in empirical studies, mixture copulas are useful in modeling the dependence in financial data. Therefore, rather than selecting a single copula based on certain criteria, we propose using a model averaging approach to estimate financial data dependence structures in a mixture copula framework. We select weights (for averaging) by a J-fold Cross-Validation procedure. We prove that the model averaging estimator is asymptotically optimal in the sense that it minimizes the squared estimation loss. Our simulation results show that the model averaging approach outperforms some competing methods when the working mixture model is misspecified. Using 12 years of data on daily returns from four developed economies’ stock indexes, we show that the model averaging approach more accurately estimates their dependence structures than some competing methods.

The asymptotic distributions of augmented Dickey–Fuller (ADF) unit root tests for autoregressive processes with a unit or near-unit root are discussed in the presence of multiple stochastic level shifts of large size occurring independently in time. The distributions depend on a Brownian motion and a Poisson-type jump process. Due to the latter, tests based on standard critical values experience power losses increasing rapidly with the number and the magnitude of the shifts. A new approach to unit root testing is suggested which requires no knowledge of either the location or the number of level shifts, and which dispenses with the assumption of independent shift occurrence. It is proposed to remove possible shifts from a time series by weighting its increments according to how likely it is, with respect to an ad hoc postulated distribution, a shift to have occurred in each period. If the number of level shifts is bounded in probability, the limiting distributions of the proposed test statistics coincide with those of ADF statistics under standard conditions. A Monte Carlo experiment shows that, despite their generality, the new tests perform well in finite samples.We are grateful to Pentti Saikkonen (the co-editor) and two referees for their helpful and constructive comments on earlier versions of this paper. We also thank Søren Johansen, Helmut Lütkepohl, Anders Rahbek, Robert Taylor, and seminar participants at the European University Institute, the University of Copenhagen, and the URCT Conference held in Faro, Portugal, September 29 to October 1, 2005, for useful comments. Both authors are grateful to the Danish Social Sciences Research Council, project 2114-04-001, for financial support.

This paper develops a new semiparametric approach for the estimation of hazard functions in the presence of unobserved heterogeneity. The hazard function is specified parametrically, whereas the distribution of the unobserved heterogeneity is indirectly estimated using the method of kernels. The semiparametric efficiency bounds are derived. The estimator obtains these bounds in large samples.The authors thank Yongmiao Chen, James Heckman, Hidehiko Ichimura, Tony Lancaster, Qi Li, Adrian Pagan, Barry Smith, two anonymous referees, and the co-editor for helpful input. We particularly thank Steven Stern, who prompted us toward this line of research. Any errors are those of the authors. Research funding for Rilstone was provided by the Social Sciences and Humanities Research Council of Canada.