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.

In this paper, we reexamine the question of statistical bias in the classic Black/Scholes option price where randomness is due to the use of the historical variance. We show that the only unbiased estimated option is an at the money option.

A semiparametric likelihood method is proposed for the estimation of sample selection models. The method is a two-step semiparametric scoring estimation procedure based on an index restriction and kernel estimation. Under some regularity conditions, the estimator is square-root n-consistent and asymptotically normal. The estimator is also asymptotically efficient in the sense that its asymptotic covariance matrix attains the semiparametric efficiency bound under the index restriction. For the binary choice sample selection model, it also attains the efficiency bound under the independence assumption. This method can be applied to the estimation of general sample selection models.

We establish [square root]n-consistency and asymptotic normality of Han's (1987a, Journal of Econometrics 35, 191–209) estimator of the parameters characterizing the transformation function in a semiparametric transformation model. We verify a Vapnik–Cervonenkis (VC) condition for the parameterizations of Box and Cox (1964, Journal of the Royal Statistical Society, Series B 34, 187–200) and Bickel and Doksum (1981, Journal of the American Statistical Association 76, 296–311). The verification establishes the VC property for a class of sets generated by a nonlinear function of the transformation parameters. We also introduce a new class of rank estimators for these parameters. These estimators require only O(n2 logn) computations to evaluate the criterion function, compared to O(n4) computations for Han's estimator. We prove that these estimators are also [square root]n-consistent and asymptotically normal. A simulation study compares two of the new estimators to Han's estimator, the fully parametric estimator of Bickel and Doksum (1981), and the nonlinear two-stage least squares estimator of Amemiya and Powell (1981, Journal of Econometrics 17, 351–381).

This note uses fixed bandwidth (fixed-b) asymptotic theory to suggest a new approach to testing cointegration parameters in a single-equation cointegration environment. It is shown that the standard tests still have asymptotic distributions that are free of serial correlation nuisance parameters regardless of the bandwidth or kernel used, even if the regressors in the cointegration relationship are endogenous.I thank Tim Vogelsang for his guidance, Pentti Saikkonen for the promptness with which he provided me with an old working paper of his, Wayne Fuller for several discussions, and an anonymous referee for many helpful comments. The usual disclaimer applies.

We present the form of the optimal instrument in a system of multiperiod conditional moment restrictions in the presence of conditional heteroskedasticity. Using Hansen's (1985, Journal of Econometrics 30, 203–228) and Hansen, Heaton, and Ogaki's (1988, Journal of the American Statistical Association 83, 863–871) work on efficiency bounds for generalized method of moments estimators, we show that this form is an autoregressive recurrence parametrized by a few auxiliary processes that are defined through a system of nonlinear stochastic restrictions, with a stability condition among them. In general, the system does not allow inversion and obtaining an explicit solution for the auxiliary parameters.This is a part of my Ph.D. dissertation (Anatolyev, 2000). I am grateful to Kenneth West for introducing me to the subject and providing advice and encouragement. I also thank Bruce Hansen, Yuichi Kitamura, seminar participants at Indiana University, Pennsylvania State University, Rutgers University, University of British Columbia, University of Virginia, and University of Wisconsin, the co-editor Richard Smith, and two referees for helpful comments and suggestions. Research for this paper was supported in part by a fellowship from the Economics Education and Research Consortium, with funds provided by the government of Sweden through the Eurasia Foundation. The opinions expressed in this paper are those of the author and do not necessarily reflect the views of the government of Sweden, the Eurasia Foundation, or any other member of the Economics Education and Research Consortium.

(a) Let , where Q is the fixed effects (FE) transformation given by Q = INT − P with P = IN [otimes ] ιTιT′/T.

Asset returns are frequently assumed to be determined by one or more common factors. We consider a bivariate factor model where the unobservable common factor and idiosyncratic errors are stationary and serially uncorrelated but have strong dependence in higher moments. Stochastic volatility models for the latent variables are employed, in view of their direct application to asset pricing models. Assuming that the underlying persistence is higher in the factor than in the errors, a fractional cointegrating relationship can be recovered by suitable transformation of the data. We propose a narrow band semiparametric estimate of the factor loadings, which is shown to be consistent with a rate of convergence, and its finite-sample properties are investigated in a Monte Carlo experiment.

We show that the order of integration of a vector autoregressive process is equal to the difference between the multiplicity of the unit root in the characteristic equation and the multiplicity of the unit root in the adjoint matrix polynomial. The equivalence with the standard I(1) and I(2) conditions (Johansen, 1996, Likelihood-Based Inference in Cointegrated Vector Auto-Regressive Models) is proved.I am very grateful to Søren Johansen for his precious insights and his continuous help throughout the development of the paper. I also thank Paolo Paruolo (the coeditor) and the referees for valuable feedback. This paper was written while the author was a Post Doc at the Department of Economics, University of Copenhagen, and the hospitality of this institution is gratefully acknowledged.

We consider a model with both a parametric global trend and a nonparametric local trend. This model may be of interest in a number of applications in economics, finance, ecology, and geology. We first propose two hypothesis tests to detect whether two nested special cases are appropriate. For the case where both null hypotheses are rejected, we propose an estimation method to capture certain aspects of the time trend. We establish consistency and some distribution theory in the presence of a large sample. Moreover, we examine the proposed hypothesis tests and estimation methods through both simulated and real data examples. Finally, we discuss some potential extensions and issues when modelling time effects.

This paper investigates how the correlations implied by a first-order simultaneous autoregressive (SAR(1)) process are affected by the weights matrix and the autocorrelation parameter. A graph theoretic representation of the covariances in terms of walks connecting the spatial units helps to clarify a number of correlation properties of the processes. In particular, we study some implications of row-standardizing the weights matrix, the dependence of the correlations on graph distance, and the behavior of the correlations at the extremes of the parameter space. Throughout the analysis differences between directed and undirected networks are emphasized. The graph theoretic representation also clarifies why it is difficult to relate properties of W to correlation properties of SAR(1) models defined on irregular lattices.

The least squares estimator for the linear regression model is shown to converge to the true parameter vector either with probability one or with probability zero. In the latter case, it either converges to a point not equal to the true parameter with probability one, or it diverges with probability one. These results are shown to hold under weak conditions on the dependent random variable and regressor variables. No additional conditions are placed on the errors. The dependent and regressor variables are assumed to be weakly dependent—in particular, to be strong mixing. The regressors may be fixed or random and must exhibit a certain degree of independent variability. No further assumptions are needed. The model considered allows the number of regressors to increase without bound as the sample size increases. The proof proceeds by extending Kolmogorov's 0-1 law for independent randomvariables to strong mixing random variables.

The martingale difference restriction is an outcome of many theoretical analyses in economics and finance. A large body of econometric literature deals with tests of that restriction. We provide new tests based on radial basis function (RBF) neural networks. Our work is based on the test design of Blake and Kapetanios (2000, 2003a, 2003b). However, unlike that work we provide a formal theoretical justification for the validity of these tests and present some new general theoretical results. These results take advantage of the link between the algorithms of Blake and Kapetanios (2000, 2003a, 2003b) and boosting. We carry out a Monte Carlo study of the properties of the new tests and find that they have very good power performance. A simplified implementation of boosting is found to have desirable properties and small computational cost. An empirical application to the S&P 500 constituents illustrates the usefulness of our new test.

A number of tests have been suggested for the test of the null of no cointegration. Under this null, correlations are spurious in the sense of Granger and Newbold (1974) and Phillips (1986). We examine a set of models local to the null of no cointegration and derive tests with optimality properties in order to examine the efficiency of available tests. We find that, for a sufficiently tight weighting over potential cointegrating vectors, commonly employed full system tests have power that can, in some situations, be quite far from the power bounds for the models examined.

This paper derives transformations for multivariate statistics that eliminate asymptotic skewness, extending the results of Niki and Konishi (1986, Annals of the Institute of Statistical Mathematics 38, 371–383). Within the context of valid Edgeworth expansions for such statistics we first derive the set of equations that such a transformation must satisfy and second propose a local solution that is sufficient up to the desired order. Application of these results yields two useful corollaries. First, it is possible to eliminate the first correction term in an Edgeworth expansion, thereby accelerating convergence to the leading term normal approximation. Second, bootstrapping the transformed statistic can yield the same rate of convergence of the double, or prepivoted, bootstrap of Beran (1988, Journal of the American Statistical Association 83, 687–697), applied to the original statistic, implying a significant computational saving.

The analytic results are illustrated by application to the family of exponential models, in which the transformation is seen to depend only upon the properties of the likelihood. The numerical properties are examined within a class of nonlinear regression models (logit, probit, Poisson, and exponential regressions), where the adequacy of the limiting normal and of the bootstrap (utilizing the k-step procedure of Andrews, 2002, Econometrica 70, 119–162) as distributional approximations is assessed.This paper is derived from my Ph.D. thesis, “Higher-Order Asymptotics for Econometric Estimators and Tests,” for which thanks for patient and helpful supervision go to Grant Hillier. Comments by Karim Abadir, Francesco Bravo, Giovanni Forchini, Soren Johansen, Paul Marriott, Mark Salmon, and Steve Satchell, participants at the conference “Differential Geometric Methods in Econometrics,” held at EUI, Florence, and by two anonymous referees proved most helpful. In particular, I thank Peter Phillips for showing interest in the paper, helping with improving the exposition, and providing me with copies of two unpublished research notes. Financial support in the form of a Leverhulme Special Research Fellowship in Economics and Mathematics is gratefully acknowledged.

The phenomenon of multiple transactions at each recording time is a common occurrence for high-frequency financial data because of the heavy trading of the market and limitation of the recording mechanism. This situation has existed for many years, but has become more common in recent years because of heavier trading. Surprisingly, there have been few studies on this important issue, in spite of some ad hoc approaches to treat multiple transactions. In this paper we investigate how to handle multiple transactions, particularly in the context of estimating the integrated volatility and integrated quarticity, which are of great interest in financial econometrics. Two approaches are proposed for this purpose, and their asymptotic properties are investigated. Their performances are confirmed by simulation studies. The estimators are also applied to some real world problems. The work represents only the first step in this direction, and some future research problems are discussed.

When values of regressors are symmetrically disposed, many M-estimators in a wide class of models have a reflection property, namely, that as the signs of the coefficients on regressors are reversed, their estimators' sampling distribution is reflected about the origin. When the coefficients are zero, sign reversal can have no effect. So in this case, the sampling distribution of regression coefficient estimators is symmetric about zero, the estimators are median unbiased and, when moments exist, the estimators are exactly uncorrelated with estimators of other parameters. The result is unusual in that it does not require response variates to have symmetric conditional distributions. It demonstrates the potential importance of covariate design in determining the distributions of estimators, and it is useful in designing and interpreting Monte Carlo experiments. The result is illustrated by a Monte Carlo experiment in which maximum likelihood and symmetrically censored least-squares estimators are calculated for small samples from a censored normal linear regression, Tobit, model.