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.

We define a three-step procedure for more efficient estimation of the nonparametric regression mean with nonparametric autocorrelated errors. The procedure is based upon a nonparametric prewhitening transformation of the dependent variable that has to be estimated from the data by a local polynomial technique. We establish the asymptotic distribution of our estimator under weak dependence conditions and show that it is more efficient than the conventional local polynomial estimator. Furthermore, we consider criterion functions based on the linear exponential family, which include the local polynomial least squares criterion as a special case. Simulation evidence suggests that significant gains can be achieved in finite samples with our approach.The authors thank Oliver Linton for his many constructive and helpful suggestions. The very insightful comments from the referees are also gratefully acknowledged. The second author gratefully acknowledges financial support from the Academic Senate, UCR.

This paper considers estimation of a sample selection model subject to conditional heteroskedasticity in both the selection and outcome equations. The form of heteroskedasticity allowed for in each equation is multiplicative, and each of the two scale functions is left unspecified. A three-step estimator for the parameters of interest in the outcome equation is proposed. The first two stages involve nonparametric estimation of the “propensity score” and the conditional interquartile range of the outcome equation, respectively. The third stage reweights the data so that the conditional expectation of the reweighted dependent variable is of a partially linear form, and the parameters of interest are estimated by an approach analogous to that adopted in Ahn and Powell (1993, Journal of Econometrics 58, 3–29). Under standard regularity conditions the proposed estimator is shown to be -consistent and asymptotically normal, and the form of its limiting covariance matrix is derived.We are grateful to B. Honoré, R. Klein, E. Kyriazidou, L.-F. Lee, J. Powell, two anonymous referees, and the co-editor D. Andrews and also to seminar participants at Princeton, Queens, UCLA, and the University of Toronto for helpful comments. Chen's research was supported by RGC grant HKUST 6070/01H from the Research Grants Council of Hong Kong.

Many empirical studies estimate the structural effect of some variable on an outcome of interest while allowing for many covariates. We present inference methods that account for many covariates. The methods are based on asymptotics where the number of covariates grows as fast as the sample size. We find a limiting normal distribution with variance that is larger than the standard one. We also find that with homoskedasticity this larger variance can be accounted for by using degrees-of-freedom-adjusted standard errors. We link this asymptotic theory to previous results for many instruments and for small bandwidth(s) distributional approximations.

This paper is concerned with the problem of identifiability of the parameters of a high frequency multivariate autoregressive model from mixed frequency time series data. We demonstrate identifiability for generic parameter values using the population second moments of the observations. In addition we display a constructive algorithm for the parameter values and establish the continuity of the mapping attaching the high frequency parameters to these population second moments. These structural results are obtained using two alternative tools viz. extended Yule Walker equations and blocking of the output process. The cases of stock and flow variables, as well as of general linear transformations of high frequency data, are treated. Finally, we briefly discuss how our constructive identifiability results can be used for parameter estimation based on the sample second moments.

We consider a positive-valued time series whose conditional distribution has a time-varying mean, which may depend on exogenous variables. The main applications concern count or duration data. Under a contraction condition on the mean function, it is shown that stationarity and ergodicity hold when the mean and stochastic orders of the conditional distribution are the same. The latter condition holds for the exponential family parametrized by the mean, but also for many other distributions. We also provide conditions for the existence of marginal moments and for the geometric decay of the beta-mixing coefficients. We give conditions for consistency and asymptotic normality of the Exponential Quasi-Maximum Likelihood Estimator of the conditional mean parameters. Simulation experiments and illustrations on series of stock market volumes and of greenhouse gas concentrations show that the multiplicative-error form of usual duration models deserves to be relaxed, as allowed in this paper.

Local linear fitting of nonlinear processes under strong (i.e., α-) mixing conditions has been investigated extensively. However, it is often a difficult step to establish the strong mixing of a nonlinear process composed of several parts such as the popular combination of autoregressive moving average (ARMA) and generalized autoregressive conditionally heteroskedastic (GARCH) models. In this paper we develop an asymptotic theory of local linear fitting for near epoch dependent (NED) processes. We establish the pointwise asymptotic normality of the local linear kernel estimators under some restrictions on the amount of dependence. Simulations and application examples illustrate that the proposed approach can work quite well for the medium size of economic time series.We thank Yuichi Kitamura and two referees for helpful comments. This research was partially supported by a Leverhulme Trust research grant, the National Natural Science Foundation of China, and the Economic and Social Science Research Council of the UK.

Under general conditions the distribution function of the first few terms in a stochastic expansion of an econometric estimator or test statistic provides an asymptotic approximation to the distribution function of the original estimator or test statistic with an error of order less than that of the limiting normal or chi-square approximation. This can be used to establish the validity of several refined asymptotic methods, including the comparison of Nagar-type moments and the use of formal Edgeworth or Edgeworth-type approximations.

This paper studies asymptotic likelihood inference on cointegration parameters in systems integrated of order two. We start with so-called triangular systems and then extend the analysis to vector autoregressions. We show that even when all unit root restrictions have been imposed, the asymptotic observed information is not (locally) ancillary, which implies that the log-likelihood ratio is not locally asymptotically mixed normal. The results are applied to inference on polynomial cointegration. Some similarities and differences with I(1) systems are also discussed.

This paper investigates the asymptotic distribution of the maximum likelihood estimator in a stochastic frontier function when the firms are all technically efficient. For such a situation the true parameter vector is on the boundary of the parameter space, and the scores are linearly dependent. The asymptotic distribution of the maximum likelihood estimator is shown to be a mixture of certain truncated distributions. The maximum likelihood estimates for different parameters may have different rates of stochastic convergence. The model can be reparameterized into one with a regular likelihood function. The likelihood ratio test statistic has the usual mixture of chi-square distributions as in the regular case.

In applied work economists often seek to relate a given response variable y to some causal parameter μ* associated with it. This parameter usually represents a summarization based on some explanatory variables of the distribution of y, such as a regression function, and treating it as a conditional expectation is central to its identification and estimation. However, the interpretation of μ* as a conditional expectation breaks down if some or all of the explanatory variables are endogenous. This is not a problem when μ* is modeled as a parametric function of explanatory variables because it is well known how instrumental variables techniques can be used to identify and estimate μ*. In contrast, handling endogenous regressors in nonparametric models, where μ* is regarded as fully unknown, presents difficult theoretical and practical challenges. In this paper we consider an endogenous nonparametric model based on a conditional moment restriction. We investigate identification-related properties of this model when the unknown function μ* belongs to a linear space. We also investigate underidentification of μ* along with the identification of its linear functionals. Several examples are provided to develop intuition about identification and estimation for endogenous nonparametric regression and related models.We thank Jeff Wooldridge and two anonymous referees for comments that greatly improved this paper.

This paper shows how fractional unit root tests originally derived under stationarity can be made robust to heteroskedasticity. This is done by using existing tests nested in a regression framework and then implementing these tests using White’s heteroskedasticity consistent standard errors (White, 1980). We show this approach is effective both asymptotically and in finite samples. We also provide some evidence on the asymptotic local power of different implementations of the tests, under both homoskedasticity and heteroskedasticity.

The p values of structural break tests, when the break date or dates are unknown, must be calculated in terms of the probability distributions of functions of Bessel processes. The literature so far has maintained that direct computation of these p values and of the corresponding critical values is too difficult and has relied on approximations based on simulations, asymptotic expansions, or curve fitting. This paper presents a fast simple method of calculating exact p values and critical values and uses the method to evaluate the accuracy of the various approximations.The author is grateful for comments and suggestions from Don Andrews, Clint Cummins, Jeff Fuhrer, Ken Garbade, Jim Mahoney, Tony Rodrigues, Josh Rosenberg, Sebastian Schich, participants in a workshop at the Federal Reserve Bank of New York, and the referees. The views expressed in this paper are those of the author and do not necessarily represent those of the Federal Reserve Bank of New York or the Federal Reserve System.

Recently Tanaka and Satchell [11] investigated the limiting properties of local maximizers of the Gaussian pseudo-likelihood function of a misspecified moving average model of order one in case the spectral density of the data process has a zero at frequency zero. We show that pseudo-maximum likelihood estimators in the narrower sense, that is, global maximizers of the Gaussian pseudo-likelihood function, may exhibit behavior drastically different from that of the local maximizers. Some general results on the limiting behavior of pseudo-maximum likelihood estimators in potentially misspecified ARMA models are also presented.

We suggest a rescaled variance type of test for the null hypothesis of stationarity against deterministic and stochastic trends (unit roots). The deterministic trend can be represented as a general function in time (e.g., nonparametric, linear, or polynomial regression, abrupt changes in the mean). Under the null, the asymptotic distribution of the test is derived, and critical values are tabulated for a wide class of stationary processes with short, long, or negative dependence structure. A simulation study examines the performance of the test in terms of size and power. The empirical performance of the test is illustrated using the S&P 500 data.The authors thank the editor, the referees, and Karim Abadir for helpful comments and Alfredas Račkauskas for drawing our attention to the criterion of Cremers and Kadelka (1986). The first author's work was supported by the ESRC grants R000238212 and R000239538. The last two authors were supported by a cooperation agreement CNRS/LITHUANIA (4714) and by a bilateral Lithuania-France research project Gilibert.

Understanding the time series dynamics of a multi-dimensional dependency structure is a challenging task. Multivariate covariance driven Gaussian or mixed normal time varying models have only a limited ability to capture important features of the data such as heavy tails, asymmetry, and nonlinear dependencies. The present paper tackles this problem by proposing and analyzing a hidden Markov model (HMM) for hierarchical Archimedean copulae (HAC). The HAC constitute a wide class of models for multi-dimensional dependencies, and HMM is a statistical technique for describing regime switching dynamics. HMM applied to HAC flexibly models multivariate dimensional non-Gaussian time series.

We apply the expectation maximization (EM) algorithm for parameter estimation. Consistency results for both parameters and HAC structures are established in an HMM framework. The model is calibrated to exchange rate data with a VaR application. This example is motivated by a local adaptive analysis that yields a time varying HAC model. We compare its forecasting performance with that of other classical dynamic models. In another, second, application, we model a rainfall process. This task is of particular theoretical and practical interest because of the specific structure and required untypical treatment of precipitation data.

Let x be a random variable whose first three moments exist. If the density of x is unimodal and positively skewed, then counterexamples are provided which show that the inequality mode ≤ median ≤ mean does not necessarily hold.I thank Andrey Vasnev for help with the graphs and Jan Magnus for various helpful discussions. I also thank Martin Bland, Paolo Paruolo, Peter Phillips, Michael Rockinger, and a referee for their comments. ESRC grant R000239538 is gratefully acknowledged.

In this paper we analyze the effects of a very general class of time-varying variances on well-known “stationarity” tests of the I(0) null hypothesis. Our setup allows, among other things, for both single and multiple breaks in variance, smooth transition variance breaks, and (piecewise-) linear trending variances. We derive representations for the limiting distributions of the test statistics under variance breaks in the errors of I(0), I(1), and near-I(1) data generating processes, demonstrating the dependence of these representations on the precise pattern followed by the variance processes. Monte Carlo methods are used to quantify the effects of fixed and smooth transition single breaks and trending variances on the size and power properties of the tests. Finally, bootstrap versions of the tests are proposed that provide a solution to the inference problem.We are grateful to Peter Phillips, a co-editor, and two anonymous referees whose comments on an earlier draft have led to a considerable improvement in the paper.

Local linear fitting is a popular nonparametric method in statistical and econometric modeling. Lu and Linton (2007, Econometric Theory23, 37–70) established the pointwise asymptotic distribution for the local linear estimator of a nonparametric regression function under the condition of near epoch dependence. In this paper, we further investigate the uniform consistency of this estimator. The uniform strong and weak consistencies with convergence rates for the local linear fitting are established under mild conditions. Furthermore, general results regarding uniform convergence rates for nonparametric kernel-based estimators are provided. The results of this paper will be of wide potential interest in time series semiparametric modeling.

This paper deals with the identification and maximum likelihood estimation of the parameters of a stochastic differential equation from discrete time sampling. Score function and maximum likelihood equations are derived explicitly. The stochastic differential equation system is extended to allow for random effects and the analysis of panel data. In addition, we investigate the identifiability of the continuous time parameters, in particular the impact of the inclusion of exogenous variables.