This paper examines several practical issues regarding the implementation of the Phillips and Hansen fully modified least squares (FMLS) method for the estimation of a cointegrating vector. Various versions of this method arise by selecting between standard and prewhitened kernel estimation and between parametric and nonparametric automatic bandwidth estimators and also among alternative kernels. A Monte Carlo study is conducted to investigate the finite-sample properties of the alternative versions of the FMLS procedure. The results suggest that the prewhitened kernel estimator of Andrews and Monahan (1992, Econometrica 60, 953–966) in which the bandwidth parameter is selected via the nonparametric procedure of Newey and West (1994, Review of Economic Studies 61, 631–653) minimizes the second-order asymptotic bias effects.

This paper reconsiders King's [12] locally optimal test procedure for first-order moving average disturbances in the linear regression model. It recommends two tests, one for problems involving positively correlated disturbances and one for negatively correlated disturbances. Both tests are most powerful invariant at a point in the alternative hypothesis parameter space that is determined by a function involving the sample size and the number of regressors. Selected bounds for the tests' significance points are tabulated and an empirical comparison of powers demonstrates the overall superiority of the new test for positively correlated moving average disturbances.

We show that the usual rank condition is necessary and sufficient to identify a vector autoregressive process whether the variables are I(0) or I(d) for d = 1,2,.... We then use this rank condition to demonstrate the interdependence between the identification of short-run and long-run relations of cointegrated process. We find that both the short-run and long-run relations can be identified without the existence of prior information to identify either relation. But if there exists a set of prior restrictions to identify the short-run relation, then this same set of restrictions is sufficient to identify the corresponding long-run relation. On the other hand, it is in general not possible to identify the long-run relations without information on the complete structure. The relationship between the identification of a vector autoregressive process and a Cowles Commission dynamic simultaneous equations model is also clarified.

This paper considers estimation based on a set of T + 1 discrete observations, y(0), y(h), y(2h),…, y(Th) = y(N), where h is the sampling frequency and N is the span of the data. In contrast to the standard approach of driving N to infinity for a fixed sampling frequency, the current paper follows Phillips [35,36] and Perron [29] and examines the “dual” asymptotics implied by letting h tend to zero while the span N remains fixed.

We suggest a way of explicitly embedding discrete processes into continuous-time processes, and using this approach we generalize the results of the above-mentioned authors and derive continuous record asymptotics for vector first-order processes with positive roots in a neighborhood of one and we also consider the case of a scalar second-order process. We illustrate the method by two examples. The first example is a near unit root model with drift and trend.

This paper studies the problem of estimating the set of finite-dimensional parameter values defined by a finite number of moment inequality or equality conditions and gives conditions under which the estimator defined by the set of parameter values that satisfy the estimated versions of these conditions is consistent in Hausdorff metric. This paper also suggests extremum estimators that with probability approaching 1 agree with the set consisting of parameter values that satisfy the sample versions of the moment conditions. In particular, it is shown that the set of minimizers of the sample generalized method of moments (GMM) objective function is consistent for the set of minimizers of the population GMM objective function in Hausdorff metric.

We consider a family of GARCH(1,1) processes introduced in He and Teräsvirta (1999a, Journal of Econometrics 92, 173–192). This family contains various popular generalized autoregressive conditional heteroskedasticity (GARCH) models as special cases. A necessary and sufficient condition for the existence of a strictly stationary solution is given.This research was financially supported by the Jan Wallander's and Tom Hedelius' Foundation, Grant J03–41. The author thanks the editor, an anonymous referee, Pentti Saikkonen, and Timo Teräsvirta for useful comments.

This paper provides two main new results: the first shows theoretically that large biases and variances can arise when the quasi-maximum likelihood (QML) estimation method is employed in a simple bivariate structure under the assumption of conditional heteroskedasticity; and the second demonstrates how these analytical theoretical results can be used to improve the finite-sample performance of a test for multivariate autoregressive conditional heteroskedastic (ARCH) effects, suggesting an alternative to a traditional Bartlett-type correction. We analyze two models: one proposed in Wong and Li (1997, Biometrika 84, 111–123) and another proposed by Engle and Kroner (1995, Econometric Theory 11, 122–150) and Liu and Polasek (1999, Modelling and Decisions in Economics; 2000, working paper, University of Basel). We prove theoretically that a relatively large difference between the intercepts in the two conditional variance equations, which leads to the two series having correspondingly different volatilities in the restricted case, may produce very large variances in some QML estimators in the first model and very severe biases in some QML estimators in the second. Later we use our bias expressions to propose an LM-type test of multivariate ARCH effects and show through simulations that small-sample improvements are possible, especially in relation to the size, when we bias correct the estimators and use the expected hessian version of the test.Both authors thank H. Wong for providing us with the Gauss program to simulate the Wong and Li (1997) model. We also thank three anonymous referees for extremely helpful comments, and we are grateful for the comments received at seminars given at Cardiff University, Michigan State University, Queen Mary London, University of Exeter, and University of Montreal. We acknowledge gratefully also the financial support from an ESRC grant (award number T026 27 1238). A previous version of this paper appeared as IVIE Working paper 2004-09.

It is well known that conventional Wald-type inference in the context of quantile regression is complicated by the need to construct estimates of the conditional densities of the response variables at the quantile of interest. This note explores the possibility of circumventing the need to construct conditional density estimates in this context with scale statistics that are explicitly inconsistent for the underlying conditional densities. This method of studentization leads conventional test statistics to have limiting distributions that are nonstandard but have the convenient feature of depending explicitly on the user’s choice of smoothing parameter. These limiting distributions depend on the distribution of the conditioning variables but can be straightforwardly approximated by resampling.

We consider the limiting behavior of the overidentifying restrictions test in the presence of neglected structural instability at a single “break point.” It is shown that the test need not be consistent against this type of misspecification. If it is consistent then it emerges that the limiting behavior of this test statistic depends on the covariance matrix estimator employed. In this paper we consider the case in which a heteroskedasticity autocorrelation covariance (HAC) is used. It is shown that (i) if the HAC estimator is based on uncentered autocovariances then the overidentifying restrictions test diverges at rate T/bT where T is the sample size and bT is the bandwidth; (ii) if the HAC estimator is based on centered autocovariances then the rate of increase of the overidentifying restrictions test is either T/bT or T depending on the form of the instability. These results are used to provide conditions for the consistency of the method of moment selection of Andrews (1999, Econometrica 67, 543–564) when certain elements of the candidate set of moments are misspecified as a result of neglected structural instability.This work was begun while Hall was a Senior Research Fellow and Peixe was a graduate student at the Department of Economics, University of Birmingham, UK, and this support is gratefully acknowledged. Peixe also gratefully acknowledges financial support from FCT under Grant PRAXIS XXI/BD/13453/97. We are very grateful for the very useful comments of Don Andrews and two anonymous referees.

This paper studies second-order properties of the empirical likelihood overidentifying restriction test to check the validity of moment condition models. We show that the empirical likelihood test is Bartlett correctable and suggest second-order refinement methods for the test based on the empirical Bartlett correction and adjusted empirical likelihood. Our second-order analysis supplements the one in Chen and Cui (2007, Journal of Econometrics141, 492–516) who considered parameter hypothesis testing for overidentified models. In simulation studies we find that the empirical Bartlett correction and adjusted empirical likelihood assisted by bootstrapping provide reasonable improvements for the properties of the null rejection probabilities.

This book is a collection of matrix results that the author found important for his own work. To make the book reasonably self-contained, the author has also added many elementary results. The book contains definitions and results only and no proofs.

We present an analytical closed-form expression for the asymptotic variance matrix in the misspecified multivariate regression model.I am grateful to Hamparsum Bozdogan of the University of Tennessee for bringing the idea of the sandwich variance matrix within the context of the misspecified multivariate regression model to my attention and to two referees for their constructive and useful comments.

We derive asymptotic expansions for semiparametric adaptive regression estimators. In particular, we derive the asymptotic distribution of the second-order effect of an adaptive estimator in a linear regression whose error density is of unknown functional form. We then show how the choice of smoothing parameters influences the estimator through higher order terms. A method of bandwidth selection is defined by minimizing the second-order mean squared error. We examine both independent and time series regressors; we also extend our results to a t-statistic. Monte Carlo simulations confirm the second order theory and the usefulness of the bandwidth selection method.

Much of modern econometrics stems directly from the post-1940 works of Haavelmo and the Cowles Commission (CC) Monograph 10. This paper examines the consolidation process of the Haavelmo-CC research program mainly during the 1950–70 period from three aspects: (i) developments of econometrics textbooks, (ii) emerging themes and trends in econometric research, and (iii) the contribution of the program to empirical modeling of real-world issues. The examination reveals that the program has gained dominance primarily through its adherence to the scientific banner and style rather than its empirical relevance. The adoption of the hard science methodology is decisive in winning over the academic community; the taxonomy of econometrics into steps involving primarily specification, identification, and estimation has played a pivotal role in generating compartmentalized research topics with manageable technical challenge and also in facilitating the educational need for compiling self-contained subjects and definitely soluble questions.

We propose an easy to use derivative-based two-step estimation procedure for semiparametric index models, where the number of indexes is not known a priori. In the first step various functionals involving the derivatives of the unknown function are estimated using nonparametric kernel estimators, in particular the average outer product of the gradient (AOPG). By testing the rank of the AOPG we determine the required number of indexes. Subsequently, we estimate the index parameters in a method of moments framework, with moment conditions constructed using the estimated average derivative functionals. The estimator readily extends to multiple equation models and is shown to be root-N-consistent and asymptotically normal.

This paper proposes a model specification testing procedure for parametric specification of the conditional mean function in a nonlinear time series model with long-range dependent. An asymptotically normal test is established even when long-range dependent is involved. To implement the proposed test in practice using a simulated example, a bootstrap simulation procedure is established to find a simulated critical value to compute both the size and power values of the proposed test.

We examine the limit properties of the nonlinear least squares (NLS) estimator under functional form misspecification in regression models with a unit root. Our theoretical framework is the same as that of Park and Phillips (2001, Econometrica 69, 117–161). We show that the limit behavior of the NLS estimator is largely determined by the relative orders of magnitude of the true and fitted models. If the estimated model is of different order of magnitude than the true model, the estimator converges to boundary points. When the pseudo-true value is on a boundary, standard methods for obtaining rates of convergence and limit distribution results are not applicable. We provide convergence rates and limit theory when the pseudo-true value is an interior point. If functional form misspecification is committed in the presence of stochastic trends, the convergence rates can be slower and the limit distribution different than that obtained under correct specification.