This paper deals with error correction models (ECM's) and cointegrated systems that are formulated in continuous time. Long-run equilibrium coefficients in the continuous system are always identified in the discrete time reduced form, so that there is no aliasing problem for these parameters. The long-run relationships are also preserved under quite general data filtering. Frequency domain procedures are outlined for estimation and inference. These methods are asymptotically optimal under Gaussian assumptions and they have the advantages of simplicity of computation and generality of specification, thereby avoiding some methodological problems of dynamic specification. In addition, they facilitate the treatment of data irregularities such as mixed stock and flow data and temporally aggregated partial equilibrium formulations. Models with restricted cointegrating matrices are also considered.

On introduit le concept de processus aleatoire vectoriel presque integre. Ce type de processus aide a travailler vers une theorie asymptotique generale de la regression pour des series temporelles multiples

This paper studies the random walk, in a general time series setting that allows for weakly dependent and heterogeneously distributed innovations. It is shown that simple least squares regression consistently estimates a unit root under very general conditions in spite of the presence of autocorrelated errors. The limiting distribution of the standardized estimator and the associated regression t statistic are found using functional central limit theory. New tests of the random walk hypothesis are developed which permit a wide class of dependent and heterogeneous innovation sequences. A new limiting distribution theory is constructed based on the concept of continuous data recording. This theory, together with an asymptotic expansion that is developed in the paper for the unit root case, explain many of the interesting experimental results recently reported in Evans and Savin (1981, 1984).

On donne la distribution d'echantillon finie exacte de la statistique de Wald pour tester des restrictions lineaires generales sur les coefficients dans le modele lineaire multivariable

This paper derives the exact finite sample distribution of the two-stage generalized least squares (GLS) estimator in a multivariate linear model with general linear parameter restrictions. This includes the seemingly unrelated regression (SUR) model as a special case and generalizes presently known exact results for the latter system. The usual classical assumptions are made concerning nonrandom exogenous variables and normally distributed errors. The theoretical results of this paper are made possible by the author's development of a matrix fractional calculus. This operator calculus is the main theoretical tool of the paper and may be used to solve a wide range of other unsolved problems in econometric distribution theory. IN THE EARLY 1960's Zellner [10] developed a two-stage GLS estimator for the coefficients in a linear multivariate system that is now popularly known as the SUR model. This two-stage procedure has since been used in many empirical applications. GLS also forms the basis of other commonly used estimators both in linear models with heteroscedastic or autocorrelated errors and in simultaneous equation systems where it leads to three stage least squares (3SLS). In spite of extensive research and perhaps surprisingly in view of the popularity of GLS methods in empirical work, the exact finite sample distribution of the SUR estimator is known only in highly specialized cases. These cases effectively restrict attention to two equation systems and models with orthogonal regressors [2]. Existing distribution theory is even more limited in the case of other commonly used GLS estimators, such as the two-stage estimator in linear models with heteroscedastic errors. Here, only low order moment formulae are known and then only in the simplest two sample setting. The research underlying the present paper is motivated by the deficiencies outlined above. Our initial object of study was the exact distribution of the SUR estimator in the general case. But the methods we have developed open the way to an exact distribution theory for econometric estimators in a much wider setting than the SUR model. The present paper will derive the exact finite sample distribution of the two-stage GLS estimator in the multivariate linear model subject to general linear parameter restrictions. This generalizes all presently known distribution theory for the SUR model itself. Two important specializations of our results will be illustrated in detail: the unrestricted multivariate linear model; and the Zellner model with pairwise orthogonal regressors. The analytical results reported here are made possible by the introduction of a fractional matrix calculus. This calculus is developed in terms of the action of

This article proposes a new approach to small sample theory that achieves a meaningful integration of earlier directions of research in this field. The approach centers on the constructive technique of approximating distributions developed recently by the author in [10]. This technique utilizes extended rational approximants (ERA's) which build on the strengths of alternative, less flexible approximation methods (such as those based on asymptotic expansions) and which simultaneously blend information from diverse analytic, numerical and experimental sources. The first part of the article explores the general theory of approximation of continuous probability distributions by means of ERA's. Existence, characterization, error bound, and uniqueness theorems for these approximants are given and a new proof is provided for the convergence result obtained earlier in [10]. Some further aspects of finding ERA's by modifications to multiple-point Pade approximants are presented and the new approach is applied to the noncircular serial correlation coefficient. The results of this application demonstrate how ERA's provide systematic improvements over Edgeworth and saddlepoint techniques. These results, taken with those of the earlier article [10], suggest that the approach offers considerable potential for empirical application in terms of its reliability, convenience, and generality.

IN THE LATE 1960's, Richardson [18] and Sawa [20] derived the exact distribution of the two-stage least squares (2SLS) estimator in a structural equation (of a simultaneous system) that contained two endogenous variables and an arbitrary number of degrees of overidentification. Their results refer to the 2SLS estimator of the coefficient of the endogenous variable included on the right hand side of the equation and were obtained under the classical assumptions (to use the term employed by Sargan [19]) of normally distributed disturbances and nonrandom exogenous variables. Very little exact finite sample theory has been published so far for estimators in structural equations containing more than two endogenous variables. Basmann et al. [4] extract the joint probability density function (p.d.f.) of the 2SLS estimator in a just identified equation containing three endogenous variables. Basmann [3] quotes a result due to Richardson for the same set up but with an 2 arbitrary number of degrees of overidentification . In Basmann's notation, this last result characterizes the subclass

Edgeworth series expansions are obtained of the finite sample distributions of the least squares estimator and the associated t ratio test statistic in the context of a first-order noncircular stochastic difference equation. General formulae are given for these expansions up to 0(Th1) where T is the sample size and explicit representations of these in terms of the true parameters are derived up to 0(12). Some numerical comparisons of the approximations and the exact distributions are made in the case of the least squares estimator.

A multiple equation nonlinear regression model with serially independent disturbances is considered. The estimation of the parameters in this model by maximum likelihood and minimum distance methods is discussed and our main subject is the relationship between these procedures. We establish that if the number of observations in a sample is sufficiently large, the iterated minimum distance procedure converges almost surely and the limit of this sequence of iterations is the quasi-maximum likelihood estimator.