Although it is only during the last decade that continuous-time models have been extensively used in applied econometric work, the development of statistical methods applicable to such models commenced over 40 years ago. The first significant contribution to the problem of estimating the parameters of continuous-time stochastic models from discrete data was made by the British statistician Bartlett [1946] only three years after the pioneering contribution of Haavelmo [1943] on simultaneous equations models. Moreover, by this time the fundamental mathematical theory of continuous-time stochastic models was already well developed, major contributions having been made by some of the leading mathematicians of the twentieth century, including Einstein, Weiner, and Kolmogorov.

Considering the contribution of the Cowles research institute to the development of econometrics, one has little choice in this address and must mainly focus attention on one major achievement, namely, the building of the simultaneous equation methodology. One does not need to demonstrate the indisputable fact that this methodology was conceived and elaborated at the Cowles Commission in the forties. Neither does one need to insist on the long standing significance of this achievement nor on its central place in any education or reflection concerning statistical inference about economic phenomena. More interesting is the question of how research at Cowles during the first 15 years of its existence led to this result and how further econometric research here during the last 30 years relates to the simultaneous equation achievement. Also relevant is the following question: how does the message that was sent out by the Cowles people to the world in 1950 stand today? Should it be replaced by another different one? Or should it simply be somewhat amended and supplemented?

To do full justice to the research work and achievements of the many econometricians that were associated with Cowles through the years would require a much longer paper than the present one. But one must at least try also to summarize here those main concerns that stood outside the simultaneous equation methodology.

Nine leading journals that publish statistical theory are used to provide a data base of institutional and individual research activity in statistics over the period 1980–1986. From this data base, we construct both institutional and individual research rankings according to standardized page counts of articles published in these journals over the stated period. The study is worldwide and we provide breakdowns of publication by country and by journal. Separate rankings are also provided for both institutions and individuals according to publication track records in the Annals of Statistics alone.

This paper demonstrates that for a finite stationary autoregressive moving average process the inverse of the covariance matrix differs from the matrix of the covariances of the inverse process by a matrix of low rank. The formula for the exact inverse of the covariance matrix of the scalar or multivariate process is provided. We obtain approximations based on this formula and evaluate some of the approximate results in the existing literature. Applications to computational algorithms and to the distributions of two-step estimators are discussed. In addition the paper contains the formula for the determinant of the covariance matrix which is useful in exact maximum likelihood estimation; it also lists the expressions for the autocovariances of scalar autoregressive moving average processes.

Building on work of McLeish, we present a number of invariance principles for doubly indexed arrays of stochastic processes which may exhibit considerable dependence, heterogeneity, and/or trending moments. In particular, we consider possibly time-varying functions of infinite histories of heterogeneous mixing processes and obtain general invariance results, with central limit theorems following as corollaries. These results are formulated so as to apply to economic time series, which may exhibit some or all of the features allowed in our theorems. Results are given for the case of both scalar and vector stochastic processes. Using an approach recently pioneered by Phillips, and Phillips and Durlauf, we apply our results to least squares estimation of unit root models.

In this paper, it will be shown that if we condition a k-variate rational-valued time series process on its entire past, it is possible to capture all relevant information on the past of the process by a single random variable. This scalar random variable can be formed as an autoregressive moving average of past observations; Since economic data are usually reported in a finite number of digits, this result applies to virtually all economic time series. Therefore, economic time series regressions generally take the form of a nonlinear function of an autoregressive moving average of past observations. This approach is applied to model specification testing of nonlinear ARX models.

Professor Bierens' approach is definitely original and interesting. We think it might become important, in particular, for testing correct specification of models (even nonlinear) in the time series context. For example, his test in Section 6 is (to our knowledge) the first test of correct specification of a model which is consistent against all stationary alternatives. Usually, specification tests checking the structure of dependence are only designed to test against a finite-dimensional alternative. The critical points we make below should be seen in the light of our above statements.

The purpose of this paper is to provide a systematic treatment of the problem of estimation in systems of linear stochastic equations where some of the disturbances are uncorrelated and where the identifying equations are equivalent to a linear system.

This paper considers the parametric estimation problem for continuous-time stochastic processes described by first-order nonlinear stochastic differential equations of the generalized Itô type (containing both jump and diffusion components). We derive a particular functional partial differential equation which characterizes the exact likelihood function of a discretely sampled Itô process. In addition, we show by a simple counterexample that the common approach of estimating parameters of an Itô process by applying maximum likelihood to a discretization of the stochastic differential equation does not yield consistent estimators.

This paper considers M-estimators of regression parameters that make use of a generalized functional form for the disturbance distribution. The family of distributions considered is the generalized t (GT), which includes the power exponential or Box-Tiao, normal, Laplace, and t distributions as special cases. The corresponding influence function is bounded and redescending for finite “degrees of freedom.” The regression estimators considered are those that maximize the GT quasi-likelihood, as well as one-step versions. Estimators of the parameters of the GT distribution and the effect of such estimates on the asymptotic efficiency of the regression estimates are discussed. We give a minimum-distance interpretation of the choice of GT parameter estimate that minimizes the asymptotic variance of the regression parameters.

We derive asymptotic expansions of the distributions of test statistics for over-identifying restrictions in a system of simultaneous equations under the null and the non-null hypotheses. We investigate the effects of the normality assumption for disturbances on the test statistics based on their asymptotic expansions. We also study the power functions of test statistics based on their asymptotic expansions. After modifying their critical regions to the same significance level, the power function of Basmann's statistic is larger than that of the likelihood ratio test when the variance of disturbances is sufficiently small. However, the difference in powers of the two test statistics disappears as the sample size grows larger.

The coefficients of the moving average (MA) representation of a vector autoregressive (VAR) process are the dynamic multipliers of the system. These quantities are often used to analyze the relationships between the variables involved. Assuming that the actual data generation process is stationary and has a VAR representation of unknown and possibly infinite order, the asymptotic distribution of the MA coefficients is derived. A computationally simple formula for the asymptotic co variance matrix is obtained.

This paper provides L1 and weak laws of large numbers for uniformly integrable L1-mixingales. The L1-mixingale condition is a condition of asymptotic weak temporal dependence that is weaker than most conditions considered in the literature. Processes covered by the laws of large numbers include martingale difference, ø(·), ρ(·), and α(·) mixing, autoregressive moving average, infinite-order moving average, near epoch dependent, L1-near epoch dependent, and mixingale sequences and triangular arrays. The random variables need not possess more than one finite moment and the L1-mixingale numbers need not decay to zero at any particular rate. The proof of the results is remarkably simple and completely self-contained.

Asymptotic expansions of the distributions of likelihood ratio and Lagrange multiplier test statistics for nonlinear restrictions on regression coefficients are derived under the null hypothesis. Nonlinear restrictions include, as a special case, the identifiability restrictions in the simultaneous equations models. Our analyses of simultaneous equations deal not only with single equations but also subsystems and complete systems. The asymptotic expansions we derive are informative about deviations of the real size of test from the nominal asymptotic size.

This paper develops a multivariate regression theory for integrated processes which simplifies and extends much earlier work. Our framework allows for both stochastic and certain deterministic regressors, vector autoregressions, and regressors with drift. The main focus of the paper is statistical inference. The presence of nuisance parameters in the asymptotic distributions of regression F tests is explored and new transformations are introduced to deal with these dependencies. Some specializations of our theory are considered in detail. In models with strictly exogenous regressors, we demonstrate the validity of conventional asymptotic theory for appropriately constructed Wald tests. These tests provide a simple and convenient basis for specification robust inferences in this context. Single equation regression tests are also studied in detail. Here it is shown that the asymptotic distribution of the Wald test is a mixture of the chi square of conventional regression theory and the standard unit-root theory. The new result accommodates both extremes and intermediate cases.

Estimation of a single structural equation when there exists structural change is considered. Equality of structural variances in different samples is shown to affect the identification condition and asymptotic efficiency of best asymptotically normal estimators when the reduced-form covariance matrices differ by the structural change. The limited information maximum likelihood estimator is presented with its asymptotic property and compared with an alternative estimator.

Second-order approximations to the variances of OLS and GLS estimators are compared when the covariance matrix is locally nonscalar. Using a result of Rothenberg, the comparison of OLS and GLS variances is shown to be asymptotically equivalent to a weighted mean square error comparison of the error covariance parameter estimators used in those two procedures. When there is only one covariance parameter, this comparison depends only on the noncentrality parameter of a classical hypothesis test for a scalar covariance matrix.

This paper is concerned with the general problem of testing one form of covariance structure against another in a normal linear regression. It is shown that all the point-optimal tests recently proposed by King and his associates can be interpreted as special cases of a Cox test for non-nested hypotheses. This provides a synthesis of a whole range of point-optimal tests as well as demonstrating that King and his associates have exposed a class of Cox tests which have an exact distribution.