This paper considers a general nonlinear econometric model framework that contains a large class of estimators defined as solutions to optimization problems. For this framework we derive several asymptotically equivalent forms of a test statistic for the local (in a way made precise in the paper) multivariate nonlinear inequality constraints test H: h(β) ≥ 0 versus K: β ∈ RK. We extend these results to consider local hypotheses tests of the form H: h1(β) ≥ 0 and h2(β) = 0 versus K: β ∈ RK. For each test we derive the asymptotic distribution for any size test as a weighted sum of χ2-distributions. We contrast local as opposed to global inequality constraints testing and give conditions on the model and constraints when each is possible. This paper also extends the well-known duality results in testing multivariate equality constraints to the case of nonlinear multivariate inequality constraints and combinations of nonlinear inequality and equality constraints.

The finite sample distributions of estimators and test statistics in ARMA time series models are generally unknown. For typical sample sizes, the approximations provided by asymptotic distributions are often unsatisfactory. Hence simulation or numerical integration methods are used to investigate the distributions. In practice only a limited part of the parameter space is examined using these methods. Thus any results which allow us to infer properties from one portion of the parameter space to another or to establish symmetry are most welcome.

For the ARMA model estimated with no intercept term, we show that the least-squares and maximum likelihood estimators have mirror-image invariant or symmetric distributions. The F, t, likelihood ratio, Wald, and Lagrange multiplier statistics are also shown to have distributions with certain mirror-image invariant or symmetry properties. The analysis is extended to misspecified models as well as to ARMA spectral densities.

These properties would have been helpful in either simplifying or extending much earlier work in this area.

Effect of nonnormality on the asymptotic property of three estimators of a single structural equation with structural change is examined. The three estimators are the limited information maximum likelihood estimator, derived under normality and equality of structural variances in different samples, given by Hodoshima, a two-stage least squares type estimator due to Barten and Bronsard, and a minimum distance estimator presented here. Normality is relaxed but the equality assumption of structural variances is retained. Under nonnormality the limited information maximum likelihood estimator is consistent but may not be efficient relative to the Barten and Bronsard's estimator. A sufficient condition is given under which the limited information maximum likelihood estimator dominates the Barten and Bronsard's estimator in terms of the asymptotic covariance matrix. The minimum distance estimator dominates other estimators.

A general framework for asymptotic tests is proposed. The framework contains as particular cases tests based on various estimation techniques: maximum likelihood methods, pseudo-maximum likelihood (PML) methods and quasi-generalized PML methods, m-estimation methods, moments or generalized moments method, and asymptotic least squares. Moreover the null hypothesis has a general mixed form, including the usual implicit and explicit form.

This paper compares the asymptotic local power properties of some tests of a null model against a single nonnested alternative and against multiple nonnested alternatives, denoted hereafter as paired and joint tests, respectively. It is demonstrated that the ranking of tests on the basis of asymptotic local powers depends on the choice of local hypothesis. When a local null hypothesis is employed, it is not possible to rank the Wald and Cox-type paired or joint tests. However, when the local hypothesis is specified with reference to one of the alternative models under consideration, a ranking of different test procedures becomes possible. Under a local alternative hypothesis, it is shown that the paired Wald test will never have greater asymptotic local power than a paired Cox-type test.

This paper continues the theoretical investigation of Park and Phillips. We develop an asymptotic theory of regression for multivariate linear models that accommodates integrated processes of different orders, nonzero means, drifts, time trends, and cointegrated regressors. The framework of analysis is general but has a common architecture that helps to simplify and codify what would otherwise be a myriad of isolated results. A good deal of earlier research by the authors and by others comes within the new framework. Special models of some importance are considered in detail, such as VAR systems with multiple lags and cointegrated variates.

Econometrics as practiced by Arthur (Art) Goldberger demonstrates extraordinary sensitivity to issues of measurement and model specification, and unusual care and caution in interpretation of results, as well as a thorough and comprehensive mastery of econometric theory. His landmark 1964 book, Econometric Theory, set a new standard of rigor in econometrics, and at the same time treated the important problems posed by limited and qualitative dependent variables years before any other text. Art Goldberger's work ranges from early contributions to macro modeling through demand analysis, multivariate modeling with latent variables, and models for sample selectivity, to his highly regarded work on important social issues of heritability of IQ, effectiveness of public versus private schools, and measurement of salary discrimination. Goldberger's influential work, especially on modeling latent or unobservable variables, is widely known and applied in sociology and psychology as well as in economics. Art has been at Wisconsin for many years, and this association is an important reason for Wisconsin's continuing reputation as a leading center for quantitative social sciences.

The quality and influence of Art Goldberger's work has earned him many professional honors. He is a Fellow of the Econometric Society, the American Statistical Association, the American Academy of Arts and Sciences, and the American Association for the Advancement of Science, and has twice been a Guggenheim Fellow. He is a Distinguished Fellow of the American Economic Association and a member of the National Academy of Sciences. Art gave the Woytinsky Lecture at the University of Michigan in 1985.

This interview took place on May 5 in Art Goldberger's office overlooking Lake Mendota. Art's remarks cover a wide range of topics, and I hope they are of interest to social scientists generally as well as to econometricians.