On the existence of involuntary unemployment equilibria

Conclusion We have proved the existence of equilibria implying involuntary unemployment of labour in non-competitive economies with fixed wage rate. We notice that model (II) and its extensions have not been presented as aninterpretation of Keynes' theory. We may define them as Keynesian models only in the sense that they admit involuntary unemployment equilibrium, which concept is basic to Keynes' theory. In any case, we think that these models are a first step in the development of more realistic microeconomic foundations for short-run macroeconomic theory.A first draft of this paper was presented in December 1972 at the Institute of Political Economy, Faculty of Economics and Banking, Siena University.     

z-Transform and identification of linear econometric models with autocorrelated errors

The paper consists of two main parts. In the first part we derive the solution of systems of linear stochastic difference equations by means of thez-transform. In the second part thisz-transform is used to treat the problem of identification of linear econometric systems (the term econometric is used to stress the special aspects of the identification problem dealt with in econometrics). It is shown, that under suitable restrictions observationally equivalent structures are related by unimodular matrices. Using this result, we state (rank-) conditions which ensure, that the unimodular matrices are constant, such that the classical econometric identification theorems can be applied. These conditions are given for stationary errors in the general case as well as in the MA, AR and ARMA case.     

Mögliche Aussagen bei Fragen der statistischen Ursachenforschung

Ohne Zusammenfassung     

On the measurability and consistency of minimum contrast estimates

Summary The concept of minimum contrast (m.c.) estimates used in this paper covers maximum likelihood (m.l.) estimates as a special case. Section 1 contains sufficient conditions for the existence of measurable m.c. estimates and for their consistency. The application of these results to m.l. estimates (section 2) yields the existence of m.l. estimates for families ofp-measures (probability measures) which are compact metric or locally compact with countable base, admitting upper semicontinuous densities, whereas the classical results refer to continuous densities. This generalization is insofar of interest as upper semicontinuous versions of the densities exist whenever the densities areμ-upper semicontinuous (whereasμ-continuity does not, in general, entail the existence of continuous versions). Under appropriate regularity conditions, consistency of asymptotic maximum likelihood estimates is proven for compact (and also locally compact) separable metric families ofp-measures with upper semicontinuous densities and for arbitrary families having uniformly continuous densities with respect to the uniformity of vague convergence. The conditions sufficient for consistency are shown “indispensable” by counterexamples. Section 3 contains auxiliary results. Besides their relevance for sections 1 and 2, some of them may also be of interest in themselves, e.g. Theorem (3.4) on the selection of semicontinuous functions from semicontinuous equivalence classes.     

Über die Approximation von Wahrscheinlichkeitsverteilungen durch Grenzverteilungen

Zusammenfassung Der Fehler, der bei der Approximation einer Wahrscheinlichkeitsverteilung durch eine Grenzverteilung entsteht, wird mit Hilfe von Differential- (bzw. Differenzen-)gleichungen asymptotisch (bezüglich eines wachsenden Parameters) entwickelt. Diese Aussagen stellen Verallgemeinerungen des vonWasow [1956] behandelten Falles auf beliebige stetige oder gitterförmige Grenzverteilungen dar. Zwei Beispiele veranschaulichen die theoretischen Aussagen.

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Summary The error, which is caused by the approximation of a probability distribution by a limit distribution, is asymptotically expanded (to an increasing parameter) by means of differential (or difference) equations. These statements are generalizations ofWasow's [1956] theorem to arbitrary continous or lattice distributions. Two examples are given to show the practical effect of the stated theorems.

     

On estimating the variance of a finite population

Summary Observing that the estimator for a finite population variance as recommended byLiu [1974a, b] can sometimes become negative, we suggest a few non-negative alternative estimators and note some of their properties. UnlikeLiu we follow the conventional Bayesian approach to get another estimator with an optimal property of uniform admissibility.This paper, however, was prepared when the author worked in the Department of Economic Statistics, University of Sydney.