Journal Article The Use of Index Numbers in Demand Analysis Get access A. R. Bergstrom A. R. Bergstrom Auckland, N.Z. Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 23, Issue 1, 1955, Pages 17–26, https://doi.org/10.2307/2296147 Published: 01 April 1955

This paper is concerned with the efficient estimation of structural parameters in closed linear systems of higher order stochastic differential equations when the data are in discrete form and the model generally includes both stock and flow variables. A general existence and uniqueness theorem for the solution of such a system is proved and used in the rigorous derivation of exact discrete models satisfied by the various types of data. It is shown how these models can be used in the computation of various asymptotically efficient estimates obtained by the maximization of the Gaussian likelihood or approximations to it.

In this article we derive the exact finite sample frequency functions of the least squares and maximum likelihood estimators of the marginal propensity to consume, assuming the basic stochastic Keynesian model. The properties of these functions are considered in more detail for particular values of the parameters and sizes of sample. It is concluded that, for samples of 10 or more observations, generated by the model considered (with realistic values of the parameters), the maximum likelihood estimator of the marginal propensity to consume is the better general purpose estimator of this parameter. ALTHOUGH THERE have been several major contributions to the large sample theory of estimators of the parameters of simultaneous equation systems,' there is, as yet, virtually no small sample theory for these estimators. With the exception of Nagar's approximations for the bias and moment matrix of k-class estimators,2 evidence concerning the small sample behaviour of various estimators is confined to two types. The first type includes studies in which the parameters of simultaneous equation models have been estimated, by various methods, from real data, so that the resulting estimates can be compared, by either referring to economic theory and other a priori evidence, or testing predictions.3 The value of such studies is limited, however, by unknown errors in both the data and the specification of the models. The second type of evidence is provided by a number of valuable Monte Carlo studies.4 But these too have certain disadvantages as compared with a mathematical study. One is that, in order to investigate the effects of variations in the sample size or the structural parameters, it would be necessary to analyse many different sets of synthetic samples. Another is that the measures obtained from Monte Carlo studies are, themselves, subject to sampling errors. Moreover, in studies based upon no more than 100 synthetic samples, these errors are not negligible.5