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

According to Wold, information on some of the variables at time t cannot be used for prediction of the values of the remaining variables at t, in a simultaneous equation system. This, however, is not the case with his causal model. This paper considers the stochastic processes underlying the two models, uses the theory of canonical correlation to discuss Wold's criticism, and suggests the type of additional information necessary to remove these objections. It further shows how both these models are complementary to each other. IN A SERIES of papers, Wold [9, 10, 11, 12] has recently proposed a new type of econometric model, which he calls the implicit causal chain model. The main incentive for this model was an attempt to combine the advantages of Tinbergen's causal model with those of the simultaneous equation systems initiated by Haavelmo [3, 4]. This latter model, which has been studied in detail by the Cowles Commission, is also known as the system. Wold has raised some objections, mainly from the point of view of prediction, against the simultaneous equation system; and in order to remedy these, he relaxed some of the restrictions on the correlational properties of the residuals in the simultaneous equation system. In this paper, these two models are discussed from an angle which offers a possibility for reconciling them. It is shown how the merits of both systems can be utilized by effecting some synthesis of them with the help of the stochastic process underlying both models. Furthermore, by using canonical analysis, it is shown how both these models are complementary. It must, of course, be added that the prediction problem has been considered purely on the basis of the stochastic model assumed, and the possible applicability of different representations of such models, for example, in economic contexts, under wider conditions is not discussed. Wold has objected to interdependent models on grounds other than prediction, especially from the point of view of the interpretation of structural parameters as elasticities, etc. The present paper, however, will not deal with this aspect of the problem. Instead, it deals primarily with the stochastic processes underlying economic models, and the demand and supply model, considered in the next section, is only for illustration.