Journal Article A Simple Method for Estimating Demand Systems under Separable Utility Assumptions Get access R. P. Byron R. P. Byron Australian National University Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 37, Issue 2, April 1970, Pages 261–274, https://doi.org/10.2307/2296418 Published: 01 April 1970 Article history Received: 01 December 1968 Accepted: 01 September 1969 Published: 01 April 1970

[The chief difficulty in applying Aitken estimators to large linear systems stems from the dimensionality of the inverse. Here the conjugate gradient algorithm is applied to the problem, leading to substantial savings in storage and some savings in time. Given that the information matrix is not computed, an inference procedure is developed which involves two easily computed statistics which straddle the conventionally estimated standard errors. The paper is intended to open the way for the application of more efficient estimation procedures to large econometric systems.]

In the first section of this paper the overidentifying restrictions on a system of linear simultaneous equations are expressed in terms of restrictions on the reduced form parameters. These restrictions provide the basis of a test of the structure using only the unrestricted reduced form parameter estimates. Under Ho the test proposed is asymptotically equivalent to a likelihood ratio test. The test may be applied as a single equation or complete system procedure and it may be presented as either a x2 or an F statistic. The case is also made here for system overidentification tests rather than single equation procedures, the arguments being drawn from the statistical literature on hypothesis testing by induction. The computational advantages of the present proposals are substantial when compared to FIML based likelihood -ratio tests and Monte Carlo experiments confirm that a system version of the test performs well in large samples. The system version of the test behaves like the FIML likelihood ratio test in large sample situations both under Ho and H1. However, the Monte Carlo studies indicate that both the single equation and system versions of the test perform poorly in small samples. THE AIM OF THIS paper is to investigate the possibility of deciding on the specification of a simultaneous equation model prior to the estimation of the structure. A well established test procedure is suggested which uses OLS estimates of the reduced form parameters; it enables the null hypothesis, that the model specified is not significantly different from the model which generated the sample, to be tested. Because of the one-to-one correspondence between the overidentified structure and the restricted reduced form, it is possible to make inferences about the structure from the observed compatibility of the reduced form restrictions with the sample information. In addition, the reduced form restrictions resulting from a particular equation may be isolated and tested separately, if desired. The principle underlying the test would appear to be due to Wald [17J; namely, that if the null hypothesis is correct and the structure postulated as the maintained hypothesis was responsible for the generation of the observed sample, then the unrestricted reduced form parameter estimates will tend, if the sample size is large enough, to satisfy the reduced form restrictions advanced under the maintained hypothesis. A number of problems relating to identification of linear simultaneous equations make life a little difficult and are discussed subsequently. Now, take the linear structure

[The parameters of a system of demand equations are estimated subject to the prior information of classical demand theory. The equations are estimated as a system using a variant of generalized least squares, the parametric restrictions being imposed by Lagrange multipliers. Tests of significance are given, both for individual restrictions and for the restrictions applied collectively. The method is applied to Barten's sixteen commodity consumer expenditure data for Holland. The work was done independently of R. H. Court's [6] similar treatment; however, there are significant differences in the method which warrant further discussion and the application is itself of some interest.]