Estimation of Large Econometric Models by Pricipal Component and Instrumental Variable Methods

Abstract

EMPIRICAL research in economics has seen the development in recent years of large simultaneous equation econometric models -large both in terms of detail and degree of disaggregation but also in their demands upon a limited of data. In the main these models have been models of macro-economic activity estimated from annual or quarterly data in the postwar period. Statistical methods for estimating simultaneous equation models were first developed by the researchers at Cowles Commission [8]. In recent years, the sheer size of such empirical models has brought a new problem to the fore, as the estimation methods previously developed cannot be used without modification. Most of those estimators -those of the kclass and three-stage-least-squares involve a first of regression or estimation using the predetermined variables of the model as regressors.1 But frequently in large models the is smaller than the number of predetermined variables, so that a meaningful first-stage regression is not possible.2 This is a sample problem of a different sort instead of needing more observations in order that the distribution of the estimates will be satisfactorily approximated by their asymptotic distributions, the is small relative to the size of the large model, to the extent that the standard simultaneous equation estimators either do not exist or are identical to ordinary least squares.3 A variety of solutions have been proposed to cope with the large-model problem, two of which are examined in detail in this paper. Each can be viewed as a modification of twostage-least-squares: 1) 2SPC (Two Stage Principal Components), originally proposed by Kloek and Mennes [11], in which a limited number of principal components of the predetermined variables are used in the first stage. 2) SOIV (Structurally Ordered Instrumental Variables), proposed by Fisher [5, 6], in which a limited number of predetermined variables are selected for the first stage by detailed use of the structure of the model. A complete assessment of the properties of these estimators in a large econometric model requires the knowledge of their small-sample distributions. These are in general unknown, although some progress has recently been made for small models by Amemiya [1], Basmann [2], Kadane [9], Mariano [12], Sawa [15], and Takeuchi [17]. Some information might be obtained by Monte Carlo techniques, except that the computational cost of systematically exploring the parameter space of a large model would be prohibitive. Still, a feasible project would be to employ a miniature model with a very few equations, but having more predetermined variables than observations. It is not clear, however, that the distributions would * This work was supported in part by National Science Foundation Grant GS-2635 and by the Brookings Institution. The initial research was undertaken during the tenure of fellowships from the Danforth Foundation and the National Science Foundation. Computations were done at the Massachusetts Institute of Technology and Stanford University computation centers. I am happy to acknowledge the numerous helpful suggestions of T. Amemiya, T. W. Anderson, P. J. Dhrymes, E. Kuh, F. M. Fisher and a referee. Much of the data was made available by G. Fromm. 1 The limited-information-maximum-likelihood method requires extraction of a characteristic root from a matrix of moments of the predetermined variables; the method can be interpreted as a first-stage regression of a synthetic endogenous variable on the predetermined variables. 2 Full information maximum likelihood estimators fail to exist when the number of parameters to be estimated in the model exceeds the size, another problem which occurs in large models. Because of computational complexity, FIML has not been a feasible estimator for models of even moderate size. 'In other cases the problem may occur in a less acute form -there may be more observations than predetermined variables, but the excess may be small, and in some sense better estimates may be obtained by using fewer variables in the first stage.