Econometric Estimation of Constrained Demand Functions for Assets

This paper is designed to present a procedure for estimating demand functions for assets subject to portfolio consistency constraints. The basic model consists of a set of equations determining the allocation of a given quantity of wealth among categories that exhaust the total. Consistency requires that the coefficients on total wealth sum to unity across all assets and that the coefficients on other explanatory variables such as rates of return sum to zero. These constraints are imposed on the coefficients by estimating the system of equations simultaneously. This is done with the use of a modified version of the technique introduced by Zellner for estimating "seemingly unrelated regressions". His estimator is obtained by first employing residuals from ordinary least squares regressions to estimate the variance-covariance matrix among the disturbances for each asset and then by applying generalized least squares. Since in the present case the variance-covariance matrix is singular, use is made of the concept of a generalized inverse to obtain the generalized least squares estimator. In view of the considerable computational burden involved in the simultaneous estimation, a method for calculating the coefficients is presented that uses the formulas for the inverse of a partitioned matrix. A considerable saving in computation can be realized by this means if a significant proportion of the explanatory variables appears in the equation for every asset. An additional problem taken up in the paper is the introduction of lagged adjustments. The familiar stock adjustment model cannot be imposed directly because it does not account consistently for the interactions among assets. A model proposed by Brainard and Tobin is discussed and a modified version of it is presented in which a common speed of adjustment is imposed on all assets. This procedure is illustrated by using an example drawn from RDX2, the quarterly econometric model of the Canadian economy developed in the Research Department of the Bank of Canada. Equations are estimated for eight assets including currency and several categories of deposits in chartered banks and in trust and mortgage loan companies.