The Estimation of Economic Relationships using Instrumental Variables

which the relationships are not exact, so that a set of ideal economic variables is assumed to be generated by a set of dynamic stochastic relationships, as in Koopmans 12, and the actual economic time series are assumed to differ from the ideal economic variables because of random disturbances or measurement errors. The asymptotic error variance matrix for the coefficients of one of the relationships is obtained in the case in which these relationships are estimated using instrumental variables. With this variance matrix we are able to discuss the problem of choice that arises when there are more instrumental variables available than the minimum number required to enable the method to be used. A method of estimation is derived which involves a characteristic equation already considered by Hotelling in defining the canonical correlation 10. This method was previously suggested by Durbin 7. The same estimates would be obtained by the maximum-likelihood limited