Determination of Linear Relations between Systematic Parts of Variables with Errors of Observation the Variances of Which Are Unknown

Given a sufficient number of instrumental variables significantly correlated with the variables, consistent estimates of the coefficients of the linear relations can be determined (if they exist), without knowledge of the disturbance variances. The estimates are discussed from the viewpoint of probability convergence. In the case of two investigational and one instrumental variable, all three variables distributed on the normal surface, the distribution of the estimate of the coefficient is found exactly for all sample sizes, on certain hypotheses. The distribution function is remarkably simple. The applicability of the theorem to economic time series is discussed by (a) comparing the probability inferences derived from this Model A with those for the simplest stationary time-series model, termed Model B, and (b) by comparing the large-sample variances on several models. It is found that the theory can be used with confidence when the series are not too short and the error variances not too large. The theory is applied to a particular time series, showing that the accuracy of the estimate of the coefficient depends on the correlation between the instrumental variable and the two investigational variables. The theory to which reference is made in Sections II, III, and IV, relating to the two-investigational-variable case, is extended to many variables and tests are given, applicable when samples are not small, for determining the significance of coefficient estimates.