MUCH APPLIED ECONOMETRIC work is based on the correlation and regression of ratio variables which have the same denominator. The denominator generally deflates the various sets of measurements in order to make them comparable. In certain cases deflation is a way of obtaining efficient estimators. Briggs [1] has studied the effect of errors in the deflating variable on the correlation between ratios. The effect of errors in the deflating variable on a scatter diagram of measurements on ratio variables is to displace each representative point along a ray from the origin passing through the representative point of the error-free measurement. This suggests that with error in the deflating variable the ordinary least squares (OLS) estimators of a bivariate regression are biased toward indicating a relationship of proportionality between the ratio variables. It can be demonstrated that this conjecture is generally valid for multiple linear regression with error in the deflator. In most cases in which deflation is used, it is reasonable to suppose that the deflator is a random variable distributed independently of the ratio variables; in this case the conditional expectation of the undeflated independent variables is proportional to the value of the deflator. Under this assumption it can be established that when there is error in the deflator the estimators of the slopes of the regression are inconsistent unless (i) the ratio regression has zero intercept, or (ii) the mean of each of the independent variables is zero, or (iii) the error in the deflator is systematic (i.e., the error term has zero variance). The estimator of the intercept may be inconsistent even when the estimators of the slopes are consistent; for the intercept estimator to be consistent it is necessary that either (i) the intercept

IN THIS PAPER we examine the concept of irreducibility in the von Neumann model, as defined by Gale [2, Ch. 9]. Since this property corresponds to a condition on the types of production in the model, we term it technological irreducibility. We show that there is a dual concept, which we call economic irreducibility, involving a condition on the price structure. The wellknown von Neumann indecomposability assumption is shown to have a very simple relationship to the property of economic irreducibility. We also show how to generalize certain results from the Perron-Frobenius theory of irreducible non-negative square matrices to the case of an irreducible von Neumann model.

T. A. Yancey, G. G. Judge, M. E. Bock, Wallace's Weak Mean Square Error Criterion for Testing Linear Restrictions in Regression: A Tighter Bound, Econometrica, Vol. 41, No. 6 (Nov., 1973), pp. 1203-1206