On Certain Methods of Estimating the Linear Structural Relation

Abstract

The first part of this paper considers two methods of estimating the linear structural relation between two variables both of which are subject to "error"; the second part of the paper comments on a recently advanced procedure for constructing the confidence region for the slope of the structural relation. In 1940 Wald [1] initiated a certain procedure for estimating the linear structural relation between two variables both of which are subject to "error." Wald's idea was extended by Nair and Banerjee [2] and later by Bartlett [3]. These procedures require some knowledge about the values of certain non-observable variables. When this knowledge is not available there is a temptation to substitute information derived from observations. One such method was considered by Wald who found sufficient conditions for the consistency of the resulting estimate. The purpose of the first part of the present paper is to find the necessary and sufficient conditions for two procedures with reference to a slightly more general case, namely, when the "errors" in the two observable variables may be correlated. The results obtained indicate that the two procedures, applied in the case of no additional knowledge about the values of the non-observable variables, will lead to consistent estimates of the slope of the structural relation in very exceptional cases only. In 1949 Hemelrijk [4] described a novel procedure for constructing the confidence region of the slope of the linear structural relation in the case when the non-observable variables have unknown fixed values and the observations are made with "error" which has the same probability distribution at each point. The present paper considers this same procedure when there is no information about the fixed non-observable variables and also when these variables are random variables, and shows that the probability that the confidence region covers the true slope is the same as before but that the probability of covering any other slope is now the same as this probability of covering the true slope.