Extensions of Hotellings Test and Other Tests for Multiple Regression Models

In simple least squares regression, formal choice of one of two candidate independent variables (IVs) may be done using a test due to Harold Hotelling. The null hypothesis is that each of the two candidate independent variables is equally correlated with the dependent. In this paper, Hotelling's test is extended so that it may also be used to test the equality of both the intercept and the regression coefficients when either independent variable is used in simple least squares. The paper presents extensions of the tests for a multiple regression model for which there are two, possibly overlapping, sets of candidate IVs. Two cases are considered. Case I occurs when there are k IVs in each of the two sets and IV j, j = 1, ..., k has a comparable meaning in each set. Case II arises when the number of IV's is not necessarily the same in each set and some or all of the IVs may be different. The paper presents procedures to test (i) whether the two sets of IVs are equally correlated with the dependent, (ii) for models where corresponding IVs are comparable, whether the regression coefficients are the same and (iii) and whether the estimated intercepts are the same. The paper also presents a test of the equality of fit of the two candidate models, which reduces to Hotelling's test when k = 1. The tests are illustrated using a number of regression models taken from the portfolio performance literature