A Cautionary Note on the Use of Linear Regression for Hypothesis Testing

This note draws researchers’ attention to the use of linear regression for the purpose of conducting a hypothesis testing. Even when multiple explanatory variables are included in a regression equation to preclude the hazard of a simple regression of omitting other factors, multicollinearity unfortunately is inherent in multiple regression simply because the included explanatory variables can share some common parameter domain; that is, they co-vary. Here we shall show that however one transforms the variance-covariance matrix of the least-squares estimation to reduce the estimation errors, the procedure amounts to affine transformations of the explanatory variables so that despite the transformation they remain to co-vary, rendering the coefficient as a partial derivative invalid. The root of this problem originates from the fact that one receives the explanatory variables’ values by observation rather than predetermining their values and then collecting the corresponding dependent variable’ values. This situation becomes especially disconcerting when a transformed explanatory variable has its estimated coefficient enjoying an exceptional degree of confidence but of no mathematical status as a partial derivative, misleading engineering or medical prescriptions and public policies.