Standard errors of partial regression coefficients

Abstract

Simple hypothesis testing, involving the statistical significance of a single regression coefficient, is conducted in the same manner in the multiple regression model as it is in the simple regression model. Indeed, the statistical test for the significance of a single partial regression coefficient is merely an extension of that for the simple regression coefficient. Specifically, the t test value for the significance of a partial regression coefficient is obtained by dividing the coefficient by its standard error as follows: byi.jk t Sb~ where byi.jk is the partial regression coefficient for the regression of the dependent variable y on the independent variable xi, controlling for the independent variables xj and Xk, and Sbi is the standard error of this partial regression coefficient. Although the t tests for simple and partial regression coefficients are identical, the equation for the standard error of a partial regression coefficient is a bit more complicated than the equation for the standard error of a simple regression coefficient. In particular, the equation for the standard error of the partial regression coefficient, byi.jk, is given as follows: