## The General Linear Model: The Basics

7.1 Invariance of the fitted values and residuals to non-singular transformations of the independent variables.

The regression model in (7.1) can be written as y = XCC-1" + u where Cisa non-singular matrix. LetX* = XC, theny = X*"* + u where "* = C-1".

a. PX* = X* (X*0X*)-1 X*0 = XC [C0X0XC]-1 C0X0 = XCC-1 (X0X)-1 c0-1 C0X0 = PX.

Hence, the regression of y on X* yields

y = X*" *ls = PX* y = PXy = X" ols which is the same fitted values as those from the regression of y on X. Since the dependent variable y is the same, the residuals from both regressions will be the same.

b. Multiplying each X by a constant is equivalent to post-multiplying the matrix X by a diagonal matrix C with a typical k-th element ck. Each Xk will be multiplied by the constant ck for k = 1,2,.., K...

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