Question

Perform joint hypothesis tests on multiple coefficients

Answer 1

Consider the following estimated model :

TestScore = A − B2(X2) + B3(X3).

  (a) (b) (c)   

A, B1 and B2 are estimated coefficients and a,b,c are the Standard Errors.

Now, to reject the hypothesis that the coefficient on X2 and the coefficient on X3 are zero, we have to resort to joint hypothesis tests.

A joint hypothesis imposes restrictions on multiple regression coefficients. This is different from conducting individual t-tests where a restriction is imposed on a single coefficient.

The homoskedasticity-only F-Statistic is given by

F=(SSR_restricted−SSR_unrestricted)/q/[SSR_unrestricted/(n−k−1)]

with SSR_restricted being the sum of squared residuals from the restricted regression, i.e., the regression where we impose the restriction. SSR_unrestricted is the sum of squared residuals from the full model, q is the number of restrictions under the null and k is the number of regressors in the unrestricted regression.