Introduction to Econometrics 3rd Edition by James Stock, Mark Watson

Exercise 5

Consider the problem of minimizing the sum of squared residuals subject to the constraint that Rb = r, where R is q × ( k + 1) with rank cj. Let
be the value of b that solves the constrained minimization problem.
a. Show that the Lagrangian for the minimization problem is L(b , ) = ( Y- Xb ) ' ( Y-Xb ) + ' ( Rb - r ), where is a q × 1 vector of Lagrange multipliers.
b. Show that

c. Show that


d. Show that F in Equation (18.36) is equivalent to the homoskeskasticity-only F -statistic in Equation (7.13).