Two-stage least-squares estimation:

(a) Suppose that the column x1 in the model matrix X also appears in the matrix Z of instrumental variables in 2SLS estimation. Explain why bx1 in the first-stage regression simply reproduces x1; that is, bx1 ¼ x1.

(b) *Show that the two formulas for the 2SLS estimator (Equations 9.32 and 9.33 on page 235) are equivalent by demonstrating that Xb0 X ¼ Xb0 Xb.

(c) *Show that when the number of instrumental variables in Z is the same as the number of columns in the model matrix X (i.e., k þ 1), the 2SLS estimator (Equation 9.32 on page 235) is equivalent to the direct IV estimator (Equation 9.28 on page 233). (Hint:

It is probably simplest to demonstrate this result using the tools of the next chapter, on the vector geometry of linear models.)