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798 Chapter 18 The Theory of Multiple Regression

The result in Equation (18.12) follows from Equations (18.15) and (18.79), the consist-
ency of X′X > n, the fourth least squares assumption (which ensures that (X′X)-1 exists),
and Slutsky’s theorem.

Appendix

18.4 Derivations of Exact Distributions of OLS Test

Statistics with Normal Errors

This appendix presents the proofs of the distributions under the null hypothesis of the
homoskedasticity-only t-statistic in Equation (18.35) and the homoskedasticity-only F-statistic
in Equation (18.37), assuming that all six assumptions in Key Concept 18.1 hold.

Proof of Equation (18.35)

If (i) Z has a standard normal distribution, (ii) W has a xm2 distribution, and (iii) Z and W
are independently distributed, then the random variable Z > 2W > m has the t-distribution
with m degrees of freedom (Appendix 17.1). To put t in this form, notice that
n bn = (snu2 >su2) Bn͉X. Then rewrite Equation (18.34) as

t = (bnj - bj,0)> 2( Bn 0X)jj, (18.80)
2W>(n - k - 1)

where W = (n – k – 1)(su2n > s2u), and let Z = (bnj - bj,0) > 2( Bn 0X)jj and m = n − k − 1.
With these definitions, t = Z> 2W>m. Thus, to prove the result in Equation (18.35), we
must show (i) through (iii) for these definitions of Z, W, and m.

i. An implication of Equation (18.30) is that, under the null hypothesis, Z =
(bnj - bj,0) > 2( Bn 0X)jj has an exact standard normal distribution, which shows (i).

ii. From Equation (18.31), W is distributed as x2n - k - 1, which shows (ii).
iii. To show (iii), it must be shown that bnj and s2un are independently distributed.

From Equations (18.14) and (18.29), Bn - B = (X′X)-1X′U and s2un = (MXU)′(MXU) >
(n - k - 1). Thus Bn - B and su2n are independent if (X′X)-1X′U and MXU are independ-
ent. Both (X′X)-1X′U and MXU are linear combinations of U, which has an N(0n * 1, s2uIn)
distribution, conditional on X. But because MXX(X′X)-1 = 0n * (k + 1) [Equation (18.26)], it
follows that (X′X)-1X′U and MXU are independently distributed [Equation (18.76)]. Con-

sequently, under all six assumptions in Key Concept 18.1,

Bn and s2un are independently distributed, (18.81)

which shows (iii) and thus proves Equation (18.35).