Econometric Analysis of Cross Section and Panel Data - Free

The Single-Equation Linear Model and OLS Estimation 55
Because EðuÞ ¼0, s 2 is also equal to VarðuÞ. Assumption OLS.3 is the weakest form
of the homoskedasticity assumption. If we write out the K K matrices in Assumption
OLS.3 element by element, we see that Assumption OLS.3 is equivalent to
assuming that the squared error, u 2 , is uncorrelated with each x j , xj 2 , and all cross
products of the form x j x k . By the law of iterated expectations, su‰cient for Assumption
OLS.3 is Eðu 2 j xÞ ¼s 2 , which is the same as Varðu j xÞ ¼s 2 when
Eðu j xÞ ¼0. The constant conditional variance assumption for u given x is the easiest
to interpret, but it is stronger than needed.
theorem 4.2 (Asymptotic Normality of OLS): Under Assumptions OLS.1–OLS.3,
pffiffiffiffi
N ð ^b bÞ @ a Normalð0; s 2 A 1 Þ ð4:9Þ
Proof: From equation (4.8) and definition of B, it follows from Lemma 3.7 and
Corollary 3.2 that
pffiffiffiffi
N ð ^b bÞ @ a Normalð0; A 1 BA 1 Þ
Under Assumption OLS.3, B ¼ s 2 A, which proves the result.
Practically speaking, equation (4.9) allows us to treat ^b as approximately normal
with mean b and variance s 2 ½Eðx 0 xÞŠ
1 =N. The usual estimator of s 2 , ^s 2 1 SSR=
ðN KÞ, where SSR ¼ P N
i¼1 ^u i
2 is the OLS sum of squared residuals, is easily shown
to be consistent. (Using N or N K in the denominator does not a¤ect consistency.)
When we also replace Eðx 0 xÞ with the sample average N 1 P N
i¼1 x0 i x i ¼ðX 0 X=NÞ, we
get
Av^arð ^bÞ ¼^s 2 ðX 0 XÞ 1
ð4:10Þ
The right-hand side of equation (4.10) should be familiar: it is the usual OLS variance
matrix estimator under the classical linear model assumptions. The bottom line of
Theorem 4.2 is that, under Assumptions OLS.1–OLS.3, the usual OLS standard
errors, t statistics, and F statistics are asymptotically valid. Showing that the F statistic
is approximately valid is done by deriving the Wald test for linear restrictions of
the form Rb ¼ r (see Chapter 3). Then the F statistic is simply a degrees-of-freedomadjusted
Wald statistic, which is where the F distribution (as opposed to the chisquare
distribution) arises.
4.2.3 Heteroskedasticity-Robust Inference
If Assumption OLS.1 fails, we are in potentially serious trouble, as OLS is not even
consistent. In the next chapter we discuss the important method of instrumental
variables that can be used to obtain consistent estimators of b when Assumption