Introduction to the Mathematical and Statistical Foundations of Econometrics
Dependent Central Limit Theorems
7.1.4. Introduction
As is true of the conditions for asymptotic normality of M-estimators in the
1.1. d. case (see Chapter 6), the crucial condition for asymptotic normality of the NLLS estimator (7.25) is that
1 n
V (dft(§0)/9§gT) ^Щ0, B], (7.32)
V” t=1
where
B = E [V2 (9f1(§o)/9§oT) (9f (§g)/9§g)] . (7.33)
It follows from (7.21) and (7.26) that
TO
ft (§o) = (во - Yg) £ во -1 Vt - j, (7.34)
j=1
which is measurable ■t-1 = о(Vt-1, Vt-2, Vt-3,...), and thus dft (§o)/9§oT
TO TO
E (во + (во - Yo)(j - 1)) в0-2 Vt-j - £ во-1 Vt-j
j=1 j=1
Therefore, it follows from the law of iterated expectations (see Chapter 3) that
 |
 |
B = a2E [(Э/НОД/Э^) (9/i(0b)/900)]
and
P (E[Vt(df (во)/дв'^):.-1] = о = 1. (7.36)
The result (7.36) makes Vt(df (0о)/30с[) a bivariate martingale difference process, and for an arbitrary nonrandom f є K2,f = о, the process Ut = Vt f T(9f (во)/дв^) is then a univariate martingale difference process:
Definition 7.4: Let Ut be a time series process defined on a common probability space {^, P}, and let. t be a sequence of sub-a-algebras of.. If for
each t,
(a) Ut is measurable. t,
(b) .-1 c.
(c) E[Ut] < oo, and
(d) P(E[Ut^t-1] = о) = 1,
then {Ut, .t} is called a martingale difference process.
If condition (d) is replaced by P(E[Ut.t-1] = Ut-1) = 1, then {Ut, .t} is called a martingale. In that case AUt = Ut - Ut-1 = Ut - E[Ut.t-1] satisfies P(E[AUt.t-1] = о) = 1. This is the reason for calling the process in Definition 7.4 a martingale difference process.
Thus, what we need for proving (7.32) is a martingale difference central limit theorem.
