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.