Asymptotic Normality
In this subsection we shall show that under certain conditions a consistent root of Eq. (4.1.9) is asymptotically normal. The precise meaning of this statement will be made clear in Theorem 4.1.3.
Theorem 4.1.3. Make the following assumptions in addition to the assumptions of Theorem 4.1.2:
(A) сР(2т/двдд’ exists and is continuous in an open, convex neighborhood of0o.
(B) Т‘і&От/Мдв’ЇРт converges to a finite nonsingular matrix A(0O) =
lim £Г_1 (0*6 7/^0% probability for any sequence 0* such that
plim s 0
(C) Г_1/2(0<2г/30)во N[0= B(0o)L where B(0O) = lim ЕТ~1(д()т/дв)во X
Let {0r} be a sequence obtained^by choosing one element from 6r defined in Theorem 4.12 such that plim 0r= 0O. (We call вт a consistent root.) Then
Vf (0r – 0o) – m A(0O)“1 B(0o)A(0o)“1 ].
Proof. By a Taylor expansion we have
dQT 
_ 3Qt 
&QT 

дв 
де 
во двдв’ 

where Q* lies between §T and в0 } Noting that the lefthand side of (4.1.10) is 0 by the definition of §T, we obtain
(4.1.11)
where+denotes the MoorePenrose generalized inverse (see note 6 of Chapter 2). Because plim в* = 60, assumption В implies
(4.1.12)
Finally, the conclusion of the theorem follows from assumption C and Eqs.
(4.1.11) and (4.1.12) by repeated applications of Theorem 3.2.7.
As we noted earlier, QT is frequently the sum of independent random variables (or, at least, of random variables with a limited degree of dependence). Therefore it is not unreasonable to assume the conclusions of a law of large numbers and a central limit theorem in assumptions В and C, respectively. However, as we also noted earlier, the following more general normalization may be necessary in certain cases: In assumption B, change Т~1&()т/двдЄ’ to Н(Т)&1()т/двдв’ЩТ), where Н(Г) is a diagonal matrix such that limr_„ H(T) = 0; in assumption C, change T~lfldQT/dd to ЩТ)д()т/дв and in the conclusion of the theorem, state the limit distribution in terms ofH(7Tl(0r – 0O).
Because assumption В is often not easily verifiable, we shall give two alternative assumptions, each of which implies assumption B. Let gT(6) — g(у, в) be a function of a Tvector of random variables у = (y,, > • • • > УтУ and a
continuous function of a АГvector of parameters в in 0, an open subset of the Euclidean ATspace, almost surely. We assume thatg(y, 0)isa random variable (that is, g& a measurable function of y). We seek conditions that ensure plim №тФт) ~ gAOo)] = 0 whenever plim 6T=60 Note that Theorem 3.2.6 does not apply here because in that theorem g( •) is a fixed function not varying with T.
Theorem 4.1.4. Assume that dgT/d0 exists for в Є 0, an open convex set, and that for any e > 0 there exists Mt such that
Р{№т/дв*<Ме)Ш 1 – e
for all T, for all в Є в, and for all z, where 6‘ is the ith element of the vector в. Then plim gT(0T) = plim gr(0o) if 0O = plim QT is in 0.
Proof. The proof of Theorem 4.1.4 follows from the Taylor expansion
where в* lies between QT and ft,
Theorem 4.1.5. Suppose gr(6) converges in probability to a nonstochastic function ^(0) uniformly in в in an open neighborhood N(60) of 60. Then plim gгФт) = g(e0) if plim §r = Qq and gifi) is continuous at Q0.
A
Proof Because the convergence of gT(fi) is uniform in в and because вт Є N(60) for sufficiently large T with a probability as close to one as desired, we can show that for any e > 0 and S > 0, there exists Tx such that for T> T{
р\8тФт)~8Фт)^<^ (4Л.13)
Because g is continuous at 60 by our assumption, gфт) converges to g(60) in probability by Theorem 3.2.6. Therefore, for any € > 0 and d > 0, there exists T2 such that for T> T2
i,[g(07.)g(0o)s]<. (4.1.14)
Therefore, from the inequalities (4.1.13) and (4.1.14) we have for T> max [Ti, T2]
Р[8тФт) ~ 8(6o) ^ €] ё 1 — A (4.1.15)
The inequality (4.1.13) requires that uniform convergence be defined by either (i) or (ii) of Definition 4.1.1. Definition 4. l. l(iii) is not sufficient, as shown in the following example attributed to A. Ronald Gallant. Let фт{в) be a continuous function with support on [— T~l, Г1] and фт(0) = 1. Define gr(m, в) = фт((о — TO) if 0 ё со, в S 1, and gT{(o, ff) = 0 otherwise. Assume that the probability measure over 0 S to S 1 is Lebesgue measure and 0r= (o/T. Then
inf P[gT(co, Q) < e] S 1 – J; > 1,
0Я6Л1 J
meaning that gT(co, в) converges to 0 semiuniformly in в. But gT(a>, вт) = 1
for all T. Note that in this example
P[ sup gT{o, в) < e] = 0. osesi
Hence, gT(a), в) does not converge uniformly in 0 in the sense of definition 4. l. l(ii).
The following theorem is useful because it gives the combined conditions for the consistency and asymptotic normality of a local extremum estimator. In this way we can do away with the condition that Q(Q) attains a strict local maximum at 0O.
Theorem 4.1.6. In addition to assumptions AC of Theorem 4.1.3, assume
(A) T~lQT(ff) converges to a nonstochastic function Q(6) in probability uniformly in 0 in an open neighborhood of 0O.
(B) A(0O) as defined in assumption В of Theorem 4.1.3 is a negative definite matrix.
(C) plim T~ ‘PQr/dOdd’ exists and is continuous in a neighborhood of 0O. Then the conclusions of Theorems 4.1.2 and 4.1.3 follow.
Proof. By a Taylor expansion we have in an open neighborhood of 0O
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