Asymptotic Normality of M-Estimators

This section sets forth conditions for the asymptotic normality ofM-estimators in addition to the conditions for consistency An estimator в of a parameter в0 є Km is asymptotically normally distributed if an increasing sequence of positive numbers an and a positive semidefinite m x m matrix £ exist such that an(в — в0) ^d Nm [0, £]. Usually, an = – Jn, but there are exceptions to this rule.

Asymptotic normality is fundamental for econometrics. Most of the econo­metric tests rely on it. Moreover, the proof of the asymptotic normality theorem in this section also nicely illustrates the usefulness of the main results in this chapter.

Given that the data are a random sample, we only need a few additional conditions over those of Theorems 6.10 and 6.11:

Theorem 6.1: Let, in addition to the conditions of Theorems 6.10 and 6.11, the following conditions be satisfied:

(a) © is convex.

(b) в0 is an interior point of ©.

(c) For each x є Kk, g(x, в) is twice continuously differentiable on ©.

(d) For each pair 6il, Qi2 of components of в,

E[supee© 92g(X1,в)/(Зв1-1 Эв;2)|] < то.

image436

Proof: I will prove the theorem for the case m = 1 only, leaving the general case as an exercise.

I have already established in Theorem 6.11 that в ^p 60. Because 0O is an interior point of ©, the probability that в is an interior point converges to 1, and consequently the probability that the first-order condition for a maximum

of Q(6) = (1 /n)YTj=ig(Xj, в) in в = в holds converges to 1. Thus,

lim P[Q'{в) = 0] = 1, (6.37)

n^TO

where, as usual, Q'(в) = d Q^)/dв. Next, observe from the mean value theo­rem that there exists a X e [0, 1] such that

VnQ 7(в) = VQ Ш + Q "(в0 + Х(в — в^)^ — в0), (6.38)

where Q"(в) = d2 0)(в)/^в)2. Note that, by the convexity of ©,

P[в0 + Х(в — в0) e ©] = 1, (6.39)

and by the consistency of в,

plim[ в0 + X(0 — в0)] = в0. (6.40)

n^TO

Moreover, it follows from Theorem 6.10 and conditions (c) and (d), with the latter adapted to the univariate case, that

plim sup | Q"(в) — Q"(в)| = 0, (6.41)

n^TO ве©

where Q"(в) is the second derivative of Q^) = E[g(X1, в)]. Then it follows from (6.39)-(6.41) and Theorem 6.12 that

plim Q"(вс + 1(0 — вс)) = Q"(вс) = 0. (6.42)

n^TO

Note that Q"(во) corresponds to the matrix A in condition (e), and thus Q"(во) is positive in the “argmin” case and negative in the “argmax” case. Therefore, it follows from (6.42) and Slutsky’s theorem (Theorem 6.3) that

plim Q"(в0 + 1(0 — в0))—1 = Q//(в0)—1 = A—1. (6.43)

n^TO

Now (6.38) can be rewritten as

– во) = -Q"(во + i(0 – 0o))-1VnQ'(во)

+ Q"(во + к(в – во))-1^'(в)

= – Q"(во + і(в – во)УХ4Пй'(во) + Op(1), (6.44)

where the op(1) term follows from (6.37), (6.43), and Slutsky’s theorem. Because of condition (b), the first-order condition for во applies, that is,

Q'(во) = E[dg(X1, во)/dво] = о. (6.45)

Moreover, condition (f), adapted to the univariate case, now reads as follows:

var[dg(Xь в^/dво] = B e (о, то). (6.46)

Therefore, it follows from (6.45), (6.46), and the central limit theorem (Theorem 6.23) that

n

VnQ ‘(во) = (1/Vn)J2 dg(Xj, во)/dво ^d N [о, B]. (6.47)

j=1

Now it follows from (6.43), (6.47), and Theorem 6.21 that

– Q "(во + і(в – во)Ух^пй ‘(во) ^d N [о, A-1BA-1]; (6.48)

hence, the result of the theorem under review for the case m = 1 follows from (6.44), (6.48), and Theorem 6.21. Q. E.D.

The result of Theorem 6.28 is only useful if we are able to estimate the asymptotic variance matrix A-1BA-1 consistently because then we will be able to design tests of various hypotheses about the parameter vector во.

Theorem 6.29: Let

Подпись: (6.49)Подпись: (6.50)A = 1 ^ d2g(Xj, 0)

n j= d0д0t ’

and

B = 1 ^ / д g(Xj, 0) /dgiXjJ) njy д0t ) у дв

Under the conditions of Theorem 6.28, plimn^TO A = A, and under the ad­ditional condition that E[sup0^^(X1, в)/двT||2] < то, plimn^TOB = B. Consequently, plimn^TO A-1 BB A-1 = A-1 BA-1.

Proof: The theorem follows straightforwardly from the uniform weak law of large numbers and various Slutsky’s theorems – in particular Theorem 6.21.

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