Consistency of Least Absolute Deviations Estimator

Consider a classical regression model

у = ХД, + u, (4.6.17)

where X is a TX К matrix ofbounded constants such that limr_«, Г_1Х’Х is a finite positive-definite matrix and u is a Г-vector of i. i.d. random variables with continuous density function /(•) such that /о/(A) dk = ^ and f(x) > 0 for all x in a neighborhood of 0. It is assumed that the parameter space В is compact. We also assume that the empirical distribution function of the rows of X, {x,’}, converges to a distribution function. The LAD estimator fi is defined to be a value of 0 that minimizes

Подпись: (4.6.18)■Sr=i>,-x,7H-£|t4

t-i сі

This is a generalization of (4.6.10). Like the median, the LAD estimator may not be unique.

We shall now prove the consistency of the LAD estimator using Theorem

4.1.1. From (4.6.18) we have

ST=^h(u,x’8), (4.6.19)

сі

where 8=fi—fi0 and h{za) is defined as

If a so,

h{za) = a

if

о

VII

N

— Oi — 2z

if

0 < z < a

= —a

if

z s a.

If a < 0,

h(zot) = a

if

Zitt

= —a + 2z

if

a < z < 0

= —a

if

z s 0.

Then A(z|x,'<5) is a continuous function of 8 uniformly in t and is uniformly bounded in t and 8 by our assumptions. Therefore h(u,x’,8) — Eh{u,x’t8) satisfies the conditions for g,(y, в) in Theorem 4.2.2. Moreover, limr_x T~l J.]LlEh(u,x’t8) can be shown to exist by Theorem 4.2.3. Therefore

Q ^ plim T~lST (4.6.21)

= 2 lim ~ £ f° A/(A) dX

– 2 lim ^ 2 [ [ /М ^ • x/^l + lim ^ 2 x’^> r-«° і,_i L Jx^f J ■* /-і

where the convergence of each term is uniform in 5 by Theorem 4.2.3.

Thus it only remains to show that Q attains a global minimum at S = 0. Differentiating (4.6.21) with respect to 6 yields

Ц = — 2 lim 2 j: [J^/U) dX • x, J + lim ^ 2 x„ (4.6.22)

image359 Подпись: (4.6.23)

which is equal to 0 at 8 = 0 because Jo /(A) dX = by our assumption. More­over, because

is positive definite at 8 = 0 because of our assumptions, Q attains a local minimum at 8 = 0. Next we shall show that this local minimum is indeed the global minimum by showing that dQ/dS Ф 0 if 8 Ф 0. Suppose dQ/dS = 0 at 8i Ф 0. Then, evaluating (4.6.22) at 8X and premultiplying it by 8[, we obtain

lim T % [l ~ m dx] x‘Sl “ °- (4-6-24)

image361 image362 Подпись: № Подпись: (4.6.25)

Tо show (4.6.24) is a contradiction, let a,, a2, and Mbe positive real numbers such that |х$!І < M for all t and /(A) § a, whenever |A| ё a2. Such numbers exist because of our assumptions. Then we have

Подпись: 8{X'X81.s aia2

~ M

Therefore (4.6.24) is a contradiction because of our assumption that lim Г_1Х’Х is positive definite. This completes the proof of the consistency of the LAD estimator by means of Theorem 4.1.1.

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