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 positivedefinite matrix and u is a Гvector 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
■Sr=i>,x,7H£t4
ti 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(zx,'<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)
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 “ ° (4624)
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
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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