The variance of a rank estimator of transformation models

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The variance of a rank estimator of transformation models

Koen Jochmans

To cite this version:

Koen Jochmans. The variance of a rank estimator of transformation models. 2011. �hal-00973007�

THE VARIANCE OF A RANK ESTIMATOR OF TRANSFORMATION MODELS

Koen Jochmans ^∗

CORE

First version: June 2010; This version: June 7, 2011

This note shows that the asymptotic variance of Chen’s [Econometrica, 70, 4 (2002), 1683–1697] two-step estimator of the link function in a linear transformation model depends on the first-step estimator of the index coefficients.

JEL classification: ^{C14, C41.}

Keywords: influence function, transformation model, two-step estimator.

The estimator

For an unspecified strictly-increasing function Λ ₀ ( · ) : R 7→ R , the linear transformation model takes the form

Λ ₀ (Y ) = Xβ + ε,

where ε is a latent disturbance, distributed independently of the covariates X, and β is an unknown coefficient vector of conformable dimension. Set Λ ₀ (y ₀ ) = 0 for a chosen baseline value y ₀ and assume that β = (1, α ^′ ₀ ) ^′ .

Let W _i ≡ (Y _i , X _i ) (i = 1, . . . , n) be observations on W ≡ (Y, X ), drawn at random from a distribution P that is supported on a set W . Let b _n ≡ (1, α ^′ _n ) ^′ be a first-step estimator of β. For fixed y, Chen (2002) proposed estimating Λ ₀ ≡ Λ ₀ (y) by Λ _n , which maximizes

1 n(n − 1)

X

i6=j

h(W _i , W _j , y, Λ, b _n ), h(W ₁ , W ₂ , y, Λ, b) ≡ [1 { Y ₁ ≥ y } − 1 { Y ₂ ≥ y ₀ } ] 1 { (X ₁ − X ₂ )b ≥ Λ } ,

with respect to Λ over a compact subset of the real line containing Λ ₀ . The influence function

Impose Assumptions 1–5 of Chen (2002). Strenghten Assumption 6 by demanding α _n to be asymptotically linear, that is, √

n(α n − α ₀ ) = n ^−1/2 P

i ψ(W i ) + o _p (1) for a function ψ( · ) that has zero mean and finite variance under P . Fix y throughout and leave the dependence of quantities on it implicit. Let τ (w, Λ, b) ≡ Eh(W, w, y, Λ, b) + Eh(w, W, y, Λ, b), V ≡ ¹ ₂ E ∇ ΛΛ τ (W, Λ 0 , β), and Ω ≡ E ∇ Λα

τ (W, Λ 0 , β). Arguments along the line of those in Sherman (1993) yield √

n(Λ _n − Λ ₀ ) = n ^−1/2 P

i I (W _i ) + o _p (1) → N ^L (0, EI (W ) ² ) for

I(w) ≡ − V ⁻¹ h

∇ Λ τ (w, Λ ₀ , β) + 1

2 Ω ψ(w) i

= J (w) − 1

2 V ⁻¹ Ω ψ(w).

Chen (2002, pp. 1687 and Theorem 1) argues that Ω = 0, so that I(w) = J(w) and the asymptotic variance of √

nΛ n is unaffected by the estimation noise in b _n .

∗

Address: Center for Operations Research and Econometrics, U.C. Louvain, Voie du Roman Pays 34, B-1348 Louvain- la-Neuve, Belgium. Tel. +32 10 474 329; E-mail: koen.jochmans@uclouvain.be. I am grateful to Songnian Chen, Geert Dhaene, Frank Kleibergen, and James Stock for comments and suggestions and to the Department of Economics of Brown University for their hospitality.

[1]

Write f and p _z for the densities of ε and Z ≡ Xβ, respectively. Arrange the components of X = (X ₁ , X) e ∈ R × X so that the distribution of scalar X ₁ given X e = x e satisfies the absolute-continuity requirement of Chen (2002, Assumption 2) for all x e in X . The calculations summarized below show that, with X (z) ≡ E[ X e | Z = z],

− V = Z _+∞

−∞

f ( − z) p _z (z + Λ ₀ ) p _z (z) dz, (1)

1 2 Ω =

Z +∞

−∞

f ( − z) p _z (z + Λ 0 ) p _z (z)

X (z + Λ 0 ) − X (z)

dz. (2)

Equation (2) reveals that Ω will generally be non-zero. Like V , it can be estimated by the cross-derivative of a smoothed version of the symmetrized objective function evaluated at (Λ _n , b _n ). Consistency follows under conditions analogous to those for the estimator of V stated in Chen (2002, pp. 1695–1696).

The conclusions drawn here extend to the case where the observations on Y are subject to random censoring. The appropriate modification to the influence function stated in Chen (2002, Theorem 2) is readily derived.

Calculations

Let τ (w) = τ (w, Λ, b) for fixed values Λ and b = (1, α ^′ ) ^′ . Write τ ₀ (w) for τ (w, Λ ₀ , β), ∇ Λ τ ₀ (w) for

∇ Λ τ (w, Λ ₀ , β), etc. Manipulate the inequalities in τ (w) to see that τ (W ) =

Z

W

(1 { y < y ₀ } − 1 { Y < y } ) 1 { xb ≤ Xb − Λ } dP (w)

− Z

W

(1 { Y < y ₀ } − 1 { y < y } ) 1 { xb < Xb + Λ } dP (w) + c ₀ for c ₀ ≡ R

(1 { Y < y ₀ } − 1 { y < y } ) dP (y), which does not depend on (Λ, b). Let p _z (z |e x) be the density of Z given X e = e x at z and let ∆ _α ( X, e e x) ≡ Z + ( X e − e x)(α − α ₀ ). By iterated expectations,

τ (W ) = − Z

X

Z _∆

( X,e e x)−Λ

−∞

S _y,y

(Y, z) p _z (z |e x) dz dP ( x) + e Z

X

Z _∆

( X,e e x)+Λ

−∞

S _y

,y (Y, z) p _z (z |e x) dz dP ( x) + e c ₀ , where S _y

,y

(Y, Z) ≡ 1 { Y < y ₁ } − F (Λ 0 (y 2 ) − Z) and F (z) ≡ R z

−∞ f (z) dz.

Use Leibniz’s rule to verify that

∇ Λ τ (W ) = Z

X

S _y,y

(Y, ∆ _α ( X, e x) e − Λ) p _z (∆ _α ( X, e x) e − Λ |e x) dP ( x) e

− Z

X

S _y

,y (Y, ∆ _α ( X, e e x) + Λ) p _z (∆ _α ( X, e e x) + Λ |e x) dP( e x)

(3)

and that ∇ Λ τ ₀ (W ) = S _y,y (Y, Z) p _z (Z − Λ) − S _y

,y

(Y, Z) p _z (Z + Λ). Notice that E ∇ Λ τ ₀ (W ) = 0 because

E[S _y,y (Y, Z) | Z = z] = 0 (4)

for any y.

[2]

Differentiate with respect to Λ under the integral sign in Equation (3), re-arrange, and evaluate at (Λ ₀ , β) to obtain

∇ ΛΛ τ ₀ (W ) = − f(Λ ₀ − Z) p _z (Z − Λ ₀ ) − f ( − Z) p _z (Z + Λ ₀ ) − c ₁

where c ₁ ≡ S _y,y (Y, Z) p ^′ _z (Z − Λ ₀ ) + S _y

,y

(Y, Z) p ^′ _z (Z + Λ ₀ ) and p ^′ _z is the derivative of p _z . Integrate and apply the moment condition in Equation (4) to dispense with c ₁ and to find that

2V = − Z +∞

−∞

f (Λ 0 − z) p _z (z − Λ 0 ) p _z (z) dz − Z +∞

−∞

f ( − z) p _z (z + Λ 0 ) p _z (z) dz.

Equation (1) follows on a change of variable from z to z − Λ ₀ in the first integral.

Follow the same steps to deduce that

∇ Λα

τ ₀ (W ) = f (Λ ₀ − Z ) p _z (Z − Λ ₀ ) [ X e − X (Z − Λ ₀ )] − f( − Z ) p _z (Z + Λ ₀ ) [ X e − X (Z + Λ ₀ )] + c ₂ for c ₂ ≡ S _y,y (Y, Z) p ^′ _z (Z − Λ ₀ )[ X e − X (Z − Λ ₀ )] − S _y

,y

(Y, Z) p ^′ _z (Z + Λ ₀ ) [ X e − X (Z + Λ ₀ )]. Because c ₂ has zero mean,

Ω = Z +∞

−∞

n

f (Λ 0 − z) p _z (z − Λ 0 )

X (z) − X (z − Λ 0 )

− f ( − z) p _z (z + Λ 0 )

X (z) − X (z + Λ 0 ) o

p _z (z) dz.

A change of variable then establishes Equation (2).

References

Chen, S. (2002). Rank estimation of transformation models. Econometrica, 70:1683–1697.

Sherman, R. P. (1993). The limiting distribution of the maximum rank correlation estimator. Econometrica, 61:123–137.

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