Nonparametric Identification and Estimation of a Transformation …cowles.yale.edu/sites/default/files/files/lunch/ichimura... · 2020-01-03 · Introduction Identiﬁcation Estimation

IntroductionIdentificationEstimation

ComputationConclusion

Nonparametric Identification and Estimation of aTransformation Model

Hidehiko Ichimura and Sokbae Lee

University of Tokyo and Seoul National University

15 February, 2012

Ichimura and Lee Nonparametric Transformation Model



Outline

1. The Model and Motivation2. Identification3. Consistency4. Asymptotic Distribution5. Computation6. Further Research




A Transformation ModelRelated Literature

The model

I Assume X and Y are observed, where X is a vector ofcovariates and Y is the dependent variable, (Y ,X ) beinggenerated by

Λ0(Y ) = m0(X ) + U,

where Λ0(·) is a strictly increasing unknown function, m0(·) isan unknown function, U is an unobserved random variable thatis independent of X with the cumulative distribution functionF0(·).

I The objective of this project is to identify and estimateunknown functions Λ0, m0, and F0.





Related Literature (1)

I Box and Cox (1964) specify all three functions parametrically;m0(X ) = X ′β, U ∼ N(0, σ2) and for y > 0,

yλ =

{(yλ − 1)/λ if λ 6= 0log y if λ = 0.

I Horowitz (1996), Ye and Duan (1997), and Chen (2002)specify m0(X ) = X ′β.

I Linton, Sperlich and Van Keilegom (2008) specifym0(X ) = G (m1(X1), . . . ,mK (XK )), where G is a knownfunction.






I Jacho-Chavez, Lewbel, and Linton (2010) studies a model:

r(x , z) = H(G (x) + F (z))

where H is a strictly monotonic function and r(x , z) isnonparametrically estimatable model assuming continuousregressors.






I The nonparametric transformation model includes the mixedproportional hazard model as a special case (Ridder, 1990).For the proportional hazard model of duration Y withunobserved heterogeneity v ,

λ(t|x , v) = λ0(t) exp[−(m0(x) + v)],

Let T (t) =´ t0 λ0(s)ds. Then

Λ0(Y ) = logT (Y ) = m0(X ) + V + ε

where ε is independent of (X ,U) and has the CDF1− exp(− exp(t)) (a Gompertz distribution).






I Ekeland, Heckman, and Nesheim (2004) present a hedonicmodel which, under suitable restrictions placed on utilityfunctions and production functions gives rise to anonparametric transformation model.

For example they assume production function F (z , x , ε) withhedonic element z , observed and unobserved control variablesx and ε, cost function C (z).






The first order condition is

Fz(z , x , ε) = C ′(z).

They assume an existence of a monotonic transformation ψ suchthat

ψ(Fz(z , x , ε)) = τ(z) + M(η(x) + ε),

where M is also monotonic. Then

ψ(C ′(z)) = τ(z) + M(η(x) + ε),

so thatM−1(ψ(C ′(z))− τ(z)) = η(x) + ε.






I Guerre, Perrigne, and Vuong (2009) study the identification ofa private value first-price auction model with risk aversebidders and showed that, writing λ(·) = U(·)/U ′(·), denotingthe number of bidders by n, and the density of the equilibriumbid by g(·|n), and bids’s α-quantile by b(α|n) we have

λ−1(

α

(n2 − 1)g(b(α|n2))

)= b(α|n1)− b(α|n2) + λ−1

(α

(n1 − 1)g(b(α|n1))

)





ObjectivesAll previous studies approach the identification via derivatives withrespect to regressors excluding discrete regressors.

In this paper weI construct an identification result which does not depend on

taking derivatives with respect to regressors,I consider identification issues when regressors are discrete, andI based on the identification results develop estimators for

functions Λ0, m0, and F0,I develop computatinal method for the estimator,I establish consistency and 1/

√n-consistency and outline the

distribution theory.I Our study helps isolate the variation intrinsic to the parameter

identification.Ichimura and Lee Nonparametric Transformation Model



Identification with Continuous RegressorsIdentification with Discrete Regressors

Point Identification (1)

I We first provide a point identification result when X includes acontinuous regressor.

I Note that for any constants a, b, and c > 0, we have

c[Λ0(Y ) + b] = c[m0(X ) + b + a] + c(U − a),

I Thus, the model is not identified without location and scalenormalization.

I Our location and scale normalization is achieved by assumingthat F0(0) = 0.5, m0(x0) = 0, and m0(x1) = 1 for some(x0, x1) such that x1 6= x0.






I Since Y = Λ−10 [m0(x) + U],

QY |X (α|x) = Λ−10 [m0(x) + F−1

0 (α)].

I As F−10 (0.5) = 0, m0(x) = Λ0[QY |X (0.5|x)].

I As m0(x0) = 0, F−10 (α) = Λ0[QY |X (α|x0)].

I Thus

Λ0[QY |X (α|x)] = Λ0[QY |X (0.5|x)] + Λ0[QY |X (α|x0)].






I We seek sufficient conditions under which the unique solution(a.s.) to the following equation is Λ0 on a compact set[y1, y2] ⊃ [0, 1]:

Λ[QY |X (α|x)] = Λ[QY |X (0.5|x)] + Λ[QY |X (α|x0)].

I Substituting the above relationships, we have

Λ[Λ−10 (m0(x) + F−1

0 (α))]

= Λ[Λ−10 (m0(x))] + Λ[Λ−1

0 (F−10 (α))].






I Examining this with s = m0(x) and t = F−10 (α) and

T (z) := Λ[Λ−10 (z)], we have

T (s + t) = T (s) + T (t).

I When s and t vary over a common interval [y1, y2], and T iscontinuous, T (z) = Cz .

I Since Λ0(QY |X (α|x)) = m0(x) + F−10 (α), the scale

normalization implies Λ0(QY |X (0.5|x1)) = 1 so thatΛ(QY |X (0.5|x1)) = 1 always via the scale normalization. Thisimplies T (1) = Λ(Λ−1

0 (1)) = 1 so that C = 1. Thus T (z) = z .I Therefore, Λ0 is identified on [y1, y2].






I Note that once Λ0(y) is identified, then m0(x) and F−10 (α) are

identified by m0(x) = Λ0[QY |X (0.5|x)] andF−1

0 (α) = Λ0[QY |X (α|x0)].I Hence, m0(x) is identified for any x satisfying

QY |X (0.5|x) ∈ [y1, y2].

I Likewise, F−10 (α) is identified for any α satisfying

QY |X (α|x0) ∈ [y1, y2].





Identification Theorem

I If the distribution of Y conditional on X is given, (Y ,X ) beinggenerated by the model specified above,

I Λ0 is a strictly increasing and continuous function,I F0(0) = 0.5 and m0(x0) = 0 and m0(x1) = 1 for two distinct

points x0 and x1, andI the marginal distribution of m0(X ) and U are continuous and

their joint support include [0, 1]2.I Then there exists a non-empty set [y1, y2], which is a strict

subset of the common support of QY |X (0.5|x) andQY |X (α|x0) over which Λ0 is identified. m0(x) is identified forany x satisfying QY |X (0.5|x) ∈ [y1, y2]. Likewise, F−1

0 (α) isidentified for any α satisfying QY |X (α|x0) ∈ [y1, y2].





Partial Identification (1)

I We next consider a discrete X taking a finite number ofprobability mass points.

I In this case s takes on a finite number of points which includes0 and 1 from our normalization.

I Note thatT (si + t) = T (si ) + T (t)

andT (sj + t) = T (sj) + T (t)

so that

T (si + t)− T (sj + t) = T (si )− T (sj).





Partial Identification (2)I Differentiating both sides we have

T ′(si + t)− T ′(sj + t) = 0

or writing ∆ij = sj − si we have

T ′(t) = T ′(t + ∆ij).

I Thus T ′(z) is a periodic function with periodicity ∆ij .I Note also that since

T ′(t + ∆ij) = T ′(t + ∆i ′j ′),

we also have

T ′(t) = T ′(t + ∆ij −∆i ′j ′),

which implies that T ′(z) is a periodic function with periodicity∆ij −∆i ′j ′ .





Partial Identification (3)I We consider the case in which there is a ∆ > 0 in all ∆ij and

differences among them and so on so that each of these termscan be written as an integer multiple of ∆.

I Clearly this is not necessarily the case, but it is also clear thatthis is always the case if all si are rational.

I We show that, in this case, we cannot point identify Λ0 butcan provide a bound and the bound becomes tight as ∆becomes small.

I Let ψ to be a non-constant periodicity ∆ function with´ ∆0 ψ(t)dt = 0.

I Then T ′(s) = ψ(s) + A for some constant A so that

T (s) =

ˆ s

0ψ(u)du + A · s + B

for some constants A and B .Ichimura and Lee Nonparametric Transformation Model





I Clearly B = 0 and that T (1) = 1 and the fact the locationnormalization implies

´ 10 ψ(u)du = 0 implies A = 1 so that

T (s) =

ˆ s

0ψ(u)du + s.

Recall that T (s) = Λ(Λ−10 (s)), taking v = Λ−1

0 (s),

Λ(v) = Λ0(v) +

ˆ Λ0(v)

0ψ(u)du.






I Let h(z) =´ z0 ψ(t)dt. Then

h(z + ∆) =

ˆ z+∆

0ψ(t)dt

= h(z) +

ˆ z+∆

zψ(t)dt,

= h(z).

so that h is also a periodicity ∆ function and

Λ(u) = Λ0(u) + h(Λ0(u)).






I Differentiating Λ(u) we observe that

Λ′(u) = Λ′0(u)[1 + h′(Λ0(u))]

so that the minimum slope of h′ is greater than −1.I Thus the supremum value h can take is ∆.I Thus the supremum deviation of Λ(u) from Λ0(u) is |∆|.




Estimation of Λ0Estimation of m0 and F0ConsistencyAsymptotic Normality

Estimating Λ0 (1)I To construct a root-n-consistent estimator of Λ0, without the

loss of generality, we normalize m0 byˆ

wX (x)m0(x)dx = 0

for some weight function wX that has a compact subset on thesupport of X and satisfies

´wX (x)dx = 1. Analogously we

normalize F0(α) byˆ

wα(a)F−10 (a)da = 0

for some weight function wα that has a compact subset on thesupport of α and satisfies

´wα(a)da = 1.





Estimating Λ0 (1)

I Then F−10 (α) =

´wX (x)Λ0[QY |X (α|x)]dx and

m0(x) =´

wα(a)Λ0[QY |X (α|x)]da, so that

Λ0[QY |X (α|x)] =

ˆwα(a)Λ0[QY |X (α|x)]da

+

ˆwX (u)Λ0[QY |X (α|u)]du.

This equation is the basis for our estimator of Λ0.





Estimating Λ0 (2)

I Our estimation procedure for Λ0 consists of two steps.I The first step is nonparametric estimation of QY |X (α|x).I Then the second step is to fit the data by minimizing an

objective function based on the identification result. We usethe least squares criterion.





Estimating Λ0 (3)

I We minimize a least-squares criterion function

Mn(Λ) :=

ˆwX (x)wα(α)

{Λ[Q̂Y |X (α|x)]

−ˆ

wα(a)Λ[Q̂Y |X (a|x)]da −ˆ

wX (u)Λ[Q̂Y |X (α|u)]du}2

dx da

over a set of possible functions for Λ, Ln, whereI Q̂Y |X (α|x) is the first-step estimator,





Estimating m0 and F0

I m0 can be estimated using the normalization´wα(a)F−1

0 (a)da = 0:

m̂(x) :=

ˆwα(a)Λ̂[Q̂Y |X (a|x)]da.

I F0 can be estimated using the normalization´wX (x)m0(x)dx = 0:

F̂−1(α) :=

ˆwX (x)Λ̂[Q̂Y |X (α|x)]dx .





Lemma for Showing Consistency

I Denote the probability limit of the objective function byM(Λ,QY |X (α|x)), the support of (X , α) by S, and for someL > 0, define

L = {Λ; |Λ′| ≤ L,ˆ ˆ

wX (u)wα(a)Λ(QY |X (a|u)dadu = 0ˆ ˆ

wX (u)wα(a)Λ2(QY |X (a|u)dadu = 1}.

I Then there is a universal constant C such that for any Λ ∈ L,

M(Λ,QY |X (α|x)) ≥ C sup(x ,α)∈S

|Λ(QY |X (α|x))− Λ0(QY |X (α|x))|2.





Asymptotic Normality: Outline of the Proof I

I We assume that the first order condition of the optimizationproblem is satisfied with equality so that

Λ̂(Q̂(α|x)) =

ˆwX (u)Λ̂(Q̂(α|u))du +

ˆwα(a)Λ̂(Q̂(a|x))da.

I Substitute t = Q̂(α|x), we obtain,

Λ̂(t) =

ˆwX (u)Λ̂(Q̂(Q̂−1(t|x)|u))du+

ˆwα(a)Λ̂(Q̂(a|x))da.

I Then by location normalization,

Λ̂(t) =

ˆwX (x)

ˆwX (u)Λ̂(Q̂(Q̂−1(t|x)|u))dudx .





Asymptotic Normality: Outline of the Proof II

I Define

T0(Λ)(t) :=

ˆwX (x)

ˆwX (u)Λ(Q0(Q−1

0 (t|x)|u))dudx ,

T̂ (Λ)(t) :=

ˆwX (x)

ˆwX (u)Λ(Q̂(Q̂−1(t|x)|u))dudx .

I Both T0 and T̂ are linear operators and the latter above canbe seen as empirical approximation to T0 using thenonparametrically estimated conditional quantile function.

I Note that Λ0 = T0Λ0, that is (I − T0)Λ0 = 0. Since Λ0 isdifferent from zero, (I − T0) must not be invertible.





Asymptotic Normality: Outline of the Proof IIII However, we write as

Λ̂− Λ0 = T̂ Λ̂− T0Λ0

= T0(Λ̂− Λ0) + (T̂ − T0)Λ0 + (T̂ − T0)(Λ̂− Λ0)

= Tm+10 (Λ̂− Λ0) + [I + T0 + · · ·+ Tm

0 ](T̂ − T0)Λ0

+ [I + T0 + · · ·+ Tm0 ](T̂ − T0)(Λ̂− Λ0),

where the last equality holds for any positive integer m.I Note that in view of the identification result,

Null(I − T0) := {φ : T0φ = φ} has the form

Null(I − T0) = {c0 + c1Λ0 : (c0, c1) ∈ R2}.





Asymptotic Normality: Outline of the Proof IVI In general, ‖T0‖ = 1; however, in view of the identification

result,

sup {‖T0φ‖ : ‖φ‖ ≤ 1 and φ /∈ Null(I − T0)} < 1.

I Hence, if√

n(Λ̂− Λ0) /∈ Null(I − T0) with probabilityapproaching one, then

limm→∞

plimn→∞Tm+10

[√n(Λ̂− Λ0)

]= 0.

I Suppose that we can show that∥∥∥T̂ − T0

∥∥∥ = op(1).





Asymptotic Normality: Outline of the Proof V

I Then

plimn→∞(T̂ − T0)[√

n(Λ̂− Λ0)] = 0.

Let

R̂m := [I + T0 + · · ·+ Tm0 ](T̂ − T0)[

√n(Λ̂− Λ0)].

Now plimn→∞R̂m = 0 for any integer m ≥ 1.





Asymptotic Normality: Outline of the Proof VII Suppose that the following strong approximation holds:

√n(T̂ − T0)Λ0 = G0 + op(1),

where G0 is a centered Gaussian process with almost surecontinuous sample paths. Then G /∈ Null(I − T0) withprobability one, so that

H := limm→∞

[I + T0 + · · ·+ Tm0 ]G0

is well defined and also is a centered Gaussian process. Insummary,

√n(Λ̂− Λ0)→d H.




Computation

I The second stage requires a constrained optimization since Λis a strictly monotone increasing function.

I We build on Ramsay (1998) who proposes a method forestimating an arbitrary twice differentiable strictly monotonefunction.




Ramsay’s method (1)

I To describe Ramsay (1998)’s method, consider a class ofmonotone functions

F = {monotone f : ln(Df ) is differentiableand D{ln(Df )} is Lebesgue square integrable},

where the notation Df refers to the first-order derivative of f .I Thus, each element of F is strictly increasing and its first

derivative is smooth and bounded almost everywhere.




Ramsay’s method (2)

I Using the same notation as in Ramsay (1998), define a partialintegration operator D−1f by

D−1f (t) =

ˆ t

0f (s)ds. (1)

Theorem 1 of Ramsay (1998) states that every function f ∈ Fcan be represented as

f (x) = C0 + C1D−1 {exp(D−1w)}

(x), (2)

where w is a Lebesgue square integrable function and C0 andC1 are arbitrary constants.




Sieve Space (1)

I Assume that Λ0 belongs to F and that w can be written asw(t) =

∑∞k=1 ckφk(t) with some basis functions

{φk : k = 1, 2, . . .}.




Sieve Space (2)I Then we consider a natural sieve space Ln by imposing the

scale normalization C1 = 1 and the location normalizationC0(c):

Ln =

{C0(c) + D−1

[exp

(Kn∑k=1

ckD−1φk

)]},

where

C0(c) = −ˆ

w0(u)

D−1

{exp

(Kn∑k=1

ckD−1φk

)}([Q̂Y |X (0.5|u)

])du

and c = (c1, . . . , cK ) for some Kn that converges to infinity asn→∞.




The Second Step with Sieve Space (1)

I Note that the location normalization ensures thatF̂−1(0.5) = 0.

I Thus, for each n, minimizing the objective function can benow viewed as an unconstrained optimization problem withKn-dimensional c.

I Specifically, our second-stage estimation consists of

minΛ∈Ln

{Mn(Λ) + λ

ˆ y2

y1

w2Kn(t)dt

}, (3)

where λ is a regularization parameter that converges to zeroand wKn(t) =

∑Knk=1 ckφk(t).




The Second Step with Sieve Space (2)

I Then (3) can be solved by some numerical optimizationalgorithm, along with the location normalization imposed on Λ.




We use the following model to conduct a simple Monte Carlosimulation:

Λ0(Y ) = X + U

where X and U are both standard normal random variables and

Λ0(t) = log t




Table 1. The Finite Sample Performance of the Estimator of ⇤0

New Estimator CS HJ KS YDy BIAS SD RMSE RMSE RMSE RMSE RMSE

0.0500.240 0.004 0.173 0.173 0.094 0.124 0.102 0.1040.430 0.013 0.129 0.130 0.069 0.069 0.069 0.0760.620 0.024 0.094 0.097 0.051 0.046 0.048 0.0580.810 0.026 0.075 0.079 0.036 0.022 0.033 0.0404.817 -0.018 0.048 0.051 0.103 0.246 0.129 0.1068.634 -0.062 0.102 0.119 0.134 0.406 0.586 0.16612.451 -0.054 0.149 0.158 0.156 0.475 0.644 0.26016.268 0.000 0.200 0.200 0.181 0.541 1.014 0.37220.086 0.069 0.247 0.256 0.195 0.572 1.332 0.489

24

CS:Songnian Chen’s estimator, HJ: Joel Horowitz’s estimatorKS: Klein-Sherman’s estimator, YD: Ye-Duan’s estimator




Summary

I We obtained new identification results, proposed a sampleanalog estimation method for the nonparametrictransformation model, and obtained some asymptoticdistribution results.

I Things to be done:I Complete the asymptotic analysis for Λ0 andI obtain results for estimation of m0 and F0.I Monte Carlo experiments.I Use the results for empirical illustration.


Documents

Nonparametric Identification and Estimation of a Transformation …cowles.yale.edu/sites/default/files/files/lunch/ichimura... · 2020-01-03 · Introduction Identiﬁcation Estimation