Nonparametric and of a Transformation Model Hidehiko Ichimura and Sokbae Lee University of Tokyo and Seoul National University 15 February, 2012
Outline 1. The Model and Motivation 2. 3. Consistency 4. Asymptotic Distribution 5. 6. Further Research
A Transformation Model Related Literature The model Assume X and Y are observed, where X is a vector of covariates and Y is the dependent variable, (Y, X ) being generated by Λ 0 (Y ) = m 0 (X ) + U, where Λ 0 ( ) is a strictly increasing unknown function, m 0 ( ) is an unknown function, U is an unobserved random variable that is independent of X with the cumulative distribution function F 0 ( ). The objective of this project is to identify and estimate unknown functions Λ 0, m 0, and F 0.
A Transformation Model Related Literature Related Literature (1) Box and Cox (1964) specify all three functions parametrically; m 0 (X ) = X β, U N(0, σ 2 ) and for y > 0, { y λ (y λ 1)/λ if λ 0 = log y if λ = 0. Horowitz (1996), Ye and Duan (1997), and Chen (2002) specify m 0 (X ) = X β. Linton, Sperlich and Van Keilegom (2008) specify m 0 (X ) = G(m 1 (X 1 ),..., m K (X K )), where G is a known function.
A Transformation Model Related Literature Related Literature (2) 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) is nonparametrically estimatable model assuming continuous regressors.
A Transformation Model Related Literature Related Literature (3) The nonparametric transformation model includes the mixed proportional hazard model as a special case (Ridder, 1990). For the proportional hazard model of duration Y with unobserved heterogeneity v, λ(t x, v) = λ 0 (t) exp[ (m 0 (x) + v)], Let T (t) = t 0 λ 0(s)ds. Then Λ 0 (Y ) = log T (Y ) = m 0 (X ) + V + ɛ where ɛ is independent of (X, U) and has the CDF 1 exp( exp(t)) (a Gompertz distribution).
A Transformation Model Related Literature Related Literature (4) Ekeland, Heckman, and Nesheim (2004) present a hedonic model which, under suitable restrictions placed on utility functions and production functions gives rise to a nonparametric transformation model. For example they assume production function F (z, x, ɛ) with hedonic element z, observed and unobserved control variables x and ɛ, cost function C(z).
A Transformation Model Related Literature Related Literature (5) The first order condition is F z (z, x, ɛ) = C (z). They assume an existence of a monotonic transformation ψ such that ψ(f z (z, x, ɛ)) = τ(z) + M(η(x) + ɛ), where M is also monotonic. Then ψ(c (z)) = τ(z) + M(η(x) + ɛ), so that M 1 (ψ(c (z)) τ(z)) = η(x) + ɛ.
A Transformation Model Related Literature Related Literature (5) Guerre, Perrigne, and Vuong (2009) study the identification of a private value first-price auction model with risk averse bidders and showed that, writing λ( ) = U( )/U ( ), denoting the number of bidders by n, and the density of the equilibrium bid by g( n), and bids s α-quantile by b(α n) we have ( ) λ 1 α (n 2 1)g(b(α n 2 )) ( ) = b(α n 1 ) b(α n 2 ) + λ 1 α (n 1 1)g(b(α n 1 ))
A Transformation Model Related Literature Objectives All previous studies approach the identification via derivatives with respect to regressors excluding discrete regressors. In this paper we construct an identification result which does not depend on taking derivatives with respect to regressors, consider identification issues when regressors are discrete, and based on the identification results develop estimators for functions Λ 0, m 0, and F 0, develop computatinal method for the estimator, establish consistency and 1/ n-consistency and outline the distribution theory. Our study helps isolate the variation intrinsic to the parameter identification.
with Continuous Regressors with Discrete Regressors Point (1) We first provide a point identification result when X includes a continuous regressor. Note that for any constants a, b, and c > 0, we have c[λ 0 (Y ) + b] = c[m 0 (X ) + b + a] + c(u a), Thus, the model is not identified without location and scale normalization. Our location and scale normalization is achieved by assuming that F 0 (0) = 0.5, m 0 (x 0 ) = 0, and m 0 (x 1 ) = 1 for some (x 0, x 1 ) such that x 1 x 0.
with Continuous Regressors with Discrete Regressors Point (2) Since Y = Λ 1 0 [m 0(x) + U], Q Y X (α x) = Λ 1 0 [m 0(x) + F 1 0 (α)]. As F 1 0 (0.5) = 0, m 0 (x) = Λ 0 [Q Y X (0.5 x)]. As m 0 (x 0 ) = 0, F 1 0 (α) = Λ 0 [Q Y X (α x 0 )]. Thus Λ 0 [Q Y X (α x)] = Λ 0 [Q Y X (0.5 x)] + Λ 0 [Q Y X (α x 0 )].
with Continuous Regressors with Discrete Regressors Point (3) We seek sufficient conditions under which the unique solution (a.s.) to the following equation is Λ 0 on a compact set [y 1, y 2 ] [0, 1]: Λ[Q Y X (α x)] = Λ[Q Y X (0.5 x)] + Λ[Q Y X (α x 0 )]. Substituting the above relationships, we have Λ[Λ 1 0 (m 0(x) + F0 1 (α))] = Λ[Λ 1 0 (m 0(x))] + Λ[Λ 1 0 (F 1 0 (α))].
with Continuous Regressors with Discrete Regressors Point (4) Examining this with s = m 0 (x) and t = F0 1 (α) and T (z) := Λ[Λ 1 0 (z)], we have T (s + t) = T (s) + T (t). When s and t vary over a common interval [y 1, y 2 ], and T is continuous, T (z) = Cz. Since Λ 0 (Q Y X (α x)) = m 0 (x) + F0 1 (α), the scale normalization implies Λ 0 (Q Y X (0.5 x 1 )) = 1 so that Λ(Q Y X (0.5 x 1 )) = 1 always via the scale normalization. This implies T (1) = Λ(Λ 1 0 (1)) = 1 so that C = 1. Thus T (z) = z. Therefore, Λ 0 is identified on [y 1, y 2 ].
with Continuous Regressors with Discrete Regressors Point (5) Note that once Λ 0 (y) is identified, then m 0 (x) and F 1 0 (α) are identified by m 0 (x) = Λ 0 [Q Y X (0.5 x)] and F 1 0 (α) = Λ 0 [Q Y X (α x 0 )]. Hence, m 0 (x) is identified for any x satisfying Q Y X (0.5 x) [y 1, y 2 ]. Likewise, F 1 0 (α) is identified for any α satisfying Q Y X (α x 0 ) [y 1, y 2 ].
with Continuous Regressors with Discrete Regressors Theorem If the distribution of Y conditional on X is given, (Y, X ) being generated by the model specified above, Λ 0 is a strictly increasing and continuous function, F 0 (0) = 0.5 and m 0 (x 0 ) = 0 and m 0 (x 1 ) = 1 for two distinct points x 0 and x 1, and the marginal distribution of m 0 (X ) and U are continuous and their joint support include [0, 1] 2. Then there exists a non-empty set [y 1, y 2 ], which is a strict subset of the common support of Q Y X (0.5 x) and Q Y X (α x 0 ) over which Λ 0 is identified. m 0 (x) is identified for any x satisfying Q Y X (0.5 x) [y 1, y 2 ]. Likewise, F0 1 (α) is identified for any α satisfying Q Y X (α x 0 ) [y 1, y 2 ].
with Continuous Regressors with Discrete Regressors Partial (1) We next consider a discrete X taking a finite number of probability mass points. In this case s takes on a finite number of points which includes 0 and 1 from our normalization. Note that T (s i + t) = T (s i ) + T (t) and T (s j + t) = T (s j ) + T (t) so that T (s i + t) T (s j + t) = T (s i ) T (s j ).
with Continuous Regressors with Discrete Regressors Partial (2) Differentiating both sides we have T (s i + t) T (s j + t) = 0 or writing ij = s j s i we have T (t) = T (t + ij ). Thus T (z) is a periodic function with periodicity ij. Note also that since we also have T (t + ij ) = T (t + i j ), T (t) = T (t + ij i j ), which implies that T (z) is a periodic function with periodicity ij i j.
with Continuous Regressors with Discrete Regressors Partial (3) 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 terms can be written as an integer multiple of. Clearly this is not necessarily the case, but it is also clear that this is always the case if all s i are rational. We show that, in this case, we cannot point identify Λ 0 but can provide a bound and the bound becomes tight as becomes small. Let ψ to be a non-constant periodicity function with 0 ψ(t)dt = 0. Then T (s) = ψ(s) + A for some constant A so that T (s) = ˆ s for some constants A and B. 0 ψ(u)du + A s + B
with Continuous Regressors with Discrete Regressors Partial (4) Clearly B = 0 and that T (1) = 1 and the fact the location normalization implies 1 0 ψ(u)du = 0 implies A = 1 so that T (s) = ˆ s 0 ψ(u)du + s. Recall that T (s) = Λ(Λ 1 0 (s)), taking v = Λ 1 0 (s), Λ(v) = Λ 0 (v) + ˆ Λ0 (v) 0 ψ(u)du.
with Continuous Regressors with Discrete Regressors Partial (5) Let h(z) = z 0 ψ(t)dt. Then h(z + ) = ˆ z+ 0 = h(z) + = h(z). ψ(t)dt ˆ z+ so that h is also a periodicity function and Λ(u) = Λ 0 (u) + h(λ 0 (u)). z ψ(t)dt,
with Continuous Regressors with Discrete Regressors Partial (6) Differentiating Λ(u) we observe that Λ (u) = Λ 0(u)[1 + h (Λ 0 (u))] so that the minimum slope of h is greater than 1. Thus the supremum value h can take is. Thus the supremum deviation of Λ(u) from Λ 0 (u) is.
of Λ 0 of m 0 and F 0 Consistency Asymptotic Normality Estimating Λ 0 (1) To construct a root-n-consistent estimator of Λ 0, without the loss of generality, we normalize m 0 by ˆ w X (x)m 0 (x)dx = 0 for some weight function w X that has a compact subset on the support of X and satisfies w X (x)dx = 1. Analogously we normalize F 0 (α) by ˆ w α (a)f 1 0 (a)da = 0 for some weight function w α that has a compact subset on the support of α and satisfies w α (a)da = 1.
of Λ 0 of m 0 and F 0 Consistency Asymptotic Normality Estimating Λ 0 (1) Then F 1 0 (α) = w X (x)λ 0 [Q Y X (α x)]dx and m 0 (x) = w α (a)λ 0 [Q Y X (α x)]da, so that ˆ Λ 0 [Q Y X (α x)] = w α (a)λ 0 [Q Y X (α x)]da ˆ + w X (u)λ 0 [Q Y X (α u)]du. This equation is the basis for our estimator of Λ 0.
of Λ 0 of m 0 and F 0 Consistency Asymptotic Normality Estimating Λ 0 (2) Our estimation procedure for Λ 0 consists of two steps. The first step is nonparametric estimation of Q Y X (α x). Then the second step is to fit the data by minimizing an objective function based on the identification result. We use the least squares criterion.
of Λ 0 of m 0 and F 0 Consistency Asymptotic Normality Estimating Λ 0 (3) We minimize a least-squares criterion function ˆ { M n (Λ) := w X (x)w α (α) Λ[ ˆQ Y X (α x)] ˆ ˆ 2dx w α (a)λ[ ˆQ Y X (a x)]da w X (u)λ[ ˆQ Y X (α u)]du} da over a set of possible functions for Λ, L n, where ˆQ Y X (α x) is the first-step estimator,
of Λ 0 of m 0 and F 0 Consistency Asymptotic Normality Estimating m 0 and F 0 m 0 can be estimated using the normalization wα (a)f0 1 (a)da = 0: ˆ ˆm(x) := w α (a)ˆλ[ ˆQ Y X (a x)]da. F 0 can be estimated using the normalization wx (x)m 0 (x)dx = 0: ˆ ˆF 1 (α) := w X (x)ˆλ[ ˆQ Y X (α x)]dx.
of Λ 0 of m 0 and F 0 Consistency Asymptotic Normality Lemma for Showing Consistency Denote the probability limit of the objective function by M(Λ, Q Y X (α x)), the support of (X, α) by S, and for some L > 0, define ˆ ˆ L = {Λ; Λ L, w X (u)w α (a)λ(q Y X (a u)dadu = 0 ˆ ˆ w X (u)w α (a)λ 2 (Q Y X (a u)dadu = 1}. Then there is a universal constant C such that for any Λ L, M(Λ, Q Y X (α x)) C sup Λ(Q Y X (α x)) Λ 0 (Q Y X (α x)) 2. (x,α) S
of Λ 0 of m 0 and F 0 Consistency Asymptotic Normality Asymptotic Normality: Outline of the Proof I We assume that the first order condition of the optimization problem is satisfied with equality so that ˆ ˆ ˆΛ( ˆQ(α x)) = w X (u)ˆλ( ˆQ(α u))du + w α (a)ˆλ( ˆQ(a x))da. Substitute t = ˆQ(α x), we obtain, ˆ ˆΛ(t) = ˆ w X (u)ˆλ( ˆQ( ˆQ 1 (t x) u))du+ w α (a)ˆλ( ˆQ(a x))da. Then by location normalization, ˆ ˆ ˆΛ(t) = w X (x) w X (u)ˆλ( ˆQ( ˆQ 1 (t x) u))dudx.
of Λ 0 of m 0 and F 0 Consistency Asymptotic Normality Asymptotic Normality: Outline of the Proof II Define ˆ T 0 (Λ)(t) := ˆ ˆT (Λ)(t) := ˆ w X (x) ˆ w X (x) w X (u)λ(q 0 (Q 1 0 (t x) u))dudx, w X (u)λ( ˆQ( ˆQ 1 (t x) u))dudx. Both T 0 and ˆT are linear operators and the latter above can be seen as empirical approximation to T 0 using the nonparametrically estimated conditional quantile function. Note that Λ 0 = T 0 Λ 0, that is (I T 0 )Λ 0 = 0. Since Λ 0 is different from zero, (I T 0 ) must not be invertible.
of Λ 0 of m 0 and F 0 Consistency Asymptotic Normality Asymptotic Normality: Outline of the Proof III However, we write as ˆΛ Λ 0 = ˆT ˆΛ T 0 Λ 0 = T 0 (ˆΛ Λ 0 ) + ( ˆT T 0 )Λ 0 + ( ˆT T 0 )(ˆΛ Λ 0 ) = T m+1 0 (ˆΛ Λ 0 ) + [I + T 0 + + T m 0 ]( ˆT T 0 )Λ 0 + [I + T 0 + + T m 0 ]( ˆT T 0 )(ˆΛ Λ 0 ), where the last equality holds for any positive integer m. Note that in view of the identification result, Null(I T 0 ) := {φ : T 0 φ = φ} has the form Null(I T 0 ) = {c 0 + c 1 Λ 0 : (c 0, c 1 ) R 2 }.
of Λ 0 of m 0 and F 0 Consistency Asymptotic Normality Asymptotic Normality: Outline of the Proof IV In general, T 0 = 1; however, in view of the identification result, sup { T 0 φ : φ 1 and φ / Null(I T 0 )} < 1. Hence, if n(ˆλ Λ 0 ) / Null(I T 0 ) with probability approaching one, then [ lim plim m n T0 n(ˆλ m+1 Λ 0 )] = 0. Suppose that we can show that ˆT T 0 = op (1).
of Λ 0 of m 0 and F 0 Consistency Asymptotic Normality Asymptotic Normality: Outline of the Proof V Then plim n ( ˆT T 0 )[ n(ˆλ Λ 0 )] = 0. Let ˆR m := [I + T 0 + + T0 m ]( ˆT T 0 )[ n(ˆλ Λ 0 )]. Now plim n ˆR m = 0 for any integer m 1.
of Λ 0 of m 0 and F 0 Consistency Asymptotic Normality Asymptotic Normality: Outline of the Proof VI Suppose that the following strong approximation holds: n( ˆT T 0 )Λ 0 = G 0 + o p (1), where G 0 is a centered Gaussian process with almost sure continuous sample paths. Then G / Null(I T 0 ) with probability one, so that H := lim m [I + T 0 + + T m 0 ]G 0 is well defined and also is a centered Gaussian process. In summary, n(ˆλ Λ0 ) d H.
The second stage requires a constrained optimization since Λ is a strictly monotone increasing function. We build on Ramsay (1998) who proposes a method for estimating an arbitrary twice differentiable strictly monotone function.
Ramsay s method (1) To describe Ramsay (1998) s method, consider a class of monotone functions F = {monotone f : ln(df ) is differentiable and D{ln(Df )} is Lebesgue square integrable}, where the notation Df refers to the first-order derivative of f. Thus, each element of F is strictly increasing and its first derivative is smooth and bounded almost everywhere.
Ramsay s method (2) Using the same notation as in Ramsay (1998), define a partial integration operator D 1 f by D 1 f (t) = ˆ t 0 f (s)ds. (1) Theorem 1 of Ramsay (1998) states that every function f F can be represented as f (x) = C 0 + C 1 D 1 { exp(d 1 w) } (x), (2) where w is a Lebesgue square integrable function and C 0 and C 1 are arbitrary constants.
Sieve Space (1) Assume that Λ 0 belongs to F and that w can be written as w(t) = k=1 c kφ k (t) with some basis functions {φ k : k = 1, 2,...}.
Sieve Space (2) Then we consider a natural sieve space L n by imposing the scale normalization C 1 = 1 and the location normalization C 0 (c): [ ( Kn )]} L n = {C 0 (c) + D 1 exp c k D 1 φ k, where ˆ C 0 (c) = D 1 {exp w 0 (u) ( Kn k=1 k=1 )} ([ ]) c k D 1 φ k ˆQ Y X (0.5 u) du and c = (c 1,..., c K ) for some K n that converges to infinity as n.
The Second Step with Sieve Space (1) Note that the location normalization ensures that ˆF 1 (0.5) = 0. Thus, for each n, minimizing the objective function can be now viewed as an unconstrained optimization problem with K n -dimensional c. Specifically, our second-stage estimation consists of { ˆ y2 } min M n (Λ) + λ wk 2 Λ L n (t)dt, (3) n y 1 where λ is a regularization parameter that converges to zero and w Kn (t) = K n k=1 c kφ k (t).
The Second Step with Sieve Space (2) Then (3) can be solved by some numerical optimization algorithm, along with the location normalization imposed on Λ.
We use the following model to conduct a simple Monte Carlo simulation: Λ 0 (Y ) = X + U where X and U are both standard normal random variables and Λ 0 (t) = log t
Table 1. The Finite Sample Performance of the Estimator of 0 New Estimator CS HJ KS YD y BIAS SD RMSE RMSE RMSE RMSE RMSE 0.050 0.240 0.004 0.173 0.173 0.094 0.124 0.102 0.104 0.430 0.013 0.129 0.130 0.069 0.069 0.069 0.076 0.620 0.024 0.094 0.097 0.051 0.046 0.048 0.058 0.810 0.026 0.075 0.079 0.036 0.022 0.033 0.040 4.817-0.018 0.048 0.051 0.103 0.246 0.129 0.106 8.634-0.062 0.102 0.119 0.134 0.406 0.586 0.166 12.451-0.054 0.149 0.158 0.156 0.475 0.644 0.260 16.268 0.000 0.200 0.200 0.181 0.541 1.014 0.372 20.086 0.069 0.247 0.256 0.195 0.572 1.332 0.489 CS:Songnian Chen s estimator, HJ: Joel Horowitz s estimator KS: Klein-Sherman s estimator, YD: Ye-Duan s estimator
Summary We obtained new identification results, proposed a sample analog estimation method for the nonparametric transformation model, and obtained some asymptotic distribution results. Things to be done: Complete the asymptotic analysis for Λ 0 and obtain results for estimation of m 0 and F 0. Monte Carlo experiments. Use the results for empirical illustration.