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n -prediction of generalized heteroscedastic transformation regression models
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√
n-prediction of generalized heteroscedastic transformation regression models✩ ∗
Songnian Chen a , , Hanghui Zhang b,c a
Department of Economics, The Hong Kong University of Science and Technology, Clearwater Bay, Kowloon, Hong Kong School of Economics, The Shanghai University of Finance and Economics, Shanghai 200433, China c Key Laboratory of Mathematical Economics (SUFE), Ministry of Education, Shanghai 200433, China b
article
info
a b s t r a c t Chen (2010) and Khan (2001) consider quantile regression estimation subject to general transformation that permits general heteroscedasticity, but the resulting conditional quantile predictors converge at rates slower than the parametric rate. In this paper, we consider the estimation of a general transformation model subject to a multiplicative form of heteroscedasticity. Our estimators for the finite dimensional parameters, the transformation function and the resulting conditional quantile predictor all converge at the parametric rate. Monte Carlo simulation experiments show that our estimators and conditional quantile predictor perform well in finite samples. © 2019 Elsevier B.V. All rights reserved.
Article history: Received 22 May 2018 Received in revised form 18 September 2019 Accepted 20 September 2019 Available online xxxx JEL classification: C14 C24 C51 Keywords: Generalized transformation regression model Heteroscedasticity √ n-prediction
1. Introduction Generalized transformation regression models have received a great deal of attention in econometrics and statistics literature. An important class of such models maintains the independence assumption between the unobservable error term and regressors, and for such models, Han (1987), Sherman (1993) and Cavanagh and Sherman (1998) developed rank estimators for the finite dimensional parametric index parameters and Horowitz (1996), Gørgens and Horowitz (1999) and Chen √ (2002) developed estimation procedures for the unknown transformation function. All these estimators converge at n-consistent parametric rate (n is the sample size), and consequently the resulting conditional quantile predictors also converge at the parametric rate. The independence assumption is highly restrictive in allowing the way the regressors to affect the outcome variable. A more flexible alternative is the transformation regression model under a conditional quantile restriction. For the transformation model under a conditional quantile restriction, Khan (2001) developed a two-step semiparametric rank estimator and Chen (2010) proposed an integrated maximum score estimator, and both ✩ We would like to thank the associate editor, two anonymous referees, conference participants at Shanghai Econometrics Workshop at the Shanghai University of Fiance and Economics and 2016 China meeting of the Econometric Society at SWUFE for their helpful comments. We also thank Jichun Si for his excellent RA on this project. Chen’s research was supported by the GRF grant 640412 from the University Grants Committee of the government of Hong Kong SAR. Zhang’s research is supported by National Natural Science Foundation of China (NSFC Project No. 71601105 and No. 71833004). Support was also provided by the 2018 Program for Changjiang Scholars and Innovative Research Team in SUFE. ∗ Corresponding author. E-mail address: [email protected] (S. Chen). https://doi.org/10.1016/j.jeconom.2019.09.003 0304-4076/© 2019 Elsevier B.V. All rights reserved.
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
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√
estimators for the finite dimension parameters are n-consistent and asymptotically normal; subsequent conditional quantile predictors, however, converge at rates slower than the parametric rate, leading to general efficiency loss in √ making prediction. The n consistent estimators of Horowitz (1996) and Chen (2002) for homoscedastic models will be inconsistent in the presence of√ heteroscedasticity. Therefore, given the widespread heteroscedastity in empirical research, it is highly desirable to have a n consistent estimator for√the transformation function for heteroscedastic transformation models, which provides the basis for the construction of n consistent conditional quantile function. This paper fills this important gap. In this paper, we consider the estimation of a general transformation model subject to a multiplicative form of heteroscedasticity with a median location normalization. By combining both rank and quantile estimation techniques, our estimators for the finite dimensional parameter and the unknown transformation function both converge at the parametric rate, and consequently, the conditional quantile predictors also converge at the parametric rate. Zhou et al. (2008) also considered the general transformation model with a multiplicative form of heteroscedasticity, but with a √ mean location normalization. They also proposed a n consistent estimator for the transformation function. However, compared with our approach, their method suffers from several serious drawbacks. First, they require a mutual exclusion restriction in a double-index setting, similar to Ichimura and Lee (1991); as a result, their estimation approach is not adaptive to the homoscedasticity setting in that their method requires the actual presence of heteroscedasticity for model identification and estimation. In other words, their method breaks down when there is no heteroscedasticity. Second, their estimation approach makes use of the mean location restriction, and thus could be sensitive to the tail behavior of the unobservable error term since estimation of the transformation function in the tail area can be quite problematic.1 Finally, their estimation procedure involves nonparametric kernel smoothing with multiple bandwidths, which could adversely affect the finite sample performance of their estimator. The paper is organized as follows. The next section presents the model and the estimators. Section 3 contains the large properties. Results from some Monte Carlo experiments are presented in Section 4. Section 5 concludes. The proofs of theorems are relegated to the Appendix. 2. The model and estimators Consider the model, g Y ∗ = X ′ β0 − σ X ′ γ0 ε
)
(
(
)
(1)
where g is an unknown strictly increasing function, Y ∗ is the latent dependent variable, X is a q + 1 dimensional vector of regressors, β0 is the corresponding vector of coefficients. ) error term ε is independent of X , with a zero median for its ( The location normalization. The scale function σ (x) = σ x′ γ0 is known up to the finite dimensional parameter γ0 . Suppose we observe only (Y , X ) where Y = y1 Y ∗ < y + Y ∗ 1 y ≤ Y ∗ ≤ y + y1 Y ∗ > y
{
}
{
}
{
}
with −∞ ≤ y < y ≤ ∞ and both y and y are known constant terms; namely, we allow two-sided censoring here. It is well known (e.g., Horowitz (1996)) that for models subject to unknown transformation, location and scale normalization is needed for identification. As in(Horowitz (1996) and Chen (2002), the location normalization consists ) of setting g (y0 ) = 0 for some finite value y0 ∈ y, y , and the zero median restriction on ε . Our location normalization approach is similar to that in Zhou et al. (2008) who assumed g (y0 ) = 0 and set the zero mean restriction on ε . To achieve scale normalization, we require X contains a component with a nonzero coefficient, whose probability distribution conditional on the remaining components is absolutely continuous with respect to the Lebesgue measure. Following Horowitz (1996) and Chen (2002), we arrange the components of the regressors so that X1 , the first component ( of X) , satisfies this condition. As in Horowitz (1992), our scale normalization consists of setting |β10 | = 1, with X =
X1 , X˜ ′
′
)′
and β0 = β10 , β˜ 0′ . To motivate our estimators, we first develop some rank conditions, similar to Han (1987), Manski (1975, and ( 1985) ) Chen (2002, 2010). Let {Yi , Xi }ni=1 be a random sample of (Y , X ) based on model (1). For given y0 , y ∈ y, y , define { } diy = 1 {Yi > y} and djy0 = 1 Yj > y0 . Then following Chen (2002), under the location normalization g (y0 ) = 0, for any pair i ̸ = j, we obtain
(
E diy − djy0 ⏐Xi , Xj ≥ 0 whenever
(
⏐
)
Xi′ β0 − g (y)
σ Xi′ γ0
(
)
≥
Xj′ β0
( ) σ Xj′ γ0
(2)
1 In general, consistent estimation of the transformation function is only possible in a bounded set, in which case the method proposed by Zhou et al. (2008) does not apply. Indeed, they suggest specifying a parametric form for the transformation function in the tail area to overcome this difficulty.
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
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If β0 and γ0 were known, Eq. (2) suggests that g (y) can be estimated by any value maximizing the following U-statistic based objective function
{
1
∑(
diy − djy0 1
n (n − 1)
Xj′ β0
Xi′ β0 − g
}
( ) ≥ ( ′ ) . σ Xi′ γ0 σ Xj γ0
)
i̸ =j
On the other hand, following Manski (1975, 1985) and Chen (2010), for any y ∈ y, y and τ ∈ (0, 1), we also have the following rank condition:
[
E diy |Xi − τ > 0 whenever
(
)
Xi′ β0 − g (y)
( ) σ Xi′ γ0
]
> c (τ )
(3)
where c (τ ) is the τ th quantile of ε . In particular, under the condition that the median of ε is zero,
(
)
E diy |Xi −
1 2
> 0 whenever Xi′ β0 − g (y) > 0
and
(
)
E diy0 |Xi −
1 2
> 0 whenever Xi′ β0 > 0
Our estimators are based on rank conditions (2) and (3). Define
Γn (y, g , b, γ ) =
{
1 n (n − 1)
∑(
)
diy − djy0 1
i ̸ =j
Xj′ b
Xi′ b − g
}
)≥ ( ′ ) ( σ Xi′ γ σ Xj γ
for any given (b, γ , g ) in the appropriate parameter space, and let gn (y, b, γ ) = arg max Γn (y, g , b, γ ) g ∈M
where M is a compact interval in R. Next, define Qn (θ ) ≡ Qn (b, γ , c ) =
∫
Qn (y, b, γ , c , gn (y, b, γ )) ω (y) dy
(4)
where n
Qn (y, b, γ , c , g ) =
p
1 ∑∑ ( n
diy − τℓ 1 Xi′ b − g − σ Xi′ γ cℓ > 0
) {
(
)
}
(5)
i=1 ℓ=1
c = (c1 , c2 , . . . , cp )′ for some fixed integer p, ω (y) is some nonnegative weighting function defined on y, y ; in particular,
[
]
we set one of the quantiles, τl0 = 0.5, with cl0 = 0, which would ensure the identification of β0 and g (Chen, 2010); once β0 and g are identified, other τ s are used to identify γ0 . Consequently, θ0 = (β0′ , γ0′ , c0′ )′ can be estimated by θˆ , which maximizes Qn (θ ). Following Chen (2010) and Koenker (2004), we integrate over y and pool the sample information over several quantiles to improve efficiency. For technical reasons, similar to Chen (2010), we adopt smoothed objective functions. With a slight abuse of notation, θˆ , the estimator of θ0 , is defined as a value that maximizes Qn (θ ) = Qn (b, γ , c ) =
∫
Qn (y, b, γ , c , gn (y, b, γ )) ω (y) dy
where n
Qn (y, b, γ , c , g ) =
p
1 ∑∑ ( n
diy − τℓ Kh1 Xi′ b − g − σ Xi′ γ cℓ
(
)
(
) )
i=1 ℓ=1
with gn (y, b, γ ) being the smoothed maximum rank estimator that maximizes
Γn (y, g , b, γ ) =
1 n (n − 1)
( ∑(
)
diy − djy0 Kh2
i ̸ =j
where k (·) is a kernel function, K (u) =
∫u −∞
Xi′ b − g
Xj′ b
)
( )− ( ′ ) σ Xi′ γ σ Xj γ
k (v) dv is the corresponding cumulative kernel function with Kh (u) = K
(u) h
ˆ γˆ ) as our estimator for g(y) for ˆ and h1 and h2 are some proper bandwidth parameters. Then we define g(y) = gn (y, β, any y ∈ (y, y). √
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Once we have obtained gˆ and θˆ , we can estimate c (τ ) for a given τ ∈ (0, 1) , τ ̸ = 0.5. Specifically, we define cˆτ as a solution to the following maximization problem,
(
ˆ γˆ , τ , c max Qn β,
)
∫ =
c ∈M
(
)
ˆ γˆ , gˆ (y) , τ , c ω (y) dy Qn y, β,
where
(
ˆ γˆ , g , τ , c Qn y, β,
)
n
=
1∑ ( n
(
) )
diy − τ Kh1 Xi′ βˆ − g − σ Xi′ γˆ c ;
)
(
i=1
and furthermore, as in Horowitz (1996), we can define yˆ ∗τ (x) such that
(
ˆ γˆ gn yˆ ∗τ (x) , β,
)
( ) = x′ βˆ − σ x′ γˆ cˆτ
(6)
as a predictor for the τ th conditional quantile of Y given X = x. Remark 1. By imposing a zero mean restriction on the error distribution, Zhou et al. (2008) are able to identify the intercept term, in addition to the slope parameters; similarly, we allow for the presence of the intercept term as well by imposing a zero median restriction on the error distribution. On the other hand, Chen (2010) does not contain the intercept term since no location restriction is imposed on the transformation function. When there is no heteroscedasticity, our model reduces to that of Horowitz (1996) and Chen (2002); however, Horowitz (1996) and Chen (2002) do not impose any location restriction on the error distribution, and thus the intercept term is absorbed in the transformation function.
√
√
Remark 2. Note that n consistent conditional quantile prediction requires n consistent estimation of the finite dimensional parameters as well as the transformation function. Chen (2010) considered the conditional median restriction without imposing any other restriction on the form of heteroscedasticity. As a result, Chen’s (2010) estimator for the transformation function only converges at a nonparametric rate. In our context, it is possible to extend Chen’s (2010) approach by including several quantiles based on (3); however, doing so does not improve the rate of convergence for the resulting estimator of the transformation function. By exploiting the pairwise rank condition (2), combined with several √ quantile conditions (3), we are able to attain n consistent estimation of the transformation function as well as the finite dimensional parameters. 3. Large sample properties In this section, we investigate the large sample properties of the estimators. We make the following assumptions. Assumption A1. {Yi , Xi }ni=1 is a random sample of (X , Y ), with Y = y1 Y ∗ < y + Y ∗ 1 y ≤ Y ∗ ≤ y + y1 {Y ∗ > y}, with (X , Y ∗ ) generated from (1).
{
}
{
}
Assumption A2. ε is independent of X and the median of ε is zero. Assumption A3. (a) The support of the distribution of X is not contained in any proper linear subspace of Rq+1 . (b) The distribution of X1 conditional on X˜ has a positive density almost everywhere with respect to the Lebesgue measure. (c) The distribution of X˜ has a bounded support. Assumption A4.(The )transformation [ ( ) function g is ] a strictly increasing function with the location normalization g (y0 ) = 0 for some y0 ∈ y, y , and g y − ϵ, g (y) + ϵ ⊆ M for a small positive number ϵ , where M is a compact interval. Furthermore, g is twice differentiable and its derivatives are uniformly bounded over M.
˜ and β˜ 0 is an interior point of B, ˜ a compact subset of Rq . G and C are some compact Assumption A5. β10 = 1, B = {1} × B, subsets of Rq+1 and Rp respectively, with γ0 and c0 being their interior points. Define S¯X = {x: x ∈ SX and x′ β0 − cτℓ σ (x′ γ0 ) ∈ [g(y), g(y¯ )] for some τℓ ̸ = 0.5, ℓ = 1, 2, . . . , p}, where SX denotes the support of X . Assumption A6. σ (·) is a known function such that the scale function σ (·) is almost everywhere positive and uniformly bounded from above and away from zero, such that,
( ) σ x′ γ = constant σ (x′ γ0 ) for any x ∈ S¯X , then γ = γ0 . √
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( ))′ )′ ( ( ( ′ ′ )′ , c0 , where θ10 = β0′ , γ0′ and c0 = c (τ1 ) , c (τ2 ) , . . . , c τp . Let Define θ0 = θ10 ] [ ri (y, θ1 , g ) = E hij (y, θ1 , g ) + hji (y, θ1 , g ) |Xi , Yi
5
(7)
and q¯ i (y, θ1 , g (y, θ1 )) =
[ ] ∂ ∂ ri (y, θ1 , g (y, θ1 )) − E ri (y, θ1 , g (y, θ1 )) ∂g ∂g
where
{ hij (y, θ1 , g ) = 1 {Yi ≥ y} − 1 Yj ≥ y0
(
{
})
1
}
Xj′ b
Xi′ b − g
( )≥ ( ′ ) σ Xi′ γ σ Xj γ
and
] ∂2 y , θ , g y , θ r )) ( ( 1 1 i 2 ∂g2 [ 2 ] [ ] ∂ ri (y,θ10 ,g (y)) Assumption A7. V (y, θ10 ) = 12 E is negative for each y ∈ y, y and uniformly bounded away from zero. ∂g2 ( ⏐ ) Let Z1 = X ′ β0 and Z2 = X ′ γ0 . Let p z1 ⏐x˜ denote the density of Z1 at z given X˜ = x˜ , F (t ) and f (t ) are the cumulative and probability density functions of ε at t. ( ⏐ ) Assumption A8. f (t ) is uniformly bounded and twice continuously differentiable with respect to t, and p z1 ⏐x˜ is twice continuously differentiable with respect[ to( z1) for any ] given x˜ , with all the derivatives uniformly bounded. f (t ) is uniformly bounded away from zero for any t ∈ c τ , c (τ ) . The weight function, ω, is twice continuously differentiable and is V (y, θ1 ) =
1
[
E
nonnegative with a bounded support. Assumption A9. The kernel function k (·) is everywhere twice differentiable with a bounded support. The kernel function k (·) and its derivatives are uniformly bounded and have bounded variation such that:
∫
k (u) du = 1 and
∫
uk (u) du = 0
and
∫
k′ (u) du = 0,
∫
uk′ (u) du ̸ = 0.
Assumption A10. The bandwidth sequences satisfy nh41 = o (1), ln n nh31
(
)−1/2
1 = o (1) and nh42 = o (1), n−1/2 h− 2 = o (1).
Define Q (θ , g (·, θ1 )) =
∫
Q (y, θ, g (y, θ1 )) ω(y)dy
where Q (y, θ , g (y, θ1 )) =
p ∑
Qℓ (y, θ, g (y, θ1 ))
ℓ=1
with
{ Qℓ (y, θ , g (y, θ1 )) = E diy − τℓ 1
(
)
Xi′ b − g (y, b, γ )
( ) σ Xi′ γ
} > cℓ .
Let H (θ0 ) =
[
∂2 ∂θ ∂θ ′
]
∫
Q (y, θ, g (y, θ1 )) ω (y) dy ⏐θ =θ0
⏐
where given (y, θ1 ), g (y, θ1 ) maximizes
Γ (y, g , θ1 ) = E
[ (
{ )
diy − djy0 1
Xi′ b − g
Xj′ b
( )≥ ( ′ ) σ Xi′ γ σ Xj γ
}] .
Assumption A11. H (θ0 ) is negative definite. √
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
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Assumption A1 describes the data generating process. Assumptions A2 and A3 are needed for identification for
β0 (e.g., Manski (1985), Horowitz (1992, 2009), Chen (2002, 2010)). Assumptions A4 and A5 contain the location normalization. Assumption A6 is needed for ( the) identification ( ) of γ , which is commonly used for identification of nonlinear regression models. For the special case σ x′ γ = exp x′ γ , then Assumption 6 holds trivially if x ∈ S¯X does not contain a constant and satisfies the full rank condition. Note that for the case with (y, y¯ ) = (−∞, +∞), then S¯X = SX . In general, we require sufficient variation of x in S¯X . Assumptions A7 and A8 contain some boundedness and smoothness conditions. ( Assumptions ) ( A9 )and A10 place some restrictions on the kernel functions and the bandwidth sequences. h1 = O n−ν1 , h2 = O n−ν2 , ν1 ∈ (−1/3, −1/4) , ν2 ∈ (−1/2, −1/4) will be some suitable choices. The matrix H (θ0 ) in Assumption A11 is analogous to the Hessian form of the information matrix in the maximum likelihood estimation. √ The following theorem establishes consistency of θˆ and its n-asymptotic normality.
)′
(
Theorem 1. Under Assumptions A1–A11, θˆ = βˆ ′ , γˆ ′ , cˆ ′ , is
√ (
n θˆ − θ0
)
√
n-consistent for θ0 and asymptotically normal
) ( d → N 0, H (θ0 )−1 Σ (θ0 ) H (θ0 )′−1
where Σ (θ0 ) = E ψ (ξi , θ0 ) ψ (ξi , θ0 )′ and ψ (ξi , θ0 ) is defined in the Appendix. The following theorem establishes uniform consistency of gn and convergence in distribution of
√
n (gn (y) − g (y)).
Theorem 2. Under Assumptions A1–A11, (a) supy∈[y,y] |gn (y) − g (y)| = op (1); (b) For any y ∈ y, y ,
[
√
]
n
) 1 ∑( 1 n (gn (y) − g (y)) = √ Jy,y0 (ξi , θ10 ) + Jy2 (ξi , θ10 ) + op (1) n i=1
and
√
n (gn (y) − g (y)) ⇒ Hg (y), where Hg (y) is a tight Gaussian Process with mean 0 and covariance function
E Hg (y) Hg y′
( ))
(
=E
)(
[(
Jy1,y0 (ξi , θ10 ) + Jy2 (ξi , θ10 )
)]
Jy1′ ,y (ξi , θ10 ) + Jy2′ (ξi , θ10 ) 0
where Jy1,y0 (ξi , θ10 ) = −V (y, θ10 )−1 q¯ i (y, θ10 , g (y, θ10 )) and Jy2 (ξi , θ10 ) =
∂ g (y, θ10 ) H(θ0 )−1 ψ (ξi , θ0 ) ∂θ1
which are defined in the Appendix. The theorem establishes uniform consistency of cˆτ and convergence in distribution of [ following ] τ ∈ τ, τ .
√ (
n cˆτ − c (τ ) for
)
Theorem 3. Under Assumptions A1–A11, (a) supτ ∈[τ ,τ ] ⏐cˆτ − c (τ )⏐ = op (1); (b) For any τ ∈ τ , τ ,
⏐
⏐
[
]
n ) 1 ∑ n cˆτ − c (τ ) = √ Λ (ξi , θ10 , c (τ )) + op (1) n
√ (
i=1
and
√ (
) n cˆτ − c (τ ) ⇒ Hcτ (τ ), where Hcτ (τ ) is a tight Gaussian Process with mean 0 and covariance function ( ( )) [ ( ( ))] E Hcτ (τ ) Hcτ τ ′ = E Λ (ξi , θ10 , c (τ )) Λ ξi , θ10 , c τ ′
where Λ (ξi , θ10 , c (τ )) is defined in the Appendix. For a given ϵ > 0, and τ ∈ τ , τ , define yτ (x) = Qτ (Y ∗ |X = x) and
[
]
Sϵτ = {x : yτ (x) ∈ [yL + ϵ, yU − ϵ ]} ̸ = φ Theorem 4. Under Assumptions A1–A11, (a) supx∈Sϵτ ⏐yˆ ∗τ (x) − y∗τ (x)⏐ = op (1); (b) For any x ∈ Sϵτ ,
⏐
√ (
⏐
n yˆ ∗τ (x) − y∗τ (x)
)
n 1 ∑
= √
n
i=1
(
[
x˜ ′ , −σ ′ x′ γ0 c (τℓ ) x′ Iθ1 Ψ (ξi , θ0 ) −
(
)
)
] ) ( ) ( ) + op (1) σ x′ γ0 Λ (ξi , θ10 , c (τ )) − Jy1∗τ (x),y (ξi , θ0 ) + Jy2∗τ (x) (ξi , θ0 ) g ′ y∗τ (x) 0 1
(
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
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7
√ (
) n yˆ ∗τ (x) − y∗τ (x) ⇒ Hyτ (x), where Hyτ (x) is a tight mean 0 Gaussian Process whose covariance function is the same
and as the covariance function of
x˜ ′ , −σ ′ x′ γ0 c (τ ) x′ Iθ1 Ψ (ξi , θ0 ) −
(
(
)
)
] ) ( ) ( ) σ x′ γ0 Λ (ξi , θ10 , c (τ )) − Jy1∗τ (x),y (ξi , θ10 ) + Jy2∗τ (x) (ξi , θ10 ) g ′ y∗τ (x) 0 [ ] where Iθ1 = I(2q−1)×(2q−1) 0 , I is a (2q − 1) × (2q − 1) identity matrix. [
1
(
For the purpose of carrying out large-sample statistical inference, consistent estimates of the asymptotic variance and covariance matrix need to be constructed. From the proof of the theorems, we can easily show that all the components in the asymptotic variance can be consistently estimated by the sample analogue replacing the true value of the parameters by their consistent sample analogues. Specifically, ( ) ( ) (i). For the statistical inference of θˆ , H (θ0 )−1 Σ (θ0 ) H (θ0 )′−1 can be consistently estimated by Hn−1 θˆ Σn θˆ
( )
Hn′−1 θˆ , where
( )
Hn θˆ
=
∂2 ∂θ 2
∫
(
(
ˆ gn y, θˆ1 Qn y, θ,
))
ω (y) dy
and n ( ) 1∑ ( ) ( ) Σn θˆ = ψn ξi , θˆ ψn′ ξi , θˆ
n
i=1
( )
the following shows the componentwise estimation of Hn θˆ
( )
(
and Σn θˆ , specifically, ψn ξi , θˆ
)
can be consistently
estimated by
( ) ( ) ( ) ψn ξi , θˆ = ψ1n ξi , θˆ + ψ2n ξi , θˆ where p ( ) ∑ ( ) ( ) ψ1n ξi , θˆ = ϕ1ni τℓ , θˆ ϕ2ni τℓ , θˆ
ℓ=1
and
( ) ∫ ( ( ) ( )) ( ) 1 ˆ y + Hn2θ g θ, ˆ y Jny ˆ ψ2n ξi , θˆ = Hn1θ g θ, ,y0 ξi , θ1 ω (y) dy in which
(
)
ϕ1ni τℓ , θˆ = −
∫
( ) ⎛ ( ) ⎞ ˆ γˆ − σ Xi′ γˆ cˆτℓ Xi′ βˆ − gn y, β, ) 1 ⎠ ω (y) dy k⎝ diy − τℓ
(
h1
h1
( ( ) ( ′ )) ′ˆ −1 ˆ ( ) ( ) ˆ β − c σ X γ ˆ ∂ g g X , β, γ ˆ n τ ( ) ( ) n i i ℓ ϕ2ni τℓ , θˆ = X˜ i′ , σ ′ Xi′ γˆ Xi′ cˆτℓ , ιℓ σ ′ Xi′ γˆ + ∂θ1 and
( ) ∂ gn y,θˆ1 ∂θ1
can be consistently estimated according to (A.3) and (A.4), and
( ) ∫ Hn1θ g θˆ , y =
) ( ) ( ⎛ ( ) ⎞ ′ˆ p ∂g ˆ γˆ ( ˆ γˆ − σ Xi′ γˆ cˆτℓ n y , β, X β − g y , β, ∑ ∑ n n ) i 1 ⎠ ω (y) dy diy − τℓ k′ ⎝ ∂θ1 h1 nh21 i=1 ℓ=1
( ( ) ) ∫ p n ⏐ ( ) ) Xi′ b − gn (y, b, γ ) − σ Xi′ γ cℓ 1 ∑∑( ∂ ⏐ diy − τℓ k θˆ , y = ω (y) dy ⏐b=β,γ ˆ =γˆ ,cℓ =ˆcτ ℓ ∂θ nh1 h1 i=1 ℓ=1 ( ( ′ )) (ii). Next we compute the covariance of gn (y), E Hg (y) Hg y , which can be consistently estimated by n ( ) ( )) ( ( ) ( )) [ ] 1 ∑( 1 Jn,y,y0 ξi , θˆ1 + Jn2,y ξi , θˆ1 Jn1,y′ ,y ξi , θˆ1 + Jn2,y′ ξi , θˆ1 , for any y ∈ y, y 0 Hn2θ g
n
i=1
where write
( lij (y, θ1 , g ) =
1 {Yi > y} − 1 Yj > y0
{
( ) σ Xi′ γ
})
1 h2
√
( k
1 h2
(
Xi′ b − g
Xj′ b
))
( )− ( ′ ) σ Xi′ γ σ Xj γ
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
8
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
(
(
Jn1,y,y0 ξi , θˆ1
Vn y, θˆ1
)
=
)−1
n [ ( ∑
(
lij y, θˆ1 , gn y, θˆ1
n−1
))
( ( ))] + lji y, θˆ1 , gn y, θˆ1
j=1,j̸ =i
with
(
Vn y, θˆ1
)
=
n n ( )) ( ( ))] ∑ ∑ ∂ [ ( lij y, θˆ1 , gn y, θˆ1 + lji y, θˆ1 , gn y, θˆ1 2n (n − 1) ∂g i=1 j=1,j̸ =i
1
and
(
Jn2,y ξi , θˆ1 where
∂ gn (y,θ1 ) ∂θ1
)
( ) ∂ gn y, θˆ1 =
∂θ1
( )
(
Hn−1 θˆ ψn ξi , θˆ
)
is defined in Eq. (A.4)–(A.5) in the Appendix.
(iii). A consistent estimator of E Hcτ (τ ) Hcτ τ ′ 1 n
( ))
(
n
∑[
(
)
(
ˆ τ ϕn ξi , θ, ˆ τ′ Vn (τ )−1 ϕn ξi , θ,
)′
will be
( )−1
Vn τ ′
]
, for any τ ∈ τ , τ , τ ̸ = 0.5
[
]
i=1
where Vn (τ ) =
( ) ∂ Qn2 θˆ1 , gn , τ , cˆτ σ
∫ =
∂c2 ) n Xi′ γˆ ∑ (
⎛
( 2
diy − τ k′ ⎝
nh21
)
(
ˆ γˆ Xi′ βˆ − gn y, β, h1
i=1
)
( ) ⎞ − σ Xi′ γˆ cˆτ ⎠ ω (y) dy
and
∫ ( ) ϕn ξi , θˆ1 , τ = S1ni + (S2ni (y) + S3ni (y)) ω (y) dy where
(
) (
S1ni = ϕ1ni τ , θˆ σ Xi′ γˆ
)
and
⎛ 2 S2ni (y) = ⎝Hnc τ θ1
(
( )⎞ ( ) ( ) ( ) ∂ gn y, θˆ1 −1 ˆ 2 ⎠ ˆ ˆ y + Hnc ˆ H θ ψ ξ , θ θ, θ, y I n i θ 1 n τg ∂θ ′ )
and
(
)(
2 ˆ y S3ni (y) = Hnc θ, τg
(
)
(
1 2 ˆ ˆ Jny ,y0 ξi , θ1 + Jny,y0 ξi , θ1
))
with 2 Hnc τ θ1
(
∂ θˆ , y = − ∂θ1
)
∫
( ) n σ Xi′ γ ∑ (
(
diy − τ k
nh1
)
Xi′ b − gn (y, b, γ ) − σ Xi′ γ cℓ
(
h1
i=1
)
)
⏐ ⏐ ω (y) dy ⏐b=β,γ ˆ =γˆ ,cℓ =ˆcτ
ℓ
and
) ( ⎛ ( ) ⎞ ˆ γˆ − σ Xi′ γˆ cˆτ n ( ) ∫ σ (X ′ γˆ ) ∑ Xi′ βˆ − gn y, β, ( ) i 2 ⎠ ω (y) dy diy − τ k′ ⎝ Hnc θˆ , y = τg 2 nh1
(
(iv). Finally E Hy∗τ (x) Hy∗τ ℓ
ℓ
h1
i=1
( ′ )) x
can be componentwise estimated using its sample analogue where g ′ (y) is replaced by
( ( ′ˆ ( ˆ ) )) ) ( Yi −y ) Xi β−gn y,β,γˆ Xj′ βˆ 1 ( ) ˆ k hy k h − i ̸ =j σ X i γ ( ) σ (Xi′ γˆ ) 2 σ Xj′ γˆ ( ) gn′ y, θˆ1 = − ( ( )) ) ˆ gn y,β, ˆ γˆ Xi′ β− ∑ ( ( Yi − y ) Xj′ βˆ 1 ′ ( ) − djy0 k h − i̸ =j K hy ′ σ (X ′ γˆ ) 2 ∑
(
′
i
σ Xj γˆ
at y = yˆ ∗τℓ (x), if hy is chosen to be some corresponding smoothing parameter that goes to zero as n goes to infinity. √
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9
4. Monte Carlo experiments In this section we will examine the finite sample performance of our estimators, in comparison with Chen’s (2010) integrated maximum score estimator (IMSC), which only imposes a conditional median restriction. We consider both homoscedastic and heteroscedastic designs. For the homoscedastic designs, data are generated from the model g Y ∗ = X1 + β˜ 00 − (1 + γ0 X1 )ε , Y ∗ = Y
(
)
where β˜ 00 = 1, X1 follows normal distribution with mean −1 and unit variance. The transformation function is of the form g(y) = y and y0 = 0, namely the identity function; For the distribution of error term, we consider two data generating processes: in DGP1, ε is drawn from a standard normal distribution, independent of X1 ; and for DGP2, ε is drawn from a median standardized mixture normal distribution, also independent of X1 , and specifically,
ε∗ ε = √ , ε∗ ∼ 0.5N (0, −2) + 0.5N (0, 2) 5
For the heteroscedastic designs, we consider the modified Box–Cox transformation model with censoring: g Y ∗ , λ = X1 + β˜ 10 X2 + β˜ 00 −
(
)
1
{ } (1 + exp (γ0 X2 )) ε , Y = max y, Y ∗
2
|y|λ sgn(y)−1
, λ = 0.6 and y0 = 1 and β˜ 10 = 1, β˜ 00 = −1 and γ0 = −1, X1 is drawn from the standard where g (y, λ) = λ normal distribution and X2 follows the chi-square distribution with one degree of freedom, which is independent of X1 and standardized to have zero mean; the scale function is an exponential function σ (u) = 12 (1 + exp (−u)). As in the first two designs, in DGP 3, ε is the standard normal and ε is a mixture normal for DGP 4. Y is censored from below and y is a known constant such that the censoring percentage is 15%. In addition to DGPs 3–4, we introduce more covariates in the scale function in next two designs as the exclusive restriction is not required for our identification: g Y ∗ , λ = X1 + β˜ 10 X2 + β˜ 00 −
(
)
1
{ } (1 + exp (γ1 X1 + γ2 X2 )) ε, Y = max y, Y ∗
2
where γ1 = 0.5 and γ2 = −1; ε follows the standard normal for GDP 5 and is drawn from a mixture normal for DGP 6. The censoring percentage is also roughly 15%. In the last design, we consider a model with discrete regressors. Even though our identification results require the presence of continuous regressors, our rank estimator may still deliver meaningful results. The model specification is same as DGP3 except both regressors are discrete random variables: X1 ∼Poisson(1.5) − 1, X2 ∼Poisson(1) − 1 and ε ∼ N (0, 1). Before describing the details of the simulation results, we first discuss some of the details related to the implementation of our proposed estimators. Similar to Chen (2010), instead of using the smoothing version, we report the results based on the non-smoothed objective functions, and indeed smoothing version produces very similar results. It turns out that our simulation results are quite robust to the choice of weight function, and we choose ω (y) = 1 {yL < y < yU }, where yU < y and yL > y, where yL is the 0.20th quantile and yU is the 0.80th quantile of Y for both designs. In the objective function (8), we sum over the following quantiles: {0.75, 0.70, 0.60, 0.50, 0.40.0.30, 0.25}. As discussed in Chen (2002), we also adopt a segmentation method in estimating the transformation function. Given β and γ , for y > y0 , the searching grid initially starts from t0 = y0 , and tj is to be chosen such that tj > tj−1 , j = 1, 2, . . . , 8, and tj−1 acts as the reference point for the estimation of g(t) for t ∈ (tj−1 , tj ]. When y < y0 , our procedure can be implemented in a very similar way. The following describes the main steps in implementing our estimators:
• Step 1. (Obtaining initial estimate for β˜ 0 ) Following Chen (2010), we obtain a consistent estimate of β˜ 0 , βˆ ∗ , under the conditional median restriction.
• Step 2. (Obtaining initial estimate for γ0 ) Given the initial estimates of β˜ 0 , we obtain an initial consistent estimate of γ0 by, γˆ ∗ , which solves ∫ ∑ p n ( ) ) } ∑ ( ) { ( ) ( 1 diy − τℓ 1 Xi′ βˆ ∗ − gn y, βˆ ∗ , γ − σ Xi′ γ cˆτℓ βˆ ∗ , γ > 0 ω (y) dy (8) max γ
n
i=1 ℓ=1
(
)
where the transformation function gn y, βˆ ∗ , γ , as a function of γ given βˆ ∗ , maximizing
Γn
(
) y, g , βˆ ∗ , γ =
{
1 n (n − 1) √
∑( i̸ =j
)
diy − djy0 1
Xi′ βˆ ∗ − g
Xj′ βˆ ∗
( ) ≥ ( ′ ) σ Xi′ γ σ Xj γ
} (9)
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
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S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx Table 1 Estimation of coefficients: DGP1. N = 250
β˜ 00 γ0 β˜ 00 γ0
CZ Chen Two-step
N = 500
Bias
STD
RMSE
Bias
STD
RMSE
−0.002
0.140 0.104 0.210 0.088
0.140 0.104 0.213 0.089
0.011 0.000 −0.018 0.005
0.097 0.062 0.178 0.057
0.097 0.062 0.179 0.057
Bias
STD
RMSE
Bias
STD
RMSE
−0.012
0.202 0.109 0.305 0.085
0.203 0.113 0.310 0.088
0.012 0.009 −0.002 0.008
0.160 0.052 0.276 0.049
0.160 0.052 0.276 0.050
0.007 −0.034 0.014
Table 2 Estimation of coefficients: DGP2. N = 250
β˜ 00 γ0 β˜ 00 γ0
CZ Chen Two-step
(
w.r.t. γ and cˆτ∗ βˆ ∗ , γ
∫
n 1∑ (
n
i=1
(
)
N = 500
0.029 −0.053 0.022
maximizes
( ) ⎧ ⎫ ′ ∗ ∗ ⎬ ) ⎨ Xi βˆ − gn y, βˆ , γ ( ′ ) > c ω (y) dy diy − τ 1 ⎩ ⎭ σ Xi γ
w.r.t. c. gn y, βˆ ∗ , γ
)
(
and cˆτ∗ βˆ ∗ , γ
)
(10)
are solved using one-dimensional grid search. The Nelder–Mead algorithm is
used to maximize the objective function (8), for which we adopt the following starting values for γ : – DGP1-2: Run quantile regression gn∗ (Yi ) − X1i − βˆ 0∗ on X1i with intercept term at τ = 0.70, collect the corresponding coefficients q∗0 and q∗1 , the starting values of γ ∗ is given by q∗1 /q∗0 . – DGP3-7: Compute
⎛
⎛ (
Y˜i∗ ≡ log ⎝max ⎝
2 gn∗ (Yi ) − X1i − βˆ 1∗ X2i − βˆ 0∗ zτ
)
⎞⎞ − 1, 0⎠⎠
then the starting values of γ ∗ is given by a quantile regression Y˜i∗ on X1i and X2i at τ = 0.70, where zτ is the τ th quantile of N (0, 1).
• Step 3. (Estimating β˜ 0 , γ0 and c0 ) Based on the initial values of βˆ ∗ , γˆ ∗ and cτ∗ , we conduct search for the estimates
of parameters according to (4) and (5) using some local optimization routines. Nelder–Mead algorithm is employed in the Monte Carlo simulation.
Let βˆ and γˆ denote the final estimates of β˜ 0 and γ0 .
• Step 4. (Estimation of( transformation function g (y)) Given the estimates of β˜ 0 and γ0 , we estimate the transformation ) ˆ function g (y) by gn y, β, γˆ , as in Step 2. We consider sample sizes of 250 and 500, with 500 replications, and report the bias, standard deviation (SD) and root-mean-square-error (RMSE) for each design. We compare the performance of our one-step estimator (CZ) with that of Chen (2010). In addition, we also report the performance of the estimate for γ0 obtained in Step 2 (Two-step), which is also computationally simpler; together with the initial estimator for β0 obtained in Step 1, it is used as the initial estimate for our joint estimation of β0 and γ0 in Step 3. This two-step procedure is about 2.5 and 3 times faster than the joint estimation in our experiments. Tables 1–7 report the results for estimates of the finite dimensional parameters. Tables 8–21 summarize the results for the estimation of transformation function for some particular values of y, along with that of Chen (2010) for comparison. Note that for the estimation of finite dimensional parameters, our estimator performs very well and significantly better than Chen (2010) with much smaller RMSEs. Regarding the estimation of γ0 , our joint estimator and the two-step version are quite comparable for DGPs 1–3 and 5, whereas the joint estimator clearly outperforms the two-step version for DGPs 4 and 6. When the sample size increases to 500, the performances of all estimators improve as expected, both in terms of biases and standard deviations. For the case when all regressors are discrete, it turns out our estimator continues to perform well, and much better than Chen (2010) for the estimation of β0 and the two-step version for the estimation of γ0 . √
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11
Table 3 Estimation of coefficients: DGP3. N = 250
CZ
Chen Two-step
β˜ 10 β˜ 00 γ0 β˜ 10 β˜ 00 γ0
N = 500
Bias
STD
RMSE
Bias
STD
RMSE
0.008 0.002 −0.090 0.013 −0.034 −0.235
0.131 0.163 0.373 0.185 0.264 0.318
0.131 0.163 0.383 0.185 0.266 0.395
0.015
0.095 0.123 0.252 0.134 0.229 0.213
0.096 0.123 0.252 0.135 0.231 0.248
Bias
STD
RMSE
0.011 0.024 −0.125 0.034 −0.040 −0.323
0.164 0.265 0.284 0.282 0.412 0.294
0.165 0.266 0.310 0.284 0.414 0.437
Bias
STD
RMSE
−0.002
0.108 0.173 0.163 0.209 0.309 0.177
0.108 0.175 0.174 0.209 0.309 0.258
Bias
STD
RMSE
Bias
STD
RMSE
0.046 −0.024 0.164 −0.206 0.011 −0.032 −0.089 −0.313
0.173 0.219 0.392 0.389 0.225 0.307 0.347 0.307
0.179 0.220 0.425 0.440 0.225 0.309 0.358 0.439
0.025 −0.021 0.029 −0.048 0.011 −0.022 −0.090 −0.197
0.113 0.156 0.184 0.245 0.182 0.282 0.257 0.233
0.116 0.158 0.187 0.250 0.183 0.283 0.272 0.305
Bias
STD
RMSE
Bias
STD
RMSE
0.029 0.004 0.094 −0.214 0.001 −0.034 −0.017 −0.394
0.224 0.342 0.296 0.312 0.337 0.467 0.357 0.351
0.226 0.342 0.311 0.379 0.337 0.468 0.357 0.528
0.001 0.018 0.028 −0.088 −0.009 0.005 −0.014 −0.274
0.140 0.219 0.142 0.195 0.262 0.399 0.260 0.237
0.140 0.219 0.145 0.214 0.262 0.399 0.260 0.362
Bias
STD
RMSE
Bias
STD
RMSE
0.011 −0.009 −0.170 −0.057 −0.270 −0.324
0.132 0.204 0.464 0.192 0.435 0.697
0.133 0.204 0.494 0.201 0.512 0.768
0.009
−0.006 −0.129 −0.065 −0.290 −0.258
0.094 0.146 0.393 0.161 0.408 0.636
0.094 0.146 0.414 0.173 0.500 0.686
−0.003 −0.014 0.017
−0.028 −0.126
Table 4 Estimation of coefficients: DGP4. N = 250
CZ
Chen Two-step
β˜ 10 β˜ 00 γ0 β˜ 10 β˜ 00 γ0
N = 500
0.029 −0.063 0.004 0.005 −0.187
Table 5 Estimation of coefficients: DGP5. N = 250
CZ
Chen Two-step
β˜ 10 β˜ 00 γ10 γ20 β˜ 10 β˜ 00 γ10 γ20
N = 500
Table 6 Estimation of coefficients: DGP6. N = 250
CZ
Chen Two-step
β˜ 10 β˜ 00 γ10 γ20 β˜ 10 β˜ 00 γ10 γ20
N = 500
Table 7 Estimation of coefficients: DGP7. N = 250
CZ
Chen Two-step
β˜ 10 β˜ 00 γ0 β˜ 10 β˜ 00 γ0
√
N = 500
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
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S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
Table 8 Estimation of transformation function: DGP1, N = 250. y
g (y)
−1.0 −0.8 −0.6 −0.4 −0.2
−1.0 −0.8 −0.6 −0.4 −0.2
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.005 0.006 −0.017 −0.023 −0.039 0.0 −0.036 −0.058 −0.047 −0.068 −0.085
0.008 0.005 0.001 0.007 0.002 0.0 −0.003 −0.003 0.001 −0.004 −0.014
0.006 0.003 −0.002 0.003 −0.000 0.0 0.001 0.004 0.011 0.008 0.003
0.324 0.325 0.305 0.279 0.215 0.0 0.171 0.263 0.294 0.286 0.305
0.145 0.132 0.117 0.099 0.079 0.0 0.080 0.099 0.127 0.148 0.177
0.153 0.132 0.119 0.099 0.083 0.0 0.081 0.103 0.133 0.151 0.183
0.324 0.325 0.305 0.280 0.219 0.0 0.174 0.269 0.298 0.294 0.317
0.145 0.132 0.117 0.100 0.079 0.0 0.080 0.099 0.127 0.148 0.177
0.153 0.132 0.119 0.099 0.083 0.0 0.081 0.103 0.134 0.152 0.183
Table 9 Estimation of transformation function: DGP1, N = 500. y
g (y)
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.000 0.001 0.001 0.003 −0.001 0.0 −0.001 −0.001 −0.001 −0.002 −0.003
−0.001 −0.002 −0.000
0.247 0.257 0.245 0.231 0.183 0.0 0.177 0.228 0.242 0.255 0.263
0.107 0.097 0.083 0.069 0.049 0.0 0.050 0.067 0.091 0.110 0.127
0.105 0.099 0.083 0.069 0.051 0.0 0.050 0.070 0.093 0.116 0.136
0.247 0.257 0.245 0.231 0.184 0.0 0.179 0.230 0.245 0.258 0.267
0.107 0.097 0.083 0.069 0.049 0.0 0.050 0.067 0.091 0.110 0.127
0.105 0.099 0.084 0.069 0.051 0.0 0.050 0.070 0.093 0.116 0.136
−1.0 −0.8 −0.6 −0.4 −0.2
−1.0 −0.8 −0.6 −0.4 −0.2
−0.010 −0.006 −0.009 −0.007 −0.013
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0 −0.028 −0.029 −0.033 −0.038 −0.046
0.003 −0.002 0.0 0.002 0.003 0.007 0.007 0.009
Table 10 Estimation of transformation function: DGP2, N = 250. y
g (y)
−1.0 −0.8 −0.6 −0.4 −0.2
−1.0 −0.8 −0.6 −0.4 −0.2
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.014 0.006 −0.004 −0.010 −0.070 0.0 −0.001 −0.046 −0.078 −0.059 −0.087
−0.004
−0.007
0.009 0.005 0.003 0.002 0.0 −0.008 −0.010 −0.018 −0.014 −0.025
0.004 0.000 0.000 −0.004 0.0 −0.002 −0.007 −0.008 −0.004 −0.017
0.449 0.426 0.396 0.349 0.274 0.0 0.248 0.340 0.403 0.448 0.461
0.167 0.147 0.129 0.107 0.082 0.0 0.085 0.120 0.155 0.182 0.212
0.175 0.145 0.129 0.106 0.085 0.0 0.089 0.127 0.168 0.201 0.233
0.449 0.426 0.396 0.349 0.282 0.0 0.248 0.344 0.411 0.452 0.469
0.167 0.148 0.129 0.107 0.082 0.0 0.085 0.120 0.156 0.182 0.213
0.175 0.145 0.129 0.106 0.086 0.0 0.089 0.127 0.168 0.201 0.234
Table 11 Estimation of transformation function: DGP2, N = 500. y
g (y)
−1.0 −0.8 −0.6 −0.4 −0.2
−1.0 −0.8 −0.6 −0.4 −0.2
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.047 0.053 0.020 −0.003 −0.043 0.0 0.021 −0.026 −0.030 −0.021 −0.036
−0.001
−0.007
0.006 0.006 0.005 0.001 0.0 −0.003 −0.009 −0.005 −0.008 −0.003
0.002 0.001 0.000 0.001 0.0 −0.002 −0.003 0.001 0.001 0.008
0.412 0.407 0.379 0.324 0.252 0.0 0.243 0.321 0.361 0.399 0.406
0.121 0.105 0.092 0.076 0.060 0.0 0.064 0.083 0.109 0.130 0.154
0.119 0.106 0.095 0.079 0.060 0.0 0.063 0.088 0.115 0.144 0.170
0.415 0.411 0.380 0.324 0.256 0.0 0.244 0.322 0.363 0.400 0.408
0.120 0.105 0.092 0.076 0.060 0.0 0.064 0.084 0.109 0.130 0.154
0.119 0.106 0.095 0.079 0.060 0.0 0.063 0.088 0.115 0.144 0.170
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
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13
Table 12 Estimation of transformation function: DGP3, N = 250. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
−1.667 −1.032 −0.705 −0.440 −0.209 0.0 0.193 0.373 0.543 0.705 0.860
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.012 −0.038 −0.011 −0.030 −0.050 0.0 −0.018 −0.078 −0.086 −0.093 −0.096
−0.016 −0.023 −0.027 −0.032 −0.027
−0.004 −0.009 −0.018 −0.017 −0.002
0.0
0.0 −0.002 −0.005 −0.006 0.003 0.006
0.417 0.412 0.366 0.337 0.263 0.0 0.208 0.279 0.335 0.352 0.372
0.258 0.198 0.170 0.157 0.125 0.0 0.106 0.136 0.160 0.180 0.191
0.257 0.187 0.159 0.143 0.110 0.0 0.103 0.135 0.153 0.163 0.180
0.417 0.414 0.366 0.338 0.268 0.0 0.209 0.290 0.346 0.364 0.384
0.258 0.199 0.171 0.161 0.128 0.0 0.109 0.140 0.163 0.182 0.192
0.257 0.187 0.160 0.144 0.110 0.0 0.103 0.135 0.153 0.163 0.181
−0.025 −0.033 −0.031 −0.024 −0.021
Table 13 Estimation of transformation function: DGP3, N = 500. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
−1.667 −1.032 −0.705 −0.440 −0.209
−0.012 −0.017 −0.006 −0.009 −0.043
−0.020 −0.015 −0.013 −0.012 −0.011
−0.019 −0.006 −0.006 −0.001 −0.001
0.0 0.193 0.373 0.543 0.705 0.860
0.0 −0.003 −0.038 −0.049 −0.058 −0.069
0.0
−0.011 −0.008 −0.014 −0.012 −0.012
0.0 0.005 0.007 0.008 0.011 0.008
0.335 0.321 0.314 0.291 0.229 0.0 0.197 0.256 0.291 0.314 0.330
0.172 0.131 0.116 0.105 0.082 0.0 0.075 0.093 0.109 0.126 0.135
0.169 0.137 0.114 0.099 0.070 0.0 0.073 0.087 0.106 0.118 0.126
0.335 0.321 0.314 0.291 0.232 0.0 0.198 0.259 0.295 0.319 0.337
0.174 0.132 0.117 0.106 0.083 0.0 0.075 0.094 0.109 0.127 0.135
0.170 0.137 0.114 0.099 0.070 0.0 0.074 0.088 0.106 0.119 0.126
Table 14 Estimation of transformation function: DGP4, N = 250. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
−1.667 −1.032 −0.705 −0.440 −0.209 0.0 0.193 0.373 0.543 0.705 0.860
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.085 −0.023 −0.026 −0.057 −0.091 0.0 0.015 −0.054 −0.109 −0.116 −0.135
−0.055 −0.057 −0.059 −0.067 −0.049
−0.025 −0.025 −0.020 −0.028 −0.016
0.0 −0.020 −0.021 −0.025 −0.032 −0.023
0.0 0.003 0.011 0.007 0.009 0.008
0.571 0.536 0.482 0.416 0.318 0.0 0.259 0.346 0.415 0.474 0.499
0.364 0.270 0.224 0.182 0.138 0.0 0.108 0.133 0.154 0.178 0.199
0.337 0.248 0.193 0.155 0.119 0.0 0.103 0.122 0.140 0.163 0.166
0.577 0.537 0.483 0.420 0.331 0.0 0.260 0.350 0.429 0.488 0.517
0.368 0.276 0.232 0.194 0.147 0.0 0.110 0.134 0.156 0.181 0.200
0.337 0.248 0.194 0.157 0.120 0.0 0.103 0.123 0.140 0.163 0.166
Table 15 Estimation of transformation function: DGP4, N = 500. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
−1.667 −1.032 −0.705 −0.440 −0.209 0.0 0.193 0.373 0.543 0.705 0.860
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.083 0.017 0.010 −0.027 −0.072 0.0 0.031 −0.014 −0.034 −0.061 −0.072
−0.036 −0.027 −0.025 −0.031 −0.024
−0.017 −0.011 −0.010 −0.009 −0.007
0.0
0.0 −0.000 0.003 0.002 0.002 0.005
0.488 0.471 0.442 0.365 0.281 0.0 0.253 0.348 0.397 0.407 0.430
0.242 0.165 0.133 0.112 0.090 0.0 0.077 0.090 0.112 0.121 0.141
0.222 0.155 0.121 0.101 0.072 0.0 0.067 0.083 0.097 0.101 0.112
0.495 0.472 0.443 0.366 0.290 0.0 0.255 0.348 0.398 0.412 0.436
0.245 0.167 0.135 0.117 0.093 0.0 0.079 0.093 0.113 0.122 0.143
0.223 0.155 0.122 0.101 0.073 0.0 0.067 0.083 0.097 0.101 0.112
−0.015 −0.020 −0.016 −0.020 −0.022
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
14
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
Table 16 Estimation of transformation function: DGP5, N = 250. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
−1.667 −1.032 −0.705 −0.440 −0.209 0.0 0.193 0.373 0.543 0.705 0.860
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.045 0.009 0.010 −0.018 −0.055 0.0 0.015 −0.041 −0.079 −0.099 −0.095
−0.016 −0.022 −0.034 −0.031 −0.031
−0.050 −0.051 −0.039 −0.022 −0.018
0.0 −0.036 −0.043 −0.051 −0.041 −0.041
0.0 0.002 0.005 0.020 0.041 0.052
0.428 0.460 0.419 0.364 0.280 0.0 0.253 0.345 0.398 0.428 0.464
0.323 0.279 0.230 0.200 0.153 0.0 0.131 0.200 0.231 0.259 0.284
0.306 0.263 0.222 0.190 0.141 0.0 0.148 0.190 0.218 0.245 0.262
0.431 0.460 0.419 0.365 0.285 0.0 0.254 0.348 0.405 0.439 0.473
0.323 0.280 0.233 0.202 0.156 0.0 0.136 0.204 0.237 0.263 0.287
0.310 0.268 0.225 0.191 0.141 0.0 0.148 0.190 0.219 0.248 0.267
Table 17 Estimation of transformation function: DGP5, N = 500. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
−1.667 −1.032 −0.705 −0.440 −0.209 0.0 0.193 0.373 0.543 0.705 0.860
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.019 0.006 0.017 0.004 −0.038 0.0 0.014 −0.024 −0.052 −0.063 −0.066
−0.013 −0.017 −0.018 −0.015 −0.012
−0.025 −0.020 −0.016 −0.009 −0.004
0.0 −0.024 −0.027 −0.031 −0.031 −0.036
0.0 0.003 0.005 0.009 0.015 0.021
0.398 0.379 0.368 0.329 0.244 0.0 0.231 0.297 0.342 0.377 0.384
0.236 0.201 0.166 0.138 0.104 0.0 0.102 0.134 0.163 0.188 0.210
0.189 0.160 0.135 0.119 0.090 0.0 0.092 0.113 0.132 0.146 0.162
0.398 0.379 0.369 0.329 0.247 0.0 0.231 0.298 0.346 0.382 0.390
0.237 0.201 0.167 0.139 0.105 0.0 0.105 0.136 0.166 0.191 0.213
0.190 0.162 0.136 0.120 0.090 0.0 0.092 0.113 0.132 0.147 0.164
Table 18 Estimation of transformation function: DGP6, N = 250. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
−1.667 −1.032 −0.705 −0.440 −0.209 0.0 0.193 0.373 0.543 0.705 0.860
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.175 0.048 0.012 −0.031 −0.099 0.0 0.049 −0.021 −0.080 −0.133 −0.166
−0.049 −0.074 −0.081 −0.078 −0.070
−0.043 −0.052 −0.051 −0.035 −0.026
0.0 −0.027 −0.042 −0.057 −0.055 −0.048
0.0 0.017 0.024 0.030 0.046 0.058
0.608 0.555 0.483 0.412 0.328 0.0 0.286 0.374 0.429 0.479 0.540
0.509 0.406 0.331 0.255 0.205 0.0 0.166 0.221 0.251 0.298 0.348
0.422 0.339 0.289 0.223 0.165 0.0 0.159 0.201 0.232 0.270 0.304
0.633 0.557 0.483 0.413 0.342 0.0 0.291 0.375 0.436 0.497 0.565
0.511 0.413 0.341 0.267 0.217 0.0 0.168 0.225 0.257 0.303 0.351
0.424 0.343 0.294 0.225 0.167 0.0 0.160 0.202 0.234 0.273 0.310
Table 19 Estimation of transformation function: DGP6, N = 500. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
−1.667 −1.032 −0.705 −0.440 −0.209 0.0 0.193 0.373 0.543 0.705 0.860
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.120 0.047 0.034 −0.024 −0.090 0.0 0.064 −0.002 −0.042 −0.070 −0.092
−0.032 −0.037 −0.037 −0.039 −0.034
−0.015 −0.014 −0.018 −0.010 −0.009
0.0
0.0 −0.000 0.007 0.008 0.009 0.013
0.536 0.502 0.452 0.376 0.310 0.0 0.277 0.347 0.418 0.477 0.504
0.356 0.262 0.210 0.175 0.143 0.0 0.121 0.163 0.195 0.221 0.260
0.278 0.204 0.169 0.132 0.107 0.0 0.101 0.121 0.138 0.151 0.170
0.550 0.505 0.453 0.377 0.322 0.0 0.284 0.347 0.420 0.482 0.512
0.357 0.265 0.213 0.179 0.147 0.0 0.124 0.167 0.199 0.225 0.264
0.279 0.205 0.170 0.132 0.108 0.0 0.101 0.121 0.138 0.151 0.170
−0.030 −0.040 −0.039 −0.042 −0.042
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
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15
Table 20 Estimation of transformation function: DGP7, N = 250. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
Bias
−1.667 −1.032 −0.705 −0.440 −0.209 0.0 0.193 0.373 0.543 0.705 0.860
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.132 −0.084 0.035 −0.120 −0.231 0.0 −0.126 −0.233 −0.364 −0.443 −0.473
−0.041 −0.089 −0.116 −0.156 −0.164
0.017 −0.004 −0.016 −0.023 −0.021 0.0 −0.004 −0.028 −0.022 −0.033 −0.037
0.452 0.516 0.445 0.397 0.395 0.0 0.175 0.252 0.275 0.352 0.427
0.336 0.322 0.292 0.248 0.263 0.0 0.114 0.169 0.216 0.255 0.271
0.282 0.215 0.214 0.208 0.167 0.0 0.140 0.185 0.198 0.194 0.208
0.471 0.522 0.446 0.415 0.457 0.0 0.215 0.343 0.456 0.566 0.637
0.339 0.334 0.314 0.293 0.309 0.0 0.133 0.224 0.277 0.314 0.334
0.283 0.215 0.215 0.210 0.168 0.0 0.140 0.187 0.199 0.196 0.212
0.0
−0.069 −0.147 −0.174 −0.183 −0.194
Table 21 Estimation of transformation function: DGP7, N = 500. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
Bias
−1.667 −1.032 −0.705 −0.440 −0.209 0.0 0.193 0.373 0.543 0.705 0.860
−1.842 −1.253 −1.0 −0.747 −0.158
τ = 0.2
RMSE
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.143 −0.122 0.074 −0.127 −0.280 0.0 −0.117 −0.233 −0.362 −0.482 −0.510
−0.006 −0.079 −0.092 −0.144 −0.159
0.012 −0.009 −0.014 −0.013 −0.020 0.0 0.001 −0.011 −0.017 −0.024 −0.028
0.424 0.519 0.401 0.382 0.376 0.0 0.177 0.241 0.284 0.314 0.392
0.290 0.292 0.243 0.201 0.225 0.0 0.111 0.158 0.188 0.216 0.238
0.238 0.178 0.163 0.155 0.142 0.0 0.115 0.148 0.151 0.154 0.159
0.447 0.534 0.408 0.402 0.470 0.0 0.212 0.336 0.460 0.575 0.643
0.290 0.302 0.260 0.248 0.276 0.0 0.127 0.212 0.254 0.278 0.301
0.238 0.178 0.164 0.155 0.144 0.0 0.115 0.148 0.152 0.155 0.162
Table 22 Quantile predictions Qτ (Y ∗ |X = x): DGP1 x
STD
Chen
τ = 0.3
0.0
−0.061 −0.141 −0.171 −0.176 −0.183
N = 250.
τ = 0.4
τ = 0.5
τ = 0.6
τ = 0.7
τ = 0.8
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
−0.032 −0.018 −0.015 −0.002
0.157 0.150 0.155 0.158 0.193
−0.019 −0.008 −0.009 −0.003
0.160 0.160 0.157 0.154 0.164
−0.002 −0.003
0.164 0.155 0.150 0.147 0.170
−0.012 (−0.040) −0.002 (−0.006) −0.003 (0.006) −0.002 (0.010) 0.007 (0.046)
0.167 (0.250) 0.157 (0.216) 0.150 (0.209) 0.145 (0.227) 0.180 (0.234)
0.004 0.016 0.010 0.010 0.015
0.166 0.153 0.145 0.138 0.165
0.001 0.013 0.011 0.004 0.015
0.174 0.160 0.151 0.144 0.165
0.008 −0.001 0.007 0.017 0.015
0.203 0.171 0.161 0.156 0.162
0.010
0.017
0.002 0.003 0.020
Note: The median predictions using Chen (2010)’s are presented in parentheses.
For the estimation of the transformation function, it is clear that both of our one-step (CZ) and two-step (Two-step) estimators dominate that of Chen (2010), which only makes use of the conditional median restriction. For DGPs 1–4 our one-step and two-step estimators are quite comparable, whereas the one-step estimator performs noticeably better for DGPs 5 and 6, and much better for DGP 7. In addition, in Tables 22–35 we report the performance of the conditional quantile predictors of our one-step estimator and Chen (2010).2 Our conditional quantile predictors perform well for a wide range of quantiles, and significantly outperform (Chen, 2010) for the conditional median prediction. 5. Conclusion In this paper we have considered the estimation of a general transformation model subject to a multiplicative form of heteroscedasticity, based on the insights of Han (1987), Sherman (1993), Manski (1975, 1985) and Chen (2002, 2010). Our estimators for the finite dimensional parameters, the transformation function, as well as the conditional quantile predictors, all converge at the parametric rate. Monte Carlo simulation experiments indicate that our procedure performs very well in finite samples. In economic duration or statistical survival analysis, censoring is a common phenomenon. Our new estimation procedure can be extended to the case with random censoring by adjusting the objective functions as in Chen (2002, 2010). In addition, endogeneity is a difficult, and yet important issue in the literature, it might be possible to extend our analysis to the endogenous setting. These are interesting topics for future research. 2 The relative performance between our one-step and two-step estimators for the conditional quantile predictors is similar to the relative performance for the estimation of the transformation function by two estimators for seven designs. To save space, we only report the one-step estimator.
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
16
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Table 23 Quantile predictions Qτ (Y ∗ |X = x): DGP1
τ = 0.2
x
−1.842 −1.253 −1.0 −0.747 −0.158
τ = 0.3
N = 500.
τ = 0.4
τ = 0.5
τ = 0.6
τ = 0.7
τ = 0.8
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
−0.017 −0.013 −0.003 −0.001
0.112 0.114 0.114 0.113 0.126
−0.014 −0.001 −0.007 −0.002
0.115 0.114 0.110 0.111 0.116
0.002 0.004 0.007 0.001 0.007
0.114 0.101 0.101 0.099 0.124
0.013 (−0.008) 0.011 (−0.007) 0.011 (0.003) 0.010 (0.005) 0.011 (0.040)
0.119 (0.178) 0.111 (0.175) 0.105 (0.163) 0.103 (0.169) 0.126 (0.205)
0.007 0.007 0.002 0.007 0.006
0.115 0.105 0.100 0.094 0.117
0.011 0.009 0.004 0.004 0.005
0.124 0.109 0.098 0.098 0.109
0.004 0.000 0.002 −0.001 0.002
0.145 0.113 0.108 0.105 0.116
0.013
0.005
Note: The median predictions using Chen (2010)’s are presented in parentheses. Table 24 Quantile predictions Qτ (Y ∗ |X = x): DGP2
τ = 0.2
x
−1.842 −1.253 −1.0 −0.747 −0.158
τ = 0.3
N = 250.
τ = 0.4
τ = 0.5
τ = 0.6
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
−0.043 −0.043 −0.036 −0.031 −0.016
0.144 0.142 0.140 0.151 0.184
−0.026 −0.037 −0.030 −0.027 −0.003
0.163 0.154 0.146 0.144 0.169
−0.088 −0.095 −0.083 −0.082 −0.056
0.213 0.207 0.198 0.194 0.221
−0.011 (−0.088) −0.018 (−0.031) −0.020 (−0.026) −0.009 (−0.004) 0.017 (0.017)
0.231 0.219 0.215 0.215 0.249
(0.385) (0.356) (0.341) (0.341) (0.389)
τ = 0.7
τ = 0.8
Bias
RMSE
Bias
RMSE
Bias
RMSE
0.072 0.068 0.069 0.074 0.098
0.237 0.226 0.219 0.208 0.238
0.009 0.010 0.011 0.011 0.039
0.174 0.163 0.162 0.155 0.164
−0.014 −0.012 −0.006 −0.008 0.016
0.181 0.152 0.134 0.131 0.145
Note: The median predictions using Chen (2010)’s are presented in parentheses. Table 25 Quantile predictions Qτ (Y ∗ |X = x): DGP2
τ = 0.2
x
−1.842 −1.253 −1.0 −0.747 −0.158
τ = 0.3
N = 500.
τ = 0.4
τ = 0.5
τ = 0.6
τ = 0.7
τ = 0.8
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
−0.001 −0.004 −0.005 −0.006 −0.002
0.100 0.093 0.095 0.098 0.133
−0.013 −0.010 −0.016 −0.016 −0.014
0.108 0.096 0.097 0.098 0.124
−0.040 −0.040 −0.038 −0.040 −0.041
0.140 0.129 0.130 0.132 0.157
0.014 (−0.051) 0.008 (0.005) 0.008 (0.018) 0.013 (0.010) 0.018 (0.039)
0.175 (0.327) 0.172 (0.289) 0.166 (0.276) 0.158 (0.287) 0.171 (0.323)
0.063 0.060 0.058 0.062 0.069
0.171 0.164 0.159 0.155 0.161
0.013 0.011 0.010 0.009 0.025
0.117 0.105 0.100 0.097 0.113
−0.001 −0.001 −0.002
0.116 0.097 0.095 0.087 0.097
0.001 0.004
Note: The median predictions using Chen (2010)’s are presented in parentheses.
Appendix. Notations p Throughout the proof, we define θ = (b, γ , c ), θ1 = (b, γ ) where c = (cℓ )ℓ=1 and correspondingly θ0 = (θ10 , c0 ), θ10 = (β0 , γ0 ), and g (y, θ10 ) = g (y), where c0 = (c (τℓ ))pℓ=1 . Let Zi1 = Xi′ β0 and Zi2 = Xi′ γ0 . ιℓ = (0, 0, . . . , 1, 0, . . . , 0)′ is a p × 1 vector with ℓ’s element to be one. Nθ10 is a small neighborhood of θ10 .
Lemma A.1. Under Assumptions A1–A7, for any given y, g (y, θ1 ) is well defined and continuously differentiable in θ1 ∈ Nθ10 and gn (y, θ1 ) − g (y, θ1 ) = −
n 1 ∑ q¯ i (y, θ1 , g (y, θ1 ))
n
V (y, θ1 )
i=1
( ) + op n−1/2
(A.1)
and
( ) ∂ gn (y, θ1 ) ∂ g (y, θ1 ) 1 −1 −2 − = Op h22 + n−1/2 h− h2 2 +n ∂θ1 ∂θ1
(A.2)
uniformly in (y, θ ) with probability approaching one as n increases to infinity, where q¯ i (y, θ1 , g (y, θ1 )) =
[ ] ∂ ∂ ri (y, θ1 , g (y, θ1 )) − E ri (y, θ1 , g (y, θ1 )) ∂g ∂g
and V (y, θ1 ) =
1 2
[ E
∂2 ri (y, θ1 , g (y, θ1 )) ∂g2
]
with ri (y, θ1 , g ) = E hij (y, θ1 , g ) + hji (y, θ1 , g ) |Xi , Yi
[
√
]
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
17
and
{ hij (y, θ1 , g ) = 1 {Yi ≥ y} − 1 Yj ≥ y0
(
{
})
1
}
Xj′ b
Xi′ b − g
( )≥ ( ′ ) . σ Xi′ γ σ Xj γ
Proof. The proof is similar to that of Lemma A in Chen (2010). Thus we only outline the main steps here. Following [Chen] (2010) an application of the inverse function theorem yields that g (y, θ1 ) is continuously differentiable in (y, θ1 ) ∈ y, y × Nθ10 and we can similarly show that gn (y, θ1 ) uniquely maximizes the smoothed objective functions for any given ( ) θ1 ∈ Nθ10 and y ∈ y, y with probability approaching one as n increases to infinity, and gn (y, θ1 ) satisfies
( ( )) ( ) ∑ diy − djy0 1 Xj′ b Xi′ b − gn (y, θ1 ) 1 ( ) ( ) − ( ′ ) k =0 n (n − 1) h2 σ Xi′ γ h2 σ Xi′ γ σ Xj γ i̸ =j 1
and differentiation yields ( i̸ =j
∂ gn (y, θ1 ) = ∂ b˜
)
diy −djy
∑
0
(
σ Xi′ γ
k′
)
(
∑
(
1 h2
(
0
(
(i ) ( (
)
diy −djy
i ̸ =j σ 2 X ′ γ i
Xi′ b−gn (y,θ) σ X′γ
)
k′
1 h2
−
)) (
X′b
(j
σ Xj′ γ
Xi′ b−gn (y,θ1 ) σ X′γ
(
i
)
X˜ i σ Xi′ γ
(
)
−
) ))
X′b
(j
σ Xj′ γ
−
)
X˜ j
(
σ Xj′ γ
)
(A.3)
)
and (
∑ ∂ gn (y, θ1 ) =− ∂γ
)
diy −djy
0
i̸ =j
σ (Xi′ γ )
′
(
k
1 h2
(
)) (
(Xi′ b−gn (y,θ1 ))σ ′ (Xi′ γ )Xi − ( ′ ) − σ 2 (Xi′ γ ) (i ) σ Xj γ ( ) ( ( )) ∑ diy −djy0 ′ 1 Xi′ b−gn (y,θ1 ) Xj′ b ( ) k − i̸ =j σ 2 (X ′ γ ) h2 σ (Xi′ γ ) σ Xj′ γ i Xi′ b−gn (y,θ1 ) σ X′γ
Xj′ b
(
) (
)
Xj′ b σ ′ Xj′ γ Xj ( ) σ 2 X′γ j
) (A.4)
Then, by Lemma A.1 in Carroll et al. (1997), it follows that gn (y, θ1 ) − g (y, θ1 ) = op (1)
(A.5)
uniform in y and θ1 ∈ N10 . To establish the uniform asymptotic linear representation of gn (y, θ1 ) for θ1 ∈ N10 , define Ln (y, θ1 , g) =
1
∑
n (n − 1)
i̸ =j
lij (y, b, γ , g)
where
(
diy − djy0
lij (y, b, γ , g) =
)
1
( ) k σ Xi′ γ h2
(
1
(
Xi′ b − gn (y, θ1 )
h2
( ) σ Xi′ γ
−
Xj′ b
))
( ) σ Xj′ γ
With standard U-statistic decomposition, we can write Ln (y, θ1 , g) = L(y, θ1 , g) + Ln1 (y, θ1 , g) + Ln2 (y, θ1 , g) where L(y, θ1 , g) = E lij (y, b, γ , g)
[
]
n
Ln1 (y, θ1 , g) =
] [ ]] 1∑ [ [ E lij (y, b, γ , g)|Xi , Yi − E lij (y, b, γ , g) n i=1
+
n 1∑ [ [
n
E lji (y, b, γ , g)|Xi , Yi − E lij (y, b, γ , g)
]
[
]]
i=1
and Ln2 (y, θ1 , g) =
1
∑
n (n − 1)
i ̸ =j
l2ij (y, b, γ , g)
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
18
N = 250.
x
τ = 0.2
Value
Bias
RMSE
Bias
τ = 0.3 RMSE
τ = 0.4 Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) 1(−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
0.018 −0.007 −0.012 −0.016 −0.024 0.020 −0.016 −0.031 −0.033 −0.030 0.023 −0.015 −0.026 −0.037 −0.033 0.034 −0.011 −0.023 −0.040 −0.033 0.036 −0.017 −0.051 −0.060 −0.030
0.196 0.154 0.146 0.157 0.217 0.258 0.205 0.201 0.204 0.240 0.295 0.227 0.211 0.218 0.268 0.324 0.249 0.236 0.237 0.297 0.425 0.332 0.320 0.322 0.360
0.028 0.015 0.008 0.002 0.002 0.035 0.019 0.007 0.001 0.008 0.032 0.016 0.009 −0.005 0.010 0.044 0.019 0.013 0.003 −0.000 0.070 0.038 0.013 0.001 0.005
0.106 0.095 0.097 0.110 0.167 0.184 0.153 0.144 0.145 0.199 0.202 0.169 0.160 0.173 0.223 0.236 0.195 0.188 0.193 0.249 0.330 0.283 0.260 0.267 0.310
−0.022 −0.004 0.012 0.015 0.017 0.035 0.024 0.024 0.023 0.030 0.040 0.033 0.028 0.024 0.025 0.049 0.035 0.026 0.016 0.035 0.054 0.050 0.045 0.039 0.035
0.088 0.064 0.059 0.074 0.142 0.119 0.118 0.126 0.133 0.179 0.158 0.149 0.148 0.150 0.197 0.186 0.173 0.171 0.176 0.222 0.270 0.249 0.247 0.240 0.299
−0.028(−0.049) −0.032(−0.066) −0.026(−0.054) 0.001(−0.005) 0.019(−0.002) −0.007(−0.021) 0.013(0.021) 0.013(0.022) 0.013(0.017) 0.019(0.013) 0.014(0.023) 0.014(0.025) 0.016(0.016) 0.013(0.001) 0.030(0.048) 0.017(0.023) 0.014(0.005) 0.014(0.005) 0.023(0.025) 0.026(0.078) 0.019(0.033) 0.025(0.031) 0.021(0.018) 0.029(0.037) 0.028(0.157)
0.152(0.182) 0.114(0.165) 0.081(0.124) 0.049(0.093) 0.131(0.204) 0.055(0.104) 0.060(0.102) 0.077(0.114) 0.106(0.144) 0.159(0.261) 0.067(0.105) 0.089(0.131) 0.108(0.143) 0.123(0.182) 0.175(0.312) 0.104(0.138) 0.121(0.152) 0.130(0.183) 0.146(0.220) 0.202(0.344) 0.169(0.249) 0.179(0.260) 0.195(0.274) 0.202(0.310) 0.286(0.451)
0.082 −0.005 −0.020 −0.019 0.022 −0.030 −0.020 0.005 0.023 0.028 −0.029 0.009 0.024 0.024 0.033 0.005 0.025 0.025 0.028 0.048 0.031 0.027 0.035 0.039 0.041
0.183 0.158 0.133 0.079 0.122 0.134 0.076 0.050 0.078 0.151 0.101 0.056 0.080 0.109 0.169 0.076 0.090 0.113 0.136 0.202 0.156 0.172 0.180 0.198 0.276
0.438 0.168 0.032 −0.016 0.023 0.040 −0.023 −0.018 0.007 0.036 −0.021 −0.022 −0.010 0.022 0.039 −0.037 −0.018 0.022 0.031 0.054 0.015 0.031 0.034 0.047 0.072
0.464 0.230 0.160 0.131 0.100 0.167 0.145 0.094 0.050 0.146 0.163 0.106 0.057 0.072 0.156 0.149 0.080 0.066 0.105 0.193 0.107 0.123 0.151 0.176 0.267
1.030 0.581 0.288 0.043 0.029 0.417 0.114 0.002 −0.010 0.046 0.217 0.017 −0.016 −0.009 0.053 0.079 −0.023 −0.021 0.024 0.051 −0.054 −0.005 0.032 0.042 0.079
1.042 0.601 0.326 0.163 0.088 0.444 0.193 0.155 0.092 0.147 0.264 0.162 0.131 0.061 0.166 0.182 0.162 0.097 0.068 0.182 0.181 0.100 0.104 0.152 0.252
Note: The median predictions using Chen (2010)’s are presented in parentheses.
τ = 0.5
τ = 0.6
τ = 0.7
τ = 0.8 S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 26 Quantile predictions Qτ (Y ∗ |X = x): DGP3
N = 500.
x
τ = 0.2
Value
Bias
RMSE
τ = 0.3 Bias
RMSE
Bias
τ = 0.4 RMSE
τ = 0.5 Bias
RMSE
τ = 0.6 Bias
RMSE
Bias
τ = 0.7 RMSE
τ = 0.8 Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) (−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
0.008 −0.003 −0.007 −0.004 0.008 −0.003 −0.008 −0.012 −0.009 −0.003 0.002 −0.010 −0.011 −0.011 −0.010 −0.002 −0.012 −0.016 −0.017 −0.010 −0.006 −0.025 −0.028 −0.029 −0.009
0.144 0.113 0.105 0,106 0.160 0.182 0.141 0.133 0.137 0.178 0.195 0.151 0.140 0.154 0.194 0.222 0.173 0.166 0.171 0.204 0.289 0.230 0.216 0.217 0.236
0.015 0.010 0.007 0.005 0.009 0.013 0.003 0.003 −0.002 0.008 0.009 0.004 0.003 −0.000 0.011 0.008 0.004 0.002 0.002 0.000 0.017 0.004 0.001 −0.005 −0.011
0.074 0.065 0.066 0.074 0.122 0.132 0.109 0.102 0.103 0.142 0.144 0.121 0.115 0.119 0.155 0.162 0.141 0.131 0.135 0.177 0.225 0.193 0.187 0.193 0.206
−0.013 −0.002 0.009 0.009 0.014 0.014 0.011 0.008 0.007 0.014 0.014 0.008 0.008 0.006 0.014 0.013 0.006 0.007 0.007 0.011 0.012 0.012 0.013 0.007 0.006
0.052 0.034 0.033 0.049 0.103 0.075 0.081 0.086 0.088 0.116 0.100 0.100 0.097 0.102 0.132 0.120 0.114 0.112 0.115 0.151 0.169 0.159 0.158 0.165 0.206
−0.019(−0.037) −0.014(−0.037) −0.011(−0.029) 0.003(0.001) 0.010(0.008) −0.005(−0.013) 0.008(0.011) 0.007(0.014) 0.006(0.006) 0.006(0.024) 0.008(0.015) 0.004(0.011) 0.003(0.003) 0.004(−0.005) 0.012(0.003) 0.003(0.009) 0.003(−0.003) 0.004(−0.004) −0.000(−0.004) 0.010(0.040) −0.003(0.001) −0.001(0.007) 0.000(0.016) 0.007(0.007) −0.005(0.069)
0.104(0.154) 0.075(0.123) 0.047(0.086) 0.026(0.059) 0.095(0.170) 0.031(0.065) 0.039(0.064) 0.063(0.079) 0.070(0.106) 0.116(0.203) 0.046(0.071) 0.059(0.090) 0.073(0.106) 0.087(0.137) 0.130(0.226) 0.069(0.095) 0.081(0.115) 0.093(0.140) 0.099(0.171) 0.146(0.290) 0.120(0.185) 0.127(0.206) 0.142(0.227) 0.150(0.227) 0.195(0.334)
0.067 −0.011 −0.014 −0.009 0.010 −0.020 −0.012 0.003 0.007 0.009 −0.017 0.005 0.008 0.007 0.010 0.004 0.009 0.007 0.005 0.016 0.007 0.005 0.005 0.004 0.005
0.130 0.114 0.095 0.051 0.083 0.094 0.047 0.029 0.046 0.108 0.058 0.033 0.047 0.069 0.117 0.039 0.054 0.073 0.090 0.136 0.104 0.119 0.126 0.135 0.187
0.444 0.156 0.019 −0.009 0.013 0.031 −0.011 −0.010 0.002 0.011 −0.011 −0.014 −0.007 0.011 0.012 −0.021 −0.010 0.011 0.009 0.020 0.012 0.011 0.009 0.009 0.015
0.455 0.187 0.113 0.093 0.073 0.121 0.105 0.066 0.027 0.100 0.121 0.075 0.034 0.045 0.111 0.104 0.043 0.041 0.067 0.128 0.059 0.083 0.102 0.119 0.183
1.040 0.589 0.286 0.027 0.014 0.421 0.096 −0.003 −0.006 0.012 0.207 0.009 −0.012 −0.008 0.012 0.063 −0.012 −0.011 0.012 0.013 −0.028 −0.001 0.015 0.010 0.021
1.046 0.598 0.304 0.112 0.056 0.433 0.143 0.108 0.065 0.094 0.232 0.115 0.092 0.034 0.102 0.133 0.111 0.061 0.040 0.121 0.123 0.053 0.069 0.098 0.166
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Note: The median predictions using Chen (2010)’s are presented in parentheses.
19
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 27 Quantile predictions Qτ (Y ∗ |X = x): DGP3
20
N = 250.
x
τ = 0.2
Value
Bias
RMSE
Bias
τ = 0.3 RMSE
Bias
τ = 0.4 RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) (−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
0.017 −0.014 −0.031 −0.037 −0.041 0.011 −0.033 −0.046 −0.063 −0.072 −0.002 −0.042 −0.065 −0.072 −0.073 0.006 −0.046 −0.076 −0.086 −0.083 0.025 −0.032 −0.067 −0.094 −0.059
0.211 0.165 0.155 0.161 0.207 0.240 0.191 0.182 0.194 0.260 0.267 0.200 0.199 0.211 0.268 0.295 0.234 0.228 0.245 0.280 0.390 0.316 0.302 0.313 0.348
0.001 −0.020 −0.027 −0.028 −0.024 −0.003 −0.030 −0.035 −0.043 −0.060 0.000 −0.027 −0.037 −0.048 −0.062 −0.011 −0.046 −0.063 −0.069 −0.060 0.011 −0.055 −0.064 −0.074 −0.053
0.163 0.140 0.136 0.135 0.185 0.207 0.177 0.171 0.183 0.227 0.229 0.201 0.192 0.191 0.256 0.254 0.222 0.217 0.225 0.268 0.351 0.312 0.300 0.302 0.330
0.002 −0.005 0.012 −0.019 −0.022 −0.011 −0.024 −0.027 −0.034 −0.046 −0.018 −0.028 −0.035 −0.042 −0.062 −0.021 −0.040 −0.039 −0.045 −0.057 −0.038 −0.057 −0.065 −0.071 −0.062
0.112 0.094 0.099 0.115 0.166 0.180 0.166 0.165 0.174 0.208 0.213 0.198 0.196 0.201 0.256 0.250 0.232 0.225 0.229 0.288 0.357 0.337 0.328 0.339 0.341
−0.012(−0.159) −0.021(−0.155) −0.021(−0.113) 0.010(−0.028)
0.181(0.304) 0.136(0.296) 0.101(0.252) 0.070(0.177) 0.167(0.311) 0.085(0.197) 0.094(0.174) 0.119(0.189) 0.145(0.250) 0.215(0.375) 0.114(0.195) 0.135(0.230) 0.158(0.256) 0.185(0.295) 0.249(0.426) 0.153(0.249) 0.181(0.286) 0.201(0.317) 0.221(0.351) 0.269(0.503) 0.272(0.393) 0.286(0.442) 0.301(0.454) 0.306(0.493) 0.352(0.632)
0.363 0.152 0.079 0.034 0.080 0.079 0.042 0.018 0.038 0.120 0.042 0.020 0.023 0.073 0.128 0.023 0.022 0.066 0.099 0.139 0.075 0.111 0.129 0.147 0.152
0.409 0.258 0.215 0.138 0.153 0.220 0.149 0.090 0.085 0.215 0.179 0.104 0.081 0.130 0.241 0.124 0.090 0.123 0.179 0.266 0.159 0.215 0.254 0.283 0.355
0.788 0.388 0.155 0.027 0.045 0.233 0.033 0.008 −0.006 0.072 0.082 −0.004 −0.004 0.005 0.076 0.002 −0.011 −0.011 0.034 0.086 −0.036 0.020 0.038 0.060 0.084
0.803 0.422 0.243 0.182 0.108 0.288 0.185 0.151 0.079 0.171 0.199 0.167 0.104 0.051 0.184 0.185 0.129 0.066 0.081 0.205 0.126 0.078 0.118 0.178 0.286
1.267 0.728 0.382 0.089 0.033 0.586 0.186 0.032 −0.004 0.048 0.345 0.055 −0.004 −0.013 0.063 0.150 −0.015 −0.016 −0.005 0.066 −0.048 −0.036 0.010 0.031 0.074
1.277 0.745 0.417 0.211 0.085 0.605 0.251 0.182 0.127 0.133 0.379 0.180 0.152 0.081 0.160 0.220 0.174 0.123 0.055 0.173 0.188 0.113 0.066 0.116 0.240
Note: The median predictions using Chen (2010)’s are presented in parentheses.
τ = 0.5
0.048(0.056) −0.002(−0.047) 0.032(0.022) 0.042(0.034) 0.043(0.060) 0.059(0.049) 0.039(0.033) 0.046(0.053) 0.048(0.062) 0.052(0.061) 0.049(0.051) 0.049(0.062) 0.053(0.070) 0.056(0.070) 0.064(0.067) 0.048(0.105) 0.072(0.084) 0.076(0.084) 0.071(0.066) 0.060(0.078) 0.057(0.241)
τ = 0.6
τ = 0.7
τ = 0.8 S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 28 Quantile predictions Qτ (Y ∗ |X = x): DGP4
N = 500.
x
τ = 0.2
Value
Bias
RMSE
τ = 0.3 Bias
RMSE
Bias
τ = 0.4 RMSE
Bias
τ = 0.5 RMSE
Bias
τ = 0.6 RMSE
τ = 0.7 Bias
RMSE
τ = 0.8 Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) (−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
0.011 −0.003 −0.007 −0.017 −0.020 −0.012 −0.006 −0.018 −0.024 −0.036 0.007 −0.010 −0.016 −0.030 −0.034 0.008 −0.017 −0.029 −0.036 −0.043 0.020 −0.009 −0.024 −0.035 −0.030
0.142 0.113 0.108 0.107 0.143 0.168 0.131 0.122 0.124 0.167 0.180 0.144 0.137 0.141 0.173 0.198 0.161 0.154 0.156 0.186 0.259 0.209 0.195 0.197 0.210
−0.001 −0.011 −0.010 −0.014 −0.017 −0.002 −0.012 −0.018 −0.026 −0.026 −0.008 −0.020 −0.020 −0.024 −0.031 −0.007 −0.016 −0.024 −0.029 −0.029 −0.006 −0.024 −0.028 −0.037 −0.027
0.115 0.100 0.095 0.098 0.128 0.143 0.121 0.116 0.118 0.148 0.158 0.134 0.127 0.131 0.165 0.170 0.145 0.138 0.142 0.178 0.222 0.192 0.186 0.187 0.206
−0.002 −0.008 −0.016 −0.020 −0.020 −0.023 −0.028 −0.025 −0.025 −0.027 −0.026 −0.028 −0.032 −0.029 −0.036 −0.031 −0.035 −0.036 −0.040 −0.038 −0.047 −0.050 −0.050 −0.055 −0.035
0.066 0.068 0.075 0.083 0.113 0.129 0.122 0.118 0.120 0.144 0.154 0.144 0.138 0.138 0.162 0.178 0.164 0.161 0.162 0.181 0.242 0.227 0.221 0.218 0.228
−0.003(−0.090) −0.008(−0.087) −0.008(−0.058) 0.012(0.002) 0.032(0.047) 0.004(−0.015) 0.024(0.031) 0.025(0.038) 0.025(0.033) 0.034(0.029) 0.027(0.037) 0.028(0.040) 0.026(0.042) 0.035(0.047) 0.041(0.041) 0.030(0.046) 0.033(0.036) 0.039(0.044) 0.040(0.048) 0.037(0.053) 0.047(0.080) 0.044(0.066) 0.049(0.054) 0.049(0.085) 0.044(0.193)
0.132(0.243) 0.094(0.215) 0.062(0.169) 0.046(0.125) 0.117(0.266) 0.051(0.139) 0.063(0.138) 0.083(0.156) 0.105(0.186) 0.146(0.283) 0.072(0.153) 0.095(0.172) 0.110(0.198) 0.127(0.251) 0.164(0.329) 0.107(0.189) 0.126(0.214) 0.136(0.257) 0.150(0.290) 0.183(0.390) 0.178(0.324) 0.191(0.328) 0.200(0.351) 0.215(0.372) 0.243(0.518)
0.328 0.117 0.056 0.027 0.047 0.059 0.036 0.017 0.023 0.073 0.041 0.017 0.013 0.043 0.080 0.023 0.013 0.043 0.060 0.088 0.049 0.068 0.078 0.094 0.104
0.350 0.194 0.162 0.102 0.099 0.169 0.113 0.064 0.050 0.139 0.132 0.069 0.044 0.080 0.157 0.089 0.047 0.078 0.115 0.180 0.093 0.136 0.157 0.182 0.243
0.789 0.371 0.122 0.007 0.020 0.206 0.007 −0.003 −0.002 0.031 0.048 −0.007 −0.006 −0.002 0.035 −0.018 −0.014 −0.008 0.016 0.036 −0.025 0.010 0.017 0.023 0.047
0.796 0.386 0.173 0.136 0.066 0.234 0.141 0.112 0.054 0.103 0.141 0.118 0.080 0.032 0.111 0.143 0.097 0.045 0.046 0.123 0.092 0.043 0.076 0.111 0.170
1.272 0.730 0.367 0.055 0.017 0.585 0.163 0.004 −0.006 0.021 0.332 0.023 −0.016 −0.007 0.027 0.129 −0.027 −0.013 −0.005 0.036 −0.043 −0.023 0.008 0.013 0.040
1.276 0.737 0.382 0.141 0.050 0.595 0.195 0.130 0.091 0.089 0.349 0.131 0.120 0.056 0.100 0.173 0.132 0.090 0.030 0.123 0.143 0.071 0.037 0.078 0.153
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Note: The median predictions using Chen (2010)’s are presented in parentheses.
21
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 29 Quantile predictions Qτ (Y ∗ |X = x): DGP4
22
N = 250.
x
τ = 0.2
Value
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) (−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
−0.017 −0.038 −0.053 −0.066 −0.075
0.192 0.161 0.158 0.161 0.219 0.365 0.273 0.259 0.259 0.288 0.530 0.365 0.314 0.294 0.339 0.929 0.565 0.428 0.370 0.408 3.018 2.275 2.641 1.210 0.564
−0.001 −0.010 −0.016 −0.023 −0.028 −0.011 −0.036 −0.051 −0.060 −0.057
0.094 0.080 0.083 0.106 0.176 0.224 0.194 0.187 0.193 0.222 0.309 0.258 0.240 0.237 0.268 0.481 0.355 0.318 0.306 0.334 1.725 1.239 0.894 0.705 0.482
−0.041 −0.021 −0.001 0.002 0.004 0.011 0.003 −0.002 −0.015 −0.007 0.241 0.002 −0.010 −0.018 −0.014 0.037 0.003 −0.012 −0.017 −0.018 0.222 0.127 0.072 0.014 −0.037
0.114 0.076 0.054 0.062 0.143 0.126 0.126 0.131 0.134 0.183 0.198 0.171 0.171 0.177 0.223 0.285 0.247 0.222 0.215 0.270 0.858 0.567 0.455 0.383 0.443
−0.028(−0.065) −0.025(−0.065) −0.019(−0.054) 0.005(−0.007) 0.034(0.006) −0.010(−0.023) 0.013(0.020) 0.017(0.021) 0.017(0.016) 0.028(0.019) 0.014(0.023) 0.017(0.024) 0.014(0.020) 0.022(0.000) 0.031(0.054) 0.017(0.023) 0.013(0.016) 0.018(0.007) 0.014(0.040) 0.024(0.092) 0.008(0.049) 0.014(0.069) 0.011(0.059) 0.010(0.084) 0.001(0.187)
0.159(0.198) 0.112(0.168) 0.071(0.121) 0.049(0.093) 0.143(0.230) 0.061(0.102) 0.064(0.099) 0.085(0.114) 0.112(0.159) 0.175(0.306) 0.073(0.108) 0.098(0.138) 0.117(0.166) 0.141(0.204) 0.205(0.361) 0.113(0.151) 0.136(0.185) 0.152(0.215) 0.172(0.265) 0.240(0.429) 0.212(0.308) 0.238(0.333) 0.246(0.360) 0.265(0.435) 0.396(0.608)
0.028 0.011 0.005 −0.001 0.068 −0.000 −0.006 0.015 0.048 0.078 −0.012 0.013 0.049 0.062 0.087 −0.005 0.049 0.064 0.074 0.094 0.073 0.087 0.089 0.100 0.085
0.215 0.175 0.123 0.067 0.157 0.149 0.086 0.066 0.103 0.196 0.120 0.079 0.101 0.140 0.219 0.110 0.120 0.145 0.179 0.257 0.211 0.241 0.267 0.289 0.371
0.192 0.057 0.035 0.020 0.072 0.025 0.010 0.000 0.019 0.095 −0.005 −0.007 −0.004 0.053 0.098 −0.036 −0.021 0.010 0.064 0.118 −0.103 −0.038 0.043 0.066 0.120
0.288 0.220 0.196 0.124 0.162 0.240 0.181 0.121 0.079 0.211 0.240 0.163 0.113 0.118 0.231 0.251 0.160 0.123 0.155 0.265 0.308 0.234 0.230 0.256 0.358
0.618 0.274 0.115 0.048 0.075 0.350 0.075 0.015 0.013 0.098 0.275 0.022 −0.017 −0.002 0.106 0.219 −0.019 −0.047 −0.008 0.108 0.190 −0.098 −0.129 −0.015 0.099
0.647 0.340 0.245 0.202 0.153 0.403 0.240 0.210 0.117 0.203 0.340 0.236 0.205 0.101 0.223 0.301 0.239 0.211 0.102 0.242 0.301 0.295 0.288 0.199 0.330
0.001
−0.067 −0.092 −0.111 −0.108 −0.082 −0.024 −0.072 −0.102 −0.136 0.237 0.053 −0.036 −0.102 −0.155 1.052 0.736 0.519 0.194 −0.092
τ = 0.3
0.016 −0.026 −0.040 −0.061 −0.070 0.073 0.009 −0.031 −0.047 −0.092 0.541 0.291 0.140 0.031 −0.076
τ = 0.4
Note: The median predictions using Chen (2010)’s are presented in parentheses.
τ = 0.5
τ = 0.6
τ = 0.7
τ = 0.8 S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 30 Quantile predictions Qτ (Y ∗ |X = x): DGP5
N = 500.
x
τ = 0.2
Value
Bias
RMSE
τ = 0.3 Bias
RMSE
Bias
τ = 0.4 RMSE
Bias
τ = 0.5 RMSE
Bias
τ = 0.6 RMSE
τ = 0.7 Bias
RMSE
τ = 0.8 Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) (−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
0.012 −0.001 −0.008 −0.016 −0.006 0.009 −0.011 −0.023 −0.030 −0.025 0.023 −0.008 −0.019 −0.032 −0.032 0.052 −0.001 −0.023 −0.037 −0.042 0.226 0.124 0.055 0.011 −0.011
0.137 0.117 0.109 0.114 0.154 0.243 0.188 0.172 0.171 0.198 0.319 0.242 0.218 0.210 0.232 0.440 0.307 0.266 0.253 0.266 0.996 0.722 0.567 0.464 0.356
0.011 0.004 −0.000 −0.004 −0.004 0.000 −0.011 −0.013 −0.017 −0.014 0.007 −0.004 −0.013 −0.018 −0.015 0.021 −0.004 −0.013 −0.019 −0.026 0.084 0.034 0.006 −0.015 −0.028
0.057 0.057 0.063 0.079 0.123 0.151 0.133 0.126 0.128 0.165 0.203 0.175 0.167 0.162 0.187 0.280 0.227 0.207 0.200 0.216 0.586 0.444 0.382 0.335 0.287
−0.018 −0.008 0.006 0.006 0.001 0.010 0.003 −0.002 −0.006 0.003 0.009 0.002 −0.001 −0.005 −0.000 0.015 0.004 −0.000 −0.006 −0.002 0.042 0.025 0.013 −0.003 −0.014
0.059 0.033 0.030 0.046 0.100 0.085 0.087 0.091 0.094 0.134 0.129 0.120 0.120 0.123 0.158 0.177 0.161 0.157 0.159 0.191 0.354 0.304 0.279 0.263 0.279
−0.021(0.043) −0.017(0.037) −0.012(0.028) 0.002(0.002) 0.011(0.002) −0.006(0.012) 0.007(0.011) 0.006(0.010) 0.002(0.008) 0.007(0.027) 0.006(0.014) 0.004(0.009) 0.000(0.011) 0.001(−0.004) 0.008(0.010) 0.003(0.010) −0.001(0.006) −0.001(0.001) 0.000(0.006) 0.002(0.065) −0.006(0.017) −0.008(0.032) −0.010(0.047) −0.004(0.039) −0.013(0.109)
0.102(0.163) 0.073(0.118) 0.046(0.084) 0.025(0.062) 0.100(0.186) 0.032(0.064) 0.039(0.068) 0.056(0.081) 0.080(0.118) 0.126(0.238) 0.046(0.077) 0.069(0.099) 0.085(0.124) 0.098(0.155) 0.146(0.266) 0.078(0.113) 0.097(0.140) 0.107(0.164) 0.123(0.214) 0.168(0.349) 0.150(0.245) 0.166(0.279) 0.173(0.293) 0.192(0.320) 0.262(0.415)
0.005 0.002 0.001 −0.003 0.028 −0.004 −0.006 0.004 0.020 0.030 −0.008 0.003 0.021 0.020 0.034 −0.009 0.026 0.026 0.031 0.037 0.044 0.044 0.039 0.034 0.036
0.152 0.111 0.080 0.042 0.094 0.097 0.052 0.034 0.057 0.120 0.075 0.042 0.056 0.081 0.135 0.063 0.067 0.085 0.110 0.164 0.132 0.158 0.174 0.186 0.241
0.144 0.016 −0.000 0.000 0.022 0.003 −0.011 −0.005 −0.001 0.024 −0.012 −0.013 −0.010 0.017 0.027 −0.024 −0.019 −0.005 0.022 0.033 −0.054 −0.015 0.021 0.021 0.030
0.197 0.151 0.131 0.086 0.087 0.173 0.134 0.077 0.039 0.121 0.174 0.115 0.058 0.052 0.136 0.177 0.103 0.053 0.076 0.155 0.196 0.100 0.099 0.142 0.220
0.610 0.238 0.059 0.002 0.023 0.333 0.045 −0.010 −0.005 0.026 0.259 0.012 −0.019 −0.010 0.031 0.210 −0.006 −0.022 −0.009 0.026 0.194 −0.039 −0.058 0.015 0.026
0.624 0.272 0.161 0.135 0.073 0.358 0.169 0.162 0.084 0.112 0.296 0.177 0.153 0.065 0.127 0.258 0.188 0.148 0.058 0.145 0.264 0.226 0.185 0.083 0.202
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Note: The median predictions using Chen (2010)’s are presented in parentheses.
23
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 31 Quantile predictions Qτ (Y ∗ |X = x): DGP5
24
N = 250.
x
τ = 0.2
Value
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) (−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
−0.006 −0.036 −0.058 −0.071 −0.082
0.218 0.187 0.183 0.184 0.236 0.337 0.260 0.253 0.268 0.336 0.493 0.338 0.311 0.328 0.395 0.816 0.530 0.432 0.405 0.455 3.321 1.811 1.433 1.114 0.627
−0.008 −0.025 −0.040 −0.049 −0.062 −0.011 −0.055 −0.074 −0.094 −0.134
0.156 0.142 0.146 0.155 0.207 0.287 0.236 0.236 0.238 0.306 0.402 0.302 0.281 0.278 0.356 0.606 0.417 0.361 0.358 0.411 1.857 1.211 0.870 0.684 0.560
−0.020 −0.016 −0.024 −0.030 −0.035 −0.032 −0.057 −0.060 −0.065 −0.091 −0.046 −0.070 −0.082 −0.091 −0.122 −0.042 −0.085 −0.106 −0.122 −0.139 0.093 −0.042 −0.093 −0.135 −0.137
0.130 0.103 0.103 0.120 0.196 0.237 0.218 0.216 0.223 0.272 0.309 0.285 0.274 0.271 0.334 0.435 0.368 0.348 0.342 0.396 1.102 0.781 0.682 0.570 0.558
0.031(−0.186) −0.033(−0.153) −0.026(−0.102) 0.012(−0.032) 0.056(0.026) −0.004(−0.040) 0.034(0.025) 0.047(0.053) 0.055(0.066) 0.059(0.026) 0.042(0.048) 0.056(0.066) 0.059(0.074) 0.060(0.080) 0.056(0.076) 0.062(0.090) 0.064(0.104) 0.066(0.097) 0.072(0.099) 0.049(0.148) 0.083(0.164) 0.089(0.190) 0.081(0.169) 0.078(0.184) 0.063(0.374)
0.210(0.360) 0.160(0.302) 0.118(0.234) 0.088(0.169) 0.179(0.327) 0.114(0.204) 0.114(0.186) 0.143(0.216) 0.179(0.271) 0.242(0.409) 0.139(0.228) 0.170(0.251) 0.194(0.289) 0.208(0.352) 0.284(0.518) 0.197(0.300) 0.217(0.356) 0.230(0.380) 0.252(0.433) 0.342(0.623) 0.311(0.546) 0.344(0.621) 0.370(0.644) 0.389(0.681) 0.535(0.922)
0.155 0.096 0.082 0.050 0.124 0.089 0.077 0.045 0.065 0.176 0.076 0.059 0.057 0.111 0.191 0.058 0.057 0.083 0.159 0.217 0.027 0.114 0.194 0.245 0.269
0.341 0.275 0.215 0.128 0.204 0.288 0.181 0.121 0.138 0.283 0.271 0.157 0.139 0.187 0.307 0.268 0.168 0.176 0.262 0.360 0.338 0.282 0.342 0.411 0.532
0.340 0.115 0.056 0.048 0.082 0.122 0.021 0.034 0.020 0.126 0.072 0.013 0.015 0.021 0.137 0.042 −0.015 −0.005 0.038 0.156 −0.010 −0.079 −0.050 0.075 0.179
0.413 0.295 0.260 0.191 0.165 0.284 0.252 0.190 0.114 0.239 0.284 0.253 0.176 0.102 0.259 0.288 0.255 0.175 0.120 0.290 0.322 0.316 0.236 0.193 0.402
0.690 0.283 0.105 0.034 0.066 0.448 0.074 −0.004 0.012 0.105 0.390 0.025 −0.035 0.003 0.109 0.361 −0.013 −0.059 −0.014 0.110 0.419 −0.060 −0.151 −0.065 0.121
0.717 0.362 0.278 0.244 0.142 0.490 0.251 0.252 0.158 0.208 0.439 0.254 0.255 0.135 0.223 0.414 0.262 0.259 0.134 0.233 0.467 0.295 0.353 0.218 0.307
0.006
−0.067 −0.103 −0.123 −0.149 0.082
−0.037 −0.103 −0.143 −0.172 0.212 0.017 −0.079 −0.143 −0.177 0.965 0.485 0.268 0.071 −0.130
τ = 0.3
0.002 −0.063 −0.096 −0.120 −0.139 0.053 −0.060 −0.101 −0.131 −0.168 0.516 0.233 0.079 −0.034 −0.125
τ = 0.4
Note: The median predictions using Chen (2010)’s are presented in parentheses.
τ = 0.5
τ = 0.6
τ = 0.7
τ = 0.8 S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 32 Quantile predictions Qτ (Y ∗ |X = x): DGP6
N = 500.
x
τ = 0.2
Value
Bias
RMSE
τ = 0.3 Bias
RMSE
Bias
τ = 0.4 RMSE
Bias
τ = 0.5 RMSE
Bias
τ = 0.6 RMSE
τ = 0.7 Bias
RMSE
τ = 0.8 Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) (−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
0.009 −0.005 −0.016 −0.022 −0.034 0.022 −0.012 −0.031 −0.038 −0.065 0.041 −0.012 −0.039 −0.059 −0.070 0.071 0.000 −0.033 −0.058 −0.071 0.257 0.132 0.049 −0.020 −0.056
0.157 0.130 0.124 0.125 0.162 0.232 0.176 0.166 0.176 0.207 0.285 0.215 0.207 0.207 0.233 0.399 0.273 0.248 0.243 0.270 0.928 0.643 0.513 0.410 0.350
0.005 −0.004 −0.012 −0.016 −0.027 0.007 −0.015 −0.023 −0.032 −0.042 0.009 −0.023 −0.029 −0.042 −0.055 0.026 −0.016 −0.036 −0.052 −0.052 0.128 0.047 0.008 −0.035 −0.054
0.112 0.106 0.104 0.107 0.143 0.194 0.166 0.155 0.161 0.185 0.245 0.203 0.189 0.185 0.221 0.321 0.251 0.234 0.232 0.251 0.660 0.505 0.432 0.379 0.344
−0.003 −0.007 −0.014 −0.018 −0.028 −0.027 −0.033 −0.038 −0.043 −0.045 −0.046 −0.049 −0.050 −0.051 −0.054 −0.051 −0.058 −0.062 −0.066 −0.066 −0.067 −0.087 −0.092 −0.096 −0.074
0.061 0.060 0.072 0.085 0.129 0.160 0.152 0.149 0.153 0.177 0.215 0.196 0.190 0.191 0.221 0.289 0.259 0.244 0.234 0.256 0.525 0.444 0.407 0.381 0.362
−0.008(−0.100) −0.013(−0.080) −0.011(−0.056) 0.011(−0.001) 0.028(0.028) 0.001(−0.014) 0.023(0.032) 0.029(0.039) 0.030(0.037) 0.037(0.034) 0.028(0.038) 0.034(0.047) 0.034(0.052) 0.035(0.055) 0.044(0.060) 0.040(0.065) 0.040(0.056) 0.043(0.074) 0.040(0.067) 0.042(0.096) 0.053(0.137) 0.058(0.109) 0.059(0.123) 0.058(0.133) 0.056(0.257)
0.140(0.271) 0.104(0.200) 0.066(0.157) 0.048(0.122) 0.122(0.288) 0.060(0.142) 0.068(0.143) 0.092(0.167) 0.117(0.206) 0.170(0.312) 0.082(0.167) 0.109(0.197) 0.126(0.226) 0.144(0.294) 0.197(0.385) 0.128(0.231) 0.145(0.262) 0.162(0.319) 0.173(0.354) 0.237(0.486) 0.222(0.427) 0.240(0.443) 0.259(0.489) 0.279(0.512) 0.344(0.691)
0.091 0.066 0.050 0.028 0.056 0.058 0.048 0.024 0.020 0.087 0.053 0.035 0.016 0.049 0.098 0.047 0.021 0.026 0.069 0.117 0.014 0.039 0.088 0.116 0.145
0.238 0.189 0.147 0.091 0.113 0.198 0.129 0.075 0.051 0.161 0.182 0.101 0.060 0.087 0.186 0.159 0.084 0.070 0.126 0.221 0.164 0.117 0.170 0.225 0.328
0.292 0.053 0.007 0.017 0.028 0.068 −0.011 0.002 −0.000 0.046 0.024 −0.024 −0.001 −0.006 0.051 −0.015 −0.030 −0.009 0.003 0.056 −0.051 −0.065 −0.030 0.024 0.078
0.325 0.194 0.187 0.131 0.083 0.187 0.192 0.136 0.073 0.130 0.196 0.188 0.116 0.052 0.137 0.206 0.186 0.102 0.046 0.155 0.249 0.215 0.100 0.086 0.237
0.673 0.242 0.037 0.007 0.026 0.431 0.041 −0.026 −0.006 0.040 0.371 −0.001 −0.035 −0.010 0.037 0.341 −0.030 −0.041 −0.013 0.047 0.402 −0.071 −0.086 −0.023 0.053
0.685 0.283 0.175 0.163 0.067 0.450 0.169 0.184 0.114 0.122 0.394 0.175 0.179 0.096 0.126 0.366 0.191 0.178 0.080 0.136 0.423 0.229 0.229 0.075 0.196
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Note: The median predictions using Chen (2010)’s are presented in parentheses.
25
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 33 Quantile predictions Qτ (Y ∗ |X = x): DGP6
26
N = 250.
x
τ = 0.2
Value
Bias
RMSE
τ = 0.3 Bias
RMSE
Bias
τ = 0.4 RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) (−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
0.041 −0.010 −0.024 −0.039 −0.024 0.039 −0.012 −0.027 −0.034 −0.052 0.046 −0.016 −0.030 −0.062 −0.020 0.042 −0.018 −0.048 −0.061 −0.050 0.087 0.001 −0.023 −0.048 −0.035
0.254 0.199 0.189 0.195 0.221 0.331 0.262 0.235 0.232 0.283 0.369 0.303 0.281 0.281 0.293 0.411 0.347 0.325 0.320 0.332 0.518 0.392 0.370 0.358 0.378
0.022 0.004 −0.008 −0.018 −0.001 0.032 −0.003 −0.022 −0.005 −0.035 0.032 0.007 0.008 −0.013 0.007 0.021 −0.008 −0.019 −0.024 −0.003 0.036 −0.014 −0.020 0.010 0.006
0.124 0.107 0.103 0.121 0.185 0.210 0188 0.190 0.185 0.232 0.248 0.216 0.206 0.204 0.255 0.255 0.229 0.219 0.241 0.285 0.344 0.316 0.308 0.293 0.317
−0.046 −0.011 0.011 0.005 0.017 0.026 0.005 −0.007 −0.010 −0.018 0.021 −0.002 0.002 0.009 −0.020 0.020 0.007 0.021 −0.001 0.009 0.023 −0.003 −0.030 0.008 0.012
0.138 0.089 0.074 0.082 0.175 0.138 0.142 0.158 0.163 0.203 0.190 0.187 0.189 0.194 0.234 0.236 0.218 0.225 0.219 0.264 0.317 0.297 0.299 0.297 0.321
−0.043(−0.144) −0.053(−0.140) −0.033(−0.036) −0.002(−0.063) 0.031(−0.135) −0.013(−0.062) 0.012(−0.061) 0.005(−0.051) 0.008(−0.024) 0.010(−0.107) 0.011(−0.016) 0.004(−0.067) 0.011(−0.035) 0.027(−0.024) 0.043(0.069) 0.010(−0.062) 0.021(−0.019) 0.029(−0.019) 0.023(−0.023) 0.041(0.055) 0.026(0.024) 0.021(−0.064) 0.011(−0.144) 0.029(0.070) 0.042(0.138)
0.164(0.212) 0.155(0.222) 0.112(0.096) 0.076(0.102) 0.164(0.197) 0.92(0.108) 0.082(0.123) 0.096(0.140) 0.123(0.120) 0.183(0.310) 0.097(0.131) 0.110(0.152) 0.133(0.133) 0.168(0.194) 0.216(0.415) 0.127(0.170) 0.112(0.128) 0.178(0.209) 0.198(0.299) 0.260(0.532) 0.231(0.308) 0.228(0.366) 0.252(0.467) 0.247(0.413) 0.281(0.390)
0.134 0.001 −0.027 −0.022 0.041 −0.045 −0.034 0.001 0.014 0.042 −0.049 0.001 0.016 0.044 0.071 −0.008 0.021 0.041 0.040 0.066 0.044 0.032 0.032 0.065 0.075
0.197 0.155 0.155 0.106 0.147 0.170 0.121 0.082 0.081 0.174 0.147 0.092 0.083 0.149 0.2134 0.110 0.112 0.153 0.180 0.243 0.217 0.217 0.0223 0.260 0.288
0.515 0.232 0.069 −0.024 0.038 0.088 −0.025 −0.039 −0.007 0.044 −0.008 −0.049 −0.033 0.032 0.072 −0.056 −0.040 0.015 0.040 0.073 0.003 0.033 0.033 0.037 0.067
0.532 0.272 0.162 0.157 0.134 0.179 0.165 0.141 0.079 0.181 0.168 0.157 0.120 0.107 0.215 0.178 0.137 0.114 0.150 0.241 0.182 0.190 0.209 0.236 0.292
1.107 0.658 0.362 0.092 0.041 0.494 0.175 0.024 −0.029 0.067 0.290 0.046 −0.030 −0.006 0.082 0.123 −0.026 −0.041 0.038 0.090 −0.058 −0.011 0.042 0.052 0.107
1.116 0.672 0.386 0.170 0.110 0.514 0.225 0.156 0.132 0.183 0.324 0.162 0.160 0.087 0.206 0.197 0.170 0.146 0.112 0.227 0.191 0.152 0.161 0.184 0.279
Note: The median predictions using Chen (2010)’s are presented in parentheses.
τ = 0.5
τ = 0.6
τ = 0.7
τ = 0.8 S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 34 Quantile predictions Qτ (Y ∗ |X = x): DGP7
N = 500.
x
τ = 0.2
Value
Bias
RMSE
τ = 0.3 Bias
RMSE
Bias
τ = 0.4 RMSE
τ = 0.5 Bias
RMSE
τ = 0.6 Bias
RMSE
Bias
τ = 0.7 RMSE
τ = 0.8 Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) (−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
0.028 −0.010 −0.024 −0.033 −0.017 0.047 0.006 −0.008 −0.012 −0.037 0.038 −0.009 −0.027 −0.047 −0.021 0.038 −0.022 −0.044 −0.049 −0.038 0.062 0.008 −0.008 −0.041 −0.006
0.200 0.161 0.147 0.153 0.165 0.267 0.197 0.187 0.175 0.243 0.284 0.230 0.217 0.223 0.230 0.321 0.282 0.274 0.276 0.285 0.391 0.324 0.304 0.306 0.302
0.013 −0.002 −0.009 −0.016 0.001 0.006 −0.013 −0.024 −0.010 −0.035 0.013 −0.001 0.002 −0.025 −0.002 0.011 −0.024 −0.032 −0.041 −0.013 0.002 −0.025 −0.018 −0.018 −0.002
0.091 0.082 0.083 0.090 0.151 0.151 0.141 0.139 0.147 0.194 0.176 0.165 0.158 0.158 0.206 0.192 0.172 0.164 0.181 0.224 0.256 0.246 0.252 0.247 0.252
−0.038 −0.017 −0.001 0.004 0.004 0.013 −0.006 −0.018 −0.017 −0.036 0.001 −0.019 −0.010 0.002 −0.001 −0.007 −0.007 0.009 −0.004 0.007 −0.002 −0.028 −0.042 −0.014 −0.009
0.096 0.057 0.054 0.071 0.133 0.110 0.108 0.115 0.122 0.158 0.140 0.136 0.140 0.158 0.183 0.172 0.167 0.174 0.171 0.215 0.234 0.221 0.221 0.224 0.237
−0.042(−0.165) −0.034(−0.139) −0.025(−0.031) −0.004(−0.063) 0.021(−0.145) −0.014(−0.063) 0.009(−0.082) 0.002(−0.044) 0.001(−0.011) −0.001(−0.072) 0.008(−0.010) 0.001(−0.061) −0.001(−0.022) 0.014(−0.048) 0.016(0.057) 0.001(−0.059) 0.003(−0.005) 0.018(−0.046) 0.019(−0.014) 0.025(0.007) 0.020(0.029) 0.016(−0.062) −0.001(−0.193) 0.002(−0.006) 0.013(0.104)
0.131(0.195) 0.107(0.207) 0.077(0.074) 0.051(0.090) 0.125(0.185) 0.059(0.092) 0.062(0.110) 0.072(0.137) 0.086(0.092) 0.143(0.270) 0.066(0.127) 0.083(0.144) 0.091(0.106) 0.125(0.144) 0.164(0.423) 0.092(0.161) 0.098(0.120) 0.131(0.152) 0.152(0.313) 0.208(0.500) 0.176(0.318) 0.184(0.358) 0.183(0.427) 0.168(0.237) 0.200(0.280)
0.134 −0.006 −0.037 −0.022 0.023 −0.047 −0.033 −0.005 0.004 0.030 −0.044 −0.003 0.004 0.023 0.031 −0.014 0.007 0.021 0.018 0.029 0.016 0.014 0.017 0.023 0.051
0.170 0.116 0.126 0.085 0.119 0.139 0.092 0.056 0.067 0.148 0.114 0.062 0.068 0.106 0.150 0.087 0.083 0.114 0.141 0.202 0.158 0.176 0.186 0.195 0.206
0.518 0.231 0.060 −0.029 0.023 0.069 −0.035 −0.037 −0.008 0.022 −0.015 −0.048 −0.027 0.017 0.043 −0.059 −0.038 0.003 0.020 0.038 −0.011 0.016 0.013 0.004 0.039
0.527 0.252 0.127 0.124 0.103 0.133 0.134 0.111 0.054 0.136 0.133 0.129 0.077 0.075 0.156 0.147 0.098 0.067 0.110 0.183 0.115 0.131 0.142 0.168 0.211
1.114 0.663 0.362 0.074 0.036 0.495 0.161 0.009 −0.032 0.034 0.280 0.025 −0.028 −0.016 0.056 0.106 −0.025 −0.040 0.019 0.062 −0.059 −0.017 0.028 0.012 0.053
1.119 0.670 0.375 0.140 0.100 0.505 0.192 0.127 0.110 0.138 0.299 0.126 0.131 0.059 0.168 0.154 0.139 0.111 0.080 0.179 0.158 0.104 0.115 0.133 0.218
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Note: The median predictions using Chen (2010)’s are presented in parentheses.
27
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 35 Quantile predictions Qτ (Y ∗ |X = x): DGP7
28
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
with l2ij (y, b, γ , g) = lij (y, b, γ , g) − E lij (y, b, γ , g)|Xi , Yi − E lij (y, b, γ , g)|Xj , Yj + E lij (y, b, γ , g)
[
[
]
[
]
[
]]
Then because of (A.5) and Corollary 4 in Sherman (1993), it is straightforward to show that 1 Ln2 (y, θ1 , gn (y, θ1 )) = Op n−1 h− . 2
(
)
Also, similar to the proof of Lemma 7A of Sherman (1994), we have Ln1 (y, θ1 , gn (y, θ1 )) − Ln1 (y, θ1 , g(y, θ1 )) = op n−1/2
(
)
with n
1∑
Ln1 (y, θ1 , g(y, θ1 )) =
n
n
( )
i=1
n 1∑
=
q¯ i (y, θ1 , g (y, θ1 )) + O h22
q¯ i (y, θ1 , g (y, θ1 )) + op n−1/2
(
)
i=1
In addition, it can be shown easily that L(y, θ1 , g(y, θ1 )) = O h22
( )
Therefore, L(y, θ1 , gn (y, θ1 )) = L(y, θ1 , gn (y, θ1 )) − L(y, θ1 , g(y, θ1 ))
+ L(y, θ1 , g(y, θ1 )) ( ) ∂ L (y, β, γ , g¯n (y, θ1 )) = (gn (y, θ1 ) − g(y, θ1 )) + O h22 ∂g ( ) = V (y, θ1 ) (gn (y, θ1 ) − g(y, θ1 )) + op n−1/2 , where g¯n (y, θ1 ) lies between gn (y, θ1 ) and g(y, θ1 ). Therefore 0 = Ln (y, θ1 , gn (y, θ1 ))
( ) ( ) 1 + Ln1 (y, θ1 , g (y, θ1 )) + op n−1/2 = L (y, θ1 , gn (y, θ1 )) + Op n−1 h− 2 ( −1/2 ) = V (y, θ1 ) (gn (y, θ1 ) − g(y, θ1 )) + Ln1 (y, θ1 , g (y, θ1 )) + op n Consequently, from the above results, we can deduce that gn (y, θ1 ) − g(y, θ1 ) =
n 1∑
n
Jy1,y0 (ξi , θ1 ) + op n−1/2
(
)
i=1
uniformly in y ∈ [y, y¯ ] and θ1 ∈ N10 , where Jy1,y0 (ξi , θ1 ) = −V (y, θ1 )−1 q¯ i (y, θ1 , g (y, θ1 )) with V (y, θ1 ) =
1 E 2
[
∂ 2 ri (y,θ1 ,g (y,θ1 )) ∂g2
]
.
We now consider the rate of convergence of
∂ gn (y,θ1 ) ∂θ1
for y ∈ y, y¯ and θ1 ∈ N10 . First write
[
]
∂ gn (y,θ1 ) ∂ b˜
as
∂ gn (y, θ1 ) Lnn = Ldn ∂ b˜ where Lnn
( ( )) ( ) ( ) ∑ diy − djy0 1 Xj′ b Xi′ b − gn (y, θ1 ) 1 X˜ i X˜ j ′ ( ) ( ) ( )− ( ′ ) k = − ( ′ ) n(n − 1) h2 σ Xi′ γ h22 σ Xi′ γ σ Xj γ σ Xi′ γ σ Xj γ i ̸ =j
Ldn
( ( )) ( ) ∑ diy − djy0 1 Xj′ b Xi′ b − gn (y, θ1 ) 1 ′ ( ) ( ) k − ( ′ ) = n(n − 1) h2 σ 2 Xi′ γ h22 σ Xi′ γ σ Xj γ i̸ =j
1
and 1
Working with the U-statistic decomposition of Lnn and Ldn , Ldn = Ld0 + Ldn1 + Ldn2 √
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
29
and Lnn = Ln0 + Lnn1 + Lnn2 . by following the above arguments, we can show that Ld0 Ln0
=
( ) ∂ g (y, θ1 ) + Op h22 ˜ ∂b
1 Ldn1 = Lnn1 = Op (n−1/2 h− 2 )
and 2 Ldn2 = Lnn2 = Op n−1 h− . 2
(
)
uniformly in y ∈ y, y¯ and θ1 ∈ N10 . Consequently, we have
[
]
) ( ∂ gn (y, θ1 ) ∂ g (y, θ1 ) 1 −1 −2 h2 − = Op h22 + n−1/2 h− 2 +n ˜ ˜ ∂b ∂b [ ] uniformly in y ∈ y, y¯ and θ1 ∈ N10 . The uniform rate of convergence for
∂ gn (y,θ1 ) ∂γ
holds similarly.
■
Proof of Theorem 1. We first establish the consistency of βˆ , gn and γˆ , cˆ in three steps. Define a class of functions:
(
FQ =
{(
)
diy − τ Kh2 Xi′ b − g − σ Xi′ γ c : y ∈ y, y , b ∈ B, γ ∈ G, c ∈ C , g ∈ M , h2 > 0
(
)
(
) )
[
]
}
Then using the arguments in the proof of Lemma 10A of Sherman (1994), we can verify that FQ is Euclidean for a constant envelope, and consequently by the uniform law of large numbers e.g. Pakes and Pollard (1989), we obtain Qn (y, θ , g ) − Q (y, θ, g ) = op (1) uniformly in (y, θ , g) and, Qn (θ , g (·, θ1 )) − Q (θ, g (·, θ1 )) =
∫
Qn (y, θ, g (y, θ1 )) ω (y) dy −
∫
Q (y, θ, g (y, θ1 )) ω (y) dy = op (1)
uniformly over θ . We now consider βˆ . First, we establish the identification of β0 . Note that τl0 = 0.5. Define Q¯ l0 (y, β ) = sup Ql0 (y, β, γ , c , g) g ∈M
and Q¯ l0 (β ) =
∫
Q¯ l0 (y, β )ω(y)dy
noting that Ql0 (y, β, γ , c , g) actually does not depend on (γ , c). Then following Manski (1975) and Chen (2010), for any ε0 > 0, max
∥β−β0 ∥≥ε0 ,β∈B
Q¯ l0 (β) = Q¯ l0 (β ∗ ) =
∫
Q¯ l0 (y, β ∗ )ω(y)dy <
∫
Ql0 (y, β0 , g(y))ω(y)dy
and let
∫
Ql0 (y, β0 , g(y))ω(y)dy −
max
∥β−β0 ∥≥ε0 ,β∈B
Q¯ l0 (β) > aε0
for some small aε0 > 0. In addition, for any other l ̸ = l0 , we have max
∥β−β0 ∥≥ε0 ,β∈B,(γ ,c)∈G×C
Q¯ l (β, γ , c ) ≤
∫
Ql (y, θ0 , g(y))ω(y)dy
Thus p ∑
max
∥β−β0 ∥≥ε0 ,β∈B,(γ ,c)∈G×C
Q¯ l (β, γ , c ) <
l=1
p ∫ ∑
Ql (y, θ0 , g(y))ω(y)dy − aε0
l=1
In particular, we have Q (θ0 , g(·, θ10 )) >
≥
sup
Q¯ (β, γ , c) + aε0
sup
Q (θ, gn (·, θ )) + aε0
∥β−β0 ∥≥ε0 ,(γ ,c)∈G×C ∥β−β0 ∥≥ε0 ,(γ ,c)∈G×C
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
30
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
On the other hand,
ˆ gn (·, θˆ1 )) + op (1) Q (θˆ , gn (·, θˆ1 )) = Qn (θ, ≥ Qn (θ0 , gn (·, θ10 )) + op (1) = Q (θ0 , gn (·, θ10 )) + op (1) = Q (θ0 , g(·, θ10 )) + op (1) Therefore, we obtain Q (θˆ , gn (·, θˆ1 )) >
sup
∥β−β0 ∥≥ε0 ,(γ ,c)∈G×C
Q (θ, gn (·, θ )) + aε0 + op (1)
Thus, with probability approaching one, we have βˆ − β0 < ε0 , which establishes the consistency of βˆ .
Next, we establish the uniform consistency of gn (y, θˆ1 ) uniformly in y ∈ [y, y¯ ]. For a continuous function g¯ (·), define supremum norm
¯ − g(y)| ∥g¯ − g ∥ = sup |g(y) y∈[y,¯y]
¯ such that ∥g¯ − g ∥ > 0, it is straightforward to show that Then for any g, Ql0 (β0 , g (·)) > Ql0 (β0 , g¯ (·)) and in particular, for any small ε0 > 0, Ql0 (β0 , g (·)) >
sup ∥g¯ −g ∥≥ε0
Ql0 (β0 , g¯ (·)) + aε0
for some positive aε0 . Then by choosing a small ε1 > 0, such that sup ∥β−β0 ∥≤ε1
⏐ ⏐ ⏐Ql (β0 , g¯ (·)) − Ql (β, g¯ (·))⏐ < aε0 0 0 2
thus Ql0 (β0 , g (·)) >
sup ∥g¯ −g ∥≥ε0 ,∥β−β0 ∥≤ε1
Ql0 (β, g¯ (·)) +
aε0 2
Consequently, we also have Q (β0 , γ0 , c0 , g (·)) >
p ∑
sup ∥g¯ −g ∥≥ε0 ,∥β−β0 ∥≤ε1 ,(γ ,c)∈G×C
Ql (β, γ , c , g¯ (·)) +
l=1
aε0 2
Given the consistency of βˆ there exists a sequence of εn → 0, such that
⏐ ⏐
⏐ ⏐
P(⏐βˆ − β0 ⏐ > εn ) → 0 as n increases Hence, for a large enough n, we obtain Q (θˆ , gn (·, θˆ )) ≥ Q (θ0 , g(·, θ0 )) + op (1)
>
sup
∥g −g0 ∥≥ε0 ,|β−β0 |≤εn ,(γ ,c)∈G×C
Q (β, γ , g) +
aε0 2
+ op (1)
which implies that ∥gn − g0 ∥ = op (1). We have thus established the uniform consistency of gn (·, θˆ1 ). Finally we establish the consistency of γˆ and cˆ . First, we show that once β0 and g (·) are identified, γ0 and c0 can be identified. For any other l such that τl ̸ = 0.5, noting that
[( ( E
[(
diy − τl 1 Xi β0 − g(y) − cτl σ Xi γ > 0
) {
′
(
′
)
}]
=E
F
Xi′ β0 − g(y)
( ) σ Xi′ γ0
)
) { − τl 1
Xi′ β0 − g(y)
( ) σ Xi′ γ
}] > cτl
Thus if (γ , cτl ) maximizes the above expression, then we have Xi′ β0 − g(y)
σ Xi′ γ
(
)
> cτl iff
Xi′ β0 − g(y)
( ) σ Xi′ γ0
> c (τl )
or Xi′ β0 − cτl σ Xi′ γ > g(y) iff Xi′ β0 − c (τl ) σ Xi′ γ0 > g(y)
(
)
(
√
)
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S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
31
for y ∈ y, y , which implies that when Xi′ β0 − c (τl ) σ Xi′ γ0 ∈ g(y), g(y) , we have
(
]
[
)
[
]
Xi′ β0 − c (τl ) σ Xi′ γ0 = Xi′ β0 − cτl σ Xi′ γ = g(y)
(
)
(
)
for some y ∈ ∈ y, y , which, in turn, implies that
[
]
( ) σ x′ γ0 cτl = σ (x′ γ ) c (τl ) for any x ∈ S¯X ; hence, by Assumption A6, we have γ = γ0 , which further implies that cτl = c (τl ). From the above identification result for γ0 and c0 we can deduce that for γ ̸ = γ0 , Ql (y, β0 , γ0 , c0 ) > Ql (y, β0 , γ , c0 ) Then, again, similar to the above argument, we can show that Q (θˆ , gn (·, θˆ )) ≥ Q (θ0 , gn (·, θ0 )) + op (1)
= Q (θ0 , g(·, θ0 )) + op (1) >
sup
∥g −g0 ∥≤εn ,|β−β0 |≤εn ,∥γ −γ0 ∥≥ε0 ,γ ∈G,c ∈C
Q (β, γ , g) +
aε0 2
+ op (1)
for a sequence of εn → 0, which implies the consistency of γˆ . The consistency of cˆ follows similarly. Now we establish the asymptotic normality of θˆ : From the consistency θˆ , it follows that the following first-order condition
( ) ˆ γˆ , cˆ ∂ Qn β, 0=
∂θ
holds with probability approaching one as n increases. A Taylor expansion yields
√
[
n(θˆ − θ0 ) = −
∂ 2 Qn (β ∗ , γ ∗ , c ∗ ) ∂θ∂θ ′
]−1 √
n∂ Qn (β0 , γ0 , c0 )
∂θ
where θ lies on the line segment of θˆ and θ0 . ∗
Note that ∂ Qn (β0 , γ0 , c0 )
∂θ
∂ Qn (y, θ0 , gn (y, θ10 )) ω (y) dy ∂θ ∫ ∂ Qn (y, θ0 , gn (y, θ10 )) ∂ gn (y, θ10 ) + ω (y) dy ∂g ∂θ = Tn1 + Tn2
∫
=
Define
∂ Qn (y, θ0 , gn (y, θ10 )) ∂ g (y, θ10 ) ω (y) dy ∂g ∂θ ) ( ∗ and we now show that Tn2 = Tn2 + op n−1/2 . ∗ Tn2 =
∫
Note that ∗ Tn2 − Tn2
( ) ∂ Qn (y, θ0 , gn (y, θ10 )) ∂ gn (y, θ10 ) ∂ g (y, θ10 ) − ω (y) dy ∂g ∂θ ∂θ ⏐ ⏐ ∂ gn (y, θ10 ) ∂ g (y, θ10 ) ⏐ ⏐ × ⏐T˜n2 ⏐ ≤ sup − ∂θ ∂θ y∈[y,y] ( ) ⏐⏐ ⏐⏐ 1 −1 −2 = Op h22 + n−1/2 h− h2 × ⏐T˜n2 ⏐ 2 +n ∫
=
For T˜n2 , with some standard arguments, e.g., Pollard (1995) and by Lemma A.1, we can show that T˜n2 =
∫ +
∫
n
p
) 1 1 ∑∑ ( diy − τ k n h1 i=1 ℓ=1 p n
) 1 1 ∑∑ ( diy − τ 2 k′ n h1
(
(
Zi1 − g (y) − σ (Z2i ) c0 (τℓ ) h1
Zi1 − g (y) − σ (Z2i ) c0 (τℓ ) h1
i=1 ℓ=1
√
)
)
ω (y) dy
(gn (y, θ10 ) − g (y))ω (y) dy
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32
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
∫ +
p
n
) 1 1 ∑∑ ( diy − τ 3 k′ n h1
(
Zi1 − g (y) − σ (Z2i ) c0 (τℓ )
)
h1
i=1 ℓ=1
(gn (y, θ10 ) − g (y))2 ω (y) dy
) 4 3 | − g y y , θ g sup + Op h− ) ( )| ( 10 n 1 y∈[y,¯y] ) ( ) ( ( )−1/2 ) ( ln n × Op n−1/2 = Op h21 + (nh1 )−1/2 ln n + Op h21 + nh31 ) ( ( ) ( ) ( )−1/2 4 ln n × Op n−1 + Op n−3/2 h− + Op h21 + nh51 1 (
Therefore, with Assumption A10, we have ∗ = op n−1/2 Tn2 − Tn2
(
)
∗ by Assumption 10. Furthermore, write Tn1 and Tn2 as
∂ Qn (y, θ0 , g (y, θ10 )) ω (y) dy ∂θ ∫ 2 ( ) ∂ Qn (y, θ0 , g (y, θ10 )) + (gn (y, θ10 ) − g (y)) ω (y) dy + op n−1/2 ∂θ∂ g
∫
Tn1 =
and
∂ Qn (y, θ0 , g (y, θ10 )) ∂ g (y, θ10 ) ω (y) dy ∂g ∂θ1 ∫ 2 ( ) ∂ Qn (y, θ0 , g (y, θ10 )) + (gn (y, θ10 ) − g (y)) ω (y) dy + op n−1/2 ∂g2
∗ Tn2 =
∫
Then, similar to the arguments in the proof of Theorem 1 in Chen (2010), we can show that n
∗ Tn1 + Tn2 =
1∑ n
( ) (ψ1 (ξi , θ0 ) + ψ2 (ξi , θ0 )) + op n−1/2
i=1
where
ψ1 (ξi , θ0 ) =
p ∑
ϕ1i (τℓ ) ϕ2i (τℓ )
ℓ=1
with
ϕ1i (τℓ ) = (1 {εi < c (τℓ )} − τℓ )
( ) ω g −1 (Zi1 − c (τℓ ) σ (Zi2 )) ( ) g ′ g −1 (Zi1 − c (τℓ ) σ (Zi2 ))
and
)′ ∂ g (g −1 (Zi1 − c (τℓ ) σ (Zi2 )) , θ10 ) ′ ′ ˜ ϕ2i (τℓ ) = Xi , c (τℓ ) σ (Zi2 ) Xi , ιℓ σ (Z2i ) + ∂θ (
and
ψ2 (ξi , θ0 ) =
∫
[
Hθ1g (θ0 , y) + Hθ2g (θ0 , y) Jy1,y0 (ξi , θ10 ) ω (y) dy
]
where Hθ1g
(θ0 , y) =
p ∑ ∂ g (y, θ10 )
ℓ=1
∂θ
Hgg (y, θ0 , τ )
Hgg (y, θ0 , τ ) = E f (c (τℓ )) Jb 0, x˜ , g (y) , τℓ p χ 0, x˜ , g (y) , τℓ ⏐x˜
[
(
) ⏐ )]
) ( (
)′
(
2′ Hθ2g (θ0 , y) = Hβ2˜ ′g (θ0 , y) , Hγ2′g (θ0 , y) , Hcg (θ0 , y)
and Hβ2˜ g (θ0 , y) =
p ∑ [
E x˜ f (c (τℓ )) Jb 0, x˜ , g (y) , τℓ p χ 0, x˜ , g (y) , τℓ ⏐x˜
(
) ( (
) ⏐ )]
ℓ=1
√
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S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
33
( ( ) ) ⎤ χ 0, x˜ , g (y) , τℓ − x˜ ′ β˜ 0 , x˜ ′ × c (τℓ ) ∑ ⎢ ( ( ( ) ) ) ⎥ E ⎣ σ ′ γ10 χ 0, x˜ , g (y) , τℓ − x˜ ′ β˜ 0 + x˜ ′ γ˜0 × ⎦ Hγ2 g (θ0 , y) = − ⏐ ( ) ( ( ) ) ℓ=1 f (c (τℓ )) Jb 0, x˜ , g (y) , τℓ p χ 0, x˜ , g (y) , τℓ ⏐x˜ [ ] ( ( ( ) ) ) σ γ10 χ 0, x˜ , g (y) , τℓ − x˜ ′ β˜ 0 + x˜ ′ γ˜0 × 2 ( ) ( ( )⏐ ) Hcτ g (θ0 , y) = −E ℓ f (c (τℓ )) Jb 0, x˜ , g (y) , τℓ p χ 0, x˜ , g (y) , τℓ ⏐x˜ ⎡
p
Consequently, we obtain
√
n∂ Qn (β0 , γ0 , c0 )
∂θ
n 1 ∑
= √
n
n 1 ∑
= √
n
(ψ1 (ξi , θ0 ) + ψ2 (ξi , θ0 )) + op (1)
i=1
ψ (ξi , θ0 ) + op (1)
i=1
Also, similar to the arguments in the proof of Theorem 1 in Chen (2010), we can show
∂ 2 Qn (b∗ , γ ∗ , c ∗ ) = H (θ0 ) + op (1) ∂θ ∂θ ′ As a result, n ) √ ( 1 ∑ n θˆ − θ0 = √ H (θ0 )−1 ψ (ξi , θ0 ) + op (1)
n
i=1
n 1 ∑
= √
n
Ψ (ξi , θ0 ) + op (1)
i=1
and thus, Theorem 1 then follows by applying the central limit theorem.
■
Proof of Theorem 2. Note that
√ (
(
)
)
n gn y, θˆ1 − g (y) =
√ [( n
(
)
(
gn y, θˆ1 − g y, θˆ1
))
( ( ) )] + g y, θˆ1 − g (y, θ10 ) .
Then from the uniform asymptotic linear representation and the consistency of θˆ1 , the stochastic equicontinuity holds for
√ (
(
)
(
)
(
n gn y, θˆ1 − g y, θˆ1
))
√ −
n (gn (y, θ10 ) − g (y, θ10 )) = op (1)
Therefore,
√ (
)
n gn y, θˆ1 − g (y)
∂ g (y, θ10 ) √ n(θˆ1 − θ10 ) + op (1) ∂θ1 ] n [ q¯ i (y, θ10 , g (y, θ10 )) ∂ g (y, θ10 ) 1 ∑ − = √ + Ψ (ξi , θ0 ) + op (1) V (y, θ10 ) ∂θ1 n √
=
n (gn (y, θ10 ) − g (y)) +
i=1
n ] 1 ∑[ 1 = √ Jy,y0 (ξi , θ10 ) + Jy2 (ξi , θ10 ) + op (1) n i=1
Finally, following Pakes and Pollard (1989), it is straightforward to show that the Jy1,y0 (ξi , θ10 ) : y ∈ y, y and {2 [ ]} Jy (ξi , θ10 ) : y ∈ y, y are both Euclidean class with a square integrable envelop, Theorem 2 follows from Theorem (19.14) of Van der Vaart (1998). ■
{
[
]}
Proof of Theorem 3. Similar of) gn (y, θˆ1 )[ we ]can prove the uniform [ to ] the arguments in proving the uniform√consistency ( consistency of cˆτ for τ ∈ τ , τ . To establish the weak convergence of n cˆτ − c (τ ) for τ ∈ τ , τ , apply a Taylor series expansion to obtain
√ (
n cˆτ − c (τ )
)
( ) ⎞−1 √ ( ) ∂ 2 Qn θˆ1 , gn , τ , c¯ n∂ Qn θˆ1 , gn , τ , c (τ ) ⎠ = ⎝− + op (1) ∂ cτ2 ∂ cτ ⎛
√
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34
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
and we have
( ) ∂ Qn θˆ1 , gn , τ , c (τ ) ∂ cτ ) ∂ Qn (θ10 , g , τ , c (τ )) ∂ 2 Qn (θ10 , g , τ , c (τ )) ( = + θˆ1 − θ10 ∂ cτ ∂ cτ ∂θ1 ∫ 2 ) ( ∂ Qn (y, θ10 , g (y) , τ , c (τ )) + (gn (y, θ10 ) − g (y)) ω (y) dy + op n−1/2 ∂ cτ ∂ g ( ) = Cn1 + Cn2 + Cn3 + op n−1/2 Using the arguments and results in Theorems 1 and 2, it is straightforward to show that Cn1 =
n 1∑
n
Si1 + op n−1/2
(
i=1 n 1∑
∫ Cn2 =
)
n
Si2 (y) ω (y) dy + op n−1/2
)
Si3 (y) ω (y) dy + op n−1/2
)
(
i=1
and n 1∑
∫ Cn3 =
n
(
i=1
where Si1 = ϕ1 (εi , Zi1 , Zi2 , τ ) σ (Zi2 ) and Si2 (y) =
(
Hc2τ θ1
(θ10 , y) +
Hc2τ g
) ∂ g (y, θ10 ) Iθ1 Ψ (ξi , θ0 ) (θ10 , y) ∂θ ′
and Si3 (y) = Hc2τ g (θ10 , y) Jy1,y0 (ξi , θ10 ) + Jy2,y0 (ξi , θ10 )
)
(
Next, following the arguments in the proof of Theorem 1, we can show that
( ) ∂ 2 Qn θˆ1 , gn , τ , c¯ ∂ cτ2
= V (τ , θ10 , g ) + op (1)
uniform in τ , where V (τ , θ10 , g ) = −
∫
[
] ( ( ( ) ) ) σ 2 γ10 χ 0, x˜ , g (y) , τ − x˜ ′ β˜ 0 + x˜ ′ γ˜0 × ( ) ( ( ) ⏐ ) ω (y) dy + op (1) E f (c (τ )) Jb 0, x˜ , g (y) , τ p χ 0, x˜ , g (y) , τ ⏐x˜
Consequently, we have n ) 1 ∑ −1 n cˆτ − c (τ ) = − √ V (τ , θ10 , g ) ϕ (ξi , θ10 , c (τ )) + op (1) n
√ (
i=1
n 1 ∑
= −√
n
Λ (ξi , θ10 , c (τ )) + op (1)
i=1
where
ϕ (ξi , θ10 , c (τ )) = Si1 +
∫
(Si2 (y) + Si3 (y)) ω (y) dy
It is straightforward to show that {Λ (·, θ1 , cτ ) : θ1 ∈ B × G, cτ ∈ C } form an Euclidean class. Then Theorem 3 follows by Theorem (19.14) of Van der Vaart (1998). ■ Proof of Theorem 4. Recall gn yˆ ∗τ (x) = x′ βˆ − σ x′ γˆ cˆτ and g y∗τ (x) = x′ β0 − σ (x′ γ0 )c (τ ), then from Theorems 2 and 3, we have
(
)
(
)
(
)
gn yˆ ∗τ (x) − g y∗τ (x) = op (1)
(
)
(
)
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
35
uniformly in x. In addition, from Lemma A.1 we have gn yˆ ∗τ (x) − g yˆ ∗τ (x) = op (1)
(
)
(
)
uniformly in x. Thus we have g yˆ ∗τ (x) − g y∗τ (x) = op (1)
)
(
)
(
uniformly in x. Then the uniform convergence of yˆ ∗τ (x) follows since 1
⏐ ⏐ ∗ ⏐(yˆ (x) − y∗ (x))⏐ ≤ max τ τ
y∈[y,¯y] g ′ (y)
⏐ ( ∗ ) )⏐ ( ⏐g yˆ (x) − g y∗ (x) ⏐ . τ
τ
Next we establish the weak convergence of
√
√ (
n yˆ ∗τ (x) − y∗τ (x) . Write
)
√
n(gn yˆ ∗τ (x) − g yˆ ∗τ (x) ) as
(
)
(
)
n(gn yˆ ∗τ (x) − g yˆ ∗τ (x) )
)
(
√
)
(
( ) ) √ ( ( ′ ) n σ x γˆ cˆτ − σ x′ γ0 c (τ ) ) ( )) √ ( ( − n g yˆ ∗τ (x) − g y∗τ (x) ) ( ( )√ ( ) ) ( )) √ ( √ ( β˜ − β˜ − σ x′ γ0 n cˆτ − c (τ ) + σ ′ x′ γ0 c (τ ) x′ n γˆ − γ0 = nx˜ ′ ˆ )) ) ( √ ( ( − n g yˆ ∗τ (x) − g y∗τ (x) + op (1) =
( ˆ
)
nx˜ ′ β˜ − β˜ 0 −
From the uniform asymptotic linear representation in Theorem 2, we have
√ (
n gn yˆ ∗τ (x) − g yˆ ∗τ (x)
(
)
(
))
=
√ (
n gn y∗τ (x) − g y∗τ (x)
(
)
(
))
+ op (1)
Thus
√ ( (
n g yˆ ∗τ (x) − g y∗τ (x)
)
(
))
) ( ( )√ ( ) )) ) ( √ ( √ ( β˜ − β˜ 0 − σ x′ γ0 n cˆτ − c (τ ) + σ ′ x′ γ0 c (τ ) x′ n γˆ − γ0 = x˜ ′ n ˆ ) ) ( ( √ − ngn y∗τ (x) − g y∗τ (x) + op (1) ( ) A Taylor expansion of g yˆ ∗τ (x) , together with some simple algebra, yields ) √ ( ∗ n yˆ τ (x) − y∗τ (x) ) ] [ )√ ( ( ) √ ′ (ˆ nx˜ β˜ − β˜ 0 − σ x′ γ0 n cˆτ − c (τ ) − 1 = ′( ∗ ) × )) + op (1) ) ) ( ( ) √ ( ( √ ( g yτ (x) σ ′ x′ γ0 c (τ ) x′ n γˆ − γ0 − n gn y∗τ (x) − g y∗τ (x) ) ] [ ( ′ ( ) )√ ( n θˆ1 − θ10 − x˜ , −σ ′ x′ γ0 c (τ ) x′ 1 = ′( ∗ ) × ( )√ ( ) √ ( ( ) ( )) + op (1) g yτ (x) σ x′ γ0 n cˆτ − c (τ ) − n gn y∗τ (x) − g y∗τ (x) ) ( ) ( ′ ] [ n x˜ , −σ ′ x′ γ0 c (τ ) x′ Iθ1 Ψ (ξi , θ0 ) 1 ∑ 1 ( ) ( ) ( ) = √ + op (1) n −σ x′ γ0 Λ (ξi , θ10 , c (τ )) − Jy1∗τ (x),y (ξi , θ10 ) + Jy2∗τ (x) (ξi , θ10 ) g ′ y∗τ (x) i=1 0
n
1 ∑
= √
n
J¯y,y0 (τ , ξi , x, θ0 )
i=1
Following Pakes and Pollard (1989), it is straightforward to show that the J¯y,y0 (τ , ξi , x, θ0 ) : τ ∈ τ , τ is a Euclidean class of functions with a square integrable envelop. Then, Theorem 4 follows by applying Theorem (19.14) of Van der Vaart (1998). ■
{
[
]}
References Carroll, R.J., Fan, J., Gijbels, I., Wand, M.P., 1997. Generalized partially linear single-index models. J. Amer. Statist. Assoc. 92 (438), 477–489. Cavanagh, C.L., Sherman, R.P., 1998. Rank estimators for monotonic index models. J. Econometrics 84 (2), 351–381. Chen, S., 2002. Rank estimation of transformation models. Econometrica 70 (4), 1683–1697. Chen, S., 2010. An integrated maximum score estimator for a generalized censored quantile regression. J. Econometrics 155 (1), 90–98. Gørgens, Tue, Horowitz, J.L., 1999. Semiparametric estimation of a censored regression model with an unknown transformation of the dependent variable. J. Econometrics 90 (2), 155–191. Han, A.K., 1987. Nonparametric analysis of a generalized regression model: The maximum rank correlation estimator. J. Econometrics 35 (2), 303–316.
√
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Horowitz, J.L., 1992. A smoothed maximum score estimator for the binary response model. Econometrica 63 (3), 505–531. Horowitz, J.L., 1996. Semiparametric estimation of a regression model with an unknown transformation of the dependent variable. Econometrica 64, 103–137. Horowitz, J.L., 2009. Semiparametric and Nonparametric Methods in Econometrics. Springer, New York. Ichimura, H., Lee, L.F., 1991. Semiparametric least squares estimation of multiple index models: Single equation. Nonparametric and Semiparametric Methods in Econometrics and Statistics: Proceedings of the Fifth International Symposium in Economic Theory and Econometrics. Khan, S., 2001. Two stage rank estimation of quantile index models. J. Econometrics 100 (2), 319–355. Koenker, R., 2004. Quantile regression for longitudinal data. J. Multivariate Anal. 91 (1), 74–89. Manski, C.F., 1975. Maximum score estimation of the stochastic utility model of choice. J. Econometrics 3 (3), 205–228. Manski, C.F., 1985. Semiparametric analysis of discrete response: Asymptotic properties of the maximum score estimator. J. Econometrics 27 (3), 313–333. Pakes, A., Pollard, D., 1989. Simulation and the asymptotics of optimization estimators. Econometrica 57, 1027–1057. Pollard, D., 1995. Uniform ratio limit theorems for empirical processes. Scand. J. Statist. 22 (3), 271–278. Sherman, B., 1993. The limiting distribution of the maximum rank correlation estimator. Econometrica 61 (1), 123–137. Sherman, B., 1994. U-processes in the analysis of a generalized semiparametric regression estimator. Econometric Theory 10 (2), 372–395. Van der Vaart, 1998. Asymptotic Statistics. Cambridge University Press, Cambridge. Zhou, X., Lin, H., Johnson, E., 2008. Non-parametric heteroscedastic transformation regression models for skewed data with an application to health Care costs. J. R. Stat. Soc. Ser. B Stat. Methodol. 70 (5), 1029–1047.
√
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√
n-prediction of generalized heteroscedastic transformation regression models✩ ∗
Songnian Chen a , , Hanghui Zhang b,c a
Department of Economics, The Hong Kong University of Science and Technology, Clearwater Bay, Kowloon, Hong Kong School of Economics, The Shanghai University of Finance and Economics, Shanghai 200433, China c Key Laboratory of Mathematical Economics (SUFE), Ministry of Education, Shanghai 200433, China b
article
info
a b s t r a c t Chen (2010) and Khan (2001) consider quantile regression estimation subject to general transformation that permits general heteroscedasticity, but the resulting conditional quantile predictors converge at rates slower than the parametric rate. In this paper, we consider the estimation of a general transformation model subject to a multiplicative form of heteroscedasticity. Our estimators for the finite dimensional parameters, the transformation function and the resulting conditional quantile predictor all converge at the parametric rate. Monte Carlo simulation experiments show that our estimators and conditional quantile predictor perform well in finite samples. © 2019 Elsevier B.V. All rights reserved.
Article history: Received 22 May 2018 Received in revised form 18 September 2019 Accepted 20 September 2019 Available online xxxx JEL classification: C14 C24 C51 Keywords: Generalized transformation regression model Heteroscedasticity √ n-prediction
1. Introduction Generalized transformation regression models have received a great deal of attention in econometrics and statistics literature. An important class of such models maintains the independence assumption between the unobservable error term and regressors, and for such models, Han (1987), Sherman (1993) and Cavanagh and Sherman (1998) developed rank estimators for the finite dimensional parametric index parameters and Horowitz (1996), Gørgens and Horowitz (1999) and Chen √ (2002) developed estimation procedures for the unknown transformation function. All these estimators converge at n-consistent parametric rate (n is the sample size), and consequently the resulting conditional quantile predictors also converge at the parametric rate. The independence assumption is highly restrictive in allowing the way the regressors to affect the outcome variable. A more flexible alternative is the transformation regression model under a conditional quantile restriction. For the transformation model under a conditional quantile restriction, Khan (2001) developed a two-step semiparametric rank estimator and Chen (2010) proposed an integrated maximum score estimator, and both ✩ We would like to thank the associate editor, two anonymous referees, conference participants at Shanghai Econometrics Workshop at the Shanghai University of Fiance and Economics and 2016 China meeting of the Econometric Society at SWUFE for their helpful comments. We also thank Jichun Si for his excellent RA on this project. Chen’s research was supported by the GRF grant 640412 from the University Grants Committee of the government of Hong Kong SAR. Zhang’s research is supported by National Natural Science Foundation of China (NSFC Project No. 71601105 and No. 71833004). Support was also provided by the 2018 Program for Changjiang Scholars and Innovative Research Team in SUFE. ∗ Corresponding author. E-mail address: [email protected] (S. Chen). https://doi.org/10.1016/j.jeconom.2019.09.003 0304-4076/© 2019 Elsevier B.V. All rights reserved.
√
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√
estimators for the finite dimension parameters are n-consistent and asymptotically normal; subsequent conditional quantile predictors, however, converge at rates slower than the parametric rate, leading to general efficiency loss in √ making prediction. The n consistent estimators of Horowitz (1996) and Chen (2002) for homoscedastic models will be inconsistent in the presence of√ heteroscedasticity. Therefore, given the widespread heteroscedastity in empirical research, it is highly desirable to have a n consistent estimator for√the transformation function for heteroscedastic transformation models, which provides the basis for the construction of n consistent conditional quantile function. This paper fills this important gap. In this paper, we consider the estimation of a general transformation model subject to a multiplicative form of heteroscedasticity with a median location normalization. By combining both rank and quantile estimation techniques, our estimators for the finite dimensional parameter and the unknown transformation function both converge at the parametric rate, and consequently, the conditional quantile predictors also converge at the parametric rate. Zhou et al. (2008) also considered the general transformation model with a multiplicative form of heteroscedasticity, but with a √ mean location normalization. They also proposed a n consistent estimator for the transformation function. However, compared with our approach, their method suffers from several serious drawbacks. First, they require a mutual exclusion restriction in a double-index setting, similar to Ichimura and Lee (1991); as a result, their estimation approach is not adaptive to the homoscedasticity setting in that their method requires the actual presence of heteroscedasticity for model identification and estimation. In other words, their method breaks down when there is no heteroscedasticity. Second, their estimation approach makes use of the mean location restriction, and thus could be sensitive to the tail behavior of the unobservable error term since estimation of the transformation function in the tail area can be quite problematic.1 Finally, their estimation procedure involves nonparametric kernel smoothing with multiple bandwidths, which could adversely affect the finite sample performance of their estimator. The paper is organized as follows. The next section presents the model and the estimators. Section 3 contains the large properties. Results from some Monte Carlo experiments are presented in Section 4. Section 5 concludes. The proofs of theorems are relegated to the Appendix. 2. The model and estimators Consider the model, g Y ∗ = X ′ β0 − σ X ′ γ0 ε
)
(
(
)
(1)
where g is an unknown strictly increasing function, Y ∗ is the latent dependent variable, X is a q + 1 dimensional vector of regressors, β0 is the corresponding vector of coefficients. ) error term ε is independent of X , with a zero median for its ( The location normalization. The scale function σ (x) = σ x′ γ0 is known up to the finite dimensional parameter γ0 . Suppose we observe only (Y , X ) where Y = y1 Y ∗ < y + Y ∗ 1 y ≤ Y ∗ ≤ y + y1 Y ∗ > y
{
}
{
}
{
}
with −∞ ≤ y < y ≤ ∞ and both y and y are known constant terms; namely, we allow two-sided censoring here. It is well known (e.g., Horowitz (1996)) that for models subject to unknown transformation, location and scale normalization is needed for identification. As in(Horowitz (1996) and Chen (2002), the location normalization consists ) of setting g (y0 ) = 0 for some finite value y0 ∈ y, y , and the zero median restriction on ε . Our location normalization approach is similar to that in Zhou et al. (2008) who assumed g (y0 ) = 0 and set the zero mean restriction on ε . To achieve scale normalization, we require X contains a component with a nonzero coefficient, whose probability distribution conditional on the remaining components is absolutely continuous with respect to the Lebesgue measure. Following Horowitz (1996) and Chen (2002), we arrange the components of the regressors so that X1 , the first component ( of X) , satisfies this condition. As in Horowitz (1992), our scale normalization consists of setting |β10 | = 1, with X =
X1 , X˜ ′
′
)′
and β0 = β10 , β˜ 0′ . To motivate our estimators, we first develop some rank conditions, similar to Han (1987), Manski (1975, and ( 1985) ) Chen (2002, 2010). Let {Yi , Xi }ni=1 be a random sample of (Y , X ) based on model (1). For given y0 , y ∈ y, y , define { } diy = 1 {Yi > y} and djy0 = 1 Yj > y0 . Then following Chen (2002), under the location normalization g (y0 ) = 0, for any pair i ̸ = j, we obtain
(
E diy − djy0 ⏐Xi , Xj ≥ 0 whenever
(
⏐
)
Xi′ β0 − g (y)
σ Xi′ γ0
(
)
≥
Xj′ β0
( ) σ Xj′ γ0
(2)
1 In general, consistent estimation of the transformation function is only possible in a bounded set, in which case the method proposed by Zhou et al. (2008) does not apply. Indeed, they suggest specifying a parametric form for the transformation function in the tail area to overcome this difficulty.
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
3
If β0 and γ0 were known, Eq. (2) suggests that g (y) can be estimated by any value maximizing the following U-statistic based objective function
{
1
∑(
diy − djy0 1
n (n − 1)
Xj′ β0
Xi′ β0 − g
}
( ) ≥ ( ′ ) . σ Xi′ γ0 σ Xj γ0
)
i̸ =j
On the other hand, following Manski (1975, 1985) and Chen (2010), for any y ∈ y, y and τ ∈ (0, 1), we also have the following rank condition:
[
E diy |Xi − τ > 0 whenever
(
)
Xi′ β0 − g (y)
( ) σ Xi′ γ0
]
> c (τ )
(3)
where c (τ ) is the τ th quantile of ε . In particular, under the condition that the median of ε is zero,
(
)
E diy |Xi −
1 2
> 0 whenever Xi′ β0 − g (y) > 0
and
(
)
E diy0 |Xi −
1 2
> 0 whenever Xi′ β0 > 0
Our estimators are based on rank conditions (2) and (3). Define
Γn (y, g , b, γ ) =
{
1 n (n − 1)
∑(
)
diy − djy0 1
i ̸ =j
Xj′ b
Xi′ b − g
}
)≥ ( ′ ) ( σ Xi′ γ σ Xj γ
for any given (b, γ , g ) in the appropriate parameter space, and let gn (y, b, γ ) = arg max Γn (y, g , b, γ ) g ∈M
where M is a compact interval in R. Next, define Qn (θ ) ≡ Qn (b, γ , c ) =
∫
Qn (y, b, γ , c , gn (y, b, γ )) ω (y) dy
(4)
where n
Qn (y, b, γ , c , g ) =
p
1 ∑∑ ( n
diy − τℓ 1 Xi′ b − g − σ Xi′ γ cℓ > 0
) {
(
)
}
(5)
i=1 ℓ=1
c = (c1 , c2 , . . . , cp )′ for some fixed integer p, ω (y) is some nonnegative weighting function defined on y, y ; in particular,
[
]
we set one of the quantiles, τl0 = 0.5, with cl0 = 0, which would ensure the identification of β0 and g (Chen, 2010); once β0 and g are identified, other τ s are used to identify γ0 . Consequently, θ0 = (β0′ , γ0′ , c0′ )′ can be estimated by θˆ , which maximizes Qn (θ ). Following Chen (2010) and Koenker (2004), we integrate over y and pool the sample information over several quantiles to improve efficiency. For technical reasons, similar to Chen (2010), we adopt smoothed objective functions. With a slight abuse of notation, θˆ , the estimator of θ0 , is defined as a value that maximizes Qn (θ ) = Qn (b, γ , c ) =
∫
Qn (y, b, γ , c , gn (y, b, γ )) ω (y) dy
where n
Qn (y, b, γ , c , g ) =
p
1 ∑∑ ( n
diy − τℓ Kh1 Xi′ b − g − σ Xi′ γ cℓ
(
)
(
) )
i=1 ℓ=1
with gn (y, b, γ ) being the smoothed maximum rank estimator that maximizes
Γn (y, g , b, γ ) =
1 n (n − 1)
( ∑(
)
diy − djy0 Kh2
i ̸ =j
where k (·) is a kernel function, K (u) =
∫u −∞
Xi′ b − g
Xj′ b
)
( )− ( ′ ) σ Xi′ γ σ Xj γ
k (v) dv is the corresponding cumulative kernel function with Kh (u) = K
(u) h
ˆ γˆ ) as our estimator for g(y) for ˆ and h1 and h2 are some proper bandwidth parameters. Then we define g(y) = gn (y, β, any y ∈ (y, y). √
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Once we have obtained gˆ and θˆ , we can estimate c (τ ) for a given τ ∈ (0, 1) , τ ̸ = 0.5. Specifically, we define cˆτ as a solution to the following maximization problem,
(
ˆ γˆ , τ , c max Qn β,
)
∫ =
c ∈M
(
)
ˆ γˆ , gˆ (y) , τ , c ω (y) dy Qn y, β,
where
(
ˆ γˆ , g , τ , c Qn y, β,
)
n
=
1∑ ( n
(
) )
diy − τ Kh1 Xi′ βˆ − g − σ Xi′ γˆ c ;
)
(
i=1
and furthermore, as in Horowitz (1996), we can define yˆ ∗τ (x) such that
(
ˆ γˆ gn yˆ ∗τ (x) , β,
)
( ) = x′ βˆ − σ x′ γˆ cˆτ
(6)
as a predictor for the τ th conditional quantile of Y given X = x. Remark 1. By imposing a zero mean restriction on the error distribution, Zhou et al. (2008) are able to identify the intercept term, in addition to the slope parameters; similarly, we allow for the presence of the intercept term as well by imposing a zero median restriction on the error distribution. On the other hand, Chen (2010) does not contain the intercept term since no location restriction is imposed on the transformation function. When there is no heteroscedasticity, our model reduces to that of Horowitz (1996) and Chen (2002); however, Horowitz (1996) and Chen (2002) do not impose any location restriction on the error distribution, and thus the intercept term is absorbed in the transformation function.
√
√
Remark 2. Note that n consistent conditional quantile prediction requires n consistent estimation of the finite dimensional parameters as well as the transformation function. Chen (2010) considered the conditional median restriction without imposing any other restriction on the form of heteroscedasticity. As a result, Chen’s (2010) estimator for the transformation function only converges at a nonparametric rate. In our context, it is possible to extend Chen’s (2010) approach by including several quantiles based on (3); however, doing so does not improve the rate of convergence for the resulting estimator of the transformation function. By exploiting the pairwise rank condition (2), combined with several √ quantile conditions (3), we are able to attain n consistent estimation of the transformation function as well as the finite dimensional parameters. 3. Large sample properties In this section, we investigate the large sample properties of the estimators. We make the following assumptions. Assumption A1. {Yi , Xi }ni=1 is a random sample of (X , Y ), with Y = y1 Y ∗ < y + Y ∗ 1 y ≤ Y ∗ ≤ y + y1 {Y ∗ > y}, with (X , Y ∗ ) generated from (1).
{
}
{
}
Assumption A2. ε is independent of X and the median of ε is zero. Assumption A3. (a) The support of the distribution of X is not contained in any proper linear subspace of Rq+1 . (b) The distribution of X1 conditional on X˜ has a positive density almost everywhere with respect to the Lebesgue measure. (c) The distribution of X˜ has a bounded support. Assumption A4.(The )transformation [ ( ) function g is ] a strictly increasing function with the location normalization g (y0 ) = 0 for some y0 ∈ y, y , and g y − ϵ, g (y) + ϵ ⊆ M for a small positive number ϵ , where M is a compact interval. Furthermore, g is twice differentiable and its derivatives are uniformly bounded over M.
˜ and β˜ 0 is an interior point of B, ˜ a compact subset of Rq . G and C are some compact Assumption A5. β10 = 1, B = {1} × B, subsets of Rq+1 and Rp respectively, with γ0 and c0 being their interior points. Define S¯X = {x: x ∈ SX and x′ β0 − cτℓ σ (x′ γ0 ) ∈ [g(y), g(y¯ )] for some τℓ ̸ = 0.5, ℓ = 1, 2, . . . , p}, where SX denotes the support of X . Assumption A6. σ (·) is a known function such that the scale function σ (·) is almost everywhere positive and uniformly bounded from above and away from zero, such that,
( ) σ x′ γ = constant σ (x′ γ0 ) for any x ∈ S¯X , then γ = γ0 . √
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( ))′ )′ ( ( ( ′ ′ )′ , c0 , where θ10 = β0′ , γ0′ and c0 = c (τ1 ) , c (τ2 ) , . . . , c τp . Let Define θ0 = θ10 ] [ ri (y, θ1 , g ) = E hij (y, θ1 , g ) + hji (y, θ1 , g ) |Xi , Yi
5
(7)
and q¯ i (y, θ1 , g (y, θ1 )) =
[ ] ∂ ∂ ri (y, θ1 , g (y, θ1 )) − E ri (y, θ1 , g (y, θ1 )) ∂g ∂g
where
{ hij (y, θ1 , g ) = 1 {Yi ≥ y} − 1 Yj ≥ y0
(
{
})
1
}
Xj′ b
Xi′ b − g
( )≥ ( ′ ) σ Xi′ γ σ Xj γ
and
] ∂2 y , θ , g y , θ r )) ( ( 1 1 i 2 ∂g2 [ 2 ] [ ] ∂ ri (y,θ10 ,g (y)) Assumption A7. V (y, θ10 ) = 12 E is negative for each y ∈ y, y and uniformly bounded away from zero. ∂g2 ( ⏐ ) Let Z1 = X ′ β0 and Z2 = X ′ γ0 . Let p z1 ⏐x˜ denote the density of Z1 at z given X˜ = x˜ , F (t ) and f (t ) are the cumulative and probability density functions of ε at t. ( ⏐ ) Assumption A8. f (t ) is uniformly bounded and twice continuously differentiable with respect to t, and p z1 ⏐x˜ is twice continuously differentiable with respect[ to( z1) for any ] given x˜ , with all the derivatives uniformly bounded. f (t ) is uniformly bounded away from zero for any t ∈ c τ , c (τ ) . The weight function, ω, is twice continuously differentiable and is V (y, θ1 ) =
1
[
E
nonnegative with a bounded support. Assumption A9. The kernel function k (·) is everywhere twice differentiable with a bounded support. The kernel function k (·) and its derivatives are uniformly bounded and have bounded variation such that:
∫
k (u) du = 1 and
∫
uk (u) du = 0
and
∫
k′ (u) du = 0,
∫
uk′ (u) du ̸ = 0.
Assumption A10. The bandwidth sequences satisfy nh41 = o (1), ln n nh31
(
)−1/2
1 = o (1) and nh42 = o (1), n−1/2 h− 2 = o (1).
Define Q (θ , g (·, θ1 )) =
∫
Q (y, θ, g (y, θ1 )) ω(y)dy
where Q (y, θ , g (y, θ1 )) =
p ∑
Qℓ (y, θ, g (y, θ1 ))
ℓ=1
with
{ Qℓ (y, θ , g (y, θ1 )) = E diy − τℓ 1
(
)
Xi′ b − g (y, b, γ )
( ) σ Xi′ γ
} > cℓ .
Let H (θ0 ) =
[
∂2 ∂θ ∂θ ′
]
∫
Q (y, θ, g (y, θ1 )) ω (y) dy ⏐θ =θ0
⏐
where given (y, θ1 ), g (y, θ1 ) maximizes
Γ (y, g , θ1 ) = E
[ (
{ )
diy − djy0 1
Xi′ b − g
Xj′ b
( )≥ ( ′ ) σ Xi′ γ σ Xj γ
}] .
Assumption A11. H (θ0 ) is negative definite. √
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Assumption A1 describes the data generating process. Assumptions A2 and A3 are needed for identification for
β0 (e.g., Manski (1985), Horowitz (1992, 2009), Chen (2002, 2010)). Assumptions A4 and A5 contain the location normalization. Assumption A6 is needed for ( the) identification ( ) of γ , which is commonly used for identification of nonlinear regression models. For the special case σ x′ γ = exp x′ γ , then Assumption 6 holds trivially if x ∈ S¯X does not contain a constant and satisfies the full rank condition. Note that for the case with (y, y¯ ) = (−∞, +∞), then S¯X = SX . In general, we require sufficient variation of x in S¯X . Assumptions A7 and A8 contain some boundedness and smoothness conditions. ( Assumptions ) ( A9 )and A10 place some restrictions on the kernel functions and the bandwidth sequences. h1 = O n−ν1 , h2 = O n−ν2 , ν1 ∈ (−1/3, −1/4) , ν2 ∈ (−1/2, −1/4) will be some suitable choices. The matrix H (θ0 ) in Assumption A11 is analogous to the Hessian form of the information matrix in the maximum likelihood estimation. √ The following theorem establishes consistency of θˆ and its n-asymptotic normality.
)′
(
Theorem 1. Under Assumptions A1–A11, θˆ = βˆ ′ , γˆ ′ , cˆ ′ , is
√ (
n θˆ − θ0
)
√
n-consistent for θ0 and asymptotically normal
) ( d → N 0, H (θ0 )−1 Σ (θ0 ) H (θ0 )′−1
where Σ (θ0 ) = E ψ (ξi , θ0 ) ψ (ξi , θ0 )′ and ψ (ξi , θ0 ) is defined in the Appendix. The following theorem establishes uniform consistency of gn and convergence in distribution of
√
n (gn (y) − g (y)).
Theorem 2. Under Assumptions A1–A11, (a) supy∈[y,y] |gn (y) − g (y)| = op (1); (b) For any y ∈ y, y ,
[
√
]
n
) 1 ∑( 1 n (gn (y) − g (y)) = √ Jy,y0 (ξi , θ10 ) + Jy2 (ξi , θ10 ) + op (1) n i=1
and
√
n (gn (y) − g (y)) ⇒ Hg (y), where Hg (y) is a tight Gaussian Process with mean 0 and covariance function
E Hg (y) Hg y′
( ))
(
=E
)(
[(
Jy1,y0 (ξi , θ10 ) + Jy2 (ξi , θ10 )
)]
Jy1′ ,y (ξi , θ10 ) + Jy2′ (ξi , θ10 ) 0
where Jy1,y0 (ξi , θ10 ) = −V (y, θ10 )−1 q¯ i (y, θ10 , g (y, θ10 )) and Jy2 (ξi , θ10 ) =
∂ g (y, θ10 ) H(θ0 )−1 ψ (ξi , θ0 ) ∂θ1
which are defined in the Appendix. The theorem establishes uniform consistency of cˆτ and convergence in distribution of [ following ] τ ∈ τ, τ .
√ (
n cˆτ − c (τ ) for
)
Theorem 3. Under Assumptions A1–A11, (a) supτ ∈[τ ,τ ] ⏐cˆτ − c (τ )⏐ = op (1); (b) For any τ ∈ τ , τ ,
⏐
⏐
[
]
n ) 1 ∑ n cˆτ − c (τ ) = √ Λ (ξi , θ10 , c (τ )) + op (1) n
√ (
i=1
and
√ (
) n cˆτ − c (τ ) ⇒ Hcτ (τ ), where Hcτ (τ ) is a tight Gaussian Process with mean 0 and covariance function ( ( )) [ ( ( ))] E Hcτ (τ ) Hcτ τ ′ = E Λ (ξi , θ10 , c (τ )) Λ ξi , θ10 , c τ ′
where Λ (ξi , θ10 , c (τ )) is defined in the Appendix. For a given ϵ > 0, and τ ∈ τ , τ , define yτ (x) = Qτ (Y ∗ |X = x) and
[
]
Sϵτ = {x : yτ (x) ∈ [yL + ϵ, yU − ϵ ]} ̸ = φ Theorem 4. Under Assumptions A1–A11, (a) supx∈Sϵτ ⏐yˆ ∗τ (x) − y∗τ (x)⏐ = op (1); (b) For any x ∈ Sϵτ ,
⏐
√ (
⏐
n yˆ ∗τ (x) − y∗τ (x)
)
n 1 ∑
= √
n
i=1
(
[
x˜ ′ , −σ ′ x′ γ0 c (τℓ ) x′ Iθ1 Ψ (ξi , θ0 ) −
(
)
)
] ) ( ) ( ) + op (1) σ x′ γ0 Λ (ξi , θ10 , c (τ )) − Jy1∗τ (x),y (ξi , θ0 ) + Jy2∗τ (x) (ξi , θ0 ) g ′ y∗τ (x) 0 1
(
√
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7
√ (
) n yˆ ∗τ (x) − y∗τ (x) ⇒ Hyτ (x), where Hyτ (x) is a tight mean 0 Gaussian Process whose covariance function is the same
and as the covariance function of
x˜ ′ , −σ ′ x′ γ0 c (τ ) x′ Iθ1 Ψ (ξi , θ0 ) −
(
(
)
)
] ) ( ) ( ) σ x′ γ0 Λ (ξi , θ10 , c (τ )) − Jy1∗τ (x),y (ξi , θ10 ) + Jy2∗τ (x) (ξi , θ10 ) g ′ y∗τ (x) 0 [ ] where Iθ1 = I(2q−1)×(2q−1) 0 , I is a (2q − 1) × (2q − 1) identity matrix. [
1
(
For the purpose of carrying out large-sample statistical inference, consistent estimates of the asymptotic variance and covariance matrix need to be constructed. From the proof of the theorems, we can easily show that all the components in the asymptotic variance can be consistently estimated by the sample analogue replacing the true value of the parameters by their consistent sample analogues. Specifically, ( ) ( ) (i). For the statistical inference of θˆ , H (θ0 )−1 Σ (θ0 ) H (θ0 )′−1 can be consistently estimated by Hn−1 θˆ Σn θˆ
( )
Hn′−1 θˆ , where
( )
Hn θˆ
=
∂2 ∂θ 2
∫
(
(
ˆ gn y, θˆ1 Qn y, θ,
))
ω (y) dy
and n ( ) 1∑ ( ) ( ) Σn θˆ = ψn ξi , θˆ ψn′ ξi , θˆ
n
i=1
( )
the following shows the componentwise estimation of Hn θˆ
( )
(
and Σn θˆ , specifically, ψn ξi , θˆ
)
can be consistently
estimated by
( ) ( ) ( ) ψn ξi , θˆ = ψ1n ξi , θˆ + ψ2n ξi , θˆ where p ( ) ∑ ( ) ( ) ψ1n ξi , θˆ = ϕ1ni τℓ , θˆ ϕ2ni τℓ , θˆ
ℓ=1
and
( ) ∫ ( ( ) ( )) ( ) 1 ˆ y + Hn2θ g θ, ˆ y Jny ˆ ψ2n ξi , θˆ = Hn1θ g θ, ,y0 ξi , θ1 ω (y) dy in which
(
)
ϕ1ni τℓ , θˆ = −
∫
( ) ⎛ ( ) ⎞ ˆ γˆ − σ Xi′ γˆ cˆτℓ Xi′ βˆ − gn y, β, ) 1 ⎠ ω (y) dy k⎝ diy − τℓ
(
h1
h1
( ( ) ( ′ )) ′ˆ −1 ˆ ( ) ( ) ˆ β − c σ X γ ˆ ∂ g g X , β, γ ˆ n τ ( ) ( ) n i i ℓ ϕ2ni τℓ , θˆ = X˜ i′ , σ ′ Xi′ γˆ Xi′ cˆτℓ , ιℓ σ ′ Xi′ γˆ + ∂θ1 and
( ) ∂ gn y,θˆ1 ∂θ1
can be consistently estimated according to (A.3) and (A.4), and
( ) ∫ Hn1θ g θˆ , y =
) ( ) ( ⎛ ( ) ⎞ ′ˆ p ∂g ˆ γˆ ( ˆ γˆ − σ Xi′ γˆ cˆτℓ n y , β, X β − g y , β, ∑ ∑ n n ) i 1 ⎠ ω (y) dy diy − τℓ k′ ⎝ ∂θ1 h1 nh21 i=1 ℓ=1
( ( ) ) ∫ p n ⏐ ( ) ) Xi′ b − gn (y, b, γ ) − σ Xi′ γ cℓ 1 ∑∑( ∂ ⏐ diy − τℓ k θˆ , y = ω (y) dy ⏐b=β,γ ˆ =γˆ ,cℓ =ˆcτ ℓ ∂θ nh1 h1 i=1 ℓ=1 ( ( ′ )) (ii). Next we compute the covariance of gn (y), E Hg (y) Hg y , which can be consistently estimated by n ( ) ( )) ( ( ) ( )) [ ] 1 ∑( 1 Jn,y,y0 ξi , θˆ1 + Jn2,y ξi , θˆ1 Jn1,y′ ,y ξi , θˆ1 + Jn2,y′ ξi , θˆ1 , for any y ∈ y, y 0 Hn2θ g
n
i=1
where write
( lij (y, θ1 , g ) =
1 {Yi > y} − 1 Yj > y0
{
( ) σ Xi′ γ
})
1 h2
√
( k
1 h2
(
Xi′ b − g
Xj′ b
))
( )− ( ′ ) σ Xi′ γ σ Xj γ
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S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
(
(
Jn1,y,y0 ξi , θˆ1
Vn y, θˆ1
)
=
)−1
n [ ( ∑
(
lij y, θˆ1 , gn y, θˆ1
n−1
))
( ( ))] + lji y, θˆ1 , gn y, θˆ1
j=1,j̸ =i
with
(
Vn y, θˆ1
)
=
n n ( )) ( ( ))] ∑ ∑ ∂ [ ( lij y, θˆ1 , gn y, θˆ1 + lji y, θˆ1 , gn y, θˆ1 2n (n − 1) ∂g i=1 j=1,j̸ =i
1
and
(
Jn2,y ξi , θˆ1 where
∂ gn (y,θ1 ) ∂θ1
)
( ) ∂ gn y, θˆ1 =
∂θ1
( )
(
Hn−1 θˆ ψn ξi , θˆ
)
is defined in Eq. (A.4)–(A.5) in the Appendix.
(iii). A consistent estimator of E Hcτ (τ ) Hcτ τ ′ 1 n
( ))
(
n
∑[
(
)
(
ˆ τ ϕn ξi , θ, ˆ τ′ Vn (τ )−1 ϕn ξi , θ,
)′
will be
( )−1
Vn τ ′
]
, for any τ ∈ τ , τ , τ ̸ = 0.5
[
]
i=1
where Vn (τ ) =
( ) ∂ Qn2 θˆ1 , gn , τ , cˆτ σ
∫ =
∂c2 ) n Xi′ γˆ ∑ (
⎛
( 2
diy − τ k′ ⎝
nh21
)
(
ˆ γˆ Xi′ βˆ − gn y, β, h1
i=1
)
( ) ⎞ − σ Xi′ γˆ cˆτ ⎠ ω (y) dy
and
∫ ( ) ϕn ξi , θˆ1 , τ = S1ni + (S2ni (y) + S3ni (y)) ω (y) dy where
(
) (
S1ni = ϕ1ni τ , θˆ σ Xi′ γˆ
)
and
⎛ 2 S2ni (y) = ⎝Hnc τ θ1
(
( )⎞ ( ) ( ) ( ) ∂ gn y, θˆ1 −1 ˆ 2 ⎠ ˆ ˆ y + Hnc ˆ H θ ψ ξ , θ θ, θ, y I n i θ 1 n τg ∂θ ′ )
and
(
)(
2 ˆ y S3ni (y) = Hnc θ, τg
(
)
(
1 2 ˆ ˆ Jny ,y0 ξi , θ1 + Jny,y0 ξi , θ1
))
with 2 Hnc τ θ1
(
∂ θˆ , y = − ∂θ1
)
∫
( ) n σ Xi′ γ ∑ (
(
diy − τ k
nh1
)
Xi′ b − gn (y, b, γ ) − σ Xi′ γ cℓ
(
h1
i=1
)
)
⏐ ⏐ ω (y) dy ⏐b=β,γ ˆ =γˆ ,cℓ =ˆcτ
ℓ
and
) ( ⎛ ( ) ⎞ ˆ γˆ − σ Xi′ γˆ cˆτ n ( ) ∫ σ (X ′ γˆ ) ∑ Xi′ βˆ − gn y, β, ( ) i 2 ⎠ ω (y) dy diy − τ k′ ⎝ Hnc θˆ , y = τg 2 nh1
(
(iv). Finally E Hy∗τ (x) Hy∗τ ℓ
ℓ
h1
i=1
( ′ )) x
can be componentwise estimated using its sample analogue where g ′ (y) is replaced by
( ( ′ˆ ( ˆ ) )) ) ( Yi −y ) Xi β−gn y,β,γˆ Xj′ βˆ 1 ( ) ˆ k hy k h − i ̸ =j σ X i γ ( ) σ (Xi′ γˆ ) 2 σ Xj′ γˆ ( ) gn′ y, θˆ1 = − ( ( )) ) ˆ gn y,β, ˆ γˆ Xi′ β− ∑ ( ( Yi − y ) Xj′ βˆ 1 ′ ( ) − djy0 k h − i̸ =j K hy ′ σ (X ′ γˆ ) 2 ∑
(
′
i
σ Xj γˆ
at y = yˆ ∗τℓ (x), if hy is chosen to be some corresponding smoothing parameter that goes to zero as n goes to infinity. √
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
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9
4. Monte Carlo experiments In this section we will examine the finite sample performance of our estimators, in comparison with Chen’s (2010) integrated maximum score estimator (IMSC), which only imposes a conditional median restriction. We consider both homoscedastic and heteroscedastic designs. For the homoscedastic designs, data are generated from the model g Y ∗ = X1 + β˜ 00 − (1 + γ0 X1 )ε , Y ∗ = Y
(
)
where β˜ 00 = 1, X1 follows normal distribution with mean −1 and unit variance. The transformation function is of the form g(y) = y and y0 = 0, namely the identity function; For the distribution of error term, we consider two data generating processes: in DGP1, ε is drawn from a standard normal distribution, independent of X1 ; and for DGP2, ε is drawn from a median standardized mixture normal distribution, also independent of X1 , and specifically,
ε∗ ε = √ , ε∗ ∼ 0.5N (0, −2) + 0.5N (0, 2) 5
For the heteroscedastic designs, we consider the modified Box–Cox transformation model with censoring: g Y ∗ , λ = X1 + β˜ 10 X2 + β˜ 00 −
(
)
1
{ } (1 + exp (γ0 X2 )) ε , Y = max y, Y ∗
2
|y|λ sgn(y)−1
, λ = 0.6 and y0 = 1 and β˜ 10 = 1, β˜ 00 = −1 and γ0 = −1, X1 is drawn from the standard where g (y, λ) = λ normal distribution and X2 follows the chi-square distribution with one degree of freedom, which is independent of X1 and standardized to have zero mean; the scale function is an exponential function σ (u) = 12 (1 + exp (−u)). As in the first two designs, in DGP 3, ε is the standard normal and ε is a mixture normal for DGP 4. Y is censored from below and y is a known constant such that the censoring percentage is 15%. In addition to DGPs 3–4, we introduce more covariates in the scale function in next two designs as the exclusive restriction is not required for our identification: g Y ∗ , λ = X1 + β˜ 10 X2 + β˜ 00 −
(
)
1
{ } (1 + exp (γ1 X1 + γ2 X2 )) ε, Y = max y, Y ∗
2
where γ1 = 0.5 and γ2 = −1; ε follows the standard normal for GDP 5 and is drawn from a mixture normal for DGP 6. The censoring percentage is also roughly 15%. In the last design, we consider a model with discrete regressors. Even though our identification results require the presence of continuous regressors, our rank estimator may still deliver meaningful results. The model specification is same as DGP3 except both regressors are discrete random variables: X1 ∼Poisson(1.5) − 1, X2 ∼Poisson(1) − 1 and ε ∼ N (0, 1). Before describing the details of the simulation results, we first discuss some of the details related to the implementation of our proposed estimators. Similar to Chen (2010), instead of using the smoothing version, we report the results based on the non-smoothed objective functions, and indeed smoothing version produces very similar results. It turns out that our simulation results are quite robust to the choice of weight function, and we choose ω (y) = 1 {yL < y < yU }, where yU < y and yL > y, where yL is the 0.20th quantile and yU is the 0.80th quantile of Y for both designs. In the objective function (8), we sum over the following quantiles: {0.75, 0.70, 0.60, 0.50, 0.40.0.30, 0.25}. As discussed in Chen (2002), we also adopt a segmentation method in estimating the transformation function. Given β and γ , for y > y0 , the searching grid initially starts from t0 = y0 , and tj is to be chosen such that tj > tj−1 , j = 1, 2, . . . , 8, and tj−1 acts as the reference point for the estimation of g(t) for t ∈ (tj−1 , tj ]. When y < y0 , our procedure can be implemented in a very similar way. The following describes the main steps in implementing our estimators:
• Step 1. (Obtaining initial estimate for β˜ 0 ) Following Chen (2010), we obtain a consistent estimate of β˜ 0 , βˆ ∗ , under the conditional median restriction.
• Step 2. (Obtaining initial estimate for γ0 ) Given the initial estimates of β˜ 0 , we obtain an initial consistent estimate of γ0 by, γˆ ∗ , which solves ∫ ∑ p n ( ) ) } ∑ ( ) { ( ) ( 1 diy − τℓ 1 Xi′ βˆ ∗ − gn y, βˆ ∗ , γ − σ Xi′ γ cˆτℓ βˆ ∗ , γ > 0 ω (y) dy (8) max γ
n
i=1 ℓ=1
(
)
where the transformation function gn y, βˆ ∗ , γ , as a function of γ given βˆ ∗ , maximizing
Γn
(
) y, g , βˆ ∗ , γ =
{
1 n (n − 1) √
∑( i̸ =j
)
diy − djy0 1
Xi′ βˆ ∗ − g
Xj′ βˆ ∗
( ) ≥ ( ′ ) σ Xi′ γ σ Xj γ
} (9)
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
10
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx Table 1 Estimation of coefficients: DGP1. N = 250
β˜ 00 γ0 β˜ 00 γ0
CZ Chen Two-step
N = 500
Bias
STD
RMSE
Bias
STD
RMSE
−0.002
0.140 0.104 0.210 0.088
0.140 0.104 0.213 0.089
0.011 0.000 −0.018 0.005
0.097 0.062 0.178 0.057
0.097 0.062 0.179 0.057
Bias
STD
RMSE
Bias
STD
RMSE
−0.012
0.202 0.109 0.305 0.085
0.203 0.113 0.310 0.088
0.012 0.009 −0.002 0.008
0.160 0.052 0.276 0.049
0.160 0.052 0.276 0.050
0.007 −0.034 0.014
Table 2 Estimation of coefficients: DGP2. N = 250
β˜ 00 γ0 β˜ 00 γ0
CZ Chen Two-step
(
w.r.t. γ and cˆτ∗ βˆ ∗ , γ
∫
n 1∑ (
n
i=1
(
)
N = 500
0.029 −0.053 0.022
maximizes
( ) ⎧ ⎫ ′ ∗ ∗ ⎬ ) ⎨ Xi βˆ − gn y, βˆ , γ ( ′ ) > c ω (y) dy diy − τ 1 ⎩ ⎭ σ Xi γ
w.r.t. c. gn y, βˆ ∗ , γ
)
(
and cˆτ∗ βˆ ∗ , γ
)
(10)
are solved using one-dimensional grid search. The Nelder–Mead algorithm is
used to maximize the objective function (8), for which we adopt the following starting values for γ : – DGP1-2: Run quantile regression gn∗ (Yi ) − X1i − βˆ 0∗ on X1i with intercept term at τ = 0.70, collect the corresponding coefficients q∗0 and q∗1 , the starting values of γ ∗ is given by q∗1 /q∗0 . – DGP3-7: Compute
⎛
⎛ (
Y˜i∗ ≡ log ⎝max ⎝
2 gn∗ (Yi ) − X1i − βˆ 1∗ X2i − βˆ 0∗ zτ
)
⎞⎞ − 1, 0⎠⎠
then the starting values of γ ∗ is given by a quantile regression Y˜i∗ on X1i and X2i at τ = 0.70, where zτ is the τ th quantile of N (0, 1).
• Step 3. (Estimating β˜ 0 , γ0 and c0 ) Based on the initial values of βˆ ∗ , γˆ ∗ and cτ∗ , we conduct search for the estimates
of parameters according to (4) and (5) using some local optimization routines. Nelder–Mead algorithm is employed in the Monte Carlo simulation.
Let βˆ and γˆ denote the final estimates of β˜ 0 and γ0 .
• Step 4. (Estimation of( transformation function g (y)) Given the estimates of β˜ 0 and γ0 , we estimate the transformation ) ˆ function g (y) by gn y, β, γˆ , as in Step 2. We consider sample sizes of 250 and 500, with 500 replications, and report the bias, standard deviation (SD) and root-mean-square-error (RMSE) for each design. We compare the performance of our one-step estimator (CZ) with that of Chen (2010). In addition, we also report the performance of the estimate for γ0 obtained in Step 2 (Two-step), which is also computationally simpler; together with the initial estimator for β0 obtained in Step 1, it is used as the initial estimate for our joint estimation of β0 and γ0 in Step 3. This two-step procedure is about 2.5 and 3 times faster than the joint estimation in our experiments. Tables 1–7 report the results for estimates of the finite dimensional parameters. Tables 8–21 summarize the results for the estimation of transformation function for some particular values of y, along with that of Chen (2010) for comparison. Note that for the estimation of finite dimensional parameters, our estimator performs very well and significantly better than Chen (2010) with much smaller RMSEs. Regarding the estimation of γ0 , our joint estimator and the two-step version are quite comparable for DGPs 1–3 and 5, whereas the joint estimator clearly outperforms the two-step version for DGPs 4 and 6. When the sample size increases to 500, the performances of all estimators improve as expected, both in terms of biases and standard deviations. For the case when all regressors are discrete, it turns out our estimator continues to perform well, and much better than Chen (2010) for the estimation of β0 and the two-step version for the estimation of γ0 . √
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
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11
Table 3 Estimation of coefficients: DGP3. N = 250
CZ
Chen Two-step
β˜ 10 β˜ 00 γ0 β˜ 10 β˜ 00 γ0
N = 500
Bias
STD
RMSE
Bias
STD
RMSE
0.008 0.002 −0.090 0.013 −0.034 −0.235
0.131 0.163 0.373 0.185 0.264 0.318
0.131 0.163 0.383 0.185 0.266 0.395
0.015
0.095 0.123 0.252 0.134 0.229 0.213
0.096 0.123 0.252 0.135 0.231 0.248
Bias
STD
RMSE
0.011 0.024 −0.125 0.034 −0.040 −0.323
0.164 0.265 0.284 0.282 0.412 0.294
0.165 0.266 0.310 0.284 0.414 0.437
Bias
STD
RMSE
−0.002
0.108 0.173 0.163 0.209 0.309 0.177
0.108 0.175 0.174 0.209 0.309 0.258
Bias
STD
RMSE
Bias
STD
RMSE
0.046 −0.024 0.164 −0.206 0.011 −0.032 −0.089 −0.313
0.173 0.219 0.392 0.389 0.225 0.307 0.347 0.307
0.179 0.220 0.425 0.440 0.225 0.309 0.358 0.439
0.025 −0.021 0.029 −0.048 0.011 −0.022 −0.090 −0.197
0.113 0.156 0.184 0.245 0.182 0.282 0.257 0.233
0.116 0.158 0.187 0.250 0.183 0.283 0.272 0.305
Bias
STD
RMSE
Bias
STD
RMSE
0.029 0.004 0.094 −0.214 0.001 −0.034 −0.017 −0.394
0.224 0.342 0.296 0.312 0.337 0.467 0.357 0.351
0.226 0.342 0.311 0.379 0.337 0.468 0.357 0.528
0.001 0.018 0.028 −0.088 −0.009 0.005 −0.014 −0.274
0.140 0.219 0.142 0.195 0.262 0.399 0.260 0.237
0.140 0.219 0.145 0.214 0.262 0.399 0.260 0.362
Bias
STD
RMSE
Bias
STD
RMSE
0.011 −0.009 −0.170 −0.057 −0.270 −0.324
0.132 0.204 0.464 0.192 0.435 0.697
0.133 0.204 0.494 0.201 0.512 0.768
0.009
−0.006 −0.129 −0.065 −0.290 −0.258
0.094 0.146 0.393 0.161 0.408 0.636
0.094 0.146 0.414 0.173 0.500 0.686
−0.003 −0.014 0.017
−0.028 −0.126
Table 4 Estimation of coefficients: DGP4. N = 250
CZ
Chen Two-step
β˜ 10 β˜ 00 γ0 β˜ 10 β˜ 00 γ0
N = 500
0.029 −0.063 0.004 0.005 −0.187
Table 5 Estimation of coefficients: DGP5. N = 250
CZ
Chen Two-step
β˜ 10 β˜ 00 γ10 γ20 β˜ 10 β˜ 00 γ10 γ20
N = 500
Table 6 Estimation of coefficients: DGP6. N = 250
CZ
Chen Two-step
β˜ 10 β˜ 00 γ10 γ20 β˜ 10 β˜ 00 γ10 γ20
N = 500
Table 7 Estimation of coefficients: DGP7. N = 250
CZ
Chen Two-step
β˜ 10 β˜ 00 γ0 β˜ 10 β˜ 00 γ0
√
N = 500
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
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S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
Table 8 Estimation of transformation function: DGP1, N = 250. y
g (y)
−1.0 −0.8 −0.6 −0.4 −0.2
−1.0 −0.8 −0.6 −0.4 −0.2
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.005 0.006 −0.017 −0.023 −0.039 0.0 −0.036 −0.058 −0.047 −0.068 −0.085
0.008 0.005 0.001 0.007 0.002 0.0 −0.003 −0.003 0.001 −0.004 −0.014
0.006 0.003 −0.002 0.003 −0.000 0.0 0.001 0.004 0.011 0.008 0.003
0.324 0.325 0.305 0.279 0.215 0.0 0.171 0.263 0.294 0.286 0.305
0.145 0.132 0.117 0.099 0.079 0.0 0.080 0.099 0.127 0.148 0.177
0.153 0.132 0.119 0.099 0.083 0.0 0.081 0.103 0.133 0.151 0.183
0.324 0.325 0.305 0.280 0.219 0.0 0.174 0.269 0.298 0.294 0.317
0.145 0.132 0.117 0.100 0.079 0.0 0.080 0.099 0.127 0.148 0.177
0.153 0.132 0.119 0.099 0.083 0.0 0.081 0.103 0.134 0.152 0.183
Table 9 Estimation of transformation function: DGP1, N = 500. y
g (y)
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.000 0.001 0.001 0.003 −0.001 0.0 −0.001 −0.001 −0.001 −0.002 −0.003
−0.001 −0.002 −0.000
0.247 0.257 0.245 0.231 0.183 0.0 0.177 0.228 0.242 0.255 0.263
0.107 0.097 0.083 0.069 0.049 0.0 0.050 0.067 0.091 0.110 0.127
0.105 0.099 0.083 0.069 0.051 0.0 0.050 0.070 0.093 0.116 0.136
0.247 0.257 0.245 0.231 0.184 0.0 0.179 0.230 0.245 0.258 0.267
0.107 0.097 0.083 0.069 0.049 0.0 0.050 0.067 0.091 0.110 0.127
0.105 0.099 0.084 0.069 0.051 0.0 0.050 0.070 0.093 0.116 0.136
−1.0 −0.8 −0.6 −0.4 −0.2
−1.0 −0.8 −0.6 −0.4 −0.2
−0.010 −0.006 −0.009 −0.007 −0.013
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0 −0.028 −0.029 −0.033 −0.038 −0.046
0.003 −0.002 0.0 0.002 0.003 0.007 0.007 0.009
Table 10 Estimation of transformation function: DGP2, N = 250. y
g (y)
−1.0 −0.8 −0.6 −0.4 −0.2
−1.0 −0.8 −0.6 −0.4 −0.2
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.014 0.006 −0.004 −0.010 −0.070 0.0 −0.001 −0.046 −0.078 −0.059 −0.087
−0.004
−0.007
0.009 0.005 0.003 0.002 0.0 −0.008 −0.010 −0.018 −0.014 −0.025
0.004 0.000 0.000 −0.004 0.0 −0.002 −0.007 −0.008 −0.004 −0.017
0.449 0.426 0.396 0.349 0.274 0.0 0.248 0.340 0.403 0.448 0.461
0.167 0.147 0.129 0.107 0.082 0.0 0.085 0.120 0.155 0.182 0.212
0.175 0.145 0.129 0.106 0.085 0.0 0.089 0.127 0.168 0.201 0.233
0.449 0.426 0.396 0.349 0.282 0.0 0.248 0.344 0.411 0.452 0.469
0.167 0.148 0.129 0.107 0.082 0.0 0.085 0.120 0.156 0.182 0.213
0.175 0.145 0.129 0.106 0.086 0.0 0.089 0.127 0.168 0.201 0.234
Table 11 Estimation of transformation function: DGP2, N = 500. y
g (y)
−1.0 −0.8 −0.6 −0.4 −0.2
−1.0 −0.8 −0.6 −0.4 −0.2
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.047 0.053 0.020 −0.003 −0.043 0.0 0.021 −0.026 −0.030 −0.021 −0.036
−0.001
−0.007
0.006 0.006 0.005 0.001 0.0 −0.003 −0.009 −0.005 −0.008 −0.003
0.002 0.001 0.000 0.001 0.0 −0.002 −0.003 0.001 0.001 0.008
0.412 0.407 0.379 0.324 0.252 0.0 0.243 0.321 0.361 0.399 0.406
0.121 0.105 0.092 0.076 0.060 0.0 0.064 0.083 0.109 0.130 0.154
0.119 0.106 0.095 0.079 0.060 0.0 0.063 0.088 0.115 0.144 0.170
0.415 0.411 0.380 0.324 0.256 0.0 0.244 0.322 0.363 0.400 0.408
0.120 0.105 0.092 0.076 0.060 0.0 0.064 0.084 0.109 0.130 0.154
0.119 0.106 0.095 0.079 0.060 0.0 0.063 0.088 0.115 0.144 0.170
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
13
Table 12 Estimation of transformation function: DGP3, N = 250. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
−1.667 −1.032 −0.705 −0.440 −0.209 0.0 0.193 0.373 0.543 0.705 0.860
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.012 −0.038 −0.011 −0.030 −0.050 0.0 −0.018 −0.078 −0.086 −0.093 −0.096
−0.016 −0.023 −0.027 −0.032 −0.027
−0.004 −0.009 −0.018 −0.017 −0.002
0.0
0.0 −0.002 −0.005 −0.006 0.003 0.006
0.417 0.412 0.366 0.337 0.263 0.0 0.208 0.279 0.335 0.352 0.372
0.258 0.198 0.170 0.157 0.125 0.0 0.106 0.136 0.160 0.180 0.191
0.257 0.187 0.159 0.143 0.110 0.0 0.103 0.135 0.153 0.163 0.180
0.417 0.414 0.366 0.338 0.268 0.0 0.209 0.290 0.346 0.364 0.384
0.258 0.199 0.171 0.161 0.128 0.0 0.109 0.140 0.163 0.182 0.192
0.257 0.187 0.160 0.144 0.110 0.0 0.103 0.135 0.153 0.163 0.181
−0.025 −0.033 −0.031 −0.024 −0.021
Table 13 Estimation of transformation function: DGP3, N = 500. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
−1.667 −1.032 −0.705 −0.440 −0.209
−0.012 −0.017 −0.006 −0.009 −0.043
−0.020 −0.015 −0.013 −0.012 −0.011
−0.019 −0.006 −0.006 −0.001 −0.001
0.0 0.193 0.373 0.543 0.705 0.860
0.0 −0.003 −0.038 −0.049 −0.058 −0.069
0.0
−0.011 −0.008 −0.014 −0.012 −0.012
0.0 0.005 0.007 0.008 0.011 0.008
0.335 0.321 0.314 0.291 0.229 0.0 0.197 0.256 0.291 0.314 0.330
0.172 0.131 0.116 0.105 0.082 0.0 0.075 0.093 0.109 0.126 0.135
0.169 0.137 0.114 0.099 0.070 0.0 0.073 0.087 0.106 0.118 0.126
0.335 0.321 0.314 0.291 0.232 0.0 0.198 0.259 0.295 0.319 0.337
0.174 0.132 0.117 0.106 0.083 0.0 0.075 0.094 0.109 0.127 0.135
0.170 0.137 0.114 0.099 0.070 0.0 0.074 0.088 0.106 0.119 0.126
Table 14 Estimation of transformation function: DGP4, N = 250. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
−1.667 −1.032 −0.705 −0.440 −0.209 0.0 0.193 0.373 0.543 0.705 0.860
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.085 −0.023 −0.026 −0.057 −0.091 0.0 0.015 −0.054 −0.109 −0.116 −0.135
−0.055 −0.057 −0.059 −0.067 −0.049
−0.025 −0.025 −0.020 −0.028 −0.016
0.0 −0.020 −0.021 −0.025 −0.032 −0.023
0.0 0.003 0.011 0.007 0.009 0.008
0.571 0.536 0.482 0.416 0.318 0.0 0.259 0.346 0.415 0.474 0.499
0.364 0.270 0.224 0.182 0.138 0.0 0.108 0.133 0.154 0.178 0.199
0.337 0.248 0.193 0.155 0.119 0.0 0.103 0.122 0.140 0.163 0.166
0.577 0.537 0.483 0.420 0.331 0.0 0.260 0.350 0.429 0.488 0.517
0.368 0.276 0.232 0.194 0.147 0.0 0.110 0.134 0.156 0.181 0.200
0.337 0.248 0.194 0.157 0.120 0.0 0.103 0.123 0.140 0.163 0.166
Table 15 Estimation of transformation function: DGP4, N = 500. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
−1.667 −1.032 −0.705 −0.440 −0.209 0.0 0.193 0.373 0.543 0.705 0.860
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.083 0.017 0.010 −0.027 −0.072 0.0 0.031 −0.014 −0.034 −0.061 −0.072
−0.036 −0.027 −0.025 −0.031 −0.024
−0.017 −0.011 −0.010 −0.009 −0.007
0.0
0.0 −0.000 0.003 0.002 0.002 0.005
0.488 0.471 0.442 0.365 0.281 0.0 0.253 0.348 0.397 0.407 0.430
0.242 0.165 0.133 0.112 0.090 0.0 0.077 0.090 0.112 0.121 0.141
0.222 0.155 0.121 0.101 0.072 0.0 0.067 0.083 0.097 0.101 0.112
0.495 0.472 0.443 0.366 0.290 0.0 0.255 0.348 0.398 0.412 0.436
0.245 0.167 0.135 0.117 0.093 0.0 0.079 0.093 0.113 0.122 0.143
0.223 0.155 0.122 0.101 0.073 0.0 0.067 0.083 0.097 0.101 0.112
−0.015 −0.020 −0.016 −0.020 −0.022
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
14
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
Table 16 Estimation of transformation function: DGP5, N = 250. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
−1.667 −1.032 −0.705 −0.440 −0.209 0.0 0.193 0.373 0.543 0.705 0.860
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.045 0.009 0.010 −0.018 −0.055 0.0 0.015 −0.041 −0.079 −0.099 −0.095
−0.016 −0.022 −0.034 −0.031 −0.031
−0.050 −0.051 −0.039 −0.022 −0.018
0.0 −0.036 −0.043 −0.051 −0.041 −0.041
0.0 0.002 0.005 0.020 0.041 0.052
0.428 0.460 0.419 0.364 0.280 0.0 0.253 0.345 0.398 0.428 0.464
0.323 0.279 0.230 0.200 0.153 0.0 0.131 0.200 0.231 0.259 0.284
0.306 0.263 0.222 0.190 0.141 0.0 0.148 0.190 0.218 0.245 0.262
0.431 0.460 0.419 0.365 0.285 0.0 0.254 0.348 0.405 0.439 0.473
0.323 0.280 0.233 0.202 0.156 0.0 0.136 0.204 0.237 0.263 0.287
0.310 0.268 0.225 0.191 0.141 0.0 0.148 0.190 0.219 0.248 0.267
Table 17 Estimation of transformation function: DGP5, N = 500. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
−1.667 −1.032 −0.705 −0.440 −0.209 0.0 0.193 0.373 0.543 0.705 0.860
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.019 0.006 0.017 0.004 −0.038 0.0 0.014 −0.024 −0.052 −0.063 −0.066
−0.013 −0.017 −0.018 −0.015 −0.012
−0.025 −0.020 −0.016 −0.009 −0.004
0.0 −0.024 −0.027 −0.031 −0.031 −0.036
0.0 0.003 0.005 0.009 0.015 0.021
0.398 0.379 0.368 0.329 0.244 0.0 0.231 0.297 0.342 0.377 0.384
0.236 0.201 0.166 0.138 0.104 0.0 0.102 0.134 0.163 0.188 0.210
0.189 0.160 0.135 0.119 0.090 0.0 0.092 0.113 0.132 0.146 0.162
0.398 0.379 0.369 0.329 0.247 0.0 0.231 0.298 0.346 0.382 0.390
0.237 0.201 0.167 0.139 0.105 0.0 0.105 0.136 0.166 0.191 0.213
0.190 0.162 0.136 0.120 0.090 0.0 0.092 0.113 0.132 0.147 0.164
Table 18 Estimation of transformation function: DGP6, N = 250. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
−1.667 −1.032 −0.705 −0.440 −0.209 0.0 0.193 0.373 0.543 0.705 0.860
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.175 0.048 0.012 −0.031 −0.099 0.0 0.049 −0.021 −0.080 −0.133 −0.166
−0.049 −0.074 −0.081 −0.078 −0.070
−0.043 −0.052 −0.051 −0.035 −0.026
0.0 −0.027 −0.042 −0.057 −0.055 −0.048
0.0 0.017 0.024 0.030 0.046 0.058
0.608 0.555 0.483 0.412 0.328 0.0 0.286 0.374 0.429 0.479 0.540
0.509 0.406 0.331 0.255 0.205 0.0 0.166 0.221 0.251 0.298 0.348
0.422 0.339 0.289 0.223 0.165 0.0 0.159 0.201 0.232 0.270 0.304
0.633 0.557 0.483 0.413 0.342 0.0 0.291 0.375 0.436 0.497 0.565
0.511 0.413 0.341 0.267 0.217 0.0 0.168 0.225 0.257 0.303 0.351
0.424 0.343 0.294 0.225 0.167 0.0 0.160 0.202 0.234 0.273 0.310
Table 19 Estimation of transformation function: DGP6, N = 500. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
−1.667 −1.032 −0.705 −0.440 −0.209 0.0 0.193 0.373 0.543 0.705 0.860
Bias
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.120 0.047 0.034 −0.024 −0.090 0.0 0.064 −0.002 −0.042 −0.070 −0.092
−0.032 −0.037 −0.037 −0.039 −0.034
−0.015 −0.014 −0.018 −0.010 −0.009
0.0
0.0 −0.000 0.007 0.008 0.009 0.013
0.536 0.502 0.452 0.376 0.310 0.0 0.277 0.347 0.418 0.477 0.504
0.356 0.262 0.210 0.175 0.143 0.0 0.121 0.163 0.195 0.221 0.260
0.278 0.204 0.169 0.132 0.107 0.0 0.101 0.121 0.138 0.151 0.170
0.550 0.505 0.453 0.377 0.322 0.0 0.284 0.347 0.420 0.482 0.512
0.357 0.265 0.213 0.179 0.147 0.0 0.124 0.167 0.199 0.225 0.264
0.279 0.205 0.170 0.132 0.108 0.0 0.101 0.121 0.138 0.151 0.170
−0.030 −0.040 −0.039 −0.042 −0.042
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
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15
Table 20 Estimation of transformation function: DGP7, N = 250. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
Bias
−1.667 −1.032 −0.705 −0.440 −0.209 0.0 0.193 0.373 0.543 0.705 0.860
STD
RMSE
Chen
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.132 −0.084 0.035 −0.120 −0.231 0.0 −0.126 −0.233 −0.364 −0.443 −0.473
−0.041 −0.089 −0.116 −0.156 −0.164
0.017 −0.004 −0.016 −0.023 −0.021 0.0 −0.004 −0.028 −0.022 −0.033 −0.037
0.452 0.516 0.445 0.397 0.395 0.0 0.175 0.252 0.275 0.352 0.427
0.336 0.322 0.292 0.248 0.263 0.0 0.114 0.169 0.216 0.255 0.271
0.282 0.215 0.214 0.208 0.167 0.0 0.140 0.185 0.198 0.194 0.208
0.471 0.522 0.446 0.415 0.457 0.0 0.215 0.343 0.456 0.566 0.637
0.339 0.334 0.314 0.293 0.309 0.0 0.133 0.224 0.277 0.314 0.334
0.283 0.215 0.215 0.210 0.168 0.0 0.140 0.187 0.199 0.196 0.212
0.0
−0.069 −0.147 −0.174 −0.183 −0.194
Table 21 Estimation of transformation function: DGP7, N = 500. y
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
g (y)
Bias
−1.667 −1.032 −0.705 −0.440 −0.209 0.0 0.193 0.373 0.543 0.705 0.860
−1.842 −1.253 −1.0 −0.747 −0.158
τ = 0.2
RMSE
Two-step
CZ
Chen
Two-step
CZ
Chen
Two-step
CZ
0.143 −0.122 0.074 −0.127 −0.280 0.0 −0.117 −0.233 −0.362 −0.482 −0.510
−0.006 −0.079 −0.092 −0.144 −0.159
0.012 −0.009 −0.014 −0.013 −0.020 0.0 0.001 −0.011 −0.017 −0.024 −0.028
0.424 0.519 0.401 0.382 0.376 0.0 0.177 0.241 0.284 0.314 0.392
0.290 0.292 0.243 0.201 0.225 0.0 0.111 0.158 0.188 0.216 0.238
0.238 0.178 0.163 0.155 0.142 0.0 0.115 0.148 0.151 0.154 0.159
0.447 0.534 0.408 0.402 0.470 0.0 0.212 0.336 0.460 0.575 0.643
0.290 0.302 0.260 0.248 0.276 0.0 0.127 0.212 0.254 0.278 0.301
0.238 0.178 0.164 0.155 0.144 0.0 0.115 0.148 0.152 0.155 0.162
Table 22 Quantile predictions Qτ (Y ∗ |X = x): DGP1 x
STD
Chen
τ = 0.3
0.0
−0.061 −0.141 −0.171 −0.176 −0.183
N = 250.
τ = 0.4
τ = 0.5
τ = 0.6
τ = 0.7
τ = 0.8
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
−0.032 −0.018 −0.015 −0.002
0.157 0.150 0.155 0.158 0.193
−0.019 −0.008 −0.009 −0.003
0.160 0.160 0.157 0.154 0.164
−0.002 −0.003
0.164 0.155 0.150 0.147 0.170
−0.012 (−0.040) −0.002 (−0.006) −0.003 (0.006) −0.002 (0.010) 0.007 (0.046)
0.167 (0.250) 0.157 (0.216) 0.150 (0.209) 0.145 (0.227) 0.180 (0.234)
0.004 0.016 0.010 0.010 0.015
0.166 0.153 0.145 0.138 0.165
0.001 0.013 0.011 0.004 0.015
0.174 0.160 0.151 0.144 0.165
0.008 −0.001 0.007 0.017 0.015
0.203 0.171 0.161 0.156 0.162
0.010
0.017
0.002 0.003 0.020
Note: The median predictions using Chen (2010)’s are presented in parentheses.
For the estimation of the transformation function, it is clear that both of our one-step (CZ) and two-step (Two-step) estimators dominate that of Chen (2010), which only makes use of the conditional median restriction. For DGPs 1–4 our one-step and two-step estimators are quite comparable, whereas the one-step estimator performs noticeably better for DGPs 5 and 6, and much better for DGP 7. In addition, in Tables 22–35 we report the performance of the conditional quantile predictors of our one-step estimator and Chen (2010).2 Our conditional quantile predictors perform well for a wide range of quantiles, and significantly outperform (Chen, 2010) for the conditional median prediction. 5. Conclusion In this paper we have considered the estimation of a general transformation model subject to a multiplicative form of heteroscedasticity, based on the insights of Han (1987), Sherman (1993), Manski (1975, 1985) and Chen (2002, 2010). Our estimators for the finite dimensional parameters, the transformation function, as well as the conditional quantile predictors, all converge at the parametric rate. Monte Carlo simulation experiments indicate that our procedure performs very well in finite samples. In economic duration or statistical survival analysis, censoring is a common phenomenon. Our new estimation procedure can be extended to the case with random censoring by adjusting the objective functions as in Chen (2002, 2010). In addition, endogeneity is a difficult, and yet important issue in the literature, it might be possible to extend our analysis to the endogenous setting. These are interesting topics for future research. 2 The relative performance between our one-step and two-step estimators for the conditional quantile predictors is similar to the relative performance for the estimation of the transformation function by two estimators for seven designs. To save space, we only report the one-step estimator.
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
16
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Table 23 Quantile predictions Qτ (Y ∗ |X = x): DGP1
τ = 0.2
x
−1.842 −1.253 −1.0 −0.747 −0.158
τ = 0.3
N = 500.
τ = 0.4
τ = 0.5
τ = 0.6
τ = 0.7
τ = 0.8
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
−0.017 −0.013 −0.003 −0.001
0.112 0.114 0.114 0.113 0.126
−0.014 −0.001 −0.007 −0.002
0.115 0.114 0.110 0.111 0.116
0.002 0.004 0.007 0.001 0.007
0.114 0.101 0.101 0.099 0.124
0.013 (−0.008) 0.011 (−0.007) 0.011 (0.003) 0.010 (0.005) 0.011 (0.040)
0.119 (0.178) 0.111 (0.175) 0.105 (0.163) 0.103 (0.169) 0.126 (0.205)
0.007 0.007 0.002 0.007 0.006
0.115 0.105 0.100 0.094 0.117
0.011 0.009 0.004 0.004 0.005
0.124 0.109 0.098 0.098 0.109
0.004 0.000 0.002 −0.001 0.002
0.145 0.113 0.108 0.105 0.116
0.013
0.005
Note: The median predictions using Chen (2010)’s are presented in parentheses. Table 24 Quantile predictions Qτ (Y ∗ |X = x): DGP2
τ = 0.2
x
−1.842 −1.253 −1.0 −0.747 −0.158
τ = 0.3
N = 250.
τ = 0.4
τ = 0.5
τ = 0.6
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
−0.043 −0.043 −0.036 −0.031 −0.016
0.144 0.142 0.140 0.151 0.184
−0.026 −0.037 −0.030 −0.027 −0.003
0.163 0.154 0.146 0.144 0.169
−0.088 −0.095 −0.083 −0.082 −0.056
0.213 0.207 0.198 0.194 0.221
−0.011 (−0.088) −0.018 (−0.031) −0.020 (−0.026) −0.009 (−0.004) 0.017 (0.017)
0.231 0.219 0.215 0.215 0.249
(0.385) (0.356) (0.341) (0.341) (0.389)
τ = 0.7
τ = 0.8
Bias
RMSE
Bias
RMSE
Bias
RMSE
0.072 0.068 0.069 0.074 0.098
0.237 0.226 0.219 0.208 0.238
0.009 0.010 0.011 0.011 0.039
0.174 0.163 0.162 0.155 0.164
−0.014 −0.012 −0.006 −0.008 0.016
0.181 0.152 0.134 0.131 0.145
Note: The median predictions using Chen (2010)’s are presented in parentheses. Table 25 Quantile predictions Qτ (Y ∗ |X = x): DGP2
τ = 0.2
x
−1.842 −1.253 −1.0 −0.747 −0.158
τ = 0.3
N = 500.
τ = 0.4
τ = 0.5
τ = 0.6
τ = 0.7
τ = 0.8
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
−0.001 −0.004 −0.005 −0.006 −0.002
0.100 0.093 0.095 0.098 0.133
−0.013 −0.010 −0.016 −0.016 −0.014
0.108 0.096 0.097 0.098 0.124
−0.040 −0.040 −0.038 −0.040 −0.041
0.140 0.129 0.130 0.132 0.157
0.014 (−0.051) 0.008 (0.005) 0.008 (0.018) 0.013 (0.010) 0.018 (0.039)
0.175 (0.327) 0.172 (0.289) 0.166 (0.276) 0.158 (0.287) 0.171 (0.323)
0.063 0.060 0.058 0.062 0.069
0.171 0.164 0.159 0.155 0.161
0.013 0.011 0.010 0.009 0.025
0.117 0.105 0.100 0.097 0.113
−0.001 −0.001 −0.002
0.116 0.097 0.095 0.087 0.097
0.001 0.004
Note: The median predictions using Chen (2010)’s are presented in parentheses.
Appendix. Notations p Throughout the proof, we define θ = (b, γ , c ), θ1 = (b, γ ) where c = (cℓ )ℓ=1 and correspondingly θ0 = (θ10 , c0 ), θ10 = (β0 , γ0 ), and g (y, θ10 ) = g (y), where c0 = (c (τℓ ))pℓ=1 . Let Zi1 = Xi′ β0 and Zi2 = Xi′ γ0 . ιℓ = (0, 0, . . . , 1, 0, . . . , 0)′ is a p × 1 vector with ℓ’s element to be one. Nθ10 is a small neighborhood of θ10 .
Lemma A.1. Under Assumptions A1–A7, for any given y, g (y, θ1 ) is well defined and continuously differentiable in θ1 ∈ Nθ10 and gn (y, θ1 ) − g (y, θ1 ) = −
n 1 ∑ q¯ i (y, θ1 , g (y, θ1 ))
n
V (y, θ1 )
i=1
( ) + op n−1/2
(A.1)
and
( ) ∂ gn (y, θ1 ) ∂ g (y, θ1 ) 1 −1 −2 − = Op h22 + n−1/2 h− h2 2 +n ∂θ1 ∂θ1
(A.2)
uniformly in (y, θ ) with probability approaching one as n increases to infinity, where q¯ i (y, θ1 , g (y, θ1 )) =
[ ] ∂ ∂ ri (y, θ1 , g (y, θ1 )) − E ri (y, θ1 , g (y, θ1 )) ∂g ∂g
and V (y, θ1 ) =
1 2
[ E
∂2 ri (y, θ1 , g (y, θ1 )) ∂g2
]
with ri (y, θ1 , g ) = E hij (y, θ1 , g ) + hji (y, θ1 , g ) |Xi , Yi
[
√
]
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
17
and
{ hij (y, θ1 , g ) = 1 {Yi ≥ y} − 1 Yj ≥ y0
(
{
})
1
}
Xj′ b
Xi′ b − g
( )≥ ( ′ ) . σ Xi′ γ σ Xj γ
Proof. The proof is similar to that of Lemma A in Chen (2010). Thus we only outline the main steps here. Following [Chen] (2010) an application of the inverse function theorem yields that g (y, θ1 ) is continuously differentiable in (y, θ1 ) ∈ y, y × Nθ10 and we can similarly show that gn (y, θ1 ) uniquely maximizes the smoothed objective functions for any given ( ) θ1 ∈ Nθ10 and y ∈ y, y with probability approaching one as n increases to infinity, and gn (y, θ1 ) satisfies
( ( )) ( ) ∑ diy − djy0 1 Xj′ b Xi′ b − gn (y, θ1 ) 1 ( ) ( ) − ( ′ ) k =0 n (n − 1) h2 σ Xi′ γ h2 σ Xi′ γ σ Xj γ i̸ =j 1
and differentiation yields ( i̸ =j
∂ gn (y, θ1 ) = ∂ b˜
)
diy −djy
∑
0
(
σ Xi′ γ
k′
)
(
∑
(
1 h2
(
0
(
(i ) ( (
)
diy −djy
i ̸ =j σ 2 X ′ γ i
Xi′ b−gn (y,θ) σ X′γ
)
k′
1 h2
−
)) (
X′b
(j
σ Xj′ γ
Xi′ b−gn (y,θ1 ) σ X′γ
(
i
)
X˜ i σ Xi′ γ
(
)
−
) ))
X′b
(j
σ Xj′ γ
−
)
X˜ j
(
σ Xj′ γ
)
(A.3)
)
and (
∑ ∂ gn (y, θ1 ) =− ∂γ
)
diy −djy
0
i̸ =j
σ (Xi′ γ )
′
(
k
1 h2
(
)) (
(Xi′ b−gn (y,θ1 ))σ ′ (Xi′ γ )Xi − ( ′ ) − σ 2 (Xi′ γ ) (i ) σ Xj γ ( ) ( ( )) ∑ diy −djy0 ′ 1 Xi′ b−gn (y,θ1 ) Xj′ b ( ) k − i̸ =j σ 2 (X ′ γ ) h2 σ (Xi′ γ ) σ Xj′ γ i Xi′ b−gn (y,θ1 ) σ X′γ
Xj′ b
(
) (
)
Xj′ b σ ′ Xj′ γ Xj ( ) σ 2 X′γ j
) (A.4)
Then, by Lemma A.1 in Carroll et al. (1997), it follows that gn (y, θ1 ) − g (y, θ1 ) = op (1)
(A.5)
uniform in y and θ1 ∈ N10 . To establish the uniform asymptotic linear representation of gn (y, θ1 ) for θ1 ∈ N10 , define Ln (y, θ1 , g) =
1
∑
n (n − 1)
i̸ =j
lij (y, b, γ , g)
where
(
diy − djy0
lij (y, b, γ , g) =
)
1
( ) k σ Xi′ γ h2
(
1
(
Xi′ b − gn (y, θ1 )
h2
( ) σ Xi′ γ
−
Xj′ b
))
( ) σ Xj′ γ
With standard U-statistic decomposition, we can write Ln (y, θ1 , g) = L(y, θ1 , g) + Ln1 (y, θ1 , g) + Ln2 (y, θ1 , g) where L(y, θ1 , g) = E lij (y, b, γ , g)
[
]
n
Ln1 (y, θ1 , g) =
] [ ]] 1∑ [ [ E lij (y, b, γ , g)|Xi , Yi − E lij (y, b, γ , g) n i=1
+
n 1∑ [ [
n
E lji (y, b, γ , g)|Xi , Yi − E lij (y, b, γ , g)
]
[
]]
i=1
and Ln2 (y, θ1 , g) =
1
∑
n (n − 1)
i ̸ =j
l2ij (y, b, γ , g)
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
18
N = 250.
x
τ = 0.2
Value
Bias
RMSE
Bias
τ = 0.3 RMSE
τ = 0.4 Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) 1(−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
0.018 −0.007 −0.012 −0.016 −0.024 0.020 −0.016 −0.031 −0.033 −0.030 0.023 −0.015 −0.026 −0.037 −0.033 0.034 −0.011 −0.023 −0.040 −0.033 0.036 −0.017 −0.051 −0.060 −0.030
0.196 0.154 0.146 0.157 0.217 0.258 0.205 0.201 0.204 0.240 0.295 0.227 0.211 0.218 0.268 0.324 0.249 0.236 0.237 0.297 0.425 0.332 0.320 0.322 0.360
0.028 0.015 0.008 0.002 0.002 0.035 0.019 0.007 0.001 0.008 0.032 0.016 0.009 −0.005 0.010 0.044 0.019 0.013 0.003 −0.000 0.070 0.038 0.013 0.001 0.005
0.106 0.095 0.097 0.110 0.167 0.184 0.153 0.144 0.145 0.199 0.202 0.169 0.160 0.173 0.223 0.236 0.195 0.188 0.193 0.249 0.330 0.283 0.260 0.267 0.310
−0.022 −0.004 0.012 0.015 0.017 0.035 0.024 0.024 0.023 0.030 0.040 0.033 0.028 0.024 0.025 0.049 0.035 0.026 0.016 0.035 0.054 0.050 0.045 0.039 0.035
0.088 0.064 0.059 0.074 0.142 0.119 0.118 0.126 0.133 0.179 0.158 0.149 0.148 0.150 0.197 0.186 0.173 0.171 0.176 0.222 0.270 0.249 0.247 0.240 0.299
−0.028(−0.049) −0.032(−0.066) −0.026(−0.054) 0.001(−0.005) 0.019(−0.002) −0.007(−0.021) 0.013(0.021) 0.013(0.022) 0.013(0.017) 0.019(0.013) 0.014(0.023) 0.014(0.025) 0.016(0.016) 0.013(0.001) 0.030(0.048) 0.017(0.023) 0.014(0.005) 0.014(0.005) 0.023(0.025) 0.026(0.078) 0.019(0.033) 0.025(0.031) 0.021(0.018) 0.029(0.037) 0.028(0.157)
0.152(0.182) 0.114(0.165) 0.081(0.124) 0.049(0.093) 0.131(0.204) 0.055(0.104) 0.060(0.102) 0.077(0.114) 0.106(0.144) 0.159(0.261) 0.067(0.105) 0.089(0.131) 0.108(0.143) 0.123(0.182) 0.175(0.312) 0.104(0.138) 0.121(0.152) 0.130(0.183) 0.146(0.220) 0.202(0.344) 0.169(0.249) 0.179(0.260) 0.195(0.274) 0.202(0.310) 0.286(0.451)
0.082 −0.005 −0.020 −0.019 0.022 −0.030 −0.020 0.005 0.023 0.028 −0.029 0.009 0.024 0.024 0.033 0.005 0.025 0.025 0.028 0.048 0.031 0.027 0.035 0.039 0.041
0.183 0.158 0.133 0.079 0.122 0.134 0.076 0.050 0.078 0.151 0.101 0.056 0.080 0.109 0.169 0.076 0.090 0.113 0.136 0.202 0.156 0.172 0.180 0.198 0.276
0.438 0.168 0.032 −0.016 0.023 0.040 −0.023 −0.018 0.007 0.036 −0.021 −0.022 −0.010 0.022 0.039 −0.037 −0.018 0.022 0.031 0.054 0.015 0.031 0.034 0.047 0.072
0.464 0.230 0.160 0.131 0.100 0.167 0.145 0.094 0.050 0.146 0.163 0.106 0.057 0.072 0.156 0.149 0.080 0.066 0.105 0.193 0.107 0.123 0.151 0.176 0.267
1.030 0.581 0.288 0.043 0.029 0.417 0.114 0.002 −0.010 0.046 0.217 0.017 −0.016 −0.009 0.053 0.079 −0.023 −0.021 0.024 0.051 −0.054 −0.005 0.032 0.042 0.079
1.042 0.601 0.326 0.163 0.088 0.444 0.193 0.155 0.092 0.147 0.264 0.162 0.131 0.061 0.166 0.182 0.162 0.097 0.068 0.182 0.181 0.100 0.104 0.152 0.252
Note: The median predictions using Chen (2010)’s are presented in parentheses.
τ = 0.5
τ = 0.6
τ = 0.7
τ = 0.8 S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 26 Quantile predictions Qτ (Y ∗ |X = x): DGP3
N = 500.
x
τ = 0.2
Value
Bias
RMSE
τ = 0.3 Bias
RMSE
Bias
τ = 0.4 RMSE
τ = 0.5 Bias
RMSE
τ = 0.6 Bias
RMSE
Bias
τ = 0.7 RMSE
τ = 0.8 Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) (−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
0.008 −0.003 −0.007 −0.004 0.008 −0.003 −0.008 −0.012 −0.009 −0.003 0.002 −0.010 −0.011 −0.011 −0.010 −0.002 −0.012 −0.016 −0.017 −0.010 −0.006 −0.025 −0.028 −0.029 −0.009
0.144 0.113 0.105 0,106 0.160 0.182 0.141 0.133 0.137 0.178 0.195 0.151 0.140 0.154 0.194 0.222 0.173 0.166 0.171 0.204 0.289 0.230 0.216 0.217 0.236
0.015 0.010 0.007 0.005 0.009 0.013 0.003 0.003 −0.002 0.008 0.009 0.004 0.003 −0.000 0.011 0.008 0.004 0.002 0.002 0.000 0.017 0.004 0.001 −0.005 −0.011
0.074 0.065 0.066 0.074 0.122 0.132 0.109 0.102 0.103 0.142 0.144 0.121 0.115 0.119 0.155 0.162 0.141 0.131 0.135 0.177 0.225 0.193 0.187 0.193 0.206
−0.013 −0.002 0.009 0.009 0.014 0.014 0.011 0.008 0.007 0.014 0.014 0.008 0.008 0.006 0.014 0.013 0.006 0.007 0.007 0.011 0.012 0.012 0.013 0.007 0.006
0.052 0.034 0.033 0.049 0.103 0.075 0.081 0.086 0.088 0.116 0.100 0.100 0.097 0.102 0.132 0.120 0.114 0.112 0.115 0.151 0.169 0.159 0.158 0.165 0.206
−0.019(−0.037) −0.014(−0.037) −0.011(−0.029) 0.003(0.001) 0.010(0.008) −0.005(−0.013) 0.008(0.011) 0.007(0.014) 0.006(0.006) 0.006(0.024) 0.008(0.015) 0.004(0.011) 0.003(0.003) 0.004(−0.005) 0.012(0.003) 0.003(0.009) 0.003(−0.003) 0.004(−0.004) −0.000(−0.004) 0.010(0.040) −0.003(0.001) −0.001(0.007) 0.000(0.016) 0.007(0.007) −0.005(0.069)
0.104(0.154) 0.075(0.123) 0.047(0.086) 0.026(0.059) 0.095(0.170) 0.031(0.065) 0.039(0.064) 0.063(0.079) 0.070(0.106) 0.116(0.203) 0.046(0.071) 0.059(0.090) 0.073(0.106) 0.087(0.137) 0.130(0.226) 0.069(0.095) 0.081(0.115) 0.093(0.140) 0.099(0.171) 0.146(0.290) 0.120(0.185) 0.127(0.206) 0.142(0.227) 0.150(0.227) 0.195(0.334)
0.067 −0.011 −0.014 −0.009 0.010 −0.020 −0.012 0.003 0.007 0.009 −0.017 0.005 0.008 0.007 0.010 0.004 0.009 0.007 0.005 0.016 0.007 0.005 0.005 0.004 0.005
0.130 0.114 0.095 0.051 0.083 0.094 0.047 0.029 0.046 0.108 0.058 0.033 0.047 0.069 0.117 0.039 0.054 0.073 0.090 0.136 0.104 0.119 0.126 0.135 0.187
0.444 0.156 0.019 −0.009 0.013 0.031 −0.011 −0.010 0.002 0.011 −0.011 −0.014 −0.007 0.011 0.012 −0.021 −0.010 0.011 0.009 0.020 0.012 0.011 0.009 0.009 0.015
0.455 0.187 0.113 0.093 0.073 0.121 0.105 0.066 0.027 0.100 0.121 0.075 0.034 0.045 0.111 0.104 0.043 0.041 0.067 0.128 0.059 0.083 0.102 0.119 0.183
1.040 0.589 0.286 0.027 0.014 0.421 0.096 −0.003 −0.006 0.012 0.207 0.009 −0.012 −0.008 0.012 0.063 −0.012 −0.011 0.012 0.013 −0.028 −0.001 0.015 0.010 0.021
1.046 0.598 0.304 0.112 0.056 0.433 0.143 0.108 0.065 0.094 0.232 0.115 0.092 0.034 0.102 0.133 0.111 0.061 0.040 0.121 0.123 0.053 0.069 0.098 0.166
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Note: The median predictions using Chen (2010)’s are presented in parentheses.
19
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 27 Quantile predictions Qτ (Y ∗ |X = x): DGP3
20
N = 250.
x
τ = 0.2
Value
Bias
RMSE
Bias
τ = 0.3 RMSE
Bias
τ = 0.4 RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) (−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
0.017 −0.014 −0.031 −0.037 −0.041 0.011 −0.033 −0.046 −0.063 −0.072 −0.002 −0.042 −0.065 −0.072 −0.073 0.006 −0.046 −0.076 −0.086 −0.083 0.025 −0.032 −0.067 −0.094 −0.059
0.211 0.165 0.155 0.161 0.207 0.240 0.191 0.182 0.194 0.260 0.267 0.200 0.199 0.211 0.268 0.295 0.234 0.228 0.245 0.280 0.390 0.316 0.302 0.313 0.348
0.001 −0.020 −0.027 −0.028 −0.024 −0.003 −0.030 −0.035 −0.043 −0.060 0.000 −0.027 −0.037 −0.048 −0.062 −0.011 −0.046 −0.063 −0.069 −0.060 0.011 −0.055 −0.064 −0.074 −0.053
0.163 0.140 0.136 0.135 0.185 0.207 0.177 0.171 0.183 0.227 0.229 0.201 0.192 0.191 0.256 0.254 0.222 0.217 0.225 0.268 0.351 0.312 0.300 0.302 0.330
0.002 −0.005 0.012 −0.019 −0.022 −0.011 −0.024 −0.027 −0.034 −0.046 −0.018 −0.028 −0.035 −0.042 −0.062 −0.021 −0.040 −0.039 −0.045 −0.057 −0.038 −0.057 −0.065 −0.071 −0.062
0.112 0.094 0.099 0.115 0.166 0.180 0.166 0.165 0.174 0.208 0.213 0.198 0.196 0.201 0.256 0.250 0.232 0.225 0.229 0.288 0.357 0.337 0.328 0.339 0.341
−0.012(−0.159) −0.021(−0.155) −0.021(−0.113) 0.010(−0.028)
0.181(0.304) 0.136(0.296) 0.101(0.252) 0.070(0.177) 0.167(0.311) 0.085(0.197) 0.094(0.174) 0.119(0.189) 0.145(0.250) 0.215(0.375) 0.114(0.195) 0.135(0.230) 0.158(0.256) 0.185(0.295) 0.249(0.426) 0.153(0.249) 0.181(0.286) 0.201(0.317) 0.221(0.351) 0.269(0.503) 0.272(0.393) 0.286(0.442) 0.301(0.454) 0.306(0.493) 0.352(0.632)
0.363 0.152 0.079 0.034 0.080 0.079 0.042 0.018 0.038 0.120 0.042 0.020 0.023 0.073 0.128 0.023 0.022 0.066 0.099 0.139 0.075 0.111 0.129 0.147 0.152
0.409 0.258 0.215 0.138 0.153 0.220 0.149 0.090 0.085 0.215 0.179 0.104 0.081 0.130 0.241 0.124 0.090 0.123 0.179 0.266 0.159 0.215 0.254 0.283 0.355
0.788 0.388 0.155 0.027 0.045 0.233 0.033 0.008 −0.006 0.072 0.082 −0.004 −0.004 0.005 0.076 0.002 −0.011 −0.011 0.034 0.086 −0.036 0.020 0.038 0.060 0.084
0.803 0.422 0.243 0.182 0.108 0.288 0.185 0.151 0.079 0.171 0.199 0.167 0.104 0.051 0.184 0.185 0.129 0.066 0.081 0.205 0.126 0.078 0.118 0.178 0.286
1.267 0.728 0.382 0.089 0.033 0.586 0.186 0.032 −0.004 0.048 0.345 0.055 −0.004 −0.013 0.063 0.150 −0.015 −0.016 −0.005 0.066 −0.048 −0.036 0.010 0.031 0.074
1.277 0.745 0.417 0.211 0.085 0.605 0.251 0.182 0.127 0.133 0.379 0.180 0.152 0.081 0.160 0.220 0.174 0.123 0.055 0.173 0.188 0.113 0.066 0.116 0.240
Note: The median predictions using Chen (2010)’s are presented in parentheses.
τ = 0.5
0.048(0.056) −0.002(−0.047) 0.032(0.022) 0.042(0.034) 0.043(0.060) 0.059(0.049) 0.039(0.033) 0.046(0.053) 0.048(0.062) 0.052(0.061) 0.049(0.051) 0.049(0.062) 0.053(0.070) 0.056(0.070) 0.064(0.067) 0.048(0.105) 0.072(0.084) 0.076(0.084) 0.071(0.066) 0.060(0.078) 0.057(0.241)
τ = 0.6
τ = 0.7
τ = 0.8 S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 28 Quantile predictions Qτ (Y ∗ |X = x): DGP4
N = 500.
x
τ = 0.2
Value
Bias
RMSE
τ = 0.3 Bias
RMSE
Bias
τ = 0.4 RMSE
Bias
τ = 0.5 RMSE
Bias
τ = 0.6 RMSE
τ = 0.7 Bias
RMSE
τ = 0.8 Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) (−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
0.011 −0.003 −0.007 −0.017 −0.020 −0.012 −0.006 −0.018 −0.024 −0.036 0.007 −0.010 −0.016 −0.030 −0.034 0.008 −0.017 −0.029 −0.036 −0.043 0.020 −0.009 −0.024 −0.035 −0.030
0.142 0.113 0.108 0.107 0.143 0.168 0.131 0.122 0.124 0.167 0.180 0.144 0.137 0.141 0.173 0.198 0.161 0.154 0.156 0.186 0.259 0.209 0.195 0.197 0.210
−0.001 −0.011 −0.010 −0.014 −0.017 −0.002 −0.012 −0.018 −0.026 −0.026 −0.008 −0.020 −0.020 −0.024 −0.031 −0.007 −0.016 −0.024 −0.029 −0.029 −0.006 −0.024 −0.028 −0.037 −0.027
0.115 0.100 0.095 0.098 0.128 0.143 0.121 0.116 0.118 0.148 0.158 0.134 0.127 0.131 0.165 0.170 0.145 0.138 0.142 0.178 0.222 0.192 0.186 0.187 0.206
−0.002 −0.008 −0.016 −0.020 −0.020 −0.023 −0.028 −0.025 −0.025 −0.027 −0.026 −0.028 −0.032 −0.029 −0.036 −0.031 −0.035 −0.036 −0.040 −0.038 −0.047 −0.050 −0.050 −0.055 −0.035
0.066 0.068 0.075 0.083 0.113 0.129 0.122 0.118 0.120 0.144 0.154 0.144 0.138 0.138 0.162 0.178 0.164 0.161 0.162 0.181 0.242 0.227 0.221 0.218 0.228
−0.003(−0.090) −0.008(−0.087) −0.008(−0.058) 0.012(0.002) 0.032(0.047) 0.004(−0.015) 0.024(0.031) 0.025(0.038) 0.025(0.033) 0.034(0.029) 0.027(0.037) 0.028(0.040) 0.026(0.042) 0.035(0.047) 0.041(0.041) 0.030(0.046) 0.033(0.036) 0.039(0.044) 0.040(0.048) 0.037(0.053) 0.047(0.080) 0.044(0.066) 0.049(0.054) 0.049(0.085) 0.044(0.193)
0.132(0.243) 0.094(0.215) 0.062(0.169) 0.046(0.125) 0.117(0.266) 0.051(0.139) 0.063(0.138) 0.083(0.156) 0.105(0.186) 0.146(0.283) 0.072(0.153) 0.095(0.172) 0.110(0.198) 0.127(0.251) 0.164(0.329) 0.107(0.189) 0.126(0.214) 0.136(0.257) 0.150(0.290) 0.183(0.390) 0.178(0.324) 0.191(0.328) 0.200(0.351) 0.215(0.372) 0.243(0.518)
0.328 0.117 0.056 0.027 0.047 0.059 0.036 0.017 0.023 0.073 0.041 0.017 0.013 0.043 0.080 0.023 0.013 0.043 0.060 0.088 0.049 0.068 0.078 0.094 0.104
0.350 0.194 0.162 0.102 0.099 0.169 0.113 0.064 0.050 0.139 0.132 0.069 0.044 0.080 0.157 0.089 0.047 0.078 0.115 0.180 0.093 0.136 0.157 0.182 0.243
0.789 0.371 0.122 0.007 0.020 0.206 0.007 −0.003 −0.002 0.031 0.048 −0.007 −0.006 −0.002 0.035 −0.018 −0.014 −0.008 0.016 0.036 −0.025 0.010 0.017 0.023 0.047
0.796 0.386 0.173 0.136 0.066 0.234 0.141 0.112 0.054 0.103 0.141 0.118 0.080 0.032 0.111 0.143 0.097 0.045 0.046 0.123 0.092 0.043 0.076 0.111 0.170
1.272 0.730 0.367 0.055 0.017 0.585 0.163 0.004 −0.006 0.021 0.332 0.023 −0.016 −0.007 0.027 0.129 −0.027 −0.013 −0.005 0.036 −0.043 −0.023 0.008 0.013 0.040
1.276 0.737 0.382 0.141 0.050 0.595 0.195 0.130 0.091 0.089 0.349 0.131 0.120 0.056 0.100 0.173 0.132 0.090 0.030 0.123 0.143 0.071 0.037 0.078 0.153
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Note: The median predictions using Chen (2010)’s are presented in parentheses.
21
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 29 Quantile predictions Qτ (Y ∗ |X = x): DGP4
22
N = 250.
x
τ = 0.2
Value
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) (−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
−0.017 −0.038 −0.053 −0.066 −0.075
0.192 0.161 0.158 0.161 0.219 0.365 0.273 0.259 0.259 0.288 0.530 0.365 0.314 0.294 0.339 0.929 0.565 0.428 0.370 0.408 3.018 2.275 2.641 1.210 0.564
−0.001 −0.010 −0.016 −0.023 −0.028 −0.011 −0.036 −0.051 −0.060 −0.057
0.094 0.080 0.083 0.106 0.176 0.224 0.194 0.187 0.193 0.222 0.309 0.258 0.240 0.237 0.268 0.481 0.355 0.318 0.306 0.334 1.725 1.239 0.894 0.705 0.482
−0.041 −0.021 −0.001 0.002 0.004 0.011 0.003 −0.002 −0.015 −0.007 0.241 0.002 −0.010 −0.018 −0.014 0.037 0.003 −0.012 −0.017 −0.018 0.222 0.127 0.072 0.014 −0.037
0.114 0.076 0.054 0.062 0.143 0.126 0.126 0.131 0.134 0.183 0.198 0.171 0.171 0.177 0.223 0.285 0.247 0.222 0.215 0.270 0.858 0.567 0.455 0.383 0.443
−0.028(−0.065) −0.025(−0.065) −0.019(−0.054) 0.005(−0.007) 0.034(0.006) −0.010(−0.023) 0.013(0.020) 0.017(0.021) 0.017(0.016) 0.028(0.019) 0.014(0.023) 0.017(0.024) 0.014(0.020) 0.022(0.000) 0.031(0.054) 0.017(0.023) 0.013(0.016) 0.018(0.007) 0.014(0.040) 0.024(0.092) 0.008(0.049) 0.014(0.069) 0.011(0.059) 0.010(0.084) 0.001(0.187)
0.159(0.198) 0.112(0.168) 0.071(0.121) 0.049(0.093) 0.143(0.230) 0.061(0.102) 0.064(0.099) 0.085(0.114) 0.112(0.159) 0.175(0.306) 0.073(0.108) 0.098(0.138) 0.117(0.166) 0.141(0.204) 0.205(0.361) 0.113(0.151) 0.136(0.185) 0.152(0.215) 0.172(0.265) 0.240(0.429) 0.212(0.308) 0.238(0.333) 0.246(0.360) 0.265(0.435) 0.396(0.608)
0.028 0.011 0.005 −0.001 0.068 −0.000 −0.006 0.015 0.048 0.078 −0.012 0.013 0.049 0.062 0.087 −0.005 0.049 0.064 0.074 0.094 0.073 0.087 0.089 0.100 0.085
0.215 0.175 0.123 0.067 0.157 0.149 0.086 0.066 0.103 0.196 0.120 0.079 0.101 0.140 0.219 0.110 0.120 0.145 0.179 0.257 0.211 0.241 0.267 0.289 0.371
0.192 0.057 0.035 0.020 0.072 0.025 0.010 0.000 0.019 0.095 −0.005 −0.007 −0.004 0.053 0.098 −0.036 −0.021 0.010 0.064 0.118 −0.103 −0.038 0.043 0.066 0.120
0.288 0.220 0.196 0.124 0.162 0.240 0.181 0.121 0.079 0.211 0.240 0.163 0.113 0.118 0.231 0.251 0.160 0.123 0.155 0.265 0.308 0.234 0.230 0.256 0.358
0.618 0.274 0.115 0.048 0.075 0.350 0.075 0.015 0.013 0.098 0.275 0.022 −0.017 −0.002 0.106 0.219 −0.019 −0.047 −0.008 0.108 0.190 −0.098 −0.129 −0.015 0.099
0.647 0.340 0.245 0.202 0.153 0.403 0.240 0.210 0.117 0.203 0.340 0.236 0.205 0.101 0.223 0.301 0.239 0.211 0.102 0.242 0.301 0.295 0.288 0.199 0.330
0.001
−0.067 −0.092 −0.111 −0.108 −0.082 −0.024 −0.072 −0.102 −0.136 0.237 0.053 −0.036 −0.102 −0.155 1.052 0.736 0.519 0.194 −0.092
τ = 0.3
0.016 −0.026 −0.040 −0.061 −0.070 0.073 0.009 −0.031 −0.047 −0.092 0.541 0.291 0.140 0.031 −0.076
τ = 0.4
Note: The median predictions using Chen (2010)’s are presented in parentheses.
τ = 0.5
τ = 0.6
τ = 0.7
τ = 0.8 S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 30 Quantile predictions Qτ (Y ∗ |X = x): DGP5
N = 500.
x
τ = 0.2
Value
Bias
RMSE
τ = 0.3 Bias
RMSE
Bias
τ = 0.4 RMSE
Bias
τ = 0.5 RMSE
Bias
τ = 0.6 RMSE
τ = 0.7 Bias
RMSE
τ = 0.8 Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) (−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
0.012 −0.001 −0.008 −0.016 −0.006 0.009 −0.011 −0.023 −0.030 −0.025 0.023 −0.008 −0.019 −0.032 −0.032 0.052 −0.001 −0.023 −0.037 −0.042 0.226 0.124 0.055 0.011 −0.011
0.137 0.117 0.109 0.114 0.154 0.243 0.188 0.172 0.171 0.198 0.319 0.242 0.218 0.210 0.232 0.440 0.307 0.266 0.253 0.266 0.996 0.722 0.567 0.464 0.356
0.011 0.004 −0.000 −0.004 −0.004 0.000 −0.011 −0.013 −0.017 −0.014 0.007 −0.004 −0.013 −0.018 −0.015 0.021 −0.004 −0.013 −0.019 −0.026 0.084 0.034 0.006 −0.015 −0.028
0.057 0.057 0.063 0.079 0.123 0.151 0.133 0.126 0.128 0.165 0.203 0.175 0.167 0.162 0.187 0.280 0.227 0.207 0.200 0.216 0.586 0.444 0.382 0.335 0.287
−0.018 −0.008 0.006 0.006 0.001 0.010 0.003 −0.002 −0.006 0.003 0.009 0.002 −0.001 −0.005 −0.000 0.015 0.004 −0.000 −0.006 −0.002 0.042 0.025 0.013 −0.003 −0.014
0.059 0.033 0.030 0.046 0.100 0.085 0.087 0.091 0.094 0.134 0.129 0.120 0.120 0.123 0.158 0.177 0.161 0.157 0.159 0.191 0.354 0.304 0.279 0.263 0.279
−0.021(0.043) −0.017(0.037) −0.012(0.028) 0.002(0.002) 0.011(0.002) −0.006(0.012) 0.007(0.011) 0.006(0.010) 0.002(0.008) 0.007(0.027) 0.006(0.014) 0.004(0.009) 0.000(0.011) 0.001(−0.004) 0.008(0.010) 0.003(0.010) −0.001(0.006) −0.001(0.001) 0.000(0.006) 0.002(0.065) −0.006(0.017) −0.008(0.032) −0.010(0.047) −0.004(0.039) −0.013(0.109)
0.102(0.163) 0.073(0.118) 0.046(0.084) 0.025(0.062) 0.100(0.186) 0.032(0.064) 0.039(0.068) 0.056(0.081) 0.080(0.118) 0.126(0.238) 0.046(0.077) 0.069(0.099) 0.085(0.124) 0.098(0.155) 0.146(0.266) 0.078(0.113) 0.097(0.140) 0.107(0.164) 0.123(0.214) 0.168(0.349) 0.150(0.245) 0.166(0.279) 0.173(0.293) 0.192(0.320) 0.262(0.415)
0.005 0.002 0.001 −0.003 0.028 −0.004 −0.006 0.004 0.020 0.030 −0.008 0.003 0.021 0.020 0.034 −0.009 0.026 0.026 0.031 0.037 0.044 0.044 0.039 0.034 0.036
0.152 0.111 0.080 0.042 0.094 0.097 0.052 0.034 0.057 0.120 0.075 0.042 0.056 0.081 0.135 0.063 0.067 0.085 0.110 0.164 0.132 0.158 0.174 0.186 0.241
0.144 0.016 −0.000 0.000 0.022 0.003 −0.011 −0.005 −0.001 0.024 −0.012 −0.013 −0.010 0.017 0.027 −0.024 −0.019 −0.005 0.022 0.033 −0.054 −0.015 0.021 0.021 0.030
0.197 0.151 0.131 0.086 0.087 0.173 0.134 0.077 0.039 0.121 0.174 0.115 0.058 0.052 0.136 0.177 0.103 0.053 0.076 0.155 0.196 0.100 0.099 0.142 0.220
0.610 0.238 0.059 0.002 0.023 0.333 0.045 −0.010 −0.005 0.026 0.259 0.012 −0.019 −0.010 0.031 0.210 −0.006 −0.022 −0.009 0.026 0.194 −0.039 −0.058 0.015 0.026
0.624 0.272 0.161 0.135 0.073 0.358 0.169 0.162 0.084 0.112 0.296 0.177 0.153 0.065 0.127 0.258 0.188 0.148 0.058 0.145 0.264 0.226 0.185 0.083 0.202
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Note: The median predictions using Chen (2010)’s are presented in parentheses.
23
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 31 Quantile predictions Qτ (Y ∗ |X = x): DGP5
24
N = 250.
x
τ = 0.2
Value
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) (−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
−0.006 −0.036 −0.058 −0.071 −0.082
0.218 0.187 0.183 0.184 0.236 0.337 0.260 0.253 0.268 0.336 0.493 0.338 0.311 0.328 0.395 0.816 0.530 0.432 0.405 0.455 3.321 1.811 1.433 1.114 0.627
−0.008 −0.025 −0.040 −0.049 −0.062 −0.011 −0.055 −0.074 −0.094 −0.134
0.156 0.142 0.146 0.155 0.207 0.287 0.236 0.236 0.238 0.306 0.402 0.302 0.281 0.278 0.356 0.606 0.417 0.361 0.358 0.411 1.857 1.211 0.870 0.684 0.560
−0.020 −0.016 −0.024 −0.030 −0.035 −0.032 −0.057 −0.060 −0.065 −0.091 −0.046 −0.070 −0.082 −0.091 −0.122 −0.042 −0.085 −0.106 −0.122 −0.139 0.093 −0.042 −0.093 −0.135 −0.137
0.130 0.103 0.103 0.120 0.196 0.237 0.218 0.216 0.223 0.272 0.309 0.285 0.274 0.271 0.334 0.435 0.368 0.348 0.342 0.396 1.102 0.781 0.682 0.570 0.558
0.031(−0.186) −0.033(−0.153) −0.026(−0.102) 0.012(−0.032) 0.056(0.026) −0.004(−0.040) 0.034(0.025) 0.047(0.053) 0.055(0.066) 0.059(0.026) 0.042(0.048) 0.056(0.066) 0.059(0.074) 0.060(0.080) 0.056(0.076) 0.062(0.090) 0.064(0.104) 0.066(0.097) 0.072(0.099) 0.049(0.148) 0.083(0.164) 0.089(0.190) 0.081(0.169) 0.078(0.184) 0.063(0.374)
0.210(0.360) 0.160(0.302) 0.118(0.234) 0.088(0.169) 0.179(0.327) 0.114(0.204) 0.114(0.186) 0.143(0.216) 0.179(0.271) 0.242(0.409) 0.139(0.228) 0.170(0.251) 0.194(0.289) 0.208(0.352) 0.284(0.518) 0.197(0.300) 0.217(0.356) 0.230(0.380) 0.252(0.433) 0.342(0.623) 0.311(0.546) 0.344(0.621) 0.370(0.644) 0.389(0.681) 0.535(0.922)
0.155 0.096 0.082 0.050 0.124 0.089 0.077 0.045 0.065 0.176 0.076 0.059 0.057 0.111 0.191 0.058 0.057 0.083 0.159 0.217 0.027 0.114 0.194 0.245 0.269
0.341 0.275 0.215 0.128 0.204 0.288 0.181 0.121 0.138 0.283 0.271 0.157 0.139 0.187 0.307 0.268 0.168 0.176 0.262 0.360 0.338 0.282 0.342 0.411 0.532
0.340 0.115 0.056 0.048 0.082 0.122 0.021 0.034 0.020 0.126 0.072 0.013 0.015 0.021 0.137 0.042 −0.015 −0.005 0.038 0.156 −0.010 −0.079 −0.050 0.075 0.179
0.413 0.295 0.260 0.191 0.165 0.284 0.252 0.190 0.114 0.239 0.284 0.253 0.176 0.102 0.259 0.288 0.255 0.175 0.120 0.290 0.322 0.316 0.236 0.193 0.402
0.690 0.283 0.105 0.034 0.066 0.448 0.074 −0.004 0.012 0.105 0.390 0.025 −0.035 0.003 0.109 0.361 −0.013 −0.059 −0.014 0.110 0.419 −0.060 −0.151 −0.065 0.121
0.717 0.362 0.278 0.244 0.142 0.490 0.251 0.252 0.158 0.208 0.439 0.254 0.255 0.135 0.223 0.414 0.262 0.259 0.134 0.233 0.467 0.295 0.353 0.218 0.307
0.006
−0.067 −0.103 −0.123 −0.149 0.082
−0.037 −0.103 −0.143 −0.172 0.212 0.017 −0.079 −0.143 −0.177 0.965 0.485 0.268 0.071 −0.130
τ = 0.3
0.002 −0.063 −0.096 −0.120 −0.139 0.053 −0.060 −0.101 −0.131 −0.168 0.516 0.233 0.079 −0.034 −0.125
τ = 0.4
Note: The median predictions using Chen (2010)’s are presented in parentheses.
τ = 0.5
τ = 0.6
τ = 0.7
τ = 0.8 S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 32 Quantile predictions Qτ (Y ∗ |X = x): DGP6
N = 500.
x
τ = 0.2
Value
Bias
RMSE
τ = 0.3 Bias
RMSE
Bias
τ = 0.4 RMSE
Bias
τ = 0.5 RMSE
Bias
τ = 0.6 RMSE
τ = 0.7 Bias
RMSE
τ = 0.8 Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) (−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
0.009 −0.005 −0.016 −0.022 −0.034 0.022 −0.012 −0.031 −0.038 −0.065 0.041 −0.012 −0.039 −0.059 −0.070 0.071 0.000 −0.033 −0.058 −0.071 0.257 0.132 0.049 −0.020 −0.056
0.157 0.130 0.124 0.125 0.162 0.232 0.176 0.166 0.176 0.207 0.285 0.215 0.207 0.207 0.233 0.399 0.273 0.248 0.243 0.270 0.928 0.643 0.513 0.410 0.350
0.005 −0.004 −0.012 −0.016 −0.027 0.007 −0.015 −0.023 −0.032 −0.042 0.009 −0.023 −0.029 −0.042 −0.055 0.026 −0.016 −0.036 −0.052 −0.052 0.128 0.047 0.008 −0.035 −0.054
0.112 0.106 0.104 0.107 0.143 0.194 0.166 0.155 0.161 0.185 0.245 0.203 0.189 0.185 0.221 0.321 0.251 0.234 0.232 0.251 0.660 0.505 0.432 0.379 0.344
−0.003 −0.007 −0.014 −0.018 −0.028 −0.027 −0.033 −0.038 −0.043 −0.045 −0.046 −0.049 −0.050 −0.051 −0.054 −0.051 −0.058 −0.062 −0.066 −0.066 −0.067 −0.087 −0.092 −0.096 −0.074
0.061 0.060 0.072 0.085 0.129 0.160 0.152 0.149 0.153 0.177 0.215 0.196 0.190 0.191 0.221 0.289 0.259 0.244 0.234 0.256 0.525 0.444 0.407 0.381 0.362
−0.008(−0.100) −0.013(−0.080) −0.011(−0.056) 0.011(−0.001) 0.028(0.028) 0.001(−0.014) 0.023(0.032) 0.029(0.039) 0.030(0.037) 0.037(0.034) 0.028(0.038) 0.034(0.047) 0.034(0.052) 0.035(0.055) 0.044(0.060) 0.040(0.065) 0.040(0.056) 0.043(0.074) 0.040(0.067) 0.042(0.096) 0.053(0.137) 0.058(0.109) 0.059(0.123) 0.058(0.133) 0.056(0.257)
0.140(0.271) 0.104(0.200) 0.066(0.157) 0.048(0.122) 0.122(0.288) 0.060(0.142) 0.068(0.143) 0.092(0.167) 0.117(0.206) 0.170(0.312) 0.082(0.167) 0.109(0.197) 0.126(0.226) 0.144(0.294) 0.197(0.385) 0.128(0.231) 0.145(0.262) 0.162(0.319) 0.173(0.354) 0.237(0.486) 0.222(0.427) 0.240(0.443) 0.259(0.489) 0.279(0.512) 0.344(0.691)
0.091 0.066 0.050 0.028 0.056 0.058 0.048 0.024 0.020 0.087 0.053 0.035 0.016 0.049 0.098 0.047 0.021 0.026 0.069 0.117 0.014 0.039 0.088 0.116 0.145
0.238 0.189 0.147 0.091 0.113 0.198 0.129 0.075 0.051 0.161 0.182 0.101 0.060 0.087 0.186 0.159 0.084 0.070 0.126 0.221 0.164 0.117 0.170 0.225 0.328
0.292 0.053 0.007 0.017 0.028 0.068 −0.011 0.002 −0.000 0.046 0.024 −0.024 −0.001 −0.006 0.051 −0.015 −0.030 −0.009 0.003 0.056 −0.051 −0.065 −0.030 0.024 0.078
0.325 0.194 0.187 0.131 0.083 0.187 0.192 0.136 0.073 0.130 0.196 0.188 0.116 0.052 0.137 0.206 0.186 0.102 0.046 0.155 0.249 0.215 0.100 0.086 0.237
0.673 0.242 0.037 0.007 0.026 0.431 0.041 −0.026 −0.006 0.040 0.371 −0.001 −0.035 −0.010 0.037 0.341 −0.030 −0.041 −0.013 0.047 0.402 −0.071 −0.086 −0.023 0.053
0.685 0.283 0.175 0.163 0.067 0.450 0.169 0.184 0.114 0.122 0.394 0.175 0.179 0.096 0.126 0.366 0.191 0.178 0.080 0.136 0.423 0.229 0.229 0.075 0.196
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Note: The median predictions using Chen (2010)’s are presented in parentheses.
25
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 33 Quantile predictions Qτ (Y ∗ |X = x): DGP6
26
N = 250.
x
τ = 0.2
Value
Bias
RMSE
τ = 0.3 Bias
RMSE
Bias
τ = 0.4 RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) (−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
0.041 −0.010 −0.024 −0.039 −0.024 0.039 −0.012 −0.027 −0.034 −0.052 0.046 −0.016 −0.030 −0.062 −0.020 0.042 −0.018 −0.048 −0.061 −0.050 0.087 0.001 −0.023 −0.048 −0.035
0.254 0.199 0.189 0.195 0.221 0.331 0.262 0.235 0.232 0.283 0.369 0.303 0.281 0.281 0.293 0.411 0.347 0.325 0.320 0.332 0.518 0.392 0.370 0.358 0.378
0.022 0.004 −0.008 −0.018 −0.001 0.032 −0.003 −0.022 −0.005 −0.035 0.032 0.007 0.008 −0.013 0.007 0.021 −0.008 −0.019 −0.024 −0.003 0.036 −0.014 −0.020 0.010 0.006
0.124 0.107 0.103 0.121 0.185 0.210 0188 0.190 0.185 0.232 0.248 0.216 0.206 0.204 0.255 0.255 0.229 0.219 0.241 0.285 0.344 0.316 0.308 0.293 0.317
−0.046 −0.011 0.011 0.005 0.017 0.026 0.005 −0.007 −0.010 −0.018 0.021 −0.002 0.002 0.009 −0.020 0.020 0.007 0.021 −0.001 0.009 0.023 −0.003 −0.030 0.008 0.012
0.138 0.089 0.074 0.082 0.175 0.138 0.142 0.158 0.163 0.203 0.190 0.187 0.189 0.194 0.234 0.236 0.218 0.225 0.219 0.264 0.317 0.297 0.299 0.297 0.321
−0.043(−0.144) −0.053(−0.140) −0.033(−0.036) −0.002(−0.063) 0.031(−0.135) −0.013(−0.062) 0.012(−0.061) 0.005(−0.051) 0.008(−0.024) 0.010(−0.107) 0.011(−0.016) 0.004(−0.067) 0.011(−0.035) 0.027(−0.024) 0.043(0.069) 0.010(−0.062) 0.021(−0.019) 0.029(−0.019) 0.023(−0.023) 0.041(0.055) 0.026(0.024) 0.021(−0.064) 0.011(−0.144) 0.029(0.070) 0.042(0.138)
0.164(0.212) 0.155(0.222) 0.112(0.096) 0.076(0.102) 0.164(0.197) 0.92(0.108) 0.082(0.123) 0.096(0.140) 0.123(0.120) 0.183(0.310) 0.097(0.131) 0.110(0.152) 0.133(0.133) 0.168(0.194) 0.216(0.415) 0.127(0.170) 0.112(0.128) 0.178(0.209) 0.198(0.299) 0.260(0.532) 0.231(0.308) 0.228(0.366) 0.252(0.467) 0.247(0.413) 0.281(0.390)
0.134 0.001 −0.027 −0.022 0.041 −0.045 −0.034 0.001 0.014 0.042 −0.049 0.001 0.016 0.044 0.071 −0.008 0.021 0.041 0.040 0.066 0.044 0.032 0.032 0.065 0.075
0.197 0.155 0.155 0.106 0.147 0.170 0.121 0.082 0.081 0.174 0.147 0.092 0.083 0.149 0.2134 0.110 0.112 0.153 0.180 0.243 0.217 0.217 0.0223 0.260 0.288
0.515 0.232 0.069 −0.024 0.038 0.088 −0.025 −0.039 −0.007 0.044 −0.008 −0.049 −0.033 0.032 0.072 −0.056 −0.040 0.015 0.040 0.073 0.003 0.033 0.033 0.037 0.067
0.532 0.272 0.162 0.157 0.134 0.179 0.165 0.141 0.079 0.181 0.168 0.157 0.120 0.107 0.215 0.178 0.137 0.114 0.150 0.241 0.182 0.190 0.209 0.236 0.292
1.107 0.658 0.362 0.092 0.041 0.494 0.175 0.024 −0.029 0.067 0.290 0.046 −0.030 −0.006 0.082 0.123 −0.026 −0.041 0.038 0.090 −0.058 −0.011 0.042 0.052 0.107
1.116 0.672 0.386 0.170 0.110 0.514 0.225 0.156 0.132 0.183 0.324 0.162 0.160 0.087 0.206 0.197 0.170 0.146 0.112 0.227 0.191 0.152 0.161 0.184 0.279
Note: The median predictions using Chen (2010)’s are presented in parentheses.
τ = 0.5
τ = 0.6
τ = 0.7
τ = 0.8 S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 34 Quantile predictions Qτ (Y ∗ |X = x): DGP7
N = 500.
x
τ = 0.2
Value
Bias
RMSE
τ = 0.3 Bias
RMSE
Bias
τ = 0.4 RMSE
τ = 0.5 Bias
RMSE
τ = 0.6 Bias
RMSE
Bias
τ = 0.7 RMSE
τ = 0.8 Bias
RMSE
(−0.342, −0.936) (−0.342, −0.725) (−0.342, −0.545) (−0.342, −0.292) (−0.342, 0.642) (0.247, −0.936) (0.247, −0.725) (0.247, −0.545) (0.247, −0.292) (0.247, 0.642) (0.500, −0.936) (0.500, −0.725) (0.500, −0.545) (0.500, −0.292) (0.500, 0.642) (0.753, −0.936) (0.753, −0.725) (0.753, −0.545) (0.753, −0.292) (0.753, 0.642) (1.342, −0.936) (1.342, −0.725) (1.342, −0.545) (1.342, −0.292) (1.342, 0.642)
0.028 −0.010 −0.024 −0.033 −0.017 0.047 0.006 −0.008 −0.012 −0.037 0.038 −0.009 −0.027 −0.047 −0.021 0.038 −0.022 −0.044 −0.049 −0.038 0.062 0.008 −0.008 −0.041 −0.006
0.200 0.161 0.147 0.153 0.165 0.267 0.197 0.187 0.175 0.243 0.284 0.230 0.217 0.223 0.230 0.321 0.282 0.274 0.276 0.285 0.391 0.324 0.304 0.306 0.302
0.013 −0.002 −0.009 −0.016 0.001 0.006 −0.013 −0.024 −0.010 −0.035 0.013 −0.001 0.002 −0.025 −0.002 0.011 −0.024 −0.032 −0.041 −0.013 0.002 −0.025 −0.018 −0.018 −0.002
0.091 0.082 0.083 0.090 0.151 0.151 0.141 0.139 0.147 0.194 0.176 0.165 0.158 0.158 0.206 0.192 0.172 0.164 0.181 0.224 0.256 0.246 0.252 0.247 0.252
−0.038 −0.017 −0.001 0.004 0.004 0.013 −0.006 −0.018 −0.017 −0.036 0.001 −0.019 −0.010 0.002 −0.001 −0.007 −0.007 0.009 −0.004 0.007 −0.002 −0.028 −0.042 −0.014 −0.009
0.096 0.057 0.054 0.071 0.133 0.110 0.108 0.115 0.122 0.158 0.140 0.136 0.140 0.158 0.183 0.172 0.167 0.174 0.171 0.215 0.234 0.221 0.221 0.224 0.237
−0.042(−0.165) −0.034(−0.139) −0.025(−0.031) −0.004(−0.063) 0.021(−0.145) −0.014(−0.063) 0.009(−0.082) 0.002(−0.044) 0.001(−0.011) −0.001(−0.072) 0.008(−0.010) 0.001(−0.061) −0.001(−0.022) 0.014(−0.048) 0.016(0.057) 0.001(−0.059) 0.003(−0.005) 0.018(−0.046) 0.019(−0.014) 0.025(0.007) 0.020(0.029) 0.016(−0.062) −0.001(−0.193) 0.002(−0.006) 0.013(0.104)
0.131(0.195) 0.107(0.207) 0.077(0.074) 0.051(0.090) 0.125(0.185) 0.059(0.092) 0.062(0.110) 0.072(0.137) 0.086(0.092) 0.143(0.270) 0.066(0.127) 0.083(0.144) 0.091(0.106) 0.125(0.144) 0.164(0.423) 0.092(0.161) 0.098(0.120) 0.131(0.152) 0.152(0.313) 0.208(0.500) 0.176(0.318) 0.184(0.358) 0.183(0.427) 0.168(0.237) 0.200(0.280)
0.134 −0.006 −0.037 −0.022 0.023 −0.047 −0.033 −0.005 0.004 0.030 −0.044 −0.003 0.004 0.023 0.031 −0.014 0.007 0.021 0.018 0.029 0.016 0.014 0.017 0.023 0.051
0.170 0.116 0.126 0.085 0.119 0.139 0.092 0.056 0.067 0.148 0.114 0.062 0.068 0.106 0.150 0.087 0.083 0.114 0.141 0.202 0.158 0.176 0.186 0.195 0.206
0.518 0.231 0.060 −0.029 0.023 0.069 −0.035 −0.037 −0.008 0.022 −0.015 −0.048 −0.027 0.017 0.043 −0.059 −0.038 0.003 0.020 0.038 −0.011 0.016 0.013 0.004 0.039
0.527 0.252 0.127 0.124 0.103 0.133 0.134 0.111 0.054 0.136 0.133 0.129 0.077 0.075 0.156 0.147 0.098 0.067 0.110 0.183 0.115 0.131 0.142 0.168 0.211
1.114 0.663 0.362 0.074 0.036 0.495 0.161 0.009 −0.032 0.034 0.280 0.025 −0.028 −0.016 0.056 0.106 −0.025 −0.040 0.019 0.062 −0.059 −0.017 0.028 0.012 0.053
1.119 0.670 0.375 0.140 0.100 0.505 0.192 0.127 0.110 0.138 0.299 0.126 0.131 0.059 0.168 0.154 0.139 0.111 0.080 0.179 0.158 0.104 0.115 0.133 0.218
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
√
Note: The median predictions using Chen (2010)’s are presented in parentheses.
27
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
Table 35 Quantile predictions Qτ (Y ∗ |X = x): DGP7
28
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
with l2ij (y, b, γ , g) = lij (y, b, γ , g) − E lij (y, b, γ , g)|Xi , Yi − E lij (y, b, γ , g)|Xj , Yj + E lij (y, b, γ , g)
[
[
]
[
]
[
]]
Then because of (A.5) and Corollary 4 in Sherman (1993), it is straightforward to show that 1 Ln2 (y, θ1 , gn (y, θ1 )) = Op n−1 h− . 2
(
)
Also, similar to the proof of Lemma 7A of Sherman (1994), we have Ln1 (y, θ1 , gn (y, θ1 )) − Ln1 (y, θ1 , g(y, θ1 )) = op n−1/2
(
)
with n
1∑
Ln1 (y, θ1 , g(y, θ1 )) =
n
n
( )
i=1
n 1∑
=
q¯ i (y, θ1 , g (y, θ1 )) + O h22
q¯ i (y, θ1 , g (y, θ1 )) + op n−1/2
(
)
i=1
In addition, it can be shown easily that L(y, θ1 , g(y, θ1 )) = O h22
( )
Therefore, L(y, θ1 , gn (y, θ1 )) = L(y, θ1 , gn (y, θ1 )) − L(y, θ1 , g(y, θ1 ))
+ L(y, θ1 , g(y, θ1 )) ( ) ∂ L (y, β, γ , g¯n (y, θ1 )) = (gn (y, θ1 ) − g(y, θ1 )) + O h22 ∂g ( ) = V (y, θ1 ) (gn (y, θ1 ) − g(y, θ1 )) + op n−1/2 , where g¯n (y, θ1 ) lies between gn (y, θ1 ) and g(y, θ1 ). Therefore 0 = Ln (y, θ1 , gn (y, θ1 ))
( ) ( ) 1 + Ln1 (y, θ1 , g (y, θ1 )) + op n−1/2 = L (y, θ1 , gn (y, θ1 )) + Op n−1 h− 2 ( −1/2 ) = V (y, θ1 ) (gn (y, θ1 ) − g(y, θ1 )) + Ln1 (y, θ1 , g (y, θ1 )) + op n Consequently, from the above results, we can deduce that gn (y, θ1 ) − g(y, θ1 ) =
n 1∑
n
Jy1,y0 (ξi , θ1 ) + op n−1/2
(
)
i=1
uniformly in y ∈ [y, y¯ ] and θ1 ∈ N10 , where Jy1,y0 (ξi , θ1 ) = −V (y, θ1 )−1 q¯ i (y, θ1 , g (y, θ1 )) with V (y, θ1 ) =
1 E 2
[
∂ 2 ri (y,θ1 ,g (y,θ1 )) ∂g2
]
.
We now consider the rate of convergence of
∂ gn (y,θ1 ) ∂θ1
for y ∈ y, y¯ and θ1 ∈ N10 . First write
[
]
∂ gn (y,θ1 ) ∂ b˜
as
∂ gn (y, θ1 ) Lnn = Ldn ∂ b˜ where Lnn
( ( )) ( ) ( ) ∑ diy − djy0 1 Xj′ b Xi′ b − gn (y, θ1 ) 1 X˜ i X˜ j ′ ( ) ( ) ( )− ( ′ ) k = − ( ′ ) n(n − 1) h2 σ Xi′ γ h22 σ Xi′ γ σ Xj γ σ Xi′ γ σ Xj γ i ̸ =j
Ldn
( ( )) ( ) ∑ diy − djy0 1 Xj′ b Xi′ b − gn (y, θ1 ) 1 ′ ( ) ( ) k − ( ′ ) = n(n − 1) h2 σ 2 Xi′ γ h22 σ Xi′ γ σ Xj γ i̸ =j
1
and 1
Working with the U-statistic decomposition of Lnn and Ldn , Ldn = Ld0 + Ldn1 + Ldn2 √
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
29
and Lnn = Ln0 + Lnn1 + Lnn2 . by following the above arguments, we can show that Ld0 Ln0
=
( ) ∂ g (y, θ1 ) + Op h22 ˜ ∂b
1 Ldn1 = Lnn1 = Op (n−1/2 h− 2 )
and 2 Ldn2 = Lnn2 = Op n−1 h− . 2
(
)
uniformly in y ∈ y, y¯ and θ1 ∈ N10 . Consequently, we have
[
]
) ( ∂ gn (y, θ1 ) ∂ g (y, θ1 ) 1 −1 −2 h2 − = Op h22 + n−1/2 h− 2 +n ˜ ˜ ∂b ∂b [ ] uniformly in y ∈ y, y¯ and θ1 ∈ N10 . The uniform rate of convergence for
∂ gn (y,θ1 ) ∂γ
holds similarly.
■
Proof of Theorem 1. We first establish the consistency of βˆ , gn and γˆ , cˆ in three steps. Define a class of functions:
(
FQ =
{(
)
diy − τ Kh2 Xi′ b − g − σ Xi′ γ c : y ∈ y, y , b ∈ B, γ ∈ G, c ∈ C , g ∈ M , h2 > 0
(
)
(
) )
[
]
}
Then using the arguments in the proof of Lemma 10A of Sherman (1994), we can verify that FQ is Euclidean for a constant envelope, and consequently by the uniform law of large numbers e.g. Pakes and Pollard (1989), we obtain Qn (y, θ , g ) − Q (y, θ, g ) = op (1) uniformly in (y, θ , g) and, Qn (θ , g (·, θ1 )) − Q (θ, g (·, θ1 )) =
∫
Qn (y, θ, g (y, θ1 )) ω (y) dy −
∫
Q (y, θ, g (y, θ1 )) ω (y) dy = op (1)
uniformly over θ . We now consider βˆ . First, we establish the identification of β0 . Note that τl0 = 0.5. Define Q¯ l0 (y, β ) = sup Ql0 (y, β, γ , c , g) g ∈M
and Q¯ l0 (β ) =
∫
Q¯ l0 (y, β )ω(y)dy
noting that Ql0 (y, β, γ , c , g) actually does not depend on (γ , c). Then following Manski (1975) and Chen (2010), for any ε0 > 0, max
∥β−β0 ∥≥ε0 ,β∈B
Q¯ l0 (β) = Q¯ l0 (β ∗ ) =
∫
Q¯ l0 (y, β ∗ )ω(y)dy <
∫
Ql0 (y, β0 , g(y))ω(y)dy
and let
∫
Ql0 (y, β0 , g(y))ω(y)dy −
max
∥β−β0 ∥≥ε0 ,β∈B
Q¯ l0 (β) > aε0
for some small aε0 > 0. In addition, for any other l ̸ = l0 , we have max
∥β−β0 ∥≥ε0 ,β∈B,(γ ,c)∈G×C
Q¯ l (β, γ , c ) ≤
∫
Ql (y, θ0 , g(y))ω(y)dy
Thus p ∑
max
∥β−β0 ∥≥ε0 ,β∈B,(γ ,c)∈G×C
Q¯ l (β, γ , c ) <
l=1
p ∫ ∑
Ql (y, θ0 , g(y))ω(y)dy − aε0
l=1
In particular, we have Q (θ0 , g(·, θ10 )) >
≥
sup
Q¯ (β, γ , c) + aε0
sup
Q (θ, gn (·, θ )) + aε0
∥β−β0 ∥≥ε0 ,(γ ,c)∈G×C ∥β−β0 ∥≥ε0 ,(γ ,c)∈G×C
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
30
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
On the other hand,
ˆ gn (·, θˆ1 )) + op (1) Q (θˆ , gn (·, θˆ1 )) = Qn (θ, ≥ Qn (θ0 , gn (·, θ10 )) + op (1) = Q (θ0 , gn (·, θ10 )) + op (1) = Q (θ0 , g(·, θ10 )) + op (1) Therefore, we obtain Q (θˆ , gn (·, θˆ1 )) >
sup
∥β−β0 ∥≥ε0 ,(γ ,c)∈G×C
Q (θ, gn (·, θ )) + aε0 + op (1)
Thus, with probability approaching one, we have βˆ − β0 < ε0 , which establishes the consistency of βˆ .
Next, we establish the uniform consistency of gn (y, θˆ1 ) uniformly in y ∈ [y, y¯ ]. For a continuous function g¯ (·), define supremum norm
¯ − g(y)| ∥g¯ − g ∥ = sup |g(y) y∈[y,¯y]
¯ such that ∥g¯ − g ∥ > 0, it is straightforward to show that Then for any g, Ql0 (β0 , g (·)) > Ql0 (β0 , g¯ (·)) and in particular, for any small ε0 > 0, Ql0 (β0 , g (·)) >
sup ∥g¯ −g ∥≥ε0
Ql0 (β0 , g¯ (·)) + aε0
for some positive aε0 . Then by choosing a small ε1 > 0, such that sup ∥β−β0 ∥≤ε1
⏐ ⏐ ⏐Ql (β0 , g¯ (·)) − Ql (β, g¯ (·))⏐ < aε0 0 0 2
thus Ql0 (β0 , g (·)) >
sup ∥g¯ −g ∥≥ε0 ,∥β−β0 ∥≤ε1
Ql0 (β, g¯ (·)) +
aε0 2
Consequently, we also have Q (β0 , γ0 , c0 , g (·)) >
p ∑
sup ∥g¯ −g ∥≥ε0 ,∥β−β0 ∥≤ε1 ,(γ ,c)∈G×C
Ql (β, γ , c , g¯ (·)) +
l=1
aε0 2
Given the consistency of βˆ there exists a sequence of εn → 0, such that
⏐ ⏐
⏐ ⏐
P(⏐βˆ − β0 ⏐ > εn ) → 0 as n increases Hence, for a large enough n, we obtain Q (θˆ , gn (·, θˆ )) ≥ Q (θ0 , g(·, θ0 )) + op (1)
>
sup
∥g −g0 ∥≥ε0 ,|β−β0 |≤εn ,(γ ,c)∈G×C
Q (β, γ , g) +
aε0 2
+ op (1)
which implies that ∥gn − g0 ∥ = op (1). We have thus established the uniform consistency of gn (·, θˆ1 ). Finally we establish the consistency of γˆ and cˆ . First, we show that once β0 and g (·) are identified, γ0 and c0 can be identified. For any other l such that τl ̸ = 0.5, noting that
[( ( E
[(
diy − τl 1 Xi β0 − g(y) − cτl σ Xi γ > 0
) {
′
(
′
)
}]
=E
F
Xi′ β0 − g(y)
( ) σ Xi′ γ0
)
) { − τl 1
Xi′ β0 − g(y)
( ) σ Xi′ γ
}] > cτl
Thus if (γ , cτl ) maximizes the above expression, then we have Xi′ β0 − g(y)
σ Xi′ γ
(
)
> cτl iff
Xi′ β0 − g(y)
( ) σ Xi′ γ0
> c (τl )
or Xi′ β0 − cτl σ Xi′ γ > g(y) iff Xi′ β0 − c (τl ) σ Xi′ γ0 > g(y)
(
)
(
√
)
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
31
for y ∈ y, y , which implies that when Xi′ β0 − c (τl ) σ Xi′ γ0 ∈ g(y), g(y) , we have
(
]
[
)
[
]
Xi′ β0 − c (τl ) σ Xi′ γ0 = Xi′ β0 − cτl σ Xi′ γ = g(y)
(
)
(
)
for some y ∈ ∈ y, y , which, in turn, implies that
[
]
( ) σ x′ γ0 cτl = σ (x′ γ ) c (τl ) for any x ∈ S¯X ; hence, by Assumption A6, we have γ = γ0 , which further implies that cτl = c (τl ). From the above identification result for γ0 and c0 we can deduce that for γ ̸ = γ0 , Ql (y, β0 , γ0 , c0 ) > Ql (y, β0 , γ , c0 ) Then, again, similar to the above argument, we can show that Q (θˆ , gn (·, θˆ )) ≥ Q (θ0 , gn (·, θ0 )) + op (1)
= Q (θ0 , g(·, θ0 )) + op (1) >
sup
∥g −g0 ∥≤εn ,|β−β0 |≤εn ,∥γ −γ0 ∥≥ε0 ,γ ∈G,c ∈C
Q (β, γ , g) +
aε0 2
+ op (1)
for a sequence of εn → 0, which implies the consistency of γˆ . The consistency of cˆ follows similarly. Now we establish the asymptotic normality of θˆ : From the consistency θˆ , it follows that the following first-order condition
( ) ˆ γˆ , cˆ ∂ Qn β, 0=
∂θ
holds with probability approaching one as n increases. A Taylor expansion yields
√
[
n(θˆ − θ0 ) = −
∂ 2 Qn (β ∗ , γ ∗ , c ∗ ) ∂θ∂θ ′
]−1 √
n∂ Qn (β0 , γ0 , c0 )
∂θ
where θ lies on the line segment of θˆ and θ0 . ∗
Note that ∂ Qn (β0 , γ0 , c0 )
∂θ
∂ Qn (y, θ0 , gn (y, θ10 )) ω (y) dy ∂θ ∫ ∂ Qn (y, θ0 , gn (y, θ10 )) ∂ gn (y, θ10 ) + ω (y) dy ∂g ∂θ = Tn1 + Tn2
∫
=
Define
∂ Qn (y, θ0 , gn (y, θ10 )) ∂ g (y, θ10 ) ω (y) dy ∂g ∂θ ) ( ∗ and we now show that Tn2 = Tn2 + op n−1/2 . ∗ Tn2 =
∫
Note that ∗ Tn2 − Tn2
( ) ∂ Qn (y, θ0 , gn (y, θ10 )) ∂ gn (y, θ10 ) ∂ g (y, θ10 ) − ω (y) dy ∂g ∂θ ∂θ ⏐ ⏐ ∂ gn (y, θ10 ) ∂ g (y, θ10 ) ⏐ ⏐ × ⏐T˜n2 ⏐ ≤ sup − ∂θ ∂θ y∈[y,y] ( ) ⏐⏐ ⏐⏐ 1 −1 −2 = Op h22 + n−1/2 h− h2 × ⏐T˜n2 ⏐ 2 +n ∫
=
For T˜n2 , with some standard arguments, e.g., Pollard (1995) and by Lemma A.1, we can show that T˜n2 =
∫ +
∫
n
p
) 1 1 ∑∑ ( diy − τ k n h1 i=1 ℓ=1 p n
) 1 1 ∑∑ ( diy − τ 2 k′ n h1
(
(
Zi1 − g (y) − σ (Z2i ) c0 (τℓ ) h1
Zi1 − g (y) − σ (Z2i ) c0 (τℓ ) h1
i=1 ℓ=1
√
)
)
ω (y) dy
(gn (y, θ10 ) − g (y))ω (y) dy
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32
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
∫ +
p
n
) 1 1 ∑∑ ( diy − τ 3 k′ n h1
(
Zi1 − g (y) − σ (Z2i ) c0 (τℓ )
)
h1
i=1 ℓ=1
(gn (y, θ10 ) − g (y))2 ω (y) dy
) 4 3 | − g y y , θ g sup + Op h− ) ( )| ( 10 n 1 y∈[y,¯y] ) ( ) ( ( )−1/2 ) ( ln n × Op n−1/2 = Op h21 + (nh1 )−1/2 ln n + Op h21 + nh31 ) ( ( ) ( ) ( )−1/2 4 ln n × Op n−1 + Op n−3/2 h− + Op h21 + nh51 1 (
Therefore, with Assumption A10, we have ∗ = op n−1/2 Tn2 − Tn2
(
)
∗ by Assumption 10. Furthermore, write Tn1 and Tn2 as
∂ Qn (y, θ0 , g (y, θ10 )) ω (y) dy ∂θ ∫ 2 ( ) ∂ Qn (y, θ0 , g (y, θ10 )) + (gn (y, θ10 ) − g (y)) ω (y) dy + op n−1/2 ∂θ∂ g
∫
Tn1 =
and
∂ Qn (y, θ0 , g (y, θ10 )) ∂ g (y, θ10 ) ω (y) dy ∂g ∂θ1 ∫ 2 ( ) ∂ Qn (y, θ0 , g (y, θ10 )) + (gn (y, θ10 ) − g (y)) ω (y) dy + op n−1/2 ∂g2
∗ Tn2 =
∫
Then, similar to the arguments in the proof of Theorem 1 in Chen (2010), we can show that n
∗ Tn1 + Tn2 =
1∑ n
( ) (ψ1 (ξi , θ0 ) + ψ2 (ξi , θ0 )) + op n−1/2
i=1
where
ψ1 (ξi , θ0 ) =
p ∑
ϕ1i (τℓ ) ϕ2i (τℓ )
ℓ=1
with
ϕ1i (τℓ ) = (1 {εi < c (τℓ )} − τℓ )
( ) ω g −1 (Zi1 − c (τℓ ) σ (Zi2 )) ( ) g ′ g −1 (Zi1 − c (τℓ ) σ (Zi2 ))
and
)′ ∂ g (g −1 (Zi1 − c (τℓ ) σ (Zi2 )) , θ10 ) ′ ′ ˜ ϕ2i (τℓ ) = Xi , c (τℓ ) σ (Zi2 ) Xi , ιℓ σ (Z2i ) + ∂θ (
and
ψ2 (ξi , θ0 ) =
∫
[
Hθ1g (θ0 , y) + Hθ2g (θ0 , y) Jy1,y0 (ξi , θ10 ) ω (y) dy
]
where Hθ1g
(θ0 , y) =
p ∑ ∂ g (y, θ10 )
ℓ=1
∂θ
Hgg (y, θ0 , τ )
Hgg (y, θ0 , τ ) = E f (c (τℓ )) Jb 0, x˜ , g (y) , τℓ p χ 0, x˜ , g (y) , τℓ ⏐x˜
[
(
) ⏐ )]
) ( (
)′
(
2′ Hθ2g (θ0 , y) = Hβ2˜ ′g (θ0 , y) , Hγ2′g (θ0 , y) , Hcg (θ0 , y)
and Hβ2˜ g (θ0 , y) =
p ∑ [
E x˜ f (c (τℓ )) Jb 0, x˜ , g (y) , τℓ p χ 0, x˜ , g (y) , τℓ ⏐x˜
(
) ( (
) ⏐ )]
ℓ=1
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
33
( ( ) ) ⎤ χ 0, x˜ , g (y) , τℓ − x˜ ′ β˜ 0 , x˜ ′ × c (τℓ ) ∑ ⎢ ( ( ( ) ) ) ⎥ E ⎣ σ ′ γ10 χ 0, x˜ , g (y) , τℓ − x˜ ′ β˜ 0 + x˜ ′ γ˜0 × ⎦ Hγ2 g (θ0 , y) = − ⏐ ( ) ( ( ) ) ℓ=1 f (c (τℓ )) Jb 0, x˜ , g (y) , τℓ p χ 0, x˜ , g (y) , τℓ ⏐x˜ [ ] ( ( ( ) ) ) σ γ10 χ 0, x˜ , g (y) , τℓ − x˜ ′ β˜ 0 + x˜ ′ γ˜0 × 2 ( ) ( ( )⏐ ) Hcτ g (θ0 , y) = −E ℓ f (c (τℓ )) Jb 0, x˜ , g (y) , τℓ p χ 0, x˜ , g (y) , τℓ ⏐x˜ ⎡
p
Consequently, we obtain
√
n∂ Qn (β0 , γ0 , c0 )
∂θ
n 1 ∑
= √
n
n 1 ∑
= √
n
(ψ1 (ξi , θ0 ) + ψ2 (ξi , θ0 )) + op (1)
i=1
ψ (ξi , θ0 ) + op (1)
i=1
Also, similar to the arguments in the proof of Theorem 1 in Chen (2010), we can show
∂ 2 Qn (b∗ , γ ∗ , c ∗ ) = H (θ0 ) + op (1) ∂θ ∂θ ′ As a result, n ) √ ( 1 ∑ n θˆ − θ0 = √ H (θ0 )−1 ψ (ξi , θ0 ) + op (1)
n
i=1
n 1 ∑
= √
n
Ψ (ξi , θ0 ) + op (1)
i=1
and thus, Theorem 1 then follows by applying the central limit theorem.
■
Proof of Theorem 2. Note that
√ (
(
)
)
n gn y, θˆ1 − g (y) =
√ [( n
(
)
(
gn y, θˆ1 − g y, θˆ1
))
( ( ) )] + g y, θˆ1 − g (y, θ10 ) .
Then from the uniform asymptotic linear representation and the consistency of θˆ1 , the stochastic equicontinuity holds for
√ (
(
)
(
)
(
n gn y, θˆ1 − g y, θˆ1
))
√ −
n (gn (y, θ10 ) − g (y, θ10 )) = op (1)
Therefore,
√ (
)
n gn y, θˆ1 − g (y)
∂ g (y, θ10 ) √ n(θˆ1 − θ10 ) + op (1) ∂θ1 ] n [ q¯ i (y, θ10 , g (y, θ10 )) ∂ g (y, θ10 ) 1 ∑ − = √ + Ψ (ξi , θ0 ) + op (1) V (y, θ10 ) ∂θ1 n √
=
n (gn (y, θ10 ) − g (y)) +
i=1
n ] 1 ∑[ 1 = √ Jy,y0 (ξi , θ10 ) + Jy2 (ξi , θ10 ) + op (1) n i=1
Finally, following Pakes and Pollard (1989), it is straightforward to show that the Jy1,y0 (ξi , θ10 ) : y ∈ y, y and {2 [ ]} Jy (ξi , θ10 ) : y ∈ y, y are both Euclidean class with a square integrable envelop, Theorem 2 follows from Theorem (19.14) of Van der Vaart (1998). ■
{
[
]}
Proof of Theorem 3. Similar of) gn (y, θˆ1 )[ we ]can prove the uniform [ to ] the arguments in proving the uniform√consistency ( consistency of cˆτ for τ ∈ τ , τ . To establish the weak convergence of n cˆτ − c (τ ) for τ ∈ τ , τ , apply a Taylor series expansion to obtain
√ (
n cˆτ − c (τ )
)
( ) ⎞−1 √ ( ) ∂ 2 Qn θˆ1 , gn , τ , c¯ n∂ Qn θˆ1 , gn , τ , c (τ ) ⎠ = ⎝− + op (1) ∂ cτ2 ∂ cτ ⎛
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
34
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
and we have
( ) ∂ Qn θˆ1 , gn , τ , c (τ ) ∂ cτ ) ∂ Qn (θ10 , g , τ , c (τ )) ∂ 2 Qn (θ10 , g , τ , c (τ )) ( = + θˆ1 − θ10 ∂ cτ ∂ cτ ∂θ1 ∫ 2 ) ( ∂ Qn (y, θ10 , g (y) , τ , c (τ )) + (gn (y, θ10 ) − g (y)) ω (y) dy + op n−1/2 ∂ cτ ∂ g ( ) = Cn1 + Cn2 + Cn3 + op n−1/2 Using the arguments and results in Theorems 1 and 2, it is straightforward to show that Cn1 =
n 1∑
n
Si1 + op n−1/2
(
i=1 n 1∑
∫ Cn2 =
)
n
Si2 (y) ω (y) dy + op n−1/2
)
Si3 (y) ω (y) dy + op n−1/2
)
(
i=1
and n 1∑
∫ Cn3 =
n
(
i=1
where Si1 = ϕ1 (εi , Zi1 , Zi2 , τ ) σ (Zi2 ) and Si2 (y) =
(
Hc2τ θ1
(θ10 , y) +
Hc2τ g
) ∂ g (y, θ10 ) Iθ1 Ψ (ξi , θ0 ) (θ10 , y) ∂θ ′
and Si3 (y) = Hc2τ g (θ10 , y) Jy1,y0 (ξi , θ10 ) + Jy2,y0 (ξi , θ10 )
)
(
Next, following the arguments in the proof of Theorem 1, we can show that
( ) ∂ 2 Qn θˆ1 , gn , τ , c¯ ∂ cτ2
= V (τ , θ10 , g ) + op (1)
uniform in τ , where V (τ , θ10 , g ) = −
∫
[
] ( ( ( ) ) ) σ 2 γ10 χ 0, x˜ , g (y) , τ − x˜ ′ β˜ 0 + x˜ ′ γ˜0 × ( ) ( ( ) ⏐ ) ω (y) dy + op (1) E f (c (τ )) Jb 0, x˜ , g (y) , τ p χ 0, x˜ , g (y) , τ ⏐x˜
Consequently, we have n ) 1 ∑ −1 n cˆτ − c (τ ) = − √ V (τ , θ10 , g ) ϕ (ξi , θ10 , c (τ )) + op (1) n
√ (
i=1
n 1 ∑
= −√
n
Λ (ξi , θ10 , c (τ )) + op (1)
i=1
where
ϕ (ξi , θ10 , c (τ )) = Si1 +
∫
(Si2 (y) + Si3 (y)) ω (y) dy
It is straightforward to show that {Λ (·, θ1 , cτ ) : θ1 ∈ B × G, cτ ∈ C } form an Euclidean class. Then Theorem 3 follows by Theorem (19.14) of Van der Vaart (1998). ■ Proof of Theorem 4. Recall gn yˆ ∗τ (x) = x′ βˆ − σ x′ γˆ cˆτ and g y∗τ (x) = x′ β0 − σ (x′ γ0 )c (τ ), then from Theorems 2 and 3, we have
(
)
(
)
(
)
gn yˆ ∗τ (x) − g y∗τ (x) = op (1)
(
)
(
)
√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.
S. Chen and H. Zhang / Journal of Econometrics xxx (xxxx) xxx
35
uniformly in x. In addition, from Lemma A.1 we have gn yˆ ∗τ (x) − g yˆ ∗τ (x) = op (1)
(
)
(
)
uniformly in x. Thus we have g yˆ ∗τ (x) − g y∗τ (x) = op (1)
)
(
)
(
uniformly in x. Then the uniform convergence of yˆ ∗τ (x) follows since 1
⏐ ⏐ ∗ ⏐(yˆ (x) − y∗ (x))⏐ ≤ max τ τ
y∈[y,¯y] g ′ (y)
⏐ ( ∗ ) )⏐ ( ⏐g yˆ (x) − g y∗ (x) ⏐ . τ
τ
Next we establish the weak convergence of
√
√ (
n yˆ ∗τ (x) − y∗τ (x) . Write
)
√
n(gn yˆ ∗τ (x) − g yˆ ∗τ (x) ) as
(
)
(
)
n(gn yˆ ∗τ (x) − g yˆ ∗τ (x) )
)
(
√
)
(
( ) ) √ ( ( ′ ) n σ x γˆ cˆτ − σ x′ γ0 c (τ ) ) ( )) √ ( ( − n g yˆ ∗τ (x) − g y∗τ (x) ) ( ( )√ ( ) ) ( )) √ ( √ ( β˜ − β˜ − σ x′ γ0 n cˆτ − c (τ ) + σ ′ x′ γ0 c (τ ) x′ n γˆ − γ0 = nx˜ ′ ˆ )) ) ( √ ( ( − n g yˆ ∗τ (x) − g y∗τ (x) + op (1) =
( ˆ
)
nx˜ ′ β˜ − β˜ 0 −
From the uniform asymptotic linear representation in Theorem 2, we have
√ (
n gn yˆ ∗τ (x) − g yˆ ∗τ (x)
(
)
(
))
=
√ (
n gn y∗τ (x) − g y∗τ (x)
(
)
(
))
+ op (1)
Thus
√ ( (
n g yˆ ∗τ (x) − g y∗τ (x)
)
(
))
) ( ( )√ ( ) )) ) ( √ ( √ ( β˜ − β˜ 0 − σ x′ γ0 n cˆτ − c (τ ) + σ ′ x′ γ0 c (τ ) x′ n γˆ − γ0 = x˜ ′ n ˆ ) ) ( ( √ − ngn y∗τ (x) − g y∗τ (x) + op (1) ( ) A Taylor expansion of g yˆ ∗τ (x) , together with some simple algebra, yields ) √ ( ∗ n yˆ τ (x) − y∗τ (x) ) ] [ )√ ( ( ) √ ′ (ˆ nx˜ β˜ − β˜ 0 − σ x′ γ0 n cˆτ − c (τ ) − 1 = ′( ∗ ) × )) + op (1) ) ) ( ( ) √ ( ( √ ( g yτ (x) σ ′ x′ γ0 c (τ ) x′ n γˆ − γ0 − n gn y∗τ (x) − g y∗τ (x) ) ] [ ( ′ ( ) )√ ( n θˆ1 − θ10 − x˜ , −σ ′ x′ γ0 c (τ ) x′ 1 = ′( ∗ ) × ( )√ ( ) √ ( ( ) ( )) + op (1) g yτ (x) σ x′ γ0 n cˆτ − c (τ ) − n gn y∗τ (x) − g y∗τ (x) ) ( ) ( ′ ] [ n x˜ , −σ ′ x′ γ0 c (τ ) x′ Iθ1 Ψ (ξi , θ0 ) 1 ∑ 1 ( ) ( ) ( ) = √ + op (1) n −σ x′ γ0 Λ (ξi , θ10 , c (τ )) − Jy1∗τ (x),y (ξi , θ10 ) + Jy2∗τ (x) (ξi , θ10 ) g ′ y∗τ (x) i=1 0
n
1 ∑
= √
n
J¯y,y0 (τ , ξi , x, θ0 )
i=1
Following Pakes and Pollard (1989), it is straightforward to show that the J¯y,y0 (τ , ξi , x, θ0 ) : τ ∈ τ , τ is a Euclidean class of functions with a square integrable envelop. Then, Theorem 4 follows by applying Theorem (19.14) of Van der Vaart (1998). ■
{
[
]}
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√
Please cite this article as: S. Chen and H. Zhang, n-prediction of generalized heteroscedastic transformation regression models. Journal of Econometrics (2019), https://doi.org/10.1016/j.jeconom.2019.09.003.