Identification in a fully nonparametric transformation model with heteroscedasticity

Abstract

An identification result for nonparametrically transformed location scale models is proven. The result is constructive in the sense that it provides an explicit expression of the transformation function.

Introduction

Let $d_X\in \mathbb{N}$. The underlying question of this article can be formulated quite easily: Given some real valued random variable $Y$ and some ${\mathbb{R}}^{d_X}$-valued random variable $\mathbf{X}$ fulfilling the heteroscedastic transformation model

$$h\left(Y\right)=g\left(\mathbf{X}\right)+\sigma \left(\mathbf{X}\right)\epsilon$$

with some error term $\epsilon$ independent of $\mathbf{X}$ and fulfilling $E\left[\epsilon \right]=0$ and $Var\left(\epsilon \right)=1$, are the model components $h:\mathbb{R}\to \mathbb{R},g:{\mathbb{R}}^{d_X}\to \mathbb{R},\sigma :{\mathbb{R}}^{d_X}\to (0,\infty )$ and the error distribution uniquely determined if the joint distribution of $(Y,\mathbf{X})$ is known? This uniqueness is called identification of a model.

Over the last years, transformation models have attracted more and more attention since they are often used to obtain desirable properties by first transforming the dependent random variable of a regression model. Applications for such transformations can reach from reducing skewness of the data to inducing additivity, homoscedasticity or even normality of the error terms. Already Box and Cox (1964), Bickel and Doksum (1981) and Zellner and Revankar (1969) introduced some parametric classes of transformation functions. Horowitz (1996) developed an identification result for a linear regression function $g$ and homoscedastic errors. Later, these ideas were extended by Ekeland et al. (2004) to general smooth regression functions $g$. The arguably most general identification results so far were provided by Chiappori et al. (2015) and Vanhems and Van Keilegom (2019), who considered general regression functions and even allowed heteroscedastic errors to some extend, but required conditional homoscedasticity of the error and some exogenous regressors, conditional on endogenous regressors or some control variable, respectively. Unfortunately, all of these approaches cannot be applied to the heteroscedastic model in (1.1) without further assumptions so that different methods are needed. Despite their practical relevance (e.g. in duration models, see Khan et al., 2011), results allowing heteroscedasticity are rare. While Zhou et al. (2009) showed identifiability in a single-index model with a linear regression function $g$ and a known variance function ${\sigma }^2$, Neumeyer et al. (2016) simply required identifiability implicitly in their assumptions. In contrast to the approaches mentioned above, it is tried here to avoid the assumption of the presence of some exogenous regressors and any parametric assumption on $h,g$ or $\sigma$. Note that the validity of the model is unaffected by linear transformations, that is, Eq. (1.1) still holds when replacing $h$, $g$ and $\sigma$ by

$$\tilde{h}\left(y\right)=ah\left(y\right)+b,\tilde{g}\left(\mathbf{x}\right)=ag\left(\mathbf{x}\right)+b,\tilde{\sigma }\left(\mathbf{x}\right)=a\sigma \left(\mathbf{x}\right)$$

for some $a,b\in \mathbb{R}$. Consequently, at least two conditions for fixing $a$ and $b$ are needed. These will be called location and scale constraints in the following.

The remainder is organized as follows. First, some assumptions are listed before the main identification result for heteroscedastic transformation models is motivated and stated. Afterwards, a short conclusion is given in Section 3. The proof can be found in the supplementary material.