Linear regression estimation in non-linear single index models

TL;DR

This paper introduces a simple regression-based method to consistently estimate the index parameter in non-linear single index models with Gaussian covariates, requiring minimal assumptions on the unknown link function.

Contribution

It demonstrates that the index parameter can be consistently estimated via regression without smoothness assumptions on the link function, under Gaussian covariates.

Findings

Estimator is asymptotically normal.

Method works without smoothness assumptions on $f_0$.

Simulation study confirms effectiveness.

Abstract

In this article, we consider the problem of estimating the index parameter $\alpha_0$ in the single index model $E[Y |X] = f_0(\alpha_0^T X)$ with $f_0$ the unknown ridge function defined on $\mathbb{R}$, $X$ a d-dimensional covariate and $Y$ the response. We show that when $X$ is Gaussian, then $\alpha_0$ can be consistently estimated by regressing the observed responses $Y_i$, $i = 1, . . ., n$ on the covariates $X_1, . . ., X_n$ after centering and rescaling. The method works without any additional smoothness assumptions on $f_0$ and only requires that $cov(f_0(\alpha_0^T X),\alpha_0^TX) \neq 0$, which is always satisfied by monotone and non-constant functions $f_0$. We show that our estimator is asymptotically normal and give the expression with its asymptotic variance. The approach is illustrated through a simulation study.