Introduction

The papers by Susanne Schennach and by Alexandre Belloni, Victor Chernozhukov, and Christian B. Hansen that appear in this volume synthesize important contributions that these authors made in different areas of econometrics: latent-variables models for the former and l1-penalized estimation methods for the latter. This discussion shows that these two areas of research are related and proposes a penalization approach to estimate models with latent variables.

Latent-Variables Models. Part of the recent work of Schennach aims at providing nonparametric identification results in latent-variables models (LVM hereafter). As an important example, Hu and Schennach (2008) provided conditions under which all latent distributions are nonparametrically identified in nonlinear LVM that satisfy conditional-independence restrictions. These results represent significant improvements in a literature that so far has focused mostly on linear models (Kotlarski 1967; Székely and Rao 2000).

Schennach's paper focuses on measurement-error models in which the true regressor is an unobserved latent variable. However, the techniques she discusses also may be used in models with a group structure and panel-data models (in which the group fixed effects are the latent variables) or in dynamic-decision models (with unobserved states). For example, using similar techniques, Hu and Shum (2012) provided conditions under which structural dynamic models are identified nonparametrically under Markovian assumptions on the dynamics of unobserved state variables.

Although the recent literature has made important progress on nonparametric point-identification of LVM under economically plausible assumptions, the estimation side is far less developed. In additive models, non-parametric estimators were proposed that rely on the use of characteristic functions (Horowitz and Markatou 1996; Bonhomme and Robin 2010). Recently, Arellano and Bonhomme (2012) and Evdokimov (2010) used similar ideas in linear and nonlinear panel-data regression models with fixed effects, respectively.