Inference on sets in finance

Abstract

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Abstract. In this paper we consider the problem of inference on a class of sets describing a collection of admissible models as solutions to a single smooth inequality. Classical and recent examples include, among others, the Hansen-Jagannathan (HJ) sets of admissible stochastic discount factors, Markowitz-Fama (MF) sets of meanvariances for asset portfolio returns, and the set of structural elasticities in Chetty (2012)'s analysis of demand with optimization frictions. We show that the econometric structure of the problem allows us to construct convenient and powerful confidence regions based upon the weighted likelihood ratio and weighted Wald (directed weighted Hausdorff) statistics. The statistics we formulate differ (in part) from existing statistics in that they enforce either exact or first order equivariance to transformations of parameters, making them especially appealing in the target applications. Moreover, the resulting inference procedures are also more powerful than the structured projection methods, which rely upon building confidence sets for the frontier-determining sufficient parameters (e.g. frontier-spanning portfolios), and then projecting them to obtain confidence sets for HJ sets or MF sets. Lastly, the framework we put forward is also useful for analyzing intersection bounds, namely sets defined as solutions to multiple smooth inequalities, since multiple inequalities can be conservatively approximated by a single smooth inequality. We present two empirical examples that show how the new econometric methods are able to generate sharp economic conclusions. Terms of use: Documents in

Empirical Likelihood for Random Sets

“…By a similar argument as used to show (6) this follows if equations (8)- (11) hold. Indeed (8) and (11) follow by the same arguments as in the proof of part (i).…”

Moment inequalities in the context of simulated and predicted variables

“…Our method "regularizes" or smoothes the objective function, and we show that this method leads to a straightforward bias correction method. While the idea of regularization appears in some other contexts, such as Haile & Tamer (2003), Chernozhukov, Kocatulum & Menzel (2015) and Masten & Poirier (2017), we formally show that this approach has uniform validity in the context of inference with simulated variables. Our regularization method is based on the class of µ-smooth approximations studied in the non-smooth optimization literature (Nesterov, 2005;Beck & Teboulle, 2012).…”

“…Using these results, we provide functional forms of smooth approximations to some of the commonly used test statistics. The idea of regularizing test statistics (or estimated bounds) also appears in related contexts (Haile & Tamer, 2003;Chernozhukov et al, 2015;Kaido, 2017;Masten & Poirier, 2017). Our contribution here is to show its uniform validity in the context of inference with simulated variables.…”

On the properties of the constrained Hansen–Jagannathan distance

“…The sample squared constrained HJ-distance can be obtained aŝ Chernozhukov, Hong, and Tamer (2007) and Chernozhukov, Kocatulum, and Menzel (2015) develop an approach to conducting inference on parameter sets, characterized by a smooth inequality constraint, which can be adapted to the setup in (37). Instead, we base our statistical analysis on the dual (conjugate) problem that gives rise to an unconstrained extremum estimator.…”