Identication robust inference in structural multivariate factor models with rank restrictions

We propose identi…cation robust inference methods for structural multivariate factor models with rank restrictions. Such models involve nonlinear reduced rank restrictions whose identi…cation may raise serious non-regularities leading to the failure of standard asymptotics. First, we prove several invariance and nuisance-parameter reduction results for commonly used eigenvalue and minimum root based statistics; we also derive useful scaleinvariance results for heteroskedasticity-autocorrelation robust multivariate Wald-type criteria. Second, we derive con…dence set estimates for structural parameters based on inverting minimum-distance type pivotal statistics. We provide analytical solutions to the latter problem which hold exactly (or asymptotically) imposing (or relaxing) Gaussian fundamentals. Simulation-based counterparts are also suggested for non-Gaussian hypotheses. Results are not restricted to the intertemporal i.i.d. setting. The statistics we invert include Hotelling’s T criterion, which is widely used in multivariate analysis for test purposes. Our proposed con…dence sets have much more informational content than Hotelling-type tests, and extend their relevance beyond reduced form speci…cations. Our approach further provides multivariate extensions of the classical Fieller problem, and may be viewed as a generalization of Dufour and Taamouti’s (Econometrica, 2005) quadrics-based set estimation method beyond the linear limited information simultaneous equations setting. Third, we provide a formal de…nition of a statistically non-informative factor and prove necessary conditions linking the presence of such factors to unbounded set estimation outcomes. We also document the perverse e¤ects of adding such factors on J-type minimum-root-based model tests. Fourth, we provide a uni…ed analytical treatment of point estimation. With reduced rank constraints, normalizations raise uniqueness issues that may not matter in some contexts, yet in many econometric or …nancial structural models, normalizations are motivated by underlying theory. Fifth, proposed inference methods are applied to a multi-factor Capital Asset Pricing type model with unobservable risk-free rates and an Arbitrage Pricing Theory based model with Fama-French factors. Results reveal dramatic di¤erences between the standard Wald-type con…dence set estimates and our proposed identi…cation robust ones and illustrate the severe implications of redundant factors. In particular, we …nd that the term structure variables and the momentum factor are statistically non-informative in many sub-periods. This causes tests for model …t to spuriously pass the underlying pricing restrictions, though associated con…dence sets are much too wide. These results document the serious pitfalls of usual asset pricing tests and illustrate the worth of our proposed con…dence set based analysis.