Identification-Robust Estimation and Testing of the Zero-Beta CAPM

Abstract

We propose exact simulation-based procedures for: (i) testing mean-variance efficiency when the zero-beta rate is unknown; (ii) building confidence intervals for the zero-beta rate. On observing that this parameter may be weakly identified, we propose likelihood-ratio-type tests as well as Fieller-type procedures based on a Hotelling-HAC statistic, which are robust to weak identification and allow for non-Gaussian distributions including parametric GARCH structures. The Fieller-Hotelling-HAC procedure also accounts (asymptotically) for general forms of heteroskedasticity and autocorrelation. We propose confidence sets for the zero-beta rate based on “inverting” exact tests for this parameter; for both procedures proposed, these sets can be interpreted as multivariate extensions of the classic Fieller method for inference on ratios. The exact distribution of likelihood-ratio-type statistics for testing efficiency is studied under both the null and the alternative hypotheses. The relevant nuisance parameter structure is established and finite-sample bound procedures are proposed, which extend and improve available Gaussian-specific bounds. Finite-sample distributional invariance results are also demonstrated analytically for the HAC statistic proposed by MacKinlay and Richardson (1991) . We study invariance to portfolio repacking for the tests and confidence sets proposed. The statistical properties of the proposed methods are analysed through a Monte Carlo study and compared with alternative available methods. Empirical results on NYSE returns show that exact confidence sets are very different from asymptotic ones, and allowing for non-Gaussian distributions affects inference results. Simulation and empirical evidence suggests that likelihood-ratio-type statistics—with p -values corrected using the Maximized Monte Carlo test method—are generally preferable to their multivariate Fieller-Hotelling-HAC counterparts from the viewpoints of size control and power.