Asymptotic variance approximations for invariant estimators in uncertain asset-pricing models

ABSTRACT

This article derives explicit expressions for the asymptotic variances of the maximum likelihood and continuously-updated GMM estimators in models that may not satisfy the fundamental asset-pricing restrictions in population. The proposed misspecification-robust variance estimators allow the researcher to conduct valid inference on the model parameters even when the model is rejected by the data. While the results for the maximum likelihood estimator are only applicable to linear asset-pricing models, the asymptotic distribution of the continuously-updated GMM estimator is derived for general, possibly nonlinear, models. The large corrections in the asymptotic variances, that arise from explicitly incorporating model misspecification in the analysis, are illustrated using simulations and an empirical application.

KEYWORDS:

• Asset pricing
• asymptotic approximation
• continuously-updated GMM
• maximum likelihood
• misspecification-robust tests
• model misspecification

JEL CLASSIFICATION:

• C12
• C13
• G12

Acknowledgment

We would like to thank the Editor, an Associate Editor, and two anonymous referees for useful comments and suggestions. The views expressed here are the authors’ and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System.

Notes

Note that p₁, the first element of p, is nonzero with probability one. This is a direct consequence of the fact that when the factors and returns are continuously distributed, the eigenvector p is also continuously distributed, and none of the elements of this eigenvector will have a probability mass at zero.

Newey and Smith (2004, p. 222) establish the equality of this CU-GMM estimator and the CU-GMM estimator based on

Newey and Smith (2004) and Antoine et al. (2007) show that , t = 1,…, T, in (40) represent the implied probability weights associated with the CU-GMM estimator.

We refer the readers to an online appendix for the CU-GMM estimation of the beta-pricing model.

In our empirical application, the degree-of-freedom parameter of the multivariate t-distribution is estimated to be 8.1.

The test asset return and factor data are obtained from Kenneth French's website.

These size distortions are somewhat expected for a small T and a relatively large N given the small number of time-series observations per moment condition.

For the GLS CSR estimator and related misspecification-robust t-tests, we refer the readers to Kan et al. (2013). For the HJD estimator and related misspecification-robust t-tests, we refer the readers to Kan and Robotti (2009) and Gospodinov et al. (2013).