**5. Conclusions**

*t* (*η*ˆ = 0.27)

Since the pioneering work by Lange et al. (1989), the *t*-distribution has proved to be a versatile and robust modeling approach in many regression models. In fact, the *t*-distribution has a parameter (*η*) modeling kurtosis, which brings more flexibility than the normal distribution.

In this paper, robust methods for statistical inference in asset pricing models with emphasis on CAPM were developed. In effect, assuming a multivariate *t*-distribution for the stock returns, ML equations for parameters were derived and statistics were proposed to test linear hypotheses of interest, in particular the hypothesis of mean-variance efficiency. Simple expressions were provided in this study for the likelihood-ratio, Wald, score and gradient statistics and for the score function and Fisher information matrix. The proposed statistics generalized results from the literature, which considered tests for mean-variance efficiency under the assumption of multivariate normality (Brandimarte 2018; Campbell et al. 1997; Chou and Lin 2002; Gibbons et al. 1989; Mazzoni 2018). In addition, statistical inference based on the *t*-distribution is simple to implement, and the computational cost is considerably low.

A simple graphical device for checking the model was implemented, and the methodology developed in this paper was illustrated with two real data sets: the Chilean Stock Market data set (a developing country), and another from the New York Stock Exchange, USA (a developed country). In both data sets, the CAPM under the *t*-distribution clearly presents a better fit than under the normal distribution. Additionally, in the application of the multifactor asset pricing model to the NYSE data set, the multivariate *t*-distribution presents a better fit than the normal distribution.

This empirical study provides new evidence for the useful application of the *t*-distribution in modeling stock returns (Kan and Zhou 2017). As we have pointed out, the log-returns frequently present some degree of skewness. We are currently working on statistical inference in the asset pricing models under the multivariate skew-elliptical distributions. We understand that a skewed *t*-distribution of Branco and Dey (2001) may be useful for returns with high levels of skewness. For previous applications of the skew-elliptical distributions in finance and actuarial science, see Harvey et al. (2010) and Adcock et al. (2015). See also Paula et al. (2011).

**Author Contributions:** Conceptualization and investigation, M.G., D.C. and R.C.; methodology, M.G. and A.M.; software and validation, M.G., D.C. and A.M.; formal analysis, resources, data curation, M.G., D.C., R.C. and A.M.; writing—original draft preparation, M.G., D.C., and A.M.; writing—review and editing, M.G., D.C., R.C. and A.M.; visualization, M.G., D.C., R.C. and A.M.; supervision, M.G., D.C., R.C. and A.M.; project administration, M.G. and A.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The first author acknowledges the partial financial support from Project Puente 001/2019, Dirección de Investigación de la Vicerrectoría de Investigación de la Pontificia Universidad Catlica de Chile, Chile. The authors are grateful to the editor and two reviewers for their helpful comments and suggestions.

**Conflicts of Interest:** The authors declare no conflict of interest.
