**1. Introduction**

The Capital Asset Pricing Model (CAPM) is one of the most important asset pricing models in financial economics. It is widely used in estimating the cost of capital for companies and measuring portfolio (or investment fund) performance, among others applications; see, for instance, Campbell et al. (1997), Amenc and Le Sourd (2003), Broquet et al. (2004), Levy (2012) and Ejara et al. (2019).

The CAPM framework provides financial practitioners with a measure of *beta* (or systematic risk) for entire stock markets, industry sub-sectors, and individual equities (Pereiro 2010).

The literature on the CAPM based on the multivariate normal distribution is vast, as seen, for instance, in the works published by Elton and Gruber (1995), Campbell et al. (1997), Broquet et al. (2004), Francis and Kim (2013), Johnson (2014), Brandimarte (2018) and Mazzoni (2018). However, multivariate normality is not required to ensure the validity of the CAPM. In fact, it is well known that the CAPM is still valid within the class of elliptical distributions, of which multivariate normal and multivariate *t*-distributions are special cases (see Chamberlain 1983; Hamada and Valdez 2008; Ingersoll 1987; Owen and Rabinovitch 1983). It is also well known that in practice, excess returns are not normally distributed. Most financial assets exhibit excess kurtosis, that is to say, returns having distributions whose tails are heavier than those of the normal distribution and present some degree of skewness; see Fama (1965), Blattberg and Gonedes (1974), Zhou (1993), Campbell et al. (1997), Bekaert and Wu (2000), Chen et al. (2001), Hodgson et al. (2002) and Vorknik (2003). Recently Bao et al. (2018) discuss estimation in the univariate CAPM with asymmetric power distributed errors. In this paper, the multivariate version of the CAPM is considered, primarily focusing on modeling non-normal returns due to excess kurtosis.

Within the class of elliptical distributions, the multivariate *t*-distribution has been widely used to model data with heavy tails. For instance, Lange et al. (1989) discuss the use of the *t*-distribution in regression and in problems related to multivariate analysis. Sutradhar (1993) has considered a score test aimed at testing if the covariance matrix is equal to some specified covariance matrix using the *t*-distribution; Bolfarine and Galea (1996) used the *t*-distribution in structural comparative calibration models, while Pinheiro et al. (2001) used the multivariate *t*-distribution for robust estimation in linear mixed-effects models. Cademartori et al. (2003), Fiorentini et al. (2003), Galea et al. (2008), Galea et al. (2010) and Kan and Zhou (2017) provide empirical evidence of the usefulness of *t*-distribution to model stock returns. In addition, statistical inference based on the *t*-distribution is simple to implement, and the computational cost is considerably low.

Following Kan and Zhou (2017), there are three main reasons for using the *t*-distribution in modeling returns of financial instruments. (i) empirical evidence shows that this distribution is appropriate for modeling non-normal returns in many situations, (ii) with the algorithms implemented in this paper, the *t*-distribution has become almost as tractable as the normal one, and (iii) the CAPM is still valid under *t*-distribution. It is clear that the *t*-distribution does not describe all the features of the return data. For instance, the volatility variation over time is one of them, for which the GARCH models are very useful. However, according to our experience (see Cademartori et al. 2003; Galea et al. 2008 2010; Galea and Giménez 2019), and as mentioned by Kan and Zhou (2017), there is little evidence of GARCH effects on the monthly data that are typically used for asset pricing and corporate studies. In addition, when we have a moderate number of assets, for example more than 10, the fit of the GARCH models requires an important computational effort, which limits its application to real data sets. For more details see Harvey and Zhou (1993) and Kan and Zhou (2017).

Thus, the main goal of this paper is to develop statistical inference tools, such as parameter estimation and hypothesis tests, in asset pricing models, with an emphasis on the CAPM, using the multivariate *t*-distribution. An extension of the CAPM, the multifactor asset pricing model (MAPM), is also discussed. The *t*-distribution incorporates an additional parameter, which allows modeling returns with high kurtosis. We consider a reparameterization of the multivariate *t*-distribution with a finite second moment. This enables a more direct comparison with the normal distribution (see Bolfarine and Galea 1996; Sutradhar 1993). Based on Fiorentini et al. (2003), who use the reparameterization of degrees of freedom suggested by Lange et al. (1989) to model financial data, this version of the multivariate *t*-distribution is adopted to test hypotheses of interest, such as the hypothesis of mean-variance efficiency. The three most widely used tests based on the likelihood function are considered; Wald tests, likelihood-ratio tests, and score tests (also known as Lagrange multiplier tests). Under the assumption of normality, these tests have been discussed in the literature, see for instance Campbell et al. (1997) and Chou and Lin (2002). Recently Kan and Zhou (2017) discuss the likelihood-ratio tests in the CAPM assuming that the excess returns follow a multivariate *t*-distribution. In this paper, the modeling of the asset returns conditional on market portfolios and the three most widely used tests are considered. Additionally, a fourth test statistic is considered, based on the likelihood proposed by Terrell (2002), the gradient test. To our knowledge this test has not been applied to test hypothesis in asset pricing models.

The article is developed as follows. In Section 2, the CAPM under the multivariate *t*-distribution, estimation of parameter and tests of mean-variance efficiency are briefly reviewed, and the Generalized Method of Moments is summarized for comparative purposes. In Section 3, the methodology developed in this paper is applied to two real data sets: the Chilean Stock Market data set and

another from the New York Stock Exchange, USA. In Section 4, multifactor asset pricing models under the *t*-distribution are discussed. In Section 5, a conclusion and final comments are included. The appendices contain technical details.
