**2. Methodology**

#### *2.1. The CAPM under the t-Distribution*

First, a set of *p* ≥ 1 assets of interest is considered, and let *Ri* denote the return for asset *i*, with *i* = 1, ... , *p*. CAPM specifies that the stock's expected return is equal to the risk-free rate return plus a risk premium; i.e.,

$$E(R\_i) = R\_f + \beta\_i \{ E(R\_m) - R\_f \}, \ i = 1, \ldots, p,\tag{1}$$

where *Rf* is the risk-free interest rate, *βi* is the systematic risk of the asset *i*, and *Rm* is the market return. This model was independently derived by Sharpe (1964), Lintner (1965) and Mossin (1966). For these *p* assets, the excess returns can be described using the following multivariate linear regression model; Gibbons et al. (1989), MacKinlay and Richardson (1991) and Campbell et al. (1997),

$$\mathbf{y}\_t = \mathfrak{a} + \mathfrak{B}\mathbf{x}\_t + \mathfrak{e}\_t, \ t = 1, \dots, n,\tag{2}$$

where *yt* = (*y*<sup>1</sup>*t*, ... , *ypt*)*<sup>T</sup>* is a *p* × 1 vector representing excess returns of the set of *p* assets of interest in period *t* such that, *yit* = *Rit* − *Rf t* denotes the excess return of asset *i* during period *t*, *α* = (*<sup>α</sup>*1, ... , *<sup>α</sup>p*)*<sup>T</sup>* is the intercept vector, *β* = (*β*1, ... , *βp*)*<sup>T</sup>* is the slope vector that corresponds to the sensitivity of the portfolio return to changes in this benchmark return; *xt* = *Rmt* − *Rf t* represents the excess return of the market portfolio during period *t* and finally  *t* is the errors vector during period *t*, with mean zero and variance-covariance matrix **Σ**, independent of *t*, for *t* = 1, ... , *n*. If the CAPM holds for this set of assets and the benchmark portfolio is mean-variance efficient, the following restriction on the parameters of model (2) should hold *<sup>E</sup>*(*yt*) = *βxt*, for *t* = 1, ... , *n*. Hence, this restriction implies a testable hypothesis:

$$H\_{\mathfrak{a}} \colon \mathfrak{a} = \mathbf{0}.\tag{3}$$

Much of the theory of the CAPM is based on the assumption that excess returns follow a multivariate normal distribution; see for instance Campbell et al. (1997), Broquet et al. (2004), Johnson (2014), Brandimarte (2018), Mazzoni (2018) and Galea and Giménez (2019). However, it has been shown that although the assumption of normality is sufficient to generate the model (1), it is not necessary. Chamberlain (1983), Owen and Rabinovitch (1983), Ingersoll (1987), Berk (1997) and most recently Hamada and Valdez (2008) show that (1) can be obtained under the assumption of elliptically symmetric return distributions. In particular, Berk (1997) showed that when agents maximize the expected utility, elliptical symmetry is both necessary and sufficient for the CAPM.

In this paper, we are interested in develop statistical inference tools, estimation and hypothesis tests in asset pricing models supposing that  *t*, the random errors vector following a multivariate *t*-distribution, has a mean zero and a covariance matrix **Σ**. In effect, we supposed that the density function of  *t* is given by

$$f(\mathfrak{e}) = |\mathfrak{\Sigma}|^{-1/2} \mathfrak{g}(\delta), \quad \delta \ge 0,\tag{4}$$

where

$$g(\delta) = k\_p(\eta) \left( 1 + c(\eta)\delta \right)^{-\frac{1}{2\eta} \left( 1 + \eta p \right)},$$

with *δ* =  *<sup>T</sup>***Σ**−<sup>1</sup>, *kp*(*η*)=(*c*(*η*)/*π*)*p*/2{Γ((<sup>1</sup> + *ηp*)/2*η*)/Γ(1/2*η*)} and *c*(*η*) = *η*/(<sup>1</sup> − <sup>2</sup>*η*), 0 < *η* < 1/2. In this case we wrote  *t* ∼ *Tp*(**<sup>0</sup>**, **Σ**, *η*). From properties of the *t*-distribution (see Appendix A), we have, given *xt*, that *yt* ∼ *Tp*(*α* + *βxt*, **Σ**, *η*) independently, *t* = 1, ... , *n*. The *t*-distribution offers a more flexible framework for modeling asset returns. In this distribution *η* is a shape parameter that can be used for adjusting the kurtosis distribution and for providing more robust procedures than the ones that use the normal distribution, with moderate additional computational effort.

Following Campbell et al. (1997), we consider the joint distribution of the excess returns given the excess return market. Specifically, we assume that the excess returns *y*1, ... , *yn*, given the excess return market, are independent random vectors with a multivariate *t*-distribution and common covariance matrix. Then, the probability density function of *yt*takes the form of

$$f(y\_t | \theta) = |\Sigma|^{-1/2} g(\delta\_t),\tag{5}$$

where, *δt* = (*yt* − *α* − *βxt*)*<sup>T</sup>***Σ**−<sup>1</sup>(*yt* − *α* − *βxt*), for *t* = 1, ... , *n*. Therefore, the density for a sample of *n* periods is given by

$$f(\boldsymbol{Y}|\boldsymbol{\theta}) = \prod\_{t=1}^{n} f(\boldsymbol{y}\_t|\boldsymbol{\theta}) = \prod\_{t=1}^{n} |\boldsymbol{\Sigma}|^{-1/2} \boldsymbol{g}(\delta\_t) \tag{6}$$

with *Y* = (*y*1, ... , *<sup>y</sup>n*) and *θ* = (*αT*, *β<sup>T</sup>*, *<sup>σ</sup>T*, *<sup>η</sup>*)*<sup>T</sup>*, where *σ* = vech(**Σ**) is the *p*(*p* + 1)/2 vector obtained from vec(**Σ**) by deleting from it all of the elements that are above the diagonal of **Σ**.
