**Appendix A. The Multivariate** *t***-Distribution**

For completeness, we present some properties of the multivariate *t*-distribution, with finite second moment, based on the parameterization given in (5).

**Property A1.** *Let y* ∼ *Tp*(*μ*, **Σ**, *η*)*, with η* < 1/2*.*

*(i) Suppose that y*|*u* ∼ *Np*(*μ*, *<sup>u</sup>*<sup>−</sup>1**Σ**), *and that u* ∼ *<sup>G</sup>*(1/2*η*, 1/2*c*(*η*))*, then y* ∼ *Tp*(*μ*, **Σ**, *η*)


$$F = \left(\frac{1}{1 - 2\eta}\right)\frac{\delta}{p} \sim F(p, 1/\eta)$$


$$
\mu = \left(\begin{array}{cc} \mu\_1 \\ \mu\_2 \end{array}\right) \\
\text{and } \Sigma = \left(\begin{array}{cc} \Sigma\_{11} & \Sigma\_{12} \\ \Sigma\_{21} & \Sigma\_{22} \end{array}\right) / \Delta
$$

*then y*1 ∼ *Tp*1 (*μ*1, **Σ**11, *η*)

*(vii) Using the same previous partition, the conditional distribution of y*1 *given y*2*,* (*y*1|*y*2) ∼ *Tp*1 (*μ*1 − **Σ**−<sup>1</sup> 11 **<sup>Σ</sup>**12(*μ*2 − *<sup>y</sup>*2), *<sup>q</sup>*(*y*2, *η*)(**<sup>Σ</sup>**11 − **<sup>Σ</sup>**12**Σ**−<sup>1</sup> 22 **<sup>Σ</sup>**21), *η*)*,*

*where δ* = (*y* − *μ*)*<sup>T</sup>***Σ**−<sup>1</sup>(*y* − *μ*)*, <sup>q</sup>*(*y*2, *η*) = {*c*<sup>−</sup><sup>1</sup>(*η*)+(*y*2 − *<sup>μ</sup>*2)*<sup>T</sup>***Σ**−<sup>1</sup> 22 (*y*2 − *<sup>μ</sup>*2)}/(*η*<sup>−</sup><sup>1</sup> + *p*2)*, and <sup>G</sup>*(*<sup>a</sup>*, *b*) *denotes the gamma distribution with probability density function f*(*x*) = *baxa*−<sup>1</sup>*exp*{−*bx*}/Γ(*a*)*, for x*, *a*, *b* > 0*.*

#### **Appendix B. ML Estimation Using the EM Algorithm**

To obtain the ML estimate using the EM algorithm, we augmented the observed data, *Y*, by incorporating latent variables to obtain *Y*com = {(*yT*1 , *<sup>u</sup>*1), ... ,(*yTn* , *un*)}. Thus, based on property *i*) we can consider the following hierarchical model *<sup>y</sup>t*|*ut* ind∼ *Np*(*α* + *βxt*, **<sup>Σ</sup>**/*ut*), *ut* ind∼ *<sup>G</sup>*(1/2*η*, 1/2*c*(*η*)), for *t* = 1, ... , *n*. The log-likelihood function of complete data, is denoted by L*c*(*θ*) = log *f*(*<sup>Y</sup>*com|*θ*). By using Property A1, the conditional expectation of the complete-data log-likelihood function can be expressed as

$$E\{\mathcal{L}\_{\varepsilon}(\theta)|\mathcal{Y}, \theta^{(\*)}\} = Q(\theta|\theta^{(\*)}) = Q\_1(\pi|\theta^{(\*)}) + Q\_2(\eta|\theta^{(\*)}),\tag{A1}$$

where *τ* = (*αT*, *β<sup>T</sup>*, *<sup>σ</sup><sup>T</sup>*)*<sup>T</sup>*, and

$$\begin{split} Q\_{1}(\boldsymbol{\pi}|\boldsymbol{\theta}^{(\ast)}) &= -\frac{n}{2}\log|\boldsymbol{\Sigma}| - \frac{1}{2}\sum\_{t=1}^{n}\omega\_{t}^{(\ast)}(\boldsymbol{y}\_{t}-\boldsymbol{\pi}-\boldsymbol{\beta}\boldsymbol{x}\_{t})^{T}\boldsymbol{\Sigma}^{-1}(\boldsymbol{y}\_{t}-\boldsymbol{\pi}-\boldsymbol{\beta}\boldsymbol{x}\_{t}), \\ Q\_{2}(\boldsymbol{\eta}|\boldsymbol{\theta}^{(\ast)}) &= n\Big{(}\frac{1}{2\eta}\log\left(\frac{1}{2c(\eta)}\right)-\log\boldsymbol{\Gamma}\left(\frac{1}{2\eta}\right)+\frac{1}{2c(\eta)}\Big{[}\boldsymbol{\Psi}\left(\frac{1/\eta^{(\ast)}+p}{2}\right) \\ &\quad -\log\left(\frac{1/\eta^{(\ast)}+p}{2}\right)+\frac{1}{n}\sum\_{t=1}^{n}(\log\omega\_{t}^{(\ast)}-\omega\_{t}^{(\ast)})\Big{]}, \end{split}$$

where *ω*(∗) *t* = *<sup>E</sup>*{*ut*|*y<sup>t</sup>*, *θ*(∗)} are the weights *ωt* defined in (8) and evaluated at *θ* = *<sup>θ</sup>*(∗), for *t* = 1, ... , *n*. Maximizing the Q-function (A1), we obtain the iterative process defined in Equation (9).

#### **Appendix C. ML Estimation under** *<sup>H</sup>β* **and** *<sup>H</sup>αβ*

In this case *<sup>H</sup>β* : *β* = **1**. To calculate the values of the statistics *Lr*, *Sc* and *Ga*, we need to estimate *θ* under *<sup>H</sup>β*. The EM algorithm leads to the following equations to obtain the ML estimates of *α*, **Σ** and *η* under *<sup>H</sup>β*:

$$\tilde{\alpha} = \frac{\sum\_{t=1}^{n} \tilde{\omega} \iota \left( y\_t - \mathbf{1}\_p \mathbf{x}\_t \right)}{\sum\_{t=1}^{n} \tilde{\omega}\_t}, \qquad \tilde{\mathbf{L}} = \frac{1}{n} \sum\_{t=1}^{n} \tilde{\omega}\_l \left( y\_t - \tilde{\mathbf{a}} - \mathbf{1}\_p \mathbf{x}\_t \right) \left( y\_t - \tilde{\mathbf{a}} - \mathbf{1}\_p \mathbf{x}\_t \right)^T \tag{A2}$$

and

$$\tilde{\eta}^{-1} = \frac{2}{a + \log a - 1} + 0.0416 \left\{ 1 + \varepsilon r f \left( 0.6594 \log \left( \frac{2.1971}{a + \log a - 1} \right) \right) \right\}.$$

where *a* = −(1/*n*) ∑*nt*=<sup>1</sup>(*vt*<sup>2</sup> − *vt*1), with *vt*1 = (1 + *pη*)/(<sup>1</sup> + *<sup>c</sup>*(*η*)˜*δt*) and *vt*2 = *ψ*1 + *pη* 2*η* − log 1 + *<sup>c</sup>*(*η*)˜*δt* 2*η* , ˜*δt* = (*yt* − *α*˜ − **<sup>1</sup>***pxt*)*<sup>T</sup>***Σ**˜ <sup>−</sup><sup>1</sup>(*yt* − *α*˜ − **<sup>1</sup>***pxt*), *<sup>ω</sup>*˜*t* = 1 + *ηp η c*(*η*) 1 + *<sup>c</sup>*(*η*)˜*δt* , for *t* = 1, . . . , *n*.

It may also be of interest to test the joint hypothesis *<sup>H</sup>αβ* : *α* = **0**, *β* = **1**. Similarly, to calculate the values of the statistics *Lr*, *Sc* and *Ga*, we need to estimate **Σ** and *η* under *<sup>H</sup>αβ*. The EM algorithm leads to the following equations to obtain the ML estimates of **Σ** and *η* under *<sup>H</sup>αβ*:

$$\mathfrak{L} = \frac{1}{n} \sum\_{t=1}^{n} \tilde{\omega}\_{t} (y\_t - \mathbf{1}\_p \mathbf{x}\_t) (y\_t - \mathbf{1}\_p \mathbf{x}\_t)^T \tag{A3}$$

and

$$\eta^{-1} = \frac{2}{a + \log a - 1} + 0.0416 \left\{ 1 + \varepsilon r f \left( 0.6594 \log \left( \frac{2.1971}{a + \log a - 1} \right) \right) \right\},$$

$$\begin{aligned} \text{where } a &= -(1/\eta) \sum\_{t=1}^{n} (v\_{t2} - v\_{t1}), \text{ with } v\_{t1} = (1 + p\eta)/(1 + c(\eta)\delta\_{t}) \text{ and } v\_{t2} = \psi \left(\frac{1 + p\eta}{2\eta}\right) - \psi \left(\frac{1 + p\eta}{2\eta}\right), \\ \log\left(\frac{1 + c(\eta)\delta\_{t}}{2\eta}\right), \delta\_{t} &= (y\_{t} - \mathbf{1}\_{p}\mathbf{x}\_{t})^{T} \mathfrak{L}^{-1} (y\_{t} - \mathbf{1}\_{p}\mathbf{x}\_{t}), \tilde{\omega}\_{l} = \left(\frac{1 + \eta p}{\eta}\right) \left(\frac{c(\eta)}{1 + c(\eta)\tilde{\delta}\_{l}}\right), \text{ for } t = 1, \dots, n. \end{aligned}$$

#### **Appendix D. Equality of the Score and Gradient Tests under Normality**

First, let us remember that the score tests, under normality, test for mean-variance efficiency can be written as (Campbell et al. 1997) *Sc* = J0 1 + J0/*n*, where J0 = *nbα*<sup>ˆ</sup> *<sup>T</sup>***Σ**<sup>ˆ</sup> −1*α*ˆ, with *b* = *s*2/(*x*¯2 + *<sup>s</sup>*<sup>2</sup>), which is the corresponding Wald tests. In this case, the Gradient test takes the form of *Ga* = *<sup>U</sup>Tα* (*θ*˜)*α*ˆ, where

$$\mathcal{U}\_a(\vec{\theta}) = \mathfrak{L}^{-1} \sum\_{t=1}^n (y\_t - \vec{\beta}x\_t),\tag{A4}$$

*β* ˜ = *β* ˆ + *cα*ˆ, **Σ** ˜ = **Σ** ˆ + *bα*<sup>ˆ</sup>*α*<sup>ˆ</sup> *T* and *c* = *x*¯/(*x*¯2 + *s*<sup>2</sup>); see Campbell et al. (1997) for details on these results. Then, by replacing ∑*nt*=<sup>1</sup>(*y<sup>t</sup>* − *β* ˜ *xt*) = *nbα*<sup>ˆ</sup> in (A4) and using the Sherman-Morrison formula to invert the matrix **Σ** ˜ , we obtain

$$\begin{split} Ga &= nb\hat{\mathfrak{a}}^T \hat{\mathfrak{L}}^{-1} \hat{\mathfrak{a}} \\ &= nb\hat{\mathfrak{a}}^T (\hat{\mathfrak{L}} + b\hat{\mathfrak{a}}\hat{\mathfrak{a}}^T)^{-1} \hat{\mathfrak{a}} \\ &= \frac{\mathcal{J}\_0}{1 + \mathcal{J}\_0/n} \\ &= \text{Sc.} \end{split}$$
