*3.3. Robustness*

Aspects of the robustness of the *T*CAPM with respect to the *N*CAPM can be illustrated perturbing some observations in the original data. Changes in the ML estimates of *β* can be evaluated using the following procedure. First, an observation can be perturbed to create an outlier by *yn* ← *yn* + <sup>Δ</sup>**1***p*, for Δ = −0.20, −0.10, 0, 0.10, 0.20. Then, we re-calculate the ML estimates of *β* under the *T*CAPM and under the *N*CAPM. Note that, for the *N*CAPM,

$$
\hat{\beta}\_{\Delta j} = \hat{\beta}\_{j} + (\mathbf{x}\_{n} - \boldsymbol{\mathfrak{x}}) \boldsymbol{\Delta} / \sum\_{t=1}^{n} (\mathbf{x}\_{t} - \boldsymbol{\mathfrak{x}})^{2}, \tag{14}
$$

for *j* = 1, ... , *p*. Finally, a graph of *β* ˆ Δ*j*, *j* = 1, ... , *p* versus Δ for each of the *p* assets, is useful to visualize changes in the estimators. Figures 6 and 7 show the curves of the estimates of *β* ˆ Δ*j*, *j* = 1, ... , 5 versus Δ for each of the 5 assets included in the two data sets considered in this paper. For this perturbation scheme, it can be observed that the influence on parameter estimation was unbounded in the *N*CAPM, see Equation (14), whereas it was obviously bounded in the *T*CAPM. This suggests that TCAPM provided an appropriate way for achieving robust statistical inference.

**Figure 6.** Perturbed ML estimates of *β* under the NCAPM (red line) and TCAPM (blue line), for the Chilean Stock Market data set.

**Figure 7.** Perturbed maximum likelihood (ML) estimates of *β* under the NCAPM (red line) and TCAPM (blue line), for the NYSE data set.

#### **4. Multifactor Asset Pricing Models under the** *t***-Distribution**

As suggested by a referee, in some cases it is necessary to use more than one factor to estimate the expected returns of the assets of interest. In this Section we briefly discuss an extension of the *T*CAPM that includes more than one factor. More details on estimation, hypothesis testing, and applications will be discussed in a separate paper. The multifactor model (MAPM) is a multivariate linear regression model with excess returns on *p* assets, as follows:

$$\mathbf{y}\_t = \mathfrak{u} + \mathfrak{B}\_1 \mathbf{x}\_{t1} + \dots + \mathfrak{B}\_q \mathbf{x}\_{tq} + \mathfrak{e}\_t, \ t = 1, \dots, n,\tag{15}$$

where (*<sup>x</sup>*1, ... , *xq*) denotes the excess returns of *q* factors (benchmark assets), *βj* is a *p* × 1 parameter vector, *j* = 1, . . . , *q*. The above regression model can be expressed as

$$\begin{aligned} y\_t &= \mathbf{a} + \mathbf{B}\_1 \mathbf{x}\_t^f + \mathbf{e}\_{t\prime} \\ y\_t &= \mathbf{B} \mathbf{x}\_t + \mathbf{e}\_{t\prime} \end{aligned} \tag{16}$$

where *B* = (*<sup>α</sup>*, *<sup>B</sup>*1), denotes the matrix (*p* × *q* + 1) of regression coefficients, *B*1 = (*β*1, ... , *βq*), *xt* = (1, *xft* )*T* and *xft* = (*xt*1,..., *xtq*)*<sup>T</sup>*, *t* = 1, . . . , *n*.

As in Section 2.1, we assume that the excess returns *yt* follows an multivariate *t*-distribution with mean vector *μt* and variance-covariance matrix **Σ**, named *yt* ∼ *Tp*(*μ<sup>t</sup>*, **Σ**, *η*), independent, *t* = 1, ... , *n*, whose density function takes the form,

$$f(y\_t | \theta) = |\Sigma|^{-1/2} g(\delta\_t),\tag{17}$$

where, *δt* = (*yt* − *<sup>μ</sup>t*)*<sup>T</sup>***Σ**−<sup>1</sup>(*yt* − *<sup>μ</sup>t*) is the square of the Mahalanobis distance, with *μt* = *Bxt*, *xt* = (1, *xt*1,..., *xtq*)*<sup>T</sup>* denoting the *t*-th row of the matrix (*n* × *q* + 1) *X* = (*<sup>x</sup>*1,..., *<sup>x</sup>n*)*<sup>T</sup>*, for *t* = 1, . . . , *n*.

Thus, in this case, the ML estimates of *B*, **Σ** and *η* are obtained as solution of the following equations (see Equation (9)):

$$\mathbf{B}^T = (\mathbf{X}^T \mathbf{W} \mathbf{X})^{-1} \mathbf{X}^T \mathbf{W} \mathbf{Y}, \quad \mathbf{\hat{z}} = \frac{1}{n} \sum\_{t=1}^n \omega\_t \mathbf{\hat{e}}\_t \mathbf{\hat{e}}\_t^T,\tag{18}$$

and

$$\dot{\eta}^{-1} = \frac{2}{a + \log a - 1} + 0.0416 \left\{ 1 + \epsilon rf \left( 0.6594 \log \left( \frac{2.1971}{a + \log a - 1} \right) \right) \right\} \eta$$

where ˆ*t* = *yt* − *B*ˆ *Txt* and *W* = diag(*<sup>ω</sup>*1, ... , *<sup>ω</sup>n*) an *n* × *n* diagonal matrix with elements *ωt* = 1 + *ηp η c*(*η*) 1 + *<sup>c</sup>*(*η*)*<sup>δ</sup>t* , for *t* = 1, . . . , *n*, and the matrix (*n* × *p*) *Y* = (*y*1,..., *<sup>y</sup>n*)*<sup>T</sup>*.

As in Section 2.3, the standard errors of the ML estimators *B*ˆ , **Σ**ˆ and *η*ˆ can be estimated using the expected information matrix. In this case, the Fisher information matrix for *θ* = (*<sup>B</sup>*, **Σ**, *η*) assumes the same form of the matrix *J* given in the Equation (10), but the information concerning to *B* is now *J*11 = *<sup>c</sup>α*(*η*)(*XTX*) ⊗ **<sup>Σ</sup>**−1, with *cα* as defined in Equation (10). To test linear hypotheses of interest, such as *H<sup>α</sup>*, we can use the same four tests discussed in Section 2.4.

As a illustration we consider the NYSE data set. We fit the following three-factor model,

$$\mathbf{y}\_t = \mathbf{z} + \boldsymbol{\mathcal{B}}\_1 \mathbf{x}\_l + \boldsymbol{\mathcal{B}}\_2 \text{SMB}\_l + \boldsymbol{\mathcal{B}}\_3 \text{HML}\_l + \boldsymbol{\varepsilon}\_{l,\prime} \ k = 1, \ldots, n,\tag{19}$$

where, *x* is the excess return of the S&P500 index, used in the NYSE data set, while SMB and HML used in the Fama-French model were from the website of Prof. Kenneth French. For details on risk factors, SMB and HML see Fama and French (1995).

In this case, the three risk factors explained between 32% and 50% of the variability of the five assets' returns, which corresponded to an increase of approximately 5% with respect to the CAPM. From Table 7, wecan see that the values in *α*ˆ were very similar to those obtained using CAPM as presented in Table 5, while estimates in *β* ˆ 1 tended to be lower than estimates in *β* ˆ , in the case of CAPM (see Table 5).

Figure 8 displays the transformed distance plots for the normal and *t* distributions. Here *N*MAPM denotes the three risk factors under normality and *T*MAPM denotes the three risk factors under multivariate *t*-distribution. These graphics show clear evidence that the *T*MAPM had a better fit than the *N*MAPM. Using the likelihood-ratio test, the hypothesis *Hα* could not be rejected (*p*-value = 0.1277), if we used the *N*MAPM. However, if we used *T*MAPM, the hypothesis (3) was rejected (*p*-value = 0.0011). Once again, there was a change in statistical inference.

Gelectric

Microsoft

Boeing

Ford

Gelectric

Microsoft

Bank of Am


 1.2836 (0.0910)

 1.1967 (0.0970)

 1.1916 (0.0634)

 1.1063 (0.0609)

 1.4170 (0.0780)

 1.2611 (0.0497)

 1.1636 (0.0524)  0.1535 (0.0849)

 0.2127 (0.1045)

−0.0050 (0.1218)

−0.1058 (0.1298)

−0.0279 (0.0815)

−0.1479 (0.0665)

−0.3077 (0.0701)  0.4815 (0.1194)

 1.2474 (0.0832)

 0.5455 (0.0799)

 0.6901 (0.1024)

 0.3387 (0.0652)

−0.6713 (0.1273)

−0.4498 (0.0688)

−0.0081 (0.0038)

−0.0017 (0.0027)

−0.0149 (0.0033)

−0.0081 (0.0021)

 0.0051 (0.0041)

 0.0056 (0.0026)

 0.0017 (0.0022)

**Table 7.** Adjustment results of Multifactor Asset Pricing Model (MAPM) using the multivariate normaland*t*distributionsfromtheNYSEdatasetandFama-Frenchdataset.Standarderrorsare

**Figure 8.** Plots of transformed distances for the *N*MAPM (**a**) and *T*MAPM (**b**), for the NYSE data set.
