# Multivariate Student versus Multivariate Gaussian Regression Models with Application to Finance

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## Abstract

**:**

## 1. Introduction

## 2. Multivariate Regression Models

#### 2.1. Literature Review

#### 2.2. Univariate Regression Case Reminder

#### 2.3. The Multivariate Regression Model

#### 2.4. Multivariate Normal Error Vector

#### 2.5. Uncorrelated Multivariate Student (UT) Error Vector

**Proposition**

**1.**

**Proposition**

**2.**

#### 2.6. Independent Multivariate Student Error Vector

**Proposition**

**3.**

## 3. Simulation Study

#### 3.1. Design

#### 3.2. Estimators of the $\beta $ Parameters

#### 3.3. Estimators of the Variance Parameters

## 4. Selection between the Gaussian and IT Models

#### 4.1. Distributions of Mahalanobis Distances

#### 4.2. Examples

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

EM | Expectation-maximization |

MLE | Maximum likelihood estimator |

N | Normal (Gaussian) model |

IT | Independent multivariate Student |

UT | Uncorrelated multivariate Student |

RB | Relative bias |

MSE | Mean squared error |

RRMSE | Root relative mean squared error |

DGP | Data-generating process |

## Appendix A

**Proof of Proposition**

**1.**

**Proof of Proposition**

**2.**

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**Figure 1.**The RRMSE of the IT estimator of ${\widehat{\beta}}_{12}$ for the UT DGP in solid line, for the IT DGP in dashed line, and for the Gaussian DGP in dotted line with ${\nu}_{DGP}=3$ (respectively, ${\nu}_{DGP}=4,\phantom{\rule{4pt}{0ex}}{\nu}_{DGP}=5$) on the left (respectively, middle, right) plot.

**Figure 3.**Toy data: scatterplots of residuals in the Gaussian DGP (respectively, the IT DGP with ${\nu}_{DGP}=3$, the IT DGP with ${\nu}_{DGP}=4$) on the first row (respectively, the second row, the third row).

**Figure 4.**Financial data: Q-Q plots of the Mahalanobis distances for the normal, IT (${\nu}_{MLE}=3$), and IT (${\nu}_{MLE}=4$) estimators.

**Figure 5.**Toy data: Q-Q plots of the Mahalanobis distances of the residuals for the normal (respectively, the IT with ${\nu}_{DGP}=3$, the IT with ${\nu}_{DGP}=4$) case empirical quantiles against the normal (respectively, the IT with ${\nu}_{MLE}=3$, the IT with ${\nu}_{MLE}=4$) case theoretical quantiles in the first row (respectively, the second row, the third row).

Model | Distribution |
---|---|

N$({\mathit{\u03f5}}_{1},\dots ,{\mathit{\u03f5}}_{n})$ | ${\mathcal{N}}_{nL}(\mathbf{0},{\mathbf{I}}_{n}\otimes {\mathsf{\Sigma}}_{N})=\prod _{i=1}^{n}{\mathcal{N}}_{L}(\mathbf{0},{\mathsf{\Sigma}}_{N})$ |

UT$({\mathit{\u03f5}}_{1},\dots ,{\mathit{\u03f5}}_{n})$ | ${\mathbf{T}}_{nL}(\mathbf{0},{\mathbf{I}}_{n}\otimes {\mathsf{\Sigma}}_{UT},\nu )$ |

IT$({\mathit{\u03f5}}_{1},\dots ,{\mathit{\u03f5}}_{n})$ | $\prod _{i=1}^{n}{\mathbf{T}}_{L}(\mathbf{0},{\mathsf{\Sigma}}_{IT},\nu )$ |

**Table 2.**Relative bias and relative root mean squared error of the estimators of $\mathit{\beta}$ (${\widehat{\mathit{\beta}}}_{N},\phantom{\rule{4pt}{0ex}}{\widehat{\mathit{\beta}}}_{UT},{\widehat{\mathit{\beta}}}_{IT}$) for the corresponding DGP (Gaussian, UT, and IT).

DGP | N | UT (${\mathit{\nu}}_{\mathit{DGP}}=3$) | IT (${\mathit{\nu}}_{\mathit{DGP}}=3$) | ||||
---|---|---|---|---|---|---|---|

Methods | Estimators | RB (%) | RRMSE | RB (%) | RRMSE | RB (%) | RRMSE |

${\widehat{\mathit{\beta}}}_{N},{\widehat{\mathit{\beta}}}_{UT}$ | ${\widehat{\beta}}_{01}$ | −0.07 | 1.00 | −0.06 | 1.00 | −0.09 | 1.48 |

${\widehat{\beta}}_{02}$ | 0.00 | 1.00 | 0.00 | 1.00 | 0.00 | 1.48 | |

${\widehat{\beta}}_{11}$ | −0.02 | 1.00 | −0.01 | 1.00 | −0.07 | 1.46 | |

${\widehat{\beta}}_{12}$ | −0.00 | 1.00 | −0.00 | 1.00 | −0.00 | 1.46 | |

${\widehat{\mathit{\beta}}}_{IT}({\nu}_{MLE}=3)$ | ${\widehat{\beta}}_{01}$ | −0.09 | 1.04 | −0.09 | 1.09 | −0.03 | 1.00 |

${\widehat{\beta}}_{02}$ | 0.00 | 1.04 | 0.00 | 1.09 | 0.00 | 1.00 | |

${\widehat{\beta}}_{11}$ | −0.04 | 1.07 | −0.02 | 1.08 | −0.03 | 1.00 | |

${\widehat{\beta}}_{12}$ | −0.00 | 1.07 | −0.00 | 1.08 | −0.00 | 1.00 |

**Table 3.**Bias and MSE of the maximum likelihood estimators of $\mathit{\beta}$ for the corresponding DGP (Gaussian, UT, and IT).

DGP | N | UT (${\nu}_{\mathit{DGP}}=3$) | IT (${\nu}_{\mathit{DGP}}=3$) | |||
---|---|---|---|---|---|---|

Estimators | Bias | MSE | Bias | MSE | Bias | MSE |

${\widehat{\beta}}_{01}$ | $-1.39\times {10}^{-3}$ | $4.57\times {10}^{-2}$ | $-1.27\times {10}^{-3}$ | $3.72\times {10}^{-2}$ | $6.65\times {10}^{-4}$ | $1.99\times {10}^{-2}$ |

${\widehat{\beta}}_{02}$ | $2.41\times {10}^{-5}$ | $2.18\times {10}^{-5}$ | $1.47\times {10}^{-5}$ | $1.76\times {10}^{-5}$ | $9.90\times {10}^{-6}$ | $9.50\times {10}^{-6}$ |

${\widehat{\beta}}_{11}$ | $-6.62\times {10}^{-4}$ | $2.16\times {10}^{-2}$ | $3.23\times {10}^{-4}$ | $2.05\times {10}^{-2}$ | $-1.02\times {10}^{-3}$ | $9.84\times {10}^{-3}$ |

${\widehat{\beta}}_{12}$ | $1.87\times {10}^{-5}$ | $1.02\times {10}^{-5}$ | $3.90\times {10}^{-6}$ | $9.60\times {10}^{-6}$ | $2.14\times {10}^{-5}$ | $4.70\times {10}^{-6}$ |

Methods | DGP | N | UT | IT | ||||
---|---|---|---|---|---|---|---|---|

RRMSE | ${\nu}_{\mathit{DGP}}=3$ | ${\nu}_{\mathit{DGP}}=4$ | ${\nu}_{\mathit{DGP}}=5$ | ${\nu}_{\mathit{DGP}}=3$ | ${\nu}_{\mathit{DGP}}=4$ | ${\nu}_{\mathit{DGP}}=5$ | ||

N | ${\widehat{\beta}}_{01}$ | 1.00 | 1.00 | 1.00 | 1.00 | 1.48 | 1.22 | 1.14 |

${\widehat{\beta}}_{02}$ | 1.00 | 1.00 | 1.00 | 1.00 | 1.48 | 1.23 | 1.14 | |

${\widehat{\beta}}_{11}$ | 1.00 | 1.00 | 1.00 | 1.00 | 1.46 | 1.22 | 1.13 | |

${\widehat{\beta}}_{12}$ | 1.00 | 1.00 | 1.00 | 1.00 | 1.46 | 1.22 | 1.13 | |

IT (${\nu}_{MLE}=3$) | ${\widehat{\beta}}_{01}$ | 1.04 | 1.09 | 1.09 | 1.08 | 1.00 | 1.00 | 1.01 |

${\widehat{\beta}}_{02}$ | 1.04 | 1.09 | 1.09 | 1.08 | 1.00 | 1.00 | 1.01 | |

${\widehat{\beta}}_{11}$ | 1.07 | 1.08 | 1.10 | 1.08 | 1.00 | 1.00 | 1.01 | |

${\widehat{\beta}}_{12}$ | 1.07 | 1.08 | 1.09 | 1.09 | 1.00 | 1.00 | 1.01 | |

IT (${\nu}_{MLE}=4$) | ${\widehat{\beta}}_{01}$ | 1.02 | 1.07 | 1.06 | 1.06 | 1.00 | 1.00 | 1.00 |

${\widehat{\beta}}_{02}$ | 1.01 | 1.06 | 1.06 | 1.05 | 1.00 | 1.00 | 1.00 | |

${\widehat{\beta}}_{11}$ | 1.04 | 1.06 | 1.07 | 1.06 | 1.00 | 1.00 | 1.00 | |

${\widehat{\beta}}_{12}$ | 1.04 | 1.05 | 1.07 | 1.06 | 1.00 | 1.00 | 1.00 | |

IT (${\nu}_{MLE}=5$) | ${\widehat{\beta}}_{01}$ | 1.00 | 1.05 | 1.05 | 1.04 | 1.01 | 1.00 | 1.00 |

${\widehat{\beta}}_{02}$ | 1.00 | 1.05 | 1.05 | 1.04 | 1.01 | 1.00 | 1.00 | |

${\widehat{\beta}}_{11}$ | 1.03 | 1.04 | 1.05 | 1.05 | 1.01 | 1.00 | 1.00 | |

${\widehat{\beta}}_{12}$ | 1.03 | 1.04 | 1.05 | 1.05 | 1.01 | 1.00 | 1.00 |

**Table 5.**The bias and the MSE of $\widehat{\rho},\phantom{\rule{4pt}{0ex}}{\widehat{\sigma}}_{1}^{2},\phantom{\rule{4pt}{0ex}}{\widehat{\sigma}}_{2}^{2}$.

Methods | DGP | N | UT (${\nu}_{\mathit{DGP}}=3$) | IT (${\nu}_{\mathit{DGP}}=3$) | |||
---|---|---|---|---|---|---|---|

Bias | MSE | Bias | MSE | Bias | MSE | ||

N | $\widehat{\rho}$ | $-4.85\times {10}^{-4}$ | $9.46\times {10}^{-4}$ | $-2.08\times {10}^{-4}$ | $7.68\times {10}^{-4}$ | $-3.99\times {10}^{-3}$ | $1.17\times {10}^{-2}$ |

${\widehat{\sigma}}_{1}^{2}$ | $-3.89\times {10}^{-3}$ | $8.33\times {10}^{-3}$ | $-1.05\times {10}^{-1}$ | 58 | $6.94\times {10}^{-3}$ | $3.17$ | |

${\widehat{\sigma}}_{2}^{2}$ | $-1.75\times {10}^{-3}$ | $2.01\times {10}^{-3}$ | $-5.17\times {10}^{-2}$ | $14.93$ | $-1.77\times {10}^{-2}$ | $2.85\times {10}^{-1}$ | |

IT${\nu}_{MLE}=3$ | $\widehat{\rho}$ | $-1.70\times {10}^{-4}$ | $8.94\times {10}^{-4}$ | $-2.18\times {10}^{-4}$ | $9.05\times {10}^{-4}$ | $-2.03\times {10}^{-4}$ | $1.07\times {10}^{-3}$ |

${\widehat{\sigma}}_{1}^{2}$ | $2.00$ | $4.06$ | $1.80$ | $244.87$ | $-1.43\times {10}^{-2}$ | $1.54\times {10}^{-2}$ | |

${\widehat{\sigma}}_{2}^{2}$ | $1.00$ | $1.02$ | $0.91$ | $64.75$ | $-7.30\times {10}^{-3}$ | $3.94\times {10}^{-3}$ |

**Table 6.**The RB of $\widehat{\rho},\phantom{\rule{4pt}{0ex}}{\widehat{\sigma}}_{1}^{2},\phantom{\rule{4pt}{0ex}}{\widehat{\sigma}}_{2}^{2}$ with $\nu =3,4,5$.

Methods | DGP | N | UT | IT | ||||
---|---|---|---|---|---|---|---|---|

RB (%) | ${\nu}_{\mathit{DGP}}=3$ | ${\nu}_{\mathit{DGP}}=4$ | ${\nu}_{\mathit{DGP}}=5$ | ${\nu}_{\mathit{DGP}}=3$ | ${\nu}_{\mathit{DGP}}=4$ | ${\nu}_{\mathit{DGP}}=5$ | ||

N | $\widehat{\rho}$ | −0.14 | −0.06 | −0.06 | −0.06 | −1.13 | −0.24 | 0.02 |

${\widehat{\sigma}}_{1}^{2}$ | −0.21 | −5.23 | −3.34 | −2.31 | 0.35 | −0.08 | −0.12 | |

${\widehat{\sigma}}_{2}^{2}$ | −0.18 | −5.17 | −3.33 | −2.20 | −1.77 | −0.30 | −0.09 | |

IT, ${\nu}_{MLE}=3$ | $\widehat{\rho}$ | −0.05 | −0.06 | −0.06 | −0.06 | −0.06 | −0.04 | −0.02 |

${\widehat{\sigma}}_{1}^{2}$ | 99.99 | 90.25 | 93.89 | 95.80 | −0.72 | 32.79 | 50.12 | |

${\widehat{\sigma}}_{2}^{2}$ | 100.05 | 90.60 | 93.90 | 96.03 | −0.73 | 32.79 | 50.13 | |

IT, ${\nu}_{MLE}=4$ | $\widehat{\rho}$ | −0.05 | −0.06 | −0.06 | −0.06 | −0.06 | −0.04 | −0.01 |

${\widehat{\sigma}}_{1}^{2}$ | 42.62 | 35.80 | 38.32 | 39.68 | −24.66 | −0.24 | 11.18 | |

${\widehat{\sigma}}_{2}^{2}$ | 42.66 | 36.01 | 38.34 | 39.85 | −24.67 | −0.23 | 11.19 | |

IT, ${\nu}_{MLE}=5$ | $\widehat{\rho}$ | −0.06 | −0.06 | −0.06 | −0.06 | −0.06 | −0.04 | −0.00 |

${\widehat{\sigma}}_{1}^{2}$ | 24.71 | 18.85 | 21.03 | 22.23 | −31.75 | −10.13 | −0.14 | |

${\widehat{\sigma}}_{2}^{2}$ | 24.74 | 19.02 | 21.04 | 22.38 | $-3$1.76 | −10.13 | −0.14 |

**Table 7.**The RRMSE of $\widehat{\rho},\phantom{\rule{4pt}{0ex}}{\widehat{\sigma}}_{1}^{2},\phantom{\rule{4pt}{0ex}}{\widehat{\sigma}}_{2}^{2}$ in the Gaussian DGP, the UT DGP (${\nu}_{\mathrm{DGP}}=3,4,5$), and the IT DGP (${\nu}_{\mathrm{DGP}}=3,4,5$).

Methods | DGP | N | UT | IT | ||||
---|---|---|---|---|---|---|---|---|

RRMSE | ${\nu}_{\mathit{DGP}}=3$ | ${\nu}_{\mathit{DGP}}=4$ | ${\nu}_{\mathit{DGP}}=5$ | ${\nu}_{\mathit{DGP}}=3$ | ${\nu}_{\mathit{DGP}}=4$ | ${\nu}_{\mathit{DGP}}=5$ | ||

N | $\widehat{\rho}$ | 1.00 | 1.00 | 1.00 | 1.00 | 3.21 | 1.91 | 1.42 |

${\widehat{\sigma}}_{1}^{2}$ | 1.00 | 1.00 | 1.00 | 1.00 | 14.33 | 2.65 | 1.64 | |

${\widehat{\sigma}}_{2}^{2}$ | 1.00 | 1.00 | 1.00 | 1.00 | 8.50 | 2.24 | 1.78 | |

IT, ${\nu}_{MLE}=3$ | $\widehat{\rho}$ | 0.97 | 1.09 | 1.09 | 1.09 | 1.00 | 1.00 | 1.01 |

${\widehat{\sigma}}_{1}^{2}$ | 22.07 | 2.05 | 2.11 | 2.16 | 1.00 | 5.89 | 9.18 | |

${\widehat{\sigma}}_{2}^{2}$ | 22.45 | 2.08 | 2.11 | 2.16 | 1.00 | 5.77 | 9.13 | |

IT, ${\nu}_{MLE}=4$ | $\widehat{\rho}$ | 0.95 | 1.06 | 1.06 | 1.06 | 1.01 | 1.00 | 1.00 |

${\widehat{\sigma}}_{1}^{2}$ | 9.49 | 1.46 | 1.47 | 1.48 | 4.04 | 1.00 | 2.31 | |

${\widehat{\sigma}}_{2}^{2}$ | 9.65 | 1.48 | 1.47 | 1.48 | 4.00 | 1.00 | 2.30 | |

IT, ${\nu}_{MLE}=5$ | $\widehat{\rho}$ | 0.94 | 1.05 | 1.05 | 1.05 | 1.01 | 1.00 | 1.00 |

${\widehat{\sigma}}_{1}^{2}$ | 5.58 | 1.27 | 1.27 | 1.28 | 5.16 | 1.99 | 1.00 | |

${\widehat{\sigma}}_{2}^{2}$ | 5.68 | 1.28 | 1.28 | 1.27 | 5.10 | 1.95 | 1.00 |

**Table 8.**All datasets: the p-values of the Mahalanobis distances tests with the null hypothesis and the corresponding estimators.

Hypothesis ${\mathit{H}}_{0}$ | Toy DGP | Financial Data | ||
---|---|---|---|---|

Methods | N | IT, ${\nu}_{\mathit{DGP}}=3$ | IT, ${\nu}_{\mathit{DGP}}=4$ | |

N | 0.546 | $2.2\times {10}^{-16}$ | $2.2\times {10}^{-16}$ | $2.2\times {10}^{-16}$ |

IT, ${\nu}_{MLE}=3$ | $2.2\times {10}^{-16}$ | 0.405 | 0.033 | 0.882 |

IT, ${\nu}_{MLE}=4$ | $2.2\times {10}^{-16}$ | 0.023 | 0.303 | 0.049 |

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## Share and Cite

**MDPI and ACS Style**

Nguyen, T.H.A.; Ruiz-Gazen, A.; Thomas-Agnan, C.; Laurent, T.
Multivariate Student versus Multivariate Gaussian Regression Models with Application to Finance. *J. Risk Financial Manag.* **2019**, *12*, 28.
https://doi.org/10.3390/jrfm12010028

**AMA Style**

Nguyen THA, Ruiz-Gazen A, Thomas-Agnan C, Laurent T.
Multivariate Student versus Multivariate Gaussian Regression Models with Application to Finance. *Journal of Risk and Financial Management*. 2019; 12(1):28.
https://doi.org/10.3390/jrfm12010028

**Chicago/Turabian Style**

Nguyen, Thi Huong An, Anne Ruiz-Gazen, Christine Thomas-Agnan, and Thibault Laurent.
2019. "Multivariate Student versus Multivariate Gaussian Regression Models with Application to Finance" *Journal of Risk and Financial Management* 12, no. 1: 28.
https://doi.org/10.3390/jrfm12010028