# Maximum Likelihood Inference for Asymmetric Stochastic Volatility Models

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## Abstract

**:**

## 1. Introduction

`stochvol`package of R (R Core Team 2020). Among the Bayesian alternatives, this method was chosen because it is easily available and is very quick.

## 2. Methods

#### 2.1. Parameter Estimation

#### 2.2. VaR Forecasting

- 1.
- Estimate the parameters $\Theta =(\varphi ,{\sigma}_{\omega},\alpha ,\rho ,{\mu}_{i},{\sigma}_{i})$ for $i=1,\dots ,m$.
- 2.
- From the Kalman filter, calculate ${\widehat{h}}_{t|t-1}$ for $t=2,\dots ,T+1$ and obtain the predicted volatilities ${\widehat{\sigma}}_{t|t-1}=\widehat{\beta}exp\left\{0.5{\widehat{h}}_{t|t-1}\right\}$, where $\widehat{\beta}=exp(\widehat{\alpha}/2)$.
- 3.
- Obtain the standardized residuals ${e}_{t}={r}_{t}/{\widehat{\sigma}}_{t|t-1}$ for $t=2,\dots ,T$ and compute the $\gamma $-quantile of ${e}_{t}$, which is called ${q}_{\gamma}$.
- 4.
- The $100(1-\gamma )$%-VaR is then equal to$$Va{R}_{T+1/T}={q}_{\gamma}{\widehat{\sigma}}_{T+1|T}.$$

#### 2.3. Implementation

`Rcpp`and

`RcppArmadillo`packages, and we use the

`nlminb`function to perform the numerical optimization needed to obtain parameter estimates.

#### 2.4. The `Stochvol` Method

`stochvol`package of R where the estimation of A-SV models is performed by Bayesian MCMC methods. Details about the priors for each parameter and other estimation steps are presented in Hosszejni and Kastner (2021).

`stochvol`package also provides the posterior distribution of the volatility, ${\sigma}_{t}$, for $t\in \{1,\dots ,T\}$. We consider the sample mean of this posterior as the point estimate of the volatility. Furthermore, the $100(1-\gamma )\%$-VaR calculated from

`stochvol`is the $\gamma $ quantile of the predictive distribution for ${r}_{T+1}$, which is also provided by the package.

## 3. Monte Carlo Experiments

`stochvol`method described in Section 2.4, specifying the distribution of ${\epsilon}_{t}$.

`stochvol`method. Overall, our proposal presents better results for $\widehat{\varphi}$ and for ${\widehat{\sigma}}_{\omega}$ when $\varphi =0.95$, and

`stochvol`was the best for $\widehat{\alpha}$ and $\widehat{\rho}$. However, these methods show similar results in several situations. For instance, for the leverage parameter, the point estimates of the proposal with $m=3$ and the

`stochvol`package are very close in cases 1, 2, and 4 when $\varphi =0.95$ and when $\varphi =0.99$ in cases 2, 4, and 5. In addition, when $\varphi =0.95$, the proposal with $m=2$ and the

`stochvol`present very close results in terms of the bias for ${\widehat{\sigma}}_{\omega}$.

## 4. Empirical Illustrations

`stochvol`method when ${\epsilon}_{t}\sim N(0,1)$6. Using the first 2500 observations, we calculated the point parameter estimates of the posterior means. The results are shown in Table 3. Compared to the proposed method, the results are similar, with the exception of the USD-MXN series. In particular, according the point estimate and precision, the

`stochvol`method accuses the presence of leverage for that series. In addition, in Figure 1, we present the

`stochvol`estimates of ${\sigma}_{t}$, as described in Section 2.4. As can be seen, these estimates are very similar to those obtained when using the proposed method.

`stochvol`method are presented in Table 5. We observed better results using the proposal with $m=3$ than the

`stochvol`method. For instance, excluding three cases (the 95%-VaR of a long position in the Nikkei, the 99%-VaR of a long position in the USD-BRL, and the 97.5%-VaR of a short position in the USD-BRL), the proportion of the violations of the proposal is much closer to the nominal values than those of the

`stochvol`method.

`stochvol`is graphically similar with that obtained using the proposal. However, even though the volatilities of both methods are close most of the time, the

`stochvol`method is affected more by outliers. This fact can clearly be observed for two selected situations shown in Figure 3. Here, we can see that the 99%-VaR of our proposal is more stable than

`stochvol`even after extreme returns. This occurs for a long position in IBOV, after 17 May 2017, when news of a major scandal triggered a political crisis in Brazil during the administration of Michel Temer. The same happens for a short position in USD-MXN after 9 November 2016, when Donald Trump was elected the president of the USA.

## 5. Conclusions

`stochvol`package using simulated data and real-life data. The Monte Carlo experiments reveal that in terms of point estimations, these methods show similar results in several situations but, overall, the proposal provides better results for estimating $\varphi $, the

`stochvol`method is better for estimating leverage parameter $\rho $, and there is no clear winner for estimating ${\sigma}_{\omega}$. The empirical analysis evidenced that even though both methods showed similar VaR forecasts for most parts of the analyzed time series, the proposal exhibited a percentage of violations closer to the nominal VaR values and that is better for accounting for the influence of outliers.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Notes

1 | In the ARCH/GARCH framework, models for accounting for the leverage effect include the EGARCH model of Nelson (1991) and the GJR model proposed by Glosten et al. (1993), among others. |

2 | The results for $\varphi =0.97$ are qualitative similar and are available upon request |

3 | The total number of observations are as follows: 6419 for IBOV, 6543 for S&P 500, 6371 for Nikkei, 6565 for FTSE, 5533 for USD-BRL, and 5977 for USD-MXN. |

4 | We tested our method for both FX series in a different period compared to Asai and McAleer (2011). Since we did not obtain significant correlation parameter estimates even when using the stochvol method, the results are not reported here. |

5 | However, $\widehat{\rho}$ is significant using the stochvol method, and so we considered this time series in our analysis. |

6 | We tested the estimation when ${\epsilon}_{t}\sim {t}_{\nu}$, but all degrees of freedom were higher than 30; thus, the results did not differ greatly from the Gaussian distribution. |

7 | As pointed out by a referee, the estimation process can be accelerated by using steady state expressions in the Kalman filter. This idea can be particularly fruitful for more complex models and deserves further investigation. |

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**Figure 1.**Estimated volatilities. The gray line denotes the returns of the first 2500 observations of each series. The blue line denotes the predicted volatility of our proposal using $m=3$, while the red line denotes the sample mean of the posterior sample of the volatility, obtained from

`stochvol`. In these graphs, all volatilities are multiplied by 3.

**Figure 2.**Backtesting results of the proposed method with $m=3$. The blue line is the predicted 99%-VaR of a long position, while the purple line is the 99%-VaR of a short position (with change in sign).

**Figure 3.**Comparison of VaR forecasting. The blue line is the forecast using the proposed method (with $m=3$) and the red line is the forecast obtained using

`stochvol`. The graph at the top shows the backtesting of 1%-quantile for IBOV during 2017 and the graph at the bottom shows the 99%-quantile for USD-MXN during 2016.

**Table 1.**Results of Monte Carlo experiments for the A-SV model ($\varphi =0.95$ and $\alpha $ around −7.36). For each case, the bias, standard deviation (SD), and root mean square error (RMSE) of the estimates are presented. Results are based on 1000 replications of a time series of size $T=2500$.

Proposal | Stochvol | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{m}=2$ | $\mathit{m}=3$ | ||||||||||||||

$\widehat{\mathit{\varphi}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{\omega}}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\rho}}$ | $\widehat{\mathit{\varphi}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{\omega}}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\rho}}$ | $\widehat{\mathit{\varphi}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{\omega}}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\rho}}$ | ||||

Case 1 | ${\epsilon}_{t}\sim N(0,1)$ | $\rho =-0.50$ ${\sigma}_{\omega}=0.15$ | Bias | −0.005 | 0.008 | −0.141 | 0.120 | −0.006 | 0.026 | 0.382 | 0.096 | −0.012 | 0.017 | −0.002 | 0.090 |

SD | 0.023 | 0.039 | 0.077 | 0.164 | 0.034 | 0.041 | 0.202 | 0.186 | 0.017 | 0.024 | 0.059 | 0.068 | |||

RMSE | 0.024 | 0.039 | 0.160 | 0.203 | 0.034 | 0.049 | 0.432 | 0.210 | 0.021 | 0.030 | 0.059 | 0.113 | |||

Case 2 | $\rho =-0.75$ ${\sigma}_{\omega}=0.15$ | Bias | −0.004 | −0.005 | −0.145 | 0.121 | −0.004 | 0.022 | 0.359 | 0.114 | −0.013 | 0.020 | −0.005 | 0.150 | |

SD | 0.016 | 0.030 | 0.067 | 0.137 | 0.025 | 0.033 | 0.205 | 0.152 | 0.013 | 0.020 | 0.050 | 0.049 | |||

RMSE | 0.016 | 0.030 | 0.160 | 0.183 | 0.025 | 0.039 | 0.413 | 0.189 | 0.018 | 0.029 | 0.050 | 0.158 | |||

Case 3 | $\rho =0.40$ ${\sigma}_{\omega}=0.25$ | Bias | −0.002 | 0.023 | −0.148 | −0.134 | −0.002 | 0.055 | 0.399 | −0.108 | −0.005 | 0.009 | 0.002 | −0.045 | |

SD | 0.012 | 0.036 | 0.106 | 0.080 | 0.012 | 0.037 | 0.139 | 0.087 | 0.011 | 0.025 | 0.095 | 0.057 | |||

RMSE | 0.013 | 0.043 | 0.182 | 0.156 | 0.012 | 0.066 | 0.423 | 0.138 | 0.012 | 0.026 | 0.095 | 0.072 | |||

Case 4 | ${\epsilon}_{t}\sim {t}_{5}$ | $\rho =-0.50$, ${\sigma}_{\omega}=0.15$ | Bias | −0.008 | 0.006 | −0.523 | 0.153 | −0.008 | 0.021 | 0.037 | 0.110 | −0.030 | 0.041 | −0.020 | 0.135 |

SD | 0.026 | 0.046 | 0.081 | 0.145 | 0.032 | 0.048 | 0.169 | 0.167 | 0.045 | 0.045 | 0.074 | 0.079 | |||

RMSE | 0.028 | 0.046 | 0.529 | 0.211 | 0.033 | 0.052 | 0.173 | 0.200 | 0.054 | 0.061 | 0.077 | 0.156 | |||

Case 5 | $\rho =-0.75$ ${\sigma}_{\omega}=0.15$ | Bias | −0.006 | −0.013 | −0.526 | 0.165 | −0.005 | 0.011 | 0.021 | 0.126 | −0.023 | 0.038 | −0.027 | 0.214 | |

SD | 0.019 | 0.038 | 0.076 | 0.176 | 0.019 | 0.038 | 0.137 | 0.149 | 0.027 | 0.032 | 0.066 | 0.060 | |||

RMSE | 0.020 | 0.040 | 0.532 | 0.241 | 0.020 | 0.039 | 0.139 | 0.195 | 0.035 | 0.050 | 0.072 | 0.222 | |||

Case 6 | $\rho =0.40$, ${\sigma}_{\omega}=0.25$ | Bias | −0.003 | 0.015 | −0.529 | −0.163 | −0.003 | 0.042 | 0.034 | −0.124 | −0.009 | 0.019 | −0.014 | −0.062 | |

SD | 0.013 | 0.039 | 0.113 | 0.083 | 0.013 | 0.041 | 0.238 | 0.096 | 0.014 | 0.033 | 0.107 | 0.066 | |||

RMSE | 0.014 | 0.042 | 0.541 | 0.183 | 0.013 | 0.058 | 0.240 | 0.157 | 0.016 | 0.039 | 0.108 | 0.091 |

**Table 2.**Results of Monte Carlo experiments for the A-SV model ($\varphi =0.99$ and $\alpha $ around −7.36). For each case, the bias, standard deviation (SD), and root mean square error (RMSE) of the estimates are presented. Results are based on 1000 replications of a time series of size $T=2500$.

Proposal | Stochvol | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{m}=2$ | $\mathit{m}=3$ | ||||||||||||||

$\widehat{\mathit{\varphi}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{\omega}}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\rho}}$ | $\widehat{\mathit{\varphi}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{\omega}}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\rho}}$ | $\widehat{\mathit{\varphi}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{\omega}}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\rho}}$ | ||||

Case 1 | ${\epsilon}_{t}\sim N(0,1)$ | $\rho =-0.50$ ${\sigma}_{\omega}=0.15$ | Bias | −0.002 | 0.022 | −0.119 | 0.149 | −0.002 | 0.052 | 0.402 | 0.133 | −0.002 | 0.007 | 0.007 | 0.078 |

SD | 0.004 | 0.020 | 0.297 | 0.097 | 0.004 | 0.022 | 0.331 | 0.099 | 0.004 | 0.015 | 0.254 | 0.064 | |||

RMSE | 0.004 | 0.030 | 0.320 | 0.178 | 0.004 | 0.056 | 0.521 | 0.166 | 0.004 | 0.017 | 0.254 | 0.100 | |||

Case 2 | $\rho =-0.75$ ${\sigma}_{\omega}=0.15$ | Bias | −0.001 | 0.007 | −0.150 | 0.174 | −0.001 | 0.043 | 0.327 | 0.175 | −0.002 | 0.008 | 0.024 | 0.123 | |

SD | 0.003 | 0.018 | 0.266 | 0.105 | 0.003 | 0.020 | 0.306 | 0.096 | 0.003 | 0.013 | 0.203 | 0.044 | |||

RMSE | 0.004 | 0.020 | 0.305 | 0.204 | 0.004 | 0.047 | 0.448 | 0.200 | 0.004 | 0.015 | 0.204 | 0.131 | |||

Case 3 | $\rho =0.40$ ${\sigma}_{\omega}=0.25$ | Bias | −0.001 | 0.043 | −0.065 | −0.151 | −0.001 | 0.087 | 0.419 | −0.137 | −0.002 | 0.005 | 0.025 | −0.040 | |

SD | 0.004 | 0.026 | 0.752 | 0.071 | 0.004 | 0.029 | 0.852 | 0.078 | 0.003 | 0.019 | 0.443 | 0.055 | |||

RMSE | 0.004 | 0.051 | 0.755 | 0.167 | 0.004 | 0.092 | 0.949 | 0.157 | 0.004 | 0.020 | 0.444 | 0.068 | |||

Case 4 | ${\epsilon}_{t}\sim {t}_{5}$ | $\rho =-0.50$, ${\sigma}_{\omega}=0.15$ | Bias | −0.002 | 0.014 | −0.522 | 0.190 | −0.002 | 0.039 | −0.001 | 0.158 | −0.003 | 0.010 | −0.007 | 0.105 |

SD | 0.004 | 0.022 | 0.275 | 0.102 | 0.004 | 0.025 | 0.370 | 0.111 | 0.004 | 0.018 | 0.260 | 0.072 | |||

RMSE | 0.005 | 0.027 | 0.590 | 0.215 | 0.005 | 0.046 | 0.370 | 0.192 | 0.005 | 0.020 | 0.260 | 0.127 | |||

Case 5 | $\rho =-0.75$ ${\sigma}_{\omega}=0.15$ | Bias | −0.002 | −0.003 | −0.537 | 0.224 | −0.002 | 0.026 | −0.059 | 0.202 | −0.003 | 0.010 | −0.006 | 0.166 | |

SD | 0.004 | 0.022 | 0.252 | 0.125 | 0.004 | 0.024 | 0.364 | 0.119 | 0.004 | 0.016 | 0.216 | 0.050 | |||

RMSE | 0.004 | 0.022 | 0.593 | 0.257 | 0.004 | 0.035 | 0.368 | 0.235 | 0.005 | 0.019 | 0.216 | 0.174 | |||

Case 6 | $\rho =0.40$, ${\sigma}_{\omega}=0.25$ | Bias | −0.001 | 0.034 | −0.520 | −0.177 | −0.001 | 0.071 | 0.011 | −0.150 | −0.002 | 0.008 | −0.008 | −0.053 | |

SD | 0.004 | 0.027 | 0.633 | 0.074 | 0.004 | 0.032 | 0.730 | 0.085 | 0.004 | 0.022 | 0.462 | 0.063 | |||

RMSE | 0.004 | 0.043 | 0.819 | 0.191 | 0.004 | 0.078 | 0.730 | 0.172 | 0.004 | 0.023 | 0.462 | 0.083 |

**Table 3.**Results of the estimation of the A-SV model for the real-life time series using the first 2500 observations. The standard errors and standard deviations for the proposal and

`stochvol`methods are in parentheses.

Proposal | Stochvol | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{m}=\mathbf{2}$ | $\mathit{m}=\mathbf{3}$ | ||||||||||||

$\widehat{\mathbf{\varphi}}$ | ${\widehat{\mathbf{\sigma}}}_{\mathbf{\omega}}$ | $\widehat{\mathbf{\alpha}}$ | $\widehat{\mathbf{\rho}}$ | $\widehat{\mathbf{\varphi}}$ | ${\widehat{\mathbf{\sigma}}}_{\mathbf{\omega}}$ | $\widehat{\mathbf{\alpha}}$ | $\widehat{\mathbf{\rho}}$ | $\widehat{\mathbf{\varphi}}$ | ${\widehat{\mathbf{\sigma}}}_{\mathbf{\omega}}$ | $\widehat{\mathbf{\alpha}}$ | $\widehat{\mathbf{\rho}}$ | ||

IBOV | Estimate | 0.975 | 0.168 | −7.950 | −0.386 | 0.968 | 0.224 | −7.431 | −0.413 | 0.956 | 0.214 | −7.850 | −0.471 |

Std. error/Std. dev. | (0.010) | (0.034) | (0.172) | (0.118) | (0.011) | (0.041) | (0.165) | (0.113) | (0.010) | (0.025) | (0.095) | (0.058) | |

Nikkei | Estimate | 0.982 | 0.136 | −8.811 | −0.345 | 0.984 | 0.157 | −8.254 | −0.380 | 0.971 | 0.165 | −8.727 | −0.463 |

Std. error/Std. dev. | (0.008) | (0.029) | (0.191) | (0.155) | (0.007) | (0.032) | (0.206) | (0.162) | (0.007) | (0.020) | (0.111) | (0.063) | |

S&P 500 | Estimate | 0.986 | 0.104 | −9.244 | −0.776 | 0.986 | 0.138 | −8.704 | −0.760 | 0.976 | 0.163 | −9.135 | −0.658 |

Std. error/Std. dev. | (0.006) | (0.024) | (0.166) | (0.170) | (0.005) | (0.026) | (0.173) | (0.123) | (0.006) | (0.020) | (0.110) | (0.047) | |

FTSE | Estimate | 0.987 | 0.137 | −9.222 | −0.728 | 0.986 | 0.178 | −8.744 | −0.752 | 0.987 | 0.135 | −9.265 | −0.627 |

Std. error/Std. dev. | (0.004) | (0.022) | (0.219) | (0.136) | (0.004) | (0.025) | (0.208) | (0.105) | (0.003) | (0.015) | (0.174) | (0.058) | |

USD-BRL | Estimate | 0.980 | 0.249 | −9.581 | 0.224 | 0.979 | 0.279 | −9.127 | 0.234 | 0.980 | 0.210 | −9.592 | 0.322 |

Std. error/Std. dev. | (0.007) | (0.039) | (0.299) | (0.104) | (0.007) | (0.041) | (0.277) | (0.111) | (0.005) | (0.019) | (0.211) | (0.062) | |

USD-MXN | Estimate | 0.968 | 0.143 | −11.037 | 0.163 | 0.967 | 0.164 | −10.463 | 0.118 | 0.862 | 0.354 | −10.926 | 0.363 |

Std. error/Std. dev. | (0.018) | (0.046) | (0.135) | (0.170) | (0.018) | (0.052) | (0.155) | (0.191) | (0.038) | (0.059) | (0.060) | (0.054) |

$\mathit{m}=2$ | $\mathit{m}=3$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Asset | Stat/Test | Long Position | Short Position | Long Position | Short Position | ||||||||

$1\%$ | $2.5\%$ | $5\%$ | $5\%$ | $2.5\%$ | $1\%$ | $1\%$ | $2.5\%$ | $5\%$ | $5\%$ | $2.5\%$ | $1\%$ | ||

IBOV | Proportion | 0.012 | 0.024 | 0.048 | 0.047 | 0.026 | 0.011 | 0.011 | 0.024 | 0.048 | 0.046 | 0.025 | 0.010 |

Kupiec | 0.287 | 0.608 | 0.558 | 0.332 | 0.682 | 0.656 | 0.547 | 0.608 | 0.558 | 0.301 | 0.920 | 0.773 | |

Christoffersen | 0.483 | 0.082 | 0.513 | 0.385 | 0.899 | 0.574 | 0.663 | 0.082 | 0.795 | 0.380 | 0.959 | 0.622 | |

Christoffersen-Pelletier | 0.029 | 0.045 | 0.421 | 0.516 | 0.699 | 0.708 | 0.029 | 0.015 | 0.668 | 0.064 | 0.953 | 0.428 | |

Nikkei | Proportion | 0.014 | 0.029 | 0.052 | 0.051 | 0.026 | 0.013 | 0.012 | 0.029 | 0.052 | 0.050 | 0.026 | 0.012 |

Kupiec | 0.029 | 0.126 | 0.489 | 0.744 | 0.666 | 0.110 | 0.195 | 0.152 | 0.536 | 0.909 | 0.741 | 0.322 | |

Christoffersen | 0.042 | 0.021 | 0.459 | 0.910 | 0.834 | 0.253 | 0.140 | 0.022 | 0.612 | 0.977 | 0.878 | 0.513 | |

Christoffersen-Pelletier | 0.016 | 0.003 | 0.115 | 0.075 | 0.718 | 0.167 | 0.030 | 0.135 | 0.252 | 0.008 | 0.470 | 0.133 | |

S&P 500 | Proportion | 0.017 | 0.034 | 0.056 | 0.052 | 0.026 | 0.013 | 0.016 | 0.032 | 0.055 | 0.050 | 0.025 | 0.011 |

Kupiec | 0.000 | 0.000 | 0.105 | 0.481 | 0.769 | 0.080 | 0.000 | 0.009 | 0.180 | 0.894 | 0.994 | 0.389 | |

Christoffersen | 0.000 | 0.002 | 0.266 | 0.029 | 0.868 | 0.201 | 0.000 | 0.022 | 0.407 | 0.065 | 0.956 | 0.406 | |

Christoffersen-Pelletier | 0.001 | 0.047 | 0.099 | 0.010 | 0.914 | 0.432 | 0.001 | 0.113 | 0.083 | 0.000 | 0.329 | 0.247 | |

FTSE | Proportion | 0.014 | 0.031 | 0.050 | 0.053 | 0.028 | 0.013 | 0.013 | 0.028 | 0.049 | 0.052 | 0.025 | 0.013 |

Kupiec | 0.022 | 0.030 | 0.900 | 0.443 | 0.188 | 0.086 | 0.045 | 0.261 | 0.815 | 0.579 | 0.970 | 0.117 | |

Christoffersen | 0.036 | 0.014 | 0.071 | 0.025 | 0.015 | 0.117 | 0.128 | 0.031 | 0.157 | 0.002 | 0.072 | 0.153 | |

Christoffersen-Pelletier | 0.461 | 0.236 | 0.478 | 0.000 | 0.027 | 0.895 | 0.916 | 0.406 | 0.756 | 0.000 | 0.057 | 0.778 | |

USD-BRL | Proportion | 0.012 | 0.026 | 0.054 | 0.048 | 0.023 | 0.010 | 0.011 | 0.025 | 0.054 | 0.049 | 0.023 | 0.010 |

Kupiec | 0.405 | 0.801 | 0.310 | 0.697 | 0.420 | 0.952 | 0.511 | 0.923 | 0.272 | 0.760 | 0.493 | 0.903 | |

Christoffersen | 0.470 | 0.773 | 0.230 | 0.927 | 0.639 | 0.740 | 0.548 | 0.990 | 0.326 | 0.549 | 0.686 | 0.721 | |

Christoffersen-Pelletier | 0.679 | 0.094 | 0.038 | 0.956 | 0.482 | 0.553 | 0.661 | 0.230 | 0.013 | 0.686 | 0.451 | 0.492 | |

USD-MXN | Proportion | 0.009 | 0.026 | 0.050 | 0.054 | 0.027 | 0.013 | 0.010 | 0.026 | 0.050 | 0.053 | 0.025 | 0.013 |

Kupiec | 0.632 | 0.660 | 0.929 | 0.312 | 0.448 | 0.131 | 0.895 | 0.585 | 0.929 | 0.390 | 0.994 | 0.095 | |

Christoffersen | 0.662 | 0.531 | 0.343 | 0.016 | 0.043 | 0.020 | 0.709 | 0.071 | 0.343 | 0.288 | 0.239 | 0.018 | |

Christoffersen-Pelletier | 0.990 | 0.952 | 0.134 | 0.085 | 0.002 | 0.001 | 0.364 | 0.352 | 0.025 | 0.384 | 0.009 | 0.035 |

Asset | Stat/Test | Long Position | Short Position | ||||
---|---|---|---|---|---|---|---|

$1\%$ | $2.5\%$ | $5\%$ | $5\%$ | $2.5\%$ | $1\%$ | ||

IBOV | Prop | 0.011 | 0.024 | 0.043 | 0.038 | 0.018 | 0.008 |

Kupiec | 0.547 | 0.760 | 0.052 | 0.000 | 0.004 | 0.124 | |

Christoffersen | 0.518 | 0.934 | 0.149 | 0.002 | 0.014 | 0.243 | |

Christoffersen-Pelletier | 0.718 | 0.990 | 0.529 | 0.356 | 0.284 | 0.568 | |

Nikkei | Prop | 0.014 | 0.032 | 0.051 | 0.043 | 0.016 | 0.005 |

Kupiec | 0.009 | 0.013 | 0.800 | 0.045 | 0.000 | 0.000 | |

Christoffersen | 0.017 | 0.044 | 0.968 | 0.134 | 0.001 | 0.000 | |

Christoffersen-Pelletier | 0.648 | 0.950 | 0.476 | 0.687 | 0.027 | 0.261 | |

S&P 500 | Prop | 0.019 | 0.037 | 0.060 | 0.044 | 0.017 | 0.005 |

Kupiec | 0.000 | 0.000 | 0.003 | 0.089 | 0.001 | 0.001 | |

Christoffersen | 0.000 | 0.000 | 0.005 | 0.120 | 0.001 | 0.005 | |

Christoffersen-Pelletier | 0.660 | 0.878 | 0.061 | 0.274 | 0.468 | 0.786 | |

FTSE | Prop | 0.016 | 0.035 | 0.057 | 0.040 | 0.017 | 0.006 |

Kupiec | 0.000 | 0.000 | 0.059 | 0.002 | 0.001 | 0.008 | |

Christoffersen | 0.000 | 0.000 | 0.162 | 0.000 | 0.001 | 0.025 | |

Christoffersen-Pelletier | 0.053 | 0.148 | 0.123 | 0.005 | 0.944 | 0.282 | |

USD-BRL | Prop | 0.009 | 0.017 | 0.044 | 0.054 | 0.026 | 0.010 |

Kupiec | 0.536 | 0.005 | 0.134 | 0.310 | 0.630 | 0.952 | |

Christoffersen | 0.415 | 0.004 | 0.009 | 0.571 | 0.888 | 0.740 | |

Christoffersen-Pelletier | 0.412 | 0.905 | 0.479 | 0.394 | 0.749 | 0.412 | |

USD-MXN | Prop | 0.008 | 0.020 | 0.043 | 0.067 | 0.034 | 0.018 |

Kupiec | 0.311 | 0.033 | 0.069 | 0.000 | 0.001 | 0.001 | |

Christoffersen | 0.469 | 0.026 | 0.102 | 0.001 | 0.002 | 0.001 | |

Christoffersen-Pelletier | 0.899 | 0.281 | 0.194 | 0.018 | 0.529 | 0.091 |

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## Share and Cite

**MDPI and ACS Style**

Abbara, O.; Zevallos, M.
Maximum Likelihood Inference for Asymmetric Stochastic Volatility Models. *Econometrics* **2023**, *11*, 1.
https://doi.org/10.3390/econometrics11010001

**AMA Style**

Abbara O, Zevallos M.
Maximum Likelihood Inference for Asymmetric Stochastic Volatility Models. *Econometrics*. 2023; 11(1):1.
https://doi.org/10.3390/econometrics11010001

**Chicago/Turabian Style**

Abbara, Omar, and Mauricio Zevallos.
2023. "Maximum Likelihood Inference for Asymmetric Stochastic Volatility Models" *Econometrics* 11, no. 1: 1.
https://doi.org/10.3390/econometrics11010001