# Does the Assumption on Innovation Process Play an Important Role for Filtered Historical Simulation Model?

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## Abstract

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## 1. Introduction

## 2. Filtered Historical Simulation Models

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- Let ${R}_{t}$ denotes the daily log-returns. The benchmark GARCH(1,1) model, introduced by Bollerslev (1986), is defined by$$\begin{array}{c}{R}_{t}=\mu +{e}_{t},\hfill \\ {e}_{t}={\epsilon}_{t}\phantom{\rule{0.166667em}{0ex}}{h}_{t},\phantom{\rule{4pt}{0ex}}{\epsilon}_{t}\sim i.i.d.\hfill \\ {h}_{t}^{2}=\omega +{\gamma}_{1}\phantom{\rule{0.166667em}{0ex}}{e}_{t-1}^{2}+{\gamma}_{2}{h}_{t-1}^{2},\hfill \end{array}$$$$\ell \left(\psi \right)={\displaystyle \sum _{t=1}^{T}}\left[\mathrm{ln}\left(f\left({\epsilon}_{t};\tau \right)\right)-\frac{1}{2}\mathrm{ln}\left({h}_{t}^{2}\right)\right]$$The standardized residuals of estimated GARCH(1,1) model are extracted as follows:$${\epsilon}_{t}=\frac{{\widehat{e}}_{t}}{{\widehat{h}}_{t}},$$Now, we can generate the first simulated residual by randomly (with replacement) draw standardized residuals from the dataset with multiplying the one-day ahead volatility forecast:$${z}_{t+1}^{\ast}={e}_{1}^{\ast}{h}_{t+1}.$$The first simulated return for period $t+1$ can be obtained as follows:$${R}_{t+1}^{\ast}={\mu}_{t+1}+{z}_{t+1}^{\ast},$$

#### 2.1. Normal Distribution

#### 2.2. Skew-Normal Distribution

#### 2.3. Student’s-t Distribution

#### 2.4. Skew-T Distribution

#### 2.5. Generalized Error Distribution

#### 2.6. Skewed Generalized Error Distribution

## 3. Evaluation of VaR Forecasts

## 4. Empirical Results

#### 4.1. Data Description

**rugarch**package in R software is used to obtain parameter estimation of normal, Student’s-t, GED and SGED models. The

**constrOptim**function in R software is used to minimize negative log-likelihood functions of GARCH-ST and GARCH-SN models.

#### 4.2. Backtesting Results

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Daily VaR forecast of GARCH models with different innovation distributions for 97.5% and 99% confidence levels.

ISE-100 | |
---|---|

Number of observations | 1092 |

Minimum | −0.048 |

Maximum | 0.027 |

Mean | $6.6\times {10}^{-5}$ |

Median | $2\times {10}^{-4}$ |

Std. Deviation | 0.006 |

Skewness | −0.603 |

Kurtosis | 4.957 |

Jarque-Bera | 1190.970 (p <0.001) |

**Table 2.**In-sample performance of GARCH models under skewed and fat-tailed innovation distributions.

Parameters | Normal | Student-T | ST | SN | GED | SGED |
---|---|---|---|---|---|---|

$\mu $ | $5.24\times {10}^{-4}$ | $5.67\times {10}^{-4}$ | $8.56\times {10}^{-4}$ | $4.05\times {10}^{-4}$ | $5.67\times {10}^{-4}$ | $3.51\times {10}^{-4}$ |

$3.41\times {10}^{-4}$ | $3.04\times {10}^{-4}$ | $3.60\times {10}^{-4}$ | $3.34\times {10}^{-4}$ | $3.77\times {10}^{-4}$ | $3.26\times {10}^{-4}$ | |

$\omega $ | $3.69\times {10}^{-6}$ | $2.01\times {10}^{-6}$ | $2.27\times {10}^{-6}$ | $3.65\times {10}^{-6}$ | $2.72\times {10}^{-6}$ | $2.63\times {10}^{-6}$ |

$1.50\times {10}^{-6}$ | $1.53\times {10}^{-6}$ | $6.71\times {10}^{-6}$ | $1.73\times {10}^{-6}$ | $1.66\times {10}^{-6}$ | $1.57\times {10}^{-6}$ | |

${\gamma}_{1}$ | 0.1194 | 0.0759 | 0.1510 | 0.1460 | 0.0930 | 0.0873 |

0.0413 | 0.0336 | 0.2940 | 0.0560 | 0.0393 | 0.0353 | |

${\gamma}_{2}$ | 0.8234 | 0.8908 | 0.8240 | 0.7950 | 0.8600 | 0.8650 |

0.0449 | 0.0485 | 0.3130 | 0.0662 | 0.0518 | 0.0486 | |

$\nu $ | - | 4.7490 | 4.8760 | - | 1.2020 | - |

- | 1.1600 | 0.5620 | - | 0.1040 | - | |

$\lambda $ | - | - | −0.2750 | −1.5050 | - | 0.8860 |

- | - | 2.4260 | 0.2630 | - | 0.0440 | |

$\kappa $ | - | - | - | - | - | 1.2200 |

- | - | - | - | - | 0.1093 | |

$-\ell $ | −1381.1100 | −1405.003 | −1402.2630 | −1388.1800 | −1401.0600 | −1402.7300 |

p = 0.05 | ||||||

Models | Mean VaR (%) | N. Of Vio. | Failure Rate | LR-uc | LR-cc | DQ |

FSH-N | −0.910 | 29 | 0.041 | 1.146 (0.284) | 1.186 (0.552) | 4.586 (0.710) |

FSH-SN | −0.911 | 29 | 0.041 | 1.146 (0.284) | 1.186 (0.552) | 4.545 (0.715) |

FSH-T | −0.897 | 32 | 0.046 | 0.278 (0.597) | 0.459 (0.794) | 2.996 (0.885) |

FSH-GED | −0.899 | 30 | 0.043 | 0.788 (0.374) | 0.863 (0.649) | 4.041 (0.775) |

FSH-SGED | −0.904 | 30 | 0.043 | 0.788 (0.374) | 0.863 (0.649) | 4.255 (0.749) |

FSH-ST | −0.897 | 32 | 0.046 | 0.278 (0.597) | 0.459 (0.794) | 3.187 (0.867) |

p = 0.025 | ||||||

Models | Mean VaR (%) | N. Of Vio. | Failure Rate | LR-uc | LR-cc | DQ |

FSH-N | −1.193 | 20 | 0.029 | 0.350 (0.554) | 0.630 (0.729) | 6.820 (0.448) |

FSH-SN | −1.196 | 20 | 0.029 | 0.350 (0.554) | 0.630 (0.729) | 6.805 (0.449) |

FSH-T | −1.177 | 20 | 0.029 | 0.350 (0.554) | 0.630 (0.729) | 4.056 (0.773) |

FSH-GED | −1.179 | 20 | 0.029 | 0.350 (0.554) | 0.630 (0.729) | 4.579 (0.711) |

FSH-SGED | −1.187 | 20 | 0.029 | 0.350 (0.554) | 0.630 (0.729) | 5.102 (0.647) |

FSH-ST | −1.177 | 20 | 0.029 | 0.350 (0.554) | 0.630 (0.729) | 4.051 (0.774) |

Models | Mean VaR (%) | N. Of Vio. | Failure Rate | LR-uc | LR-cc | DQ |

FSH-N | −1.546 | 9 | 0.013 | 0.529 (0.466) | 0.764 (0.682) | 15.479 (0.030) |

FSH-SN | −1.549 | 9 | 0.013 | 0.529 (0.466) | 0.764 (0.682) | 16.338 (0.022) |

FSH-T | −1.526 | 9 | 0.013 | 0.529 (0.466) | 0.764 (0.682) | 13.185 (0.067) |

FSH-GED | −1.530 | 8 | 0.011 | 0.137 (0.710) | 0.323 (0.851) | 16.115 (0.024) |

FSH-SGED | −1.538 | 9 | 0.013 | 0.529 (0.466) | 0.764 (0.682) | 14.620 (0.041) |

FSH-ST | −1.526 | 9 | 0.013 | 0.529 (0.466) | 0.764 (0.682) | 12.893 (0.075) |

**Table 4.**Loss functions results of FHS models for long position ($p=0.05$, $p=0.025$, and $p=0.01$).

p = 0.05 | ||||||

Models | ARLF | Min.-Max. ARLF | UL | Min.-Max. UL | FLF | Min.-Max. FLF |

FSH-N | 0.0172063 | ($1\times {10}^{-4}$, 5.133) | −0.0179110 | (−2.265, −0.010) | 0.0283988 | ($1\times {10}^{-4}$, 5.133) |

FSH-SN | 0.0171900 | ($1\times {10}^{-4}$, 5.121) | −0.0178643 | (−2.263, −0.011) | 0.0283877 | ($1.37\times {10}^{-4}$, 5.121) |

FSH-T | 0.0173740 | ($5.84\times {10}^{-8}$, 5.150) | −0.0181385 | (−2.269, $-2\times {10}^{-4}$) | 0.0284271 | ($5.84\times {10}^{-8}$, 5.150) |

FSH-GED | 0.0172983 | ($1.89\times {10}^{-6}$, 5.135) | −0.0180858 | (−2.266, −0.001) | 0.0283820 | ($1.89\times {10}^{-6}$, 5.135) |

FSH-SGED | 0.0173162 | ($2.05\times {10}^{-5}$, 5.145) | −0.0179867 | (−2.268, −0.004) | 0.0284463 | ($2.05\times {10}^{-5}$, 5.145) |

FSH-ST | 0.0173437 | ($9.28\times {10}^{-6}$, 5.127) | −0.0181591 | (−2.264, −0.003) | 0.0283897 | ($9.28\times {10}^{-6}$, 5.127) |

p = 0.025 | ||||||

Models | ARLF | Min.-Max. ARLF | UL | Min.-Max. UL | FLF | Min.-Max. FLF |

FSH-N | 0.0097984 | (0.003, 3.896) | −0.0108364 | (−1.974, −0.058) | 0.0228029 | (0.003, 3.896) |

FSH-SN | 0.0097622 | (0.001, 3.863) | −0.0107498 | (−1.965, −0.033) | 0.0227883 | (0.001, 3.863) |

FSH-T | 0.0098350 | (0.002, 3.886) | −0.0108600 | (−1.971, −0.049) | 0.0226889 | (0.002, 3.886) |

FSH-GED | 0.0097898 | (0.004, 3.876) | −0.0108694 | (−1.968, −0.063) | 0.0226529 | (0.004, 3.876) |

FSH-SGED | 0.0098099 | (0.002, 3.893) | −0.0107848 | (−1.973, −0.040) | 0.0227502 | (0.002, 3.893) |

FSH-ST | 0.0098287 | (0.002, 3.884) | −0.0108752 | (−1.971, −0.049) | 0.0226755 | (0.002, 3.884) |

p = 0.01 | ||||||

Models | ARLF | Min.-Max. ARLF | UL | Min.-Max. UL | FLF | Min.-Max. FLF |

FSH-N | 0.0052475 | ($7.17\times {10}^{-5}$, 2.885) | −0.0051635 | (−1.698, −0.008) | 0.0212668 | ($7.17\times {10}^{-5}$, 2.885) |

FSH-SN | 0.0052483 | ( $8.22\times {10}^{-5}$, 2.893) | −0.0051776 | (−1.700, −0.009) | 0.0213011 | ($8.22\times {10}^{-5}$, 2.893) |

FSH-T | 0.0052399 | ($1\times {10}^{-4}$, 2.914) | −0.0051979 | (−1.707, −0.011) | 0.0210625 | ($1.15\times {10}^{-4}$, 2.914) |

FSH-GED | 0.0052125 | ($3\times {10}^{-4}$, 2.882) | −0.0051546 | (−1.697, −0.017) | 0.0210964 | ($2.81\times {10}^{-4}$, 2.882) |

FSH-SGED | 0.0052148 | ($4\times {10}^{-4}$, 2.883) | −0.0051849 | ( −1.697, −0.019) | 0.0211543 | ($3.98\times {10}^{-4}$, 2.883) |

FSH-ST | 0.0052135 | ($3.32\times {10}^{-5}$, 2.895) | −0.0051634 | (−1.701, −0.006) | 0.0210319 | ($3.32\times {10}^{-5}$, 2.895) |

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**MDPI and ACS Style**

Altun, E.; Tatlidil, H.; Ozel, G.; Nadarajah, S. Does the Assumption on Innovation Process Play an Important Role for Filtered Historical Simulation Model? *J. Risk Financial Manag.* **2018**, *11*, 7.
https://doi.org/10.3390/jrfm11010007

**AMA Style**

Altun E, Tatlidil H, Ozel G, Nadarajah S. Does the Assumption on Innovation Process Play an Important Role for Filtered Historical Simulation Model? *Journal of Risk and Financial Management*. 2018; 11(1):7.
https://doi.org/10.3390/jrfm11010007

**Chicago/Turabian Style**

Altun, Emrah, Huseyin Tatlidil, Gamze Ozel, and Saralees Nadarajah. 2018. "Does the Assumption on Innovation Process Play an Important Role for Filtered Historical Simulation Model?" *Journal of Risk and Financial Management* 11, no. 1: 7.
https://doi.org/10.3390/jrfm11010007