# Monitoring Volatility Change for Time Series Based on Support Vector Regression

^{*}

## Abstract

**:**

## 1. Introduction

## 2. CUSUM Monitoring Procedure

## 3. Monitoring Procedure via SVR-GARCH Model

#### 3.1. Support Vector Regression

#### 3.2. Particle Swarm Optimization

Algorithm 1 Standard PSO algorithm |

1: procedure PSO($N,{w}_{\mathrm{max}},{w}_{\mathrm{min}},{c}_{1},{c}_{2},{T}_{max}$) |

2: $\mathrm{Accelerated}\phantom{\rule{4.pt}{0ex}}\mathrm{factor}=({c}_{1},{c}_{2}),\mathrm{max}\phantom{\rule{4.pt}{0ex}}\mathrm{generation}\phantom{\rule{4.pt}{0ex}}={T}_{\mathrm{max}}$ |

3: $\mathrm{Initialize}\phantom{\rule{4.pt}{0ex}}\mathrm{location}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\mathrm{velocity};$ |

4: while $t<{T}_{\mathrm{max}}$ do |

5: $t\leftarrow t+1$; |

6: $w\left(t\right)\leftarrow {w}_{max}-\frac{{w}_{max}-{w}_{min}}{{T}_{max}}$ |

7: for $i=1,\dots ,N$ do |

8: update ${v}_{i}\left(t\right)$; |

9: ${x}_{i}\left(t\right)\leftarrow {x}_{i}(t-1)+{v}_{i}\left(t\right)$; |

10: update ${p}_{i}\left(t\right);$ |

11: end for |

12: update $g\left(t\right);$ |

13: end while |

14: end procedure |

#### 3.3. Monitoring Nonlinear Time Series via SVR

**Remark**

**1.**

- 1.
- Estimate $\omega ,\alpha ,\beta $ with $\widehat{\omega},\widehat{\alpha},\widehat{\beta}$ from training sample ${y}_{1},\dots ,{y}_{m}$;
- 2.
- Estimate ${\sigma}_{t}^{2}$ recursively with ${\widehat{\sigma}}_{t}^{2}=\widehat{\omega}+\widehat{\alpha}{y}_{t-1}^{2}+\widehat{\beta}{\widehat{\sigma}}_{t-1}^{2}$ and some initial values ${y}_{0}$ and ${\widehat{\sigma}}_{0}$;
- 3.
- Generate iid standard normal random variables ${\eta}_{t}^{b}$, $t=1,\dots ,n$, $b=1,\dots ,B$, and construct a bootstrap sample ${y}_{t}^{b}={\widehat{\sigma}}_{t}{\eta}_{t}^{b}$;
- 4.
- Based on ${y}_{t}^{b}$, $t=1,\dots ,n$, estimate $\omega ,\alpha ,\beta $ with ${\widehat{\omega}}^{*},{\widehat{\alpha}}^{*},{\widehat{\beta}}^{*}$, and calculate the bootstrapped residuals ${\widehat{\u03f5}}_{t}^{b*}={y}_{t}^{b}/{\widehat{\sigma}}_{t}^{b*}$ with ${\widehat{\sigma}}_{t}^{b*}$ obtained recursively by ${\widehat{\sigma}}_{t}^{b*2}={\widehat{\omega}}^{*}+{\widehat{\alpha}}^{*}{y}_{t-1}^{b2}+{\widehat{\beta}}^{*}{\widehat{\sigma}}_{t-1}^{b*2}$;
- 5.
- Based on these residuals, construct the monitoring process ${\widehat{T}}_{n}^{max,b}\left(k\right)$, $k=1,\dots ,n$, $b=1,\dots ,B$, similarly to ${\widehat{T}}_{n}^{max}$ in (8) with ${\widehat{W}}_{k}^{b}$ analogously defined to ${\widehat{W}}_{k}$;
- 6.
- Finally, the critical value c is determined as the $100\alpha \%$ upper quantile of ${\widehat{T}}_{n}^{max,b}={max}_{1\le k\le n}{\widehat{T}}_{n}^{max,b}\left(k\right)$ for $b=1,\dots ,B$.

## 4. Simulation Experiments

- GARCH(1,1): $\omega =0.3,\phantom{\rule{4pt}{0ex}}\alpha =0.3,\phantom{\rule{4pt}{0ex}}\beta =0.3$
- AGARCH(1,1): $\omega =0.1,\phantom{\rule{4pt}{0ex}}\alpha =0.1,\phantom{\rule{4pt}{0ex}}\beta =0.8,\phantom{\rule{4pt}{0ex}}b=1$
- GJR-GARCH(1,1): $\omega =0.1,\phantom{\rule{4pt}{0ex}}{\alpha}_{1}=0.3,\phantom{\rule{4pt}{0ex}}{\alpha}_{2}=0.1,\phantom{\rule{4pt}{0ex}}\beta =0.5$
- BCTT-GARCH(1,1): $\omega =0.3,\phantom{\rule{4pt}{0ex}}{\alpha}_{1}=0.4,\phantom{\rule{4pt}{0ex}}{\alpha}_{2}=0.2,\phantom{\rule{4pt}{0ex}}\beta =0.3,\phantom{\rule{4pt}{0ex}}\delta =0.8$.

## 5. Real Data Analysis

## 6. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CUSUM | cumulative sum |

SPC | statistical process control |

ARL | average run length |

ARMA | autoregressive and moving average |

ARCH | autoregressive conditionally heteroskedasticity |

GARCH | generalized autoregressive conditionally heteroskedasticity |

SVR | support vector regression |

SVM | support vector machine |

PSO | particle swarm optimization |

iid | independent and identically distributed |

MAE | mean absolute error |

EWMA | exponentially weighted moving average |

AGARCH | asymmetric GARCH |

GJR-GARCH | Glosten, Jagannathan and Runkle-GARCH |

BCTT-GARCH | Box-Cox transformed threshold GARCH |

ARMA | autoregressive and moving average |

QMLE | quasi-maximum likelihood estimator |

KOSPI | Korea Composite Stock Price Index |

ACF | autocorrelation function |

PACF | partial ACF |

## References

- Page, E.S. Continuous inspection schemes. Biometrika
**1954**, 41, 100–115. [Google Scholar] [CrossRef] - Page, E.S. A test for a change in a parameter occurring at an unknown point. Biometrika
**1955**, 42, 523–527. [Google Scholar] [CrossRef] - Wu, Z.; Jiao, J.; Yang, M.; Liu, Y.; Wang, Z. An enhanced adaptive CUSUM control chart. IIE Trans.
**2009**, 41, 642–653. [Google Scholar] [CrossRef] - Regier, P.; Briceño, H.; Boyer, J.N. Analyzing and comparing complex environmental time series using a cumulative sums approach. MethodsX
**2019**, 6, 779–787. [Google Scholar] [CrossRef] - Contreras-Reyes, J.E.; Idrovo-Aguirre, B.J. Backcasting and forecasting time series using detrended cross-correlation analysis. Phys. A Stat. Mech. Appl.
**2020**, 560, 125109. [Google Scholar] [CrossRef] - Montgomery, D.C. Statistical Quality Control; Wiley Global Education: Hoboken, NJ, USA, 2012. [Google Scholar]
- Gombay, E.; Serban, D. Monitoring parameter change in AR (p) time series models. J. Multivar. Anal.
**2009**, 100, 715–725. [Google Scholar] [CrossRef] - Huh, J.; Oh, H.; Lee, S. Monitoring parameter change for time series models with conditional heteroscedasticity. Econ. Lett.
**2017**, 152, 66–70. [Google Scholar] [CrossRef] - Na, O.; Lee, Y.; Lee, S. Monitoring parameter change in time series models. Stat. Methods Appl.
**2011**, 20, 171–199. [Google Scholar] [CrossRef] - Vapnik, V. Statistical Learning Theory; John Wiley and Sons: New York, NY, USA, 1998. [Google Scholar]
- Fernandez-Rodriguez, F.; Gonzalez-Martel, C.; Sosvilla-Rivero, S. On the profitability of technical trading rules based on artificial neural networks: Evidence from the Madrid stock market. Econ. Lett.
**2000**, 69, 89–94. [Google Scholar] [CrossRef] - Cao, L.; Tay, F. Financial forecasting using support vector machines. Neural Comput. Appl.
**2001**, 10, 184–192. [Google Scholar] [CrossRef] - Pérez-Cruz, F.; Afonso-Rodriguez, J.; Giner, J. Estimating GARCH models using SVM. Quant. Financ.
**2003**, 3, 163–172. [Google Scholar] [CrossRef] - Chen, S.; Härdle, W.K.; Jeong, K. Forecasting volatility with support vector machine-based GARCH model. J. Forecast.
**2010**, 29, 406–433. [Google Scholar] [CrossRef] - Bezerra, P.; Albuquerque, P. Volatility forecasting via SVR–GARCH with mixture of Gaussian kernels. Comput. Manag. Sci.
**2017**, 14, 179–196. [Google Scholar] [CrossRef] - Vapnik, V.N. The Nature Of Statistical Learning Theory; Springer: New York, NY, USA, 2000. [Google Scholar]
- Smola, A.; Schölkopf, B. A tutorial on support vector regression. Stat. Comput.
**2004**, 14, 199–222. [Google Scholar] [CrossRef] [Green Version] - Lee, S.; Lee, S.; Moon, M. Hybrid change point detection for time series via support vector regression and CUSUM method. Appl. Soft Comput.
**2020**, 89, 106101. [Google Scholar] [CrossRef] - Lee, S.; Kim, C.; Lee, S. Hybrid CUSUM change point test for time series with time-varying volatilities based on support vector regression. Entropy
**2020**, 22, 578. [Google Scholar] [CrossRef] - Lee, S.; Tokutsu, Y.; Maekawa, K. The cusum test for parameter change in regression models with ARCH errors. J. Jpn. Stat. Soc.
**2004**, 34, 173–188. [Google Scholar] [CrossRef] [Green Version] - Oh, H.; Lee, S. Modified residual CUSUM test for location-scale time series models with heteroscedasticity. Ann. Inst. Stat. Math.
**2019**, 71, 1059–1091. [Google Scholar] [CrossRef] - Lee, S. Location and scale-based CUSUM test with application to autoregressive models. J. Stat. Comput. Simul.
**2020**, 90, 2309–2328. [Google Scholar] [CrossRef] - Lee, S.; Ha, J.; Na, O.; Na, S. The CUSUM test for parameter change in time series models. Scand. J. Stat.
**2003**, 30, 781–796. [Google Scholar] [CrossRef] - Kennedy, J.; Eberhart, R. Particle swarm optimization. In Proceedings of the ICNN’95-International Conference on Neural Networks, Perth, Australia, 27 November–1 December 1995; IEEE: Piscataway, NJ, USA, 1995; Volume 4, pp. 1942–1948. [Google Scholar]
- Ozcan, E.; Mohan, C.K. Analysis of a simple particle swarm optimization system. Intell. Eng. Syst. Through Artif. Neural Netw.
**1998**, 8, 253–258. [Google Scholar] - Trelea, I.C. The particle swarm optimization algorithm: Convergence analysis and parameter selection. Inf. Process. Lett.
**2003**, 85, 317–325. [Google Scholar] [CrossRef] - Yasuda, K.; Ide, A.; Iwasaki, N. Adaptive particle swarm optimization. In SMC’03 Conference Proceedings, Proceedings of the 2003 IEEE International Conference on Systems, Man and Cybernetics, Washington, DC, USA, 8 October 2003; IEEE: Piscataway, NJ, USA, 2003; Volume 2, pp. 1554–1559. [Google Scholar]
- Zhang, Y.; Wang, S.; Ji, G. A comprehensive survey on particle swarm optimization algorithm and its applications. Math. Probl. Eng.
**2015**, 2015, 1–38. [Google Scholar] [CrossRef] [Green Version] - Wang, D.; Tan, D.; Liu, L. Particle swarm optimization algorithm: An overview. Soft Comput.
**2018**, 22, 387–408. [Google Scholar] [CrossRef] - Qian, W.; Li, M. Convergence analysis of standard particle swarm optimization algorithm and its improvement. Soft Comput.
**2018**, 22, 4047–4070. [Google Scholar] [CrossRef] - Billingsley, P. Convergence of Probability Measure; Wiley: New York, NY, USA, 1968. [Google Scholar]
- Bollerslev, T. Generalized autoregressive conditional heteroskedasticity. J. Econom.
**1986**, 31, 307–327. [Google Scholar] [CrossRef] [Green Version] - Francq, C.; Zakoian, J.M. GARCH Models: Structure, Statistical Inference and Financial Applications; John Wiley & Sons: New York, NY, USA, 2019. [Google Scholar]
- Cortes, C.; Vapnik, V. Support-vector networks. Mach. Learn.
**1995**, 20, 273–297. [Google Scholar] [CrossRef] - Abe, S. Support Vector Machines for Pattern Classification; Springer: New York, NY, USA, 2005; Volume 2. [Google Scholar]
- Marini, F.; Walczak, B. Particle swarm optimization (PSO). A tutorial. Chemom. Intell. Lab. Syst.
**2015**, 149, 153–165. [Google Scholar] [CrossRef]

**Table 1.**Empirical size and power of the SVR-GARCH monitoring procedure for the GARCH(1,1), AGARCH(1,1), GJR-GARCH(1,1), and BCTT-GARCH(1,1) models.

4-17 | $\mathit{n}=1000$ | $\mathit{n}=2000$ | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Change location | $0.1n$ | $0.2n$ | $0.3n$ | $0.4n$ | $0.5n$ | $0.6n$ | $0.7n$ | $0.1n$ | $0.2n$ | $0.3n$ | $0.4n$ | $0.5n$ | $0.6n$ | $0.7n$ | ||

GARCH(1,1) | size | 0.038 | 0.045 | |||||||||||||

power | $\omega \to 1$ | 0.903 | 0.893 | 0.879 | 0.867 | 0.824 | 0.778 | 0.737 | 0.953 | 0.945 | 0.934 | 0.916 | 0.893 | 0.845 | 0.805 | |

$\omega \to 0.1$ | 0.954 | 0.942 | 0.898 | 0.824 | 0.701 | 0.475 | 0.206 | 0.971 | 0.963 | 0.959 | 0.916 | 0.840 | 0.616 | 0.317 | ||

$\alpha \to 0.6$ | 0.961 | 0.955 | 0.940 | 0.924 | 0.882 | 0.830 | 0.726 | 0.995 | 0.990 | 0.976 | 0.973 | 0.951 | 0.939 | 0.870 | ||

$\beta \to 0.6$ | 0.974 | 0.958 | 0.955 | 0.946 | 0.907 | 0.871 | 0.832 | 0.996 | 0.992 | 0.988 | 0.981 | 0.975 | 0.940 | 0.920 | ||

AGARCH(1,1) | size | 0.039 | 0.037 | |||||||||||||

power | $\omega \to 1$ | 0.993 | 0.989 | 0.987 | 0.985 | 0.968 | 0.959 | 0.939 | 0.996 | 0.997 | 0.988 | 0.995 | 0.989 | 0.980 | 0.975 | |

$\beta \to 0.2$ | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.992 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | ||

$\alpha ,\beta \to 0.8,0.1$ | 0.967 | 0.927 | 0.924 | 0.900 | 0.856 | 0.767 | 0.672 | 0.995 | 0.992 | 0.983 | 0.978 | 0.950 | 0.899 | 0.832 | ||

$b\to 3$ | 0.989 | 0.986 | 0.981 | 0.978 | 0.962 | 0.943 | 0.920 | 0.994 | 0.995 | 0.994 | 0.984 | 0.987 | 0.988 | 0.966 | ||

GJR-GARCH(1,1) | size | 0.042 | 0.033 | |||||||||||||

power | $\omega \to 1$ | 1.000 | 0.999 | 1.000 | 0.998 | 0.999 | 0.998 | 0.996 | 1.000 | 1.000 | 1.000 | 1.000 | 0.999 | 1.000 | 0.999 | |

$\omega \to 0.01$ | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | ||

${\alpha}_{2}\to 0.8$ | 1.000 | 1.000 | 0.999 | 0.999 | 0.998 | 0.997 | 0.967 | 1.000 | 1.000 | 1.000 | 1.000 | 0.999 | 1.000 | 0.998 | ||

$\beta \to 0.75$ | 0.996 | 0.993 | 0.994 | 0.986 | 0.986 | 0.977 | 0.954 | 1.000 | 1.000 | 1.000 | 0.998 | 0.998 | 0.997 | 0.989 | ||

BCTT-GARCH(1,1) | size | 0.048 | 0.039 | |||||||||||||

power | $\omega \to 1$ | 0.978 | 0.973 | 0.973 | 0.957 | 0.948 | 0.934 | 0.897 | 0.997 | 0.996 | 0.993 | 0.989 | 0.979 | 0.975 | 0.955 | |

$\omega \to 0.01$ | 0.995 | 0.995 | 0.993 | 0.978 | 0.951 | 0.855 | 0.556 | 0.999 | 0.997 | 0.997 | 0.997 | 0.987 | 0.958 | 0.819 | ||

${\alpha}_{1}\to 1$ | 0.997 | 0.995 | 0.988 | 0.992 | 0.962 | 0.947 | 0.903 | 1.000 | 0.999 | 1.000 | 1.000 | 0.996 | 0.989 | 0.968 | ||

${\alpha}_{2}\to 0.9$ | 0.987 | 0.971 | 0.968 | 0.941 | 0.921 | 0.875 | 0.817 | 0.995 | 0.993 | 0.988 | 0.992 | 0.980 | 0.958 | 0.911 | ||

$\beta \to 0.6$ | 0.986 | 0.983 | 0.984 | 0.967 | 0.970 | 0.941 | 0.897 | 0.998 | 0.994 | 0.997 | 0.992 | 0.990 | 0.992 | 0.972 | ||

$\delta \to 0.2$ | 0.999 | 0.998 | 0.999 | 0.992 | 0.969 | 0.891 | 0.585 | 0.999 | 0.999 | 0.998 | 0.999 | 0.996 | 0.978 | 0.854 | ||

$\delta \to 2$ | 0.850 | 0.814 | 0.795 | 0.749 | 0.671 | 0.585 | 0.502 | 0.945 | 0.919 | 0.890 | 0.846 | 0.800 | 0.728 | 0.602 |

**Table 2.**Summary statistics and the results of the preliminary retrospective test of the training set, and the result of the monitoring test regarding S&P500, KOSPI, and Microsoft.

S&P500 | KOSPI | Microsoft | ||
---|---|---|---|---|

Summary statistics (training set) | Observations | 1640 | 1016 | 1417 |

Mean | 0.0604 | 0.0096 | 0.0428 | |

Standard deviation | 0.6931 | 0.7728 | 1.4408 | |

Minimum | −3.7272 | −3.1429 | −12.1033 | |

Median | 0.0352 | 0.0070 | 0.03145 | |

Maximum | 3.6642 | 2.9124 | 7.0330 | |

Skewness | −0.1064 | −0.0264 | −0.6141 | |

Excess kurtosis | 2.2428 | 1.3848 | 6.8293 | |

Retrospective test (training set) | Test statistic | 0.8069 | 1.2876 | 0.5897 |

Monitoring test | Location | 97/10/28 | 20/03/11 | 20/03/13 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lee, S.; Kim, C.K.; Kim, D.
Monitoring Volatility Change for Time Series Based on Support Vector Regression. *Entropy* **2020**, *22*, 1312.
https://doi.org/10.3390/e22111312

**AMA Style**

Lee S, Kim CK, Kim D.
Monitoring Volatility Change for Time Series Based on Support Vector Regression. *Entropy*. 2020; 22(11):1312.
https://doi.org/10.3390/e22111312

**Chicago/Turabian Style**

Lee, Sangyeol, Chang Kyeom Kim, and Dongwuk Kim.
2020. "Monitoring Volatility Change for Time Series Based on Support Vector Regression" *Entropy* 22, no. 11: 1312.
https://doi.org/10.3390/e22111312