# Hybrid CUSUM Change Point Test for Time Series with Time-Varying Volatilities Based on Support Vector Regression

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Support Vector Regression for the GARCH Model

- Step 1. Prescribe the points to be evaluated within this space, then divide the given time series into training and validation time series of size n and ${n}^{{}^{\prime}}$, respectively. This preliminary procedure is required for the subsequent task of validating the fitted SVR-GARCH model, which determines the best tuning parameter sets.
- Step 2. Note that the conditional variance ${\sigma}_{t}^{2}$ of (9) is unknown. As a remedy, replace ${\sigma}_{t}^{2}$ with the initial estimates ${\tilde{\sigma}}_{t}^{2}$, which plays the role of a proxy of ${\sigma}_{t}^{2}$. The estimate ${\tilde{\sigma}}_{t}^{2}$ is based on the training time series using a moving average method (Niemira [43]):$${\tilde{\sigma}}_{t}^{2}=\frac{1}{m}\sum _{j=1}^{m}{y}_{t-j+1}^{2},$$
- Step 3. Given a set of tuning parameters, we estimate g in (1) with $\widehat{g}$ using the SVR with ${\sigma}_{t}^{2}$ replaced by ${\tilde{\sigma}}_{t}^{2}$. Then, the estimate ${\widehat{\sigma}}_{t}^{2}$ of ${\sigma}_{t}^{2}$ is obtained as:$${\widehat{\sigma}}_{t}^{2}=\widehat{g}({y}_{t-1}^{2},\dots ,{y}_{t-p}^{2},{\tilde{\sigma}}_{t-1}^{2},\dots ,{\tilde{\sigma}}_{t-q}^{2}).$$
- Step 4. Applying the estimated SVR-GARCH model and using the same proxy formula as in Step 2 for the validation time series, the mean absolute error (MAE) is computed as follows:$$\mathrm{MAE}=\frac{1}{{n}^{{}^{\prime}}}\sum _{t=1}^{{n}^{{}^{\prime}}}|{\widehat{\sigma}}_{t}^{2}-{\tilde{\sigma}}_{t}^{2}|.$$The MAE escalates the robustness of the model against outliers and therefore provides more flexibility in a model fitting than the root mean squared error.
- Step 5. Repeat Steps 2 to 4 for all the tuning parameter sets selected in Step 1 and choose the combination that minimizes the MAE. Then, perform Steps 2 and 3 using the training and validation time series together to determine the final model, which is used in obtaining the residuals.

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

## 3. Hybrid CUSUM Test via the SVR-GARCH Model

## 4. Simulation Results

- Step 1. Generate a time series of length $2n$ from a prescribed GARCH model.
- Step 2. Follow the estimation scheme described in Section 3 with $m=5$. In this procedure, the first 0.7n number of time series constitute the training set, and the following $0.3n$ number of time series constitute the validation set.
- Step 3. Conduct the CUSUM of squares test described in Section 3. We utilize the remaining n number of time series as a testing set.
- Step 4. Repeat Steps 1 to 3 1000 times iteratively, and then, compute the empirical sizes and powers.

- GARCH(1,1) model:$$\begin{array}{cc}\hfill {y}_{t}& ={\sigma}_{t}{\u03f5}_{t},\hfill \\ \hfill {\sigma}_{t}^{2}& =\omega +\alpha {y}_{t-1}^{2}+\beta {\sigma}_{t-1}^{2},\hfill \\ \hfill \alpha & \ge 0,\beta \ge 0.\hfill \end{array}$$
- AGARCH(1,1) model:$$\begin{array}{cc}\hfill {y}_{t}& ={\sigma}_{t}{\u03f5}_{t},\hfill \\ \hfill {\sigma}_{t}^{2}& =\omega +\alpha {({y}_{t-1}-b)}^{2}+\beta {\sigma}_{t-1}^{2},\hfill \\ \hfill \alpha & \ge 0,\beta \ge 0.\hfill \end{array}$$
- GJR-GARCH(1,1) model:$$\begin{array}{cc}\hfill {y}_{t}& ={\sigma}_{t}{\u03f5}_{t},\hfill \\ \hfill {\sigma}_{t}^{2}& =\omega +{\alpha}_{1}{{y}_{t-1}^{+}}^{2}+{\alpha}_{2}{{y}_{t-1}^{-}}^{2}+\beta {\sigma}_{t-1}^{2},\hfill \\ \hfill {y}_{t}^{+}& =max({y}_{t},\phantom{\rule{4pt}{0ex}}0),\phantom{\rule{4pt}{0ex}}{y}_{t}^{-}=-min({y}_{t},\phantom{\rule{4pt}{0ex}}0),\hfill \\ \hfill {\alpha}_{1}& \ge 0,{\alpha}_{2}\ge 0,\beta \ge 0.\hfill \end{array}$$
- TGARCH(1,1) model:$$\begin{array}{cc}\hfill {y}_{t}& ={\sigma}_{t}{\u03f5}_{t},\hfill \\ \hfill {\sigma}_{t}& =\omega +\alpha |{y}_{t-1}|+\beta {\sigma}_{t-1},\hfill \\ \hfill \alpha & \ge 0,\beta \ge 0.\hfill \end{array}$$
- Log-linear GARCH(1,1) model (a specific variation of the EGARCH($p,q$) model):$$\begin{array}{cc}\hfill {y}_{t}& ={\sigma}_{t}{\u03f5}_{t},\hfill \\ \hfill log{\sigma}_{t}^{2}& =\omega +\alpha log{y}_{t-1}^{2}+\beta log{\sigma}_{t-1}^{2}.\hfill \end{array}$$

- GARCH model: $\omega =0.3,\phantom{\rule{4pt}{0ex}}\alpha =0.3,\phantom{\rule{4pt}{0ex}}\beta =0.3$;
- AGARCH model: $\omega =0.3,\phantom{\rule{4pt}{0ex}}\alpha =0.3,\phantom{\rule{4pt}{0ex}}\beta =0.4,\phantom{\rule{4pt}{0ex}}b=1$;
- GJR-GARCH model: $\omega =0.3,\phantom{\rule{4pt}{0ex}}{\alpha}_{1}=0.3,\phantom{\rule{4pt}{0ex}}{\alpha}_{2}=0.4,\phantom{\rule{4pt}{0ex}}\beta =0.3$;
- TGARCH model: $\omega =0.3,\phantom{\rule{4pt}{0ex}}\alpha =0.3,\phantom{\rule{4pt}{0ex}}\beta =0.3$;
- log-linear GARCH model: $\omega =0.3,\phantom{\rule{4pt}{0ex}}\alpha =0.3,\phantom{\rule{4pt}{0ex}}\beta =0.3$.

- GARCH(1,1) changes to log-linear GARCH(1,1);
- log-linear GARCH(1,1) changes to GARCH(1,1);
- TGARCH(1,1) changes to AGARCH(1,1);
- AGARCH(1,1) changes to GJR-GARCH(1,1).

## 5. Real Data Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CUSUM | cumulative sum |

SVR | support vector regression |

SVM | support vector machine |

GARCH | generalized autoregressive conditionally heteroscedastic |

EGARCH | exponential GARCH |

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

TGARCH | threshold GARCH |

APARCH | asymmetric power ARCH |

NN | neural network |

ARMA | autoregressive and moving average |

QMLE | quasi-maximum likelihood estimator |

KOSPI | Korea Composite Stock Price Index |

KRW | Korean Won |

VaR | value at risk |

ES | expected shortfall |

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**Figure 1.**Plot of ACF and partial autocorrelation function (PACF) up to lag 25 of log-returns of the (

**a**) S&P 500, (

**b**) KOSPI, and (

**c**) KRW/USD indices.

**Figure 2.**Plot of (

**a**) the raw index of S&P 500 and (

**b**) its log-returns with the detected change point.

**Figure 3.**Plot of (

**a**) the raw index of KOSPI and (

**b**) its log-returns with the detected change point.

$\mathit{\omega}=0.3$ | ||||
---|---|---|---|---|

$\mathit{\alpha}=\mathbf{0.3}$ | $\mathit{n}=\mathbf{500}$ | $\mathit{n}=\mathbf{1000}$ | $\mathit{n}=\mathbf{2000}$ | |

$\mathit{\beta}=\mathbf{0.3}$ | ||||

size | 0.023 | 0.038 | 0.055 | |

change of $\omega $ | $\to \phantom{\rule{4pt}{0ex}}\omega =1$ | 0.761 | 0.826 | 0.956 |

$\to \phantom{\rule{4pt}{0ex}}\omega =0.1$ | 0.612 | 0.792 | 0.97 | |

change of $\alpha ,\phantom{\rule{4pt}{0ex}}\beta $ | $\to \phantom{\rule{4pt}{0ex}}\alpha =0.1,\phantom{\rule{4pt}{0ex}}\beta =0.2$ | 0.355 | 0.651 | 0.949 |

$\to \phantom{\rule{4pt}{0ex}}\alpha =0.4,\phantom{\rule{4pt}{0ex}}\beta =0.5$ | 0.649 | 0.802 | 0.952 | |

change of mixed parameters | $\to \phantom{\rule{4pt}{0ex}}\omega =0.7,\phantom{\rule{4pt}{0ex}}\alpha =0.1$ | 0.871 | 0.969 | 0.981 |

$\to \phantom{\rule{4pt}{0ex}}\omega =0.1,\phantom{\rule{4pt}{0ex}}\beta =0.1$ | 0.848 | 0.952 | 0.964 |

$\mathit{\omega}=0.3$ | ||||
---|---|---|---|---|

$\mathit{\alpha}=\mathbf{0.3},\phantom{\rule{4pt}{0ex}}\mathit{\beta}=\mathbf{0.4}$ | $\mathit{n}=\mathbf{500}$ | $\mathit{n}=\mathbf{1000}$ | $\mathit{n}=\mathbf{2000}$ | |

$\mathit{b}=\mathbf{1}$ | ||||

size | 0.031 | 0.04 | 0.03 | |

change of $\omega $ | $\to \phantom{\rule{4pt}{0ex}}\omega =1$ | 0.881 | 0.951 | 0.975 |

$\to \phantom{\rule{4pt}{0ex}}\omega =0.1$ | 0.26 | 0.613 | 0.904 | |

change of $\alpha ,\phantom{\rule{4pt}{0ex}}\beta $ | $\to \phantom{\rule{4pt}{0ex}}\alpha =0.1,\phantom{\rule{4pt}{0ex}}\beta =0.1$ | 0.783 | 0.939 | 0.975 |

$\to \phantom{\rule{4pt}{0ex}}\alpha =0.6$ | 0.762 | 0.863 | 0.941 | |

change of b | $\to \phantom{\rule{4pt}{0ex}}b=0$ | 0.591 | 0.897 | 0.976 |

$\to \phantom{\rule{4pt}{0ex}}b=3$ | 0.898 | 0.936 | 0.957 | |

change of mixed parameters | $\to \phantom{\rule{4pt}{0ex}}\omega =0.1,\phantom{\rule{4pt}{0ex}}\alpha =0.5,\phantom{\rule{4pt}{0ex}}\beta =0.4,\phantom{\rule{4pt}{0ex}}b=0$ | 0.565 | 0.726 | 0.846 |

$\to \phantom{\rule{4pt}{0ex}}\omega =0.1,\phantom{\rule{4pt}{0ex}}\beta =0.6,\phantom{\rule{4pt}{0ex}}b=2$ | 0.879 | 0.926 | 0.966 |

$\mathit{\omega}=0.3$ | ||||
---|---|---|---|---|

${\mathit{\alpha}}_{\mathbf{1}}=\mathbf{0.3},\phantom{\rule{4pt}{0ex}}{\mathit{\alpha}}_{\mathbf{2}}=\mathbf{0.4}$ | $\mathit{n}=\mathbf{500}$ | $\mathit{n}=\mathbf{1000}$ | $\mathit{n}=\mathbf{2000}$ | |

$\mathit{\beta}=\mathbf{0.3}$ | ||||

size | 0.021 | 0.029 | 0.028 | |

change of $\omega $ | $\to \phantom{\rule{4pt}{0ex}}\omega =1$ | 0.851 | 0.912 | 0.947 |

$\to \phantom{\rule{4pt}{0ex}}\omega =0.1$ | 0.644 | 0.787 | 0.866 | |

change of ${\alpha}_{i},\phantom{\rule{4pt}{0ex}}\beta (i=1,2)$ | $\to \phantom{\rule{4pt}{0ex}}{\alpha}_{1}=0.5,\phantom{\rule{4pt}{0ex}}{\alpha}_{2}=0.8$ | 0.418 | 0.695 | 0.879 |

$\to \phantom{\rule{4pt}{0ex}}{\alpha}_{2}=0.1,\phantom{\rule{4pt}{0ex}}\beta =0.1$ | 0.421 | 0.751 | 0.888 | |

change of mixed parameters | $\to \phantom{\rule{4pt}{0ex}}\omega =0.5,\phantom{\rule{4pt}{0ex}}{\alpha}_{2}=0.6$ | 0.657 | 0.863 | 0.928 |

$\to \phantom{\rule{4pt}{0ex}}\omega =0.1,\phantom{\rule{4pt}{0ex}}\beta =0.7$ | 0.717 | 0.857 | 0.925 |

$\mathit{\omega}=0.3$ | ||||
---|---|---|---|---|

$\mathit{\alpha}=\mathbf{0.3}$ | $\mathit{n}=\mathbf{500}$ | $\mathit{n}=\mathbf{1000}$ | $\mathit{n}=\mathbf{2000}$ | |

$\mathit{\beta}=\mathbf{0.3}$ | ||||

size | 0.037 | 0.049 | 0.059 | |

change of $\omega $ | $\to \phantom{\rule{4pt}{0ex}}\omega =1$ | 0.718 | 0.795 | 0.879 |

$\to \phantom{\rule{4pt}{0ex}}\omega =0.1$ | 0.647 | 0.84 | 0.902 | |

change of $\alpha ,\phantom{\rule{4pt}{0ex}}\beta $ | $\to \phantom{\rule{4pt}{0ex}}\alpha =0.1,\phantom{\rule{4pt}{0ex}}\beta =0.2$ | 0.805 | 0.886 | 0.913 |

$\to \phantom{\rule{4pt}{0ex}}\alpha =0.4,\phantom{\rule{4pt}{0ex}}\beta =0.5$ | 0.735 | 0.838 | 0.897 | |

change of mixed parameters | $\to \phantom{\rule{4pt}{0ex}}\omega =0.7,\phantom{\rule{4pt}{0ex}}\alpha =0.1$ | 0.907 | 0.97 | 0.994 |

$\to \phantom{\rule{4pt}{0ex}}\omega =0.1,\phantom{\rule{4pt}{0ex}}\beta =0.1$ | 0.499 | 0.674 | 0.772 |

$\mathit{\omega}=0.3$ | ||||
---|---|---|---|---|

$\mathit{\alpha}=\mathbf{0.3}$ | $\mathit{n}=\mathbf{500}$ | $\mathit{n}=\mathbf{1000}$ | $\mathit{n}=\mathbf{2000}$ | |

$\mathit{\beta}=\mathbf{0.3}$ | ||||

size | 0.047 | 0.039 | 0.037 | |

change of $\omega $ | $\to \phantom{\rule{4pt}{0ex}}\omega =1$ | 0.906 | 0.984 | 0.997 |

$\to \phantom{\rule{4pt}{0ex}}\omega =0.1$ | 0.228 | 0.382 | 0.507 | |

change of $\alpha ,\phantom{\rule{4pt}{0ex}}\beta $ | $\to \phantom{\rule{4pt}{0ex}}\alpha =-0.1,\phantom{\rule{4pt}{0ex}}\beta =-0.2$ | 0.868 | 0.946 | 0.976 |

$\to \phantom{\rule{4pt}{0ex}}\alpha =-0.4,\phantom{\rule{4pt}{0ex}}\beta =0.5$ | 0.917 | 0.985 | 1 | |

change of mixed parameters | $\to \phantom{\rule{4pt}{0ex}}\omega =0.7,\phantom{\rule{4pt}{0ex}}\beta =0.5$ | 0.82 | 0.973 | 0.998 |

$\to \phantom{\rule{4pt}{0ex}}\omega =0.1,\phantom{\rule{4pt}{0ex}}\alpha =-0.1$ | 0.862 | 0.944 | 0.971 |

$\mathit{\omega}=\mathbf{0.3}$ | ||||

$\mathit{\alpha}=\mathbf{0.3}$ | $\mathit{n}=\mathbf{500}$ | $\mathit{n}=\mathbf{1000}$ | $\mathit{n}=\mathbf{2000}$ | |

$\mathit{\beta}=\mathbf{0.3}$ | ||||

GARCH → log-GARCH | $\to \phantom{\rule{4pt}{0ex}}\omega =1$ | 0.879 | 0.946 | 0.956 |

$\to \phantom{\rule{4pt}{0ex}}\alpha =0.1,\phantom{\rule{4pt}{0ex}}\beta =0.1$ | 0.668 | 0.918 | 0.969 | |

log-GARCH → GARCH | $\to \phantom{\rule{4pt}{0ex}}\omega =1$ | 0.907 | 0.97 | 0.994 |

$\to \phantom{\rule{4pt}{0ex}}\alpha =0.1,\phantom{\rule{4pt}{0ex}}\beta =0.1$ | 0.499 | 0.674 | 0.772 | |

TGARCH → AGARCH | $\to \phantom{\rule{4pt}{0ex}}\omega =1,\phantom{\rule{4pt}{0ex}}b=1$ | 0.728 | 0.795 | 0.879 |

$\to \phantom{\rule{4pt}{0ex}}\alpha =0.1,\phantom{\rule{4pt}{0ex}}\beta =0.1,\phantom{\rule{4pt}{0ex}}b=1$ | 0.851 | 0.891 | 0.919 | |

$\mathit{\omega}=\mathbf{0.3}$ | ||||

$\mathit{\alpha}=\mathbf{0.3},\phantom{\rule{4pt}{0ex}}\mathit{\beta}=\mathbf{0.3}$ | $\mathit{n}=\mathbf{500}$ | $\mathit{n}=\mathbf{1000}$ | $\mathit{n}=\mathbf{2000}$ | |

$\mathit{b}=\mathbf{1}$ | ||||

AGARCH → GJR-GARCH | $\to \phantom{\rule{4pt}{0ex}}\omega =0.7,\phantom{\rule{4pt}{0ex}}{\alpha}_{1}=0.3,\phantom{\rule{4pt}{0ex}}{\alpha}_{2}=0.7,\phantom{\rule{4pt}{0ex}}b=0$ | 0.549 | 0.861 | 0.958 |

$\to \phantom{\rule{4pt}{0ex}}{\alpha}_{1}=0.1,\phantom{\rule{4pt}{0ex}}{\alpha}_{2}=0.3,\phantom{\rule{4pt}{0ex}}\beta =0.1,\phantom{\rule{4pt}{0ex}}b=0$ | 0.802 | 0.931 | 0.976 |

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## Share and Cite

**MDPI and ACS Style**

Lee, S.; Kim, C.K.; Lee, S.
Hybrid CUSUM Change Point Test for Time Series with Time-Varying Volatilities Based on Support Vector Regression. *Entropy* **2020**, *22*, 578.
https://doi.org/10.3390/e22050578

**AMA Style**

Lee S, Kim CK, Lee S.
Hybrid CUSUM Change Point Test for Time Series with Time-Varying Volatilities Based on Support Vector Regression. *Entropy*. 2020; 22(5):578.
https://doi.org/10.3390/e22050578

**Chicago/Turabian Style**

Lee, Sangyeol, Chang Kyeom Kim, and Sangjo Lee.
2020. "Hybrid CUSUM Change Point Test for Time Series with Time-Varying Volatilities Based on Support Vector Regression" *Entropy* 22, no. 5: 578.
https://doi.org/10.3390/e22050578