# Improving Many Volatility Forecasts Using Cross-Sectional Volatility Clusters

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Garch-Type Specifications Incorporating Cross-Sectional Volatility Clusters

#### 2.1. The CW-GARCH Specification

#### 2.2. Cross-Sectional Cluster Model

## 3. Estimation

**Step****1:****Step****2:**

## 4. Empirical Study

#### 4.1. In-Sample Results

#### 4.2. Forecasting Experiments

- Using observation from 1 to 1500, fit a CW-GARCH, sCW-GARCH, GARCH and GJR-GARCH model to the time series of returns on asset s
- Generate one-step-ahead forecasts of conditional variance for the subsequent 50 days, that is for times 1501 to 1550
- Re-estimate the models using an updated estimation window from 51 to 1550
- Iterate steps 2-3 until the end of the series.

## 5. Conclusions and Final Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GARCH | Generalized autoregressive conditional heteroskedasticity |

GJR-GARCH | GJR model proposed by Glosten et al. (1993) |

CW-GARCH | Clusterwise generalized autoregressive conditional heteroskedasticity |

sCW-GARCH | Smooth CW-GARCH |

NYSE | New York Stock Exchange |

ML | Maximum Likelihood |

RMSPE | Root mean square prediction error |

QLIKE | Loss function proposed by Patton (2011) |

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**Figure 1.**For the data set introduced in Section 4 it is shown the kernel density estimate of the cross-sectional distribution of realized volatility in two different time points: panel (

**a**) refers to 27 August 1998; panel (

**b**) refers to 28 August 1998.

**Figure 2.**Parametric density estimate of the cross-sectional distribution for that data sets of the example in Figure 1. (

**a**) Realized volatility on 27/Aug/1998; fitted density based on model (7) with $G=3$, under the restriction (8) and that ${\pi}_{0}=0$ (that is the uniform component is inactive). (

**b**) Realized volatility on 27/Aug/1998; fitted density based on model (7) with $G=3$, under the restriction (8). (

**c**) Realized volatility on 28/Aug/1998; fitted density based on model (7) with $G=3$, under the restriction (8) and that ${\pi}_{0}=0$ (uniform component is inactive). (

**d**) Realized volatility on 28/Aug/1998; fitted density based on model (7) with $G=3$, under the restriction (8).

**Figure 3.**(

**a**) Time series of the realized variance of asset JNJ from August 2004 to August 2005. (

**b**) Sequence of fitted class labels memberships for the asset JNJ from August 2004 to August 2005. (

**c**) Time series of the realized variance of asset JNJ from March 2001 to March 2002. (

**d**) Sequence of fitted class labels memberships for the asset JNJ from March 2001 to March 2002.

**Figure 4.**(

**a**) Time series of the realized variance of asset LSI from August 2004 to August 2005. (

**b**) Sequence of fitted class labels memberships for the asset LSI from August 2004 to August 2005. (

**c**) Time series of the realized variance of asset LSI from March 2001 to March 2002. (

**d**) Sequence of fitted class labels memberships for the asset LSI from March 2001 to March 2002.

**Figure 5.**(

**a**) Time series of the realized variance of asset NOVL from August 2004 to August 2005. (

**b**) Sequence of fitted class labels memberships for the asset NOVL from August 2004 to August 2005. (

**c**) Time series of the realized variance of asset NOVL from March 2001 to March 2002. (

**d**) Sequence of fitted class labels memberships for the asset NOVL from March 2001 to March 2002.

**Figure 6.**(

**a**) Boxplots of the distribution (across the portfolio) of the ratio (BIC achieved by a given model)/(BIC achieved by the GARCH(1,1) model), where the ratio is scaled in percentage value. Each data point is a BIC-ratio for model (1) or (4) achieved by one of the $S=123$ assets in the sample. (

**b**) Boxplots of the distribution (across the portfolio) of the ratio (BIC achieved by a given model)/(BIC achieved by the GJR-GARCH(1,1) model), where the ratio is scaled in percentage value. Each data point is a BIC-ratio for model (1) or (4) achieved by one of the $S=123$ assets in the sample.

**Figure 7.**p-values of the Likelihood Ratio Test (LRT) test of CW-GARCH vs. GARCH for all the 123 assets included in our panel. The y-axis of the plot is scaled in terms of logit(p-value), y-axis labels are shown as percentage p-values.

**Figure 8.**For each loss and each method, it is shown the distribution (across the $S=123$ assets) of the percentage difference between the loss of the method and the loss of the top performer, relative to the size of the loss of the top performer. A large relative difference indicates a large performance gap from the top performer. (

**a**) RMSPE percentage relative difference to top performer. (

**b**) QLIKE percentage relative difference to top performer.

**Figure 9.**Distribution (across the $S=123$ assets) of RMSPE ($\times {10}^{3}$), that is each point in these scatters is the RMSPE ($\times {10}^{3}$) for a given asset. The dashed line is the 45-degree line. (

**a**) RMSPE for the classical GARCH(1,1) model against the RMSPE for the CW-GARCH specification. (

**b**) RMSPE for the classical GARCH(1,1) model against the RMSPE for the sCW-GARCH specification. (

**c**) RMSPE for the GJR-GARCH(1,1) model against the RMSPE for the CW-GARCH specification. (

**d**) RMSPE for the GJR-GARCH(1,1) model against the RMSPE for the sCW-GARCH specification.

**Figure 10.**Distribution (across the $S=123$ assets) of QLIKE, that is each point in these scatters is the time QLIKE for a given asset. The dashed line is the 45-degree line. (

**a**) QLIKE for the classical GARCH(1,1) model against the QLIKE for the CW-GARCH specification. (

**b**) QLIKE for the classical GARCH(1,1) model against the QLIKE for the sCW-GARCH specification. (

**c**) QLIKE for the GJR-GARCH(1,1) model against the QLIKE for the CW-GARCH specification. (

**d**) QLIKE for the GJR-GARCH(1,1) model against the QLIKE for the sCW-GARCH specification.

**Table 1.**Results of Monte Carlo simulations for CW-GARCH and sCW-GARCH models. Key to table: $\mathit{\psi}$ = “true” parameter values; $\widehat{E}\left({\widehat{\mathit{\psi}}}_{A}\right)$ = Monte Carlo average of fitted parameters for model A ($A\in \{c,s\})$ where c and s refer to CW-GARCH and sCW-GARCH models, respectively; $se\left(E\left({\widehat{\mathit{\psi}}}_{A}\right)\right)$ = simulated standard error of Monte Carlo mean of ${\widehat{\mathit{\psi}}}_{A}$; $RMSE\left({\widehat{\mathit{\psi}}}_{A}\right)$ = simulated RMSE of ${\widehat{\mathit{\psi}}}_{A}$.

${\mathit{\omega}}_{0}^{*}$ | ${\mathit{\alpha}}_{0}$ | ${\mathit{\beta}}_{0}$ | ${\mathit{\omega}}_{1}^{*}$ | ${\mathit{\alpha}}_{1}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\omega}}_{2}^{*}$ | ${\mathit{\alpha}}_{2}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\omega}}_{3}^{*}$ | ${\mathit{\alpha}}_{3}$ | ${\mathit{\beta}}_{3}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\psi}$ | 0.240 | 0.020 | 0.950 | 1.600 | 0.090 | 0.680 | 0.390 | 0.110 | 0.870 | 0.030 | 0.020 | 0.960 |

CW-GARCH | ||||||||||||

$\widehat{E}\left({\widehat{\mathit{\psi}}}_{c}\right)$ | 0.318 | 0.016 | 0.940 | 1.315 | 0.085 | 0.721 | 0.415 | 0.110 | 0.869 | 0.093 | 0.017 | 0.953 |

$se\left(\widehat{E}\left({\widehat{\mathit{\psi}}}_{c}\right)\right)$ | 0.015 | 0.001 | 0.003 | 0.015 | 0.002 | 0.003 | 0.013 | 0.001 | 0.002 | 0.007 | 0.001 | 0.001 |

$RMSE\left({\widehat{\mathit{\psi}}}_{c}\right)$ | 0.078 | 0.004 | 0.010 | 0.285 | 0.005 | 0.041 | 0.025 | 0.000 | 0.001 | 0.063 | 0.003 | 0.007 |

sCW-GARCH | ||||||||||||

$\widehat{E}\left({\widehat{\mathit{\psi}}}_{s}\right)$ | 0.323 | 0.015 | 0.941 | 1.314 | 0.081 | 0.725 | 0.438 | 0.110 | 0.864 | 0.097 | 0.016 | 0.953 |

$se\left(\widehat{E}\left({\widehat{\mathit{\psi}}}_{s}\right)\right)$ | 0.017 | 0.001 | 0.003 | 0.014 | 0.002 | 0.003 | 0.016 | 0.001 | 0.003 | 0.008 | 0.001 | 0.002 |

$RMSE\left({\widehat{\mathit{\psi}}}_{s}\right)$ | 0.083 | 0.005 | 0.009 | 0.286 | 0.009 | 0.045 | 0.048 | 0.000 | 0.006 | 0.067 | 0.004 | 0.007 |

GARCH | GJR-GARCH | CW-GARCH | sCW-GARCH | |
---|---|---|---|---|

Average BIC | −29,218.33 | −29,488.13 | −29,901.60 | −29,942.67 |

Median BIC | −29,579.29 | −29,782.38 | −29,851.66 | −29,923.45 |

Average loglik | 14,620.90 | 14,759.71 | 14,997.74 | 15,018.28 |

Median loglik | 14,801.38 | 14,906.84 | 14,972.77 | 15,008.67 |

Component | GARCH | GJR-GARCH | CW-GARCH | sCW-GARCH | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Low | Mid | High | Noise | Low | Mid | High | Noise | |||

$\mathit{\omega}$ * | 56.1997 | 31.3277 | 6.7061 | 4.8966 | 4.3016 | 619.17 | 5.1024 | 4.0491 | 5.4529 | 620.44 |

$\mathit{\alpha}$ | 0.1283 | 0.0692 | 0.1365 | 0.1030 | 0.0803 | 0.0311 | 0.1419 | 0.0984 | 0.0706 | 0.0315 |

$\mathit{\beta}$ | 0.6525 | 0.7262 | 0.8405 | 0.8310 | 0.8742 | 0.7992 | 0.8535 | 0.8423 | 0.8730 | 0.7992 |

$\gamma $ | 0.0826 |

Component | GARCH | GJR-GARCH | CW-GARCH | sCW-GARCH | ||||||
---|---|---|---|---|---|---|---|---|---|---|

low | mid | high | noise | low | mid | high | noise | |||

$\mathit{\omega}$ * | 0.1905 | 0.1785 | 0.1238 | 0.1273 | 0.0033 | 0.3170 | 0.1707 | 0.1469 | 0.0342 | 0.5252 |

$\mathit{\alpha}$ | 0.0484 | 0.0278 | 0.0755 | 0.0617 | 0.0399 | 0.0087 | 0.0741 | 0.0574 | 0.0427 | 0.0075 |

$\mathit{\beta}$ | 0.9232 | 0.9367 | 0.9269 | 0.9280 | 0.9538 | 0.9283 | 0.9268 | 0.9370 | 0.9485 | 0.9283 |

$\gamma $ | 0.0366 |

**Table 5.**Percentage frequencies (across the $S=123$ assets) of models resulting the best performer according to Root Mean Square Prediction Error (RMSPE) and QLIKE.

GARCH | GJR-GARCH | CW-GARCH | sCW-GARCH | |
---|---|---|---|---|

RMSPE | 20.34% | 24.58% | 33.05% | 22.03% |

QLIKE | 17.80% | 32.20% | 29.66% | 20.34% |

**Table 6.**Median and IQR of the distribution (across the $S=123$ assets) of the RMSPE ($\times {10}^{3}$) and the QLIKE.

GARCH | GJR-GARCH | CW-GARCH | sCW-GARCH | |
---|---|---|---|---|

Median of RMSPE | 0.44 | 0.46 | 0.27 | 0.27 |

IQR of RMSPE | 0.64 | 0.83 | 0.37 | 0.43 |

Median of QLIKE | −7.17 | −7.21 | −7.31 | −7.32 |

IQR of QLIKE | 1.13 | 1.20 | 0.89 | 0.90 |

**Table 7.**Forecasting performance of CW-GARCH and sCW-GARCH models vs. GARCH (left panel) and GJR-GARCH (right panel), respectively: signs of average MSPE differentials (relative frequencies) and Diebold-Mariano rejection frequencies.

Benchmark: GARCH | Benchmark: GJR | |||||||
---|---|---|---|---|---|---|---|---|

CW-GARCH | sCW-GARCH | CW-GARCH | sCW-GARCH | |||||

$\%L$ | $\%W$ | $\%L$ | $\%W$ | $\%L$ | $\%W$ | $\%L$ | $\%W$ | |

$\underset{\left(80.0\right)}{28.5}$ | $\underset{\left(81.8\right)}{71.5}$ | $\underset{\left(63.6\right)}{26.8}$ | $\underset{\left(86.7\right)}{73.2}$ | $\underset{\left(73.2\right)}{33.3}$ | $\underset{\left(85.4\right)}{66.7}$ | $\underset{\left(59.5\right)}{30.1}$ | $\underset{\left(82.6\right)}{69.9}$ |

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## Share and Cite

**MDPI and ACS Style**

Coretto, P.; La Rocca, M.; Storti, G.
Improving Many Volatility Forecasts Using Cross-Sectional Volatility Clusters. *J. Risk Financial Manag.* **2020**, *13*, 64.
https://doi.org/10.3390/jrfm13040064

**AMA Style**

Coretto P, La Rocca M, Storti G.
Improving Many Volatility Forecasts Using Cross-Sectional Volatility Clusters. *Journal of Risk and Financial Management*. 2020; 13(4):64.
https://doi.org/10.3390/jrfm13040064

**Chicago/Turabian Style**

Coretto, Pietro, Michele La Rocca, and Giuseppe Storti.
2020. "Improving Many Volatility Forecasts Using Cross-Sectional Volatility Clusters" *Journal of Risk and Financial Management* 13, no. 4: 64.
https://doi.org/10.3390/jrfm13040064