## 1 Introduction

^{1}Secondly, Hansen and Huang (2016) use a measurement equation where the log of the realized measure of volatility is a function of the conditional volatility of return in the same period. In our model, we use a HEAVY-RM structure where the realized measure of volatility is a lagged function of realized measure. Therefore, the realized EGARCH model is used to estimate/predict the conditional variance of return, while the EHEAVY model can also model and predict realized variance as in HEAVY-type models. Thirdly, we derive a joint quasi-maximum likelihood estimation approach for the EHEAVY model and a closed-form multi-step ahead forecasting procedure.

## 2 The exponential HEAVY models

^{2}for the conditional mean of the squared returns (or the realized measures).

^{3}Their representation is more like an EGARCH-X model, where both \(\alpha _{rr}|e_{rt-1}|\) and \(\alpha _{rR}|e_{Rt-1}|\) are included. Consistent with the evidence in the HEAVY literature, we find that the estimated \(\alpha _{rr}\) is very small or insignificant,

^{4}which implies that the informative absolute (or squared) return about future volatility is small. So \(\alpha _{rr}|e_{rt-1}|\) is excluded in the EHEAVY model. The EHEAVY-r equation has the same number of parameters as the EGARCH model.

## 3 Representation and forecasting

### 3.1 Vector multiplicative error representation

### 3.2 Multiple-step ahead forecasting

^{5}

## 4 Estimation

^{6}Therefore, it is currently beyond the scope of this article to fully derive the asymptotic theory for the estimators. However, it is worth mentioning that Hansen and Huang (2016) did provide the asymptotic distribution of the QML estimator for the realized EGARCH model, but they did not establish the conditions under which this distribution holds. As a result, the asymptotic distribution cannot be verified.

\(T=2000\) | \(T=5000\) | |||||||
---|---|---|---|---|---|---|---|---|

True | RB | RMSE | N.test | True | RB | RMSE | N.test | |

\(w_r\) | \(-\)0.30 | \(-\)1.347 | 15.265 | 0.000 | \(-\)0.30 | \(-\)0.491 | 1.490 | 0.990 |

\(w_R\) | \(-\)0.30 | \(-\)0.878 | 10.842 | 0.000 | \(-\)0.30 | \(-\)0.166 | 1.704 | 0.882 |

\(\alpha _{rR}\) | 0.30 | 0.236 | 6.055 | 0.000 | 0.30 | 0.127 | 1.648 | 0.535 |

\(\alpha _{RR}\) | 0.40 | 0.513 | 6.681 | 0.000 | 0.40 | \(-\)0.271 | 2.282 | 0.968 |

\(\beta _r\) | 0.96 | \(-\)0.392 | 6.252 | 0.000 | 0.96 | \(-\)0.041 | 0.408 | 0.013 |

\(\beta _R\) | 0.95 | \(-\)0.414 | 6.259 | 0.000 | 0.95 | \(-\)0.054 | 0.572 | 0.014 |

\(\gamma _{rr}\) | \(-\)0.10 | 1.317 | 2.706 | 0.098 | \(-\)0.10 | 0.650 | 1.111 | 0.032 |

\(\gamma _{Rr}\) | \(-\)0.10 | 0.631 | 3.451 | 0.278 | \(-\)0.10 | \(-\)0.078 | 1.770 | 0.798 |

## 5 Empirical application

### 5.1 Data

Symbol | \(r_{t,oc}^2\) | \(r_{t,cc}^2\) | \(rk_t\) | |||
---|---|---|---|---|---|---|

Mean | StDev | Mean | StDev | Mean | StDev | |

AEX | 19.14 | 58.66 | 30.20 | 90.48 | 18.94 | 41.00 |

AORD | 11.37 | 33.18 | 14.18 | 40.78 | 11.43 | 33.62 |

BFX | 16.91 | 53.84 | 24.97 | 82.66 | 15.56 | 33.11 |

BSESN | 26.52 | 73.62 | 34.11 | 113.67 | 21.74 | 55.93 |

BVLG | 10.33 | 27.44 | 20.37 | 63.63 | 9.61 | 18.94 |

BVSP | 32.46 | 84.07 | 52.22 | 154.24 | 30.25 | 59.08 |

DJI | 19.23 | 62.21 | 22.57 | 87.40 | 17.04 | 40.12 |

FCHI | 20.99 | 55.16 | 31.85 | 89.56 | 21.01 | 40.85 |

FTMIB | 21.79 | 66.42 | 39.34 | 140.98 | 18.84 | 28.39 |

FTSE | 21.33 | 64.90 | 21.73 | 65.81 | 20.27 | 51.21 |

GDAXI | 24.64 | 66.65 | 33.99 | 98.28 | 24.44 | 48.37 |

GSPTSE | 15.27 | 59.91 | 17.63 | 76.25 | 12.04 | 42.55 |

HSI | 17.00 | 60.45 | 33.57 | 104.67 | 15.53 | 32.19 |

IBEX | 23.02 | 65.02 | 33.73 | 101.83 | 23.17 | 41.90 |

IXIC | 27.46 | 84.76 | 40.05 | 118.11 | 23.86 | 57.06 |

KS11 | 20.55 | 62.85 | 34.64 | 106.81 | 17.60 | 33.17 |

KSE | 24.59 | 58.84 | 29.44 | 73.34 | 19.17 | 33.58 |

MXX | 23.59 | 64.79 | 25.83 | 70.04 | 17.69 | 32.61 |

N225 | 19.72 | 71.59 | 35.04 | 100.69 | 17.60 | 49.25 |

NSEI | 22.07 | 70.98 | 33.46 | 115.24 | 18.65 | 52.81 |

OMXC20 | 21.14 | 63.40 | 26.12 | 74.09 | 19.12 | 48.68 |

OMXHPI | 21.90 | 61.17 | 28.30 | 77.33 | 20.08 | 44.57 |

OMXSPI | 21.60 | 67.17 | 27.65 | 85.35 | 19.03 | 53.95 |

OSEAX | 27.16 | 77.85 | 30.13 | 89.10 | 21.63 | 60.62 |

RUT | 23.10 | 68.98 | 38.58 | 117.01 | 18.65 | 45.35 |

SMSI | 22.77 | 69.16 | 32.52 | 107.83 | 21.71 | 47.22 |

SPX | 19.90 | 64.16 | 24.46 | 86.94 | 16.68 | 40.08 |

SSEC | 30.76 | 74.33 | 37.52 | 99.60 | 26.41 | 43.12 |

SSMI | 14.76 | 52.42 | 21.43 | 68.79 | 13.74 | 35.12 |

STI | 10.50 | 24.38 | 18.40 | 78.33 | 9.05 | 15.51 |

STOXX50E | 26.55 | 75.92 | 32.56 | 90.84 | 25.59 | 57.26 |

Average | 21.23 | 62.72 | 29.89 | 92.57 | 18.91 | 42.49 |

### 5.2 Estimation results

Open-to-close return | Close-to-close return | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Min | Med | Max | No sig | Min | Med | Max | No sig | |||

Penal A: EGARCH model estimation results | ||||||||||

\(w_r\) | \(-\)0.111 | \(-\)0.059 | \(-\)0.022 | 28 | \(-\)0.087 | \(-\)0.043 | 0.005 | 24 | ||

\(\alpha _{rr}\) | 0.118 | 0.160 | 0.227 | 31 | 0.103 | 0.158 | 0.214 | 31 | ||

\(\beta _{r}\) | 0.960 | 0.978 | 0.988 | 31 | 0.956 | 0.976 | 0.985 | 31 | ||

\(\gamma _{rr}\) | \(-\)0.143 | \(-\)0.09 | \(-\)0.023 | 31 | \(-\)0.152 | \(-\)0.106 | \(-\)0.025 | 31 | ||

LL | \(-\)24,723 | \(-\)21,025 | \(-\)8199 | 1 | \(-\)22,191 | \(-\)18,853 | \(-\)6740 | 0 | ||

BIC | 16430 | 41983 | 49480 | 1 | 13510 | 37739 | 44416 | 0 | ||

Penal B: EGARCH-X model estimation results | ||||||||||

\(w_r\) | \(-\)0.414 | \(-\)0.273 | \(-\)0.115 | 30 | \(-\)0.468 | \(-\)0.316 | \(-\)0.038 | 30 | ||

\(\alpha _{rr}\) | \(-\)0.032 | 0.021 | 0.142 | 13 | \(-\)0.013 | 0.044 | 0.205 | 23 | ||

\(\alpha _{rR}\) | 0.221 | 0.366 | 0.602 | 31 | 0.051 | 0.195 | 0.619 | 30 | ||

\(\beta _{r}\) | 0.919 | 0.965 | 0.980 | 31 | 0.939 | 0.949 | 0.998 | 31 | ||

\(\gamma _{rr}\) | \(-\)0.158 | \(-\)0.106 | 0.001 | 28 | \(-\)0.169 | \(-\)0.138 | \(-\)0.013 | 29 | ||

\(\gamma _{rR}\) | \(-\)0.072 | 0.009 | 0.059 | 4 | \(-\)0.068 | \(-\)0.028 | 0.087 | 2 | ||

LL | \(-\)24,663 | \(-\)20,894 | \(-\)8130 | 19 | \(-\)22,078 | \(-\)18,673 | \(-\)6631 | 20 | ||

BIC | 16,307 | 41,840 | 49,378 | 8 | 13,308 | 37,397 | 44,207 | 6 | ||

Penal C: EHEAVY model estimation results | ||||||||||

\(w_r\) | \(-\)0.432 | \(-\)0.282 | \(-\)0.158 | 31 | \(-\)0.478 | \(-\)0.268 | \(-\)0.104 | 31 | ||

\(\alpha _{rR}\) | 0.275 | 0.406 | 0.588 | 31 | 0.277 | 0.394 | 0.675 | 31 | ||

\(\beta _{r}\) | 0.921 | 0.966 | 0.983 | 31 | 0.935 | 0.967 | 0.990 | 30 | ||

\(\gamma _{rr}\) | \(-\)0.150 | \(-\)0.093 | \(-\)0.023 | 30 | \(-\)0.154 | \(-\)0.108 | \(-\)0.052 | 31 | ||

\(w_R\) | \(-\)0.504 | \(-\)0.327 | \(-\)0.204 | 31 | \(-\)0.523 | \(-\)0.305 | \(-\)0.186 | 31 | ||

\(\alpha _{RR}\) | 0.301 | 0.448 | 0.696 | 31 | 0.298 | 0.430 | 0.685 | 31 | ||

\(\beta _{R}\) | 0.887 | 0.962 | 0.981 | 31 | 0.919 | 0.962 | 0.989 | 31 | ||

\(\gamma _{Rr}\) | \(-\)0.149 | \(-\)0.093 | \(-\)0.026 | 30 | \(-\)0.146 | \(-\)0.105 | \(-\)0.033 | 31 | ||

\(\rho\) | 0.818 | 0.845 | 0.890 | 31 | 0.773 | 0.815 | 0.886 | 31 | ||

LL | \(-\)24,674 | \(-\)20,901 | \(-\)8136 | 11 | \(-\)22,071 | \(-\)18,679 | \(-\)6636 | 10 | ||

BIC | 16,303 | 41,832 | 49,372 | 21 | 13,302 | 37,393 | 44,176 | 24 |

### 5.3 News impact curve

### 5.4 Forecasting comparison

Panel A: Ratio of the losses | |||||||
---|---|---|---|---|---|---|---|

s=1 | s=5 | s=22 | |||||

(Average) | MSE | QLIK | MSE | QLIK | MSE | QLIK | |

HEAVY | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | |

AHEAVY | 0.9651 | 0.8811 | 0.9916 | 0.9813 | 1.0003 | 0.9857 | |

REGARCH | 0.9440 | 0.8249 | 0.9635 | 0.9288 | 0.9620 | 0.9182 | |

EHEAVY | 0.9486 | 0.8266 | 0.9660 | 0.9359 | 0.9630 | 0.9118 |

Panel B: MCS test | ||||||
---|---|---|---|---|---|---|

s=1 | s=5 | s=22 | ||||

MSC | MSE | QLIK | MSE | QLIK | MSE | QLIK |

HEAVY | 31 | 2 | 30 | 10 | 28 | 5 |

AHEAVY | 30 | 8 | 31 | 13 | 26 | 11 |

REGARCH | 31 | 29 | 31 | 30 | 31 | 25 |

EHEAVY | 31 | 28 | 31 | 30 | 31 | 27 |

### 5.5 Portfolio exercise

^{7}

^{8}

Panel A: Ratio of portfolio performances | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

s=1 | s=5 | s=22 | ||||||||||||||

(Average) | R | SR | CER | EU | R | SR | CER | EU | R | SR | CER | EU | ||||

HEAVY | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | ||||

AHEAVY | 0.997 | 0.981 | 1.054 | 1.044 | 1.021 | 1.024 | 1.181 | 1.049 | 1.094 | 1.012 | 1.152 | 1.020 | ||||

REGARCH | 1.011 | 1.202 | 1.554 | 1.673 | 1.094 | 1.120 | 1.276 | 1.221 | 1.155 | 1.029 | 1.220 | 1.167 | ||||

EHEAVY | 1.081 | 1.579 | 2.052 | 1.773 | 1.189 | 1.441 | 1.652 | 1.315 | 1.291 | 1.045 | 1.458 | 1.179 |

Panel B: MCS test | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

s=1 | s=5 | s=22 | ||||||||||

R | SR | CER | EU | R | SR | CER | EU | R | SR | CER | EU | |

HEAVY | 24 | 0 | 5 | 23 | 1 | 8 | 25 | 2 | 20 | |||

AHEAVY | 23 | 0 | 5 | 29 | 1 | 9 | 27 | 5 | 22 | |||

REGARCH | 25 | 20 | 28 | 30 | 16 | 29 | 30 | 13 | 30 | |||

EHEAVY | 30 | 26 | 29 | 31 | 22 | 31 | 31 | 22 | 31 |