# Risk Model Validation: An Intraday VaR and ES Approach Using the Multiplicative Component GARCH

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- A rigorous model validation, both in terms of in-sample fit and out-sample performance for the Multiplicative Component Generalised Autoregressive Heteroskedasticity (MC-GARCH) model under five error distributions is provided. Statistical and graphical tests are conducted to validate the models.
- (2)
- One component of the MC-GARCH model is the daily variance forecast. For this purpose, the GARCH(1,1) and EGARCH(1,1) under the five error distributions are compared and the best model among the 10 GARCH models is used to forecast the daily variance.
- (3)
- The modelling and forecasting performance of the MC-GARCH model under different distributional assumptions is assessed in this study.
- (4)
- The 99% intraday VaR is forecasted and three backtesting procedures are used. This is the first study to assess the VaR predictive ability of the MC-GARCH models by using an asymmetric VaR loss function.
- (5)
- This is the first study to forecast the intraday expected shortfall under different distributional assumptions for the MC-GARCH model. Again, three backtests are used including the recently proposed ES regression backtest of Bayer and Dimitriadis (2018).

## 2. Past Studies on MC-GARCH Model

## 3. Methodology

#### 3.1. Model Specification

#### 3.1.1. Models for the Daily Variance Component

**GARCH(1,1)**

**EGARCH(1,1) Model**

#### 3.1.2. Model for Intraday Returns

**MC-GARCH(1,1) Model**

- ${h}_{t}$ denotes the daily variance component
- ${s}_{i}$ denotes the diurnal/calendar variance component in each intraday period
- ${q}_{t,i}$ denotes the intraday variance component
- ${\epsilon}_{t,i}$ is an error term following a specified distribution

#### 3.2. Parameter Estimation

#### 3.3. Value-at-Risk and Expected Shortfall Evaluation

- ${z}_{t+1}=\frac{{X}_{t+1}-{\mu}_{t+1}}{{\sigma}_{t+1}}$
- ${F}^{-1}\left(\alpha \right)=\frac{Va{R}_{t+1}^{\alpha}-{\mu}_{t+1}}{{\sigma}_{t+1}}$

#### 3.4. Backtesting

#### 3.4.1. Value-at-Risk Backtesting

**Kupiec’s Unconditional Coverage Test**

**A Duration-Based Approach to VaR Backtesting**

**Asymmetric VaR Loss Function**

**Model Confidence Set Procedure**

#### 3.4.2. Expected Shortfall Backtesting

**The Bivariate ES Regression Backtest**

**Exceedance Residual (ER) Backtest**

**V-Test for the Expected Shortfall**

## 4. Estimation Results

#### 4.1. Data Description

#### 4.2. Heteroskedasticity and Normality Tests of the Return Series

#### 4.3. Identifying the Conditional Mean Equation

#### 4.4. Model Checking for the Mean Equation

#### 4.5. Estimation of Daily Variance Forecast

#### 4.6. Fitting Performance

**Model Validation: In-Sample Fit:**

#### 4.7. Intraday VaR Forecast

#### 4.7.1. Kupiec’s Test

#### 4.7.2. VaR Duration Test

#### 4.7.3. Backtesting VaR Using an Asymmetric Loss Function

#### 4.8. Intraday ES Forecast

#### 4.8.1. A Regression-Based ES Backtesting Procedure: the Bivariate ES Regression Backtest

#### 4.8.2. Exceedance Residual (ER) Backtest

#### 4.8.3. V-Tests

## 5. Conclusions

#### 5.1. Recommendations for Practitioners

#### 5.2. Further Studies

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

1-min Returns | Daily Returns | |
---|---|---|

Mean | 1.23 × 10^{−7} | 4.62 × 10^{−5} |

Standard deviation | 0.00017 | 0.00633 |

Maximum | 0.00193 | 0.02781 |

Minimum | −0.00332 | −0.03733 |

Skewness | −0.38839 | −0.08208 |

Kurtosis | 18.31108 | 4.90965 |

Observations | 28,289 | 3159 |

Test Statistic | p-Value | Decision | |
---|---|---|---|

1-min returns | 277,030 | 0 | Reject ${H}_{0}$ |

Daily returns | 483.55 | 0 | Reject ${H}_{0}$ |

Test Statistic | Lag Order | p-Value | |
---|---|---|---|

1-min returns | −30.596 | 30 | 0.01 |

Daily returns | −14.03 | 14 | 0.01 |

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**Figure 2.**Correlogram of the absolute returns for the 1-min EUR/USD returns. ACF: Auto Correlation Function.

**Figure 3.**ACF and PACF plots for intraday and daily return series. PACF: Partial Auto Correlation Function.

**Figure 6.**Diurnal component, daily volatility component, intradaily volatility component, and total composite volatility.

GARCH(1,1) | |||||
---|---|---|---|---|---|

Normal | Student’s-t | Skewed Student’s-t | JSU | GED | |

AIC | −7.4566 | −7.4665 | −7.4659 | −7.4662 | −7.4704 |

BIC | −7.449 | −7.4569 | −7.4543 | −7.4547 | −7.4608 |

Log-likelihood | 11,781.8 | 11,798.33 | 11,798.32 | 11,798.9 | 11,804.5 |

EGARCH(1,1) | |||||
---|---|---|---|---|---|

Normal | Student’s-t | Skewed Student’s-t | JSU | GED | |

AIC | −7.4602 | −7.4695 | −7.4689 | −7.4693 | −7.4734 |

BIC | −7.4506 | −7.458 | −7.4555 | −7.4559 | −7.4619 |

Log-likelihood | 11,788.4 | 11,804.14 | 11,804.15 | 11,804.8 | 11,810.2 |

MC-GARCH(1,1) | |||||
---|---|---|---|---|---|

Normal | Student’s-t | Skewed Student’s-t | JSU | GED | |

$\mu $ | 0 (0.62316) | 0 (0.94151) | 0 (0.53047) | 0 (0.59145) | 0 (0.98539) |

$\omega $ | 0.011999 (0) | 0.008613 (0) | 0.008651 (0) | 0.008727 (0) | 0.009911 (0) |

${\alpha}_{1}$ | 0.037484 (0) | 0.043874 (0) | 0.043774 (0) | 0.043762 (0) | 0.041275 (0) |

${\beta}_{1}$ | 0.950441 (0) | 0.949255 (0) | 0.949335 (0) | 0.949197 (0) | 0.949529 (0) |

shape, $\mathsf{\nu}$ | - | 6.893944 (0) | 6.894106 (0) | 1.878735 (0) | 1.340094 (0) |

skewness | - | - | 1.012434 (0) | 0.037765 (0) | - |

MC-GARCH(1,1) | |||||
---|---|---|---|---|---|

Normal | Student’s-t | Skewed Student’s-t | JSU | GED | |

AIC | −15.021 | −15.046 | −15.046 | −15.048 | −15.057 |

BIC | −15.019 | −15.045 | −15.044 | −15.046 | −15.055 |

Log-Likelihood | 212,463 | 212,826 | 212,827.4 | 212,849.3 | 212,976.5 |

Rank | 5 | 4 | 3 | 2 | 1 |

Normal | Student’s-t | Skewed Student’s-t | JSU | GED | |
---|---|---|---|---|---|

Expected VaR Exceedances | 15 | 15 | 15 | 15 | 15 |

Actual VaR Exceedances | 27 | 21 | 22 | 21 | 20 |

Actual % | 1.80% | 1.40% | 1.50% | 1.40% | 1.30% |

p-value | 0.005 | 0.142 | 0.089 | 0.142 | 0.217 |

Model | b | p-Value |
---|---|---|

MC-GARCH_norm | 0.877439 | 0.397975 |

MC-GARCH_std | 0.85151 | 0.392917 |

MC-GARCH_sstd | 0.85151 | 0.392917 |

MC-GARCH_jsu | 0.85151 | 0.392917 |

MC-GARCH_ged | 0.85151 | 0.392917 |

Superior Set of Model | ||
---|---|---|

Model | Rank | Loss (× 10^{−6}) |

MC-GARCH_std | 2 | 4.61995 |

MC-GARCH_sstd | 1 | 4.615442 |

MC-GARCH_jsu | 4 | 4.744222 |

MC-GARCH_ged | 3 | 4.639826 |

Model | p-Value | Boot p-Value |
---|---|---|

MC-GARCH_std | 0.806 | 0.580 |

MC-GARCH_sstd | 0.763 | 0.527 |

MC-GARCH_jsu | 0.755 | 0.492 |

MC-GARCH_ged | 0.868 | 0.664 |

Model | Expected Exceedances | Actual Exceedances | p-Value |
---|---|---|---|

MC-GARCH_std | 15 | 21 | 0.1845 |

MC-GARCH_sstd | 15 | 22 | 0.1322 |

MC-GARCH_jsu | 15 | 21 | 0.1302 |

MC-GARCH_ged | 15 | 20 | 0.1077 |

Model | ${\mathit{V}}_{1}$ | ${\mathit{V}}_{2}$ | V |
---|---|---|---|

MC-GARCH_std | 0.0004419 | 0.0015778 | 0.0010099 |

MC-GARCH_sstd | 0.0004391 | 0.0015690 | 0.0010041 |

MC-GARCH_jsu | 0.0004383 | 0.0015667 | 0.0010025 |

MC-GARCH_ged | 0.0004243 | 0.0015206 | 0.0009724 |

© 2019 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/).

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**MDPI and ACS Style**

Summinga-Sonagadu, R.; Narsoo, J.
Risk Model Validation: An Intraday VaR and ES Approach Using the Multiplicative Component GARCH. *Risks* **2019**, *7*, 10.
https://doi.org/10.3390/risks7010010

**AMA Style**

Summinga-Sonagadu R, Narsoo J.
Risk Model Validation: An Intraday VaR and ES Approach Using the Multiplicative Component GARCH. *Risks*. 2019; 7(1):10.
https://doi.org/10.3390/risks7010010

**Chicago/Turabian Style**

Summinga-Sonagadu, Ravi, and Jason Narsoo.
2019. "Risk Model Validation: An Intraday VaR and ES Approach Using the Multiplicative Component GARCH" *Risks* 7, no. 1: 10.
https://doi.org/10.3390/risks7010010