# Do We Need Stochastic Volatility and Generalised Autoregressive Conditional Heteroscedasticity? Comparing Squared End-Of-Day Returns on FTSE

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## Abstract

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## 1. Introduction

## 2. Previous Work and Econometric Models

#### 2.1. Stochastic Volatility

#### 2.2. ARCH and GARCH

#### 2.3. Realised Volatility

#### 2.4. Historical-Volatility Model

#### 2.5. Heterogeneous Autoregressive Model (HAR)

#### 2.6. Quantile Regression

## 3. Analysis Results

#### 3.1. Preliminary Analysis

#### 3.2. SV and GARCH Estimates

#### 3.3. Quantile-Regression Results

#### 3.4. Rolling-Regression Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 10.**Prediction plots for last 20 observations by SV and GARCH, and values of lagged DMSQFTSERET and RV5.

Summary statistics using 27 April 2009–16 April 2019 observations for FTSERET variable (2454 valid observations) | |||
---|---|---|---|

Mean | Median | Minimum | Maximum |

0.00022266 | 0.00050198 | −0.047798 | 0.050323 |

Std. Dev. | C.V. | Skewness | Ex. kurtosis |

0.0097220 | 43.663 | −0.13603 | 2.1367 |

5% | 95% | IQ Range | Missing obs. |

−0.015687 | 0.016056 | 0.010418 | 0 |

Summary Statistics, using27 April 2009–16 April 2019 observationsfor RV5 variable (2454 valid observations) | |||

Mean | Median | Minimum | Maximum |

8.6261 × 10${}^{-5}$ | 5.1600 × 10${}^{-5}$ | 1.3300 × 10${}^{-6}$ | 0.0057390 |

Std. Dev. | C.V. | Skewness | Ex. kurtosis |

0.00016097 | 1.8661 | 19.805 | 633.01 |

5% | 95% | IQ Range | Missing obs. |

1.4275 × 10${}^{-5}$ | 0.00025567 | 6.8100 × 10${}^{-5}$ | 0 |

Summary of 1000 Markov chain Monte Carlo (MCMC) draws after burn-in of 1000 | |||||
---|---|---|---|---|---|

Prior distributions | |||||

$mu\sim normal$ | mean = 0 | S.D. = 100 | |||

$(phi+1)/2\sim beta$ | ${a}_{0}=5$ | ${b}_{0}=1.5$ | |||

$sigm{a}^{2}\sim 1\ast Chisq(df=1)$ | |||||

Posterior draws thinning = 1 | |||||

Mean | S.D. | 5% | 50% | 95% | |

$mu$ | −9.5938 | 0.11914 | −9.7831 | −9.5944 | −9.3963 |

$phi$ | 0.9577 | 0.01051 | 0.9389 | 0.9586 | 0.9734 |

$sigma$ | 0.2342 | 0.02868 | 0.1915 | 0.2321 | 0.2851 |

$exp(mu/2)$ | 0.0083 | 0.00049 | 0.0075 | 0.0083 | 0.0091 |

$sigm{a}^{2}$ | 0.0557 | 0.01378 | 0.0367 | 0.0539 | 0.0813 |

Summary statistics using 27 April 2009–16 April 2019 observations for SV variable (2454 valid observations) | |||
---|---|---|---|

Mean | Median | Minimum | Maximum |

0.87152 | 0.80112 | 0.37146 | 2.1388 |

Std. Dev. | C.V. | Skewness | Ex. kurtosis |

0.32600 | 0.37405 | 1.2754 | 1.9519 |

5% | 95% | IQ Range | Missing obs. |

0.45723 | 1.5217 | 0.38324 | 0 |

Summary statistics using 27 April 2009–16 April 2019 observationsfor garchh variable_t (2454 valid observations) | |||

Mean | Median | Minimum | Maximum |

0.92405 | 0.84738 | 0.51430 | 2.3305 |

Std. Dev. | C.V. | Skewness | Ex. kurtosis |

0.30490 | 0.32996 | 1.6654 | 3.3669 |

5% | 95% | IQ Range | Missing obs. |

0.59248 | 1.5934 | 0.32905 | 0 |

Summary statistics using 27 April 2009–16 April 2019 observationsfor SQRV5L variable (2454 valid observations) | |||

Mean | Median | Minimum | Maximum |

0.81989 | 0.71837 | 0.11543 | 7.5756 |

Std. Dev. | C.V. | Skewness | Ex. kurtosis |

0.43644 | 0.53231 | 3.3736 | 29.966 |

5% | 95% | IQ Range | Missing obs. |

0.37737 | 1.5990 | 0.44676 | 0 |

**Table 4.**Generalised Autoregressive Conditional Heteroscedasticity (GARCH) (1,1) fitted to FTSE returns.

Coefficients | Standard Error | T Statistic | |
---|---|---|---|

mu | 1.360 × 10${}^{-4}$ | 1.601 × 10${}^{-4}$ | 0.850 |

omega | 3.406 × 10${}^{-6}$ | 8.087 × 10${}^{-7}$ | 4.211 *** |

alpha1 | 1.199 × 10${}^{-1}$ | 1.728 × 10${}^{-2}$ | 6.939 *** |

beta1 | 8.443 × 10${}^{-1}$ | 2.240 × 10${}^{-2}$ | 37.688 *** |

**Table 5.**Regression analysis of three volatility models: SV, GARCH(1,1), and lagged demeaned squared returns.

Ordinary least squares (OLS) using 27 April 2009–16 April 2019 observations (T = 2454). Dependent variable: SQRV5L. | ||||
---|---|---|---|---|

Coefficient | Std. Error | t-ratio | p-value | |

const | 0.305752 | 0.0225825 | 13.54 | 0.0000 |

SV | 0.589930 | 0.0242700 | 24.31 | 0.0000 |

Mean dependent var | 0.819888 | S.D. dependent var | 0.436435 | |

Sum squared resid | 376.5133 | S.E. of regression | 0.391859 | |

${R}^{2}$ | 0.194171 | Adjusted ${R}^{2}$ | 0.193842 | |

$F(1,2452)$ | 590.8287 | p-value(F) | 4.1 × 10${}^{-117}$ | |

Log-likelihood | −1182.037 | Akaike criterion | 2368.075 | |

Schwarz criterion | 2379.686 | Hannan–Quinn | 2372.294 | |

$\widehat{\rho}$ | 0.503365 | Durbin–Watson | 0.991133 | |

OLS using 27 April 2009–16 April 2019 observations ($\mathit{T}$ = 2454). Dependent variable: SQRV5L. | ||||

Coefficient | Std. Error | t-ratio | p-value | |

const | 0.223766 | 0.0251093 | 8.912 | 0.0000 |

garchh_t | 0.645116 | 0.0258051 | 25.00 | 0.0000 |

Mean dependent var | 0.819888 | S.D. dependent var | 0.436435 | |

Sum squared resid | 372.3346 | S.E. of regression | 0.389678 | |

${R}^{2}$ | 0.203114 | Adjusted ${R}^{2}$ | 0.202789 | |

$F(1,2452)$ | 624.9784 | p-value(F) | 4.6 × 10${}^{-123}$ | |

Log-likelihood | −1168.344 | Akaike criterion | 2340.687 | |

Schwarz criterion | 2352.298 | Hannan–Quinn | 2344.906 | |

$\widehat{\rho}$ | 0.490865 | Durbin–Watson | 1.015052 | |

OLS using 4 June 2009–16 April 2019 observations ($\mathit{T}$ = 2426). Dependent variable: SQRV5L. | ||||

Coefficient | Std. Error | t-ratio | p-value | |

const | 0.544629 | 0.0111890 | 48.68 | 0.0000 |

LSQDMFTSE_1 | 0.433934 | 0.414977 | 1.046 | 0.2958 |

LSQDMFTSE_2 | 0.515250 | 0.419102 | 1.229 | 0.2190 |

LSQDMFTSE_3 | 1.18029 | 0.420051 | 2.810 | 0.0050 |

LSQDMFTSE_4 | 0.839649 | 0.421807 | 1.991 | 0.0466 |

LSQDMFTSE_5 | 1.35951 | 0.423013 | 3.214 | 0.0013 |

LSQDMFTSE_6 | 0.102526 | 0.422970 | 0.2424 | 0.8085 |

LSQDMFTSE_7 | 0.663505 | 0.423177 | 1.568 | 0.1170 |

LSQDMFTSE_8 | 0.242477 | 0.421954 | 0.5747 | 0.5656 |

LSQDMFTSE_9 | 1.58351 | 0.423249 | 3.741 | 0.0002 |

LSQDMFTSE_10 | 0.715148 | 0.422994 | 1.691 | 0.0910 |

LSQDMFTSE_11 | 1.72472 | 0.421790 | 4.089 | 0.0000 |

LSQDMFTSE_12 | 2.43111 | 0.422496 | 5.754 | 0.0000 |

LSQDMFTSE_13 | 2.11699 | 0.422470 | 5.011 | 0.0000 |

LSQDMFTSE_14 | 0.667181 | 0.422379 | 1.580 | 0.1143 |

LSQDMFTSE_15 | 2.04957 | 0.422271 | 4.854 | 0.0000 |

LSQDMFTSE_16 | 0.438659 | 0.422071 | 1.039 | 0.2988 |

LSQDMFTSE_17 | 0.994273 | 0.421786 | 2.357 | 0.0185 |

LSQDMFTSE_18 | 0.430727 | 0.420974 | 1.023 | 0.3063 |

LSQDMFTSE_19 | 1.06033 | 0.420799 | 2.520 | 0.0118 |

LSQDMFTSE_20 | 0.0170270 | 0.421031 | 0.04044 | 0.9677 |

LSQDMFTSE_21 | 1.03750 | 0.419658 | 2.472 | 0.0135 |

LSQDMFTSE_22 | 0.303899 | 0.420813 | 0.7222 | 0.4703 |

LSQDMFTSE_23 | 0.842299 | 0.420028 | 2.005 | 0.0450 |

LSQDMFTSE_24 | 1.12694 | 0.419905 | 2.684 | 0.0073 |

LSQDMFTSE_25 | 0.834749 | 0.418637 | 1.994 | 0.0463 |

LSQDMFTSE_26 | 0.746605 | 0.416243 | 1.794 | 0.0730 |

LSQDMFTSE_27 | 1.92539 | 0.415125 | 4.638 | 0.0000 |

LSQDMFTSE_28 | 2.02831 | 0.411577 | 4.928 | 0.0000 |

Mean dependent var | 0.812434 | S.D. dependent var | 0.432184 | |

Sum squared resid | 312.0871 | S.E. of regression | 0.360831 | |

${R}^{2}$ | 0.310989 | Adjusted ${R}^{2}$ | 0.302940 | |

$F(28,2397)$ | 38.63918 | p-value(F) | 6.2 × 10${}^{-171}$ | |

Log-likelihood | −954.8254 | Akaike criterion | 1967.651 | |

Schwarz criterion | 2135.677 | Hannan–Quinn | 2028.745 | |

$\widehat{\rho}$ | 0.429918 | Durbin–Watson | 1.139917 |

**Table 6.**Quantile-regression results of SQRV5L regression on lagged SV, lagged GARCH and lagged DMSQFTSERET Quantile estimates, using 27 April 2009–16 April 2019 observations (T = 2454). Asymptotic standard errors assuming independent and identically distributed (IID) errors.

Variable | Tau | Coefficient | Std. Error | t-Ratio |
---|---|---|---|---|

SV(-1) | 0.05 | 0.00179384 | 0.000194807 | 9.20827 *** |

SV(-1) | 0.25 | 0.00386922 | 0.000179883 | 21.5097 *** |

SV(-1) | 0.50 | 0.00558958 | 0.000199581 | 28.0066 *** |

SV(-1) | 0.75 | 0.00746445 | 0.000279415 | 26.7146 *** |

SV(-1) | 0.95 | 0.0117519 | 0.000798124 | 14.7245 *** |

GARCHh_t(-1) | 0.05 | 0.00159755 | 0.000195806 | 8.15884 *** |

GARCHh_t(-1) | 0.25 | 0.00420579 | 0.000191819 | 21.9259 *** |

GARCHh_t(-1) | 0.50 | 0.00627205 | 0.000204113 | 30.7283 *** |

GARCHh_t(-1) | 0.75 | 0.00873417 | 0.000334193 | 26.1351 *** |

GARCHh_t(-1) | 0.95 | 0.0135944 | 0.000860269 | 15.8025 *** |

DMSQFTSERET(-1) | 0.05 | 0.0132912 | 0.00310350 | 4.28266 *** |

DMSQFTSERET(-1) | 0.25 | 0.0322416 | 0.00350232 | 9.20578 *** |

DMSQFTSERET(-1) | 0.50 | 0.0402209 | 0.00311836 | 12.8981 *** |

DMSQFTSERET(-1) | 0.75 | 0.0522737 | 0.00566966 | 9.21991 *** |

DMSQFTSERET(-1) | 0.95 | 0.0737864 | 0.0140149 | 5.26486 *** |

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## Share and Cite

**MDPI and ACS Style**

Allen, D.E.; McAleer, M.
Do We Need Stochastic Volatility and Generalised Autoregressive Conditional Heteroscedasticity? Comparing Squared End-Of-Day Returns on FTSE. *Risks* **2020**, *8*, 12.
https://doi.org/10.3390/risks8010012

**AMA Style**

Allen DE, McAleer M.
Do We Need Stochastic Volatility and Generalised Autoregressive Conditional Heteroscedasticity? Comparing Squared End-Of-Day Returns on FTSE. *Risks*. 2020; 8(1):12.
https://doi.org/10.3390/risks8010012

**Chicago/Turabian Style**

Allen, David E., and Michael McAleer.
2020. "Do We Need Stochastic Volatility and Generalised Autoregressive Conditional Heteroscedasticity? Comparing Squared End-Of-Day Returns on FTSE" *Risks* 8, no. 1: 12.
https://doi.org/10.3390/risks8010012