Do We Need Stochastic Volatility and Generalised Autoregressive Conditional Heteroscedasticity? Comparing Squared End-Of-Day Returns on FTSE
Abstract
: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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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 observations for RV5 variable (2454 valid observations) | |||
Mean | Median | Minimum | Maximum |
8.6261 × 10 | 5.1600 × 10 | 1.3300 × 10 | 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 | 0.00025567 | 6.8100 × 10 | 0 |
Summary of 1000 Markov chain Monte Carlo (MCMC) draws after burn-in of 1000 | |||||
---|---|---|---|---|---|
Prior distributions | |||||
mean = 0 | S.D. = 100 | ||||
Posterior draws thinning = 1 | |||||
Mean | S.D. | 5% | 50% | 95% | |
−9.5938 | 0.11914 | −9.7831 | −9.5944 | −9.3963 | |
0.9577 | 0.01051 | 0.9389 | 0.9586 | 0.9734 | |
0.2342 | 0.02868 | 0.1915 | 0.2321 | 0.2851 | |
0.0083 | 0.00049 | 0.0075 | 0.0083 | 0.0091 | |
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 observations for 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 observations for 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 |
Coefficients | Standard Error | T Statistic | |
---|---|---|---|
mu | 1.360 × 10 | 1.601 × 10 | 0.850 |
omega | 3.406 × 10 | 8.087 × 10 | 4.211 *** |
alpha1 | 1.199 × 10 | 1.728 × 10 | 6.939 *** |
beta1 | 8.443 × 10 | 2.240 × 10 | 37.688 *** |
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 | |
0.194171 | Adjusted | 0.193842 | ||
590.8287 | p-value(F) | 4.1 × 10 | ||
Log-likelihood | −1182.037 | Akaike criterion | 2368.075 | |
Schwarz criterion | 2379.686 | Hannan–Quinn | 2372.294 | |
0.503365 | Durbin–Watson | 0.991133 | ||
OLS using 27 April 2009–16 April 2019 observations ( = 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 | |
0.203114 | Adjusted | 0.202789 | ||
624.9784 | p-value(F) | 4.6 × 10 | ||
Log-likelihood | −1168.344 | Akaike criterion | 2340.687 | |
Schwarz criterion | 2352.298 | Hannan–Quinn | 2344.906 | |
0.490865 | Durbin–Watson | 1.015052 | ||
OLS using 4 June 2009–16 April 2019 observations ( = 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 | |
0.310989 | Adjusted | 0.302940 | ||
38.63918 | p-value(F) | 6.2 × 10 | ||
Log-likelihood | −954.8254 | Akaike criterion | 1967.651 | |
Schwarz criterion | 2135.677 | Hannan–Quinn | 2028.745 | |
0.429918 | Durbin–Watson | 1.139917 |
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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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
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 StyleAllen, 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