Information-Theoretic Measures and Modeling Stock Market Volatility: A Comparative Approach
Abstract
:1. Introduction
2. Methodology
2.1. Volatility
2.2. GARCH Models
- The process is called GARCH (p, q) process if and ;
- The random variables are identical and independent, is the residual series and its conditional variance;
- , are real parameters and ensures that at all times.
- where and. and represent sign effect and leverage effect;
- For leverage effect must be statistically significant and negative;
- Returns are stationary if and EGARCH captures serial dependence and leverage effects in returns;
- If is small, then decreases. For large , the value of increases;
- Due to log-transformation of variance, it guarantees positivity of variance without any restriction on parameters.
- where andindicates asymmetry of returns;
- assumes value equals to 1 for (negative-shock), and zero otherwise;
- For positive and significant , leverage effect exists.
- where are real parameters;
- The variable is strictly positive and denotes the conditional standard deviation of ;
- The current volatility depends on both the modulus and the sign of the past returns through and ;
- For 0, the effect of the bad news is greater than those of the good news;
- The GJR-GARCH model due to Glosten et al. (1993) is a version of TGARCH, which corresponds to squaring the variables involved in Equation (8).
Conditional Distributions and GARCH Modeling
- whereand are location, scale, and shape parameters, respectively, and is a Gamma function.
- where are location and scale parameters, respectively, and represents the tail-thickness parameter;
- For the distribution converges to the standard normal distribution, and for , it has thicker tails than the normal distribution.
- where denoted skewness and shape parameters, respectively. For negative skewness , and for positive skewness . The parameter ;
- For the skewed-GED distribution converges to the GED;
- The sign function equals to −1 for negative values of its argument and equals to 1 for positive values;
- The values of , and
- where denotes the asymmetry coefficient, are mean and the standard deviation, respectively.
- The density is skewed to the right if and skewed to the left if .
2.3. Information Theoretic Measures and Volatility Modeling
2.3.1. Shannon Entropy Measure
- where, the convention holds, and , represents the probability of , for . therefore, and
- The entropy reaches to its maximum value if all events follow the equally likely assumption;
- The entropy corresponding to an event with probability less than one has a positive sign.
- (1)
- Maximum Likelihood (ML);
- (2)
- effreys: Bayesian estimate with a = 1/2;
- (3)
- Laplace: Bayesian estimate with a = 1;
- (4)
- Schurmann–Grassberger (SG): Bayesian estimate with a = 1/(length of underlying asset prices series);
- (5)
- Minimax with a = sqrt (sum (underlying asset prices series))/(length of underlying asset prices series);
- (6)
- Shrink entropy: Uses James–Stein-type shrinkage at the level of cell frequencies.
2.3.2. Tsallis Entropy
- where and Rare events of interests denote , and frequently encountered have ;
- The q-exponential function , whose inverse is the q-logarithmic function ;
- Gell-Mann and Tsallis (2004) suggested q1.4 for high-frequency financial returns;
- The value of the parameter q decreases to 1 as the frequency of the data decreases. The values emphasize highly volatile signals.
2.3.3. Approximate and Sample Entropy
- Suppose the underlying time series, of length ;
- For , let and be two vectors of length and denotes distance between the two vectors. Therefore,
- Two vectors and are called similar if , where denotes the specified tolerance. Now compute the relative frequency for each of the .
- where denote number of vectors similar to for a fixed and Now computing the average frequency
- Finally, ApEn can be computed by using the following statistics.
- where denotes number of vectors pairs of length and with ;
- denotes total number of templates equals to length with
3. Empirical Analysis
3.1. Stationarity and Normality Tests
3.2. Testing the ARCH Effect
3.3. Volatility Estimation Using GARCH Models
3.4. Volatility Assessment Using Shannon and Tsallis Entropy
3.5. Regularity and Randomness Using Approximate Entropy and Sample Entropy
4. Discussion and Conclusions
- Both best-fitted models suggested the remarkable existence of the leverage effect in the Pakistan Stock Exchange;
- A high variation in the estimated volatility of the pandemic period has observed, and extreme downturns in prices are expected;
- The behavior of the variance is asymmetric for PSX-100 closing returns of both examined periods;
- The GARCH volatility modeling shows a usual behavior of the Pakistani market response towards bad news.
- We have pointed out that no remarkable difference between the randomness of volatility series before and during the COVID-19 pandemic, based on Shannon and Tsallis entropy estimates;
- All values of entropies are positive on both periods, and volatility shows non-linear dynamics;
- The overall relative difference of estimated entropies is not significant;
- The overall relative difference of estimated entropies is not significant under both the Shannon and Tsallis entropies;
- In the case of closing prices of the PSX-100, the Shannon entropy shows almost similar behavior under all estimation methods;
- In the case of ApEn and SampEn, the market shows mixed behavior, and the results are more sensitive. The sample entropy results reported more randomness in the pre-pandemic period for the Pakistani Stock Market. We detected both entropies are very sensitive to the selection of parameters.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Model | LOGLIK. | BIC | SIC | HQIC | AIC |
---|---|---|---|---|---|
Student T | |||||
GJRGARCH | 745.2045 | −5.9935 | −5.8795 | −5.9956 | −5.9476 |
EGARCH | 741.4358 | −5.9629 | −5.8489 | −5.9649 | −5.917 |
TGARCH | 740.9011 | −5.9585 | −5.8446 | −5.9606 | −5.9126 |
Asymmetric Student T | |||||
GJRGARCH | 746.4149 | −5.9952 | −5.867 | −5.9978 | −5.9436 |
EGARCH | 742.1684 | −5.9607 | −5.8325 | −5.9633 | −5.9091 |
TGARCH | 741.6314 | −5.9564 | −5.8281 | −5.9589 | −5.9047 |
Generalized Error | |||||
GJRGARCH | 745.4326 | −5.9954 | −5.8814 | −5.9974 | −5.9495 |
EGARCH | 742.1894 | −5.969 | −5.855 | −5.971 | −5.9231 |
TGARCH | 741.8362 | −5.9661 | −5.8522 | −5.9682 | −5.9202 |
Skewed GED | |||||
GJRGARCH | 746.4959 | −5.9959 | −5.8677 | −5.9985 | −5.9443 |
EGARCH | 742.675 | −5.9648 | −5.8366 | −5.9674 | −5.9132 |
TGARCH | 742.3509 | −5.9622 | −5.834 | −5.9648 | −5.9106 |
Model | LOGLIK. | BIC | SIC | HQIC | AIC |
---|---|---|---|---|---|
Student T | |||||
GJRGARCH | 611.1641 | −6.1547 | −6.0209 | −6.1579 | −6.1006 |
EGARCH | 611.0953 | −6.1438 | −5.9933 | −6.1478 | −6.0829 |
TGARCH | 611.8439 | −6.1617 | −6.0279 | −6.1648 | −6.1075 |
Asymmetric Student T | |||||
GJRGARCH | 612.1173 | −6.1543 | −6.0037 | −6.1582 | −6.6851 |
EGARCH | 615.0001 | −6.1939 | −6.0601 | −6.197 | −6.1075 |
TGARCH | 614.4997 | −6.1786 | −6.028 | −6.1825 | −6.0933 |
Generalized Error | |||||
GJRGARCH | 616.2783 | −6.2069 | −6.0731 | −6.2101 | −6.1075 |
EGARCH | 616.9883 | −6.204 | −6.0534 | −6.2079 | −6.0933 |
TGARCH | 612.136 | −6.1647 | −6.0309 | −6.1678 | −6.1397 |
Skewed GED | |||||
GJRGARCH | 612.2701 | −6.1558 | −6.0053 | −6.1598 | −6.0933 |
EGARCH | 612.6194 | −6.1696 | −6.0358 | −6.1727 | −6.1397 |
TGARCH | 613.2739 | −6.1661 | −6.0155 | −6.1700 | −6.1176 |
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Returns | Before COVID-19 | COVID-19 | ||||||
---|---|---|---|---|---|---|---|---|
Closing | Opening | High | Low | Closing | Opening | High | Low | |
Minimum | −0.0268 | −0.0268 | −0.0335 | −0.0367 | −0.071 | −0.071 | −0.0666 | −0.0655 |
1st Quartile | −0.0071 | −0.0078 | −0.0068 | −0.0065 | −0.0054 | −0.0054 | −0.0064 | −0.0054 |
Median | 0.0001 | 0.0004 | 0.0001 | 0.0012 | 0.0012 | 0.0012 | 0.0011 | 0.0022 |
Mean | 0.0003 | 0.0004 | 0.0003 | 0.0004 | 0.0002 | 0.0002 | 0.0002 | 0.0002 |
3rd Quartile | 0.0075 | 0.0079 | 0.007 | 0.0075 | 0.0085 | 0.0086 | 0.0075 | 0.0082 |
Maximum | 0.0351 | 0.0427 | 0.0337 | 0.0374 | 0.0468 | 0.0468 | 0.056 | 0.0498 |
Skewness | 0.1326 | 0.1736 | 0.1744 | −0.0276 | −1.151 | −1.1479 | −0.9461 | −1.1668 |
Kurtosis | 3.1178 | 3.206 | 3.3096 | 3.4331 | 8.0603 | 8.0203 | 9.1404 | 7.7247 |
SD | 0.0116 | 0.0122 | 0.0107 | 0.0118 | 0.0155 | 0.0156 | 0.0138 | 0.0152 |
CV | 0.0243 | 0.0327 | 0.0282 | 0.0323 | 0.0125 | 0.0147 | 0.0144 | 0.015 |
Range | 0.0619 | 0.0696 | 0.0672 | 0.0741 | 0.1179 | 0.1179 | 0.1226 | 0.1153 |
Inter−Quartile range | 0.0145 | 0.0157 | 0.0138 | 0.0141 | 0.0139 | 0.014 | 0.0139 | 0.0136 |
JB Stats | 0.7971 | 1.5441 | 2.0296 | 1.6715 | 303.1570 | 298.9304 | 404.2784 | 272.5008 |
Lilliefors Stats | 0.0479 | 0.0336 | 0.0298 | 0.0375 | 0.1281 | 0.1276 | 0.0986 | 0.1193 |
Pearson Chi Sq. Test | 19.8455 | 10.5772 | 13.3577 | 17.6829 | 66.4959 | 66.1870 | 43.9431 | 70.3577 |
Period of Variables | Stationary Tests | ARCH Effect | |||
---|---|---|---|---|---|
Variable | ADF Stats | Philip Perron Stats | KPSS Stats | LM Stats | LB Stats |
Closing Returns—Before COVID-19 | −12.9559 (0.0001) | −12.9191 (0.0001) | 0.4042 (0.0753) | 36.728 (0.0002) | 8.3691 (0.0038) |
Closing Returns—COVID-19 | −13.4422 (0.0100) | −13.7433 (0.0001) | 0.3249 (0.0100) | 73.918 (0.0000) | 5.8386 (0.0157) |
GJR-GARCH-SGED COVID-19 Period | ||||||||
mu | ar1 | ma1 | skew | shape | ||||
0.0002 (0.0010) | 0.8163 (0.1413) | −0.7490 (0.1661) | 0.0000 (0.0000) | 0.0000 (0.0126) | 0.8222 (0.0312) | 0.2437 (0.0820) | 0.8801 (0.1162) | 1.6361 (0.3015) |
EGARCH-STD before-COVID-19 Period | ||||||||
mu | ar1 | ma1 | shape | |||||
−0.0013 (0.0000) | −0.1308 (0.0000) | 0.3022 (0.0000) | −0.1310 (0.0000) | −0.1792 (0.0000) | 0.9854 (0.0001) | −0.1452 (0.0000) | 18.7972 (0.0030) |
Estimation | Volatility | Prices | ||
---|---|---|---|---|
Method | COVID-19 | Before COVID-19 | COVID-19 | Before COVID-19 |
ML | 5.3661 | 5.4902 | 5.5037 | 5.5050 |
Jeffreys | 5.5052 | 5.5053 | 5.5037 | 5.5050 |
Laplace | 5.5053 | 5.5053 | 5.5037 | 5.5050 |
SG | 5.4207 | 5.4971 | 5.5037 | 5.5050 |
minimax | 5.4439 | 5.4994 | 5.5037 | 5.5050 |
Shrink | 5.5053 | 5.5053 | 5.5037 | 5.5050 |
Parameter | Volatility | Prices | ||
---|---|---|---|---|
q | COVID-19 | Before COVID-19 | COVID-19 | Before COVID-19 |
1 | 5.3661 | 5.4902 | 5.5037 | 5.5050 |
1.2 | 3.2792 | 3.3314 | 3.3365 | 3.3370 |
1.4 | 2.1998 | 2.2212 | 2.2232 | 2.2234 |
1.6 | 1.5958 | 1.6045 | 1.6052 | 1.6053 |
1.8 | 1.2309 | 1.2344 | 1.2346 | 1.2347 |
2 | 0.9944 | 0.9958 | 0.9959 | 0.9959 |
Returns | Volatility | Prices | |||||
---|---|---|---|---|---|---|---|
COVID-19 | Before COVID-19 | COVID-19 | Before COVID-19 | COVID-19 | Before COVID-19 | ||
0.05 | 1 | 2.0077 | 1.9918 | 0.5665 | 1.051 | 0.8225 | 0.876 |
2 | 0.6347 | 0.5192 | 0.475 | 0.7457 | 0.5948 | 0.6834 | |
3 | 0.0958 | 0.047 | 0.3587 | 0.3605 | 0.4043 | 0.4813 | |
0.1 | 1 | 1.9011 | 2.0217 | 0.4579 | 0.9009 | 0.5764 | 0.5923 |
2 | 0.9052 | 0.7616 | 0.3918 | 0.7500 | 0.482 | 0.565 | |
3 | 0.2774 | 0.1014 | 0.3073 | 0.5177 | 0.3998 | 0.4872 | |
0.15 | 1 | 1.7757 | 1.9621 | 0.4091 | 0.7828 | 0.4231 | 0.4143 |
2 | 0.995 | 0.9847 | 0.3325 | 0.6667 | 0.3639 | 0.4258 | |
3 | 0.4658 | 0.222 | 0.2761 | 0.5267 | 0.3085 | 0.3828 | |
0.2 | 1 | 1.6367 | 1.8465 | 0.3535 | 0.6795 | 0.3358 | 0.3074 |
2 | 1.0500 | 1.1772 | 0.3105 | 0.6127 | 0.3141 | 0.3251 | |
3 | 0.6265 | 0.3374 | 0.2585 | 0.5123 | 0.2757 | 0.3164 |
r | l | Returns | Volatility | Prices | |||
---|---|---|---|---|---|---|---|
COVID-19 | Before COVID-19 | COVID-19 | Before COVID-19 | COVID-19 | Before COVID-19 | ||
0.05 | 1 | 2.4511 | 2.6077 | 0.3638 | 1.0016 | 0.7022 | 0.8336 |
2 | 2.3931 | 2.4849 | 0.3316 | 1.0452 | 0.613 | 0.8029 | |
3 | 2.9444 | 1.7047 | 0.3039 | 0.8627 | 0.5625 | 0.8052 | |
0.1 | 1 | 2.0316 | 2.4002 | 0.2712 | 0.7824 | 0.4747 | 0.5476 |
2 | 1.875 | 2.4022 | 0.2397 | 0.8176 | 0.4061 | 0.5454 | |
3 | 1.9879 | 2.3514 | 0.2128 | 0.7807 | 0.3757 | 0.5301 | |
0.15 | 1 | 1.7655 | 2.165 | 0.2215 | 0.6359 | 0.3431 | 0.385 |
2 | 1.5924 | 2.1313 | 0.19 | 0.6476 | 0.2913 | 0.3878 | |
3 | 1.678 | 2.7515 | 0.1632 | 0.6107 | 0.2723 | 0.3644 | |
0.2 | 1 | 1.5554 | 1.9543 | 0.1939 | 0.5431 | 0.271 | 0.2969 |
2 | 1.434 | 2.0363 | 0.1655 | 0.5518 | 0.2335 | 0.2982 | |
3 | 1.5666 | 2.3026 | 0.143 | 0.5125 | 0.2184 | 0.2895 |
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Sheraz, M.; Nasir, I. Information-Theoretic Measures and Modeling Stock Market Volatility: A Comparative Approach. Risks 2021, 9, 89. https://doi.org/10.3390/risks9050089
Sheraz M, Nasir I. Information-Theoretic Measures and Modeling Stock Market Volatility: A Comparative Approach. Risks. 2021; 9(5):89. https://doi.org/10.3390/risks9050089
Chicago/Turabian StyleSheraz, Muhammad, and Imran Nasir. 2021. "Information-Theoretic Measures and Modeling Stock Market Volatility: A Comparative Approach" Risks 9, no. 5: 89. https://doi.org/10.3390/risks9050089