Modeling Electricity Price Dynamics Using Flexible Distributions
Abstract
:1. Introduction
2. Literature Review
3. Methodology
4. Estimation Results
5. Discussion
6. Conclusions and Policy Implications
Funding
Acknowledgments
Conflicts of Interest
Nomenclature
Mean parameter | |
Standard deviation parameter | |
Shape parameter | |
Skewness parameter | |
Peakedness parameter | |
Gamma function defined as | |
Set of real numbers | |
-test | Goodness of fit test |
p-value | Probability value of a test statistic |
(if p-value is less than 0.10, then the null hypothesis is rejected) | |
Natural logarithm of SMP from the peak-demand period of day t | |
Residuals from the mean equation (not standardized) | |
Conditional variance or volatility (based on notation in [23]) | |
Normal distribution with zero mean and conditional variance | |
Information set at time (based on notation in [23]) | |
The indicator function equal to 1 if and 0 otherwise | |
Standardized residuals |
Abbreviations
AIC | Akaike Information Criterion |
AR | Autoregressive |
ARCH | Autoregressive Conditional Heteroscedasticity |
BDS-test | Brock–Dechert–Scheinkman test of i.i.d. |
Coef of Var | Coefficient of Variation |
COVID-19 | Corona Virus Disease 2019 |
GARCH | Generalized Autoregressive Conditional Heteroscedasticity |
GED | Generalized Error Distribution |
GHYP | Generalized Hyperbolic Distribution |
i.i.d. | independent and identically distributed |
JB-test | Jarque–Bera normality test [31] |
JSU | Johnson’s distribution |
Obs | Number of Observations |
Regime 1 | April 1990–March 1993 (Coal contracts), Reference period |
Regime 2 | April 1993–March 1994 |
Regime 3 | April 1994–March 1996 (Price-cap regulation) |
Pre-Regime 4 | April 1996–July 1996 |
Regime 4 | July 1996–July 1999 (Divestment 1 introduced on 1 July 1996) |
Regime 5 | July 1999–March 2001 (Divestment 2 introduced on 20 July 1999) |
SGED | Skew Generalized Error Distribution |
SMP | System Marginal Price |
SST | Skew Student’s t distribution |
St Dev | Standard Deviation |
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Price | Regime 1 | Regime 2 | Regime 3 | Pre-Regime 4 | Regime 4 | Regime 5 |
---|---|---|---|---|---|---|
Apr 90–Mar 93 | Apr 93–Mar 94 | Apr 94–Mar 96 | Apr 96–July 96 | July 96–July 99 | July 99–Mar 01 | |
Coal Contracts | Price-Cap Reg | Divestment 1 | Divestment 2 | |||
Mean | 25.66 | 32.90 | 37.22 | 35.25 | 41.99 | 35.91 |
Min | 14.78 | 14.94 | 7.88 | 17.17 | 14.54 | 12.15 |
Max | 62.97 | 55.95 | 211.24 | 76.74 | 105.09 | 77.89 |
St Dev | 4.69 | 6.52 | 17.64 | 11.39 | 19.29 | 11.95 |
Coef of Var (%) | 18.26 | 19.81 | 47.40 | 32.30 | 45.95 | 33.28 |
Obs | 1096 | 365 | 731 | 91 | 1114 | 616 |
Assumed Distribution Intercepts and Variables | SGED | SST | GHYP | JSU | |||||
---|---|---|---|---|---|---|---|---|---|
Coef | Std Err | Coef | Std Err | Coef | Std Err | Coef | Std Err | ||
Mean Equation | −0.3728 *** | 0.0147 | −0.4892 *** | 0.0872 | −0.4976 *** | 0.0936 | −0.4792 *** | 0.0803 | |
0.3110 *** | 0.0091 | 0.3066 *** | 0.0069 | 0.3061 *** | 0.0107 | 0.3069 *** | 0.0019 | ||
0.1388 *** | 0.0060 | 0.1422 *** | 0.0195 | 0.1417 * | 0.0730 | 0.1413 *** | 0.0306 | ||
0.0362 *** | 0.0022 | 0.0381 | 0.0258 | 0.0380 | 0.0278 | 0.0382 ** | 0.0186 | ||
0.0577 *** | 0.0119 | 0.0569 *** | 0.0214 | 0.0573 *** | 0.0165 | 0.0572 *** | 0.0165 | ||
0.1306 *** | 0.0051 | 0.1304 *** | 0.0192 | 0.1303 *** | 0.0258 | 0.1303 *** | 0.0270 | ||
0.1759 *** | 0.0061 | 0.1705 *** | 0.0308 | 0.1708 *** | 0.0082 | 0.1713 *** | 0.0192 | ||
−0.0486 *** | 0.0096 | −0.0468 | 0.0310 | −0.0468 * | 0.0282 | −0.0472 ** | 0.0193 | ||
−0.0484 *** | 0.0065 | −0.0445 ** | 0.0217 | −0.0441 ** | 0.0175 | −0.0446 *** | 0.0172 | ||
0.1320 *** | 0.0052 | 0.1275 *** | 0.0179 | 0.1276 *** | 0.0204 | 0.1281 *** | 0.0206 | ||
−0.0421 *** | 0.0076 | −0.0422 * | 0.0248 | −0.0421 ** | 0.0198 | −0.0421 ** | 0.0165 | ||
0.0988 *** | 0.0043 | 0.1020 *** | 0.0168 | 0.1022 *** | 0.0185 | 0.1017 *** | 0.0186 | ||
−0.0534 *** | 0.0063 | −0.0546 *** | 0.0191 | −0.0550 *** | 0.0162 | −0.0551 *** | 0.0163 | ||
−0.0449 *** | 0.0090 | −0.0435 *** | 0.0151 | −0.0438 *** | 0.0131 | −0.0437 *** | 0.0127 | ||
0.0372 *** | 0.0040 | 0.0386 ** | 0.0151 | 0.0390 *** | 0.0129 | 0.0388 *** | 0.0121 | ||
−0.0286 *** | 0.0048 | −0.0264 ** | 0.0123 | −0.0265 | 0.0177 | −0.0267 *** | 0.0065 | ||
0.0669 *** | 0.0107 | 0.0655 *** | 0.0111 | 0.0655 *** | 0.0112 | 0.0654 *** | 0.0102 | ||
0.0616 *** | 0.0015 | 0.0723 *** | 0.0028 | 0.0731 *** | 0.0051 | 0.0714 *** | 0.0006 | ||
Regime 2 | 0.0206 *** | 0.0055 | 0.0201 ** | 0.0082 | 0.0203 | 0.0144 | 0.0202 *** | 0.0056 | |
Regime 3 | 0.0353 *** | 0.0097 | 0.0344 *** | 0.0114 | 0.0354 * | 0.0196 | 0.0352 *** | 0.0087 | |
Pre-Regime 4 | 0.0246 | 0.0325 | 0.0273 | 0.0334 | 0.0287 | 0.0377 | 0.0277 | 0.0321 | |
Regime 4 | 0.0315 *** | 0.0087 | 0.0286 ** | 0.0127 | 0.0294 | 0.0237 | 0.0295 *** | 0.0060 | |
Regime 5 | −0.0037 | 0.0097 | −0.0023 | 0.0121 | −0.0019 | 0.0187 | −0.0020 | 0.0084 | |
0.0080 ** | 0.0033 | 0.0079 ** | 0.0039 | 0.0079 * | 0.0043 | 0.0079 ** | 0.0040 | ||
−0.0136 *** | 0.0021 | −0.0141 *** | 0.0040 | −0.0138 *** | 0.0040 | −0.0139 *** | 0.0040 | ||
−0.0111 *** | 0.0029 | −0.0117 *** | 0.0037 | −0.0115 *** | 0.0036 | −0.0114 *** | 0.0036 | ||
−0.0117 *** | 0.0033 | −0.0115 *** | 0.0035 | −0.0114 *** | 0.0036 | −0.0115 *** | 0.0036 | ||
0.0105 *** | 0.0033 | 0.0105 *** | 0.0035 | 0.0106 *** | 0.0034 | 0.0104 *** | 0.0034 | ||
Volatility Equation | 0.0037 *** | 0.0005 | 0.0037 *** | 0.0006 | 0.0037 *** | 0.0005 | 0.0036 *** | 0.0005 | |
0.1571 *** | 0.0367 | 0.1653 *** | 0.0386 | 0.1674 *** | 0.0395 | 0.1646 *** | 0.0383 | ||
0.0810 *** | 0.0205 | 0.0903 *** | 0.0219 | 0.0900 *** | 0.0216 | 0.0888 *** | 0.0215 | ||
0.0314 ** | 0.0156 | 0.0229 | 0.0166 | 0.0218 | 0.0164 | 0.0242 | 0.0162 | ||
0.0412 ** | 0.0194 | 0.0378 * | 0.0195 | 0.0388 * | 0.0208 | 0.0385 ** | 0.0193 | ||
0.0761 *** | 0.0244 | 0.0776 *** | 0.0245 | 0.0785 *** | 0.0246 | 0.0780 *** | 0.0246 | ||
0.0903 *** | 0.0247 | 0.1091 *** | 0.0271 | 0.1101 *** | 0.0280 | 0.1066 *** | 0.0261 | ||
0.0961 *** | 0.0224 | 0.0941 *** | 0.0226 | 0.0922 *** | 0.0225 | 0.0933 *** | 0.0223 | ||
0.0802 | 0.0522 | 0.0882 | 0.0539 | 0.0829 | 0.0540 | 0.0845 | 0.0534 | ||
Regime 2 | 0.0038 *** | 0.0011 | 0.0041 *** | 0.0012 | 0.0041 *** | 0.0012 | 0.0040 *** | 0.0012 | |
Regime 3 | 0.0287 *** | 0.0038 | 0.0277 *** | 0.0040 | 0.0283 *** | 0.0040 | 0.0280 *** | 0.0040 | |
Pre-Regime 4 | 0.0547 *** | 0.0179 | 0.0559 *** | 0.0183 | 0.0570 *** | 0.0184 | 0.0559 *** | 0.0182 | |
Regime 4 | 0.0336 *** | 0.0042 | 0.0324 *** | 0.0043 | 0.0326 *** | 0.0044 | 0.0325 *** | 0.0043 | |
Regime 5 | 0.0280 *** | 0.0043 | 0.0275 *** | 0.0045 | 0.0279 *** | 0.0045 | 0.0275 *** | 0.0044 | |
0.0031 *** | 0.0006 | 0.0034 *** | 0.0006 | 0.0033 *** | 0.0007 | 0.0033 *** | 0.0006 | ||
1.4254 *** | 0.0459 | 6.8967 *** | 0.7803 | 0.0290 ** | 0.0130 | 1.9448 *** | 0.1241 | ||
1.0539 *** | 0.0225 | 1.0431 *** | 0.0253 | 0.9946 *** | 0.0007 | 0.1625 ** | 0.0759 | ||
−3.4734 *** | 0.3959 |
SGED | SST | GHYP | JSU | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Dimension | 2 | 0.46 | 0.13 | 0.15 | 0.23 | 0.30 | 0.09 | 0.09 | 0.11 | 0.31 | 0.09 | 0.09 | 0.15 | 0.32 | 0.10 | 0.10 | 0.15 |
3 | 0.30 | 0.15 | 0.12 | 0.17 | 0.13 | 0.08 | 0.06 | 0.07 | 0.15 | 0.08 | 0.06 | 0.11 | 0.15 | 0.09 | 0.06 | 0.10 | |
4 | 0.12 | 0.18 | 0.12 | 0.20 | 0.04 | 0.08 | 0.06 | 0.10 | 0.04 | 0.09 | 0.06 | 0.13 | 0.05 | 0.10 | 0.07 | 0.13 | |
5 | 0.07 | 0.32 | 0.17 | 0.32 | 0.03 | 0.18 | 0.10 | 0.19 | 0.03 | 0.21 | 0.10 | 0.24 | 0.03 | 0.22 | 0.11 | 0.23 | |
6 | 0.10 | 0.68 | 0.40 | 0.71 | 0.05 | 0.46 | 0.30 | 0.52 | 0.06 | 0.51 | 0.30 | 0.57 | 0.05 | 0.52 | 0.32 | 0.55 |
Assumed Theoretical Distribution | ||||
---|---|---|---|---|
SGED | SST | GHYP | JSU | |
Sign Bias | 0.45 | 0.12 | 0.13 | 0.14 |
Negative Sign Bias | 1.28 | 1.35 | 1.33 | 1.33 |
Positive Sign Bias | 1.55 | 1.91 * | 1.90 * | 1.88 * |
Joint Effect | 4.68 | 5.75 | 5.70 | 5.62 |
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Tashpulatov, S.N. Modeling Electricity Price Dynamics Using Flexible Distributions. Mathematics 2022, 10, 1757. https://doi.org/10.3390/math10101757
Tashpulatov SN. Modeling Electricity Price Dynamics Using Flexible Distributions. Mathematics. 2022; 10(10):1757. https://doi.org/10.3390/math10101757
Chicago/Turabian StyleTashpulatov, Sherzod N. 2022. "Modeling Electricity Price Dynamics Using Flexible Distributions" Mathematics 10, no. 10: 1757. https://doi.org/10.3390/math10101757