# Electricity Price Instability over Time: Time Series Analysis and Forecasting

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## Abstract

**:**

_{2}prices), internal (consumption and generation) and external (net import between neighboring bidding zones) electricity flows. Based on the SARIMAX model, we tried to combine all these factors to forecast electricity prices in the single bidding zone. It was found that the SARIMAX (1, 1, 2) × (3, 1, 0, 7) model with exogenous prices, internal and external electricity flows, which has the lowest AIC and MAPE values, is the best-fitted model for the DE-LU bidding zone. Anonymous trading and unpredictable individual bidding strategies lead to persistent price volatility, which causes electricity prices to deviate from fundamental trends. To reveal the risk factors, the SARIMAX model of electricity prices needs to be supplemented with a GARCH model of the residual returns. For forecasting electricity price residual volatility in the DE-LU bidding zone, the SARIMAX model with exogenous prices, internal and external electricity flows must be accompanied with the GARCH (7, 0) model.

## 1. Introduction

_{2}prices did so on 8 December 2021 at a level of 87 Euro/t CO

_{2}eq. In Q1 2022 (7 March 2022–8 March 2022), new price maximums were set at 486 €/MWh, 227 €/MWh and 403 Euro/t for electricity, gas and coal, respectively, while the maximum for CO

_{2}prices was updated a month earlier (8 February 2022) and amounted to 97 Euro/t CO

_{2}eq. [3,33,34,35]. All this requires taking into consideration new situations in the functioning of electricity markets and forecasting of electricity prices.

## 2. Materials and Methods

#### 2.1. Methodological Approach

#### 2.2. Data Collection and Preparation

_{2}price) [35] were also used as exogenous variables.

#### 2.3. Data Analysis

#### 2.4. Model Development and Forecasting

- (1)
- ARIMA (p, d, q), developed by Box and Jenkins (1970) [49]:$${\u2206}^{d}{Y}_{t}=c+{\sum}_{i=1}^{p}{\phi}_{i}{\u2206}^{d}{Y}_{t-i}+{\sum}_{j=1}^{q}{\theta}_{j}{\epsilon}_{t-j}+{\epsilon}_{t}$$
- (2)
- SARIMA (p, d, q) (P, D, Q, s), first suggested by Chatfield and Prothero (1973) [50]:$${\u2206}^{d}{Y}_{t}=c+{\sum}_{i=1}^{p}{\phi}_{i}{\u2206}^{d}{Y}_{t-i}+{\sum}_{j=1}^{q}{\theta}_{j}{\epsilon}_{t-j}+{\sum}_{l=1}^{P}({\mathsf{\Phi}}_{l}\times \sum {\u2206}^{d}{Y}_{t-{s}_{l}})+{\sum}_{k=1}^{Q}({\Theta}_{k}\times \sum {\u2206}^{d}{\epsilon}_{t-{s}_{k}})+{\epsilon}_{t}$$
- (3)
- ARIMAX (p, d, q)×, first used by Young and Whitehead (1977) [51]:$${\u2206}^{d}{Y}_{t}=c+{\sum}_{i=1}^{p}{\phi}_{i}{\u2206}^{d}{Y}_{t-i}+{\sum}_{j=1}^{q}{\theta}_{j}{\epsilon}_{t-j}+\beta {X}_{t}+{\epsilon}_{t}$$
- (4)
- SARIMAX (p, d, q) (P, D, Q, s)×, developed by Holst et al. (1988) [52]:$${\u2206}^{d}{Y}_{t}=c+{\sum}_{i=1}^{p}{\phi}_{i}{\u2206}^{d}{Y}_{t-i}+{\sum}_{j=1}^{q}{\theta}_{j}{\epsilon}_{t-j}+{\sum}_{l=1}^{P}({\mathsf{\Phi}}_{l}\times \sum {\u2206}^{d}{Y}_{t-s}{}_{l})+{\sum}_{k=1}^{Q}({\Theta}_{k}\times \sum {\u2206}^{d}{\epsilon}_{t-s}{}_{k})+\beta {X}_{t}+{\epsilon}_{t}$$
_{t}, Y_{t−i}are the values in the current period and i periods ago, respectively; ∆^{d}is the difference between the values in the current period and d periods ago; ${\epsilon}_{t},{\epsilon}_{t-j}$ are error terms for the current period and j periods ago; c is the baseline constant factor; ${\phi}_{i}$ is the part of the value i periods ago relevant in explaining the current one; ${Y}_{t-{s}_{l}}$ is the seasonal lagged value s × l periods ago; ${\epsilon}_{t-{s}_{k}}$ is the seasonal lagged error for s × k periods ago; ${\theta}_{j}$ is the part of the error for j periods ago relevant in explaining the current value; ${\mathsf{\Phi}}_{l}$ is the part of the seasonal lagged value l periods ago relevant in explaining the current one; ${\Theta}_{k}$ is the part of the seasonal lagged error for k periods ago relevant in explaining the current value; X is the exogeneous factor; $\beta $ is the slope coefficient for the exogeneous factor.

_{2}prices), (ii) internal (consumption and generation) electricity flows, and (iii) external (net import) electricity flows. At each stage, we demonstrated how the SARIMAX models evolved as additional groups of factors were included.

## 3. Results

#### 3.1. Data Analysis of Electricity Prices

_{2}(+170%) was observed. In 2018, electricity prices in the DE-LU bidding zone were among the lowest in the EU.

_{2}prices (+57% p.a.), which led to a coal-to-gas switching in power generation.

_{2}prices fell in the 1st half of 2020 but recovered in the 2nd half of 2020, while the profitability of coal generation dropped below 0.

_{2}(69 €/tCO

_{2}in Q4 2021) and, as a consequence, growth in the electricity prices (178.9 €/MWh in Q4 2021). Such price trends caused the reverse switching of electricity generation from gas to coal and an increase in the share of nuclear energy.

_{2}price being 68.10 €/tCO

_{2}.

- The time series of the daily electricity prices are non-stationary. Thus, the null hypothesis has failed to be rejected. The electricity prices have a partly time-dependent structure with no constant variance over time. In this regard, they cannot be used as raw data for the analysis and forecasting;
- Whereas the time series of electricity price returns and electricity price differences are stationary, their current values are conditional and can be used for forecasting,the ACF and PACF plots for electricity price returns show no significant lags, i.e., immediate and secondary impacts of the past values of the time series on the current periods;
- The ACF and PACF plots for electricity price differences show numerous impacts, which proves the expediency of using the ARIMA model based on exact difference–stationary time series;
- The ACF plot of electricity price differences also indicates the presence of seasonality every 7 lags, which corresponds to a weekly cycle of electricity consumption and, accordingly, fluctuations in the electricity prices.

#### 3.2. Time Series Analysis of Electricity Prices and Their Forecasting

_{2}(coef. = 0.27) have an immediate directly proportional impact on the changes in the electricity price, while coal prices have an immediate inversely proportional impact (coef. = −0.02), but this factor is insignificant (p-value = 0.25). The standard error and MAPE of the first model accounts for 408 €/MWh and 34%, respectively. Therefore, forecasting electricity prices based on the SARIMAX model with only exogenous prices is quite risky.

_{2}prices have an insignificant immediate positive impact (coef. = 0.06, p-value = 0.55). Oil, nuclear, hydro, solar and biofuel generation, the p-value for which exceeds a significant level, are also insignificant for pricing in this bidding area. However, coal and gas generation and consumption exert a directly proportional impact (with the coefficients of 0.008, 0.014 and 0.03, respectively); wind and nuclear power generation, and pumped storage exert an inversely proportional impact (with the coefficients of −0.011, −0.024 and −0.085, respectively). At the same time, this model shows no immediate impact of previous prices but only a secondary directly proportional impact of the 3-day random deviations that have a damping effect (with the coefficients of 0.58, 0.15 and 0.08, respectively). Furthermore, an immediate impact of the 3-week seasonal effects (with the coefficients of 0.17, 0.04 and 0.13, respectively) is observed. The standard error and MAPE of the second model are still significant and account for 272 €/MWh and 24%. Therefore, we cannot rely on the SARIMAX with exogenous prices and internal flows, and above all because its best-fitted model by Auto Arima Package is integrated in the order of 0 and the data are non-stationary.

_{2}prices exert a directly proportional impact (with the coefficients of 0.92 and 0.58, respectively), while the coal prices have an inversely proportional impact (coef. = −0.10). All types of generation, except oil, hydro and biofuel, are statistically significant. At the same time, wind and solar generation as well as pumped storage have an inversely proportional impact (with the coefficients of −0.007, −0.011 and −0.05, respectively), while coal, gas and nuclear power generation have a directly proportional impact (with the coefficients of 0.017, 0.013 and 0.028, respectively). The excess of the electricity imports over its exports in case of such bidding zones as AT, BE, CH, DK1, NL, NO2 and SE4 will cause an increase in the electricity prices in the DE-LU bidding zone, while the opposite situation of an excess of electricity exports over imports will cause a decrease. Trading with such zones as CZ, DK2, FR and PL does not affect the changes in the pricing in the DE-LU bidding zone. The day-ago and 3-weeks-ago prices have a directly proportional impact on the formation of the electricity prices (with the coefficients of 0.37, 0.13, 0.07 and 0.13, respectively), the 2-days-ago random effects have an inversely proportional impact (with the coefficients of −0.73 and −0.18, respectively). The standard error and MAPE of the third model are significantly lower than in the previous two and account for 154 €/MWh and 17%, respectively. Thus, the SARIMAX model with exogenous prices, internal and external flows can be defined as the best-fitted model of electricity prices to use for their forecasting.

#### 3.3. Residual Forecasting

## 4. Discussion and Conclusions

- SARIMAX modelling of the influence of fundamental factors of the internal and external related markets on the changes in electricity prices over time. These factors are considered to be: (i) exogenous prices (gas, coal and CO
_{2}prices), (ii) internal (consumption and generation) electricity flows, and (iii) external (net import) electricity flows; - GARCH modelling of the volatility of the electricity price residuals, which allows tracking and forecasting the risks of price distortions under the influence of individual trading strategies of electricity market participants.

_{2}prices.

_{2}prices), conventional (coal, gas and nuclear) and intermittent RES (wind and solar) power generation, pumped storage and external electricity flows between seven neighboring bidding zones (AT, BE, CH, DK1, NL, NO2 and SE4). This model enables tracking price fluctuations over time. Thus, for April 2022, four price drops (4 April 2022, 7 April 2022, 9 April 2022 and 23 April 2022) were forecasted. The application of this model showed an average forecast error of 17% for the testing period and 24% for the forecast period.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ACF | autocorrelation function |

ADF test | augmented Dickey–Fuller test |

AIC | Akaike information criterion |

AR component | autoregressive component |

ARCH | autoregressive conditional heteroscedastic |

ARIMA | autoregressive integrated moving average |

ARIMAX | autoregressive integrated moving average with exogenous factors |

DE-LU bidding zone | German/Luxemburg bidding zone |

EDA | exploratory data analysis |

ENTSO-E | European Association for the Cooperation of Transmission System Operators for Electricity |

GARCH | generalized autoregressive conditional heteroscedastic |

LLF | log likelihood function |

MA component | moving average component |

NEMO | nominated electricity market operator |

PACF | partial autocorrelation function |

SARIMA | seasonal autoregressive integrated moving average |

SARIMAX | seasonal autoregressive integrated moving average with exogenous factors |

TSO | transmission system operators |

## Appendix A

## Appendix B

**Table A1.**Re-fitting the SARIMAX model with exogenous prices, internal and external flows for May 2022.

Parameter | Coef. | Std. Err. | p-Value | Parameter | Coef. | Std. Err. | p-Value |
---|---|---|---|---|---|---|---|

SARIMAX (1, 1, 2) × (3, 1, 0, 7) LLF = −5199 AIC = 10,469 MAPE = 17.42% | |||||||

Gas_price | 0.963 | 0.04 | 0 | CH_net_flows | 0.0032 | 0.011 | 0.777 |

Coal_price | 0.1746 | 0.024 | 0 | CZ_net_flows | −0.1234 | 0.038 | 0.001 |

CO_{2}_price | 0.0066 | 0.115 | 0.955 | DK1_net_flows | 0.1938 | 0.01 | 0 |

Consumption | 0.0023 | 0.001 | 0.005 | DK2_net_flows | −0.0447 | 0.024 | 0.059 |

Coal_gen | 0.0096 | 0.002 | 0 | FR_net_flows | −0.0212 | 0.029 | 0.467 |

Gas_gen | 0.0054 | 0.006 | 0.335 | NL_net_flows | 0.0141 | 0.01 | 0.142 |

Oil_gen | 0.0018 | 0.111 | 0.102 | NO2_net_flows | 0.1937 | 0.013 | 0 |

Hydro_gen | 0.0211 | 0.018 | 0.24 | SE4_net_flows | 0.3047 | 0.126 | 0.015 |

Pumped_ stor | −0.0376 | 0.029 | 0.2 | ar.L1 | 0.3891 | 0.04 | 0 |

Wind_gen | −0.0018 | 0.001 | 0.147 | ma.L1 | −0.8082 | 0.042 | 0 |

Solar_gen | −0.0031 | 0.004 | 0.389 | ma.L2 | −0.0996 | 0.032 | 0.002 |

Biofuels_gen | 0.0225 | 0.016 | 0.164 | ar.S.L7 | 0.1624 | 0.021 | 0 |

Nuclear_gen | 0.023 | 0.013 | 0.076 | ar.S.L14 | 0.108 | 0.025 | 0 |

AT_net_flows | 0.1573 | 0.016 | 0 | ar.S.L21 | 0.1023 | 0.024 | 0 |

BE_net_flows | 0.0941 | 0.017 | 0 | sigma2 | 182.075 | 3.934 | 0 |

**Figure A2.**Electricity price forecasting by the SARIMAX model with exogenous prices, internal and external flows in the DE-LU bidding zone for the testing and forecast periods.

**Table A2.**Re-fitting the GARCH model of residual return volatility in the DE-LU bidding zone for May 2022.

Parameter | Coef. | Std. Err. | p-Value | Parameter | Coef. | Std. Err. | p-Value |
---|---|---|---|---|---|---|---|

GARCH (7, 0) LLF = −739 AIC = 1497 | |||||||

omega | 0.0907 | 2.83 × 10^{−2} | 1.34 × 10^{−3} | alpha [4] | 0 | 1.06 × 10^{−3} | 1 |

alpha [1] | 2.19 × 10^{−4} | 2.58 × 10^{−4} | 0.395 | alpha [5] | 0 | 2.79 × 10^{−3} | 1 |

alpha [2] | 5.17 × 10^{−9} | 1.58 × 10^{−3} | 1 | alpha [6] | 0 | 2.33 × 10^{−4} | 1 |

alpha [3] | 0 | 2.93 × 10^{−3} | 1 | alpha [7] | 0.9998 | 0.603 | 9.75 × 10^{−2} |

**Figure A3.**Forecasting the residual return volatility in the DE-LU bidding zone for the testing and forecast periods. Source: constructed by the authors based on the results of previous research.

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**Figure 3.**Autocorrelation and stationarity of the daily electricity prices in the DE-LU bidding zone. Source: constructed by the authors based on [3].

**Figure 4.**EDA of electricity price differences in the DE-LU bidding zone. Source: constructed by the authors based on [3].

**Figure 6.**Forecast of the electricity prices in the DE-LU bidding zone for the next 30 days. Source: constructed by authors based on the results of previous research.

**Figure 7.**Results of modelling and forecasting the volatility of the electricity price residuals. Source: constructed by the authors based on the results of previous research.

**Table 1.**Descriptive statistics of electricity prices in the DE-LU bidding zone in October 2018–March 2022.

Parameters | Q4 2018 | 2019 | 2020 | 2021 | Q1 2022 | October 2018–March 2022 |
---|---|---|---|---|---|---|

Mean (µ) | 52.58 | 37.67 | 30.47 | 96.85 | 184.60 | 63.93 |

Median (Me) | 55.05 | 39.21 | 31.59 | 74.52 | 177.85 | 43.43 |

Standard Deviation (std.dev.) | 13.30 | 11.87 | 13.92 | 66.93 | 89.18 | 61.67 |

Variation (var.) | 25.3% | 31.5% | 45.7% | 69.1% | 48.3% | 96.5% |

Kurtosis (K) | 1.5 | 8.1 | 1.6 | 4.2 | 1.1 | 9.0 |

Skewness (S) | −0.8 | −1.4 | −0.4 | 1.8 | 0.9 | 2.7 |

Min | 9.16 | −39.75 | −24.99 | −15.39 | 40.89 | −39.75 |

Max | 80.47 | 85.91 | 75.04 | 435.09 | 486.49 | 486.49 |

Volatility (vol.) | 48.5% | 142.1% | 194.4% | 79.1% | 58.2% | 137.4% |

Parameter | Coef. | Std. Err. | p-Value | Parameter | Coef. | Std. Err. | p-Value |
---|---|---|---|---|---|---|---|

SARIMAX (2, 1, 1) × (1, 1, 3, 7) model, LLF = −5657, AIC = 11,335, MAPE = 34.04% | |||||||

Gas price | 1.6238 | 0.041 | 0 | ma.L1 | 0.7325 | 0.055 | 0 |

Coal price | −0.0218 | 0.019 | 0.245 | ar.S.L7 | 0.9946 | 0.005 | 0 |

CO_{2}_price | 0.2717 | 0.097 | 0.005 | ma.S.L7 | −0.9264 | 0.018 | 0 |

ar.L1 | −0.108 | 0.06 | 0.047 | ma.S.L14 | −0.13 | 0.024 | 0 |

ar.L2 | 0.3069 | 0.036 | 0 | ma.S.L21 | 0.1321 | 0.017 | 0 |

sigma2 | 408.0554 | 7.141 | 0 |

**Table 3.**Results of building the SARIMAX model with exogenous prices and internal electricity flows.

Parameter | Coef. | Std. Err. | p-Value | Parameter | Coef. | Std. Err. | p-Value |
---|---|---|---|---|---|---|---|

SARIMAX (0, 0, 3) × (3, 0, 0, 7) model, LLF = −5262, AIC = 10,564, MAPE = 24.26% | |||||||

Gas_price | 1.9303 | 0.042 | 0 | Solar_gen. | −0.0031 | 0.005 | 0.491 |

Coal_price | −0.1481 | 0.021 | 0 | Biofuels_gen. | 0.0337 | 0.027 | 0.216 |

CO_{2}_price | 0.0571 | 0.096 | 0.553 | Nuclear_gen. | −0.0238 | 0.011 | 0.026 |

Consumption | −0.0026 | 0.001 | 0 | ma.L1 | 0.5816 | 0.016 | 0 |

Coal_gen. | 0.0081 | 0.003 | 0.006 | ma.L2 | 0.146 | 0.017 | 0 |

Gas_gen. | 0.0138 | 0.006 | 0.023 | ma.L3 | 0.083 | 0.015 | 0 |

Oil_gen. | 0.1478 | 0.119 | 0.216 | ar.S.L7 | 0.1717 | 0.018 | 0 |

Hydro_gen. | 0.0236 | 0.019 | 0.219 | ar.S.L14 | 0.0377 | 0.021 | 0.076 |

Pumped_stor. | −0.0853 | 0.029 | 0.003 | ar.S.L21 | 0.1369 | 0.021 | 0 |

Wind_gen. | −0.0112 | 0.002 | 0 | sigma2 | 271.6839 | 7.106 | 0 |

Parameter | Coef. | Std. Err. | p-Value | Parameter | Coef. | Std. Err. | p-Value |
---|---|---|---|---|---|---|---|

SARIMAX (1, 1, 2) × (3, 1, 0, 7) model, LLF = −4896 AIC = 9853 MAPE = 16.94% | |||||||

Gas_price | 0.9207 | 0.041 | 0 | CH_net_flows | 0.0525 | 0.011 | 0 |

Coal_price | −0.1041 | 0.023 | 0 | CZ_net_flows | 0.0079 | 0.044 | 0.857 |

CO_{2}_price | 0.5839 | 0.159 | 0 | DK1_net_flows | 0.2263 | 0.012 | 0 |

Consumption | 0.002 | 0.001 | 0.001 | DK2_net_flows | 0.019 | 0.031 | 0.544 |

Coal_gen. | 0.0173 | 0.003 | 0 | FR_net_flows | 0.0136 | 0.028 | 0.629 |

Gas_gen. | 0.0127 | 0.005 | 0.013 | NL_net_flows | 0.0177 | 0.01 | 0.066 |

Oil_gen. | 0.0009 | 0.102 | 0.32 | NO2_net_flows | 0.1934 | 0.014 | 0 |

Hydro_gen. | 0.0185 | 0.016 | 0.243 | SE4_net_flows | 0.2243 | 0.135 | 0.097 |

Pumped_stor. | −0.0498 | 0.026 | 0.06 | ar.L1 | 0.3704 | 0.037 | 0 |

Wind_gen. | −0.0072 | 0.002 | 0 | ma.L1 | −0.7313 | 0.041 | 0 |

Solar_gen. | −0.0112 | 0.004 | 0.003 | ma.L2 | −0.1781 | 0.031 | 0 |

Biofuels_gen. | 0.0314 | 0.048 | 0.517 | ar.S.L7 | 0.1315 | 0.021 | 0 |

Nuclear_gen. | 0.0275 | 0.011 | 0.012 | ar.S.L14 | 0.0682 | 0.024 | 0.005 |

AT_net_flows | 0.1537 | 0.018 | 0 | ar.S.L21 | 0.1269 | 0.023 | 0 |

BE_net_flows | 0.0723 | 0.018 | 0 | sigma2 | 154.4615 | 4.001 | 0 |

**Table 5.**Comparison of the actual and forecasted values of electricity prices in the DE-LU bidding zone for April 2022.

Statistics | Current Electricity Price, €/MWh | Forecasted Electricity Prices, €/MWh | ||
---|---|---|---|---|

SARIMAX Model with Exogenous Prices | SARIMAX Model with Exogenous Prices and Internal Flows | SARIMAX Model with Exogenous Prices, Internal and External Flows | ||

mean | 166.0 | 183.2 | 151.8 | 143.5 |

std.dev. | 56.8 | 30.3 | 15.9 | 44.7 |

min | 54.3 | 122.9 | 121.4 | 47.1 |

Q1 (25%) | 116.7 | 163.8 | 140 | 92.7 |

Q2 (50%) | 182.3 | 189.4 | 150 | 159.4 |

Q3 (75%) | 220.1 | 208.9 | 161.2 | 173.9 |

Q4 (max) | 237.9 | 231.7 | 182 | 205.8 |

MAPE | - | 37% | 34% | 24% |

Max error | - | 213% | 155% | 55% |

Statistics | SARIMAX Model with Exogenous Prices | SARIMAX Model with Exogenous Prices and Internal Flows | SARIMAX Model with Exogenous Prices, Internal and External Flows | |||
---|---|---|---|---|---|---|

€/MWh | % | €/MWh | % | €/MWh | % | |

mean | −0.397 | −9.12 | −0.23 | 5.46 | 0.365 | 3.69 |

std.dev. | 20.23 | 182.07 | 15.82 | 69.8 | 11.97 | 44.51 |

min | −157.548 | −558.09 | −128.3 | −387.6 | −107.93 | −290.61 |

Q1 (25%) | −6.122 | −13.29 | −6.55 | −13.24 | −4.23 | −8.38 |

Q2 (50%) | 0.57 | 1.68 | −0.25 | −0.2 | 0.33 | 0.9 |

Q3 (75%) | 6.18 | 12.79 | 5.73 | 13.01 | 4.8 | 10.49 |

Q4 (max) | 112.23 | 116.21 | 117.67 | 182.28 | 108.42 | 90.44 |

**Table 7.**Results of building the GARCH models of electricity price residual returns by the SARIMAX model with exogenous prices.

Parameter | Coef. | Std. Err. | p-Value | Parameter | Coef. | Std. Err. | p-Value |
---|---|---|---|---|---|---|---|

GARCH (7, 7) model, LLF = −2383, AIC = 4798 | |||||||

omega | 0.0621 | 7.07 × 10^{−2} | 0.38 | beta [1] | 1.72 × 10^{−5} | 1.04 × 10^{−2} | 0.999 |

alpha [1] | 2.54 × 10^{−5} | 5.05 × 10^{−3} | 0.996 | beta [2] | 1.34 × 10^{−4} | 2.19 × 10^{−3} | 0.951 |

alpha [2] | 2.35 × 10^{−4} | 6.94 × 10^{−4} | 0.734 | beta [3] | 1.45 × 10^{−4} | 1.30 × 10^{−3} | 0.911 |

alpha [3] | 2.83 × 10^{−4} | 6.31 × 10^{−4} | 0.654 | beta [4] | 1.13 × 10^{−4} | 9.37 × 10^{−4} | 0.904 |

alpha [4] | 2.21 × 10^{−4} | 6.28 × 10^{−4} | 0.724 | beta [5] | 9.91 × 10^{−5} | 1.83 × 10^{−3} | 0.957 |

alpha [5] | 1.96 × 10^{−4} | 5.36 × 10^{−4} | 0.715 | beta [6] | 1.22 × 10^{−5} | 2.18 × 10^{−2} | 1 |

alpha [6] | 2.07 × 10^{−3} | 1.91 × 10^{−3} | 0.279 | beta [7] | 0.9514 | 1.39 × 10^{−2} | 0 |

alpha [7] | 0.0482 | 1.59 × 10^{−2} | 2.40 × 10^{−3} |

**Table 8.**Results of building the GARCH models of electricity price residual returns by the SARIMAX model with exogenous prices and internal flows.

Parameter | Coef. | Std. Err. | p-Value | Parameter | Coef. | Std. Err. | p-Value |
---|---|---|---|---|---|---|---|

GARCH (1, 7) model, LLF = −732, AIC = 1485 | |||||||

omega | 0.0244 | 1.31 × 10^{−2} | 6.36 × 10^{−2} | beta [3] | 1.08 × 10^{−9} | 5.45 × 10^{−3} | 1 |

alpha [1] | 0.4812 | 0.16 | 2.55 × 10^{−3} | beta [4] | 1.24 × 10^{−9} | 4.72 × 10^{−3} | 1 |

beta [1] | 2.56 × 10^{−10} | 4.78 × 10^{−3} | 1 | beta [5] | 0.0347 | 2.77 × 10^{−2} | 0.21 |

beta [2] | 1.09 × 10^{−9} | 1.62 × 10^{−2} | 1 | beta [6] | 0.0198 | 1.85 × 10^{−2} | 0.284 |

beta [7] | 0.4643 | 7.04 × 10^{−2} | 4.16 × 10^{−11} |

**Table 9.**Results of building the GARCH models of electricity price residual returns by the SARIMAX model with exogenous prices, internal and external flows.

Parameter | Coef. | Std. Err. | p-Value | Parameter | Coef. | Std. Err. | p-Value |
---|---|---|---|---|---|---|---|

GARCH (7, 0) model, LLF = −556 AIC = 1130 | |||||||

omega | 0.0782 | 4.62 × 10^{−2} | 9.05 × 10^{−2} | alpha [4] | 0 | 2.31 × 10^{−3} | 1 |

alpha [1] | 0.1138 | 0.237 | 0.63 | alpha [5] | 0 | 2.48 × 10^{−3} | 1 |

alpha [2] | 0 | 7.04 × 10^{−3} | 1 | alpha [6] | 1.96 × 10^{−4} | 7.25 × 10^{−4} | 0.787 |

alpha [3] | 0 | 2.39 × 10^{−3} | 1 | alpha [7] | 0.886 | 0.512 | 8.38 × 10^{−2} |

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## Share and Cite

**MDPI and ACS Style**

Wang, D.; Gryshova, I.; Kyzym, M.; Salashenko, T.; Khaustova, V.; Shcherbata, M.
Electricity Price Instability over Time: Time Series Analysis and Forecasting. *Sustainability* **2022**, *14*, 9081.
https://doi.org/10.3390/su14159081

**AMA Style**

Wang D, Gryshova I, Kyzym M, Salashenko T, Khaustova V, Shcherbata M.
Electricity Price Instability over Time: Time Series Analysis and Forecasting. *Sustainability*. 2022; 14(15):9081.
https://doi.org/10.3390/su14159081

**Chicago/Turabian Style**

Wang, Diankai, Inna Gryshova, Mykola Kyzym, Tetiana Salashenko, Viktoriia Khaustova, and Maryna Shcherbata.
2022. "Electricity Price Instability over Time: Time Series Analysis and Forecasting" *Sustainability* 14, no. 15: 9081.
https://doi.org/10.3390/su14159081