Signs of Fluctuations in Energy Prices and Energy Stock-Market Volatility in Brazil and in the US

Abstract

Volatility reflects the degree of variation in a time series, and a measurement of the stock performance in the energy sector can help one understand the pattern of fluctuations within this industry, as well as the factors that influence it. One of these factors could be the COVID-19 pandemic, which led to extreme volatility within the stock market in several economic sectors. It is essential to understand this regime of volatility so that robust financial strategies can be adopted to handle it. This study used stock data from the Yahoo! Finance API and data from the energy-price database from the US Energy Information Administration to conduct a comparative analysis of the volatility in the energy sector in Brazil and in the United States, as well as of the energy prices in California. The volatility in these time series were modeled using GARCH. The stock volatility regimes, both before and after COVID-19, were identified with a Markov switching model; the spillover index between the energy markets in the USA and in Brazil was evaluated with the Diebold–Yilmaz index; and the causality between the energy stock price and the energy prices was measured with the Granger causality test. The findings of this study show that (i) the volatility regime introduced by COVID-19 is still prevalent in Brazil and in the USA, (ii) the changes in the energy market in the US affect the Brazilian market significantly more than the reverse, and (iii) there is a causality relationship between the energy stock markets and the energy prices in California. These results may assist in the achievement of effective regulation and economic planning, while also supporting better market interventions. Also, acknowledging the persistent COVID-19-induced volatility can help with developing strategies for future crisis resilience.

1. Introduction

Financial liberalization was one of the key national policies in the period from 1980 to 2000, and it was determined by the impact of a country’s level of financial development on its growth rate (Bailliu 2000). As a result, there was a significant integration of national financial markets with global financial markets, which led to an acceleration of globalization and a greater diversification of asset investment in international markets, which culminated in significant changes, e.g., greater economic union (Stiglitz 2000).

The globalization of financial markets has become a worldwide trend, especially with the advent of Industry 4.0 (Bispo et al. 2024). Domestic stock markets have become increasingly globalized owing to the significant rise in the number of foreign investors, technological advances, and the removal of cross-border restrictions on the flow of capital in most countries. Thus, the volatility transmission between international markets affects investors since they must constantly monitor and evaluate fluctuations in their stock so that they can diversify their portfolios and spread their risks.

The volatility of international markets and the need to diversify portfolios has brought about reforms leading to a greater degree of liberalization. This has had a significant impact on the electric power sector, including the way in which it is organized; moreover, it has implications on new types of investment and their effects on industrial transactions (Bekaert and Harvey 2003). In other words, the new commercial strategies have enabled the industry to be more flexible. These strategies include the development of an electricity market. This is a spot market that fulfills some essential conditions in a deregulated industry: it means transactions can be more flexible, it allows adjustments to be made to the amount of the energy that is contracted (and the power generated), and it provides a reference point for long-term contracts. In other words, a spot market is an essential mechanism for adjusting supply and demand (Entezari and Fuinhas 2024; Stiglitz 2000).

A volatility model, such as Generalized Auto-Regressive Conditional Heteroskedasticity (GARCH), predicts volatility (Capelli et al. 2021). Nearly all financial applications of volatility models involve forecasting future returns. While a volatility model is typically designed to predict the absolute magnitude of returns, it can also be utilized to estimate quantiles or even the entire return distribution. These kinds of forecasts are employed in various financial activities, such as risk management, market timing, and portfolio selection (Chen et al. 2023). An essential requirement of each is the volatility’s predictability. A risk manager must know if it is likely if their portfolio will abruptly change. An option trader needs to anticipate the expected volatility over the future life of a contract and understand the potential variability of this forecast to effectively hedge the contract.

As far as the Brazilian electricity sector is concerned, it should be noted that, unlike in other countries, there is still a lack of a real market that can efficiently fulfill the requirements mentioned above. In Brazil, there has been a significant increase in the number of free consumers, which has been partly stimulated by short-term price patterns. Thus, owing to the considerable growth of the economy in recent years and the consequent tightening of the relationship between demand and supply, the cost of settling differences (in the PLD) has experienced considerable volatility and unpredictability, and this has made the short-term electricity market an environment that is characterized by a high degree of uncertainty (Ang et al. 2022; Caldeira et al. 2016; Chiou-Wei et al. 2008; Entezari and Fuinhas 2024).

For this reason, one of the biggest concerns of agents in the Brazilian electricity sector, especially those operating in the Free Contracting Environment (ACL), is the volatility and unpredictability of the Price of Settlement of Differences (PLD), which is the Brazilian proxy for the spot price. These factors represent a significant tariff and financial risk for agents in the sector, and it also affects a number of sectors of the Brazilian economy.

Furthermore, the COVID-19 pandemic triggered waves of significant disruptions in the financial market around the world. Several studies were published to deepen the understanding of the energy sector market during this period (Padhan and Prabheesh 2021; Setiawan et al. 2021). However, an event of this proportion is expected to trigger disturbances for many years to come (Padhan and Prabheesh 2021), so the short post-pandemic period would not allow for a real analysis of all the facts and effects within the energy financial sector. It is relevant to evaluate if, years after the declaration of the pandemic, these disturbances are still reflected in the volatility of the energy sector. The complexity of forecasting due to the influence of rapid developments is impacted by several factors that are often unknown, thus making this forecasting process even more difficult (Castle et al. 2022). Years after this fact, countries with emerging and developed economies are still looking for strategies to adapt to this new normal in the financial market (Setiawan et al. 2021).

The situation is similar in Brazil, where the initial effects of COVID-19 was first experienced in early 2020 (Padhan and Prabheesh 2021). However, the lack of data and clarity regarding energy prices and policies involving the definition of these post-COVID-19 values still represents a gap in the literature for the development of the Brazilian industry. Therefore, this study aims to (i) verify the suitability of the GARCH model for analyzing the stock market volatility in Brazil and in the USA with the Akaike Information Criterion and Bayesian Information Criterion; (ii) to determine if there are financial causal relationships between the stock market prices and the energy-price values; (iii) to compare the volatility in the energy stock markets in the pre- and post-pandemic periods using the Markov switching model; and to (iv) evaluate the transmission of shocks between the energy markets in the studied regions using the Diebold–Yilmaz index.

This paper compares the energy market in Brazil and in the United States, measuring the volatilities and causalities between the stock prices in the companies that operate in the industry sector in these countries. The fluctuation in energy price charged to customers, based on California, was also evaluated. As a consequence, this paper contributes by conducting a volatility-based risk assessment on the energy sector in Brazil and in the United States. Several statistical tests were performed on the studied time series, revealing patterns of the stock prices in this sector. Ultimately, a causality relationship between the energy stock market fluctuation and the energy price in California was statistically proved.

This article is organized as follows. Section 2 contains a review of the literature on GARCH volatility, as well as the models and their variations that were applied to the financial market. Next, Section 3 presents the materials and methods that were used to carry out this research to analyze the energy-price data from Brazil and the USA. Section 4 presents the results that were found with the application of the GARCH model and the other statistical tests used in the data related to the energy industry sector, the patterns and factors that influence its volatility, and a discussion of the implications of the results. Finally, Section 5 concludes this study with the main conclusions and contributions that the research brought to the literature.

2. Literature Review

Volatility is defined as the conditional variation of a time series. In the case of the financial time series, the volatility of returns was modeled. It should be noted that although the underlying series was stationary and had constant variation, it could still fluctuate over a short period. Recognizing this allows one to understand the conditional variation in the study of volatility, which is of particular interest to short-term forecasting. Due to the variation in the volatility over time, classical time series models were not suitable for modeling it. Still, one of their assumptions was that the variation is constant (Capelli et al. 2021).

Volatility is not directly observable if there is only one observation. Financial series show long periods of high volatility followed by periods of low volatility, indicating heteroskedasticity. Variations conditioned on past information (pronounced for the short term) are more interesting than unconditioned media and variations (pronounced for the long term) (Kindalkar et al. 2022).

In general, volatility models are used to make projections and estimates, such as predicting the absolute value of the magnitude of an asset’s price returns, estimating quantities, or even the entire probability density function of the returns (Bollerslev et al. 2020). These forecasts and estimates are used in various financial activities, such as risk management, portfolio selection, and the short and long positions in holding an asset (Almeida and Gonçalves 2022).

Furthermore, it was not expected that financial asset prices will evolve independently of the market around them, but, contrarily, it was assumed that the other variables may contain relevant information for the volatility of a series (Alqahtani et al. 2020).

The volatility in the energy sector is measured in the Wilder Hill New Energy Global Innovation (NEX) Index (Geng et al. 2021) in China (He et al. 2021) and in the USA (Mensi et al. 2021). However, to the best of our knowledge, there is little research regarding the transmission of the volatility specifically between Brazil, a developing country with promising renewable energy resources, and the USA, a reference in the energy sector, especially when investigating the volatility regimes in these countries before and after COVID-19.

The pandemic is relevant in this context because the uncertainty raised by the global health crisis creates volatility in market prices, including in the energy sector (Szczygielski et al. 2022). Researchers have assessed the impact of the COVID-19 pandemic on stock market prices (Baek et al. 2020; Khan et al. 2023). However, the duration of time that this volatility may endure is undetermined, and statistical techniques should be used to examine if the effects of this shock are still prevalent.

It has been observed that volatility sometimes tends to cluster in time, meaning that greater volatility periods are commonly followed by high volatility, and lower volatility tends to be succeeded by lower volatility values (Nikolova et al. 2020). This has also been observed in relation to the COVID-19 pandemic (Lúcio and Caiado 2022).

Another factor that may create uncertainty in an economic sector, thus generating volatility, is the occurrence of cyber attacks. In this regard, the energy sector in the U.S. has been, primarily, breached by malicious insiders and in incidents involving portable assets (Rodrigues et al. 2024). These breaches can lead to significant financial losses, reputational damage, and decreased consumer confidence (Pimenta Rodrigues et al. 2024), thus increasing the volatility of the stock values of affected companies.

As many volatility models assume that positive and negative innovations affect the asset’s conditional volatility symmetrically, ref. Bhowmik and Wang (2020) conducted a literature review on stock market volatility analysis after selecting 50 significant works. Their study concluded that GARCH accurately captures volatility and the returns in situations with symmetric information, while asymmetric GARCH models are more suitable for asymmetric information. Examples of asymmetric GARCH models include Exponential Generalized Autoregressive Conditional Heteroskedasticity (EGARCH), Glosten-Jagannathan-Runkle (GJR) GARCH, and Threshold Generalized Autoregressive Conditional Heteroskedasticity (TGARCH).

3. Methodology

This research involved collecting data, which were further processed and analyzed for relevant patterns and causal effects. This study used stock data from the Yahoo! Finance API and the California energy-price data from the U.S. Energy Information Administration, as presented in Section 3.1.

These data were analyzed using Python version 3.10.12, to select the most adequate volatility model, along with additional statistical tests, as presented in Section 3.2. Then, the chosen model was applied, providing information on the volatility of the time series. Figure 1 depicts the general methodology that was used for conducting this work.

3.1. Materials Used

This work used stock data obtained from Yahoo! Finance, via its API, and energy prices in California, which were obtained from the Energy Information Administration. These data sets are described in this section.

3.1.1. Stock Price Data

The criteria used for selecting the energy companies in each region were based on their market share. The stock data were gathered from Yahoo! Finance.This includes the daily closing values of energy companies’ stocks in Brazil (AESB3, ALUP11, AURE3, CEBR3, CMIG4, COCE5, CPEE3, CPLE6, EGIE3, ELET3, ENEV3, ENGI11, EQTL3, LIGT, NEOE3, SRNA3, and TAEE11), as well as data on energy company shares in the United States (AEP, AES, CMS, D, DUK, ETR, EXC, FE, NRG, PEG, SO, VST, WEC, XEL, EIX, NEE, PCG, RUN, SPWR, and SRE), six of which are from the state of California (EIX, NEE, PCG, RUN, SPWR, and SRE).

The US market was selected due to its global influence on the industry and the market, including the energy sector. Moreover, California was selected because it is a prominent state in this country with time-variant retail tariffs (Varghese and Sioshansi 2020), as opposed to the prices in other states and in Brazil, which do not vary significantly in time. The intention was to assess whether the volatility in the stock prices was related to the volatility in the energy prices and to compare the scenarios in variable- and fixed-energy pricing.

These stock values are stored in a database, along with California energy prices. Daily energy prices were unavailable for Brazil, and the lack of clarity and structure in the energy sector are gaps that this research aims to address.

3.1.2. Energy-Price Data in California

The general energy prices in Brazil do not vary hourly, like in California. For that reason, the Californian energy price was chosen for comparison to evaluate the consequence of the stock market’s volatility on the energy price.

These data were retrieved from the U.S. Energy Information Administration, who gathered the data from the Intercontinental Exchange. The published data included the date, the high, the low, and the weighted average energy price (USD/MWh) and the price hub to which they were associated. The energy hubs include Indiana Hub, Mid C, New England Mass Hub, PJM West, California NP-15, California Palo Verde, and California SP-15. As the scope of this work was restricted to California’s energy prices, only the hubs in this state were considered. The energy prices in these hubs were aggregated into a single time series by a simple daily average operation. This study used the weighted average energy price, which ranged from January 2014 to December 2023.

3.2. Data Analysis

As suggested by Bhowmik and Wang (2020), evaluating the symmetry of a time series is a fundamental step before volatility modeling, as it is a relevant feature that for selecting the appropriate model. Hence, this work assessed the symmetry of the aggregated stock market values in Brazil, the USA, and California. Thus, the symmetry of the data were initially evaluated to determine which volatility model was adequate. Then, statistical tests were conducted to identify patterns in the time series, as described in Section 3.2.1. Ultimately, the volatility was modeled as shown in Section 3.2.2.

3.2.1. Statistical Tests Used

Several volatility models, such as GARCH, require that the time series is stationary, that is, its statistical properties, such as mean and variance, are constant over time Creamer and Ben-Zvi (2021); Gujarati (2009). If a trend was observed in the data, thus implying it is in a non-stationary time series, it must be differentiated until it presents no unit root. To evaluate this, the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test was used. Its hypotheses, which were accepted or rejected according to the test’s statistics, were as follows: $H_0$ = the time series is stationary or $H_1$ = the time series is not stationary.

As the properties of the time series may switch between different regimes over time, possibly according to a Markov process, the Markov switching model (MSM) was used to assess the presence and patterns of such regimes, such as shifts from low to high volatility. These regime changes in economic data may be due to new policies or to external shocks, and, in this work, we used MSM to evaluate the stock market regimes before and after the COVID-19 pandemic.

The Diebold–Yilmaz (DY) index quantifies the spillover effects between the energy market variables. In this work, it was used to understand the transmission of shocks across the energy sector between Brazil, California, and United States as a whole.

The index, measured using Equation (1), ranged between 0% and 100%, with higher values indicating stronger spillover effects and implying that changes in one variable significantly impact the other variable. The index value was not symmetric, that is, $D{Y}_{ij}$ may not be equal to $D{Y}_{ji}$, with the directionality of the index indicating which variable is transmitting the spillover effects and which variable is receiving them.

$$D{Y}_{ij}=\sqrt{{\displaystyle \frac{1}{T}}\sum _{t=1}^T{\left({\displaystyle \frac{FEV{D}_{it}}{FEV{D}_{jt}}}-1\right)}^2}\times \sqrt{{\displaystyle \frac{V_i}{V_j}}},$$

(1)

where T is the total number of periods in the sample; $FEV{D}_{it}$ is the forecast-error variance decomposition of variable i at time t, representing the proportion of the forecast-error variance of variable i that is attributable to its shocks at time t; $FEV{D}_{jt}$ is the forecast-error variance decomposition of variable j at time t, representing the proportion of the forecast-error variance of variable j attributable to its shocks at time t; $V_i$ is the total variance of variable i over the sample period; and $V_j$ is the total variance of variable j over the sample period.

The Granger causality test was used to evaluate the causality relationship between the energy stock market and the energy prices in California. This test was used to determine whether the energy stock market in California can predict energy prices in the state and, conversely, whether energy prices can predict movements in the California energy stock market. Its hypotheses, which were accepted or rejected according to the test’s p-values, were as follows: $H_0$ = X does not Granger-cause Y or $H_1$ = X Granger-causes Y.

3.2.2. Generalized Auto-Regressive Conditional Heteroskedasticity Model (GARCH)

One of the first approaches for calculating volatility was the Black–Scholes model, which was created to assist in pricing European options. Importantly, as it is considered the starting point in measuring volatility, this model has a significant disadvantage because it considers the conditional variance as constant over time. Furthermore, it is also assumes that prices follow a normal distribution. Furthermore, economic instability does not allow volatility to be considered constant over time in most situations. Because of this, one of the most common approaches currently is calculating volatility from daily returns using GARCH models. These models consider it as a non-constant measurement over time, and they allow for flexibility in choosing the probability distributions that best represent the data set studied Black and Scholes (1973).

When dealing with the share prices of electricity-generating companies, there is a volatile behavior (where periods of large fluctuations and then more stable behavior, which is followed again by periods of significant volatility, etc.) because the variables are subject to behavior influenced by economic factors, such as the 2023 Energy Balance (EPE 2023), stock increases, and taxes, among other factors that can cause variation in the price of electricity assets. Thus, one of the first approaches to calculating volatility was the Black–Scholes model, which was created to help price European options. Although relevant, as it is considered the starting point for measuring volatility, this model has a significant disadvantage in evaluating the conditional variance constant over time. Considering the conditional variance, prices were suggested as following a normal distribution.

Because of this, one of the most common approaches currently is calculating volatility from daily returns using the GARCH models proposed by Bollerslev (1986). These models consider volatility a non-constant measure over time and allow for flexibility in choosing probability distributions that best represent the data set studied. Being a series of returns, the conditional mean and the conditional variance of the process are given by Equation (2), with the simplest model, of order $(p.q)$, being provided by GARCH$(1,1)$ (Gujarati 2009; Khan et al. 2023; Riady and Apriani 2023; Sarkodie 2017):

$${\sigma }_t^2={\alpha }_0+{\alpha }_1{\mu }_{t-1}^2+{\alpha }_2{\sigma }_{t-1}^2,$$

(2)

where ($\alpha$) represents the GARCH parameter, (${\mu }_t$) represents the estimated residual of the regression (squared error of period t), and (${\sigma }_t^2$) represents the conditional variance of period t. The parameters of the GARCH$(p.q)$ model are, respectively, the necessary lag terms of the squared error term (p) and the conditional variance terms referring to the lags (q). As the GARCH model is a variation of the ARCH model, this relationship can be given as GARCH$(p,q)$ being the same as the Autoregressive Conditional Heteroscedastic model ARCH$(p,q)$, which is represented by Equation (3) (Gujarati 2009):

$${\sigma }_t^2={\alpha }_0+{\alpha }_1{\mu }_{t-1}^2+\dots +{\alpha }_m{\sigma }_{t-m}^2.$$

(3)

As was noted, Equations (2) and (3) are similar. Their difference lies in that the GARCH model considers their variances in addition to past data. Therefore, the GARCH model would be the ARCH model with additional parameters. As such, in general terms, GARCH$(p,q)$ is given by ${\mu }_t={\sigma }^t\epsilon$, thus rewriting the model according to Equation (4):

$${\sigma }_t^2={\alpha }_0+\sum _{i=1}^m{\alpha }_i{\mu }_{t-i}^2+\sum _{j=1}^s{\beta }_j{\sigma }_{t-j}^2,$$

(4)

where ${\alpha }_0>0$, ${\alpha }_i\ge 0$, ${\beta }_j\ge 0$ and ${\sum }_{i=1}^{max(m,s)}({\alpha }_i+{\beta }_i)<1$. As such, a value $v_t$ was considered, which is given by the relation $v_t={\alpha }_t^2-{\sigma }_t^2$, so ${\sigma }_t^2={\alpha }_t^2-v_t$. When looking at past periods, ($t-1$) can be rewritten with ${\sigma }_{t-i}^2={\alpha }_{t-i}^2-v_{t-i}$ given that $(i=1,.\dots ,n)$, so Equation (4) can be rewritten as follows (given by Equation (5)):

$${\sigma }_t^2={\alpha }_0+\sum _{i=1}^{max(m,s)}({\alpha }_i+{\beta }_i){\mu }_{t-i}^2+v_t+\sum _{j=1}^s{\beta }_j{v}_{t-j}^2.$$

(5)

Hence, the GARCH model given by Equation (5) is the same as the ARMA-squared model. Although there are many steps that require exhaustive statistical and mathematical tests, there are also many tools that perform these analyses automatically currently available (Chen et al. 2023).

Information criteria are commonly used to choose the least complex model, which leads to the lowest residual variance. Each additional regressor included in a model is associated with a penalty; if it is less than the decrease in the sum of residuals, the regressor must be incorporated into the model. If the penalty is greater than the reduction in the sum, then adding the regressor is not justifiable. In other words, the information criteria are used to define which candidate models are more parsimonious and which, therefore, should lead to more reasonable modeling. Two of the most widely used criteria are Akaike and Schwarz (Ahmad and Chen 2020).

To define the parameters, the auto.arima() function was used to identify the best model, where the parameters were adjusted using the AIC, AICc, and BIC indices to indicate the best model. This function can also help in the parameter-selection process, considering that this is a dynamic process with several dependencies (Ahmad and Chen 2020).

4. Analysis and Results

This study evaluated energy companies’ stock market time-series volatility and its causality to the energy price in variable areas. To achieve this, this work used the data on the stock market and the energy prices in the United States, California, and Brazil, and it extracted the historical market data for each company using Yahoo! Finance.

4.1. Stock-Price Data Aggregation

Table 1. Statistical summary of the stock price companies operating in Brazil.

Ticker	Count	Mean	Std	Min	25%	50%	75%	Max	Weight	Skewn	Kurt
AESB3	772	11.33	1.53	9.08	10.22	10.95	12.14	17.20	0.022	1.02	0.58
ALUP11	2568	19.63	5.45	8.61	14.70	20.32	24.32	30.30	0.053	0.03	−1.30
AURE3	524	14.03	0.92	11.32	13.60	14.11	14.52	16.60	0.020	−0.51	0.73
CEBR3	2568	10.52	8.67	2.80	4.44	6.56	13.78	45.60	0.002	1.66	2.49
CMIG4	2568	8.36	2.66	2.76	5.68	8.84	10.47	13.42	0.048	−0.13	−1.20
COCE5	2568	48.02	8.23	30.50	41.75	48	54.36	67.50	0.011	0.02	−0.92
CPFE3	2568	26.54	6.35	13.53	20.73	27.11	31.71	39.20	0.083	−0.20	−1.05
CPLE6	2568	4.92	2.24	1.77	2.99	3.75	6.80	10.41	0.050	0.51	−0.96
EGIE3	2568	35.53	7.22	23.80	28.40	37.56	41.83	54.10	0.067	0.14	−1.37
ELET3	2568	25.33	13.72	4.57	12.84	25.76	36.89	51.57	0.177	−0.17	−1.39
ENEV3	2568	9.47	6.73	2.13	3.50	8.99	13.46	49.20	0.057	1.93	7.55
ENGI11	2568	33.04	14.33	9.97	18.69	39.03	45.41	59.84	0.161	−0.27	−1.47
EQTL3	2568	17.04	8.80	3.66	10.10	16.17	24.36	35.83	0.099	0.23	−1.11
LIGT3	2568	14.10	5.60	1.92	9.52	15.07	18.76	24.68	0.015	−0.32	−0.97
NEOE3	1203	18.10	2.58	13.63	16.17	17.70	19.74	27.53	0.087	0.92	0.92
SRNA3	586	10.12	1.13	7.60	9.19	10.06	10.91	13.01	0.019	0.27	−0.73
TAEE11	2568	27.38	7.79	15.10	20.40	25.25	35.36	45.75	0.028	0.44	−1.24
Aggr	2568	1.19	0.48	0.44	0.79	1.35	1.64	1.95	-	−0.18	−1.56

and

Table 2. Statistical summary of the stock price of companies operating in the United States. The gray rows indicate companies that operate in California.

Ticker	Count	Mean	Std	Min	25%	50%	75%	Max	Weight	Skewn	Kurt
AEP	2609	75.47	14.29	46.08	63.85	77.10	86.69	105.18	0.06	−0.10	−1.00
AES	2609	16.59	5.26	8.54	12.18	14.90	20.60	29.27	0.03	0.67	−0.75
CMS	2609	50.67	11.90	26.12	41.70	53.35	60.83	73.56	0.02	−0.34	−1.08
D	2609	71.54	9.75	39.53	69.00	73.54	77.88	89.80	0.06	−1.27	1.23
DUK	2609	87.00	10.85	64.15	78.41	86.65	94.77	115.43	0.11	0.31	−0.64
ETR	2609	91.19	16.42	60.52	77.01	90.09	104.56	134.78	0.03	0.20	−1.02
EXC	2609	30.86	7.03	18.16	25.21	29.71	35.56	50.25	0.06	0.55	−0.61
FE	2609	36.48	4.86	26.56	32.62	36.11	39.35	52.27	0.03	0.65	0.12
NRG	2609	31.85	11.37	8.98	25.28	33.56	38.68	83.65	0.02	0.33	1.57
PEG	2609	52.36	9.62	31.33	43.52	52.82	60.69	74.73	0.04	−0.10	−1.19
SO	2609	55.69	10.23	40.40	46.87	52.23	64.75	80.16	0.10	0.43	−1.12
VST	1914	23.54	9.67	12.60	18.52	22.30	24.64	93.42	0.03	3.72	17.19
WEC	2609	74.15	18.70	40.31	58.57	76.42	91.55	108.28	0.03	−0.10	−1.39
XEL	2609	52.70	13.75	27.35	40.86	53.19	65.05	77.41	0.04	−0.13	−1.33
EIX	2608	64.97	7.57	44.47	59.12	64.27	70.19	82.64	0.05	0.18	−0.51
NEE	2609	50.89	21.39	21.06	29.95	47.08	72.81	93.36	0.16	0.23	−1.41
PCG	2209	32.45	20.88	3.80	12.26	19.40	52.35	71.56	0.06	0.30	−1.55
RUN	2609	20.64	17.27	4.63	7.77	14.67	24.92	96.50	0.01	1.57	1.94
SPWR	2609	12.98	8.73	1.88	5.07	10.70	20.28	54.01	0.00	0.85	0.60
SRE	2609	61.78	10.04	43.50	53.46	59.32	70.11	87.83	0.17	0.42	−0.86
Aggr	2511	2.42	0.56	1.46	1.94	2.31	2.95	3.61	-	0.19	−1.30
Aggr CA	2511	7.59	1.62	4.80	6.29	7.18	9.07	11.34	-	0.35	−1.11

present the summary of statistics for the companies operating in Brazil and the US, respectively, considered in this work.

For analyzing the energy sector within each country, the time series for the considered companies are aggregated into a single time series for each company. For this, a weighted average by the enterprise value, as of June 2024, for each company was used. The “Weight” columns in Table 1 and Table 2 represent the weight used for each company for this average calculation and its contribution to the calculated index. The shares of the companies with the highest average values in Brazil were COCE5, EGIE3, ENGI11, TAEE11, CPFE3, and ELET3, while, in the USA, the companies with the highest representativeness in value were ETR, DUK, AEP, WEC, D, and EIX.

A large part of the kurtosis values of the shares showed that the values of the shares primarily had a platykurtic distribution. This suggests that the data have smoother peaks compared to other kurtosis distributions. Also, a high kurtosis value was observed in shares with lower average values, indicating that low-value shares have high peak moments, which were not sustained over time.

The “Close” price column was used for this analysis, and the time range was restricted from January 2014 to June 2024. The currency used in the time series was USD, as well as BRL for the Brazilian companies. A conversion was performed according to the quotation of June 2024. The stock-price time series for companies operating in California was also aggregated to evaluate the volatility of the energy stock prices compared to the state’s energy cost.

After aggregating the stock market data for Brazil, the United States, and California, the resulting time series were used for the analysis. These time series are presented in Figure 2. The vertical dashed lines represent a drop in the price of all these time series near the date the World Health Organization (WHO) declared COVID-19 a pandemic. This drop suggests that a higher volatility was introduced by the pandemic into the energy sector, as was also observed in other sectors of the industry (Baek et al. 2020; Khan et al. 2023). Figure 2b shows that the values of Brazilian stocks demonstrated greater instability compared to the general US and California stocks.

Further analysis was conducted on these time series, as shown in the following sections. In Section 4.2, the presence of a unit root is evaluated, while Section 4.3 estimates the symmetry of the data. Section 3.2.2 presents a discussion and predictions regarding the volatility of the energy stock market using GARCH, and Section 4.5 investigates the regime-switching pattern before and after the COVID-19 pandemic. Section 4.7 discusses the causality relationship between the energy prices in California and the energy stock market fluctuation.

4.2. Unit Root Testing

Determining whether a time series is stationary is a fundamental step before estimating volatility, as several models assume that the statistical properties of the time series are constant over time.

Therefore, the series must be transformed into stationary data by removing its differentiations, thus satisfying the assumptions of the stationarity of the GARCH model.

The Kwiatkowski–Phillips–Schmidt–Shin, or KPSS, test was used to test for the presence of a unit root in the time series. The test statistics were compared against a critical value associated with a significance level of 5%. If the test statistics exceeded the critical value, the null hypothesis was rejected, suggesting that the series was non-stationary. Conversely, failure to reject the null hypothesis implied that the series was stationary, as evidenced by the test statistics below the critical value.

As seen in

Table 3. The KPSS test statistics and critical value of 0.46 for a 5% significance level for the stock prices.

Time Series	Original	Differentiated Once
BRA	7.03	0.05
USA	7.81	0.02
USA_CA	7.14	0.02

, the KPSS test statistics for the original time series were above the critical value (5%) of 0.46, thus suggesting that they were non-stationary. After one differentiation, however, the test statistics were below this value, indicating that they had achieved stationarity.

As shown in Figure 3, a peak was noted shortly after January 2020 in all time series in the resultant stationary time series for each region after removing one difference. This peak could possibly be associated with the COVID-19 pandemic. It was also observed that the variation in the amplitude of these time series was greater to the right of the peak (post-pandemic) than to the left (pre-pandemic). This was evidenced by the California time series, which more frequently crossed or nearly reached the 0.25 reference line after 2020. This suggests that the volatility in the energy sector in California had not reached its pre-pandemic values, which is more deeply explored in Section 4.5.

4.3. Symmetry Evaluation

Before volatility estimations, each time series must be evaluated in terms of its symmetry as this helps in selecting the appropriate model specifications (Bhowmik and Wang 2020). In symmetric information environments, where volatility is expected to react similarly to positive and negative shocks, a standard GARCH model may be sufficient to capture volatility dynamics. On the other hand, in scenarios of asymmetric information, where volatility reacts differently to positive and negative shocks, asymmetric GARCH models like EGARCH, GJR-GARCH, or TGARCH may provide more accurate representations.

To appropriately select the volatility model in this work, we evaluated the symmetry of the distribution of the time series of stock prices for energy companies operating in Brazil, the USA, and California, as shown in Figure 4.

Upon evaluating the distributions of the aggregated stock prices, as depicted in Figure 4, along with the skewnesses that were close to zero, as represented in Table 1 and Table 2, it was noted that these time series were approximately symmetrical. Hence, a GARCH model was adequate for estimating their volatility, and this model was used for the volatility estimation in these time series, as seen in Section 4.4.

4.4. Volatility Modeling

As shown in this section, we employed the GARCH model to estimate the volatility of the stock prices within the energy sector in Brazil, in the United States, and in California. The aim was to evaluate the dynamic price movements of the energy sector that were possibly influenced by COVID-19. Equation (6) was used to normalize the data prior to the estimation of the volatility of the time series.

$$\mathrm{New}\mathrm{ts}=\left({\displaystyle \frac{\mathrm{Max}-\mathrm{Min}}{\mathrm{value}-\mathrm{Min}}}\right)\times (\mathrm{New}\max -\mathrm{New}\min )+\mathrm{New}\min .$$

(6)

With the data normalized, tests were carried out to determine the values for the parameters $(p,q)$.

4.4.1. Optimal GARCH Parameter Selection

The GARCH model was run, for each aggregated time series, using the varied parameters p,q ∈ [1, 6]. Each model was evaluated with the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC), also known as the Schwarz Criterion. The parameters that minimized AIC and BIC were selected. These values are presented in

Table 4. Optimal GARCH parameters and their AIC and BIC.

Time Series	p	q	AIC	BIC
BRA	2	2	27,988	28,023
USA	5	2	25,459	25,512
USA_CA	5	4	25,218	25,282

.

4.4.2. Volatility Prediction with GARCH

According to Engle and Patton (2001), volatility exhibits persistence, and a significant price fluctuation in an asset tends to be followed by further significant changes. In contrast, similarly, minor changes usually follow minor fluctuations. The results found for the volatility estimated by the GARCH model for each time series and their respective measured standard deviation of the volatility presented by the model can be seen in

Table 5. Estimated GARCH volatility for each time series.

Time Series	std_dev	garch_volatility
BRA	0.10	3773.98
USA	0.56	1134.72
USA_CA	1.63	775.67

.

4.5. Volatility Regimes Pre- and Post-Pandemic

The Markov switching model examines the variation of regimes within a time series. In this study, it was used to evaluate the states of the energy stock market volatility. This analysis considered the stationary and normalized version of each aggregated time series.

The regime shift is demonstrated in Figure 5, in which the unpainted areas of the graphs represent the periods that were more likely to be associated with State 1, and the green-painted regions represent those that were more likely to be related to State 2. The red dashed vertical line represents the date the WHO announced the COVID-19 pandemic.

From Figure 5, it may be noted that State 1, with its lower volatility, was more frequent before the pandemic. Conversely, State 2, with its greater volatility, was more predominant after it. In Brazil, as represented in Figure 5a, the predominance of State 2 started around two years before the pandemic but was more consistent after it.

Before the pandemic, the dominance of State 1 may suggest a period that could be characterized by relative stability, reflecting lower volatility in the stock values.

However, the emergence of State 2 as the predominant state following the announcement of the pandemic indicated a shift in market dynamics. This transition may indicate heightened uncertainty and increased market volatility (Engle and Patton 2001).

This state of higher volatility and uncertainty (State 2) is still predominant in Brazil, in the U.S., and in California, even four years after the pandemic declaration. This indicates that the energy stock market has not entirely recovered its stability from the pre-pandemic years.

To further this analysis, we examined the time-series structural characteristics through the Bai–Perron test using a Radial Basis Function and a penalty parameter of 10. This statistical test detected multiple structural breaks, which are significant changes in time series, such as the mean, variance, or trend changes.

This test resulted in two structural breakpoints for the stock market data of the U.S. companies, which were registered on 23 April 2020 and 21 February 2020. The structural breakpoint was registered on 13 February 2020 for the companies operating in California. No structural breakpoint was observed for the Brazilian market.

These dates are near the COVID-19 pandemic declaration, which is seen as the transition region between the persistence of States 1 and 2 in Figure 5b,c. This reinforces the change in the time-series patterns due to the disease.

This finding emphasizes the relationship between exogenous shocks, such as global health crises, and the evolution of financial markets.

4.6. Spillover Effect between the Regions

The Diebold–Yilmaz spillover index is a metric for assessing a volatility’s directional transmission between different time-series variables. As illustrated in

Table 6. DY indexes, where each row represents the transmitting asset and where each column represents the receiving asset.

	BRA	USA	USA_CA
BRA	-	5.98%	6.34%
USA	64.05%	-	0.50%
USA_CA	76.0%	28.16%	-

, each cell represents the index value that indicates the magnitude of spillover effects from the row variable to the column variable.

Upon computing the spillover dynamics between Brazil, the United States, and the state of California, the extent to which volatility propagated across these regions was observed. A higher index value signifies a more substantial spillover effect, suggesting that changes in one region’s variable significantly influence the volatility of another. For instance, the relatively high values of the DY index between the USA and Brazil (64.05) and between California and Brazil (76.01) indicate substantial spillover effects from the USA and California to Brazil, respectively. Conversely, the Brazilian energy stock market does not have a significant spillover effect on the United States (5.98) or California (6.34).

4.7. Causality between Energy Price and Energy Stock Market

To evaluate whether the volatility of the energy stock market is reflected in the stipulated energy price, the Granger causality test was conducted between the close values of the energy stock market of the companies in California and the energy-price history data of this U.S. state. The relative values of the normalized energy prices correlated with the time series of the stock prices, as shown in Figure 6.

Figure 6 illustrates the cross-correlation heatmap between the normalized and aggregated energy stock-price time series (differentiated once) and the normalized energy price. The maximum cross-correlation values were found between 83 to 90 and 98 to 102 lags, as visualized in Figure 6, suggesting that changes in the energy stock prices preceded similar changes in the energy prices over a daily interval period. This relationship indicates a predictive lag where fluctuations in the stock market prices of energy companies can forecast subsequent changes in the energy prices within this period.

Figure 7 presents the normalized and aggregated time series of the energy stock prices and energy prices when differentiated once. It shows increased volatility in the energy sector’s stock market values since the COVID-19 pandemic. The greater concentration of peaks in the energy-price series corresponds to the same time range, indicating that the volatility in the stock market is reflected in the energy prices. This period of increased volatility highlights the energy prices’ sensitivity to California stock market fluctuations.

By analyzing these figures, it is evident that the energy prices were significantly influenced by the volatility in the stock market, with a noticeable lag of 83 to 90 and 98 to 102 days. This relationship is crucial for stakeholders and investors as it provides insight into how market dynamics in the energy sector respond to economic and health events, such as the COVID-19 pandemic.

As seen in Section 4.5, since the COVID-19 pandemic, the stock market values of the energy sector in California have been more volatile. Furthermore, as observed in Figure 7, this period also corresponds to the time range that had a greater concentration of peaks in the energy prices in California.

As depicted in Figure 8a,b, the lag values extracted from the windowed cross-correlation analysis yielded a more representative alignment of both time series after the pandemic. In other words, it was perceived that the stock market’s volatility was reflected in the volatility of energy prices for consumers in California after approximately 100 days.

The Granger causality test was conducted to investigate the causality relationship between California’s energy stock market and the energy prices. Figure 9 shows the p-values for the test between the different time series for 0 to 300 lags.

Figure 9a,b evidence a reciprocal causality relationship between the energy stock market prices and the energy prices in California. This was due to the p-values being below the significance level of 5%, especially between 50 and 250 lags, thus suggesting acceptance of $H_1$. This indicated the existence of a causality relationship. This means that the past values of both time series contained helpful information to predict the future values of the other.

To serve as a baseline, the test was replicated with a noisy stationary time series of the null mean and standard deviation that were equal to 1. Figure 9c,d evidence that neither time series presented a Granger causality with this noise, as displayed by the p-values being consistently above the significance value. This reinforces the relationship between the energy stock market and the energy prices in California.

As seen in Section 4.5, the energy stock market in California is currently under a predominance of State 2, which signifies greater volatility, thus suggesting that the energy prices in California are also under a volatile state.

The persistence of high volatility in the energy stock market, especially in the post-pandemic period, has significant implications. For investors, this volatility requires more robust risk management strategies and a greater emphasis on volatilization. Additionally, policymakers must consider the underlying causes of this sustained volatility, including the ongoing economic uncertainties and energy-market dynamics.

4.8. Discussion of the Results

These results may be valuable for investors, regulators, and economic studies, specifically for the energy sector. The presence of unit root in the time series of both countries and of California, as seen in Section 4.2, suggests that shocks to stock prices can have permanent effects, creating persistent movements in the values rather than short-term fluctuations around a stable mean. Such shocks could arise from policies and regulatory changes, for example.

The symmetrical distribution of the values in the time series, as observed in Section 4.3, may indicate that positive and negative price movements are equally likely, reflecting balanced market sentiment and reactions to information. This is also a desirable feature in volatility modeling with GARCH, as shown in Section 4.4, where a greater volatility was predicted in the Brazilian market when compared to the US and California.

A financial shock could also come as consequence of a global health crisis, such as COVID-19. The pandemic resulted in significant disruptions in the global market, including the energy sector. This disturbance led to a significant spike in volatility, with an initial expectation of normalization as economies reopened. Section 4.5, however, shows that the high volatility state induced by the disease is still in effect in the USA, including California, and in Brazil. This long-lasting state may be a consequence of the presence of unit roots, which are implied in more persistent effects.

Section 4.6 demonstrates that the Diebold–Yilmaz index indicated a significant spillover effect between USA and Brazil and between California and Brazil. This implies that financial shocks in the USA and California influence volatility in the Brazilian market. However, this influence is asymmetrical, as the DY index from Brazil to the USA and to California is low.

Ultimately, Section 4.7 used the Granger causality test to evidence a bidrectional causal relationship between the energy stock market and energy prices in California, implying that movements in one variable contain predictive information about future movements in the other. Thus, incorporating lagged variables from both energy stock market prices and energy prices in California allows for a better anticipation of market movements and for the adaptation of pricing strategies in response to market fluctuations.

5. Conclusions

Estimating and predicting volatility becomes more urgent in periods of significant market turbulence when asset prices exhibit higher volatility. Traditionally, a high level of variation in the indices and prices of financial assets has been a sign of periods of crisis. Thus, the capacity to analyze current volatilities and to predict the future allows agents to choose the best strategies for maximizing their positions depending on their level of risk aversion.

Thus, a certain degree of stability is necessary for some sectors of the economy, such as electricity, where investments have long-term maturity periods and require significant amounts of capital. This is because there is an urgent need for these companies to have access to capital in the market, and equity investors have low liquidity in these assets. In light of this, when there is a greater volatility in electricity company shares, it leads to a better perception of risk in the sector. However, this significantly restricts its growth capacity as financial market agents and institutions begin to charge higher rates of return on these investments because of the greater degree of uncertainty.

This study provides a detailed analysis of the volatility in the energy stock markets of Brazil and the U.S., particularly in the aftermath of the COVID-19 pandemic. The findings suggest that the pandemic has resulted in a lasting regime of high volatility, which remains high even years after its onset. Since the COVID-19 pandemic, the volatility regime has not consistently switched back to the pre-pandemic regime in either the Brazilian or American markets.

The application of the GARCH model proved to be effective in predicting this volatility of returns, which means that it is useful when conducting a financial analysis. In addition, the Granger causality test revealed that there is a significant relationship between the stock market volatility and energy prices in California, which suggests that fluctuations in energy stock prices can predict changes in the energy prices, with a lag of approximately 83–90 and 98–102 days.

In general, the results of the adjustments obtained with the aid of the models showed the relationships to be as expected. The results obtained from the causality and spillover tests can assist in improving the estimates of the short-term energy stock prices in the Brazilian market and can be used by companies operating in the sector to enhance energy-oriented production planning and marketing (based on previous observations of the market in the USA). The proposed model helps make more accurate estimates of the PLD by taking into account the presence and implications of the parameter variation method throughout the analyzed sample.

Future research should explore the specific factors that lead to this sustained volatility and should design more refined models to predict market behavior under varying economic conditions. We also seek to assess the volatility in the energy market through using other models such as EGARCH and TGARCH, and we seek to compare the obtained results in the energy sector with larger stock markets, such as Bovespa and Dow Jones. Additionally, we also suggest exploring how cyber-security incidents can impact the volatility of stock prices with a similar approach to that used on this work. The volatility of the stock prices of a company before and after it has been breached should be compared and the persistence of the shock in reaction to an incident should be evaluated.