2.1 Renewable energy markets

The renewable energy market is expected to expand further as global commitments to mitigate climate change intensify. This has contributed to the growth of the clean energy finance literature, which examines renewable energy markets in relation to financial, economic, technological, and policy-related factors. Existing studies investigate renewable energy assets alongside technology stocks, crude oil prices, broad equity markets, and macroeconomic conditions to better understand the drivers and risk characteristics of renewable energy market dynamics.

A substantial strand of the literature examines the interrelationships between renewable energy markets, conventional energy markets, and technology stocks. Existing studies document significant dependence, causality, spillovers, and transmission mechanisms across these markets [10,44–49]. Empirical evidence suggests that renewable energy equity returns are influenced by both oil price dynamics and technology sector performance, while nonlinear and tail-dependent relationships are also widely documented through copula-based and multi-horizon analyses [46,47]. Additional studies report substantial spillover effects between clean energy, technology, crude oil, and broader equity markets, highlighting the strong interconnectedness of renewable energy assets with global financial and energy systems [48,49].

Another important body of research focuses on the political, economic, and financial risks associated with renewable energy investments. Existing evidence suggests that renewable energy projects are particularly sensitive to financing conditions, policy uncertainty, and market-related risks due to their capital-intensive and innovation-driven nature [50–53]. While some studies report that socially responsible and renewable energy investments do not significantly underperform conventional benchmarks [50], others argue that renewable energy assets may offer relatively modest risk-adjusted returns [51]. Additional evidence identifies policy design risk and financing constraints as important barriers to renewable energy deployment, particularly during the early stages of investment [52,53].

Uncertainty arising from broader economic and financial conditions has also been shown to affect renewable energy markets. Using conditional VaR-based frameworks, some studies find that uncertainty originating from financial and oil markets exerts a stronger influence on energy stock markets than policy uncertainty, while policy uncertainty itself appears more pronounced in renewable energy markets than in traditional energy markets [54]. Additional research documents bidirectional causal relationships between geopolitical risk and renewable energy markets, highlighting the sensitivity of clean energy investments to global political developments [55].

More recent contributions examine heavy-tailedness and tail risk properties of stock returns across different markets, with particular emphasis on the role of financial crises. [36] analyze excess volatility and tail behavior in developed and emerging stock markets and show that, while developed markets often conform to the Cubic Law of Stock Returns, emerging markets exhibit heavier tails. Importantly, after the 2008 financial crisis, both developed and emerging markets experienced a decline in their tail indices, indicating more pronounced heavy-tailedness. Focusing on the Chinese A- and H-share markets, [56] compare tail risk properties under different institutional environments and find that tail indices are similar across markets, with some returns potentially exhibiting infinite second moments. Further analysis of the 2015 Chinese stock market crash reveals structural breaks in heavy-tailedness and identifies liquidity and firm size as key determinants of tail risk, while also showing that tail properties did not revert immediately after the crisis [57].

Traditional asset pricing frameworks such as the Capital Asset Pricing Model (CAPM) and multifactor models have also been applied to renewable energy equities to assess their risk-return characteristics [58]. Empirical findings remain mixed. Some studies report relatively strong risk-adjusted performance and evidence of a green equity premium [59,60], whereas others document periods of weak or negative returns [51,59]. These mixed results underscore the heterogeneity of performance outcomes in renewable energy markets and suggest that renewable energy assets cannot be characterized uniformly from a risk-return perspective.

Against this background, the existing literature provides important insights into the drivers, interdependencies, and risk characteristics of renewable energy markets. At the same time, differences in geographic coverage, sectoral composition, and index construction across studies raise questions regarding the comparability of empirical findings. These issues are addressed in the following subsection, which reviews the renewable energy indices commonly employed in empirical research.

2.2 Renewable energy indices

A range of renewable energy equity indices has been employed in the literature to analyze clean energy markets. These indices differ in geographic coverage, sectoral composition, and weighting methodology, which may affect their comparability and the interpretation of empirical findings, particularly in the context of distributional properties and extreme market behavior.

One of the most widely used renewable energy indices is the WilderHill Clean Energy Index (ECO). This equal-dollar-weighted index tracks publicly traded U.S. firms operating in clean energy sectors such as wind, ethanol, solar, and hydrogen fuel cells [61]. As one of the earliest indices designed to capture clean energy and climate-related solutions, ECO remains an important benchmark in the literature and has been employed in studies examining renewable energy market dynamics, uncertainty transmission, causality, and volatility behaviour [62–64]. Owing to its U.S.-centric composition, ECO primarily reflects developments in the North American clean energy market.

The S&P Global Clean Energy Select Index (SPGCE) provides broader international exposure by tracking firms engaged in clean energy-related activities across developed and emerging markets [65]. Its global scope has motivated its use in studies examining dependence structures, safe-haven properties, and interactions between clean energy and commodity markets [66–68]. The inclusion of firms from emerging economies may expose SPGCE to additional sources of regulatory, geopolitical, and market-related risk, potentially influencing its tail behavior relative to regionally concentrated indices.

The European Renewable Energy Index (ERIX) focuses on renewable energy firms operating within Europe, including companies active in wind, biomass, marine, water, and geothermal energy. ERIX has been used to examine the effects of non-renewable energy shocks, carbon risk management, and renewable energy market dynamics within the European institutional environment [69,70]. Its regional focus reflects regulatory and market conditions specific to European energy markets, which may shape both return dynamics and extremal risk characteristics.

In contrast to the broader clean energy indices discussed above, the MAC Global Solar Energy Index (SUN) focuses specifically on firms operating in the solar energy sector, including equipment manufacturers and suppliers of services and materials to solar technology producers. This index has been employed in several empirical studies of renewable energy markets [66,71–73]. Due to its sector-specific composition, SUN may be particularly sensitive to technology cycles, supply-chain disruptions, and policy interventions, potentially generating tail behavior distinct from that of more diversified clean energy indices.

To facilitate comparison with conventional energy markets, the present study also considers the Dow Jones U.S. Energy Index (DJUSEN) as a benchmark. DJUSEN tracks large U.S.-based firms engaged in traditional energy activities, including oil and gas exploration, production, and related services. Due to its U.S.-centric composition and focus on conventional energy production, DJUSEN reflects exposure to different regulatory frameworks, commodity price dynamics, and market structures than renewable energy indices, making it a suitable benchmark for assessing differences in tail behavior between clean and conventional energy markets [74].

The indices considered in this study differ materially in terms of geographic exposure, sectoral focus, and weighting methodology. ECO is U.S.-focused and equal-weighted, SPGCE represents a global renewable energy universe and is market-capitalization weighted, SUN is sector-specific with a focus on solar energy firms, and DJUSEN serves as a conventional U.S. oil and gas benchmark. This heterogeneity enhances the external relevance of the analysis by capturing alternative market representations of energy assets. Differences in weighting methodology may also influence the degree of exposure to smaller and potentially more volatile firms, which could affect observed tail behavior across indices. At the same time, it limits direct causal attribution of cross-index differences in tail behavior to any single index characteristic.

2.3 Heavy-tailedness and inference

Heavy-tailed distributions refer to probability distributions whose tails are not exponentially bounded, in contrast to thin-tailed distributions such as the normal distribution [75]. Compared with thin-tailed distributions, heavy-tailed distributions exhibit slower tail decay and therefore assign relatively higher probability to extreme realizations [21]. The presence of such extremes affects sample statistics and their asymptotic properties, often resulting in inflated estimates of higher-order moments. Examples include Pareto, Student-t, stable, and other power-law distributions [15,21,23,76].

A broad empirical literature across economics, finance, insurance, risk management, and related disciplines documents that many variables of interest, including financial returns and exchange rates, deviate from Gaussianity and exhibit heavy-tailed behavior, often in the form of power-law tails [15–21,23,76]. Early contributions by [77] and [78] established the presence of heavy-tailed characteristics in financial returns and commodity prices, motivating a large subsequent literature. Later theoretical and empirical studies, particularly [18,79,80], link the emergence of power-law tails to economic mechanisms associated with the Zipf distribution of firm and investor sizes and their trading impact. Heavy-tailedness therefore plays a central role in assessing the likelihood of extreme market fluctuations and is commonly quantified through tail indices.

The degree of heavy-tailedness is measured by tail indices, which characterize both (i) the rate of power-law decay of distributional tails and (ii) the highest order of finite moments of the underlying stochastic variable [21]. For a return or economic variable (e.g., a measure of risk) that follows a heavy-tailed power-law distribution, the tail behavior can be expressed as follows:

(1)

(2)

where , and . Defining , one obtains

(3)

with as . The parameters , , and are referred to as the tail index and the right and left tail indices, respectively. Smaller values of imply heavier tails, slower decay of extreme probabilities, and a lower order of finite moments. Consequently, lower tail index values correspond to a higher likelihood of observing extreme outcomes [21].

The tail index directly determines moment existence. If , the mean is finite; if , the variance is finite; and if , the fourth moment and kurtosis are finite. Empirical evidence from financial and commodity markets indicates that tail indices for stock returns and exchange rates in developed markets often lie in the range of , implying finite means and variances but infinite fourth moments [18–20,24,80]. At the same time, returns in both developed and emerging markets may exhibit tail indices below 2, corresponding to infinite variances [21,25,26,56,57]. Related evidence from wealth and income distributions suggests tail index values around for wealth and for income, indicating potentially infinite variances and higher moments [19–21].

The literature proposes several approaches for estimating tail indices of heavy-tailed distributions, with comprehensive reviews provided in [15,21,81]. Two of the most widely applied methods are (i) Hill’s estimator [82], also known as a quasi-maximum likelihood estimator, and (ii) OLS estimation based on LLRS regressions [21,38]. Although alternative and more complex estimation techniques exist, Hill’s estimator and LLRS regressions remain popular due to their relative simplicity and widespread applicability in empirical work.

An important practical consideration in tail index estimation concerns the choice of the tail threshold or truncation level. Diagnostic tools such as mean excess plots and the stability of Hill estimates across alternative threshold choices are commonly discussed in the EVT literature. At the same time, many empirical studies adopt fixed tail fractions, such as 5% or 10%, to facilitate comparability across assets and to mitigate small-sample instability in financial return data. Consistent with this established practice, the present study employs fixed truncation levels while emphasizing statistical inference based on CI rather than reliance on point estimates alone. Lower truncation levels may capture more extreme observations but can also increase estimation uncertainty due to the smaller effective number of tail observations, particularly in finite samples.

In empirical applications, observed tail behavior may deviate from an exact Pareto specification due to slowly varying components and nonlinear dependence structures typical of financial time series. Importantly, LLRS regression with the optimal rank shift and corrected SE has been shown to exhibit favorable finite-sample performance and robustness under such deviations, including in the presence of volatility clustering and GARCH-type dynamics, as documented in [38]. In this setting, tail index estimation is interpreted as characterizing effective tail thickness and moment implications rather than testing a literal Pareto data-generating process.

Overall, statistically justified tail index estimation is essential, as inappropriate modeling of tail behavior may lead to misleading inference and economic conclusions. These considerations motivate the emphasis on robust inference procedures and CI-based interpretation when assessing heavy-tailedness and extremal risk in financial markets.

3.1 Hill’s estimator

Hill’s estimator is a semi-parametric method for estimating the tail index of a heavy-tailed distribution. It approximates the upper tail of the return distribution by a Pareto law and estimates the corresponding tail index using a maximum likelihood principle. Owing to its conceptual simplicity and long-standing use in empirical finance, Hill’s estimator provides a natural benchmark for assessing the degree of tail risk in asset returns.

Consider a sample drawn from a population whose tail behavior follows a power-law distribution as defined in Eqs. (1)-(3) with tail . Let and define the ordered absolute returns as:

(4)

where denotes the -th largest absolute return. In empirical applications, the truncation level is typically selected as a fixed fraction of the total sample size, such as 5% or 10% of . This choice reflects a trade-off between bias and variance in tail index estimation. Lower truncation levels focus more directly on the most extreme observations but may also increase estimation uncertainty due to the smaller effective number of tail observations, particularly in finite samples. This effect is typically reflected in wider CIs at lower truncation levels. Further details on truncation rules are provided in [21,25].

Under these conditions, Hill’s estimator of the tail index associated with the absolute-tail power-law model in Eq. (3) is given by:

(5)

The corresponding SE is:

(6)

which yields the following 95% CI for :

(7)

Hill’s estimator possesses desirable efficiency properties under the assumption of independent and identically distributed (i.i.d.) observations drawn from an exact power-law distribution. In such settings, it coincides with the maximum likelihood estimator (MLE) and exhibits standard asymptotic behavior [21,83]. However, empirical studies have shown that Hill-based inference is sensitive to finite samples, heterogeneous observations, and serial dependence, all of which may lead to biased tail index estimates and understated SE. These limitations are particularly relevant in financial applications, where return series exhibit volatility clustering and temporal dependence in extreme observations, violating the i.i.d. assumption [15,19,21,38,83]. Illustrations of poor finite-sample performance under dependence and deviations from exact power-law behavior are provided by the so-called Hill horror plots in Chapters 4–6 of [15].

To address these shortcomings, several alternative tail index estimators have been proposed, including weighted variants of Hill’s estimator [84] and more sophisticated nonlinear estimation procedures [15,37]. Despite these developments, the LLRS regression framework remains one of the most widely used approaches for tail index estimation in empirical research. Its continued popularity reflects both its analytical simplicity and its robustness properties when appropriately implemented. In particular, LLRS regressions with an optimal rank shift and corrected SE have been shown to perform well in settings characterized by dependence and deviations from idealized power-law assumptions, as documented in [38] and discussed further in [21].

3.2 OLS LLRS regression

While Hill’s estimator provides a widely used benchmark for tail index estimation, an alternative and equally prominent approach is based on LLRS regressions. This method exploits the linear relationship implied by power-law tails between the logarithm of an observation’s rank and the logarithm of its magnitude and has been extensively employed in empirical studies of heavy-tailed financial and economic variables due to its simplicity and flexibility.

To estimate the tail index using the conventional LLRS approach, one typically considers the following OLS regression with shift parameter , as discussed in [21,38]:

(8)

where denotes the rank of the ordered absolute return , defined as in Eq. (4), and is the tail truncation level. Equivalently, interpreting as the rank and as the size of the observation, the estimated regression can be written as:

(9)

Under the power-law model for absolute returns given in Eq. (3), the slope coefficient obtained from Eqs. (8) or (9) provides an estimate of the tail index . As with Hill’s estimator, the LLRS framework can be applied separately to positive and negative returns in order to estimate the right and left tail indices and defined in Eqs. (1) and (2), respectively.

The LLRS regression approach to tail index estimation has been widely adopted in the empirical literature on heavy-tailed phenomena, including applications in finance, insurance, and macroeconomic risk analysis. Representative examples include [85–87], among many others. Its appeal stems from its transparent interpretation, ease of implementation, and direct link to the underlying power-law structure of tail distributions.

Despite these advantages, it is well known that the standard LLRS estimator based on may suffer from substantial finite-sample bias, particularly when the number of tail observations is limited. This limitation motivates refinements of the LLRS framework that improve its small-sample performance and robustness properties.

3.3 OLS LLRS regression with optimal shift

As established in [38], the OLS estimator obtained from the standard LLRS regression in Eq. (8) with is consistent for the tail index under idealized conditions. However, this estimator exhibits pronounced finite-sample bias, which is particularly relevant in empirical applications where the effective number of tail observations is limited. To address this limitation, [38] propose a simple yet effective correction based on introducing an optimal rank shift of into the LLRS regression.

Specifically, the modified LLRS approach consists of estimating the following OLS regression:

(10)

As in the standard LLRS framework, the slope coefficient from Eq. (10) provides an estimate of the tail index . Throughout the paper, the resulting bias-corrected estimator is denoted by .

The rank-shift of 1/2 (implemented as ) is optimal in the sense that it eliminates the leading-order finite-sample bias of the LLRS estimator. Moreover, [38] derive the correct asymptotic SE for , given by:

(11)

Based on these corrected SE, the corresponding 95% CI for the tail index is:

(12)

Importantly, the order statistics used in tail index estimation are dependent even when the underlying sample is i.i.d., since the ranking procedure itself induces cross-sectional dependence across observations. For this reason, the asymptotic SEs derived in [38] differs from conventional OLS SE and incorporate the correction in Eq. (11), yielding valid CIs for tail index inference. As demonstrated in their theoretical analysis and numerical evidence, the shifted LLRS estimator with corrected SE remains robust under empirically relevant dependence, including volatility clustering and GARCH-type dynamics.

An important advantage of the LLRS regression with optimal rank shift is its robustness to deviations from exact power-law behavior, including slowly varying components that may arise in empirical tail data. As documented in [38], the estimator exhibits favorable finite-sample performance under such departures from idealized Pareto specifications. These properties make the shifted LLRS framework particularly suitable for empirical tail-risk analysis in financial markets. The LLRS regression with optimal rank shift and corrected SE has subsequently been employed in a wide range of empirical studies, including [25–27,39,56,57,88,89], among others. Owing to its favorable finite-sample properties and theoretical robustness, this approach constitutes the primary tail index estimation method used in the present study.

To complement these theoretical robustness properties, we assess serial dependence and volatility clustering in the return series using Ljung-Box diagnostics (Appendix Table A1 S1 Table), apply GARCH(1,1) filtering to extract standardized return innovations (Appendix Fig A1 S1 Fig), and re-estimate tail indices using both raw and filtered returns as a robustness check (Appendix Table A2 S2 Table). Comparing tail index estimates obtained from raw returns and GARCH-filtered residuals provides additional insight into the sources of heavy-tailedness in renewable energy equity returns. While filtering reduces the influence of volatility clustering and conditional heteroskedasticity, the persistence of heavy-tailed behavior in standardized residuals indicates that extremal risk cannot be attributed solely to time-varying volatility dynamics. The filtered-return analysis is therefore interpreted as a robustness assessment of whether heavy-tailedness primarily reflects conditional volatility effects or broader unconditional distributional properties.

Throughout the paper, and denote the Hill and shifted LLRS estimates of the tail index associated with the power-law model in Eq. (3). The corresponding SE and 95% CI are denoted by , , 95% and 95% , as defined in Eqs. (6), (11), (7), and (12), respectively. Estimates of the right and left tail indices are denoted by , and , with analogous notation used for their corresponding SE and CI.

3.4 Data

This study analyzes four renewable energy equity indices and one conventional energy index to examine tail risk and heavy-tailedness properties across clean and traditional energy markets. The renewable energy indices considered are the WilderHill Clean Energy Index (ECO), the S&P Global Clean Energy Index (SPGCE), the European Renewable Energy Index (ERIX), and the MAC Global Solar Energy Index (SUN). The Dow Jones U.S. Oil and Gas Index (DJUSEN) is employed as a benchmark representing conventional energy assets. Together, these indices provide coverage across different geographic regions (United States, Europe, and global markets) and sectoral scopes (broad renewable energy versus solar-specific activity), allowing for meaningful cross-index comparisons.

Daily index price data are obtained from the Bloomberg Terminal. The price series cover the period from 1 April 2005–25 February 2025. Daily log returns are computed as where denotes the index price at time .[All main empirical conclusions remain qualitatively unchanged when arithmetic returns are used instead.] Log returns are commonly used in extreme value and tail-risk analysis and provide a convenient representation for asymptotic tail inference. This transformation ensures scale invariance, facilitates comparability across indices with different price levels, and aligns the empirical framework with standard practice in financial econometrics.

Due to the log-return transformation, the effective return sample begins on 4 April 2005 and ends on 25 February 2025. The number of available observations differs slightly across indices because of exchange-specific trading calendars, public holidays, and index-level data availability. The final sample sizes are 5,008 observations for ECO, 5,176 for SPGCE, 5,112 for ERIX, 5,169 for SUN, and 5,021 for DJUSEN. All data transformations and subsequent empirical analyses were implemented using R.

A summary description of the indices and their defining characteristics is provided in Table 1. These indices are widely employed in empirical studies of renewable and conventional energy markets and offer a suitable basis for examining tail risk and heavy-tailedness across alternative geographic and sectoral representations of energy assets.

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Table 1. Renewable energy and conventional energy indices.

https://doi.org/10.1371/journal.pone.0351106.t001

4.1 Return plots and summary statistics

Fig 1 presents the evolution of price levels (Panel A) and daily log returns (Panel B) for renewable energy indices (ECO, SPGCE, ERIX, SUN) and the conventional energy index (DJUSEN) over the full sample period. The price series exhibit pronounced boom-bust dynamics, particularly during major global stress episodes, while the return series display clear volatility clustering and episodic extreme movements. Visual inspection suggests broadly similar volatility patterns across renewable and conventional energy markets, especially during periods of systemic financial stress.

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Fig 1. Price and return dynamics of renewable and conventional energy indices.

Notes: Panel A shows price levels, and Panel B shows daily log returns for ECO, SPGCE, ERIX, SUN, and DJUSEN over the sample period. Grey shaded areas in Panel B denote periods associated with major global stress events, namely the 2007–2008 global financial crisis (GFC) and the COVID-19 market shock in 2020–2021.

https://doi.org/10.1371/journal.pone.0351106.g001

Two major episodes of heightened volatility are apparent as highlighted in Fig 1 Panel B. The first corresponds to the 2007−2008 GFC, during which global financial markets and banking systems experienced severe distress and exceptionally elevated volatility (e.g., [90]). These conditions affected all major asset classes, including renewable and conventional energy markets. The second episode occurs in early 2020 following the outbreak of the COVID-19 pandemic. The unprecedented collapse in economic activity, combined with disruptions in energy demand and financialized commodity trading, culminated in extreme price and return movements, including the historically negative settlement price of West Texas Intermediate (WTI) crude oil futures in April 2020. Existing literature documents strong interdependencies between oil prices and renewable energy markets, implying that extreme return realizations in renewable energy indices during this period are consistent with broader energy-market dynamics (see Section 2.1).

Table 2 reports summary statistics for daily log returns of all indices considered. Average returns are close to zero across all series, which is typical for high-frequency financial data. Among renewable energy indices, SUN exhibits the highest return volatility, as measured by the standard deviation (SD), while SPGCE displays the lowest. Skewness estimates are uniformly negative, indicating a tendency toward larger downside than upside return realizations.

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Table 2. Summary statistics of daily log returns.

https://doi.org/10.1371/journal.pone.0351106.t002

Kurtosis and Jarque-Bera statistics strongly reject the null hypothesis of normality for all return series at conventional significance levels, confirming substantial departures from Gaussian behavior. These statistics are reported as descriptive diagnostics of non-normality and distributional asymmetry. However, in the presence of volatility clustering, dependence, and potentially infinite higher-order moments, kurtosis is not informative about tail heaviness. Accordingly, tail behavior and moment existence are formally evaluated using tail index estimates and CI-based inference, as reported in Section 4.2.

Augmented Dickey-Fuller (ADF) test results indicate that all price series are non-stationary in levels, while their corresponding return series are stationary across alternative deterministic specifications. These results, consistent with standard financial time-series properties, are reported in Appendix Table A3 S3 Table.

4.2 Tail risk and tail index estimates

Building on the descriptive evidence in Section 4.1 and the heavy-tailed inference framework outlined in Section 2.3, this section provides a formal assessment of tail risk in renewable and conventional energy markets using tail index estimation. The analysis focuses on overall tail behavior and potential gain-loss asymmetries, with statistical inference conducted using CI-based methods rather than reliance on point estimates alone.

Table 3 reports tail index estimates obtained using the LLRS regression approach with the optimal rank shift of −1/2, together with corresponding estimates based on Hill’s estimator. Results are presented for two truncation levels, 10% and 5%, retaining the largest and extreme observations, respectively, from a total sample of size . These truncation choices are standard in the EVT and financial tail-risk literature and facilitate comparability across assets and studies (see [15,25,56,91,92]). For each estimator and truncation level, standard errors and 95% CIs are reported.

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Table 3. The tail index estimates for renewable energy and conventional energy indexes.

https://doi.org/10.1371/journal.pone.0351106.t003

To support the choice of truncation levels and assess robustness, Fig 2 presents truncation-stability plots of LLRS-based tail index estimates across a wider range of tail fractions, from 2.5% to 15%, together with their corresponding CIs. CIs widen at lower truncation levels because the effective number of tail observations decreases as estimation focuses more narrowly on the most extreme realizations, thereby increasing estimation uncertainty in finite samples. The estimates exhibit a relatively stable pattern across truncation levels, and the substantial overlap of CIs indicates that differences across thresholds are not statistically distinguishable. This suggests that the inferred power-law tail behavior of renewable energy returns is robust to the choice of truncation level and that the core conclusions regarding the (in)finiteness of moments are not driven by a specific threshold selection. Across this range, the effective number of tail observations varies from approximately 125 to over 750 observations, ensuring that asymptotic approximations are meaningful while avoiding undue contamination from the central part of the return distribution.

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Fig 2. LLRS-based tail index estimates across alternative truncation levels.

Notes: Fig 2 displays LLRS-based tail index estimates for renewable energy indices (ECO, SPGCE, ERIX, SUN) and the conventional energy index (DJUSEN) across truncation levels ranging from 2.5% to 15%. Black lines denote point estimates of the tail index, while red lines indicate the corresponding upper and lower 95% CI. Truncation levels represent the fraction of the largest absolute return observations retained in the tail estimation.

https://doi.org/10.1371/journal.pone.0351106.g002

Differences in estimated tail indices across renewable energy indices should therefore be interpreted as descriptive evidence of heterogeneous tail risk profiles rather than as outcomes driven by a single underlying factor. While broader geographic exposure, sectoral breadth, or weighting schemes may plausibly contribute to heavier tails, particularly for global indices such as SPGCE, the present analysis does not seek to disentangle these effects. Accordingly, cross-index comparisons are framed in terms of robustness and CI-based inference rather than causal attribution.

Based on Table 3, tail index estimates for both renewable and conventional energy indices are consistent with heavy-tailed return distributions characterized by power-law decay. Across indices and estimation methods, estimated tail indices exceed the threshold associated with infinite means and are generally above the boundary for finite second moments (i.e., ), while remaining below values typically associated with thin-tailed behavior. These findings align with extensive evidence documenting heavy-tailed returns in financial and commodity markets [21,25,93].

A formal comparison based on the reported CIs indicates that tail index estimates for renewable and conventional energy indices are not statistically distinguishable at conventional significance levels. Specifically, the CIs overlap across indices, truncation levels, and estimation approaches, implying that numerical differences in point estimates cannot be interpreted as statistically significant. Accordingly, CI-based inference does not support statistically meaningful differences in tail heaviness between renewable and conventional energy markets.

The tail index estimates in Table 3 further allow inference on the existence of statistical moments implied by power-law behavior. Under a power-law distribution, the magnitude of the tail index determines the highest finite moment, with implying a finite mean, implying a finite variance, and implying a finite fourth moment (see Section 2.3). Based on CI-based inference, the results provide strong support for the existence of finite first and second moments for both renewable and conventional energy indices. In particular, for all indices and estimation approaches, the LBs of the reported CIs lie above the threshold , indicating finite variances. At the same time, CIs frequently extend below the threshold , indicating that fourth moments and kurtosis may not be finite for several indices and specifications.

Inference regarding higher-order moments is less conclusive. In several cases, CIs extend across the threshold , preventing definitive conclusions regarding the finiteness of third moments. These findings are consistent with a broad body of empirical evidence documenting financial return distributions characterized by finite variances but potentially infinite higher-order moments [21,25,93].

In addition to overall heavy-tailedness, asymmetry between upward and downward extremes is a well-documented stylized feature of financial return distributions (see [14,21,25] and references therein). Gain-loss asymmetry reflects the tendency for large negative returns to occur more frequently or with greater magnitude than large positive returns, particularly during periods of market stress. Empirical evidence further indicates that adverse shocks often exert a disproportionate impact on the lower tail of return distributions, while the upper tail remains comparatively less affected [94]. Despite its relevance for downside risk assessment, tail asymmetry in the context of power-law behavior has received limited attention in renewable energy markets. To examine potential asymmetries in tail behavior, Tables 4 and 5 report estimates of right-tail (positive returns) and left-tail (negative returns) indices based on the power-law specifications in equations (1) and (2). Results are presented for both the LLRS regression approach and Hill’s estimator at the 10% and 5% truncation levels, together with their corresponding CIs.

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Table 4. The right tail index estimates for renewable and conventional energy indices.

https://doi.org/10.1371/journal.pone.0351106.t004

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Table 5. The left tail index estimates for renewable and conventional energy indices.

https://doi.org/10.1371/journal.pone.0351106.t005

Across indices and truncation levels, point estimates of right- and left-tail indices differ in magnitude, indicating variation in the relative thickness of upper and lower tails. In several cases, point estimates can indicate that downward return movements are associated with heavier tails than upward movements, while for other indices and truncation levels the opposite pattern is observed. These patterns are not uniform across assets or estimation approaches, highlighting heterogeneity in tail behavior. However, formal comparisons based on CIs indicate that right- and left-tail indices are generally not statistically distinguishable at conventional significance levels. In particular, CIs for positive and negative tail indices overlap across indices, truncation levels, and estimation methods. As a result, differences implied by point estimates should be interpreted as directional tendencies rather than statistically established asymmetries. Overall, the asymmetry analysis provides suggestive but statistically inconclusive evidence regarding differences between extreme positive and negative return movements, consistent with existing findings for stock returns and exchange rates and with the broader volatility literature, where asymmetric volatility responses do not necessarily translate into distinct tail decay rates [25,57].

To further explore the time variation in tail risk, we analyze the dynamic evolution of tail indices using a recursive expanding-window estimation approach. Recursive estimation is widely used in forecasting and spillover analyses to capture changes in distributional properties over time, particularly in the presence of persistent extreme events (see, e.g., [95,96]). Following [35], the recursive procedure begins with an initial estimation window of 500 daily observations and then expands the sample one observation at a time until the end of the sample period. Once major extreme events occur, their effects carry over into subsequent estimation windows, allowing recursive tail indices to capture the evolving degree of tail heaviness through time. Unlike rolling-window approaches, recursive expanding-window estimation permanently retains the influence of past extreme events in subsequent estimation windows. Consequently, recursive tail indices primarily capture the persistence and long-run influence of crisis-period extremes rather than short-term mean reversion in tail behavior. A rolling-window framework may therefore produce more transitory fluctuations in estimated tail risk, whereas the recursive approach adopted here is better suited for assessing the lasting impact of systemic shocks on tail dynamics.

Fig 3 reports recursive tail index estimates for renewable and conventional energy indices based on a fixed truncation level of 10%, using the LLRS regression approach in Panel A and Hill’s estimator in Panel B. Across indices and estimation methods, the recursive estimates exhibit a pronounced decline during the 2007–2008 GFC, indicating a substantial increase in tail heaviness during this period of systemic financial stress. For certain indices, recursive tail estimates temporarily fall to levels consistent with , implying that variance may not be finite during crisis periods. Such episodes highlight that standard finite-variance risk models may become unreliable during severe market stress, even if tail behavior stabilizes in subsequent periods.

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Fig 3. Recursive tail index estimates for renewable and conventional energy indices.

Notes: Fig 3 reports recursive tail index estimates based on daily log returns using a recursive expanding window with an initial estimation window of 500 observations and a fixed truncation level of 10%. Panel A reports LLRS-based estimates using the optimal rank shift of 1/2, while Panel B reports estimates based on Hill’s estimator. Black lines denote point estimates and red lines indicate the corresponding 95% CIs. Grey shaded areas indicate the 2007–2008 GFC and the COVID-19 market shock in 2020–2021. Y-axes are standardized within each index to facilitate comparison of tail dynamics over time.

https://doi.org/10.1371/journal.pone.0351106.g003

After the GFC, recursive tail indices stabilize and evolve more gradually, indicating that the heightened tail risk induced by crisis-period extremes persists over time but does not continue to intensify. The post-crisis dynamics are broadly similar across renewable and conventional energy markets, suggesting that tail risk in clean energy assets responds to major global shocks in a manner comparable to traditional energy assets. A more modest change in tail behavior is observed around the onset of the COVID-19 pandemic in early 2020, particularly for the conventional energy index. The comparatively more modest tail-risk response during the COVID-19 period may reflect important differences in the underlying nature of the two crises. Whereas the GFC originated within the financial system and was characterized by severe banking distress, prolonged deleveraging, and widespread uncertainty regarding financial solvency, the COVID-19 shock was largely exogenous and accompanied by rapid monetary and fiscal policy interventions. These measures may have limited the persistence of systemic financial stress and contributed to a comparatively smaller deterioration in tail behavior across energy markets. While renewable energy indices also exhibit mild changes during this period, these shifts are considerably smaller than those observed during the GFC, consistent with broader evidence indicating the COVID-19 shock did not generate tail risk effects comparable to earlier systemic crises [42].

Taken together, the recursive analysis complements the static tail and asymmetry results by emphasizing the time-varying and state-dependent nature of tail risk in energy markets. Renewable energy investments display heavy-tailed dynamics similar to those of conventional energy assets, with tail risk responding strongly to major global shocks while remaining comparatively stable during more recent stress episodes. These results motivate a broader discussion of their implications for risk measurement, portfolio construction, econometric modeling, and policy considerations in renewable energy markets.