Regime-dependent dynamics of nuclear and coal energy in high-income economies: evidence from a Markov switching analysis

Introduction: Nuclear energy remains a key low-carbon source in high-income economies, whereas coal continues to account for a substantial share of greenhouse-gas emissions.

Methods: This study examines the regime-dependent behaviour of per capita nuclear and coal energy production from 1965 to 2023 using a Hidden Markov Model (HMM). Structural breaks, volatility shifts, and persistent production regimes are estimated via the Expectation–Maximisation algorithm and smoothed state probabilities.

Results: The results indicate strong persistence in the medium- and high-production regimes of nuclear energy, while coal exhibits a gradual yet sustained transition from high toward low regimes, consistent with long-term decarbonisation efforts.

Conclusion: Overall, regime-switching dynamics highlight differing adjustment paths: nuclear production shows cyclical shifts linked to policy/technology cycles, whereas coal reflects a long-run structural decline across high-income economies.

1 Introduction

Energy systems play a critical role in supporting economic development and societal welfare. There is a clear relationship between a country’s energy consumption and its level of economic development. Coal, a non-renewable energy source that played a significant role during the Industrial Revolution, contributes to the global energy supply and stands out as a critical transitional resource in the shift to renewable energy systems (Balat, 2007). However, countries’ climatic, political, social, economic, and technical conditions influence energy decisions. Nuclear energy production is considered a sustainable option for meeting national energy demand and contributes to the broader energy transition process (Gralla et al., 2017).

Throughout centuries of industrialization, many national economies remained dependent on fossil fuels such as coal, petroleum, and natural gas. The first nuclear power plants that became operational in the 1950s served as an alternative to fossil fuels and provided many countries with cleaner energy options. The sudden surge in oil prices during the 1970s oil crisis led numerous countries to accelerate their nuclear energy programs. The shift toward renewable energy sources has only gained momentum in recent years. Nuclear energy is often regarded as safer than fossil fuels. Although not an alternative or renewable energy source, the materials used in nuclear power plants are based on renewable energy (Usman et al., 2022).

Due to their unique characteristics, nuclear power plants have become a preferred choice in countries’ energy production. Pursuing reliable, safe, competitive, sustainable, and accessible energy sources has elevated nuclear plants above other alternatives. One of the most significant advantages of nuclear power plants is their lack of carbon emissions during operation. This feature makes nuclear energy indispensable in combating the increasingly severe climate crisis. The signing of the Paris Climate Agreement has further underscored the importance of carbon-free energy sources. From a safety perspective, nuclear power plants contribute only about 1% to natural radiation levels. This minimal impact does not adversely affect agriculture, fishing, tourism activities, or the daily lives of local communities near the plants. Thanks to advanced safety systems, nuclear power plants can operate with minimal environmental impact (T. C. Energy and Natural Resources Ministry, 2025). In addition to these advantages, nuclear energy also poses challenges, including waste management, safety risks, high upfront investment costs, and public acceptance concerns, which are now acknowledged to ensure a more balanced discussion.

Recent literature increasingly emphasises that energy transitions in high-income economies do not occur through discrete structural shifts, but rather through technological change, regulatory shocks, and institutional capacity. Studies highlight that nuclear and coal follow distinct transition paths: nuclear often demonstrates long-term stability and low-carbon baseload potential, whereas coal is associated with carbon lock-in and environmental pressure (Nathaniel et al., 2021; Danish et al., 2022; Zhang et al., 2023). Moreover, nonlinear approaches-particularly Markov-switching models-have been widely applied to detect structural breaks, volatility changes, and regime-dependent behaviour in energy markets and emissions trajectories (Cevik et al., 2020; Murshed et al., 2022). Despite this growing body of work, existing studies rarely compare nuclear and coal production within a unified Markov-switching framework over a long historical period, creating a gap that the present study addresses.

The Markov Switching Models introduced by Hamilton (1989), significantly contribute to analyzing regime shifts in time series data. These models assume that the behavior of variables such as nuclear and coal energy production can be represented by different regimes that change stochastically over time. The defining feature of the Markov-switching model is that an unobserved state variable drives regime changes. This variable follows a stochastic process, specifically a first-order Markov chain, meaning future states depend solely on the current state. Consequently, the system can remain in a particular regime for a random duration before transitioning to another regime based on probability rules (Kuan, 2002).

In conclusion, despite the lack of renewable energy, nuclear energy production has emerged as a significant alternative to coal production for high-income countries. Markov switching models enable researchers to reveal the dynamics of nuclear and coal energy production, facilitating more informed decision-making in light of these findings. This study examines per capita nuclear and coal energy production in high-income countries¹ between 1965 and 2023. The core research question of this study is to identify and characterise the regime shifts in nuclear and coal energy production over time, and to determine whether these regimes exhibit different levels of persistence and transition dynamics across high-income countries. A Markov switching model was employed to capture regime-switching processes. Focusing particularly on the sustainability impacts of nuclear energy and coal, the study aims to contribute to the literature by investigating regime shifts in nuclear and coal production in high-income countries.

This study makes three fundamental contributions by providing a regime-based perspective on per capita nuclear and coal energy production through a robust nonlinear modeling framework. Unlike existing studies that employ MS-VAR, nonlinear ARDL, or panel approaches, this paper presents the first long-horizon univariate Markov-switching analysis that jointly evaluates nuclear and coal production regimes, offering a precise structural classification of long-term regime behavior. First, it provides a long-term empirical assessment of per capita nuclear and coal energy production in high-income countries for the period 1965–2023, accounting for regime shifts. This approach has not been applied in this context before. Second, while previous studies typically examine nuclear and coal energy production separately, this research analyzes both energy sources under a comparable Markov Switching Model structure, enabling a clearer understanding of regime-dependent dynamics and persistence characteristics. Third, the six countries selected for the analysis represent economies with mature energy systems, advanced technological infrastructures, and significant policy transitions-including market liberalization, nuclear expansion or phase-out decisions, and firm decarbonization commitments. Therefore, by focusing on high-income countries, the study addresses both the structural and policy-driven aspects of transitions in nuclear and coal energy production.

The study consists of five main sections: the first section discusses nuclear and coal energy production; the second presents a literature review; the third explains the model and data and provides information about Markov switching models; the fourth discusses empirical findings; and the final section offers policy recommendations for countries regarding nuclear and coal energy production.

2 Literature review

The substantial petroleum and natural gas reserves in certain countries, along with the non-renewable nature of these resources, have driven many nations to seek alternative energy sources. The carbon intensity and environmental impacts of traditional fossil fuels such as coal, petroleum, and natural gas have rendered nuclear energy an increasingly attractive option. Although nuclear energy also falls under the category of non-renewable resources, it is commonly recognized as a clean energy technology due to its zero carbon dioxide emissions during operation. With these characteristics, nuclear energy is one of today’s and tomorrow’s most strategic energy sources.

Due to the high emissions caused by coal production and its adverse effects on climate change, countries have begun gradually moving away from coal, with nuclear energy becoming a significant energy supply during this transition. While coal remains widely used in steel, electricity, and cement production, nuclear energy’s clean electricity generation capacity has become an increasingly important factor in shaping energy policies.

Optimally balancing nuclear and coal production dynamics during the transition to sustainable energy systems requires innovative approaches. This section summarizes research analyzing trends in nuclear and coal energy production using Markov-switching models and advanced econometric methods.

Recent studies indicate that energy transitions in high-income economies unfold through nonlinear, regime-dependent models. Nuclear energy is generally associated with long-term stability and low-carbon baseload potential (Nathaniel et al., 2021; Danish et al., 2022), while coal production is linked to carbon dynamics and environmental stress (Zhang et al., 2023; Aimon et al., 2023). Furthermore, researchers have increasingly used nonlinear models, particularly Markov transition frameworks, to detect structural breaks, volatility shifts, and asymmetric adjustments in energy production and consumption processes (Cevik et al., 2020; Murshed et al., 2022).

Despite growing research on Markov transition models, existing studies rarely analyze nuclear and coal production within an integrated, long-term framework of regime transition. Therefore, this study fills a significant gap by examining the regime dynamics of both energy sources together, providing a more comprehensive understanding of long-term transition models in high-income economies.

In the studies summarized in Table 1, it is observed that the periods addressed and the econometric analysis techniques used regarding nuclear energy and coal production differ. When the studies are evaluated as a whole, it is concluded that nuclear energy positively affects economic growth and generally reduces CO₂ emissions. It is also noteworthy that nuclear energy has emerged as a significant alternative in terms of environmental sustainability. On the other hand, coal production is seen to negatively impact ecological sustainability by increasing CO₂ emissions. In this context, it can be stated that there are apparent differences between nuclear energy and coal production in terms of their environmental and economic impacts.

Table 1

Table 1. Some research on nuclear energy and coal production and their results.

3 Model and data

This study employs a nonlinear time series framework-namely, the Markov Switching Model (MSM)-to investigate regime-dependent patterns in per capita nuclear and coal energy production across high-income countries. These models are particularly well suited for analysing economic and energy data characterized by structural breaks and state-dependent volatility, which linear models often fail to capture.

Markov-regime models are based on a series of chain-dependent experiments conducted by Andrey Andreyevich Markov in 1906–1907. In this study, In this work, Markov mathematically described the behaviour of gas molecules (Brownian motion) in a closed container. The first correct mathematical formulation was developed by N. Wiener in 1923. Later, the general theory of the process was produced between 1930 and 1940 by Andrey Nikolaevich Kolmogorov, William Feller, Wolfgang Doeblin, Paul Lévy, and Joseph Leo Doob. Markov analysis is a method of examining the current movement of a variable to predict its future behaviour (Thierauf, 1970).

Markov regime-switching models are classified among nonlinear time series models and were developed as an alternative to the fixed-parameter linear time series modeling approach proposed by Box and Jenkins (1970). Unlike other nonlinear time-series models, this approach considers economic expansions, contractions, and growth–recession phases as distinct regimes, and it probabilistically evaluates which regime the economic cycle is in. Transitions between regimes are determined based on unobservable variables. These unobservable variables can be modeled using a Markov chain with discrete stochastic regime variables with fixed or time-varying transition probabilities (Dursun, 2023). Each process is linear within the Markov-regime model, but their combination is not. Therefore, it falls under the category of nonlinear models (Hondroyiannis and Papapetrou, 2016).

Although early studies such as those by Burns and Mitchell (1946) and Neftci (1984) drew attention to asymmetries in time series behavior, Hamilton’s pioneering study in 1989 allowed for regime transitions and asymmetric behavior, enabling the modeling of nonlinear characteristics of variables in different phases. Markov-switching models lead to changes in parameter values because they allow the modeling of periodic changes in model parameters through the transition process between various stages. In this model, an unobservable stochastic state variable drives the regime-switching process. Therefore, the path followed by the state variable and the time-varying parameters are inferred from the data (Baycan and Yildirim, 2017).

This study aims to investigate the asymmetric circulation dynamics in per capita nuclear energy and coal production using Markov regime-switching models. This model is effective in analyzing time series with state-dependent dynamics.

Let us assume that yt, representing per capita nuclear energy and coal production in high-income countries, is expressed as the output of two different components.

$y_t=n_t+z_t(1)$

here the variable n_t represents the Markov trend, while the variable z_t indicates the Gaussian components used in state-space models, which rely on Gaussian possibility distributions.

The Markov trend (n_t) is consists of,

$n_t=\alpha \left(s_t\right)+n_{t-1}(2)$

here, s_t ∈ {1, … ,M} represents a potential Markov process dictating the current economic situation, and let α(s_t)= α_i when s_t = i, with i ∈ {1, … ,M}.

Consequently, the dynamics of Markov regime-switching provide a stochastic framework for governing transitions among states. The unobserved regime variable follows a first-order k-state Markov process, in which the current regime depends solely on the regime that was active in the previous period.

The probability rule is shown as:

$P\left[s_t=j\left.\right|s_{t-1}=i,s_{t-2}=k,\dots \right]=P\left[s_t=j\left.\right|s=i\right]=p_{ij}(3)$

The transition probability from state i to state j is denoted by pij, as shown in Equation 3. Where i, j, k are elements of the set {1, … , M} where k represents the state and M represents the total number of situation in the unnoticed situation variable.

According to the law of possibility, ${\displaystyle \sum _{j-1}^M}{p}_{ij}$ = 1.

The second premises in Equation 2 represents the Gaussian constituent. This is clarified in particular in reference (Hamilton, 1989).This component is represented as:

$z_t=z_{t-1}+{\phi }_1\left(z_{t-1}-z_{t-2}\right)+\dots +{\phi }_r\left(z_{t-r}-z_{t-r-1}\right)+{\epsilon }_t(4)$

here, the term εt is characterized as a Gaussian white noise duration, denoted by εt/σ (s_t) ∼ NID(0,1) where σ (s_t) represents the standard deviation associated with the state variable. It is important to note that εt is distinct of n_t+h, ∀h ≥ 0 *. NID here refers to a sequence of generally scattered stochastic random variables.

Taking the difference from Equation 1and substituting Equation 4, we obtain the first-difference specification, as given in Equation 5.

$\Delta y_t=\alpha \left(s_t\right)+{\phi }_1\left(z_{t-1}-z_{t-2}\right)+···+{\phi }_r\left(z_{t-r}-z_{t-r-1}\right)+{\epsilon }_t(5)$

This model is effective at distinguishing between regimes characterized by different means, variances, and transition probabilities. To examine potential structural breaks in nuclear and coal production arising from sudden policy changes, this study employs a hidden-Markov specification.

Hidden Markov models are particularly useful for distinguishing between genuine regime shifts and temporary collapses, especially when structural changes occur in the data production process due to frequent political fluctuations or sudden external shocks. This is clearly observed in the global economy. Given that such structural disruptions can have a significant impact on regime inference, the application of hidden Markov models provides robust forecasting results that are not affected by sensitivities associated with potential structural disruptions (Baycan and Yıldırım, 2017).

Following Hamilton’s (1990), foundational work, we employ the Expectation-Maximization (EM) algorithm coupled with nonlinear filtering to obtain maximum likelihood estimates of the model parameters. This methodology proves particularly valuable for time series analysis involving latent variables, as it allows the underlying state to be inferred directly from the data’s natural evolution-without imposing restrictive a priori assumptions on parameter values. For theoretical foundations of the EM algorithm, we refer to Dempster et al. (1977), while Krolzig (1997) provides key insights into its application within Markov-Switching (MS) model frameworks.

The dataset comprises annual observations of per capita nuclear and coal energy production, measured in terawatt-hours (TWh), spanning the period 1965–2023. To ensure interpretability, all production variables were rescaled into per capita kilowatt-hours (kWh) and this unit is used consistently throughout the analysis, tables, and interpretation. Country classifications are based on income levels defined by the World Development Indicators (WDI), and this study exclusively focuses on high-income economies. Energy production data are sourced from the Our World in Data platform, which compiles long-term, internationally harmonised energy statistics. All variables are adjusted for population to ensure comparability across countries and time.

Although many countries are included in the World Bank’s high-income country group, this study focuses on six countries that provide continuous and comparable data on nuclear and coal production since 1965. These six countries were selected because they provide uninterrupted annual data for both nuclear and coal production since 1965, allowing a consistent long-run regime analysis. These countries provide a suitable sample for regime change analysis due to their long-term data and the significant transformations they have undergone in nuclear and coal production. Major producers such as the United States and France were excluded due to missing or inconsistent long-term production series, and this limitation is acknowledged explicitly.

This selected sample ensures continuous time-series data coverage required by Markov Transition models, comparability between countries, and the representativeness of key high-income economies that play an active role in nuclear and coal energy transitions. Although the analysis is limited to six countries due to data constraints, the findings provide meaningful insights into broader trends observed among high-income economies. Because the dataset reflects production rather than consumption, the findings should not be interpreted as direct indicators of territorial emissions or final energy use.

4 Empirical results

This section presents the empirical findings of the Markov Switching Models applied to per capita nuclear and coal energy production in high-income countries over the period 1965–2023. The analysis focuses on identifying the number of regimes, estimating regime-dependent means and variances, and evaluating the transition dynamics across states. First, we analyze the unit root process of the nuclear energy and coal production series. The Augmented Dickey-Fuller and Phillips-Perron tests indicate that per capita nuclear energy and coal production satisfy the stationarity condition at the first difference. Then, by examining the nonlinear structure of the series and determining the number of regimes, we highlight regime-specific variances, transition probabilities, and smoothed probabilities. The findings provide detailed insights into the different operational states of nuclear energy production and the conditions that drive transitions between them. A three-regime model is selected as the AIC, SC, and HQ information criteria yield the lowest values. Subsequently, we investigate whether time-varying variance plays a significant role in this selection and conclude that the model is more meaningful when time-varying variance is assumed. Before estimating the Markov-switching models, descriptive statistics and exploratory time-series plots were examined to visually assess nonlinearity and potential regime shifts.The findings presented in Table 2; Figure 1 summarize the descriptive statistics and trend graphs for per capita nuclear energy and coal production.

Table 2

Table 2. Descriptive statistics.

Figure 1

Figure 1. Trend analysis.

When descriptive statistics and trend graphs are considered together, they reveal that nuclear and coal energy production exhibited distinctly different dynamics during the 1965–2023 period. The skewness, kurtosis, and deviations from normality observed in both series indicate that energy production processes do not exhibit a single-regime linear structure, supporting the use of a nonlinear, regime-sensitive methodology such as the Markov Switching model. Table 3 presents the ADF and PP unit root test results and critical values.

Table 3

Table 3. ADF and PP unit root test results.

The ADF and PP test statistics for nuclear energy and coal production per capita were well below the critical values at all significance levels (1%, 5%, 10%). The ADF test statistic for the nuclear energy variable is −15.00, the PP test statistic is −14.64; the ADF and PP test statistics for the coal variable are −7.27 and −7.26, respectively. The probability values of 0.0000 indicate that the unit root tests are stationary at the 1% significance level and that the null hypothesis is rejected. These findings confirm that both series are I (1) stationary in the first difference. Therefore, the stationarity condition required for the application of Markov Switching models is satisfied, and the use of a nonlinear modeling approach is statistically valid. The findings presented in Table 4 summarize the performance and classification of per capita nuclear energy and coal production across regimes. Since the model is estimated in first differences, the identified regimes reflect growth dynamics rather than absolute production levels.

Table 4

Table 4. MSM(3)–AR (0) Results for the high income nuclear energy and coal production.

Table 4 presents the results of a Markov Switching model applied to annual nuclear energy and coal production data. Three distinct regimes were identified: low, medium, and high production. The log-likelihood values of −743.248 for nuclear energy production and −421.488 for coal production indicate a strong fit of the model to the data. Furthermore, the likelihood ratio test for both classifications (LRP = 0.0000) confirms the statistical significance of regime switching, indicating different production patterns influenced by varying economic or market conditions.

It is observed that nuclear energy and coal production in high-income countries occur under three different regimes. The nuclear energy production regime reveals a starting level of 444.438 units per capita, with the low production regime representing periods of maintenance or temporary shutdowns at 87.828 units and the high production regime representing periods of intense production at full capacity at 709.715 units. In contrast, coal production is characterized by stable periods of production at a medium level of 1.844 units, maintenance or shutdown periods at a low level of 1.015 units, and full capacity production periods at a high level of 3.218 units. The larger differences between nuclear-energy production regimes compared with coal indicate that nuclear plants tend to operate either at full capacity or at significantly reduced levels. By contrast, the smaller differences between coal-production regimes suggest that coal output is relatively easier to sustain.

When examining transition probabilities between regimes, marked stability is evident in nuclear energy and coal production. In high-income countries, the probabilities of remaining in the medium (Regime 0) and high (Regime 2) production regimes for nuclear energy are 94.98% and 94.75%, respectively. In contrast, for coal production, these probabilities are 90.80% and 96.42%, respectively. Transitions to low-production regimes are relatively rare for both energy types. The findings indicate that high-income countries maintain strong production stability in both energy types, although coal production regimes are more distinctly separated.

The model selection criteria, including the AIC, SC, and HQ values, further confirm the robustness of the analysis. For nuclear energy production in high-income countries, AIC, SC, and HQ values of 25.499, 25.816, and 25.624, respectively, and for coal production, values of 14.558, 14.840, and 14.669, confirm a strong model fit while capturing the complexity of production dynamics. Alternative specifications (AR (0)-AR (2), two- and three-regime models) were compared using AIC, SC, and HQ, and the preferred specification was selected based on overall information-criterion superiority.

Figure 2 shows the smoothed probabilities for nuclear energy production states in high-income countries. Regime 0 (Medium Production), represented by the blue line, dominates most periods and reflects the advanced economies’ stable and consistent production characteristics of advanced economies. Regime 1 (Low Production), shown in gray, is short-lived and indicates resilience against prolonged declines in production. Regime 2 (High Production), displayed in yellow, emerges under optimal conditions and highlights the capacity of high-income countries to achieve high production levels with long-term stability. Transitions between regimes are infrequent, emphasizing that nuclear energy production in these economies is generally stable and efficient.

Figure 2

Figure 2. Smoothed probabilities associated with high-income nuclear energy production regimes. The figure is generated using the PcGive econometric modelling software.

Figure 3 shows the smoothed probabilities for coal production situations in high-income countries. Regime 2 (High Production), shown in yellow, dominates the early period from 1965 to the end of 1992. This situation indicates that coal production development in these countries is the primary situation. Moderate production, shown in blue, appears briefly in 1993, reflecting temporary declines or stagnation in production during a transition phase. Regime 1 (Low Production), shown in gray, becomes prominent after 2000, indicating a transition to decreasing production levels.

Figure 3

Figure 3. Smoothed probabilities associated with high-income coal production regimes. The figure is generated using the PcGive econometric modelling software.

It is seen that medium and high production regimes dominated nuclear energy and coal production in high-income countries throughout the period. Low production regimes were experienced in both productions, albeit briefly. While it is seen that the medium regime production of nuclear energy production remains longer than coal production, this situation shows that nuclear energy production is preferred for economic development in high-income countries.

The transition from medium-to high-level nuclear production reflects periods of intensified investment in nuclear infrastructure and efforts to reduce dependence on fossil fuels, particularly during the oil crises of the 1970s and early 1980s. Declines in nuclear output, by contrast, align with significant safety concerns and subsequent regulatory tightening following events such as Three President’s Commission on the Accident at Three Mile Island (Kemeny Commission) (1979), International Atomic Energy Agency (IAEA) (1986), and Government of Japan (2011). For coal, the gradual shift from high to medium and eventually low production regimes corresponds to long-term decarbonisation strategies, expanding environmental regulations, and the increasing role of renewable and nuclear energy in the power mix. To more clearly connect these patterns with historical developments, dominant regime periods have been explicitly outlined: nuclear Regime 2 (high) is most evident approximately in 1974–1985 and 1995–2010; Regime 0 (medium) characterises 1965–1973 and 1986–1994; and Regime 1 (low) emerges around major safety-related disruptions (1979, 1986, 2011). For coal, Regime 2 (high) spans 1965–1992, Regime 0 (medium) appears around 1993–2000, and Regime 1 (low) dominates after 2000 as climate and air-quality policies intensified. This clarified timeline demonstrates that the estimated regime shifts closely align with key historical, technological, and policy events shaping energy transitions in high-income economies. A detailed mapping of regime durations to historical and policy events is provided in Table 5.

Table 5

Table 5. Mapping of nuclear and coal production regimes to historical events.

This study examines the persistence and duration of the identified regimes using transition probabilities calculated for nuclear power and coal generation for high-income countries. These results are presented in Tables 6, 7. The regime classification provides critical insights into the behavior of nuclear power and coal generation under different conditions, helping to identify periods of stability and transition. Such findings are essential for policymakers and investors and help them develop strategies that increase production efficiency or reduce regime shifts.

Table 6

Table 6. Estimated Markov probabilities of staying in the same state.

Table 7

Table 7. Transition probability matrix for nuclear and coal production regimes.

Table 6 presents the probabilities persistence for nuclear and coal production regimes and highlights their stability. For nuclear power production in high-income countries, Regime 0 (Medium Production) has a probability of persistence of 0.9498, indicating a 94.98% probability of remaining in this state. Regime 1 (Low Production) is surprisingly stable, with a high likelihood of persistence of 0.9501, indicating that low production states dominate most of the observed period and that exit is difficult. Regime 2 (High Production) has a probability of persistence of 0.9475, indicating that they continue without reverting to lower states once high production levels are reached.

For coal production in high-income countries, Regime 0 (Medium Production) has a probability of persistence of 0.9080, indicating a 90.80% probability of remaining in this state. Regime 1 (Low Production) has a perfect value of 1.000, indicating complete stability with no exits from this regime. Regime 2 (High Production) has a probability of persistence of 0.9641, suggesting that they continue without reverting to lower states once high production levels are reached. The results show significant persistence in the high-production regime, while the low-production regime represents a period of complete self-sustaining calm. The value of 1.000 for the low-coal regime is a rounding artifact; when estimated in a two-regime specification, this state is not absorbing.

In addition, Table 7 reports the full transition probability matrix p= [pij] for nuclear and coal regimes, including the off-diagonal elements such as p₀₂ and p₂₀, which allow a more complete characterisation of regime switches.

The average duration of staying in the same condition and the percentage results are presented in Table 8.

Table 8

Table 8. Average durations and percentages of staying in the same state.

Table 8 reports the duration and percentage distribution of nuclear and coal production regimes in high-income countries. For nuclear energy, Regime 0 (medium production) accounts for 35.59% of the sample, with an average duration of 10.5 years and stable, sustained nuclear output. Regime 1 (low production) accounts for 32.20% of the observations and displays a relatively long average duration of 19 years, suggesting that low-production episodes are not short-term fluctuations but reflect extended periods of reduced capacity, maintenance cycles, or policy-driven slowdowns. Regime 2 (high production) likewise accounts for 32.20% of the sample, with an average duration of 19 years, showing that high-production phases tend to persist once achieved, consistent with long-term planning and operational stability in nuclear systems.

For coal production, Regime 0 (medium production) accounts for 18.64% of the sample, with an average duration of 11 years, reflecting transitional phases rather than dominant production states. Regime 1 (low production) accounts for 35.59% of the observations and persists on average for 21 years, indicating prolonged periods of reduced coal output that align with tightening environmental regulations and structural declines in coal use. Regime 2 (high production) accounts for 45.76% of the observations with an average duration of 27 years, the longest among all regimes. This demonstrates that coal remained in high production for multiple decades despite decarbonisation efforts, highlighting the inertia and entrenched nature of coal-dependent energy systems.

Overall, the results show that both nuclear and coal production regimes exhibit considerable persistence. However, coal’s high-production regime lasts significantly longer than its nuclear counterpart, underscoring the slower adjustment of coal systems compared to nuclear production’s more flexible structure.

Table 9 shows that the diagnostic test results adequately capture the nonlinear dynamics in both nuclear energy and coal production variables. The normality test indicates limited deviation from normality in nuclear energy production, while residuals in coal production are perfectly normally distributed. This situation is acceptable for regime-switching models. The ARCH test being significant for both variables confirms the existence of time-varying volatility and methodologically supports the use of a regime-dependent variance structure. Portmanteau Q test results indicate the presence of autocorrelation in both variables. This is a common occurrence in Markov Switching models due to the continuity of regime transitions. Overall, these diagnostic tests indicate that the model is robust and stable. Table 9 presents a resilience analysis of nuclear energy and coal production.

Table 9

Table 9. Diagnostic tests results.

Table 10 shows that the robustness analysis demonstrates that the base models are preserved for alternative delay structures and regime counts. In nuclear energy production, two- and three-regime models are highly robust against regime structure, delay count, and regime count. In coal production, the persistence of high regimes and the limited transitions to low regimes are observed similarly across all alternative models. This indicates that the model results are highly robust. Overall, the robustness analysis confirms the stability and reliability of the MSM results. Additional robustness checks-including linear AR benchmark models, alternative sample start dates, and log-transformed specifications-were performed and confirm the stability of the main results.

Table 10

Table 10. Robustness analysis for nuclear energy and coal production.

Figure 4 shows the combined representation of the smoothed probabilities for nuclear energy (yellow) and coal (black) production under Regime 2. The figure clearly shows that coal production remained in the high-production regime during the 1965–1990 period, while nuclear energy production entered the high regime much later and maintained it until approximately 2010. This combined figure allows a direct comparison of the timing, continuity, and structural characteristics of the high-production regimes across the two energy sources.

Figure 4

Figure 4. A comparative and combined display of smoothed probabilities related to nuclear energy and coal production regimes in high-income countries. The figure was created using PcGive econometric modeling software.

5 Discussion and conclusion

5.1 Discussion

The findings reveal a clear structural contrast between the long-run behaviour of nuclear and coal production in high-income economies. Nuclear production displays movements across three regimes that can reverse over time, while coal production follows a gradual shift toward lower levels without any indication of returning to its earlier high-production state. This contrast reflects the different roles these two energy sources have played in national energy strategies and in the broader transition toward low-carbon systems.

The three nuclear regimes have distinct meanings. The low-production state lasts for extended periods and indicates that nuclear output can remain subdued due to maintenance cycles, policy hesitation, regulatory reviews or operational constraints. The medium regime corresponds to stable capacity use in mature nuclear systems. The high-production regime reflects periods when nuclear facilities operate close to full capacity. These long-lasting patterns confirm that nuclear output can adjust to changing policy and technological conditions without becoming locked in a single path.

The timing of regime changes aligns with major events in nuclear policy history. The shift toward high production during the 1970s coincides with the oil crises that led many advanced economies to expand their nuclear fleets. Reversions to medium or low production match periods of heightened safety concerns after the accidents at Three Mile Island in 1979, Chernobyl in 1986 and Fukushima in 2011. These developments show that nuclear production responds to shifts in public policy and safety regulation as much as to market conditions.

Coal production follows a different pattern. It remains in a high-production regime from the mid-1960s until the early 1990s. A short transition period follows, after which the low-production regime becomes dominant. The probability of staying in this state is estimated at one. This near-absorbing behaviour is strong evidence of a structural decline shaped by climate policy, environmental regulation and improvements in renewable and nuclear technologies. Once coal production falls to low levels it does not return to previous highs. This irreversible pattern reflects long-standing lock-in forces such as reliance on coal-based industrial structures, employment patterns and existing infrastructure that slowed transition efforts in earlier decades.

These distinct dynamic profiles carry implications for energy planning. The reversibility of nuclear regimes suggests that nuclear output can be raised or stabilised when long-term decarbonisation goals become a priority. At the same time, the persistence of the low-nuclear regime shows that nuclear production can remain limited if investment and policy support weaken. Coal’s one-way decline indicates that regulatory tools and climate policies have already changed its long-run trajectory, yet the long duration of past high-production regimes highlights the structural resistance that once characterised coal transitions. Effective coal-phase-out strategies therefore require steady policy coordination, sectoral restructuring and social support mechanisms.

The combined interpretation of both sources suggests that high-income economies tend to phase down coal while adjusting nuclear capacity around regulatory and technological cycles. This pattern implies that the two energy sources respond to different types of pressures. Coal responds to long-term structural and policy forces, while nuclear responds more directly to safety regulation, investment decisions and public acceptance.

Several limitations should be recognised. Because the analysis relies on first-difference data, the regimes reflect changes in production rather than production levels. National-level data may also conceal differences across sectors or individual facilities. The sample includes only countries with continuous long-run data, which limits broader generalisation. Future work could incorporate country-level heterogeneity, examine renewable energy regimes alongside nuclear and coal or use multivariate regime-switching models to study interactions among different energy sources.

Overall, the results provide an integrated view of how nuclear and coal production evolve over time. Nuclear production moves through cyclical adjustments shaped by policy and technology, while coal production follows a structural and irreversible decline. These findings help clarify the dynamic conditions that shape long-term energy transitions in advanced economies.

5.2 Conclusion

This study investigates the regime-dependent and nonlinear dynamics of per capita nuclear and coal energy production in high-income economies using a hidden Markov framework. By applying a hidden Markov process coupled with the Expectation-Maximisation algorithm and nonlinear filtering techniques, the analysis reveals distinct and persistent production regimes for both energy sources over the period 1965–2023.

The results indicate three statistically significant regimes-low, medium, and high production-for both nuclear and coal energy. Nuclear energy production exhibits high regime persistence, particularly in the medium and high states, reflecting its strategic role in the long-term energy portfolios of high-income nations. In contrast, coal production shows a gradual shift from high to low regimes, suggesting the impact of decarbonisation policies and structural economic transformations.

The model also uncovers meaningful differences in volatility and duration across regimes. Nuclear energy production demonstrates greater variance between regimes, likely due to periods of plant outages or capacity upgrades. Coal production, however, displays more consistent variance across states, though with longer average durations in high-production regimes, highlighting the inertia in coal phase-out processes.

Empirical findings reveal regime transitions that coincide with historical coal production and nuclear energy events. Historical events-such as the increase in nuclear investments during the oil crises of the 1970s and the transition to stricter safety standards after Chornobyl-have led to significant structural changes in coal production and nuclear energy regimes. The stability shown by the probability of coal production remaining in Regime 1 (Low Production) emphasizes the strategic importance of this resource in energy supply security. In contrast, a more stable regime structure in nuclear energy production is more suitable for long-term planning.

From a policy perspective, the findings highlight the need to preserve stable and efficient nuclear production regimes while accelerating the transition away from coal. This requires sustained investment in nuclear infrastructure, safety improvements, and technological innovation to ensure operational resilience and support long-term emission reductions. At the same time, coal transition policies must overcome structural barriers-especially in energy-intensive industries-that keep coal production locked in high-regime states. The regime shifts identified in this study also correspond closely with major historical policy events. The sharp changes in coal production observed in the early 1990s coincide with stricter air-quality regulations and the restructuring of energy markets across Europe. Similarly, transitions in nuclear regimes align with key policy developments such as post-Chornobyl safety reforms, increased nuclear investment following the 1970s oil crises, and more recent commitments to low-carbon electricity under international climate agreements. These alignments suggest that the smoothed regime probabilities reflect not only statistical dynamics but also policy-driven structural transformations within national energy systems.

For energy planners and investors, the regime classification framework presented herein provides valuable insights into the stability and transition probabilities of key energy sources. Understanding these latent dynamics enhances strategic decision-making in energy infrastructure, risk assessment, and climate policy alignment.

Future research could extend this framework by incorporating renewable energy production regimes and analysing their interactions with both fossil fuels and nuclear energy. Integrating macroeconomic and geopolitical variables would also provide a more complete picture of how external shocks influence regime shifts. Such an expanded approach could help reveal the underlying dynamics shaping transitions between energy sources and offer insights into achieving an optimal balance between fossil, nuclear, and renewable systems in the context of global energy transformation. Furthermore, future studies could apply multivariate regime-switching models or comparative analyses across a broader set of countries to deepen understanding of cross-source interactions and structural adjustments within energy systems. Future work could employ multivariate MS-VAR or panel Markov-switching models to capture cross-source interactions that cannot be identified in a univariate framework.

This study offers important insights into the persistence, volatility, and structural changes characterising nuclear and coal production by estimating regime-specific heteroskedasticity and transition probabilities. These findings contribute to long-term energy planning by demonstrating the relevance of latent regime dynamics for policy and investment decisions. However, the analysis is subject to several limitations. First, the dataset covers only six high-income countries due to data availability, which may restrict generalisability. Second, the MSM framework used here examines univariate production series and does not capture interactions with macroeconomic, geopolitical, or technological factors. Third, aggregate national data may conceal sector-level divergences in energy production behaviours. Future research could address these limitations by expanding the country sample, integrating policy or geopolitical shocks, and employing complementary econometric techniques to reveal a more nuanced understanding of energy transition processes.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

ME: Conceptualization, Validation, Data curation, Supervision, Writing – review and editing, Methodology, Project administration, Funding acquisition, Investigation, Writing – original draft, Resources, Formal Analysis, Visualization, Software. İB: Supervision, Data curation, Conceptualization, Investigation, Writing – review and editing, Software, Methodology, Funding acquisition, Resources, Formal Analysis, Validation, Visualization, Project administration, Writing – original draft.

Funding

The author(s) declared that financial support was received for this work and/or its publication. This study was supported by the Scientific Research Coordination Unit of Anadolu University under the project number 2894.

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declared that generative AI was used in the creation of this manuscript. During the preparation of this work, we used AI in order to correct the grammatical errors in the original manuscript. After using it, we reviewed and edited the content as needed and take full responsibility for the content of the publication.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Footnotes

¹This study selected high-income countries engaged in nuclear and coal energy production, specifically Canada, Germany, Japan, South Korea, Spain, and the United Kingdom

References

Aimon, H., Kurniadi, A. P., Sentosa, S. U., and Rahman, N. A. (2023). Production, consumption, export and carbon emission for coal commodities: cases of Indonesia and Australia. IJEEP 13 (5), 484–492. doi:10.32479/ijeep.14798

CrossRef Full Text | Google Scholar

Akdag, S., and Iskenderoglu, O. (2018). The impact of non-renewable energy, renewable energy, and nuclear energy consumption on economic growth in European union member states and candidate countries. J. Turkish Stud. 13 (30), 1–14. doi:10.7827/TurkishStudies.14423

CrossRef Full Text | Google Scholar

Anser, M. K., Godil, D. I., Khan, M. A., Nassan, A. A., Zaman, K., and Abro, M. M. Q. (2021). The impact of coal combustion, nitrous oxide emissions, and traffic emissions on COVID-19 cases: a Markov-switching approach. Environ. Sci. Pollut. Res. 28, 64882–64891. doi:10.1007/s11356-021-15494-x

PubMed Abstract | CrossRef Full Text | Google Scholar

Balat, H. (2007). Role of coal in sustainable energy development. Energy Explor. Exploitation 25 (3), 151–174. doi:10.1260/014459807772009169

CrossRef Full Text | Google Scholar

Baycan, I. O., and Yildirim, G. (2017). A nonlinear markov switching approach to characterize the cyclical asymmetric dynamics of the German manufacturing industry. Roma: Econworld. Italy.

Google Scholar

Baycan, I. O., and Yildirim, G. (2017). Examining the dynamics of the Turkish manufacturing industry: a hidden markov model approach. Chin. J. Phys. 55 (3), 703–709. doi:10.1016/j.cjph.2017.03.020

CrossRef Full Text | Google Scholar

Bildirici, M. E. (2020). Environmental pollution, hydropower and nuclear energy generation before and after catastrophe: bathtub-weibull curve and MS-VECM methods. Nat. Resour. Forum 44 (4), 289–310. doi:10.1111/1477-8947.12194

CrossRef Full Text | Google Scholar

Box, G. E. P., and Jenkins, G. M. (1970). Time series analysis: forecasting and control. San Francisco, CA: Holden-Day

Google Scholar

Burns, A. F., and Mitchell, W. C. (1946). Measuring business cycles. New York: National Bureau of Economic Research

Google Scholar

Cevik, E. I., Yildirim, D. C., and Dibooglu, S. (2020). Renewable and non-renewable energy consumption and economic growth in the U.S.: a Markov-switching VAR analysis. Energy Environ. 32 (3), 519–541. doi:10.1177/0958305X20944035

CrossRef Full Text | Google Scholar

Coglianese, J., Gerarden, T. D., and Stock, J. H. (2020). The effects of fuel prices, environmental regulations, and other factors on U.S. coal production, 2008–2016. Energy J. 41 (1), 55–82. doi:10.5547/01956574.41.1.jcog

CrossRef Full Text | Google Scholar

Danish, , Ulucak, R., and Erdogan, S. (2022). The effect of nuclear energy on the environment in the context of globalization: consumption vs. production-based CO2 emissions. Nucl. Eng. Technol. 54 (4), 1312–1320. doi:10.1016/j.net.2021.10.030

CrossRef Full Text | Google Scholar

Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B 39, 1–38. doi:10.1111/j.2517-6161.1977.tb01600.x

CrossRef Full Text | Google Scholar

Duran, M. S., Bozkaya, S., Onifade, S. T., and Kaya, M. G. (2022). Nuclear energy consumption and energy-driven growth nexus: a system GMM analysis. Environ. Sci. Pollut. Res. 29, 70564–70572. doi:10.1007/s11356-022-22951-8

PubMed Abstract | CrossRef Full Text | Google Scholar

Dursun, F. (2023). Markov regime switching models: theory and application. Seruven Yayinevi.

Google Scholar

Gao, W., Ullah, S., Zafar, S. M., and Usman, A. (2024). How does nuclear energy consumption contribute to or hinder green growth? Prog. Nucl. Energy 170, 10511. doi:10.1016/j.pnucene.2024.105111

CrossRef Full Text | Google Scholar

Government of Japan (2011). Report of japanese government to the IAEA ministerial conference on nuclear safety: the accident at TEPCO’s fukushima nuclear power stations. Vienna: IAEA

Google Scholar

Gralla, F., Abson, D. J., Moller, A. P., Lang, D. J., and Wehrden, H. V. (2017). Energy transitions and national development indicators: a global review of nuclear energy production. Renew. Sustain. Energy Rev. 70, 1251–1265. doi:10.1016/j.rser.2016.12.026

CrossRef Full Text | Google Scholar

Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57 (2), 357–384. doi:10.2307/1912559

CrossRef Full Text | Google Scholar

Hamilton, J. D. (1990). Analysis of regime switching. Econometrics 45, 39–70. doi:10.1016/0304-4076(90)90093-9

CrossRef Full Text | Google Scholar

Hondroyiannis, G., and Papapetrou, E. (2016). Stock returns and inflation in Greece: a Markov switching approach. Rev. Financial Econ. 15 (1), 76–94. doi:10.1016/j.rfe.2005.02.002

CrossRef Full Text | Google Scholar

Islam, S., Raihan, A., Paul, A., Ridwan, M., Rahman, M. S., Rahman, J., et al. (2024). Dynamic impacts of sustainable energies and technological innovation. JEEE 3 (2), 1–14. doi:10.56946/jeee.v3i2.448

CrossRef Full Text | Google Scholar

International Atomic Energy Agency (IAEA) (1986). Summary report on the post-accident review meeting on the chernobyl accident. Vienna: IAEA (Safety Series No. 75-INSAG-1)

Google Scholar

Krolzig, H. M. (1997). Markov-switching vector autoregressions. Springer.

Google Scholar

Kuan, C. M. (2002). Lecture on the Markov switching model. Inst. Econ. Acad. Sinica, 1–30. Available online at: https://homepage.ntu.edu.tw/∼ckuan/pdf/Lec-Markov_note.pdf.

Google Scholar

Li, P., Yu, H., Zhang, J., Du, M., and Xiong, J. (2021). Coal supply sustainability in China. Front. Energy Res. 9, 701719. doi:10.3389/fenrg.2021.701719

CrossRef Full Text | Google Scholar

Lin, B., and Ullah, S. (2024). Modeling the impacts of changes in nuclear energy, natural gas, and coal. Energy 287, 129572. doi:10.1016/j.energy.2023.129572

CrossRef Full Text | Google Scholar

Mbarek, M. B., Khairallah, R., and Feki, R. (2015). Causality relationships between renewable energy, nuclear energy and economic growth in France. Environ. Syst. Decis. 35, 133–142. doi:10.1007/s10669-015-9537-6

CrossRef Full Text | Google Scholar

Mohseni, R., and Modallal, L. S. (2017). Estimating stock price in the energy market. Iran. J. Manag. Stud. 10 (3), 715–728. doi:10.22059/ijms.2017.221553.672398

CrossRef Full Text | Google Scholar

Murshed, M., Saboori, B., Cohen, Y., Wang, H., and Dogan, B. (2022). Nuclear energy, renewable energy and CO2 in the G7. Renew. Energy 190, 664–674. doi:10.1016/j.renene.2022.03.121

CrossRef Full Text | Google Scholar

Mushtaq, Z., Wei, W., and Li, Z. W. (2024). Technological innovations and energy structure in China’s coal consumption. Environ. Sci. Pollut. Res. 31 (6), 9461–9476. doi:10.1007/s11356-023-31785-x

PubMed Abstract | CrossRef Full Text | Google Scholar

Nathaniel, S. P., Alam, M. S., Murshed, M., Mahmood, H., and Ahmad, P. (2021). Nuclear and renewable energy in G7 CO2 abatement. Environ. Sci. Pollut. Res. 28, 47957–47972. doi:10.1007/s11356-021-13728-6

PubMed Abstract | CrossRef Full Text | Google Scholar

President’s Commission on the Accident at Three Mile Island (Kemeny Commission) (1979). Report of the president’s commission on the accident at three mile island. Washington, DC: U.S. Government Printing Office

Google Scholar

Neftci, S. N. (1984). Are economic time series asymmetric over the business cycle?. J. Pol. Econ. 92 (2), 307–328.

CrossRef Full Text | Google Scholar

Rehman, A., Ma, H., Ozturk, I., and Radulescu, M. (2022). Dynamic effects of energy sources on economic growth. Environ. Sci. Pollut. Res. 29 (32), 48784–48794. doi:10.1007/s11356-022-19317-5

PubMed Abstract | CrossRef Full Text | Google Scholar

Syed, A. A., Kamal, M. A., and Tripathi, R. (2021). Nuclear energy and environmental pollution nexus in India. Environ. Sci. Pollut. Res. 28, 54744–54755. doi:10.1007/s11356-021-14365-9

PubMed Abstract | CrossRef Full Text | Google Scholar

T. C. Energy and Natural Resources Ministry (2025). Nuclear energy. Available online at: https://enerji.gov.tr.

Google Scholar

Thierauf, R. J. (1970). Decision making through operations research. Wiley.

Google Scholar

Usman, M., Jahanger, A., Makhdum, M. S., Radulescu, M., Balsalobre-Lorente, D., and Jianu, E. (2022). Ecological footprint and nuclear energy. Energies 1515 (17), 6442. doi:10.3390/en15176442

CrossRef Full Text | Google Scholar

Wojciechowski, W., Streimikiene, D., Wojciechowski, A., and Bilan, Y. (2023). Nuclear energy in low-carbon transition. Environ. Sci. Pollut. Res. 30 (59), 124353–124373. doi:10.1007/s11356-023-30827-8

PubMed Abstract | CrossRef Full Text | Google Scholar

Xue, Y., and Huang, Y. (2016). Coal price co-movement using MS-VEC. Appl. Econ. 49 (7), 693–701. doi:10.1080/00036846.2016.1205720

CrossRef Full Text | Google Scholar

Zhang, Y., Li, L., Sadiq, M., and Chien, F. (2023). Non-renewable energy and carbon emissions in China. Energy Environ. 35 (4), 2248–2269. doi:10.1177/0958305x221150432

CrossRef Full Text | Google Scholar