Abstract
To develop effective and equitable emission reduction policies in the European Union, it is necessary to consider not only the level of environmental inequality but also its persistence over time. Therefore, examining the stationarity properties of inequality indicators is important for determining whether such inequalities are temporary or permanent. Accordingly, this study analyzes the stationarity characteristics of ecological footprint inequality in the European Union for the period 1961–2024. Ecological footprint inequality is measured using three complementary inequality indices: the Gini coefficient, the Theil-T index, and the Atkinson index. In addition to traditional unit root tests such as ADF, PP, and KPSS, the analysis employs Fourier-based tests, structural break unit root tests, and the artificial intelligence-based ARNN-ADF test to examine the stationarity structure of the series. Furthermore, the ARFIMA model is employed as a robustness check to assess the long-memory properties of ecological footprint inequality. The findings of this study are test-dependent. Conventional unit root tests suggest level stationarity, whereas Fourier-based, structural-break and nonlinear tests provide stronger evidence of persistence. ARFIMA results further indicate a stationary long-memory process operating close to the non-stationarity boundary. These findings suggest that shocks to ecological footprint inequality are not purely transitory but tend to persist over long periods. Overall, the results highlight the need for structural and long-term policies to reduce ecological footprint inequalities across European Union countries.
Keywords: Ecological footprint inequality, European Union, Unit root tests
Introduction
The increase in fossil fuel consumption and rapidly rising carbon dioxide emissions contribute to an expanding ecological footprint and the degradation of natural ecosystems, thereby threatening sustainable development goals. [41]. Global greenhouse gas emissions rose by 1.1% in 2022, reaching 50.6 gigatons of CO₂ equivalents, and began to rise again following the temporary decline during the pandemic. In contrast, European Union countries have managed to reduce greenhouse gas emissions by about one-third while maintaining economic growth. However, as of 2023, the per capita ecological footprint had reached 2.6 global hectares, with approximately 60% of this figure attributable to carbon-related components [19]. In this regard, international agreements such as the United Nations Sustainable Development Goals and the Paris Agreement provide a global framework for reducing environmental pollution and ensuring long-term ecological balance [20]. However, one of the main challenges to achieving these goals is that shocks to greenhouse gas emissions can have long-lasting effects. This persistence is shaping the political economy of climate change and making the balance between economic growth and climate action more delicate. Indeed, shocks to greenhouse gas emissions, and particularly CO₂ emissions, can produce temporary or permanent effects [18].
In general, CO₂ and greenhouse gas emissions are among the key indicators used to assess the environmental impact of production and consumption activities. These indicators reveal the amount of gases that directly contribute to climate change and provide an important reference for policy-making processes. However, assessing environmental pressure solely through emission levels may not fully reflect the total impact of human activities on nature. In this regard, the Ecological Footprint stands out as a powerful indicator that more comprehensively measures the total impact of human activities on nature [27]. The Ecological Footprint measures the total demand on nature in a holistic manner, taking into account the productive land and water areas necessary for human survival [40]. Therefore, the ecological footprint, which is central to the sustainable development approach, is becoming an increasingly important indicator for both measuring environmental pressure and assessing environmental inequalities [21].
In recent years, researchers have begun to apply tools such as the Gini coefficient, which is used to measure income inequality, to the analysis of environmental inequalities. In this context, List [31] and Heil and Wodon [24] made significant contributions to the literature, particularly by using the environmental Gini coefficient to measure inequalities in the spatial distribution of carbon dioxide and other pollutants [33]. Discussions regarding global inequalities in carbon emissions were first systematically addressed in the 1996 report by the Intergovernmental Panel on Climate Change. The report in question classifies the greenhouse gas emissions of countries with different income levels. In the period following this debate, Heil and Wodon ([23, 24]) analyzed per capita CO₂ emissions using the Gini coefficient and a group-based decomposition method and found that inequalities stemmed largely from intergroup income differences [11].
Although the level and distribution of environmental inequalities have been examined in numerous studies, the persistence of these inequalities over time has received limited attention in the literature. However, the policy implications of environmental inequality depend not only on its magnitude but also on whether deviations from its long-term trajectory are temporary or permanent. The presence of a unit root in a time series means that the indicator in question has not spontaneously returned to its long-term equilibrium level after the shock [32]. Therefore, if ecological footprint inequality follows a stationary process, the shocks in the distribution of environmental pressure among countries are expected to subside over time. Conversely, if the series contains a unit root, it is possible for these shocks to have long-lasting effects. This indicates that the inequality in ecological pressure should be considered not as a temporary fluctuation, but as a structural and permanent distribution problem.
This study analyzes the stationarity properties of per capita ecological footprint inequality across European Union countries over the period 1961–2024. The European Union consists of countries operating under common environmental policy frameworks. However, significant differences between member states in terms of income level, production structure, energy system, and ecological pressure make the European Union a suitable area for institutional analysis. Policy frameworks such as the European Green Deal and the Eighth Programme for Environmental Action demonstrate the European Union’s commitment to environmental sustainability goals at the institutional level European Commission [16]. However, the European Union’s enlargement process necessitates that environmental inequality be assessed within a changing institutional membership structure over time, rather than within a fixed geographical group of countries. Therefore, following Conyon’s [12] institutional inequality framework, countries are incorporated into the analysis from the year they became members of the European Union. Thus, the ecological footprint inequality series is developed using a dynamic institutional panel structure that captures the actual composition of European Union membership.
To address the sensitivity of inequality measurement to the choice of indicator, this study does not rely exclusively on the Gini coefficient. Instead, ecological footprint inequality is measured using three complementary inequality indices: the Gini coefficient, the Theil-T index, and the Atkinson index 
This multi-index approach is consistent with the logic followed by current institutional inequality studies such as Conyon [12], which assess the robustness of distribution patterns using Gini, Theil and Atkinson indices together. In this respect, the study reduces the limitations that relying on a single measure of inequality might create and offers a broader empirical basis for examining the persistence of ecological footprint inequality.
Considering structural breaks is important for reliably analyzing the long-term statistical properties of time series [9]. Economic, environmental, and institutional shocks can alter the stationarity structure of time series and undermine the effectiveness of traditional unit root tests. For this reason, the study employs both traditional and new-generation unit root tests. The ADF, PP, and KPSS tests are administered as baseline comparison tests. Fourier-based unit root tests, Lee–Strazicich structural break tests, and the ARNN-ADF test are used to account for gradual structural changes, abrupt breaks, and nonlinear adjustment dynamics, respectively. The Fourier approach allows for the modeling of various types of structural changes without requiring prior knowledge of the number or form of breaks [45]. The ARNN-ADF test, on the other hand, enables the evaluation of linear, quadratic, and cubic relationships within the same framework thanks to the hidden layer structure of the artificial neural network architecture [17, 44]. This distinction is important because the ecological footprint inequality series spans a long period characterized by European Union enlargements, energy shocks, institutional transformations, and changes in environmental policies.
The contribution of this study to the literature can be summarized under three headings. First, the study expands the literature on environmental inequality by examining the persistence of this inequality rather than focusing solely on the level or trend of ecological footprint inequality. Second, it strengthens its measurement strategy by using the Gini coefficient, the Theil-T index, and the Atkinson index together. Third, the study adopts a comprehensive set of unit root tests to compare findings from traditional approaches with those from tests that incorporate structural breaks and nonlinear dynamics. This framework allows for a more cautious and methodologically transparent assessment of whether ecological footprint inequality in the European Union is temporary or permanent.
The remainder of the article is organized as follows. The second section provides a summary of the relevant literature. The third section describes the dataset and econometric methods used. The fourth section presents and discusses the empirical findings. Finally, the fifth section evaluates the results, policy implications, and research suggestions for future studies.
Literature review
Various methods are used in the literature to measure inequalities in per capita CO₂ and ecological footprint. These methods are generally derived from the Gini, Theil, Generalized Entropy, and Atkinson indices, which are commonly used to analyze income inequality ([13, 22, 24, 34, 42]; Padilla and Duro [14, 7, 14]). The main aim of these studies is to measure the level of environmental inequality and to explain the determinants behind this inequality.
In the context of applying these methods, studies examining spatial inequalities at the national or regional level are frequently encountered. Chen et al., [10] assessed energy-related spatial inequality for Chinese provinces using the Gini coefficient of CO₂ emissions and found a downward trend in sectoral inequality using the resource increase decomposition method. Padilla and Duro [14], using the Theil index for a 19-year time series, show that the per capita CO₂ emission inequality among EU countries has decreased over time. This situation is explained by the fact that energy intensity, which initially caused large differences between countries, has become more balanced over time. Clarke-Sather et al., [11] measured the inequality of per capita CO₂ emissions among provinces in China using the Gini and Theil indices. According to their findings, they stated that inequality is more pronounced in provinces with high income levels, and that this is related to the more intensive use of greenhouse gas-producing fuels in wealthy regions. In addition, Zhao et al., [46], who analyzed regional inequality in agricultural energy-related CO2 emissions using the Kaya-Theil decomposition approach, observed that CO2 increased significantly but inequality tended to decrease. Analysis of the Kaya-Theil decomposition reveals that energy intensity is the primary cause of total CO2 inequality, followed by the level of agricultural development and CO₂ emission intensity. Ata et al., [2] used the Theil index and Kaya decomposition method to analyze the inequality in CO₂ emissions in the Iranian household sector and found that the inequality in CO₂ emissions decreased significantly during the 2000–2017 period. The Kaya decomposition results show that the main determinant of this reduction is the improvement in energy efficiency, and that energy efficiency stands out as the primary factor in reducing CO₂ inequality.
Analyses of ecological footprint inequality also use similar inequality measures. White [42], examined the inequality in the distribution of resource consumption and per capita ecological footprint at the global level using the Gini and Atkinson indices and found that this inequality was high. One of the main reasons for this inequality is the difference in energy consumption intensity between countries. Similarly, Remuzgo and Sarabia [37], examined the CO₂ emission inequality in IEA regions during the 1990–2010 period using Theil decomposition based on rock factors and showed that the inequality decreased by approximately 22%. Teixidó-Figueras and Duro [39], compared various inequality indices to measure ecological footprint inequality and stated that the most suitable indices for environmental data are CV² (Coefficient of Variation Square) and GE(2) (Generalized Entropy Index) due to their neutral properties. The study reveals that inequality in ecological footprint components has decreased over time, but overall ecological footprint inequality has increased, and this increase is largely due to structural differences between countries. Zheng et al., [47] analyzed both carbon footprint intensity and carbon footprint inequality by dividing 43 major economies into income groups. The findings reveal that carbon intensity has decreased across all income groups in developed countries, but inequality has increased, while in developing countries carbon intensity has increased and the level of inequality varies among groups. The limited reduction in carbon footprint among the highest income group in developed countries indicates that these groups are resistant to adopting cleaner lifestyles. On the other hand, in developing countries, the carbon footprint is rapidly increasing due to the Western lifestyle adopted by high-income groups. Consistent with these findings, other studies assessing ecological footprint inequality by income group show that the highest income groups have the highest carbon footprints at both individual and national levels [25, 26, 43, 48]. On a global scale, Chancel [8], found that the bottom 50% of the world’s population is responsible for only 12% of total emissions, while the top 10% accounts for 48% of global emissions. Andretti et al., [1] explain this situation by stating that increased prosperity leads to increased consumption, thereby increasing the ecological footprint. Li et al., [30], who took a different approach to carbon footprint inequality, evaluated in their study for China how much household consumption patterns, expenditures, age, and urbanization affect the difference in per capita carbon footprint among income groups. They also concluded that consumption patterns contribute very similarly to carbon footprint inequality, that carbon footprint consumption is more balanced in older households, and that carbon emissions are higher in urban areas than in rural areas.
Materials and methods
Data
This study aims to examine the stationarity properties of inequality in the distribution of per capita ecological footprint across European Union countries over the period 1961–2024. The main reason for analyzing European Union countries is that these countries have long implemented environmental sustainability policies. In addition, the availability of long-term and comparable datasets in European Union countries, together with their commitment to common environmental policy objectives, makes this sample appropriate for the analysis. The per capita ecological footprint data used in the study were obtained from the Global Footprint Network database.
The study follows a two-stage methodology. In the first stage, inequality in the distribution of per capita ecological footprint across European Union countries is calculated. For this purpose, the analysis does not rely on a single inequality indicator; instead, the Gini coefficient, the Theil-T index, and the Atkinson index (ε = 1) are used together. This approach allows for the assessment of whether the inequality findings are sensitive to the chosen criterion. The ecological footprint value per person for each year is denoted by 
. Here, 𝑖 represents the country and 𝑡 represents the year. 𝑁𝑡 represents the number of countries included in the calculation for the relevant year, and 
represents the average per capita ecological footprint of these countries. Because the European Union membership structure has changed over time, 𝑁𝑡 is used instead of the constant 𝑁 in the formulas. In this context, the Gini coefficient is calculated as follows:
 |
1 |
The Theil-T index is defined as follows:
 |
2 |
The Atkinson index is calculated for 
using the following limit formula:
 |
3 |
Considering the historical enlargement process of the European Union, countries are incorporated into the inequality calculation beginning from their year of membership. In this context, founding members have been included from 1961 onwards; and countries that joined in subsequent waves of enlargement have been included from 1973, 1981, 1986, 1995, 2004, 2007, and 2013. The United Kingdom was excluded from the series from 2020 onwards due to its departure from the European Union. Therefore, the inequality series are generated each year to reflect the actual membership composition of the European Union. In this dynamic membership approach, the number and composition of countries change over time. These changes can create mechanistic effects on inequality indices. Therefore, the dates obtained from structural break tests are not considered as direct causal evidence of specific historical events, but rather as indicators of structural instability in the ecological footprint inequality process. The trend of the inequality indices created within this framework over time is presented in Fig. 1. During the period examined, the Gini coefficient decreased from 0.271 in 1961 to 0.180 in 2024. Similarly, the Theil-T index decreased from 0.134 to 0.055, and the Atkinson index decreased from 0.118 to 0.051. This pattern indicates a long-term downward trend in per capita ecological footprint inequality, but also shows that the series have followed a fluctuating course, especially after the European Union enlargement years.
Fig. 1.
Evolution of ecological footprint inequality indices in European Union countries
Descriptive statistics for the Gini, Theil-T, and Atkinson indices, calculated to measure the inequality in the per capita ecological footprint distribution during the study period, are presented in Table 1. The mean value of the Gini coefficient is 0.168, while the mean of the Theil-T index is 0.055 and the mean of the Atkinson index is 0.049. The fact that the mean values are higher than the median values in all three indices indicates that the series have a right-skewed distribution. This reveals that in some years during the period studied, the ecological footprint inequality exceeded the general trend. The skewness and kurtosis values also support the conclusion that the series deviate from a normal distribution. Indeed, Jarque–Bera statistics and their probability values show that the assumption of normal distribution is rejected for all three inequality indices. However, these descriptive findings do not directly lead to conclusions about the stationarity properties of the series. Therefore, in the following stage, the stationarity structure of these inequality indices is examined in detail through traditional and complementary unit root tests.
Table 1.
Descriptive statistics of ecological footprint inequality indicators
| Gini-EFI | Theil-EFI | Atkinson-EFI | |
|---|---|---|---|
| Mean | 0.168069 | 0.055 | 0.049 |
| Median | 0.161063 | 0.050 | 0.044 |
| Maximum | 0.271257 | 0.134 | 0.118 |
| Minimum | 0.136927 | 0.033 | 0.031 |
| Std. Dev. | 0.027451 | 0.019 | 0.017 |
| Skewness | 1.699326 | 1.886 | 2.070 |
| Kurtosis | 6.176469 | 7.413 | 7.935 |
| Jarque-Bera | 57.70878 (0.000) | 89.86645 (0.000) | 110.6503 (0.000) |
In the second phase of the study, the stationarity properties of the ecological footprint inequality indices calculated for the period 1961–2024 are analyzed using traditional and next-generation unit root tests. The econometric framework of these tests is presented in the following subsection.
Methods
In time series analysis, which covers the change of a variable from the past to the present, stationarity means that the variance and mean of the variables do not change over time. In such series, the impact of external shocks diminishes over time, and the variable returns to its long-run equilibrium level. Therefore, temporary policies only create short-term effects. If a lasting impact is desired, that is, if the goal is to change the long-term level of the series, then a permanent policy change is required. In contrast, with non-stationary series, since there is no return to the mean, even temporary policies can have long-term and lasting consequences. Therefore, both the stationarity of the series and the persistence of the policy should be considered when evaluating policy effects [4, 35, 36].
In this study, in order to analyze the stationarity structure of time series, both traditional unit root tests and new generation unit root tests were preferred. While traditional unit root tests focus particularly on abrupt structural breaks [28, 49], the new generation of unit root tests, Fourier-based tests, often fail to adequately reflect the nonlinear and time-gradual structural transformations observed in economic series. Furthermore, traditional structural break unit root tests may require the predetermination or modeling of the number, type, or structure of breaks under specific parametric constraints. This can increase the risk of model misspecification and weaken test power, especially when the number, timing, and type of breaks are unknown. To overcome these limitations, unit root tests based on Fourier functions have been developed [3, 15]. The Fourier approach can flexibly model soft structural changes of unknown forms without the need to predetermine break dates and types. Fourier approaches can implicitly model nonlinear and smooth structural breaks in time series without requiring prior knowledge of the number and form of breaks by integrating low-frequency sine and cosine terms into the deterministic component. This increases the power of the test while preventing spurious results. In this context, FADF, FKPSS, and FGLS tests were applied in the study.
The FADF test generalizes the classic ADF structure by adding sine and cosine components to the deterministic terms. In this way, structural breaks in the series can be represented indirectly without being directly included in the model. The FKPSS test, on the other hand, operates based on the stationarity hypothesis and compensates for the insensitivity of the classical KPSS test to structural breaks using Fourier components. In both tests, the Fourier frequency (k) is chosen data-wise to minimize the total squared errors. The FGLS test, on the other hand, combines the generalized least squares detrending method with Fourier components, offering a more flexible testing framework that can simultaneously model both trends and structural breaks [38].
The basic assumptions of Fourier-based unit root tests vary depending on the test used. In FADF and FGLS tests, when the absolute value of the test statistic is greater than the critical value, the null hypothesis (H₀) is rejected and the alternative hypothesis (H₁) is accepted. This indicates that the series does not contain a unit root and is stationary. In contrast, the hypothesis structure is reversed in the FKPSS test. Here, the null hypothesis (H₀) suggests that the series is stationary, while the alternative hypothesis (H₁) suggests that it contains a unit root. In this test, if the calculated statistic falls below the critical value, H₀ is not rejected; instead, the conclusion is reached that the series is stationary. These tests offer advantages over traditional structural break unit root tests, as they do not require prior knowledge of break times and allow breaks to occur in a nonlinear and smooth manner. They make stationarity analysis more flexible by modeling structural transformations, which are frequently observed in economic time series, using Fourier terms.
In addition, this study employs the Lee and Strazicich [29] LM unit root test to account for possible abrupt structural breaks in the ecological footprint inequality series. The main advantage of this test is that structural breaks are incorporated into the deterministic component of the model under both the null and alternative hypotheses. This feature is important because ignoring structural breaks may lead conventional unit root tests to produce misleading stationarity results.
In this study, both the one-break and two-break versions of the Lee-Strazicich LM test are applied. Model A allows for a break in the intercept and therefore captures a level shift in the series, whereas Model C allows for breaks in both the intercept and the trend and therefore captures simultaneous level and trend shifts. Applying both models provides a more comprehensive assessment of whether the persistence of ecological footprint inequality remains valid after accounting for different forms of structural change.
In this context, the following regression equation was used for the LM unit root test:
 |
4 |
where Zₜ denotes the vector of exogenous variables and εₜ is assumed to be independently and identically distributed with zero mean and constant variance, that is, εₜ ~ iid(0, σ²). The null hypothesis of the test corresponds to the unit root case, H₀: β = 1, while the alternative hypothesis is H₁: β < 1, indicating stationarity around structural breaks. The LM unit root test statistic is obtained from the following regression:
 |
5 |
where S̃ₜ₋₁ denotes the detrended series used in the LM testing regression. In this framework, the significance of the coefficient φ is used to evaluate whether the unit root null hypothesis can be rejected after allowing for structural breaks in the deterministic component of the model.
Another complementary method employed in the study is the ARNN-ADF unit root test, which is based on artificial intelligence-based autoregressive neural networks. This test models the time-dependent structure of the series using an artificial neural network architecture, a nonlinear learning algorithm, and generates a stationarity decision independently of the assumptions of classical tests. The fact that it doesn’t require prior knowledge about the shape, direction, and timing of transformations makes the ARNN-ADF method more flexible and powerful compared to traditional tests. In series dominated by nonlinear transformation processes such as economic crises, structural reforms, and policy changes, the ARNN-ADF test plays a complementary role to Fourier-based and structural break tests. The key advantage of this approach lies in the hidden layer of the neural network, which effectively captures the complex nonlinear structure underlying macroeconomic dynamics. At the same time, the model accommodates both linear and nonlinear behaviors by integrating the “linear”, “quadratic”, and “cubic” terms into a single nonlinear test framework. Autoregressive neural networks (ARNN-ADF) unit root test is described by the regression model expressed below [6].
 |
7 |
The null hypothesis (H₀) of the unit root test in question is tested in a similar way to the ADF unit root test. The null hypothesis of the ARNN-ADF test corresponds to the presence of a unit root. Similar to the conventional ADF test, if the ARNN-ADF test statistic is more negative than the bootstrapped critical values obtained from Monte Carlo simulations, the null hypothesis is rejected. This indicates that the series under investigation is stationary. Conversely, failure to reject the null hypothesis suggests that the series exhibits non-stationary or persistent dynamics [5, 44].
Finally, the Autoregressive Fractionally Integrated Moving Average (ARFIMA) model is used as a robustness check to examine the long-memory properties of ecological footprint inequality. Unlike conventional unit root tests, which are based on integer orders of integration, the ARFIMA framework allows the integration parameter to take fractional values. This makes it possible to assess whether shocks to ecological footprint inequality disappear rapidly or persist over long periods. In this study, the ARFIMA (0, d, 0) specification is estimated separately for the Gini-EFI, Theil-EFI, and Atkinson-EFI series as follows:
 |
8 |
where L denotes the lag operator, d is the fractional integration parameter, 
is the mean of the series, and
is a white-noise error term. The value of d provides information about the memory and persistence properties of the series. The interpretation of the fractional integration parameter is summarized in Table 2.
Table 2.
Interpretation of the fractional integration parameter
| Fractional integration parameter | Interpretation |
|---|---|
| d = 0 | Short-memory process |
| 0 < d < 0.5 | Stationary long-memory process |
| d = 0.5 | Boundary case between stationarity and non-stationarity |
| 0.5 < d < 1 | Non-stationary but mean-reverting long-memory process |
| d = 1 | Unit root process |
| d > 1 | Non-stationary and non-mean-reverting process |
Accordingly, values of d between 0 and 0.5 indicate that the series is stationary but exhibits long-memory behavior. In this case, shocks to the series decay over time, but the adjustment process is slow. Values of d close to 0.5 imply that the series is near the non-stationarity boundary and therefore displays a high degree of persistence.
Results and discussion
In this study, traditional unit root tests were first applied to comprehensively evaluate the stationarity properties of the series. Then, next-generation unit root tests were used, taking into account possible structural breaks, nonlinear variations, and regime transitions. Furthermore, to improve the methodological robustness of the findings, the AI-based ARNN-ADF unit root test, used increasingly more often in the literature, was also included in the analysis. Finally, the ARFIMA model was applied as a robustness analysis to evaluate the long-term memory and persistence properties of the series. This multi-methodological framework aims to evaluate the stationarity, structural break, nonlinear fit and long-term persistence properties of ecological footprint inequality series together.
The results of traditional unit root tests are reported in Table 3. The ADF and PP test results show that the null hypothesis of unit root is rejected at the 1% significance level for all three ecological footprint inequality indicators. This finding indicates that the Gini-EFI, Theil-EFI, and Atkinson-EFI series are stationary at the levels according to the ADF and PP tests. For the KPSS test, the null hypothesis is stationarity. Since the KPSS statistics are below the 1% critical value, the null hypothesis cannot be rejected. The KPSS test, which takes stationarity as the null hypothesis, provides complementary evidence. The KPSS statistics for all three series remain below the 1% critical value of 0.739, indicating that the null hypothesis of stationarity cannot be rejected. Therefore, traditional unit root tests together demonstrate that the ecological footprint inequality indices are stationary at the levels. Considering the possible limitations of traditional tests, the analysis was extended with Fourier-based tests, structural break tests, ARNN-ADF, and ARFIMA approaches.
Table 3.
Traditional unit root test results
| ADF test stat. | PP test stat. | KPSS test stat. | |
|---|---|---|---|
| Gini-EFI |
-4.218*** (0.001) |
-4.218*** (0.000) |
0.333 |
| Theil-EFI |
-4.9065*** (0.000) |
-4.9065*** (0.000) |
0.2569 |
| Atkinson-EFI |
-4.9515*** (0.000) |
-5.3647*** (0.000) |
0.3267 |
Note: Parentheses indicate probability values and ***, **, and * indicate significance at the 1%, 5%, and 10% levels, respectively. The 1% critical value for the KPSS test statistic is 0.739
Table 4 presents the Fourier-based unit root test results for the Gini-EFI, Theil-EFI, and Atkinson-EFI series. The FADF test rejects the unit root null only for the Theil-EFI series at the 10% significance level, while no evidence of stationarity is found for the Gini-EFI and Atkinson-EFI series. The FGLS results do not reject the unit root null hypothesis for any of the three series. Moreover, the FKPSS statistics exceed the 1% critical value in all cases, implying rejection of the stationarity null hypothesis. Overall, Fourier-based tests reveal that ecological footprint inequality series do not display strong stationarity once smooth structural changes and nonlinear dynamics are incorporated.
Table 4.
Fourier unit root test results
| Constant + trend | ||||||
|---|---|---|---|---|---|---|
| FADF test | Frequency | FGLS test | Frequency | FKPSS test | Frequency | |
| Gini-EFI | -0.995 | 3 | -3.252 | 1 | 0.207 | 1 |
| Theil-EFI | -3.705 | 1 | -2.641 | 3 | 0.176 | 1 |
| Atkinson-EFI | -2.7540 | 3 | -2.7247 | 1 | 0.1919 | 1 |
Note: Critical values at the 1%, 5%, and 10% levels are − 4.45, − 3.78, and − 3.44 for FADF; −4.77, − 4.17, and − 3.87 for FGLS; and 0.07, 0.05, and 0.04 for FKPSS, respectively. The null hypothesis is a unit root for FADF and FGLS, and stationarity for FKPSS
Table 5 reports the results of the Lee-Strazicich single-break LM unit root test for the Gini-EFI, Theil-EFI, and Atkinson-EFI series. Model A allows for level shift, while Model C allows for both level and trend shift. The results show that the test statistics are above the 5% critical values in all cases. Therefore, the structural break unit root hypothesis cannot be rejected for any of the three ecological footprint inequality indicators. This finding demonstrates that the Gini-EFI, Theil-EFI, and Atkinson-EFI series do not provide evidence of stationarity when a single structural break is considered.
Table 5.
Lee–Strazicich one-break LM unit root test results
| Gini-EFI | Theil-EFI | Atkinson-EFI | ||||
|---|---|---|---|---|---|---|
| Model A | Model C | Model A | Model C | Model A | Model C | |
| Test Statistic | -0.710 | -3.021 | -0.492 | -3.188 | -0.342 | -3.228 |
| Lag Length | 2 | 2 | 2 | 2 | 2 | 2 |
| Break Date | 1982 | 1973 | 2002 | 1974 | 1972 | 1973 |
| Critical Value (%5) | -3.566 | -4.47 | -3.566 | -4.47 | -3.566 | -4.47 |
Table 6 presents the results of the Lee–Strazicich two-break LM unit root test. This test allows for the consideration of multiple structural changes in the series, compared to the single-break model. According to the results, the test statistics obtained for Model A and Model C remain above the 5% critical values in all series. Therefore, the unit root hypothesis cannot be rejected for the Gini-EFI, Theil-EFI, and Atkinson-EFI series. This finding reveals that the ecological footprint inequality series do not show stationarity even when two structural breaks are allowed.
Table 6.
Lee–Strazicich two breaks LM unit root test results
| Gini-EFI | Theil-EFI | Atkinson-EFI | ||||
|---|---|---|---|---|---|---|
| Model A | Model C | Model A | Model C | Model A | Model C | |
| Test Statistic | -0.920 | -4.271 | -0.644 | -4.494 | -0.522 | -4.305 |
| Lag Length | 2 | 2 | 2 | 2 | 2 | 2 |
| First Break Date | 1979 | 1977 | 1972 | 1970 | 1972 | 1970 |
| Second Break Date | 1982 | 1983 | 2002 | 2010 | 2002 | 2010 |
| Critical Value (%5) | -3.842 | -5.59 | -3.842 | -5.59 | -3.842 | -5.59 |
The break dates obtained from the table show that the structural changes in the EFI series are mainly concentrated in the 1970s, early 1980s, and post-2000 period. However, since the EFI indices were created with the changing composition of EU membership in mind, these dates are not directly attributed to specific historical events. Instead, they are considered as statistical indicators of structural instability in the ecological footprint inequality process. In this context, the break dates demonstrate that the EFI series is sensitive to changes in sample composition, institutional structure, and the distribution of ecological pressure over time. More importantly, the inability to reject the unit root null hypothesis even after allowing for two structural breaks indicates that ecological footprint inequality persists despite structural changes.
The ARNN-ADF unit root test results are presented in Table 7. The ARNN-ADF test is a complementary test that extends the ADF type unit root test using an artificial neural network approach and takes into account nonlinear dynamics. In this test, the null hypothesis is that the series is not stationary. If the test value is lower than the bootstrapped critical values, the null hypothesis is rejected, and it is concluded that the series is stationary. In this context, Table 7 shows the ARNN-ADF findings regarding ecological footprint inequality indices.
Table 7.
ARNN-ADF unit root test results
| ARNN-ADF Test Statistic |
CV | |||
|---|---|---|---|---|
| Bootstrapped 1% Critical Value | Bootstrapped 5% Critical Value |
Bootstrapped 10% Critical Value | ||
| Gini-EFI | -5.765 | -11.616 | -7.504 | -6.242 |
| Theil-EFI | -7.527 | -11.118 | -7.194 | -5.668 |
| Atkinson-EFI | -5.849 | -10.780 | -6.954 | -5.655 |
The ARNN-ADF results provide mixed evidence regarding the stationarity properties of ecological footprint inequality. For the Gini-EFI series, the ARNN-ADF statistic is not more negative than the bootstrapped critical values; therefore, the unit root null hypothesis cannot be rejected. This implies that Gini-based ecological footprint inequality displays persistent, non-stationary dynamics under the ARNN-ADF framework. By contrast, the unit root null is rejected for Theil-EFI at the 5% significance level and for Atkinson-EFI at the 10% significance level. These findings suggest that the stationarity properties of ecological footprint inequality are sensitive to the inequality measure used.
Table 8 reports the ARFIMA(0,d,0) estimates used to examine the long-memory properties of ecological footprint inequality. The estimated d coefficients are positive, statistically significant, and close to 0.491 for all three EFI indicators. Since the point estimates lie within the 0 < d < 0.5 interval, the results are consistent with stationary long-memory behavior under the parsimonious ARFIMA (0,d,0) specification. However, their proximity to the 0.5 threshold indicates a high degree of persistence and suggests that the series are close to the non-stationarity boundary. Thus, shocks to ecological footprint inequality do not appear to be permanent in the strict unit-root sense, but they dissipate only slowly over time. This finding provides additional evidence that ecological footprint inequality displays persistent dynamics.
Table 8.
ARFIMA (0,d,0) long-memory robustness results
| Model: ARFIMA (0, d, 0) | |||||||
|---|---|---|---|---|---|---|---|
| Indicator | d coefficient | Std. Error | t-statistic | Prob. | AIC | SIC | Interpretation |
| Gini-EFI | 0.491*** | 0.035 | 13.992 | 0.000 | -5.411 | -5.310 | Stationary long memory; near the non-stationarity boundary |
| Theil-EFI | 0.491*** | 0.030 | 16.352 | 0.000 | -6.071 | -5.970 | Stationary long memory; near the non-stationarity boundary |
| Atkinson-EFI | 0.491*** | 0.029 | 16.414 | 0.000 | -6.384 | -6.283 | Stationary long memory; near the non-stationarity boundary |
Note: d denotes the fractional integration parameter. AIC and SIC refer to the Akaike and Schwarz information criteria, respectively. Values of 0 < d < 0.5 indicate stationary long-memory behavior, while values close to 0.5 suggest high persistence and proximity to the non-stationarity boundary. *** denotes statistical significance at the 1% level
The divergence between conventional and more flexible unit root tests should be interpreted in light of their different assumptions about the data-generating process. Conventional unit root tests generally evaluate stationarity under a linear and structurally stable adjustment framework. Therefore, they may indicate level stationarity when the series follows a declining long-run tendency but contains structural changes that are not explicitly modelled. As emphasized by Perron [36], ignoring structural breaks may lead to misleading inference about the presence or absence of a unit root. Similarly, structural-break unit root tests show that allowing for changes in the deterministic component of the series can substantially alter stationarity conclusions [28, 29, 49]. In contrast, Fourier-based tests are designed to capture smooth and gradual structural shifts without requiring prior information about the number, timing, or exact form of breaks [3, 15, 38]. This is particularly relevant for ecological footprint inequality, because the series may be affected by gradual energy transitions, changes in industrial structure, EU enlargement processes, and institutional transformations in environmental policy. Moreover, nonlinear approaches such as the ARNN-ADF test allow for more complex adjustment dynamics that may not be captured by standard linear tests [5, 17, 44]. Therefore, the findings should not be interpreted as showing that all ecological footprint inequality series are strictly non-stationary in a simple unit-root sense. Rather, they suggest that although ecological footprint inequality may appear stationary under linear conventional tests, it displays a high degree of persistence once structural change, nonlinear adjustment, and long-memory dynamics are taken into account. This interpretation is also supported by the ARFIMA results, which indicate stationary long-memory behavior close to the non-stationarity boundary, implying that shocks to ecological footprint inequality dissipate only slowly over time [4].
All these findings are of particular significance in the context of the EU’s environmental and climate policies, as the EU represents one of the most institutionally advanced political unions in terms of climate-related commitments. The results are partly consistent with Padilla and Duro [14], who showed that environmental inequality among EU countries tends to decline over time. Similarly, the descriptive evidence in this study indicates a downward trend in ecological footprint inequality. However, the present study goes beyond the analysis of the direction of inequality by examining its time-series properties. The unit root and structural break tests indicate that ecological footprint inequality exhibits persistent dynamics, while the ARFIMA results further reveal stationary long-memory behavior close to the non-stationarity boundary. Therefore, although ecological footprint inequality has declined over time, shocks to this inequality appear to dissipate only slowly. This finding indicates that environmental inequalities in the EU are not temporary; rather, they constitute a persistent distribution problem that requires long-term and structural policy interventions.
Conclusion and policy recommendations
This study examines the stationarity properties of per capita ecological footprint inequality in 27 EU countries during the period 1961–2024, based on Gini, Theil, and Atkinson indices. For this purpose, traditional, Fourier-based, structural break, and nonlinear unit root tests were applied. First, Fourier-based tests were employed, which are based on the assumption that structural changes can occur nonlinearly and gradually over time. In this context, FADF, FKPSS, and FGLS tests were applied. In addition, Lee and Strazicich’s [29] LM unit root tests, which can internally include level and trend breaks in the series, were also included in the analysis. Both single-break and double-break versions were considered in these tests; Model A considered the level break, while Model C considered both the level and trend breaks. Finally, the ARNN-ADF test, which models nonlinear dependencies with an artificial neural network architecture, was also included in the calculations. The findings show that the test results regarding the stationarity structure of ecological footprint inequality indicators are sensitive to the method used. While traditional unit root tests suggest stationarity, tests that consider structural breaks and nonlinear dynamics support the persistence of ecological footprint inequality. The fundamental reason for this difference is that traditional unit root tests assume the series follows a linear and structurally stable process throughout the entire period. However, ecological footprint inequality is an indicator that can be affected by long-term energy transitions, changes in industrial structure, EU enlargement processes, and institutional transformations in environmental policies. Thus, without considering such structural changes, traditional tests may not fully reflect the persistence characteristics of the series.
Therefore, the overall conclusion of the study is that the EFI series exhibits persistent characteristics when evaluated under more flexible test structures, rather than simple linear tests. A significant portion of Fourier-based tests, structural break LM tests, and the nonlinear ARNN-ADF test support the finding of a unit root in ecological footprint inequality. While some FADF results offer limited evidence of stationarity, this finding is generally not supported by other Fourier-based tests, LM tests, and the ARNN-ADF test. Therefore, it can be said that the impact of shocks to ecological footprint inequality in the 27 European Union countries is not entirely temporary, and the series exhibits persistent dynamics under certain conditions. In other words, increases or decreases in EFI over time do not disappear spontaneously, and the effects of these changes can persist in the long term.
Policy recommendations
The findings indicate that ecological footprint inequality among EU countries cannot be explained solely by transitory shocks but may also be associated with more persistent and structural dynamics. Although traditional unit root tests provide evidence in favor of stationarity, tests that account for structural breaks and nonlinear dynamics strengthen the evidence of persistence in ecological footprint inequality. Therefore, the main policy implication of this study is that environmental policies should not focus solely on reducing total emissions or the average ecological footprint. These policies should also be designed to account for how environmental burdens are distributed across countries. Within this framework, it is important that the EU’s climate and environmental policies be supported by indicators that capture cross-country differences. The possibility that ecological footprint inequality may exhibit persistent characteristics suggests that some countries may face higher adjustment costs or more limited institutional and technological capacity during the environmental transition process. Ecological footprint inequality, in particular, should be monitored more carefully in countries with energy-intensive production structures, high dependence on fossil fuels, or more limited capacity for environmental transformation.
EU countries are not homogeneous in terms of energy sources, production structures, and environmental technology capacity. While some countries adapt more rapidly to renewable energy and low-carbon production technologies, reliance on fossil fuels and energy-intensive sectors may persist for a longer period in others. Therefore, designing environmental transition policies in a way that takes country heterogeneity into account is important for reducing ecological footprint inequality.
The links between ecological footprint inequality and income, consumption, and production structures should also be considered. Higher income levels may increase consumption-related ecological pressure, while lower-income countries may have more limited access to environmental technologies and a weaker capacity for transformation. Therefore, reducing environmental inequalities should not be addressed solely through environmental policies, but should also be considered together with energy efficiency, technological transformation, the modernization of production structures, and the promotion of sustainable consumption patterns. However, this study does not test the direct causal effects of specific policy instruments. Therefore, rather than proposing specific fiscal mechanisms or institutional tools, the findings point to general directions that should be considered in policy design. These directions can be summarized as long-term environmental monitoring, policy design sensitive to country heterogeneity, support for the energy transition, and greater visibility of environmental burden-sharing.
In conclusion, ecological footprint inequality in EU countries is not only an environmental indicator, but also an important policy dimension in terms of sustainable development, just transition, and environmental burden-sharing. Therefore, the success of environmental policies should be evaluated not only by reductions in total emissions or the average ecological footprint, but also by a more balanced distribution of environmental pressure across countries. This approach may contribute to establishing a stronger link between the EU’s long-term climate goals and the principles of environmental justice and just transition.
Study limitations and future directions
By using the Gini, Theil-T and Atkinson indices, this study examines ecological footprint inequality across EU countries from different perspectives. However, these indices are aggregate measures. They show the general level of inequality, but they do not explain which countries or country groups contribute most to this inequality. Therefore, future studies may use decomposition-based methods or population-weighted inequality measures to identify the main sources of ecological footprint inequality more clearly.
Future research may also examine the components of the ecological footprint separately. This study focuses on total ecological footprint per capita, but ecological footprint includes different components such as carbon footprint, cropland, grazing land, forest products, fishing grounds and built-up land. Analyzing these components separately would help explain whether inequality is mainly related to carbon emissions, land use, resource consumption or other environmental pressures.
In addition, future studies may compare different country groups within the EU. Countries may differ in terms of income level, energy structure, industrial development and environmental policies. Considering these differences would help produce more country-specific and effective policy recommendations.
Author contributions
S.Y: Conceptualization, Formal analysis; B.A.: Software, Methodology; N.O.B.: Data curation, Writing original draft; M.R.: Resources, Validation, Writing original draft; L.T.: Investigation, Reviewing and editing.
Funding
This research received no funding.
Data availability
Data will be available on request from the corresponding author.
Declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Data Availability Statement
Data will be available on request from the corresponding author.