Stochastic convergence of ecological footprint: new insights from a unit root test based on smooth transitions and nonlinear adjustment

The ecological footprint has currently become a highly popular environmental performance indicator. It provides the basis for setting goals, identifying options for action, and tracking progress toward stated goals. Because the examination of the existence of convergence is important for the climate change protection of the earth, the convergence of ecological footprint and its subcomponents are a major concern for scholars and policymakers. To this end, this study aims to investigate the stochastic convergence of ecological footprint and its subcomponents. We employ the recently developed Hepsag (2021) unit root test that allows nonlinearity and smooth structural change simultaneously to study stochastic convergence in per-capita ecological footprint over the period 1961–2018 for the most polluting countries. The results provide mixed evidence of the presence of stochastic convergence in conventional unit root tests such as ADF, KPSS and Fourier KPSS. According to the Hepsag (2021) unit root test results for all countries, built-up land footprint converges except Australia, Malaysia, Poland, and Turkey. Carbon footprint converges for Indonesia, Malaysia, Mexico, South Africa, Thailand, Turkey, the UK, and the USA. Cropland footprint converges for Australia, Canada, China, France, Indonesia, Italy, Japan, Korea, Malaysia, Mexico, Poland, South Africa, the UK, and Vietnam. Fishing grounds footprint converges in Brazil, France, Germany, Indonesia, Italy, Mexico, South Africa, and Vietnam. Forest product footprint converges in Australia, Canada, France, Germany, India, Korea, Mexico, Poland, Turkey, and Vietnam. Grazing land footprint converges in Canada, France, India, Indonesia, Japan, Korea, Poland, South Africa, Thailand, and Vietnam. And lastly, the total ecological footprint converges in Canada, France, Korea, Malaysia, Mexico, South Africa, the UK, and the USA.

long-term consequences. The climate change is almost entirely caused by global warming, which is defined as the increase in the average surface temperature of the world due to the increase in greenhouse gases released into the atmosphere. The harmful effects of human-induced climate change, such as the excessive use of natural resources, destruction of forests, the pollution of clean water, rapid population growth and urbanization, and increased chemical production, along with the rapid industrialization process, have also increased environmental degradation.
According to the Intergovernmental Panel on Climate (IPCC), it is estimated that human activities have caused the world to warm in the range of about 1 °C, 0.8 °C, and 1.2 °C, compared to the pre-industrial period. The IPPC stated that if the current increase in greenhouse gas emissions continues, global warming will exceed 1.5 °C between 2030 and 2052. It is also stated that this situation has critical importance in sustainable development and poverty prevention. In order not to exceed this limit, it is necessary to reduce global emissions by 45% by 2030 compared to 2010 and to reach net zero emissions by 2050. In this context, countries need to renew their commitments as soon as possible with a rapid and comprehensive transformation in the fields of agriculture, energy, industry, and transportation. Therefore, Haider and Akram (2019) highlight that the convergence of environmental degradation indicators is essential in order to put forward and implement policies to reduce climate change.
The first basis for the convergence of countries in terms of environmental quality is based on the environmental Kuznets curve (EKC)-based environmental capture hypothesis proposed by Brock and Scott Taylor (2003). The EKC, which shows the relationship between economic growth and environmental pollution, was first put forward by Grossman and Krueger (1995). It is reported that first, the degree of environmental degradation increases with the increase in per-capita income, and then the quality of the environment begins to increase after the per-capita income reaches a certain threshold. In other words, environmental pollution increases first and then starts to decrease after reaching a certain level of income. The EKC shows an inverted U-shaped relationship between environmental indicators and economic growth. According to the EKC, countries that reach a certain level of income reduce their emissions. In this case, the increase in income will converge with percapita emissions (Strazicich and List 2003). As the income levels of the countries increase, it is expected that the growth rate of their emissions will slow down and cause the emissions to converge (Acar and Lindmark 2017). As the second reason, convergence is based on efforts to limit carbon emissions within the framework of international agreements such as the IPCC, Kyoto protocol, and Paris agreement to prevent global warming and climate change (Aldy 2006).
CO 2 emission has the largest share of greenhouse gas emissions. Fossil fuels are the main source of this CO 2 emission to spread to the atmosphere. The efforts to reduce CO 2 emissions are among the most important issues on the world's agenda. Despite widespread support for the Paris Agreement and the possibility to provide energy demand from clean, affordable and sustainable energy sources, energy-related CO 2 emissions increased by an average of 1.3% annually over 2014-2019 (Gielen et al. 2021). It decreased by 7% in 2020 only due to the pandemic. According to the International Energy Agency (2021), carbon emissions increased by 6% in 2021, reaching a record level of 36.3 billion tons. The report states that existing policy measures are not sufficient to achieve these goals. It is also stated that these countries should make more efforts to reach their net zero emission targets by 2050 (World Energy Outlook 2021).
In this context, since greenhouse gas emissions are focused on as the cause of global warming, the studies on CO 2 convergence has been examined more by both policymakers and researchers (Apergis and Payne 2017;Acar et al. 2018;Erdogan and Solarin 2021). However, the CO 2 emissions make up a part of the total (Al-Mulali et al. 2015). Environmental quality is multidimensional. For an effective climate change policy, it has become necessary to analyze a comprehensive indicator that reveals different aspects of environmental quality across countries (Bilgili and Ulucak 2018). Thus, the ecological footprint which measures ecological sustainability was defined by Rees (1992) and Wackernagel (1994) in the early 1990s. It is used as a comprehensive measure of anthropogenic pressure on the environment. The ecological footprint is more comprehensive than CO 2 emissions to set targets for climate change and monitor progress in achieving those targets (Al-Mulali et al. 2015;Ulucak and Lin 2017;Solarin 2019).
The ecological footprint was developed to calculate the effect of a population has on nature as a result of its production and consumption activities (Mancini et al. 2016). According to the World Wide Fund for Nature (WWF), this measurement indicates the bio-capacity required to regenerate the natural resources consumed by a community or human activity and neutralize or eliminate the resulting waste through existing technology and resources. The WWF divides the ecological footprint into six subcomponents. These are cropland, grazing land, fishing grounds, forest land, built-up land, and carbon footprint. The effect of the carbon footprint in these components is more than the effects of all other components. The carbon footprint is the fastest growing component and also accounts for around 60% of all degradation.
The ecological footprint indicates the extent to which people have crossed the environmental limit and the natural limit in assessing how countries use ecological resources. Therefore, the long-term impact of the link between the use of natural resource and environmental degradation become different between countries (Sarkodie 2021). In this line, the ecological footprint per capita varies considerably (Nguyen 2005), and therefore it becomes very important to examine the magnitude of inequality and trends in environmental degradation levels across countries. In addition, because the industrial characteristics of countries can affect the convergence of environmental degradation indicators (Isik et al. 2021b), the convergence of countries' CO 2 emissions and ecological footprints admit researchers and policymakers to monitor in assessing the efficacy and performance of environmental policies (Bilgili and Ulucak 2018). Within this framework, the convergence of countries can agree on the level of emission reduction targets and meet obligations to mutual environmental degradation by implementing common environmental policies, as indicated in international negotiations (Solarin et al. 2019a). In this context, the aim of this study is to test the stochastic convergence of ecological footprint (for six components) using both traditional unit root tests such as ADF and KPSS and linear and nonlinear unit root tests such as Hepsag's (2021) unit root test, which allowed structural changes, for 20 countries that accounted for 75% of the world's CO 2 emissions in 2020.
The contribution of this study is to the existing literature in several ways. First, while on line of a vast literature in this area investigates the convergence of CO 2 emission as an indicator of environmental degradation, another line of analaysis, such as Bilgili and Ulucak (2018), Isik et al. (2021a) and Erdogan and Okumus (2021), Tillaguango et al. (2021), and Yilanci et al. (2022b), focus on the convergence of the ecological footprint, which is a more comprehensive indicator of environmental degradation (Baabou et al. 2017). Accordingly, this study analyzes the convergence of the total and six subcomponents of the ecological footprint separately so that the different characteristics of the subcomponents are not ignored. Second, unlike most previous studies, we analyze the convergence of total and subcomponents of ecological footprint per capita for large sample countries. It has been previously stated that the carbon footprint has a very high share in the ecological footprint. Therefore, in the analysis, the top 20 countries with the highest CO 2 emissions for 2021 according to BP Stat Rev. are considered, such as China, the USA, India, Japan, Germany, Korea, Canada, Indonesia, South Africa, Mexico, Brazil, France, Italy, Malaysia, Poland, Thailand, Turkey, the UK, Vietnam, and Australia. These countries make 75% of the total emissions. Third, our contribution is methodological. Most studies in the empirical literature (such as Solarin et al. 2019a andUlucak et al. 2020) use panel methods, and this leads to neglect of the country-specific effect. In order to capture these heterogeneities, unlike previous studies in the empirical literature using panel methods, we apply several traditional as well as more recent time-series methodologies. At the same time, this study takes into account a specification which allows for unknown structural break and nonlinearity simultaneously. Hereby, it allows to comprehend how the stances and policies of different countries contribute to sustainable development (Bigerna et al. 2022). Finally, to the best of our knowledge, this is the first study in the stochastic convergence literature to examine the ecological footprint and its subcomponents with a unit root test that allows for both structural change and nonlinearity simultaneously.
The rest of this paper is structured as follows: the "An overview of the convergence literature" section is devoted to a brief review of relevant literature. The "Methodology" section presents data and empirical strategy. The "Data analysis and findings" section discusses the estimation results while the "Concluding remarks" section concludes.

An overview of the convergence literature
The concept of environmental convergence investigates whether the environmental policies of countries tend to become similar over time (Ulucak et al. 2020). Upon methodologically examining the empirical studies on environmental convergence, it is seen that different convergence concepts are used. The first of these is beta (β)-convergence. β-Convergence, which stemmed from the Neoclassical growth literature developed by Solow (1956) to test income convergence and concentrated on the linear relationship between the initial and later levels of emissions, is divided into unconditional (absolute)-conditional (relative) convergence. The inverse association between the initial levels of emissions and the subsequent growth rates indicates that absolute convergence is valid (Erdogan and Solarin 2021). In other terms, it means that countries would converge to a similar steady-state emission level regardless of their initial conditions. Conditional convergence, however, indicates that each country attains its own steady-state equilibrium and would converge to its own steady-state equilibrium, taking into account the differences across countries (Bilgili and Ulucak 2018;Erdogan and Okumus 2021). The second is the sigma (σ)-convergence. In the environmental convergence literature, σ-convergence indicates that the cross-section variance of per-capita emissions tends to decrease over time across the compared countries (Acar et al. 2018;Churchill et al. 2020). The third concept of convergence used in empirical studies is club convergence. Club convergence indicates that, when countries with similar initial conditions and common characteristics are grouped, each group has the same steady-state equilibrium and would converge to its own equilibrium (Erdogan and Okumus 2021;Islam 2003). Lastly, stochastic convergence states that shocks to a country's per-capita emissions are temporarily relative to the average of the per-capita emissions of another country or a group of countries, and would return to a long-term equilibrium state. As a result of the conducted analyses by performing different unit root or stationarity tests, it is concluded that stochastic convergence is valid if the series obtained by dividing the per-capita emissions for the ith country by the average per-capita emissions of a country group is found to be stationary (Erdogan and Solarin 2021;Nazlioglu et al. 2021;Payne 2020).
The introduction of new unit root tests with different features (such as structural break, and nonlinearity) to the literature, especially in time series analysis, has led to an increase in the number of studies testing the environmental convergence hypothesis. These studies can be divided into two groups. In the first group of studies, the carbon dioxide (CO 2 ) emission variable is mainly utilized as an indicator of environmental degradation, although different variables such as sulfur dioxide ( SO 2 ), environmental performance index, and CO 2 emission intensity may also be used. The second group of studies concentrates on the ecological footprint (EF), which is highly comprehensive and ideal indicator of environmental degradation, as well as its sub-components (Ozturk et al. 2016;Ulucak and Lin 2017). Upon examining the studies in the literature, it is seen that panel unit root tests with different qualities and the panel club convergence test coined by Sul (2007, 2009) have been performed extensively in testing the convergence hypothesis. Although some studies obtained findings that supported convergence, some other studies concluded that no convergence existed due to the differences in both the used variables used, the employed methods, and the selected country or country groups. Strazicich and List (2003), as one of the first groups of studies, examined the environmental convergence hypothesis using CO 2 emissions in 21 industrialized countries by performing panel unit root tests and cross-section regression analysis. In their study, utilizing the data obtained over the period 1960-1997, the authors concluded that stochastic and conditional convergence were valid for the examined countries. Lanne and Liski (2004), by using the CO 2 emissions of 16 industrialized countries over the period 1870-2028, investigated the existence of environmental convergence by performing the panel unit root test which considers structural breaks and concluded that no convergence existed. Nguyen (2005) examined whether or not a convergence existed among the countries by using the per-capita CO 2 emissions of 100 countries between the years 1966-1996. By conducting a nonparametric distribution analysis, the author stated that no environmental convergence existed for the sample group of 100 countries; nonetheless, β-convergence was valid for the sample group consisting of 23 industrialized countries. Aldy (2006) investigated whether environmental convergence existed using the per-capita CO 2 emissions of 88 countries over the period 1960-2000, for both the whole sample group of 88 countries and the sub-sample group consisting of 23 Organization for Economic Co-operation and Development (OECD) member countries. The results of the study, in which unit root and Markov transition matrix analyses were conducted, indicated that convergence was valid for merely 13 countries out of the sample group consisting of 23 OECD-member countries as well as other 88 countries. Lee and Chang (2009) explicated whether per-capita CO 2 emissions converged stochastically for 21 OECD-member countries by performing Carrion-i-Silvestre et al. 's (2005) panel unit root test that allows structural breaks. In their study, by using the data obtained over the period 1950-2002, the authors concluded that stochastic convergence was valid for the investigated countries. Jobert et al. (2010), by using the data obtained over the period 1971-2006 and employing the Bayesian shrinkage estimation method, investigated whether or not per-capita CO 2 emissions in 22 European Union (EU) member countries converged, and they found that such convergence existed. Herrerias (2012), who investigated the convergence of per-capita CO 2 emissions in 25 EU countries over the period 1920-2007 using the distribution dynamics approach, concluded that the convergence was valid for all of those countries. Yavuz and Yilanci (2013) examined whether or not per-capita CO 2 emissions in G7 countries converged between the years 1960-2005 performing the threshold autoregressive (TAR) panel unit root test and concluded that such convergence existed. Payne et al. (2014) performing the residual augmented least squares-Lagrange multiplier (RALS-LM) unit root test and per-capita SO 2 emissions investigated whether or not stochastic conditional convergence was valid for the USA between 1900 and 1998. The results of the analysis indicated the existence of stochastic conditional convergence for the investigated states. Wang et al. (2014) tested whether the Chinese provinces converged by using per-capita CO 2 emissions between 1995 and 2011 by employing the time-varying factor model developed by Phillips and Sul (2007). They concluded that convergence did not occur for the entire sample, whereas the formation of convergence clubs was suggested. By performing nonlinear time series and panel unit root tests, Tiwari et al. (2016) investigated whether or not the convergence hypothesis was valid, based on the stationarity of per-capita CO 2 emissions of 35 sub-Saharan African countries over the period 1960-2009. In their study, by performing nonlinear time series and panel unit root tests, the authors obtained results that supported convergence for 27 countries in the time series analysis and 15 countries in the panel data analysis. Ahmed et al. (2017) analyzed the convergence of annual per-capita CO 2 emissions of 162 countries consisting of different income groups over the period 1960-2010 by employing the wavelet unit root method, and they obtained results that supported convergence for 38 countries, whereas concluded that divergence existed for 124 countries. Apergis and Payne (2017) investigated whether the per-capita CO 2 emissions of 50 US states were convergent by employing the time-varying factor model approach developed by Sul (2007, 2009). In their study, by using the annual data obtained over the period 1980-2013, the authors concluded that although some states exhibited convergence, some other states exhibited divergence. Unlike previous studies, Brännlund and Karimu (2018) investigated whether or not convergence existed by developing an environmental performance index for 94 countries. The results of the study, in which the data obtained over the period 1971-2008 were utilized and three different panel unit root tests were performed, supported the β convergence. Presno et al. (2018) investigated whether percapita CO 2 emissions indicated stochastic convergence for 28 OECD-member countries by performing a nonlinear unit root test. By utilizing the data obtained over the period 1901-2009, the authors found little evidence of the validity of stochastic convergence for those countries. The study, which also investigated whether the β-convergence was valid, found that the β-convergence was valid for developed countries, whereas it was not valid for developing countries. Emir et al. (2019) examined the convergence properties of CO 2 intensity of 28 EU countries over the period 1990-2016 by employing the time-varying factor model approach developed by Phillips and Sul (2007). By employing the Phillips and Sul (2007) clustering method, the countries included in the analysis were categorized by the similarities in the data matrix and it was tested whether convergence existed within each group. The results indicated that no general convergence club existed for all countries, whereas it indicated that convergence clubs took place in some subgroups of different countries. Erdogan and Acaravci (2019) investigated the convergence of per-capita CO 2 emissions in 28 OECD-member countries by performing the Fourier panel Kwiatkowski-Phillips-Schmidt-Shin (PANKPSS) unit root test. The results of the study, in which the data obtained over the period 1960-2014 were used, indicated that convergence existed in the examined countries. Churchill et al. (2020) investigated whether or not the environmental convergence hypothesis is valid, by using per-capita CO 2 emissions obtained from 17 emerging markets. By using data obtained over the period 1921-2014 and performing the Lagrange Multiplier and RALS-LM unit root tests, the authors concluded that shocks toward per-capita CO 2 emissions in merely 11 countries were temporary, whereas stochastic convergence was valid for those 11 countries. Solarin and Tiwari (2020) investigated whether or not the convergence hypothesis was valid for 32 OECD countries by using percapita SO 2 emissions. The results of the data obtained over the period 1850-2005 and the Fourier PANKPSS unit root test indicated that the convergence hypothesis was valid in the mentioned countries. Erdogan and Solarin (2021), by performing the Fourier-based wavelet augmented Dickey-Fuller unit root test and using per-capita CO 2 emissions, investigated whether or not stochastic convergence was valid for 151 countries of different income groups over the period 1960-2016. The analysis results revealed that per-capita CO 2 emissions in 35 high-income countries, 27 upper-middleincome countries, 30 low-middle-income countries, and 13 low-income countries followed a stationary process; in other words, stochastic convergence was valid across different income groups. Nazlioglu et al. (2021), using the annual data obtained over the period 1960-2016, examined whether or not CO 2 per-capita emissions exhibited stochastic convergence in 13 Organization of Petroleum Exporting Countries (OPEC) countries and 18 petroleum-producing non-OPEC member countries. By performing new panel stability tests that take into account cross-correlations and structural breaks, the authors concluded that no convergence existed regarding per-capita CO 2 emissions across countries. Payne and Apergis (2021), by using the per capita CO 2 emissions of 65 developing countries over the period 1972-2014, investigated whether the stochastic and club convergence hypotheses were valid for those countries. The authors performed the cross-sectional ADF and Bai and Carrion-i-Silvestre (2009) panel unit root tests to detect stochastic convergence and employed the time-varying factor model approach developed by Sul (2007, 2009) to detect club convergence. The analysis results revealed that both stochastic convergence and convergence clubs were valid for the entire countries, as well as subgroups constituted by different countries. By performing panel unit root tests and employing the Phillips and Sul (2007) time-varying factor model approach, Tiwari et al. (2021), in contrast to convergence, obtained results that supported divergence across the states. Ulucak and Apergis (2018), one of the studies that used EF and its subcomponents as the indicators of environmental degradation, examined whether or not per-capita EF converged for 20 EU countries, by employing the time-varying factor model developed by Sul (2007, 2009). In their study, where they considered the period 1961-2013, the authors concluded that convergence clubs existed for some of the subgroups. Bilgili and Ulucak (2018) examined whether or not the convergence of per-capita EF existed for G20 countries by performing the bootstrap-based panel KPSS and club convergence tests developed by Sul (2007, 2009). The results of the study, in which the period 1961-2014 was considered, indicated that stochastic and deterministic convergence of per-capita EF existed. Haider and Akram (2019), by performing the panel convergence test developed by Sul (2007, 2009), investigated whether or not the annual per-capita EF and per-capita carbon footprint of 77 countries between 1961 and 2014 indicated club convergence, detected no convergence club on a sample basis, and concluded that convergence clubs existed for some of the subgroups. Solarin (2019) analyzed whether or not per-capita CO 2 emissions, per-capita carbon footprint, and per-capita EF converged in 27 OECD-member countries over the 1961-2013 period, by performing the LM and RALS-LM unit root tests. Results revealed that per-capita CO 2 emissions converged for 12 countries, per-capita carbon footprint converged for 15 countries, and per-capita EF converged for 13 countries. Solarin et al. (2019a), who investigated the convergence of per capita EF and its components by employing club convergence approaches developed by Sul (2007, 2009) and Schnurbus et al. (2017), concluded that several convergence clubs existed in the examined countries. Solarin et al. (2019b) analyzed whether or not the shocks to carbon footprints in 92 countries with different income levels were temporary by performing various unit root tests. The results obtained from the study, in which the data obtained from the period 1961-2014 were used, indicated that the carbon footprint tended to be mean-reverting for merely 25 out of 92 countries; in other words, the shocks to the carbon footprint in those countries were temporary and the carbon footprints were convergent. Yilanci et al. (2019) investigated whether or not EF and its subcomponents contained unit roots by performing the stationarity test, which allows both soft and sharp breaks, developed by Bahmani-Oskooee et al. (2014). The analysis results revealed that only the fishery footprint of the used ecological indicators contained unit root (not stationary); in other words, it did not tend to be mean-reverting and did not exhibit convergence. Ulucak et al. (2020) investigated whether or not the environmental convergence hypothesis was valid, by using the EF and its components over the period 1961-2014 for 23 countries located in the Sub-Saharan Africa region. The authors employed the time-varying factor model approach introduced by Sul (2007, 2009) in their study and concluded that no overall convergence club for the entire countries, but some convergence clubs existed for certain sub-components. Yilanci and Pata (2020) examined the convergence of per-capita EF across 5 members of the Association of Southeast Asian Nations (ASEAN-5) over the period 1961-2016. The results of the study performing the two-regime TAR panel unit root test provided strong evidence for the existence of absolute convergence of per-capita EF across the ASEAN-5 countries. Erdogan and Okumus (2021), by performing the Fourier PANKPSS and club convergence tests, examined whether or not the annual per-capita EF exhibited stochastic and club convergence for 89 countries consisting of low-, middle-, and high-income groups over the period 1961-2016. The panel stationarity test results indicate that, contrary to convergence, divergence existed in the mentioned countries, whereas club convergence test results indicated that several club convergences existed across different income groups. Isik et al. (2021a) examined whether or not per-capita EF converged in the North American Free Trade Agreement (USMCA) member countries. In their study, where they utilized the data obtained between 1961 and 2016 and performed the TAR panel unit root test, the authors concluded that the differences in the per-capita EF of the countries decreased over time; thus, convergence existed. Yildirim et al. (2021) examined whether or not the EF and its four components (cropland footprint, grazing land footprint, fishing footprint, and forest footprint) converged in 16 European countries by performing the Fourier panel Kapetanios-Shin-Snell test. In their study, where they utilized the data obtained over the period 1961-2016, the authors concluded that no convergence existed in the examined countries. Yilanci et al. (2022a) investigated whether or not the shocks toward the annual per-capita EF of 13 Mediterranean countries between 1961 and 2014 were temporary, by performing the LM and RALS-LM unit root tests, and detected that the shocks toward per-capita EF of 13 Mediterranean countries were temporary. Those results supported the existence of convergence for these countries. In their study examining the convergence of carbon footprint and per-capita EF for G7 countries over the period 1961-2016, by performing the panel Fourier threshold unit root test, Yilanci et al. (2022b) concluded that absolute convergence was valid for G7 countries and that policy authorities might have implemented common environmental policies against environmental degradation in those countries.
Briefly, there are various studies in the literature using different environmental degradation indicators and employing different econometric methods. Upon examining the results obtained from the studies, it is stated that no consensus exists on environmental convergence; it is seen that the results of some studies supported the environmental convergence hypothesis across countries, whereas the results of some studies did not support the validity of the environmental convergence hypothesis. There may be several reasons why the results obtained from those studies are inconsistent. The first of these may be due to the employment of conventional econometric methods that do not take into account nonlinearity and/or structural changes, and the second may stem from the preference of panel data analysis methods in which countries with different environmental structures and characteristics are used as the sample group in the studies. According to Hsiao (1985), the power of panel data analysis method depends on the reliability and scope of the information it covers, as well as the validity of the constraints on which statistical methods are built. Maddala (1999) criticized the performance of panel unit root tests in testing the theory of convergence across countries. Solarin et al. (2019a) and Haider and Akram (2019) concluded that there was no environmental convergence between countries for the entire sample group in their studies, even though they employed the panel data analysis method; however, the convergence hypothesis was valid in subgroups when countries were classified by certain characteristics for the same sample groups. The distinguishing feature of this study in comparison to previous studies and its contribution to the existing literature involve the performance of the unit root test, which considers both nonlinearity and smooth transition structural change, in testing the existence of stochastic convergence of ecological footprint and its sub-components, and the employment of the time-series method to avoid aggregation bias that can be caused by the aforementioned panel data analysis method.

Data collection process
The purpose of our econometric analysis is to test the validity of stochastic convergence in the 20 countries with the highest carbon emissions, accounting for approximately 75% of the world's carbon emissions according to the BP Statistical Review of World Energy 2021. For this purpose, this manuscript collected data of total per-capita ecological footprint and subcomponents of ecological footprint per capita provided by Global Footprint Network for the period 1961-2018.
For each country i, we study the natural logarithm of the ratio of per-capita total ecological footprint and subcomponents relative to the average across all countries as follows, where RPEF is the relative per capita ecological footprint (or subcomponent) for country i at time t. If the RPEF follows a stationary process, it will be determined that stochastic convergence is valid.

Empirical methodology
In conventional unit root tests, the existence of a constant mean reversion rate is accepted. Thus, for such tests to conclude that y t is stationary, the convergence process must be linear. As claimed by Datta (2003), the process may not be linear or it may have structural changes along with technology and/or policy shocks (King and Ramlogan-Dobson 2011). In the real world, various time-series data exhibit nonlinearities, outliers, and structural breaks in either the mean or variance. All these properties, such as random walk, that cannot be accurately captured by models reduce the explanatory power of conventional unit root tests. Many economic and financial time series, such (1) RPEF = ln( PEF it average PEF t ) as inflation, unemployment rate, and nominal and real interest rates, may be a trend stationary with a structural break in the unconditional mean that causes fixed coefficient models to perform weakly in practice. Perron (1989;1990) and Perron and Vogelsang (1992) stated that structural breaks could have caused a stationary time series at the level to exhibit difference stationarity, and as a result, those breaks would have affected the strength of standard unit root tests. Appropriate consideration of deviations such as parameter shifts, trend breaks, and nonlinearities requires the development of robust unit root tests. In practice, it is troublesome to notice whether a time series exhibiting unit root-like behavior is actually difference stationary or it is a monotonic nonlinear transformation of a difference stationary series. Specifying the true timeseries model incorrectly can influence the divergence rate of the test statistic with standard unit root tests, causing it to exhibit inconsistency (Aparicio et al. 2006).
Studies aiming to test stochastic convergence, in general, prefer to perform conventional unit root tests such as the ADF, Phillips-Perron, and KPSS. However, these tests accept the assumption of linearity of the variables and there are many reasons to question this assumption. Enders and Granger (1998) stated that the explanatory power of conventional unit root tests would decrease in the presence of an asymmetric adaptation process. Besides the nonlinearity assumption, the potential impacts of various events (1974 energy crisis, 2008 global economic crisis, etc.) on the series are not taken into account in conventional unit root tests due to the use of long-span data in the study, which also leads to a decline in the explanatory power of these tests. In the presence of structural breaks and nonlinearity in time series data, the power of conventional unit root tests that do not allow these two impacts simultaneously would decline. Therefore, the probability of rejecting the null hypothesis implying the existence of unit root according to the ADF test would decrease and it would not be possible to distinguish the stationary process from the nonstationary process (Hepsag 2021).
In this study, Hepsag's (2021) unit root test, which is an ESTAR type test, is employed to fill the gap in the stochastic convergence analyses performed by concurrently considering both nonlinearity and structural breaks in the literature. In Hepsag's (2021) unit root test, structural breaks among different regimes are taken into account with the logistic smooth transition function, and nonlinearity is considered through the ESTAR model proposed in Kruse (2011). The test was developed as an alternative to Leybourne et al. (1998) and Kruse's (2011) unit root tests. Hepsag's (2021) unit root test procedure was established by following the study of Leybourne et al. (1998) and defining the three logistic smooth transition models specified in Eqs. 2, 3, and 4 (Hepsag 2021). v t denotes the error term; and S t ( , ) represents the logistic smooth transition function determined according to the sample number T.
denotes the timing of the midpoint of the transition, and the velocity of the transition is determined by the coefficient .
Assuming that v t represents a zero-mean I(0) process, model A represents a stationary process around the mean that ranges from the initial value of 1 to the final value of 1 + 2 . Model B, similar to model A, expresses a changing process from the initial value of 1 to the final value of 1 + 2 with the constant slope term.
And finally, while model C ranges from the constant term 1 to 1 + 2 , the slope simultaneously ranges from 1 to 1 + 2 at the same transition rate (Hepsag 2021). In the first stage of Hepsag's (2021) unit root test; models A, B, and C are estimated by the nonlinear least-squares method, and residuals are obtained.
In the second stage, Kruse's (2011) unit root test is performed on these residues. Then, as in Eq. 9, the Kruse (2011) ESTAR model is modified to allow a nonzero position parameter c. v t denotes residuals estimated in the first stage. In his study, Kruse (2011) suggested applying a first-order Taylor approximation to Eq. 9 and obtaining the auxiliary regression equation specified in Eq. 10.
In Hepsag's (2021) unit root test, the null hypothesis implies the existence of a unit root, whereas the alternative hypothesis implies ESTAR stationarity with the smooth break.

Data analysis and findings
For comparison purposes, the ADF and KPSS tests, which are conventional unit root tests that do not take into account nonlinearity and structural change, and the Fourier KPSS (hereafter FKPSS) test results, which merely consider structural change, are presented in Table 1.
According to the ADF stationarity test result, the null hypothesis is rejected for built-up land, carbon, and cropland footprints at Australia; forest products and total EF footprints for Brazil; cropland and grazing land footprints for France; carbon footprint for Italy; grazing land footprint for Japan; fishing grounds and grazing land footprints for Korea; built-up land and cropland footprints for Malaysia; grazing land footprint for Mexico; forest product footprint for South Africa; fishing grounds footprint for Thailand and Turkey; and lastly grazing land footprint for the UK and determined to be stationary at the level so stochastic convergence hypothesis is valid.
According to the KPSS test result, it is determined that the stochastic convergence is valid at built-up land, carbon, and cropland footprints for Australia; forest products and total EF footprints for Brazil; forest product footprint for China; built-up land, cropland, and grazing land footprints for France and Germany; cropland and forest product footprints for India; total EF for Indonesia; carbon and cropland footprints for Italy; carbon, cropland, grazing land, and total EF footprints for Japan; built-up land and cropland footprints for Malaysia; built-up land, forest products, and grazing land footprints for Mexico; forest products and total EF for South Africa; built-up land, cropland, fishing grounds, and grazing land footprints for Turkey; built-up land and grazing land footprints for the UK; built-up land, cropland, and forest product footprints for the USA; and cropland footprint for Vietnam.
According to the FKPSS test results, the null hypothesis cannot be rejected for built-up land and forest product footprints for Australia; forest products and total EF for Brazil; built-up land, cropland, forest products, and grazing land footprints for France; built-up land and cropland footprints for Germany and Malaysia; cropland, forest products, and grazing land footprints for India; total EF for Indonesia; carbon and cropland footprints for Italy; cropland, grazing land, and total EF footprints for Japan; forest products, grazing land, and total EF footprints for Mexico; fishing grounds and total EF footprints for Poland; builtup land, carbon, and forest product footprints for South Africa; grazing land footprint for Thailand and the USA; built-up land, cropland, fishing grounds, and grazing land footprints for Turkey; and built-up land footprint for the UK, so the stochastic convergence hypothesis is determined valid. It is estimated that the main reason for the difference in results among the ADF, KPSS, and FKPSS stationary tests involves the fact that these tests have limitations on nonlinearity and/or structural change, and these limitations negatively affect their explanatory powers. Hepsag's (2021) test results, which may yield more reliable results compared to the ADF, KPSS, and FKPSS tests, are presented in Table 2

Concluding remarks
On November 11, 2014, a bilateral agreement was signed by the USA and China to minimize the emission of greenhouse gases, which is one of the main causes of global warming. Following the agreement, at the Paris Climate Change Summit on December 12, 2015, attended by the leaders of 195 countries, it was announced that the agreement known today as the Paris Climate Agreement was signed. The main objective of the signed agreement was to limit future greenhouse gas emissions along with much more drastic measures compared to previous attempts. In this regard, it was aimed that future global temperatures would increase by 1.5 o C or at most 2 o C compared to the pre-industrial period. The accomplishment of these goals merely depends on the countries that signed the agreement to act together and fulfill their obligations.
In this study, it is investigated whether or not the top 20 pollutant countries (Australia, Brazil, Canada, China, France, Germany, India, Indonesia, Italy, Japan, Korea, Malaysia, Mexico, Poland, South Africa, Thailand, Turkey, the UK, the USA, and Vietnam), which account for approximately 75% of the worldwide CO 2 emissions according to the BP Statistical Review of World Energy 2021, converge depending on their implemented policies  Hepsag's (2021) unit root test yields more reliable results by considering the structural changes and nonlinearity in the series simultaneously. To this end, the data of the ecological footprint, as well as subcomponents such as built-up land, carbon, cropland, grazing land, fishing grounds, and forest products obtained over the period 1961-2018, are utilized in the analysis. The FKPSS test, which allows only structural change, as well as the conventional unit root tests, namely, the ADF and KPSS unit root tests, tends to yield mixed results. It is anticipated that the differences in the explanation parameters account for the result differences across the tests. Hepsag's (2021) unit root test results can be summarized as follows: • According to built-up land footprint data, all 16 countries converge, except for Australia, Malaysia, Poland, and Turkey. • According to carbon footprint data, only Indonesia, Malaysia, Mexico, South Africa, Thailand, Turkey, the UK, and the USA converge. • According to the cropland footprint, countries other than Brazil, Germany, India, Thailand, Turkey, and the USA stochastically converge. • Brazil, France, Germany, Indonesia, Italy, Mexico, South Africa, and Vietnam stochastically converge according to the fishing grounds' footprint. • According to forest products' footprint data, Australia, Canada, France, Germany, India, Korea, Mexico, Poland, Turkey, and Vietnam stochastically converge. • According to grazing land footprint data, Canada, France, India, Indonesia, Japan, Korea, Poland, South Africa, Thailand, and Vietnam converge. • Consequently, according to the total ecological footprint, Canada, France, Korea, Malaysia, Mexico, South Africa, the UK, and the USA stochastically converge.
The main policy recommendations that can be made as a result of the study are as follows: • As a result of the analysis, a series of measures such as technology transfer, foreign direct investment in pollution-reducing clean industries, and incentives to R&D centers for the development of technologies that would enhance efficiency in energy usage, are required to be taken by policymakers in order for the mean divergent countries to fulfill the goals of the Paris Agreement. • Another important finding obtained as a result of the study involves the necessity of international cooperation in order to prevent the adverse impacts of climate change, especially global warming.
• Another important policy outcome is government regulations as well as the quality of institutions that would implement these regulations. In order to ensure sustainable economic growth, price-based and rights-based regulations should be made and institutions should monitor the economic units within the framework of these regulations.
The obtained results of the study are consistent with the findings of Solarin et al. (2019a) and Haider and Akram (2019) who examined the convergence of ecological footprint and carbon footprint, as well as Solarin et al. (2019b) who investigated the convergence of carbon footprint. Nonetheless, Erdogan and Acaravci (2019), who examined the convergence of carbon emissions, yielded contradictory results. It is thought that the employed econometric method accounts for the difference between the findings of these two studies.
There are several avenues for future research. This paper focused on stochastic convergence in per-capita ecological footprint and subcomponents among the top 20 pollutant countries. It would be interesting to study other forms of convergence such as club convergence.