A Survey of Econometric Approaches to Convergence Tests of Emissions and Measures of Environmental Quality
A Survey of Econometric Approaches to Convergence Tests of Emissions and Measures of Environmental Quality
 Junsoo Lee, Junsoo LeeDepartment of Economics, Finance, and Legal Studies, University of Alabama
 James E. PayneJames E. PayneCollege of Business Administration, University of Texas at El Paso
 and Md. Towhidul IslamMd. Towhidul IslamDepartment of Economics, Finance, and Legal Studies, University of Alabama
Summary
The analysis of convergence behavior with respect to emissions and measures of environmental quality can be categorized into four types of tests: absolute and conditional βconvergence, σconvergence, club convergence, and stochastic convergence. In the context of emissions, absolute βconvergence occurs when countries with high initial levels of emissions have a lower emission growth rate than countries with low initial levels of emissions. Conditional βconvergence allows for possible differences among countries through the inclusion of exogenous variables to capture countryspecific effects. Given that absolute and conditional βconvergence do not account for the dynamics of the growth process, which can potentially lead to dynamic panel data bias, σconvergence evaluates the dynamics and intradistributional aspects of emissions to determine whether the crosssection variance of emissions decreases over time. The more recent club convergence approach tests the decline in the crosssectional variation in emissions among countries over time and whether heterogeneous timevarying idiosyncratic components converge over time after controlling for a common growth component in emissions among countries. In essence, the club convergence approach evaluates both conditional σ and βconvergence within a panel framework. Finally, stochastic convergence examines the time series behavior of a country’s emissions relative to another country or group of countries. Using univariate or panel unit root/stationarity tests, stochastic convergence is present if relative emissions, defined as the log of emissions for a particular country relative to another country or group of countries, is trendstationary.
The majority of the empirical literature analyzes carbon dioxide emissions and varies in terms of both the convergence tests deployed and the results. While the results supportive of emissions convergence for large global country coverage are limited, empirical studies that focus on country groupings defined by income classification, geographic region, or institutional structure (i.e., EU, OECD, etc.) are more likely to provide support for emissions convergence. The vast majority of studies have relied on tests of stochastic convergence with tests of σconvergence and the distributional dynamics of emissions less so. With respect to tests of stochastic convergence, an alternative testing procedure accounts for structural breaks and crosscorrelations simultaneously is presented. Using data for OECD countries, the results based on the inclusion of both structural breaks and crosscorrelations through a factor structure provides less support for stochastic convergence when compared to unit root tests with the inclusion of just structural breaks.
Future studies should increase focus on other air pollutants to include greenhouse gas emissions and their components, not to mention expanding the range of geographical regions analyzed and more robust analysis of the various types of convergence tests to render a more comprehensive view of convergence behavior. The examination of convergence through the use of ecoefficiency indicators that capture both the environmental and economic effects of production may be more fruitful in contributing to the debate on mitigation strategies and allocation mechanisms.
Subjects
 Econometrics, Experimental and Quantitative Methods
 Economic Theory and Mathematical Models
 Environmental, Agricultural, and Natural Resources Economics
Relevance of Emissions and Environmental Quality Convergence
The growing concerns regarding the global impact of greenhouse gas emissions and their impact on climate change have drawn the attention of both researchers and policymakers, as demonstrated through the actions generated from the Framework Convention on Climate Change in 1992, the Kyoto Protocol in 1997, and the Paris Agreement in 2015. While efforts toward the expansion of renewable energy sources, conservation measures, and advanced technologies that improve energy efficiency proceed in hopes of reducing the growth in greenhouse gas emissions, fossil fuels continue to be a primary energy source for many countries. Indeed, the energy mix of countries, level of economic development, economic structure, natural resource endowments, and other considerations contribute at various levels to the generation of greenhouse gas emissions across countries (Timilsina, 2016). This is relevant for discussions that pertain to emission allocation approaches with respect to issues of fairness and equity associated with the distribution of per capita emissions—that is, countries with lower per capita emissions may very well expect countries with higher per capita emissions to shoulder more of the burden for the mitigation and reduction in emissions (Payne & Apergis, 2021; Zhou & Wang, 2016). In such discussions, emission allocation strategies become less of a concern if there is convergence in per capita emissions. On the other hand, if per capita emissions fail to converge, then a per capita emissions allocation approach may drive the relocation of emissionsintensive industries and resource transfers through international trading of carbon allowances. Moreover, an underlying key assumption in climate change models resides in the convergence of emissions, which also facilitates projecting future emissions.
Though convergence studies have been undertaken across different types of emissions and environmental quality measures, the majority of the literature on emissions convergence has focused on carbon dioxide emissions. This is not surprising given the prominence of fossil fuels in the global energy consumption mix and the contribution of carbon dioxide emissions to greenhouse gas emissions. Given that emissions are a byproduct of the production and consumption activities associated with income generation, emissions convergence tests parallel the econometric approaches utilized for income convergence. This link between income and emissions is captured by the environmental Kuznets curve (EKC). The EKC hypothesizes that as per capita income levels increase due to economic growth, pollution emissions increase and environmental quality diminishes until an inflection point is reached. Thus, beyond the inflection point, as per capita income increases, environmental quality improves.
In this regard, Brock and Taylor (2003, 2010) and Stern (2017) highlight the role of scale, technique, and composition effects underlying the EKC hypothesis. The scale effect notes that as income increases, emissions increase as well. The technique effect captures the transfer and adoption of more modern and cleaner technologies in response to global competition that will alter the emissions intensity of production. Finally, the composition effect reflects the ability of countries to exploit their comparative advantage due to differences in their factor endowments, institutions, and regulatory environment. The combination of the changes in the input mix related to production due to the substitution of more environmentally friendly inputs in the production process and the reduction in emissions from technological advances alongside the changes in the output mix of industries with different pollution intensities (i.e., the composition effect) underlies the EKC hypothesis (Stern, 2017). Brock and Taylor (2003, 2010) note the role of technological progress in the production process and the abatement of emissions to demonstrate the relationship between the EKC hypothesis and the convergence of emissions through their green Solow model.
Since the early empirical work of Grossman and Krueger (1991), Shafik and Bandyopadhyay (1992), and Panayotou (1993) on the EKC hypothesis, the results from the empirical literature have remained inconclusive. Dinda (2004) provides an excellent discussion of the empirical explanations associated with the EKC hypothesis and a summary of the empirical studies through the 1990s and early 2000s. An important aspect highlighted by Dinda (2004) is the role of international trade with respect to the scale, technique, and composition effects associated with the EKC hypothesis. In the context of international trade, the displacement and pollution haven hypotheses suggest that pollutionintensive industries in countries with strong environmental regulations may migrate to those countries with a weaker regulatory environment (Antweiller et al., 2001; Cole, 2006; Cole & Elliott, 2003). Indeed, the regulatory environment is a relevant consideration in order to prevent a “race to the bottom” scenario with respect to the movement toward weaker environmental standards. Also, foreign direct investment and the diffusion of technology could enhance the adoption of cleaner technologies that could reduce pollution and improve environmental quality (Cole et al., 2008).
Kijima et al. (2010) note that the reducedform nature of the EKC hypothesis raises questions regarding the dynamics between economic growth and environmental quality and the need to develop theoretical models that explain the relationship. Likewise, MullerFurstenberger and Wagner (2007) and Wagner (2015) discuss the theoretical and econometric approaches to testing the EKC hypothesis. In a series of studies, Kaika and Zervas (2013a, 2013b) provide a summary of empirical studies and factors other than economic growth that may lead to the inverted Ushaped pattern of the EKC hypothesis. Reiterating the points set forth by Kijima et al. (2010), the empirical literature raises doubts about whether gross domestic product (GDP) adequately captures the transition of the type of production taking place (i.e., agricultural, industrial, and service sectors) over time. Sarkodie and Strezov (2019) undertake a bibliometric metaanalysis to examine the trends in the studies of the EKC hypothesis. For those studies validating the inverted Ushape relationship between emissions and GDP, Sarkodie and Strezov (2019) reveal that the average turning point is an annual income of US$8,910 with a great deal of heterogeneity among turning points mainly due to differences in the period analyzed and econometric approaches pursued. In general, they find that low and middleincome countries are below the turning point thresholds of annual income, whereas highincome countries are above.
In light of the debate on the validity of the EKC hypothesis, this survey focuses attention on the econometric approaches undertaken to test the standard measures of convergence: absolute and conditional βconvergence, σconvergence and distributional dynamics, club convergence, and stochastic convergence. As noted previously, much of the econometric modeling associated with emissions convergence draws from the income convergence literature.^{1} While a number of studies on emissions convergence have evaluated emissions intensity in terms of GDP, the vast majority of studies measure emissions on a per capita basis. Therefore, the survey of the tests of emissions convergence is set in per capita terms. In addition, convergence studies related to other measures of environmental quality beyond various types of emissions to include ecological footprint data and ecoefficiency indicators are referenced. Moreover, the emissions convergence literature has primarily been focused at the country level. However, many studies have applied convergence analysis of emissions at the disaggregated level defined by regions, states, metropolitan areas, provinces and at the industrial sector level.^{2}
This study is organized in sections that review tests of absolute and conditional βconvergence, σconvergence and the distributional dynamics of emissions, club convergence analysis, and stochastic convergence, which includes an extension of the tests for stochastic convergence that jointly incorporates structural breaks and crosscorrelation within a panel setting. Concluding remarks and directions for future research are also provided.
Emissions Convergence Test: Absolute and Conditional βConvergence
Following the early work of Baumol (1986), βconvergence occurs when countries with high initial levels of per capita emissions have a lower emission growth rate than countries with low initial levels of per capita emissions. Within a crosssectional framework, βconvergence can be examined based on the following specification:
The dependent variable is the logged growth rate of per capita emissions ($\mathit{em}$) between period 0 and t, denoted as $\mathit{ln}\left(\frac{{\mathit{em}}_{\mathit{it}}}{{\mathit{em}}_{i0}}\right)$, with the initial logged per capita emissions, $\mathit{ln}\left({\mathit{em}}_{i0}\right)$, as the independent variable along with the error term, ${\epsilon}_{i}$, and i represents a country. This specification allows one to test the null hypothesis of divergence ${H}_{0}:\beta =0$ for all i against the alternative hypothesis ${H}_{0}:\beta <0$ for some i. Thus, rejection of the null hypothesis supports convergence with the speed of convergence given by $\lambda =\left(\frac{\mathit{ln}\left(1\beta \right)}{t}\right)$.^{3}
The above crosssectional tests for βconvergence (or absolute convergence) rest with the assumption that all countries follow the same steady state for per capita emissions. Unlike absolute convergence, conditional convergence allows for possible differences among countries. Conditional βconvergence can be examined with the inclusion of exogenous variables in (1):
where ${Z}_{i}$ is a vector of exogenous variables to capture countryspecific effects. In this framework, the null hypothesis of absolute convergence is given as ${H}_{0}:\delta =0\phantom{\rule{0.25em}{0ex}}\text{and}\phantom{\rule{0.25em}{0ex}}\beta <0$ for all i against the alternative hypothesis of conditional convergence as ${H}_{1}:\delta \ne 0\phantom{\rule{0.25em}{0ex}}\text{and}\phantom{\rule{0.25em}{0ex}}\beta <0$ for some i.
As cited by Pettersson et al. (2014), both crosssection and even panel data regressions may suffer from Galton’s fallacy of regression toward the mean in terms of evaluating emissions convergence (Quah, 1993b). Specifically, the estimates of $\widehat{\beta}$ and $\widehat{\delta}$ from equation (2) cannot be used to predict $\beta $ and $\delta $ given it is likely the error term, ${\epsilon}_{i}$, is correlated with the initial emissions level, ${\mathit{em}}_{i0}$. Evans and Karras (1996) demonstrate that the issue of contemporaneous correlation can be prevented only in the case in which countries have identical firstorder autoregressive processes or all permanent crosscountry differences in per capita emissions are completely controlled for. As such, the system generalized method of moments estimation and instrumental variables provides a valid estimation of the parameters. In addition, tests for absolute and conditional βconvergence of emissions have been extended to a panel framework along several dimensions through panel fixed and random effects models and through the analysis of crosssectional dependence and dynamic panel models, as in the case of generalized method of moments.
Table 1. Tests of Absolute and Conditional βConvergence
Study 
Time Period 
Variable 
Region 
Approach 
Results 

List (1999) 
1929–1994 
Per capita NO_{x} and SO_{2} emissions 
10 US BEA regions 
Absolute βconvergence (crosssectional) 
Support for absolute βconvergence of SO_{2} emissions, but not for NOx emissions. 
Strazicich and List (2003) 
1960–1997 
Per capita CO_{2} emissions 
21 OECD countries 
Conditional βconvergence (crosssectional) 
Support for conditional βconvergence of per capita CO_{2} emissions. 
Nguyen Van (2005) 
1966–1996 
Per capita CO_{2} emissions 
100 countries 
Absolute βconvergence (crosssectional and panel) 
Absence of convergence for entire sample of countries, but convergence for 26 industrialized countries of per capita CO_{2} emissions. 
Bimonte (2009) 
1970–2006 
Percentage of protected areas 
19 OECD countries 
Absolute βconvergence (crosssectional) 
Support for absolute βconvergence of the percentage of protected areas. 
Brock and Taylor (2010) 
1960–1998 
Per capita CO_{2} emissions 
173 countries 
Absolute and conditional βconvergence (crosssectional) 
Support for absolute and conditional βconvergence of per capita CO_{2} emissions. 
Jobert et al. (2010) 
1971–2006 
Per capita CO_{2} emissions 
22 EU countries 
Absolute and conditional βconvergence (Bayesian shrinkage estimator) 
Support for absolute and conditional βconvergence of per capita CO_{2} emissions. 
Ordas Criado et al. (2011) 
1980–2005 
Per capita SO_{2} and NO_{x} emissions 
25 European countries 
Conditional βconvergence (panelparametric, semiparametric, and nonparametric estimators) 
Support for conditional βconvergence of per capita SO_{2} and NO_{x} emissions. 
Huang and Meng (2013) 
1985–2008 
Per capita CO_{2} emissions 
urban areas of 30 Chinese provinces 
Conditional βconvergence (spatialtemporal) 
Support for conditional βconvergence of per capita CO_{2} emissions. 
Li and Lin (2013) 
1971–2008 
Per capita CO_{2} emissions 
110 countries by income classification 
Absolute and conditional βconvergence (panel) 
Absolute βconvergence reveals divergence among the full sample of countries, but evidence of convergence of per capita CO_{2} emissions within income classification. Conditional βconvergence of per capita CO_{2} emissions within income classification. 
Solarin (2014) 
1960–2010 
Per capita CO_{2} emissions 
39 African countries 
Absolute βconvergence (panel) 
Mixed evidence on absolute βconvergence of per capita CO_{2} emissions. 
Wang and Zhang (2014) 
1966–2010 
Per capita CO_{2} emissions 
Six sectors across 28 Chinese provinces 
Absolute and conditional βconvergence (panel) 
Support for absolute βconvergence of per capita CO_{2} emissions for two sectors and conditional βconvergence for the other four sectors. 
Brannlund et al. (2015) 
1990–2008 
CO_{2} emissions intensity 
14 Swedish manufacturing sectors 
Conditional βconvergence (panel fixed effects) 
Convergence of CO_{2} emissions intensity for manufacturing sectors to different steady states and capitalintensive sectors converge more slowly. 
Hao et al. (2015a) 
1995–2011 
CO_{2} emissions intensity 
29 Chinese provinces 
Absolute and conditional βconvergence (panel) 
Support for absolute and conditional βconvergence of CO_{2} emissions intensity. 
Hao et al. (2015b) 
2002–2012 
Per capita SO_{2} emissions 
113 Chinese cities 
Absolute and conditional βconvergence (panel) 
Support for absolute and conditional βconvergence of per capita SO_{2} emissions. 
Zhao et al. (2015) 
1990–2010 
CO_{2} emissions intensity 
30 Chinese provinces 
Absolute and conditional βconvergence (spatial panel model) 
Support for absolute and conditional βconvergence of CO_{2} emissions intensity. 
Acar and Lindmark (2016) 
1973–2004 
CO_{2} emissions intensity from oil combustion 
86 countries 
Conditional βconvergence (crosssectional) 
Support for unconditional βconvergence of CO_{2} emission intensity due to oil combustion in the subperiods 1973–1979 and 1979–1991 with no evidence of convergence for the post1991 preKyoto period. 
Burnett (2016) 
1960–2010 
Per capita CO_{2} emissions 
48 contiguous U.S. states 
Conditional βconvergence (panel) 
Support for conditional βconvergence of per capita CO_{2} emissions. 
Acar and Lindmark (2017) 
1973–2010 
Per capita CO_{2} emissions from coal and oil 
28 OECD countries 
Conditional βconvergence (panel) 
Support for conditional βconvergence in both per capita CO_{2} coal and oil emissions across the three periods 1973–2010, 1973–1991, and 1992–2010. 
Apergis et al. (2017) 
1997–2013 
CO_{2} emissions intensity 
50 U.S. states and District of Columbia 
Absolute βconvergence (crosssectional) 
Support for absolute βconvergence of CO_{2} emissions intensity. 
de Oliveira and Bourscheidt (2017) 
1996–2007 
Per capita greenhouse gas emissions (from energy use, CO_{2}, CH_{4}, CO) 
33 sectors across a panel of 39 countries 
Conditional βconvergence (panel) 
Support for convergence in CH_{4} emissions in agriculture, food, and service sectors. Moderate evidence of convergence in CO_{2} emissions in agriculture, food, nondurable goods manufacturing, and services along with convergence in per capita greenhouse gas emissions from energy use in the extractive industry sector. 
Brannlund et al. (2017) 
1985–2010 
Per capita CO_{2} emissions 
124 countries 
Conditional βconvergence (parametric and nonparametric panel) 
Support for conditional βconvergence of per capita CO_{2} emissions for the entire sample and subsamples: OECD countries, nonOECD countries, lowincome countries, and highincome countries. 
Long et al. (2017) 
2005–2010 
Ecoefficiency measures 
Cement manufacturers, China 
Absolute and conditional βconvergence (panel) 
Support for absolute βconvergence in ecoefficiency measures for the nation as a whole along with east and west regions. Evidence of conditional βconvergence in ecoefficiency measures for the nation as a whole along with the east, middle, and west regions. 
Tiwari and Mishra (2017) 
1972–2010 
CO_{2} emissions 
18 Asian countries 
Absolute βconvergence (crosssectional) 
Support for absolute βconvergence of CO_{2} emissions. 
Acar and Yeldan (2018) 
1995–2013 
CO_{2} sectoral emissions and CO_{2} sectoral emissions intensity based on valueadded 
Turkey 
Conditional βconvergence (panel) 
Evidence of conditional βconvergence for CO_{2} sectoral emissions and CO_{2} sectoral emissions intensity based on valueadded. Based on classification as low, medium, and high technology sectors, the evidence of conditional βconvergence is mixed. 
Kounetas (2018) 
1970–2010 
CO_{2} emissions intensity 
23 EU countries 
Absolute βconvergence (crosssectional) 
Support for absolute βconvergence with respect to energy consumption, CO_{2} emissions, energy intensity, CO_{2} emissions intensity, and carbonization index. 
Rios and Gianmoena (2018) 
1970–2014 
Per capita CO_{2} emissions 
141 countries 
Absolute and conditional βconvergence (spatial models convergence clubs) 
Support for spatial convergence clubs of per capita CO_{2} emissions that allows for spillover effects from neighboring countries more so than conditional βconvergence 
Yu et al. (2018) 
1995–2015 
CO_{2} emissions intensity 
24 Chinese industrial sectors 
Conditional βconvergence (panel) 
Support for conditional βconvergence of CO_{2} emissions intensity. 
Zang et al. (2018) 
2003–2015 
Per capita CO_{2} emissions and CO_{2} emissions intensity 
201 countries 
Absolute βconvergence (panel) 
Support for absolute βconvergence in per capita CO_{2} emissions and CO_{2} emissions intensity for entire sample of countries. Categorized by income level, absolute βconvergence in high income countries for per capita CO_{2} emissions and in low, middle, and income countries for CO_{2} emissions intensity. Categorized by region, absolute βconvergence in Europe and Central Asia for per capita CO_{2} emissions and in East Asia and the Pacific, Europe and Central Asia, Latin America and the Caribbean, and the Middle East and North Africa for CO_{2} emissions intensity. 
FernandezAmador et al. (2019) 
1997–2014 
Per capita and valueadded CO_{2} emissions 
66 countries and 12 regions 
Conditional βconvergence (Bayesian structural model) 
Support for countryspecific conditional βconvergence of per capita and valueadded CO_{2} emissions with slower convergence associated with global conditional convergence. 
Karakaya et al. (2019) 
1960–2013 
Per capita CO_{2} emissions 
16 OECD countries 
Absolute and conditional βconvergence (crosssectional) 
Support for absolute and conditional βconvergence of per capita CO_{2} emissions. 
Solarin (2019) 
1960–2013 
Per capita CO_{2} emissions, per capita carbon footprint, and per capita ecological footprint 
27 OECD countries 
Absolute βconvergence (crosssectional) 
Support for absolute βconvergence for per capita CO_{2} emissions in 12 countries, per capita carbon footprint in 15 countries, and per capita ecological footprint in 13 countries. 
Yu et al. (2019) 
2005–2015 
Per capita CO_{2} emissions 
74 cities in Yangtze River Economic Belt of China 
Conditional βconvergence (spatial dynamic model) 
Support for conditional βconvergence of per capita CO_{2} emissions for the whole sample and three subgroups 
Note: BEA, Bureau of Economic Analysis.
Table 1 summarizes the empirical studies that have examined crosssectional and panel estimation approaches of absolute and conditional βconvergence, which have largely focused on per capita CO_{2} emissions with some studies utilizing CO_{2} emissions intensity and other environmental quality measures. Large multicountry studies by Nguyen Van (2005), Brock and Taylor (2010), Li and Lin (2013), Acar and Lindmark (2016), Brannlund et al. (2017), Rios and Gianmoena (2018), Zang et al. (2018), and FernandezAmador et al. (2019) provide mixed evidence on absolute and conditional βconvergence. In addition to the large multicountry studies, several studies focus exclusively on OECD or EU countries. Strazicich and List (2003), Bimonte (2009), Jobert et al. (2010), Ordas Criado et al. (2011), Acar and Lindmark (2017), Kounetas (2018), Karakaya et al. (2019), and Solarin (2019) provide overwhelming support for absolute and conditional βconvergence. However, studies that emphasize other geographical regions are limited. Solarin (2014) presents mixed evidence of absolute βconvergence for African countries, while Tiwari and Mishra (2017) lend support for absolute βconvergence for Asian countries.
Studies have also explored absolute and conditional βconvergence at the subnational level and, in some cases, across industry sectors. List (1999) examines per capita SO_{2} and NO_{x} emissions across 10 U.S. regions (as defined by the Bureau of Economic Analysis) to find support for absolute βconvergence for SO_{2} emissions, but not for NO_{x} emissions. In the case of U.S. states, Burnett (2016) shows evidence for per capita CO_{2} emissions conditional βconvergence, whereas Apergis et al. (2017) reveals absolute βconvergence for CO_{2} emissions intensity. A number of studies investigate absolute and conditional βconvergence of CO_{2} emissions at the subnational level for China. Huang and Meng (2013), Hao et al. (2015a, 2015b), Zhao et al. (2015), and Yu et al. (2019) examine cities and provinces in China to find support for absolute and conditional βconvergence. Studies by Wang and Zhang (2014), Brannlund et al. (2015), de Oliveira and Bourscheidt (2017), Long et al. (2017), Acar and Yeldan (2018), and Yu et al. (2018) at the industrial sector level reveal some variation in support of absolute and conditional βconvergence across sectors.
Emissions Convergence Test: σConvergence
Quah (1993a, 1993b), Evans (1996), and Evans and Karras (1996) note that absolute and conditional βconvergence approaches do not account for the dynamics of the growth process, which can create the potential for dynamic panel data bias if there is insufficient time series data. To address this issue, Quah (1993a, 1996a, 1996b, 1997) proposes evaluating the dynamics and intradistributional aspects of the crosssectional units, known as σconvergence. Defined in the context of emissions, σconvergence indicates a decrease in the crosssection variance of per capita emissions over time. Given that βconvergence is a necessary but not a sufficient condition for σconvergence, Quah (1993b, 1996b) suggests using a nonparametric approach to investigate the crosssectional distribution of relative per capita CO_{2} emissions over time to infer convergence behavior.
Nguyen Van (2005) and Pettersson et al. (2014) illustrate the use of a nonparametric distributional approach to examine σconvergence in analyzing the crosssectional distribution of per capita emissions over time through intradistributional mobility.^{4} This approach allows for the possibility of capturing the distributional dynamics such as twin peaks reflecting a bimodal distribution of emissions or the clustering among groups of countries through polarization and stratification. In the distribution approach, relative per capita emissions, ${\mathit{em}}^{r}$, is defined relative to the sample average with the crosscountry distribution of relative per capita emissions given as ${\phi}_{t}\phantom{\rule{0.12em}{0ex}}\left({\mathit{em}}^{r}\right)$ at time t. As per capita emissions evolve over time, the density at time $t+\tau $ for positive $\tau $ is ${\phi}_{t+\tau}\left({\mathit{em}}_{\tau}^{r}\right)$, where $\left({\mathit{em}}_{\tau}^{r}\right)$ denotes relative per capita emissions at time $t+\tau $. The conditional density of ${\mathit{em}}_{\tau}^{r}$, denoted ${f}_{\tau}\left({\mathit{em}}_{\tau}^{r}{\mathit{em}}^{r}\right)$, provides the link between the two distributions in capturing the distribution dynamics of per capita emissions between $t$ and $\tau $.
Let ${f}_{i,t+\tau}\left(\mathit{EM}\right)\phantom{\rule{0.12em}{0ex}}$ represent the joint distribution of $\mathit{EM}$, where $\mathit{EM}=\left[{\mathit{em}}^{r},{\mathit{em}}_{\tau}^{r}\right]$. The joint distribution at the point ${\mathit{EM}}_{0}$ is given as follows:
where N is the number of countries, K(.) is a kernel function, and h is the bandwidth parameter. In practice, the Epanechnikov or Gaussian kernel functions are typically employed. Equation (3) provides kernelsmoothed crosscountry densities for per capita emissions. The kernels can then be plotted for various time periods to show the evolution of the shape of the crosscountry densities over time. These kernel plots cannot provide insight on the intradistributional dynamics, but by knowing that ${\phi}_{t}\left({\mathit{em}}^{r}\right)=\int {f}_{i,t+\tau}\left(\mathit{EM}\right){\mathit{dem}}_{\tau}^{r}$ can be estimated and the conditional distribution for relative per capita emissions can be obtained as follows:
In turn, convergence can be examined by computing the surface and contour plots of this conditional density in providing information on the distribution dynamics within a continuous framework.^{5}
Alternatively, drawing from the work of Barro and SalaiMartin (1992), a time series plot or trend analysis to determine whether the crosssectional variance (standard deviation or coefficient of variation) of per capita emissions decreases over time can be used to illustrate σconvergence. In addition, the evaluation of the interquartile range (IQR), defined as the difference between the third and first quartiles, can determine whether the interquartile range of per capita emissions decreases from the initial year. If the IQR decreases from the initial year, the distribution of per capita emissions converges (Aldy, 2006, 2007). Another approach is to plot the Gini coefficient as a measure of statistical dispersion to infer whether the inequality among countries decreases over time (Bimonte, 2009; Zang et al., 2018).
Table 2. Tests of σConvergence
Study 
Time Period 
Variable 
Region 
Approach 
Results 

Nguyen Van (2005) 
1966–1996 
Per capita CO_{2} emissions 
100 countries 
σconvergence (nonparametric distributional dynamics approach) 
Absence of per capita CO_{2} emissions convergence for entire sample of countries, but convergence for 26 industrialized countries. 
Aldy (2006) 
1960–2000 
Per capita CO_{2} emissions 
88 countries and 23 OECD countries 
σconvergence (nonparametric distributional dynamics approach) 
Absence of per capita CO_{2} emissions convergence for 88 country samples. Evidence of convergence in CO_{2} emissions for 23 OECD country sample. 
Aldy (2007) 
1960–1999 
Per capita CO_{2} emissions from consumption and production 
50 US states 
σconvergence (nonparametric distributional dynamics approach) 
Divergence with respect to per capita CO_{2} emissions from production. 
Ezcurra (2007a) 
1960–1999 
CO_{2} emissions 
140 countries 
σconvergence (nonparametric distributional dynamics approach) 
Probability mass concentrated around the average of CO_{2} emissions increased over time. 
Ezcurra (2007b) 
1960–1999 
Per capita CO_{2} emissions 
87 countries 
σconvergence (nonparametric distributional dynamics approach) 
Decrease in crosscountry disparities in per capita CO_{2} emissions with the probability mass concentrated around the average of per capita CO_{2} emissions increasing over time. Spatial differences in per capita CO_{2} emissions explained by per capita income and climate conditions. 
Bimonte (2009) 
1970–2006 
Percentage of protected areas 
19 OECD countries 
σconvergence (Gini coefficient) 
Support for σconvergence of percentage of protected areas. 
Ordas Criado and Grether (2011) 
1960–2002 
Per capita CO_{2} emissions 
166 countries 
σconvergence (nonparametric distributional dynamics approach) 
Different convergence dynamics depending on the use of per capita CO_{2} emissions in levels or relative terms. 
Herrerias (2012) 
1920–2007 
Per capita CO_{2} emissions 
25 EU countries 
σconvergence (nonparametric distributional dynamics approach) 
Support for convergence of per capita CO_{2} emissions across all 25 EU countries. Unweighted per capita CO_{2} emissions reveal differences in convergence patterns pre and post World War II with convergence the strongest after the 1970s. Weighted per capita CO_{2} emissions by GDP and population reveal faster convergence rates. 
Moutinho et al. (2014) 
1996–2009 
CO_{2} emissions intensity 
16 Portuguese industry and energy sectors 
σconvergence (trend analysis of the coefficient of variation) 
σconvergence for CO_{2} emissions intensity, CO_{2} emissions from fossil fuel consumption, energy intensity and economic structure except for fossil fuel intensity. 
Wang and Zhang (2014) 
1966–2010 
Per capita CO_{2} emissions 
6 sectors across 28 Chinese provinces 
σconvergence (trend analysis of the standard deviation) 
Support for σconvergence of per capita CO_{2} emissions. 
Wu et al. (2016) 
2002–2011 
Per capita CO_{2} emissions 
286 Chinese cities 
σconvergence (nonparametric distributional dynamics approach) 
Support for σconvergence of per capita CO_{2} emissions. 
Apergis et al. (2017) 
1997–2013 
CO_{2} emissions intensity 
50 US states and District of Columbia 
σconvergence (trend analysis of the standard deviation) 
Support for σconvergence of CO_{2} emissions intensity. 
Tiwari and Mishra (2017) 
1972–2010 
CO_{2} emissions 
18 Asian countries 
σconvergence (nonparametric distributional dynamics approach) 
Support for σconvergence of CO_{2} emissions. 
Acar and Yeldan (2018) 
1995–2013 
CO_{2} sectoral emissions and CO_{2} sectoral emissions intensity based on value added 
Turkey 
σconvergence (trend analysis of the standard deviation) 
Some evidence to support σconvergence for CO_{2} sectoral emissions and CO_{2} sectoral emissions intensity based on value added. 
Kounetas (2018) 
1970–2010 
CO_{2} emissions intensity 
23 EU countries 
σconvergence (nonparametric distributional dynamics approach) 
Support for σconvergence with respect to energy consumption, CO_{2} emission, energy intensity, CO_{2} emissions intensity, and carbonization index. Distributional dynamics do not support convergence. 
Yu et al. (2018) 
1995–2015 
CO_{2} emissions intensity 
24 Chinese industrial sectors 
σconvergence (trend analysis of the coefficient of variation) 
Mixed results for σconvergence of CO_{2} emissions intensity. 
Zang et al. (2018) 
2003–2015 
Per capita CO_{2} emissions and CO_{2} emissions intensity 
201 countries 
σconvergence (Gini coefficient and trend analysis of the standard deviation) 
Both per capita CO_{2} emissions and CO_{2} emissions intensity provide support for σconvergence. 
Solarin (2019) 
1961–2013 
Per capita CO_{2} emissions, per capita carbon footprint, and per capita ecological footprint 
27 OECD countries 
σconvergence (trend analysis of coefficient of variation) 
Support for σconvergence across all three indicators: per capita CO_{2} emissions, per capita carbon footprint, and per capita ecological footprint. 
Yu et al. (2019) 
2005–2015 
Per capita CO_{2} emissions 
74 cities in Yangtze River Economic Belt of China 
σconvergence (trend analysis of the coefficient of variation) 
Support for σconvergence of per capita CO_{2} emissions. 
Apergis and Payne (2020) 
1971–2014 
CO_{2} emissions intensity 
Canada, Mexico, and US 
σconvergence (trend analysis of the standard deviation) 
Support for σconvergence of CO_{2} emissions intensity. 
Table 2 presents the studies that have examined σconvergence. Many of the studies use the nonparametric distribution approach grounded in Quah (1993a, 1993b). The other studies rely on trend analysis of the coefficient of variation (or standard deviation) and the Gini coefficient. The vast majority of studies focus on per capita CO_{2} emissions or CO_{2} emissions intensity with respect to σconvergence. Large multicountry studies by Nguyen Van (2005), Aldy (2006), Ezcurra (2007a, 2007b), Ordas Criado and Grether (2011), and Zang et al. (2018) provide mixed evidence of σconvergence. Studies for OECD and EU countries by Aldy (2006), Bimonte (2009), Herrerias (2012), Kounetas (2018), and Solarin (2019) provide more evidence of σconvergence than large multicountry studies. Similar to the studies testing for absolute and conditional βconvergence, the number of studies focused on other geographical regions is limited, with the exception of Tiwari and Mishra (2017) in the case of Asian countries and Apergis and Payne (2020) for North American Free Trade Agreement (NAFTA) member countries, both of whom find support for σconvergence.
Studies have also investigated σconvergence at the subnational level and across industry sectors. Aldy (2007) presents evidence of divergence of per capita CO_{2} emissions from production across U.S. states, whereas Apergis et al. (2017) find σconvergence. Wu et al. (2016) yield support for σconvergence in per capita CO_{2} emissions in Chinese cities. Studies at the industrial sector level by Moutinho et al. (2014), Wang and Zhang (2014), Acar and Yeldan (2018), and Yu et al. (2018) reveal some variations in support of σconvergence for CO_{2} emissions across sectors.
Emissions Convergence Test: Club Convergence
The PhillipsSul (2007) approach examines both conditional σconvergence and βconvergence within a panel framework.^{6} In terms of conditional σconvergence, the PhillipsSul approach tests the decline in the crosssectional variation in emissions among countries over time, and, with respect to conditional βconvergence, whether heterogeneous timevarying idiosyncratic components converge over time to a constant once the common growth component among countries is accounted for.
Within a timevarying common factor framework, the Phillips and Sul (2007) approach is given as follows:
where i = 1, …, N and t = 1, …, T. Here, ${\mathit{em}}_{\mathit{it}}$ denotes per capita emissions comprised of a common component (${\mu}_{t}$) and an idiosyncratic component (${\delta}_{\mathit{it}}$), both of which are timevarying. Note the idiosyncratic component, ${\delta}_{\mathit{it}}$ is a measure of the distance between ${\mathit{em}}_{\mathit{it}}$ and the common component, ${\mu}_{t}$. Phillips and Sul (2007) use the relative transition parameter ${h}_{\mathit{it}}$ as follows:
Equation (6) captures the loading coefficient, ${\delta}_{\mathit{it}}$, relative to the panel average, thus the transition path with respect to per capita emissions for country i relative to the panel average. As the factor loadings, ${\delta}_{\mathit{it}}$, converge to a constant, the crosssectional mean of the relative transition path, ${h}_{\mathit{it}}$, for per capita emissions i converges to unity and the crosssection variation, ${H}_{t}$, of the relative transition path converges to zero as $t\to \infty $:
The semiparametric form of ${\delta}_{\mathit{it}}$ follows as:
where δ_{i} is fixed; ξ_{it} ~ iid(0,1) varies across per capita emissions for country i = 1, 2, …, N; σ_{i} is an idiosyncratic scale parameter; L(t) is a slowly varying function where L(t)→∞ and t→∞; and α represents the decay rate. In equation (8), ${\delta}_{\mathit{it}}$ converges to δ_{i} for α ≥ 0. Hence, the null hypothesis of convergence is ${H}_{0}:{\delta}_{i}=\delta $ and α ≥ 0, against the alternative hypothesis ${H}_{A}:{\delta}_{i}\ne \delta $ for some i and/or α < 0.
Following Phillips and Sul (2007), L(t) = log t in the decay model, so the empirical log t regression can be used to test for convergence along with a clustering algorithm to identify convergence clubs as follows:
for t = rT, rT+1, …, T where r > 0 is set on the interval [0.2, 0.3]. For $b=2\alpha $, the null hypothesis is considered a onesided test of $b\ge 0$ against $b<0$.^{7} In turn, the clustering algorithm to identify convergence clubs requires multiple steps. The first step requires ordering per capita emissions of the countries in the panel based on the final values of per capita emissions for the respective countries. Next, starting from the highestorder country in terms of per capita emissions, sequentially estimate equation (9) on the k highest country per capita emissions to identify a core group of countries using the cutoff point criterion ${k}^{\ast}={\mathit{\text{ArgMax}}}_{k}\left\{{t}_{{b}_{k}}\right\}$, subject to ${\mathit{Min}}_{k}\left\{{t}_{{b}_{k}}\right\}>1.65,k=2,3,\dots N$. In the next step, add one country at a time from the remaining countries to the core group and reestimate equation (9) using the sign criterion ($b\ge 0$) to determine whether to add a country to the core group. Then, repeat the above steps iteratively for the remaining countries until clubs can no longer be formed.
Table 3. Tests of Club Convergence
Study 
Time Period 
Variable 
Region 
Approach 
Results 

Panopoulou and Panteldis (2009) 
1960–2003, 1975–2003 
Per capita CO_{2} emissions, CO_{2} emissions per GDP 
128 countries and 84 countries 
Club convergence 
Presence of two convergence clubs; 1960–1985 full panel convergence among all countries; 1975–2003 four convergence clubs. 
Camarero, PicazoTadeo, and Tamarit (2013) 
1960–2008 
CO_{2} emissions intensity 
23 OECD countries 
Club convergence 
Four convergence clubs for CO_{2} emissions intensity with four countries nonconvergent. 
Camarero, Castillo, et al. (2013) 
1980–2008 
Ecoefficiency indicators, CO_{2} emissions 
22 OECD countries 
Club convergence 
Emergence of four convergence clubs with respect to ecoefficiency indicators and CO_{2} emissions. 
Herrerias (2013) 
1980–2009 
Per capita CO_{2} emissions by energy source (petroleum, coal, and natural gas) 
162 countries 
Club convergence 
Club convergence tests reveal multiple convergence clubs associated with per capita CO_{2} emissions from petroleum, coal, and natural gas. 
Camarero et al. (2014) 
1990–2009 
Ecoefficiency indicators, greenhouse gas emissions, CO_{2}, N_{2}O, and CH_{4} emissions 
27 EU countries 
Club convergence 
Support for multiple convergence clubs associated with ecoefficiency indicators for GHG, CO_{2}, N_{2}O, and CH_{4} emissions. 
Wang et al. (2014) 
1995–2011 
CO_{2} emissions intensity 
29 Chinese provinces 
Club convergence 
Support for three convergence clubs of CO_{2} emissions intensity. 
Burnett (2016) 
1960–2010 
Per capita CO_{2} emissions 
48 contiguous U.S. states 
Club convergence 
Support for one convergence club of per capita CO_{2} emissions of 26 states. 
RobalinoLopez et al. (2016) 
1980–2010 
Per capita CO_{2} emissions (Kaya components) 
10 South American countries 
Club convergence 
Two convergence clubs for per capita GDP, CO_{2} emissions intensity, per capita CO_{2} emissions, energy intensity, and full Kaya convergence. Colombia and Peru are nonconvergent. 
Apergis and Payne (2017) 
1980–2013 
Per capita CO_{2} emissions, aggregate, by sector, and by fossil fuel source 
50 US states and District of Columbia 
Club convergence 
Multiple convergence clubs of per capita CO_{2} emissions at the aggregate level, by sector, and for natural gas and coal with full panel convergence for petroleum. 
Biligili and Ulucak (2018) 
1961–2014 
Per capita ecological footprint 
G20 countries 
Club convergence 
Support for two convergence clubs of per capita ecological footprint. 
Liu et al. (2018) 
2003–2015 
Per capita SO_{2} emissions and per capita industrial soot emissions 
285 Chinese cities 
Club convergence 
Emergence of four convergence clubs for per capita industrial SO_{2} emissions and three convergence clubs for per capita industrial soot emissions. 
Ulucak and Apergis (2018) 
1961–2013 
Per capita ecological footprint 
20 EU countries 
Club convergence 
Support for three convergence clubs of per capita ecological footprint. 
Yu et al. (2018) 
1995–2015 
CO_{2} emissions intensity 
24 Chinese industrial sectors 
Club convergence 
Two strong convergence clubs for CO_{2} emissions intensity among 20 industrial sectors with the remaining four sectors exhibiting weaker convergence. 
Haider and Akram (2019a) 
1980–2016 
Per capita CO_{2} emissions and its components: petroleum, coal, and natural gas 
53 countries 
Club convergence 
No support for convergence of per capita CO_{2} emissions for the full sample. Two convergence clubs related to total emissions and emissions from natural gas and petroleum and three convergence clubs related to emissions from coal. 
Haider and Akram (2019b) 
1961–2014 
Per capita ecological footprint and per capita carbon footprint 
77 countries 
Club convergence 
Support for two convergence clubs based on per capita ecological footprint and per capita carbon footprint, respectively. 
HamitHaggar (2019) 
1990–2014 
Per capita greenhouse gas emissions 
10 Canadian provinces and territories and two sectors, residential and transportation 
Club convergence 
Multiple convergence clubs of per capita greenhouse gas emissions at the aggregate level and by residential and transportation sectors. 
Solarin (2019) 
1961–2014 
Per capita ecological footprint and its six components 
92 countries 
Club convergence 
Support for multiple convergence clubs: 10 convergence clubs for per capita ecological footprint, four convergence clubs for per capita builtup footprint, five convergence clubs for per capita carbon footprint, seven convergence clubs for per capita cropland footprint, two convergence clubs for per capita fishing ground footprint, two convergence clubs for per capita grazing land footprint, and whole panel convergence for per capita forest land footprint. 
Apergis and Garzon (2020) 
1990–2017 
Per capita greenhouse gas emissions 
19 Spanish regions 
Club convergence 
Support for four convergence clubs of per capita greenhouse gas emissions. 
Apergis et al. (2020) 
1971–2014 
CO_{2} emissions intensity and its components 
6 Central American countries 
Club convergence 
Support for two convergence clubs for CO_{2} emissions intensity, two convergence clubs for energy intensity, and one convergence club for the carbonization index with Panama nonconvergent. 
Apergis and Payne (2020) 
1971–2014 
CO_{2} emissions intensity 
Canada, Mexico, and US 
Club convergence 
Full panel convergence of CO_{2} emissions intensity. 
Ivanovski and Churchill (2020) 
1990–2017 
Per capita components of greenhouse gas emissions, CO_{2}, CH_{4}, and N_{2}O 
8 Australian states and territories 
Club convergence 
Support for multiple convergence clubs associated with per capita CO_{2}, CH_{4}, and N_{2}O emissions. 
Payne and Apergis (2021) 
1972–2014 
Per capita CO_{2} emissions 
65 low and lowermiddle income countries 
Club convergence 
For lowincome countries three convergence clubs with Haiti nonconvergent. For lowermiddle income countries five convergence clubs with Cabo Verde, Comoros, Mongolia, Papua New Guinea, Sao Tome and Principe, Solomon Islands, and Vanuatu nonconvergent. 
Ulucak et al. (2020) 
1961–2014 
Per capita ecological footprint and its six components 
23 subSaharan African countries 
Club convergence 
Support for multiple convergence clubs: four convergence clubs for per capita cropland footprint, three convergence clubs for per capita ecological footprint, three convergence clubs for per capita carbon footprint, two convergence clubs for per capita fishing ground footprint, two convergence clubs for per capita grazing land footprint, and whole panel convergence for per capita builtup footprint and per capita forest land footprint, respectively. 
As shown in Table 3, the large multicountry studies by Panopoulou and Pantelidis (2009), Herrerias (2013), Haider and Akram (2019a, 2019b), Solarin (2019), and Payne and Apergis (2021) primarily focus on per capita CO_{2} emissions, but they also include evaluating ecological and carbon footprint data in examining club convergence. In general, these studies identify multiple convergence clubs, which reflects the emergence of multiple steady states. Likewise, studies on OECD, EU, and G20 countries by Camarero, Castillo, et al. (2013), Camarero, PicazoTadeo, and Tamarit (2013), Camarero et al. (2014), Biligili and Ulucak (2018), and Ulucak and Apergis (2018) also show distinct convergence clusters. The number of studies focused on other geographical regions is limited. RobalinoLopez et al. (2016), in the case of South American countries, identifies two distinct convergence clubs for per capita CO_{2} emissions. Likewise, Apergis et al. (2020) find two convergence clubs concerning CO_{2} emissions intensity for Central American countries. Ulucak et al. (2020) discover multiple convergence clubs for per capita ecological footprint data and its components for subSaharan African countries. In a study of Canada, Mexico, and the United States with respect to the implementation of NAFTA, Apergis and Payne (2020) reveal full panel convergence of CO_{2} emissions intensity.
Tests of club convergence have also been conducted at the subnational level for several countries. In an analysis of per capita CO_{2} emissions across U.S. states, Burnett (2016) finds one convergence club of 26 states while Apergis and Payne (2020) discover multiple convergence clubs at the aggregate level, by sector, and for natural gas and coal emissions, but full panel convergence for petroleum emissions. With respect to CO_{2} emission intensity, Wang et al. (2014) and Yu et al. (2018) show multiple convergence clubs across Chinese provinces and industrial sectors, respectively. Similarly, for per capita SO_{2} emissions and industrial soot emissions across Chinese cities, Liu et al. (2018) find multiple convergence clubs. HamitHaggar (2019) examines per capita greenhouse gas emissions for the residential and transportation sectors across provinces and territories in Canada and reveals multiple convergence clubs. Similarly, Ivanovski and Churchill (2020) find multiple convergence clubs for components of greenhouse gas emissions across the states and territories in Australia. In the case of Spain, Apergis and Garzon (2020) also discover multiple convergence clubs for per capita greenhouse gas emissions across 19 regions.
Emissions Convergence Test: Stochastic Convergence
Paralleling the work of Carlino and Mills (1993, 1996) and Bernard and Durlauf (1995, 1996) in regard to income convergence, unit root/stationarity tests have been used to test for stochastic convergence of emissions. Specifically, stochastic convergence of a country’s per capita emissions is present if relative per capita emissions, defined as the log of per capita emissions for country i relative to another country (or the average of a group of countries), is trendstationary, whereby shocks will be transitory in nature.^{8}
For example, the following augmented DickeyFuller (ADF) unit root test of Dickey and Fuller (1979) and Said and Dickey (1984) is one of the many unit root or stationarity tests that have been employed to test for stochastic convergence.
where ${\mathit{em}}_{t}^{r}$ denotes relative per capita emissions, usually defined as the log of per capita emissions in country i relative to the average per capita emissions among a particular group of countries, and t is the time trend. The null hypothesis of a unit root is ${H}_{0}:\phi =1$ against the alternative hypothesis of stationarity, ${H}_{1}:\phi <1$. In this framework, the rejection of the null hypothesis would support stochastic convergence. Such unit root/stationarity tests employed without consideration for structural breaks may lead to biased results. List (1999) was the first study to explore stochastic convergence of emissions, specifically for sulfur dioxides and nitrogen oxides, incorporating structural breaks. Additional studies using univariate unit root/stationarity tests of stochastic convergence include Aldy (2006), Bulte et al. (2007), Barassi et al. (2008, 2011), Lee et al. (2008), Lee and Chang (2009), Li et al. (2014), Moutinho et al. (2014), Payne et al. (2014), Solarin (2014), Sun et al. (2016), Churchill et al. (2018), Lin et al. (2018), Presno et al. (2018), Karakaya et al. (2019), and Solarin et al. (2019).^{9}
Alternatively, researchers have employed panel unit root/stationarity tests to exploit the significant power gain of panel tests in leveraging both the crosssection and time dimensions inherent in panel data. Studies by Strazicich and List (2003), Aldy (2007), Camarero et al. (2008), Moutinho et al. (2014), Wang and Zhang (2014), Hao et al. (2015a), Long et al. (2017), Acar and Yeldan (2018), and Yu et al. (2019) deploy firstgeneration panel unit root tests, which assumes crosssectional independence across individual units, to examine stochastic convergence. On the other hand, studies by Barassi et al. (2008), Lee and Chang (2008), RomeroAvila (2008), Westerlund and Basher (2008), Li et al. (2014), Acaravci and Erdogan (2016), Apergis et al. (2017), Acar and Yeldan (2018), Biligili and Ulucak (2018), Yu et al. (2018), Erdogan and Acaravci (2019), Karakaya et al. (2019), Apergis and Payne (2020), Payne and Apergis (2021), Solarin and Tiwari (2020), and Ulucak et al. (2020) implement secondgeneration panel unit root tests, which allow for crosssectional dependence in the error structure based on common factor modeling through the use of principal component analysis or crosssectional averages in order to determine the presence of stochastic convergence.
Rather than using average per capita emissions in defining relative per capita emissions to test stochastic convergence, studies by Nourry (2009), Herrerias (2013), and ElMontasser et al. (2015) have entertained the pairwise approach of Pesaran (2007) to examine country pairs to determine the stochastic convergence patterns between countries. In addition, studies by Barassi et al. (2011) and Barassi et al. (2018) test for stochastic convergence using fractional integration techniques.^{10} Furthermore, tests of stochastic convergence have been undertaken using wavelet analysis (Ahmed et al., 2017), quantile unit root tests (Lin et al., 2018), and asymmetric (nonlinear) unit root tests (Presno et al., 2018; Yavuz & Yilanci, 2013; Yilanci & Pata, 2020).^{11}
Table 4. Tests of Stochastic Convergence
Study 
Time Period 
Variable 
Region 
Approach 
Results 

List (1999) 
1929–1994 
Per capita NO_{x} and SO_{2} emissions 
10 US BEA regions 
Stochastic convergence (univariate unit root tests) 
Some evidence of stochastic convergence of per capita NO_{x} and SO_{2} emissions across US BEA regions. 
Strazicich and List (2003) 
1960–1997 
Per capita CO_{2} emissions 
21 OECD countries 
Stochastic convergence (panel unit root tests) 
Support for stochastic convergence of per capita CO_{2} emissions. 
Aldy (2006) 
1960–2000 
Per capita CO_{2} emissions 
88 countries and 23 OECD countries 
Stochastic convergence (univariate unit root tests) 
Mixed evidence on stochastic convergence of per capita CO_{2} emissions. 
Aldy (2007) 
1960–1999 
Per capita CO_{2} emissions from consumption and production 
50 US states 
Stochastic convergence (panel unit root tests) 
Some evidence of stochastic convergence related to per capita CO_{2} emissions from consumption. 
Bulte et al. (2007) 
1929–1999 
Per capita NO_{x} and SO_{2} emissions 
48 contiguous US states 
Stochastic convergence (univariate unit root tests) 
Stronger stochastic convergence of per capita NO_{x} and SO_{2} emissions over the federal pollution control period (1970–1999) than during the local control period (1929–1969). 
Barassi et al. (2008) 
1950–2002 
Per capita CO_{2} emissions 
21 OECD countries 
Stochastic convergence (univariate and panel unit root/stationarity tests) 
Absence of stochastic convergence of per capita CO_{2} emissions. 
Camarero et al. (2008) 
1971–2002 
Environmental performance indicators 
22 OECD countries 
Stochastic convergence (panel unit root tests) 
Support for stochastic convergence in all 22 countries based on the production measure and 15 of 22 countries based on the ratio of CO_{2} emissions to GDP. 
Lee et al. (2008) 
1960–2000 
Per capita CO_{2} emissions 
21 OECD countries 
Stochastic convergence (univariate unit root tests) 
Support for stochastic convergence of per capita CO_{2} emissions in 13 of the 21 countries. 
Lee and Chang (2008) 
1960–2000 
Per capita CO_{2} emissions 
21 OECD countries 
Stochastic convergence (univariate and panel unit root tests) 
Support for stochastic convergence of per capita CO_{2} emissions in 7 of the 21 countries. 
RomeroAvila (2008) 
1960–2002 
Per capita CO_{2} emissions 
23 OECD countries 
Stochastic convergence (panel stationarity tests) 
Support for stochastic convergence of per capita CO_{2} emissions. 
Westerlund and Basher (2008) 
1870–2002 
Per capita CO_{2} emissions 
16 developed and 12 developing countries 
Stochastic convergence (panel unit root tests) 
Support for stochastic convergence of per capita CO_{2} emissions. 
Lee and Chang (2009) 
1950–2002 
Per capita CO_{2} emissions 
21 OECD countries 
Stochastic convergence (panel unit root/stationarity tests) 
Support for stochastic convergence of per capita CO_{2} emissions. 
Nourry (2009) 
1950–2003, 1950–1990 
Per capita CO_{2} emissions and Per capita SO_{2} emissions 
29 OECD countries, 127 countries and 29 OECD countries, 81 countries 
Stochastic convergence (pairwise tests) 
Absence of stochastic convergence for per capita CO_{2} emissions for the 127 countries and OECD country sample. Absence of stochastic convergence of per capita SO_{2} emissions for the 81 countries and OECD country sample. 
Barassi et al. (2011) 
1870–2004 
Per capita CO_{2} emissions 
18 OECD countries 
Stochastic convergence (fractional integration tests) 
Relative per capita CO_{2} emissions are fractionally integrated in 13 of the 18 countries with relative per capita CO_{2} emissions integrated of order one and not mean reverting for the other 5 countries. 
Herrerias (2013) 
1980–2009 
Per capita CO_{2} emissions by energy source (petroleum, coal, and natural gas) 
162 countries 
Stochastic convergence (pairwise tests) 
Evidence of divergence across developed and developing countries with respect to per capita CO_{2} emissions from petroleum, coal, and natural gas. 
Yavuz and Yilanci (2013) 
1960–2005 
Per capita CO_{2} emissions 
G7 countries 
Stochastic convergence (threshold autoregressive panel unit root test) 
Results indicate nonlinearity and regime shift with convergence in the first regime but the absence of convergence in the second regime for per capita CO_{2} emissions. 
Li et al. (2014) 
1990–2014 
CO_{2} emissions 
50 US states 
Stochastic convergence (univariate and panel unit root tests) 
Support for stochastic convergence of CO_{2} emissions in only 12 of the 50 US states 
Moutinho et al. (2014) 
1996–2009 
CO_{2} emissions intensity 
16 Portuguese industry and energy sectors 
Stochastic convergence (univariate and panel unit root tests) 
Absence of stochastic convergence for CO_{2} emissions intensity. 
Payne et al. (2014) 
1900–1998 
Per capita SO_{2} emissions 
50 US states and District of Columbia 
Stochastic convergence (univariate unit root tests) 
Support for stochastic convergence of per capita SO_{2} emissions across the 50 US states and the District of Columbia. 
Solarin (2014) 
1960–2010 
Per capita CO_{2} emissions 
39 African countries 
Stochastic convergence (univariate unit root tests) 
Support for stochastic convergence of per capita CO_{2} emissions in 31 of the 39 countries. 
Wang and Zhang (2014) 
1966–2010 
Per capita CO_{2} emissions 
Six sectors across 28 Chinese provinces 
Stochastic convergence (panel unit root tests) 
Support for stochastic convergence of per capita CO_{2} emissions. 
El Montasser et al. (2015) 
1990–2011 
Per capita greenhouse gas emissions 
G7 countries 
Stochastic convergence (pairwise and panel unit root/stationarity tests) 
Absence of stochastic convergence for per capita greenhouse gas emissions. 
Hao et al. (2015a) 
1995–2011 
CO_{2} emissions intensity 
29 Chinese provinces 
Stochastic convergence (panel unit root tests) 
Support for stochastic convergence of CO_{2} emissions intensity. 
Acaravci and Erdogan (2016) 
1960–2011 
Per capita CO_{2} emissions 
Seven global regions 
Stochastic convergence (panel unit root/stationarity tests) 
Support for stochastic convergence of per capita CO_{2} emissions with allowance for structural breaks across the seven regions. 
Sun et al. (2016) 
1971–2010 
CO_{2} emissions 
10 largest global economies 
Stochastic convergence (univariate unit root tests) 
Mixed results on stochastic convergence of CO_{2} emissions. 
Ahmed et al. (2017) 
1960–2010 
Per capita CO_{2} emissions 
162 countries 
Stochastic convergence (wavelet tests) 
Support for stochastic convergence of per capita CO_{2} emissions for 38 countries with relative per capita CO_{2} emissions exhibiting nonstationary behavior for the remaining 124 countries. 
Apergis et al. (2017) 
1997–2013 
CO_{2} emissions intensity 
50 US states and District of Columbia 
Stochastic convergence (panel unit root tests) 
Absence of stochastic convergence for CO_{2} emissions intensity. 
Long et al. (2017) 
2005–2010 
Ecoefficiency measures 
Cement manufacturers, China 
Stochastic convergence (panel unit root tests) 
Evidence of stochastic convergence in ecoefficiency measures in the east, middle, and west regions. 
Acar and Yeldan (2018) 
1995–2013 
CO_{2} sectoral emissions and CO_{2} sectoral emissions intensity based on value added 
Turkey 
Stochastic convergence (panel unit root tests) 
Absence of stochastic convergence for CO_{2} sectoral emissions and CO_{2} sectoral emissions intensity based on value added. 
Barassi et al. (2018) 
1950–2013 
Per capita CO_{2} emissions 
28 OECD countries 
Stochastic convergence (fractional integration tests) 
Weak support for stochastic convergence of per capita CO_{2} emissions with only between 30 and 40% of the countries exhibiting convergence. 
Biligili and Ulucak (2018) 
1961–2014 
Per capita ecological footprint 
G20 countries 
Stochastic convergence (panel unit root/stationarity tests) 
Support for stochastic convergence of per capita ecological footprint. 
Churchill et al. (2018) 
1900–2014 
Per capita CO_{2} emissions 
44 developed and developing countries 
Stochastic convergence (univariate unit root tests) 
Support for stochastic convergence of per capita CO_{2} emissions. Stochastic convergence is more prevalent in the post–World War II period relative to the pre–World War II period. 
Lin et al. (2018) 
1950–2013 
Per capita CO_{2} emissions 
G18 countries 
Stochastic convergence (univariate and quantile unit root tests) 
Stochastic convergence of per capita CO_{2} emissions in only five of the 18 countries with 13 countries exhibiting convergence in certain quantiles. 
Presno et al. (2018) 
1901–2009 
Per capita CO_{2} emissions 
28 OECD countries 
Stochastic convergence (univariate and panel unit root tests) 
Support for stochastic convergence of per capita CO_{2} emissions for the countries as a whole, but some dispersion among developed countries. 
Erdogan and Acaravci (2019) 
1960–2014 
Per capita CO_{2} emissions 
28 OECD countries 
Stochastic convergence (panel unit root/stationarity tests) 
Support for stochastic convergence of per capita CO_{2} emissions at the country level, but less support at the panel level. 
Karakaya et al. (2019) 
1960–2013 
Per capita CO_{2} emissions 
16 OECD countries 
Stochastic convergence (univariate and panel unit root tests) 
Support for stochastic convergence of per capita CO_{2} emissions. 
Solarin (2019) 
1961–2013 
Per capita CO_{2} emissions, per capita carbon footprint, and per capita ecological footprint 
27 OECD countries 
Stochastic convergence (univariate unit root tests) 
Majority of the countries exhibit stochastic convergence with respect to per capita CO_{2} emissions, per capita carbon footprint, and per capita ecological footprint. 
Yu et al. (2019) 
2005–2015 
Per capita CO_{2} emissions 
74 cities in Yangtze River Economic Belt of China 
Stochastic convergence (panel unit root tests) 
Absence of stochastic convergence for per capita CO_{2} emissions. 
Apergis and Payne (2020) 
1971–2014 
CO_{2} emissions intensity 
Canada, Mexico, and US 
Stochastic convergence (panel unit root tests) 
Support for stochastic convergence of CO_{2} emissions intensity. 
Payne and Apergis (2021) 
1972–2014 
Per capita CO_{2} emissions 
65 low and lowermiddle income countries 
Stochastic convergence (panel unit root tests) 
Support for stochastic convergence of per capita CO_{2} emissions. 
Solarin and Tiwari (2020) 
1850–2005 
SO_{2} emissions 
32 OECD countries 
Stochastic convergence (panel stationarity tests) 
Support for stochastic convergence of SO_{2} emissions. 
Ulucak et al. (2020) 
1961–2014 
Per capita ecological footprint and its six components 
23 subSaharan African countries 
Stochastic convergence (panel unit root tests) 
Absence of stochastic convergence for per capita ecological footprint and its six components. 
Yilanci and Pata (2020) 
1961–2016 
Per capita ecological footprint 
5 ASEAN countries 
Stochastic convergence (asymmetric panel unit root test) 
Absence of convergence for per capita ecological footprint in the first regime, but convergence in the second regime. 
Note: BEA, Bureau of Economic Analysis.
As shown in Table 4, the number of studies that test for stochastic convergence far exceeds other convergence tests discussed. The majority of the studies examine per capita CO_{2} emissions in tests of stochastic convergence, with close to half of the studies focused exclusively on OECD, G20, G18, and G7 countries. The results from these studies provide mixed evidence of stochastic convergence. Large multicountry studies by Aldy (2006), Westerlund and Basher (2008), Nourry (2009), Herrerias (2013), Acaravci and Erdogan (2016), Sun et al. (2016), Ahmed et al. (2017), Churchill et al. (2018), and Payne and Apergis (2021) also find mixed evidence in support of stochastic convergence. In regard to other geographical regions, Solarin (2014) reveals that most African countries support stochastic convergence in per capita CO_{2} emissions, whereas Ulucak et al. (2020) show the absence of stochastic convergence among subSaharan African countries in terms of per capita ecological footprint data. In the case of ASEAN countries, Yilanci and Pata (2020) provide mixed evidence of stochastic convergence with respect to per capita ecological footprint data.
At the subnational level for the United States, List (1999), Bulte et al. (2007), and Payne et al. (2014), in the case of per capita SO_{2} and NO_{x} emissions, find some support for stochastic convergence. With respect to CO_{2} emissions at the state level, Aldy (2007), Li et al. (2014), and Apergis et al. (2017) find mixed results on stochastic convergence. Studies by Wang and Zhang (2014), Hao et al. (2015a), and Long et al. (2017) are supportive of stochastic convergence at the province and sectoral level in China, while Yu et al. (2019) fail to support stochastic convergence of per capita CO_{2} emissions among the cities in the Yangtze River economic region of China. Other studies at the sectoral level include Moutinho et al. (2014) for Portugal and Acar and Yeldan (2018) for Turkey, both of which show the absence of stochastic convergence in CO_{2} emissions.
Stochastic Convergence with Breaks and CrossCorrelations: Alternative Approach
While the issues of structural breaks and crosscorrelations have been entertained separately in the emissions convergence literature, the possibility of jointly controlling for both issues remains unexplored. In this regard, we provide new test results that control for both structural breaks and crosscorrelations jointly. We consider two complementary approaches. First, one may adopt the procedure of Bai and CarrioniSilvestre (2009), which utilizes dummy variables to control for structural changes and the panel analysis of nonstationarity in idiosyncratic and common components (PANIC) approach of Bai and Ng (2004) to deal with crosssectional dependence:
where ${d}_{i,t}$ denotes the deterministic terms that include dummy variables for structural changes; ${F}_{t}$ is a r x 1 vector representing the unobserved common factors; and ${\pi}_{i}$ denotes factor loadings that capture the responses of each crosssection unit to the common factors. The PANIC procedure facilitates estimating the factor terms. This procedure requires the estimation of the number of breaks and their locations. One limitation of this procedure is that it relies on a nonparametric estimate of the longrun variance to correct for serial correlation, as in Phillips and Perron (1988), which often yields size distortions.
Alternatively, the procedure of Nazlioglu et al. (2020), which employs the Fourier function for structural changes with the PANIC approach, is another viable option. Nazlioglu et al. (2020) consider a panel version model of Equation (11) with unknown forms of nonlinear breaks,
where ${d}_{i,t}$ now denotes the flexible Fourier function used in Enders and Lee (2012a, 2012b). The Fourier function fits well for various nonlinear breaks in economic analysis. More importantly, it provides a parsimonious model specification to avoid losing power when allowing for multiple breaks. The implementation of such a modeling framework may provide useful insights with respect to stochastic convergence tests of emissions.
One issue is how to estimate the parameters for structural changes, either dummy variables in (11) or Fourier terms in (12), and the factor terms jointly. In the above two approaches, an iterative procedure adopted in Bai and CarrioniSilvestre (2009) and Nazlioglu et al. (2020) can be deployed. These tests have a nice feature when adopting the PANIC procedure. In both cases, the asymptotic distributions of these tests are unaffected by the presence of unknown forms of crosscorrelations; therefore, the same critical values of the corresponding tests that do not allow for the factor terms can be used.
To illustrate this approach, we use annual data from 1960 to 2016 on per capita CO_{2} emissions (in metric tons) obtained from the World Bank Development Indicators for 28 OECD countries: Australia, Austria, Belgium, Canada, Chile, Colombia, Denmark, Finland, Greece, Hungary, Iceland, Ireland, Israel, Japan, Luxembourg, Mexico, the Netherlands, New Zealand, Norway, Poland, Portugal, South Korea, Spain, Sweden, Switzerland, Turkey, the United Kingdom, and the United States.^{12} Figure 1 presents the time series plot of relative per capita CO_{2} emissions. A cursory view of Figure 1 suggests the tendency toward the convergence of relative per capita CO_{2} emissions across countries.
Table 5. Results of Bai and CarrioniSilvestre Tests (Breaks and Factors)

Break, Factor One Break 
Breaks, Factor Two Breaks 
Breaks, Factor Three Breaks 


Country 
MSB 
pval 
brk1 
MSB 
pval 
brk1 
brk2 
MSB 
pval 
brk1 
brk2 
brk3 
Australia 
0.129 
0.83 
1972 
0.006^{a} 
0.00 
1974 
1982 
0.003^{a} 
0.00 
1974 
1982 
1992 
Austria 
0.079 
0.21 
2005 
0.011^{a} 
0.01 
1972 
1980 
0.006^{a} 
0.00 
1972 
1980 
1991 
Belgium 
0.017^{a} 
0.01 
2007 
0.039 
0.24 
1980 
1988 
0.119 
0.99 
1980 
1988 
1998 
Canada 
0.057 
0.17 
1975 
0.000^{a} 
0.00 
1981 
1991 
0.000^{a} 
0.00 
1968 
1981 
1991 
Chile 
0.037 
0.15 
1987 
0.072 
0.91 
1987 
1997 
0.077 
0.95 
1971 
1987 
1997 
Colombia 
0.123 
0.46 
2004 
0.043 
0.21 
1994 
2004 
0.009^{a} 
0.00 
1973 
1994 
2004 
Denmark 
0.113 
0.61 
1996 
0.103 
0.95 
1969 
1996 
0.030 
0.28 
1969 
1981 
1996 
Finland 
0.256 
0.87 
1973 
0.103 
0.99 
1973 
2004 
0.036 
0.31 
1980 
1993 
2003 
Greece 
0.243 
0.92 
1977 
0.184 
0.97 
1977 
2008 
0.040 
0.61 
1977 
1989 
2008 
Hungary 
0.171 
0.91 
1983 
0.029^{b} 
0.04 
1984 
1992 
0.099 
0.97 
1973 
1983 
1994 
Iceland 
0.099 
1.00 
1969 
0.080 
0.99 
1969 
1977 
0.023 
0.27 
1969 
1977 
1985 
Ireland 
0.062 
0.72 
2001 
0.021^{c} 
0.06 
1993 
2001 
0.023 
0.16 
1971 
1984 
2001 
Israel 
0.053 
0.19 
1995 
0.066 
0.67 
1980 
1995 
0.006^{a} 
0.00 
1970 
1980 
1995 
Japan 
0.136 
0.98 
1970 
0.026 
0.34 
1970 
1987 
0.051 
0.69 
1970 
1987 
1995 
Luxembourg 
0.037 
0.55 
1998 
0.015^{a} 
0.00 
1998 
2006 
0.061 
0.63 
1990 
1998 
2006 
Mexico 
0.200 
0.99 
1982 
0.237 
1.00 
1973 
1982 
0.121 
1.00 
1973 
1982 
1995 
Netherlands 
0.064 
0.52 
1972 
0.035 
0.57 
1979 
1988 
0.049 
0.74 
1969 
1979 
1988 
New Zealand 
0.034^{c} 
0.09 
1970 
0.009^{a} 
0.00 
1979 
1988 
0.031 
0.75 
1970 
1979 
1988 
Norway 
0.161 
0.94 
1986 
0.053 
0.29 
1986 
1996 
0.046 
0.64 
1986 
1996 
2008 
Poland 
0.082 
0.53 
1983 
0.011^{a} 
0.00 
1987 
2003 
0.008^{a} 
0.00 
1973 
1983 
2003 
Portugal 
0.022 
0.11 
1999 
0.006^{a} 
0.00 
1985 
1999 
0.001^{a} 
0.00 
1975 
1985 
1999 
South Korea 
0.112 
0.81 
1995 
0.232 
1.00 
1997 
2006 
0.033 
0.75 
1979 
1997 
2006 
Spain 
0.007^{a} 
0.00 
1976 
0.070 
0.90 
1976 
2007 
0.030 
0.71 
1976 
1988 
2007 
Sweden 
0.595 
1.00 
1969 
0.160 
1.00 
1970 
1991 
0.174 
1.00 
1969 
1991 
2002 
Switzerland 
0.030 
0.61 
1968 
0.040 
0.67 
1968 
1986 
0.081 
0.98 
1971 
1979 
1988 
Turkey 
0.019^{a} 
0.00 
1977 
0.010^{a} 
0.00 
1987 
2001 
0.069 
0.64 
1979 
1987 
2004 
United Kingdom 
0.277 
0.91 
1974 
0.001^{a} 
0.00 
1974 
2008 
0.009 
0.22 
1974 
1986 
2008 
United States 
0.030 
0.11 
1982 
0.041 
0.39 
1982 
2000 
0.051 
0.85 
1982 
2000 
2008 
# of rejections 

4 


11 



7 



# of factors 

5 


5 



5 



Note: The superscripts a, b, and c denote the rejection of the null hypothesis at the 1%, 5%, and 10% significance levels, respectively. The pvalues are obtained from the response surface estimates provided in Bai and CarrioniSilvestre (2009). The Gauss codes for finding the critical values and computing the pvalues are provided at the website: Junsoolee codes.
Table 5 presents the unit root tests with structural breaks and factors for the ADFPANIC tests of Bai and CarrioniSilvestre (2009). In column two, we find the null hypothesis of a unit root is rejected for four countries (Belgium, New Zealand, Spain, and Turkey) when one structural break is allowed. With two structural breaks, the null hypothesis of a unit root is rejected for 11 countries (Australia, Austria, Canada, Hungary, Ireland, Luxembourg, New Zealand, Poland, Portugal, Turkey, and the United Kingdom). And with three structural breaks, the null hypothesis of a unit root is rejected for seven countries (Australia, Austria, Canada, Colombia, Israel, Poland, and Portugal). Thus, the results provide very limited support for stochastic convergence of relative per capita CO_{2} emissions among OECD countries.
Table 6. Results of Nazlioglu et al. Tests (Fourier Breaks and Factors)

Fourier (m = 1) 
Fourier (m = 2) 
Fourier (m = 3) 
Fourier (m = estimated) 


LM 
pval 
lag 
LM 
pval 
lag 
LM 
pval 
lag 
LM 
pval 
m 
lag 

Australia 
−3.21 
0.55 
7 
−3.68 
0.81 
7 
−4.50 
0.74 
8 
−3.97^{a} 
0.01 
0 
2 
Austria 
−4.02^{c} 
0.06 
0 
−4.31 
0.47 
3 
−4.66 
0.65 
8 
−4.74^{a} 
0.00 
0 
3 
Belgium 
−3.92 
0.20 
6 
−4.08 
0.61 
6 
−5.05 
0.46 
6 
−3.59^{b} 
0.03 
0 
5 
Canada 
−3.60 
0.33 
5 
−4.20 
0.53 
5 
−4.93 
0.54 
5 
−1.90 
0.54 
0 
1 
Chile 
−2.10 
0.97 
8 
−4.66 
0.29 
5 
−6.03^{c} 
0.08 
5 
−3.26 
0.99 
3 
8 
Colombia 
−3.36 
0.45 
5 
−4.67 
0.28 
8 
−4.43 
0.77 
8 
−2.99^{c} 
0.10 
0 
4 
Denmark 
−2.60 
0.66 
0 
−4.30 
0.19 
0 
−3.65 
0.97 
8 
−0.89 
0.99 
0 
7 
Finland 
−2.09 
0.97 
7 
−4.55 
0.34 
5 
−6.02^{c} 
0.08 
5 
−1.22 
0.93 
0 
7 
Greece 
−1.19 
1.00 
8 
−4.68 
0.28 
4 
−4.69 
0.67 
6 
−1.53 
1.00 
3 
8 
Hungary 
−4.16 
0.13 
6 
−3.89 
0.71 
6 
−4.71 
0.62 
8 
−1.42 
0.80 
0 
0 
Iceland 
−3.70 
0.28 
3 
−4.42 
0.41 
7 
−5.84 
0.11 
8 
−2.71 
0.16 
0 
1 
Ireland 
−2.86 
0.75 
8 
−3.41 
0.90 
8 
−4.22 
0.85 
8 
−2.71 
0.17 
0 
8 
Israel 
−2.74 
0.58 
0 
−4.45 
0.39 
5 
−4.38 
0.84 
5 
−2.72 
0.17 
0 
8 
Japan 
−2.39 
0.92 
2 
−4.07 
0.61 
7 
−5.03 
0.45 
7 
−1.95 
1.00 
2 
7 
Luxembourg 
−3.67 
0.30 
4 
−3.79 
0.76 
8 
−4.72 
0.62 
8 
−3.86 
0.94 
3 
8 
Mexico 
−2.62 
0.84 
7 
−4.04 
0.63 
5 
−5.19 
0.36 
7 
−1.31 
0.86 
0 
0 
Netherlands 
−3.24 
0.52 
6 
−3.52 
0.87 
6 
−5.04 
0.49 
2 
−2.69 
1.00 
3 
1 
New Zealand 
−2.39 
0.92 
7 
−3.75 
0.78 
8 
−3.70 
0.96 
8 
−2.94^{c} 
0.06 
0 
0 
Norway 
−2.88 
0.72 
5 
−3.68 
0.82 
5 
−4.75 
0.64 
6 
−3.12 
1.00 
3 
1 
Poland 
−2.70 
0.80 
5 
−4.59 
0.32 
5 
−6.05^{c} 
0.08 
5 
−4.48 
0.37 
2 
8 
Portugal 
−2.06 
0.98 
3 
−4.93^{b} 
0.05 
0 
−4.52 
0.79 
4 
−1.23 
1.00 
3 
3 
South Korea 
−3.27 
0.49 
4 
−4.11 
0.59 
4 
−4.30 
0.88 
4 
−3.65 
0.99 
3 
4 
Spain 
−3.76 
0.26 
5 
−4.02 
0.64 
6 
−4.63 
0.71 
6 
−2.92 
0.98 
2 
5 
Sweden 
−2.74 
0.79 
5 
−5.24^{c} 
0.10 
4 
−5.48 
0.26 
4 
−1.71 
0.67 
0 
6 
Switzerland 
−2.05 
0.97 
8 
−1.96 
1.00 
8 
−4.44 
0.81 
5 
−1.23 
0.94 
0 
4 
Turkey 
−3.84 
0.23 
5 
−5.27^{c} 
0.09 
5 
−5.57 
0.19 
7 
−2.85 
0.13 
0 
1 
United Kingdom 
−2.67 
0.83 
8 
−5.85^{b} 
0.02 
2 
−6.30^{c} 
0.06 
2 
−2.21 
1.00 
3 
7 
United States 
−3.87 
0.22 
3 
−3.94 
0.68 
3 
−3.79 
0.98 
3 
−4.21^{a} 
0.01 
0 
1 
# of rejections 

1 


4 


4 


6 


# of factors 

5 


5 


5 


5 


Note: The superscripts a, b, and c denote the rejection of the null hypothesis at the 1%, 5%, and 10% significance levels, respectively. Here, m is the number of cumulative frequencies. The pvalues are obtained from the response surface estimates provided in Nazlioglu et al. (2020). The Gauss codes for finding the critical values and computing the pvalues are provided at the website: Junsoolee codes.
Table 6 presents the results for the Lagrange MultiplierPANIC (LMPANIC) tests of Nazlioglu et al. (2020), who employ smooth breaks using the Fourier function with a factor structure. The number of rejections for the null hypothesis of a unit root is small across the results using three different cumulative frequencies. When the number of cumulative frequencies (m) is 1, 2, and 3, the number of rejections are 1, 4, and 4, respectively. When m is estimated as the value that minimizes the Bayesian information criteria for each series, the null hypothesis of a unit root is rejected in six countries (Australia, Austria, Belgium, Colombia, New Zealand, and the United States; see the last column of Table 6). These results reveal that the evidence supporting stochastic convergence is even more limited compared to the results reported in Table 5 based on the ADFPANIC unit root tests.
Table 7. Results from Some Benchmark Tests (Breaks, No Factor)
Country 
Dummy Break, No Factor Lee and Strazicich (2003) 
Smooth Breaks, No Factor Enders and Lee (2012b) LM with Fourier 


LM One Trend Break 
LM Two Trend Breaks 
m = 1 
m = 2 
m = 3 

LM 
pval 
brk 
LM 
Pval 
brk1 
brk2 
LM 
pval 
p 
LM 
pval 
p 
LM 
pval 
p 

Australia 
−4.25^{a} 
0.01 
1978 
−6.58^{a} 
0.00 
1977 
1990 
−3.55 
0.13 
7 
−3.93 
0.30 
7 
−4.72 
0.23 
8 
Austria 
−4.93^{a} 
0.00 
1984 
−7.02^{a} 
0.00 
1978 
1989 
−4.79^{a} 
0.01 
0 
−4.88^{b} 
0.05 
6 
−5.4^{c} 
0.08 
6 
Belgium 
−4.50^{a} 
0.00 
1974 
−7.52^{a} 
0.00 
1980 
1996 
−3.76^{c} 
0.09 
6 
−4.05 
0.26 
6 
−5.00 
0.16 
7 
Canada 
−4.85^{a} 
0.00 
1983 
−5.85^{a} 
0.00 
1977 
1987 
−4.16^{b} 
0.04 
0 
−3.51 
0.58 
5 
−4.19 
0.57 
5 
Chile 
−3.54^{c} 
0.06 
1976 
−5.83^{a} 
0.00 
1979 
1994 
−1.77 
0.95 
6 
−4.21 
0.21 
5 
−6.34^{a} 
0.01 
5 
Colombia 
−5.20^{a} 
0.00 
1992 
−6.58^{a} 
0.00 
1985 
1992 
−3.35 
0.18 
8 
−4.29 
0.14 
8 
−3.69 
0.70 
8 
Denmark 
−4.08^{b} 
0.02 
2001 
−5.60^{a} 
0.00 
1970 
1993 
−2.21 
0.88 
3 
−3.86 
0.33 
7 
−4.24 
0.48 
7 
Finland 
−4.09^{b} 
0.02 
1970 
−5.88^{a} 
0.00 
1971 
2001 
−1.88 
0.92 
7 
−3.39 
0.59 
7 
−4.89 
0.20 
7 
Greece 
−3.11 
0.17 
1974 
−6.77^{a} 
0.00 
1981 
2007 
−0.96 
0.99 
8 
−4.81^{c} 
0.07 
4 
−3.77 
0.67 
8 
Hungary 
−3.84^{b} 
0.03 
1985 
−5.71^{a} 
0.00 
1980 
1989 
−4.02^{b} 
0.05 
6 
−3.96 
0.30 
6 
−5.13 
0.12 
8 
Iceland 
−5.92^{a} 
0.00 
1977 
−5.75^{a} 
0.00 
1974 
1986 
−5.27^{a} 
0.00 
7 
−5.08^{b} 
0.03 
7 
−6.77^{a} 
0.00 
8 
Ireland 
−3.57^{c} 
0.06 
1991 
−4.82^{a} 
0.01 
1973 
1999 
−2.43 
0.68 
8 
−3.68 
0.48 
0 
−4.57 
0.28 
8 
Israel 
−4.07^{b} 
0.02 
1999 
−5.15^{a} 
0.01 
1973 
1990 
−2.34 
0.72 
8 
−4.64^{c} 
0.09 
5 
−4.68 
0.32 
5 
Japan 
−4.18^{a} 
0.01 
1975 
−5.36^{a} 
0.00 
1975 
1993 
−2.29 
0.77 
7 
−4.18 
0.19 
7 
−5.42^{c} 
0.07 
7 
Luxembourg 
−3.87^{b} 
0.03 
2001 
−5.13^{a} 
0.01 
1989 
1996 
−3.20 
0.32 
1 
−3.27 
0.73 
1 
−5.14 
0.14 
6 
Mexico 
−4.34^{a} 
0.01 
1987 
−5.13^{a} 
0.01 
1987 
2002 
−2.81 
0.47 
7 
−3.83 
0.34 
7 
−4.53 
0.33 
7 
Netherlands 
−4.33^{a} 
0.01 
1972 
−6.79^{a} 
0.00 
1979 
1990 
−3.34 
0.25 
0 
−3.56 
0.52 
6 
−5.00 
0.22 
2 
New Zealand 
−4.43^{a} 
0.01 
1985 
−6.51^{a} 
0.00 
1975 
1982 
−3.39 
0.22 
2 
−4.92^{b} 
0.04 
8 
−4.77 
0.21 
8 
Norway 
−3.66^{b} 
0.05 
1978 
−7.07^{a} 
0.00 
1987 
2005 
−3.25 
0.29 
0 
−4.24 
0.17 
7 
−3.84 
0.72 
6 
Poland 
−3.35^{c} 
0.10 
1987 
−5.72^{a} 
0.00 
1979 
1995 
−2.20 
0.79 
8 
−4.55 
0.11 
5 
−5.85^{b} 
0.03 
6 
Portugal 
−3.50^{c} 
0.07 
1978 
−6.07^{a} 
0.00 
1996 
2007 
−2.48 
0.64 
8 
−5.99^{a} 
0.00 
4 
−6.68^{a} 
0.00 
4 
South Korea 
−3.28 
0.12 
1991 
−5.50^{a} 
0.00 
1990 
2001 
−2.62 
0.65 
4 
−3.93 
0.35 
4 
−4.22 
0.58 
4 
Spain 
−3.84^{b} 
0.03 
1994 
−4.71^{b} 
0.02 
1973 
2006 
−2.58 
0.68 
3 
−4.04 
0.29 
0 
−4.49 
0.44 
3 
Sweden 
−2.97 
0.23 
1982 
−5.53^{a} 
0.00 
1976 
1997 
−2.45 
0.74 
5 
−7.29^{a} 
0.00 
4 
−7.38^{a} 
0.00 
4 
Switzerland 
−4.30^{a} 
0.01 
1974 
−5.50^{a} 
0.00 
1974 
1991 
−4.66^{a} 
0.01 
0 
−5.42^{a} 
0.02 
0 
−4.86 
0.25 
5 
Turkey 
−4.30^{a} 
0.01 
1998 
−6.03^{a} 
0.00 
1992 
2004 
−3.66 
0.11 
5 
−5.09^{a} 
0.03 
5 
−5.72^{b} 
0.04 
6 
United Kingdom 
−3.61^{b} 
0.05 
1997 
−5.79^{a} 
0.00 
1973 
1982 
−3.10 
0.28 
8 
−5.95^{a} 
0.00 
2 
−6.33^{a} 
0.01 
2 
United States 
−4.26^{a} 
0.01 
1977 
−5.16^{a} 
0.01 
1984 
2005 
−3.67^{c} 
0.10 
7 
−3.02 
0.78 
7 
−3.09 
0.93 
7 
# of rejections 

25 


28 



7 


10 


9 

# of factors 

n/a 


n/a 



n/a 


n/a 


n/a 

Note: The superscripts a, b, and c denote the rejection of the null hypothesis at the 1%, 5%, and 10% significance levels, respectively. Here, m is the number of cumulative frequencies. The pvalues are obtained from the response surface estimates provided in Nazlioglu and Lee (2020) for the Lee and Strazicich (2003) tests and Nazlioglu et al. (2020) for the Enders and Lee (2012a, 2012b) tests, respectively. The Gauss codes for finding the critical values and computing the pvalues are provided at the website: Junsoolee codes.
To evaluate the effects of ignoring crosscorrelations, we compare the above results with the benchmark tests where no factor terms are allowed (i.e., crosscorrelations are not considered). Table 7 provides the results using the Lee and Strazicich (2003) one trend break model to test for a unit root without factors shown in the left panels. The results reveal that the null hypothesis of a unit root is rejected for 25 countries. The findings from the Lee and Strazicich (2003) tests with two trend breaks without factors show rejections of the null hypothesis of a unit root for all countries, thus providing overwhelming support for stochastic convergence.
In the right panels of Table 7, we present the results from the Enders and Lee (2012b) LM tests with smooth breaks using a Fourier function at three different cumulative frequencies but without factors. We find that the number of rejections of the null hypothesis of a unit root is 7, 10, and 9 when m = 1, 2, and 3, respectively. The LM tests with smooth breaks yield less support for stochastic convergence than the tests with dummy variables.
Thus, in comparing the results in Tables 5 and 6 with those in Table 7, we discover that the null hypothesis of a unit root is rejected quite less often from the tests accounting for crosscorrelations than the corresponding tests not accounting for them. Since all of these tests already include structural changes, the differences are due to crosscorrelations. In general, the effects of ignoring crosscorrelations are unknown: they can yield more or fewer rejections of the null hypothesis. In our application using OECD data on relative per capita CO_{2} emissions, we find the inclusion of crosscorrelations yields different results. As such, this application illustrates the importance of controlling for crosscorrelations in addition to structural changes. This methodological approach can be easily extended to test for stochastic convergence of other types of emissions and measures of environmental quality.
Discussion and Directions for Future Research
As noted in the survey studies of Pettersson et al. (2014), Acar et al. (2018), and Payne (2020), the findings related to emissions convergence vary considerably, depending on the number and scope of countries examined and the econometric approaches, as shown in Tables 1–4, which summarize the empirical studies to date.^{13} Though results supportive of emissions convergence for large multicountry coverage are limited, empirical studies more focused on country groupings defined by income classification, geographic region, or institutional structure (i.e., EU, OECD, etc.) are more likely to provide support for emissions convergence. A review of Tables 1–4 reveals that the vast majority of studies have investigated stochastic convergence. Few studies have examined σconvergence through distributional dynamics. With respect to tests of stochastic convergence, we present an alternative testing procedure that accounts for structural breaks and crosscorrelations simultaneously. Using data for OECD countries, we show that the inclusion of both structural breaks and crosscorrelations through a factor structure provides fewer rejections of the null hypothesis of a unit root in relative per capita emissions compared to unit root tests with the inclusion of just structural breaks.
Moreover, a majority of studies have focused on CO_{2} emissions, with less attention given to other air pollutants or greenhouse gas emissions, not to mention geographical regions. In this regard, future studies should expand the empirical analysis on convergence to include other air pollutants and a more robust analysis of the various types of convergence tests to render a more holistic view of the convergence behavior of emissions and measures of environmental quality. With the debate often centered on the appropriate mitigation strategies and the allocation mechanisms associated with the reduction in the growth rate of greenhouse gas emissions and its components, future empirical work should also incorporate the economic impact through the use of environmental performance and ecoefficiency indicators as set forth by Camarero et al. (2008), Camarero, Castillo, et al. (2013), and Camarero et al. (2014). As more countries are able to produce more goods and services that have less impact on the environment and natural resources, examining convergence through the use of ecoefficiency indicators that capture both the environmental and economic effects of production may be more fruitful in contributing to the debate on mitigation strategies and allocation mechanisms.
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Notes

1. For surveys of the literature on emissions convergence, see Pettersson et al. (2014), Acar et al. (2018), and Payne (2020).

2. Our outline of econometric approaches parallels to some extent the presentation by Pettersson et al. (2014); however, with more references to studies specific to the econometric approaches pursued.

3. Speed of convergence required for per capita emissions to move from its initial level at time t to halfway to the sample mean.

4. This presentation parallels Nguyen Van (2005) and Pettersson et al. (2014).

5. As noted by Pettersson et al. (2014), Quah (1993b) also suggests the use of stochastic kernels or Markov chain transition matrices.

6. Remember the issue raised by Quah (1993b), Evans (1996), and Evans and Karras (1996) that tests of crosssectional βconvergence do not consider the possibility of multiple steady states.

7. In estimating equation (9), heteroskedasticity and autocorrelation consistent standard errors are employed in the least squares estimates of b.

8. Studies by Heil and Selden (1999), Lanne and Liski (2004), Lee and List (2004), Christidou et al. (2013) Yamazaki et al. (2014), Barros et al. (2016), Tiwari et al. (2016), Belbute and Pereira (2017), GilAlana et al. (2017), Ulucak and Lin (2017), Cai et al. (2018), Solarin and Bello (2018), GilAlana and Trani, (2019), and Yilanci et al. (2019) are relevant to understanding the time series behavior of emissions and other measures of environmental quality over time; however, these studies do not explicitly test stochastic convergence as defined by using a relative measure of emissions or environmental quality measures.

9. Castle and Hendry (in press) highlight the fact that climate and human behavior are interconnected and constantly evolving over time. As such, Castle and Hendry (in press) characterize the time series behavior of such interactions as widesense nonstationary processes, in which the modeling of time series behavior of such process is inherently complex as the datagenerating process is unknown. Moreover, widesense nonstationary processes do not have a constant distribution and contain stochastic trends and sudden shifts. For more information on the modeling strategy in this regard, see Castle et al. (2019, 2020), and Castle and Hendry (2020).

10. Studies by Barros et al. (2016), Belbute and Pereira (2017), GilAlana et al. (2017), and GilAlana and Trani (2019) apply the fractional integration approach to determine the degree of persistence and long memory behavior in the levels of carbon dioxide emissions, but not explicitly testing for stochastic convergence in the use of relative emissions.

11. Cai et al. (2018) employ quantilebased unit root tests of the levels of per capita carbon dioxide emissions.

12. The 28 OECD countries were constrained to those countries with a complete data series for per capita CO_{2} emissions over the entire period 1960–2016 as reported by the World Bank Development Indicators.

13. Tables 1–4 draw upon Payne’s (2020) survey of the literature on convergence of carbon dioxide emissions with additional studies incorporated for other pollutants and more recent studies.