Economic Growth, Innovation, and CO₂ Emissions: Analyzing the Environmental Kuznets Curve and the Innovation Claudia Curve in BRICS Countries

Abstract

This study explores the dynamic relationship between economic growth, technological innovation, and $C{O}_2$ emissions in BRICS nations, integrating the Environmental Kuznets Curve (EKC) and Innovation Claudia Curve (ICC) frameworks. Using a panel ARDL approach on data from 1991 to 2023, we investigate the long-run and short-run interactions between GDP, renewable energy consumption (RENC), foreign direct investment (FDI), urbanization (URB), and patent applications (PAs) in shaping environmental outcomes. The findings confirm the EKC hypothesis, revealing an N-shaped relationship between GDP and emissions, indicating that while economic growth initially leads to higher $C{O}_2$ emissions, this trend reverses at a critical threshold before a secondary increase occurs at higher income levels. The ICC framework identifies a cubic relationship between innovation and emissions, where technological advancements initially drive higher emissions before contributing to sustainability at later stages, though an excessive scale of innovation may reintroduce environmental pressures. RENC is found to significantly mitigate emissions, while URB and FDI display dual and context-dependent effects, highlighting the multidimensionality of sustainable transitions in emerging economies. These results underscore the importance of targeted policy interventions, such as scaling renewable energy infrastructure, promoting green innovation, guiding urban expansion, and aligning FDI with environmental objectives.

1. Introduction

Economic growth has long been associated with increased environmental degradation due to industrialization and energy consumption. The growth–emissions nexus refers to the complex dependence between economic growth and environmental degradation, sometimes measured by $C{O}_2$ emissions [1,2,3,4]. This relationship is quantified by the Environmental Kuznets Curve (EKC), introduced by Grossman and Krueger [5]. The EKC theory asserts that the relationship between economic growth and $C{O}_2$ emissions follows an inverted U-shaped curve [6,7,8]. In the early stages of economic growth, industrialization, energy consumption, and resource use cause an increase in $C{O}_2$ emissions [9,10,11,12]. As economies evolve, they begin to adopt new technologies, and a transition to service-based structures takes place.

BRICS countries collectively account for over 40% of the world’s population and contribute significantly to global $C{O}_2$ emissions [13,14]. Their economic trajectories and environmental policies play an important role in determining global sustainability efforts. However, in emerging economies such as those in the BRICS group (Brazil, Russia, India, China, and South Africa), this transition is not uniform and can be influenced by factors such as rapid industrialization, energy policies, and the pace of green technology adoption [15,16,17].

In parallel, the Innovation Claudia Curve (ICC) theory explores how innovation influences economic and environmental sustainability [18,19]. As economies adopt advanced technologies and improve energy efficiency, emissions can decline even in the early stages of economic development, partially challenging the EKC hypothesis. This interaction between innovation, economic growth, and environmental impact raises critical questions about optimal policies that can ensure a balance between sustainable development and economic progress. By incorporating a cubic function for technological innovation, this study aims to provide a nuanced understanding of how different stages of innovation adoption affect environmental sustainability in emerging economies.

The growth–emissions nexus has become more complex due to recent technological advancements and the need for sustainable development [20,21]. According to endogenous growth theory, technological innovation (TI) has an essential role in driving global economic growth [22]. Klewitz and Hansen [23] assert TI provides a new approach to efficient resource use. The main drivers of economic growth are the expansion of input factors, the speed of innovation, and efficient resource allocation [24]. The ICC describes the interaction between TI and environmental outcomes. TI is represented by patent applications. Even if TI is a driver of sustainability, its implementation can lead to higher $C{O}_2$ emissions due to resource-intensive processes. However, as TI progresses, a turning point is reached where further TI contributes to a decrease in $C{O}_2$ emissions [19].

This study focuses on BRICS countries, which are important contributors to $C{O}_2$ emissions and TI. We analyzed the interplay between GDP, renewable energy consumption (RENC), foreign direct investment (FDI), urbanization (URB), and the number of patent applications (PAs).

The selection of BRICS countries is motivated by several key factors. These countries represent some of the world’s fastest-growing economies and play a substantial role in global GDP. They are interesting cases for examining the ICC and the EKC since they are major contributors to global $C{O}_2$ due to their reliance on fossil fuels and their rapid urbanization. They exhibit various levels of economic development and technological advancement. They are emerging hubs for TI due to increased PAs and investments in renewable energy.

While previous studies have analyzed the EKC and the role of TI in $C{O}_2$ reduction, the cubic relationship proposed in the ICC remains underexplored, particularly in emerging economies such as BRICS. This study aims to bridge this gap by examining the complex interplay between economic growth, innovation, and environmental outcomes using a robust econometric approach.

This study aims to address the following research questions (RQs):

➢

RQ1: Does economic growth in BRICS countries follow the EKC hypothesis, and if so, what shape does it take (inverted-U, N, or W)?

➢

RQ2: How does technological innovation, measured through patent applications, influence $C{O}_2$ emissions in BRICS countries?

➢

RQ3: What role do foreign direct investment, renewable energy consumption, and urbanization play in influencing environmental sustainability in BRICS economies?

In this research, we employed a panel ARDL model to examine the long-run and short-run relationships between economic growth, innovation, and $C{O}_2$ emissions in BRICS nations from 1991 to 2023. Unlike static models, the panel ARDL approach allowed us to capture both short-run and long-run dynamics, making it particularly suitable for analyzing economic and environmental interactions over time in heterogeneous economies such as BRICS. The findings provide valuable insights for policymakers aiming to balance economic growth with environmental sustainability. They highlight the need for targeted strategies to promote renewable energy adoption, encourage green foreign investments, and develop sustainable urban planning initiatives in BRICS nations. Unlike previous studies that focus primarily on linear or quadratic relationships in the EKC framework, this research introduces a cubic approach to the ICC, allowing for a more refined analysis of how different phases of technological progress impact $C{O}_2$ emissions in BRICS countries.

Despite extensive literature on the EKC and the role of innovation in $C{O}_2$ emissions dynamics, gaps remain. Most existing studies focus on linear and quadratic forms, ignoring the potential for complex and non-monotonic patterns. There is limited empirical evidence on how technological innovation impacts $C{O}_2$ emissions beyond a threshold, particularly in emerging economies. This study addresses these gaps by exploring both the EKC and the less-studied ICC in a unified framework. The novelty consists of applying a cubic specification to reflect the N-shaped and W-shaped dynamics and using a panel ARDL model to uncover both the long-run and the short-run relationships. By focusing on BRICS countries, major polluters, and innovation hubs, the research offers fresh insights into sustainable development trajectories in the Global South.

Our study advances existing EKC and innovation–sustainability research by integrating the EKC and the ICC into a unified empirical framework, using a cubic specification for both economic growth and TI. Our work captures the complex, non-monotonic dynamics of emissions, especially in emerging economies. The panel ARDL model allows simultaneous short- and long-run effects, distinguishing between temporary shocks and structural trends. In doing so, we go beyond prior studies by (i) identifying the existence of N- and potential W-shaped patterns in emissions trajectories, (ii) highlighting the rebound effects at high levels of innovation, and (iii) revealing that not all innovation is environmentally beneficial unless strategically guided. This dual-curve model provides new insights for policymakers seeking to balance economic growth with climate goals in complex, fast-changing economies.

The structure of the paper is organized into several key sections. Section 2 provides an overview of relevant studies on the ICC and the EKC and discusses relationships between TI, economic activities, and environmental outcomes across different regions and contexts. Section 3 presents the theoretical framework and discusses the foundational theories of the ICC and the EKC. Their mathematical models are detailed, together with hypothesized relationships between the variables. Also, in this section, we introduce the panel unit root tests and panel ARDL model to investigate the short-run and long-run relationships. Section 4 presents the findings from statistical analysis, testing the validity of the EKC and ICC hypotheses. The results were analyzed in the context of circular economy (CE) principles, highlighting sustainability, resource efficiency, and environmental impact mitigation. Section 5 discusses how the findings align with CE strategies. It highlights the importance of sustainable urbanization, renewable energy integration, and innovation-driven solutions in reducing $C{O}_2$ emissions. In the last section, Section 6, we summarize the main insights, confirming the validity of the EKC and ICC hypotheses in BRICS countries. It also highlights the critical roles of renewable energy, technological innovation, and foreign direct investment in sustainable development. Additionally, policy recommendations are proposed, including fostering green innovation, promoting sustainable urban planning, and implementing strategies to mitigate $C{O}_2$ emissions.

2. Literature Review

The scientific literature addressing the relationship between economic growth, innovation, and environmental sustainability in emerging economies is extensive yet often fragmented. In this section, we synthesize key contributions that inform both the theoretical foundations and the empirical approach of this study. We distinguish between two major research streams: (i) studies investigating the EKC and its shape dynamics in the context of BRICS and comparable economies and (ii) recent works exploring the environmental impact of technological innovation, including those proposing cubic relationships such as the ICC. Our objective is to position the current research within this dual framework and highlight how our study builds upon, extends, or challenges previous findings. In addition, we review econometric approaches used in similar contexts, including panel ARDL and Fourier ARDL, to justify our methodological choices. The integration of these strands of literature allows us to propose a unified empirical model that examines the non-linear and dynamic interplay between $C{O}_2$ emissions, economic growth, and innovation in BRICS countries.

2.1. Economic and Environmental Factors

Several studies have been carried out to investigate factors that influence the environment. Sadiq et al. [25] explore the effects of economic policy uncertainty, GDP, population, and renewable energy consumption on $C{O}_2$ emissions in BRIC countries over the period of 1990–2020. By means of fully modified ordinary least squares (FMOLS) and dynamic ordinary least squares (DOLS), the authors show that GDP growth and population expansion substantially drive $C{O}_2$ emissions, whereas renewable energy consumption reduces them. Economic policy uncertainty intensifies $C{O}_2$ emissions, highlighting the need for stable and consistent economic policies to support environmental sustainability. Our research extends this by adopting a cubic EKC specification, enabling us to test for N- and W-shaped relationships while also incorporating additional explanatory variables (RENC, URB, and FDI) within a panel ARDL framework.

Vo and Vo [26] explore the causal relationships between renewable energy use, population, economic growth, and $C{O}_2$ emissions in seven ASEAN economies from 1990 to 2020. Using panel VAR and Granger causality, the findings reveal that population growth drives renewable energy use and $C{O}_2$ emissions. Our study validates this finding within a multivariate panel context for BRICS, confirming the short- and long-run impact of urbanization on CO₂ emissions as a key structural factor.

The study by Hasan et al. [27] examines the EKC hypothesis in BRICS countries from 1990 to 2020, focusing on the relationship between industrial growth and $C{O}_2$ emissions. The roles of financial development, trade openness, renewable energy, and fossil fuel consumption in environmental outcomes are also explored. The study finds that fossil fuel use exacerbates $C{O}_2$ emissions, while renewable energy, financial growth, and trade openness improve environmental quality. The EKC hypothesis is validated, and a feedback loop is identified among industrial growth, trade, and emissions. We contribute by applying a cubic EKC model that better captures emissions dynamics and integrating policy-related variables such as technological innovation and renewable energy.

Georgescu et al. [28] explore the main determinants of $C{O}_2$ emissions in Latin American countries. The study tests the validity of the pollution halo hypothesis using a panel ARDL model to analyze both short-run and long-run dynamics. The research also investigates the Energy Kuznets Curve and shows that $C{O}_2$ emissions are a monotonically decreasing function of renewable energy sources. Our study confirms this relationship in BRICS as well but goes further by quantifying the impact of RENC within both the EKC and ICC models.

Agozie et al. [29] examine the relationship between FDI, economic complexity, renewable energy, natural resources, urbanization, and $C{O}_2$ emissions in BRICS economies from 1990 to 2019. Both inverted-U and N-shaped EKC relationships exist, indicating varying dynamics between economic complexity and $C{O}_2$ emissions. The Pollution Haven Hypothesis (PHH) is supported, showing that FDI contributes to environmental degradation. Renewable energy and the interaction between economic complexity and urbanization reduce $C{O}_2$ emissions, while natural resources and urbanization increase them. We contribute by modeling this relationship explicitly using a cubic functional form and identifying two turning points in relation to both GDP and innovation.

Kartal et al. [30] explore the effects of environmental policy stringency (EPS), the energy transition index (ETI), and income (GDP) on environmental quality, focusing on BRICS countries from 2000/Q1 to 2020/Q4. Using the wavelet local multiple correlation (WLMC) model, the analysis examines the relationships between these variables under time- and frequency-based frameworks. We offer a complementary but more structurally grounded approach using ARDL with cointegration testing, capturing long-run dynamics more directly.

Nica et al. [31] examine the effects of FDI, GDP per capita, renewable energy, and urbanization on India’s $C{O}_2$ emissions from 1990 to 2023. By means of the ARDL model, the existence of an N-shaped EKC is proved. Our study generalizes this relationship across all BRICS nations and extends the analysis through the inclusion of the Innovation Claudia Curve.

Yilanci et al. [32] investigate the environmental impact of FDI and energy consumption using a Fourier bootstrap ARDL model. Their country-level results show mixed FDI effects. Our study builds on this by modeling FDI’s average effect in a panel structure and jointly testing its interaction with innovation and GDP within the EKC and ICC frameworks.

Li et al. [33] employ a Fourier ARDL model to explore the link between health expenditure, $C{O}_2$ emissions, and GDP in BRICS. While their focus is on cointegration and causality patterns, our contribution complements this with a panel ARDL approach that incorporates renewable energy and innovation, targeting emission-reduction dynamics. Joo and Shawl [34] analyze the FDI–growth nexus in BRICS using panel ARDL, finding long-run cointegration with key macroeconomic variables. Our study links this macroeconomic context to environmental outcomes by examining how FDI interacts with $C{O}_2$ emissions and sustainability in a broader ecological–economic model.

Furthermore, recent bibliometric studies offer valuable macro-level insights that complement our empirical model. Xu et al. [35] conduct a bibliometric analysis of carbon neutrality research, identifying research hotspots such as renewable energy, carbon capture technologies, and climate policy frameworks. Their findings emphasize the growing importance of carbon neutrality strategies and highlight China and the US as key contributors. While their study is global in scope, the highlighted relevance of renewable energy and technological transitions reinforces the importance of integrating RENC and innovation in our BRICS-focused analysis.

Gou et al. [36] provide a complementary bibliometric and systematic review, exploring how fuzzy set theory (FST) supports CE strategies in the context of Industry 4.0. They show that FST enhances decision-making under uncertainty in all CE stages, from design to recycling. Their highlighting of the synergies between technological innovation, sustainability, and digital transformation aligns with our study’s theoretical foundation, particularly in exploring how innovation (PAs) and RENC contribute to long-term emission reduction in emerging economies like BRICS.

These contributions highlight the relevance of linking the EKC and ICC frameworks to broader sustainability transitions, particularly in the context of CE and carbon neutrality. Our research builds on this emerging literature by offering a unified empirical model that integrates economic growth, innovation, and renewable energy in the specific context of BRICS economies.

2.2. Technology Innovation

Jones [37] claims that R&D investment and technology innovation (TI) contribute to reducing environmental pollution. This aligns with broader economic and environmental theories, such as the Porter Hypothesis and the EKC. We validate this effect but also contribute by modeling the cubic relationship between innovation and $C{O}_2$ emissions, capturing potential rebound effects.

The study by Kumail et al. [38] analyzes the relationship between tourism development, green TI, and $C{O}_2$ emissions in Asia’s major tourist destinations between 1990 and 2022. Using the CS-ARDL model and Dumitrescu–Hurlin causality tests, the environmental impact of tourism growth and TI is investigated. Their findings reveal that while both tourism and TI initially lead to increased $C{O}_2$ emissions, they eventually result in reductions once a critical innovation threshold is exceeded. Our study confirms this pattern within a cubic ICC framework, using PAs as a proxy for innovation and identifying a possible reintensification of emissions at advanced stages.

Khan et al. [18] investigate the role of TI and renewable energy consumption in reducing $C{O}_2$ emissions in OECD countries during the period of 2004 to 2019, using the two-step system generalized method of moments (GMM). While economic growth and innovation both significantly contribute to increased carbon emissions, the study confirms the presence of the ICC while finding no evidence to support the EKC hypothesis. Additionally, FDI has a negative impact on carbon emissions, validating the PHH. We contribute by simultaneously testing both curves (the EKC and the ICC) in BRICS, offering a unified and comprehensive empirical model.

Wang et al. [39] examine the relationship between energy technology patents and $C{O}_2$ emissions in 30 Chinese provinces during 1997–2008. Their results indicate that patents for fossil-fueled technologies do not significantly reduce $C{O}_2$ emissions, while patents for carbon-free energy technologies are effective, particularly in eastern China, but not at central, western, or national levels. Our study expands this view by showing that in BRICS countries, innovation (measured via PAs) does not automatically reduce emissions, supporting the need for the strategic alignment of innovation policies.

Adebayo and Kirikkaleli [40] use wavelet statistical tools to study the relationship between $C{O}_2$ emissions, TI, globalization, renewable energy, and GDP growth in Japan during 1990–2015. Globalization, GDP growth, and TI increase $C{O}_2$ emissions, while renewable energy reduces emissions in the short and medium terms. Our research provides additional support for this conclusion in the BRICS context, underscoring the importance of integrating RENC into innovation-led sustainability strategies.

Udeagha and Ngepah [41] examine how green finance, fintech, and energy innovation affect environmental sustainability in BRICS. Their findings support the EKC and show that innovation, GFN, and fintech promote $C{O}_2$ reduction. Our contribution expands this perspective by integrating innovation into a cubic ICC framework and analyzing its interaction with FDI and economic growth while emphasizing long-run structural dynamics.

To summarize, the existing scientific literature confirms the relevance of economic growth, renewable energy, urbanization, foreign direct investment, and innovation in shaping environmental outcomes in BRICS countries. However, most studies either focus on individual countries, apply linear or quadratic specifications, or treat the EKC and the ICC separately. Moreover, few studies jointly examine the role of innovation, FDI, and renewable energy within a unified panel model. Our study fills this gap by proposing a cubic panel ARDL approach that allows for the simultaneous testing of the Environmental Kuznets Curve and the Innovation Claudia Curve across BRICS, offering new insights into the complex dynamics between growth, technology, and environmental quality.

3. Methodology

Panel unit root tests, including Levin, Lin, and Chu (LLC) [42], Im, Pesaran, and Shin (IPS) [43], and ADF–Fisher Chi-square [44,45], were used in this paper to determine the stationarity of variables in panel data. Each test operates under distinct assumptions, making them complementary tools for the robust analysis of panel datasets. These panel unit root tests were selected due to their complementary strengths. The LLC test is useful when panel units are expected to behave similarly. Recognizing the heterogeneity among BRICS countries, we also employed the IPS and ADF–Fisher Chi-square. This is important in our context, as the BRICS countries exhibit structural and institutional differences that could influence their macroeconomic and environmental dynamics.

The LLC test assumes a common unit root process across all panel units. In contrast, the IPS test allows for heterogeneity across units by estimating individual ${\rho }_i$ parameters, thus accommodating panel-specific unit root processes. The panel unit root tests used in this study, namely, the LLC and the IPS, both rely on the following general ADF specification [42]:

$${∆Y}_{it}=\rho Y_{it-1}+{\sum }_{j=1}^p{\varphi }_j{\Delta Y}_{it-j}+X_{it}^{\prime }\beta +{\epsilon }_{it}$$

(1)

where $Y_{it}$ is the variable of interest for cross-section i at time t. $\rho$ is the parameter of interest, with the null hypothesis $H_0: \rho =0$ (presence of a unit root).

${∆Y}_{it}=Y_{it}-Y_{it-1}$, $X_{it}^{\prime }$ represents transposed deterministic components (e.g., trends or fixed effects), and ${\epsilon }_{it}$ is the error term.

The IPS test statistic is described in Equation (2):

$$t_{IPS}={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{t}_i$$

(2)

where $t_i$ are the individual ADF statistics. The ADF–Fisher test combines p-values from individual ADF tests using Fisher’s formula:

$${\chi }^2=-2{\sum }_{i=1}^N\mathrm{l}\mathrm{n}\left(p_i\right)$$

(3)

where, in Equation (3), $p_i$ represents the p-value of the individual ADF test for panel unit $i$, ${\chi }^2$ follows a chi-square distribution with 2N degrees of freedom under the null hypothesis $H_0$ (unit root exists for all panel units).

The panel Autoregressive Distributed Lag (ARDL) model by Pesaran et al. [46] is a robust econometric technique used to analyze long-term and short-term dynamics between variables in panel data. It is particularly useful when dealing with non-stationarity variables that are integrated in different orders (I(0) or I(1) but not I(2)). Given the varying integration properties of the dataset, this flexibility is essential. For this study, the panel ARDL model allowed for the investigation of the relationship between variables such as $C{O}_2$ emissions, GDP, FDI, urbanization, renewable energy consumption, and patent applications across BRICS countries during 1991–2023. This is important for understanding how the effects of economic or policy shocks persist or dissipate across BRICS countries. The model also supports heterogeneous short-run dynamics across countries, allowing each country to respond differently to short-term fluctuations in economic activity or innovation. At the same time, it assumes long-term homogeneity, which is appropriate given the increasing convergence in environmental objectives and international agreements among BRICS countries.

$${\Delta Y}_{it}={\varphi }_i\left(Y_{it-1}-{\beta }_0-{\beta }_1{X}_{it-1}\right)+{\sum }_{j=1}^p{\alpha }_{ij}{\Delta Y}_{it-j}+{\sum }_{k=0}^q{\gamma }_{ik}{\Delta X}_{it-k}+{\epsilon }_{it}$$

(4)

In Equation (4), $Y_{it}$ is the dependent variable for entity i at time t, and $X_{it}$ is a vector of independent variables. ${\varphi }_i$ measures the speed of adjustment back to the long-run equilibrium, ${\beta }_0$ and ${\beta }_1$ represent the long-run coefficients, ${\alpha }_{ij}$ and $y_{ik}$ capture short-run dynamics, and ${\epsilon }_{it}$ is the error term. The panel ARDL model simultaneously estimates the long-run equation, according to Equation (5):

$$Y_{it-1}={\beta }_0+{\beta }_1{X}_{it-1}+u_{it}$$

(5)

and the short-run dynamics, described in Equation (6):

$${\Delta Y}_{it}={\varphi }_i{u}_{it-1}+{\sum }_{j=1}^p{\alpha }_{ij}{\Delta Y}_{it-j}+{\sum }_{k=0}^q{\gamma }_{ik}{\Delta X}_{it-k}+{\epsilon }_{it}$$

(6)

The error correction term (ECT) represents deviations from the long-run equilibrium. A significant and negative coefficient of the ECT $\left({\varphi }_i\right)$ between −2 and 0 [47] confirms the presence of cointegration.

3.1. Theoretical Framework: Environmental Kuznets Curve

The EKC assumes an inverted U-shaped relationship between environmental degradation and economic growth. In the early stages of development, economic growth leads to environmental degradation as industrialization, urbanization, and energy consumption intensify. Beyond a certain income threshold, economic growth contributes to environmental improvement due to structural economic shifts, technological innovation, and increased environmental awareness.

The classic EKC is modeled as a quadratic relationship. The cubic EKC becomes more complex, allowing for more nuanced interpretations of the relationship. The cubic EKC has the cubed income as an independent variable, and this enables the identification of a potential N-relationship. This extends the theoretical framework. In the early stage of economic development, resources are extensively used, and environmental degradation increases rapidly [48]. In this phase, industries rely on resource extraction and pollutive technologies. Lax environmental regulations are driven by limited financial capacity to invest in environmental conservation efforts [49]. In this phase, the linear relationship between income and environmental degradation is positive. Beyond the turning point, the level of income is associated with environmental improvements. TI, regulatory initiatives, and investments in renewable energy drive a decline in environmental degradation, forming the downward slope of an inverted U-shaped EKC. This phase of economic development is characterized by a shift toward a service-oriented economy with information- and technology-driven industries. Stricter environmental regulations and the adoption of cleaner and more efficient technologies lead to reduced environmental degradation.

The cubic term allows for the possibility of a second turning point. In highly developed economies, overconsumption, rebound effects, or diminishing returns from TI may cause a second increase in environmental degradation. This creates an upward slope following the initial inverted U-shape. For studies discussing cubic EKC cases explicitly, we considered some references. Dinda [50] provides a comprehensive review of the EKC literature and discusses the possibility of more complex shapes, such as N- and W-shaped curves. The study by Kaufmann et al. [51] explores the potential for higher-order polynomials, including cubic terms, to better capture the relationship between economic activity and environmental indicators. The work by Shafik and Bandyopadhyay [52] examines the relationship between economic growth and multiple environmental indicators and considers non-linear patterns. Kijima et al. [53] survey various forms of the EKC, including cubic models that allow for N-shaped curves and other complex relationships.

The cubic EKC model is expressed as:

$${CO}_2={\beta }_0+{\beta }_{1}GDP+{\beta }_2{GDP}^2+{\beta }_3{GDP}^3+\epsilon$$

(7)

where ε is the error term.

The intercept ${\beta }_0$ represents the baseline level of $C{O}_2$ emissions. The linear term ${\beta }_1$ represents the initial effect of economic growth on environmental degradation. A positive coefficient ${\beta }_1$ can be interpreted as economic growth contributing to increased $C{O}_2$ emissions at low-income levels. The quadratic term ${\beta }_2$ captures the curvature of the relationship. A negative coefficient ${\beta }_2$ can be interpreted as the potential for a turning point, where economic growth initially increases $C{O}_2$ emissions but eventually reduces them. A positive coefficient ${\beta }_2$ can be interpreted as an increase in $C{O}_2$ emissions simultaneously with an increase in economic growth, potentially at an accelerating rate.

The cubic term ${\beta }_3$ indicates the N-shaped pattern of EKC. A positive cubic term ${\beta }_3$ can be interpreted as after an initial reduction in $C{O}_2$ emissions, emissions, economic growth at very high levels may lead to increased $C{O}_2$ emissions again. This suggests a potential reversal of the downward trend in emissions as income continues to rise. A detailed discussion of the cubic EKC can be found in Mikayilov et al. [54]. The cubic EKC is more flexible in capturing different dynamics of the relationship between economic growth and environmental degradation. In the following, we analyze the signs and the significance of the EKC coefficients in Equation (7). This determines the number of turning points of the EKC or its linearity. Each case reflects different stages of development, technological changes, policy interventions, and economic activities that influence $C{O}_2$ emissions:

(i)

Case 1—inverted U-shape with no secondary rise (simple U-shape): ${\beta }_1>0,{\beta }_2<0,{\beta }_3=0$. When the cubic term ${\beta }_3$ is not statistically significant or is omitted, the EKC is an inverted U-shaped curve. If the linear coefficient ${\beta }_1$ is positive, it shows that economic growth drives $C{O}_2$ emissions up. If the quadratic coefficient ${\beta }_2$ is negative, it shows that $C{O}_2$ emissions start to slow and eventually decrease as the economies mature, possibly due to TI, more efficient production processes, or environmental policies;

(ii)

Case 2—N-shaped curve (multiple U-shapes with two turning points): ${\beta }_1>0,{\beta }_2<0,{\beta }_3>0$. This case depicts a situation where $C{O}_2$ emissions first increase, then decrease, and increase again after a second turning point. The EKC has an N-shape. This happens because economic growth leads to both increases and decreases in $C{O}_2$ emissions over time. The positive coefficient for the cubic term ${\beta }_3$ implies that after an initial decline in $C{O}_2$ emissions, a rebound effect may take place when $C{O}_2$ emissions increase due to higher energy consumption and the industrialization of resource-intensive growth. This situation may occur in countries where the transition to greener technologies and policies is slow, and economic growth at higher levels leads to increased consumption and industrial activity that temporarily raise emissions before they eventually decline again [48].

The secondary rise in $C{O}_2$ emissions at higher income levels, observed in the N-shaped EKC, can be explained by several structural and behavioral mechanisms. As economies reach advanced stages of development, consumption patterns shift toward energy-intensive lifestyles, high-tech products, and expansive infrastructure. The marginal environmental benefits of early innovations and regulations may begin to decline, leading to a saturation effect. Rebound effects, where efficiency gains lower energy costs and stimulate increased consumption, further contribute to the reintensification of emissions. In the BRICS context, this pattern is particularly relevant due to rapid urbanization, increasing vehicle ownership, growing digital infrastructure, and energy demands from expanding service sectors. Without continuous green innovation and stringent environmental policies, economic maturity alone may not be sufficient to sustain emission reductions beyond the first turning point. Thus, the final upward slope of the N-shaped EKC highlights the need for sustained and adaptive sustainability strategies even in high-income phases;

(iii)

Case 3—W-shaped curve (multiple turning points with fluctuating emissions): ${\beta }_1>0,{\beta }_2<0,{\beta }_3>0$ and potentially higher-order terms $({\beta }_3<0)$. A W-shaped EKC is an extension of the N-shaped EKC. A W-shaped EKC suggests that $C{O}_2$ emissions rise and fall twice during economic development. This scenario occurs as a cyclical pattern of environmental degradation and recovery. This pattern can be driven by factors causing fluctuations in $C{O}_2$ emissions over time, such as increased consumption patterns or the limitations of the previous pollution control strategies [55]. For example, periods of rapid industrialization may cause emissions to rise, followed by a decline as cleaner technologies are adopted. Then, emissions may rise again as consumption or industrial activity increases before they eventually stabilize or decline due to sustainability initiatives;

(iv)

Case 4—inverted N-shaped curve (after initial rise, emissions consistently decrease): ${\beta }_1>0,{\beta }_2<0,{\beta }_3<0$. In this case, $C{O}_2$ emissions first increase, then continuously decrease without a second rise, indicating a consistent and sustained decline after the turning point. The negative cubic term ${\beta }_3<0$ indicates that after reaching a peak, $C{O}_2$ emissions do not rise again but instead keep decreasing as the economy develops further. Although this pattern is less frequent, it may indicate certain policy changes or technology breakthroughs at different phases of economic growth [55]. This situation can arise when an economy shifts to a sustainable development path, in which TI and regulations continuously lower $C{O}_2$ emissions while maintaining economic growth;

(v)

Case 5—constant growth (no turning point): ${\beta }_1>0,{\beta }_2\approx 0,{\beta }_3\approx 0$. In the constant growth case, the relationship between economic growth and $C{O}_2$ emissions is linear or monotonically increasing. There are no turning points, and $C{O}_2$ emissions continuously rise as the economy grows. This might occur in economies where environmental degradation is tightly linked to economic activities, such as heavy reliance on fossil fuels, high industrial activity, or unsustainable consumption patterns [55].

3.2. Theoretical Framework for the Innovation Claudia Curve

The ICC depicts the relationship between $C{O}_2$ emissions and PA, grounded in the economic, environmental, and technological theories surrounding innovation. This relationship can be analyzed through the EKC, the Porter Hypothesis [56], and theories of TI.

The cubic specification of the ICC reflects a non-linear relationship between TI and environmental impact. In the early stages of technological development, innovation often supports industrial expansion, which may increase resource consumption and fossil fuel use, thereby raising $C{O}_2$ emissions [57]. As TI matures, it tends to shift toward cleaner technologies and energy efficiency, leading to $C{O}_2$ emission reduction [58]. However, at very high levels of TI, a second rise in $C{O}_2$ emissions may occur due to rebound effects, such as increased energy consumption driven by efficiency gains or the carbon footprint of large-scale digital infrastructures [59]. The cubic relationship is thus theoretically justified and empirically suitable for capturing these sequential phases in the TI–$C{O}_2$ emissions trajectory.

Innovation in green technologies, renewable energy, and sustainable manufacturing processes can mitigate the environmental impacts of economic growth. The Porter Hypothesis asserts that stricter environmental regulations can stimulate TI, leading to the development of cleaner technologies that improve environmental quality. The rise in PAs reflects the adoption of TI, which can lead to cleaner production, reduced waste, and lower $C{O}_2$ emissions. The relationship between TI and $C{O}_2$ emissions becomes more pronounced in developed economies that have the infrastructure to implement new technologies.

There is a feedback loop between PAs and $C{O}_2$ in the ICC. Initially, PAs are driven by economic growth, since firms and governments invest in TI. Over time, TI contributes to $C{O}_2$ emissions by offering cleaner alternatives to traditional production methods. As economies advance, the adoption of clean technologies accelerates, creating a positive feedback loop that decreases $C{O}_2$ emissions.

In the long run, the $C{O}_2$ emission reduction due to TI may lead to a stabilization or even a decline in the environmental impact, reinforcing the adoption of green technologies. This is the negative feedback loop.

The ICC may include critical thresholds at which structural shifts in the economy or TI can completely change the relation between PAs and $C{O}_2$ emissions. Advances in carbon-neutral energy production or carbon capture and storage can reduce $C{O}_2$ emissions, even in economies that expand quickly.

The ICC also aligns with endogenous growth theory [60,61] by which economic growth is driven by internal factors, including TI. PAs are a proxy for innovation and represent an endogenous factor that can shift the relationship between economic growth and $C{O}_2$ emissions.

The ICC framework provides insights into how TI should be managed to serve sustainability goals. Rather than assuming that TI always reduces emissions, it emphasizes that TI must be directed and regulated. Early innovation may require environmental safeguards, while mid-level TI benefits from targeted support for green R&D. At advanced stages, avoiding rebound requires systemic policy integration, such as carbon pricing and lifecycle assessments [62]. Therefore, a phased innovation policy is required to align with the stages of technological maturity and account for potential trade-offs between TI intensity and environmental quality.

A stronger theoretical justification of the cubic ICC framework lies in recognizing the sequential phases through which TI impacts environmental outcomes. Initially, TI drives industrial growth and technological scaling, often increasing emissions. As economies advance, the innovation landscape moves toward cleaner, more efficient technologies, resulting in reduced emissions. At high levels of technological maturity, emissions may rise again due to the rebound effect [63]. The rebound effect occurs as follows: efficiency gains lower costs and stimulate additional energy use, while the operation of high-tech systems, such as cloud computing, data centers, or AI infrastructure, requires substantial energy, often from mixed sources. This leads to a secondary rise in emissions, especially if renewable energy supply lags behind demand. Thus, the ICC’s cubic structure captures this full cycle of innovation’s environmental impact: acceleration, mitigation, and potential reintensification. This non-linear pattern is relevant for BRICS countries, where innovation ecosystems are expanding rapidly.

To ensure the validity and robustness of the model, we conducted a comprehensive series of diagnostic tests. First, panel unit root tests (LLC, IPS, and ADF–Fisher) confirm that all variables were integrated in the order I(0) or I(1), justifying the use of the panel ARDL approach. This method is appropriate for exploring both short-run and long-run dynamics across heterogeneous units, such as BRICS countries. Furthermore, the presence of cointegration relationships was confirmed through the significance and expected sign of the error correction term (ECT), indicating the existence of stable long-term equilibria. All regressions included lag structures and significance levels that allowed us to differentiate between short-run shocks and long-term trends. The interpretation of results was grounded in statistically significant coefficients and supported by the theoretical and empirical literature. Additionally, we complemented the regression results with derivative-based turning point analysis and graphical representations to verify the functional form of the cubic relationships identified in both the EKC and ICC models.

4. Results

This study is based on the variables in

Table 1. Variable specifications.

Variable	Acronym	Measurement Unit	Source
$C{O}_2$ emissions per capita	$C{O}_2$	Tonnes per person	Our World in Data
Patent applications	Pas	Number	World Bank
Gross domestic product	GDP	Constant 2015 USD	World Bank
Renewable energy consumption	RENC	% of total final energy	World Bank
Urban population	URB	% of total population	World Bank
Foreign direct investment	FDI	% of GDP	World Bank

collected from the World Bank and Our World in Data and extracted for the period of 1991–2023.

From now on, we work with natural logarithmic transformation of the data. According to Lütkepohl and Xu [64], log transformations are beneficial for stabilizing variance, linearizing non-linear relationships, and improving the interpretability of forecast results.

Figure 1 displays the trend plots for BRICS countries. The trends in $C{O}_2$ emissions reveal significant heterogeneity across BRICS countries. China and Russia have the highest levels of emissions, reflecting their historical reliance on fossil fuels and energy-intensive industries. China’s emissions increased consistently until the early 2010s, followed by a decline, which can be attributed to a combination of factors, including improved energy efficiency and shifts in energy policies aimed at reducing carbon intensity. While the proportion of renewable energy in the energy mix declined after 2005, China’s Five-Year Plans played an important role in driving energy-saving measures and restructuring the industrial and energy sectors [65]. This restructuring, along with improvements in energy efficiency, led to a significant reduction in $C{O}_2$ emissions per unit of GDP, especially after 2011 [66]. The reduction was also influenced by the government’s focus on low-carbon energy policies, including the implementation of high-efficiency coal-fired plants and energy-efficient technologies, which resulted in reduced coal consumption and $C{O}_2$ emissions per unit of electricity produced [67]. Thus, the decline in emissions should be seen as the result of a broader set of measures, not exclusively an increase in renewable energy use. India shows a persistent upward trend, consistent with its rapid economic development and industrialization in recent decades. Brazil and South Africa have relatively stable emissions, although South Africa remains above Brazil in terms of emission intensity. All BRICS countries exhibit an overall upward trend in GDP at varying speeds. China has shown the steepest and most consistent growth, proving its role as a global economic powerhouse since the early 2000s. India also displays steady economic expansion. Russia and South Africa demonstrate more volatile patterns, with visible slowdowns and fluctuations, especially after 2014. Brazil’s GDP remains relatively stable, with moderate growth over time.

The FDI patterns are considerably volatile. South Africa has significant fluctuations, especially in the early 1990s and post-2015, possibly due to political and economic uncertainty. Brazil and Russia display variations, although Brazil shows a more consistent upward trajectory in the mid-2000s. India and China have more stable and growing FDI inflows, which aligns with their roles as destinations for foreign capital due to market size, policy reforms, and expanding infrastructure.

Urbanization rates steadily increase for all BRICS countries, with China and Brazil maintaining the highest levels. India starts from the lowest base but shows a continuous rise, indicating accelerating migration to urban centers. Russia and South Africa maintain mid-range urbanization levels with gradual growth. It follows that urbanization is a shared structural transformation in emerging economies, often linked with industrial growth, infrastructure development, and rising $C{O}_2$ emissions.

Innovation activity, as proxied by patent applications, is dominated by China, which shows an increase starting in the early 2000s. China leads the BRICS in technological advancement, reflecting its investment in R&D and high-tech manufacturing. India and Russia also exhibit growth in patent filings. Brazil and South Africa show relatively flat or stagnant trends, suggesting weaker innovation capacity or under-reporting.

Brazil consistently exhibits the highest level of renewable energy consumption, likely due to its long-standing reliance on hydropower and biofuels. China and India show moderately fluctuating trends, reflecting their efforts to integrate renewables alongside fossil fuels. Russia ranks the lowest, highlighting its ongoing dependence on traditional energy sources. South Africa’s trend has been slightly declining since the mid-2000s, suggesting a need for renewed commitment to green energy.

The violin plot in Figure 2 displays the distribution of six variables: $C{O}_2$ emissions, GDP, FDI, URB, PAs, and RENC. All variables are log-transformed to allow a more accurate visualization of their internal variability. Each plot illustrates the density and variability of the respective variables, providing insights into their statistical characteristics and patterns. The violin plots do not aim to compare absolute levels across variables but rather to illustrate the shape, dispersion, and skewness of each distribution individually.

The distribution of $C{O}_2$ emissions is relatively symmetric, meaning that the values are clustered around the mean with no extreme outliers. This symmetry suggests consistent patterns of $C{O}_2$ emissions across the dataset, which may reflect stable emission trends over time or among BRICS nations.

For GDP, the violin plot shows a distribution with notable dispersion but not necessarily a wider spread compared to other variables. The point at the increase indicates that most GDP values are clustered within a certain range, while the wider distribution reveals substantial disparities in economic performance across the data points. This variability could reflect disparities in the economic development levels.

The plot for FDI shows a distinct lower tail, suggesting the presence of outliers or smaller values on the lower end of the distribution. This asymmetry implies that while many observations fall within a central range, certain instances of low FDI deviate from the overall pattern. These deviations might indicate periods or regions with reduced foreign investment. Specifically, the notably low FDI values in South Africa in 1992 can be attributed to the political and economic instability following the end of apartheid in 1991. Prior to 1994, South Africa’s economy was intensely dependent on a highly concentrated and extractive financial market, particularly in sectors like mining, with a limited number of participants and limited access to global financial markets due to international sanctions [68,69]. The transition to democracy in 1994 allowed South Africa to rejoin the international financial system, but this period of political change and market liberalization resulted in a temporary decline in foreign investment as the nation adapted to new democratic and economic structures. The challenges faced during this transition period, combined with the high levels of economic inequality, likely deterred foreign investment in the immediate post-apartheid years [70]. The economic inequality further deepened as the newly democratic system allowed only a small number of market participants to gain from significant capital and privatized assets [69,71].

URB exhibits notable variability across the BRICS countries, as shown in both Figure 1 and Figure 2. While China and India demonstrate rapid urbanization growth, with values gradually increasing [72,73], Brazil, Russia, and South Africa display more gradual and varied urbanization rates [74]. The violin plot in Figure 2 highlights this significant variability, with Brazil and South Africa showing more consistent and lower urbanization levels, while China and India exhibit broader distributions due to faster urbanization trends. This variability suggests that urbanization rates differ substantially between these countries, likely driven by diverse demographic, economic, and policy factors within each nation.

For PAs, the violin plot shows a widespread pronounced upper density, indicating a concentration of higher values. This reflects significant variation in innovation levels, with certain regions or periods experiencing a high number of PAs, likely driven by increased focus on research and development. For example, according to Figure 1 and Figure 3, it is evident that the significant increase in PAs can largely be attributed to the upward trend observed in China. China has shown an intense increase in PAs from the early 2000s onward, reflecting the country’s growing highlighting of innovation, particularly in technology and industry [75,76]. This trend is the primary driver of the widespread and pronounced upper density in the violin plot, where China’s sharp increase in PAs stands out compared to other BRICS nations. While other countries, such as Brazil and India, show some increase in PAs, these trends are much less pronounced. South Africa and Russia, on the other hand, have shown relatively small growth in PAs.

RENC displays a distribution with moderate variability. The absence of extreme tails suggests that RENC remains consistent across the dataset, though the spread indicates differences in adoption rates or usage levels.

To better illustrate the variability across countries and support the validity of panel estimation while acknowledging cross-country heterogeneity, individual violin plots for each BRICS country are provided in Appendix A, Figure A1, Figure A2, Figure A3, Figure A4 and Figure A5. These plots depict the log-transformed distributions of the main variables, GDP, FDI, URB, PAs, and RENC. The visualizations reveal significant differences in scale and dispersion. For instance, China exhibits the widest spread in PAs, reflecting its dominant role in innovation output, while India shows greater dispersion in FDI and urbanization compared to Russia, where the distributions are more concentrated. These findings justify the use of a panel ARDL model with country-fixed effects while also reinforcing the relevance of interpreting certain results at the individual country level.

Figure 3 shows the comparison of six variables (GDP, FDI, URB, PAs, RENC, and $C{O}_2)$ for the five countries (Brazil, Russia, India, China, and South Africa) over the years. China’s GDP has grown gradually, while Brazil and India have experienced slower growth with fluctuations. These differences may suggest that emerging economies have distinct economic development trajectories. China has attracted a significant volume of FDI as it has increasingly integrated into the global economy, especially since 2000. Brazil and Russia have recorded growth, but with more fluctuations, while India and South Africa have experienced slower yet steady growth. This highlights the variability in economic environments and FDI attraction among the BRICS countries. The increase in urbanization is evident in China and India, with notable growth starting from the 2000s. Brazil and South Africa have shown a more consistent rate of urbanization, though at a slower speed compared to China and India. These developments suggest that rapid urbanization, especially in China and India, may be linked to economic growth and infrastructure development. China has seen a significant increase in the number of patent applications, reflecting its dominant role in global innovation. India and Russia have seen a steady increase in applications but on a much smaller scale, while Brazil and South Africa have exhibited more moderate activity in technological innovation. These differences suggest that China plays a key role in global technological advancement, and countries with less innovation activity may face challenges in achieving sustainable economic development.

Although China and India have made progress in renewable energy use, consumption remains low compared to Brazil and South Africa, which have a smaller share of renewable energy in their energy mix. These differences suggest that the transition to renewable energy sources varies significantly across the BRICS countries. China has seen a significant increase in $C{O}_2$ emissions, parallel to its economic growth and rapid urbanization. While India and South Africa have emitted less, Russia and Brazil show relatively stable emissions with moderate increases. This graph suggests that $C{O}_2$ emissions are directly linked to economic growth and industrialization.

According to

Table 2. Descriptive statistics (log variables).

Statistics	$\mathit{C}{\mathit{O}}_2$	GDP	FDI	URB	PAs	RENC
Mean	1.37	8.26	0.33	3.99	10.23	2.84
Median	1.59	8.66	0.59	4.12	10.12	2.92
Maximum	2.77	9.40	2.26	4.47	14.27	3.95
Minimum	$-$0.36	6.27	$-$6.08	3.24	8.05	1.16
Std. dev.	0.87	0.90	1.14	0.39	1.45	0.96
Skewness	$-$0.27	$-$0.65	$-$0.20	$-$0.60	1.22	$-$0.52
Kurtosis	1.68	2.05	10.44	1.88	4.31	1.87

, $C{O}_2$ emissions have a mean of 1.37 and a standard deviation of 0.87, indicating a moderate degree of variation across the BRICS countries. The maximum value of 2.78 points to significantly higher emissions in some cases, while the minimum value of -0.36 suggests near-zero emissions in others. The distribution is slightly positively skewed, meaning that a few higher values raise the average, but most countries fall within a more modest emissions range. The kurtosis value of 1.68 indicates a relatively flat distribution with fewer outliers than a normal distribution. The GDP values average around 8.27, corresponding to approximately USD 3900 in real terms. The standard deviation of 0.90 shows some degree of economic heterogeneity among BRICS countries. The GDP distribution shows slight positive skewness, indicating that while most countries are clustered around the mean, some have notably higher GDP. The kurtosis value of 2.05 suggests the distribution is close to normal. The FDI indicator has a mean of 0.33. However, the standard deviation is quite high at 1.15, indicating that FDI inflows vary widely across the BRICS countries. The minimum value of -6.08 suggests significant capital outflows in some instances, while the maximum of 2.27 reflects strong FDI inflows in others. The high skewness of 2.07 and the very high kurtosis of 10.45 show that the distribution is heavily influenced by a few extreme cases, possibly reflecting large inflows in countries such as China or Brazil during certain periods. The urbanization rate has a mean of 3.99, which translates to an average urbanization level of about 54 percent. The relatively low standard deviation of 0.39 indicates that the level of urbanization is fairly consistent among BRICS countries. The slight positive skewness implies that more countries are on the lower end of the urbanization spectrum, while a few have significantly higher rates. The kurtosis of 1.88 suggests a relatively even distribution without many extreme values. PAs have a mean of 10.23. The standard deviation of 1.45 suggests that the number of patents filed varies significantly among BRICS nations. The PA distribution is moderately right-skewed, with a skewness of 1.22, and has a kurtosis of 4.31. RENC has a mean of 2.85. The standard deviation is 0.96, showing that there is some variability in RENC across BRICS countries. The RENC distribution is slightly right-skewed, with a skewness of 0.53, suggesting that a few countries have higher renewable energy use. The kurtosis of 1.88 implies a somewhat flat distribution, meaning fewer extreme values than would be expected under normal distribution.

Before estimating the long-run and short-run relationships, the optimal lag structure for the panel ARDL model was determined using the Akaike Information Criterion (AIC). The selection process is visualized in Appendix B (Figure A6 and Figure A7), which displays the AIC values for different lag specifications. The best model for the EKC equation was identified as ARDL (3, 2, 2, 2, 2), while for the ICC model, the optimal structure was also ARDL (3, 2, 2, 2, 2). These specifications were subsequently used to estimate the dynamic interactions among $C{O}_2$ emissions, GDP, FDI, URB, PAs, and RENC. The consistent AIC minimization across models ensures the validity and robustness of our results.

In the analysis of

Table 3. Panel unit root tests.

At Levels
	$C{O}_2$	GDP	GDP2	GDP3	FDI	URB	PA	RENC
Unit root (common unit root process)
LLC	$-0.28$ (0.38)	$-$2.57 *** (0.00)	$-$1.73 *** (0.04)	$-0.29$ (0.38)	$-1.61$ * (0.05)	$-$1.60 * (0.05)	0.72 (0.78)	1.77 (0.96)
Unit root (individual unit root process)
IPS	0.96 (0.83)	0.87 (0.80)	1.64 (0.95)	2.71 (0.99)	$-$1.99 ** (0.02)	1.97 (0.97)	1.66 (0.95)	1.96 (0.99)
ADF–Fisher Chi-square	6.29 (0.79)	6.15 (0.80)	3.76 (0.95)	2.07 (0.99)	26.58 *** (0.00)	5.46 (0.85)	1.54 (0.31)	8.42 (0.58)
At first difference
Unit root (common unit root process)
LLC	$-$2.53 *** (0.00)	$-$1.90 ** (0.02)	$-$1.78 ** (0.03)	$-$1.90 ** (0.02)	$-$3.68 *** (0.00)	$-$3.52 *** (0.00)	$-$11.61 *** (0.00)	$-$8.54 *** (0.00)
Unit Root (individual unit root process)
IPS	$-$4.36 *** (0.00)	$-$2.83 *** (0.00)	$-$2.72 *** (0.00)	$-$2.71 *** (0.00)	6.72 *** (0.00)	$-$8.15 *** (0.00)	$-$9.59 *** (0.00)	$-$7.58 *** (0.00)
ADF–Fisher Chi-square	38.22 *** (0.00)	27.17 *** (0.00)	26.37 *** (0.00)	25.81 *** (0.00)	61.39 *** (0.00)	281.72 *** (0.00)	82.57 *** (0.00)	78.69 *** (0.00)

*, **, and ***: Significant at 10%, 5%, and 1% levels.

, the LLC test indicates non-stationarity for several variables (e.g., $C{O}_2$ and RENC) at levels. However, at first differences, the null hypothesis H0 was rejected, confirming stationarity ($\rho <0$).

The IPS test similarly found that most variables were non-stationary at levels but stationary after first differencing. This highlights heterogeneity among BRICS countries, as countries may follow distinct dynamic processes. The ADF–Fisher test results confirm the LLC and IPS findings, i.e., non-stationarity at levels and stationarity after first difference.

First, we tested the existence of the cubic EKC for BRICS countries by means of an ARDL model starting from the dependence relation (8):

$$C{O}_2=f(GDP, GDP2, GDP3, FDI, URB, PA, RENC)$$

(8)

where, in Equation (8), GDP2 and GDP3 represent the squared and cubic GDP, respectively.

The long-run equation of the panel ARDL model captures the stable relationships between $C{O}_2$ emissions and the independent variables.

From

Table 4. Panel ARDL (3, 2, 2, 2, 2, 2, 2, 2) results for EKC model.

Variable	Coefficient	Std. Error	t-Statistic	Prob.
Long run equation
GDP	58.14	7.52	7.73	0.00 ***
GDP2	$-$7.53	0.95	$-$7.88	0.00 ***
GDP3	0.32	0.04	8.07	0.00 ***
FDI	$-$0.02	0.01	$-$2.06	0.04 **
URB	1.03	0.19	5.33	0.00 ***
PAs	0.09	0.01	6.21	0.00 ***
RENC	$-$0.96	0.05	$-$16.97	0.00 ***
Short-run equation
Cointeq01	$-$0.43	0.17	$-$2.45	0.01 **
$\mathrm{D}(C{O}_2$$(-$1))	$-$0.38	0.17	$-$2.28	0.02 **
$\mathrm{D}(C{O}_2$$(-$2))	$-$0.20	0.09	$-$2.11	0.01 **
D(GDP)	$-$279.03	293.58	$-$0.95	0.02 **
$\mathrm{D}\left(\mathrm{GDP}\right(-1$))	$-$71.00	93.06	$-$0.76	0.03 **
D(GDP2)	30.07	33.37	0.90	0.34
$\mathrm{D}\left(\mathrm{LGDP}2\right(-$1))	7.41	11.43	0.64	0.44
D(GDP3)	$-$1.06	1.27	$-$0.84	0.37
$\mathrm{D}\left(\mathrm{GDP}3\right(-$1))	$-$0.24	0.47	$-$0.51	0.51
D(FDI)	0.02	0.003	7.59	0.40
$\mathrm{D}\left(\mathrm{FDI}\right(-$1))	0.01	0.005	2.76	0.60
D(URB)	$-$6.33	20.43	$-$0.31	0.00 ***
$\mathrm{D}\left(\mathrm{URB}\right(-$1))	40.94	22.62	1.81	0.00 ***
D(PA)	$-$0.05	0.03	$-$1.68	0.75
$\mathrm{D}\left(\mathrm{PA}\right(-$1))	$-$0.05	0.00	$-$11.21	0.07 *
D(RENC)	0.12	0.07	1.65	0.09 *
$\mathrm{D}\left(\mathrm{RENC}\right(-$1))	$-$0.02	0.08	$-$0.29	0.00 ***
C	$-$65.61	26.48	$-$2.47	0.10
Validation metrics
Root MSE	0.02	Mean dependent variable		0.01
S.D. dependent var.	0.04	S.E. of regression		0.01
Akaike information criterion	$-$4.38	Sum squared residuals		0.03
Schwarz criterion	$-$2.54	Log likelihood		454.37
Hannan–Quinn criterion	$-$3.64

*, **, and ***: Significant at 10%, 5%, and 1% levels.

, one can see that a 1-unit increase in GDP leads to a long-run increase in $C{O}_2$ emissions by 58.14. Economically, this reflects the carbon-intensive nature of growth in BRICS countries, where industrialization and energy production remain heavily dependent on fossil fuels. Sectors such as manufacturing, construction, and heavy industries play a dominant role in these economies, leading to a significant increase in environmental degradation during the growth process.

The squared term of GDP has a significant negative coefficient, $-$7.53, suggesting that the EKC hypothesis holds for BRICS countries. The negative coefficient reflects the transition toward sustainability as economies mature and environmental policies become more stringent.

The cubic term of GDP has a positive and significant coefficient (0.32). This indicates an N-shaped EKC, where, after the decline in $C{O}_2$ emissions at higher income levels, further economic growth may again lead to rising $C{O}_2$ emissions. This pattern could be driven by increased consumption, infrastructure expansion, and diminishing returns from TI, highlighting the complex dynamics between growth and environmental outcomes.

A one-unit increase in FDI leads to a long-run decrease in $C{O}_2$ emissions by 0.02. This suggests that FDI in BRICS countries is likely associated with technology transfer and more efficient production methods, helping to reduce $C{O}_2$ emissions in the long run. This indicates the potential for green investment and sustainable industrial growth through global partnerships.

A one-unit increase in URB leads to a long-run increase in $C{O}_2$ emissions by 10.3. This reflects the environmental challenges associated with urban expansion, such as higher energy demand, transportation emissions, and industrial activities in urban centers. Urbanization drives growth by concentrating on economic activities, but the environmental costs highlight the need for sustainable urban planning and energy efficiency measures.

A one-unit increase in PAs leads to a long-run increase in $C{O}_2$ emissions by 0.09. This may reflect that innovation activities in BRICS countries are currently concentrated in industries or technologies that are energy-intensive or reliant on carbon-intensive processes. This result highlights a potential disconnect between TI and environmental sustainability. It suggests that not all TI is geared toward green or eco-friendly solutions. For sustainable development, fostering innovation that targets clean energy, energy efficiency, and low-carbon technologies should be promoted.

A one-unit increase in RENC leads to long run negative impact on $C{O}_2$ emissions by −0.96. This proves that renewable energy can be used to decouple economic growth from environmental degradation. Increased investment in renewables can reduce dependency on fossil fuels, leading to a more sustainable growth trajectory for the BRICS nations.

ECT has a negative and significant coefficient, indicating a rapid adjustment of about 43% of the deviations from the long-run equilibrium annually. This highlights that while shocks to $C{O}_2$ emissions occur, the system quickly returns to its long-run path, proving the stability of the underlying relationships. In the short run, lagged $C{O}_2$ emissions (D(CO₂(−1)) and D(CO₂(−2))) have significant negative effects, confirming that past $C{O}_2$ emission reductions contribute to subsequent declines. However, economic growth (D(GDP)) does not significantly impact $C{O}_2$ emissions in the short term. This implies that the effects of GDP changes take time to materialize in $C{O}_2$ emission patterns.

D(FDI) has a significant positive short-term effect on $C{O}_2$ emissions. A one-unit increase in D(FDI) leads to a short-run increase in $C{O}_2$ emissions by 0.02. This indicates that while FDI reduces emissions in the long run, its immediate impact may be an increase due to industrial activities associated with new investments. Similarly, URB shows a delayed but marginally significant effect, reflecting the gradual nature of urbanization’s impact on emissions.

In the short run, PAs have a weak but negative coefficient for their immediate effect. A one-unit increase in PAs leads to a short-run decrease in $C{O}_2$ emissions by 0.05. However, the lagged value of PAs shows a stronger and significant negative effect (−0.05). This implies that TI takes time to yield environmental benefits, as the adoption and implementation of new technologies occur with delays.

The EKC hypothesis in the context of BRICS countries is supported by the cubic relationship between economic growth and $C{O}_2$ emissions, as indicated by the long-run results of the Panel ARDL model in Table 3. The estimated EKC equation for $C{O}_2$ emissions in BRICS countries, according to Figure 2, based on the long-term coefficients is given by Equation (9):

$$C{O}_2=58.14GDP-7.53GD{P}^2+0.32GD{P}^3-0.02FDI+1.03URB+0.09PA-0.96RENC$$

(9)

To find the turning points, we took the derivative of $C{O}_2$ with respect to GDP and set it to zero, according to Equation (10):

$$0.96GD{P}^2-15.06GDP+58.14=0$$

(10)

The two solutions are $GD{P}_1\approx 6.85$ and $GD{P}_2\approx 8.83$. To interpret these GDP values in real terms, we take the exponential, and we use approximations: $GD{P}_1\approx \$946M$, $GD{P}_2\approx \$68B$.

The two turning points of the EKC for BRICS countries in Figure 4 have distinct economic, social, and environmental implications on the relationship between $C{O}_2$ and GDP. In the initial stages of economic development, GDP growth is heavily tied to rising $C{O}_2$ emissions since countries prioritize industrialization and urbanization to boost economic performance. This phase involves extensive reliance on fossil fuels, resource-intensive industries, and minimal environmental regulations. For instance, China and India have historically experienced significant $C{O}_2$ emissions growth due to coal dependency and rapid industrialization aimed at meeting their developmental goals [27].

As economic growth continues, BRICS nations approach a turning point (GDP₁ = 6.85), as shown in Figure 4. At this stage, awareness of environmental issues rises, and economies begin to adopt cleaner technologies and energy-efficient practices. This transition is in line with findings suggesting that economic complexity and renewable energy consumption play key roles in mitigating environmental degradation as economies evolve [77]. For example, Brazil’s focus on renewable energy sources like hydroelectric power and biofuels exemplifies a shift toward sustainability. Similarly, South Africa has started diversifying its energy mix to reduce its dependence on coal. Research also indicates that policies promoting energy efficiency and innovations in the energy sector are critical to reducing emissions in this stage of economic growth [78]. This inflection represents a critical transition from unregulated growth to a more balanced approach that incorporates environmental considerations.

In the later stages of development, as GDP reaches higher levels (GDP2 = 8.83) in Figure 4, $C{O}_2$ emissions begin to decline. Economic activities shift from resource-intensive manufacturing to less carbon-intensive services and advanced technologies. Urbanization, while initially contributing to $C{O}_2$ emissions, starts to generate efficiency benefits through smart infrastructure and sustainable urban planning. This is consistent with findings suggesting that the implementation of technological innovations and urban planning contribute to emission reductions in the advanced stages of economic development [79]. For instance, China’s smart city initiatives and green urban policies reflect this transition. Moreover, FDI introduces cleaner and more efficient technologies, aiding in emission reductions, while innovation in renewable energy and green technologies becomes a driving force for sustainable growth. Studies also highlight that FDI plays a significant role in introducing cleaner technologies and fostering economic growth without compromising environmental quality [80,81].

The additional variables in this EKC provide deeper insights into its dynamics. URB positively impacts $C{O}_2$ emissions during early phases but later supports efficiency gains as economies adopt more sustainable urbanization practices [82,83,84,85]. RENC, with its strong negative impact on $C{O}_2$ emissions, highlights its critical role in achieving sustainability. For example, Brazil’s leadership in biofuels and China’s investments in solar and wind energy showcase how renewable energy drives emission reductions. These investments align with the growing body of research showing that renewable energy consumption is essential for mitigating environmental impacts [86,87,88]. Conversely, while PAs indicate innovation, their positive impact on emissions could suggest a lag between TI and widespread implementation.

The N-shaped EKC curve observed for the BRICS countries, compared to the individual behavior of each country, can be explained by the non-linear relationship identified between GDP and $C{O}_2$ emissions. This shape makes sense both economically and ecologically, and the factors influencing it may vary between each country, as well as collectively, for BRICS. In Figure 4, we observe that as the economy develops, GDP rises due to the increase in industrial activities that may raise the demand for energy, leading to an exponential rise in $C{O}_2$ emissions. As the economies mature and invest in cleaner technologies and energy efficiency, $C{O}_2$ emissions may start to decrease, even though GDP continues to rise. This is where the N-shaped curve appears and the growth of GDP no longer necessarily brings a large impact on the environment. The overall EKC curve for BRICS can show a similar trend to the individual EKC curve for each country but with an average of the behavior of all these economies. This could reflect a transition from rapid growth to a phase of stagnating $C{O}_2$ emissions as economies become “greener”. However, individual countries may be at different stages of this transition, with some having a higher level of economic development and a much steeper curve, while others are still in the early stages of development. This is the stage where developed economies, like many of the BRICS countries, are transitioning toward green economies. For example, in India, GDP is still at a lower level compared to many developed economies, as shown in Figure A10 of Appendix C. This explains why the curve for India is relatively linear at certain periods, as $C{O}_2$ emissions have not risen at an accelerated rate, and development is happening at a relatively slower pace. In contrast, China, with a much higher GDP, is rapidly undergoing industrialization and has a much steeper upward curve, as shown in Figure A11 of Appendix C. China is a large developing economy with intensive use of energy and industrial resources. Brazil and Russia exhibit different behaviors: Brazil, with an economy partially based on natural resources, and Russia, with a significant base of energy resources, follow different stages of the curve depending on their economic and political factors (Figure A8 and Figure A9 of Appendix C).

As shown in

Table 5. Panel Johansen–Fisher cointegration test results for ARDL-EKC model.

Hypothesized No. of CE(s)	Fisher Stat. (Trace Test)	Prob. (Trace Test)	Fisher Stat. (Max-Eigen Test)	Prob. (Max-Eigen Test)
None	446.3	0	160.6	0
At most 1	239.2	0	97.99	0
At most 2	169.7	0	59.54	0
At most 3	119.5	0	48.79	0
At most 4	80.46	0	29.73	0.0009
At most 5	62.25	0	37.29	0
At most 6	40.03	0	28.85	0.0013
At most 7	24.94	0	24.94	0.0055

, we performed the Johansen–Fisher panel cointegration test as a robustness test to check for the existence of long-run relationships among multiple non-stationary variables. The test results provide strong evidence in favor of cointegration among the included variables. The trace and maximum eigenvalue statistics both yield extremely low p-values (all below 0.01 for up to seven cointegrating vectors), which decisively reject the null hypothesis of no cointegration. The cointegration results enhance the credibility of the EKC hypothesis for BRICS countries by proving that the relationship is not spurious and holds consistently over the long term. This further strengthens the theoretical foundation of the panel ARDL estimations and supports the policy implications.

Next, we checked the existence of a cubic ICC by means of an ARDL model starting from the dependence relation (11):

$$C{O}_2=f(GDP, FDI, URB, PA, PA2, PA3, RENC)$$

(11)

In Equation (11), PA2 and PA3 are squared and cubic PAs, respectively.

From

Table 6. Panel ARDL (3, 2, 2, 2, 2, 2, 2, 2) for ICC model.

Variable	Coefficient	Std. Error	t-Statistic	Prob.
Long-run equation
GDP	$-$0.27	0.05	$-$4.96	0.00 ***
FDI	0.04	0.01	4.20	0.00 ***
URB	1.16	0.32	3.63	0.00 ***
PAs	32.72	5.89	5.55	0.00 ***
PA2	$-$3.64	0.62	$-$5.84	0.00 ***
PA3	0.14	0.02	6.15	0.00 ***
RENC	$-$0.29	0.04	$-$6.76	0.00 ***
Short-run equation
Cointeq01	$-$0.42	0.21	$-$2.00	0.04 *
$\mathrm{D}(C{O}_2$$(-$1))	$-$0.14	0.18	$-$0.79	0.43
$\mathrm{D}(C{O}_2$$(-$2))	0.00	0.11	0.01	0.98
D(GDP)	0.12	0.51	0.23	0.81
$\mathrm{D}\left(\mathrm{GDP}\right(-1$))	0.61	0.62	0.98	0.33
D(FDI)	0.00	0.01	0.10	0.92
$\mathrm{D}\left(\mathrm{FDI}\right(-$1))	0.01	0.00	1.59	0.11
D(URB)	$-$25.98	17.80	$-$1.46	0.14
$\mathrm{D}\left(\mathrm{URB}\right(-$1))	39.84	31.24	1.28	0.20
D(PA)	30.22	40.52	0.75	0.45
$\mathrm{D}\left(\mathrm{PA}\right(-$1))	$-$20.41	53.78	$-$0.38	0.70
D(PA2)	$-$3.41	4.55	$-$0.75	0.45
$\mathrm{D}\left(\mathrm{PA}2\right(-$1))	1.52	5.53	0.27	0.78
D(PA3)	0.13	0.17	0.75	0.45
$\mathrm{D}\left(\mathrm{PA}3\right(-$1))	$-$0.03	0.19	$-$0.17	0.86
D(RENC)	$-$0.22	0.16	$-$1.35	0.18
$\mathrm{D}\left(\mathrm{RENC}\right(-$1))	$-$0.11	0.06	$-$1.99	0.05 *
C	$-$40.98	20.44	$-$2.00	0.04 **
Validation metrics
Root MSE	0.01	Mean dependent variable		0.02
S.D. dependent var.	0.05	S.E. of regression		0.02
Akaike information criterion	$-$4.28	Sum squared residuals		0.03
Schwarz criterion	$-$2.44	Log likelihood		445.78
Hannan–Quinn criterion	$-$3.53

*, **, and ***: Significant at 10%, 5%, and 1% levels.

, one can see that a one-unit increase in GDP contributes to a long-run reduction in $C{O}_2$ emissions by 0.27. This reflects the decoupling phenomenon, where higher GDP enables cleaner technologies, investments in green infrastructure, and stricter environmental policies. For BRICS countries, sustained economic growth can support transitions toward low-carbon economies, particularly as governments focus on green energy systems.

The positive coefficient of FDI (0.04) suggests that, in the long term, FDI increases emissions. This could be due to foreign companies establishing carbon-intensive operations in BRICS countries with less stringent environmental regulations, aligning with the “pollution haven” hypothesis. To address this, BRICS nations need policies that attract green FDI while enforcing compliance with sustainable practices.

URB (1.16) is a significant long-term driver of $C{O}_2$ emissions. Rapid urban expansion increases energy demand, industrial activity, and transportation needs, leading to higher emissions. In the initial phase, at lower innovation levels, emissions rise due to industrial intensification. In the middle phase, as innovation advances, emissions decrease due to cleaner technologies and eco-friendly practices. During the advanced phase, excessive innovation leads to higher emissions, possibly due to the energy demands of advanced technologies. This relationship highlights the need to focus on green patents and ensure innovation aligns with sustainability goals.

The negative coefficient (−0.29) of RENC underscores the long-term environmental benefits of renewable energy. A higher share of renewable energy in the energy mix reduces reliance on fossil fuels, significantly lowering emissions.

BRICS countries should continue prioritizing renewable energy expansion, particularly China and India, which have high coal dependency.

ECT (COINTEQ01=−0.42) shows moderate adjustment speed, with 42% of deviations from the long-run equilibrium corrected annually. This indicates that structural changes take time but will eventually align emissions with economic and policy shifts.

The short-run impact of GDP on emissions is statistically insignificant. This suggests that immediate economic fluctuations do not significantly alter $C{O}_2$ emissions, as structural changes require time to influence environmental outcomes. Similarly, short-run changes in FDI show negligible impacts on emissions. This reflects the lagged effects of foreign investments on industrial activities and environmental degradation. The short-run impact of URB is insignificant, suggesting that the effects of population shifts and urban expansion on emissions take longer to materialize. However, rapid and unplanned urbanization may still lead to environmental pressures in the medium term.

The short-term coefficients for PAs are not significant, indicating that TI’s immediate effects on $C{O}_2$ emissions are limited. This reflects the time lag between R&D activities, technological adoption, and observable environmental impacts. In the short run, RENC has a weak but negative impact on emissions (−0.11), with some significance at lagged levels (−0.22). This shows that transitions to renewables begin reducing emissions in the short term, but the effects are not as pronounced as in the long run.

The cubic relationship between PAs and emissions indicates the existence of the ICC and is expressed in Equation (12):

$$C{O}_2=32.72PA-3.64P{A}^2+0.14P{A}^3$$

(12)

We set the derivative of $C{O}_2$ w.r.t. PA to zero to find critical points, and we obtained the quadratic Equation (13):

$$0.42P{A}^2-7.28PA+32.72=0$$

(13)

The discriminant of this quadratic curve is negative, indicating that there are no real roots. Thus, there are no turning points in the real domain for this curve.

The curve in Figure 5 illustrates the relationship between PAs and $C{O}_2$ emissions for BRICS countries based on the cubic Equation (13). When the number of PAs is low (close to zero), $C{O}_2$ emissions are relatively low and gradually increase. This suggests that innovation activities, at a small scale, contribute modestly to $C{O}_2$ emissions. As PAs increase, $C{O}_2$ emissions begin to rise more sharply, indicating a stronger contribution of innovation activities to $C{O}_2$ emissions in this range. This might be linked to increased industrial activities and energy usage associated with scaling innovation. At very high levels of PAs, $C{O}_2$ emissions grow rapidly in an almost exponential fashion. This could reflect the environmental costs of maintaining large-scale innovation ecosystems in BRICS countries. Brazil’s position on the ICC curve indicates a moderate relationship between patent applications and $C{O}_2$ emissions, reflecting relatively small-scale innovation activities and an industry less focused on energy-intensive innovations. Russia shows a significant increase in $C{O}_2$ emissions as patent applications rise, suggesting a higher interdependence between innovation and energy consumption. The country’s industrial base is more energy-intensive, and its innovation activities seem to be more closely tied to these sectors. India follows a similar trend, with $C{O}_2$ emissions increasing notably as innovation activities intensify. This highlights the rapid development of key sectors that are becoming more energy-intensive as innovation scales up. China, with an advanced innovation industry, demonstrates a rapid increase in $C{O}_2$ emissions as patent applications rise. This illustrates the environmental costs of technological and industrial development in the country, where scaling innovation has a significant impact on energy consumption and $C{O}_2$ emissions. With a relatively lower number of patent applications compared to other BRICS countries, South Africa remains on a more linear trajectory in terms of $C{O}_2$ emissions associated with innovation. While the relationship exists, the growth rate is not as steep, reflecting a more moderate measure of industrial and technological development.

Table 7. Panel Johansen–Fisher cointegration test results for ARDL-ICC model.

Hypothesized No. of CE(s)	Fisher Stat. (Trace Test)	Prob. (Trace Test)	Fisher Stat. (Max-Eigen Test)	Prob. (Max-Eigen Test)
None	303.7	0	124.3	0
At most 1	164.9	0	65.2	0
At most 2	109.6	0	43.9	0
At most 3	70.7	0	29.9	0.0009
At most 4	46.8	0	17.6	0.0616
At most 5	35.8	0.0001	19.7	0.0296
At most 6	25.5	0.0045	21.9	0.0158
At most 7	17	0.0756	17	0.0756

presents the panel-level Johansen–Fisher cointegration test results, which assess the existence of long-run equilibrium relationships among the variables included in the ICC model. Both the trace and maximum eigenvalue statistics show extremely low p-values (p < 0.01) for the first few hypothesized cointegrating relationships, indicating strong rejection of the null hypothesis of no cointegration. The results suggest that at least three, and possibly more, cointegrating relationships may exist in the panel. While the trace test supports up to six cointegrating vectors at the 1% level, the more conservative maximum eigenvalue test confirms up to three with high confidence. This supports the validity of long-run relationships among the ICC variables and justifies the inclusion of long-run coefficients in the panel ARDL framework.

5. Integration of Circular Economy Concepts into the Theoretical Framework

The theoretical framework presented offers significant parallels with the principles of a CE, which seeks to create a sustainable economic model by decoupling economic activity from environmental degradation. This section explores how the EKC and ICC frameworks align with CE objectives and how BRICS countries can leverage circular economy strategies to enhance environmental sustainability.

The EKC and ICC concepts emphasize the role of economic development and technological advancements in reducing environmental degradation. These insights resonate with CE principles, which prioritize resource efficiency and waste minimization. The N-shaped EKC found in BRICS countries suggests that while economic growth initially worsens $C{O}_2$ emissions, targeted interventions, such as CE initiatives, can reduce environmental harm. However, the secondary rise in emissions at higher GDP levels indicates that CE policies must be reinforced to avoid the rebound effects caused by unsustainable consumption. Additionally, a CE highlights the transition toward closed-loop systems, which minimize waste generation, maximize resource use, and promote recycling and reuse. These strategies can further mitigate the rise in emissions observed at higher GDP levels in the EKC, providing a more sustainable path forward.

The EKC framework highlights the transition from an initial phase of resource-intensive growth to a phase where economic activities become less environmentally harmful. This transition is central to CE strategies, particularly in BRICS countries, where rapid industrialization has led to environmental challenges. The findings suggest that policies encouraging cleaner production methods, recycling initiatives, and industrial symbiosis can help countries move from the peak of the EKC toward a sustainable development path. A shift toward product life-cycle management and sustainable manufacturing in BRICS could reduce waste and lower carbon emissions, aligning with the CE principles of extending the life of products and materials.

The ICC framework illustrates the relationship between innovation, as measured by PAs, and environmental outcomes. In the context of a CE, technological innovation should prioritize sustainable solutions such as renewable energy, waste-to-energy processes, and carbon-capture technologies. However, the findings indicate that innovation in BRICS countries currently has a mixed impact on emissions, implying that not all technological advancements are environmentally friendly. As a CE advances, the focus should be on fostering innovation that not only drives economic growth but also reduces environmental impact through sustainable technologies and practices.

The findings on FDI and URB further reinforce CE principles. While FDI reduces emissions in the long run, its short-term impact is linked to industrial expansion, potentially increasing environmental stress. Therefore, green FDI policies should be implemented to attract sustainable investment. Similarly, urbanization is identified as a key driver of $C{O}_2$ emissions, highlighting the importance of circular urban planning that incorporates smart infrastructure, green buildings, and efficient waste management systems. Urban areas should adopt circular economy models by integrating renewable energy, circular supply chains, and sustainable waste management to minimize their ecological footprint.

The findings on RENC and FDI within the theoretical framework highlight critical policy levers for promoting CE principles. Policies encouraging renewable energy adoption and attracting green FDI align with CE goals by fostering cleaner production methods and reducing dependency on non-renewable resources. Furthermore, the negative relationship between renewable energy consumption and $C{O}_2$ emissions highlights the importance of integrating renewable energy strategies into CE policies. Promoting green innovation through R&D in renewable technologies can further facilitate the transition to a CE by providing alternative, low-carbon energy sources.

The framework’s findings on URB underscore its dual role as a driver of economic growth and a source of environmental pressure. CE practices, such as sustainable urban planning, waste-to-resource systems, and circular construction methods, can mitigate these pressures. Urban centers are key to implementing CE solutions, serving as hubs for innovation and sustainable development. As urbanization continues in BRICS, the adoption of circular urban planning models is essential to ensure that cities remain sustainable, resilient, and resource-efficient.

The econometric analyses in the framework, including panel unit root tests and ARDL models, provide valuable insights into the dynamics of economic growth, innovation, and environmental impact. These methodologies can be applied to evaluate the effectiveness of CE policies in real-world scenarios, offering a data-driven approach to guide policy decisions. By integrating CE strategies into econometric models, policymakers can better assess the long-term impacts of CE practices and refine their approaches to sustainability.

6. Conclusions and Policy Recommendations

This study confirms the validity of both the EKC and Innovation Claudia Curve (ICC) hypotheses in BRICS countries, revealing complex interactions between economic growth, technological innovation, and $C{O}_2$ emissions.

The findings demonstrate that the relationship between economic growth and emissions follows an N-shaped EKC, indicating that while emissions initially rise with GDP growth, they decline after reaching a certain income threshold, only to increase again at higher GDP levels (RQ1). This suggests that although technological advancements and policy interventions contribute to reducing environmental impact, economic expansion at later stages may counteract these gains due to increased consumption and industrial activity. These results are consistent with the cubic GDP term being statistically significant and positive, confirming the third-phase reversal. The existence of an N-shaped EKC highlights the necessity for sustained regulatory efforts to ensure that the transition toward a low-carbon economy is maintained beyond the initial turning point.

Furthermore, the study provides empirical support for the ICC, illustrating a cubic relationship between patent activity and $C{O}_2$ emissions (RQ2). At lower levels of technological innovation, emissions tend to rise due to the resource-intensive nature of industrial processes. As innovation advances, emissions decline, reflecting the role of clean technologies and improved energy efficiency. However, at very high levels of patent activity, emissions increase once again, likely due to the significant energy demands of large-scale technological ecosystems. This U-shaped segment in the relationship is confirmed by the significance and sign of the second- and third-order coefficients of the PA variable in our model. This underscores the need to align technological progress with sustainability goals, ensuring that innovation efforts are directed toward renewable energy, circular production models, and low-carbon technologies rather than merely expanding industrial capacity.

The study also explores the role of foreign direct investment (FDI), renewable energy consumption (RENC), and urbanization (URB) in shaping environmental outcomes in BRICS countries (RQ3). FDI is found to have a dual effect, contributing to emissions in the short term due to industrial expansion but leading to long-term reductions through technology transfer and efficiency gains. This is supported by the short-run positive and long-run negative coefficients of FDI in our ARDL model. Renewable energy consumption emerges as a key driver of emissions reduction, reinforcing its potential to decouple economic growth from environmental degradation. Urbanization, however, presents a significant challenge, as it is associated with rising emissions due to increased energy demand, transportation, and industrialization. In our estimations, URB consistently displays a significant positive effect on $C{O}_2$ emissions across models, both in the short and long run. This highlights the urgent need for sustainable urban planning that prioritizes green infrastructure, energy-efficient housing, and low-carbon transportation solutions.

Given these findings, several policy recommendations emerge. Expanding investments in renewable energy infrastructure is imperative to reduce fossil fuel dependency and lower emissions. Governments should incentivize businesses and industries to transition toward sustainable energy sources, integrating clean technologies across sectors. In addition, strengthening environmental regulations can encourage green innovation by directing research and development efforts toward eco-friendly solutions. Stricter policies promoting sustainable patenting and tax incentives for companies investing in green technologies will be critical to ensuring that innovation contributes to environmental sustainability. Furthermore, urbanization strategies should be redesigned to incorporate circular economy principles, fostering waste reduction, smart city initiatives, and resource-efficient construction. The findings also suggest that FDI must be strategically managed, with policies aimed at attracting sustainable investments while preventing pollution-intensive industries from relocating to BRICS economies under weak environmental standards. Lastly, stabilizing economic policies is important to reduce uncertainty and encourage long-term green investments, fostering a regulatory environment that promotes low-carbon growth while ensuring economic stability. These policy directions are not generic but directly informed by our empirical results, particularly the observed dynamics in the EKC and ICC curves and the significance of FDI, RENC, and URB variables across models. Therefore, the recommendations are grounded in evidence and respond specifically to the non-linear relationships identified.

These recommendations are not general suggestions but are grounded in the empirical results of the panel ARDL analysis and the observed patterns specific to the BRICS context. The identification of N-shaped and cubic relationships in the regression results justifies differentiated strategies across development phases, linking each policy proposal to a specific turning point found in the model.

Policymakers in BRICS countries can leverage these findings by adopting targeted strategies aligned with the stages of the EKC and the ICC. First, green investment should be scaled through public–private partnerships focusing on renewable infrastructure, especially in high-emitting sectors like energy and manufacturing. Urban planning must prioritize compact, transit-oriented development and circular construction to reduce the environmental footprint of rapid urbanization. In parallel, technology incentives, such as tax credits for clean patents, green R&D grants, and fast-track approval for eco-innovations, can redirect innovation toward low-carbon pathways. Introducing carbon pricing mechanisms and green certification for incoming FDI can prevent pollution havens and ensure that investment supports sustainability goals. Also, regional cooperation among BRICS can promote green finance instruments, harmonize environmental standards, and support innovation sharing through climate technology hubs. Hence, the proposed policies are derived from and justified by the empirical findings of this study, offering country-relevant pathways to manage the trade-offs between economic expansion, technological progress, and environmental sustainability.

This study is constrained by data limitations, including inconsistencies and variations in data quality across BRICS countries. Such limitations may affect the robustness and generalizability of our findings. Moreover, discrepancies in the measurement of variables like renewable energy consumption and patent classifications introduce potential biases. The reliance on cubic EKC and ICC models assumes specific relationships between economic growth, innovation, and emissions, which may not fully capture real-world complexities. Additionally, the short-term impacts of innovation and urbanization remain less evident, suggesting the need for further investigation into immediate policy effects.

Future research should consider a disaggregated analysis to explore the sector-specific impacts of economic growth and innovation on emissions, enabling more targeted policy interventions. Integrating circular economy principles into the analysis could provide insights into how resource efficiency and waste minimization influence the EKC and ICC dynamics. Investigating consumer behavior and cultural factors would offer a deeper understanding of how societal trends shape the relationship between economic growth, innovation, and emissions. Employing advanced econometric models, including non-linear and machine learning techniques, could better capture complex relationships and feedback loops in the data. Comparative studies between BRICS and other regions, such as OECD countries, could help validate theories like the EKC and the ICC across diverse development contexts. These further research directions will provide a more comprehensive understanding of the factors driving $C{O}_2$ emissions and the pathways to sustainable development.