Equilibrium bifurcation and extreme risk in the EU carbon futures market

Abstract

Considering the long-term memory and volatility clustering of the European Union (EU) Carbon Emission Allowances (EUA) futures returns, based on the economy–energy–environment system perspective and the assumption of investors' heterogeneity, this study proposes a joint modeling approach combining the fractionally integrated generalized autoregressive conditional heteroscedasticity model (FIGARCH) and the stochastic cusp catastrophe model (SCC) to examine the equilibrium bifurcations and extreme risks in the EU carbon futures market. The relevant results are threefold. (1) The SCC model has good fitting effect and interpretability, and is an effective method for investigating catastrophe reactions under time-varying volatility conditions. (2) In the EUA futures market, chartists are mainly affected by short-term price and trading volume changes, which leads to the emergence of equilibrium bifurcations, while fundamentalists make investment decisions based on the economy, the energy market, and market supply–demand, which affects the asymmetry of equilibrium bifurcations. (3) Using the catastrophe criterion (i.e., Cardan's discriminant of the equilibrium surface equation), we identify148 equilibrium bifurcation time points in the EUA futures market from December 3, 2009 to September 16, 2020, most of which are concentrated in two upward periods with an average scale of extreme risks is about 32.51 %. Our analysis provides theoretical support for regulatory authorities to stabilize the carbon futures market and build a collaborative extreme risk management framework covering energy and macroeconomics, also proposing suggestions for traders to effectively prevent extreme risks.

1. Introduction and literature reviews

Since the 1997 Kyoto Protocol established a mechanism to reduce emissions that was applicable to countries at different levels of development, the world's carbon emissions trading system has boomed, becoming a significant environmental economic policy to control greenhouse gas emissions through the market mechanism. As a quasi-financial market, carbon price fluctuations do not evolve in a stable manner and is usually accompanied by abnormal fluctuations or extreme risks [1,2]. Compared with general risks, extreme risks occur less frequently; however, when they do occur, the consequences can be severe. First, for carbon market participants, extreme price volatility may cause significant financial risks for businesses and investors, reducing their enthusiasm for compliance and potentially suppressing investments in low-carbon technologies and sustainable projects. Second, regarding market pricing efficiency, extreme risks prevent carbon emissions rights prices from reflecting their true value, which impacts market pricing efficiency. Finally, in terms of emissions reduction policy implementation, extreme risks in the carbon market can diminish governments' confidence and capability in enforcing climate policies, with negative impacts on the overall societal goal of emissions reduction [3]. Therefore, thoroughly analyzing the price fluctuation characteristics and dynamic mechanism of the carbon trading market and accurately understanding the time points and scale of extreme risks is of considerable theoretical and practical significance.

Abnormal fluctuations and jump phenomena are two significant occurrences in financial markets, that are closely associated with extreme risks, and such occurrences in the carbon market have garnered widespread research attention. Daskalakis et al. (2009) were the first to introduce the jump-diffusion process to examine carbon emissions rights spot prices, determining that the returns series of the European Union Carbon Emission Allowances (EUA) exhibit significant jump characteristics [4]. Subsequently, relevant studies on the carbon futures market have also emerged such as Pan et al. (2022), who proposed an optimized LM jump test method to determine the direction and time points of jumps in price series of the EUA futures and discovered that significant information shocks can cause substantial and asymmetric continuous jumps [5]. Guo et al. (2022) conducted volatility predictions of EUA carbon futures price series, finding that GARCH-MIDAS-JUMP and GARCH-MIDAS-JUMP-LJ models that were constructed by introducing short and long-term jump components into the basic GARCH-MIDAS model had better predictive performance and demonstrated that jump characteristics have a significant impact on carbon futures price volatility [6]. Bei et al. (2021) introduced an ARJI class model to describe jump processes with dynamic, time-varying characteristics to investigate the jump problem in EUA futures prices, and found that the ARJI-GARCH, $\text{ARJI}-R_{t-1}^2$, and $\text{ARJI}-h_t$ models further improved the fitting effect of the model with constant jump parameters [7]. Abnormal fluctuations and extreme risks in the carbon market primarily stem from traders' complex practices in the market such as speculation, heterogeneous assumptions, herd behavior, and other phenomena. Several studies have examined the complex interconnections of extreme risks between the carbon market and other markets such as stocks and energy. Liu et al. (2023) estimated time-varying skewness by constructing a GARCH-S model as a proxy variable of extreme risk to examine the spillover effects of extreme risks in China's carbon and stock markets. The two-way spillover effect revealed between carbon and industrial stock markets confirmed that the carbon market is affected by other markets in the evolution of extreme risks [8]. Du et al. (2022) examined the extreme risks of EU ETS carbon futures, Brent crude oil futures, and the S&P 500 index based on the value at risk (VaR) method and a time-varying spillover index model (TVP-VaR-DY), revealing significant extreme risk spillover effects among the three markets [9]. Yuan and Yang (2020) studied the extreme risk spillover between the EU carbon market, financial market, and crude oil market using the generalized autoregressive scored dynamic conditional copula (GAS-DCS-copula) model, finding that the financial market had a more evident extreme risk spillover effect on the carbon market compared with the oil market [10]. These studies demonstrated that introducing jumps in the continuous diffusion process can reveal abnormal fluctuations and extreme risks more accurately. In terms of generation mechanism, the studies also confirmed that abnormal fluctuations and extreme risks are attributable to the endogenous mechanism of the market and are influenced by complex risk transmission mechanisms between different markets. Examination of the inherent characteristics of a system based on catastrophe theory provides a novel perspective for investigating extreme risks in financial markets. Catastrophe theory examines how continuous, small control variables changes in a system lead to noncontinuous, “sudden” changes in state variables [[11], [12], [13]]. With the widespread application of catastrophe theory in deterministic dynamic systems, Cobb et al. extended the catastrophe theory of deterministic dynamic systems to stochastic dynamic systems from 1980 to 1985, proposing statistical catastrophe theory [14,15]. In the field of statistical catastrophe theory, the most widely applied technique is statistical cusp catastrophe (SCC) model. Cobb et al. determined the stationary density function of the SCC model and established a robust and consistent parameter estimation method [16,17]. Representative studies using the SCC model for financial markets include Barunik et al. (2015, 2009), who constructed SCC models through returns that were adjusted by realized volatility and short-time window modeling respectively to examine the catastrophe behaviors of the US stock market. The results demonstrated that these modeling strategies can capture the catastrophe characteristics of the market and explain stock market crashes more accurately [18,19]. Lin (2016) applied Barunik's time-varying volatility estimation method, using the SCC model to study the catastrophe characteristics of China's stock market. The results revealed that this model has application potential for describing the differentiated characteristics of market trading behaviors and market sentiment in special periods [20]. However, no research has applied statistical catastrophe theory to examine the carbon emissions trading market.

Compared with existing studies on jump phenomena and extreme risk spillover effects between different markets, the SCC model has three notable advantages. First, the SCC model has multiple equilibria, heterogeneous expectations, and nonlinear stochastic characteristics, breaking through the assumptions of homogeneous expectations, single equilibrium, and linear trajectories in existing research. Second, the SCC model can be used to explain extreme risks through bifurcation of the stable equilibrium state of the system state variable, which fits the inherent evolutionary laws of complex systems and has significant theoretical advantages. Finally, the SCC model can incorporate heterogeneous investor assumptions and different markets’ impact into a unified framework to examine market crashes and extreme risks. However, certain shortcomings in the research on the application of the SCC model in financial markets or carbon markets remain. First, the system state variable should satisfy the assumption of constant volatility to obtain a consistent estimate of the parameters. Although Barunik et al. (2015) proposed a creative approach to address this limitation by adjusting returns through realized volatility, the modeling method to resolve the assumption requires further expansion. Second, current studies remain limited in the selection of control variables to identify indicators within a single market, neglecting the impact of other related markets on extreme risks. Finally, further examination of the time points and scale of extreme risk occurrences through catastrophe criteria and multiequilibrium states are lacking.

Considering issues above, this study draws on Barunik et al.’s (2015) SCC modeling approach by combining the advantages of the classic fractionally integrated generalized autoregressive conditional heteroskedasticity (FIGARCH) model to estimate time-varying volatility when financial asset returns exhibit long memory and volatility clustering. We propose this combination modeling strategy to investigate equilibrium bifurcations and extreme risks in the EUA futures market. The primary contributions of this study are as follows. First, the study employs the FIGARCH model to measure the time-varying volatility of EUA carbon futures returns and constructs state variables using the returns adjusted by volatility as state factors, expanding the modeling method to solve the assumption of constant state variable volatility. Second, we retain the heterogeneous investor hypothesis and construct control variables within the complex economy–energy–environment system, introducing economic and energy subsystems' impact on the carbon market into the SCC model, enriching the literature examining the catastrophe behaviors of the carbon market using catastrophe theory. Third, the study explores the application of the SCC model catastrophe criteria, using it to measure equilibrium bifurcation points and the scale of extreme risks of the EUA futures market, providing a novel method for measuring extreme risks.

The remainder of this paper is arranged as follows. Section 2 introduces the theoretical framework and research hypotheses for the SCC model. Section 3 presents the research design, including variable selection, empirical models, and parameter estimation strategies. Section 4 fits and tests the SCC model using the full sample data and the rolling window method. Section 5 further employs catastrophe criteria to measure equilibrium bifurcation time points and extreme risks scale of the EUA futures market. Section 6 conducts robustness tests; and section 7 presents conclusions.

2. Theoretical framework and research hypotheses

2.1. Theoretical framework

Catastrophe theory is a basic theory for examining the bifurcation, singularity, and structural stability of complex dynamic systems. The theory focuses on the evolution of state variables' behaviors in the system under the continuous action of control variables. Referencing Cobb's statistical catastrophe theory, the stochastic differential equation of the SCC model can be expressed as follows [14,15]:

$$\mathrm{d}{y}_t=-\frac{\mathrm{d}V\left(y_t;{\alpha }_x,{\beta }_x\right)}{\mathrm{d}{y}_t}\mathrm{d}t+{\sigma }_{y_t}\mathrm{d}{W}_t$$ (1)

where the drift term $-\left[\mathrm{d}V\left(y_t;{\alpha }_x,{\beta }_x\right)/\mathrm{d}{y}_t\right]\mathrm{d}t$ preserves the dynamic properties of deterministic systems; that is, the internal mechanism by which the system state variable $y_t$ moves toward an equilibrium position under the action of the potential function $V\left(y_t;{\alpha }_x,{\beta }_x\right)$. ${\sigma }_{y_t}\mathrm{d}{W}_t$ is the diffusion term of the random process, W_t is the Wiener process, and ${\sigma }_{y_t}$ is the diffusion function. Introducing the diffusion term characterizes the inherent randomness of the system. The SCC model is extended to stochastic dynamic systems.

In Equation (1), the potential function is determined by the state variable $y_t$ and the control variables ${\alpha }_x$, ${\beta }_x$ as follows:

$$V\left(y_t;{\alpha }_x,{\beta }_x\right)=\left(-1/4\right)y_t^4+\left({\beta }_x/2\right)y_t^2+{\alpha }_x{y}_t$$ (2)

where if $-\mathrm{d}V\left(y_t;{\alpha }_x,{\beta }_x\right)/\mathrm{d}{y}_t=0$, then the potential function is minimum, the system state variable is in equilibrium. The equilibrium surface equation of the state variable can be obtained from Equation (2) as follows:

$$-\frac{\mathrm{d}V\left(y_t;{\alpha }_x,{\beta }_x\right)}{\mathrm{d}{y}_t}=-y^3+{\beta }_{x}y+{\alpha }_x=0$$ (3)

The equation of equilibrium surface is a cubic polynomial of the state variable, and the Cardin discriminant of Equation (3) is as follows:

$$\Delta =\frac{1}{4}{\alpha }_x^2-\frac{1}{27}{\beta }_x^3$$ (4)

which is also known as the catastrophe criterion. The control variables ${\alpha }_x$ and ${\beta }_x$ constitute the control space that determines whether the potential function incurs a catastrophe. ${\alpha }_x$ is called the asymmetric factor, and ${\beta }_x$ is the bifurcation factor. When $\Delta >0$, the equilibrium equation exhibits one real root. Corresponding to the non-folding region of the equilibrium surface in Fig. 1(a), in the control space (a two-dimensional plane of ${\alpha }_x$ and ${\beta }_x$), the control variables are located in the larger region outside the sharp point polyline. When $\Delta <0$, the equilibrium equation has three real roots. Corresponding to the folded region of the equilibrium surface in Fig. 1(a), the upper and lower parts of the region are the upper and lower lobes, respectively, representing the stable equilibrium points (i.e., the Morse critical points of the potential function, $y_s$ in Fig. 1(a)). The middle section of the folded region, the middle lobe, presents the degenerate critical points (i.e., the non-Morse critical points of the potential function, $y_{ns}$ in Fig. 1(a)), which are unstable equilibrium positions. Therefore, the folded state of the equilibrium surface indicates that the state variable produces two stable equilibrium states under the influence of the control variable. At these points, the control variables are located in the cusp region of the control space. In particular, the boundary of the cusp region is defined by Δ = 0.

Fig. 1.

(a) Equilibrium surface and control space of the stochastic cusp catastrophe model (left).Fig. 1(b)Control space and conditional density function (right).

Note: In Fig. 1(a), $y_s$ are located on the upper and lower lobes, representing stable equilibrium positions. $y_{ns}$ is located in the middle of the lobe, which is an unstable equilibrium position.

The catastrophe criterion provides a basic tool for locating multiple equilibrium positions. When the control variable satisfies the condition Δ < 0, the system exhibits equilibrium bifurcations. In a stochastic dynamic system, state variables fluctuate around the equilibrium surface. Let the two stable solutions of the equilibrium surface equation be $y_{s1}$ and $y_{s2}$. The expected catastrophe scale is as Equation (5):

$$CP=\max \left(y_{s1},y_{s2}\right)-\min \left(y_{s1},y_{s2}\right)$$ (5)

In practice, the state and control variables of the SCC model are generally obtained by smooth transformations of system state factors $z_{i,t}$ and independent variables $x_{i,t}$.

$$\{\begin{array}{c}{y}_t={\omega }_0+{\omega }_1{z}_{1,\mathrm{t}}+{\omega }_2{z}_{2,\mathrm{t}}+\cdots +{\omega }_l{x}_{l,t}\\ {\alpha }_x={\alpha }_0+{\alpha }_1{x}_{1,\mathrm{t}}+{\alpha }_2{x}_{2,t}+\cdots +{\alpha }_m{x}_{m,t}\\ {\beta }_x={\beta }_0+{\beta }_1{x}_{1,t}+{\beta }_2{x}_{2,t}+\cdots +{\beta }_n{x}_{n,t}\end{array}$$ (6)

According to Cobb's statistical catastrophe theory, when ${\sigma }_{y_t}=\sigma$, indicating that the diffusion function is constant, and the current measurement scale is not to be nonlinearly transformed, the state variable $y_t$ has the following stable conditional density function [14,18,19]:

$$f_s\left(y_t|x_{1,t},x_{2,t},\cdots \right)=\psi \mathit{exp}\left[\frac{\left(-1/4\right)y_t^4+\left({\beta }_x/2\right)y_t^2+{\alpha }_x{y}_t}{\sigma }\right]$$ (7)

where $\psi$ is a standardized parameter, ensuring that the integral value between the density function and the horizontal axis equals 1. The density function has a configurational correspondence with the potential function in the catastrophe model. As shown in Fig. 1(b), the density function illustrates a bimodal feature in the cusp region, and the symmetry of the bimodal state is primarily determined by the asymmetric factor. The density function retains the unimodal distribution characteristic of the exponential distribution family outside the cusp region. Based on Equation (7), Cobb proposed a maximum likelihood method for parameter estimation (MLE) of the SCC model [15].

2.2. Research hypotheses

The carbon market has the general characteristics of a financial market but also has typical anomalies that traditional financial theory cannot explain. These anomalies have primarily been attributed to the nonlinear properties of the carbon trading system. Yang et al. (2020, 2017) examined the price behaviors of the EUA, confirming that the market is a nonlinear dynamic system with fractal and chaotic characteristics [21,22]. Within the framework of a nonlinear system, the carbon market does not evolve in isolation and will exchange information, energy, and material with related systems in various forms. The carbon market has an open and dissipative structure. As indicated in section 1, the relationships between the energy market, economic systems, and the carbon market are particularly close; therefore, this study focuses on the complex economy–energy–environment system, considering the carbon market as a subsystem within it. Additionally, as a policy-driven market, the carbon market's extreme risks are subjected to impact from numerous external factors, such as economic policy uncertainty, climate policy uncertainty, and geopolitical risks. These factors represent external influences on the entire economy–energy–environment system [23,24]. Consequently, when examining the occurrence of catastrophe within the carbon market subsystem, such factors are omitted. Based on the above analysis, this study proposes research hypothesis 1.

Hypothesis 1

Considering the complex dynamic correlation between the carbon market and energy economic systems, the carbon emissions trading system is considered to be a subsystem of the complex economy–energy–environment system, and its catastrophe responses will be affected by economic and energy system variables.

Carbon market returns have long-term memory and multi-scale statistical self-similarity (with fractal characteristics) [25]. This means that the synchronicity of information transmission and the assumption of investors' homogeneous expectations do not match the reality of carbon markets. In related studies, the basic hypothesis of Zeeman et al. (1976) has generally been retained. Two main types of investors are involved in financial markets; namely, fundamentalists and chartists. Fundamentalists focus more on the intrinsic value of financial assets, which determines the price demand in the market and represents changes in asymmetric control variables. Chartists predominantly engage in short-term arbitrage activities, paying attention to short-term price fluctuations. They simultaneously inject a certain amount of liquidity into the market, which can lead to market instability that reflects the changes of market bifurcation variables [26]. Based on Zeeman's basic hypothesis, this study proposes hypothesis 2.

Hypothesis 2

Chartists in the carbon market are primarily affected by short-term volume and price changes within the market, which generates equilibrium bifurcations. Fundamentalists make investment decisions based on energy markets, the functioning of the economy, and market supply and demand, predominantly affecting asymmetric factors.

3. Research design

3.1. Variable selection and data sources

This study uses the settlement prices of the EUA futures contracts and continuous trading volumes (Fig. 2) as the sample for constructing the state factors to be adjusted and the independent variables of the carbon futures subsystem. In particular, considering that chartists primarily make trading decisions based on short-term price changes, referencing Lin Li (2016) [20] and Jiang et al. (2017) [27], we introduce two-day price momentum into the independent variables of the carbon futures subsystem. The exogenous variables for the carbon futures subsystem include energy market and economic indicators. The relationship between the carbon market and the fossil energy market is closest, with high correlations between different fossil energy prices. Referencing Ji et al. (2018) [28] and Chevalier (2019) [29], this study uses the returns of Brent crude oil futures settlement price to represent energy market indicators. The economic indicator is FTSE 100 returns, which is representative of the EU region. All data were sourced from the WIND database, with a sample size of 3030 observations from June 3, 2009 to March 25, 2021 (Table 1). Considering the characteristics of the model, we winsorized the variables’ tails by 1 % and 99 % to eliminate the influence of extreme values. We execute data processing and modeling using Stata15 and MATLAB R2016b.

Fig. 2.

EUA futures' prices and volumes.

Table 1.

Variable descriptions.

Variable types		Variable names	Variable descriptions
State factor to be adjusted $r_t$		Returns of EUA futures	The logarithmic returns of the settlement price of European Climate Exchange (ECX) EUA futures.
Independent variables of Carbon futures subsystem	${\mathrm{d}V}_t$	Changes in EUA futures trading volume	The logarithmic difference in trading volumes of ECX EUA futures.
	$M_2$	Two-day price momentums	The settlement of ECX EUA futures price changes over two trading days, i.e., $M_2=\left(p_t-p_{t-2}\right)/p_{t-2}$, which is highly sensitive to real-time market information and short-term market sentiment.
Exogenous variables of the carbon futures subsystem	${\mathrm{d}Brent}_t$	Returns of Brent crude oil futures	The logarithmic returns of the settlement prices for Brent crude oil futures of the Intercontinental Exchange. Brent crude oil serves as one of the significant benchmark international oil prices, particularly in regions such as Europe.
	${\mathrm{d}Ftse}_t$	Returns of FTSE 100 Index	The logarithmic FTSE 100 returns index of the London Stock Exchange. This index is a significant indicator of the economic condition in the European region.

3.2. Constructing the state factor under time-varying volatility

The state variable should be determined prior to constructing the SCC. According to Equation (6), the state variable is usually a smooth transformation of the state factor. Referencing Barunik et al. (2015) [18], we use the returns adjusted by time-varying volatility as the state factor as follows:

$$z_t=\frac{r_t-{\hat{\mu }}_t}{{\hat{\sigma }}_t}$$ (8)

where $r_t$ is the returns of EUA futures. ${\hat{\mu }}_t$ and ${\hat{\sigma }}_t$ are the estimates of conditional mean and conditional heteroskedasticity, respectively. For the carbon futures market, Equation (8) can be regarded as a proxy measure of excess returns per risk. A standardized residuals sequence ($z_t$) usually has constant volatility, satisfying SCC model condition (i.e., ${\sigma }_{y_t}$ is a constant). Based on the volatility clustering and long-term memory characteristics of carbon futures returns [30,31], the study applies the FIGARCH (p, d, q) model proposed by Baillie et al. (1996) to measure time-varying volatility. Introducing the integral order d controls the autocorrelation function of the conditional heteroskedasticity series to decay in the form of double curvature to describe the long-term memory characteristics of finance time series more accurately [32,33]. In practice, the FIGARCH (1, d, 1) model with a simple structure is usually adopted as Equation (9):

$$\{\begin{array}{c}{r}_t={\mu }_t+u_t\\ u_t={\sigma }_t{z}_t\\ {\sigma }_t^2=Var\left(r_t|F_{t-1}\right)=\omega +\left[1-\beta L-\left(1-\phi L\right){\left(1-L\right)}^d\right]u_t^2+\beta {\sigma }_{t-1}^2\end{array}$$ (9)

where L is the lag operator, and the constraints of FIGARCH (1, d, 1) parameters are $0<\mathrm{\omega },0\le \beta \le \phi +d,0\le d\le 1-2\phi$. According to Bollerslev and Mikkelsen (1996), the conditional variance of FIGARCH (1, d, 1) can be expressed in the following ARCH ($\infty$) form [34]:

$${\sigma }_t^2=\omega /\left(1-\beta \right)+{\sum }_{j=1}^{\infty }{\lambda }_j{u}_{t-j}^2$$ (10)

where ${\lambda }_j$ represents the weights of FIGARCH. To improve the algorithm speed, this study adopts the fast fourier transform algorithm proposed by Nielsen et al. (2020) to calculate weights of Equation (10) [35].

FIGARCH model parameter estimations cannot use the traditional MLE method because MLE usually assumes that $z_t$ follows a particular distribution, such as normal distribution, t distribution, or Skew-t distribution. In this study, $z_t$ is taken as the state factor. We assume that the linear function $y_t$ (state variable) of $z_t$ follows the distribution defined by Equation (7). FIGARCH parameter estimations adopt the nonlinear least squares method (NLS), and parameter estimates are obtained when the criterion function $\mathrm{S}\left(\hat{\mathit{b}}\right)=\frac{1}{2}{\sum }_{t=1}^T{\left[{\left(r_t-{\hat{\mu }}_t\right)}^2-{\hat{\sigma }}_t^2\right]}^2$ reaches its minimum. The parameter estimation results can ensure that the model best fits the sample data.

3.3. SCC model considering the influences of economy and energy systems

Under the economy–energy–environment system framework, as the external influencing factors of the carbon market subsystem, economic and energy subsystems primarily affect the asymmetric control factor ${\alpha }_x$, and time lags may occur in information transfer between subsystems. In this study, we include lagged variables of the economy and energy market in the model. In addition to market prices, volume changes are also significant market information that reflects the whole market's supply and demand circumstances, and can also provide trade reference for chartists. Therefore, volume changes exert a common influence on ${\alpha }_x$ and ${\beta }_x$. In addition, short-term price changes within the carbon market can significantly affect chartists' trading strategies, which is an important component variable of the bifurcation factor. In summary, the SCC model constructed in this study is as follows:

$$\{\begin{array}{c}\mathrm{d}{y}_t=\left(-y_t^3+{\beta }_x{y}_t+{\alpha }_x\right)\mathrm{d}t+\mathrm{d}{W}_t\\ y_t={\omega }_0+{\omega }_1{z}_t\\ {\alpha }_x={\alpha }_0+{\alpha }_1\mathrm{d}{V}_t+{\sum }_{i=0}^p{\gamma }_i\mathrm{d}{Brent}_{t-i}+{\sum }_{j=0}^q{\rho }_j\mathrm{d}{Ftse}_{t-j}\\ {\beta }_x={\beta }_0+{\beta }_1\mathrm{d}{V}_t+{\beta }_2{M}_2\end{array}$$ (11)

where $y_t$ is the state variable, which is obtained by smooth transformation of state factor $z_t$. In this study, we use the MLE method proposed by Cobb et al. (1980) and Grasman et al. (2009) to estimate parameters $\left[{\hat{\omega }}_0,{\hat{\omega }}_1;{\hat{\alpha }}_0,{\hat{\alpha }}_1,\widehat{{\gamma }_i},\widehat{{\rho }_j};{\hat{\beta }}_0,{\hat{\beta }}_1,{\hat{\beta }}_2\right]$ of the SCC model. We determine the lagged order of economic and energy market indicators in the model by employing Bayesian information criterion (BIC). Referencing Barunik et al. (2015) and Lin (2016), we compare the BIC of the SCC model with the information criteria of general linear regression and nonlinear logistic models to evaluate the explanatory power of the model [18,20]. The nonlinear logistic model is as follows:

$$y_t=\frac{1}{1+\mathit{exp}\left(-{\alpha }_x/{\beta }_x^2\right)}+{\epsilon }_t$$ (12)

The meaning of the variables in Equation (12) remains the same as previous models. We compare the pseudo-R² of the SCC model with the goodness of fit of linear regression and nonlinear logistic models to evaluate the models’ fitting effect.

$$\text{pseudo}-R^2=1-\frac{{\sum }_{t=1}^T{\hat{e}}_t^2}{{\sum }_{t=0}^T{\left(y_t-{\bar{y}}_t\right)}^2}$$ (13)

where ${\hat{e}}_t$ is the residuals estimated by the model. The $\text{pseudo}$-R² of Equation (13) can approximately represent the proportion of the data fluctuations explained by the model in relation to the original data fluctuations.

4. Empirical results and analysis

4.1. Descriptive statistics

Table (2)reveals that all variables have significant leptokurtosis and fat tail characteristics. The Ljung-Box Q (20) statistic of EUA futures returns rejects the null hypothesis at the 1 % level, indicating that the series has a clear long-term correlation. The PP test results show that all variables are stationary series. The ARCH-LM test of EUA futures returns rejects the null hypothesis at the 5 % level, indicating that the series has conditional heteroskedasticity.

Table 2.

Variables’ descriptive statistics.

Variables	Mean	Min	Max	SD	Skewness	Kurtosis	Q (20)	PP (8)	ARCH-LM(8)
$r_t$	0.0004	$-$ 0.0875	0.0827	0.0284	$-$ 0.1087	4.1910	65.2933***	$-$ 56.6889***	431.5106***
${\mathrm{d}V}_t$	0.0003	$-$ 1.2900	1.3823	0.4614	0.1454	3.6432	458.4244***	$-$ 100.4978***	251.7253***
$M_2$	0.0017	$-$ 0.1180	0.1248	0.0398	0.0562	4.3024	724.3997***	$-$ 31.7239***	550.1706***
${\mathrm{d}Brent}_t$	0.0001	$-$ 0.0620	0.0578	0.0195	$-$ 0.1651	4.3373	19.0027	$-$ 55.9959***	466.2315***
${\mathrm{d}Ftse}_t$	0.0002	$-$ 0.0304	0.0276	0.0097	$-$ 0.2008	4.1728	36.7929**	$-$ 52.3566***	501.0309***

Note: ** and *** indicate significance at 5 % and 1 % levels, respectively. Q (20) is the Ljung-Box Q statistic with a lagged order of 20. Considering the heteroskedastic characteristics of the data, we adopt a PP stability test adjusted by the heteroskedasticity and the Newey-West default lagged order.

4.2. Measurement results of carbon futures market state factor

In this study, when NLS is used to estimate the FIGARCH (1, d, 1) model, we employ an adaptive differential evolution with external archive (JADE) and quasi-Newton algorithm (QN) to cross-check the accuracy of the results. JADE is an efficient swarm intelligence optimization algorithm that can solve complex optimization problems through initialization, mutation, crossover, and selection operations [36]. The algorithm uses a new mutation strategy, DE/current-to-Pbest, to automatically adjust the control parameters during the iteration process for improved search quality that has strong robustness and global optimization capabilities and has been widely used in high-dimensional function optimization problems and parameter estimations of GARCH class models [37,38]. When the FIGARCH (1, d, 1) model is used, the truncation parameter is set to 1000, the population size of JADE is 300, $c=0.1,Pbest=0.1,{\mu }_{CR}=0.5,{\mu }_F=0.5$, and the maximum number of iterations is 100. The results obtained are presented in Table 3.

Table 3.

Estimation results of FIGARCH (1, d, 1) model.

Parameters/statistics	FIGARCH (MLE-Norm)	FIGARCH (MLE-T)	FIGARCH (MLE-Skew-t)	FIGARCH (NLS-JADE)	FIGARCH (NLS-QN)
$\mu$ ( × 10−4)	9.3765** (3.8140)	7.8262* (4.1285)	8.1932* (4.5486)	−5.8015 (20.3651)	−5.7930 (20.3600)
$\omega$ ( × 10−5)	2.3431 (1.7191)	1.4394 (9.4070)	2.0152*** (6.6482)	5.3110** (2.5040)	5.3019** (2.5019)
$\phi$	0.1014 (0.1882)	0.1520 (0.1141)	0.0667 (0.0547)	0.0525 (0.0900)	0.0525 (0.0899)
$d$	0.4092*** (0.0705)	0.5041*** (0.0938)	0.4589*** (0.0467)	0.3427*** (0.0339)	0.3429*** (0.0339)
$\beta$	0.4228* (0.2374)	0.5604*** (0.1786)	0.4380*** (0.0804)	0.2807*** (0.1083)	0.2809*** (0.1082)
$\upsilon$	–	9.0244*** (1.2636)	9.1046*** (1.2398)	–	–
$\lambda$	–	–	0.0112 (0.0102)	–	–
RMSE ( × 10−3)	1.3253	1.3272	1.3263	1.3210	1.3210
Q(20)	25.6216	24.8332	24.9378	27.5145	27.5092
Q2(20)	15.7451	16.8267	15.4977	19.2262	19.1989
ARCH-LM (8)	10.6196	13.2673	11.6238	8.1936	8.1898

Note: *, **, and *** indicate significance at 1 %, 5 %, and 10 % levels, respectively. Standard errors for parameter estimates are in parentheses. Norm, T, GED, and Skew-t represent the normal distribution, t distribution, and Skew-t distribution hypotheses in MLE estimation. $\upsilon$ is the degree of freedom. $\lambda$ is the coefficient of skewness. The numerical algorithm of MLE estimation is a quasi-Newton algorithm. Q (20) and Q² (20) are the Ljung-Box Q statistics of standardized residuals and the square of standardized residuals with a 20 order lag, respectively. ARCH-LM (8) is the heteroskedasticity test of standardized residuals. We calculate the hysteresis order using the Newey-West method.

Table 3 shows that the NLS estimates using JADE and QN algorithms are very close and have the smallest root mean squared errors (RMSE). The NLS estimation does not depend on the distribution assumptions of standardized residuals and has good numerical stability and high fitting accuracy. The Ljung-Box Q test of the standardized residuals reveals no long-term correlation. The Ljung-Box Q test for the square of standardized residuals and the ARCH-LM test for standardized residuals do not reject the null hypothesis. These findings indicate that the standardized residuals have no heteroskedasticity. Using standardized residuals as the state factor of the carbon futures market can meet the conditional limitations of the SCC model for ${\sigma }_{y_t}=\sigma$. Fig. 3 presents the FIGARCH (NLS-JADE) estimation results for EUA futures returns ($r_t$), conditional heteroskedasticity series (${\hat{\sigma }}_t^2$), and state factor ($z_t$). $r_t$ has obvious volatility clustering and time-varying variance structures however, the standardized residuals ($z_t$) adjusted for conditional heteroskedasticity exhibit no obvious heteroskedasticity.

Fig. 3.

Returns of EUA futures prices (top), conditional heteroskedasticity series (middle), and adjusted returns (bottom).

4.3. Full sample fitting of SCC model

We apply BIC to determine the order of Equation (11). As shown in Fig. 4, when $p=1,q=1$, the model has a minimum BIC value. The form of the asymmetric factor is determined to be ${\alpha }_x={\alpha }_0+{\alpha }_1\mathrm{d}{V}_t+{\gamma }_0\mathrm{d}{Brent}_t+{\gamma }_1\mathrm{d}{Brent}_{t-1}+{\rho }_0\mathrm{d}{Ftse}_t+{\rho }_1\mathrm{d}{Ftse}_{t-1}$.

Fig. 4.

Lag order determination results from the stochastic cusp catastrophe model.

After determining the form of the model, we fit the full sample data. In addition, we construct the other three SCC models for comparison with Model (11) (i.e., $r_t$ is taken as the state factor, set to ${\gamma }_0={\gamma }_1=0$ and ${\rho }_0={\rho }_1=0$). The results are presented in Table 4.

Table 4.

Full sample estimation results of stochastic cusp catastrophe models.

SCC parameters/statistics	Unconstrained model with $r_t$ as the state factor	$z_t$ as the state factor
		Unconstrained model (Model 11)	${\gamma }_0={\gamma }_1=0$	${\rho }_0={\rho }_1=0$
${\mathit{\alpha }}_{\mathbf{0}}$ [Intercept]	4.9991*** (0.3471)	−4.9992*** (0.1751)	−4.9992*** (0.4237)	−4.9992*** (0.4227)
${\mathit{\alpha }}_{\mathbf{1}}$ [$\mathrm{d}{V}_t$]	−0.5484*** (0.1567)	0.7758*** (0.0446)	0.8059*** (0.0415)	0.7943*** (0.0456)
${\mathit{\gamma }}_{\mathbf{0}}$ [$\mathrm{d}{Brent}_t$]	0.1798*** (0.0461)	0.1701*** (0.0352)	–	0.2199*** (0.0337)
${\mathit{\gamma }}_{\mathbf{1}}$ [$\mathrm{d}{Brent}_{t-1}$]	−0.2828 (0.3058)	−0.1846*** (0.0280)	–	−0.2149*** (0.0290)
${\mathit{\rho }}_{\mathbf{0}}$ [$\mathrm{d}{Ftse}_t$]	0.1698 (0.1708)	0.1503*** (0.0544)	0.1985*** (0.0178)	–
${\mathit{\rho }}_{\mathbf{1}}$ [$\mathrm{d}{Ftse}_{t-1}$]	−0.2091 (0.3249)	−0.1266*** (0.0348)	−0.1802*** (0.0378)	–
${\mathit{\beta }}_{\mathbf{0}}$ [Intercept]	4.9990*** (0.9120)	−2.0232*** (0.1563)	−2.0558*** (0.1131)	−2.0239*** (0.2215)
${\mathit{\beta }}_{\mathbf{1}}$ [$\mathrm{d}{V}_t$]	0.2390*** (0.0509)	0.7010*** (0.0305)	0.7290*** (0.0360)	0.7182*** (0.0362)
${\mathit{\beta }}_{\mathbf{2}}$ [$\mathrm{d}{M}_2$]	0.8154*** (0.1414)	−1.0947*** (0.0302)	−1.0882*** (0.0277)	−1.0886*** (0.0264)
${\mathit{\omega }}_{\mathbf{0}}$ [Intercept]	2.5861*** (0.1376)	−1.2734*** (0.0222)	−1.2691*** (0.0737)	−1.2738*** (0.0508)
${\mathit{\omega }}_{\mathbf{1}}$ [$r_{t}or{z}_t$]	0.2023*** (0.0099)	0.4231*** (0.0113)	0.4193*** (0.0112)	0.4208*** (0.0125)
LL	−3348	−3330	−3369	−3355
${pseudo-R}_{SCC}^2$	0.4760	0.4961	0.4813	0.4868
$R_{Linear}^2$	0.4723	0.4631	0.4458	0.4537
$R_{Logistic}^2$	0.4888	0.4937	0.4822	0.4834
${BIC}_{SCC}$	6784	6754	6811	6782
${BIC}_{Linear}$	11756	9457	9526	9487
${BIC}_{Logistic}$	9902	7606	7657	7651
F-test	–	–	16.2476***	10.6864***
LR-test	–	–	72.7249***	43.7944***

Note: *** indicates that parameter estimates are significant at 1 % level. Standard errors are in parentheses. Both F-test and LR-test are tests for nonlinear constraints, $F=\left[\left(R_r^2-R_u^2\right)/J\right]/\left[R_u^2/\left(T-K\right)\right]\dot{\backsim }F\left(J,T-K\right),LR=-2\left({LL}_r-{LL}_u\right)\dot{\backsim }{\mathrm{\chi }}^2\left(J\right)$ that are used to verify whether the constraint conditions are valid; J is the number of constraint conditions, and K is the number of unconstrained parameters. $R_r^2$ and $R_u^2$ are the respective $pseudo-R^2$ of the constrained and unconstrained models. ${LL}_r$ and ${LL}_u$ are logarithmic likelihood values of the constrained and unconstrained models.

Table 4 reveals four notable findings. (1) The pseudo-R² of the SCC model constructed with $z_t$ as the state factor is 0.4961 and the BIC value is 6754. The fitting effect is better than that of the SCC model with $r_t$ as the state factor, alleviating the influence of the time-varying volatility of state variables to some extent. (2) The unconstrained SCC model constructed with $z_t$ as the state factor has a higher pseudo-R² and a lower BIC value. Both are superior to the linear regression model and the nonlinear logistic model with the characteristics of pseudo-catastrophe. The SCC model can describe the dynamic characteristics of the carbon futures market more accurately, and state variables may exhibit equilibrium bifurcation behaviors at certain time points. (3) The parameter estimations of the SCC model constructed with $z_t$ as the state factor are all significant at the 1 % level, indicating that variable selection is reasonable. Both the F-test and LR-test of model constraint reject the null hypothesis that the constraint condition equals 0. The impact of the economy and energy markets cannot be ruled out, and hypothesis 1 is plausible. Changes in the trading volumes of carbon futures will affect bifurcation and asymmetry factors; market signals that influence the investment decisions of both fundamentalists and chartists alike. In addition, the two-day price momentum is also one of the main reasons affecting bifurcation effects, verifying hypothesis 2.

4.4. Rolling window estimations of stochastic cusp catastrophe model

Referencing the research methodologies of [18,20], we introduce rolling window analysis, which sets a window width using the samples within the window to fit the SCC model, and then continuously scrolling forward with a certain step width until the entire sample time is traversed. This process has two main purposes. First, it allows for investigation of the explanatory ability and fitting effect of the SCC model at different periods through rolling window estimations. Second, the dynamic characteristics of the state variable's equilibrium state evolutionary behaviors can be accurately captured on a short-time scale. The rolling window estimation sets a window width of 250 trading days, and the step width is 10 trading days. The SCC model, linear regression model, and logistic model are respectively fitted within the full sample data intervals, obtaining 278 test windows. The results are presented in Fig. 5, Fig. 6.

Fig. 5.

Pseudo-R² or R² of the stochastic cusp catastrophe, logistic, and linear regression models under rolling window. Note: The gray background area is the pseudo-R² that is greater than 0.5 of the stochastic cusp catastrophe model.

Fig. 6.

Bayesian information criterion values of the stochastic cusp catastrophe, logistic, and linear regression models under rolling window.

As shown in Fig. 5, Fig. 6, the pseudo-R² of the SCC model is more accurate than that of the linear regression and nonlinear logistic models in most periods. The SCC model has lower BIC values than the other two models in almost all instances. The SCC model has approximately 193 fitting intervals with a pseudo-R² that is greater than 0.5, accounting for approximately 69.42 % of the rolling intervals. Notably, the R² or pseudo-R² of the three models have similar change trends, indicating that the overall influence of subsystem variables of the economy, energy market, and carbon market on the state variable will not change systematically due to changes in the models. The higher explanatory power of the SCC model is partially due to more detailed capture of carbon futures market structures. Overall, the SCC model's fitting effect and explanatory capability are the best among the three models.

Fig. 7 presents the parameter estimations and significance test results using the SCC model under the rolling window, with three notable findings. (1) The coefficients of volumes ${\hat{\alpha }}_1$ and ${\hat{\beta }}_1$ are significant in most periods of the full dataset. The trading volumes comprehensively reflect market activity and liquidity in addition to key information such as market sentiment. It continues to affect the asymmetric and bifurcation of the state variable in the EUA futures market. (2) The coefficient ${\hat{\gamma }}_0$ of the current value of Brent crude oil futures returns was significant from March 2012 to January 2013, March 2016 to September 2017, and October 2019 to March 2021, indicating that this variable contributed to exacerbating the differentiated behaviors of the carbon futures market during these periods. The impact of Brent crude oil futures returns lagged by one period is more persistent (${\hat{\gamma }}_1$). (3) The impact of FTSE 100 index returns on the asymmetric effect of the carbon futures market is weaker than that of Brent crude oil futures returns. The significant impact was concentrated from June 2009 to July 2012, December 2015 to November 2016, January 2018 to April 2018, and September 2019 to March 2021, implying that the energy market is more closely linked to the carbon market. In addition, the $M_2$ of EUA futures prices is significant for explaining the bifurcation effects of the system. The results show that short-term price changes of EUA futures have an important influence on the bifurcation effects of the state variable. Certainly, we also recognize that the SCC model constructed in this study is essentially a data-driven prediction model, and detailed structural analysis of the results might be inappropriate; however, the satisfactory fitting effect does not preclude the use of this model for further analysis of equilibrium bifurcation time points and measuring extreme risks.

Fig. 7.

Parameter estimations and significance tests of the stochastic cusp catastrophe model under a rolling window. Note: The red line in the figures represents the estimated values of the parameters. The blue scatter plots are p-values. The black dotted line is the 0.05 significance level. The intercept terms ${\hat{\alpha }}_0$, ${\hat{\beta }}_0$, and ${\hat{\omega }}_0$ are highly significant on almost all windows, which is omitted here for brevity.

5. Further research: equilibrium bifurcation time points and the scale of extreme risks

5.1. Equilibrium bifurcation time points

As noted previously, the equilibrium bifurcation points of the EUA futures state variable can be identified through the catastrophe criterion. Let the estimated values of the asymmetric and bifurcation variables be ${\hat{\alpha }}_x$ and ${\hat{\beta }}_x$, respectively.

$$\{\begin{array}{c}{\hat{\alpha }}_x={\hat{\alpha }}_0+{\hat{\alpha }}_1{dV}_t+{\hat{\gamma }}_0{dBrent}_t+{\hat{\gamma }}_1{dBrent}_{t-1}+{\hat{\rho }}_0{dFtse}_t+{\hat{\rho }}_1{dFtse}_{t-1}\\ {\hat{\beta }}_x={\hat{\beta }}_0+{\hat{\beta }}_1{dV}_t+{\hat{\beta }}_2{M}_2\end{array}$$ (14)

To ensure computational accuracy, we use the rolling window method to estimate ${\hat{\alpha }}_x$ and ${\hat{\beta }}_x$ with a window width of 250 trading days, and a step width of 10 trading days. The method steps are as follows. For each rolling window step, we calculate the values of the 10 control variables located at the center of the window based solely on the estimated values of the parameters under the current window, discarding the 120 values before and after the window period. Continuously rolling the window can produce continuous estimations of control variables. This method ensures that the estimated values of the control variables are always in the center of the window and avoid the influence of different window divisions. It also reduces the data length of ${\hat{\alpha }}_x$ and ${\hat{\beta }}_x$ by 240 compared with the sample size. After obtaining ${\hat{\alpha }}_x$ and ${\hat{\beta }}_x$, we calculate the catastrophe criterion ($\Delta$) according to Equation (4). The results are presented in Fig. 8.

Fig. 8.

Time series of asymmetric variable ${\hat{\alpha }}_x$, bifurcated variable ${\hat{\beta }}_x$ and catastrophe criterion $\Delta$ estimated using the rolling window method.

We identified 148 equilibrium bifurcation time points (Δ < 0) of EUA futures state variables from December 3, 2009 to September 16, 2020 using catastrophe criterion. According to Fig. 8, two relatively concentrated bifurcation periods emerge, i.e., April 2015 to August 2015 and February 2018 to January 2019. In the first period, 14 equilibrium bifurcation points occur, 82 equilibrium bifurcation points occur in the second period, corresponding to two significant crest positions of bifurcated variable ${\hat{\beta }}_x$. At these points, due to drastic changes in the bifurcation variable, the state variable enters a cusp bifurcation region, and two equilibrium positions are generated under the action of the asymmetric variable. Restoring ${\hat{\alpha }}_x$ and ${\hat{\beta }}_x$ and the estimates of the state variable ${\hat{y}}_t$ to three-dimensional space (Fig. 9) among all the equilibrium bifurcation points, 73 are located near the upper lobe of the equilibrium surface (i.e., ${\hat{y}}_{s1}$), and the remaining 75 are located near the lower lobe of the equilibrium surface (i.e., ${\hat{y}}_{s2}$). According to stochastic catastrophe theory, the state variable may fluctuate near the stable equilibrium position or suddenly change to another equilibrium position under the influences of the control variables. Bifurcation of the equilibrium state means that the system has a nonsingle steady state. Referencing related concepts of financial risks, this study defines potential risks of carbon futures market state variable equilibrium bifurcations as extreme risks. This means that when the state variable undergoes a catastrophe, the state factor and returns will also change accordingly, leading to abnormal fluctuations and even crashes.

Fig. 9.

Equilibrium surface of the EUA futures market state variable and its equilibrium bifurcations. Note: Pink points are the projections of the state variable (${\hat{y}}_t$) of the equilibrium bifurcation points into the control space.

5.2. Quantifying extreme risks

After determining the equilibrium bifurcation time points, the potential expected catastrophe scale between state variable equilibrium states can be further investigated based on Equation (5). Let the estimated values of the control variables at the equilibrium bifurcation time points be ${\hat{\alpha }}_c$ and ${\hat{\beta }}_c$, respectively. Solving equation ${-y_s^3+{{\beta }_{c}y}_s+\alpha }_c=0$ of the equilibrium surface can obtain the upper lobe (${\hat{y}}_{s1}$) and lower lobe (${\hat{y}}_{s2}$) of the equilibrium surface of the state variable. The points in the cusp region in Fig. 9 can be expanded in the temporal dimension. Furthermore, according to the linear correspondence between the state variable and the state factor, when state variable changes between the stable equilibrium points ${\hat{y}}_{s1}$ and ${\hat{y}}_{s2}$, the corresponding state factor $z_t$ changes between ${\hat{z}}_{s1}$ and ${\hat{z}}_{s1}$, which causes the returns ($r_t$) to change between ${\hat{r}}_{s1}$ and ${\hat{r}}_{s2}$. Finally, combined with the estimated parameters, we can calculate the extreme risks scale of the carbon futures market. The results are presented in Fig. 10 and Table 5.

Fig. 10.

Estimations of state variable ${\hat{y}}_t$ (top), state factor $z_t$ (middle), and returns $r_t$ (bottom) at equilibrium bifurcation points (dual-equilibrium points).

Table 5.

Statistical characteristics of state variables, state factors, and extreme risks scale of returns at equilibrium bifurcation points.

Variable	Mean	Median	0.25 quantile	0.75 quantile	Min	Max	SD
${\hat{y}}_t$	4.4973	4.5749	4.2025	4.8799	2.0228	5.8984	0.6452
$z_t$	11.6793	12.4156	9.4965	14.1039	4.0524	18.1408	3.1000
$r_t$	0.3251	0.3074	0.1912	0.4162	0.0750	0.7795	0.1554

When equilibrium bifurcations occur in the carbon futures market, the distance between the upper and lower lobes is between 2.0228 and 5.8984. Reflected in the returns, this ranges from 7.50 % to 77.95 %. The average extreme risks scale is 32.51 %. Notably, 64.86 % of the equilibrium bifurcation time points were concentrated in April 2015 to August 2015, and February 2018 to January 2019, with a large scale. Based on the analysis of catastrophe theory, the carbon market, the crude oil market, and international economic circumstances, the possible reasons are as follows. (1) Both of these time points are stages of EUA futures price fluctuation and rise, particularly from February 2018 to January 2019, when carbon futures prices are rapidly fluctuating and rising. The market is accompanied by significant speculation and dramatic changes in trading volume that inject instability into the market, resulting in the bifurcations of market equilibrium. (2) The energy and economic subsystems exacerbate the asymmetric effect of equilibrium bifurcations of state variable in the EUA futures market. During these two periods, the crude oil market fluctuated intensely and significantly, reaching its lowest level since 2008 in 2015. At the same time, many factors of instability arose in the global economy during these two periods. In 2015, the global economy continued to suffer from sluggish growth in developed economies, while emerging economies committed to structural adjustment which slowed down economic growth. In 2018, black swan events such as the China-U.S. trade war, currency crisis, and geopolitical uncertainty occurred frequently, with significant impact on the global economy. These results generally align with the discussion of financial “bubbles1” by Fama (2014) and Greenwood (2019). Financial “bubbles” are usually accompanied by price rises and speculation, and are also influenced by exogenous factors [39,40].

6. Robustness tests

To further investigate the predictive capabilities and robustness of the SCC model, we use the returns of the S&P Global 1200 index, WTI crude oil futures contract, ICE natural gas futures contract and Newcastle thermal coal futures contract as economic and energy market indicators to reconstruct asymmetric control factors. The S&P Global 1200 index is a global index of representative companies including those in Europe that reflects the level of world economic development. The WTI crude oil futures are the most market-indicative benchmark of crude oil next to Brent crude oil futures. Similarly, ICE natural gas futures and Newcastle thermal coal futures contracts are influential in the international fossil fuel market, representing natural gas and coal, respectively. Based on Equation (11), when the returns of the FTSE 100 index are replaced with the returns of S&P 1200 index, the model is denoted as SCC-(1). When the returns of the Brent crude oil futures contract are replaced with the returns of the WTI crude oil futures contract, the model is denoted as SCC-(2). When both returns are replaced with those of the S&P 1200 index and WTI crude oil futures contract, the model is denoted as SCC-(3). When replacing the returns of Brent crude oil futures contract with the returns of ICE natural gas futures contract and Newcastle thermal coal futures contract, the models are denoted as SCC-(4) and SCC-(5), respectively. The parameter estimation results for the above five models are presented in Table 6, revealing that the estimation results of variable coefficients do not produce significant changes, and their significance remains consistent with our baseline model. Although the pseudo-$R_{SCC}^2$ values of SCC-(4) and SCC-(5) models are slightly lower than that of the logistic model, overall, the fitting performance of the SCC model is more accurate than that of the linear regression and nonlinear logistic models. This indicates that the results of the SCC model are robust.

Table 6.

Full sample data estimation results of stochastic cusp catastrophe models.

SCC Parameters/Statistics	SCC-(1)	SCC-(2)	SCC-(3)	SCC-(4)	SCC-(5)
${\mathit{\alpha }}_{\mathbf{0}}$ [Intercept]	−4.9992a	−4.9992a	−4.9992a	−4.9992a	−4.9992a
${\mathit{\alpha }}_{\mathbf{1}}$ [$\mathrm{d}{V}_t$]	0.7654a	0.7859a	0.7721a	0.7809a	0.7850a
${\mathit{\gamma }}_{\mathbf{0}}$ [$\mathrm{d}{Brent}_t$, $\mathrm{d}{Wti}_t$, $\mathrm{d}{Gas}_t$or$\mathrm{d}{Coal}_t$]	0.1463a	0.1656a	0.1415a	0.3567a	0.2120a
${\mathit{\gamma }}_{\mathbf{1}}$ [$\mathrm{d}{Brent}_{t-1}$, $\mathrm{d}{Wti}_{t-1}$, $\mathrm{d}{Gas}_{t-1}$or$\mathrm{d}{Coal}_{t-1}$]	−0.1711a	−0.1758a	−0.1585a	−0.3781a	−0.2524a
${\mathit{\rho }}_{\mathbf{0}}$ [$\mathrm{d}{Ftse}_t$or$\mathrm{d}{Sp1200}_t$]	0.1852a	0.1559a	0.1933a	0.1960a	0.1834a
${\mathit{\rho }}_{\mathbf{1}}$ [$\mathrm{d}{Ftse}_{t-1}$or$\mathrm{d}{Sp1200}_{t-1}$]	−0.1325a	−0.1333a	−0.1408a	−0.1770a	−0.1659a
${\mathit{\beta }}_{\mathbf{0}}$ [Intercept]	−2.0524a	−2.0291a	−2.0629a	−2.1674a	−2.1571a
${\mathit{\beta }}_{\mathbf{1}}$ [$\mathrm{d}{V}_t$]	0.6916a	0.7070a	0.6954a	0.7246a	0.7219a
${\mathit{\beta }}_{\mathbf{2}}$ [$\mathrm{d}{M}_2$]	−1.0983a	−1.0942a	−1.0979a	−1.1120a	−1.1040a
${\mathit{\omega }}_{\mathbf{0}}$ [Intercept]	−1.2687a	−1.2726a	−1.2672a	−1.2489a	−1.2512a
${\mathit{\omega }}_{\mathbf{1}}$ [$z_t$]	0.4233a	0.4225a	0.4227a	0.4277a	0.4249a
LL	−3330	−3339	−3335	−3302	−3326
${pseudo-R}_{SCC}^2$	0.4966	0.4935	0.4945	0.5010	0.4963
$R_{Linear}^2$	0.4644	0.4604	0.4620	0.4739	0.4650
$R_{Logistic}^2$	0.4925	0.4919	0.4910	0.5043	0.4982
${BIC}_{SCC}$	6748	6766	6758	6626	6673
${BIC}_{Linear}$	9446	9472	9459	9349	9401
${BIC}_{Logistic}$	7613	7616	7622	7481	7519

Note.

^aindicates significance at the 1 % level.

Based on the above five models, we adopt the rolling window method detailed in section 5.1 to determine the multiequilibrium bifurcation time points of the carbon futures market and measure the scale of extreme risks, presenting the results in Table 7. After replacing the asymmetric factor variables, the numbers of equilibrium bifurcation time points, the dual-equilibrium positions of the state variable, and the mean extreme risks of the carbon futures market, the returns did not present significant changes. This confirms that the measured results of the equilibrium bifurcation time points and the scale of extreme risks are robust. However, we also note that fewer equilibrium bifurcation time points are evident during the period from April 2015 to August 2015 for SCC-(4) and SCC-(5). The main reason may be that in the global energy trade, natural gas and coal trade exhibits obvious regional characteristics and price fluctuations differ from those of the crude oil market during certain periods. This is also one of the reasons why we chose crude oil as a variable of the energy market in this study. Additionally, this conclusion also provides a direction for us to introduce more representative energy market variables in future research.

Table 7.

Robustness tests of rolling window estimation results.

Models	Number of equilibrium bifurcation time points (upper lobe/lower lobe)	04-2015–08-2015 Number of equilibrium bifurcation time points	02-2018–01-2019 Number of equilibrium bifurcation time points	$r_t$ Mean scale of extreme risks
SCC-(1)	137(63/74)	12	76	0.3268
SCC-(2)	148(71/77)	21	72	0.2869
SCC-(3)	142(64/78)	18	79	0.3097
SCC-(4)	138(66/72)	9	87	0.3299
SCC-(5)	137(66/71)	5	79	0.3092

7. Conclusion

In this study, FIGARCH and SCC models are combined to consider the long-term memory and volatility clustering characteristics of EUA futures returns. Based on Zeeman's hypothesis of heterogeneous investors, we introduce the influence of economic and energy subsystems on carbon market subsystems to examine equilibrium bifurcations and the extreme risks of the carbon market. The main conclusions are as follows.

• (1)We validate the combination of FIGARCH and SCC models as a feasible modeling strategy for investigating the time-varying volatility of returns. First, the study conducts NLS estimation of FIGARCH model parameters employing the JADE algorithm to obtain time-varying volatility. Second, we construct the SCC model using the returns adjusted by time-varying volatility as the state variable. The SCC model based on this modeling strategy is shown to have good interpretability and fitting effect, providing a novel approach to address the constraint of constant state variable volatility for the SCC model.
• (2)From the perspective of the economy–energy–environment system, chartists are primarily affected by trading volume changes and short-term price momentum within the subsystems, which causes equilibrium bifurcations in the system, whereas fundamentalists make investment decisions based on changes in trading volume and economic and energy market performance, which introduces an asymmetric effect of equilibrium bifurcations. Considering these findings, regulatory authorities should strengthen supervision over various types of traders, particularly when approaching the performance period. Frequent market transactions by various entities may cause excessive speculative sentiment and short-term price fluctuations, potentially generating equilibrium bifurcations and extreme risks in the market. Moreover, market participants should develop comprehensive risk hedging strategies based on specified investment goals, combined with market trading volume and short-term price performance, while considering the energy markets and macroeconomic circumstances, striving to strategically move between buying and selling positions to mitigate the impact of potential extreme risks in the market.
• (3)The application of catastrophe criterion offers a valuable approach for measuring extreme risks. This study measures 148 equilibrium bifurcation time points in the EUA futures market, which were mainly concentrated in price increase ranges of April 2015 to August 2015 and February 2018 to January 2019. The average scale of extreme risks is about 32.51 %. Consequently, it is advisable for regulatory authorities to establish risk early-warning technologies that encompass energy and macroeconomic indicators to effectively identify and manage potential extreme risks. For various market participants, when market prices continuously rise, they often also harbor more points of instability. Therefore, it is essential to understand market circumstances to avoid the potential losses caused by indiscriminately following trends.

This article serves as a continuation of Barunik et al.’s research, applying the SCC model to financial markets by including statistical catastrophe theory to the study of the carbon market. To reveal the mechanism and evolution of the law of extreme risks in the carbon market under system theory more accurately, the main directions for further research are twofold. First, future studies can achieve estimations of time-varying volatility more accurately such as the likelihood function construction based on $I\left(x\right)=f\left(x\right)\sigma \left(x\right)$ instead of the density function for the estimation of SCC model parameters as noted by Wagenmakers et al. (2005) [17]. Second, subsequent studies can incorporate a combination of more independent variables that reflect market characteristics, heterogeneous assumptions, and influencing factors into the model to extend the research.

Funding

This work was supported by the Key Project of the National Social Science Foundation of China [21AGJ009]. The authors thank the NSSFC for its financial support and the comments of the anonymous evaluation experts, and are responsible for the consequences of this article.

Data availability statement

Data will be made available on request.

CRediT authorship contribution statement

Junlong Mi: Writing – review & editing, Writing – original draft, Software, Methodology, Formal analysis, Data curation. Xing Yang: Writing – review & editing, Project administration, Methodology, Funding acquisition, Conceptualization. Jiawen Li: Writing – review & editing, Resources, Formal analysis. Zhihua Yang: Writing – review & editing, Writing – original draft, Investigation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.