The global energy landscape is undergoing a profound transformation, shaped by overlapping forces such as climate change mitigation efforts [1, 2], geopolitical realignments [3, 4], and technological innovation [5]. Central to this transformation is the interplay between economic policy uncertainty (EPU) and the strategic behavior of listed oil companies [6, 7]. As governments grapple with decarbonization targets, energy security concerns, and macroeconomic stabilization, the resultant policy volatility creates both risks and opportunities for oil firms [8]. Conversely, the operational decisions of these firms—ranging from capital expenditure allocations to lobbying activities—significantly influence policy trajectories [9, 10]. This paper explores the bidirectional dynamical impacts between EPU and listed oil companies in China, addressing a critical gap in the literature by analyzing the dynamic feedback mechanisms that characterize their interactions.

Economic policy uncertainty, defined as the ambiguity surrounding future policy actions [11], has surged in recent decades due to factors such as trade wars, regulatory reforms, and the transition to net-zero economies [12, 13]. For the oil sector, which is capital-intensive, politically sensitive, and exposed to long-term demand risks, EPU exacerbates operational challenges [14]. For example, the EU’s evolving carbon pricing framework and the US’s vacillating stance on shale oil subsidies have created a climate of uncertainty that disrupts investment planning (BP, 2022). Conversely, oil company responses—such as divesting from fossil fuels or expanding into renewable energy—can alter policymakers’ perceptions of industry viability, thereby shaping subsequent regulatory choices [13].

Existing studies on EPU and oil firm behavior primarily focus on directional impacts [15, 16]. For instance, a large body of research documents how EPU reduces corporate investment [17], increases financing costs [18], and distorts innovation trajectories [19]. In the energy sector, scholars have examined how policy uncertainty affects renewable energy adoption (Aklin & Urpelainen, 2012) and oil price volatility [20, 21]. However, these studies often treat policy as an exogenous shock, neglecting the role of firms in shaping policy outcomes. The findings are mixed regarding the impacts between EPU and oil firms. Some scholars report negative causal impacts between EPU and firm behavior [22]. For example, some scholars suggest that policy uncertainty may increase uncertainty as it leads to increased costs and, therefore, negatively affects investment behavior. Bloom [23] has reported that increased uncertainty raises the “real option value to wait,” which forces companies to delay their investment until the uncertainty is reduced. In contrast, some literature provides evidence of positive causal impacts. Belderbos et al. [24] reported that firms in across-country investments improve the value of growth options, and this often occurs when high market uncertainty exists in host countries. In summary, the impacts between EPU and listed oil companies are debatable and mixed concerning the directional and bidirectional as well as positive and negative impacts, requiring new techniques to clarify the complex connections between EPU and oil firms.

The interplays between EPU and listed oil companies may occur from various channels, as shown in Figure 1 [25–27]. 1) For investment decision-making. Policy uncertainty increases the unpredictability of a company’s future cash flows, leading oil companies to postpone or reduce long-term capital expenditures such as exploration and development and refining capacity expansion [28]. 2) Financing cost channel. The rise in market risk premiums caused by policy fluctuations is transmitted through debt financing costs and equity financing costs [22]. 3) Expectation management [29]. The ambiguity of policy signals leads to deviations in management’s forecasts of future demand. OPEC + member countries adjusted their monthly production quotas repeatedly due to the repeated release policy of the US Strategic Petroleum Reserve. 4) Supply chain reconfiguration. Changes in trade policies (such as the US sanctions on Venezuelan crude oil) force oil companies to adjust their procurement networks, increasing logistics costs [30]. 5) Tax policy. Changes in policies such as windfall profit taxes directly affect corporate profits (for example, the UK’s Energy Profits Levy in 2022 led to a 12% decline in BP’s quarterly net profit.) 6) Technological innovation. The uncertainty of the carbon pricing mechanism affects the R&D investment of oil companies in carbon capture, utilization, and storage (CCUS) technologies [31]. 7) Market access. The tightening of environmental protection policies (such as the EU’s plan to ban the sale of fuel-powered vehicles by 2030) forces oil companies to adjust their business structures. 8) Consumer expectation [32]. The expected fluctuations in energy prices triggered by policy uncertainty affect terminal demand [33]. 9) Political risk channel [21, 34, 35]. Geopolitical conflicts (such as the Russia–Ukraine war) combined with the uncertainty of energy policies led to the revaluation of the assets of oil companies. These channels form a dynamic feedback mechanism. For example, oil companies influence policy-making through political donations, thus forming a two-way interaction between policies and corporate behavior. It is obvious that some channels may pass positive impacts between economic policy and listed oil companies, while negative impacts flow through other channels. These facts suggest not simply a single causal impact between economic policy and listed oil companies. Moreover, the fluctuation trends between variables are often neither the same nor opposite. In addition to the positive and negative impact patterns, more complex patterns may exist between EPU and listed oil companies.

Some scholars aim to detect the interplay and volatility spillover between economic policy and listed oil companies [36, 37]. Another popular approach is causality analysis [38–40]. The Granger causality test is one of the popular causality analysis methods, and it is widely used in economics and finance [41]. Due to the nonlinearity in many real-world systems, the nonlinear Granger causality test was developed to measure nonlinear causal impacts [42]. However, traditional Granger causality methods are based on statistic regression and are reliable for stable series, but they show limitations for unstable series. Moreover, they also show limitations when we must quantify the strength of causal impacts. Transfer entropy is a nonparametric tool that investigates causality by measuring the information flowing between observations. As certified by previous works, transfer entropy is equal to the Granger causality test when the sample datasets are Gaussian distributed [43]. Unlike previous statistical approaches, cross-convergent mapping (CCM) is developed from the dynamics embedded in the state space [44]. It reports that the time series variables are causally linked if they are generated from the same dynamical system and they share a common attractor. The CCM method has a satisfactory ability to detect causality in complex nonlinear systems, which solves the limitation of the Granger causality test.

Although several causal detection techniques are widely used in financial systems, like Granger causality, transfer entropy, and CCM, these methods can only distinguish a single causal impact. For example, only positive or negative causal impacts can be obtained between two observations via traditional methods. This is very limited for practical situations. If there is only a single causal impact (e.g., positive impact) from variable X to variable Y, then the increase (decrease) of X will drive the increase (decrease) of variable Y. In other words, they are more likely to evolve in similar directions. However, these rules are often rare for many real-world observations; their fluctuation associations are complex, neither the same nor opposite directions. The complex causality between them can be partially explained by the aforementioned various channels. Some channels propagate positive impacts, while others flow through negative causal impacts. Thus, we speculate that there may be more than one kind of causal impact between economic policy and listed oil companies and possibly multiple impacts. Thus, more reliable methods are needed to reveal the complex interactions in real-world systems. Stavroglou et al. [45] proposed the pattern causality algorithm (PC) based on the state space reconstruction theory, which can not only detect causality, but it can also quantify causal strength. More importantly, PC can distinguish both positive and negative causal impacts as well as a more complex causal interaction, namely, dark causality.

Thus, this work employs pattern causality methods to examine multiple causality impacts between economic policy and listed oil companies. The study makes three contributions. First, it utilizes a new theoretical framework that captures the complex dynamical causalities between EPU and listed oil firms in China. Second, in addition to the previously reported positive and negative causality, it reveals more complex causal interactions (dark causality) between EPU and the stock prices of listed companies, both positive and negative. Third, it analyzes the dominant causal interactions between economic policy and listed oil companies by quantifying the magnitude of multiple causalities.

The remainder of this paper is organized as follows: Section 2 describes the methods. Section 3 depicts the sample data. In Section 4, we investigate the complex causal impacts between EPU and the stock prices of listed oil companies in China. Section 5 summarizes our main findings.

According to dynamical systems theory, in a dynamic system, if we can estimate the state of component Y based on the dynamics of component X, we can infer causality from variable Y to variable X [46]; the more accurate the estimation can be, and the stronger the causality is. On the other hand, inaccurate estimation indicates variable Y has no dynamics information left in variable X. We speculate that there is no causal interaction from Y to X. According to [44], it is possible to test estimations by the mappings between reconstructed dynamical structures from time series variables. In particular, we can measure the causality by mapping among neighbor points in the reconstructed dynamical structure. The dynamical structure is reconstructed based on the delay embedding theory. Thus, we can test the dynamical causal interactions between variables based on the accuracy of mutual estimations. As we know, the causal interactions between variables can be positive or negative, even in more complex ways. In addition, the strengths of impacts are different for different samples, thus requiring not only causality inference but also the causal strength and causal type.

A dynamics-based causality method is proposed to solve the above issues, namely, pattern causality, which examines causality from X to Y by quantifying how consistently average patterns in the state space of X correspond to average patterns in the state space of Y in contemporaneous neighborhoods. According to pattern causality, there may exist three heterogeneous causal impacts: positive, negative, and dark causalities. A positive causality (i.e., “mutualism”) drives two variables to evolve in the same direction, while opposite forces exist for negative causality (i.e., “competition”). Dark causality suggests more complex interactions that are neither “mutualism” nor “competition,” and it forces two variables to evolve with fluctuations.

Specifically, the main processes of pattern causality are given by four fundamental steps [40, 45]. First, we must make a state space reconstruction from time series [47]. Based on Takens’ theory, we can reconstruct a multidimensional state space from time series using two faithful parameters, that is, embedding dimension E and lag τ [48]. Specifically, we reconstruct E-dimensional state space A X from time series X t , see Equation 1. A X = X 1 X 1 + τ ⋯ X 1 + E − 1 τ X 2 X 2 + τ ⋯ X 2 + E − 1 τ ⋮ ⋮ ⋱ ⋮ X h X h + τ ⋯ X l , (1) where l is the length and h = l − E − 1 τ . The row vectors ( x t ) on A X denote the states evolving with time.

Once we get the reconstructed state space for both variables X and Y, we must then determine the average pattern of each point in the state space. The average pattern shows the fluctuation trend in state space. The average pattern of x t is estimated by its nearest neighbors. Specifically, we must find the weighted combination from its E+1 nearest points based on the Euclidean distance, see Equation 2: w i x = e − d x t , x i ∑ i e − d x t , x i , (2) where d denotes the Euclidean distance, and x i is the nearest neighbor of a given x t .

The average pattern P x t can be determined by the coarse process; see Equations 3–5. h i x = X t x i + τ − X t x i X t x i , ⋯ , X t x i + E − 1 τ − X t x i + E − 2 τ X t x i + E − 2 τ , (3) H x t = ∑ i = 1 E + 1 w i x h i x , (4) P x t = P a t t e r n H x t . (5)

Nine patterns exist in a 3-dimensional state space: ↗ , ↗ , ↗ , → , ↗ , ↘ , → , ↗ , → , → , → , ↘ , ↘ , ↗ , ↘ , → ↘ , ↘ .

Third, we distinguish three kinds of causality based on the average pattern in the state space A X and A Y . Positive causality is inferred when A Y has a similar average pattern on A X . Negative causality is determined when opposite patterns are coupled; for example, a ↗ , ↗ pattern causes the ↘ , ↘ ) pattern. Moreover, a dark causality is inferred when neither similar nor opposite patterns (e.g., pattern ↗ , ↗ causes the pattern ↘ , ↗ ).

After identifying the type of causality, we must quantify the causal strength. The causal strength is measured by the accuracy of cross estimation between state spaces. In particular, we estimate the average pattern of y t contemporaneously with x t by a mapping that keeps the weights w i x on A X , see Equations 6-9: y ⌢ t x i = Y t x i , Y t x i − τ , ⋯ , Y t x i − E − 1 τ , y ⌢ t x i ∈ N ⌢ y t , (6) h i y ⌢ = Y t x i − τ − Y t x i Y t x i , ⋯ , Y t x i − E − 1 τ − Y t x i − E − 2 τ Y t x i − E − 2 τ , (7) H ⌢ y t = ∑ i = 1 E + 1 w i x h i y ⌢ , (8) P ⌢ y t = P a t t e r n H ⌢ y t . (9)

By examining all state points on A X and A Y , we obtain all the estimated average patterns of y t based on the dynamics of x t and the real average patterns of y t . Then, accuracy can be used as a metric of causal strength where the estimated average pattern of y t is equal to the real average pattern. Thus, for each possible pattern P x t , we calculate the percentage of occasions for which the contemporaneously estimated average pattern P ⌢ y t equals the true average pattern P y t . Because there are three kinds of causalities, we can measure the causal strength for positive, negative, and dark causalities (more details can be seen in [40]).

This work aims to detect the impact of economic policy on listed oil companies. Thus, we collect the sample data about economic policy and listed oil companies. For the economic policy, we use a widely used index, namely, the economic policy uncertainty index (EPU), which is a newspaper-based index for policy uncertainty in China (More details are given at https://www.policyuncertainty.com/china_monthly.html). The monthly series from 2010/6/1 to 2025/2/1 are used. The length of the observed series is 177.

We consider the ten most popular listed oil companies: China National Petroleum Corporation, China Petrochemical Corporation, China National Offshore Oil Corporation, China Oilfield Services Limited, COSCO SHIPPING Energy Transportation, China Merchants Group, China Merchants Energy Transportation, Shenergy Company Limited, Guanghui Energy, and Jereh Group. Their monthly stock price series from 2010/6/1 to 2025/2/1 are used as sample. Their corresponding stock codes are given in Table 2. All the datasets are free at https://cn.investing.com/. The datasets are depicted in Figure 2.

From the statistical analysis, the stock price of China National Petroleum Corporation shows relatively higher fluctuations with a standard deviation of 20.3. Meanwhile, the stock price for China National Offshore Oil Corporation remains stable and has the lowest standard deviation. It is obvious the EPU has the highest volatility due to the highest standard deviation (see Table 1). Based on the correlation analysis, many stock prices show positive connections, while negative correlations are discovered between the EPU index and stock prices of other listed oil companies; see Figure 3.

This work employs the pattern causality method to reveal complex causality between economic policy and listed oil companies. Unlike previous findings that only one kind of causal impact between them, our study suggests that three different kinds of causal impacts simultaneously exist: positive, negative, and dark causality. Our work may have several implications for practical application and theoretical analysis.

i) Three different kinds of causal impacts between economic policy and listed oil companies suggest more complex impacts between them, which gives new inspiration to understanding complex associations between them.

As discussed in this work, dark causality may drive two variables to fluctuate in a nonlinear manner, moving in neither perfect unison nor in complete opposition, indicating a more intricate relationship between them. The underlying economic implications of dark causality can be understood from the following aspects:

First, dark causality offers a more realistic explanation for phenomena observed in real-world economic systems. In these systems, the fluctuations between variables rarely exhibit simple, unidirectional patterns. Instead, they often display complex, intertwined trends that deviate from straightforward positive or negative correlations.

Second, it provides novel evidence for the inherent complexity of economic systems. By uncovering these hidden causal relationships, we gain deeper insights into the intricate mechanisms that govern economic behavior, enhancing our understanding of how economic systems operate and evolve.

Third, dark causality reveals an even more convoluted nature of the relationships among economic variables. This complexity gives rise to increased uncertainty, and when dark causality dominates, it may signal elevated risks. As such, dark causality has the potential to be linked to systemic risks within economic systems.

Despite several interesting findings being discovered, there are also some limitations that can be solved in the future. In particular, dark causality is a newly defined interaction, and its underlying explanation needs to be discussed further. In addition, it is also difficult to distinguish impacts from other external factors, such as other related markets.