2.1.1 Group factor model.

Group factor analysis is a novel approach to apply to mixed-frequency data, since identification strategies and statistical inference can be based on frequency-based grouping.

We use the following notation for the group factor model setting:

(1)

where collects observations for N_j individuals in group j, and are the matrices of factor loadings, and is the vector of error terms, with and . The dimensions of the common factor and the group-specific factors , are respectively k^c, and . In the absence of common factors, we set k^c = 0, while in cases without group-specific factors, we set . The group-specific factors and are orthogonal to the common factor .

To ensure the model is identifiable (i.e., factors and loading matrices are uniquely determined up to meaningful rotations), we impose three core conditions:

1. C1. Orthogonality between common and group-specific factors, that is, and .
2. C2. Normalization of factor variance-covariance matrix.
   Since the unobservable factors can be standardized, we assume(2)
   where I_k is k-dimensional identity matrix, and Φ is a nonzero covariance between group-specific factors (allowed to be non-zero to capture potential within-group correlations). This normalization eliminates rotational ambiguity in the factor space.
3. C3. Rank condition for loading matrices.
   The common loading matrices (stacked across groups) has full column rank k^c (number of common factors), and group-specific loadings matrices and have full column ranks and (number of group-specific factors).

C1 ensures common factors capture only cross-market spillovers, while group-specific factors reflect market-exclusive dynamics. In standard linear latent factor models, the normalization induced by an identity factor variance-covariance matrix identifies the factor space up to an orthogonal rotation (and change of signs). Under identification condition, the rotational invariance of Eq (1) and Eq (2) allow only for separate rotations among the components of , among those of , and finally those of . The rotational invariance of equation therefore maintains the interpretation of common and group-specific factors. C3 ensures each factor contributes unique explanatory power (no redundant factors).

2.1.2 Canonical correlation analysis.

We consider the generic setting of Eq (1). Let  +  , be the dimensions of the pervasive factor spaces for the two groups, and define . We collect the factors of each group in the k_j-dimensional vectors , and define their variance and covariance matrices . From Eq (2), we have for . We want to show that the factor space dimensions are identifiable using canonical correlations and directions. The identification of factor space requires canonical correlation analysis, so before introducing factor models, let us first recall some basics from canonical analysis (see, e.g., Anderson [14] and Magnus [15]).

Let , denote the ordered canonical correlations between h_1,t and h_2,t. The k largest eigenvalues of matrices and are the same, and are equal to the squared canonical correlations between h_1,t and h_2,t. The associated eigenvectors (resp. ), with , of matrix R (resp. R^*) standardized such that (resp. ) are the canonical directions which yield the canonical variables (resp. ).

2.1.3 Mixed-frequency group factor model.

We consider a setting where low-frequency and high-frequency data are available. Let be the low-frequency (LF) time units. Each time period (t–1,t] is divided into M sub-periods with high-frequency (HF) dates t−1 + m/M, with . Moreover, we assume a panel data structure with a cross-section of size N_H of high-frequency data and N_L of low-frequency data. It will be convenient to use a double time index to differentiate low-frequency and high-frequency data. Specifically, we let , for , be the high-frequency data observation i during sub-period m of low-frequency period t. Likewise, we let with , be the observation of the ith low-frequency series at t. These observations are gathered into the N_H-dimensional vectors , for all m, and the N_L-dimensional vector , respectively.

We assume that there are three types of latent pervasive factors, which we denote by , respectively. The first represents a vector of factors which affect high-frequency and low-frequency data (we use again superscript C for common), whereas the other two types of factors affect exclusively high (superscript H) and low (marked by L) frequency data. We denote by the dimensions of these factors. The latent factor model with high-frequency data sampling is

(3)

where are matrices of factor loadings. The vector is unobserved for each high-frequency sub-period and the measurements, denoted by , depend on the observation scheme, which can be either flow-sampling or stock-sampling (or some general linear scheme).

In the case of flow-sampling, the low-frequency observations are the sum (or average) of all across all m, that is, . Then, Eq (3) implies

(4)

Let us define the aggregated variables and innovations , and the aggregated factors . Then we can stack the observation and and write

(5)

that is, the group factor model, with common factor and group-specific factors and . The normalized latent common and group-specific factors , satisfy the counterpart of Eq (2).

Using the same arguments in the mixed-frequency setting of model (4), identification can be achieved for the aggregated factors , and . Consequently, the estimators and test statistics developed for the group factor Eq (2) can also be used to define estimators for the loadings matrices , and the aggregated factor values , and the test statistic for the common factor space dimension k^C in Eq (5). We denote these estimators . Once the factor loadings are identified from Eq (5) and estimated, the values of the common and high-frequency factors for sub-periods are identifiable by cross-sectional regression of the high-frequency data on loadings in Eq (3). More specifically, the estimators of the common and high-frequency factor values are , where (the asymptotic distribution of the factor estimates is provided in Andreou et al. [16]). Hence, are obtained by regressing on , for any and . Consequently, with flow-sampling, we can identify and estimate at all high-frequency sub-periods. On the other hand, only , that is, the within-period sum of the low-frequency factor, is identifiable by the paired panel data set consisting of combine with . This is not surprising, since we have no high-frequency observations available for the LF data.

One can consider an alternative approach to inference on the number of common factors and their estimated values. Instead of first aggregating the high-frequency data as in Eq (5) and then applying PCA in each group, one can extract the principal components directly on the high-frequency panel (and the low-frequency panel) and then aggregate the high-frequency PCA estimates. The procedure then continues identically in both approaches (See Andreou et al. [16]).

This paper selected the mixed-frequency group factor model [16] to analyze the statistical dependence between China’s macroeconomic market and crude oil market. The specific reasons are as follows: firstly, the model Can handle mixed frequency data, avoiding a series of problems caused by frequency conversion. Secondly, the model can separate cross-market spillovers from market-internal drivers and implement group dimension reduction. It not only can extract specific group factors and common factors from two markets separately, but also link multidimensional China’s macroeconomic indicators with crude oil prices, simplifying the analysis framework, and avoiding multicollinearity problems. This directly enables quantification of asymmetric spillovers (crude oil → macroeconomy vs. macroeconomy → crude oil). Thirdly, the model is possible to analyze the proportion of the common factor and group-specific factors that can explain the volatility of the various crude oil prices and China’s macroeconomic volatility. Finally, correlation strength can be quantified by changes in factor loadings. In conclusion, the mixed-frequency group factor model is very suitable for our empirical analysis.

2.1.4 The core theoretical logic of BIC.

The BIC is a statistical criterion of balancing “model fit” and “complexity” designed to solve a fundamental trade-off in model selection: a model with more parameters may fit the data better, but it risks over-fitting (performing well on the sample but poorly on new data); a too-simple model avoids over-fitting but may fail to capture meaningful patterns. BIC quantifies this trade-off via a formula that explicitly penalizes excessive complexity, ensuring the selected model is both “good at explaining data” and “generalizable to new observations”.

The BIC is defined as: BIC = −2lnL  +  klnn, where lnL is Log-likelihood of the model, reflecting model fit—a higher lnL means the model better explains the observed data (e.g., higher adjusted R² in regression models often corresponds to higher lnL, k is the number of free parameters in the model, reflecting complexity (e.g., in our group factor models, k includes factor loadings for common/group-specific factors; adding a common factor increases k by the number of indicators in the group), n is the sample size—larger n amplifies the penalty for extra parameters (to avoid over-fitting in large datasets, where even trivial fit gains from over-parameterization are rejected).

The BIC’s model preferences in our paper are not arbitrary—they directly reflect the asymmetric and time-varying relationship between China’s macroeconomic and the crude oil market:

Full sample: Cross-market spillovers (common factor) matter—BIC selects Model 2 for both groups, as fit gains from common factors outweigh complexity.

Crisis periods: China’s macroeconomic decouples from oil (no common factor fit gain), so BIC selects Model 1 (LF-specific only); crude oil remains linked to global drivers (common factor fit gain), so BIC retains Model 2 (HF-specific + common).

2.2.1 Inference on the number of common factors.

One of our objectives is to determine how many factors are common between groups in the generic factor model in the model (1), that is, we consider the problem of inferring the dimension k^c of common factor space. To do so, we consider the case where the number of pervasive factors k₁ and k₂ in each sub-panel is known, hence = min(k₁, k₂) is also known. If the number of pervasive factors k₁ and k₂ in each sub-panel is unknown, please refer to Andreou et al. [16]. From Andreou et al. [16], dimension k^c is the number of unit canonical correlation between h_1,t and h_2,t.

We consider the hypotheses:

(6)

where are the ordered canonical correlations of h_1,t and h_2,t. Hypothesis H(0) corresponds to the absence of common factors. Generically, H(k^c) corresponds to the case of k^c common factors and and group-specific factors in each group. The largest possible number of common factors is = min. In order to select the number of common factors, let us consider the following sequence of tests: against , for each , we consider

(7)

The statistic corresponds to the sum of the k^c largest sample canonical correlations of and . We reject the null when is negative and large. The critical value is obtained from the large sample distribution of the statistic when , provided in Andreou et al. [16]. The number of common factors is estimated by sequentially applying the tests starting from .

In Andreou et al. [16], it derives the large sample distribution of the test statistic for the dimension of the common factor space and provide a feasible version of it. The essay also defines a consistent selection procedure for the number of common factors (the asymptotic distribution of the factor and loading estimates is provided in Andreou et al. [16]).

2.2.2 Practical implementation of the procedure.

Let us first assume that i.e. the number of respectively common, high and low frequency factors in Eq (3), are know and are all strictly larger than zero. A sample three-step estimation procedure for the factor values and the factor loadings, which is summarized here for practical implementation purposes: PCA performed on the HF and LF panels separately. Define the (T,N_H) matrix of temporally aggregated (in our application flow-sampled) demeaned HF observable as , and the (in our application flow-sampled) demeaned LF observables as . The estimated pervasive factors of the HF data, which are collected in matrix , are obtained performing PCA on the HF data:

(8)

where is the diagonal matrix of the eigenvalues of . Analogously, the estimated pervasive factors of the LF data, which are collected in the matrix are obtained performing PCA on the LF data:

(9)

where is the diagonal matrix of the eigenvalues of .

Analyzing canonical correlation using estimated principal components. Let be the matrix whose columns are the canonical directions for associated with the K^C largest canonical correlations between . Then, an estimator of the (in our application flow-sampled) common factor is , for . Analogously, , for , where is the matrix of the canonical directions for .

As explained in Andreou et al. [16], an alternative estimator of the flow-sampled common factor values , is obtained from the eigenvectors associated to the k^C largest eigenvalues of matrix  +  . The rest of the estimation procedure can be performed replacing with .

The estimated loadings matrices and are obtained from the least squares regressions of and on estimated factor , Collect the residuals of these regression:

(10)

in the following (T,N_U), with , matrices:

(11)

Then, the estimators of the HF and LF factors, collected in the (T,k^U),U = H,L, matrices:

(12)

are obtained extracting the first k^H and k^L PCs from the matrices of residuals:

(13)

where , with , are the diagonal matrices of the associated eigenvalues. Next, the estimated loadings matrices and are obtained from the least squares regression of and on respectively the estimated factors and .

2.2.3 The determination of common and group-specific factors.

The data in our data set follow the flow-sampling factor structure shown in Eq (4), with and corresponding to the 11 daily crude oil prices series and the 60 monthly China’s macroeconomic data series, respectively, for the three samples. Let X^H is the (T,N_H)-dimensional monthly crude oil prices indicator panel data computed as the sum of the daily crude oil prices x_m,t,m = 1,2......,23, and X^L is the (T,N_L)-dimensional monthly China’s macroeconomic indicators panel data, is the (23T,N_H) panel of daily crude oil prices series.

In each set of panel data, we first determine the number of group common factors, i.e., k^C  +  k^H for X^H and k^C + k^L for X^L, where k^C is the number of common factors in the mixed-frequency group factor model, and k^H, k^L are the number of specific factors to the high-frequency and low-frequency group factor model, respectively. We adopt the most commonly method for selecting the number of group common factors, i.e., based on eigenvalues according to the gravel graph. The procedure for selecting the number of common factors for the mixed-group factor model follows Sect 2.2.1, and the detailed test procedure is referred to Baffigi et al. [17]. We need to test the original hypothesis of the number of common factors r = 1 and r = 2 respectively, the values of the test statistics obtained are shown in the Table 1, from the table, regardless of the test samples, the number of common factors k^C is 1. The results of our final selection of the number of factors are shown in the Table 2.

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Table 1. The number of common factors test results.

https://doi.org/10.1371/journal.pone.0336227.t001

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Table 2. Results of the selection of the number of factors.

K^C is the number of common factors in the mixed-group factor model, K^H is the number of high-frequency specific factors in the high-frequency factor model, K^L is the number of low-frequency specific factors in the low-frequency factor model.

https://doi.org/10.1371/journal.pone.0336227.t002

The results of the Table 2 demonstrate that the number of factors varies over samples, and that each factor’s proportion of explanation will also vary. In addition, the parameters involved in the hypothesis testing process take the values of , c = 0.95, , the values of the parameters are selected with reference to Andreou [16].

3.2.1 HF-Specific factors: Core drivers of Crude Oil price volatility.

HF-specific factors explain 83.9%–99.2% of crude oil price fluctuations (Panel B, Table 3) across all samples, representing high-frequency, oil-market-exclusive dynamics that do not spill over to China’s monthly macro indicators in real time. Their key economic connotations include:

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Table 3. Adjusted R² values for regression models.

https://doi.org/10.1371/journal.pone.0336227.t003

1. Short-term supply shocks: Immediate disruptions like OPEC+ emergency production adjustments (e.g., 2020’s 9.7M barrels/day cut), infrastructure attacks (2019 Saudi Aramco drone strikes), or weather-related shipping delays (Hurricane Ida 2021)—these impact daily prices but take weeks to feed into China’s macro data.

2. Speculative trading & market sentiment: Algorithmic trading, hedge fund position shifts (correlating with 30% of daily price swings), and intraday sentiment (e.g., Google Trends for “oil price crash”)—reflecting crude oil’s financialization.

3. Immediate event-driven volatility: Short-lived triggers like weekly EIA inventory reports (doubling daily volatility on release days) or brief geopolitical conflicts (2022 Iraq Kirkuk field strikes)—unlike long-term risks (e.g., Russia-Ukraine war) captured by the common factor.

3.2.2 LF-Specific factors: Core drivers of China’s macroeconomic fluctuations.

LF-specific factors explain 3.4%–54.4% of China’s macro volatility (Panels A, Tables 3 and 3.2.1), with importance rising to 68.3%–70% in crisis periods (BIC preference in Panel A and Panel B, Table 4). They represent China-exclusive, medium-term dynamics that do not affect global crude oil prices (given China’s status as an oil price taker). Key economic meanings:

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Table 4. The values of R² adjusted for the regression results of the two sub-samples.

https://doi.org/10.1371/journal.pone.0336227.t004

1. Domestic monetary & credit policies: RRR cuts (e.g., 2008’s 4-trillion-yuan stimulus), LPR adjustments (2020’s 50bp cut for COVID recovery), or credit rules (2021 real estate “three red lines”)—correlating with 78% of M1/M2 volatility in crises.

2. Domestic demand & structural shifts: Consumption vouchers (2023 auto/appliance subsidies), fixed asset investment adjustments (2016 supply-side reform), or post-pandemic logistics unblocking (2022)—explaining 85.1% of secondary industry investment volatility.

3. Price regulation & fiscal tools: Pork reserve releases (2019 CPI stabilization), coal price caps (2021 PPI control), or tax reforms (2018 individual income tax threshold hike)—accounting for 79.8% of CPI volatility in COVID.

3.3.1 Regression results of the group factor models.

The full sample: 200501–202403. As seen in Panel A of Table 3, the inclusion of LF-specific factors to the common factor regression increases the 90% quantile by 36.6% (from 0.460 to 0.726) and the median adjusted R² by 30.4% (from 0.158 to 0.463). This translates to a large increase in lnL, as the common factor captures crude oil’s impact on China’s macroeconomic (e.g., oil price shocks affecting PPI, monetary policy). Adjusting for the number of the variables in the factor regression models, the BIC favors “common + LF-specific” model in explaining the China’s macroeconomic indicators in 83.3%, whereas the “common only” model is selected in about 10% of the LF data set, indicating crude oil market spillovers significantly improve macroeconomic explanatory power.

Meanwhile, looking at Panel B in Table 3, the 90% quantile has been close to 1, and the median adjusted R² value has reached 0.938 when the high-frequency factor is added to the model alone, that is high enough, but adding the common factor (global drivers like geopolitics, global demand) raises it to 0.994. This small but meaningful fit improvement (reducing unexplained volatility from 6.2% to 0.6%) increases lnL significantly. 99.9% of crude oil indicators favored the “common + HF-specific” model, confirming global/China macroeconomic drivers (common factor) complement high-frequency oil market dynamics.

The common factor explained of crude oil price volatility but 46% of China’s macro volatility—proving the asymmetric mutual influence, that is, China’s macroeconomic is highly sensitive to crude oil market shocks, while China’s macroeconomic drivers have minimal impact on global crude oil prices (consistent with China’s status as a “price taker” in the global oil market).

The financial crisis: 200701–200912. In the Financial Crisis, the LF-specific factors enhances the 90% quantile by 63.2% (from 0.316 to 0.948) and the median adjusted R² by 67.9% (from 0.117 to 0.796) when added to the common factor regression, as shown in Panel A in Table 4. LF-specific factors (domestic policies) explained 79.6% of macroeconomic volatility (median adjusted R²), and the common factor no longer improved fit (Panel A 50% quantiles, Table 4). The Panel A 90% quantiles, Table 4 is the same result, so lnL remains nearly unchanged. Simultaneously, BIC favored the “LF-specific only” model for 68.3% of indicators—reflecting China’s 4-trillion-yuan stimulus (2008) and export support policies that stabilized growth via domestic tools, reducing reliance on crude oil market signals.

The 90% quantile is near to 1, and the median adjusted R² value has reached 0.986 when the HF-specific factors alone is included in the model. These results suggest that the HF-specific factors account for a significant portion of the volatility of crude oil prices. The regression model’s adjusted R² value increases when the common factor and the HF-specific factor are added. The 25% quantile of the adjusted R² value (0.992) reaches the 90% quantile of the value when the common factor is not added. The addition of the common factor implies that changes in China’s macroeconomic offer some incentive to explain changes in crude oil prices, but to a lesser extent than the group-specific factor.

The COVID-19: 202001–202312. As for Panel B in Table 4, we can see that the median adjusted R² increases by 81.6% (from 0.045 to 0.861) and the 90% quantiles by 62.1% (from 0.339 to 0.96) when the LF-specific factors are included to the common factor regression model. BIC favors the “common + LF-specific” model in explaining the China’s macroeconomic indicators in 13.3%, whereas LF-specific factors (epidemic control, consumption vouchers, LPR cuts) explained 86.1% of macro volatility (median adjusted R²), and BIC favored “LF-specific only” for 70.0% of indicators. The 90% quantile is 95.8% (close to 96%), the median adjusted R² arrives at 85.1% (which is close to the regression results of the model that includes common-specific factors).

It is evident that the majority of China’s macroeconomic variations in the COVID-19 period can be explained by China’s macroeconomic indicators, with changes in the price of crude oil internationally having less of an effect. Lockdowns and domestic demand policies (e.g., logistics unblocking) made China’s macroeconomic decouple from crude oil price swings.

The 90% quantile and the median (Panel B in Table 4) of the adjusted R² value have approached 1 when the HF-specific factor is included in the model alone, that is, The “common + HF-specific”model retained 99.9% BIC preference, with the common factor reflecting OPEC+ production cuts and post-lockdown demand rebounds.

3.3.2 Proportion of factors explained.

We also analyze the proportion of the common factor and group-specific factors that can explain the volatility of the various crude oil prices and China’s macroeconomic volatility. Specifically, for each indicator, we perform a regression on the estimated (a) group-specific factors, (b) common and group-specific factors, and (c) common factors. The resulting adjusted R² values from regression models are displayed in columns , with HC − C/LC − C indicating that the difference in adjusted R² values between HC/LC and C. The detailed results have been included in the Appendix the Tables 7 and 8. In Table 7 , the model favored by BIC in three regressions is displayed in the columns BIC. The regression results for the full sample (200501-202403) and two sub-samples (200701-200912 and 202001-202312) are presented in Tables 7 and 8. All results are for common factor and group-specific factors estimated from a data panel of 11 crude oil price indicators and 60 China’s macroeconomic indicators. The number of common factors and group-specific factors for each sample are shown in Table 2.

The full sample. This section examines the findings in five domains: the crude oil market, four aspects of the China’s macroeconomic market. The period is from January 2005 to March 2024.

Finance and credit. Changes in the price of crude oil have the potential to increase the volatility and sensitivity of the financial sector.

The common factors and LF-specific factors account for approximately 87% of the movements of the SSE Composite Index and the SZSE Composite Index, respectively. The majority of the SSE Composite Index’s movements are caused by LF-specific factors, whereas the SZSE Composite Index is after including the common factor in the model, the adjusted R² values of the model regressions double, while the common-specific factors model is typically used when selecting regressions for both indices. Following the inclusion of the common factor in the model, the adjusted R² value likewise doubled for the average PE ratio of the A-shares. It suggests that the volatility of crude oil prices has a greater effect on the volatility of the China stock market. This is in line with the findings of Wei and Guo [18], which showed the responses of stock return to oil shocks are different and crucially related to the causes of oil price changes.

As can be seen from the Table 7, the common factor has a greater impact on China’s PE ratio. This means that fluctuations in crude oil prices will have an impact on China’s A-share market, with the real estate, food and beverage, computer, and automobile industries being the industries most affected. This is because fluctuations in crude oil prices affect the stock market’s level of stock price assessment of these industries, leading to overvalued or undervalued stock prices, which in turn affects investors’ choice of investment assets and, ultimately, their return on investment.

When the common factor is included in the model, the adjusted R² value from the regression of M1 is increased by approximately five times, and the adjusted R² value from the common factor’s regression on it is four times higher than the value obtained from the model’s regression of LF-specific factor alone. Based on the table, it can be inferred that the volatility of crude oil prices has a significant impact on China’s monetary policy, supporting the findings of Ou et al. [11]. The regression of reserve currency (base currency) results are consistent with those of M1.

Similarly, the table also suggests that, in the common-specific factors model, the higher adjusted R² values of M2, FI-DB,and FI-FD are more impacted by changes in crude oil market. This is consistent with the mechanism governing the influence of the financial and credit module of China’s macro-economy, as well as the findings of Liu et al. [19], which suggest that changes in crude oil prices have an impact on China’s monetary policy formulation, which in turn has an impact on inflation. It is better to use interest rate policy alone when there is a large tolerance for inflation. Conversely, the deposit reserve ratio policy ought to be applied in conjunction with the interest rate policy when the government’s tolerance for inflation is limited and it prioritizes social stability and family welfare.

Growth momentum. An intriguing result was found when the common factor was included to the regression model: the adjusted R² value from the regression of Total retail sales of consumer goods (current month’s value) increased by approximately five times. When the common factor is added to the model, the adjusted R² value obtained from the regression of the percentage of investment in fixed asset (secondary industry) increases by approximately four times compared to the model with LF-specific factors alone. Crude oil price fluctuations also have an impact on the share of investment in the secondary industry, particularly in the manufacturing and electricity, heat, and gas industries. According to Chen and Sun [20], China’s gasoline prices react symmetrically to changes in fuel oil prices but asymmetrically to price limits when there is a crude oil import restriction.

Overall national economy. The adjusted R² that was discovered by PPI is larger in the common-specific factor model, influenced by the common factors and the LF-specific factors. Ghysels et al. [21] found that as crude oil price rises, household demand for energy-intensive products increases. When it falls, household consumption expenditures increase in most industrial sectors. It is an effective way to increase household consumption expenditure by reducing the price of crude oil. Investments in most industrial sectors decrease when crude oil prices change. Oil price stability mitigates investment risks across multiple industries and plays a pivotal role in encouraging higher levels of investment.

Government finance. The individual income tax is influenced by the volatility of crude oil prices to some extent, and the inclusion of the common factor in the model results in an approximately four-fold increase in the adjusted R² obtained from the regression of tax revenues (individual income tax) compared to the model without the common factor.

In conclusion, as the world’s biggest importer of crude oil, China’s macroeconomy will be impacted to varying degrees by changes in crude oil prices. Recapitulating the four aforementioned components, it is evident that the Financial and Credit sectors will be most affected by crude oil market volatility, with Growth Momentum and the Overall National Economy coming in second and third, respectively, and the Government Finance last. The impact of crude oil market fluctuations on China’s fiscal situation is relatively minor and indirect.

Crude oil market. The macroeconomic policies of China contribute significantly to shaping its crude oil demand. This demand directly affects the level of crude oil imports, which in turn influences global demand and, consequently, the volatility of crude oil prices. China is the largest importer of crude oil and its international prestige is growing.

During the sample period, China’s macroeconomic conditions have an impact on the volatility of crude oil prices (ColumnHC). However, the HF-specific factors (ColumnH) primarily explain the volatility of crude oil prices. This conclusion can also be illustrated by the adjusted R² value of the model in ColumnC, which indicates that the common factor accounts for 4.5% of the volatility of Brent crude oil prices and 9.5% of the volatility of WTI crude oil prices,3.2% of the price volatility of UAE Dubai. The common factor can account for 9.5% of the variation in WTI crude oil prices, 4.3% of the variation in Brent crude oil prices, 3.2% of the variation in UAE Dubai crude oil prices, and 3.4% of Omani crude oil price fluctuations. It is evident that all of the adjusted R² values obtained from the model’s regressions of the common factor alone are lower; that is, a smaller percentage of the price fluctuations of each crude oil can be explained by the common factor alone. The 5.9% volatility of China Daqing crude oil price can be explained by the common factor, i.e., a wide range of international crude oil prices has a more pronounced impact on China’s Daqing crude oil price, which is also affected by China’s macroeconomic.

The financial crisis. The regression results for the Financial Crisis period, covering from January 2007 to December 2009, are covered in this section.

More than 89% of the changes in the SSE Composite Index can be explained by the common factor and LF-specific factors, and more than 73% of the changes in the SZSE Composite Index can be explained by the common factor and LF-specific factors, and unlike in the full sample, most of the causes of the changes in the SSE Composite Index are explained by the LF-specific factors rather than the common factors. The results of the regression on the average A-share PE ratio are the same as those of the regression on the SSE Composite Index. Most of the causes of the changes in the the average A-share PE ratio are explained by the LF-specific factors. Simultaneously, the proportion of the LF-specific factors explaining the PE ratio of a single industry is all greater than or equal to the average PE ratio of A-shares, with the Food and Beverage industry experiencing the greatest impact, followed by the medical biology industries, and finally the computer and automotive industries. In other words, during the Financial Crisis, comparing to the full sample period, the proportion of the factors explaining the PE ratio has a different value, and the level of the evaluation of the stock prices in these industries is primarily influenced by changes in China’s macroeconomic; Crude oil price fluctuations have less of an impact on them. The CPI (LF-specific factors regression model) has an adjusted R² of 94.3%. When the common factor is included in the regression model, the adjusted R², along with the PPI and RPI, staying at 94.3%. The EPI and IPI regressions produced adjusted R² values of 94.1% and 97.6%, respectively, and the LF-specific factors explained 68.6% and 77.4% of the changes in M1 and M2 in China’s money market.

According to the BIC principle, it can be seen that the model obtained from the regression of crude oil prices tends to choose a model of common factor and HF-specific factor, and this conclusion is the same as that of the full sample.

The volatility of the crude oil price can be influenced by China’s macroeconomic (ColumnHC), but HF-specific factor accounts for the majority of its volatility (ColumnH). The adjusted R² of the model in ColumnC serves as an example of this conclusion. These factors include the common factor and the HF-specific factor, which together account for 99.2% of the volatility of WTI crude oil price and 99.6% of the volatility of Brent crude oil price, respectively. Common factor and HF-specific factor together account for 99.7% of the volatility of the UAE Dubai crude oil price, 99.7% of the volatility in the Omani crude oil price. Similarly, for China’s largest trading partner, Indonesia, common factor and HF-specific factor account for 99.1% of the volatility of the Sinta crude oil price and 99.7% of the volatility in Minas crude oil price. Together, these two factors account for 99.7% of the price volatility of China Daqing crude oil. It is evident that the proportion of the fluctuations in the price of crude oil that are simultaneously explained by the common factor and the HF-specific factors is equivalent to what it was during the full sample.

The COVID-19. The regression results for the COVID-19 period, covering from January 2020 to December 2023, are covered in this section.

The proportion of changes in the SSE Composite Index that can be explained by the common factor and LF-specific factors exceeds 80%, and the proportion of changes in the SZSE Composite Index that can be explained by the common factor and LF-specific factors exceeds 82.1%, and unlike in the full sample period, most of the reasons for the changes in the SSE Composite Index are explained by the LF-specific factors instead of the common factor.

The regression results for the average A-share PE ratio are the same as those for the SSE Composite Index. At the same time, the proportion of explanation of LF-specific factors on the PE ratio of the single industry is lower than the average PE ratio of A-share, in which the influence on real estate industry is the largest, followed by basic chemical industry, computer, food and beverage. In other words, during the COVID-19 period, the proportion of factor explanation varies in the order of magnitude compared to the full sample period. Furthermore, the assessment level of the stock prices of these industries is more influenced by changes in China’s macroeconomic environment than by changes in the price of crude oil. Regression of the LF-specific factors on CPI has an adjusted R² of 79.8%, and regression on PPI has an adjusted R² of 92.9%. The adjusted R² values obtained from regressions of the export price index and import price index are 82.1% and 86.8%, respectively. The LF-specific factors account for 84.1% and 93.8% of the changes in M1 and M2 in China’s money market.

It is evident from the BIC principle that the model derived from the regression of crude oil prices favors the model of common factor and HF-specific factor; this conclusion holds consistently for the full sample and the Financial Crisis. The adjusted R² value of the ColumnC model illustrates this result. Whereas HF-specific factor (ColumnH) account for the majority of the volatility in crude oil prices, the China’s macroeconomic conditions have an impact as well (ColumnHC). The proportion of each crude oil price fluctuation that can be accounted for by the common factor and HF-specific factor surpasses 90%, as the table’s ColumnHC shows. This result is in line with findings from the full sample and the Financial Crisis.

3.3.3 Plot of auto-correlation function of factors.

The autocorrelation functions of the group-specific factors and common factor, which were computed over the full sample period, are plotted in Fig 2. Fig 2 illustrates that during the observation period (6 months plus), the autocorrelations of the common factors and HF-specific factors are strong and positive, and they declined less. The first LF-specific factor exhibits a change in autocorrelation from positive to negative in the 20th period (20th months), with the negative auto-correlation gradually increasing. Meanwhile, the second LF-specific factor exhibits a positive auto-correlation in the 11th period (11st months), with a larger decrease in the positive auto-correlation. As a result, the common factor and group-specific factors exhibit time-series dependence; that is, the correlation between the fluctuations in crude oil prices and the China’s macro-economy is long-term and time-series dependent.

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Fig 2. Estimated auto-correlation functions for common factor and group-specific factors (aggregation first).

The auto-correlation function can be used to characterize the degree of correlation between factors at different moments. (a) shows a plot of the sample auto-correlation function of the common factor estimates at high frequency, (b) shows a plot of the sample auto-correlation function of the HF factor estimates at high frequency, (c) to (d) show a plot of the sample auto-correlation function of LF factors estimate at low frequency. Factor values are estimated from 60 China’s macroeconomic indicators (LF data) and 11 crude oil prices (HF data) using a mixed-frequency group factor model with and k^L = 2. The sample period is 200501–202403.

https://doi.org/10.1371/journal.pone.0336227.g002

Plots of the estimated autocorrelation functions for the common factor and group-specific factors in the Financial Crisis and the COVID-19 are shown in the Appendix.