Multivariate empirical mode decomposition (MEMD)

MEMD (Multivariate Empirical Mode Decomposition) is an extension of EMD. The key superiority of this method rests with its ability to simultaneously decompose multivariate time series data into different frequency domain data self-adaptively, which ensures an equal number of intrinsic mode functions (IMFs). Moreover, these IMFs are effectively aligned based on a frequency scale, facilitating a matched composition of IMF components in terms of both quantity and scale. This research analyzes the risk dependence between different variables and SCFI at different time scales. In order to make the relationship between each variable and SCFI comparable, it is necessary to standardize the time scale. MEMD can decompose multiple variables within a unified framework, which perfectly solves this problem. Rehman and Mandic [39] proposed the basic steps of MEMD, which are as follows.

1. (1) Select a suitable point set on the n − 1 sphere and sample it.
2. (2) For all k groups (the entire set of direction vectors), the set is used as the projection set to calculate the input signal along the direction vector , where and .
3. (3) Find the corresponding projection signal maximum time point .
4. (4) For to obtain the multivariate envelope curve .
5. (5) For a group of direction vectors K, calculate the average value m(T) of the envelope curve, and its calculation formula is given as:(1)
6. (6) Find d(t) by d(t) = x(t) − m(t). If d(t) meets the stop criteria of multiple IMF, it will be regarded as an intrinsic mode function. Otherwise, return to Step 2. Among them, the stop criterion is that after continuous multiple screening, the mean value of the upper envelope and the lower envelope must be approximately equal to zero or meet some requirements.

Finally, the original time series can be decomposed into several IMF components and one residual, which can be written as:

(2)

Where n is the number of IMFs, x(t) indicates one of the time series in the multivariate data.

Least absolute shrinkage and selection operator (LASSO)

LASSO was first proposed by Tibshirani [40]. Its essence is the least squares estimation with penalty factor, which compresses the coefficient with small absolute value to 0 through the penalty term, so as to achieve the purpose of variable selection and parameter estimation. The LASSO method has been found to effectively identify influential variables and those that can offer additional valuable information [41]. LASSO achieves model sparsity through L₁ regularization, thereby improving the interpretability and generalization ability of the model. It has advantages in multivariate selection and is particularly suitable when it is necessary to clarify the importance of variables. This method helps us focus on essentials when analyzing 12 potential factors. Suppose the dependent variable is and the independent variable is , where , is the coefficient vector, and the following linear model is considered:

(3)

The variable selection and parameter estimation of LASSO are obtained by Eq (4):

(4)

The solution of Eq (4) can be transformed into the following optimization problem with penalty term, where is the adjustment parameter, and corresponding to:

(5)

Copula

Copula analysis consists of two steps: determining marginal distribution and analyzing dependency structure. Traditional time series models, such as ARMA and GARCH family models, are used for fitting marginal distributions. The Copula function is used to construct tail dependency.

Marginal distribution.

In this paper, the ARMA-EGARCH model with the skewd Student-t, Student-t, generalized error and normal distributions are employed to model the marginals to capture the asymmetry and leverage effects in the recombined series. For a stationary time series r_t, an ARMA(p, q) is applied to model the conditional mean model. It is represented by Eq (6).

(6)

where c is the constant term, , and are autoregressive parameters. is the random error of t moment. p and q indicate the lag orders which can be determined by AIC value. The restricted variance is modeled with the EGARCH method through the following equations:

(7)(8)(9)

where and are a nonrandom real scalar sequence, and . It can be seen that when , under the same fluctuation size, the increase in future conditional variance under negative fluctuations is greater than that under positive fluctuations, reflecting asymmetry and facilitating the description of financial price fluctuations. The EGARCH process not only ensures the nonnegativity of h_t, but also eliminates the nonnegative restriction on the coefficients of the GARCH process, and can reflect the asymmetry of volatility.

Static and time-varying Copulas.

A copula function can connect multiple distribution functions with their respective marginal distribution functions. In recent years, many researchers have devoted themselves to extending and applying the copula method in shipping markets [13,17,42–44]. It has been suggested by Sklar [45] that any joint distribution function F(x₁, x₂) with the marginal distribution of F(x₁, x₂) could be represented as:

(10)

If F₁(x₁) and F₂(x₂) are continuous and the joint distribution function is given, then

(11)

where u₁ = F₁(x₁),u₂ = F₂(x₂).

This study considers different static copula specification, and then according to the best fitted static copula specification, this study further examines the tail-dependence dynamically with corresponding time-varying copula models. In building high dependence among variables this study considers the copulas which examine the asymmetric and symmetric. The four copula functions are: Gaussian and Student-t (elliptical copulas), Clayton and Rotated Clayton (Archimedean copulas). The details can be found in Bai [13]. AIC is usually applied to determine which copula specification is the best-fitted one. So that the features of the tail dependence between the two studied variables are obtained.

In a short summary, this study’s research process is shown in Fig 1. In the initial phase, the MEMD-based methodology is employed to decompose and structurally reconfigure the return series into distinct subsets representing short-term to medium- and long-term components. Subsequently, LASSO is utilized for the purpose of identifying pivotal drivers influencing SCFI. Following this, the third stage involves investigating the tail dependence relationships between SCFI and these identified influential factors using Copula models. Finally, comprehensive insights and conclusions are derived based on the findings of the analysis.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 1. The flowchart of the MEMD-based copula analysis.

https://doi.org/10.1371/journal.pone.0344386.g001

Decomposition and reconstruction

In this paper, SCFI and potential influence drivers are decomposed into 11 groups of IMFs and one group of residuals through MEMD. Considering the complexity of the data, the permutation entropy proposed by Bandt and Pompe [47] is applied in component reconstruction. The calculation of permutation entropy can be found in Bandt and Pompe [47] and Chen et al. [48]. This paper refers to Zheng et al. [49] and set equal to 0.5. However, one threshold can only clarify the IMF components into two timescales, which may overlook pivotal time scales and lack accuracy. Moreover, our dataset contains data from over 15 years, so it is necessary to divide the data into different timescales. According to the permutation entropy values displayed in Table 2, the permutation entropy values of all the first four IMFs are larger than 0.5. The first four IMFs can construct a short-term timescale (from 5.18 to 7.03 weeks). Then, it can be found that the permutation entropy values of all variables’ IMF7, IMF8, IMF9, IMF10, and IMF11 are almost within the scope of 0.2 to 0.3. So, this paper sets 0.3 as another threshold to divide the last six IMF components into two timescales, namely medium-term (from 29.84 to 35.35 weeks) and long-term (from 99.99 to 178.65 weeks). The periods are consistent with the findings of Chen et al. [49] in analyzing the cycles of CCFI with EMD.

Download:

• PNG
  larger image
• TIFF
  original image

Table 2. Permutation entropy of each variable’s IMFs.

https://doi.org/10.1371/journal.pone.0344386.t002

The plots for different timescales of different variables are displayed in Figs A-1–A-3 in S1 Appendix, respectively. It becomes evident that the short-term sequence group exhibits numerous small and irregular fluctuations. These short-term sequences maintain the same structural characteristics as the original data, as they accurately capture the extracted short-term fluctuations. Consequently, the curves representing the medium- and long-term sequence groups display a smooth trend, devoid of such rapid fluctuations. It should be noticed that all timescale components are stationary.

Estimation results of LASSO

LASSO, a variable selection and parameter estimation method, is chosen in this paper to effectively screen and estimate the parameters of the selected variables, considering short-, medium-, and long-term perspectives. Table 3 shows the results of LASSO estimation.

Download:

• PNG
  larger image
• TIFF
  original image

Table 3. Selection of variables.

https://doi.org/10.1371/journal.pone.0344386.t003

LASSO analysis reveals that developments in other shipping sectors notably influence container freight rates. The dry bulk market, in particular, has a stronger and more consistent impact on the SCFI than the tanker market. The BDI affects the SCFI as a full-frequency band (short-medium to long-term), as its market structure and demand fluctuations are closely related to container transport. In contrast, the tanker shipping market’s short-term impact is limited due to differing demand structures, though long-term fluctuations can still indirectly influence the SCFI through competition for shipping resources. In the short- and medium- term, the “siphon effect” of capacity resources is manifested in the negative relationship between the rising BDI and the falling SCFI; a prolonged rise in BDI tends to reflect strong global demand for commodities, which usually corresponds to periods of global economic recovery or growth. In this context, foreign trade is also synchronized with the growth, which in turn promotes the rise in demand for container transport, and SCFI rise correspondingly. Therefore, in the long run, the BDI has a positive impact on the SCFI.

The second finding pertains to the stock market impact on container transport. In the short-term, the parameters associated with the SHCI show significance. This implies that fluctuations in the Chinese stock market can influence container freight rates in the short-term. In the medium-term, LASSO keeps both SP500 and SHCI, suggesting that not only the Chinese but also the US stock market impacts container transportation. However, MSCI_E is selected only in the long-term. Due to the close relationship between the shipping market and the economic situation, the US stock market is relatively more eye-catching compared to the European stock market, and the US stock market is still the core indicator of the world economy. In the long-term, the parameters associated with the three stock markets exhibit significance. This indicates that stock markets play a part in shaping container freight rates over the long term. Short-term stock market volatility mainly reflects changes in capital market sentiment, and its impact on SCFI is relatively indirect and limited. In the container shipping industry, especially in China as a major global manufacturing and exporting country, stock market volatility usually reflects changes in the health of the Chinese economy and foreign trade demand. European stock markets, which represent a part of the global shipping demand structure, affect the container transportation market mainly through their long-term indirect linkages with the global economy. As the stock market of one of the world’s major economies, the long-term performance of the U.S. stock market directly reflects global economic growth, consumer demand, investment levels, and international trade activity. The rise or fall of the United States stock market in the medium- to long-term is often accompanied by a boom or bust in the global economy, which in turn affects the level of demand for container transportation and freight rates.

The impact of the crude oil market on the container transport market is mainly reflected in the short- and medium-term, the former is reflected in the direct impact of fuel costs, the latter indirectly through the strategic adjustment of shipping companies and policy response to the level of freight, and in the long-term its role will be gradually weakened by technological progress and structural changes.

The impact of the metal market is more important mainly in the short-term; however, the impact of the gold market is more important in the medium- and long-term. In the short-term, the metal market directly affects container demand through trade volume fluctuations. The gold market, as an indicator of macro-risk and global economic expectations, influences freight movements through investment confidence, policy environment, and other medium- to long-term paths.

Lastly, the geopolitical risks, agricultural products, carbon finance, and foreign exchange markets show enduring impacts on the container transport market from the short-term to the long-term. As global environmental awareness grows, fluctuations in the carbon market, especially in the context of carbon emission taxes and green shipping policies, may affect container freight rates by increasing or decreasing transportation costs. Geopolitical risks such as trade disputes, regional conflicts, and other factors may have an impact on the demand for container transport by disrupting the global supply chain and increasing transportation costs, etc. Against the backdrop of agricultural products being the main export cargo, price fluctuations in the agricultural products market have a direct impact on the demand for container transport. Geopolitics, agricultural markets, carbon finance and foreign exchange markets, and other factors will not only trigger short-term fluctuations in freight rates, but also through the global supply chain layout, shipping cost structure and international trade pattern of long-term shaping, in a number of points in time continue to affect the container transport market.

These conclusions provide key drivers for the container transport industry in the short-, medium- as well as long-term, especially in the face of a large number of potential predictors.

Estimation results of copula models

This section further conducts a dependency structural analysis on the important factors identified through LASSO screening at different time intervals. The estimation results for the ARMA-EGARCH specifications are presented in Table B1 in S1 Appendix. The residual diagnosis test confirms the adequacy of the selected marginal distribution models. The estimated static copula results for different time horizons are presented in Table 4. The optimal copula is identified by cross-validation. As depicted in Figs 3–5, the time-varying nature of dynamic equivalent correlations between SCFI and selected powerful drivers can be observed across different time scales. To enhance the reliability of our results, this study excludes the initial 10% time-varying parameters from each group, and the optimal copula is identified using AIC, BIC, and maximum likelihood approaches (See Appendix C in S1 Appendix).

Download:

• PNG
  larger image
• TIFF
  original image

Table 4. Constant copula estimation results.

https://doi.org/10.1371/journal.pone.0344386.t004

Download:

• PNG
  larger image
• TIFF
  original image

Fig 3. Time-varying dependence parameters in the short run.

https://doi.org/10.1371/journal.pone.0344386.g003

Download:

• PNG
  larger image
• TIFF
  original image

Fig 4. Time-varying dependence parameters in the medium run.

https://doi.org/10.1371/journal.pone.0344386.g004

Download:

• PNG
  larger image
• TIFF
  original image

Fig 5. Time-varying dependence parameters in the long run.

https://doi.org/10.1371/journal.pone.0344386.g005

The initial findings reveal that no tail dependence or asymmetric tail dependence is observed in the short-term (as shown in Fig 5). The short-term dependence between factors and SCFI is limited and can be most accurately characterized by a Rotated Clayton or Gaussian distribution. Only Metal-SCFI and SP_Agri-SCFI pairs show significant results. These findings indicate that short-term SCFI is influenced by factors that either have no or higher tail dependence with them. Consequently, it is worth mentioning that upper extreme events in metal and agricultural markets could weigh on the SCFI in the short-term, pushing it higher. Because the rise in prices of these two commodities will trigger investors’ concerns about inflation, the market panic caused by rising costs will lead to a joint increase in SCFI in the short-term.

Based on the short-term, no tail dependence exists in EUA-SCFI, GPRD-SCFI. Upper tail dependence exists in BDI-SCFI, SHCI-SCFI, USD-SCFI, Oil-SCFI, Metal-SCFI, and SP_Agri-SCFI. Based on the medium-term, symmetric tail dependence emerges in BDTI-SCFI and SHCI-SCFI(as shown in Table 4). This highlights the necessity for stakeholders to recognize that this correlation extends beyond the conventional linear relationship typically observed during normal market conditions. The expansion of manufacturing industry drives the synchronous growth of crude oil demand (BDTI increase) and industrial product exports (SCFI increase). A large part of the Chinese economy is driven by foreign trade, and the Chinese stock market is closely related to SCFI. The short-term market dependence of SCFI and SHCI may not be significant due to market disturbances, but they show a significant synchronous upward and downward trend in the medium-term. In the medium-term, USD-SCFI indicates lower tail dependence, because the US dollar index declines and trade demand decreases, when the expectation of a global economic recession intensifies. But in the long-term USD-SCFI indicates upper tail dependence, which indicate the public’s response to economic downturns is more rapid, while their response to economic upswings is relatively slower. Gold is a widely recognized safe haven asset, and its trend often contradicts the USD index. So the tail dependence between Gold and SCFI is exactly opposite to that between SCFI and USD in the medium- and long-term. The medium-term time-varying copula results show that the dependence in extreme risk events is more powerful than that in normal periods, especially in 2012, 2016, 2020, 2022 and 2024 (as shown in Fig 4). These years were closely associated with major events that affect the economy and politics, i.e., the European debt crisis and the U.S. fiscal crisis in 2012; Brexit in 2016; the COVID-19 outbreak in 2020, geopolitical conflict in 2022, and 2024.

Moreover, the long-term dependence is more powerful than the short- and medium-term dependence(as shown in Fig 5). Overall, the results demonstrate that there is an average upper tail dependence between the carbon finance market and the container shipping market in the long-term, which is consistent with the recent research by Meng et al. [7]. An upper tail dependence can also be observed between the bulker/tanker shipping markets and the container shipping market in the long-term. In the long-term, due to the impact of shipping cycles, the three freight rate indices tend to show an upward trend. In the long-term, the global influence of the US stock market is also reflected, with SP500-SCFI showing systematic tail dependence. The influence of European stock markets is limited. There is an upper tail dependence between Carbon price (EUA) and SCFI, as shipping companies currently use surcharges to compensate for the incremental cost caused by the shipping industry’s participation in carbon trading. However, it is commonly recognized that it is easy to raise prices, but difficult to lower them. This highlights the necessity for stakeholders to recognize that the extreme upside risk in the associated shipping market, exchange market, and carbon finance market deserves to be flagged.

The dependence between SHCI and SCFI is best represented by the Student-t copula in medium-term (as shown in Table 4). Comparable patterns can be observed between SP500 and SCFI in the long-term. These results suggest there is symmetric tail dependence between the Chinese/American stock markets and the container shipping market at longer timescales. This is further supported by similar patterns observed in the long-term GPRD-SCFI pair. The dependence structure suggests that a surge in global geopolitical risks would lead to a corresponding increase in the SCFI. Geopolitical crises can affect waterway safety and trade relation, subsequently freight rates changes. Considering the market lag, so symmetrical tail dependence exists in the long-term. From the time-varying Figs 3 to 5, it is evident that the dependency structure between these two variables is relatively weak in the short-term. However, in 2022, with the escalation of the Ukraine-Russia conflict, there was an improvement observed in the long-term dependency structure between GPRD and SCFI. This crisis triggers changes in trade flows (such as economic sanctions, trade relations), which in turn lead to the reallocation of shipping capacity. Capacity adjustment is not a task that can be accomplished overnight, and Russia’s influence on the global situation is important. So on the dynamic graph, an improvement is observed. The outbreak of the Syrian War in 2011 and the tense situation in the Middle East from 2016 to 2017 posed a threat to the passage of container ships in the surrounding waters, so improvements are also observed. Subsequently, the ongoing geopolitical conflict has further enhanced the long-term dependency structure.

When assessing the relationship between the gold market and the container shipping market, it is found that the Rotated Clayton copula is best fitted for describing the tail dependence in the medium-term, while the Clayton copula is more suitable for the long-term (as shown in Table 4. This suggests that participants in container transport market need to be concerned about both the medium-term extreme upside risk and the long-term extreme downside risk to the gold market.

Implications

The findings of this paper are useful for various stakeholders in the shipping industry and financial markets. Specifically, the whole-time horizon fluctuations in various markets, such as the dry bulk transport market, the Chinese stock market, the foreign exchange market, the agricultural market, the carbon market, and geopolitical risk, can offer valuable insights for assessing the trajectory of SCFI. In addition, short-term speculators need to pay attention to the impact of the crude oil and metals markets; medium- to long-term investors also need to pay attention to the U.S. stock market and gold market.

When engaging in tail risk management and control activities, stakeholders holding Shanghai Containerized Freight Index (SCFI) assets should particularly focus on the potential upper tail risks stemming from short-term metal and agricultural markets. In the medium-term, stakeholders should particularly focus on the potential upper tail risks stemming from gold and foreign exchange markets. Unlike the short-term, in the medium-term, risk managers in the container shipping market should be cautious of both upper and lower tail risks resulting from extreme fluctuations in the tanker transport market and the Chinese stock market. In the long-term, risk managers in the container shipping market should be cautious of upper tail risks from dry bulk transport, foreign exchange market, and carbon market, as well as lower tail risks from the gold market. Particular vigilance should be exercised against the upward and downward tail risks posed by the U.S. stock market and geopolitical risk factors. Timely precautions should be taken to avoid the risks associated with extreme volatility.

Besides, the multiscale analysis offers different implications to the stakeholders who are working in different sectors of the shipping market. Market participants can use multi-scale analysis to identify risk patterns at different timescales and adopt flexible risk response strategies. Investors need to closely follow the developments in the above markets and anticipate risks, especially in the major political and economic events on a global scale. These risks can be mitigated, for example, by hedging strategies, building a diversified portfolio, or using derivative financial instruments. In addition, risk managers in the container transport market should ensure that they have emergency response mechanisms in place to adjust their strategies promptly in extreme risk events to minimize potential losses. The conclusion of this article is also helpful for traders of other financial assets to consider Shanghai Containerized Freight Index future when constructing investment portfolios.