Research on the Dependence Structure and Risk Spillover of Internet Money Funds Based on C -Vine Copula and Time-Varying t -Copula

Internet money funds (IMFs) are the most widely involved products in the Internet ﬁnancial products market. This research utilized the C -vine copula model to study the risk dependence structure of IMFs and then introduces the time-varying t -copula model to analyze the risk spillover of diverse IMFs. The results show the following: (1) The risks of Internet-based IMFs, bank-based IMFs, and fund-based IMFs have obvious dependence structure, and the degree of risk dependence among diﬀerent categories of IMFs is signiﬁcantly diﬀerent. (2) There are risk spillover eﬀects among diverse IMFs, and their risk dependence relationship is characterized by cyclical feature. (3) The risk spillover eﬀect among diverse IMFs is pronounced, and dynamic risk dependence between IMFs is characterized by synchronization.


Introduction
In recent years, China has paid due attention to the digital transformation of finance and actively promoted the further improvement of the modern digital financial system, which has also stimulated the rapid rise of China's Internet financial products market. Internet financial products as "disintermediated" investment transactions are employing network information technology [1]. Among the Internet financial products, Internet money funds (IMFs) are the most widely participated products. IMFs, also known as Internet money market funds, usually gather idle funds of individual investors and then invest the idle funds by the fund management companies to obtain profit. e traditional money market funds usually implement the "T + 2" mode, while the IMFs mostly use "T + 0" or "T + 1" subscription and redemption mode. us, users of IMFs proliferate. Taking Yu'E Bao as an example, its size was up to about 972.415 billion yuan as of the end of the first quarter of 2021.
However, some IMF platforms still have problems such as liquidity risk, maturity mismatch, and APP security loopholes, which further aggravate the uncertainty of IMFs market risk. At the beginning of 2020, the sudden outbreak of the COVID-19 pandemic caused a severe impact on both offline and online finance, making the Internet financial products market more vigorous and vulnerable. Due to the risk dependence relationship, the accumulation and superposition of risks in the Internet financial products market are accelerated, and it also quickly leads to the spread of individual risks to other financial products' markets and forms risk spillovers. In the post-COVID-19 era, it is essential for governments to pay full attention to the prevention and control of risks in the IMFs market.
In this context, does the market risk of IMFs have a dependence structure? Are there risk spillover effects between different categories of IMFs? How does the dependence of IMFs change dynamically? e above questions still need to be further explored and discussed.
is research used the Canonical vine copula (C-vine copula) model to examine the risk spillover effect and dynamic dependence among IMFs based on the time-varying t-copula model to provide valuable references and suggestions for the risk prevention and control of the Internet financial products market.

Literature Review
Appropriate monetary liquidity is the primary concern for IMFs. Yang et al. [2] demonstrated that liquidity risk was considered as the main factor in Yu'E Bao's investment strategy. By employing the detailed portfolio holdings of US money market funds, Aftab and Varotto [3] found that these essential players in the shadow banking sector were vulnerable to liquidity shocks. Dong et al. [4] and Chen et al. [5] investigated the linkage effect between Internet finance and commercial banks. Dong et al. [4] found that the development of Internet finance has a negative impact on banks' liquidity. Meanwhile, Chen et al. [5] testified that Internet finance's innovation significantly increased commercial banks' risk-taking behavior.
Some research analyzed the impact of risks presented in the Internet financial products market. Tan et al. [6] conducted an in-depth revelatory case study on Yu'E Bao. Fernandes et al. [7] adopted the autoregressive distributed lag (ARDL) model and analyzed the contribution of digital financial services to financial inclusion in Mozambique. Wang and Ben [8] examined the relationship between online shopping and investment in e-commerce money market funds based on the data from the China Household Finance Survey dataset.
In terms of the relationship between risk and the Internet financial products market, Sung et al. [9] argued that the panic in the IMFs market might be triggered by distrust in the operation of fund managers. Qi et al. [10] found that credit risk and personal information risk were crucial elements that affected the development of Internet finance. From the perspective of the Internet financial products market risk, Xiong et al. [11] proposed a reasonable Internet financial products market portfolio plan for individual or family investment. From the perspective of complex systems, Xu et al. [12] explored the contagion relationship between different risk factors in Internet finance and concluded that risk was transmitted outward through the internal cycle of Internet finance. Fan et al. [13] considered credit risk as an essential issue in the development of Internet finance and conducted an in-depth study on online lending in China.
Copula can effectively measure nonlinear correlation and tail dependence. However, regular copula models cannot build multivariate models. erefore, Bedford and Cooke [14,15] proposed the vine copula model to solve this problem. e vine copula model can describe the pairwise correlation structure between variables and enhance the flexibility of modeling. erefore, the vine copula has been widely used to research risk dependence structure and risk spillover in financial markets. Pourkhanali et al. [16] used Cvine copula and drawable vine copula (D-vine copula) to study the correlation between international financial institutions, and they analyzed the complex dependence among borrowers with an intuitive systematic risk model. Syuhada and Hakim [17] took cryptocurrency as the research object and carried out a risk portfolio on investment according to the risk dependence structure. Hadded et al. [18] and Xiao [19] both studied the risk dependence structure in the stock market using the vine copula, and Xiao [19] further looked at the risk spillovers of stock markets during periods of volatility and depression.
Considering the time-varying characteristics of variables, some researchers used time-varying copula models to study the dynamic dependence structure and spillover effects among financial markets. Yan et al. [20] studied the tail dependence of financial markets with the time-varying tcopula model and gave the optimal portfolio choice. Some researchers have also used the time-varying copula model to study a particular financial market. Duong and Huynh [21] and Wu et al. [22] studied the risk in the stock market. e latter focused on the impact of RMB exchange rate and equity spillover effects and found a positive relationship between them. Han et al. [23] used a time-varying copula to analyze the dynamic dependence between financial assets and constructed a value-at-risk (VaR) portfolio model. Rehman et al. [24,25] studied the extreme dependence and risk spillover relationship between Bitcoin and precious metals using time-varying copula and later studied the dependence structure and found the existence of risk spillover effect between Bitcoin and Islamic stocks.
Existing research studies have mainly focused on analyzing single risk or portfolio risk in the financial market, and few quantitative analyses and empirical studies have been conducted on the risk dependence of different categories of IMFs and the dynamic risk spillover between them. In this context, it is of practical significance to study the risk dependence of IMFs and analyze the direction and intensity of risk spillover of IMFs for the stable, sustainable development of the Internet financial products market.

Data Selection.
e sample data were collected from the Wind database, which divided IMFs into three categories, including Internet-based IMFs (INTE), bank-based IMFs (BANK), and fund-based IMFs (FUND). INTE mainly refers to IMFs docked by the third-party payment institutions; BANK refers to IMFs docked by banks, while FUND refers to IMFs docked by the fund companies. Our research followed the categories in Wind. Five representative funds of each category were selected, respectively. e 15 sample IMFs were chosen according to their category, fund size, year of establishment, industry representativeness, and so on. e basic information of sample IMFs is shown in Table 1.
To ensure the continuity of the data, the seven-day annualized returns (%) of the 15 IMFs are recorded as A i (i � 1, . . . , 15). e data covers January 31, 2016, to January 31, 2020. ere are 1,462 observations of each fund return series after removing the invalid values, totaling 21,930 observations.

Descriptive Statistics.
e first-order difference of the original data was used to obtain the logarithmic seven-day annualized return series of the sample IMFs, denoted as 2 Complexity 15) to reflect the fluctuation of fund returns. We calculated B i as follows: (1) Table 2 presents the results of the descriptive analysis. e average yield series of INTE, BANK, and FUND was used as the return series and subjected to first-order differencing. e descriptive statistics after first-order differencing are shown in Table 3.
According to Tables 2 and 3, the mean value of the logreturn series B i (i � 1, . . . , 15) and three categories of IMFs are close to 0 and have the characteristics such as "fat-tail" and "nonnormality," so the t-distribution can be considered to fit the log-return series of the three categories.

Stability Test.
Heteroskedasticity and autocorrelation are common features of the time series of IMFs. erefore, the stability test was performed for B i (i � 1, . . . , 15). According to Table 4, it can be found that the ADF test statistics are statistically significant, indicating that B i (i � 1, . . . , 15) and the log-return series of three categories of IMFs are stable.

ARCH Effect Test.
Before the ARCH effect test, INTE, BANK, and FUND should be tested for autocorrelation. Taking the BANK as an example, firstly, the BANK series were tested for autocorrelation at lagged 36th order until the absolute value of Q-Stat at 36th order was greater than 0. e P value of Q-Stat results showed that it passed the significance test, indicating the existence of autocorrelation in the BANK series. According to the results of the BANK series autocorrelation test, AR (1) and AR (2) were established for comparison, and the orders were determined by AIC and SC minimum criteria. e results showed that AR (2) had the better results and the regression coefficients of AR (1) and AR (2) were significant. Finally, the ARCH-LM test was performed, and the length of the lag was set to 2. e results showed that the P value of the F-statistic was 0.002. erefore, the BANK series had an ARCH effect, and the GARCH model could be applied later.
Similarly, the above tests were performed with the INTE and FUND series. e results showed autocorrelation and ARCH effects in INTE, BANK, and FUND, which provided the preconditions for constructing a model using the marginal distribution to describe the risk dependence among IMFs.

Model Design
e vine copula model was introduced to portray the risk dependence structure among multiple IMFs, forming a multilayer tree structure diagram and then realizing the measurement of multiple dependence structures. Subsequently, a time-varying t-copula model was introduced to calculate the risk spillover ΔCoVaR and analyze the changes of dynamic dependence among diverse IMFs.

Edge Distribution Model.
e data tests reveal that the selected INTE, BANK, and FUND series are biased, nonnormal, peak fat-tail, autocorrelated, and volatility aggregated. erefore, when modeling and analyzing the logreturn, it is essential to eliminate the autocorrelation, volatility aggregation, and so on. erefore, the AR model and GARCH model can be used. Katsiampa [26] and Ma et al. [27] pointed out that the GARCH model was more accurate for finance-related time series, and the t-distribution could better portray the nonnormal characteristics of finance-related time series data. Owing to that fact, the marginal  Table 5.
According to the results in Table 5, the AIC and SC values of the log-return series model of the IMFs are relatively small, and the model can be considered as a better fit for the data. To estimate the residual series of B i (i � 1, . . . , 15), the standardized residual series were derived, and the new series was obtained by MATLAB. According to the K-S test results, it can be considered that the marginal distribution sequence of IMFs B i (i � 1, . . . , 15) is independent and identically distributed in the standard uniform distribution. en, the new residual series was analyzed by the copula model.

C-Vine Structure and Modeling.
e vine structure overcomes the limitation that traditional copula cannot accurately measure the different dependence structures among multiple variables. It divides the multivariable into binary structures and selects the appropriate copula function to establish the joint distribution according to the specific    ere are two common vine structures: C-vine copula and D-vine copula. e Cvine copula is suitable for the situation of primary variables leading to other variables, and the D-vine copula is ideal for the case that the relationship between variables is relatively independent [14,28]. e parameters were estimated by the C-vine copula and D-vine copula, respectively (for results, see Table 6).
According to Table 6, the AIC and BIC values in C-vine are smaller than those in D-vine. Considering the likelihood and the model selection criterion of minimizing AIC and BIC, this research selected the C-vine copula to analyze the risk dependence structure of the three categories of IMFs. e decomposition of the C-vine copula is specified as where F is the conditional distribution, f is the density function, and c is the conditional density.

Time-Varying t-Copula Model and ΔCoVaR.
Considering the different dependence and time-varying characteristics of INTE, BANK, and FUND, we established a time-varying t-copula model to describe the dynamic dependence relationship between INTE, BANK, and FUND more accurately and evaluate their linkage correlation and contagion correlation. Considering the dynamic and complex nature of the risk linkage process, the idea of Tse and Tsui [29] was introduced here to portray the dynamic change of the dependence coefficient of the time-varying t-copula model: where ρ is the linear correlation coefficient of the two probabilistically integrated transformed random variables, r t−1 is the correlation coefficient of the samples within the rolling window period, R is the covariance of the sample series, and m and n are the unknown parameters to be estimated in the equation. e GARCH model was used to calculate the VaR to predict the volatility of INTE, BANK, and FUND and to model their volatility patterns. e GARCH (1, 1) model is specified as where y t is the time series of the rate of return, a t is the disturbance of rate of return, σ t 2 is the conditional variance, ξ t is the independent identically distributed white noise sequence, and α 0 , α 1 , β 1 are model parameters, α 0 > 0, α 1 > 0, β 1 > 0.
CoVaR refers to the risk that other IMFs are affected during a certain confidence level when certain IMFs generate risk in a certain time period. e equation of CoVaR is as follows: where q is the confidence level and I a,b is the VaR of fund a and fund b. Adrian and Brunnermeier [30] captured the tail dependence between the financial system as a whole and specific institutions by using ΔCoVaR. Based on the research of Adrian and Brunnermeier, the risk-added value ΔCoVaR was used as an index to measure risk spillover. e calculation for ΔCoVaR can be summarized as where ΔCoVaR a,b q is the risk spillover from IMF a to IMF b and VaR a q is the unconditional VaR of IMF a.

Analysis of Risk Dependence
Structure. e C-vine copula was used to model the dependence structure among IMFs. Kendall's rank correlation coefficient τ between two variables was calculated by the R language (see Table 7).
Each layer of the C-vine has a key node, which has a dominant influence on other nodes. According to the results in Table 7, FUND was selected as the pivotal variable in the C-vine copula structure. Figure 1 shows the dependence structure based on the C-vine copula.
As can be seen in Figure 1, there are two trees: T j (j � 1, 2). e main pivot point in the first layer is FUND 1, which is connected to the BANK 2 and the INTE 3, with each edge corresponding to the pair-copula density.
In order to choose a suitable copula model to measure the dependence structure of the IMFs, it is necessary to observe the distribution sequence scatter plot and frequency diagram. Taking the first layer structure as an example, the plots and diagrams are shown in  According to Figures 2 and 4, although the scatter distribution area is wide, the distribution is more obvious on the main diagonal. Figures 3 and 5 can visually show the tail correlation between the two sequences, showing an upper and lower tail correlation between BANK and FUND. Complexity e t-copula, Gaussian copula, and Clayton copula were modeled, respectively. e parameter estimation results (see Table 8) show that t-copula was the most appropriate model in this study by the AIC criterion.
FUND is the critical node in the relationship among IMFs. As seen from the first layer in Table 8, each category of IMFs shows high unconditional dependence. In the second layer, the INTE-BANK|FUND indicates a conditional correlation, which means that the FUND must be used as known information for the C-vine copula when INTE is fitted with the BANK.
Among them, the correlation coefficients of INTE-FUND and BANK-FUND in the first layer are positive, and the correlation coefficient of BANK-FUND is the highest, which indicates that the return rate of BANK-FUND is more likely to move in the same direction. In the second layer, the correlation coefficient of INTE-BANK is positive when FUND is taken as a known condition, and the return of BANK and INTE will also move in the same direction.

Measure of Risk Spillover
Effects. Based on the above results, the time-varying t-copula model was introduced as a way to calculate the CoVaR values and the VaR values between IMFs. en, the ΔCoVaR values of spillover effects were calculated to analyze the direction and intensity of risk spillover among INTE, BANK, and FUND.    Table 9.
e mean VaR value of the FUND is 0.2034, which is much larger than the INTE (0.1169) and the BANK (0.0830), indicating that the FUND products are exposed to the most significant risk. e possible reason is that the FUND's asset allocation is much more prominent in bonds and securities, and its cash holdings are smaller than the INTE and BANK. From a temporal perspective, the VaR values of the BANK and FUND both show an overall upward trend from 2016 to 2019. Meanwhile, the INTE shows a fluctuating decline, indicating that the risk regulation measures for INTE have played a specific role in recent years. e parameters were estimated by the time-varying tcopula model. Results are shown in Table 10.
Monte Carlo simulation was carried out based on the results in Table 10. e results of ΔCoVaR are shown in Table 11. e sequence diagrams of the pair-to-pair risk spillover relationship among IMFs based on the ΔCoVaR results are shown in Figures 6-8. us, changes in macroeconomic policy play an influential role in the risk spillover of the IMFs market. e risk spillover among diverse IMFs is directional. From the perspective of the year-by-year risk spillover effect, the ΔCoVaR from BANK to FUND is enormous and increasing year by year from 2016 to 2018. While the ΔCoVaR from FUND to BANK is relatively small, indicating that when the BANK produces risks, they are more likely to infect the FUND products. Nevertheless, when FUND's risk spillover occurs, it will not have a significant impact on BANK.
e empirical study of Dong et al. [4] and Chen et al. [31] demonstrated that there was mutual causality between Internet finance and the banking industry. On the whole, our empirical results on the risk spillover effect between INTE and BANK are similar to their research, but still, there are differences.
In terms of spillover intensity, the absolute value of ΔCoVaR from INTE to BANK is greater than that from BANK to INTE, which indicates that the volatility spillover effect from INTE to BANK is more substantial. e ΔCoVaR from INTE to FUND is relatively small, while the risk spillover from FUND to INTE is relatively large. at result demonstrates that INTE would be affected by FUND when FUND is at risk. But on the contrary, FUND is not obviously affected by INTE. In 2019, although the two-way risk spillover value between BANK and INTE was similar, the spillover direction between BANK and FUND changed. When the risk of FUND occurs, it is likely to be transmitted to BANK.
In general, there is indeed a risk spillover phenomenon between diverse IMFs in recent years. (1) FUND has a significant influence on both BANK and INTE. ere is a clear trend risk spillover from the FUND to other IMFs, indicating that once a certain risk was generated by the FUND products, it would easily affect the whole IMFs market. Figure 8 also shows that the spillover peaked around October 2019, indicating that both INTE and BANK are vulnerable to FUND. FUND products share part of the risk generated by INTE and BANK products and simultaneously increase the probability of risk occurrence for INTE and BANK products. (2) e mean value of risk spillover from BANK to FUND is the largest, while the risk spillover effect of INTE to BANK shows a fluctuating downward trend.

Dynamic Dependence Analysis of Risk Spillover.
In order to show the changes of dependence among INTE, BANK, and FUND, time-varying t-copula was used for the dynamic dependence coefficient sequences (see .   of parameter 3 in Table 10, and the trend of the dynamic correlation coefficient changes more smoothly when the value of parameter 3 is closer to 1. e estimated values of parameter 3 in the time-varying t-copula model are 0.8705, 0.7623, and 0.9301, indicating that the trend of the dependence change between the BANK and the FUND is stable. e obtained dynamic correlation coefficients were subjected to descriptive statistics. e results are shown in Table 12.         In summary, from January 31, 2016, to January 31, 2020, the INTE-BANK and the INTE-FUND show a significant positive correlation in most of the trading time. As a whole, the dependence relationship fluctuates a lot, and those IMFs' profits and losses are synchronous. Among them, the BANK-FUND maintains a positive correlation during the trading process, and its degree of fluctuation is almost zero, which means that the trend of dependence and association between them is the most stable.

Conclusions
Our research selected 15 IMFs for empirical analysis. e Cvine copula model was chosen to analyze the dependence structure of INTE, BANK, and FUND. en the timevarying t-copula model was introduced to calculate the risk spillover between them. e conclusions obtained are as follows.
Firstly, there is a well-defined risk dependence structure among INTE, BANK, and FUND. Secondly, risk spillovers do exist among the IMFs, their risk spillovers are similar in periodicity, and the risk spillover among different categories has directionality.
irdly  both fluctuate a lot, while BANK-FUND has maintained a significant positive correlation during the trading process and has a more stable dependence relationship. is research sheds some light on the research on the dependence structure and risk spillover of IMFs, and the findings imply that investors should clearly understand that no IMFs can guarantee an absolute return. ey should pay attention to the return scale and various risk indicators of IMFs. And investors can further optimize their investment portfolios based on the risk dependence relationship between different IMFs.
Still, there are some limitations to this research. For instance, as we used 15 IMFs for the sample, further expansion of the sample size would be considered to obtain more accurate research results. In addition, we used the Cvine copula and D-vine copula in the empirical test, and we would take more copula functions into account in future research and select the best fitting copula model.

Data Availability
All data used to support the findings of this study are downloaded from the Wind database, and the data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this research.