# Economic Policy Uncertainty and Conditional Dependence between China and U.S. Stock Markets

Authors: Xinyu Wu; Meng Zhang; Mengqi Wu; Hao Cui
Journal: Complexity
 (2022)

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## Abstract

In this paper, we investigate the impact of economic policy uncertainty (EPU) on the conditional dependence between China and U.S. stock markets by employing the Copula-mixed-data sampling (Copula-MIDAS) framework. In the case of EPU, we consider the global EPU (GEPU), the American EPU (AEPU), and the China EPU (CEPU). The empirical analysis based on the Shanghai Stock Exchange Composite (SSEC) index in China and the S&P 500 index in the U.S. shows that the tail dependence between China and U.S. stock markets is symmetrical, and thet Copula outperforms alternative Copulas in terms of in-sample goodness of fit. In particular, we find that the t Copula-MIDAS model with EPU dominates the traditional time-varying t Copula in terms of in-sample fitting. Moreover, we observe that both the GEPU and AEPU have a significantly positive impact on the conditional dependence between China and U.S. stock markets, whereas CEPU has no significant impact. The tail dependence between China and U.S. stock markets exhibits an increasing trend, particularly in the recent years.


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## Body

## 1. Introduction

The economic activities of many countries and regions are intertwined due to the financial globalization. Financial globalization plays an important role in global economic growth, which not only facilitates the international capital flow but also promotes the deployment of global resources. However, financial turbulence intensifies as economic and financial globalization develops, putting the financial system’s stability under great strain. Financial crises have erupted frequently since the 1990s and spread quickly from one market to another. The subprime mortgage crisis in the U.S. in 2008, for example, swiftly spread to Europe and East Asia, wreaking havoc on global economic growth.In recent years, the Chinese financial market has experienced rapid development and played an increasingly vital role in the world economy, which enhanced its dependence with other countries’ financial markets. Because the U.S. is the world’s largest economy with the highest financialization, it is critical to investigate the conditional dependence between China and U.S. stock markets.Numerous studies have provided evidence that economic policy uncertainty (EPU) in various countries is interrelated (see, e.g., [1, 2]). In particular, Jiang et al. [3] showed that there is a close interrelationship between the policy uncertainties of U.S. and China, and the interrelationship is mostly affected by bilateral trade, exchange rate, and investor sentiment. Moreover, many researchers have investigated the relation between EPU and international stock markets (see, e.g., [4–7]). Li and Peng [8] found that American EPU will lead to changes in the co-movements of China stock market with U.S. stock market. Further, Li et al. [9] showed that the American EPU affects China stock market in the long term. Zhang et al. [10] found that China’s global influence has increased, but the U.S. has maintained its dominance.Since 2016, the global black swan events such as the Brexit, the U.S. interest rate hike, the U.S.-China trade war, and the COVID-19 pandemic have affected global stock markets significantly, which has forced governments around the world to make frequent changes to their policies in order to limit the economic impact of these events. As a result, the current level of EPU is at extremely elevated levels. The issue of whether the high EPU is likely to affect the conditional dependence between China and U.S. stock markets requires immediate research, as it has important implications for investors, regulators, and risk managers.The rest of the paper is organized as follows. In Section2, we present a literature review. Section 3 describes the methodology, including the marginal distribution model, the Copula-MIDAS framework, and the estimation method. Section 4 describes the data. Section 5 presents the empirical results. Section 6 concludes the paper.


## 2. Literature Review

The study of the correlation between international stock markets has attracted a great deal of attention in the literature. Many previous studies use the co-integration test, VAR model, and Granger causality test to investigate the correlation between the stock markets (see, e.g., [11–13]). However, considering the weak autocorrelation and conditional heterogeneity of stock market returns, it is unreasonable to use traditional regression approach, since it imposes the strong conditions of independent homogeneity. To overcome this problem, Engle [14] proposed a simple class of multivariate GARCH models referred to as dynamic conditional correlation, which is capable of capturing linear correlation between stock markets. In practice, however, the correlation between the stock markets is nonlinear, suggesting that it is unreasonable to use traditional linear models to describe the dependence structure between the stock markets. As a result, the Copula approach is suggested, which has been widely used in the literature for describing the nonlinear dependence structures. In fact, the Copula approach provides a general framework and can better capture the tail dependence. Ning [15] used the Copula to investigate the symmetrical tail dependence between the stock and foreign exchange markets. Aloui et al. [16, 17] constructed a Copula-GARCH model to explore the conditional dependence between the crude oil price, the U.S. dollar exchange rate, and the natural gas price.As the static Copula approach cannot capture the time-varying correlation between the data, some scholars suggested the time-varying Copula approach. Manner and Reznikova [18] provided a survey over existing Copula models allowing for time-varying dependencies that have been proposed in recent years. Reboredo [19] investigated the correlation between crude oil markets using the time-varying Copula approach. Wang et al. [20] used the time-varying Copula model to explore the dependence structure between Chinese and other major world economies’ stock markets. Wu et al. [21] suggested that dynamic Student’s t Copula is superior to static Student’s t and other dynamic Copulas when exploring the correlation between oil prices and exchange rates. Jammazi et al. [22] investigated the dynamic dependence between the stock and long-term government bond returns over the past two decades using the time-varying Copula approach.Recently, several scholars incorporated low-frequency macrovariables into the Copula framework to explore if macrofactors have an impact on the dependence structure between financial markets. In particular, Gong et al. [23, 24] developed a Copula-mixed-data sampling (Copula-MIDAS) model to investigate the impact of macrofundamentals, market uncertainty, and liquidity on the interdependence of Chinese stock and bond markets. Jiang et al. [25] constructed a TVM-Copula-MIDAS model to explore the interdependence between Chinese stock, bond, and fund markets. The model considers the impact of exogenous explanatory variables on dynamic dependence.However, to the best of our knowledge, there are few studies to investigate the impact of EPU on the conditional dependence between China and U.S. stock markets. Against this backdrop, this paper aims to investigate how EPU affects the conditional dependence between China and U.S. stock markets. To do so, we employ the Copula-MIDAS framework and the EPU index proposed by Baker et al. [26], which is constructed based on newspapers’ coverage frequency. Specially, for the EPU, we consider the global EPU (GEPU), the American EPU (AEPU), and the China EPU (CEPU). We investigate how different EPU indices affect the conditional dependence between China and U.S. stock markets.


## 3. Methodology

### 3.1. Marginal Distribution Model

The Copula approach allows us to separately specify models for marginal distributions. Thus, we have a lot of flexibility in selecting marginal distribution models. Considering the characteristics of the conditional heteroscedasticity of stock market returns, this paper employs the popular GARCH model to model and fit the return series. The GARCH model can be written as(1)rt=μ+εt,εt=σtzt,zt∼N0,1,σt2=ω+αεt−12+βσt−12,where rt is the return, μ is the conditional mean of the return, which is usually set to 0, σt2 is the conditional variance of the return, ϵt is the return innovation, and zt is the standardized return innovation.


### 3.2. Copula

The Copula is a multivariate marginal distribution function, which describes a flexible dependence structure between two variables and is commonly referred to as a connection function or dependence function. According to Sklar’s theorem, for a joint distribution function, the marginal distributions and the dependence structure represented by a Copula function can be separated. Letr1t and r2t be the SSEC index return and S&P 500 index return, respectively. We assume that r1t and r2t are modeled by the GARCH model presented in Section 3.1. The standardized return innovations are obtained as z1t and z2t. We assume that the conditional cumulative distribution functions of z1t and z2t are F1z1t and F2z2t, respectively, which are estimated via their empirical cumulative distribution functions F^1z1t and F^2z2t. Also, the joint distribution function of z1t and z2t is Hz1t,z2t. Then, according to Sklar’s theorem, there exists a Copula function C.,.2⟶ [0, 1] such that(2)Hz1t,z2t=Cut,vtθ,where ut=F^1z1t and vt=F^2z2t follow (0, 1) uniform distribution and θ is the dependence parameter of the Copula. Therefore, for any multidimensional distribution H, it can be divided into marginal distributions Fi and a Copula function.The Copula has good modeling characteristics, which is able to describe the tail dependence between the variables in extreme cases. It can describe the upper-tail dependence and the lower-tail dependence. The upper-tail dependence coefficientλU and the lower-tail dependence coefficient λL are defined as(3)λU=limu⟶1−ProbF2z2t≥u|F1z1t≥u=limu⟶1−1−2u+Cu,u|θ1−u,λL=limu⟶0+ProbF2z2t≤u|F1z1t≤u=limu⟶0+Cu,u|θu.IfλLλU=0, i.e., the lower (upper) tail dependence coefficient is equal to zero, there is no lower (upper) tail dependence. Also, if λLλU∈0,1, there is a lower (upper) tail dependence. Commonly used Copula functions are the binary normal Copula function, binary t Copula function, binary Archimedes Copula function, i.e., the Gaussian Copula, Student-t Copula, Gumbel Copula, Survival Gumbel Copula, Clayton Copula, and Survival Clayton Copula, which are presented in Table 1.Table 1
Copula functions.
λLλUGaussian Copula00Student-t Copula2tν+1−ν+1⋅1−θ/1+θ2tν+1−ν+1⋅1−θ/1+θGumbel Copula0λU=2−21/θSurvival Gumbel CopulaλL=2−21/θ0Clayton CopulaλL=21/θ0Survival Clayton Copula0λU=21/θAs can be seen from Table1, the Gaussian Copula cannot describe the tail dependence, while the Student-t Copula can describe the symmetrical tail dependence but cannot describe the asymmetric dependence. The Gumbel Copula and Survival Clayton Copula can only describe the upper-tail dependence, but not the lower-tail dependence. The Survival Gumbel Copula and Clayton Copula can describe the lower-tail dependence rather than the upper-tail dependence.


### 3.3. Copula-MIDAS

In this paper, we aim to investigate the impact of EPU on the conditional dependence between China and U.S. stock markets. As the frequency of EPU index (monthly) is different from that of the U.S. and China stock market returns (daily), the traditional Copula approach cannot be used for our purpose. In order to solve the mixed-frequency data problem, this paper employs the Copula-MIDAS model that can directly incorporate EPU index into the dynamic process of the dependence structure.The Copula-MIDAS model can be written as(4)ut,vt∼Cut,vt|θt,θt=Ψλt,(5)λt=Cτ+βλt−1+α1S∑s=1SF1−1ut−sF2−1vt−s,(6)Cτ=c¯+γ∑k=1KcϕkωcEPUτ−k,where EPUt denotes the GEPU, AEPU, or CEPU, which may affect the dependence between the China and U.S. stock markets, and Ψ⋅ is an appropriate transformation function to ensure that the parameter always remains in its domain.The basic idea of the Copula-MIDAS model is to decompose the dependence structure described by Copula into long-term and short-term components using the MIDAS approach, which is similar to that of GARCH-MIDAS model.In equation (5), we assume that β<1, which ensures that λt is a stationary process. The lag order S is determined by the Akaike information criterion (AIC) and Schwarz information criterion (SC).In equation (6), the long-term trend of λt, Cτ, is modeled by smoothing the monthly variable EPUτ in the spirit of MIDAS regression approach. Kc is the number of MIDAS lags. Following Colacito et al. [27], the weighting function ϕkωp,c is chosen as the beta polynomial function:(7)ϕkωc=1−i/Kcωc−1∑i=1Kc1−i/Kcωc−1.To implement the Copula and Copula-MIDAS models, we adopt a two-step procedure for the estimation, namely, the inference functions for the margins (IFM) method by Joe and Xu [28]. This method breaks the estimation into two steps. To be specific, in the first step, the marginals are obtained by estimating the GARCH model via the quasi-maximum likelihood method. In the second step, given the estimated marginal functions, we estimate the parameters of the Copula via the quasi-maximum likelihood method.


## 3.1. Marginal Distribution Model

The Copula approach allows us to separately specify models for marginal distributions. Thus, we have a lot of flexibility in selecting marginal distribution models. Considering the characteristics of the conditional heteroscedasticity of stock market returns, this paper employs the popular GARCH model to model and fit the return series. The GARCH model can be written as(1)rt=μ+εt,εt=σtzt,zt∼N0,1,σt2=ω+αεt−12+βσt−12,where rt is the return, μ is the conditional mean of the return, which is usually set to 0, σt2 is the conditional variance of the return, ϵt is the return innovation, and zt is the standardized return innovation.


## 3.2. Copula

The Copula is a multivariate marginal distribution function, which describes a flexible dependence structure between two variables and is commonly referred to as a connection function or dependence function. According to Sklar’s theorem, for a joint distribution function, the marginal distributions and the dependence structure represented by a Copula function can be separated. Letr1t and r2t be the SSEC index return and S&P 500 index return, respectively. We assume that r1t and r2t are modeled by the GARCH model presented in Section 3.1. The standardized return innovations are obtained as z1t and z2t. We assume that the conditional cumulative distribution functions of z1t and z2t are F1z1t and F2z2t, respectively, which are estimated via their empirical cumulative distribution functions F^1z1t and F^2z2t. Also, the joint distribution function of z1t and z2t is Hz1t,z2t. Then, according to Sklar’s theorem, there exists a Copula function C.,.2⟶ [0, 1] such that(2)Hz1t,z2t=Cut,vtθ,where ut=F^1z1t and vt=F^2z2t follow (0, 1) uniform distribution and θ is the dependence parameter of the Copula. Therefore, for any multidimensional distribution H, it can be divided into marginal distributions Fi and a Copula function.The Copula has good modeling characteristics, which is able to describe the tail dependence between the variables in extreme cases. It can describe the upper-tail dependence and the lower-tail dependence. The upper-tail dependence coefficientλU and the lower-tail dependence coefficient λL are defined as(3)λU=limu⟶1−ProbF2z2t≥u|F1z1t≥u=limu⟶1−1−2u+Cu,u|θ1−u,λL=limu⟶0+ProbF2z2t≤u|F1z1t≤u=limu⟶0+Cu,u|θu.IfλLλU=0, i.e., the lower (upper) tail dependence coefficient is equal to zero, there is no lower (upper) tail dependence. Also, if λLλU∈0,1, there is a lower (upper) tail dependence. Commonly used Copula functions are the binary normal Copula function, binary t Copula function, binary Archimedes Copula function, i.e., the Gaussian Copula, Student-t Copula, Gumbel Copula, Survival Gumbel Copula, Clayton Copula, and Survival Clayton Copula, which are presented in Table 1.Table 1
Copula functions.
λLλUGaussian Copula00Student-t Copula2tν+1−ν+1⋅1−θ/1+θ2tν+1−ν+1⋅1−θ/1+θGumbel Copula0λU=2−21/θSurvival Gumbel CopulaλL=2−21/θ0Clayton CopulaλL=21/θ0Survival Clayton Copula0λU=21/θAs can be seen from Table1, the Gaussian Copula cannot describe the tail dependence, while the Student-t Copula can describe the symmetrical tail dependence but cannot describe the asymmetric dependence. The Gumbel Copula and Survival Clayton Copula can only describe the upper-tail dependence, but not the lower-tail dependence. The Survival Gumbel Copula and Clayton Copula can describe the lower-tail dependence rather than the upper-tail dependence.


## 3.3. Copula-MIDAS

In this paper, we aim to investigate the impact of EPU on the conditional dependence between China and U.S. stock markets. As the frequency of EPU index (monthly) is different from that of the U.S. and China stock market returns (daily), the traditional Copula approach cannot be used for our purpose. In order to solve the mixed-frequency data problem, this paper employs the Copula-MIDAS model that can directly incorporate EPU index into the dynamic process of the dependence structure.The Copula-MIDAS model can be written as(4)ut,vt∼Cut,vt|θt,θt=Ψλt,(5)λt=Cτ+βλt−1+α1S∑s=1SF1−1ut−sF2−1vt−s,(6)Cτ=c¯+γ∑k=1KcϕkωcEPUτ−k,where EPUt denotes the GEPU, AEPU, or CEPU, which may affect the dependence between the China and U.S. stock markets, and Ψ⋅ is an appropriate transformation function to ensure that the parameter always remains in its domain.The basic idea of the Copula-MIDAS model is to decompose the dependence structure described by Copula into long-term and short-term components using the MIDAS approach, which is similar to that of GARCH-MIDAS model.In equation (5), we assume that β<1, which ensures that λt is a stationary process. The lag order S is determined by the Akaike information criterion (AIC) and Schwarz information criterion (SC).In equation (6), the long-term trend of λt, Cτ, is modeled by smoothing the monthly variable EPUτ in the spirit of MIDAS regression approach. Kc is the number of MIDAS lags. Following Colacito et al. [27], the weighting function ϕkωp,c is chosen as the beta polynomial function:(7)ϕkωc=1−i/Kcωc−1∑i=1Kc1−i/Kcωc−1.To implement the Copula and Copula-MIDAS models, we adopt a two-step procedure for the estimation, namely, the inference functions for the margins (IFM) method by Joe and Xu [28]. This method breaks the estimation into two steps. To be specific, in the first step, the marginals are obtained by estimating the GARCH model via the quasi-maximum likelihood method. In the second step, given the estimated marginal functions, we estimate the parameters of the Copula via the quasi-maximum likelihood method.


## 4. Data

For our empirical analysis, we use the daily returns of the SSEC index and the S&P 500 index for the sample period from January 4, 2005, to January 5, 2021, resulting in a total of 3770 observations. The data were obtained from the Wind Database. For the EPU indices, we choose the GEPU, AEPU, and CEPU, for the sample period from January 2005 to January 2021, resulting in a total of 192 monthly data.Table2 presents the descriptive statistics of the daily SSEC and S&P 500 index returns as well as the monthly GEPU, AEPU, and CEPU indices. As can be seen from Table 2, the means of the SSEC and S&P 500 index returns are larger than zero, indicating that the two stock markets have positive returns in the long term. Comparing the standard deviations of the SSEC index and S&P 500 index returns, we can observe that China stock market has a higher volatility than the U.S. stock market. The skewness and kurtosis of the index returns reveal that the two return series exhibit a leptokurtic and heavy-tailed distribution. In addition, it can be seen from the Jarque–Bera statistics that the return series do not follow the normal distribution. The means for the EPU indices from high to low are CEPU, GEPU, and AEPU, implying that Chinese government makes more frequent changes to its economic policies than others.Table 2
Descriptive statistics of daily SSEC and S&P 500 index returns as well as monthly GEPU, AEPU, and CEPU indices.
SSECS&P 500GEPUAEPUCEPUMean0.00030.0003145.2330127.0769150.5792Min.−0.1276−0.137848.819657.202623.7000Max.−0.13780.1096429.5147350.4598661.8000Std.0.01640.012972.934249.5498114.4801Skewness−0.6053−0.72261.24061.31541.9207Kurtosis8.322619.05934.45445.53536.9297JB4681.677240851.152966.1778106.7857241.5982Note. Std. denotes the standard deviation, and JB denotes the Jarque–Bera statistics.Figure1 presents the time series plots of the daily SSEC index and S&P 500 index returns. It is obvious that the two return series exhibit well-known behaviors of volatility clustering. Figure 2 plots the monthly GEPU, AEPU, and CEPU indices. As can be seen from Figure 2, the GEPU, AEPU, and CEPU exhibit an increasing trend, particularly in recent years.Figure 1
Daily SSEC index and S&P 500 index returns. (a) SSEC: returnr1,t. (b) S&P 500 return: r2,t.
(a)(b)Figure 2
Monthly GEPU, AEPU, and CEPU indices. (a) Global EPU. (b) American EPU. (c) Chinese EPU.
(a)(b)(c)


## 5. Empirical Results

In this section, we present the empirical results. In Section5.1, we present the estimation results for the marginal distribution model and obtain the standardized residual (return innovation) series. In Section 5.2, various static Copulas are compared based on the empirical cumulative distribution functions computed from the standardized residual (return innovation) series, and the best fitting Copula is selected. In Section 5.3, we employ the Copula-MIDAS framework based on the best fitting Copula to investigate the impact of EPU (including the GEPU, AEPU, and CEPU) on the dynamic tail dependence between China and U.S. stock markets.


### 5.1. Estimation Results for Marginal Distribution Model

Before modeling the index returns, the ARCH test is conducted. The ARCH test results are presented in Table3. It can be seen from Table 3 that the ARCH effects for both the SSEC and S&P 500 index returns are significant, suggesting that the index returns should be modeled by the GARCH model.Table 3
ARCH test results for the SSEC index and S&P 500 index returns.
NameValueStandard errort statisticP valueSSECConstant1.1105e − 063.2666e − 073.39970.0007ARCH (1, 1)0.060330.003716.48604.6519e − 61S&P 500Constand2.7209e − 064.0015e − 076.79971.0482e − 11ARCH (1, 1)0.13990.008516.39102.2015e − 6Table4 presents the parameter estimation results for the marginal distribution model, namely, the GARCH model. As can be seen from the table, the estimates of the persistence coefficient, α+β, are close to 1, suggesting that the volatilities of China and U.S. stock markets exhibit high persistence. In particular, we find that the volatility persistence of China stock market is higher than that of U.S. stock market.Table 4
Estimation results for the marginal distribution model.
μωβαLog-likSSEC0.00031.1081E − 060.93880.05991.0696e + 04(0.0002)(3.1475E − 07)(0.0029)(0.0036)S&P 5000.00063.0093E − 060.82970.14911.2234e + 04(0.0001)(4.1251E − 07)(0.0097)(0.0094)Note. The number in parenthesis is the standard error.Based on the parameter estimation results reported in Table4, we obtain the standardized residual (return innovation) series, which are presented in Figure 3. Figure 4 presents the QQ plots of the standardized residuals z1t and z2t. It is clear that the residuals do not follow the standard normal distribution. Thus, to estimate the marginal distributions, we use the empirical cumulative distribution functions of the standardized residuals z1t and z2t, which are presented in Figure 5.Figure 3
Standardized residual series for the SSEC index and S&P 500 index returns. (a)z1t. (b) z2t.
(a)(b)Figure 4
QQ plots of the standardized residualsz1t and z2t. (a) QQ plot of z1t. (b) QQ plot of z2t.
(a)(b)Figure 5
Marginal distributions ofz1t and z2t. (a) Marginal distribution of z1t (ut). (b) Marginal distribution of z2t (vt).
(a)(b)


### 5.2. Estimation Results for Static Copulas

Based on the estimated marginals, various static Copulas are estimated via the quasi-maximum likelihood method. The estimation results of various Copulas are reported in Table5. It can be seen from Table 5 that the parameter estimates of various Copulas are significant, indicating that there is a certain correlation between China and U.S. stock markets. In particular, we find that the Student-t Copula is the best fitting Copula for describing the dependence between China and U.S. stock markets in terms of the Log-lik and AIC values.Table 5
Estimation results for various Copulas.
GaussianStudent-tGumbelSurvival GumbelClaytonSurvival Claytonθ0.11910.11651.06271.06510.12500.1099(0.0160)(0.0166)(0.0113)(0.0110)(0.0198)(0.0199)ν31.1864(17.5890)Log-lik26.734928.435019.663727.517724.894318.1982AIC−51.4698−52.8700−37.3274−52.0123−47.7887−34.3964Note. Log-lik denotes the log likelihood, and AIC denotes the Akaike information criterion.


### 5.3. Estimation Results for Dynamic Student-t Copulas

To capture the dynamic dependence between China and U.S. stock markets, the dynamic Copula approach should be used. Traditionally, the time-varying Copula model is used. In the time-varying Copula model, we have(8)λt=ω+βλt−1+α1S∑s=1SF1−1ut−sF2−1vt−s.It should be noted that in the time-varying Copula model, the EPU cannot be incorporated. As a consequence, the traditional time-varying Copula model cannot capture the impact of EPU on the correlation between China and U.S. stock markets.In this section, we compare the time-varying Copula model to the Copula-MIDAS model incorporating EPU for describing the tail dependence between China and U.S. stock markets. The specific Copula is chosen as the best fitting Copula, i.e., Student-t Copula. Estimation results for the two types of dynamic Student-t Copulas, the time-varying Student-t Copula and the Student-t Copula-MIDAS, are presented in Table 6.Table 6
Estimation results for dynamic Student-t Copulas.
Time-varying Student-t CopulaStudent-t Copula-MIDASGEPUAEPUCEPUω0.00111.14331.65451.0295(0.0022)(0.5502)(1.2871)(0.3407)α0.01050.0899−0.00630.0107(0.0066)(0.1235)(0.1582)(0.0157)β1.9930−1.8775−1.86471.7766(0.0255)(0.2469)(0.4247)(0.3783)ν3.43623.44563.49173.5096(0.6345)(0.6308)(0.6598)(0.6692)c¯−2.5240−2.6225−0.1229(0.7195)(0.9822)(0.2000)γ0.61620.64780.0320(0.1499)(0.2078)(0.0522)Log-lik31.895136.379133.502338.4785Note. Log-lik represents the log likelihood.As can be seen from Table6, compared to the time-varying Student-t Copula model, the Student-t Copula-MIDAS model with EPU indices yields higher log-likelihood values, indicating that the EPU affects the correlation between China and U.S. stock markets. The positive estimates of the coefficient γ suggest that EPU has a positive impact on the conditional tail dependence of China and U.S. stock markets. It should also be noted that the estimates of coefficient γ for the GEPU and AEPU are significant, while they are not significant for the CEPU, suggesting that both the GEPU and AEPU have significant impacts on the correlation between China and U.S. stock markets, while CEPU has no significant impact. The possible explanation is that although China’s influence in the world economy is growing, as an emerging market, it is difficult for China’s economic policies to influence the U.S. stock market. Therefore, the correlation between China and U.S. stock market will be more significantly affected by GEPU and AEPU.Figure6 presents the estimated tail dependences from the Student-t Copula-MIDAS model with GEPU, AEPU, and CEPU. It is clear that the tail dependence between China and U.S. stock markets varies greatly over time, indicating that the extreme risk between China and U.S. stock markets is changing dynamically. In particular, we observe that the tail dependence between China and U.S. stock markets exhibits an increasing trend, particularly in recent years.Figure 6
Estimated tail dependences from Student-t Copula-MIDAS model with GEPU, AEPU, and CEPU.


## 5.1. Estimation Results for Marginal Distribution Model

Before modeling the index returns, the ARCH test is conducted. The ARCH test results are presented in Table3. It can be seen from Table 3 that the ARCH effects for both the SSEC and S&P 500 index returns are significant, suggesting that the index returns should be modeled by the GARCH model.Table 3
ARCH test results for the SSEC index and S&P 500 index returns.
NameValueStandard errort statisticP valueSSECConstant1.1105e − 063.2666e − 073.39970.0007ARCH (1, 1)0.060330.003716.48604.6519e − 61S&P 500Constand2.7209e − 064.0015e − 076.79971.0482e − 11ARCH (1, 1)0.13990.008516.39102.2015e − 6Table4 presents the parameter estimation results for the marginal distribution model, namely, the GARCH model. As can be seen from the table, the estimates of the persistence coefficient, α+β, are close to 1, suggesting that the volatilities of China and U.S. stock markets exhibit high persistence. In particular, we find that the volatility persistence of China stock market is higher than that of U.S. stock market.Table 4
Estimation results for the marginal distribution model.
μωβαLog-likSSEC0.00031.1081E − 060.93880.05991.0696e + 04(0.0002)(3.1475E − 07)(0.0029)(0.0036)S&P 5000.00063.0093E − 060.82970.14911.2234e + 04(0.0001)(4.1251E − 07)(0.0097)(0.0094)Note. The number in parenthesis is the standard error.Based on the parameter estimation results reported in Table4, we obtain the standardized residual (return innovation) series, which are presented in Figure 3. Figure 4 presents the QQ plots of the standardized residuals z1t and z2t. It is clear that the residuals do not follow the standard normal distribution. Thus, to estimate the marginal distributions, we use the empirical cumulative distribution functions of the standardized residuals z1t and z2t, which are presented in Figure 5.Figure 3
Standardized residual series for the SSEC index and S&P 500 index returns. (a)z1t. (b) z2t.
(a)(b)Figure 4
QQ plots of the standardized residualsz1t and z2t. (a) QQ plot of z1t. (b) QQ plot of z2t.
(a)(b)Figure 5
Marginal distributions ofz1t and z2t. (a) Marginal distribution of z1t (ut). (b) Marginal distribution of z2t (vt).
(a)(b)


## 5.2. Estimation Results for Static Copulas

Based on the estimated marginals, various static Copulas are estimated via the quasi-maximum likelihood method. The estimation results of various Copulas are reported in Table5. It can be seen from Table 5 that the parameter estimates of various Copulas are significant, indicating that there is a certain correlation between China and U.S. stock markets. In particular, we find that the Student-t Copula is the best fitting Copula for describing the dependence between China and U.S. stock markets in terms of the Log-lik and AIC values.Table 5
Estimation results for various Copulas.
GaussianStudent-tGumbelSurvival GumbelClaytonSurvival Claytonθ0.11910.11651.06271.06510.12500.1099(0.0160)(0.0166)(0.0113)(0.0110)(0.0198)(0.0199)ν31.1864(17.5890)Log-lik26.734928.435019.663727.517724.894318.1982AIC−51.4698−52.8700−37.3274−52.0123−47.7887−34.3964Note. Log-lik denotes the log likelihood, and AIC denotes the Akaike information criterion.


## 5.3. Estimation Results for Dynamic Student-t Copulas

To capture the dynamic dependence between China and U.S. stock markets, the dynamic Copula approach should be used. Traditionally, the time-varying Copula model is used. In the time-varying Copula model, we have(8)λt=ω+βλt−1+α1S∑s=1SF1−1ut−sF2−1vt−s.It should be noted that in the time-varying Copula model, the EPU cannot be incorporated. As a consequence, the traditional time-varying Copula model cannot capture the impact of EPU on the correlation between China and U.S. stock markets.In this section, we compare the time-varying Copula model to the Copula-MIDAS model incorporating EPU for describing the tail dependence between China and U.S. stock markets. The specific Copula is chosen as the best fitting Copula, i.e., Student-t Copula. Estimation results for the two types of dynamic Student-t Copulas, the time-varying Student-t Copula and the Student-t Copula-MIDAS, are presented in Table 6.Table 6
Estimation results for dynamic Student-t Copulas.
Time-varying Student-t CopulaStudent-t Copula-MIDASGEPUAEPUCEPUω0.00111.14331.65451.0295(0.0022)(0.5502)(1.2871)(0.3407)α0.01050.0899−0.00630.0107(0.0066)(0.1235)(0.1582)(0.0157)β1.9930−1.8775−1.86471.7766(0.0255)(0.2469)(0.4247)(0.3783)ν3.43623.44563.49173.5096(0.6345)(0.6308)(0.6598)(0.6692)c¯−2.5240−2.6225−0.1229(0.7195)(0.9822)(0.2000)γ0.61620.64780.0320(0.1499)(0.2078)(0.0522)Log-lik31.895136.379133.502338.4785Note. Log-lik represents the log likelihood.As can be seen from Table6, compared to the time-varying Student-t Copula model, the Student-t Copula-MIDAS model with EPU indices yields higher log-likelihood values, indicating that the EPU affects the correlation between China and U.S. stock markets. The positive estimates of the coefficient γ suggest that EPU has a positive impact on the conditional tail dependence of China and U.S. stock markets. It should also be noted that the estimates of coefficient γ for the GEPU and AEPU are significant, while they are not significant for the CEPU, suggesting that both the GEPU and AEPU have significant impacts on the correlation between China and U.S. stock markets, while CEPU has no significant impact. The possible explanation is that although China’s influence in the world economy is growing, as an emerging market, it is difficult for China’s economic policies to influence the U.S. stock market. Therefore, the correlation between China and U.S. stock market will be more significantly affected by GEPU and AEPU.Figure6 presents the estimated tail dependences from the Student-t Copula-MIDAS model with GEPU, AEPU, and CEPU. It is clear that the tail dependence between China and U.S. stock markets varies greatly over time, indicating that the extreme risk between China and U.S. stock markets is changing dynamically. In particular, we observe that the tail dependence between China and U.S. stock markets exhibits an increasing trend, particularly in recent years.Figure 6
Estimated tail dependences from Student-t Copula-MIDAS model with GEPU, AEPU, and CEPU.


## 6. Conclusions

This paper employs the Copula-MIDAS framework to investigate the impact of EPU on the tail dependence between China and the U.S. stock markets. For the EPU, we consider the GEPU, AEPU, and CEPU. Our empirical results demonstrate that the tail dependence between China and U.S. stock markets is symmetrical, and thet Copula outperforms alternative static Copulas in terms of in-sample goodness of fit. In particular, we find that the t Copula-MIDAS model with EPU dominates the traditional time-varying t Copula in terms of in-sample fitting. Moreover, we observe that both the GEPU and AEPU have a significantly positive impact on the conditional dependence between China and U.S. stock markets, whereas CEPU has no significant impact. The tail dependence between China and U.S. stock markets exhibits an increasing trend, particularly in recent years.It is worth pointing out that the approach adopted in the paper could be extended. While the simple GARCH model is used as the marginal distribution model in the paper, it would be interesting to use the realized GARCH model instead of the GARCH model, which exploits the intraday high-frequency data. This would lead to a better fitting result. Moreover, future research could extend our work by investigating the role of EPU in risk management and portfolio decision using the Copula-MIDAS approach. We leave these for future research.



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*Source: 8137932-2022-01-07.xml*