The Volatility of High-Yield Bonds Using Mixed Data Sampling Methods

It is well known that economic policy uncertainty prompts the volatility of the high-yield bond market. However, the correlation between economic policy uncertainty and volatility of high-yield bonds is still not clear. In this paper, we employ GARCHMIDAS models to investigate their correlation with US economic policy uncertainty index and S&P high-yield bond index. The empirical studies show that mixed volatility models can effectively capture the realized volatility of high-yield bonds, and economic policy uncertainty and macroeconomic factors have significant effects on the long-term component of high-yield bonds volatility.

uncertainty of monetary policy which implies a significant positive correlation between the volatility of high-yield bonds and US economic policy uncertainty. Moreover, we have tested the full sample and sub-ample sizes in the proposed models and results of the full sample test are consistent with subsample tests. This result shows that GARCH-MIDAS models are robust for characterizing the correlation between the volatility of high-yield bonds and US economic policy uncertainty. The contribution of our work is twofold. First, while the traditional financial model can only used to research the same frequency financial data, GARCH-MIDAS model provides a way to measure the relationship between low-frequency macroeconomic data and high-frequency financial data. As a result, the GARCH-MIDAS model effectively avoids the influence of high-frequency noise and the loss of information. Second, GARCH-MIDAS model provides a practical investment method on high-yield bonds market. Our empirical results have shown that economic policy uncertainty has significant impacts on the volatility of high-yield bonds and are positively correlated. Therefore, the proposed algorithms based on GARCH-MIDAS models can be used as investment strategies for risk management in practice. The rest of the paper is organized as follows. Section 2 provides a literature review for the studies of high-yield bonds and economic policy uncertainty. Section 3 describes the algorithms based on the GARCH-MIDAS model. Section 4 describes the experimental studies. Finally, Section 5 presents our conclusions.

Literature review
Studies of long-term low-grade bonds between 1977 and 1989 indicated that the returns of high-yield bonds were between those of stocks and high-rated bonds, and their fluctuations were lower than those of high-grade bonds. As shown in [Downing, Underwood and Xing (2009) ;Hong, Lin and Wu (2012)], the behavior and structure of high-yield bonds are much closer to stocks than to bonds. Downing et al. [Downing, Underwood and Xing (2009)] found that those bonds with rating BBB or lower have positive correlations with stocks, and stock returns can be used to predict unconvertible junk and BBB-rated bonds. Hong et al. [Hong, Lin and Wu (2012)] further indicated that the relationship between stocks and high-yield bonds is very strong. The stock market can influence on high-yield bonds [Zhang and Wu (2014)]. As many portfolios include high-yield bonds issued by energy companies, studies in [Gormus, Nazlioglu and Soytas (2018)] found that oil and ethanol markets have price transmission in the high-yield bond market. On the other hand, it has been shown that economic policy uncertainty plays an increasingly important role in the high-yield bond market [Colombo (2013); Dakhlaoui and Aloui (2016)]. Economic policy uncertainty has significant implications for many markets, such as stock market [Arouri, Estay, Rault et al. (2016)], bond market [Wisniewski and Lambe (2015)] and crude oil market [Conrad, Loch and Rittler (2014)]. Moreover, Engle et al. [Engle, Ghysels and Sohn (2013)] proposed the GARCH-MIDAS to deal with different frequency data. In this model, the high-frequency return of highyield bonds was separated into high-frequency short-term components and low-frequency long-term components using MIDAS technology. The short-term components filter out the noise in the high-frequency return, and the macroeconomics variables describe the long-term components. The idea of decomposing volatility into short-term and long-term components can be traced back to Ding et al. [Ding and Granger (1996)], both of which satisfy the GARCH setting. Subsequently, to relax the assumption that the unconditional variance is constant, Engle et al. [Engle and Rangel (2008)] considered the long-term component as the setting of the time-varying variance, and proposed the Spline-GARCH model, but the long-term component and the short-term component keep the same frequency in the model setting. Engle et al. [Engle, Ghysels and Sohn (2013)] combined the mixed data sampling (MIDAS) technique and the volatility model [Ghysels, Santa-Clara and Valkanov (2006); Ghysels, Sinko and Valkanov (2007)] into the GARCH-MIDAS model to separate the long-term low-frequency components and short-term highfrequency components, and the new model allowed the use of low-frequency macroeconomic factors to characterize long-term components. Nieto et al. [Nieto, Novales and Rubio (2015)] used the GARCH-MIDAS model to study the impact of macroeconomics on the volatility of corporate bonds. In addition, machine learning techniques have been widely used in forecasting a variety of complex data, such as deep learning [Tu, Lin, Wang et al. (2018)]. Liu et al. [Liu, Xu, Yang et al. (2018)] proposed an efficient and secure arbitrary N-Party quantum key agreement protocol.

GARCH-MIDAS models and algorithms
The GARCH-MIDAS model [Engle, Ghysels and Sohn (2013)] allowed us to distinguish short-and long-run sources of volatility and link them directly to economic variables. Denote by , i t r the return of high-yield bonds on the i th day in the t th month, and by t N the number of trading days in the t th month. Therefore, , i t r can be expressed as , , , , 1, 2, , is the information set containing the past Then, short-term component , i t g follows the GARCH (1,1) process: where m and θ are parameters, K represents the lag order of a realized volatility t RV , and a weighted Beta function with two parameters is defined by Eqs.
(1) to (5) are mixed volatility models with the realized volatility, named as GARCH-MIDAS-RV models with the parameter space To study the impact of economic policy uncertainty (EPU) on the return variance, let EPU X be economic policy uncertainty factor. We consider a logarithmic form of the long-term component t τ as follows: where X m and X θ are parameters, and X K is the lag order of ( ) Eqs. (1), (2), (5) and (6) are mixed volatility models based on EPU, named as GARCH-MIDAS-X models. The detail algorithms of two GARCH-MIDAS models are given as follows. if Model does not converge 8: Go to step 4 and proceed to the next cycle 9: end if 10: Select the model (6)      Parentheses report t-statistics. *** indicates a significance at the 1% level and LLF is a log likelihood function.
It can be seen from Tab. 2 that the values of µ in the mixed volatility model with different explanatory variables are 0.5%, which implies that these models effectively capture the fluctuation of high-yield bonds. From the value of θ , it can be found that realized volatility of high-yield bonds, economic policy uncertainty and other four macroeconomic variables are positively correlated with the long-term components of the volatility of high-yield bonds. Moreover, these regression results show that the volatility of high-yield bonds is higher with the fluctuation of economic policy uncertainty, monetary policy, taxes, government spending and financial regulation, respectively.

Figure 2: Total volatility and long-term compositions of high-yield bond return fitted by mixed volatility models in full samples
In Fig. 2, the dotted line indicates the total volatility of the high-yield bonds, and the solid line indicates the secular volatility denoted by long-term components volatility of highyield bonds estimated by the mixed volatility model, i.e., t τ . It can be seen that the longterm component of the volatility has the same trend as the total volatility, which implies that the long-term component can reflect the overall trend of the total volatility. Particularly, economic policy uncertainty and monetary policy can fit better than other variables.

Robust
As can be seen from Fig. 1  Parentheses report t-statistics. *** indicates significance at the 1% level, and LLF is a log likelihood function. T is larger, which implies that after financial crisis, the long-term components of taxes and government spending for the volatility of high-yield bonds is smaller. Moreover, the long-term effects of economic policy uncertainty, taxes, government spending and financial regulation on high-yield bonds are respectively consistent in the two subsamples, while the impact of monetary policy on high-yield bonds after financial crisis is stronger than that on highyield bonds before financial crisis. In addition, the total volatility and long-term component volatility of high-yield bond return estimated by the mixed volatility model in the two subsamples are plotted, as shown in Fig. 3.

Figure 3:
Total volatility and long-term components of high-yield bonds return fitted by mixed volatility models in two subsamples In Fig. 3, the dotted line and the solid line represent the total volatility and the secular volatility denoted by long-term component volatility of high-yield bonds fitted by the mixed volatility model, respectively. It can be found that for economic policy uncertainty index, monetary policy, taxes, government spending and financial regulation, the total volatility of high-yield bonds is almost consistent with the long-term component volatility in two different subsamples.

Conclusion
In order to avoid the information leakage caused by the same frequency data, we have used GARCH-MIDAS models with a decomposition of a volatility into short-term highfrequency components and long-term low-frequency components to examine the relationship between low-frequency macroeconomic variables and high-frequency financial market volatility. Moreover, based on two types of GARCH-MIDAS models, we have investigated the impact of economic policy uncertainty on the volatility of highyield bond market. The empirical results show that both the realized volatility of highyield bonds and the macroeconomic factors are positively correlated with the long-term components of high-yield bonds. This result indicates that investors can predict the volatility of high-yield bonds from the macroeconomic information.