Volatility Asymmetry of Scale Indexes-Taking China as an Example

This study focused on the volatility asymmetry of scale indexes in China’s stock market. A total of 12 indexes in four categories were studied during the study period, which lasted from January 1, 2012 to September 30, 2018. The study results showed that large-scale volatility asymmetry almost never occurred but small-scale volatility asymmetry was very significant, which could result from the status of information reception in China. It was easier for large companies to receive information than small companies; therefore, compared with small companies, information asymmetry rarely happened to large companies, leading to the situation where small companies were more likely to have fiercer negative responses to information. However, this study did not divide the study period into bull markets and bear markets, and the addition and subtraction of these two periods could dilute the causes of the disappearance of large-scale volatility asymmetry.


INTRODUCTION
In the past, the studies on asset prices and risks usually used returns to explain risks, however risks have become the factor to explain returns since Fama (1971) indicated the relationship between risks and returns. Risk is a variable that cannot be directly quantified by the market; hence, it is necessary to find a suitable risk proxy variable. The traditional financial literature mostly supports the viewpoint of using volatility as a risk proxy variable. According to the capital asset pricing model (CAPM) proposed by Sharpe (1964), Lintner (1965) and Mossin (1966), it is more certain that system risks are an important factor in explaining returns on assets. Since then, the arbitrage pricing theory (APT) proposed by Ross (1976), the three-factor model (market, scale, and book-to-market ratio) proposed by Fama andFrench (1992, 1993), and the four factors (market, scale, book-to-market ratio, and momentum effect) proposed by Carhart (1997) have been used to predict returns. The volatility of these factors is the main explanatory variable, therefore studies on volatility have been a major field of financial commodity prices. Additionally, with the development of measuring instruments, volatility has been extended from static estimations to dynamic models for prediction. Therefore, this study proposed using dynamic models to investigate the changes in risk (volatility) of real estate in Taiwan.
Volatility asymmetry attracted considerable attention in academic circles in the past. For example, studies conducted by Chelley-Steeley and Steeley (1996), Laopodis (1997), Hung (1997), Yang (2000), and other scholars showed the phenomenon of volatility asymmetry. The so-called phenomenon of volatility asymmetry refers to new information causing price volatility. If the new information is positive, the future price volatility will be reduced. This Journal is licensed under a Creative Commons Attribution 4.0 International License On the contrary, if the new information is negative, the future price volatility will rise. The phenomenon of volatility asymmetry first appeared in studies on stick markets, such as when Black (1976) verified the negative relationship between the current returns and future volatility of the stock market. Christie (1982), Schwert (1990), and Koutmos and Saidi (1991) also found the same results. From the above studies, it can be deduced that companies' financial leverage ratios will rise when new information causes stock prices to fall; in other words, the risks of holding stocks will increase (the volatility of future returns is high). On the contrary, when new information causes stock prices to rise, the companies' financial leverage ratios will fall and the volatility of future returns will be low. This phenomenon is called the leverage effect. Nevertheless, Lo and MacKinlay (1988) believed that this phenomenon originates from nonsynchronous trading. Sentana and Wadhwani (1992) deduced that the phenomenon of volatility asymmetry is caused by traders' herding behavior. Shih-Yung et al. (2012) found that government policies also have significant effects on risk changes in asset investment. Hence, no unanimous conclusion has been reached on whether the volatility asymmetry of stock returns is caused by the leverage effect.
On the other hand, prices and trading volumes in the stock market can be considered high-frequency financial time series data. Brooks (2002) believed that the phenomena related to financial time series, such as leptokurtosis and volatility clustering, cannot be processed by using linear models, therefore it is necessary to consider using non-linear models. On finance, the two series of models most commonly used are ARCH (Autoregressive Conditional Heteroskedasticity) models and switching models. Among them, ARCH models are the most widely used. The ARCH models were proposed by Engle (1982) and were extended to GARCH models (Generalized ARCH) by Bollerslev (1986) to describe the volatility clustering of returns. Studies on volatility mainly focus on volatility asymmetry and the long-term and shortterm effects of volatility. GARCH models cannot differentiate the different effects of positive and negative information on the degree of volatility (volatility asymmetry), therefore Nelson (1991) developed the exponential GARCH model (EGARCH) for differentiation, while Campbell and Hentschel (1992) created the Quadratic GARCH model (QGARCH) to fit the phenomenon of volatility asymmetry. However, through a comparison of both series of models, Engle et al., (1993) found that the fitness of the EGARCH model was better. Hafner et al., (1998) also used empirical data to verify the advantages of the EGARCH model in fitting the volatility of high-frequency data.
About the volatility asymmetry of scale indexes, Cheung and Ng (1992) first proposed that the volatility asymmetry of company scale indexes will change over time. Duffee (1995), in a study of the NYSE and AMEX, found that the volatility of small-scale indexes is higher than that of large-scale indexes. Shih-yung et al. (2014) studied Taiwan and found that the volatility asymmetry of small-scale indexes is less likely to exist and that the volatility asymmetry of large-scale indexes was reduced significantly after the financial tsunami. So far, it has been developed for 30 years since the pilot project was started in 1989, and the compilation of these indexes, such as industry indexes and scale indexes, has matured. Hence, this study explored the volatility asymmetry of the scale indexes based on scale indexes in China.
This study consists of four sections. Section 1 is the introduction, which introduces the literature on volatility asymmetry. Section 2 shows the research method and further introduces the methods used in this study. Section 3 shows the data and empirical analysis, including brief descriptions of the data used in the study and analysis of the volatility asymmetry of 12 major scale indexes in China. The last section is the conclusion, which summarizes the analysis results and provides suggestions.

RESEARCH METHODOLOGY
This study explored the effects of volatility in different company scales. EGARCH was adopted in this paper to discuss volatility asymmetry. According to the suggestions of Bollerslev et al. (1992), models should be designed according to the principle of simplicity. This study used the EGARCH (1,1) models developed by Nelson (1991) for analysis, and the models are described below.
In the above models, Eq.
(1) represents that, based on the information set (I t-1 ) in the Period t-1, the return (R t ) in period t is subject to a distribution in which both the expected values and variances change with time. Eq. (2) describes the auto-correlated behavior of a return, namely, the mean equation. In Eq. (3), the variance equation is the key formula of the EGARCH models. It shows that the variance is also exponentially auto-correlated and that the residual will affect the future variance, where, z h t t t

{ H
is the value gained after the residual is standardized by the conditional variance. In Eq. (3), the effects of the standardized residual in the previous period on the current variance can be seen from the coefficient If the value of α 1 is positive, it indicates that the new information will increase the future volatility. however, positive or negative information will cause different increases in volatility. If z t-1 of the positive information is positive and z t-1 of the negative information is negative, when the coefficient δ is negative, the increase in future volatility caused by negative information will be greater than that caused by positive information; when the coefficient δ is positive, the increase in future volatility caused by positive information will be greater than that caused by negative information, which is volatility asymmetry.

Research Data
The purpose of this study was to investigate the effects of scale on volatility asymmetry. Hence, large-cap, mid-cap and small-cap indexes were selected from all the indexes in China's stock market for comparison with the composite indexes selected. After that, whether there was volatility asymmetry of all the indexes during the sample period was investigated. A total of 12 indexes in four categories were selected, as shown in Table 1, and the study period was from January 1, 2012 to September 30, 2018. The data were sourced from the Shanghai Stock Exchange, the Shenzhen Stock Exchange, and China Securities Index Co., Ltd.
As shown in Table 1, this study divided the indexes into composite, large-cap, mid-cap, and small-cap indexes. The composite indexes included the SSE composite index and the SZSE composite index; the large-cap indexes included the mega-cap index, the SSE 50 index, the CSI 100 index, and the SZSE 100 index; the mid-cap indexes included the SSE mid-cap index, the CSI 200 index, and the SZSE 200 index; and the small-cap indexes included the SSE small-cap index, the CSI 500 index, and the SZSE 700 index. The scale index division and index abbreviations of this study are shown in Table 2.
The trend charts of all indexes are shown in Figures 1-12, which indicate that all the peaks of the stock indexes were concentrated in June 2015 as well as from 2012 to 2014, and that the stocks entered a consolidation phase after 2016.

Descriptive Statistics
Firstly, the basic statistics (including sample sizes, means, standard deviations, skewness coefficients, and kurtosis coefficients in all periods), the Jarque-Bera statistics, and the ADF and PP unit root test statistics of the return sequences of the four major categories of indexes were observed, and the data were collected and recorded as shown in Tables 3-5. According to Engle and Clive, (1987), the stationary variable series in the system is required to be verified before co-integration test of the multivariables, and the variable series must be corrected through difference. Therefore, the stationary distribution of the     According to the time-series literature by Pagan and Wickens (1989), it can be found that the unit root test includes Dickey-Fuller (DF), Augmented Dickey-Fuller (ADF) and Phillips and Perron (PP). ADF is stronger and more stable than DF. ADF and PP can solve white noise problem caused by the moving average term. The Engle and Clive (1987) pointed out that ADF is better than PP. Thus, ADF is used to test stability of the time series in this paper.
In ADF, the regression analysis is conducted for the series lagging one period and the first-order differential lag term after the firstorder difference of the non-stationary data series. First, the time tendency is not considered, and the regression equation is applied: In Eq.(5), ε t is white noise process, and the appropriate lag period n is selected to make white noise uncorrelated between error terms. From Eq. (4), β 1 = 0 if Y t is not stationary, and β 1 ≠ 0 if Y t is stationary. Thus, the statistical test is assumed to be.
H 0 : β 1 = 0 (when Y t data series has unit root, it is non-stationary time series).
H 1 : β 1 ≠ 0 (when Y t data series has no unit root, it is stationary time series).
If the time series Y t cannot reject null hypothesis H 0 after ADF, the data series needs further difference and then is substituted into the ADF model to test whether it is stationary time series. The equation is as follows: If the time series Y t rejects the null hypothesis, the data of the time series are stationary and conform to ARMA. (Y t ) is I(1) data series, and most of economic variables often show properties of I (1). I (d) shows the stationary state after d-order difference of the data.  Table 4 shows that the samples in this study were not unit roots after the first order difference.
This study investigated the GARCH models. Bollerslev (1986) indicated that ARCH/ GARCH has a leptokurtic distribution and conditional variance heterogeneity. It could be known from Table 3 that the sample size of this study showed a leptokurtic distribution, therefore it was necessary to consider conditional variance heterogeneity.
In order to understand whether the data in this study had ARCH effects, this study used the Q statistics of the Ljung-Box test for observation. The Q statistics were tested as shown in Table 5 (only LB(2) 2~L B(5) 2 ) and LB(2) 2~L B(12) 2 are listed), and most of them were significant and under the 1% significance level, which indicated that the squared residuals of all series may have autocorrelations. Therefore, they all had the phenomenon of conditional heteroscedasticity (CH). The mean equation of the GARCH models can process autocorrelations of a series and the allowable variances of the variance equation are determined by the previous variances and distractions, therefore conditional variance heterogeneity can be acceptable. Hence, the GARCH models were the appropriate choice because all the series in this study showed conditional variance heterogeneity and the samples in this study were suitable for GARCH model analysis.
In order to establish the GARCH models that would be applicable to the correlation study on the return volatility of the stock price index for all stocks, the optimal lag period in the model formulas mentioned in the above paragraphs needed to be determined; that is, the optimal lag period of the mean equation and the variance equation needed to be determined. On the lag period of the variance equation, Brooks (2002) believed that, generally, GARCH (1,1) models can capture the volatility clustering effects in data by taking the conditional variances and the squared residuals of the first lag period. Based on the purpose of parsimonious parameters, in this study, GARCH (1,1) models were first established for empirical analysis. Moreover, after parameter estimation, a test was conducted on the standardized residuals and their square to verify the existence of the autocorrelation effect and determine the applicability of the GARCH (1,1) models.
In addition, regarding the optimal lag period of the mean equation, generally speaking, AIC (Akaike's Information Criterion) is the most suitable for small samples (<100), HQC (the Hannan-Quinn Criterion) is the most suitable for medium samples (100~600), and the smallest model of SBC (the Schwartz Bayesian Criterion) is the most suitable for large samples (>600). With 1,641 samples in this study, the SBC values of all indexes in all periods were adopted, as shown in Table 6. In this study, the optimal mean equation adopted for all indexes was the data in the first lag period, as shown below.

Empirical Analysis
The purpose of this study was to explore the effects of scale on volatility asymmetry, and EGARCH was used to discuss the scale indexes and investigate the changes in return volatility. Therefore, the study focused on a comparison of the absolute values of the coefficient δ.
This study first selected large-cap, mid-cap, and small-cap indexes from all the indexes in China's stock market for comparison with the selected composite indexes, and then investigated whether there was volatility asymmetry in all the indexes during the sample period. The above-mentioned EGARCH models were mainly adopted to analyze the phenomenon of volatility asymmetry, with the results as shown in Table 7. By observing whether the value of δ had any significant difference during the two sub-periods, the degree of  volatility asymmetry in the two sub-periods before and after the luxury tax was analyzed for comparison.
Based on the significant conditions of the Jarque-Bera test shown in Table 3, this study was suitable for generalized error distribution models. Hence, the results of the AR(1)-EGARCH(1,1) models of the SZSE composite index were estimated as below: As previously mentioned, the so-called phenomenon of volatility asymmetry refers to positive or negative information that causes different increases in volatility. When the coefficient δ is negative, the increase in future volatility caused by negative information will be greater than that caused by positive information; when the coefficient δ is positive, the increase in future volatility caused by positive information will be greater than that caused by negative information.
As can be seen from Table 7, the coefficients δ of the indexes in this study were all negative, revealing that the major composite scale indexes in China showed that the increase in future volatility caused by negative information would be greater than that caused by positive information. However, only eight indexes had

CONCLUSION AND RECOMMENDATIONS
This study focused on the long-term (January 1, 2012 to September 30, 2018) volatility asymmetry of the major composite scale indexes of China (12 in total). The study results showed that longterm volatility asymmetry may not occur.
However, the study results also showed that volatility asymmetry is more likely to happen to small-scale indexes than to large-scale indexes. The results indicated that situations in which the increase in future volatility caused by negative information will be greater than that caused by positive information are more likely to happen to small-scale indexes. The results differed from previous studies which showed that volatility asymmetry is more likely to happen to large-scale indexes. The difference in results may be related to the different shareholder structure in China.
In China's stock market, volatility asymmetry is less likely to happen to large-scale indexes, which may indicate that shareholders in large companies are better informed than those in small companies. Hence, the factors (positive and negative information, and financial leverage ratio) leading to volatility asymmetry of large-scale indexes have relatively small effects. Generally speaking, volatility asymmetry is more likely to happen in a bear market (Shih-yung et al. (2014). This study covered bull markets, bear markets, and consolidation periods, and the study data may have diluted the results. Further studies conducted in different periods of time may produce more specific results.