Uncovering The China’s Stock Market Variance

Prediction

Hang Cheng, Hui Guo, and Yongdong Shi ∗

This version: March 13, 2019

ABSTRACT

Including 25 potential variables we present a comprehensive study on China’s stock market

variance prediction with sample from 1995 to 2018. Contrary to many previous studies, we

find most economic activity variables have neglected forecasting power and scaled price ratio

have a nonlinear correlation with future variance. Based on the evidence provided by the

Bayesian model averaging and the out of sample test, the stock market turnover provides ap-

propriate additional information while illiquidity proposed by Pastor and Stambaugh (2003)

has a strong prediction ability in the quarterly data.

JEL classification: G12; C22

Keyword: China’s stock market; Time-Varying Stock Market Variance; Conditional Vari-

ance; Realized Variance; Bayesian model averaging; Out-Of-Sample

∗Cheng is at the Research Center of Applied Finance, Dongbei University of Finance and Economics;Email: chenghangch@qq.com. Guo is at the Department of Finance and Real Estate, University of Cincin-nati; Email: hui.guo@uc.edu. Shi is at the Research Center of Applied Finance, Dongbei University ofFinance and Economics; Email: ydshi@263.net. We thank Xiaoman Li, Mingyong Song, Chao Wang, JinleWang, Sanfa Wang, Tongtong Wang, Shijie Zheng and seminar participants at the University of DongbeiUniversity of Finance and Economics. Shi acknowledges the financial support of the Fundamental ResearchFunds for the National Natural Science Foundation of China [Grant Nos. 71471031, 71772030, 71702025],the major project of the National Social Science Foundation of China [Grant No. 14AZD089], DistinguishedProfessor Support Plan of Liaoning Province [Grant No. [2018]35].

Stock market volatility has a pivotal role in academic research on finance, in particular, it

is critical to understanding the time seires pattern of the stock market return. In Merton

(1973)’s ICAPM model, market risk, also known as conditional market volatility is important

determinants of expected stock market return. However, a major problem with the condi-

tional market volatility is it can not be observed directly so subsequent studies use some

potential economic variables to predict volatility. Literature suggests that economic activity

is among the most important factors for stock market volatility prediction. After Schwert

(1989) finds little correlation between economic activity and stock market volatility, scholars

use different economic variables and econometric methods to reach different conclusions.

A recent study by Paye (2012) involved, they use abundant macroeconomic and financial

variables to investigate the stock market volatility forecasting. Similar with Schwert (1989)’s

finding, they argue that lagged volatility covers a lot of future economic condition which

means the predicted benefits are small when additional economic activity variables are in-

cluded. Nonejad (2017) investigates whether information from financial and macroeconomic

variables is helpful in predicting volatility in a comprehensive Bayesian model averaging

framework. Wang, Wei, Wu, and Yin (2018) mention that crude oil volatility can predictive

stock volatility and provides different information from traditional macro variables.

In this paper we attempt to uncover the predictability of China’s stock market volatility.

Chen, Jiang, Li, and Xu (2016) draws our attention to U.S. economic variables which can

forecast the future monthly volatilities of the Chinese stock market. The study by Cai, Chen,

Hong, and Jiang (2017) offers probably the most comprehensive empirical analysis of China’s

stock market variance prediction. To be specific, they find some of the 13 variables, such

as, dividend-price ratio, inflation, turnover and changes in the M1 money supply positively

and significantly forecast the Chinese stock market volatilities. Perhaps the most serious

disadvantage of their research is that the sample spans from January 1997 to December

2012. Based on the framework of their research, we employ 25 potential economic variables

commonly used in the literature and extend the sample, from January 1995 to December

2018. To investigate the stable relation between potential economic variables and stock

market volatility, two sub-samples results in monthly and quarterly data frequency are also

reported, from 1995 to 2007, and from 2008 to 2018, respectively. Following Nonejad (2017),

we use Bayesian model averaging to find the most appropriate predictor in the 25 potential

economic variables.

When the auto-correlation of stock returns caused by artificial market mechanism is

high, the original calculation method of volatility underestimates the real volatility. Since

December 26 of 1996, the Chinese stock market has imposed a daily price limit of 10%,

we highly emphasize the impact of the original calculation on market volatility. Adjusted

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volatility measure tends to be more extreme in particular market moments. In subsequent

empirical studies, we use the adjusted volatility measurement.

The most obvious finding to emerge from this study is that most economic variables

have neglect forecasting power of future stock market variance, including inflation, money

supply growth shock, GDP growth shock, e.g. The only difference is the illiquidity measure

proposed by Pastor and Stambaugh (2003) have nearly 6% extra explanatory power over

the lagged variance in quarterly China’s data. Our finding is consistent with Chen, Eaton,

and Paye (2018)’s discussion that most aggregate illiquidity proxies contain a component

reflecting aggregate volatility.

Previously published studies on the effect of scaled price ration on market volatility are

not consistent. Campbell and Cochrane (1999) and Bansal and Yaron (2004) imply that

the scaled price ratio is monotone negative related with volatility which is consisted with

Cai et al. (2017)’s finding of China’s data. However, David and Veronesi (2013) come to

a different conclusion that volatility of stock returns is non-linearly related to the scaled

price ratio. In our findings, when we expand the data sample till 2018, we find that the

relationship in the early stage is exactly as Cai et al. (2017) present, but the results in

the later data were just the opposite. More precisely, the price dividend ratio is significant

negative related with one-quarter-ahead stock market variance at 1% levels in the sample

spans from 1995 to 2007 while it significant positive forecast next period variance from 2008

to 2018. Time-varying relationships lead to insignificant correlation across the full sample

which is similar with Beeler and Campbell (2012)’s conclusion. Both the Bayesian model

averaging and the out of sample test have proved that they have little effect on the prediction

of volatility in China’s data.

Consistent with Cai et al. (2017)’s conclusions, we also find that stock market turnover

have additional information of one-month-ahead stock market variance beyond the lagged

variance while it has 100% posterior probability in the BMA model. In the quarterly data,

the BMA posterior probability, 52.8%, imply it still provides considerable information. In-

terestingly, log(TO) is significant at 1% levels after control illquidity measure proposed by

Pastor and Stambaugh (2003) but not before.

Lastly, although this paper focuses on simple linear forecasting models, Bayesian model

averaging and out of sample testing also provide sufficient evidence. Robust empirical results

show significant variables in the in-sample regression also have a high posterior probability

in the Bayesian model averaging and play a strong role in the out-of-sample test.

The remainder of the paper is organized as follows. Section I discusses the variables con-

struction and data sources of China’s stock market. Section II describes the main empirical

findings including in-sample regression, Bayesian model averaging and out of sample testing.

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Section III offers some concluding remarks.

I. Variables

A. Stock Market Volatility

The sum of daily return’s square is one of the most common procedures for determining

the stock market volatility. Following Schwert (1989), Paye (2012) and Cai et al. (2017)

we calculate the monthly stock market volatility using Chinese data with the same method.

Accounting for the positive auto-correlation of daily stock market return as was done by

French, Schwert, and Stambaugh (1987), they measure stock market volatility as realized

variance of month t as

MVt (k) =Dt∑d=1

e2m,d,t + 2

k∑j=1

Dt∑d=j+1

em,d,tem,d−j,t, (1)

where em,d,t is the value-weighted daily excess stock market return in day d of month t and

Dt is the number of trading days in month t. The daily value-weighted stock market return is

from CSMAR and the daily risk rate is from RESSET. k is the order of serial correlations due

to non-synchronous trading; and French et al. (1987) set it to 1 for the U.S. data. Perhaps

the most serious disadvantage of setting k = 1 is that higher order positive serial correlations

in China’s daily excess stock market returns makes the volatility estimated by this method

underestimated. Due to the 10% daily return limit since December 26, 1996 (Hu, Pan, and

Wang (2018b)), artificial rule create a high degree of autocorrelation in stock market daily

returns.1

The differences between monthly realized market variances with k = 0 (MV 0, dashed

line) and with k = 3 (MV 3, solid line) are highlighted in figure 1. It was suggested that MV 3

is noticeably higher than MV 0 during turbulent periods of December 1996, June 1999, May

2006, June 2007, and August 2015 after the daily price limit was established in December 26,

1996. As expected, a high positive correlation was found between MV 3 and MV 0 as 81%.

Interestingly, MV 4 (k = 4) and MV 5 (k = 5) were observed to more similar to MV 3, with

a correlation coefficient of 96% and 93%, respectively. Comparing the different setting of k,

it suggests that it is important to adjust for high order auto-correlations when constructing

realized A-share stock market variance.

Same as the US data, Chinese stock market realized variance is positively skewed and

1The 1st-order to 5th-order autocorrelations are 4.4%, -1.3%, 1.3%, 4.3%, and 0.6%, respectively, overthe 1995 to 2018 period.

4

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Jan-95

Aug-95

Mar-96

Oct-9

6

May-97

Dec-9

7

Jul-9

8

Feb-99

Sep-99

Apr-0

0

Nov-0

0

Jun-01

Jan-02

Aug-02

Mar-03

Oct-0

3

May-04

Dec-0

4

Jul-0

5

Feb-06

Sep-06

Apr-0

7

Nov-0

7

Jun-08

Jan-09

Aug-09

Mar-10

Oct-1

0

May-11

Dec-1

1

Jul-1

2

Feb-13

Sep-13

Apr-1

4

Nov-1

4

Jun-15

Jan-16

Aug-16

Mar-17

Oct-1

7

May-18

Dec-1

8

MV MV3

Figure 1. Realized Market Variances MV 0 (Dashed Line) and MV 3 (Solid Line):The figure plots two monthly realized market variance measures constructed using equation (1)over the Jan 1995 to Dec 2018 period. We set k to 0 in equation (1) for MV 0 and to 3 for MV 3.

leptokurtotic.2 Following Andersen, Bollerslev, Diebold, and Labys (2003), Paye (2012), Cai

et al. (2017) and Nonejad (2017), we use the natural logarithm value of realized variance,

log(MV 3), as the explained variable of stock market volatility. The first auto-correlation

coefficient in table I shows the log(MV 3) is persistent, which is same as previous literature.

Finally, we use a similar method for quarterly calculations.

B. Forecasting Variables

In previous studies on forecasting volatility, different variables have been found to be

related to it, such as many macroeconomics and financial variables (Schwert (1989),Campbell

and Cochrane (1999), Bansal and Yaron (2004), Paye (2012), Christiansen, Schmeling, and

Schrimpf (2012), Girardin and Joyeux (2013), Cai et al. (2017), Nonejad (2017)). The main

purpose of this study is to assess the extent to which these factors can predict the volatility

of Chinese stock market. Next, all explanatory variables from the literature are introduced:3

2The skewness and kurtosis is 5.017 and 34.97 respectively for the monthly sample from January 1995 toDecember 2018. The skewness and kurtosisnatural of natural logarithm value of MV 3 is −0.309 and 0.387which is mentioned in table I.

3We try to obtain monthly and quarterly results for all indicators who are based on the non-ST stocksin China’s stock market.

5

• Turnover (TO): The aggregate TO is calculated as the ratio between sum of individual

stock’s trading shares to sum of individual stock’s floating shares.4

• Illiquidity measure (Pastor): This illiquidity measure Pastor is from the Pastor and

Stambaugh (2003). In contrast to their method, however, we scale the aggregate

Pastor measure by 2.93 from 2007 on.5 We exclude the stocks that have less than 7

days in each month and share prices less than 5 RMB yuan at the end of the previous

month.6 In order to solve the non-synchronous trading issue, we follow Cheng et al.

(2018) to construct the quarterly measure.

• Illiquidity measure (Amihud): We construct Amihud following Brennan, Huh, and

Subrahmanyam (2013) and Lou and Shu (2017) based on turnover instead of trading

volume in Amihud (2002). After stripping out stocks below the 7 (45) trading days,

below the 1% percentile and above the 99% percentile in each month (quarter), we use

the floating value-weighted value as Amihud.

• Scaled price ratios (pd, pb, pe): In most recent Chinese data empirical asset pricing

studies, such as Cai et al. (2017) and Liu, Stambaugh, and Yuan (2018), scaled price

ratios of Chinese stock market are consistent with the standard approaches used in the

literature for the US data. However, there are certain drawbacks associated with the

use of traditional method. Hu, Chen, Shao, and Wang (2018a) point out that only

floating A-shares can be invested by mainland investors and their market prices are

negotiated rather than traded. On the basis of Hu et al. (2018a), we follow Cheng et al.

(2018) to improve the measurement of these variables. For example, the numerator

of price to book value ratio (pb) is the market value of the folating A-shares in the

last day of each month and the denominator is the total book value from the latest

accounting statements belongs to the floating part (floating A-shares divided by the

total shares including A, B and H shares). We construct these three variables (pirce to

dividend ratio (pd), pirce to book value ratio (pb), pirce to earnings ratio (pb)) monthly

and quarterly in similar way.

• Firm-level variance (FV 3): Similar to the MV 3 construction, we mesure monthly

4The details about floating shares in Chinese stock market are mentioned in Hu et al. (2018b).5Cheng, Guo, and Shi (2018) pointed out that the Split-Share Structure Reform during the April 2005 to

December 2006 allows non-floating shares to be converted into floating shares which has increased outstandingfloating shares substantially. The ratio of the floating share market capitation by 2007Q1 to that by 2005Q1is 2.93.

6The largest trading day in February 1999 was 7 days and it is done to ensure the integrity of time seriesdata.

6

realized firm-level variance as:

FVi,t (k) =

Di,t∑d=1

e2i,d,t + 2

k∑j=1

Di,t∑d=j+1

ei,d,tei,d−j,t, (2)

The value-weighted firm-level variance is:

FVt (k) ≡Nt∑i=1

ωi,tFVi,t (k) , ωi,t =vi,t−1∑Nt

j=1 vj,t−1

, (3)

where ei,d,t is the daily excess return on stock i in day d of month t, Di,t is the number

of trading days for stock i in month t, k is the order of serial correlations due to non-

synchronous trading, and Nt is the number of stocks in month t, ωi,t is the weight of

stock i in month t, and vi,t−1 is the market capitalization of stock i’s floating shares

at the end of month t − 1. Be consistent with the above we exclude stocks that

have less than 7 trading days in month t; and results are qualitatively similar when

including all normally traded A-share stocks. To ensure a positive realized variance,

we replace FVi,t (k) with FVi,t (k) > 0 for 0 ≤ k < k when FVi,t (l) < 0 for l =

k+ 1, ..., k. We add another additional filtering criteria, while results are qualitatively

similar using unfiltered data. That is we remove the daily returns of which absolute

values exceed 10.3448% because they are associated with special events mentioned in

Hu et al. (2018b). Quarterly data are structured the same way, except that we exclude

stocks that have less than 45 trading days in each quarter.

• Idiosyncratic Variance (IV 3): In Cheng et al. (2018)’s study, IV 3 was constructed

quarterly according to China’s specific trading conditions, the non-synchronous trading

issue, and we use their methodology directly to construct quarterly data. Here we

construct the monthly value-weighted idiosyncratic variance in a similar way with

FV 3 and Guo and Savickas (2006), accompanied by the same filters. First, we regress

the individual stock’s daily excess returns on a constant and daily excess stock market

returns:

ei,d,t = α + β ∗ em,d,t + ηi,d,t, (4)

And then construct quarterly realized idiosyncratic variance as:

IVi,t (k) =

Di,t∑d=1

η2i,d + 2

k∑j=1

Di,t∑d=j+1

ηi,dηi,d−j. (5)

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The value-weighted idiosyncratic variance is

IVt (k) =Nt∑i=1

ωi,tIVi,t (k) . (6)

And just like we did before, we set k equal to 3.

• Stochastically detrended risk-free rate (RREL): We measure RREL as the difference

between the risk-free rate and its average over the past 12 months. We use the last

month of each quarter as a quarterly measure.

• Economic policy uncertainty (EPU): We get the monthly Chinese economic policy

uncertain index from the website www.policyuncertainty.com and use the natural

logarithm of the raw value as the EPU measure. Similarly, the last month of each

quarter is used as a quarterly measure.

• Stock market excess return (RET ): We obtain the floating value weighted Chinese A-

shares market return form database CSMAR and minus the risk free rate from database

RESSET as RET . The compound rate of months in each quarter is the quarterly stock

market excess return.

• Inflation (CPI): The consumer price index is from national bureau of statistics of

China. We use the natural logarithm of 1 plus growth of consumer price index year-

on-year as CPI and lag it by one month because of its delayed release. The quarterly

CPI is the natural logarithm of 1 plus the sum raw vaule of 3 months in each quater.

• Money supply (M2,M1,M0): We obtain monthly money supply data from the People’s

Bank of China directly. M0t is the shock of the M0 growth rate in month t, and we

construct M1 and M2 in a similar way. The sum of monthly value is the quarterly

measure.

• IPO first day return (IPOR) and IPO number (IPON): Similar to Baker and Wurgler

(2006) and Guo (2011), IPOR is the mean value of return between IPO first day close

price to offering price while IPO number is the number of companies that go public

that month. We adjust the offering price as closing price of the first day when the

price does not hit the limit imposed by the CSRC after 2014. In each quarter, we still

calculate the mean value of individual IPO first day return and numbers as IPOR and

IPON .

• Industrial production growth rate shock (IP): We obtain monthly industrial production

growth rate year-on-year from national bureau of statistics of China. IPt is the growth

rate of month t minus the value of month t − 1. Three months growth rate values in

each quarter add up to a quarterly growth rate and the shock is the quarterly measure.

• Consumer confidence index shock (CCI): Consumer confidence index is from the na-

8

tional bureau of statistics. We calculate it as the shock of monthly value. In each

quarter, the mean value of monthly is what we need.

• Macroeconomic Leading Index (MLI): MLI is published monthly by the national

bureau of statistics. We chose the value of the last month of each quarter as the

quarterly measurement.

• Financial leverage (FL and FL2): Consisting with Christie (1982), Schwert (1989)

and scaled price ratio, FL (FL2) is the natural logarithm ratio of the book value of

debts belongs to the floating part in the latest available accounting statements to the

market value of floating equities (debts plus market value of floating equities).

• GDP growth shock (GDP ): We use the first-order difference data of GDP growth from

national bureau of statistics only quarterly.

• Oil volatility (Voil): Following Wang et al. (2018), we get West Texas Intermediate

(WTI) crude oil daily spot price data from the website of Energy Information Admin-

istration www.eia.gov. The Voil is the sum of daily return squares in each month or

quarter.

II. Empirical Findings

A. Descriptive statistics

Table I provides the preliminary statistics of the variables mentioned over the sample

from January 1995 to December 2018 in panel A of monthly data, and from 1995Q1 to

2018Q4 for quarterly frequency in panel B. This table is quite revealing in several ways.

First, most of the predictors are persistent with the fisrt autocorrelation ρ1 greater than 0.6.

Second the unit root test proposed by Dickey and Fuller (1979) and the p value reported in

last columns show that the test rejects the null assumption that most variables have unit

roots. We exclude the unstationary time-series variables in the following regression analysis.

In Paye (2012) discussion, although Stambaugh (1999) mention that high autocorrelation

can lead to estimation bias in forecasting regressions, it is not serious enough to affect our

conclusions.

B. In sample forecasting regression

Data from several China’s empirical studies, such as Girardin and Joyeux (2013) and

Cai et al. (2017), suggest that the lagged variance have significant forecasting power for the

China’s stock market variance since realized variance is persistent with first autocorrelation

coefficient 0.417 in table I. Base on the preceding research, we use the forecasting regression

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form as:

log(MV 3)t+1 = α + ρ ∗ log(MV 3)t + β ∗Xt + εt+1, (7)

where log(MV 3)t is natural logarithm of stock market variance in equation 1 setting k = 3

in month or quarter t, Xt denotes additional predictive variable. The table II present results

with different Xt in panle A and B, accompanied by monthly and quarterly sample frequency.

There is a large volume of published studies describing that the realized variance is

positive autocrrelated in international or China’s data, such as Bollerslev, Chou, and Kroner

(1992), Cai et al. (2017), e.g.. For comparing, the first row of panel A and B in table II is

the of equation 7 regression have no Xt. It shows that there is a significant forecasing power

of lagged realized variance at 1% level in both sample frequency. It have 17.2% and 29.4%

adjusted R2 in monthly and quarterly results. The relation is stable in both sub-sample

periods, spans from 1995 to 2007 and from 2008 to 2018. Interestingly, the latter sample is

observed to have significantly higher explanatory power, no matter what the frequency of

the data. Particularly the adjusted R2 is almost attain to 47.5% in the second sub-sample

of quarterly frequency data in panel B of table II. Although lagged variance already has

strong explanatory power, next we test whether adding other variables can bring additional

explanatory power.

A number of studies have postulated a positive relation between turnover and conditional

market variance, such as Lamoureux and Lastrapes (1990) Gallant, Rossi, and Tauchen

(1992) and Harris and Raviv (1993). In the recent research of Hu et al. (2018b), they find in

China’s stock market turnover and realized variance is observed have a similar time trend.

In our forecasting regression, natural logarithm turnover is significant with one-month-ahead

log(MV 3) at 1% level and have additional around 4% adjusted R2 after control the lagged

log(MV 3) in monthly data. Although log(TO) is insignificant in the bivariate regression of

quaterly panel in B, unreported results show that raw value turnover drives MV 3 out in

the forecasting regression with one-quarter-ahead MV 3. After all, consisting with preceding

research, we find that turnover is one of the important factors of conditional market variance

in China’s stock market.

Studies such as that conducted by Stoll (2000), Watanabe and Watanabe (2005) and

Ait-Sahalia and Saglam (2017) have shown that the conditional market variance is positive

correlated with illiquidity measure, for example bid-ask spread and Pastor proposed by

Pastor and Stambaugh (2003). Chen et al. (2018) presents illiquidity contain real economic

activity infromation, which is suggested by Schwert (1989) that macro economic activity

has a deep connection with market volatility. It may imply that illiquidity have the future

information about the stock market volatility. In our empirical results, Amihud measure

based on Brennan et al. (2013) have neglectable forecasting power of ahead log(MV 3) in

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both sample frequency. In special, it is statistically significant at 10% level in the second

quarterly sample but have puzzling negative relation. A possible explanation for this might

be that turnover, which is positive correlated with log(MV 3), constitute the denominator

of Amihud based on Brennan et al. (2013).

There is a surprising remarkable outcome of Pastor measure. In the panel A of monthly

data Pastor is positive correlated to one-month-ahead log(MV 3) but insignificant. But the

relation is unstable in two sub-samples, negative for the first sub-sample and positive for the

second while the coefficient vary a lot. However, we find a positive significant and stable

correlation in the quarterly data and get nearly 6% additional adjusted R2. This discrepancy

could be attributed to the measure of return reversal proposed by Pastor and Stambaugh

(2003), which may not applicable to China’s monthly data. Longer sample of regression may

get more accurate estimation due to the daily price limit of 10%. After all while the price

impact Amihud have neglected forecasting power of market variance, the return reversal

Pastor have positive correlation with future variance in low frequency data.

Interestingly, there are also differences in the forecasting regression using scaled price

ratio. In the leading asset pricing model of Campbell and Cochrane (1999) and Bansal

and Yaron (2004), scaled market price is a monotone negative linear function of conditional

variance. Contrary to this conclusion and the empirical results from Cai et al. (2017), we

observe a positive correlation between pd and next period log(MV 3) in the all sample, while

these two variables have exact opposite relationship in two sub-samples whatever the sample

frequency is, negative in the first sample and positive in another. Especially in the quarterly

results, the inverse relationship in the two sub-samples was significant at least 5% level. This

finding is consistent with Schwert (1989) and David and Veronesi (2013), who argues that

the correlation between volatility and the price valuations change stochastically over time.

The positive correlation between one period ahead log(MV 3) and scaled price ratio is robust

when we use pb and pe in the full sample. After all our finding is contrary to US data result

and preceding China’s empirical results.

In reviewing the literature, such as Guo and Savickas (2006) and Guo, Lin, and Pai

(2018), value weighted firm level variance FV 3 and value weight idiosyncratic variance IV 3

are found to be high correlated with market variance and a dominator variable of conditional

equity premium in US stock market. Similarly, we find that although ther are correlated with

each other highly in China’s stock market, what is expected is that the prediction power

of log(FV 3) and log(IV 3) are lost as the sample frequency decreases. After controlling

the lagged log(MV 3), both of them have neglected forecasting power of next quarter stock

market variance.

The stochastically detrended risk-free rate RREL is negative with one period ahead

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variance log(MV 3) but no significant relation are found between them.

There are many empirical studies suggest an association between stock market volatil-

ity and macroeconomic activity variables (Schwert (1989), Christiansen et al. (2012), Paye

(2012), Girardin and Joyeux (2013), Mittnik, Robinzonov, and Spindler (2015), Chen et al.

(2016), Cai et al. (2017), Nonejad (2017), Wang et al. (2018) Wei, Yu, Liu, and Cao (2018)).

On the basis of previous work, we provide a more extensive and comprehensive study of this

connection and the sample interval is expanded from 1995 to the present (2018).

We include macroeconomic activity variables commonly used in literature and available in

China’s stock market.7 Our finding is similar with Schwert (1989) and Paye (2012), which is

that predictive power most macroeconomic variables are economically small when controlling

for lagged realized market variance. It is important to note that the CPI is not significant

in the post-1997 sample no matter what the frequency is. A possible explanation for this

might be that the extreme values in the 1995 and 1996 samples make the OLS estimation

results biased.

To summary, all the macroeconomic activity variables used here, such as economic pol-

icy uncertainty, inflation, money supply growth shock, IPO first day return8, IPO numbers,

industrial production growth rate shock, consumer confidence index shock, macroeconomic

leading index, financial leverage, GDP growth shock, oil volatility, have neglected forecasting

power of one-period-ahead stock market variance while controlling the one lagged variance.

log(TO) and log(Pastor) contain unstable significant posititve relation to the stock market

variance, while the scaled price raion have opposite relationship in two sub-samples. Untab-

ulated results show that IP and log(IV 3) is no longer significant when two lagged variance

are controlled and the other three variables remain the same in quarter frequency. Although

RREL is significant in the full sample, it provides around 1% of the additional explanation.

The predictive power of the remaining unmentioned indicators can be ignored.

C. Multivariate Selection regression

As mentioned above, we find most variables including macroeconomic activity variables

lost significant forecasting power after controlling the lagged stock market variance. The

debate about parameter uncertainty and model uncertainty of volatility prediction always

exists. Following Nonejad (2017) we apply Bayesian model averaging (BMA) to solve model

uncertainty of variance prediction.

7As mentioned in Cai et al. (2017), since there is no corporate bond data, the commonly used creditspread measurement in the literature is not applicable to the Chinese market.

8IPOFDR is not stationary in the full sample. Unreported results show that it has no explanatory powerin the pre-2013 sample while it is stationary.

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In table III, we present the marginal importance of the potential variables based on the

posterior probability using BMA. We exclude the log(FV 3) and log(IV 3) because of their

high correlation with log(MV 3) and only one variable of each type is retained.9

In the panel A of table III, log(MV 3), log(TO) and intercept have 100% posterior proba-

bility and all the best 5 models contain these three variables. In the model 1, both log(MV 3)

and log(TO) are significant at 1% level and have 23.1% adjusted R2 while it have the high-

est posterior probability with 27.5%, accompanied by lowest BIC value with −60.29. The

other four models, model 2 to model 5, contain another additional variable, such as Amihud,

RREL, Voil, IP . Although these additional variables are significant, they have almost neg-

ligible additional adjusted R2 comparing to model 1. On the other hand, these four models’

BIC are higher and posterior probability are lower. Moreover, these variables has minor

posterior probability of 32.6%, 18%, 8.5%, 12.3%, respectively. Variables not mentioned

have a lower probability which seems that they do not seem to matter much. Therefore, it

seems that log(MV 3) and log(TO) are the dominant forecasting variables of stock market

variance in China’s stock market monthly data.

As panle B of table III showing, similarly to the monthly frequency, log(Pastor) and

intercept have 100% posterior probability while log(MV 3) have 97.9% and all the best 5

models contain these three variables in the quarterly data. In the rest of potential variables,

log(TO), which is contained in model 1 and model 4, have the highest posterior probability

with 52.8%. It is apparent from this table that model 1 is the best model with highest

adjusted R2, posterior probability and lowest BIC. In contrast, Amihud has an intermediate

posterior probability of 40.7%, while the other covariates do not seem to matter much. The

difference between the model 1 and 2 is that Amihud replaces log(TO), which reduces the

explanatory power slightly. The observed high correlation of -0.66 between Amihud and

log(TO) might be explained this finding. Taking posterior probability and BIC as criterion,

we choose log(TO) as the last variable. log(Pastor) is not selected in the monthly data,

which is probably caused by the error of monthly data estimation.

D. Out-of-sample Test

Out of sample test have been used to investigate the mechanical properties of time-series

forecasting model, such as Welch and Goyal (2008), Campbell and Thompson (2008), Cai

et al. (2017). In this section, we perform the out of sample test by report the R2oos, ENC-

NEW proposed by Clark and McCracken (2001) and MSE-F which is the equal forecast

9For example, we retain pd for scaled price ratio, m2g for money supply growth shock. Our results arerobust for different measure.

13

accuracy test developed by McCracken (1999). For detail, R2oos is calculated as follows:

R2oos = 1 −

∑T−1n (MVa −MV )2∑T−1n (MVb −MV )2

(8)

where MVa and MVb are the predicted value of augmented model and benchmark model

respectively. MV is the realized value of stock market variance. T is the length of the entire

sample while n is the number of in-sample. If the augmented model performs better than

benchmark model, R2oos is going to be greater than 0. The calculation formula of ENC–NEW

and MSE-F tests are as follows:

ENC–NEW = (T − n)

∑T−1n [(MVb −MV )2 − (MVa −MV ) (MVb −MV )]∑T−1

n (MVa −MV )2(9)

MSE − F = (T − n)

∑T−1n [(MVb −MV )2 − (MVa −MV )2]∑T−1

n (MVa −MV )2(10)

We use the first 96 observations, spans from January 1995 to December 2003, for the initial

in-sample estimation and make out of sample forecasts for the rest period , from January

2004 to November 2018, using an expanding sample. We report the out of sample test

results in table IV. This also accords with our earlier in sample findings, which showed that

in the univariate forecast model, lagged log(MV 3) have statistically significant out-of-sample

forecasting power for one-month-ahead stock market variance, whether using the ENC–NEW

statistic or the MSE-F statistic along with the asymptotic critical value provided by Clark

and McCracken (2001) and McCracken (1999).

The evidence is consistent with that from the in-sample regressions and Bayesian model

averaging results. In the monthly results, lagged log(MV 3) have the highest explanatory

power and have the highest posterior probability as 100%. While the lagged log(TO) still

significant at 1% level in the in-sample regression and have the same posterior probability

with log(MV 3), the three out of sample statistics increases substantially, the out-of-sample

R2oos to 27.3% from 22.1%, after we include lagged log(TO) in the forecast model. To compare

the variables that are shown to be useless by in-sample regressions and Bayesian model

averaging, we include lagged log(Pator) in the forecast model. No significant differences are

found after lagged log(Pator) is included. The three out of sample statistics are almost the

same with the model without lagged log(Pator).

We find similar results in quarterly data in panel B of table IV. Here we use the first

32 observations, spans from 1995 Q1 to 2003 Q4, for the initial in-sample estimation and

make out of sample forecasts for the rest period , from 2004 Q1 to 2018 Q3. What stands

14

out in panel B of table IV is there is no obvious difference in the statistics with or without

lagged log(TO), who is not significant in the quarterly regression and have 52.8% posterior

probability. But three out of sample statistics are higher when lagged log(Pastor) is included

while it has 100% posterior probability in the BMA model. What is striking about the table

IV is no evidence is found for scaled price ratio pd have forecasting power for stock market

variance. The R2oos decrease to 31.9% from 39.1% while pd is included as an additional

variable. This conclusion is consistent with our previous findings but different from the

theory of Campbell and Cochrane (1999) and Bansal and Yaron (2004) and the empirical

finding of Cai et al. (2017).

To summarize, our out-of-sample evidence and in-sample evidence are consistent. Unre-

ported results show that a variable also has no predictive power in out-of-sample prediction

if it does not perform well in sample. The finding of lagged log(TO) and log(Pastor) have

significant forecasting power is robust to different statistical methods. Lagged log(TO) is

also significant determinants of stock market variance beyond lagged variance in China’s

monthly data. Moreover, log(Pator) forecasts stock market variance out of sample, espe-

cially in quarterly frequency sample.

III. Conclusion

This study explores the prediction of China’s stock market variance. We use 25 candidate

variables including not only macroeconomic indicators but also financial indicators commonly

used in a considerable amount of literature. Contrary to Cai et al. (2017)’s findings, we find

most macroeconomic variables have neglected forecasting power of China’s stock market

variance spans the period from 1995 to 2018, whether the sample frequency is monthly or

quarterly.

Cai et al. (2017) find that dividend-price ratio positively and significantly forecast the

Chinese stock market volatility in the earlier sample period extends from January 1997 to

December 2012. However, while the sample period is extended to 2018, we find time-varying

correlation between scaled price ratio, such as pd, pb, pe, and future stock market variance.

In the early stage, pd significant negative forecast one-quarter-ahead stock market variance

which is consistent with Cai et al. (2017), but in the second sub-sample, their relationship

turned out to be exactly the opposite of positive significance. The consequence of this is

that the results of the overall sample are not significant.

While they conclude the shocks to economic fundamentals, such as inflation, and money

supply lead to a high future stock market volatility, we find insignificant relationship be-

tween these economic variables and next period variance. Apart from these variables, other

15

macroeconomic variables, such as economic policy uncertainty, industrial production growth

shock, consumer confidence index shock, GDP growth rate shock and volatility of crude

oil yield have not significant forecasting power. In addition, financial leverage proposed by

Schwert (1989) seems have time-varying correlation to future stock market volatility sim-

ilarly with scaled price ratio. While the volatility of crude oil yield can predict US stock

market variance mentioned by Wang et al. (2018), it can not forecast China’s stock market

variance which may imply these two markets are different.

In a study conducted by Chen et al. (2018), it is shown that stock market illiquidity

measure has a profound interaction with volatility. We find stock market turnover and

illiquidity measure proposed by Pastor and Stambaugh (2003) has significant forecasting

power beyond the lagged stock market variance, while Cai et al. (2017) reports that turnover

has strong forecasting power for the future Chinese market volatility. Amihud measure

proposed by Brennan et al. (2013) seems have no significant correlation with stock market

variance.

Our main findings are consistent with those documented by Schwert (1989) in the U.S.

market. Most variables that reflect economic fundamentals are not driver of stock market

volatility. In general, therefore, it seems that illiquidity has the greatest impact on the

volatility of China’s stock market.

16

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21

Table I Summary of All Variables

Variable Name Mean SD Skwness Kurtosis ρ1 ρ2 Dickey-Fuller p-value

Panel A: Monthly frequency

log(MV 3) log Market Volatility −5.586 1.275 −0.309 0.387 0.417 0.376 −3.692 0.025MV 3 Market Volatility 0.784a 1.264a 5.017 34.97 0.221 0.173 −4.330 0.000TO Turnover 0.337 0.279 2.141 5.317 0.801 0.672 −3.918 0.014

Pastor Illiquidity measure 0.167a 0.401a 5.701 42.38 0.650 0.389 −5.181 0.000Amihud Illiquidity measure 2.558 1.242 0.908 0.943 0.668 0.473 −4.516 0.000

pd Price to dividend ratio 4.520 0.613 0.429 −1.019 0.978 0.948 −3.605 0.033pb Price to book value ratio 0.918 0.421 0.373 −1.085 0.972 0.942 −3.556 0.038pe Price to earnings ratio 3.096 0.495 0.236 −1.309 0.976 0.949 −3.584 0.035FV 3 Firm-level variance 0.018 0.019 4.053 25.03 0.396 0.327 −3.767 0.021IV 3 Idiosyncratic variance 0.660a 0.524a 2.927 14.06 0.600 0.450 −3.560 0.037RREL Stochastically detrended risk-free rate −6.126b 68.03b −0.452 1.348 0.936 0.831 −5.421 0.000EPU Economic policy uncertainty 4.719 0.738 0.073 0.306 0.696 0.620 −3.386 0.057RET Stock market excess return 0.513a 8.686a −0.072 1.511 0.090 0.145 −5.267 0.000CPI Inflation 2.789a 3.863a 2.437 7.933 0.941 0.882 −4.593 0.000M2 Money supply growth shock of M2 −7.754a 1.002 0.500 3.320 −0.009 0.060 −5.065 0.000M1 Money supply growth shock of M1 −5.543a 2.309 0.139 2.435 −0.217 0.084 −4.563 0.000M0 Money supply growth shock of M0 −1.667a 7.496 0.390 6.669 −0.561 0.050 −8.798 0.000IPOR IPO first day return 1.264 1.230 1.871 4.944 0.680 0.569 −2.963 0.170IPON IPO number 11.55 11.32 1.196 8.637 0.759 0.657 −3.848 0.017IP Industrial production growth rate shock −4.444a 4.106 −0.219 7.141 −0.577 0.096 −7.802 0.000CCI Consumer confidence index shock 3.507a 1.846 −0.546 5.084 −0.025 −0.063 −8.441 0.000MLI Macroeconomic Leading Index 100.76 1.663 0.494 0.230 0.948 0.881 −3.311 0.070FL Financial leverage −1.091a 1.047 −0.294 −1.386 0.991 0.981 −3.014 0.149FL2 Financial leverage −0.806 0.545 −0.637 −1.068 0.993 0.986 −2.558 0.341Voil Oil Volatility 1.229a 1.329a 3.705 19.58 0.613 0.511 −4.849 0.000

Panel B: Quarterly frequency

log(MV 3) log Market Volatility −4.258 1.011 −5.361a −0.567a 0.548 0.397 −3.666 0.031MV 3 Market Volatility 2.313a 2.637a 2.472 7.162 0.298 0.291 −3.414 0.057TO Turnover 1.011 0.780 2.021 4.661 0.705 0.497 −3.906 0.017

Pastor Illiquidity measure 96.05b 0.138a 3.930 23.05 0.287 0.097 −8.786 0.000Amihud Illiquidity measure 2.472 1.029 0.506 −0.329 0.474 0.211 −4.405 0.000

pd Price to dividend ratio 4.515 0.616 0.420 −1.018 0.907 0.789 −4.772 0.000pb Price to book value ratio 0.926 0.424 0.382 −1.053 0.901 0.782 −4.169 0.000pe Price to earnings ratio 3.103 0.499 0.238 −1.325 0.914 0.813 −4.715 0.000FV 3 Firm-level variance 0.051 0.042 2.135 5.344 0.418 0.353 −3.472 0.049IV 3 Idiosyncratic variance 0.021 0.013 1.508 2.055 0.551 0.394 −3.883 0.018RREL Stochastically detrended risk-free rate −6.148b 69.25b −0.593 0.910 0.736 0.383 −4.746 0.000EPU Economic policy uncertainty 4.759 0.713 0.521 0.085 0.683 0.574 −3.415 0.057RET Stock market excess return 0.030 0.180 0.863 0.668 0.206 0.033 −5.171 0.000CPI Inflation 8.177a 0.111 2.253 6.534 0.850 0.685 −3.738 0.025M2 Money supply growth shock of M2 −0.233 1.743 0.635 3.803 0.287 0.072 −4.639 0.000M1 Money supply growth shock of M1 −0.166 3.104 0.445 −0.198 0.331 0.243 −4.752 0.000M0 Money supply growth shock of M0 −0.050 2.767 −0.104 −0.546 −0.442 0.162 −4.721 0.000IPOR IPO first day return 1.336 1.130 1.245 1.674 0.782 0.761 −2.203 0.493IPON IPO number 34.65 31.21 1.101 0.569 0.726 0.521 −4.005 0.012IP Industrial production growth rate shock −0.515 5.626 −0.634 3.410 0.146 −0.010 −6.171 0.000CCI Consumer confidence index shock 0.104 2.652 −0.884 3.321 0.066 −0.247 −4.379 0.000MLI Macroeconomic Leading Index 100.74 1.644 0.518 0.123 0.828 0.602 −2.174 0.505FL Financial leverage −2.683a 1.054 −0.307 −1.398 0.968 0.938 −3.360 0.066FL2 Financial leverage −0.811 0.549 −0.640 −1.082 0.977 0.949 −3.195 0.093GDP GDP growth shock −0.058 0.960 0.287 0.411 0.038 −0.057 −5.351 0.000Voil Oil Volatility 3.687a 3.369a 3.119 12.68 0.555 0.221 −4.121 0.000

Note: The table reports the univariate summary statistics of selected variables used in the paper. Different statistics such asmean, standard deviation, skewness, kurtosis, first order autocorrelation coefficient and second order autocorrelation coefficient arereported. The last two columns provide the Dickey and Fuller (1979) test and its p value. Panel A and B report the montly andquarterly sample, respectively. The monthly sample spans the January 1995 to December 2018 period except the January 1995to December 2017 period for MLI and January 1996 to December 2018 period for M2, M1 and M0. The quarter sample spansthe 1995Q1 to 2018Q4 period except the 1995Q1 to 2017Q4 period for MLI and 1996Q1 to 2018Q4 period for M2, M1 and M0.Superscript a indicates being scaled by 100, and Superscript b indicates being scaled by 100000.

22

Table II In-sample forecasting regressions

Variable Name 1995.01-2018.12 1995.01-2007.12 2008.01-2018.12

ρ β adjR2 ρ β adjR2 ρ β adjR2

Panel A: Monthly frequency

0.418∗∗∗ 0.172 0.287∗∗∗ 0.077 0.566∗∗∗ 0.318(5.753) (3.331) (7.623)

log(TO) Turnover 0.294∗∗∗ 0.453∗∗∗ 0.213 0.162∗∗ 0.430∗∗∗ 0.116 0.431∗∗∗ 0.605∗∗∗ 0.362(3.866) (4.513) (2.293) (2.886) (3.784) (3.402)

log(Pastor) Illiquidity measure 0.418∗∗∗ 0.994 0.169 0.286∗∗∗ −2.377 0.071 0.568∗∗∗ 134.2 0.315(5.759) (0.274) (3.379) (−0.549) (7.870) (0.965)

Amihud Illiquidity measure 0.415∗∗∗ −0.051 0.171 0.283∗∗∗ −0.053 0.074 0.568∗∗∗ −0.051 0.314(5.934) (−0.841) (3.477) (−0.875) (7.597) (−0.497)

pd Price to dividend ratio 0.411∗∗∗ 0.120 0.172 0.283∗∗∗ −0.159 0.076 0.512∗∗∗ 0.529∗∗∗ 0.341(5.919) (1.012) (3.183) (−0.945) (5.727) (3.261)

pb Price to book value ratio 0.401∗∗∗ 0.300 0.178 0.288∗∗∗ −0.122 0.072 0.472∗∗∗ 1.032∗∗∗ 0.367(6.004) (1.645) (3.296) (−0.471) (5.351) (4.572)

pe Price to earnings ratio 0.415∗∗∗ 0.105 0.171 0.272∗∗∗ −0.382 0.087 0.512∗∗∗ 0.853∗∗∗ 0.348(5.933) (0.691) (2.902) (−1.620) (5.087) (3.141)

log(FV 3) Firm level variance 0.125 0.617∗∗∗ 0.208 −0.058 0.710∗∗∗ 0.131 0.312∗ 0.546∗∗ 0.337(0.877) (3.148) (−0.582) (3.878) (1.672) (2.080)

log(IV 3) Idiosyncratic variance 0.286∗∗∗ 0.468∗∗∗ 0.208 0.148∗∗ 0.466∗∗∗ 0.119 0.414∗∗∗ 0.567∗∗∗ 0.354(3.106) (3.061) (1.987) (2.777) (3.341) (2.726)

RREL Stochastically detrended risk-free rate 0.378∗∗∗ -2.913a 0.192 0.261∗∗∗ −2.949a 0.839 0.520∗∗∗ -2.472∗∗a 0.338(5.255) (-2.700) (2.848) (−1.303) (7.033) (-2.534)

EPU Economic policy uncertainty 0.407∗∗∗ −0.112 0.173 0.288∗∗∗ 0.018 0.071 0.554∗∗∗ −0.131 0.317(6.183) (−1.076) (3.420) (0.108) (8.258) (−0.983)

RET Market excess return 0.414∗∗∗ 0.938 0.173 0.253∗∗∗ 1.928∗ 0.087 0.580∗∗∗ 0.888 0.316(5.666) (1.155) (2.954) (1.839) (7.768) (0.804)

CPI Inflation 0.392∗∗∗ 4.010∗∗ 0.183 0.237∗∗∗ 4.991∗∗∗ 0.105 0.561∗∗∗ 2.788 0.314(5.211) (2.420) (2.652) (2.627) (7.641) (0.490)

m2g Money supply growth shock of M2 0.402∗∗∗ −0.326b 0.161 0.297∗∗∗ −0.080 0.080 0.565∗∗∗ 0.046 0.314(5.129) (−0.050) (3.111) (−0.798) (7.607) (0.657)

m1g Money supply growth shock of M1 0.427∗∗∗ 0.005b 0.177 0.300∗∗∗ 0.028 0.078 0.571∗∗∗ −0.023 0.315(5.674) (0.002) (3.193) (0.620) (7.640) (−0.666)

m0g Money supply growth shock of M0 0.427∗∗∗ 0.096b 0.177 0.301∗∗∗ −0.926b 0.079 0.566∗∗∗ 0.974b 0.316(5.687) (0.135) (3.108) (−1.124) (7.617) (0.892)

IPON IPO number 0.410∗∗∗ −0.536b 0.171 0.277∗∗∗ 0.018 0.081 0.546∗∗∗ −0.592b 0.316(5.910) (−0.605) (3.457) (1.195) (8.241) (−0.667)

IP Industrial production growth rate shock 0.419∗∗∗ −0.011 0.170 0.289∗∗∗ −0.888b 0.071 0.566∗∗∗ −0.011 0.314(5.775) (−0.759) (3.367) (−0.501) (7.616) (−0.467)

CCI Consumer confidence index shock 0.413∗∗∗ −0.026 0.170 0.287∗∗∗ −0.042 0.072 0.566∗∗∗ −0.048b 0.313(5.696) (−0.891) (3.365) (−0.714) (7.567) (−0.016)

MLI Macroeconomic leading index 0.418∗∗∗ 0.060 0.182 0.266∗∗∗ 0.129∗ 0.091 0.593∗∗∗ 0.013 0.345(5.651) (1.380) (3.124) (1.945) (8.181) (0.319)

Voil Oil Volatility 0.403∗∗∗ 0.125 0.175 0.270∗∗∗ −0.193 0.081 0.485∗∗∗ 0.224∗∗∗ 0.334(5.660) (1.578) (2.973) (−1.452) (5.046) (2.732)

23

Variable Name 1995Q1-2018Q4 1995Q1-2007Q4 2008Q1-2018Q4

ρ β adjR2 ρ β adjR2 ρ β adjR2

Panel B: Quarterly frequency

0.548∗∗∗ 0.294 0.349∗∗∗ 0.104 0.694∗∗∗ 0.475(6.395) (3.144) (6.352)

log(TO) Turnover 0.450∗∗∗ 0.242 0.299 0.310∗ 0.082 0.087 0.477∗∗∗ 0.700∗∗∗ 0.526(3.760) (1.419) (1.910) (0.305) (3.204) (3.043)

log(Pastor) Illiquidity measure 0.533∗∗∗ 1.868∗∗∗a 0.352 0.380∗∗∗ 1.704∗∗∗a 0.188 0.662∗∗∗ 1.618a 0.470(6.721) (3.403) (3.505) (3.156) (6.239) (0.947)

Amihud Illiquidity measure 0.543∗∗∗ −0.034 0.287 0.359∗∗∗ 0.053 0.089 0.678∗∗∗ -0.256∗ 0.502(6.421) (−0.338) (3.139) (0.455) (5.772) (-1.661)

pd Price to dividend ratio 0.542∗∗∗ 0.049 0.287 0.338∗∗∗ -0.311∗∗ 0.128 0.612∗∗∗ 0.552∗∗∗ 0.505(6.376) (0.411) (3.133) (-1.975) (5.657) (2.640)

pb Price to book value ratio 0.526∗∗∗ 0.206 0.293 0.352∗∗∗ −0.278 0.101 0.563∗∗∗ 0.998∗∗∗ 0.530(6.244) (1.112) (3.212) (−1.206) (5.434) (3.828)

pe Price to earnings ratio 0.545∗∗∗ 0.041 0.286 0.312∗∗∗ -0.554∗∗∗ 0.156 0.618∗∗∗ 0.803∗∗∗ 0.508(6.442) (0.265) (2.763) (-2.282) (5.233) (2.666)

log(FV 3) Firm level variance 0.538∗∗ 0.016 0.286 0.235 0.162 0.087 0.448∗ 0.457 0.476(2.316) (0.050) (0.527) (0.270) (1.912) (1.316)

log(IV 3) Idiosyncratic variance 0.473∗∗∗ 0.198 0.293 0.289 0.132 0.089 0.475∗∗∗ 0.642∗∗ 0.509(3.446) (0.823) (1.517) (0.409) (3.215) (2.523)

RREL Stochastically detrended risk-free rate 0.494∗∗∗ -2.307∗a 0.309 0.310∗∗ −1.968a 0.096 0.639∗∗∗ −2.186a 0.489(5.082) (-1.792) (2.407) (−0.691) (4.963) (−1.598)

EPU Economic policy uncertainty 0.518∗∗∗ −0.200 0.304 0.332∗∗∗ −0.189 0.093 0.675∗∗∗ −0.206 0.478(6.113) (−1.498) (2.918) (−0.719) (6.418) (−1.095)

RET Market excess return 0.540∗∗∗ 0.241 0.288 0.296∗∗∗ 0.537 0.095 0.717∗∗∗ 0.765 0.474(6.142) (0.483) (2.734) (0.918) (6.088) (0.914)

CPI Inflation 0.494∗∗∗ 1.588∗∗ 0.314 0.204∗ 2.259∗∗∗ 0.189 0.680∗∗∗ 1.561 0.469(5.450) (2.147) (1.708) (2.596) (6.639) (0.877)

m2g Money supply growth of M2 0.555∗∗∗ 0.254b 0.292 0.333∗∗∗ −0.062 0.093 0.692∗∗∗ 0.671b 0.462(6.230) (0.069) (2.785) (−0.840) (6.076) (0.163)

m1g Money supply growth of M1 0.544∗∗∗ 0.014 0.294 0.339∗∗∗ 0.022 0.087 0.691∗∗∗ 0.314b 0.462(5.790) (0.504) (2.784) (0.475) (5.904) (0.102)

m0g Money supply growth of M0 0.546∗∗∗ 0.044 0.307 0.344∗∗∗ 0.062∗ 0.127 0.692∗∗∗ 0.010 0.463(6.076) (1.354) (2.914) (1.655) (6.161) (0.173)

IPON IPO number 0.534∗∗∗ −0.185b 0.289 0.333∗∗∗ 0.388b 0.091 0.695∗∗∗ 0.002b 0.462(6.407) (−0.538) (2.828) (0.679) (5.891) (0.005)

IP Industrial production growth rate shock 0.545∗∗∗ -0.031∗∗ 0.316 0.362∗∗∗ -0.040∗ 0.147 0.685∗∗∗ −0.016 0.469(6.447) (-2.138) (3.168) (-1.942) (6.450) (−1.136)

CCI Consumer confidence index shock 0.575∗∗∗ 0.037 0.295 0.357∗∗∗ 0.066∗ 0.118 0.745∗∗∗ 0.039 0.470(6.294) (1.156) 3.113 (1.649) (6.189) (0.861)

FL Financial leverage 0.496∗∗∗ -0.141∗∗ 0.311 0.350∗∗∗ 0.157 0.090 0.531∗∗∗ -0.421∗∗∗ 0.534(5.777) (-2.085) (3.185) (0.747) (4.815) (-3.677)

FL2 Financial leverage 0.529∗∗∗ −0.393 0.293 0.344∗∗∗ 0.825 0.102 0.539∗∗∗ -4.025∗∗∗ 0.541(6.210) (−1.087) (3.234) (1.336) (4.96) (-5.178)

GDP GDP growth shock 0.546∗∗∗ −0.027 0.287 0.349∗∗∗ 0.022 0.086 0.679∗∗∗ −0.126 0.471(6.417) (−0.330) (3.171) (0.219) (6.550) (−1.101)

Voil Oil Volatility 0.531∗∗∗ 0.089 0.290 0.327∗∗∗ −0.123 0.090 0.695∗∗∗ −0.159b 0.462(5.934) (0.950) (2.913) (−0.676) (4.699) (−0.011)

Note: This table presents the in sample forecasting regression results using all sorts of variables. The OLS regression form is:log(MV 3)t+1 = α + ρ ∗ log(MV 3)t + β ∗ Xt. We report the coefficients and the adjusted R2 of the regression, and Newey andWest (1987) t-statistics with 6 lags and 2 lags in parentheses, of panle A and B respectively. The results of three different durationsamples: from January 1995 to December 2018, from January 1995 to December 2007 and from January 2008 to December 2018.(The sample of money supply starts from Jan 1996, and MLE end at Dec 2017. The time of the quarterly sample is the same asthe corresponding monthly data.) ***, **, and * denote significance at the 1%, 5%, and 10% levels. We use bold fonts to highlightthe significance of at least the 10% level. Superscript a indicates being scaled by 0.01. Superscript b indicates being scaled by 100.

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Table III Marginal evidence of importance

probne0 postmean postsd model 1 model 2 model 3 model 4 model 5

Panel A: Monthly frequency

Intercept 100.000 −3.882 2.082 −3.289 −3.691 −3.446 −3.482 −3.267log(MV 3) 100.000 0.280 0.070 0.302∗∗∗ 0.250∗∗ 0.290∗∗∗ 0.287∗∗∗ 0.301∗∗∗

log(TO) 100.000 0.541 0.173 0.476∗∗∗ 0.698∗∗∗ 0.418∗∗∗ 0.475∗∗∗ 0.498∗∗∗

log(Pastor) 1.800 0.035 0.976 . . . . .pd 1.800 −0.001 0.018 . . . . .

RREL 18.000 −35.870 89.816 . . -1.973∗a . .EPU 1.700 0.000 0.014 . . . . .m2g 1.800 0.000 0.010 . . . . .IPON 5.500 0.000 0.002 . . . . .IP 12.300 −0.003 0.011 . . . . -0.027∗

CCI 2.300 −0.001 0.007 . . . . .MLI 7.500 0.004 0.020 . . . . .

Amihud 32.600 0.053 0.088 . 0.161 . . .Voil 8.500 0.662 2.655 . . . 8.284∗ .R2 0.237 0.250 0.247 0.245 0.244

adjR2 0.231 0.241 0.238 0.236 0.236BIC −60.291 −59.231 −58.138 −57.326 −57.221

post prob 0.275 0.162 0.094 0.062 0.059

Panel B: Quarterly frequency

Intercept 100.000 −2.739 1.931 −2.977 −1.843 −2.147 −2.999 −2.022log(MV 3) 97.900 0.413 0.145 0.340∗∗∗ 0.501∗∗∗ 0.545∗∗∗ 0.334∗∗∗ 0.466∗∗∗

log(TO) 52.800 0.258 0.288 0.483∗∗∗ . . 0.501∗∗∗ .log(Pastor) 100.000 266.700 74.214 2.574∗∗∗a 2.923∗∗∗a 2.124∗∗∗a 2.517∗∗∗a 2.813∗∗∗

pd 4.300 −0.003 0.036 . . . . .RREL 8.300 −11.960 55.266 . . . . −1.487a

EPU 5.000 −0.002 0.037 . . . . .m2g 3.800 0.000 0.009 . . . . .IPON 3.900 0.000 0.001 . . . . .IP 8.600 −0.002 0.007 . . . -0.019∗ .CCI 5.800 0.002 0.011 . . . . .MLI 6.600 0.003 0.017 . . . . .

Amihud 40.700 −0.093 0.130 . -0.237∗∗ . . -0.224∗∗

GDP 4.500 −0.002 0.021 . . . . .Voil 4.000 0.034 0.550 . . . . .R2 0.431 0.428 0.384 0.441 0.437

adjR2 0.411 0.408 0.370 0.414 0.410BIC −36.178 −35.714 −33.746 −33.200 −32.626

post prob 0.196 0.155 0.058 0.044 0.033

Note: This table presents the marginal evidence of importance of the potential explanatory variables based on Bayesian modelaveraging (BMA). The second column probne0 represents the posterior probability that each variable is non-zero (in percent). Thethird column postmean displays the posterior mean of each coefficient (from model averaging). The column postsd reports theposterior standard deviation of each coefficient (from model averaging). And we report 5 best models based on the post probabilityand BIC. ***, **, and * denote significance at the 1%, 5%, and 10% levels with unreport Newey and West (1987) t-value adjustedwith 6 lags and 2 lags in parentheses, of panle A and B respectively. We use bold fonts to highlight the significance of at least the10% level. Superscript a indicates being scaled by 0.01.

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Table IV Out-of-Sample Forecasts

model R2oos ENC-NEW 1% Critical Value MSE-F 1% Critical Value

Panel A: Monthly Frequency

log(MV 3) 0.221 40.677 4.134 54.141 3.951log(MV 3)&log(TO) 0.273 53.418 5.107 71.863 4.250

log(MV 3)&log(TO) & log(Pastor) 0.273 54.122 5.805 71.804 4.184

Panel B: Quarterly Frequency

log(MV 3) 0.391 31.702 4.134 40.482 3.951log(MV 3)&pd 0.319 25.234 5.107 29.531 4.250

log(MV 3)&log(TO) 0.402 31.962 5.107 42.416 4.250log(MV 3)&log(TO) & log(Pastor) 0.419 38.058 5.805 45.408 4.184

Note: The table reports the out-of-sample forecast results for stock market variance log(MV 3). R2oos and ENC-NEW are the out-

of-sample R2 and Clark and McCracken (2001)’s ENC-NEW test statistic, respectively. MSE-F is the equal forecast accuracy testdeveloped by McCracken (1999). We also reproduce the corresponding 1% asymptotic critical values of the ENC-NEW and MSE-Ftest in the 4 and 6 columns. Panel A and B perform forecasting regression monthly and quarterly using the lagged variables. Weuse the first 96 observations for the initial in-sample estimation and make out-of-sample forecasts recursively using an expandingsample over the 97 to 287. The quarterly initial in-sample is 32 while the expanding sample is from 33 to 95. We use bold fonts tohighlight the significance of at least the 1% level.

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