uncovering the china’s stock market variance...
TRANSCRIPT
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: [email protected]. Guo is at the Department of Finance and Real Estate, University of Cincin-nati; Email: [email protected]. Shi is at the Research Center of Applied Finance, Dongbei University ofFinance and Economics; Email: [email protected]. 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.
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• 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.
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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-
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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
REFERENCES
Ait-Sahalia, Yacine, and Mehmet Saglam, 2017, High frequency market making: Implications
for liquidity, Available at SSRN .
Amihud, Yakov, 2002, Illiquidity and stock returns: cross-section and time-series effects,
Journal of Financial Markets 5, 31–56.
Andersen, Torben G., Tim Bollerslev, Francis X. Diebold, and Paul Labys, 2003, Modeling
and forecasting realized volatility, Econometrica 71, 579–625.
Baker, Malcolm, and Jeffrey Wurgler, 2006, Investor sentiment and the cross-section of stock
returns, The Journal of Finance 61, 1645–1680.
Bansal, Ravi, and Amir Yaron, 2004, Risks for the long run: A potential resolution of asset
pricing puzzles, The Journal of Finance 59, 1481–1509.
Beeler, Jason, and John Y. Campbell, 2012, The long-run risks model and aggregate asset
prices: An empirical assessment, Critical Finance Review 1, 141–182.
Bollerslev, Tim, Ray Y. Chou, and Kenneth F. Kroner, 1992, Arch modeling in finance: A
review of the theory and empirical evidence, Journal of Econometrics 52, 5–59.
Brennan, Michael, Sahn-Wook Huh, and Avanidhar Subrahmanyam, 2013, An analysis of
the amihud illiquidity premium, The Review of Asset Pricing Studies 3, 133–176.
Cai, Weixian, Jian Chen, Jimin Hong, and Fuwei Jiang, 2017, Forecasting chinese stock
market volatility with economic variables, Emerging Markets Finance and Trade 53, 521–
533.
Campbell, John Y., and Samuel B. Thompson, 2008, Predicting excess stock returns out of
sample: Can anything beat the historical average?, The Review of Financial Studies 21,
1509–1531.
17
Campbell, John Y., and John H. Cochrane, 1999, By force of habit: A consumption-based
explanation of aggregate stock market eehavior, Journal of Political Economy 107, 205–
251.
Chen, Jian, Fuwei Jiang, Hongyi Li, and Weidong Xu, 2016, Chinese stock market volatility
and the role of u.s. economic variables, Pacific-Basin Finance Journal 39, 70–83.
Chen, Yong, Gregory W. Eaton, and Bradley S. Paye, 2018, Micro(structure) before macro?
the predictive power of aggregate illiquidity for stock returns and economic activity, Jour-
nal of Financial Economics 130, 48–73.
Cheng, Hang, Hui Guo, and Yongdong Shi, 2018, Uncovering china’s stock market risk return
relation: Crazy casino punters or risk averse investors?, Available at SSRN .
Christiansen, Charlotte, Maik Schmeling, and Andreas Schrimpf, 2012, A comprehensive
look at financial volatility prediction by economic variables, Journal of Applied Economet-
rics 27, 956–977.
Christie, Andrew A., 1982, The stochastic behavior of common stock variances: Value,
leverage and interest rate effects, Journal of Financial Economics 10, 407–432.
Clark, Todd E., and Michael W. McCracken, 2001, Tests of equal forecast accuracy and
encompassing for nested models, Journal of Econometrics 105, 85–110.
David, Alexander, and Pietro Veronesi, 2013, What ties return volatilities to price valuations
and fundamentals?, Journal of Political Economy 121, 682–746.
Dickey, David A., and Wayne A. Fuller, 1979, Distribution of the estimators for autore-
gressive time series with a unit root, Journal of the American Statistical Association 74,
427–431.
French, Kenneth R., G. William Schwert, and Robert F. Stambaugh, 1987, Expected stock
returns and volatility, Journal of Financial Economics 19, 3–29.
18
Gallant, A. Ronald, Peter E. Rossi, and George Tauchen, 1992, Stock prices and volume,
The Review of Financial Studies 5, 199–242.
Girardin, Eric, and Roselyne Joyeux, 2013, Macro fundamentals as a source of stock market
volatility in china: A garch-midas approach, Economic Modelling 34, 59–68.
Guo, Hui, 2011, Ipo first-day return and ex ante euity premium, Journal of Financial and
Quantitative Analysis 46, 871–905.
Guo, Hui, Qian Lin, and Yu Jou Pai, 2018, On the stock market variance-return or price
relations: A tale of two variances, Available at SSRN .
Guo, Hui, and Robert Savickas, 2006, Idiosyncratic volatility, stock market volatility, and
expected stock returns, Journal of Business & Economic Statistics 24, 43–56.
Harris, Milton, and Artur Raviv, 1993, Differences of opinion make a horse race, The Review
of Financial Studies 6, 473–506.
Hu, Grace Xing, Can Chen, Yuan Shao, and Jiang Wang, 2018a, Fama-french in china: Size
and value factors in chinese stock returns, International Review of Finance .
Hu, Grace Xing, Jun Pan, and Jiang Wang, 2018b, Chinese capital market: An empirical
overview, Available at SSRN .
Lamoureux, Christopher G., and William D. Lastrapes, 1990, Heteroskedasticity in stock
return data: Volume versus garch effects, The Journal of Finance 45, 221–229.
Liu, Jianan, Robert F. Stambaugh, and Yu Yuan, 2018, Size and value in china, Journal of
Financial Economics, Forthcoming. Available at SSRN .
Lou, Xiaoxia, and Tao Shu, 2017, Price impact or trading volume: Why is the amihud (2002)
measure priced?, The Review of Financial Studies 30, 4481–4520.
19
McCracken, Michael W, 1999, Asymptotics for out of sample tests of causality, manuscript,
Louisiana State University .
Merton, Robert C., 1973, An intertemporal capital asset pricing model, Econometrica 41,
867–887.
Mittnik, Stefan, Nikolay Robinzonov, and Martin Spindler, 2015, Stock market volatility:
Identifying major drivers and the nature of their impact, Journal of Banking & Finance
58, 1–14.
Newey, Whitney K, and Kenneth D West, 1987, A simple, positive semi-definite, het-
eroskedasticity: An autocorrelation consistent covariance matrix, Econometrica 55, 703–
708.
Nonejad, Nima, 2017, Forecasting aggregate stock market volatility using financial and
macroeconomic predictors: Which models forecast best, when and why?, Journal of Em-
pirical Finance 42, 131–154.
Paye, Bradley S., 2012, ‘deja vol’: Predictive regressions for aggregate stock market volatility
using macroeconomic variables, Journal of Financial Economics 106, 527–546.
Pastor, Lubos, and Robert F. Stambaugh, 2003, Liquidity risk and expected stock returns,
Journal of Political Economy 111, 642–685.
Schwert, G. William, 1989, Why does stock market volatility change over time?, The Journal
of Finance 44, 1115–1153.
Stambaugh, Robert F., 1999, Predictive regressions, Journal of Financial Economics 54,
375–421.
Stoll, Hans R., 2000, Presidential address: Friction, The Journal of Finance 55, 1479–1514.
Wang, Yudong, Yu Wei, Chongfeng Wu, and Libo Yin, 2018, Oil and the short-term pre-
dictability of stock return volatility, Journal of Empirical Finance 47, 90–104.
20
Watanabe, Akiko, and Masahiro Watanabe, 2005, Liquidity and conditional heteroscedas-
ticity in stock returns, Available at SSRN .
Wei, Yu, Qianwen Yu, Jing Liu, and Yang Cao, 2018, Hot money and china’s stock market
volatility: Further evidence using the garch–midas model, Physica A: Statistical Mechanics
and its Applications 492, 923–930.
Welch, Ivo, and Amit Goyal, 2008, A comprehensive look at the empirical performance of
equity premium prediction, The Review of Financial Studies 21, 1455–1508.
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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