1. finance
2. investing
3. stocks

The Usefulness of Cross-sectional Dispersion for
Forecasting Aggregate Stock Price Volatility∗
Sungje Byun†
November, 2014
Abstract
Does cross-sectional dispersion in the returns of different stocks help forecast
aggregate stock volatility? This paper develops a model of stock returns where dispersion in returns across different stocks is modeled jointly with aggregate volatility. Although specifications that allow for feedback from cross-sectional dispersion
to aggregate volatility have a better fit in sample, they prove not to be robust for
purposes of out-of-sample forecasting. Using a full cross-section of stock returns
jointly, however, I find that use of cross-sectional dispersion can help improve parameter estimates of a GARCH process for aggregate volatility to generate better
forecasts both in sample and out of sample. Given this evidence, I conclude that
cross-sectional information helps predict market volatility indirectly rather than
directly entering in the data-generating process.
*Keywords: Stock market volatility, Cross-sectional dispersion, Estimation of
large panel data, Forecasting accuracy
∗
I am very grateful to James D. Hamilton, Allan Timmermann, Valerie Ramey, Alexis Toda and
Thomas Baranga for their helpful comments and suggestions. I also thank the participants of the UCSD
macroeconomic workshop and summer empirical macroeconomics lunch seminar.
†
Economics Department, University of California at San Diego, [email protected]
1
1
Introduction
Modeling and forecasting volatility is an important task and a popular research
agenda in financial markets. Volatility models play key roles in academic literature
for testing the fundamental tradeoff between risk and return of financial assets and
for investigating causes and consequences of the volatility dynamics in the economy.
Volatility forecasts have many practical applications as well. For example, volatility
forecasts are used for market timing decisions, portfolio selections, risk managements
and pricings of financial derivatives such as options and forward contracts since risks
are measured by the volatility of the financial asset returns.
Given its importance, there are a growing number of models and approaches
for forecasting volatility in the financial assets. Since Engle (1982) and Bollerslev
(1986), Autoregressive Conditional Heteroskedasticity (ARCH) models are the most
popular by formulating the volatility forecasts of a return as a function of known
variables. Adopting the specific functional form and/or alternative explanatory variables in the volatility forecasts, there have been numerous extensions of ARCH models focusing on highlighted characteristics such as volatility persistence, asymmetry,
long memory properties and a leptokurtic distribution of financial asset returns.
While earlier researchers use variation over time in variables of interests, crosssectional information began to be recognized as an important source for improving
volatility forecasts. Campbell et al. (2001) and Connor et al. (2006) suggested
cross-sectional dispersion across individuals as an important source of individual
stock volatility1 . In a similar context, Hwang and Satchell (2005) tested whether
cross-sectional dispersion helps forecast volatility of individual stock returns using
1
The contribution of cross-sectional dispersion is referred to as firm-specific (idiosyncratic) volatility
in Campbell et al. (2001) and common heteroscedasticity in asset specific returns in Connor et al. (2006).
2
the so-called GARCH-X models, where X refers to “cross-sectional dispersion”. Although better specified, they found a trivial improvement from GARCH-X for outof-sample volatility forecasts, concluding that GARCH-X models do not necessarily
outperform than GARCH models in forecasting individual stock volatility.
In this paper, I investigate potential channels by which cross-sectional information might help predict aggregate volatility. I develop a model of individual returns
that could be applied to the study of volatility in any financial assets, though the interest in this paper is in stock-market volatility. The approach includes a GARCH-X
model as a special case in which measures of cross-sectional dispersion appear in the
equation for predicting aggregate volatility, an approach previously investigated by
Hwang and Satchell (2005). I find that although such GARCH-X specifications can
improve in-sample forecasting accuracy, cross-sectional dispersion does not appear
to be useful for out-of-sample forecasting.
Next, I investigate another channel in which cross-sectional information helps
predict aggregate volatility by providing accurate parameter estimates. After jointly
modeling the full cross-section of individual stock returns, I estimate population parameters in the bivariate GARCH process for aggregate volatility and cross-sectional
dispersion. While sharing the same GARCH process with univariate GARCH, I
find improved forecasting accuracies that are statistically significant both in sample and out of sample for all nine loss criteria considered. Using Giacomini and
White (2006)’s conditional predictive ability tests, I show that by jointly utilizing
cross-sectional information, it also provides more accurate out-of-sample volatility
forecasts in times of recessions as well as during bear markets. I conclude that crosssectional information helps predict market volatility indirectly insofar as it helps to
obtain accurate parameter estimates for volatility forecasts.
3
This paper is organized as follows. The following section describes a model of
stock returns and clarifies the relation between models of aggregate stock volatility
and the volatility of individual returns. In Section 3, I investigate two potential
channels whereby cross-sectional information could improve volatility forecasts of
the market index return: 1) cross-sectional dispersion as an additional explanatory variable as in GARCH-X, 2) cross-sectional dispersion as an aid in parameter
estimation when individual stock returns are jointly modeled. Section 4 provides robustness checks by using the Hwang and Satchell (2005)’s measure of cross-sectional
dispersion. I also consider alternative measures for cross-sectional dispersion within
the GARCH-X model, all confirming results in Section 3. Lastly, Section 5 concludes.
2
Model
Let ri,t denote the monthly return on individual stock i measured in percent. For
example, ri,t = −1.5 means that stock i fell 1.5% from month t−1 to t. My interest is
in characterizing aggregate market volatility as measured by some weighted average
of individual returns,2
Nt−1
rt =
X
wi,t−1 ri,t
(1)
i=1
where wi,t−1 is a predetermined weight of a stock i0 s return in the evolution of the
−1
stock index return in period t. For example, wi,t−1 = Nt−1
for an equal-weighted
index and wi,t−1 = pi,t−1 si,t−1 /
NP
t−1
pj,t−1 sj,t−1 for a value weighted index where
j=1
pi,t−1 is stock i’s price and si,t−1 is its number of outstanding shares at time t − 1.
One approach would be to fit a univariate GARCH-X(1,1) model to the aggregate
2
When a stock i is newly added in the stock index in period t, there is no contribution of a stock i’s
return ri,t on the stock index return rt .
4
return:
rt = φ0 + φ1 rt−1 + ut ,
(2)
ut = σt · εt ,
(3)
εt ∼ i.i.d. (0,1),
2
σt2 = $ + αu2t−1 + βσt−1
+ πxt−1 .
(4)
Here xt−1 is a measure of the cross-sectional dispersion of stocks at time t − 1.
Note that equation (4) includes the standard univariate GARCH(1,1) as a special
case when π = 0.
One question is what measure of dispersion to use for xt−1 and what kind of
model for individual stock returns would be consistent with a process like (4) for
aggregate returns. Having an explicit answer to the latter question will also clarify
the way in which data on individual stock returns might be helpful for estimating
the parameters of equation (4). Consider an AR(1) forecasting model of stock i’s
return of the form,
ri,t = φ0i + φ1i ri,t−1 + ui,t .
(5)
Let vt be a shock to the level of stock returns, distributed as N 0, σt2 conditional
on available observations. Denoting κt a separate shock, the forecasting error of
stock i’s return (ui,t ) is modeled as
ui,t = λi,t−1 [vt + κt ηi,t ] ,
(6)
where λi,t−1 denotes a predetermined loading of stock i on the aggregate shock
5
vt , κt governs the cross-section dispersion of stock returns,3 and ηi,t is a stock i’s
idiosyncratic forecasting error that is a martingale difference sequence with unit
variance
2
E ηi,t
|Ft−1 = 1,
where Ft−1 denotes all observed variables through t − 1.
One special case of interest comes from the idea that small stocks tend to be
more risky, for which I might specify λi,t−1 as,
λi,t−1 =
λi
Nt−1 wi,t−1
(7)
where λi captures the different degree of a stock i responding to the two shocks vt
and κt . Conditional on λi , a stock with below-average weight (wi,t−1 < 1/Nt−1 )
is treated in (7) as being more exposed to the aggregate shock as well as having
a higher weight on the dispersion shock κt . Support for such a specification can
be found for examples in the results of Schwert and Seguin (1990) and Ang et al.
(2006).
Since time variation in the cross-sectional dispersion in this model is driven by
the value of κt , I would use κ2t−1 in place of xt−1 in (4) if I had direct observations
available on κt−1 .
Note that the aggregate forecasting error ut is related to the errors in forecasting
3
Connor et al. (2006) and Jones (2001) suggested a strong commonality in asset specific volatilities,
so that the average squared asset-specific return across a large number of stocks varies over time.
6
individual returns ui,t through the identity
Nt−1
ut =
X
wi,t−1 ui,t
(8)
i=1
Nt−1
= vt
X
Nt−1
wi,t−1 λi,t−1 + κt
X
i=1
wi,t−1 λi,t−1 ηi,t
i=1
For Nt large, it is reasonable to assume that
NP
t−1
wi,t−1 λi,t−1 → 1. For example,
i=1
with a constant-sized sample of equal-weighted stocks, wi,t = 1/N and N −1
N
P
λi =
i=1
1 by construction,4 in which case
NP
t−1
wi,t−1 λi,t−1 would always exactly equal 1.
i=1
Likewise for Nt−1 large, it would typically be the case that
NP
t−1
(wi,t−1 λi,t−1 )2 → 0;
i=1
for the above example
NP
t−1
(wi,t−1 λi,t−1 )2 = N −2
i=1
as N −1
N
P
i=1
N
P
i=1
λ2i which goes to zero as long
2
λ2i converges to some finite constant λ . Hence

Nt−1

X
2
E
(wi,t−1 λi,t−1 ηi,t )
i=1
implying κt
NP
t−1
Nt−1
=
X
(wi,t−1 λi,t−1 )2 → 0
i=1
p
wi,t−1 λi,t−1 ηi,t → 0. Thus when Nt−1 is large the aggregate return
i=1
ut gives a direct observation on the common shock vt :
p lim ut = vt
(9)
Nt →∞
Consider the simple case for the equal-weighted index return (wi,t = N −1 for ∀i and ∀t). Then,
a consistent estimate of λi for each i can be obtained by a univariate regression of stock i on the
bOLS =
equal-weighted index;, ui,t = λi · ut + ei,t estimated by OLS for t = 1, . . . , T . It is clear that λ
i
P
−1 P
P
T
T
N bOLS
2
−1
= 1. A modification to the value-weighted index
i=1 λi
t=1 ut
t=1 ui,t ut , satisfying N
is same except that a consistent estimate of λi is obtained by regressing Nt−1 wi,t−1 ui,t on ut .
4
7
Note further from (6) that
Nt−1
X
Nt−1
wi,t−1 (ui,t − λi,t−1 vt )2 = κ2t
i=1
X
wi,t−1 (λi,t−1 ηi,t )2
i=1
→ κ2t · λ
2
p
provided
NP
t−1
i=1
2
wi,t−1 λ2i,t → λ . Hence under these conditions, I could use the magni-
tude
Nt−2
c2t−1
=
X
wi,t−2 (ui,t−1 − λi,t−2 ut−1 )2
(10)
i=1
directly for xt−1 in (4) to explore whether cross-sectional dispersion at time t − 1
as summarized by the value of κ2t−1 contributes to aggregate market volatility σt2 .
Note that the above cross-sectional dispersion differs from cross-sectional market volatility of Hwang and Satchell (2005) in two ways5 . First, I use individual
forecasting errors instead of individual stock returns. More importantly, this formulation allows heterogeneous responses of individual stocks to the common market
shock through the term λi,t−2 .
One can fit a GARCH-X model to the aggregate return with the measure of crosssectional dispersion given in (10). This provides a natural test of whether crosssectional dispersion directly enters into the data-generating process for aggregate
volatility.
3
Results
In this section, I provide empirical evidence for the role of cross-sectional dispersion
in predicting market volatility. After describing the sample, I provide empirical evi5
Two minor differences are 1) time-varying Nt−2 , 2) predetermined individual weigths wi,t−2 as opposed to wi,t−1 in Hwang and Satchell (2005). Section 4.1. provides robustness checks using crosssectional market volatility of Hwang and Satchell (2005).
8
dence of improved specifications in GARCH-X models by allowing for feedback from
cross-sectional dispersion to aggregate volatility, showing better in-sample forecasting performance. While GARCH-X with cross-sectional dispersion proves not to
be robust for purposes of out-of-sample forecasts, I show that use of cross-sectional
dispersion can help improve parameter estimates of a GARCH process for aggregate
volatility by using individual stock returns jointly, generating better forecasts both
in sample and out of sample.
3.1
Additional explanatory variable in GARCH
The dataset contains individual stocks in CRSP value-weighted index for three major U.S. markets such as NYSE, ASE and NASDAQ. I use prices and the number
of outstanding shares of individuals stocks for calculating returns and weights of
individual stocks. Since individual returns do not include cash dividends, the aggregate volatility of my interests is equivalent to the volatility in monthly CRSP
value-weighted index without cash dividends6 (henceforth, value-weighted index).
There are total 21, 523 individual stocks in the dataset while the number of stocks
in the value-weighted index (Nt ) varies over time7 . Summary statistics of individual stock returns and historical changes in the number of stocks are provided in
Appendix A.1. I also use daily CRSP value-weighted returns for calculating realized variance as a proxy for latent volatility, which is going to be discussed later
in this section. The time period for the empirical analysis ranges from February
1954 (t = 1) to December 2013 (t = 719), corresponding to 719 monthly periods for
6
In CRSP dataset, acronyms are “vwretx” for value-weighted index return, and “retx” for that of an
individual stock return. More precisely, I use the adjusted stock price and the adjusted number of shares
outstanding for calculating individual stock returns for capturing effects from stock events except the
cash-dividend.
7
The number of stocks in the value-weighted index is 8,402 (maximum), 5,093 (median), and 989
(minimum).
9
volatility forecasts.
To begin with, I describe the empirical procedure for constructing cross-sectional
dispersion using individual stocks. First, for each individual stock i, I calculate a
monthly return ri,t for t = 1, . . . , 719, and estimate an individual forecasting error
ui,t by regressing a stock i’s return (ri,t ) on a constant and its lagged return (ri,t−1 )
for t = 2, . . . , 719. Second, a stock i’s weight in period t − 1, wi,t−1 is calculated
as wi,t−1 = pi,t−1 si,t−1 /
PN
j=1 pj,t−1 sj,t−1 ,
where pi,t−1 is a price and si,t−1 is the
number of outstanding shares of a stock i in perod t − 1 for t = 1, . . . , 719. Third,
an aggregate forecasting error ut is calculated from (8) given ui,t and wi,t−1 for
t = 1, . . . , 719 and for all i. Assuming λi = 1 for ∀i for simplicity,8 I obtain a
measure for cross-sectional dispersion provided by (10).
Figure 1 plots historical cross-sectional volatility, that is the square root of c2t in
(10). Notice that cross-sectional volatility measures average percentage deviation
of individual forecasting errors from their aggregates, and reflects individual stocks’
heterogeneity in two ways. First, it takes into account different degrees by which
individual stocks respond to the aggregate shock provided through the term λi,t−1 .
Second, each stock’s squared deviation contributes to the evolution of cross-sectional
volatility proportional to its relative share in the stock index given by wi,t−1 . The
shaded areas represent the ten NBER recession periods in the sample. Three points
are worth noting9 . First, cross-sectional dispersion itself is time-varying and also exhibits high persistence with a few clusterings, as documented in Hwang and Satchell
8
This restriction will be relaxed in the next revision. In principle, λi is obtained from a univariate
regression of Nt−1 wi,t−1 ui,t on ut for t = 1, . . . , 719. One empirical difficulty associated with time-varying
Nt is that the average of estimated λi is not necessarily equal to 1, violating the condition for internal
consistency. For dealing with this empirical issue, one could normalize the average λi at each period t.
After obtaining OLS estimates for λi , I define λ∗i,t = λi if a stock i appears in the index, and 0 otherwise,
ei,t = λ∗ / N −1 PNt λ∗ so that the internal consistency is guaranteed by
and replace λi in (7) by λ
i,t
t
i=1
i,t
construction.
9
Though not reported, the cross-sectional volatility measured from CRSP equal-weighted index return
exhibits upward trend as is consistent with the observation in Campbell et al. (2001).
10
(2005) and Connor et al. (2006). Second, I find increasing cross-sectional dispersion
during recession periods, indicating its potential role in helping to predict market
volatility during recession periods. Lastly, cross-sectional dispersion is larger in
magnitude than time-series market volatility (See Figure 2 for realized volatility),
confirming the observation in Hwang and Satchell (2005). The contemporaneous
correlation between cross-sectional volatility and realized volatility is 0.3318, providing a rationale for considering the GARCH-X model with cross-sectional dispersion.
Given cross-sectional dispersion, I fit a GARCH-X model to CRSP value-weighted
index return as described by (2), (3) and (4). The second column of Table 1 displays
maximum likelihood estimates and asymptotic standard errors10 for model parameters. For comparison, the third column reports those from a univariate GARCH(1,1)
model with π = 0 in (4). In the numerical estimation procedure, the first observations (r1 and c21 ) are given and the initial value for aggregate volatility (σ12 ) is
jointly estimated with other model parameters although not reported.
While all parameter estimates for GARCH-X except π are statistically significant
at any conventional size, I find that the statistical significance for the coefficient
estimate of cross-sectional dispersion (π) is low; p-value for the two-sided hypothesis
test is 0.47, indicating the contribution of cross-sectional dispersion in predicting
market volatility is rather low. The likelihood ratio test of H0 : π = 0 has a p-value
of 0.42,11 confirming the weak evidence for the role of cross-sectional dispersion in
predicting aggregate market volatility.
Using in-sample volatility forecasts implied by parameter estimates in Table 1,
I evaluate forecasting performance by comparing forecasting accuracies of volatility
10
Asymptotic standard errors are estimated by approximating the second derivative of the log-likelihood
functions at maximum likelihood estimates. See details for numerical MLE estimation and calculation of
asymptotic standard errors in Hamilton (1994) pp. 133-148.
11
1 degree of freedom log-likelihood ratio test statistics is 2 × {−2, 040.41 − (−2, 040.73)} = 0.6391.
11
forecasts from GARCH-X and GARCH models. I adopt realized variance as a proxy
for latent volatility following Brailsford and Faff (1996), Hansen and Lunde (2006)
2
and Patton (2011)12 . Denoting by σRV,t
realized variance in period t, it is measured
by aggregating squared daily index returns within month t as
2
σRV,t
=
mt
X
2
rd,t
(11)
d=1
where rd,t is a daily CRSP value-weighted index return at day d of month t and mt
is a number of trading days in month t.
Figure 2 plots realized volatility, which is a square root of realized variance in
(11). Since observations for historical realized variance (or volatility) are widely
documented in the earlier literature,13 I suppress further explanations and proceed
to the evaluation of volatility forecasts using realized variance.
Table 2 reports average losses under nine loss criteria that has been used in the
literature. I provide definitions for each loss criteria at the second column. Column 3 and 4 report average losses of in-sample volatility forecasts from GARCH-X
and GARCH models. Last column reports the percentage differences in forecasting
accuracies of GARCH-X relative to GARCH, where for each loss function, the nagative difference implies that GARCH-X has smaller average losses than GARCH on
average. I find that GARCH-X yields lower average losses under eight out of nine
loss criteria, and the largest improvement in forecasting accuracies is found under
MSE-LOG loss fuction: GARCH-X yields 2.50% smaller average losses compared
to the univariate GARCH model.
12
Appendix C.1 provides the forecasting performance evaluation using an alternative proxy for latent
volatility such as squared return. In general, results are similar to those with realized variance.
13
For example, Poon and Granger (2003) provide explanations for market volatility including definition,
measurement and stylized facts about financial market volatility.
12
For investigating whether cross-sectional dispersion is useful for purposes of
volatility forecasting in practice, I proceed to the evaluation of out-of-sample volatility forecasts. This is intended to address the potential over-fitting issue insofar as
improved forecasting accuracy in GARCH-X may be provided by introducing an
additional parameter in the data-generating process. I adopt a rolling fixed estimation period method14 following Brownlees et al. (2012). With 6 years of estimation
window, I fit models to a sample of 6 years, generate one-step ahead volatility forecasts and drop the oldest observation from the sample when adding the new data. I
repeat this process and evaluate the performance of 645 monthly out-of-sample forecasts from April 1960 to December 2013. Similarily, I also generate out-of-sample
forecasts using 12 years of estimation window, evaluating 573 monthly forecasts
from April 1966 to December 2013.
Table 3 compares out-of-sample forecasting accuracies of GARCH-X and GARCH
models under 6 and 12 years of estimation window size. For each loss function, column 2 and 3 (5 and 6) report average losses for out-of-sample volatility forecasts
from GARCH-X and GARCH when one–step-ahead out-of-sample forecasts are obtained from parameter estimates using 6 years (12 years) observations. Column 4
and 7 report the percentage differences in forecasting accuracies of GARCH-X relative to univariate GARCH models. Here, results are contrary to the comparison
of in-sample volatility forecasts in Table 2. Average losses of GARCH-X are larger
than GARCH for eight out of nine loss functions with 6 years of estimation window,
and for all loss functions with 12 years of estimation window.
For statistical inference for average loss differentials, I perform tests for equal
14
Many researchers documented a few merits for using a rolling fixed estimation period method among
alternatives beside the ease of statistical inference. For example, Dunis et al. (2001), Giacomini and
White (2006) and Brownlees et al. (2012) noted that a rolling estimation period method is robust in the
presence of nonstationarity. West and Cho (1995) showed that its forecasting accuracy is no worse than
an expanding sample window method.
13
forecasting accuracy suggested by Diebold and Mariano (1995) and West (1996)
(henceforth, DMW). Denoting by dt the loss differential among competing forecasts
in period t, an asymptotic pairwise test statistic for testing the null hypothesis of
no difference in the forecasting accuracy is given by,
DM W =
where d = T −1
T
P
d
avar d
dt is the sample mean loss differentials and avar d is asymptotic
t=1
variance of loss differentials. Following standard practice, I obtain a consistent esti
mate for avar d by taking a weighted sum of the available sample autocovariances
using a Bartlett kernel.
For each loss function, the statistical significance of DMW test statistics is denoted by using an asterisk on the percentage differences in column 4 and 8. With
6 years of estimation window, I find that average losses of GARCH-X are larger
than GARCH for eight out of nine loss functions, where average loss differentials
for five loss functions such as MSE, MSE-SD, MSE-prop, MAE and MAE-SD are
statistically significant at 10%. With 12 years of estimation window, it becomes
even worse: average losses of GARCH-X are larger than GARCH for all loss criteria
and six loss differentials are statistically significantly larger at 10%.
So far, I investigate one potential channel by which cross-sectional information might help predict aggregate volatility. As an additional explanatory variable
in GARCH process, cross-sectional dispersion helps predict aggregate volatility in
sample under some loss criteria although the statistical significance for the coefficient estimate of cross-sectional dispersion (π) is low. However, I find that such
improvement in volatility forecasts is not robust for purposes of out-of-sample fore-
14
casting, indicating that cross-sectional dispersion does not enter the data-generating
process directly.
3.2
Aid in parameter estimation
Next, I investigate an alternative possibility that cross-sectional information improves volatility forecasts. A convenient model for incorporating cross-sectional
information is the factor-ARCH model developed in Engle et al. (1990). Although
this model has been used in hundreds of studies, it has not been successfully applied
to a cross-section of thousands of stocks due to computational difficulties15 . In this
section, I model a bivariate GARCH process for the aggregate volatility (σt2 ) and
the cross-sectional dispersion (κ2t ), and estimate model parameters by jointly using
a full cross-section of stock returns.
Using the same dataset containing prices and outstanding shares of stocks in
monthly CRSP value-weighted index, I estimate parameters in the following model:
ui,t =
ri,t = φ0i + φ1i ri,t−1 + ui,t ,
(12)
1
κt
σ t εt +
ηi,t ,
Nt−1 wi,t−1
Nt−1 wi,t−1
(13)
εt , ηi,t ∼ i.i.d. (0,1),





σt2





u2t−1



2
σt−1

  β11 0  
  $1   α11 α12  



+
=
+


 
 
 

c2t−1
0 β22
κ2t−1
α21 α22
κ2t
$2
(14)
15
To my knowlegde, the largest number of cross-sectional observations used within the factor-ARCH
model is 50 in Engle and Sheppard (2008), where authors evaluate the performance of the class of
covariance models including factor GARCH, restricted vector GARCH, dynamic conditional correlation
GARCH models and extensions of these models.
15
where $1 , $2 > 0 and α11 , α12 , α21 , α22 , β11 , β22 ≥ 0 are model parameters. c2t−1 is
a lagged cross-sectional dispersion measure provided in (10).
Four points are worth noting. First, the equation (13) is a special case of the
earlier model given by (6) and (7) for the purpose of estimating the common volatility across individuals. With a particular restriction such as λi = 1 for ∀i in (13),
individual forecasting errors are treated as if they are affected by the aggregate market and idiosyncratic shocks differently only through weight differentials16 . Second,
aggregate volatility process in (13) is same as the GARCH-X model in (4) with
$1 = $, α11 = α, α12 = π, β11 = β. Hence this suggests the statistical test for
the direct role of cross-sectional dispersion in predicting aggregate volatility by testing H0 : α12 = 0. Third, cross-sectional dispersion process is estimated along with
aggregate volatility process, where α12 and α21 capture the dynamic dependence between two volatility processes. Lastly, the above bivariate GARCH model includes
Panel-ARCH provided by Byun and Jo (2014) as a special case: with κt = τ for ∀t in
(13), Panel-ARCH estimates a univariate ARCH process for the aggregate volatility
with α12 = β11 = 0 in (14). In such a case, τ captures the time-series average of
cross-sectional dispersion across a large number of cross-sectional observations. See
also Byun and Jo (2014) for the estimation of quarterly profit uncertainty using
industry-level sales revenues. Appendix B.1 describes the empirical procedure for
estimating parameters in the above model.
Table 4 displays maximum likelihood estimates and asymptotic standard errors
for model parameters. Parameter estimates from the bivariate GARCH process (14)
are reported in the second column (Full Model). For comparison, the preceding
columns report parameter estimates from restricted models such as α21 = 0 (Model
16
This restriction will be relaxed in the next revision.
16
1) and α12 = α21 = 0 (Model 2). Parameter estimates for individual stocks such
as φ0i and φ1i , are not reported as they are obtained separately by a univariate
regression for each i.
Four points are worth noting. First, parameter estimates for describing aggregate
volatility process such as $1 , α11 and β11 are quantitatively similar to those of
univariate GARCH reported in the third column in Table 1. Second, the statistical
significance for the coefficient estimate of cross-sectional dispersion (α12 ) is low:
p-value for the two-sided hypothesis test is 0.44, confirming the weak evidence for
the direct contribution of cross-sectional dispersion in predicting aggregate market
volatility. This further implies the weak dynamic dependence between two volatility
processes when it is combined with statistically insignificant α21 17 . Third, crosssectional volatility process is shown to be non-stationary in sample: the persistence
of the process implied from coefficient estimates for α22 and β22 is 1.0695, which is
greater than 1. Lastly, given the weak dynamic dependence, Model 2 is sufficient for
describing the bivariate GARCH process for aggregate volatility and cross-sectional
dispersion: 1 degree of freedom likelihood ratio test statistic for H0 : α21 = 0 is
0.6804 with p-value being 0.41. In other words, both Full Model and Model 1 do
not improve the specification for the bivariate GARCH model in statistical sense.
Next, I evaluate forecasting performance by comparing forecasting accuracies of
volatility forecasts from univariate GARCH and above bivariate GARCH models.
More specifically, I compare forecasting accuracies of volatility forecasts from Model
1 and Model 2 with those from univariate GARCH. Though not providing additional
prediction power in sample, I include Model 1 in the comparison of the forecasting performance in order to capture the possibility that cross-sectional dispersion
17
For testing H0 : α21 = 0, p-value is 1 from the two-sided hypothesis test as well as the likelihood
ratio test by comparing maximized log-likelihood values between the full model and the model 1.
17
could improve volatility forecasts out of sample when cross-sectional stock returns
are jointly used for parameter estimation, which differs from the previous exercise
using GARCH-X. Furthermore, it also enables me to infer the relative size of the
direct contribution from the cross-sectional dispersion in Model 1 by comparing the
forecasting performance of Model 1 with Model 2. It is because the improved forecasting accuracies of Model 2 relative to univariate GARCH can be viewed as being
obtained indirectly by using cross-sectional stock returns jointly.
Table 5 compares the forecasting accuracies of two bivariate GARCH models.
For comparison, column 2 reports average losses of in-sample volatility forecasts
from univariate GARCH displayed in Table 2. While column 3 and 5 report average losses from Model 1 and Model 2 respectively, the adjacent columns report
the percentage differences in forecasting accuracies from Model 1 (column 4) and
Model 2 (column 7) respectively. There are two lines of empirical evidence supporting the improved in-sample forecasting accuracies by utilizing cross-sectional
information. First, I find improved forecasting accuracies from both Model 1 and
Model 2 across all loss criteria. In particular, the improved forecasting accuracies
from Model 2 provides the evidence on the indirect contribution of cross-sectional
information when cross-sectional stock returns are jointly used for estimating model
parameters. Second, I find that Model 1 provides larger improvements in forecasting
accuracies than Model 2, confirming the enhanced in-sample forecasting accuracies
by directly using cross-sectional dispersion as an additional explanatory variable in
the aggregate volatility process. From Model 1, the largest improvement is found
under MSE-SD loss function, yielding 6.49% smaller average losses compared to
univariate GARCH. Under MSE-LOG loss criteria of which GARCH-X provides
the largest improvement in forecasting accuracies by 2.50% (See the last column in
18
Table 2), Model 1 has 5.57% smaller average losses than univariate GARCH, that
is larger in magitude compared to GARCH-X.
Table 6 compares forecasting accuracies of one-step-ahead out-of-sample volatility forecasts from above two models when using 6 years of estimation window.
Again, I report the percentage differences relative to forecasts from GARCH as
well as the statistical significance of DMW equal predictability test statistics using
asterisks. In contrast to the failure of GARCH-X in Table 3, I find the improved
forecasting accuracies from the bivariate GARCH models. By jointly using crosssectional stock returns for estimating model parameters, Model 2 has statistically
significantly smaller average losses than GARCH for all loss functions at 10%18 .
When extending Model 2 by including cross-sectional dispersion in the aggregate
volatility process, however, Model 1 has statistically significant improvements under
three loss functions such as QLIKE, MSE-prop and MAE-prop19 .
Table 7 provides the comparison of out-of-sample forecasting accuracies under
12 years of estimation window. While confirming improved forecasting accuracies
by jointly using cross-sectional stock returns, I find that the improved forecasting
accuracies originate both from the direct and indirect contribution of cross-sectional
dispersion. To see this, recall that GARCH-X fails to provide accurate out-of-sample
volatility forecasts when using 12 years of estimation window in Table 3. By jointly
using cross-sectional stock returns for estimating parameters in the volatility process, Model 2 yields statistically smaller average losses than GARCH under seven
among nine loss functions. Furthermore, the additional improvements can be found
18
One exception is MAE-LOG loss function, where Model 2 still has smaller average losses than
GARCH.
19
The potential explanation can be found from the non-stationarity of the cross-sectional dispersion
during the sample period. Although cross-sectional dispersion is moderately correlated with aggregate
market volatility (correlation is 0.3 in the sample), it undermines the explanatory power of the crosssectional dispersion, especially when using short estimation window.
19
from Model 1 by including cross-sectional dispersion as an additional explanatory
variable in the aggregate volatility process. Model 1 has smaller average losses
(equivalently larger differences in absolute value) than Model 2 under most loss criteria expect QLIKE and MSE-prop. Here Model 1 has the statistically significantly
smaller average losses than GARCH under eight out of nine loss functions. These
contrast with the failure of GARCH-X in predicting accurate volatility forecasts out
of sample.
I further explore improved forecasting abilities provided by the cross-sectional
information by testing whether the bivariate GARCH Models also provides more
accurate volatility forecasts in particular periods when accurate volatility forecasts
are of great interest. More specifically, I focus on the second channel of the indirect
contribution provided by using a full cross-section of stock returns, and test whether
Model 2 outperforms univariate GARCH more during recessions when accurate
volatility forecasts are of great interest. Let dt be the loss differential between Model
2 and GARCH for predicting one-step-ahead out-of-sample volatility forecasts in
period t. Using an indicator variable for NBER recession periods ItR , I perform
tests for conditional predictive ability developed in Giacomini and White (2006) for
testing H0 : E [dt |Ft−1 ] = 0, which contrasts to H0 : E [dt ] = 0 in DMW equal
(unconditional) predictability tests.
Let ht be a vector of variables that are thought to be important for relative
R
forecast performance; hR
t ≡ 1, dt , It
0
in this case. Given the conditional moment
restriction, 3 degrees of freedom Wald-type test statistic (GW) is provided by,
0 b −1
GW = (T − 1) Z Ω
Z
20
where Z ≡ (T − 1)−1
PT −1
t=1
0
−1 PT −1
R
R
b
hR
t dt+1 and Ω ≡ (T − 1)
t=1 ht dt+1 × ht dt+1
is a 3 × 3 matrix that consistently estimates the variance of hR
t dt+1 . Under the
null hypothesis, the test statistic is asymptotically chi-squared distributed with 3
degrees of freedom.
Table 8 reports GW statistics using hR
t with 6 years of (column 2) and 12 years
of estimation windows (column 3) respectively. During NBER recession periods, I
find that Model 2 provides more accurate out-of-sample volatility forecasts with 6
years of estimation window (column 2), that are statistically significant under eight
loss criteria except MSE-prop. With a larger estimation window such as 12 years
(column 3), Model 2 provides statistically significantly accurate forecasts under
seven loss criteria except MSE and MSE-prop. The preceding two columns report
GW statistics for testing whether it outperforms univariate GARCH more during
bear markets. I use another indicator variable ItN for periods with negative market
N . Results are similar to
returns, and calculate GW statistics with hN
t ≡ 1, dt , It
those using NBER recession periods. I find that Model 2 yields more accurate
forecasts during periods of negative stock returns, providing conditionally accurate
volatility forecasts that are statistically significant under most loss criteria.
Although the forecasting equation is the same as univariate GARCH, I find
the improved forecasting performance by jointly using cross-sectional information:
volatility forecasts from Model 2 are more accurate than those from univariate
GARCH both in sample and out of sample. In particular, Model 2 provides more
accurate out-of-sample volatility forecasts in times of recessions as well as during
bear markets.
The potential explanation for such improvement comes from the basic insight in
Stock and Watson (2002) that when the number of cross-sectional observations is
21
large, any aggregate factors can be uncovered essentially perfectly using the cross
section. By jointly using the full cross section of stock returns, one can come up
with better estimates of the population parameters. In other words, cross-sectional
dispersion might help to estimate parameters in the aggregate volatility process.
To see this, consider the log-likelihood function of bivariate GARCH described
2
by (12), (13) and (14). Let w
ei,t−1 ≡ Nt−1 wi,t−1
with wi,t−1 being a stock i’s weight
in (1). For expositional simplicity, I define an alternative measure of cross-sectional
dispersion using w
ei,t−1 in parallel with (10). It becomes in period t,
Nt−1
e
c2t
=
X
i=1
w
ei,t−1
1
ut
ui,t −
Nt−1 wi,t−1
2
(15)
which is a special case of (10) with λi = 1 for ∀i and with replacing weights relevant
for squared deviations of individuals by w
ei,t−1 .
Using (8) and (15), the closed-form log-likelihood of bivariate GARCH is given
by,
T X
1
Nt−1
1
Nt−1 1
2
2
L =
−
×e
ct + 2
× ut
(16)
log 2π − log Jt −
2
2
2
κ2t
κt + Nt−1 σt2
t=1
N −1 QNt−1
where Jt ≡ κ2t + Nt−1 σt2 × κ2t t−1 × i=1
(wi,t−1 Nt−1 )−2 is a determinant of
a Nt−1 × Nt−1 variance matrix of individual forecasting errors.
The above log-likelihood function (16) shows that cross-sectional information
enters in the log-likelihood function through e
c2t , helping squared errors (u2t ) to estimate model parameters. In particular, it is weighted by 1/κ2t , proportional to an
inverse of conditional cross-sectional volatility. Lastly, I provide the log-likelihood of
univariate GARCH, of which only squared forecasting errors are used for parameter
22
estimation,
T X
1
1
1 u2t
2
L =
− log 2π − log σt − × 2
2
2
2 σt
t=1
4
(17)
Robustness checks
In this section, I provide robustness checks for the empirical analysis in Section
3.1. First, I revisit tests for the role of cross-sectional dispersion in predicting
aggregate volatility using 1) GARCH-cross-sectional (GARCH-XC) model and 2)
cross-sectional market volatility measure, suggested in Hwang and Satchell (2005)20 .
Next, I address concerns for highly dispersed cross-sectional returns and corresponding cross-sectional kurtosis by considering alternative cross-sectional dispersion measures.
4.1
Revisit Hwang and Satchell (2005)
Hwang and Satchell (2005) investigate whether cross-sectional dispersion can improve conditional heteroskedastic models for volatilities in the individual stock returns. While squared market returns are highly noisy to be used in the GARCH
forecasting model, they propose to use dispersion of individual stock returns with
respect to the market return, namely, cross-sectional market volatility21 :
Nt−1
2
σC,mt−1
=
X
wi,t−2 (ri,t−1 − rt−1 )2
(18)
i=1
20
For expositional simplicity, I deviate from Hwang and Satchell (2005): GARCH-XC in this paper
corresponds to GARCHX in Hwang and Satchell (2005). For notational consistency throughout this
paper, I also use $, α, β and π in place of αi,0 , αi,1 , αi,2 and αi,3 respectively.
21
From the cross-sectional market volatility in Hwang and Satchell (2005), I made two adjustments for
reflecting the time-varying number of stocks (Nt−1 ), and for making weights of individual stocks to be
predetermined (wi,t−2 ) in period t − 1. While the former adjustment is crucial in the analysis, I find that
the effect from the latter adjustment is trivial.
23
where wi,t−2 is a weight of an individual stock i in period t − 2.
Figure 3 plots historical cross-sectional market volatility, that is the square root
2
of σC,mt−1
in (18). Note that the cross-sectional market volatility becomes smaller in
size compared to the cross-sectional dispersion in Figure 1. This is mainly due to the
misspecification in (18): for large Nt , it fails to take into account different degrees by
which individual stocks respond to the aggregate market return. Furthermore, the
cross-sectional market volatility exhibits peak during the burst of dot-com bubble
in early 2001. This contrasts to the previous observation in Connor et al. (2006)
documenting the hightened idiosyncratic volatility during the stock market crash in
October 1987.
For addressing the restrictive nature of GARCH-X model,22 Hwang and Satchell
(2005) propose GARCH-cross-sectional (GARCH-XC) model by excluding the con2
stant coefficient from GARCH-X in (4). After replacing xt−1 by σC,mt−1
, it becomes
2
2
σt2 = αu2t−1 + βσt−1
+ πσC,mt−1
(19)
where π > 0 and α, β ≥ 0.
With the cross-sectional market volatility and the GARCH-XC model proposed
by Hwang and Satchell (2005), I test the direct role of cross-sectional market volatility in predicting aggregate volatility under two specifications. After replacing xt−1
2
by the cross-sectional market volatility σC,mt−1
, I fit a GARCH-X to CRSP value-
weighted index return as described by (2), (3) and (4), and I also fit a GARCH-XC
provided by (2), (3) and (19). Coefficient estimates are qualitatively similar to those
22
The non-negativity conditions for the GARCH-X model are $ > 0, α, β, π ≥ 0, that are frequently violated in empirical applications. Though $ ≤ 0, Hwang and Satchell (2005) noted that
PT
2
T −1 t=1 $ + πσC,mt−1
is likely to be positive, finding that the conditional volatility process becomes
to be always positive under this condition.
24
reported in Table 1. In particular, the coefficients estimates for the cross-sectional
market volatility (π) are statistically significant at 1% level when using GARCH-XC
as well as GARCH-X models, confirming the improved volatility specification due
to the cross-sectional market volatility.
Table 9 reports average losses of volatilty forecasts from GARCH-X and GARCHXC models when using cross-sectional market volatility. While column 2 and 3
report average in-sample losses from two models, column 4 and 5 (column 6 and
7) report average losses of out-of-sample volatility forecasts from GARCH-X and
GARCH-XC models when using 6 years (12 years) of estimation window. Boldface
entries represent more accurate volatility forecasts from GARCH-X (or GARCHXC) than GARCH for each loss function. For out-of-sample volatilty forecasts, the
statistical significance of the equal predictive abilitity (DMW) test is denoted by a
asterisk. Results are consistent with those in Table 2 and Table 3. When including cross-sectional market volatility in the aggregate volatility process, I find the
improved forecasting accuracies in sample from both GARCH-X and GARCH-XC
models. However, I confirm the weak evidence for the role of the cross-sectional
market volatility in predicting aggregate volatility out-of-sample.
4.2
Alternative measures
In this section, I provide robustness checks by using alternative cross-sectional dispersion measures within the GARCH-X model. This comes from the recognition in
Hwang and Satchell (2005) that cross-sectional returns are highly dispersed, especially for a large number of stocks considered. For investigating whether empirical
results are affected by cross-sectional kurtosis, I consider three alternative measures:
1) 5% trimmed estimator, 2) cross-sectional dispersion across 10 CRSP Cap-Based
25
Portfolios and 3) squared interquartile range. While cross-sectional dispersion across
10 CRSP Cap-Based Portfolios is self-explanatory, the other two alternatives are
1/2
constructed as follows. Let yi,t−1 ≡ wi,t−2 (ui,t−1 − λi,t−1 ut−1 ) for t = 2, . . . , T .
Rewriting the equation for cross-sectional dispersion in (10) as,
Nt−1
c2t−1
=
X
2
yi,t−1
i=1
Then, the 5% trimmed estimator is obtained from the above equation by discarding 5% extreme yi,t−1 from both tails at each period t − 1. Similarily, the squared
interquartile range estimator is calculated by squaring interquartile range of yi,t−1
at each period t − 1.
Figure 4 plots cross-sectional volatility from three alternative measures, displayed as a square root of cross-sectional dispersion. This confirms that alternative
measures are less volatile than the baseline measure in Figure 1.
I estimate a GARCH-X model again, this time replacing c2t−1 in (10) by the
above three alternatives. To conserve on space, I do not report parameter estimates
that are generally similar to those reported in Table 1. However, it is worth noting
that none of coefficient estimates of cross-sectional dispersion (π) are statistically
significant at 10% when using three alternative measures23 .
Table 10 reports average losses of in-sample volatility forecasts and boldface
entries indicate smaller average losses than univariate GARCH for each loss function. Using alternative cross-sectional dispersion measures, I find smaller average
losses under all loss criteria except MSE-prop when using 5% trimmed estimator,
confirming the improved in-sample forecasting accuracy in Table 2.
23
Using 5% trimmed estimator, the coefficient estimate of cross-sectional dispersion (π) is 0.0012 with
asymptotic standard errors being 0.0015. With estimators using 10 CRSP Cap-Based Portfolios and
squared interquartile range, estimates are close to 0.
26
Table 11 reports average losses of out-of-sample forecasts using three alternative
measures. As before, I denote out-of-sample volatility forecasts with smaller average losses than univariate GARCH using boldface character, where the statistical
significance of DMW test statistics is denoted by a asterisk. Under 6 years of estimation window, I find the statistically significantly smaller losses from 5% trimmed
estimator for MAE-prop, and from interquartile range estimator for four loss functions such as QLIKE, MSE-LOG, MAE-LOG and MAE-prop among ten cases with
smaller losses than univariate GARCH. Under 12 years of estimation window, however, I find that evidence for improved forecasting accuracy becomes to be weaker;
CRSP Cap-Based Portfolio estimator yields the statistically smaller losses only for
MAE-LOG and interquartile range estimator provides the statistically smaller losses
for two loss functions such as MSE-LOG and MAE-LOG.
To sum up, I confirm the improved forecasting performance when using three
alternative measures in sample. However, I also confirm the weak evidence for crosssectional dispersion being helpful in forecasting volatility out-of-sample, implying
that cross-sectional dispersion does not directly enter the data-generating process
given by GARCH-X.
5
Conclusion
This paper investigates the role of cross-sectional information in predicting aggregate volatility. Given a large number of individual stocks, I develop a model of
stock returns by reflecting a natural idea that individual stocks respond to the common aggregate shock at different degrees. The model is simple, but it also provides
a natural measure for cross-sectional dispersion whose effects on the stock mar-
27
ket volatility and cyclical variations in macroeconomic variables have been popular
research topics.
Using individual stocks in the CRSP value-weighted index from 1954 to 2013, I
test the direct contribution of cross-sectional dispersion in predicting stock market
volatility. Although helpful for in-sample volatility forecasts, GARCH-X with crosssectional dispersion fails to provide more accurate out-of-sample volatility forecasts
than GARCH. In other words, I provide empirical evidence that cross-sectional
dispersion does not enter the data-generating process for market volatility.
I further explore another possibility for cross-sectional dispersion contributing
to accurate estimates for model parameters. Using full cross-section of individual
stocks jointly, I estimate parameters in the bivariate GARCH model of aggregate
volatility and cross-sectional dispersion. I find that the cross-sectional dispersion
improves the accuracies of aggregate volatility forecasts both in sample and out of
sample, and for all nine loss criteria. Furthermore, out-of-sample volatility forecasts
from the bivariate GARCH model are shown to be more accurate than those from
GARCH in times of NBER recessions as well as during the periods with negative
stock index returns.
Given improved forecasting accuracies when using cross-sectional stock returns
jointly, I conclude that cross-sectional dispersion does help predict volatility forecasts indirectly by helping to estimate parameters. On the other hand, empirical
evidence from GARCH-X indicates that it does not enter in the data-generating
process directly.
28
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Figure 1 : Cross-sectional dispersion
160
cross−sectional volatility (%)
140
120
100
80
60
40
20
0
54
58
62
66
70
74
78
82
86
90
94
98
02
06
10
time period
Figure 1 plots historical cross-sectional volatility which is a square root of cross-sectional dispersion across individual stocks following (10). Shaded areas represent NBER recession periods. The cross-sectional dispersion is shown to be time-varying and highly persistent. The
cross-sectional dispersion exhibits peak during the stock market crash in October 1987, that is
commonly observed from alternative measures considered in earlier literature.
31
Figure 2 : Realized volatility
25
realized volatility (%)
20
15
10
5
0
54
58
62
66
70
74
78
82
86
90
94
98
02
06
10
time period
Figure 2 plots historical realized volatility which is a square root of realized variance calculated
by aggregating CRSP daily value-weighted index returns following (11). Shaded areas represent
NBER recession periods.
32
Figure 3 : Cross-sectional market volatility
cross−sectional market volatility (%)
25
20
15
10
5
0
54
58
62
66
70
74
78
82
86
90
94
98
02
06
10
time period
Figure 3 plots cross-sectional market volatility proposed by Hwang and Satchell (2005). The
plotted is the square root of the cross-sectional market volatility calculated from (18). Shaded
areas represent NBER recession periods.
33
Figure 4 : Alternative cross-sectional dispersion
5% Trimmed
150
100
50
0
54
58
62
66
70
74
78
82
86
90
94
98
02
06
10
90
94
98
02
06
10
90
94
98
02
06
10
CRSP Cap−Based Portfolios
60
40
20
0
54
58
62
66
70
74
78
82
86
Interquartile range
2
1
0
54
58
62
66
70
74
78
82
86
Figure 4 plots alternative cross-sectional dispersion measures. Top panel displays cross-sectional
volatility after removing 5% extreme observations from both tails. Middle panel displays crosssectional volatility constructed from 10 CRSP Cap-Based Portfolios. Bottom panel displays
interquartile range in (10).
34
Table 1: Parameter estimates
GARCH-X
GARCH
Parameters
MLE
(s.e.)
MLE
(s.e.)
$
1.1848
(0.5104)
1.0554
(0.4355)
α
0.1070
(0.0306)
0.1176
(0.0287)
β
0.8206
(0.0412)
0.8318
(0.0354)
π
0.0004
(0.0005)
φ0
0.7197
(0.1500)
0.7298
(0.1485)
φ1
0.0629
(0.0402)
0.0634
(0.0402)
Likelihood
−2, 040.41
−2, 040.73
Table 1 reports MLE estimates (asymptotic standard errors) of model parameters in GARCH-X
and GARCH models. Asymptotic standard errors are estimated by approximating the second
derivative of the log-likelihood functions at MLE estimates. The last row reports the maximized
log-likelihood values under two volatility forecasting models.
35
Table 2: Comparison of in-sample forecasting accuracy
Criteria
2
L (σRV
, σ2)
M SE
2
(σRV
− σ2)
GARCH-X
GARCH
Difference (%)
1, 303.23
1, 312.00
−0.67
−1
3.77
3.78
−0.19
2
− log σ 2 )
(log σRV
2
0.81
0.84
−2.50
(σRV − σ)2
4.03
4.12
−2.19
2
−1
2.76
2.75
0.42
2
− σ2|
|σRV
13.57
13.74
−1.19
2
− log σ 2 |
|log σRV
0.75
0.75
−1.05
M AE − SD
|σRV − σ|
1.41
1.43
−1.23
M AE − prop
σ
RV − 1
σ
0.69
0.69
−0.87
QLIKE
M SE − LOG
M SE − SD
M SE − prop
M AE
M AE − LOG
2
σRV
σ2
− log
σRV
σ2
2
2
σRV
σ2
Table 2 provides the comparison of in-sample volatility forecasts from GARCH-X and GARCH
models. As a proxy for an unobservable volatility, historical realized variance is calculated
from the daily CRSP value-weighted index return without cash dividends. Column 2 provides
definitions of loss functions for measuring volatility forecasting accuracies. Next two columns
report average losses of in-sample volatility forecasts from GARCH-X and GARCH models.
Last column reports the percentage differences in forecasting accuracies of GARCH-X relative
to GARCH, where the negative difference implies that GARCH-X has smaller average losses
than GARCH.
36
Table 3: Comparison of out-of-sample forecasting accuracy
6 year window
Criteria
12 year window
GARCH-X
GARCH
Difference (%)
GARCH-X
GARCH
Difference (%)
1, 556.23
1, 419.27
9.65∗∗∗
1, 743.40
1, 593.86
9.38∗
QLIKE
3.96
3.95
0.24
4.04
4.00
0.81∗
M SE − LOG
0.96
0.95
1.25
0.88
0.85
3.20
M SE − SD
5.30
4.85
9.26∗∗∗
5.57
5.02
10.88∗∗∗
M SE − prop
6.37
5.04
26.43∗
4.97
3.60
38.10∗
M AE
16.26
15.40
5.56∗∗
17.42
16.29
6.99∗∗
M AE − LOG
0.80
0.80
−0.05
0.78
0.77
0.99
M AE − SD
1.60
1.56
2.30∗
1.66
1.60
3.58∗
M AE − prop
0.87
0.87
0.06
0.83
0.81
2.92
M SE
Table 3 provides the comparison of out-of-sample volatility forecasts from GARCH-X and
GARCH models. A rolling fixed estimation period method was used for calculating out-of-sample
volatility forecasts under two estimation window size: 6 years (column 2-3) and 12 years (column
5-6). Column 4 and 7 report the percentage differences in forecasting accuracies of GARCH-X
relative to univariate GARCH. While the negative difference implies that GARCH-X has smaller
average losses than univariate GARCH, asterisk represents the statistical significance of DMW
equal predictability test suggested by Diebold and Mariano (1995) and West (1996). Given critical values being 1.28 (90%), 1.65 (95%) and 2.33 (99%) respectively, */**/*** represent the
statistical significance at 90%, 95% and 99% respectively.
37
Table 4: Parameter estimates
Full Model
Model 1
Model 2
Parameters
MLE
(s.e.)
MLE
(s.e.)
MLE
(s.e.)
$1
1.2661
(0.5288)
1.2662
(0.5287)
1.1594
(0.4725)
$2
98.5860
(0.5757)
98.5860
(0.5329)
98.5858
(0.5329)
α11
0.1043
(0.0338)
0.1043
(0.0338)
0.1207
(0.0296)
α12
0.0004
(0.0005)
0.0004
(0.0005)
−
−
α21
0.0000
(0.0319)
−
−
−
−
α22
0.2933
(0.0014)
0.2933
(0.0012)
0.2933
(0.0012)
β11
0.8145
(0.0412)
0.8145
(0.0412)
0.8211
(0.0381)
β22
0.7762
(0.0009)
0.7762
(0.0008)
0.7762
(0.0008)
Likelihood
−24, 164, 572.03
−24, 164, 572.03
−24, 164, 572.37
Table 4 reports MLE estimates (asymptotic standard errors) of the bivariate GARCH model.
Asymptotic standard errors are estimated by approximating the second derivative of the loglikelihood functions at MLE estimates. For comparison, it also reports estimation results under
two restricted models: α21 = 0 (Model 1) and α12 = α21 = 0 (Model 2). The last row reports
the maximized log-likelihood values under three models.
38
Table 5: Comparison of in-sample forecasting accuracy
Criteria
GARCH
Model 1
Difference (%)
Model 2
Difference (%)
M SE
1, 312.00
1, 265.80
−3.52
1, 282.10
−2.28
QLIKE
3.78
3.76
−0.55
3.77
−0.29
M SE − LOG
0.84
0.79
−5.57
0.81
−2.78
M SE − SD
4.12
3.86
−6.49
3.98
−3.55
M SE − prop
2.75
2.70
−2.10
2.74
−0.49
M AE
13.74
13.20
−3.88
13.44
−2.17
M AE − LOG
0.75
0.73
−2.98
0.74
−1.72
M AE − SD
1.43
1.38
−3.72
1.40
−2.11
M AE − prop
0.69
0.67
−3.23
0.68
−1.86
Table 5 provides the comparison of in-sample volatility forecasts from univariate GARCH and
two bivariate GARCH models provided in Section 3.2.. As a proxy for an unobservable volatility, historical realized variance is calculated from the daily CRSP value-weighted index return
without cash dividends. Column 2 reports average losses of in-sample volatility forecasts from
GARCH provided in Table 2. Column 3 and 5 report average losses of in-sample volatility forecasts from two bivariate GARCH models, and adjacent columns report the percentage differences
in forecasting accuracies of two bivariate GARCH models relative to univariate GARCH model,
where the negative difference implies that the bivariate GARCH model has smaller average losses
than GARCH.
39
Table 6: Comparison of out-of-sample volatility forecasts (6 years)
Criteria
Model 1
Difference (%)
Model 2
Difference (%)
M SE
1, 388.92
−2.14
1, 358.07
−4.31∗∗∗
QLIKE
3.92
−0.84∗∗
3.92
−0.94∗∗∗
M SE − LOG
0.95
−0.55
0.92
−3.73∗
M SE − SD
4.76
−1.74
4.58
−5.59∗∗∗
M SE − prop
4.03
−20.15∗∗
4.07
−19.17∗∗
M AE
15.45
0.27
15.02
−2.51∗∗∗
M AE − LOG
0.80
0.23
0.79
−1.07
M AE − SD
1.57
0.24
1.54
−1.77∗∗
M AE − prop
0.82
−5.55∗∗∗
0.83
−5.32∗∗∗
Table 6 provides average losses of out-of-sample volatility forecasts from two bivariate GARCH
models introduced in Section 3.2. Given out-of-sample volatility forecasts using 6 years of
estimation window, column 2 (column 4) reports average losses of one-month-ahead volatility
forecasts from Model 1 (Model 2). Column 3 (column 5) reports the percentage differences in
forecasting accuracies of Model 1 (Model 2) relative to univariate GARCH reported in Table 3.
While the negative difference implies that Model 1 (Model 2) has smaller average losses than
univariate GARCH, asterisk represents the statistical significance of DMW equal predictability
test suggested by Diebold and Mariano (1995) and West (1996). Given critical values being 1.28
(90%), 1.65 (95%) and 2.33 (99%) respectively, */**/*** represent the statistical significance at
90%, 95% and 99% respectively.
40
Table 7: Comparison of out-of-sample volatility forecasts (12 years)
Criteria
Model 1
Difference (%)
Model 2
Difference (%)
M SE
1, 505.45
−5.55∗∗
1, 520.94
−4.58∗
QLIKE
3.97
−0.78∗∗
3.97
−0.90∗∗
M SE − LOG
0.82
−3.81∗∗
0.83
−3.03∗∗
M SE − SD
4.68
−6.77∗∗
4.75
−5.38∗
M SE − prop
3.39
−5.92
3.14
−12.83
M AE
15.60
−4.21∗∗
15.94
−2.11
M AE − LOG
0.75
−3.08∗∗∗
0.76
−1.41∗
M AE − SD
1.54
−3.73∗∗
1.57
−1.77∗
M AE − prop
0.75
−7.30∗∗∗
0.75
−6.45∗∗
Table 7 provides average losses of out-of-sample volatility forecasts from two bivariate GARCH
models introduced in Section 3.2. Given out-of-sample volatility forecasts using 12 years of
estimation window, column 2 (column 4) reports average losses of one-month-ahead volatility
forecasts from Model 1 (Model 2). Column 3 (column 5) reports the percentage differences in
forecasting accuracies of Model 1 (Model 2) relative to univariate GARCH reported in Table 3.
While the negative difference implies that Model 1 (Model 2) has smaller average losses than
univariate GARCH, asterisk represents the statistical significance of DMW equal predictability
test suggested by Diebold and Mariano (1995) and West (1996). Given critical values being 1.28
(90%), 1.65 (95%) and 2.33 (99%) respectively, */**/*** represent the statistical significance at
90%, 95% and 99% respectively.
41
Table 8: Tests for conditional predictive ability
Recession
Bear markets
Criteria
6 years
12 years
6 years
12 years
M SE
11.95∗∗∗
5.62
7.28∗
5.50
QLIKE
11.21∗∗
7.06∗
11.26∗∗
7.18∗
M SE − LOG
12.42∗∗∗
12.35∗∗∗
8.12∗∗
9.66∗∗
M SE − SD
19.13∗∗∗
8.59∗∗
19.71∗∗∗
8.52∗∗
M SE − prop
5.29
2.75
5.09
6.31∗
8.36∗∗
14.26∗∗∗
8.20∗∗
12.10∗∗∗
M AE − LOG
25.33∗∗∗
20.14∗∗∗
22.99∗∗∗
19.26∗∗∗
M AE − SD
20.07∗∗∗
18.33∗∗∗
19.88∗∗∗
18.64∗∗∗
M AE − prop
11.32∗∗
9.71∗∗
11.35∗∗∗
9.06∗∗
M AE
Table 8 reports test statistics for conditional predictive ability proposed by Giacomini and
White (2006). Under the null hypothesis of no conditional loss differentials, the test statistic
is asymptotically chi-squared distributed with 3 degrees of freedom. Column 2 and 3 reports
results for testing whether Model 2 outperforms univariate GARCH during recession. Column 4
and 5 reports results for testing whether Model 2 outperforms univariate GARCH conditional on
negative return on the value-weighted stock market index. Given critical values with 3 degrees
of freedom being 6.25 (90%), 7.81 (95%) and 11.34 (99%) respectively, */**/*** represent the
statistical significance at 90%, 95% and 99% respectively.
42
Table 9: Robustness checks - Hwang and Satchell
In-sample
6 years
12 years
Criteria
GARCH-X
GARCH-XC
GARCH-X
GARCH-XC
GARCH-X
GARCH-XC
M SE
1, 276.87
1, 273.91
1, 541.30
1, 549.40
1, 679.30
1, 677.90
QLIKE
3.75
3.74
3.98
3.98
4.05
4.05
M SE − LOG
0.74
0.73
0.92∗∗∗
0.92∗∗∗
0.87
0.87
M SE − SD
3.78
3.81
5.04
5.09
5.28
5.27
M SE − prop
2.71
2.59
7.40
7.44
5.82
5.86
M AE
13.03
13.22
15.41
15.44
16.59
16.58
M AE − LOG
0.72
0.71
0.78∗∗∗
0.78∗∗∗
0.78
0.77
M AE − SD
1.35
1.36
1.55∗
1.54∗
1.62
1.61
M AE − prop
0.66
0.66
0.91
0.90
0.87
0.86
Table 9 reports average losses of volatility forecasts from GARCH-X and GARCH-XC. Following
Hwang and Satchell (2005), I consider the cross- sectional market volatility calculated from
(18), and the GARCH-cross-sectional (GARCH-XC) model specified as in (19). Using the
cross-sectional market volatility as a measure for cross-sectional dispersion, column 2 and 3
report average in-sample losses from GARCH-X and GARCH-XC models. Column 4 and 5
(column 6 and 7) report average losses of out-of-sample volatility forecasts from GARCH-X and
GARCH-XC models when using 6 years (12 years) estimation window. For each loss function,
boldface entries represent more accurate volatility forecasts from GARCH-X (or GARCH-XC)
than GARCH. For out-of-sample volatility forecasts, the statistical significance of the equal
predictive ability test (DMW) is denoted by a asterisk. Given critical values being 1.28 (90%),
1.65 (95%) and 2.33 (99%) respectively, */**/*** represent the statistical significance at 90%,
95% and 99% respectively.
43
Table 10: Robustness checks - alternative measures (in-sample)
5%
Cap
IQR
1, 304.17
1, 311.92
1, 311.92
QLIKE
3.77
3.78
3.78
M SE − LOG
0.81
0.83
0.83
M SE − SD
4.04
4.12
4.12
M SE − prop
2.79
2.75
2.75
M AE
13.58
13.73
13.73
M AE − LOG
0.75
0.75
0.75
M AE − SD
1.41
1.43
1.43
M AE − prop
0.69
0.69
0.69
M SE
Table 10 reports robustness checks for in-sample volatility forecasts from GARCH-X. Given
concerns for a noisy cross-sectional dispersion measure when using all individual stock returns, we
consider three alternative measures as a covariate in the GARCH-X model; in (10), we calculate
1) the trimmed cross-sectional dispersion (5%) where 5% extreme observations are removed from
both tails at each t, 2) cross-sectional dispersion using 10 CRSP Cap-Based Portfolios (Cap),
3) squared interquartile range (IQR). Boldface entries have lower average losses than univariate
GARCH for each loss function. Here, we find smaller average losses with alternative measures
under all loss criteria except MSE-prop when using 5% trimmed cross-sectional dispersion.
44
Table 11: Robustness checks - alternative measures (out-of-sample)
6 years
12 years
5%
Cap
IQR
5%
Cap
IQR
1, 583.42
1, 865.05
1, 517.78
1, 780.54
1, 956.36
1, 602.34
QLIKE
3.95
3.96
3.94∗∗∗
4.03
4.02
4.01
M SE − LOG
0.95
0.91
0.93∗∗∗
0.88
0.84
0.83∗
M SE − SD
5.31
5.69
5.05
5.63
5.85
5.03
M SE − prop
5.72
5.49
5.38
4.45
4.29
3.82
M AE
16.31
16.73
15.74
17.57
17.60
16.22
M AE − LOG
0.80
0.77
0.79∗∗
0.78
0.75∗∗∗
0.76∗
M AE − SD
1.60
1.57
1.56
1.67
1.62
1.58
M AE − prop
0.86∗∗
0.90
0.86∗
0.83
0.82
0.82
M SE
Table 11 reports robustness checks for out-of-sample volatility forecasts from GARCH-X using
three alternative cross-sectional dispersion measures. While boldface entries have lower average losses than univariate GARCH for each loss function, the statistical significance of DMW
test statistics is denoted by asterisk. Given critical values being 1.28 (90%), 1.65 (95%) and
2.33 (99%) respectively, */**/*** represent the statistical significance at 90%, 95% and 99%
respectively.
45
Appendix
A.1. Summary Statistics
In this section, I provide summary statistics of individual stock returns that are
used for constructing cross-sectional dispersion across stock returns following (10).
For describing cross-sectional distribution of historical individual stock returns, I
begin by constructing time-series summary statistics of individual stock returns,
and obtain cross-sectional summary statistics across those of individual stocks in
the universe. Table A1 reports cross-sectional distribution summarized by range
statistics such as Min, Q1, Median, Q3 and Max, cross-sectional average and crosssectional standard deviation of each time-series individual summary statistics.
On average, individual stocks have appeared in the monthly stock index for 13
years (158 months) while the median corresponds to about 10 years (117 months).
Of note, there are only 118 individual stock returns spanning 60 years of the total sample period since firms listed on NASDAQ began to be included in CRSP
database at the beginning of 1973. Next, it is common to observe the positive historical average returns (16,775 stocks or 77.94%), and excess kurtosis (20,611 stocks
or 95.76%) across individual stock returns. On the other hand, cross-sectional distributions of returns and kurtosis exhibit the large heterogeniety across individual
stock returns. While the historical individual stock return is about 0.89% on average (median of about 1.03%), for example, there are a few firms with either large
postivie or large negative historical average returns. Lastly, there are 18,310 firms
(85.07%) having positively skewed historical returns, which is seemingly inconsistent with the frequently documented negatively skewed stock index return. Though
interesting, I do not pursue this finding further as it is irrelevant to the goal of this
46
paper.
Table A1 : Descriptive Statistics
Min
Q1
Median
Q3
Max
Mean
Stdev
Time periods
36
68
117
204
720
157.92
125.91
Mean
-14.00
0.13
1.03
1.81
27.63
0.89
1.82
Standard deviation
0.23
10.12
14.88
20.87
204.78
16.66
9.75
Skewness
-8.14
0.27
0.77
1.44
14.81
0.99
1.29
Kurtosis
1.81
4.23
5.67
8.75
248.63
8.54
10.40
Table A1 provides the summary statistics for individual stock returns used for constructing a
series of the monthly stock index returns.
A.2. Changes in stock index universe
Let Nt be a total number of stocks, NtA and NtD respectively be the number
of stocks that are added in and dropped from the stock index universe in period t.
Denoting by NtR the number of stocks that are carried over from t − 1 to t, law of
D . Here the former
motions are given by 1) Nt = NtR + NtA and 2) NtR = Nt−1 − Nt−1
indicates stocks in the stock index universe in period t are either those remained
from the previous period t − 1 or those newly introduced in the universe. The
latter indicates that stocks carried over from the previous period are leftovers after
excluding stocks dropped at the end of period t − 1.
Figure A1 plots the transition of the stock index universe. There are two periods
with large number of stocks being added in the stock universe: 737 stocks (39.29%)
were newly introduced during August 1962 and 2,281 stocks (46.51%) were added in
during January 1973. Apart from these two big events, the total number of stocks
has been gradually changed over time.
47
Figure A1: The dynamics of the Stock index universe
10000
Total number of Stocks (Nt)
Added Stocks (NA)
t
Dropped Stocks (ND
)
t
number of stocks
8000
6000
4000
2000
0
54
58
62
66
70
74
78
82
86
90
time period
94
98
02
06
10
Note: This figure plots the total number of stocks (solid line), number of stocks added in (long
dashed line) and dropped from the stock index universe (short dashed line) from February 1954
to December 2013.
B.1. Empirical Procedure
In this section, I describe the parameter estimation procedure of the proposed
model which is skipped in Section 3.2. For clarifying dimensions associated with
vectors and matrices, I use one underline below a variable for representing a vector
and two underlines for representing a matrix.
To begin with, consider individual forecasting errors obtained from AR(1) forecasting model of returns (12): for each stock i, an individual forecasting error ui,t is
obtained by regressing ri,t on a constant and its lagged return ri,t−1 for t = 2, . . . , T .
Let ut ≡ [u1,t , . . . , uNt ,t ]0 be a collection of individual forecasting errors at period t.
Denoting by Ωt ≡ E [ut u0t |Ft−1 ] a Nt × Nt variance-covariance matrix of individual
forecasting errors in period t, the joint log-likelihood function of individual stock
48
returns becomes,
T 1
X
Nt−1
1
0 −1
L =
−
log 2π − log Ωt − ut Ωt ut
2
2
2
t=1
In general, the numerical maximization of above log-likelihoods by iterative
methods can be quite costly since it requires an inversion and a determinant calculation of Nt × Nt matrix Ωt for each period t.
Here I overcome this empirical intractability issue by modeling an individual
forecasting error using a factor structure provided by equation (13). Since Ωt is
a symmetric matrix that is factored by a vector of individual weights, analytical
forms of inversion and determinant are given by,

Nt−1
Y
2 (Nt−1 −1)
2
2

Ω
=
κ
·
κ
+
N
σ
·
t
t−1 t
t
t
i=1
(i, j) =
Ω−1
t







2 w2
2
2
Nt−1
i,t−1 ·(κt +(Nt−1 −1)σt )
κ2t ·(κ2t +Nt−1 σt2 )
−
2 w
2
Nt−1
i,t−1 wj,t−1 σt
κ2t ·(κ2t +Nt−1 σt2 )

1

2 w2
Nt−1
i,t−1
for j = i
for j 6= i
(i, j) is an (i, j)th element in the inverse matrix Ω−1
.
where Ω−1
t
t
Then, the closed-form log-likelihood of Panel-GARCH is given by,
L =
T
X
t=1

− Nt−1
2

Nt
X
2
N
1
σt2
log 2π − log Jt − t−1
(wi,t−1 ui,t )2 − 2
2
2
2κt  i=1
κt + Nt−1 σt2
Nt
X
i=1
!2 


wi,t−1 ui,t

(Nt−1 −1) QNt−1
where Jt ≡ κ2t + Nt−1 σt2 × κ2t
× i=1 (wi,t−1 Nt−1 )−2 is a determinant
of a Nt−1 × Nt−1 variance matrix of individual forecasting errors. This can be
numerically evaluated along with the bivariate GARCH process, providing MLE
estimates for parameters.
49
C.1. Squared return proxy
In this subsection, I provide the comparison of forecasting accuracies across
models using squared returns instead of realized variance.
For evaluating the performance of volatility forecasting models, monthly squared
returns have been widely adopted as a proxy for latent volatility. See, for example,
Pagan and Schwert (1990), Day and Lewis (1992) and Franses and Van Dijk (1996)
as previous applications using squared returns.
Squared return proxy is obtained by squaring residuals from the AR(1) forecasting model of the aggregate return in (2). Given estimates for φ0 and φ1 , one can
obtain squared returns as: for each t,
2
u2t = rt − φb0 − φb1 rt−1
Using squared return proxy, I provide the comparison of forecasting accuracies
across univariate GARCH, GARCH-X and two bivariate models proposed in Section
3.2, that are in parallel with Table 2 and 5 (Table C1, in-sample), Table 3 (columns
2 - 4) and Table 6 (Table C2, out-of-sample using 6 years of estimation window),
and Table 3 (columns 5 - 7) and Table 7 (Table C3, out-of-sample using 12 years of
estimation window).
50
Table C1 : Comparison of in-sample forecasting accuracy
GARCH
GARCH-X
Model 1
Model 2
Criteria
Average
Difference (%)
Difference (%)
Difference (%)
M SE
1, 345.70
0.13
−0.67
−0.68
QLIKE
3.86
−0.02
−0.14
−0.13
M SE − LOG
6.44
−0.16
−0.62
−0.48
M SE − SD
8.38
−0.03
−1.18
−1.07
M SE − prop
3.81
0.56
3.17
3.61
M AE
18.87
−0.13
−1.10
−0.92
M AE − LOG
1.77
−0.10
−0.57
−0.49
M AE − SD
2.29
−0.15
−1.00
−0.84
M AE − prop
1.01
0.04
−0.08
−0.12
Table C1 provides the comparison of in-sample volatility forecasts from GARCH, GARCH-X,
and two bivariate GARCH models provided in Section 3.2.. As a proxy for an unobservable
volatility, squared return is used. While column 2 reports average losses of in-sample volatility forecasts from GARCH, the preceeding three columns report the percentage differences in
forecasting accuracies of GARCH-X and two bivariate GARCH models relative to univariate
GARCH model. The negative difference implies that the considered model has smaller average
losses than GARCH.
51
Table C2 : Comparison of out-of-sample forecasting accuracy (6 years)
GARCH
GARCH-X
Model 1
Model 2
Criteria
Average
Difference (%)
Difference (%)
Difference (%)
M SE
1, 586.90
8.68∗∗∗
−0.47
−1.54
QLIKE
3.99
0.22
−0.69∗∗
−0.65∗∗
M SE − LOG
6.21
1.24∗∗∗
1.40∗
0.87
M SE − SD
9.61
6.43∗∗∗
−0.06
−1.79
M SE − prop
5.44
9.53
−13.55∗∗∗
−11.61∗∗∗
M AE
20.88
4.38∗∗∗
1.15∗
−0.95
M AE − LOG
1.77
0.43
0.54
−0.25
M AE − SD
2.41
1.86∗∗
0.76
−0.78
M AE − prop
1.17
0.47
−4.13∗∗∗
−3.74∗∗∗
Table C2 provides the comparison of out-of-sample volatility forecasts from GARCH, GARCHX, and two bivariate GARCH models provided in Section 3.2.. One-month-ahead volatility
forecasts are calculated from parameter estimates using 6 years of estimation window. As a
proxy for an unobservable volatility, squared return is used. While column 2 reports average
losses of in-sample volatility forecasts from GARCH, the preceeding three columns report the
percentage differences in forecasting accuracies of GARCH-X and two bivariate GARCH models
relative to univariate GARCH model. The negative difference implies that the considered model
has smaller average losses than GARCH.
52
Table C3 : Comparison of out-of-sample forecasting accuracy (6 years)
GARCH
GARCH-X
Model 1
Model 2
Criteria
Average
Difference (%)
Difference (%)
Difference (%)
M SE
1, 687.60
4.86∗∗∗
−1.44∗∗∗
−0.85∗∗∗
QLIKE
4.06
−0.20
−0.70
−0.86∗
M SE − LOG
6.57
−0.19
0.30
−0.40∗
M SE − SD
10.03
4.20∗∗∗
−1.09∗
−1.51∗∗∗
M SE − prop
5.51
3.48
−2.20
−17.47∗∗
M AE
21.63
3.43∗∗
0.29
0.25
M AE − LOG
1.80
−0.11
0.01
0.01
M AE − SD
2.49
1.17
0.03
0.00
M AE − prop
1.14
−1.87
−4.38∗∗∗
−4.04∗∗∗
Table C3 provides the comparison of out-of-sample volatility forecasts from GARCH, GARCHX, and two bivariate GARCH models provided in Section 3.2.. One-month-ahead volatility
forecasts are calculated from parameter estimates using 12 years of estimation window. As a
proxy for an unobservable volatility, squared return is used. While column 2 reports average
losses of in-sample volatility forecasts from GARCH, the preceeding three columns report the
percentage differences in forecasting accuracies of GARCH-X and two bivariate GARCH models
relative to univariate GARCH model. The negative difference implies that the considered model
has smaller average losses than GARCH.
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