Are common factors useful in forecasting international
stock market realized variances?
Juri Marcucci∗
Bank of Italy
Via Nazionale 91, 00184 Rome, Italy
E-mail: juri@sssup.it
September 2008
Abstract
Apparently the answer is yes! We compare standard univariate time series models and multivariate factor models in terms of their ability to forecast the weekly realized variances of 33
international stock exchanges. We also try to assess to which extent adopting common pure
variance factors - which exploit the comovements across markets - helps in forecasting the
univariate realized volatilities of each single stock market. Our results show that models with
regional variance factors constructed giving equal weight to countries within the same region
tend to fare the best out of sample. In addition, models that take as variance factors the canonical variates tend to outperform those that utilize other multivariate statistical techniques, thus
confirming the predictive power of canonical correlation analysis.
JEL Classification: C52, C32
Keywords: International stock market volatilities, Forecasting, Pure Variance Common Features, Approximate factor models, Variance factors.
∗
I would like to thank Gianluca Cubadda, Graham Elliott, Robert F. Engle, Bruce Lehmann, Francesca Lotti,
Giuseppe Storti and Allan Timmermann for helpful discussions and their valuable comments. I also would like to
thank seminar participants at the University of Rome ‘Tor Vergata’ and the University of Salerno and participants at
the conference “Factor Structures for Panel and Multivariate Time Series Data” for their comments. All remaining
errors are, of course, my own responsibility. The usual disclaimers apply.
1
1
Introduction
Volatility represents a measure of risk and therefore it is a key ingredient for pricing financial
instruments, for portfolio allocation and for modern risk management, where measures of extreme
shifts in the conditional distribution of returns (such as Value-at-Risk) are fundamental for both
regulators and firms.
While a huge number of univariate models has been put forward since the seminal paper on
ARCH by Engle (1982), multivariate volatility models are still underexplored. This is certainly due
to the curse-of-dimensionality problem since the number of parameters tends to grow exponentially
with the number of assets included in a portfolio. This has led many researchers (for example, Engle et al., 1990 and Ng et al. 1992) to link the evolution of each single asset’s conditional variance
to a reduced number of common factors with time-varying volatilities. This approach is justified
by modern asset pricing theories (such as APT or CAPM) where it is standard to either assume
or motivate a low-dimensional factor structure for returns. Motivated by this idea and by the empirical fact that there are comovements in volatilities both across assets and markets, Engle et al.
(1990) introduce a Factor ARCH model for the term structure of interest rates where the dynamics in the conditional variance of each interest rate are driven by a time-varying variance factor.
King et al. (1994) propose a multifactor model for the aggregate stock returns of sixteen countries,
where the factors can be observable or not, and some of them depend on macroeconomic variables.
These models completely specify the conditional distribution of the returns so that they can deliver
predictions for their means, variances and covariances. Another strand of literature suggests factor
models directly for the variances. For example, Harvey et al. (1994) develop multivariate models
for the logarithms of the squared returns and consider the possibility of common random factors in
the variances of four exchange rate returns. In a similar spirit, Engle and Marcucci (2006) propose
a Long-Run Pure Variance Common Features model, where only the forecasts of the variances are
of interest and they are modeled jointly assuming that a logarithmic transformation of the squared
returns has a common long-run feature. Along the same lines, Anderson and Vahid (2007) suggest
factor models for the realized volatilities of 21 stocks traded in the Australian exchange showing
that using factors can improve the volatility forecasts of each single asset.
The factor approach is also justified by the evidence that volatilities tend to comove as if there
was a common factor driving the whole return process (see for example Andersen et al, 2001
who show that the realized volatilities of the 30 stocks of the Dow Jones are characterized by
strong comovements). This fact is particularly true for stock market indices of different countries
where there is also an interest in finding possible links and determinants of financial contagion or
interdependence (see Forbes and Rigobon, 2002).
2
In this paper we study the volatility processes of 33 international stock markets as measured by
the weekly realized volatilities computed using the daily MSCI indices. Our focus is on predicting
each country’s volatility. Our main goal is to assess whether the use of common variance factors
does really help in forecasting each country’s volatility. In fact, common variance factors capture
the comovements across stock markets and summarize all the relevant information to predict the
volatility of a single stock market.
We restrict our attention to multivariate factor models where the factors can be interpreted
as leading indicators, thus summarizing all the relevant information in both regional and world
stock markets. Our goal is twofold. First, we test for the possible presence of either a common
world ARCH factor or a few common regional ARCH factors by comparing a number of countries
larger than that of Engle and Susmel (1993). This represents a preliminary but necessary step
to understand the features of the volatility processes of international markets and it is important
to interpret the factors used in the following forecasting exercise. Second, we compare different
models to predict the weekly realized volatility of our set of international stock market returns. On
one side, we consider pure variance models where we directly model the weekly realized variances
both with standard univariate time series techniques and with factor models, treating volatility as if
it were observed rather than latent (thus following Andersen et al.’s (2003) approach). On the other
side, we compare the forecasting performances of univariate and multivariate volatility models
based on returns sampled at a lower frequency. In particular, we are interested in the forecasting
performances of factor models, where the variance factors summarize all the information contained
in a portfolio of volatilities. We also allow for some idiosyncratic residual correlation in our models
of the realized variances, thus modeling time-varying volatility also in the idiosyncratic factors.
Our approach is therefore in the same spirit of Stock and Watson (1998, 1999, 2002) who use
principal components in a macroeconomic context to summarize the information contained in a
very large number of covariates in a few diffusion indices to be then used for forecasting purposes.
Artis et al. (2005) use this approach to forecast 81 key UK macro variables with just six factors.
In this context, it is sensible to consider indices composed of predictor variables, and to choose
those with the larger variances since they are more likely to represent the common factors that
characterize the variables to be forecast.
Two results emerge from the empirical application. First, as in Engle and Susmel (1993),
we find no evidence of a common world ARCH factor but only of some regional ARCH factors.
Second, we find that factor models, where the common variance factors are given by the equally
weighted portfolios of single regions or time-zones, tend to outperform all the other factor models. In particular, we find that models adopting regional factors tend to be superior (in a mean
3
squared error sense) to those with a global factor and to those with statistical factors. This is in line
with previous research in the financial econometrics literature such as Anderson and Vahid (2007).
Moreover, we also find that models that use the variance factors obtained from the canonical variates tend to fare better than the others that utilize different multivariate techniques, thus confirming
the predictive power of canonical correlation analysis.
The plan of the paper is as follows. Section 2 illustrates the pure variance multivariate factor
models adopted to forecast the realized variances of our set of international stock markets’ returns.
This section also develops the econometric specification and the problems involved in the detection
of common long run volatility factors. Section 3 explores the features of the international stock
market data focusing on the presence of common ARCH factors. The empirical relevance of pure
variance factor models is discussed in the forecasting exercise of Section 4, where both univariate
and multivariate models are compared in terms of their ability to forecast the weekly realized
volatility of each stock market. Section 5 sketches some concluding remarks giving also possible
directions for further research.
2
Pure Variance Factor Models
2.1
Approximate Factor Models for Realized Variances and number of variance factors
The notion of a low-dimensional factor structure is essential in modern asset pricing theory. Traditional asset pricing theories such as APT or CAPM suggest that a small number of factors drive
the movements and in particular the comovements of large sets of asset returns.
Following the idea of Engle and Marcucci (2006), we assume the following approximate factor
structure for our portfolio of realized volatilities
Yt
(N ×1)
= Λ
Ft
(N ×r)(r×1)
+ ut
(1)
(N ×1)
where Yt is an N × 1 vector of realized volatilities, Ft is the r × 1 vector of common variance
factors (with r N ), Λ is the N × r matrix of variance factor loadings, while ut is the N × 1
vector of idiosyncratic factors independent of Ft . Approximate factor models were first introduced
by Chamberlain and Rothschild (1983). These authors show that, if the approximate factor strucT
P
ture holds, the r largest eigenvalues of the covariance matrix T −1
Yt Yt0 will tend to infinity as
t=1
both the cross section dimension N and the time dimension T go to infinity, while the (r + 1)-th
eigenvalue remains bounded. This prompts an immediate visual method to determine the number
4
of variance factors by examining the behavior of the largest eigenvalues as N grows large.
The intuition behind this model is clear. As the number of cross-sections grows large, eigenvector analysis is asymptotically equivalent to factor analysis. This means that each cross-section
provides additional information only about the pervasive factors Ft and just local information
about the idiosyncrasies ut . From a finance standpoint, as the number of cross-sections increases,
the proportion of total variation explained by the non-pervasive sources of risk must approach zero.
Thus, as N grows large, the information in the data about the common factors will be of order N ,
whereas the information on the idiosyncrasies will remain finite.
Bai and Ng (2002) suggest using the method of principal components to consistently estimate
both factors and factor loadings. Let λ̂k and F̂ k denote the estimates of the factor loadings and
factors when k is the hypothesized number of common factors. These estimates are obtained
P PT
k k 2
, that leads to the estimates
F
X
−
λ
by solving the problem minΛ,F k (N T )−1 N
it
i t
i=1
t=1
√
F̃ k or F̄ k of the common factors depending on the normalization. F̃ k is given by T times
the eigenvectors associated with the k largest eigenvalues of the T × T covariance matrix X 0 X.
0
Given F̃ k , the matrix of factor loadings is computed as Λ̃k = F̃ k X/T . On the other hand, to
√
get F̄ k we have to first estimate the matrix of factor loadings Λ̄k as N times the eigenvectors
corresponding to the k largest eigenvalues of the N × N matrix XX 0 . Then, F̄ k = X Λ̄k /N . In
their assumptions, Bai and Ng (2002) allow for time series and cross-section dependence, along
with heteroskedasticity. In addition, they allow for a weak dependence between the factors and
the idiosyncrasies. These principal components estimators are consistent for the space spanned by
the factors and not for the factors themselves. Nevertheless, this lack of identification does not
constitute a problem when the researcher is interested in forecasting. In fact, the researcher has to
take the identification issue into account only if she wants to give a structural interpretation to the
factors.
The other fundamental issue in factor models is how to determine the exact number of factors,
which is also central in finance and in particular in Arbitrage Pricing Theory (APT). Brown (1989)
and Connor and Korajczyk (1993) study the eigenstructure of the covariance matrix of a portfolio
of assets to determine the exact number of common factors in an approximate factor structure.
Brown (1989) concludes that small sample properties of the ranked eigenvalues of the covariance
matrix of asset returns can lead to false conclusions that there are too many factors because all the
eigenvalues in general tend to grow as N increases. This is a consequence of small/finite sample
biases. However, Connor and Koraijczyk (1993) argue that if all the sample eigenvalues increase
with the number of cross-sections, this should not be interpreted as evidence of a large number of
factors.
5
As remarked in Bai and Ng (2002), this technique may be severely biased, due to a limited
number of cross sections (low N ). These authors use constructively the large cross-sectional dimension to develop model selection criteria that consistently estimate the number of common
factors without making any assumption about the dependence of the idiosyncratic factors. These
criteria consistently estimate the number of factors in approximate factor models where both N
and T diverge. They are usual model selection criteria that try to find the best balance between
fit and parsimony using penalty terms that depend on both the time series and the cross section
dimensions. Their expressions are the following
NT
N +T
ln
P C1 (r) = V r, F̂ + rσ̂
NT
N +T
N +T
P C2 (r) = V r, F̂ r + rσ̂ 2
min {N, T }
NT
N +T
NT
r
IC1 (r) = ln V r, F̂
+r
ln
NT
N +T
N +T
min {N, T }
IC2 (r) = ln V r, F̂ r + r
NT
where
and σ̂ =
r
2
T
N
2
1 XX
Yit − λ̂0i F̂tr
V r, F̂ r =
N T t=1 i=1
1
NT
PT PN t=1
i=1
Yit −
max
λ̂0i F̂tr
2
(2)
(3)
(4)
(5)
(6)
, i.e. the estimate of the residual sum of squares for the
model with the maximum number of factor rmax . In practice, the criteria P C1 and P C2 compare
the decrease in the residual sum of squares with respect to a benchmark unrestricted model with
an increasing r. On the other hand, the criteria IC1 and IC2 compare the percentage improvement
in the residual sum of squares as the number of factor increases. The advantage of the panel
information criteria IC is that they do not depend on the choice of the maximum number of factors
rmax . Therefore, Bai and Ng (2002) choose penalty functions that strongly penalize the addition to
the fit generated by modeling the idiosyncrasies as the number of cross sections diverges, i.e. they
avoid an increase of the rank of the coefficient matrix that links LHS and RHS variables beyond
the number of common factors.
Anderson and Vahid (2007) use the same criteria to select the number of common variance
factors in their portfolio of 21 Australian weekly realized volatilities. However, they show that
in samples with small N and with series characterized by many jumps (as is their case), these
selection criteria often do not work because principal components tend to fit the idiosyncratic noise
rather than the signal of the common factors. To deal with this problem these authors suggest
6
an instrumental variable (IV) approach to estimate the factors where principal components are
extracted from the covariance matrix of the linear orthogonal projections of the realized volatilities
onto their most recent past.
Under the assumptions that N, T → ∞ with N < T and denoting the N × (T − p) matrix of
realized volatilities as Y = (Yp+1 , . . . , YT ) and as Y−1 the N p×(T −p) matrix of lagged values, a
consistent estimator of the forecastable common factor is given by the eigenvectors corresponding
to the r largest eigenvalues of ŶŶ0 where Ŷ is the linear projection of Y on Y−1 , i.e. Ŷ = YP,
−1
0
0
with P = Y−1
Y−1 Y−1
Y−1 for any p > 0. Under the usual normalization B 0 B = Ir , the
estimator B̂ is also the OLS reduced rank regression (RRR) estimator of B in Y = BCY−1 + U
that minimizes trace(UU0 ). Anderson and Vahid (2007) interpret this setup as instrumental variable estimation, where they take Yt−p as instruments for the common factors. In such framework,
the idiosyncrasies are, on average, uncorrelated with the instruments as N diverges, while the
instruments are, on average, correlated with the common factors. In addition, under appropriate
regularity conditions, since the approximate factor structure is preserved under linear projections
and does not depend on the number of lags, the principal components of the variance covariance
matrix of Yt|t−1,...,t−p produce consistent estimates of the common factors and the loadings B, as
N grows large. Then, since Ŷt is a consistent estimator of Yt|t−1,...,t−p , the eigenvectors associated
with the largest eigenvalues of ŶŶ0 are consistent estimators of B. In addition, since Ŷ is the orthogonal projection matrix onto the space spanned by Ŷ−1 , it follows that ŶŶ0 = YPY0 . Then,
the eigenvectors corresponding to the largest eigenvalues of this latter matrix are the estimates of B
in the RRR of Y on Y−1 . The leading indicator for the realized variances becomes ŶIV LI = ĈY−1
which Anderson and Vahid (2007) call instrumental variable leading indicator (IVLI).
The number of forecastable factors can be determined with model selection criteria à la Bai
and Ng (2002) where the residual sum of squares with r factors will be given by trace(YY0 ) −
PN
Pr
0
k=r+1 λk , where λk is the k-th largest eigenvalue of (Ŷ Ŷ ). The residual sum of
k=1 λk =
squares with the maximum number of factors would be the error sum of squares of the full rank
P
0
0
max
regression, i.e. trace (YY ) − trace ŶŶ = N
k=rmax +1 λk , where Ŷ is computed with r
factors.
More recently, Harding (2008) shows that in finite samples it is not always possible to distinguish all the common factors from the idiosyncratic noise. This has led too many researchers in
finance to identify just one single dominant factor usually called the market factor. Relying on
random matrix theory, Harding (2008) shows that the method of principal components is only a
substitute of the more laborious factor analysis and provides estimates biased towards one unique
factor.
7
Another problem that might emerge is the possible serial correlation in the errors U. Letting ut
denote the t-th column of U, we have that Eut u0t−s could be neither diagonal nor block-diagonal. A
factor structure implies that Eut u0t−s with s > 0 is strictly diagonal but we can allow for some contemporaneous correlation between countries in the same region, making it block-diagonal. Thus,
an appropriate GLS correction would take into account all these features by completely specifying
the entire structure of the model to jointly estimate it.
A partial GLS correction that ignores the serial correlation in the errors and only corrects for
cross-sectional heteroskedasticity and contemporaneous correlation among the errors, leads to the
canonical covariate estimators of the common components. If there is no serial correlation in
the idiosyncrasies and the errors are normally distributed, then this estimator would be the ML
estimator of the common factor and the best predictor given the information in Y−1 .
If the idiosyncrasies are not serially correlated, the variables in Yt have r SCCF. Then, first the
GLS estimator of the reduced rank regression is MLE under the assumption of normality of the
errors; second this estimator minimizes |UU0 |. Besides, under the usual normalization AA0 = Ir ,
the columns of A are the eigenvectors corresponding to the r largest eigenvalues of
(YY0 )
−1/2
0
0
YY−1
Y−1 Y−1
−1
−1/2
0
Y−1
Y0 (YY0 )
(7)
which correspond to the largest eigenvalues of
−1
(YY0 )
0
0
YY−1
Y−1 Y−1
−1
0
Y−1
Y0
(8)
These eigenvalues are the squared sample canonical correlation coefficients between Yt and
(Yt−1 , Yt−2 , . . . , Yt−p ). Anderson and Vahid (2007) argue that if the approximate factor structure is
correct but the idiosyncrasies have serial correlation, then the canonical variates corresponding to
the r largest eigenvalues of (8) might not provide a consistent estimator of B and in particular of
the common factors. In fact the eigenvalues of (7) are, by construction, between zero and one, and
they give the squared canonical correlations between every linear combination of Yt and their past.
−1 0
0
0
Y−1 Y−1
Y−1 Y0 diUnder the assumed factor structure the r largest eigenvalues of T1 YY−1
verge at rate N and at least the first r eigenvalues of (T −1 YY0 )−1 go to zero at the same rate.
Unfortunately, this cannot tell us if the largest eigenvalues of the product of these matrices still
reflect any information about the common factors. Therefore, canonical correlations can in theory only reveal if the dimension of the dynamic system is smaller than N , but a common factor
structure with predictable idiosyncratic components need not have reduced rank dynamics.
However, when we are interested in forecasting, we should focus not only on the contempora-
8
neous movements - that are captured by principal components analysis - but also on movements
at most leads and lags, because these might result more helpful in capturing the dynamics of the
series. Therefore, we believe that canonical correlation analysis can still be useful in identifying
the variance factors and we will adopt it in the forecasting exercise that follows. In this case, to
determine the number of variance factors we can use common features tests, even though Engle
and Marcucci (2006) show that non-normality and heteroskedasticity may weaken tests based upon
canonical correlations. In fact, if the common factors and the idiosyncrasies share the same statistical features, Engle and Kozicki’s (1993) common feature test does not work properly. The only
way to determine the number of factors is by comparing the fit of models with a different number
of factors.
2.2
Pure Variance Common Features Model
Engle and Marcucci (2006) suggest the Pure Variance Common Features (PVCF) model that assumes a linear factor structure for the N conditional variances ht of a portfolio of assets
ht = Γξt + ωt
where Γ is the N ×K matrix of variance factor loadings, ξt is the K ×1 vector of common variance
factors and ωt is the N ×1 matrix of idiosyncratic variances. In the PVCF it is also assumed that the
idiosyncratic variances follow a low-order ARCH process, and therefore, they are time-varying.
To identify the common variance factors, Engle and Marcucci (2006) use different approaches.
In one model, they use the GARCH conditional volatilities of the K largest principal components
of the returns. In the other one, they use the canonical variates between log-transformed squared
returns and their most recent past. In this case the authors take the squared returns as a proxy for the
actual volatilities of the portfolio to then obtain the linear combinations of the past squared returns
in logs that are most correlated with the actual volatilities. These linear combinations represent the
best linear predictors of the variances and can be interpreted as leading indicators since they are
a particular combination of the past variances that are most correlated with the actual and future
variances.
As shown by Otter (1990), canonical correlation analysis is a multivariate technique with good
predictive properties along with dimensionality-reducing properties. Such predictive features are
summarized in the canonical variates that are the most predictable linear combinations of the original variables. Therefore, since we are not only interested to the contemporaneous movements
between the volatilities, but also to their comovements at most leads and lags, we employ canoni-
9
cal correlations analysis that produces the most predictable linear combinations of the variables to
be forecast.
2.3
Forecasting Realized Variances with Factor Models
To provide further intuition on the use of factor models for forecasting, we consider the following general form of forecasting equation for one-step-ahead forecasts of each country’s realized
variance
ŷi,T +1|T = α̂ŶTF+1|T + φ̂(L)yi,T +1|T + ϕ̂(L)ε̂T +1|T
(9)
where ŶTF+1|T are the forecasts of the variance factors at time T + 1 based on the information up
to time T . Bai and Ng (2006) show that in the more general setting, estimation of the parameters
adds O (T −1 ) uncertainty to the forecast, while the estimation of the factors adds O (N −1 ) uncertainty. In sum, the forecast error variance for the most general case when both the factors and the
parameters have to be estimated is σε2 + O (T −1 ) + O (N −1 ).
In what follows, we will estimate a set of factor models for realized variances and we will use
the estimates of the common factors in two ways. In the first, once the variance factor is computed
we will employ ARMA models to its time series and the fitted model will give the variance factor
forecasts. In the second, we will interpret the common variance factors as leading indicators and
plug them in directly into the forecasting equation, without any further modeling. In both cases
we will also model the idiosyncrasies as ARMA to take into account possible residual correlation. This somehow generalizes the PVCF model by Engle and Marcucci (2006) by allowing the
idiosyncrasies to be modeled as GARCH-like processes.
3
Common Volatilities in International Stock Markets: Empirical Evidence
3.1
Data and Descriptive Statistics
The data we analyze in this paper consist of time series of daily and weekly stock market indices, in
local currency1 , of the major countries in the world. In particular, we downloaded from Datastream
data on MSCI price indices for the 33 major countries around the globe which have a continuous
1
The same data in US dollars are also available, but we decided to perform our analysis in local currency to avoid
accounting for exchange rates behavior. Actually, exchange rate movements may add extra noise to each market index,
making our analysis of common volatilities more difficult.
10
series starting from January 1, 1993. The complete list of the countries2 grouped by region is
given in Table 1. These indices are value-weighted and cover at least the 80% of each country’s
stock market capitalization. In addition, they are constructed in a way to avoid double counts of
multiple-listed stocks.
[Insert Table 1 about here]
The sample period goes from January 6, 1993 to April 29, 2005. For each country, we have
a total of 3216 daily price indices and 643 weekly price indices. Daily and weekly continuous
returns are calculated as the log differences of the corresponding contiguous price indices, i.e.
d
d
d
w
w
w
d
ri,t
= log(Pi,t
) − log(Pi,t−1
) for daily (d) returns and ri,t
= log(Pi,t
) − log(Pi,t−1
) = log(Pi,t
)−
d
log(Pi,t−5
) for weekly (w) returns. Daily returns have an average mean of 0.04% and an annualized
standard deviation of 23%. For Finland, Poland, Argentina, Brazil, China, Korea and Thailand the
annualized volatility is above the 30%. Daily returns exhibit the usual properties of non-normality
(the Jarque-Bera test is always significant) with a kurtosis well above the normal value of 3 and a
slightly positive skewness for some countries. As expected, daily returns also show strong evidence
of ARCH effects, both from the Ljung-Box test on the standardized squares and the ARCH LM
test until the 15-th lag.3 Within each region, we can notice some different features of the data.
For example, all the Asian markets show positive skewness along with the Latin American ones,
whereas Europe is characterized by a negative skewness. The annualized standard deviation is
around 20% for all the three macro regions. However, the average daily return mean is 0.04% for
Europe, 0.07% for America and 0.01% for Asia. Thus, American stock markets are characterized
by the highest average returns (almost twice the European ones and more than ten times the Asian
ones).
In Table 2 the summary statistics for the weekly returns are reported. They exhibit an average
mean of 0.17% and an average annualized standard deviation of 24%. The unconditional distribution of the returns is clearly non-normal for all the stock markets with a kurtosis significantly
higher than 3 and a Jarque-Bera test significant at any reasonable level. Both the Q2 and the LM
test show evidence of ARCH effects also corroborated by both high GARCH(1,1) parameters (0.87
on average) and high persistence of the shocks (0.97 on average)4 . Weekly returns have similar
features within each region, except for the mean which resembles the same behavior found in
2
From the original sample of 37 countries, we have excluded Turkey, Indonesia, Taiwan and Venezuela because
of the high number of missing data. In particular, Venezuela has more than two months of missing data, due to the
general strike at the end of 2002. Argentina and Malaysia have two and one missing weeks respectively, that have
been replaced with the realized variance computed for the previous week.
3
This and other tables are not reported for the sake of brevity but are available upon request from the author.
4
The only exception is Japan for which we cannot reject the null hypothesis of no ARCH effects.
11
daily returns, with an average of 0.19%, 0.36% and 0.03% for the European, American and Asian
stock markets, respectively. One thing to notice is the obvious improved smoothing obtained with
weekly data. Actually, while daily returns present minima and maxima more spread over (between
-8 and -12% for the minima, and between 8 and 13% for the maxima), their weekly averages are
around -15% and 15%.
5
[Insert Table 2 about here]
3.2
Correlation Analysis
Correlation analysis is the workhorse of all the literature on international stock market linkages and
financial contagion. Forbes and Rigobon (2002) have recently shown that, especially for contagion
studies, correlation analysis might be misleading because of heteroskedasticity.
Table 3 provides the correlations of the weekly returns both in levels and squares. It is evident that international stock markets are highly correlated not only within the same region but also
between different areas. For example, the levels are highly positively correlated within the three
macro regions (i.e. Europe, America and Asia) with almost all the correlations greater than 0.40.
However, also outside these regions the correlations between returns in levels are quite high, especially with the major stock markets such as US, UK, Germany, France, Spain, the Netherlands,
Sweden, Hong Kong and Japan. This suggests the existence of common regional factors along
with a world common factor that links internationl stock markets.
[Insert Table 3 about here]
The upper-right triangle of Table 3 reports the correlations between squared weekly returns.
As for the levels, there are strong positive correlations within the same regions, in particular for
the European markets that seem the most integrated. This is true especially for the European
Monetary Union group where some correlations are above 0.80. This can also be noticed for
UK and Switzerland. Another striking feature is the high correlation between the US and all the
European markets. This might also be due to the fact that these stock markets overlap. Actually,
we do not notice any high correlation between US and the Asian markets and the same holds for
the relationships between Asia and Europe. Therefore, we have evidence of regional comovements
that could be related to the presence of common regional factors. In addition, there is less evidence
of a common global variance factor.6
5
The data in US dollars - whose tables are not reported for the sake of brevity - exhibit similar features except for
the average mean returns. Now the European stock markets have a higher mean than the American ones with a ratio
of 1.5:1. In addition, the European average return is fifteen times higher than the Asian one. Austria is the only stock
market that does not show any ARCH effects at 15 lags, while Australia gives some mixed evidence.
6
The correlations of daily returns in levels and squares (not reported) show similar patterns, with strong correlations
12
3.3
ARCH and Common ARCH Tests
In Table 4 some univariate and multivariate ARCH tests for the weekly returns are reported. These
are Lagrange Multiplier (LM) tests calculated using two different information sets. The first set
of ARCH tests uses a univariate information set as in Engle (1982). Each return is squared and
used as a proxy for each country’s volatility. Thus, the own-squared returns are regressed against
a constant and 1, 4, 8 and 12 lags and the statistic is obtained by multiplying the uncentered R2 by
the sample size. This test has an asymptotic χ2 distribution with degrees of freedom equal to the
number of lags.
The second group of tests (MARCH and MARCHC) is built upon a multivariate information
set as in Engle and Susmel (1993). In addition to the own-squared returns, in the auxiliary regression we also use the squared returns of other countries in the same region (MARCH) and, along
them, those of particularly important extra-continental countries (MARCHC). As an example, for
European markets we use the lagged squared returns of Germany, France, Italy and United Kingdom in the MARCH tests, while in the MARCHC tests we add to them those of United States
and Japan. For American markets, we take the lagged squared returns of all the other countries
in the region in the MARCH test, and we add those of Germany, UK and Japan in the composite
MARCH tests. For the Asian markets, we adopt all lagged squared returns of Australia, China,
Hong Kong, Japan and Singapore in the MARCH tests, along with those of Germany, UK and US
for the MARCHC tests. The multivariate ARCH tests are calculated with both 1 and 4 lags and
have asymptotically a χ2 distribution with degrees of freedom equal to the number of regressors
used.
[Insert Table 4 about here]
All the European stock markets show strong evidence of ARCH effects except for Finland at
one lag. In America, both univariate and multivariate tests agree on the presence of ARCH effects.
In Asian markets, instead, univariate ARCH tests show evidence of no ARCH for Australia and
New Zealand at shorter lags, and for Japan at longer lags. Nevertheless, this is not confirmed
by the multivariate tests, for which all Asian markets exhibit conditional heteroskedasticity. As
in Engle and Susmel (1993), we apply the common features (CF) test suggested by Engle and
Kozicki (1993) to assess the validity of a Factor ARCH model. The intuition behind CF is similar
in spirit to cointegration and the original idea was to further develop the latter for all the possible
within each region and less evidence outside. The same happens for the returns in US dollars (also not reported). In
particular, for the weekly returns in US dollars there is more evidence of a global common factor that is probably due
to the US common currency. Both levels and squares exhibit high correlations, not only within the same region, but
also with countries located in different time zones. For weekly returns, we still notice low correlations in the squares
between Europe and Asia.
13
features a time series can present. If two series exhibit a particular feature, there might be a
linear combination of the series which does not have the feature, which therefore is said to be
common. The CF test seeks those linear combinations of the series that eliminate ARCH effects
by minimizing the usual ARCH test over the cofeature vector. The information set includes lags
of both series plus their cross-products. The CF test is a general method-of-moments-type of test.
For more details on the computation of this test, see Engle and Susmel (1993).
In our case, we are interested in the possible presence of a common ARCH factor that drives
the world volatility process. Previous analysis has shown that all international stock returns have
strong ARCH effects and it could be useful knowing if there is either a regional common ARCH
factor (as found in Engle and Susmel, 1993) or a world common factor. Given the previous results
on the correlations among squared returns, we believe that there are groups of countries within
the same region with a similar volatility behavior. It is important to examine carefully the ARCH
tests on the single time series, because if one country does not have heteroskedasticity, then the
common ARCH test would place a nil weight to the other country with ARCH.
Table 5 shows the results from common ARCH testing for the weekly returns with four lags
for both series. The results are quite similar to Engle and Susmel (1993) who use similar data but
with less countries and a shorter sample. The table displays the combination (or portfolios) of two
countries which do not have ARCH effects. The weight of the first country is normalized to one,
and the weight on the second is reported along with the minimum value of the MARCH test on
the combination. We also report the MARCH tests for both series with the information set given
by all the instruments used, i.e. the own-lagged squared returns, those from the other country and
their cross-products. Using 4 lags for all these tests, the 5% critical value of a χ2 (11) is 19.68.
[Insert Table 5 about here]
There are common ARCH factors both within Europe and Asia and many common factors
between Asia on one side and America and Europe on the other. Among the countries that share
a common ARCH factor within the same region, only some pairs have shown a high correlation
in their squared returns. For example, France and the Netherlands have a correlation of 0.88 and
Belgium and Switzerland of 0.84. In all the other cases of common ARCH, the correlations of the
squares are quite low (for example, Brazil and Japan have a correlation of 0.04). Not surprisingly,
New Zealand seems to share a common ARCH factor with almost all other countries in the world.
We believe that this is due to the fact that we have used four lags in the common ARCH test and
with four lags New Zealand does not show ARCH with the univariate tests, even though it has
heteroskedasticity with the multivariate ones. This result is not new since also Engle and Susmel
14
(1993) find that although Belgium has no ARCH only in the univariate tests, there are some linear
combinations of Belgium with other countries that share a common ARCH. Actually, in all the
portfolios without New Zealand, the MARCH test of no-ARCH portfolio is highly reduced if
compared to the test on the individual series. This gives further support to our conjecture.
Summarizing our findings, there are only few common factors with a time-varying variance
and most of them are in the European and Asian markets. In the former we get no ARCH effects
when we combine Belgium, Sweden, Denmark, Finland, France and Ireland while in the latter
Australia, Japan, Korea and New Zealand display no heteroskedasticity when combined. Beyond
these neighboring countries, we have a few other portfolios where Japan, Brazil, Austria, Australia,
Greece and Finland, combined do not have ARCH.
As Engle and Susmel (1993) point out, the common ARCH test is based on a model with only
one ARCH factor plus a constant idiosyncratic noise variance. The presence of additional factors,
as documented by King et al. (1994) or Engle and Marcucci (2006) along with a time-varying
idiosyncratic variance could make the common ARCH factor test unable to correctly select noARCH portfolios even if present. However, from a forecasting point of view, knowing that there
are only regional factors is extremely important for the correct identification of the variance factors
that help in predicting each country’s realized variance.
4
Forecasting International Stock Market Volatility
4.1
Realized Variances
Nowadays, it is quite standard to use data sampled at high frequencies to compute volatilities at
lower frequencies. This idea goes back to Taylor (1986) who directly models observable volatility
proxies, such as absolute and squared returns. After, French et al. (1987) and Schwert (1989)
utilize daily returns to estimate the variance of monthly returns. The motivation is quite straightforward: any monthly return is the sum of daily returns within the month, and the variance of this
sum can be estimated from daily returns. This idea has been further and further developed starting
from Andersen and Bollerslev (1998) and their co-authors. In particular, the advances in the theory of continuous-time finance, along with the greater and greater availability of high frequency
data, have made the calculation of realized volatilities from data sampled at high frequencies very
popular. For a recent survey see for example Andersen et al. (2003) or Andersen et al. (2004) and
the references therein.
In this literature, the logarithm of an asset price is assumed to follow a continuous time simple
diffusion process with no conditional mean dynamics
15
dp(t) = σ(t)dW (t)
(10)
where p(t) is the logarithm of the instantaneous price, dW (t) is a standard Brownian process and
σ(t) is a strictly positive volatility process independent of dW (t). For this diffusion process, the
integrated variance associated with the lower frequency w (that can be one day, one week, one
month, etc.) is the integral of the instantaneous volatility over the relevant interval (t − 1w, t)
(w)
σt
Z
t
σ 2 (ω)dω
=
(11)
t−1w
Merton (1980) shows that the integrated volatility of a Brownian motion can be approximated
using the sum of intra-daily squared returns. More recently, Andersen et al. (2001), using quadratic
variation theory, generalize this result to the broader class of finite mean semimartingales. This
class encompasses many processes used in standard arbitrage-free asset pricing applications, such
as, Ito diffusions or jump processes. As a matter of fact, under such conditions, the sum of intradaily squared returns converges to the integrated volatility of the prices allowing us to construct a
model-free estimate of the actual volatility. This also represents a non-parametric estimator called
realized volatility or realized variance.
If we have discretely sampled 4-period returns calculated as rt−j4 = p(t − j4) − p(t − (j +
1)4) where 4 = 1w/W is the small time interval, the realized volatility over the time interval w
is
RVtw
=
W
−1
X
2
rt−j4
(12)
j=0
where rt−j4 is the continuously compounded 4-frequency return. However, in what follows we
will compute the weekly realized volatilities as the sum of daily squared returns because of the
difficulty to get high-frequency returns for such a large number of international stock markets as
those analyzed in this paper.
Dealing with international stock market data, we also face the problem of different closing and
opening times. Stock exchanges in the Far and South East, such as Japan or Singapore, are open
when the European and American markets are closed and viceversa. American stock exchanges
open when the European ones are in their final working hours. This non-synchronous trading
creates staleness problems that can affect the way in which information spreads from one market
to the others. Moreover, non-synchronous trading may also create problems of autocorrelation
in the returns. This is the reason why we use weekly returns and we construct weekly realized
volatilities from daily returns following French et al. (1987) as follows:
16
RV
Ctw
=
W
−1
X
2
rt−j4
+2
j=0
W
−1
X
rt−j4 rt−(j+1)4
(13)
j=1
If the autocorrelation of returns is less then -0.5, then the second term dominates making the
variance estimate negative.7 Thus, in these cases we have adopted a hybrid measure of the realized
volatility in which we only sum the squared daily returns.8
To correct for the possible presence of jumps, we also construct the weekly bi-power variation
(see Barndorff-Nielsen and Shephard, 2004) from our daily returns using
BVtw
=
W
−1
X
|rt−j4 | rt−(j+1)4 (14)
j=0
To study the forecasting performances of different models we use both the weekly returns and
the weekly realized variances computed from daily returns. We use all the three different measures
of realized variance, i.e. RVt as the sum of squared daily returns, RV Ct given by the former plus
the French et al.’s (1987) correction, and the weekly bi-power version BVt . In what follows, we
comment only the results from RV Ct and BVt since RVt is very similar to RV Ct .
Both RV Ct and BVt display a high degree of non-normality in their descriptive statistics (not
reported). All the time series of realized volatilities are highly and positively skewed and their kurtosis is well above the normal value. The same holds for the realized volatilities in USD. Instead,
the logarithmic realized variances are much closer to normality. Only Australia, New Zealand,
France, Italy, Argentina, Switzerland, Norway, Portugal, the Netherlands, Malaysia, Singapore
and Thailand show evidence of non-normality at 1% in log(RV Ct ) in local currency.
Table 6 reports some properties of the realized variances log(RV Ct ) and log(BVt ). For each
measure of realized variance, we report the estimated coefficients9 of the ARMA(1,1) model yt =
c + φyt−1 − θεt−1 + εt (where yt = RV Ct , BVt ) and the estimated degree of fractional integration
d according to the log-periodogram regression of Geweke and Porter-Hudak (GPH) (1983).
[Insert Table 6 about here]
7
This occurs on average for five per cent of the times. For these cases, we have calculated the realized volatility as
the sum of daily squared returns only, i.e. (12).
8
Nevertheless, these two measures of weekly realized variance with and without correction are almost similar.
Their correlations among countries are all almost unity and we have therefore decided to report our results only for
RV Ct .
9
All the estimates of Table 6 are referred to the in-sample period: 1/1/1993-1/1/2003. We have also analyzed
the serial correlation properties of the realized variances of the 33 countries. According to the LM tests for the
null hypothesis of no serial correlation in the realized variances against the alternatives of first to fourth order serial
correlation, we find an overwhelmingly significant evidence of serial correlation for all the realized variances. At the
1% significance level, we reject the null of absence of serial correlations for all countries.
17
The results from the fitted ARMA(1,1) models are quite encouraging. All the estimated autoregressive parameters are large (on average 0.94 for RV Ct and 0.95 for BVt ), while the MA polynomials have roots smaller than the AR polynomials (on average 0.72 for RV Ct and 0.78 for BVt ).
An ARMA model with such characteristics implies autocorrelations that are small but quite persistent. We have also plotted the long-lag autocorrelations for the log-transformed realized variances
and volatilities. From the graphs, we find some evidence of slow decaying autocorrelations for
almost all the countries in our sample with a positive dependence that lasts for many weeks. From
a visual inspection, only Japan, Philippines, Argentina, Norway, Poland and Austria have an autocorrelation that decays more quickly towards zero. This pattern has been modeled with fractionally
integrated processes by many researchers (see for example Ballie et al., 1996) that, nevertheless,
is not pursued in this paper. A fractionally integrated process I(d), with 0 < d < 1/2 is characterized by a slow hyperbolic decay of the autocorrelations. Let I[ωj ] denote the sample periodogram
, with j = 1, 2, . . . , dT /2e. The
estimate of the spectrum at the j-th Fourier frequency ωj = 2πj
T
estimate dˆ can be obtained from the log-periodogram regression of GPH
log (I [ωj ]) = β0 + β1 log (ωj ) + ut
(15)
where j = 1, 2, . . . , m, and dˆ = −.5β1 is asymptotically normal with standard error of π(24m)−1/2 .
For the estimates in Table 6 we have taken m = 521, with a standard error of 0.0281. The 5 and
1% critical values of the null hypothesis H0 : d > 0 are respectively 0.046 and 0.065. From the
estimated fractional integration parameters d in Table 6, it is evident that such parameters are always significant but never greater than 0.168 for RV Ct and 0.133 for BVt . Actually, since all the
long-memory parameters are numerically small, we prefer modeling the persistence with ARMA
models. Moreover, the great similarity among the estimated autoregressive and moving average
coefficients leads us to think that a multivariate modeling can be extremely useful to characterize
and forecast the weekly realized variances of the international stock markets.
The correlations of weekly realized volatilities RV Ct (not reported) show many high correlations far above 0.5 in particular among European stock markets. However, all the countries in the
three regions seem very highly correlated with the neighboring countries. In addition, there are
many comovements among the three macro regions with the United States, Hong Kong, and Singapore that show high correlations with almost all the other stock markets. This clearly suggests
the presence of common regional factors and a world factor that links the major stock markets.10
10
We find similar patterns for the correlations of the other measure of realized volatility, i.e. BV and for both the
measures in US dollars. To better understand the relationships among the realized volatilities of different countries
we have also computed the cross-correlations between one country and the first 12 lags of all the other countries. The
unreported tables show that at lag one, each country’s realized volatility is significantly related to the past of all the
18
Figure 1 shows the weekly realized volatilities for the 33 countries in the sample. There are
not evident spikes or jumps if not for a few countries. However, also the BVt measure of realized
volatilities (not shown), which should be more robust to jumps, resembles similar patterns.
11
[Insert Figure 1 about here]
4.2
Univariate Forecasting Models
Taking each single time series of the realized volatilities, we start our forecasting exercise estimating univariate ARMA models, single exponential smoothing models and three pooled models.
We select the first 521 observations (i.e. about ten years of data from 1/1/1993 to 12/31/2002)
as our in-sample period and we leave the last 122 observations (from 1/1/2003 to 4/29/2005) as
our out-of-sample period. To compute the one-step-ahead forecasts of the realized volatilities,
we utilize a rolling window estimating each model with an in-sample period of ten years. In
practice, at each point, we delete the oldest observation adding a new one, estimating the model
and computing the volatility forecast. Therefore, we use no information from the out-of-sample to
build our forecasts.
For the ARMA models we start from a maximum order pmax and q max for the autoregressive
and the moving average part to then select the model that gives the lowest BIC. Almost all the
selected models are ARMA(1,1) except for a few cases (such as Finland, Switzerland and Singapore) where more AR lags are required. In Table 6 we present the in-sample estimates for the
selected ARMA models. We already noticed that the AR parameters are all quite large and very
close to each other, almost as if there were a unique common value. In addition, the roots of the
AR and MA polynomials are also close to each other. As we have already said commenting this
table, this would suggest small but persistent autocorrelations, thus implying a near long-memory
behavior. Since the GPH estimates of the fractional integration parameter d are small, we have
decided to model the univariate realized volatilities with ARMA models, leaving the analysis with
fractionally integrated processes for further research.
Single exponential smoothing models are considered the most effective methods for forecasting
a large number of time series. They are local level models parameterized in terms of a smoothing
parameter α where the forecast function is constructed by placing more weight to the most recent
other countries (both neighboring and not). From the second through the fourth lag, only Poland and New Zealand
seem not to share relationships with the other countries. At higher lags, many other countries display less and less
cross-correlations. However, the cross-correlation structure looks particularly complex, since at more than 10 lags
some countries, such as, for example, US, Singapore, Korea, Hong Kong, Canada, Norway, Denmark, Austria, Italy
or Greece still have significant relationships with almost all the other countries’ volatilities.
11
In addition, BVt and RV Ct are highly correlated, showing that they are close measures.
19
observations, i.e. ŷt+1|t = α
t−1
P
(1 − α)j yt−j where 0 < α ≤ 1.
j=0
Instead, the pooled AR models estimate each country equation by imposing the joint restriction
that all the parameters but the mean are the same. Therefore, the pooled model imposes that all the
parameters are common across equations. Obviously, were such restriction true, the cross-sectional
variation would make the estimates more precise. We have adopted three AR pooled models. In
the first one, all the parameters are constant across all countries. Since this could lead to too much
pooling, the second model imposes similar restrictions across countries in the same region. Thus,
for example, we impose that all the AR parameters of the European Monetary Union group are
equal and the same for all the other regions. The third pooled model, instead, imposes common
AR parameters for all the countries in the same time zone (Europe, America and Asia). If we have
pooled too much with the first model, we should be able to notice some improvements with the
second and third model12 .
To model the dynamics of each series, we select the order of the autoregression parameters in
such a way to minimize the Schwartz criterion, starting from a maximum order of 12. The first
pooled model is thus estimated with 11 lags and all the AR parameters from lag five to eleven are
quite small and close to each other. The second model requires different numbers of lags: one for
Australia, eight for European Monetary Union, eleven for the Non-European Monetary Union, six
for North America, four for Latin America and eight for East Asia. In all cases, the first few AR
estimated parameters are large, while the last ones are small and equal to each other. The third
model requires twelve, four and eight AR parameters for, respectively, Europe, America and Asia,
and presents the same feature in the estimated parameters.
Table 7 reports the root mean squared error (RMSE) of these univariate models for each country’s realized volatility in the out-of-sample period. We can notice that with the only exceptions of
Norway, Brazil, Philippines and Thailand, the simple exponential smoothing model always gives
better out-of-sample results. Moreover, there are no cases for which the forecast from one of the
three pooled models produces better results than ARMA or SES models. In particular, pooling
all the countries seems to be a better strategy than pooling only single regions or macro regions.
This suggests the presence of a unique global volatility factor with some temporal dependence and,
obviously, a unique loading.
[Insert Table 7 about here]
12
It could also be worthy using a data-driven pooling procedure, but we do not pursue this here.
20
4.3
Multivariate Forecasting and Factor models
Dealing with factor models, it becomes crucial to correctly select the number of factors. In the
literature on approximate factor models, the eigenstructure of the covariance matrix of the returns
has always been used. This might suffer of small-sample bias as shown by Brown (1989). A more
formal treatment of this problem has been recently made by Bai and Ng (2002) who propose a set
of selection criteria that asymptotically (when both the cross-section dimension N and the time
series dimesion T diverge) select the right number of factors. Assuming an approximate factor
model for the realized variances, we can employ such criteria to obtain the right number of factors.
From the application of the selection criteria P Cr and ICr of Bai and Ng (2002) to the realized
variances, we always find evidence of one variance factor. This happens with both measures of the
realized variances and with their log-transformations13 .
We estimate many multivariate models. The first set is given by factor models with one to three
equally weighted market portfolios. In this case, the variance factors are chosen as the equally
weighted portfolio of the whole world and each time zone (Europe, America and Asia). We then
estimate different models with one factor both global and regional (Eq. W. World, Europe, America
and Asia), and with two to three time-zone factors (Europe-America, Europe-Asia, and AsiaAmerica). The World factor is thus given by the simple average of the 33 log(RV Ct ) series. These
variance factors are plotted along with other market factors to be explained later in Figure 2. We
can notice that the four Eq. W. factors are quite similar but there is a closer similarity between the
world and the European factor.
14
[Insert Figure 2 about here]
If the true DGP has only one factor, the Eq. W. world portfolio should provide a consistent estimate of it. The multivariate model for each country’s volatility is therefore given by the common
factor plus ARMA terms to model residual serial correlation in the idiosyncrasies. The order of
the AR and MA terms has been chosen by minimizing the Schwartz criteria in sample. In most
cases, the orders selected are (1,1) and anyhow consistent with those taken in the estimation of the
univariate models without the market factor. This model is denoted Eq.W. model.
We have also estimated one to three factor models where the factors are given by the equally
weighted portfolios of each time zone. For example, the American factor is calculated as the
simple average of the six countries in that time zone. To these factors, we have added serially
13
We should also notice that other criteria considered by Bai and Ng (2002) as possible lousy competitors of ICr
and P Cr , such as AIC2, AIC3, BIC2 and BIC3, support the presence of just one factor.
14
As a matter of fact, the regional Eq.W. market factor that is mostly correlated with the global factor is given by the
simple average of the European markets with a correlation of 0.94. The Asian factor has a correlation of 0.83 while
the American is the least correlated with 0.82.
21
correlated idiosyncratic components modeled as ARMA. The order for each country has been
selected according to the BIC criterion.
All the forecasting performances of these models with equally weighted portfolios as common
factors are reported in Table 8. Quite surprisingly, the model that adopts the Eq. W. portfolio as
the common world market factor fares worse than all the others. Even when it is compared with
models with only one factor, we notice that the lowest RMSE is almost always obtained from the
model that uses the regional Eq. W. portfolio as the factor, except for Greece and Poland where the
world Eq. W. portfolio fares the best. This fact was somehow expected since each regional Eq.W.
portfolio exploits most information about that particular region. Nevertheless, when more Eq.W.
regional factors are put in the two- and three-factor models, we can notice that for more than half
of the countries the forecasts produced are even better.
[Insert Table 8 about here]
The other set of models tries to exploit the information about the common factors from multivariate models of the realized variances. One model takes the principal components of the realized
volatilities as the factors. Also in this case we have taken two factors instead of one to deal with
possible small-sample biases in the selection of the number of factors à la Bai and Ng (2002).
The time series of the two largest principal components of the realized volatilities have been then
modeled with two different ARMA. For the first PC the BIC selects the orders (2,1) while for the
second one (1,2) gives the best fit in-sample.
The idiosyncratic components have been also modeled with ARMA. To obtain the volatility
forecasts, we add to the market factor forecasts multiplied by their loadings estimated in-sample
the forecast idiosyncrasies. The two factors are plotted in Figure 2. Here, we can notice a close
similarity of the first PC factor with the Eq.W. world market factor with which it shares a correlation of 0.79. The second PC factor seems to be connected with some of the financial crises of the
nineties. This model is called Factor PC model.
Another model uses principal components analysis of the linear projection of the vector of log
realized variances Yt onto their most recent past Yt−p . Applying the selection criteria P Cr and ICr
of Bai and Ng (2002) to this model that uses Yt−p as instruments, with p = 1, 2, . . . , 8, both criteria
support the presence of one variance factor in the logarithmic realized variances. We have again
estimated two factors even though the selection criteria indicate only one. This kind of modeling
provides leading indicators for the common factors which are linear combinations of the realized
variances. As in Anderson and Vahid (2007), we do not further model such leading indicators.
Only the idiosyncratic components are modeled as before with an ARMA whose orders are chosen through the BIC. This model is called the instrumental variable leading indicator model and
22
denoted IVLI. Figure 2 also depicts these variance factors. The first IVLI factor closely resembles
the Eq. W. market factor, showing a correlation of 0.75. The second IVLI factor is instead much
harder to interpret and has a 0.78 correlation with the second PC factor.
We also adopt a model that resembles the PVCF model of Engle and Marcucci (2006) where
we use as market factors the canonical variates associated with the largest canonical correlations
between the realized variances and their most recent past (up to four weeks). We call this model
the Factor CC model. We take the canonical variates of the lagged realized variances as leading
indicators. Actually, these represent the linear combinations of the past realized variances that
are maximally correlated with the current ones. As in the IVLI model, we use low-order ARMA
models for the idiosyncratic components, where the order is chosen through the lowest BIC. The
CC variance factors are plotted in Figure 2 where we can notice how difficult the interpretation of
these factors can be. None of them resembles patterns with the Eq. W. factors. Another thing to
notice is that the first CC variance factor has a correlation of only 0.38 with the Eq.W. world factor.
The variance factors are normally not easy to interpret since there is an inherent issue of unidentifiability. However, this lack of identification is not problematic for forecasting purposes. It
does however matters when we try to identify the factors in a structural way. Therefore, the only
way to do this is to regress the estimated factors on each realized variance to then interpret the
corresponding R2 . The variables with the highest R2 load more heavily on the factor and therefore
they should be identified with them. Figure 3 depicts the R2 for the regression of each factor
on each country’s log(RV Ct ). The non-shaded areas correspond (from left to right) to Australia,
Non-European Monetary Union and Latin America. Conversely, the shaded areas refer (from left
to right) to European Monetary Union, North America and East Asia. We can see that the Eq.W.
world factor loads heavily on Europe and North America while the Eq.W. regional factors load
particularly on the respective time-zones plus some other countries outside them. For example,
only the Asian factor clearly loads on East Asia, while the other two load indifferently both on
Europe and America. In particular, the American factor loads not only on American countries, but
also on some others in Europe and Asia, such as Spain, Italy, Germany, Austria and Singapore.
[Insert Figure 3 about here]
The first IVLI variance factor closely mimics the Eq.W. world factor, while the second IVLI
factor loads heavily on Germany, Belgium, Spain, Switzerland, Sweden, Netherlands, Denmark,
Portugal, UK, Canada, Hong Kong, Korea, Malaysia and Singapore. The first PC factor is substantially a factor for Europe while the second one loads on particular countries such as Brazil and
Malaysia without a clear-cut interpretation. The CC factors are now more easily interpretable. The
23
first CC factor resembles a world factor where only few big European and American countries are
heavily loaded (Germany, Belgium, Finland, Netherlands, UK and US). The second CC factor is
substantially a factor for the Asian markets.
The final model is a reduced rank VAR for the realized variances. To choose the order and
the rank of this VAR where we allow for some serial correlation common features, we follow the
procedure suggested by Vahid and Issler (2002) that contemporaneously selects the order and the
rank. Since in their Monte-Carlo study the Hannan and Quinn (HQ) criterion has better properties,
we adopt it to select a rank of six and a lag of one. This model is called reduced-rank VAR15
(RR-VAR).
All these multivariate models for the realized variances are compared out-of-sample in Table
9. Here we can see that the CC models fare the best in twenty-four countries, while the IVLI and
PC factor models produce the best forecasts only in nine cases. Therefore, the predictive power of
canonical correlation analysis is clearly exploited since it gives the best leading indicators for the
variance factors. If we only compare one-factor models, the CC model produces the best forecasts
for twenty countries, while the IVLI and PC models outperform only in ten and three countries
respectively. This suggests that the CC model works better in forecasting the realized variances of
international stock markets.
[Insert Table 9 about here]
Overall, the models that produce the best one-step-ahead forecasts are those that adopt the
Eq.W. regional factors except for Poland and New Zealand for which the best models are the
simple exponential smoothing and the CC model with one factor.
In Table 10 we report the out-of-sample results from some univariate and multivariate models
that use the weekly returns. We have estimated both an EGARCH(1,1) on the weekly returns and
two PVCF models that use principal components (PVCF-PC) and canonical correlations (PVCFCC) to obtain the variance factors. For the PVCF models on the weekly returns, we select the
number of variance factors as in Engle and Marcucci (2006) by determining the number of significant canonical correlations between the set of log-transformed squared returns and their most
recent past (until lag four). Both for the returns in local currency and those in US dollars the
canonical correlation procedure suggests the existence of only one variance factor.
[Insert Table 10 about here]
Both these models allow for one and two factors. It is quite evident that the best model out-ofsample is the EGARCH (for 16 countries) followed by the PVCF-PC with two factors (9 countries)
15
Anderson and Vahid (2007) call this model the canonical correlation model.
24
and the one with just one factor (8 countries). If we consider only one-factor models and the
EGARCH(1,1), the PVCF-PC and the EGARCH fare equally well. However, as can be expected,
none of these models ever outperforms those based on the realized variances. It is actually not
surprising at all that models built directly for the realized variances produce superior forecasts to
those obtained from less direct methods (see for example Andersen et al., 2003).
If we only compare the forecasts from one-factor models, we notice that the best models are
those that utilize the regional Eq. W. portfolio. The European one-factor model is the best one for
fifteen countries out of seventeen, the Asian is the best for nine out of ten and the American one
for five out of six. The global Eq. W. portfolio is the best only for Greece and US. The one CC
factor model outperforms the others only for New Zealand. Thus, as one could expect, when the
factor is given by the equally weighted portfolio of the own time-zone, we get the best volatility
forecasts of all the countries in that particular region.
When the comparison is made only between the Eq.W. world portfolio with all the other onefactor models excluding the one regional factor models, the Eq. W. world model outperforms in
all countries except for Poland (SES), Argentina and Chile (one CC factor model).
Therefore, we have evidence that regional factors are more helpful in forecasting the realized
variances of the major international stock markets.
5
Conclusion
This paper deals with the issue of forecasting international stock market volatility from three different perspectives. First, we estimate univariate volatility models for the weekly realized volatilities,
taking into account their serial correlation properties. Second, we build multivariate forecasting
models in the same spirit of the diffusion index forecasting literature. This is done by assuming
an approximate factor structure for the whole set of international stock market realized variances.
Different methods are used to isolate the volatility factors. Third, we try to forecast volatility from
more traditional models of returns sampled at a lower frequency. Along with the usual GARCH
models, we employ the Pure Variance Common Features model that assumes a factor structure for
the single conditional variances of a portfolio of asset returns and we compare their forecasts with
those of the realized volatilities built with the returns at higher frequencies.
In our empirical exercise we use both daily and weekly data on MSCI indices of 33 countries.
From the daily returns we calculate several measures of weekly realized variances and we model
them with different time series techniques as if they were observables. We also look for a few
common ARCH factors that should drive all the volatilities of the major international stock mar-
25
kets. The results of our empirical exercise suggest the presence of only common regional ARCH
factors, thus confirming Engle and Susmel’s (1993) results. These findings can be due either to the
inability of common factor ARCH tests to distinguish if there is more than one common factor with
similar statistical features or to the fact that the idiosyncratic components might be time-varying
or share the same statistical properties with the common factor(s).
From the forecasting exercise we conclude that factor models that utilize equally weighted regional portfolios outperform all the other models in terms of RMSE. Our forecasting results also
show that those models where the factors are built from canonical correlation analysis produce
better forecasts than those based on other multivariate statistical techniques such as principal components. This confirms the predictive features of canonical correlations analysis. The model with
an equally weighted world factor does not produce the best forecast, thus supporting the hypothesis
that regional factors are more important. Further research is however needed to formally exploit
the predictive power of canonical correlations. It could also be of interest to check if our results
hold also for multi-step ahead forecasts or with other loss functions.
26
References
Andersen, T. G., and T. Bollerslev (1998) ‘Answering the Critics: Yes ARCH Models Do Provide
Good Volatility Forecasts.’ International Economic Review 39(4), 885–905
Andersen, T. G. and T. Bollerslev and F. X. Diebold (2004) ‘Parametric and Non-Parametric Measurement of Volatility.’ In ‘Handbook of Financial Econometrics, Y. Ait-Sahalia and L.P.
Hansen (eds)’ (Amsterdam: North-Holland)
Andersen, T. G. and T. Bollerslev and F. X. Diebold and H. Ebens (2001) ‘The distribution of
realized stock return volatility.’ Journal of Financial Economics 61, 43–76
Andersen, T. G. and T. Bollerslev and F. X. Diebold and P. Labys (2001) ‘The Distribution of
Exchange Rate Volatility.’ Journal of The American Statistical Association 96, 42–55
Andersen, T. G., T. Bollerslev, F. X. Diebold, and P. Labys (2003) ‘Modeling and Forecasting
Realized Volatility.’ Econometrica 71(2), 579–625
Anderson, H. M., and F. Vahid (2007) ‘Forecasting the Volatility of Australian Stock Returns: Do
Common Factors Help?’ Journal of Business and Economic Statistics 25(1), 76–90
Artis, M. J., A. Banerjee, and M. Marcellino (2005) ‘Factor Forecasts for the UK.’ Journal of
Forecasting 24, 279–298
Bai, J., and S. Ng (2002) ‘Determining the Number of Factors in Approximate Factor Models.’
Econometrica 70(1), 191–221
(2006) ‘Confidence Intervals for Diffusion Index Forecasts and Inference for FactorAugmented Regressions.’ Econometrica 74(4), 1133–1150
Ballie, R. T., T. Bollerslev, and H. O. Mikkelsen (1996) ‘Fractionally Integrated Generalized Autoregressive Conditional Heteoroskedasticity.’ Journal of Econometrics 74, 3–30
Barndorff-Nielsen, O. E. and N. Shepard (2004) ‘Power and Bipower Variation with Stochastic
Volatility and Jumps.’ Journal of Financial Economics 2, 1–37
Brown, S. J. (1989) ‘The Number of Factors in Security Returns.’ Journal of Finance 44(5), 1247–
1262
Chamberlain, G. and M. Rothschild (1983) ‘Arbitrage, FActor Structure and Mean-Variance Analysis in Large Asset Markets.’ Econometrica 51, 1305–1324
Connor, G., and R. A. Korajczyk (1993) ‘A Test for the Number of Factors in an Approximate
Factor Model.’ Journal of Finance 48(4), 1263–1291
Engle, R. F. (1982) ‘Autoregressive Conditional Heteroscedasticity with Estimates of U.K. Inflation.’ Econometrica 50(4), 987–1008
Engle, R. F., and R. Susmel (1993) ‘Common Volatility in International Equity Markets.’ Journal
of Business and Economic Statistics 11(2), 167–176
Engle, R. F., and S. Kozicki (1993) ‘Testing for Common Features.’ Journal of Business and
Economic Statistics 11(4), 369–380
27
Engle, R. F., V. K. Ng, and M. Rothschild (1990) ‘Asset Pricing with a Factor-ARCH Covariance
Structure: Empirical Estimates for Treasury Bills.’ Journal of Econometrics 45, 213–237
Engle, R.F., and J. Marcucci (2006) ‘A Long-Run Pure Variance Common Features Model for the
Common Volatilities of the Dow Jones.’ Journal of Econometrics 132(1), 7–42
Forbes, K. J., and R. Rigobon (2002) ‘No Contagion, Only Interdependence: Measuring Stock
Market Comovements.’ Journal of Finance 57(5), 2223–2261
French, K. R., G. W. Schwert, and R. F. Stambaugh (1987) ‘Expected Stock Returns and Volatility.’
Journal of Financial Economics 19, 3–30
Geweke, J., and S. Porter-Hudak (1983) ‘The Estimation and Applications of Long Memory Time
Series Models.’ Journal of Time Series Analysis 4, 221–238
Harding, M. C. (2008) ‘Explaining the Single Factor Bias of Arbitrage Pricing Models in Finite
Samples.’ Economic Letters 99(1), 85–88
Harvey, A. C, E. Ruiz, and N. Shephard (1994) ‘Multivariate Stocharstic Variance Models.’ Review
of Economic Studies 61, 247–264
King, M., E. Sentana, and S. Wadhwami (1994) ‘Volatility and Links Between National Stock
Markets.’ Econometrica 62, 901–934
Merton, R. C. (1980) ‘On Estimating the Expected Return on the Market: An Exploratory Investigation.’ Journal of Financial Economics 8, 323–361
Ng, V., R. F. Engle, and M. Rothschild (1992) ‘A Multi-Dynamic-Factor Model for Stock Returns.’
Journal of Econometrics 52, 245–266
Otter, P. W. (1990) ‘Canonical correlation in multivariate time series analysis with an application
to one-year-ahead and multiyear-ahead macroeconomic forecasting.’ Journal of Business and
Economic Statistics 8(4), 453–457
Schwert, G. W. (1989) ‘Why Does Stock Market Volatility Change Over Time?’ Journal of Finance 44, 1115–1153
Stock, J. H. and M. W. Watson (1998) ‘Diffusion indexes.’ NBER working paper n. 6702
(1999) ‘Forecasting Inflation.’ Journal of Monetary Economics 44, 293–335
(2002) ‘Macroeconomic Forecasting Using Diffusion Indexes.’ Journal of Business and Economic Statistics 20(2), 147–162
Taylor, S. (1986) Modeling Financial Time Series (New York, NY: Wiley)
Vahid, F., and J. V. Issler (2002) ‘The Importance of Common Cyclical Features in VAR Analysis:
A Monte-Carlo Study.’ Journal of Econometrics 109(2), 341–363
28
Table 1: Region and Country Symbols
Country
Symbol
Country
A. Europe
B. America
A1. Euro Area (EMU)
Austria
Belgium
Finland
France
Germany
Greece
Ireland
Italy
Netherlands
Portugal
Spain
Symbol
B1. North America (NAM)
OE
BG
FN
FR
BD
GR
IR
IT
NL
PT
ES
Canada
USA
CN
US
B2. Latin America (LAM)
Argentina
Brazil
Chile
Mexico
AG
BR
CL
MX
C. Asia
C1. Australia (AUS)
A2. Non-Euro Area (NEMU)
Denmark
Poland
Sweden
United Kingdom
Norway
Turkey
Switzerland
Australia
New Zealand
DK
PO
SD
UK
NW
TK
SW
AU
NZ
C2. East Asia (EAS)
China
Hong Kong
Japan
Korea
Malaysia
Philippine
Singapore
Thailand
29
CH
HK
JP
KO
MY
PH
SG
TH
Table 2: Summary Statistics of MSCI Weekly Returns in Local currency
Mean
SD
Min
Max
Sk
Q2 (15)
LM(15)
328.48**
1680.13**
103.90**
249.68**
269.96**
75.20**
219.44**
62.10**
882.12**
147.84**
117.56**
210.73**
145.81**
622.24**
150.87**
434.59**
409.23**
205.77**
272.24**
129.36**
158.99**
294.71**
63.84**
77.59**
51.38**
314.25**
63.73**
145.54**
71.91**
182.04**
381.83**
191.77**
241.89**
221.06**
99.92**
149.78**
68.28**
84.38**
141.65**
45.57**
49.59**
38.08**
139.66**
47.50**
61.74**
47.99**
83.95**
159.33**
90.56**
117.04**
114.68**
0.879
0.730
0.895
0.913
0.808
0.915
0.905
0.900
0.809
0.875
0.863
0.905
0.828
0.860
0.914
0.819
0.815
0.987
0.937
0.981
0.997
0.963
0.994
0.983
0.979
0.975
0.924
0.967
0.988
0.933
0.974
0.994
0.962
0.963
137.06**
114.51**
146.73**
501.11**
294.87**
39.82**
169.06**
193.80**
111.59**
246.58**
94.84**
82.87**
78.71**
88.16**
51.45**
97.63**
57.45**
43.27**
0.882
0.910
0.808
0.831
0.829
0.912
0.977
0.989
0.918
0.993
0.948
0.988
36.83**
119.01**
98.38**
121.07**
33.80**
57.44**
1240.95**
63.25**
75.46**
138.31**
45.21**
54.93**
119.74**
103.99**
23.69
194.76**
257.78**
92.18**
172.01**
118.92**
37.63**
37.46**
59.98**
55.31**
21.89
89.12**
84.66**
62.40**
75.82**
55.53**
0.859
0.946
0.881
0.878
0.906
0.947
0.853
0.864
0.882
0.919
0.958
0.990
0.986
0.969
0.963
0.992
0.995
0.941
0.991
0.989
Ku Jarque-Bera
β α+β
Europe
BD
BG
ES
FN
FR
GR
IR
IT
NL
OE
PT
DK
NW
PO
SD
SW
UK
0.0012
0.0014
0.0023
0.0033
0.0014
0.0022
0.0016
0.0018
0.0013
0.0013
0.0016
0.0021
0.0017
0.0041
0.0024
0.0017
0.0008
0.032
0.027
0.031
0.053
0.030
0.039
0.028
0.032
0.030
0.022
0.026
0.024
0.028
0.055
0.036
0.026
0.022
-0.150
-0.145
-0.135
-0.216
-0.129
-0.137
-0.147
-0.129
-0.158
-0.110
-0.104
-0.147
-0.146
-0.323
-0.169
-0.135
-0.102
0.165
0.184
0.134
0.255
0.166
0.172
0.096
0.112
0.180
0.059
0.092
0.103
0.109
0.268
0.138
0.146
0.133
-0.47 6.38**
-0.17 10.93**
-0.46 4.75**
-0.5 5.89**
-0.08 6.18**
0 4.68**
-0.52 5.68**
-0.31 4.39**
-0.48 8.67**
-0.71 4.87**
-0.37 4.97**
-0.21 5.78**
-0.52 5.09**
0.17* 7.82**
-0.46 5.20**
-0.29 7.00**
0.14 6.91**
America
CN
US
AG
BR
CL
MX
0.0018
0.0015
0.0020
0.0120
0.0017
0.0030
0.022
0.022
0.050
0.057
0.028
0.037
-0.111
-0.091
-0.213
-0.266
-0.148
-0.119
0.083
0.103
0.242
0.304
0.127
0.149
-0.45
-0.14
-0.01
0.17*
-0.12
0.13
5.08**
5.05**
5.35**
7.32**
6.32**
4.19**
Asia
AU 0.0014 0.018 -0.091 0.055 -0.25
NZ 0.0008 0.024 -0.128 0.076 -0.29
CH -0.0021 0.050 -0.196 0.216 -0.15
HG 0.0010 0.036 -0.166 0.135 -0.48
JP -0.0002 0.027 -0.091 0.095 0.25**
KO 0.0013 0.046 -0.192 0.185 -0.01
MY 0.0007 0.040 -0.202 0.274 0.42**
PH 0.0002 0.038 -0.166 0.163
0.05
SG 0.0006 0.030 -0.108 0.118 -0.05
TH -0.0004 0.050 -0.145 0.259 0.45**
4.06**
5.03**
4.90**
4.90**
4.01**
4.47**
9.77**
4.54**
4.68**
5.09**
Note: The sample is from January 1, 1993 to April 29, 2005. Q2 (q) is the Ljung-Box test for
the null of serial correlation in the standardized squares until lag q, while LM (q) is the ARCH
LM test until the same lag. Both test have an asymptotic χ2 distribution with q degrees of
freedom under the null of no ARCH. β is the coefficient of the lagged conditional variance in
a GARCH(1,1) estimation, while α + β is the estimated persistence of the shocks. * and **
indicate significance at 1 and 5%.
30
31
NZ
BD
BG
ES
FN
FR
GR
IT
NL
OE
PT
DK
NW
0.05 0.15**
0.07 0.09* 0.22** 0.18** 0.23** 0.09* 0.24** 0.17** 0.08* 0.19**
0.06 0.21** 0.13** 0.28** 0.15** 0.15** 0.33** 0.25**
PO
SD
SW
UK
CN
US
AG
CL
MX
CH
JP
KO
MY
PH
SG
0.01
0.05
0.07
0.05 0.08* 0.13**
0.06
0.03 0.09* 0.12** 0.12** 0.13** 0.13**
0.03 0.63** 0.84** 0.86** 0.34** 0.57**
0.01 0.14**
0.03 0.09* 0.16**
0.04
0.04 0.08*
0.08
0.03 0.11** 0.14** 0.10*
0.03
0.08 0.14** 0.18** 0.09* 0.11** 0.14** 0.10*
0 0.56** 0.61** 0.61** 0.24** 0.45**
0.04
0.05
0.07 0.16**
0.06
0.07
0.06
0.07
0.03 0.34** 0.41** 0.30** 0.27** 0.18** 0.12** 0.08* 0.16** 0.22** 0.10* 0.23** 0.15** 0.15**
0.04 0.29** 0.36** 0.24** 0.33** 0.27** 0.18** 0.17** 0.25** 0.19** 0.10* 0.11** 0.15** 0.16**
0.01 0.56** 0.86** 0.87** 0.24** 0.50** 0.10**
-0.02
0.04
-0.03
0
0
-0.02
0.01
0.02
0.07
-0.02
0
0.07 0.16** 0.15** 0.09* 0.09*
0.05 0.32** 0.48** 0.52** 0.26** 0.36** 0.15** 0.14** 0.17** 0.20** 0.12** 0.18** 0.17** 0.14**
0.05 0.15** 0.08*
0.06 0.15** 0.08* 0.13** 0.11** 0.17** 0.13** 0.15**
0.07
0.03 0.54** 0.26** 0.28** 0.23** 0.29** 0.09*
0.03 0.48** 0.62** 0.58** 0.29** 0.44** 0.26** 0.22** 0.24** 0.34** 0.11** 0.14** 0.17** 0.17**
0.05
0 0.58** 0.74** 0.72** 0.37** 0.56** 0.17** 0.08* 0.09* 0.21** 0.09* 0.18** 0.22** 0.17**
-0.02 0.54** 0.84** 0.83** 0.25** 0.54**
1
0.03
0.02
0.03
0.02
0.07
0 0.19**
0.02
-0.01
0.02
0.03
0.02
0.02
0.01 0.14**
0.03
0.01 0.16** 0.08*
0.07 0.18** 0.17**
0.02 0.10*
-0.01 0.19**
0.03 0.31** 0.09*
0.05 0.17** 0.08*
0.01 0.13**
0.02 0.12**
0.03 0.17** 0.09*
-0.02 0.11**
0.02
-0.02
-0.03
0.07 0.15** 0.14** 0.13**
0.08 0.16** 0.14** 0.13**
0.02
0.07 0.15**
0.05 0.11** 0.20** 0.14**
0.07 0.13** 0.25** 0.10* 0.14** 0.16** 0.14**
0.05 0.11** 0.11**
0.03
0.06 0.13** 0.17** 0.14**
-0.01
0.01
0.04
0.04
0.01
0
-0.03
0.02
0.05
0.04
0
-0.02
0.01 0.29**
0.06
0.07 0.21** 0.09*
0.03 0.18**
0.06 0.14**
0.01 0.17**
0.01
0.04
0.07 0.23** 0.12** 0.15** 0.09*
0.05 0.10* 0.11**
0.06
0.07 0.19** 0.42** 0.23** 0.33** 0.17**
0.04
0.07 0.27** 0.16**
0.07 0.26** 0.09*
TH 0.28** 0.25** 0.29** 0.20** 0.28** 0.17** 0.24** 0.16** 0.24** 0.24** 0.26** 0.26** 0.14** 0.20** 0.27** 0.17** 0.24** 0.24** 0.28** 0.25** 0.24** 0.24** 0.19** 0.24** 0.26** 0.36** 0.44** 0.21** 0.39** 0.44** 0.43** 0.47**
1
1 0.33**
1 0.27** 0.25**
SG 0.38** 0.30** 0.38** 0.29** 0.33** 0.30** 0.36** 0.25** 0.30** 0.32** 0.34** 0.31** 0.21** 0.28** 0.37** 0.21** 0.38** 0.34** 0.38** 0.32** 0.36** 0.21** 0.18** 0.23** 0.26** 0.46** 0.60** 0.34** 0.36** 0.52** 0.40**
PH 0.26** 0.26** 0.21** 0.13** 0.19** 0.12** 0.16** 0.20** 0.20** 0.18** 0.18** 0.21** 0.17** 0.16** 0.23** 0.17** 0.16** 0.21** 0.21** 0.20** 0.20** 0.21** 0.18** 0.19** 0.24** 0.32** 0.35** 0.19** 0.25** 0.39**
1 0.39** 0.43** 0.30**
1 0.09*
MY 0.28** 0.29** 0.21** 0.13** 0.16** 0.15** 0.18** 0.18** 0.18** 0.15** 0.17** 0.24** 0.13** 0.17** 0.26** 0.18** 0.18** 0.19** 0.21** 0.26** 0.22** 0.10* 0.09* 0.19** 0.16** 0.38** 0.43** 0.22** 0.24**
KO 0.29** 0.24** 0.36** 0.24** 0.34** 0.30** 0.31** 0.20** 0.25** 0.29** 0.33** 0.16** 0.23** 0.25** 0.30** 0.17** 0.33** 0.31** 0.31** 0.28** 0.31** 0.21** 0.21** 0.22** 0.27** 0.33** 0.38** 0.33**
1 0.27**
1 0.18** 0.21** 0.38** 0.26** 0.49** 0.24**
0.39** 0.23** 0.41** 0.28** 0.35** 0.34** 0.39** 0.27** 0.30** 0.34** 0.37** 0.25** 0.19** 0.25** 0.33** 0.18** 0.36** 0.33** 0.35** 0.37** 0.36** 0.14** 0.22** 0.21** 0.29** 0.21** 0.32**
HG 0.44** 0.30** 0.45** 0.31** 0.43** 0.35** 0.43** 0.23** 0.34** 0.38** 0.41** 0.26** 0.29** 0.32** 0.36** 0.23** 0.42** 0.38** 0.45** 0.42** 0.41** 0.28** 0.26** 0.25** 0.37** 0.59**
1 0.39**
1 0.12** 0.17** 0.18** 0.15**
CH 0.28** 0.17** 0.30** 0.21** 0.26** 0.23** 0.26** 0.17** 0.21** 0.18** 0.28** 0.23** 0.21** 0.21** 0.28** 0.17** 0.30** 0.25** 0.28** 0.32** 0.29** 0.18** 0.20** 0.21** 0.28**
MX 0.39** 0.28** 0.47** 0.34** 0.52** 0.39** 0.45** 0.26** 0.29** 0.37** 0.46** 0.26** 0.32** 0.31** 0.36** 0.24** 0.43** 0.40** 0.41** 0.50** 0.50** 0.53** 0.45** 0.42**
1 0.26** 0.17** 0.11** 0.08* 0.15** 0.14** 0.12** 0.17** 0.11**
1 0.39** 0.41** 0.15** 0.23**
CL 0.32** 0.26** 0.29** 0.25** 0.36** 0.24** 0.28** 0.23** 0.28** 0.24** 0.29** 0.23** 0.23** 0.23** 0.34** 0.15** 0.30** 0.29** 0.29** 0.32** 0.33** 0.39** 0.44**
BR 0.30** 0.25** 0.29** 0.20** 0.36** 0.26** 0.25** 0.19** 0.20** 0.22** 0.27** 0.18** 0.28** 0.20** 0.28** 0.27** 0.30** 0.26** 0.24** 0.31** 0.31** 0.46**
1 0.37** 0.42** 0.31** 0.12** 0.14** 0.12** 0.16** 0.12** 0.08* 0.19** 0.17**
1 0.09*
AG 0.26** 0.23** 0.29** 0.23** 0.37** 0.22** 0.30** 0.16** 0.18** 0.27** 0.30** 0.21** 0.21** 0.19** 0.32** 0.16** 0.27** 0.25** 0.27** 0.31** 0.30**
US 0.51** 0.27** 0.70** 0.59** 0.61** 0.53** 0.66** 0.31** 0.55** 0.51** 0.67** 0.38** 0.38** 0.49** 0.47** 0.25** 0.64** 0.63** 0.67** 0.73**
1 0.57** 0.11**
1 0.26** 0.57** 0.08*
CN 0.50** 0.27** 0.60** 0.47** 0.53** 0.47** 0.58** 0.29** 0.43** 0.46** 0.57** 0.37** 0.39** 0.43** 0.47** 0.28** 0.59** 0.54** 0.55**
UK 0.51** 0.30** 0.74** 0.72** 0.70** 0.54** 0.79** 0.33** 0.62** 0.64** 0.80** 0.42** 0.43** 0.58** 0.52** 0.25** 0.67** 0.75**
0.04
0.06 0.15**
0.05 0.14** 0.14**
0.03
-0.02 0.14**
1 0.84** 0.32** 0.57** 0.11**
1 0.57** 0.55** 0.35** 0.55**
0.05
SW 0.50** 0.30** 0.77** 0.72** 0.69** 0.48** 0.76** 0.37** 0.57** 0.62** 0.80** 0.43** 0.48** 0.53** 0.53** 0.21** 0.64**
Note: * and ** indicate significance at 1 and 5%.
TH
0.05 0.09* 0.14** 0.14** 0.22** 0.22** 0.23** 0.27** 0.14** 0.22** 0.12** 0.23** 0.19** 0.32** 0.12** 0.11** 0.31** 0.18**
SD 0.48** 0.30** 0.75** 0.57** 0.69** 0.71** 0.75** 0.32** 0.51** 0.59** 0.69** 0.39** 0.51** 0.57** 0.55** 0.31**
PO 0.23** 0.17** 0.26** 0.16** 0.28** 0.32** 0.26** 0.18** 0.22** 0.21** 0.23** 0.14** 0.29** 0.18** 0.33**
JP
HG
1 0.09* 0.42** 0.46** 0.42** 0.25** 0.40** 0.15** 0.14** 0.30** 0.22** 0.12** 0.11** 0.17** 0.16** 0.11** 0.09* 0.28** 0.14**
1 0.54**
NW 0.44** 0.35** 0.56** 0.49** 0.56** 0.46** 0.55** 0.29** 0.49** 0.48** 0.59** 0.44** 0.39** 0.52**
DK 0.40** 0.22** 0.60** 0.54** 0.57** 0.45** 0.59** 0.29** 0.50** 0.50** 0.60** 0.36** 0.41**
1 0.35** 0.37**
1 0.29** 0.28** 0.38**
PT 0.32** 0.25** 0.53** 0.42** 0.55** 0.43** 0.50** 0.29** 0.37** 0.42** 0.49** 0.29**
OE 0.33** 0.30** 0.44** 0.47** 0.44** 0.24** 0.42** 0.26** 0.39** 0.37** 0.44**
1 0.27** 0.41** 0.62** 0.45**
1 0.56** 0.24** 0.34** 0.37** 0.29**
NL 0.49** 0.30** 0.83** 0.79** 0.74** 0.55** 0.84** 0.34** 0.57** 0.67**
0.44** 0.29** 0.70** 0.57** 0.67** 0.49** 0.72** 0.29** 0.47**
BR
0.07 0.22** 0.33** 0.30** 0.21** 0.28** 0.23** 0.21** 0.10* 0.20** 0.13** 0.26** 0.23** 0.26** 0.08* 0.10* 0.30** 0.10**
1 0.32** 0.36** 0.39** 0.28** 0.37** 0.47** 0.08* 0.39** 0.43** 0.40** 0.33** 0.51** 0.12**
1 0.18** 0.19** 0.12** 0.18** 0.19** 0.11** 0.22**
IR 0.45** 0.26** 0.59** 0.53** 0.54** 0.40** 0.55** 0.34**
GR 0.30** 0.22** 0.42** 0.35** 0.35** 0.29** 0.36**
1 0.11** 0.38** 0.55** 0.88** 0.26** 0.36** 0.61** 0.41**
1 0.34**
FR 0.49** 0.27** 0.83** 0.71** 0.77** 0.61**
FN 0.40** 0.24** 0.59** 0.39** 0.54**
1 0.26** 0.67** 0.17** 0.38** 0.53** 0.69** 0.40** 0.43** 0.46** 0.44**
1 0.60** 0.23** 0.88** 0.10* 0.35** 0.51** 0.89** 0.24** 0.34** 0.62** 0.37**
ES 0.52** 0.31** 0.76** 0.63**
BG 0.45** 0.25** 0.72**
1 0.79** 0.69** 0.27** 0.83** 0.19** 0.40** 0.57** 0.81** 0.29** 0.40** 0.48** 0.37**
1 0.19**
BD 0.52** 0.30**
IT
IR
1 0.57** 0.33** 0.25** 0.28** 0.08* 0.26** 0.14** 0.26** 0.33** 0.25** 0.19** 0.17** 0.20** 0.24**
AU
NZ 0.48**
AU
Table 3: Correlations of MSCI Weekly Returns in Local Currency. Lower Left Triangle: Levels. Upper Right Triangle: Squares
Table 4: ARCH tests of Weekly Returns in Local currency
Lags
AU
NZ
BD
BG
ES
FN
FR
GR
IR
IT
NL
OE
PT
DK
NW
PO
SD
SW
UK
CN
US
AG
BR
CL
MX
CH
HG
JP
KO
MY
PH
SG
TH
1
1.78
5.86*
40.65**
98.03**
33.60**
3.78
88.83**
14.46**
23.76**
26.47**
94.19**
13.87**
4.21*
15.51**
10.80**
33.71**
23.29**
82.91**
79.04**
6.62*
24.75**
15.81**
80.36**
21.16**
11.08**
12.46**
13.85**
7.45**
6.45*
22.47**
18.25**
29.61**
6.91**
ARCH
4
8
18.10** 25.05**
8.85 26.97**
82.70** 96.38**
131.36** 141.92**
50.51** 60.90**
33.74** 46.33**
116.64** 132.65**
20.50** 38.56**
29.05** 35.09**
29.80** 36.33**
110.06** 134.57**
22.43** 27.71**
31.88** 45.26**
28.38** 38.45**
74.11** 78.74**
130.06** 143.40**
52.97** 69.45**
90.52** 104.49**
87.57** 108.82**
44.31** 58.92**
54.15** 66.09**
42.82** 49.98**
96.45** 109.00**
37.68** 42.72**
29.02** 33.90**
37.82** 51.28**
35.82** 41.22**
14.64**
16.71*
32.73** 79.85**
62.35** 76.30**
27.70** 51.13**
49.01** 59.75**
33.53** 46.01**
12
31.66**
34.23**
101.10**
146.47**
71.31**
67.81**
138.20**
41.98**
45.49**
38.07**
139.80**
42.16**
53.67**
42.20**
81.91**
153.29**
83.82**
112.19**
110.22**
77.17**
85.33**
51.28**
112.48**
56.50**
35.84**
56.39**
53.73**
20.13
85.66**
81.26**
58.28**
72.08**
54.62**
MARCH
1
4
63.70** 123.87**
60.45** 118.45**
53.53** 106.34**
107.07** 160.18**
58.43** 86.15**
9.67 50.25**
98.76** 143.62**
23.51**
35.55*
45.84** 62.52**
32.06** 40.52**
103.21** 148.04**
19.35** 40.56**
33.99** 67.85**
46.55** 71.10**
32.46** 108.46**
36.18** 135.79**
32.46** 78.15**
101.38** 141.69**
106.62** 129.06**
6.62* 61.33**
28.10** 78.62**
43.26** 79.70**
96.48** 130.53**
40.55** 61.12**
21.07** 57.29**
31.50** 57.67**
22.06** 87.51**
18.17** 66.47**
28.80** 78.47**
68.60** 148.64**
32.34** 81.18**
81.43** 148.82**
21.90** 71.26**
MARCHC
1
4
70.22** 142.57**
62.35** 123.07**
64.65** 121.24**
114.22** 168.10**
71.92** 113.07**
13.61 66.73**
106.69** 153.34**
25.42**
43.71*
56.92** 80.77**
42.47** 51.68**
106.95** 157.50**
26.35** 56.68**
37.06** 80.93**
54.82** 84.91**
41.47** 121.87**
36.62** 142.07**
33.99** 90.64**
110.50** 156.28**
116.72** 142.05**
25.87** 83.65**
102.81** 172.28**
45.68** 90.82**
98.81** 148.95**
41.54** 77.75**
24.12** 74.85**
38.51** 72.10**
23.22** 97.52**
22.75** 92.22**
29.53** 88.47**
85.31** 166.31**
33.33** 84.18**
99.26** 175.25**
23.80** 80.97**
Note: The table presents univariate ARCH LM tests at lags 1, 4, 8 and 12 and the ARCH
LM test with multivariate information sets (MARCH with information set within the same area
and MARCHC with a composite information set). For example, for the Asian markets in the
information set of MARCH we have lags of AU, HG, SG, JP and CH. For the European markets
the information set has BD, FR, IT, UK, while for American ones all the other countries’ lagged
returns. In the MARCHC tests’ information sets we have the same lags as in MARCH along
with BD, UK, US for Asian markets, US and JP for Europe and BD, JP and UK for American
markets. * and ** indicate significance at 1 and 5%.
32
Table 5: Common ARCH Factors: Estimation
Countries Cofeature LM-min MARCH1 MARCH2
Countries Cofeature LM-min MARCH1 MARCH2
Within same region
Outside the region
Asia
Europe/Asia
AU/NZ
AU/JP
NZ/JP
NZ/MY
NZ/TH
JP/KO
-0.635
0.334
0.447
-0.09
-0.16
-0.372
10.566
16.67
10.504
18.761
14.995
14.053
37.68**
29.56**
14.95
20.8
18.6
22.06*
30.74**
29.61**
35.17**
117.50**
45.84**
36.26**
148.14**
150.80**
207.51**
45.81**
40.89**
162.15**
134.15**
51.45**
55.22**
38.65**
68.99**
51.53**
76.65**
180.21**
128.86**
32.96**
153.00**
63.39**
70.62**
83.39**
62.95**
111.86**
Europe
BG/IR
BG/DK
BG/SW
FN/FR
FN/IT
FR/NL
FR/DK
IR/DK
IR/SD
IT/SD
SD/UK
-2.415
-1.881
-1.29
-1.328
-1.779
-1.007
-2.137
-1.127
-0.816
-0.841
-1.416
17.152
18.839
15.753
19.432
15.111
11.149
17.074
17.301
15.756
8.381
15.767
AU/FN
AU/GR
NZ/BD
NZ/BG
NZ/FN
NZ/FR
NZ/GR
NZ/IR
NZ/IT
NZ/NL
NZ/PT
NZ/DK
NZ/NW
NZ/PO
NZ/SD
NZ/SW
NZ/UK
ES/JP
FN/HG
GR/HG
GR/PH
IR/HG
IT/JP
OE/JP
PO/JP
0.174
-0.249
0.216
-0.204
0.132
0.186
0.028
0.495
-0.214
0.023
0.34
0.214
-0.006
0.102
0.124
-0.077
0.245
-2.983
-1.519
-1.75
-1.682
-0.98
1.308
-0.907
5.263
14.589
19.424
12.173
10.648
17.618
11.43
12.155
14.893
17.707
16.136
17.884
16.874
16.061
13.506
18.554
10.042
14.101
19.406
14.47
14.952
12.551
17.967
15.144
14.411
11.653
30.11**
28.40**
11.34
10.44
16.22
9.85
11.64
16.84
16.56
11.37
21.71*
18.9
15.54
12.54
11.8
10.36
12.71
56.10**
42.09**
45.52**
33.31**
41.34**
43.49**
27.28**
148.89**
36.73**
49.33**
95.74**
142.36**
49.57**
123.80**
46.36**
43.67**
40.96**
114.85**
47.00**
38.13**
99.67**
139.72**
63.44**
95.30**
96.20**
25.21*
46.32**
40.47**
33.05**
41.08**
20.62
24.12*
20.3
America/Asia
NZ/CN
NZ/US
NZ/AG
BR/JP
MX/JP
0.463
-0.193
0.228
-6.884
1.438
16.916
13.44
10.983
13.742
17.956
15.99
13.58
17.58
132.30**
35.21**
64.94**
82.50**
52.81**
23.60*
17.76
Europe/America
OE/AG
0.258
19.451
47.42**
56.76**
Note: Parameters that minimize the ARCH LM test of linear combinations of weekly returns. The first
country has a coefficient normalized to 1. LM-min is the minimum of the ARCH LM test and MARCH1
and MARCH2 are the multivariate information ARCH tests of the first and second series, respectively. The
multivariate information set is given by four own lags, four lags of the other series and their cross products.
The 5% critical value of a χ2 (11) is 19.68. * and ** indicate significance at 1 and 5%.
33
Table 6: Properties of Realized Volatilities
log(RV C)
Country
φ
θ
d
φ
AU
0.8699 0.7045 0.074**
0.8920
NZ
0.9478 0.7766 0.103**
0.9505
BD
0.9692 0.7108 0.135**
0.9610
BG
0.9536 0.6538 0.155**
0.9640
ES
0.9452 0.6988 0.109**
0.9546
FN
0.9891 0.8410 0.111**
0.9854
FR
0.9670 0.7765 0.118**
0.9540
GR
0.9499 0.7521 0.094**
0.9631
IR
0.9625 0.8001 0.085**
0.9742
IT
0.8948 0.6588 0.103**
0.8700
NL
0.9624 0.6421 0.168**
0.9674
OE
0.8634 0.5994 0.116**
0.8876
PT
0.9512 0.6918 0.130**
0.9590
DK
0.9382 0.7384 0.089**
0.9207
NW
0.8952 0.7022 0.104**
0.8947
PO
0.8627 0.5712 0.117**
0.9103
SD
0.9695 0.7473 0.118**
0.9658
SW
0.9108 0.5745 0.152**
0.9333
UK
0.9587 0.7038 0.126**
0.9663
CN
0.9770 0.7831 0.094**
0.9849
US
0.9721 0.7623 0.099**
0.9676
AG
0.8685 0.5513 0.115**
0.8858
BR
0.9550 0.6951 0.134**
0.9673
CL
0.9042 0.6677 0.100**
0.9177
MX
0.9260 0.7407 0.092**
0.9386
CH
0.9191 0.6721 0.112**
0.9368
HG
0.9612 0.7843 0.083**
0.9627
JP
0.8871 0.6418 0.112**
0.8695
KO
0.9835 0.8340 0.096**
0.9797
MY
0.9578 0.7221 0.111**
0.9677
PH
0.7956 0.4611 0.112**
0.8720
SG
0.9731 0.8357 0.098**
0.9795
TH
0.9382 0.7327 0.112**
0.9605
log(BV )
θ
0.7777
0.8105
0.7068
0.7650
0.7781
0.8471
0.7877
0.8230
0.8573
0.6550
0.7400
0.6698
0.7660
0.7324
0.7149
0.7987
0.7657
0.6809
0.7766
0.8560
0.7505
0.6413
0.7957
0.7362
0.7958
0.7526
0.8054
0.6633
0.8523
0.7843
0.6699
0.8934
0.8288
d
0.051*
0.070**
0.133**
0.093**
0.095**
0.093**
0.081**
0.076**
0.081**
0.090**
0.133**
0.096**
0.107**
0.099**
0.092**
0.073**
0.091**
0.117**
0.107**
0.084**
0.124**
0.078**
0.081**
0.065*
0.068**
0.100**
0.085**
0.094**
0.076**
0.083**
0.078**
0.081**
0.079**
Note: φ and θ are the AR and MA parameters, respectively of the ARMA(1,1) model:
RVt = c + φRVt−1 − θεt−1 + εt . d is the estimated degree of fractional integration
according to the GPH procedure. All estimates are in the in-sample period 1/6/19931/1/2003. * and ** indicate significance at 1 and 5%.
34
35
96
96
96
97
97
97
99
00
99
00
98
99
00
PT_RVC
98
FR_RVC
98
01
01
01
02
02
02
03
03
03
04
04
04
.000
.004
.008
.012
.016
.020
.024
.028
.000
.005
.010
.015
.020
.025
.030
.000
.001
.002
.003
.004
.005
.006
.007
.008
.000
.001
93
93
93
94
94
94
95
95
95
96
96
96
97
97
97
99
00
99
00
98
99
00
PH_RVC
98
MX_RVC
98
UK_RVC
01
01
01
02
02
02
03
03
03
04
04
04
93
93
93
94
94
94
95
95
95
96
96
96
97
97
97
99
00
99
00
98
99
00
DK_RVC
98
GR_RVC
98
01
01
01
02
02
02
03
03
03
04
04
04
.004
.005
.006
.007
.008
.009
.000
.002
.004
.006
.008
.010
.012
.000
96
96
97
97
99
00
98
99
00
CH_RVC
98
01
01
02
02
03
03
04
04
.02
.03
.04
.05
.000
.004
.008
.012
.016
.020
93
94
95
96
97
98
99
00
SG_RVC
01
02
03
04
.00
.01
.02
.03
.04
.05
.06
.00
95
95
.00
94
94
.000
.001
.002
.003
.004
.005
.006
.007
.008
.01
93
93
CN_RVC
.000
.001
.01
.02
.03
.04
.05
.000
.002
.004
.006
.008
.010
.000
.002
.004
.006
.008
.010
.000
.004
.008
.012
.016
.020
.00
.002
.004
.006
.008
.010
.012
.014
.002
95
95
95
NZ_RVC
.003
94
94
94
.01
.02
.03
.04
.002
93
93
93
AU_RVC
.003
.004
.005
.006
.007
.008
.009
.000
.002
.004
.006
.008
.010
.012
.000
.001
.002
.003
.004
.005
.006
.007
.008
.009
93
93
93
93
93
93
94
94
94
94
94
94
95
95
95
95
95
95
96
96
96
96
96
96
99
00
98
99
00
IR_RVC
98
97
97
97
97
99
00
99
00
99
00
98
99
00
TH_RVC
98
HG_RVC
98
US_RVC
98
NW_RVC
97
97
BD_RVC
01
01
01
01
01
01
02
02
02
02
02
02
03
03
03
03
03
03
04
04
04
04
04
04
96
96
96
96
97
97
97
97
99
00
99
00
99
00
.000
.001
.002
.003
.004
.005
.006
.007
.008
.009
.000
.004
.008
.012
.016
.020
.024
.028
93
94
95
96
97
99
00
98
99
00
JP_RVC
98
AG_RVC
98
PO_RVC
98
IT_RVC
98
01
01
01
01
02
02
02
02
03
03
03
03
04
04
04
04
.000
.004
.008
.012
.016
.000
.002
.004
.006
.008
.010
.012
.000
01
02
03
04
.000
.004
.008
.012
.016
.020
.024
.00
.01
.02
.03
.04
.05
.06
.07
95
95
95
95
.032
94
94
94
94
.002
.004
.006
.008
.010
.08
93
93
93
93
BG_RVC
.036
.000
.005
.010
.015
.020
.025
.030
.000
.002
.004
.006
.008
.010
.012
.000
.004
.008
.012
.016
.020
93
93
93
93
93
94
94
94
94
94
Figure 1: Weekly Realized Volatilities (RVC) in local currency
95
95
95
95
95
96
96
96
96
96
97
97
97
97
97
99
00
99
00
99
00
99
00
98
99
00
KO_RVC
98
BR_RVC
98
SD_RVC
98
NL_RVC
98
ES_RVC
01
01
01
01
01
02
02
02
02
02
03
03
03
03
03
04
04
04
04
04
.00
.02
.04
.06
.08
.10
.12
.000
.002
.004
.006
.008
.010
.012
.014
.000
.001
.002
.003
.004
.005
.006
.007
.008
.009
.000
.002
.004
.006
.008
.010
.00
.01
.02
.03
.04
.05
.06
93
93
93
93
93
94
94
94
94
94
95
95
95
95
95
96
96
96
96
96
98
99
00
98
99
00
OE_RVC
97
97
97
99
00
99
00
98
99
00
MY_RVC
98
CL_RVC
98
SW_RVC
97
97
FN_RVC
01
01
01
01
01
02
02
02
02
02
03
03
03
03
03
04
04
04
04
04
Table 7: Out-of-Sample Univariate Results for log(RV Ct )
POOLING
Country
AU
NZ
BD
BG
ES
FN
FR
GR
IR
IT
NL
OE
PT
DK
NW
PO
SD
SW
UK
CN
US
AG
BR
CL
MX
CH
HG
JP
KO
MY
PH
SG
TH
ARMA
RMSE
p
q
α
SES
RMSE
All
RMSE
Regions
RMSE
Macro Regions
RMSE
0.9098
0.8914
0.7469
0.9143
0.8690
0.9301
0.8623
0.8681
0.9589
0.8701
0.8555
0.8353
0.8800
0.8137
0.7284
0.7986
0.8110
0.9157
0.8135
0.7958
0.6888
0.9338
0.7045
0.9476
0.8740
0.7652
0.8142
0.8493
0.8056
0.8920
0.8914
0.8667
0.7938
1
1
1
1
1
3
1
1
1
1
1
1
2
1
1
1
1
3
1
1
1
1
1
1
1
1
1
1
1
1
1
3
3
0.100
0.114
0.242
0.194
0.164
0.132
0.166
0.106
0.110
0.208
0.234
0.220
0.164
0.144
0.178
0.104
0.192
0.248
0.186
0.128
0.202
0.228
0.174
0.182
0.134
0.150
0.156
0.168
0.120
0.182
0.154
0.096
0.102
0.8459
0.8895
0.7071
0.8867
0.8217
0.9073
0.8251
0.8656
0.9553
0.7837
0.8035
0.8133
0.8593
0.8096
0.7322
0.7725
0.7846
0.8804
0.7798
0.7772
0.6736
0.9335
0.7095
0.9438
0.8342
0.7517
0.8065
0.8301
0.7848
0.8651
0.8965
0.8437
0.7991
1.0849
1.2335
1.1506
1.4704
1.7012
1.2658
0.9747
0.9052
1.0742
1.6677
1.0500
1.6170
1.0883
1.4340
0.7757
0.9406
0.8971
1.0514
1.2478
1.4096
0.8325
1.5706
0.7919
1.2842
0.9755
1.1074
0.9567
1.4062
0.7937
0.9506
0.9006
0.9847
1.2305
1.0286
1.2121
5.7765
7.1561
6.6070
7.3087
6.6495
6.8097
7.9391
6.6063
6.5976
6.5630
6.9195
7.0688
6.1272
6.3577
6.5038
7.2508
6.6413
0.9894
0.8114
7.7172
6.2478
7.7510
7.2723
5.2246
5.7337
5.4423
5.3666
6.3946
6.2059
6.0252
5.5531
5.8783
6.5373
5.9290
7.5509
6.7377
7.6076
6.8513
6.9990
8.1239
6.5889
6.7331
6.6550
7.0600
7.0927
6.0336
6.4302
6.4014
7.4997
6.5448
7.1426
6.1256
7.7221
6.0636
7.7691
7.0195
5.3928
5.9204
5.6026
5.4612
6.5907
6.4204
6.2151
5.7460
1
2
1
1
1
4
1
2
1
1
1
1
1
1
1
2
1
6
1
1
1
1
1
2
1
2
2
1
1
1
2
5
2
Note: p and q are the orders of the AR and MA terms, respectively. α is the exponential
smoothing estimate. In the pooled models, we pool the AR terms both within the same
region (e.g. European Monetary Union) and within the same macro region (Europe).
36
37
q
1
3
1
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
5
1
1
1
RMSE
0.7919
0.9047
0.4022
0.5940
0.5650
0.8196
0.4336
0.7827
0.8314
0.5271
0.4379
0.7143
0.7568
0.7218
0.6012
0.8181
0.5417
0.5688
0.4679
0.7626
0.5632
0.9148
0.6663
0.9193
0.7592
0.7772
0.7645
0.8021
0.7690
0.8573
0.8580
0.8162
0.7775
1 Factor model
EU
AS
p q
RMSE p
2 1
0.7118 1
1 3
0.8881 1
1 1
0.6770 1
1 1
0.8461 3
1 1
0.7932 1
1 1
0.8849 1
3 3
0.7673 1
2 2
0.8276 2
1 1
0.9535 1
3 4
0.7705 2
1 1
0.7460 1
2 1
0.7736 6
1 4
0.8339 1
1 1
0.8367 3
1 1
0.7124 1
6 5
0.8021 3
1 1
0.7398 1
1 1
0.8275 1
1 1
0.7466 1
1 1
0.7671 1
1 1
0.6351 1
1 1
0.8781 4
4 3
0.6665 4
3 1
0.9117 3
1 1
0.8235 1
2 1
0.6366 1
1 1
0.6180 1
1 1
0.6848 3
1 1
0.6419 1
1 1
0.7625 1
2 1
0.7698 1
1 2
0.6224 2
2 1
0.7138 1
q
1
3
1
1
1
1
1
1
1
1
1
3
2
1
1
3
1
1
1
1
1
1
3
1
1
1
1
1
1
1
1
1
1
US
RMSE p
0.8584 1
0.8974 1
0.6364 1
0.8135 3
0.7723 4
0.9212 1
0.7471 1
0.8546 2
0.9208 1
0.7465 3
0.7250 1
0.7450 2
0.8035 1
0.7934 1
0.6719 2
0.8246 6
0.7022 1
0.8397 1
0.6987 1
0.6315 1
0.5631 1
0.7742 1
0.5405 6
0.8030 2
0.6285 2
0.7652 2
0.7834 1
0.8151 1
0.7691 1
0.8554 1
0.8285 2
0.8248 1
0.7836 2
q
1
3
1
1
4
1
1
1
1
1
1
3
5
1
3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
EU, US
RMSE p
0.8014 6
0.8973 1
0.4092 1
0.5892 1
0.5729 1
0.8359 1
0.4348 3
0.7801 2
0.8438 1
0.5295 3
0.4420 1
0.7371 2
0.7675 1
0.7182 1
0.5978 1
0.8067 6
0.5391 1
0.5584 1
0.4730 1
0.6505 1
0.5419 1
0.7817 1
0.5532 6
0.8099 2
0.6418 2
0.7533 1
0.7590 1
0.8062 1
0.7862 2
0.8591 1
0.8520 2
0.8017 1
0.7744 2
q
2
3
1
1
1
1
3
1
1
4
1
1
4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
1
1
2 Factor model
EU, AS
RMSE p q
0.6997 1 1
0.8891 1 3
0.4005 1 1
0.5915 1 1
0.5697 1 1
0.8174 1 1
0.4386 3 3
0.7819 2 1
0.8357 1 1
0.5255 3 1
0.4384 1 1
0.7427 1 5
0.7483 1 4
0.7225 1 1
0.6026 1 1
0.7831 6 1
0.5443 1 1
0.5692 1 1
0.4690 1 1
0.7590 1 1
0.5599 1 1
0.8936 1 1
0.6586 1 1
0.9167 3 1
0.7685 1 1
0.6375 1 1
0.6186 1 1
0.6986 1 1
0.6577 3 2
0.7659 1 1
0.7631 2 1
0.6323 1 2
0.7227 1 1
AS, US
RMSE p
0.7145 2
0.8931 1
0.6369 3
0.8132 3
0.7599 1
0.8926 2
0.7297 1
0.8153 5
0.9286 1
0.7498 6
0.7075 1
0.7565 1
0.8022 1
0.8260 3
0.6658 1
0.8075 6
0.6939 1
0.8162 1
0.6961 1
0.6363 1
0.5610 1
0.7754 1
0.5388 6
0.7988 2
0.6322 2
0.6366 1
0.6142 1
0.6873 3
0.6795 3
0.7662 1
0.7698 1
0.6295 2
0.7220 1
q
1
3
4
1
1
1
1
5
1
3
1
5
5
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
3
1
1
1
1
3 Factor model
EU, US, AS
RMSE p q
0.7124 1 1
0.8964 1 3
0.4080 1 1
0.5883 1 1
0.5765 1 1
0.8337 1 1
0.4406 3 3
0.7712 2 1
0.8455 1 1
0.5314 3 6
0.4426 1 1
0.7782 1 5
0.7674 1 4
0.7189 1 1
0.5963 1 1
0.7855 6 1
0.5443 1 1
0.5581 1 1
0.4739 1 1
0.6537 1 1
0.5428 1 1
0.7812 1 1
0.5497 6 1
0.8109 2 1
0.6460 2 1
0.6609 1 1
0.6189 1 1
0.7032 1 1
0.6983 3 5
0.7770 1 1
0.8061 1 1
0.6354 2 1
0.7114 1 1
Note: p and q are the AR and MA orders in the ARMA model for the idiosyncrasies. EU, AS and US stay for Europe, Asia and America,
respectively.
Country
AU
NZ
BD
BG
ES
FN
FR
GR
IR
IT
NL
OE
PT
DK
NW
PO
SD
SW
UK
CN
US
AG
BR
CL
MX
CH
HG
JP
KO
MY
PH
SG
TH
World
RMSE p
0.7510 1
0.8955 1
0.4936 1
0.6925 1
0.6383 1
0.8471 1
0.5402 1
0.7757 2
0.8693 1
0.6338 1
0.5419 1
0.7495 1
0.7630 1
0.7338 1
0.6236 1
0.7839 6
0.5730 1
0.6485 1
0.5553 1
0.7384 1
0.5431 1
0.9108 1
0.6605 4
0.9204 3
0.7419 1
0.7428 1
0.7048 1
0.7549 1
0.7269 1
0.8349 4
0.8520 2
0.7340 1
0.7464 1
Table 8: Out-of-Sample Results: Factor models of log(RV Ct ) with Equally Weighted portfolio(s)
38
93
94
95
96
97
98
99
00
01
02
03
04
-8
-8
-3
-2
-1
0
1
2
3
-8
-4
0
4
8
12
93
93
94
94
95
95
97
98
99
00
01
02
96
97
98
99
00
01
02
First CC of log(RVC)
96
First PC of RVC
03
03
04
04
-4
-3
-2
-1
0
1
2
3
-4
-3
-2
-1
0
1
2
3
4
-9
-7
-7
-10
-6
-6
-9
-5
-5
-10
-4
Eq. W. World Market Factor
-4
93
93
93
94
94
94
97
98
99
00
01
96
97
98
99
00
01
Second PC of RVC
96
02
02
95
96
97
98
99
00
01
02
Second CC of log(RVC)
95
95
03
03
03
Eq. W. Asian Market Factor
04
04
04
-24
-22
-20
-18
-16
-14
-12
-11
-10
-9
-8
-7
-6
-5
-4
93
93
94
94
96
97
98
99
00
01
02
95
96
97
98
99
00
01
02
First IV Leading Indicator
95
03
03
Eq. W. European Market Factor
Figure 2: Variance Factors
04
04
-6
-4
-2
0
2
4
6
-10
-9
-8
-7
-6
-5
-4
93
93
94
94
96
97
98
99
00
01
02
03
95
96
97
98
99
00
01
02
03
Second IV Leading Indicator
95
Eq. W. American Market Factor
04
04
39
20
25
30
.00
.00
5
5
5
15
20
25
15
20
25
10
15
20
25
Second CC of log(RVC)
10
Second IV Leading Indicator
10
Eq. W. Asian Market Factor
30
30
30
.0
.1
.2
.3
.4
.0
.1
.2
.3
.4
.5
5
5
15
20
10
15
20
First PC of RVC
10
25
25
30
30
Eq. W. European Market Factor
.00
.04
.08
.12
.16
.20
.24
.06
.08
.10
.12
.14
.16
.18
.20
.22
5
5
15
20
25
10
15
20
25
Second PC of RVC
10
Eq. W. American Market Factor
30
30
Note: The plots show the R2 from the regression of each country’s log(RV Ct ) on each variance factor. The shaded areas are (from left to right): EMU, NAM and EAS. The
non-shaded areas are (from left to right): AUS, NEMU, LAM.
.05
.05
30
.10
.10
25
.15
.15
20
.20
.20
15
.25
.25
10
.30
.30
5
.35
.35
First CC of log(RVC)
.00
.04
25
.05
.08
20
.10
.12
15
.15
.20
.20
.16
.25
10
30
.24
5
15
First IV Leading Indicator
10
.00
.05
.10
.15
.20
.25
.30
.30
5
Eq. W. World Market Factor
.28
.04
.08
.12
.16
.20
.24
.28
.32
.36
Figure 3: Interpretation of the Variance Factors
40
RMSE
p
q
RMSE
p
q
RMSE
p
q
RMSE
p
q
4
3
1
3
1
1
1
2
1
2
1
2
6
4
2
2
1
1
1
1
1
3
1
2
1
2
1
2
1
2
2
2
2
q
AU
0.8385 1 1
0.9411 0 0
0.9491 0 1
0.9449 0 0
0.8512 1 1
0.8069
NZ
1.0926 0 0
0.8979 1 1
1.1179 0 0
0.9890 0 1
0.8821 3 1
0.9323
BD
0.6959 1 1
0.7803 0 0
0.8178 0 1
0.7075 0 0
0.6845 1 1
0.6833
BG
1.0909 0 0
0.8385 1 3
0.9828 0 0
0.8688 0 1
0.8561 1 1
0.8295
ES
0.8083 1 1
0.8957 0 0
0.8305 0 1
0.8451 0 0
0.8046 1 1
0.7877
FN
1.0011 0 0
0.8818 1 1
0.8939 0 0
0.9068 0 1
0.8815 1 1
0.8683
FR
0.7990 4 5
0.8843 0 0
0.8463 0 1
0.8123 0 0
0.8042 1 1
0.7849
GR
0.8306 0 0
0.8318 1 1
0.8628 0 0
0.8686 0 1
0.8620 2 1
0.8447
IR
0.9283 1 1
0.9329 0 0
0.9471 0 1
0.9444 0 0
0.9450 1 1
0.9433
IT
1.0251 0 0
0.7496 4 4
0.9918 0 0
0.9190 0 1
0.7797 2 1
0.7781
NL
0.7806 1 1
0.8224 0 0
0.8823 0 1
0.7684 0 0
0.7841 1 1
0.7817
OE
0.8102 0 0
0.7699 1 1
0.8266 0 0
0.7917 0 1
0.7755 2 1
0.7931
PT
0.8476 1 1
0.8686 0 0
0.8483 0 1
0.8635 0 0
0.8423 1 2
0.8217
DK
0.8299 0 0
0.8044 1 1
0.7893 0 0
0.7984 0 1
0.8183 1 1
0.8098
NW
0.7228 6 4
0.7529 0 0
0.7443 0 1
0.7401 0 0
0.7230 2 1
0.7243
PO
0.9380 0 0
0.7891 6 4
0.9902 0 0
0.8585 0 1
0.7868 5 6
0.7895
SD
0.7514 1 3
0.7607 0 0
0.7458 0 1
0.7713 0 0
0.7638 1 1
0.7335
SW
0.9867 0 0
0.8306 2 3
0.8931 0 0
0.8502 0 1
0.8552 2 1
0.8225
UK
0.7614 2 1
0.8245 0 0
0.7947 0 1
0.7640 0 0
0.7620 1 1
0.7434
CN
0.8846 0 0
0.7938 2 1
0.9353 0 0
0.8969 0 1
0.7700 1 1
0.7906
US
0.6640 1 1
0.6933 0 0
0.7061 0 1
0.6822 0 0
0.6446 1 1
0.6531
AG
0.9364 0 0
0.9093 1 1
0.9594 0 0
0.9111 0 1
0.9019 3 3
0.9080
BR
0.7104 3 1
0.7441 0 0
0.8306 0 1
0.7025 0 0
0.7131 5 3
0.6976
CL
0.9460 0 0
0.8931 3 1
0.9845 0 0
0.9201 0 1
0.9197 2 2
0.8829
MX
0.8264 2 6
0.8767 0 0
0.9256 0 1
0.8464 0 0
0.8184 2 1
0.8242
CH
0.8837 0 0
0.7619 2 6
0.9109 0 0
0.8514 0 1
0.7649 2 1
0.7708
HG
0.8021 1 1
0.8527 0 0
0.8948 0 1
0.8159 0 0
0.8033 1 1
0.7786
JP
0.8658 0 0
0.8143 2 1
0.8398 0 0
0.8296 0 1
0.8263 2 2
0.7988
KO
0.7712 1 2
0.7890 0 0
0.7536 0 1
0.7537 0 0
0.7669 1 1
0.7477
MY
1.0272 0 0
0.8498 1 1
1.1216 0 0
0.8711 0 1
0.8707 1 1
0.8319
PH
0.8796 5 5
0.9223 0 0
0.8719 0 1
0.9330 0 0
0.8599 6 3
0.8725
SG
0.8918 0 0
0.8390 5 1
0.8423 0 0
0.8536 0 1
0.8390 2 1
0.8445
TH
0.7896 2 2
0.8067 0 0
0.8231 0 1
0.7862 0 0
0.8020 2 1
0.7986
Note: IVLI is the Instrumental Variable Leading Indicator model. RR-VAR is the reduced-rank VAR on the log variances.
p
Factor CC model
1 factor
2 factors
p
RMSE
Factor PC model
1 factor
2 factors
RMSE
Country
Factor IVLI model
1 factor
2 factors
Table 9: Out-of-Sample Results: Factor models of weekly realized volatilities
3
1
1
1
1
1
1
1
1
1
1
5
2
2
1
1
1
1
1
1
1
3
1
1
1
1
1
2
1
2
1
1
1
q
0.9555
1.0449
0.7960
0.8948
0.8899
0.9773
0.8477
0.9159
0.9540
1.0598
0.8088
0.9435
0.8957
0.8409
0.8866
0.8610
0.7491
0.8615
0.7992
0.9730
0.7428
0.9717
0.7758
0.9952
0.8656
0.8880
0.8753
0.9044
0.8112
1.0241
1.0411
0.8903
0.8560
RMSE
RR-VAR
Table 10: Out-of-Sample Results: PVCF models and EGARCH of weekly returns
Country
PVCF-PC1
RMSE
p
PVCF-PC2
RMSE
p
PVCF-CC1
RMSE
p
PVCF-CC2
RMSE
p
AU
NZ
BD
BG
ES
FN
FR
GR
IR
IT
NL
OE
PT
DK
NW
PO
SD
SW
UK
CN
US
AG
BR
CL
MX
CH
HG
JP
KO
MY
PH
SG
TH
1.1517
1.6339
0.8511
1.0867
1.1970
1.2746
1.0116
1.1883
1.1914
1.5347
1.0060
0.9288
1.1757
1.0241
0.8824
1.2323
1.0969
0.9910
0.9445
1.1013
0.8185
1.2121
1.1152
1.3649
1.3261
1.1461
1.1549
0.9798
1.0294
1.5179
1.1824
1.1666
1.1920
1.0978
1.5590
0.8520
1.0838
1.1915
1.3114
1.0286
1.1687
1.1956
1.4697
1.0094
0.9212
1.1430
0.9891
0.8833
1.2624
1.1250
0.9748
0.9106
1.1007
0.8155
1.1651
1.1425
1.3343
1.3179
1.1395
1.2021
0.9391
0.9790
1.6150
1.1558
1.0950
1.1696
1.1732
1.9034
1.0920
1.2794
1.2755
1.5145
1.0849
1.2881
1.3185
1.7948
1.1814
1.1189
1.5763
1.1788
1.0713
1.4447
1.3588
1.1674
1.0826
1.3918
1.0881
1.3333
1.2158
1.6084
1.4235
1.1879
1.3217
1.2197
1.1465
1.9701
1.3833
1.8763
1.3390
1.5805
1.9152
1.2271
1.4050
1.6446
1.5939
1.4575
1.6236
1.4141
1.9923
1.2717
1.3743
1.5137
1.4457
1.1488
1.9158
1.5849
1.2615
1.2658
1.4926
1.2704
1.6640
1.5204
1.6286
1.6724
1.6452
2.0217
1.4247
1.5562
2.1141
1.5831
1.4605
1.5768
5
1
3
1
3
5
5
7
11
1
9
1
5
5
9
9
5
1
3
11
9
5
11
5
9
11
10
1
3
11
11
1
9
5
1
3
1
3
5
5
7
9
1
9
3
5
5
9
9
5
1
3
11
9
5
11
5
9
11
1
5
1
5
11
1
9
5
3
9
5
9
5
7
7
11
1
11
3
7
5
9
9
11
11
9
11
9
9
11
7
9
11
11
11
9
11
11
3
9
5
1
3
1
3
5
5
7
9
1
9
3
5
5
9
9
5
1
3
11
9
5
11
5
9
11
1
5
1
5
11
1
9
EGARCH
RMSE
1.0636
2.5790
1.2329
0.9972
0.9366
1.6848
1.0297
1.1929
0.9840
0.8053
1.1057
0.8681
0.8878
0.8011
0.9574
1.3193
1.1066
0.9183
0.8405
0.8553
0.7062
1.6953
1.4714
1.0305
3.2587
1.5182
1.0257
1.0956
1.6476
0.8919
1.3279
0.9472
1.4971
Note: The Pure Variance Common Feautures models have been estimated with principal components (PC)
and with canonical correlations (CC). p is the order of the ARCH process for the idiosyncrasies chosen by
minimizing the BIC.
41