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Asymmetry and nonlinearity in forecasting multivariate stock market volatility

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Abstract

This cumulative dissertation studies various approaches to improve stock market volatility forecasts based on nonlinearity and asymmetric dependence modeling as well as new innovative data sources. Studying multivariate dependence patterns using a vine copula approach and incorporating Google search data as measure of investor attention in a framework of empirical similarity significantly improves volatility forecasts based on different statistical and economic measures. The importance of accurate volatility forecasts in portfolio- and risk management is highlighted in several economic applications and empirical studies.
Asymmetry and nonlinearity in
forecasting multivariate stock market
volatility
Kumulative Dissertation
der Wirtschaftswissenschaftlichen Fakult¨at
der Universit¨at Augsburg
zur Erlangung des akademischen Grades eines Doktors
der Wirtschaftswissenschaften
(Dr. rer. pol)
vorgelegt von
Herrn Dipl. Finanz¨okonom math. Moritz Daniel Heiden
Erstgutachter: Prof. Dr. Yarema Okhrin
Zweitgutachter: Prof. Dr. Dr. h.c. G¨unter Bamberg
Vorsitzender der m¨undlichen Pr¨ufung: Prof. Dr. Marco Wilkens
Tag der m¨undlichen Pr¨ufung: 21.09.2015
ii
Contents
Contents i
1 Introduction 1
1.1 Decompositions of the realized covariance matrix . . . . . . . . . . . . . . 3
1.2 A vine copula approach for predicting multivariate realized volatility . . . 5
1.3 Investor attention and stock market volatility . . . . . . . . . . . . . . . . 7
2 Article 1: Pitfalls of the Cholesky decomposition for forecasting mul-
tivariate volatility 14
2.1 Introduction................................... 15
2.2 Decomposition of the realized covariance matrix . . . . . . . . . . . . . . . 17
2.2.1 Cholesky decomposition . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Matrix exponential transformation . . . . . . . . . . . . . . . . . . 19
2.2.3 HARmodel............................... 20
2.2.4 Forecasting and bias correction . . . . . . . . . . . . . . . . . . . . 22
2.2.5 Loss functions and the MCS . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Empiricalstudy................................. 25
2.3.1 Data and descriptive statistics . . . . . . . . . . . . . . . . . . . . 25
2.3.2 Optimalodering ............................ 27
2.3.3 Modeling and forecasting procedure . . . . . . . . . . . . . . . . . 28
2.3.4 Statistically testing forecast performance . . . . . . . . . . . . . . 31
2.4 Conclusion ................................... 32
Appendix 33
3 Article 2: A multivariate volatility vine copula model 39
4 Article 3: Forecasting volatility with empirical similarity and Google
Trends 40
5 Summary and outlook 41
i
Chapter 1
Introduction
Performing accurate forecasts for financial assets is not only a very recent topic in
light of the ongoing financial crisis, but one of the great challenges of statistics and
economics in general. In the past years, increased computing power and the availability
of novel data sources lead to a wide range of new methodologies and increased the
awareness for the necessity of high dimensional models, as most applications in asset
and risk management require both, uni- and multivariate forecasts.
Predictability of asset returns has long been one of the most prominent topics in
empirical finance, arising vivid debates as to which extent it violates the efficient market
hypothesis (Fama, 1995) and random walk theories (Bachelier, 1900; Cootner, 1964).
Short-term return forecasting is mostly viewed as impossible or at best difficult, see
Christoffersen and Diebold (2006); Cenesizoglu and Timmermann (2012). At longer
horizons time-varying risk premia or low frequent time series movements are sometimes
used as an explanation for theoretical return predictability, see Barberis (2000); Ferreira
and Santa-Clara (2011). Lettau and Nieuwerburgh (2008) provide an overview on the
topic, pointing out, that changes in the mean of the return process pose a problem for
prediction. While a general consent on the topic has not been reached, the dependence
and variability of asset returns, usually captured by the covariance matrix, has been of
rising interest in recent years. Motivated by studies of univariate volatility, in which
the evidence on predictability is large (e.g. Andersen and Bollerslev (1998a); Forsberg
and Ghysels (2007)), similar results were obtained for the multivariate case, see amongst
others Engle (2002); Barndorff-Nielsen and Shephard (2004b); de Pooter et al. (2008).
Sophisticated methods that directly estimate and forecast the latent volatility process
have spread since the availability and improved accessibility of high-frequency data. Re-
search was, amongst others, initially triggered by Olsen (2011) and Dacorogna (2001).
Particularly, the use of squared returns to estimate volatility has become a standard
method in empirical finance. Originating from stochastic calculus, where Dol´eans-Dade
(1967) showed that the sum of squared returns is a consistent estimator for the quadratic
1
Chapter 1. Introduction 2
variation and Jacod (1994) derived the corresponding limit theory, the concept was soon
applied in econometrics. There, the working assumption of the price process being a
semi-martingale could be justified under no-arbitrage assumptions (Andersen et al.,
2003). For both, the uni- and multivariate setting, the corresponding asymptotic theory
was derived by Barndorff-Nielsen (2002) and Barndorff-Nielsen and Shephard (2004a).
The obtained measures are so called realized volatilities and its multivariate counter-
part realized covariances, which can be refined and robustified by various estimation
methods, see Zhang et al. (2005); Zhang (2006); Jacod et al. (2009); Zhang (2011). The
advantage of these realized measures lies in them being observable time series and as
a result, a large variety of approaches for modeling and forecasting their dynamics has
developed. Since empirical applications, such as asset pricing, portfolio optimization
and evaluation of risks are mostly implemented in the multivariate case, forecasts of the
required covariance matrix are a crucial ingredient and represent the central topic of
this thesis.
On the one hand, multivariate volatility forecasting bears the same difficulties as
its univariate counterpart, where a main focus lies on modeling so called stylized facts
(see Andersen, Bollerslev, Christoffersen, et al. (2006)), such as long-memory, cluster-
ing, leverage effects and mean reversion. On the other hand, matters are complicated
by the requirement of the forecasted covariance matrix to be symmetric and positive
semi-definite. In chapter 2 the Cholesky decomposition, a widely used method to guar-
antee both properties is studied. A special focus lies on its often neglected pitfalls
and possible solutions in empirical application. Additionally, understanding, measuring
and forecasting the dependence structure in the multivariate context is vital. As the
interconnectedness of economies has strongly increased in recent time, financial assets
become more dependent particularly during extremely negative economic phases and
asset market volatility linkages tighten during periods of financial turmoil, as Cappiello
et al. (2006) highlight. This asymmetric dependence has important implications, e.g.
for portfolio allocation, as the variance of a financial portfolio return depends not only
on the variances of the individual assets but also on the correlations between the assets.
To advance from the strict Gaussian framework to more flexible structure, dependence
is commonly studied using the concept of copulas (Joe, 1997; Nelsen, 2006), which are
also suitable for capturing the interdependencies between asset volatilities (Mendes and
Accioly, 2012). Chapter 3 introduces a new model for forecasting multivariate volatility
that utilizes vine copulas to account for nonlinear dependence and asymmetry. The
models predictive accuracy is compared to conventional models using statistical as well
as economic measures.
Another often discussed topic is the role of external factors and variables in fore-
casting volatility, see e.g. Andersen and Bollerslev (1998b); Engle and Patton (2001).
Traditionally, volatility is a measure of uncertainty of market participants regarding the
Chapter 1. Introduction 3
current and future state of the market. Chapter 4 analyzes the benefits of using a simi-
larity based model, comparing current volatility characteristics and investors attention
to the stock market for forecasting univariate volatility. The general ideas and findings
of the articles, which constitute the main part of this dissertation are shortly described
in the remainder of this introduction.
1.1 Decompositions of the realized covariance matrix
A major complication in forecasting multivariate volatility is the requirement of sym-
metry and positive semi-definiteness of the predicted covariance matrix, while preserving
parsimony in the modeling procedure. In the literature, two alternative methods are usu-
ally applied. Restrictions on the model parameters or a decomposition of the covariance
matrix. While the first one is easily applicable, model parsimony suffers especially in
large dimensions. Hence, the latter approach is preferred in the literature, where a vari-
ety of decompositions exist, each of them with specific advantages and disadvantages. A
typical modeling procedure starts with a measure of multivariate volatility, e.g. a time
series of realized covariance matrices. At each point of time, the time series is decom-
posed and the resulting vector or matrix of the decomposition is modeled and forecasted
using uni- or multivariate time series models. In the last step, the decomposition is
reversed and ideally leads to a forecast of the realized covariance (RCOV) matrix, which
is symmetric and positive semi-definite.
One of the most prominent methods is the Cholesky decomposition (CD), which has
previously been applied by Halbleib and Voev (2011); Chiriac and Voev (2011); Becker
et al. (2010). Despite its popularity, it suffers from the drawback that a change in the
order of the elements in the original covariance matrix leads to a different decomposition
and as a result influences modeling and forecasting. The problem is well known in the
literature on vector autoregression (VAR), see Keating (1996), where some authors,
e.g. Kl¨oßner and Wagner (2014) suggest a brute force approach, combining the results
from potentially all orderings of the elements. However, as the number of elements
increases, the number of possible orderings grows in a factorial sequence. Consequently,
this approach gets computationally burdensome even in relatively small dimensions.
The goal of the first article of this dissertation in chapter 2 is to analyze the impact
of the ordering on forecasts for the RCOV matrix, if a CD is used in the modeling
approach. For each of the 720 possible permutations of an exemplary six dimensional
data set of asset returns ranging from January 1, 2000 to July 30, 2008, a modeling
procedure as mentioned before is performed: First, the CD is applied at each point of
time on the RCOV belonging to the corresponding permutation. Second, the time series
of Cholesky elements are modeled based on the heterogeneous autoregressive (HAR)
model of Corsi (2009) and one-step ahead out-of-sample forecasts are generated. Third,
Chapter 1. Introduction 4
the decomposition is reversed to obtain a forecast of the RCOV matrix. Due to the
nonlinear transformation in the last step, a bias is induced, which we study using a
bias correction similar to Chiriac and Voev (2011) and Bauer and Vorkink (2011). To
evaluate predictive accuracy, the multivariate mean squared error and quasi likelihood
loss functions are calculated for each permutation and across time. Analyzing the loss
distributions, we find differences of up to 18% in predictive accuracy between the average
loss of the best and worst model. Yet, the best and worst models are not consistent over
time, so that a clear recommendation to which order to use for the next point of time
based on previous performance is not at hand.
A detailed analysis of the loss differences based on the model confidence set (MCS)
framework of Hansen, Lunde, and Nason (2011) reveals that the forecasts of the order-
ing with the smallest average loss are indeed significantly better than the forecasts of
the ordering with the largest average loss. Hence, choosing an arbitrary and possibly
“wrong” ordering may lead to misjudgment of the models forecasting ability in general.
Furthermore, we show that an ex-ante analysis of the correlation structure of the assets,
as it is sometime proposed in the VAR literature, does not yield significantly better
forecasting results. In case of applying a bias correction for the forecasts, the differ-
ences between best and worst model even worsen. While the bias correction in general
improves forecasting accuracy, the loss distribution over all permutations widens and
differences between largest and smallest average loss increase for both loss functions. A
possible solution to the ordering problem comes in the form of another decomposition,
the matrix exponential transformation (MET), which was first applied in forecasting
multivariate volatility by Bauer and Vorkink (2011). On the one hand, the MET suffers
from biased forecasts, similar to the CD. On the other hand, the elements of the MET
are not explicitly linked to the elements of the matrix it is applied on, making the MET
order invariant. Comparing forecasts of both decompositions, we find that after bias
correction, predictive accuracy does not significantly differ between the CD with the
smallest average loss and the MET. Thus, for empirical application, two conclusions can
be drawn. If a reasonable order can be imposed on the elements of the covariance matrix
or if the connections between the elements of the decomposed covariance matrix are of
interest, the CD is a rational choice. Otherwise, the application of the MET together
with a bias correction is advised, be it for comparative reasons or simply to avoid the
time consuming process of estimating all possible permutations of the CD.
Chapter 1. Introduction 5
1.2 A vine copula approach for predicting multivariate re-
alized volatility
While the article in the previous section pointed out drawbacks from using the CD in
forecasting the RCOV matrix, the decomposition is still irreplaceable if the structure and
interconnectedness of its entries is of importance. As pointed out in section 1.1, RCOV
matrices are often modeled and forecasted using a step-wise procedure, where different
time series models are applied on the whole matrix, its individual elements or a favorable
decomposition. Directly modeling the components of the RCOV matrix with univariate
processes is possible (e.g. as described in Andersen, Bollerslev, Christoffersen, et al.
(2006)), but does not guarantee positive definite forecasts and dynamic linkages among
the series, such as volatility spillovers, might be neglected (Voev, 2008). Latest multivari-
ate approaches that ensure symmetry and positive semi-definiteness of the RCOV matrix
include the Wishart autoregressive (WAR) model proposed by Gouri´eroux et al. (2009)
and its dynamic generalization, the conditional autoregressive Wishart model (CAW) by
Golosnoy et al. (2012). Chiriac and Voev (2011) choose the way of RCOV transformation
and base their vector autoregressive fractionally integrated moving average (VARFIMA)
model on a CD of the covariance matrix. Bauer and Vorkink (2011) instead transform
the covariance matrix by using the MET and a factor model approach for the individual
components. Disadvantages of these multivariate approaches are the lack of flexibility in
the parameters and the inability to conveniently model non-Gaussianity and conditional
heteroskedasticity in the volatility series itself. In contrast, the univariate framework
offers a wide range of possibilities to tackle these problems, for example models based on
fractionally integrated ARMA (ARFIMA) (Andersen et al. (2003)) or HAR processes
(Corsi, 2009) can be estimated under skewed error distributions and various general-
ized autoregressive conditional heteroscedasticity (GARCH) augmentations (see Engle
(2002); Corsi et al. (2008)). In combination with a CD modeling procedure, symme-
try and positive semi-definiteness of the forecasts of the RCOV matrix can be ensured.
Additionally, applying the CD bears the advantages of naturally interconnected entries
within the matrix. Due to the nature of the decomposition, the relation between the
elements is not linear and characteristic dependence patterns can be observed. As An-
dersen et al. (1999) point out, these patterns are subject to the high correlation between
realized correlations and realized volatility, which can be attributed to the increased
interconnectedness of economies. Copulas are a convenient and meanwhile established
way to account for a variety of nonlinear dependence patterns among the realized co-
variances (see Mendes and Accioly (2012)). However, the choice of multivariate copulas
is limited. In contrast, the bivariate case offers a rich variety of different copulas with
flexible dependence patterns, based upon a steadily growing literature especially in the
Chapter 1. Introduction 6
GARCH framework, see e.g. Aas and Berg (2009); Liu and Luger (2009); Fischer et al.
(2009).
A corresponding dynamic framework for modeling and forecasting RCOV matrices
using vine copulas to account for more flexible dependencies between assets is studied
in the second article of this dissertation, see chapter 3. Using the same six-dimensional
data set as in section 1.1, we introduce a stepwise modeling procedure based on various
time series models, such as the ARFIMA and HAR model. Similar to the previous
article, we apply these models on the individual elements of the CD of the RCOV
matrix to guarantee symmetry and positive semi-definiteness of the forecasts. Following
Corsi et al. (2008), we extend the models by including a GARCH component to account
for the so called “volatility of realized volatility”. As shown in Bai et al. (2003), the
common GARCH with Gaussian innovations is not able to account for very high values
of kurtosis of the dependent variable, as it is only controlled by two parameters, the
kurtosis of the error distribution and the persistence of the GARCH itself. Hence,
observing excess skewness and kurtosis in our data, we estimate the models based on
the class of skewed generalized error (SGED) distributions (see e.g. Fernandez and Steel
(1998)). To select an appropriate vine structure for the elements of the CD, or more
precisely for the i.i.d. residuals of the elements after marginal time series filtering, the
correlation pattern between the elements is studied. We compare two different structural
models, for which we estimate various bivariate copulas covering both tail dependence
and tail asymmetry. Given the problem of the ordering of the assets as pointed out in
section 1.1, we repeat the modeling procedure for all 720 possible orderings. Due to the
computational burden, we focus on the ordering with the highest average log likehood
over all time series for further analysis. Analogously, we choose the vine structure which
performs best compared to an arbitrary structure, which is selected and fitted according
to the maximum spanning tree principle as proposed by Dißmann et al. (2013). While
we find that tail asymmetries as implied by Clayton and Gumbel copulas are present,
preliminarily deciding to use only Gaussian or Student’s tcopulas significantly simplifies
the model selection step and only slightly decreases the model’s log likelihood. Finally,
the models can be applied in an one-day ahead out-of-sample forecasting exercise. After
performing the same bias correction procedure as in section 1.1, we assess the usefulness
of our method, comparing it to recent types of models for the RCOV matrix based on
a MCS approach. However, as Laurent et al. (2013) point out sometimes the model
with the smallest statistical loss function may not be the one preferred in the evaluation
by economic consideration. Hence, we also focus on economic evaluation by means
of conventional portfolio optimization and Value-at-Risk (VaR) forecasting. We find
that using a vine structure leads to significant improvements for HAR models regarding
statistical loss, mean-variance efficient portfolios and VaR predictions. For ARFIMA
based vine models, results are not as unambiguous except for forecasting daily VaRs.
Chapter 1. Introduction 7
There, compared to conventional models, the vine structure leads to smaller capital
requirements while providing significantly more accurate forecasts and avoiding large
exceedances of the forecasted VaR. These results are in line with the vine models ability
to model tail events due to their assumption of non-normality. Hence, especially in
combination with easily applicable conventional models, such as the HAR, our modeling
approach offers a flexible and promising way of using the advantages of copulas for
forecasting multivariate realized volatility.
1.3 Investor attention and stock market volatility
While the previous sections introduced models based on the autoregressive nature of
volatility, a large amount of research includes exogenous variables to form predictions
for its future, as asset prices are most likely dependent on other factors, see Engle and
Sheppard (2001). While Andersen and Bollerslev (1998b) analyze the impact of news
announcements of US macroeconomic data and its influence on volatility, more recent
studies, such as Barber et al. (2009), directly focus on the interest investors take in the
market. Traditionally, so called investor attention is measured by indirect proxies like
volume, turnover and news. While volume might be the natural candidate for forecast-
ing purposes, several studies, e.g. Brooks (1998) and Donaldson and Kamstra (2005)
suggest that it does not improve the accuracy of volatility predictions. News as an al-
ternative measure are mostly irregular and may underly a considerable publication lag.
Recent publications use internet message postings (S.-H. Kim and D. Kim, 2014), Face-
book users sentiment data (Siganos et al., 2014) or search frequencies (Vozlyublennaia,
2014) to assess the influence of retail investors attention on the stock market. Several
studies, among them Da et al. (2011), Vlastakis and Markellos (2012) and Andrei and
Hasler (2015), suggest that Google search volume is a driver of future volatility. While
most of the previous studies focus on analyzing the in-sample relationship of volatil-
ity and investor attention, the last article in this dissertation explicitly concentrates on
predictability in an out-of-sample forecasting framework.
We suggest including Google search data (via Google Trends) in the framework of
empirical similarity (ES) introduced by Gilboa et al. (2006), augmenting an autoregres-
sive (AR) model by a time-varying coefficient determined by the empirical similarity
between last periods Google data and realized volatility. This approach has previously
been suggested by Lieberman (2012) and resembles an autoregressive process with dy-
namic parameters. As a result, the model is able to depict stationary, non stationary
and explosive behavior, which can often be found in time series of realized volatilities,
see Chen et al. (2010); Hansen and Lunde (2014). The unique assumption behind the
model is, that investors seek information about the market before they actively trade,
Chapter 1. Introduction 8
which allows us to draw inference for different states of investor attention and volatil-
ity. For example, if investor attention is high and volatility is low, future volatility is
expected to rise due to increased participation of investors in the market. On the other
hand, if the previous level of volatility was high, low attention indicates a change point
of volatility dynamics, meaning investors are losing interest in the market. In this case,
due to decreased participation, future volatility should decrease, too. Based on weekly
realized volatility of the Dow Jones Industrial Average (DJIA) ranging from January 16,
2004 to October 18, 2013, we find that the model shows significantly better performance
compared to traditional models in an in-sample comparison as well as an out-of-sample
forecasting study. By including two alternative time-varying models, we highlight that
forecasting performance is indeed driven by the use of Google Trends data in combina-
tion with the ES framework. Furthermore, we test the robustness of the out-of-sample
study by using the realized kernel suggested in Barndorff-Nielsen, Hansen, et al. (2008)
as an alternative proxy for volatility.
Our results confirm the findings of Vlastakis and Markellos (2012) and Vozlyublennaia
(2014), who state that investor attention is a driver of volatility on short horizons. As
described by Andrei and Hasler (2015), this relationship is strongest in phases of high
volatility, where investor attention tends to be high. Additionally to evaluating the fore-
casts based on a MCS approach, we highlight the practical application by predicting the
weekly VaR. Here, the ES model produced significantly better VaR forecasts in terms
of overall accuracy and required capital, while providing an adequate number of VaR
violations. Furthermore, the inclusion of Google Trends data as simple additive term
in classical realized volatility models, such as the ARFIMA and HAR model, did not
improve forecasting accuracy. Hence, while linear models can be useful for assessing
the correlation of volatility and investor attention and studying their dependence in an
in-sample framework, these models are not flexible enough when it comes to forecasting.
However, one drawback of our model is the availability of Google Trends data. Google
standardizes the data and restricts the access for daily data to windows of 90 days,
which cannot be merged into one meaningful time-series. Other issues include the lack
of search data for certain assets or the problem that certain search terms are ambiguous.
Nevertheless, given a certain quality of the data, our model of empirical similarity is easy
to interpret, parsimonious and shows superior predictive ability, which makes the model
attractive for economic reasoning as well as practical application
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Chapter 2
Article 1: Pitfalls of the Cholesky
decomposition for forecasting
multivariate volatility
Abstract This paper studies the pitfalls of applying the Cholesky decomposition
for forecasting multivariate volatility. We analyze the impact of one of the main issues
in empirical application of using the decomposition: The sensitivity of the forecasts
to the order of the variables in the covariance matrix. We find that despite being
frequently used to guarantee positive semi-definiteness and symmetry of the forecasts,
the Cholesky decomposition has to be used with caution, as the ordering of the variables
leads to significant differences in forecast performance. A possible solution is provided
by studying an alternative, the matrix exponential transformation. We show that in
combination with empirical bias correction, forecasting accuracy of both decompositions
does not significantly differ. This makes the matrix exponential a valuable option,
especially in larger dimensions.
Keywords: realized covariances; realized volatility; cholesky; decomposition; fore-
casting
JEL Classification Numbers: C1, C53, C58
14
Chapter 2. Pitfalls of the Cholesky decomposition 15
2.1 Introduction
Forecasts of the covariance matrix are a crucial ingredient of many economic appli-
cations in asset and risk management. The concept of using realized covariances as a
proxy for the unobservable volatility process has largely spread since the availability and
improved accessibility of high-frequency data in finance. Recent approaches to forecast
the realized covariance (RCOV) matrix are based upon uni- or multivariate time series
models. To ensure mathematical validity of the forecasted RCOV matrix, such as sym-
metry and positive semi-definiteness, either parameter restrictions or decompositions are
used. The latter one is preferred to guarantee parsimony, especially if dimensions are
large.
Latest multivariate approaches that ensure symmetry and positive semi-definiteness
of the RCOV matrix include the Wishart Autoregressive (WAR) model proposed by
Gouri´eroux et al. (2009) and its dynamic generalization, the Conditional Autoregressive
Wishart by Golosnoy et al. (2012). Chiriac (2010) shows that the WAR estimation
is very sensitive to assumptions on the underlying data, causing degenerate Wishart
distributions and affecting the estimation results. Consequently, Chiriac and Voev (2011)
choose the way of transformation and base their Vector Autoregressive Fractionally
Integrated Moving Average (VARFIMA) model on a Cholesky decomposition (CD) of
the covariance matrix. Bauer and Vorkink (2011) instead transform the covariance
matrix by using the matrix exponential transformation (MET) and use a factor model
approach for the individual components. Andersen, Bollerslev, Christoffersen, et al.
(2006) and Colacito et al. (2011) modify the Dynamic Conditional Correlation (DCC)
model of Engle (2002), splitting up variances and covariances in the modeling process.
A similar approach is implemented in Halbleib and Voev (2014). The authors suggest a
mixed data sampling method based on low-frequent estimators for the correlations.
Usually, the choice of the method can be motivated by the unique properties of the
respective decomposition. For example, while the elements of the CD are explicitly
linked to the entries of original RCOV matrix, this is not the case for the MET. On
the other hand, both, CD and MET, do not separate variances and covariances. This
can be achieved by applying a DCC type decomposition, allowing for more flexibility in
the modeling process. Moreover, as Halbleib and Voev (2014) point out, high-frequency
Chapter 2. Pitfalls of the Cholesky decomposition 16
estimators for the whole covariance matrix are often noisy and simple methods to reduce
microstructure noise, such as sparse-sampling (see Andersen, Bollerslev, Diebold, et al.
(2003)), are not applicable in large dimensions. Overall, due to its simplicity, the CD
remains the most frequently used method in the literature, beside the problem that each
permutation of the elements in the original matrix yields a different decomposition. The
problem is well known in the literature on Vector Autoregression (VAR), where the VAR
is usually identified using the corresponding CD to derive the dynamic response of each
variable to an orthogonal shock (see e.g. Sims (1980)). Keating (1996) show, that the
ordering of the variables is crucial to obtain structural impulse responses, which is only
possible if the system of equations is partially recursive. The gravity of the problem
for VAR based approaches has also been pointed out by Kl¨oßner and Wagner (2014),
who analyze the extent to which measuring spillovers is influenced by the order of the
variables.
In this paper, we focus on the impact of the ordering of the assets in the origi-
nal covariance matrix on the forecasts, if a CD is used. Since the amount of possible
permutations grows very fast with increasing number of assets, it is computationally bur-
densome to calculate and compare forecasts from each ordering. Therefore, we evaluate
the predictive accuracy of all 720 permutations based on a small data set of six assets.
Analyzing the loss distributions of two established loss functions, we find differences of
up to 18% between the average loss of the best and worst model. Using the Model
Confidence Set framework of Hansen, Lunde, and Nason (2011), we show that these loss
differences are indeed statistically significant, meaning that an arbitrary ordering may
result in suboptimal forecasts and hence poor model choices. Furthermore, the applica-
tion of an ex-ante analysis of the correlation structure of the assets to obtain a specific
ordering, as it is sometime proposed in the VAR literature, does not improve forecasting
results significantly. Additionally, we take a look at the impact of a simple empirical bias
correction, as the forecasts from both, CD and MET are biased by construction. We
show, that using the ordering invariant MET and applying the bias reduction provides
a possible solution to the ordering problem, as forecasts are not significantly different
from the best CD.
Chapter 2. Pitfalls of the Cholesky decomposition 17
2.2 Decomposition of the realized covariance matrix
Let Rtbe the N×1 vector of log returns over each period of Tdays. For a portfolio
consisting of Nstocks:
Rt=p(t)p((t1)),
p(t)=(p1t, . . . ,pNt) being the log price at time t[1, . . . ,T ].
Assume, there are Mequally spaced intra-day observations, the i-th intra-day return
for the t-th period is:
ri,t p((t1) + i1
M)p((t1) + (i1) 1
M),(2.1)
with i= 1, . . . , M . According to Barndorff-Nielsen and Shephard (2002), the N×N
RCOV matrix for the t-th period is then defined as
Yt=
M
X
i=1
ri,tr0
i,t,(2.2)
which is a consistent estimator for the conditional variance-covariance matrix of the log
returns, V ar [Rt|Ft1] = Σt. The estimator can be refined to reduce market microstruc-
ture noise (e.g. Hayashi and Yoshida (2005); Zhang et al. (2005)) and account for jumps
(Christensen and Kinnebrock, 2010). The issue of asynchronicity of the data can be
addressed by methods such as linear or previous-tick interpolation (Dacorogna, 2001)
and subsampling (Zhang, 2011), which are easy to implement in empirical work. More
complex procedures are often based on the use of multivariate realized kernels (see e.g.
Barndorff-Nielsen, Hansen, et al. (2011)). However, as Halbleib and Voev (2014) point
out, these methods are still limited in application as they may lead to data loss or do
not guarantee positive definiteness.
Chapter 2. Pitfalls of the Cholesky decomposition 18
2.2.1 Cholesky decomposition
The CD, decomposes a real, positive definite1matrix into the product of a real upper
triangular matrix and its transpose (Brezinski, 2006).
The Cholesky decomposition of the naturally symmetric and positive semi-definite
Yt, with Ptbeing an upper triangular matrix, yields:
Yt=
y11,t y12,t · · · y1N,t
y12,t y22,t · · · y2N,t
.
.
..
.
.....
.
.
y1N,t · · · · · · yN N ,t
(2.3)
=
p11,t 0· · · 0
p12,t p22,t · · · 0
.
.
..
.
.....
.
.
p1N,t p2N,t · · · pN N ,t
p11,t p12,t · · · p1N,t
0p22,t · · · p2N,t
.
.
..
.
.....
.
.
0 0 · · · pN N ,t
(2.4)
=P0
tPt.
The elements pij,t,i, j = 1, . . . , N ,i<jare real and can be calculated recursively by
pij,t =
1
pii,t yij,t Pi1
k=1 pki,tpkj,tfor i<j
qyjj,t Pj1
k=1 p2
kj,t for i=j
0 for i>j
(2.5)
In reverse, the realized covolatilities can be expressed in terms of the Cholesky elements
yij,t =
min{i,j}
X
`=1
p`i,tp`j,t.(2.6)
Since in practice, modeling is carried out on the elements of the CD, one of the
problems depicted in equation 2.5 is the influence of the ordering of the variables in
the covariance matrix. Consider for example, that we swap the position of the first and
second asset in the return vector. As a result, the elements in the first and second row
1Or positive semi-definite if the condition of strict positivity for the diagonal elements of the trian-
gular matrix is dropped
Chapter 2. Pitfalls of the Cholesky decomposition 19
of the matrix in equation 2.3 will change its positions. Due to the recursive calculation
of the elements in Pt, the corresponding Cholesky elements in the first and second row
of Ptwill not merely be swapped, but completely change magnitude. Using the CD
for a N×Nportfolio, there are N! possible permutations of the stocks in the matrix,
resulting in different decompositions that are nonlinearly related to each other. Hence,
the resulting time-series of Cholesky elements pij,t differ between the decompositions.
For all model based on the CD, this may lead to varying model choices, parameter
estimates and also forecasts.
Another issue arises in obtaining forecasts for b
Yt+1. Being a quadratic transformation
of the forecast for b
Pt+1, the forecast b
Yt+1 may not be unbiased, even if the forecasts for
b
Pt+1 are. This problem is further illustrated in section 2.2.4.
Furthermore, a often desirable feature of covariance forecasting, namely the separa-
tion of variances and covariance dynamics can not be achieved by using the CD directly
on the covariance matrix. However, it is possible to first apply a DCC decomposition
approach and a CD on the correlation matrix thereafter. In general, the nonlinear depen-
dence of the elements in the decomposition can also be an advantage, as the dependency
structure between the Cholesky elements can be studied and used for forecasting, see
e.g. Brechmann et al. (2015).
2.2.2 Matrix exponential transformation
For the covariance matrix, the matrix exponential transformation (MET) was intro-
duced together with the matrix logarithm function by Chiu et al. (1996). In mathemat-
ics, both operators are frequently used for solving first-order differential systems, see
e.g. Bellman (1997).
For any real, symmetric matrix At, the matrix exponential transformation performs
a power series expansion, resulting in a real, positive (semi-)definite matrix, in our case
Yt,
Yt=Exp(At) =
X
s=0 1
s!As
t,(2.7)
with A0
tbeing the identity matrix of size N×N, and As
tbeing the s-times the standard
matrix multiplication of At.
Chapter 2. Pitfalls of the Cholesky decomposition 20
In reverse, a real, symmetric matrix Atcan be obtained from Ytby the inverse of
the matrix exponential function, the matrix logarithm function, logm(·),
At=
a11,t a12,t · · · a1N,t
a12,t a22,t · · · a2N,t
.
.
..
.
.....
.
.
a1N,t · · · · · · aNN ,t
=logm(Yt).(2.8)
Again, a reasonable practical approach would be to model and forecast the elements aij,t,
i, j = 1, . . . , N and obtain valid covariance forecasts by equation 2.7. However, due to the
power series the expansion, the relationship between Ytand Atis not straightforward to
interpret (see e.g. Asai et al. (2006)) and similar to the CD in section 2.2.1, covariances
and variances cannot be estimated separately. By applying models to At, therefore doing
the estimation and forecasting in the log-volatility space, the retransformed forecasts for
Yt+1, will be biased by Jensen’s inequality. The problem and possible solutions are
illustrated in section 2.2.4.
Nevertheless, the MET has several advantages, especially related to factor models
where a certain factor structure is analyzed by principal component methods. It can be
shown that under several conditions, as for example in our case symmetry and positive
semi-definiteness of Yt, applying the matrix logarithm function yielding Atcorresponds
to decomposing Ytinto its eigenvalues and eigenvectors (see Chiu et al. (1996)). Hence,
the As
tcan be obtained more easily via the eigenvectors than by matrix multiplication
as in equation 2.7. Further, as principal component analysis of the matrix Ytis also
based upon eigenvalue decomposition, restrictions on the structure of the covariance
matrix models can be directly implemented while constructing the As
t, see e.g. Chiu
et al. (1996); Bauer and Vorkink (2011).
2.2.3 HAR model
One of the most simple and yet successful univariate models for volatility forecasting
is the Heterogeneous Autoregressive (HAR) model of Corsi (2009). It is inspired by the
Chapter 2. Pitfalls of the Cholesky decomposition 21
Heterogeneous Market Hypothesis (M¨uller et al., 1993), which amongst other things
assumes that market participants act on different time horizons (dealing frequencies)
due to their individual preferences, and therefore create volatility specifically on these
horizons. Since in practice, volatility over longer time intervals has stronger influence on
those over shorter time intervals than conversely (Corsi, 2009), the HAR models volatility
by an additive cascade of components of volatilities in an autoregressive framework..
This leads to the following model for the daily realized volatilities xt
xt=c+β(d)x(d)
t1+β(w)x(w)
t1+β(m)x(m)
t1+εt, εtiid
(0, σ2),(2.9)
where x(·)
tis the realized volatility over the corresponding periods of interest, one day
(1d), one week (1w) and one month (1m), which are defined as: : x(d)
t=xt1,x(w)
t=
51P5
i=1 xti+1 and x(m)
t= 221P22
i=1 xti+1.
The main advantages of the HAR are that it is easy to estimate within an OLS
framework, parameters are directly interpretable and it reproduces volatility character-
istics such as long-memory without a fractionally integration component. The latter is
especially interesting, as the long-memory property could also stem from multifractal
scaling2, which can be captured by an additive component model as the HAR, whereas
fractionally integrated models imply univariate scaling (Andersen and Bollerslev, 1996).
Under the HMH hypothesis, multifractal scaling possesses clear economic justification
which is directly interpretable in the HAR framework due to the simple parameter struc-
ture (Corsi, 2009).
Regarding forecasting, standard methods for a general ARMA framework can be
used to produce direct or iterated forecasts of the conditional volatility. In contrast to
the above conventional HAR model, which is directly applied on a time-series of realized
volatilities, we use the model on the time-series of the elements of the CD or the MET,
by replacing the components xtand x(·)
twith the respective pij,t or aij,t from equations
2.4 and 2.8.
2The underlying process scales differently for various time horizons.
Chapter 2. Pitfalls of the Cholesky decomposition 22
2.2.4 Forecasting and bias correction
To obtain forecasts for the RCOV matrix b
Yt+1, the forecasts bpij,t or baij,t are generated
by the HAR model in section 2.2.3 and retransformed by equations 2.4 respectively 2.7.
This last transformation is nonlinear and induces a theoretical bias. For the CD it
is derived in Chiriac and Voev (2011) and can be expressed by the covariances of the
forecast errors u·,t+1 of the HAR model
E[ˆyij,t+1 yij,t+1] =
max{i,j}
X
`=1
E[u`i,t+1u`j,t+1].(2.10)
However, since we estimate the models independently of each other, the expression
is not feasible as we cannot consistently estimate the covariance matrix of the forecast
errors. A heuristic approach to obtain unbiased predictions is suggested in Chiriac and
Voev (2011) and further studied in Halbleib and Voev (2011). In the original approach,
due to the larger distortion of volatilities, bias correction is only carried out on the
series of realized volatilities ˆyii,t,i1, . . . , N . However, as implied by equation 2.10,
all elements of b
Yt+1 will be biased. Hence, an adaption of the approach of Chiriac and
Voev (2011), that corrects volatility and covariance forecasts can be obtained by:
ˆy(corrected),ij,t+1 = ˆyij,t+1 ·median yij,t
ˆyij,t t=1,...,n
.
Note, that the window length non which the median is estimated, controls for the
trade-off between the bias and the precision of the correction. Since we are interested
in the general relation between bias correction, we simply estimate the median in the
bias correction factor on a window length equal to our estimation window for the HAR
model in section 2.3.
In case of the MET, the analytical bias correction is more complicated but can be
derived if b
Atand the estimated residuals ˆεtare both normally distributed, see Bauer
and Vorkink (2011) for a detailed discussion. However, since normality is empirically
often not satisfied, Bauer and Vorkink (2011) suggest a similar approach to Chiriac and
Voev (2011). Their method decomposes the forecasted matrix of realized covariances
b
Yt+1 into correlations and volatilities, bias correcting the latter ones only and leaving
Chapter 2. Pitfalls of the Cholesky decomposition 23
the correlations intact. For comparative reasons, we apply our method in equation 2.2.4
which works well in our empirical application for both, CD and MET, see section 2.3.
Note that bias correcting not only the volatilities but also the covariances bears the risk
of the corrected RCOV matrix forecast no longer being positive semi-definite. However,
in our application, this is never the case.
2.2.5 Loss functions and the MCS
According to Patton and Sheppard (2009), two issues are of major importance when
comparing forecasts of the covariance matrix. First, tests have to be robust to noise in
the volatility proxy and second, they should only require minimal assumptions on the
distribution of the returns. Therefore, we rely on the method of Hansen, Lunde, and
Nason (2011) using a model confidence set (MCS) approach based upon different loss
functions to evaluate the multivariate volatility forecasts. This framework fulfills the
requirements of Patton and Sheppard (2009) and has the advantage that we can conve-
niently compare forecasts from many models without using a benchmark. Furthermore,
the MCS does not necessarily select a single best model but it allows for the possibil-
ity of equality of the models forecasting ability. Hence, a model is only removed from
the MCS if it is significantly inferior to other models, making the MCS more robust in
comparing volatility forecasts.
For our approach, we choose two loss functions that satisfy the conditions of Hansen
and Lunde (2006) for producing a consistent ranking in the multivariate case. Consis-
tency in the context of loss functions means, that the true ranking of the covariance
models is preserved, regardless if the true conditional covariance or an unbiased covari-
ance proxy is used (Hansen and Lunde, 2006). For the comparison of forecasts of the
whole covariance matrix, Laurent et al. (2013) present two families of loss functions that
yield a consistent ordering. The first family, called p-norm loss functions can be written
as
LYt,b
Ytp=
N
X
i,j=1
|yij,t byij,t|p
1/p
,(2.11)
where b
Ytis the forecast from our model for the actual RCOV matrix Yt, which we use
as a proxy for the unobservable covariance matrix Σt. The respective elements of the
Chapter 2. Pitfalls of the Cholesky decomposition 24
matrices are denoted by yij,t and byij,t. From this class, we consider the commonly used
multivariate equivalent of the mean squared error (MSE) loss: L Yt,b
Yt2
2.
The second family, called eigenvalue loss functions is based upon the square root of
the largest eigenvalue of the matrix (Ytb
Yt)2. We will consider a special case of this
family, the so called James-Stein loss (James and Stein, 1961), which is usually referred
to as the Multivariate Quasi Likelihood (QLIKE) loss function:
LYt,b
Yt=tr(b
YtYt)ln b
YtYtN , (2.12)
where Nis the number of assets.
While both, the MSE and the QLIKE loss function determine the optimal forecasts
based on conditional expectation, Clements et al. (2009) point out, that compared to the
MSE, the QLIKE has greater power in distinguishing between volatility forecasts based
on the MCS framework. As pointed out in Laurent et al. (2013), the QLIKE penalizes
underpredictions more heavily than overpredictions. West et al. (1993) show, that this is
also relevant from an investor’s point of view, as an underestimation of variances leads to
lower expected utility than an equal amount of overestimation. Hence, for a risk averse
investor, punishing underpredictions more heavily seems to be rationale when evaluating
forecasting accuracy.
For the MCS approach, we start with the full set of candidate models M0=
{1, . . . ,m0}. For all models, the loss differential between each model is computed based
upon one of our loss functions Lk,k= 1 (MSE),2 (QLIKE), so that for model iand j,
i,j = 1, . . . ,m0and every time point t= 1, . . . ,T we get:
dij,kt = LkYit ,b
YitLkYjt,b
Yjt.(2.13)
At each step of the evaluation, the hypothesis
H0:E[dij,kt ]=0,i>j∈ M,(2.14)
is tested for a subset of models M∈M0, where M=M0for the initial step. If the
H0is rejected at a given significance level α, the worst performing model is removed
Chapter 2. Pitfalls of the Cholesky decomposition 25
from the set. To give an impression on the scale of rejection, for each loss function and
model, the respective αat which the model would be removed from the MCS can be
computed.
This process continues until a set of models remains that cannot be rejected. Similar
to Hansen, Lunde, and Nason (2011), we use the range statistics to evaluate the H0,
which can be written as:
TR= max
i,j∈M |tij,k|= max
i,j∈M ¯
dij,k
qcvar( ¯
dij,k )
,(2.15)
where ¯
dij,k =1
TPT
t=1 dij,k and cvar( ¯
dij,k ) is obtained from a block-bootstrap procedure,
see Hansen, Lunde, and Nason (2011), which we implement with 10000 replications and
a block length varying from 20 to 50 to check the robustness of the results.
The worst performing model to be removed from the set Mis selected as model i
with
i= arg max
i∈M
¯
di,k
qcvar( ¯
di,k)
,(2.16)
where ¯
di,k =1
m1Pj∈M ¯
dij,k and mbeing the number of models in the actual set M.
2.3 Empirical study
2.3.1 Data and descriptive statistics
The dataset stems from the New York Stock Exchange (NYSE) Trade and Quota-
tions (TAQ) and corresponds to the one used in Chiriac and Voev (2011). It was obtained
from the Journal of Applied Econometrics Data Archive. The original data file consists
of all tick-by-tick bid and ask quotes on six stocks listed on the NYSE, American Stock
Exchange (AMEX) and the National Association of Security Dealers Automated Quo-
tation system (NASDAQ). The sample ranges from 9:30 EST until 16:00 EST over the
period January 1, 2000 to July 30, 2008 and consists of (2156 trading days). Included
individual stocks are American Express Inc. (AXP), Citigroup (C), General Electric
(GE), Home Depot Inc. (HD), International Business Machines (IBM) and JPMorgan
Chase&Co (JPM). The original tick-by-tick data has previously been transformed as
Chapter 2. Pitfalls of the Cholesky decomposition 26
follows. To obtain synchronized and regularly spaced observations, the previous-tick
interpolation method of Dacorogna (2001) is used.
Then, log-midquotes are constructed from the bid and ask quotes by geometric av-
eraging. M= 78 equally spaced 5-minute return vectors ri,t are computed from the
log-midquotes. Daily open-to-close returns are computed as the difference in the log-
midquote at the end and the beginning of each day.
For each daily period t= 1,...,2156, the series of daily RCOV matrices is constructed
as in section 2.2 by summing up the squared 5-minute return vectors:
Yt=
M
X
i=1
ri,tr0
i,t.(2.17)
This approach is further refined by a subsampling procedure to make the RCOV esti-
mates more robust to microstructure noise and non-synchronicity (see Zhang (2011)).
From the original data, 30 regularly ∆-spaced subgrids are constructed with ∆ = 300
seconds, starting at seconds 1,11,21,...,291. For each subgrid, the log-midquotes are
constructed and the RCOV matrix is obtained on each subgrid according to equation
2.17. Then, the RCOV matrices are averaged over the subgrids. To avoid noise by
measuring overnight volatilities, all computations are applied to open-to-close data. For
the descriptive statistics and estimation purposes, all daily and intradaily returns are
scaled by 100, so that the values refer to percentage returns.
At each point of time t, we apply either the CD or the MET on the obtained RCOV
matrix. Additionally, we take the logarithm of the elements on the diagonal, to ensure
positivity of the elements of the decomposition when applying the time-series models.
Since the ordering of the assets in the original RCOV matrix is relevant for the CD, we
refer to the basic alphabetic ordering of the individual stocks in section 2.3.1 for the
initial descriptive analysis of the elements of the CD.
In general, the elements of both decompositions exhibit the same characteristics as
the realized covariances, such as volatility clustering, right skewness, excess kurtosis and
high levels of autocorrelation, see tables A.1 and A.2. All series seem to be stationarity
based on the Augmented Dickey-Fuller test.
Chapter 2. Pitfalls of the Cholesky decomposition 27
2.3.2 Optimal odering
If the ordering might indeed be crucial for the forecast performance, the question
arises if there is any possibility to determine the optimal position of an asset in the
original return vector before evaluating all permutations. According to equation 2.6, the
forecasts in column j,{byij }i=1,...,j only depend on the forecasted entries of the Cholesky
matrix Pup to column 1, . . . , j,{bp`i }ij. Hence, if the asset is for example moved from
position i= 1 in the return vector to a position i > 1 the number of forecasted Cholesky
elements that enter calculation of the covolatility forecast increases with every increase
in the position. Intuitively, assets that are more correlated with each other should be
placed after assets that are less correlated so that their dependence is picked up by the
Cholesky elements. Similarly, in the estimation of structural VARs, variables are often
ordered by their degree of exogeneity from most to least exogenuous, see e.g. Bernanke
and Blinder (1992); Keating (1996). However, the CD is only useful for identifying the
structural relationship under rather restrictive conditions, e.g. in case of VAR modeling
if the underlying relationship is recursive. Based on our data set of equity returns we
cannot impose a structural relationship by means of economic theory. Nevertheless, we
analyze the correlation structure of the realized variances of the six assets to identify
possible linkages that might be helpful in ordering the assets. The full-sample correlation
matrix of the time-series of realized variances for the natural alphabetic ordering of the
assets are given in figure 2.1.
On the left side the ordering of the elements in the return vector matrix is used,
while on the right side the correlations are ordered based on the angular positions of
the eigenvectors of the correlation matrix. This method is sometimes called “correlation
ordering” (Friendly and Kwan, 2003) and places similar variables contiguously. The
correlation matrix on the right shows which assets should be grouped together. Note that
the correlations are not sorted by size, eg. from highest to lowest average correlation.
We now proceed to analyze two questions. First, do different orderings indeed yield
significantly different forecasts? Second, does ordering the variables in the returns vector
similar to the rule of correlation ordering produce superior forecasts?
Chapter 2. Pitfalls of the Cholesky decomposition 28
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
AXP
C
GE
HD
IBM
JPM
AXP
C
GE
HD
IBM
JPM
100 77
100
77
69
100
81
71
79
100
70
60
70
69
100
71
91
71
66
55
100
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
IBM
HD
GE
AXP
C
JPM
IBM
HD
GE
AXP
C
JPM
100 69
100
70
79
100
70
81
77
100
60
71
69
77
100
55
66
71
71
91
100
Figure 2.1: Correlation matrix of the original time-series of realized variances. On
the left, correlations are ordered by the alphabetic order in the return vector. On the
right, correlations are ordered based on the angular positions of the eigenvectors of the
correlation matrix. The estimate of the corresponding correlation coefficient is given
inside the square.
2.3.3 Modeling and forecasting procedure
For each decomposition and permutation of the assets, we apply the HAR model from
section 2.2.3 on each time-series of CD or MET elements. Since the MET is independent
of the chosen permutation, we can use the resulting model as a benchmark model. For
the CD, we obtain 21 different models for each one of the 720 permutations. We retain
the last 200 observations of the dataset for out-of-sample one-step ahead forecasting and
estimate the models based on the a moving window of 1956 observations. Forecasts of
the RCOV matrix are then generated according to section 2.2.4.
First, for each permutation we evaluate the forecasts by means of the multivariate
loss functions from section 2.2.5. For the CD and both loss functions, we take the average
loss over time for each permutation to obtain a distribution of losses, see figure 2.2. The
corresponding descriptive statistics are given in the upper half of table A.3.
The loss density of the MSE is multimodal and left skewed with an average loss
of 271.71. In comparison, the average loss of the MET is 334.21 which is 16% larger
than the maximum MSE loss of the CD. The difference between largest and smallest
average loss is 8%. The QLIKE loss density is more symmetric and only slightly right
skewed, with an average loss of 0.60, compared to an average loss of 0.71 for the MET.
Again, the MET QLIKE loss is 16% larger than the largest QLIKE loss of the CD. The
Chapter 2. Pitfalls of the Cholesky decomposition 29
0.00
0.05
0.10
0.15
265 270 275 280 285 290
loss
density
0
25
50
75
100
0.58 0.59 0.60 0.61
loss
density
Figure 2.2: Average (over time) MSE (left) and QLIKE (right) density for all permu-
tations. Red line is the mean value.
worst best ex-ante ex-ante vs best
MSE “3 2 1 6 4 5” “5 4 3 1 6 2” “5 4 3 1 2 6” 1.002
QLIKE “3 1 5 4 2 6” “6 2 3 4 5 1” “5 4 3 1 2 6” 1.506
Table 2.1: Orderings with the highest and lowest average losses (without bias cor-
rection) based on the respective loss function. Ex-ante gives the order proposed by
the method of correlation ordering. Additionally the average loss of the ex-ante model
relative to the best model is listed.
standard deviation of the QLIKE losses is significantly smaller (p < 0.01) than for the
MSE loss, based on the Brown-Forsythe test. Still, the difference between largest and
smallest average loss is roughly 5%. Ranking the models from best to worst (smallest to
largest average losses over time), we find that the ordering is not consistent across the
loss functions. Evaluating the model performance over time instead of taking averages,
the most frequent best model is identical in 4 of the 200 out-of-sample forecasts for both
loss functions. The most frequent worst model on the other hand differs between both
loss function. For the MSE, one certain ordering is the worst model in 12 out of the 200
forecasts. In case of the QLIKE, the most frequent worst ordering has the highest loss
in 3 out of 200 times.
To come back to the question if the method of correlation ordering in section 2.3.2
is helpful in determining the best model ex-ante, we list the worst and best orderings
based on the average loss for both loss function in table 2.1. To simplify the notation, we
rename the assets by their position in the alphabetic return vector, namely AXP= 1,
C= 2, GE= 3, HD= 4, IBM= 5 and JPM= 6.
Surprisingly, the best model under the MSE loss function nearly coincides with the
model suggested by the method of correlation ordering, with only asset 2 and 6 switching
Chapter 2. Pitfalls of the Cholesky decomposition 30
positions. For the QLIKE loss, only asset 3 is on the same position in the best model
compared to the ex-ante ordering. Regarding the average loss size, the ex-ante models
losses are only 0.2% larger than the best model based on the MSE loss, whereas for the
QLIKE loss function the ex-ante losses are 50% larger. We statistically evaluate these
differences in section 2.3.4. However, based on the mixed results from both loss functions
we cannot unambiguously establish a link between correlation ordering and forecasting
results. Additionally, as pointed out before, the model ranking is highly time-varying.
Evaluating the model at every point of time reveals that the ex-ante model has the
lowest loss at exactly one point of time for both loss functions. Again, it seems that
neither the ex-ante nor any other ordering is consistently delivering the best forecasts.
0.00
0.05
0.10
0.15
0.20
130 135 140 145 150
loss
density
0
20
40
60
80
0.19 0.20 0.21 0.22
loss
density
Figure 2.3: Average (over time) MSE (left) and QLIKE (right) density for all permu-
tations with bias correction. Red line is the mean value.
In case of the bias correction, the average loss densities for all permutations are
significantly different (p < 0.01) from the ones without bias correction based on the
Kolmogorov-Smirnov (KS) test. In general, the bias correction does decrease the average
loss, see figure 2.3. Descriptive statistics are given in the lower half of table A.3. Most
notable, the standard deviation does increase for the MSE, while in case of the QLIKE
the distribution becomes more right skewed. As a result, the difference between largest
and smallest average loss increases for both loss functions to 17% (MSE), respectively
18% (QLIKE) percent. The bias corrected average MET loss is 112.64 for the MSE and
0.18 for the QLIKE. Hence, the MET heavily benefits from the bias correction, making
it a possible alternative to the CD to circumvent the ordering problem.
For each permutation, we test the distribution of losses over time of the bias corrected
vs the non-bias corrected forecasts using the KS test. In all cases, the loss distributions
are significantly different at a level p < 0.01 and the mean loss (over time) of the bias
Chapter 2. Pitfalls of the Cholesky decomposition 31
corrected distribution is smaller than the one of the non-bias corrected. For the MSE,
the worst model with bias correction is also the same as the worst model without bias
correction. Otherwise, we find that the best and worst model are not the same as in
the case of no bias correction. As before, the ranking of the average losses from best to
worst is not consistent across the loss functions. Comparing the losses over time reveals
a similar behavior as before, where the most frequent best and worst model varies across
time.
2.3.4 Statistically testing forecast performance
To evaluate the significance of the loss differences across time, we test the losses of the
permutations using the MCS procedure introduced in section 2.2.5. We are interested
in several questions. First of all, are the forecasts from the models which are best and
worst based upon the average loss significantly different from each other? Second, how
well does the bias adjusted MET model perform compared to the best ordering and
third, is the ex-ante ordering significantly worse than the best model?
Starting with the first questions, we find that for both loss functions the worst
model can be rejected from the MCS at a α= 1% level of significance. In case of bias
correction, αfurther decreases. As mentioned in the literature the QLIKE is also more
discerning, leading to slightly lower levels of significance in both cases if compared to
the MSE. Comparing the non-corrected vs the bias corrected forecasts, we find that the
bias correction leads to significantly better forecasts for both loss functions (α= 1%).
Overall, since the differences between the forecasts are indeed statistically significant,
choosing the “wrong” ordering may lead to poor forecast performance, no matter which
loss function is chosen.
Next, we only consider the case of bias correction. As we have seen, the MET average
losses where well within the range of the average CD losses. If the MET losses are not
significantly different from the best CD model, the MET with bias correction could be a
valid alternative to avoid the ordering problem of the CD. The MET forecasts can only
be rejected from the MCS at a α= 50% significance level for the MSE and a α= 69%
significance level for the QLIKE. Hence, the forecasts from the best CD model and the
MET are not significantly different from each other at a reasonable level of confidence.
Chapter 2. Pitfalls of the Cholesky decomposition 32
Comparing the losses of the ex-ante ordering with the best model under the respective
loss function, we find that for the QLIKE the losses are significantly different (α < 1%),
while for the MSE the ex-ante model can not be rejected from the MCS (α= 9%). Hence,
initially deciding upon the ordering does not yield a clear recommendation. The danger
of arbitrarily choosing an ordering that might lead to poor forecasts and hence model
choices cannot be assessed ex-ante based on the methodology of correlation ordering.
2.4 Conclusion
In this paper, we empirically analyzed several issues arising from using the Cholesky
decomposition (CD) for forecasting the realized covariance (RCOV) matrix. We studied
the impact of the order of the variables in the covariance matrix on volatility forecast-
ing, finding that different orderings do indeed lead to significantly different forecasts
based on a MCS approach. Initially deciding upon the ordering based on the angular
positions of the eigenvectors of the correlation matrix does not lead to unambiguously
better results in forecasting. Further, we find that the best and worst models are not
consistent over time so that a clear recommendation to which order to use is not at
hand, even if forecasts are performed stepwise. A frequently used method of bias correc-
tion improves forecasting accuracy, but on the other hand widens the difference between
best and worst model so that the ordering problem worsens. On the other hand, bias
corrected forecasts from another decomposition, the matrix exponential transformation
(MET) show equal predictive ability and do not suffer from the ordering problem. Thus,
for empirical application two conclusions can be drawn. If a reasonable order can be
imposed on the elements of the covariance matrix or if the connection between the ele-
ments of the decomposed covariance matrix are of interest the CD is a rational choice.
Otherwise, the application of the MET together with a bias correction is advised, be it
for comparative reasons or simply to avoid the time consuming process of estimating all
possible permutations of the CD.
Appendix
min max mean sd skew kurt pval ADF acf l=1 acf l=2
p11,t -1.31 2.03 0.30 0.58 0.10 2.23 0.01 0.88 0.86
p12,t -0.21 7.63 0.71 0.66 2.83 16.71 0.01 0.76 0.70
p22,t -1.13 2.16 0.20 0.53 0.29 2.33 0.01 0.89 0.87
p13,t -0.50 3.99 0.52 0.43 2.17 11.41 0.01 0.65 0.59
p23,t -0.28 2.68 0.37 0.30 1.99 9.69 0.01 0.61 0.57
p33,t -1.14 1.73 0.04 0.48 0.31 2.40 0.01 0.86 0.82
p14,t -0.7 3.71 0.55 0.46 2.01 9.44 0.01 0.64 0.59
p24,t -0.38 2.40 0.36 0.30 1.77 8.79 0.01 0.46 0.46
p34,t -0.53 2.76 0.28 0.26 1.69 10.30 0.01 0.50 0.45
p44,t -0.97 1.75 0.29 0.42 0.36 2.74 0.01 0.81 0.77
p15,t -0.43 3.09 0.45 0.35 1.96 9.95 0.01 0.53 0.49
p25,t -1.22 4.68 0.31 0.27 3.42 39.36 0.01 0.47 0.39
p35,t -0.43 2.13 0.26 0.22 1.82 10.20 0.01 0.44 0.42
p45,t -0.48 1.71 0.15 0.17 1.36 10.53 0.01 0.18 0.14
p55,t -1.11 1.56 0.00 0.46 0.58 2.88 0.01 0.86 0.82
p16,t -0.29 8.16 0.72 0.66 2.88 18.55 0.01 0.73 0.65
p26,t -0.19 5.77 0.58 0.43 2.26 15.50 0.01 0.64 0.60
p36,t -0.36 2.22 0.22 0.22 1.86 10.20 0.01 0.32 0.30
p46,t -0.91 1.19 0.14 0.18 0.63 6.63 0.01 0.17 0.11
p56,t -0.87 1.37 0.14 0.19 1.25 8.48 0.01 0.18 0.15
p66,t -1.21 2.20 0.14 0.52 0.30 2.39 0.01 0.89 0.86
Table A.1: Descriptive statistics for the time-series of the elements of the (alphabetic)
Cholesky decomposition. Diagonal (log) time-series are written in bold. Additionally,
p-value of the ADF test and magnitude of the first and second autocorrelation coeffi-
cient.
33
Chapter 2. Pitfalls of the Cholesky decomposition 34
min max mean sd skew kurt pval ADF acf l=1 acf l=2
a11,t -2.72 3.63 0.32 1.13 0.13 2.19 0.01 0.88 0.86
a12,t -0.35 0.94 0.30 0.16 0.26 3.32 0.01 0.44 0.43
a22,t -2.28 4.35 0.30 1.07 0.33 2.36 0.01 0.90 0.87
a13,t -0.24 0.67 0.24 0.14 -0.08 3.06 0.01 0.28 0.26
a23,t -0.24 0.71 0.27 0.14 -0.02 2.90 0.01 0.38 0.29
a33,t -2.36 3.52 0.09 0.96 0.30 2.43 0.01 0.85 0.82
a14,t -0.34 0.73 0.20 0.13 0.04 3.08 0.01 0.24 0.20
a24,t -0.25 0.66 0.22 0.13 0.07 3.00 0.01 0.25 0.27
a34,t -0.27 0.66 0.22 0.13 -0.09 3.13 0.01 0.30 0.28
a44,t -2.01 3.63 0.65 0.84 0.35 2.77 0.01 0.81 0.76
a15,t -0.34 0.62 0.21 0.13 -0.18 3.27 0.01 0.22 0.18
a25,t -0.28 0.80 0.23 0.13 -0.04 3.17 0.01 0.29 0.23
a35,t -0.19 0.67 0.26 0.14 -0.06 2.79 0.01 0.31 0.29
a45,t -0.31 0.65 0.21 0.13 -0.08 3.18 0.01 0.21 0.17
a55,t -2.21 3.51 0.10 0.91 0.57 2.93 0.01 0.85 0.81
a16,t -0.16 0.99 0.29 0.16 0.55 3.70 0.01 0.45 0.39
a26,t -0.11 1.13 0.42 0.18 0.50 3.50 0.01 0.53 0.51
a36,t -0.32 0.63 0.23 0.13 0.02 2.97 0.01 0.21 0.20
a46,t -0.32 0.75 0.20 0.13 0.05 3.25 0.01 0.24 0.20
a56,t -0.33 0.62 0.21 0.13 0.01 3.07 0.01 0.20 0.17
a66,t -2.35 4.85 0.47 1.07 0.25 2.42 0.01 0.89 0.85
Table A.2: Descriptive statistics for time-series of the elements of the matrix exponen-
tial transformation. Diagonal (log) elements are written in bold. Additionally, p-value
of the ADF test and magnitude of the first and second autocorrelation coefficient.
Chapter 2. Pitfalls of the Cholesky decomposition 35
min max mean sd skew kurt median max/ min MET alphabetic ex-ante
without bias correction
MSE 265.82 287.43 271.71 4.65 0.75 2.92 271.08 1.08 334.21 269.27 282.71
QLIKE 0.58 0.61 0.60 0.01 0.28 3.47 0.60 1.05 0.71 0.59 0.60
with bias correction
MSE 130.29 152.47 136.43 5.30 0.79 2.62 134.48 1.17 112.64 130.52 149.19
QLIKE 0.18 0.22 0.21 0.01 0.97 3.54 0.21 1.18 0.18 0.21 0.19
Table A.3: Descriptive statistics for the CD losses over all permutations. Max/min is the ratio of the average loss of the best model vs the average
loss of the worst model. As a comparison, the average losses of the (ordering invariant) MET and the average losses of the alphabetic and ex-ante
correlation ordering are given.
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Chapter 5
Summary and outlook
The previous chapters of this dissertation outlined several topics in predicting uni-
and multivariate stock market volatility. Based on the improved availability of high-
frequency data and the econometric methods to estimate the volatility process, new
possibilities to study predictability have gained attention in the recent years. Especially
in the multivariate context, where forecasts of the covariance matrix are used in portfolio
and risk management, the requirements of the forecasted covariance matrix to be sym-
metric and positive semi-definite pose a problem. A prominent method of guaranteeing
both properties, the Cholesky decomposition (CD), was studied in chapter 2. Beside
analyzing the main pitfall, namely the influence of the ordering of the variables on the
forecasts, a possible solution for empirical application was suggested. In combination
with a bias correction, the matrix exponential transformation (MET) provided a suitable
alternative to the CD. Being one of the first studies of its kind, several extensions for
further research are still to be assessed. First, we only studied two out of several possible
decompositions. For example, it is possible to model realized covariances (RCOV) in
the popular dynamic conditional correlation (DCC) framework of Engle (2002). Similar
to the MET, a DCC type decomposition is invariant to the order of the variables. Addi-
tionally, it does allow to separate the dynamics of variances and correlations, allowing for
more flexible model specifications, see Halbleib and Voev (2014). Due to computational
efforts to evaluate the forecasts of all possible permutations of the assets, we only studied
the heterogeneous autoregressive (HAR) model of Corsi (2009) in this setting. While the
ordering problem as well as the bias are independent of the model, some approaches are
explicitly linked to a certain decomposition, e.g. the vector autoregressive fractionally
integrated moving average (VARFIMA) of Chiriac and Voev (2011) is usually combined
with the CD. In this case simply changing the decomposition to the MET may not be
applicable, as one of the main advantages of using the CD is the traceability of impulse
responses. Further research is necessary to solve the ordering problem in cases where
a change of the decomposition is either not possible or rational. Additionally, we have
41
Chapter 5. Summary and outlook 42
only analyzed one very simple data driven case of bias correction, which worked well for
our data set and model choice. However, a similar method applied in Halbleib and Voev
(2011) did not lead to improved volatility forecasts for their VARFIMA model. Hence, a
more detailed analysis or sophisticated methods, e.g. simulation based bias corrections
as proposed in Weigand (2014), might be a valuable extension of our study.
A new approach to modeling and forecasting the RCOV matrix based on the CD
was introduced in chapter 3. We proposed a model that makes use of vine copulas to
account for often neglected nonlinearities and asymmetries in conventional models for
multivariate volatility. We showed that using a vine copula structure can lead to sig-
nificant improvements in forecasting, regarding statistical loss, mean-variance efficient
portfolios and Value-at-Risk (VaR). Still, several areas of improvement of the general
framework seem fruitful for further research. First, our initial model requires the esti-
mation of a large number of parameters. Therefore, we applied a successful strategy to
decrease the number of parameters by restricting the choice of copulas without severely
impairing the models performance. For larger dimensions, the model could be extended
by truncation approaches as discussed in Brechmann et al. (2012) to further simplify
the estimation procedure. Second, while the CD is useful for our model as it explic-
itly links the time series of realized volatilities and covariances, the ordering problem
remains an issue. While in sample our analysis revealed that differences based in the
ordering are relatively small, this might not be the case for the out-of-sample forecasts,
as pointed out in chapter 2. An alternative approach of applying vine copulas for multi-
variate volatility modeling could be the use of partial correlations. For Gaussian vines,
there exists a one to one relationship between the partial correlation, which is identified
by the vine, and the correlation parameters of the joint distribution, see Baba et al.
(2004); Kurowicka and Cooke (2006). By modeling realized volatilities and the partial
correlation elements of a Gaussian regular vine, a joint model for the RCOV matrix
can be derived, which guarantees positive semi-definiteness and symmetry without the
drawbacks and restrictions of an initial decomposition. To avoid parameter restrictions
when modeling the partial correlations, a Fisher z-transformation (Fisher, 1915) can be
applied on the time series of partial correlations. The resulting values on the real line can
again be modeled by conventional time series models, similar to our previous approach.
A third extension for the model could be to allow for time-varying effects and jumps.
Several studies (amongst others Patton (2004); Cappiello et al. (2006)) argue, that the
dependence structure between assets is rather time-varying than constant, e.g. to reflect
turmoil periods. Based on a dynamic framework with locally constant parameters, a
more flexible modeling procedure is possible, see e.g. Okhrin et al. (2013). The partial
correlation approach on the other hand could be used to disentangle jump and contin-
uous variation similar to Audrino and Hu (2011), as it can directly be applied on both
Chapter 5. Summary and outlook 43
components. For the Cholesky based model, this is not possible due to the fact that the
time series of Cholesky elements is modeled instead of the original realized volatilities.
Chapter 4 approached univariate volatility from a different angle by using Google
search volume as a measure of investor attention to the stock market in the frame-
work of empirical similarity (ES) by Gilboa et al. (2006). We augmented the model of
Lieberman (2012) based on the main assumption that increased search volume leads to
higher participation in the stock market and subsequently rising volatility. While our
model showed superior predictive ability compared to several other volatility models,
we also highlighted that using Google search volume as an additional linear regressor
in standard time series models does not accurately reflect the dynamics of the data.
Based on our approach and results, a number of extensions for further research come to
mind. First of all, the model itself can be extended. Up to now, the level of investor
attention is compared to the contemporaneous level of volatility to perform forecasts
for the subsequent period. However, investors might cause volatility not exclusively on
this frequency, but also on longer horizons. This intuition is consistent with the fractal
market hypothesis of M¨uller et al. (1997) and one of the basic assumptions of the HAR
model of Corsi (2009). Hence, extending the model by components regarding different
horizons similar to Golosnoy et al. (2014) might give insights in the low frequent trading
patterns of investors and further improve predictability. Second, for simplicity we have
only analyzed the impact of Google search volume in the US on the volatility of the Dow
Jones. Obviously, since investors are not limited to the domestic market and investing
abroad is widely suggested by the theory of diversification, cross country effects of in-
vestor attention on volatility seem plausible. Similar to the work of Dimpfl and Jung
(2012), a structural multivariate model based on the concept of ES and Google search
volume could be estimated equation wise and compared to the conventionally used vec-
tor autoregressive (VAR) framework. Last but not least, the model can be applied to a
variety of other data sets. Similar to Google search volume, other measure for investor
attention, such as household survey data (D. Li and G. Li, 2014), internet message post-
ings (S.-H. Kim and D. Kim, 2014) or Facebook users sentiment data (Siganos et al.,
2014) could be analyzed.
As a conclusion, volatility seems likely to remain a worthwhile and promising field of
research. Including the behavior of investors directly into the modeling process creates
challenges and new opportunities at a time, where the importance of alternative data
sources is slowly recognized. For modeling itself, non normality of the data and devi-
ations from standard approaches can not be disregarded. Meanwhile, basic questions
such as the ordering problem in multivariate volatility models still deserve attention.
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