78733745: Long run forecast of the covariance matrix
Abstract4
Chapter 1: Introduction6
1.1 Introduction6
1.2 Background information and company context9
1.3 Problem Statement11
1.4 Rationale for the study12
1.5 Study objectives13
1.6 Scope of study14
1.7 Research design14
1.8 Limitations of the study15
Chapter 2: Literature Review
1 Introduction
The dynamics of the time-varying volatility of financial assets play a main
role in diverse fields, such as derivative pricing and risk management. Consequently,
the literature focused on estimating and forecasting conditional
variance is vast. The most popular method for modelling volatility belongs
to the family of GARCH models (see Bollerslev et al. 1992 for a review of
this topic), although other alternatives (such as stochastic volatility models)
also provide reliable estimates. The success of GARCH processes is
unquestionably tied to the fact that they are able to fit the stylized features
exhibited by volatility in a fairly parsimonious and convincing way, through
quite a feasible method. The seminal models developed by Engle (1982)
and Bollerslev (1986) were rapidly generalized in an increasing degree of
sophistication to reflect further empirical aspects of volatility.
One of the more complex features that univariate GARCH-type models
have attempted to fit is the so-called long-memory property. The volatility
of many financial assets exhibits a strong temporal dependence which is
revealed through a slow decay to zero in the autocorrelation function of
the standard proxies of volatility (usually squared and absolute valued
returns) at long lags. The basic GARCH model does not succeed in
fitting this pattern because it implicitly assumes a fast, geometric decay
in the theoretical autocorrelations. Engle and Bollerslev (1986) were
the first concerned with this fact and suggested an integrated GARCH
model (IGARCH) by imposing unit roots in the conditional variance.
The theoretical properties of IGARCH models, however, are not entirely
satisfactory in fitting actual financial data, so further models were later
developed to face temporal dependence. Ballie, Bollerslev and Mikkelsen
(1996) proposed the so-called fractionally integrated GARCH models
(FIGARCH) for volatility in the same spirit as fractional ARIMA models
which were evolved for modelling the mean of time series (see Baillie, 1996).
These models imply an hyperbolic rate of decay in the autocorrelation
function of squared residuals, and generalize the basic framework by still
using a parsimonious parameterization.
There has been a great interest in modelling the temporal dependence
in the volatility of financial series, mostly in the univariate framework1.
The analysis of the long-memory property in the multivariate framework,
however, has received much less attention, even though the estimation
of time-varying covariances between asset returns is crucial for risk
management, portfolio selection, optimal hedging and other important
applications. The main reason is that modelling conditional variance in
1An alternative approach for modelling long-memory through GARCH-type models is
based on the family of stochastic volatility (see Breidt, Crato and de Lima, 1998). An
extension of FIGARCH models has been considered in Ding, Granger and Engle (1993).
2 The multivariate modelling of long-memory
Although long-memory has been observed in the volatility of a wide range
of assets, the literature on the topic is mainly focused on foreign exchange
rate time series (FX hereafter). There exists a great deal of empirical
literature focused on modelling and forecasting the volatility of exchangerate
returns in terms of the FIGARCH models in the univariate framework.
An exhaustive review of the literature is beyond the aim of this paper.
Some recent empirical works on this issue can be found in Vilasuso (2002)
and Beine et al. (2002). On the other hand, the literature dealing with the
multivariate case is scarce.
The modelling of long-memory in the multivariate framework was firstly
studied by Teyssière (1997), who implemented several long memory volatility
processes in a bivariate context, focusing on daily FX time series. He
used an approach initially based on the multivariate constant conditional
correlation model (Bollerslev, 1990), which allows for long-memory ARCH
dynamics in the covariance equation. He also weakened the assumption
of constant correlations and estimated time-varying patterns. Teyssière
(1998) estimated several trivariate FIGARCH models on some intraday FX
rate returns. This author finds a common degree of long-memory in the
marginal variances, while the covariances do not share the same level of
persistence with the conditional variances. More recently, Pafka and Mátyás
(2001) analyzed a multivariate diagonal FIGARCH model on three FX timeseries
through quite a complex computational procedure. The multivariate
modelling on other time series has focused on the crude oil returns (Brunetti
and Gilbert, 2001). A bivariate constant correlation FIGARCH model is
fitted on these data to test for fractional cointegration in the volatility
of the NYMEX and IPE crude oil markets2. To our knowledge, there is
no other literature concerned with modelling temporal dependences in the
multivariate context.
The previous research affords a valuable contribution to the better
understanding of long-run dependences in multivariate volatility. A major
shortcoming in applying these approaches in practice, however, lies in
the overwhelming computational burden involved, which simply makes the
straightforward extension of these methods to large portfolios unfeasible
(note that only two or three assets are considered in the empirical
applications of these methods). The procedure we shall discuss is specifically
2.1 The orthogonal multivariate model
We firstly introduce notation and terminology. Consider a portfolio of K
financial assets and denote by rt = (r1t, r2t, …, rKt)????, t = 1, …,T, a weaklystationary
random vector with each component representing the return of
each portfolio asset at time t. Denote by Ft the set of relevant information
up to time t, and define the conditional covariance matrix of the process
by E(rtr????t|Ft−1) = Et−1 (rtr????t) = Ht. Denote as E(rtr????t) = Ω the (finite)
unconditional second order moment of the random vector. Note that only
second-order stationarity is required, which is the basic assumption in the
literature concerned with estimating covariance matrices of asset returns.
Other procedures proposed for estimating the covariance matrix require
much stronger assumptions (see, for instance, Ledoit and Wolf, 2003), as the
existence of higher-order moments and even iid-ness in the driving series.
As the covariance matrix Ω is positive definite, it follows by the spectral
decomposition that Ω = PΛP????, where P is an orthonormal K×K matrix of
eigenvectors, and Λ is a diagonal matrix with the corresponding eigenvalues
of Ω in its diagonal. Lastly, assume that the columns of P are ordered by
size of the eigenvalues of Λ, so the first column is the one related to the
highest eigenvalue, and so on.
The orthogonal model by Alexander is based on applying the principal
component analysis (PCA) to generate a set of uncorrelated factors from
the original series3. The PCA analysis is a well-known method widely used
in practice, and several investment consultants, such as Advanced Portfolio
Technologies, use procedures based on principal components. The basic
strategy in the Alexander model consists of linearly transforming the original
data into a set of uncorrelated latent factors so-called principal components
whose volatility can then be modelled in the univariate framework. With
these estimations, the conditional matrix Ht is easily obtained by the inverse
map of the linear transformation.
The set of principal components, yt = (y1t, y2t, …, yKt)????, is simply
defined through the linear application yt = P????rt. It follows easily that
E(yt) = 0 and E(yty???? t ) = Λ by the orthogonal property of P. The columns
of the matrix P were previously ordered according to the corresponding
eigenvalues size, so that ordered principal components have a decreasing
ability to explain the total variability and the main sources of variability.
