\ufeff78733745: Long run forecast of the covariance matrix<\/p>\n
Abstract4<\/p>\n
Chapter 1: Introduction6<\/p>\n
1.1 Introduction6<\/p>\n
1.2 Background information and company context9<\/p>\n
1.3 Problem Statement11<\/p>\n
1.4 Rationale for the study12<\/p>\n
1.5 Study objectives13<\/p>\n
1.6 Scope of study14<\/p>\n
1.7 Research design14<\/p>\n
1.8 Limitations of the study15<\/p>\n
Chapter 2: Literature Review<\/p>\n
1 Introduction<\/p>\n
The dynamics of the time-varying volatility of financial assets play a main<\/p>\n
role in diverse fields, such as derivative pricing and risk management. Consequently,<\/p>\n
the literature focused on estimating and forecasting conditional<\/p>\n
variance is vast. The most popular method for modelling volatility belongs<\/p>\n
to the family of GARCH models (see Bollerslev et al. 1992 for a review of<\/p>\n
this topic), although other alternatives (such as stochastic volatility models)<\/p>\n
also provide reliable estimates. The success of GARCH processes is<\/p>\n
unquestionably tied to the fact that they are able to fit the stylized features<\/p>\n
exhibited by volatility in a fairly parsimonious and convincing way, through<\/p>\n
quite a feasible method. The seminal models developed by Engle (1982)<\/p>\n
and Bollerslev (1986) were rapidly generalized in an increasing degree of<\/p>\n
sophistication to reflect further empirical aspects of volatility.<\/p>\n
One of the more complex features that univariate GARCH-type models<\/p>\n
have attempted to fit is the so-called long-memory property. The volatility<\/p>\n
of many financial assets exhibits a strong temporal dependence which is<\/p>\n
revealed through a slow decay to zero in the autocorrelation function of<\/p>\n
the standard proxies of volatility (usually squared and absolute valued<\/p>\n
returns) at long lags. The basic GARCH model does not succeed in<\/p>\n
fitting this pattern because it implicitly assumes a fast, geometric decay<\/p>\n
in the theoretical autocorrelations. Engle and Bollerslev (1986) were<\/p>\n
the first concerned with this fact and suggested an integrated GARCH<\/p>\n
model (IGARCH) by imposing unit roots in the conditional variance.<\/p>\n
The theoretical properties of IGARCH models, however, are not entirely<\/p>\n
satisfactory in fitting actual financial data, so further models were later<\/p>\n
developed to face temporal dependence. Ballie, Bollerslev and Mikkelsen<\/p>\n
(1996) proposed the so-called fractionally integrated GARCH models<\/p>\n
(FIGARCH) for volatility in the same spirit as fractional ARIMA models<\/p>\n
which were evolved for modelling the mean of time series (see Baillie, 1996).<\/p>\n
These models imply an hyperbolic rate of decay in the autocorrelation<\/p>\n
function of squared residuals, and generalize the basic framework by still<\/p>\n
using a parsimonious parameterization.<\/p>\n
There has been a great interest in modelling the temporal dependence<\/p>\n
in the volatility of financial series, mostly in the univariate framework1.<\/p>\n
The analysis of the long-memory property in the multivariate framework,<\/p>\n
however, has received much less attention, even though the estimation<\/p>\n
of time-varying covariances between asset returns is crucial for risk<\/p>\n
management, portfolio selection, optimal hedging and other important<\/p>\n
applications. The main reason is that modelling conditional variance in<\/p>\n
1An alternative approach for modelling long-memory through GARCH-type models is<\/p>\n
based on the family of stochastic volatility (see Breidt, Crato and de Lima, 1998). An<\/p>\n
extension of FIGARCH models has been considered in Ding, Granger and Engle (1993).<\/p>\n
2 The multivariate modelling of long-memory<\/p>\n
Although long-memory has been observed in the volatility of a wide range<\/p>\n
of assets, the literature on the topic is mainly focused on foreign exchange<\/p>\n
rate time series (FX hereafter). There exists a great deal of empirical<\/p>\n
literature focused on modelling and forecasting the volatility of exchangerate<\/p>\n
returns in terms of the FIGARCH models in the univariate framework.<\/p>\n
An exhaustive review of the literature is beyond the aim of this paper.<\/p>\n
Some recent empirical works on this issue can be found in Vilasuso (2002)<\/p>\n
and Beine et al. (2002). On the other hand, the literature dealing with the<\/p>\n
multivariate case is scarce.<\/p>\n
The modelling of long-memory in the multivariate framework was firstly<\/p>\n
studied by Teyssi\u00e8re (1997), who implemented several long memory volatility<\/p>\n
processes in a bivariate context, focusing on daily FX time series. He<\/p>\n
used an approach initially based on the multivariate constant conditional<\/p>\n
correlation model (Bollerslev, 1990), which allows for long-memory ARCH<\/p>\n
dynamics in the covariance equation. He also weakened the assumption<\/p>\n
of constant correlations and estimated time-varying patterns. Teyssi\u00e8re<\/p>\n
(1998) estimated several trivariate FIGARCH models on some intraday FX<\/p>\n
rate returns. This author finds a common degree of long-memory in the<\/p>\n
marginal variances, while the covariances do not share the same level of<\/p>\n
persistence with the conditional variances. More recently, Pafka and M\u00e1ty\u00e1s<\/p>\n
(2001) analyzed a multivariate diagonal FIGARCH model on three FX timeseries<\/p>\n
through quite a complex computational procedure. The multivariate<\/p>\n
modelling on other time series has focused on the crude oil returns (Brunetti<\/p>\n
and Gilbert, 2001). A bivariate constant correlation FIGARCH model is<\/p>\n
fitted on these data to test for fractional cointegration in the volatility<\/p>\n
of the NYMEX and IPE crude oil markets2. To our knowledge, there is<\/p>\n
no other literature concerned with modelling temporal dependences in the<\/p>\n
multivariate context.<\/p>\n
The previous research affords a valuable contribution to the better<\/p>\n
understanding of long-run dependences in multivariate volatility. A major<\/p>\n
shortcoming in applying these approaches in practice, however, lies in<\/p>\n
the overwhelming computational burden involved, which simply makes the<\/p>\n
straightforward extension of these methods to large portfolios unfeasible<\/p>\n
(note that only two or three assets are considered in the empirical<\/p>\n
applications of these methods). The procedure we shall discuss is specifically<\/p>\n
2.1 The orthogonal multivariate model<\/p>\n
We firstly introduce notation and terminology. Consider a portfolio of K<\/p>\n
financial assets and denote by rt = (r1t, r2t, …, rKt)????, t = 1, …,T, a weaklystationary<\/p>\n
random vector with each component representing the return of<\/p>\n
each portfolio asset at time t. Denote by Ft the set of relevant information<\/p>\n
up to time t, and define the conditional covariance matrix of the process<\/p>\n
by E(rtr????t|Ft\u22121) = Et\u22121 (rtr????t) = Ht. Denote as E(rtr????t) = \u03a9 the (finite)<\/p>\n
unconditional second order moment of the random vector. Note that only<\/p>\n
second-order stationarity is required, which is the basic assumption in the<\/p>\n
literature concerned with estimating covariance matrices of asset returns.<\/p>\n
Other procedures proposed for estimating the covariance matrix require<\/p>\n
much stronger assumptions (see, for instance, Ledoit and Wolf, 2003), as the<\/p>\n
existence of higher-order moments and even iid-ness in the driving series.<\/p>\n
As the covariance matrix \u03a9 is positive definite, it follows by the spectral<\/p>\n
decomposition that \u03a9 = P\u039bP????, where P is an orthonormal K\u00d7K matrix of<\/p>\n
eigenvectors, and \u039b is a diagonal matrix with the corresponding eigenvalues<\/p>\n
of \u03a9 in its diagonal. Lastly, assume that the columns of P are ordered by<\/p>\n
size of the eigenvalues of \u039b, so the first column is the one related to the<\/p>\n
highest eigenvalue, and so on.<\/p>\n
The orthogonal model by Alexander is based on applying the principal<\/p>\n
component analysis (PCA) to generate a set of uncorrelated factors from<\/p>\n
the original series3. The PCA analysis is a well-known method widely used<\/p>\n
in practice, and several investment consultants, such as Advanced Portfolio<\/p>\n
Technologies, use procedures based on principal components. The basic<\/p>\n
strategy in the Alexander model consists of linearly transforming the original<\/p>\n
data into a set of uncorrelated latent factors so-called principal components<\/p>\n
whose volatility can then be modelled in the univariate framework. With<\/p>\n
these estimations, the conditional matrix Ht is easily obtained by the inverse<\/p>\n
map of the linear transformation.<\/p>\n
The set of principal components, yt = (y1t, y2t, …, yKt)????, is simply<\/p>\n
defined through the linear application yt = P????rt. It follows easily that<\/p>\n
E(yt) = 0 and E(yty???? t ) = \u039b by the orthogonal property of P. The columns<\/p>\n
of the matrix P were previously ordered according to the corresponding<\/p>\n
eigenvalues size, so that ordered principal components have a decreasing<\/p>\n
ability to explain the total variability and the main sources of variability. <\/p>\n","protected":false},"excerpt":{"rendered":"
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