7.4.4 Splicing Methods for Obtaining Large Covariance Matrices

This subsection describes the use of PCA to generate large-dimensional covariance matrices. It has many advantages over the RiskMetrics approach, for example:

> Very large covariance matrices that are generated using this method will be very much more robust than the covariance matrices obtained by applying EWMA to each returns series separately.

> There is no need to reduce dimensions by using linear interpolation along the yield curve.

>• The method will conform to regulators requirements on historic data if at least one year of data is used to compute the principal components and their factor weights.

>• Positive defmiteness can be assured without having to use the same EWMA smoothing constant for all markets.

The splicing method amounts to a two-stage orthogonal model (Alexander, 1997). First all risk factors, such as equity market indices, exchange rates, commodities, government bond and money market rates, must be grouped into reasonably highly correlated categories. These categories will normally reflect geographic locations and instrument types. Principal component analysis is then used to extract the key risk factors from each category and the orthogonal GARCH model or the orthogonal EWMA model is applied to generate the covariance matrix for each category. Then, in the second stage, the factor weights from the principal component analysis are used to splice together a large covariance matrix for the original system.

The method is explained for just two categories; the generalization to any number of categories is straightforward. Let the first category be European equity indices, and let the second be European exchange rates. Suppose there are n variables in the first and m variables in the second. It is not the dimensions that matter. What does matter is that each category of risk factors is suitably co-dependent, so that it justifies the categorization as a separate and coherent category.

Stage 1: Find the principal components of each category, P = (Pu . . ., Pr), and separately Q = (2 • • •> Qs) where r and s are number of principal components that are used in the representation of each category. Generally r will be much less than n and s will be much less than m. Denote by A (n x r) and (m x s) the normalized factor weights matrices obtained in the PCA of the European equity and exchange rate categories, respectively. Then the within-factor covariances, that is, the covariance matrix for the equity category and for the exchange rate category separately, are given by AD, A and BD2B, respectively. Here Di and D2 are the diagonal matrices of the univariate GARCH or EWMA variances of the principal components of each system.

-12 \-1-1-1-1-1-1-1-1-1-1-1-1-1

Jan-93 Apr-93 Aug-93 Dec-93 Mar-94 Jul-94 Nov-94 Feb-95 Jun-95 Oct-95 Jan-96 May-96 Sep-96 Dec-96

(b) CAC volatility

-BEKK - --Vech O-GARCH

Jan-93 Apr-93 Aug-93 Dec-93 Mar-94 Jul-94 Nov-94 Feb-95 Jun-95 Oct-95 Jan-96 May-96 Sep-96 Dec-96 (c) DAX volatility

-BEKK

-Vech ........0-GARCH

Jan-93 Apr-93 Aug-93 Dec-93 Mar-94 Jul-94 Nov-94 Feb-95 Jun-95 Oct-95 Jan-96 May-96 Sep-96 Dec-96 (d) FTSE 100 volatility

-BEKK

-Vech ........0-GARCH