The Mathematics of Financial Modelingand Investment Management

18-MultiFactorModels Page 544 Wednesday, February 4, 2004 1:10 PM

544 The Mathematics of Financial Modeling and Investment Management

weights of cointegrating vectors. The PCA-based approach can also be

applied in the frequency domain. The analysis in the frequency domain

is an alternative way of analyzing time series. It is based on constructing

a transform of the time series which is the discrete equivalent of a Fou-

rier transform discussed in Chapter 4.^17

An alternative estimation methodology which is suitable for large

sets is the subspace-space algorithm introduced by Aoki (ref. cited) in

the context of stationary systems and extended by Bauer and Wagner

(ref. cited) to integrated systems and to polynomial cointegration.^18

Cointegration and Financial Time Series

Cointegration is an important technique for portfolio management: It

allows an investor to detect mispricings and thus sources of profit. In

fact, if a set of price processes exhibit cointegration, relative returns are

autocorrelated and therefore predictable. In other words, as we will see

in Chapter 19, although individual price processes might be unpredict-

able random walks, there are portfolios which exhibit a stationary,

mean-reverting behavior. For this reason cointegration has attracted the

attention of both academics and practitioners, especially in the areas of

index tracking and hedge fund management.

However, cointegration technology was initially developed in the

area of macroeconomics where only a small number of variables, gener-

ally less than 10, are used. Extending the concepts of cointegration to a

large number of equity prices or return processes is difficult both from

the numerical and theoretical standpoints. Assume, for example, that

one is working on a large set of equity log-price processes such as those

in the S&P 500. Standard cointegration estimation and testing methods

such as the Johansen procedures do not work for sets of processes of

this size.

Consider also that in finite samples of sets of processes such as those

found in the S&P 500, spurious cointegrating relationships will be

detected. This happens because in a large set of independent processes a

cointegration test run on a relatively small sample of points will ran-

domly test positive for many cointegrating relationship. For example,

one finds a significant number, in the range of a few percentage points,

of cointegrated pairs of processes in computer-generated independent

arithmetic random walks.

(^17) P.C.B. Phillips and S. Ouliaris, “Testing for Cointegration Using Principal Com -
ponents Methods,” Journal of Economic Dynamics and Control 12 (1988), pp. 205–
230.
(^18) The subspace algorithm is quite complex and technical. The interested reader
should consult the papers by Bauer and Wagner.