The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 327 Wednesday, February 4, 2004 12:59 PM

Financial Econometrics: Model Selection, Estimation, and Testing 327

introduced the notion of efficient markets which makes it reasonable to

assume that log price processes evolve as random walks.

The question of the empirical adequacy of the random walk model

is very important from the practical point of view. Whatever notion or

tools for financial optimization one adopts, a stock price model is a

basic ingredient. Therefore substantial efforts have been devoted to

proving or disproving the random walk hypothesis.^9

There are many statistical tests aimed at testing the random walk

hypothesis. A typical test takes the random walk as a null hypothesis.

The number of runs (that is, consecutive sequences of positive or nega-

tive returns) and the linear growth of the variance are parameters used

in classical random walk tests. More recent tests are based on the work

of Aldous and Diaconis^10 on the distribution of sequences of positive

and negative returns.

There is no definite response. Typical tests fail to reject the null

hypothesis of random walk behavior with a high level of confidence on

a large percentage of equity price processes. This does not mean that the

random walk hypothesis is confirmed, but only that it is a reasonable

first approximation. As we will see in the following sections, other mod-

els have been proposed.

CORRELATION

Before moving on to more sophisticated models, let’s consider random

walk models of portfolios of equities as opposed to single price pro-

cesses. Let’s therefore consider a multivariate random walk model of an

equity portfolio assuming that each log price process evolves as an

arithmetic random walk. We will consider a set of n time series pi,t, i =

1, ..., n that represent log price processes. Suppose that each time series

is a random walk written as

pit, =pit , – 1 ++ μi εit,

A multivariate random walk can be represented in vector form as fol-

lows:

(^9) See John Y. Campbell, Andrew W. Lo, and A. Craig MacKinley, The Econometrics
of Financial Markets (Princeton, NJ: Princeton University Press, 1997).
(^10) David Aldous and Persi Diaconis, “Shuffling Cards and Stopping Times,” Ameri-
can Mathematical Monthly 8 (1986), pp. 333–348.