The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 340 Wednesday, February 4, 2004 12:59 PM

340 The Mathematics of Financial Modeling and Investment Management

n if it can be transformed into a stationary series differencing n times. In

particular, a univariate time series X is integrated of order 1 if it can be

represented as follows:

Xt + 1 =ρXt ++b εt

ρ = 1

εt stationary possibly autocorrelated

The key feature of an integrated time series is that random innova-

tions never decay. Most economic variables are integrated variables. In

particular, testing for integration in log price processes one finds that

the null of integration cannot be rejected in most cases. For instance,

testing the log price processes in the S&P 500 using a standard test such

as the ADF test, the null of integration cannot be rejected in about 90%

of time series as shown in Exhibit 12.3. Nor can the null hypothesis of

integration be rejected for economic time series such as the monetary

mass (M3) or the Gross Disposable Product.

Suppose that a set of time series integrated of order 1 is given.

Though each series is integrated of order 1, for instance they are arith-

metic random walks, there might be linear combinations of the series

which are stationary. If this happens, the series are said to be cointe-

grated. The financial meaning of cointegration is the following. Indi-

vidual log price processes can be arithmetic random walks but there are

portfolios, in general long-short portfolios, which are stationary, and

thus mean reverting around a constant mean. In other words, individ-

ual securities might be totally unpredictable random walks but portfo-

lios might be more predictable. We will come back to the question of

the empirical findings of cointegration in real-world economic time

series and price processes. First, we need to define cointegration mathe-

matically.

EXHIBIT 12.3 Integratedness of the S&P 500

Number Type

Period of Series of Test Integratedness Percentage

From Jan. 1, 487 series Augmented Dickey- 422 series I(1) 87%

2001 to in the Fuller test with two 65 series I(0) integrated

Dec. 31, 2003 S&P 500 lags, 95% confi-

dence level.