18-MultiFactorModels Page 545 Wednesday, February 4, 2004 1:10 PM
Multifactor Models and Common Trends for Common Stocks 545When testing for cointegration on a large set, one has therefore to take
an ensemble view. In analyzing macroeconomic series, the question is
whether they are cointegrated or not; in analyzing a large number of finan-
cial time series, the problem is not if there are cointegrating relationships
but if the number of cointegrating relationships found in the sample is high
enough to warrant the belief that the system has a cointegration structure.
Another important issue—strictly related to the above—is the struc-
ture of cointegration. Cointegration can be found within highly cointe-
grated market segments (i.e., subsets of processes) that exhibit a high
number of cointegrating relationships. Alternatively, cointegration can be
found between market segments—perhaps on a different time scale. This
cointegration structure will be reflected in the structure of common trends.
These issues are presently inadequately addressed in the literature,
although much proprietary empirical and analytical work has been done
by some asset management firms. Studies of cointegration in financial
processes has been performed at the level of indexes or broad aggre-
gates. Evidence of cointegration have been found between stock indexes
in different countries and between different indexes in the same country.
One of the most quoted studies on cointegration in equity prices is the
1992 study by Kenneth Kasa.^19 who found evidence of cointegration
between stock indexes in five different countries. Using models with
from 1 to 14 lags, Kasa found that the number of lags plays an impor-
tant role: Cointegration is revealed more clearly with many lags. In a
critical review of this and other studies on cointegration on various
assets, Godbout and van Norden^20 concluded that the size of the sample
might be responsible for significant distortions.
Carol Alexander^21 and coworkers at the ISMA Center in Reading,
United Kingdom, found cointegration within small-size high-capitaliza-
tion liquid indexes such as the Dow Jones Industrial Average (DJIA).
Their empirical findings corroborate the intuition that equity prices are
in some way mean-reverting around one or more common stochastic
trends. Alexander has developed trading strategies used for both index
tracking and long-short equity portfolios based on replicating the first
common factor of the market.(^19) Kenneth Kasa, “Common Stochastic Trends in International Stock Markets,”
Journal of Monetary Economics 29 (1992), pp. 95–124.
(^20) Marie-Josee Gobbout and Simon van Norden, “Reconsidering Cointegration in
International Finance: Three Case Studies of Size Distortion in Finite Samples,”
Working Paper 97-1, Bank of Canada, 1997.
(^21) Carol Alexander and Anca Dimitriu, “The Cointegration Alpha: Enhanced Index
Tracking and Long - Short Equity Market Neutral Strategies,” Working Paper, April
2002.