The Mathematics of Financial Modelingand Investment Management

18-MultiFactorModels Page 546 Wednesday, February 4, 2004 1:10 PM

546 The Mathematics of Financial Modeling and Investment Management

Special cointegration models have been described in the literature.

In particular, the well-documented lead-lag effect described by Andrew

Lo and Craig MacKinlay^22 leads to a cointegration model. The lead-lag

effect is the strong correlation which exists between the returns at time t

of portfolios of small firms, the laggards, and the return at time t – 1 of

portfolios of large firms, the leaders. This effect can be tested either as

direct correlation between returns or as autocorrelation of portfolios

that include firms of different sizes.

In the original formulation of Lo and MacKinlay, the model is sim-

ple as there is only one exogenous factor. Consider the returns of the

two portfolios of large firms and small firms; the Lo and MacKinlay

model is written as follows, with the return of small firms a regression

on the lagged factor:

rLt = μL + β 1 Lft + εLt

rSt = μS + β 1 Sft +β 2 Sft – 1 + εSt

Angelos Kanas and Georgios Kouretas^23 have cast the lead-lag effect

of size-sorted portfolios into a cointegration framework using state-space

modeling in the form of ARDL models for prices. Summing up returns to

get prices and solving, they arrive at the following ARDL equation:

pSt = abt + + βpLt – 1 + et

where e is an autocorrelated process that includes the single common

factor.

In summary, cointegration and/or state-space modeling are powerful

modeling techniques whose applicability to real price processes has been

empirically tested. However, the practical implementation of state-space

models of large portfolios presents significant challenges given that

cointegration is largely unstable.

NONLINEAR DYNAMIC MODELS FOR PRICES AND RETURNS

While the models for portfolio management discussed above are linear

models, the linearity of equity price processes has been challenged by

(^22) Andrew Lo and Craig MacKinlay, “When Are Contrarian Profits Due to Stock
Market Overreaction?” Review of Financial Studies 3 (1990), pp. 175–206.
(^23) Angelos Kanas and Georgios Kouretas, “A Cointegration Approach to the Lead-
Lag Effect Among Size-Sorted Equity Portfolios,” Working Paper, 2001