18-MultiFactorModels Page 549 Wednesday, February 4, 2004 1:10 PM
Multifactor Models and Common Trends for Common Stocks 549a state-space model. One thereby creates a dynamic model of the factors
that drive a regressive model. As of this writing, however, the statistical
properties of these models have not been thoroughly investigated.SUMMARY
■ Multifactor models are linear regressions over a number of variables
called factors.
■ Factors can be exogenous variables or abstract variables formed by
portfolios.
■ The Arbitrage Pricing Theory (APT) asserts that each asset’s return is
equal to the risk-free rate plus a linear combination of factors.
■ The APT can be tested with maximum likelihood methods.
■ Exogenous factors can be determined with fundamental analysis.
■ Abstract factors can be determined with factor analysis or principal
component analysis.
■ Principal component analysis identifies the largest eigenvalues of the
variance-covariance matrix or the correlation matrix.
■ The largest eigenvalues correspond to eigenvectors that identify the
entire market and sectors that correspond to industry classification.
■ Multifactor models allow the decomposition of risk into systematic
risk and residual risk.
■ The most general formulation of the portfolio selection problem is util-
ity maximization in a multiperiod setting.
■ In a multiperiod setting, agents make a decision between consumption
and investment at each date; the Consumption CAPM is obtained by
aggregating all agents in a single representative agent and imposing
consumption optimality conditions.
■ Factor models can be extended in a dynamic environment as state-
space models.
■ Error correction models and state-space models are equivalent.
■ Through cointegration and state space-models it is possible to repre-
sent large portfolios through dynamic factor models.
■ There is empirical evidence of cointegration in stock prices.
■ Nonlinear models of stock prices have been proposed, ARCH/GARCH
and Markov switching models being two examples.