The Mathematics of Financial Modelingand Investment Management

18-MultiFactorModels Page 538 Wednesday, February 4, 2004 1:10 PM

538 The Mathematics of Financial Modeling and Investment Management

Dynamic market models of this type can be used to create meaning-

ful scenarios for multistage stochastic optimization. The VAR part of

the model might describe the evolution of macroeconomic variables. If

the objective is to apply multistage optimization and stay within the

domain of linear models of returns, state-space models are the models of

choice. As we have seen in Chapter 11, any stationary or asymptotically

stationary linear model can be represented in this form.

Estimation of State-Space Models

Methods for the estimation of state-space models were originally devel-

oped for engineering applications. State-space systems can be estimated

using MLE methods.^5 In 1990 Masanao Aoki^6 introduced a methodol-

ogy called the subspace algorithm to estimate state-space models; Diet-

mar Bauer and Martin Wagner^7 subsequently showed how to apply

subspace algorithms to cointegrated systems.

It was R. E. Kalman^8 who, in 1960, introduced a recursive methodol-

ogy for making forecasts based on state-space models. Known as the Kal-

man filter, the methodology proved very successful in engineering before

being applied more recently in economics and finance. Given a state-space

model, a Kalman filter computes recursively the best estimate of state:

tt ˆz = Ez[ tr^0 , ..., rt]

Kalman filters are now implemented in many software packages.

DYNAMIC MODELS FOR PRICES

The models discussed above are single factor or multifactor linear mod-

els of returns; the risk-return trade-off entailed by these models leads to

price processes that diverge exponentially. To see this point, consider

that, given log prices, returns are approximately differences of log-

prices. Therefore, log-prices are obtained by adding returns (i.e., they

are a random-walk) and the real prices are then obtained taking the

(^5) See, for instance, Helmut Luetkepohl, Introduction to Multiple Time Series Analy -
sis (New York: Springer, 1991).
(^6) Masanao Aoki, State Space Modelling of Time Series (New York: Springer, 1990).
(^7) D. Bauer and M. Wagner, “Estimating Cointegrated Systems Using Subspace Algo -
rithms,” Journal of Econometrics 11 (2002), pp. 47–84.
(^8) R.E. Kalman, “A New Approach to Linear Filtering and Prediction Problems,”
Transactions of the ASME-Journal of Basic Engineering (March 1960), pp. 35–45.