The Mathematics of Financial Modelingand Investment Management

18-MultiFactorModels Page 548 Wednesday, February 4, 2004 1:10 PM

548 The Mathematics of Financial Modeling and Investment Management

variate GARCH. Because multivariate GARCH becomes rapidly

unmanageable with the number of assets, simplified forms have been

proposed.

GARCH models are not necessarily stationary insofar as their sta-

tionarity depends on the coefficients of the ARMA process. If the

ARMA process is not stationary, then the process is called IGARCH.

While ARCH and GARCH models model volatility, asset pricing

models require that returns depend on volatility as higher volatility

commands a higher return. To capture the dependence of returns on vol-

atility, Engle, Lilien, and Robins^24 suggested adding an expected return

term to the GARCH equations. Equations then become

rt = μt + σtεt

2

μt = γ 0 + γ 1 σt

m q

σ^22

t = ∑αiσti– + ∑βirtj–

i = 1 j = 1

This model is called M-ARCH or ARCH in mean. Recall that M-ARCH

is also a way to represent the conditional CAPM.

While ARCH and GARCH models are based on empirical findings

of volatility clustering, Markov-switching models are based on a gener-

alization of the idea that a model’s parameters cannot be considered sta-

ble for long periods of time. If our objective is to retain linear models as

the basic DGP, then we have to accept that parameters will change in

time. Markov switching models use a Markov chain to drive the param-

eters of a basic linear model. The Hamilton model, for example, uses a

Markov chain to drive the parameters of a random walk. In a more gen-

eral Markov-switching VAR, a Markov chain drives the parameters of a

VAR model. Continuous-state autoregressive models might replace

Markov chains, thus originating multiplicative state-space models.

ARCH and GARCH models follow this modeling strategy.

If the objective is to model a large collection of price processes, for

example the price processes in some broad index, then dimensionality

reduction techniques must be applied. Envisage an outer driver, be it a

Markov chain or an autoregressive model, that drives the parameters of

(^24) R. Engle, D. Lilien, and R. Robins, “Estimating Time-Varying Risk Premia in the
Term Structure: the ARCH-M Model,” Econometrica 55 (1987), pp. 391–407.