The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 348 Wednesday, February 4, 2004 12:59 PM

348 The Mathematics of Financial Modeling and Investment Management

The Hamilton model can be extended to an arbitrary number of

states and to more general VAR models. In a Markov switching context,

a VAR model

xt = [A 1 ()stL + A 2 ()st L^2 + ...+ An()stLN]xt + m () st + εεεεt

has parameters that depend on a set of hidden states that are governed

by a discrete-state, discrete-time Markov chain with transition probabil-

ity matrix:

pij, = Pr(st + 1 = ist = i)

M

∑pij, =^1

j = 1

Estimation of Markov switching VAR models can be done within a

general maximum likelihood framework. The estimation procedure is

rather complex as approximate iteration techniques are used. Hamil-

ton^35 made use of the Expectation Maximization (EM) algorithm which

had been proposed earlier in a broader context.^36 Other numerical tech-

niques are available and are now implemented in commercial software

packages.

Markov switching VAR models have been applied to macroeco-

nomic problems, in particular to the explanation of business cycles.

Applications to the modeling of large portfolios present significant

problems of estimation given the large number of data necessary.

Markov switching models are, in fact, typically estimated over long

periods of time, say 20 or 30 years. If one wants to construct coherent

data sets for broad aggregates such as the S&P 500, one rapidly runs

into problems as many firms, over periods of that length, undergo signif-

icant change such as mergers and acquisitions or stock splits. As one can-

not simply exclude these firms as doing so would introduce biases in the

estimation process, ad hoc adjustment procedures are needed to handle

change. Despite these difficulties, however, Markov switching models

can be considered a promising technique for financial econometrics.

(^35) J.D. Hamilton, “Analysis of Time Series Subject to Changes in Regime,” Journal
of Econometrics 45 (1990), pp. 39–70.
(^36) A.P. Dempster, N.M. Laird, and D.B. Rubin, “Maximum Likelihood Estimation
From Incomplete Data Via the EM Algorithm,” Journal of the Royal Statistical So-
ciety 39 (1977), Series B, 1–38.