# 4.10 Regime-Switching Processes

A stochastic process is said to be **regime-switching** if its behavior is determined by different models—different regimes—during different periods. We are interested in processes that switch randomly between regimes.

Consider an *n*-dimensional white noise ** W** constructed with

*m*joint-normal regimes. Let stochastic process

**indicate the regime in force at any time**

*Z**t*, so terms

*are random variables taking on values in the set {1, 2, … ,*

^{t}Z*m*}.

**has constant probabilities of switching from one regime to another at ay time**

*Z**t*. These transition probabilities are indicated with a matrix:

[4.86]

Component *p _{i}*

_{,j}indicates the probability of transitioning from regime

*j*in force at any time

*t*– 1 to regime

*i*at time

*t*. Each column sums to 1. At any time

*t*– 1, the regime that will be in force at time

*t*is uncertain, so

*is conditionally mixed joint-normal,*

^{t}**W**[4.87]

where

[4.88]

Exhibit 4.18 indicates a realization of the univariate three-regime process

[4.89]

[4.90]

[4.91]

[4.92]

See Goldfeld and Quandt (1973) and Hamilton (1993) for estimation techniques.

###### Properties

Regime-switching models offer an alternative to GARCH processes for modeling conditional heteroskedasticity. In the multivariate case, covariance matrices directly specify any relationship between correlations and standard deviations. As dimensionality *n* increases, the number of parameters that must be specified for a completely general model becomes unmanageable. Research is ongoing.