1.3 Examples
1.3.6 Switching Processes with Markov Regime
We now consider several examples that are not HMMs but belong to the class of Markov-switching models already mentioned in Section 1.2. Perhaps the most famous example of Markov-switching processes is the switching autore- gressive process that was introduced by Hamilton (1989) to model econometric data.
1.3.6.1 Switching Linear Models
A switching linear autoregression is a model of the form
Yk = µ(Ck) + d
X
i=1
ai(Ck)(Yk−i− µ(Ck−i) + σ(Ck)Vk, k ≥ 1 , (1.23)
where {Ck}k≥0, called the regime, is a Markov chain on a finite state space
C = {1, 2, . . . , r}, and {Vk}k≥0 is white noise independent of the regime; the
functions µ : C → R, ai : C → R, i = 1, . . . , r, and σ : C → R describe
the dependence of the parameters on the realized regime. In this model, we change only the scale of the innovation as a function of the regime, but we can of course more drastically change the innovation distribution conditional on each state.
30 1 Introduction
Remark 1.3.14. A model closely related to (1.23) is
Yk = µ(Ck) + d
X
i=1
ai(Ck)Yk−i+ σ(Ck)Vk, k ≥ 1 . (1.24)
In (1.23), µ(Ck) is the mean of Yk conditional on the sequence of states
C1, . . . , Ck, whereas in (1.24) the shift is on the intercept of the autoregressive
process.
A model like this is not an HMM because, given {Ck}, the Yk are not
conditionally independent but rather form a non-homogeneous autoregression. Hence it is a Markov-switching model. Obviously, the conditional distribution of Yk does not only depend on Ck and Yk−1 but also on other lagged Cs and
Y s back to Ck−d and Yk−d. By vectorizing the Y s and Cs, that is, stacking
them in groups of d elements, we can obtain a process whose conditional distribution depends on one lagged variable only, as in Figure 1.2.
This model can be rewritten in state-space form. Let Yk = [Yk, Yk−1, . . . , Yk−d+1]t,
Ck = [Ck, Ck−1, . . . , Ck−d+1]t,
µ(Ck) = [µ(Ck), . . . , µ(Ck−d+1)]t,
Vk = [Vk, 0, . . . , 0]t,
and denote by C(c) the d × d companion matrix associated with the autore- gressive coefficients of the state c,
A(c) = a1(c) a2(c) . . . ad(c) 1 0 0 0 1 0 ... .. . . .. . .. . .. ... 0 . . . 0 1 0 . (1.25)
The stacked observation vector Yk then satisfies
Yk= µ(Ck) + A(Ck) (Yk−1− µ(Ck−1)) + σ(Ck)Vk. (1.26)
Interestingly enough, switching autoregressive processes have a rather rich probabilistic structure and have proven to be useful in many different contexts. We focus here on applications in econometrics and finance, but the scope of potential applications of these models span many different areas.
Example 1.3.15 (Regime Switches in Econometrics). The Hamilton (1989) model for the U.S. business cycle fostered a great deal of interest in Markov-switching autoregressive models as an empirical vehicle for character- izing macro-economic fluctuations. This model provides a formal statistical
1.3 Examples 31
representation of the old idea that expansion and contraction constitute two distinct economic phases: Hamilton’s model assumes that a macro-economic aggregate (real output growth, country’s gross national product measured per quarter, annum, etc.) follows one of two different autoregressions depending on whether the economy is expanding or contracting, with the shift between regimes governed by the outcome of an unobserved Markov chain. The simple business cycle model advocated by Hamilton takes the form
Yk = µ(Ck) + d
X
i=1
ai(Yk−i− µ(Ck−i)) + σVk , (1.27)
where {Vk}k≥0 is white Gaussian noise with zero mean and unit variance,
and {Ck}k≥0 is the unobserved latent variable that reflects the state of the
business cycle (the autoregressive coefficients do not change; only the mean of the process is effectively modulated). In the simplest model, {Ck} takes
only two values; for example, Ck = 0 could indicate that the economy is in
recession and Ck= 1 that it is in expansion. When Ck= 0, the average growth
rate is given by µ(0), whereas when Ck = 1 the average growth rate is µ(1).
This simple model can be made more sophisticated by making the variance a function of the state Ck as well,
Yk = µ(Ck) + d
X
i=1
ai(Yk−i− µ(Ck−i)) + σ(Ck)Vk .
The Markov assumption on the hidden states basically says that if the econ- omy was, say, in expansion the last period, the probability of going into re- cession is a fixed constant that does not depend on how long the economy has been in expansion or other measures of the strength of the expansion. This assumption, though rather naive, does not appear to be a bad representation of historical experience, though several researchers have suggested that more complicated specifications of the transition matrix ought to be considered.
Further reading on applications of switching linear Gaussian autoregres- sions in economics and finance can be found in, for instance, Krolzig (1997),
Kim and Nelson (1999), Raj (2002), and Hamilton and Raj (2003).
It is possible to include an additional degree of sophistication by consid- ering instead of a linear autoregression, linear state-space models (see for in- stance Tugnait, 1984; West and Harrison, 1989; Kim and Nelson, 1999; Doucet et al., 2000a; Chen and Liu, 2000):
Wk+1= µW(Ck+1) + A(Ck+1)Wk+ R(Ck+1)Uk,
Yk = µY(Ck) + B(Ck)Wk+ S(Ck)Vk, (1.28)
where {Ck}k≥0 is a Markov chain on a discrete state space, {Uk}k≥0 and
32 1 Introduction
and µW, µY, A, B, R, and S are vector- and matrix-valued functions of ap-
propriate dimensions. Each state of the underlying Markov chain is then asso- ciated with a particular regime of the dynamic system, specified by particular values of (µW, µY, A, B, R, S) governing the behavior the state and observa-
tions. Switching linear state-space models approximate complex non-linear dynamics with a dynamic mixture of linear processes. This type of model has found a broad range of applications in econometrics (Kim and Nelson, 1999), in engineering including control (hybrid system, target tracking), sig- nal processing (blind channel equalization) and communications (interference suppression) (Doucet et al., 2000b, 2001b).
Example 1.3.16 (Maneuvering Target). Recall that in Example 1.3.12, we considered the motion of a single target that evolves in 2-D space with (almost) constant velocity. To represent changes in the velocity (either speed or direction or both), we redefine the model that describes the evolution of the state Wk= (Px,k, ˙Px,k, Py,k, ˙Py,k) by making it conditional upon a maneuver
indicator Ck = ck ∈ {1, . . . , r} that is assumed to take only a finite number
of values corresponding to various predefined maneuver scenarios. The state now evolves according to the following conditionally Gaussian linear equation
Wk= A(Ck+1)Wk+ R(Ck+1)Uk , Uk∼ N(0, I) ,
where A(c) and R(c) describe the parameters of the dynamic system char- acterizing the motion of the target for the maneuver labeled by c. Assuming that the observations are linear, Yk = BWk+ Vk, the system is a switching
Gaussian linear state-space model.
1.3.6.2 Switching Non-linear Models
Switching autoregressive processes with Markov regime can be generalized by allowing non-linear autoregressions. Such models were considered in particular by Francq and Roussignol (1997) and take the form
Yk = φ(Yk−1, . . . , Yk−d, Xk) + σ(Yk−1, . . . , Yk−d, Xk)Vk, (1.29)
where {Xk}k≥0, called the regime, is a Markov chain on a discrete state space
X, {Vk} is an i.i.d. sequence, independent of the regime, with zero mean and
unit variance, and φ : Rd×X → R and σ : Rd×X → R+are (measurable) func-
tions. Of particular interest are the switching ARCH models (Francq et al., 2001),
Yk =
q
ζ0(Xk) + ξ1(Xk)Yt−12 + ξd(Xk)Yt−d2 Vk .
Krishnamurthy and Ryd´en (1998) studied an even more general class of switching autoregressive processes that do not necessarily admit an additive decomposition; these are characterized by