Page 262 - Contributed Paper Session (CPS) - Volume 4

CPS2220 David Degras et al.
the EM algorithm for switching SSMs and outlines our numerical contributions.
Section 4 presents a new parametric bootstrap of the MLE for statistical
inference.

2. Linear state-space models with regime switching
Linear state-space models with regime switching can be viewed as a
combination of linear dynamical systems and hidden Markov models:
= + (1)

= +

−1

where at time 1 ≤ ≤ , is the measurement vector of size , is the

hidden state vector of size ( ≤ ), represents measurement errors, and

denotes random innovations to the state process. The first and second

equations are called observation equation and state equation, respectively.
The switching variable indicates the regime under which the system (1)

operates at time . The sequence ( ) is a homogeneous Markov chain
1≤≤
taking values in a finite set of regimes, say {1, … , } , with initial state
probabilities = ( = ) and transition probabilities
1

= ( = | −1 = ) for 1 ≤ , ≤ . Under regime = , is the

transition matrix that governs the dynamics of the state vector and is the

observation matrix that maps the hidden state vector to the observed

measurements . Conditionally on the regimes ( ) , the measurement

1≤≤
errors are independent over time and have normal distribution (0. ).

Similarly, the innovations are independent over time, mutually independent

with the and have normal distribution (0. ) conditionally on ( ) .
1≤≤

Hence at time , if = , the parameters at play in model (1) are ( , , , ).

Note that regular (non-switching) linear SSMs correspond to the case = 1.

Model (1) is very general and must be specialized for practical purposes. A first
specification is
= +

(2)
= ∑ ℓ , −ℓ +

ℓ=1
This model, which we call the switching dynamics model as in [10], posits a
common observation matrix and error covariance for all regimes 1, … , .
In other words the observation equation does not depend on the regime .

On the other hand, the dynamics of the state equation switch with . At time

, conditional on , is a vector autoregressive (VAR) process of order with

transition matrices , and innovation covariance (1 ≤ ℓ ≤ denotes the
ℓ ,
lag). The dependencies between observations, state vectors, and regimes
under this model are depicted in Figure 1 (left panel).
Another possible specification of model (1) is:
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