Page 266 - Contributed Paper Session (CPS) - Volume 4

CPS2220 David Degras et al.
(Alternatively, the ̂ can be set to zero and the can be made diagonal
̂

for numerical stability.)
4. Divide the time range {1, . . . , } into subintervals = { + 1, … , +1 }

where = ⌊/⌋ for 0 ≤ ≤ and ⌊⋅⌋ denotes the integer part. Fit a

VAR(p) model to each subseries ( ) by ordinary least squares (OLS)
∈
̂ ()
̂
and call () and the corresponding estimates of the transition
matrices and innovation covariances.
̂
5. Partition the ( () ̂ () ), 1 ≤ ≤ , into clusters with the K-means
,
algorithm. Take the initial estimates ( , ), 1 ≤ ≤ , as the cluster
̂ ̂

centers. Alternatively, refit the VAR() model by OLS to each of the
subseries associated with the clustering and take the resulting ( , ) as
̂ ̂

the initial estimates.
̂
̂
̂
6. Let be the estimated regime at time : = if and ( () ,

̂
̂ ()
) belongs to cluster . Set ̂ = 1 if = , ̂ = 0.01 otherwise, and

̂
rescale so that ∑ ̂ = 1. Set ̂ = #{: ̂ −1 = , = }/#{: ̂ −1 = }

=1

for 1 ≤ , ≤ . For each , replace any ̂ less than 0.01
(̂ , … , ̂ ) by this value and rescale so that ∑ ̂ = 1.

=1
1

Initialization for the switching dynamics model (method 2)
Method 2 is identical to Method 1 in all aspects except for step 4 which
uses a more sophisticated segmentation algorithm for the time series (x̂ ),

namely binary segmentation. Initially, a VAR() model is fitted to (x̂ ) over its

entire range {1, . . . , }, yielding a sum of squared errors SSE(1, ). For each
time point 1 ≤ ≤ , one fits a VAR() over each of the subintervals {1, . . . , }
and { + 1, . . . , }, yielding a total sum of squares SSE(1, ) + SSE( + 1, ).
One then selects the time = argmin 1≤≤ {SSE(1, ) + SSE( + 1, )} as a
candidate change point. If the reduction in SSE is sufficient, say, (SSE(1, ) +
SSE( + 1, )) ≤ (1 − )SSE(1, ) for some small > 0, is accepted as a
change point and the initial time range {1, . . . , } is split in two subintervals
{1, . . . , } and { + 1, . . . , } . The process is then iterated for each new
subinterval and so on so forth until no new change points are found. The
resulting change points are denoted by < ⋯ < −1 with = 0 and =
0
1

as before. In practice, the tolerance can be selected by trial and error until a
reasonable number κ of segments has been obtained. One may also impose a
minimal distance between successive change points for faster computations
and better interpretability of results.
The initialization method for the switching observations model builds on
the above initializations. For reasons of space, we do not present it in this
paper.

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