Page 346 - Special Topic Session (STS) - Volume 3

STS550 Pierre Guérin et al.
= ∅ ( ) + ∅ ( ) + , = 1, … , , (5)

,

2
where ~ (0, ( )), and keep the (variable specific) estimates of

̂
̂
∅ ( ), denoted by ∅ ( ), where ∅ ( ) is a 1 × vector, for =

1, … . All regime-switching regressions are estimated via (pseudo) maximum
likelihood, which is why we have made a normality assumption for that is

instead not required when estimating by OLS the linear version of the 3PRF.
Note also that the Markov chains in (5) are the same as in (3). We also define
̂
̂
for later use in the second step of the algorithm the variables ∅ , and ∅ , .
̂
The variables ∅ , are a weighted average of the estimated regime-specific
factor loadings:

̂
̂
∅ , = ∑ ∅ ( = )( = |Ω ), (6)

=1
where ( = |Ω ) is the smoothed probability of being in regime j

̂
given the full sample information Ω . the variables ∅ , are instead
defined as selected factor loadings:

̂
̂ , = ∑ ∅ ( = )( = |Ω ), (7)
∅

=1
where (. ) is an indicator function that selects the regime with the
highest smoothed probability, ( = |Ω ), at time t.

̂
̂
• Step 2: Cross-section regressions of the on either ∅ , or ∅ , .

Hence, we run linear regressions
+ , = 1, … , , (8)
̂
= 0, + ∅ ,

~ (0, ), with t = 1,..., , = or = , and we keep (for
2

̂
each ) the OLS estimates , where is a × 1 vector.
̂

• Step 3: Time-series regression of yt on t-1. hence, we run one
̂
Markov-switching regression:
̂
= ( ) + ( ) −1 + , = 1,. . . , , (9)

0

~(0, ( )), and we keep the maximum likelihood estimates
2

̂
̂
( ) and ( ). We calculate the forecast ̂ +1| as:

0

) ̂ ( +1 = ) + ( +1 = |Ω ) ̂ ( +1 = ) ̂ ), (10)
̂ +1| = ∑(( +1 = |Ω 0
=1
where ( +1 = |Ω ) is the predicted probability of being in regime

in period + 1 given the information available up to time , Ω .

3. Empirical Application
In this forecasting exercise, we construct factors from a cross-section of
nominal bilateral U.S. dollar (USD) exchange rates against a panel of 26
335 | I S I W S C 2 0 1 9