It is easily checked that these restrictions matrices exclude all the regressors
involving τ∕j-1 from Pr {j∕∣ t ∣ yι.t-∖, Xi,t-ι} , and all the regressors involving yj~1
from Pr{y'fi I yi,t-↑,Xi,t-↑}. As for the simultaneous independence condition
(13), it remains unchanged, since pjjj must be identically equal to zero for all
(t,t) in order to factor out the joint distribution (7) into the product of the
marginal (9) and (10), which requires that pjjj does not depend on covariates.
Relaxing the Markov assumption. Let us now relax the Markov assumption.
For the sake of simplicity, we go back to the assumption that the information set
available in t is Vt-ι∙ Consider first the case where the relevant information for
the transition probabilities is given by the last two states visited by an individ-
ual, rather than the last one only. There are therefore 16 possible paths followed
in (t — 2), (t — 1), at the end of which the individual may choose among 4 states.
Hence, the most general model one may use to describe Pr {t∕jji ∣ yt.t-1,yi,t-2}
requires 16 × (4 — 1) = 48 transition probabilities.7
This model may be written in the form (7) by replacing st-ι by
2 „
sj.J-l — Sjι⅛-1 ® Sj,J-2
i.e. using the saturated model with Xi,t-ι = Sij-2∙ To generalize to the case in
which the last £ states visited are relevant for the transition probabilities, then
Sj,j-χ = Sjjj-i 0 SjjJ-2 ® . . . ® Sij—£
has to be used in (7) instead of Sjjj-χ. It seems natural to refer to this model as
bivariate Vector AutoRegressive Probit model of order £, or VAP(£). We will
call VAPX (£) the model where exogenous covariates are also included. Notice
that the number of parameters does increase very rapidly, since B and 7 will be
of dimension 4i × 2 and 4i × 1. The dynamic structure of the process may be
simplified by using the unsaturated model rather than the saturated one, which
would dramatically reduce the number of parameters to 3 (3£ + 1), although the
interpretation of the ensuing model is unclear. A further simplification could
be based on the following underlying latent regression:
e
Vi,t = At 4^ y AjVi,t—3 + εit
3=i
where y*t = (yi*t,yj*t), Aj (j = 1,...£) are 2 × 2 parameter matrices, μ is a
2 × 1 parameter vector, and:
ε.t√⅛L≡√[0l [ɪ p
τ' ∖ε2it) Ц 0 J ’ [_ p 1
Notice that this model implies that
t
l≡{yi,i I Vi,t-ι,∙ ∙∙ ,Vi,t-t} = μ + YjAjyi,-j
j=i
7It is easily shown that this non-Markov 4 states model may be rewritten as a Markov
16 states model, where 192 out of the 162 transition probabilities are set to zero, while 16
transition probabilities may be written as linear functions of the remaining 48.
10