where x*t is the part of x∙iιt which is linearly independent on si>t (typically,
if the constant is in both xiιt and Sjι⅛ it has to be dropped from xiιt to avoid
perfect collinearity). If we denote by к the dimension of x∙iιt, and by k* the
dimension of x*t, then B and 7 will be now of dimension (4 + fc*) × 2 and
(4 + ⅛*) × 1. It is important to point out that this way to include the covariates
amounts to assuming that the impact on the transition probabilities is the same
irrespective of Sj,t-ι, so that the effect of the covariates is the same whatever
state the individual belongs to in (t — 1). A more general model, allowing for
interaction among the covariates and the state of the process in (t — 1), i.e.
s∙iιt-ι, ensues from using in (7), instead of Sj,t-ι,
Zi1t — 1 — Si,t-1 ® %i1t-l∙ (ɪθ)
Notice that, in this case, B and 7 will be of dimension 4⅛ × 2 and 4fc × 1, so
that many more parameters have to be estimated. We will refer to the model
deriving from (16) as saturated model, while the model deriving from (15) will
be referred to as unsaturated. Notice that the unsaturated model is nested in
the saturated one, and therefore the decision about which one is convenient for
describing the data may be empirically based on testing. A simple example
may help understanding the difference between the two models. Assume that
each individual i belongs, at any time t to either one or the other two mutually
exclusive and exhaustive classes C↑ ad C2. Define
Γ)l -i
ιyi,t — 1IindividuaUECiattimet)
τ-)2 -i
jyi,t — 1{individuaU∈C2 at time t}
so that D∖t + £>?t = 1. Let A√ι⅛ = Djt)'. In this setting, one may take
x*t = d1it, and therefore
zi,t-ι = [si,i-1’ ⅛t-ι]
Zi,t-l = 8i,t-1 ®Xi,t-l = [⅛i-l⅛i-l,⅛i-l⅛i-lJ,
The most striking difference between the saturated and unsaturated model in
this case is that, in the unsaturated model, ¾ιi (г.е. belonging to class Ci or
C2) has the same impact (positive or negative or none) on the probability of Y∕t
and Ki2t irrespective of the state in (t — 1). Conversely, in the saturated model,
Xitt might have, say, a positive effect on the probability of Y^t if Y-ι.t-∖ = (0,0)z,
and no effect on the probability of Y^t if Yi1t-1 = (1,0)'.
The conditions for Granger поп-causality in the presence of covariates are
formally identical to (11) and (12), but the restriction matrices are defined as
follows for the unsaturated model:
TT * _ ʃfɪ θ TT * _ θ
1 - [ 0 ιk. ∖ ` 2 - [ о 4.
while, for the saturated model the matrices are
H* = Ik ® Hi , H* = Ik ® H2