Similarly, Til-l does not strongly cause Y2 one step ahead, given , briefly
Y1 →Y2, if
y2t ± TiL1 I C1 ∀i ∈ I (2)
Definition 2 - Strong simultaneous independence: Yf and Y2 are strongly
simultaneously independent given {J^t}, briefly Y1 φY2, if
Tt1 ± y21Λ-1 ∀t ∈ I (3)
Notice that the term simultaneous in the latter definition has exactly the
same meaning as instantaneous in Geweke (1984) and Granger (1988). A differ-
ent term is suggested here since, rigorously, Florens and Fougère (1996) observe
that one step ahead поп-causality in discrete time has an analogue in continu-
ous time when the time distance between “cause” and “effect” goes to zero, a
circumstance that they define as instantaneous causality. Therefore, in discrete
time, they use instantaneous as a synonym for one step ahead, while they do
not give any definition similar to 3. Moreover, for the simultaneous condition
(3), the term dependence is proposed instead of causality (as in Granger, 1988)
or feedback (as in Geweke, 1984), since the notion is completely а-directional in
nature (not even bi-directional).
In Economics, the notion of поп-causality has been mainly used in modelling
macroeconomic variables, and hence {lzil} and {lzi2} are usually assumed to be
continuous processes, the exogenous processes {Vi} are often not included in
the information set (so that Q} and Qf do coincide with yf and У2 respectively),
and one single realization of {Vi} is observed. In this framework, non-causality
is usually tested assuming that {Yi} belongs to the class of Vector ARIMA pro-
cesses. In microeconometric applications, where the variables are often quali-
tative and longitudinal data are usually available, the VARIMA framework is
not appropriate, and including covariates to account for individual heterogene-
ity becomes an essential aspect of modelling. Therefore, a set of ad hoc tools
has to be developed in order to make the above definitions of non-causality
operational.
In this paper the case where {Vi} is a bivariate discrete time binary process
will be addressed.3 This case is shortly discussed in Chamberlain (1982). We will
assume that N individual realizations (г = 1,..., AT) of the process are observed,
with t = 1,..., T. As we will see, depending on the dynamic structure of the
model, N might have to be large with respect to T, but if very simple dynamic
structures are assumed, a small N, or even N = 1, might be enough if T is large.
Notice that, in this case, {Vi} is needed to model individual heterogeneity, and
might well include some time fixed variables.
In our framework, at any time t ∈ {1,...,T,}, the state space of Yt =
(Yf,Y2) is given by the following states: {(0,0) ; (1,0) ; (0,1) ; (1,1)}.
Essentially the model may be represented by the diagram in Figure 1, where
each box represents one of the four states where the process could belong at time
(t — 1), and the arrows represent the transitions which might occur at time t.
3The case where {Kt1} and {V2} are continuous time counting processes (t ∈ I C R+) is
addressed in the literature (Florens and Fougère, 1996; Schweder, 1970; Aalen, 1987).