Non-causality in Bivariate Binary Panel Data

in the construction, estimation, interpretation and application of econometric
models.

In a general setting (see Florens and Fougère, 1996), a mathematically rigor-
ous definition of поп-causality based on predictability requires the specification
of the stochastic process to be predicted, the available information set, and the
reduced information set. Although several generalizations exist, we will briefly
review here the concept of discrete time one step ahead strong поп-causality (the
terminology is drawn from Florens and Fougère,1996). Here one step ahead (as
opposed to global) is referred to the prediction horizon, whereas strong (as op-
posed to weak) means that the focus is on predicting the whole distribution,
rather than just the mean. Notice that Granger’s (1969) original definition is
stated in terms of the mean. Chamberlain (1982) and Florens and Mouchart
(1982) propose the definition involving the whole distribution (see also Granger,
1988).

Let {y_i = (Y_t¹,Yf) ,t ∈ I C N = {0,1,...}}, or {K_i} for short,¹ be a discrete
time vector stochastic process. This means that, for any positive integer t ∈ I,
Yt is a vector random variable on a probability space (Ω, A, P). P is an element
of a family of probability measures, and the statistical problem of non-causality
is to test whether P satisfies поп-causality conditions.

The available information is described by P_t, which is a sub-σ-field of A. It
is assumed that the family {Pt,t ∈ I}, briefly {P_t} is a filtration, i.e. Pt C Pf
for t <t'. For simplicity, we will assume here that Pt is the canonical filtration
associated to the multivariate stochastic process {(K_i, ¾)} = {(Y^t, Yfl, Xt) },²
where each of {1√¹}, {V_i²} and {¾} may either be scalar or vector processes.
This of course implies that {F)} is adapted to {Pt}, i-e. Yt is .^-measurable for
any t ∈ I.

The reduced information set is represented by the filtrations {<√_i' } and {<√²}∙
We will assume that {Q) } is the canonical filtration of {(Yfl , Xt)}, and {(√²} is
the canonical filtration of {(y_i²,A⅛)}, which implies that Yfl is ^-measurable
and Yfl is t/²-measurable for any t ∈ I. Let then {T_i^l}, {(F_i²} and {(⅛} be the
canonical filtration associated to the processes {F^¹}, {V_i²} and {F)} respec-
tively. Notice that yj C Qf C Pt,Vt ∈ I, and similarly yfl Q Qf — Xt,Vt ∈ ʃ-

In the paper, we will adopt the following definitions, stated in terms of
conditional independence of sub-σ-fields of A (see Florens and Mouchart, 1982,
Appendix, for the relevant results about conditional independence):

Definition 1 - Strong one step ahead non-causality (Granger поп-causality):
yf-ι does not strongly cause Yfl one step ahead, given Qf~₁, briefly Y¹ ∙÷÷ У²,
if

½¹±½²-ιl⅞¹-ι Vtel                (1)

¹The following notation is used through the paper: {Zt} denotes a stochastic process, Zt
being the value of the process at time t; {zt} and Zt represent the corresponding realizations.
Moreover, Pr{¾ ∣ w⅛} is adopted as a short notation for Pr{Z⅛ = zt ∣ Wt = w⅛}.

²The canonical (or self exciting) filtration associated to the process {Z⅛} defined on
(Ω,√4, P) is a family {7⁷t} of sub-σ-fields of A, whose element Pt is the σ-field generated
by the family of Z_s, O ≤ s ≤ t. Intuitively, Pt embodies the knowledge of the history of {Zt}
up to time t.

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