Non-causality in Bivariate Binary Panel Data

1)i,t — μ + Ajiji_tt-j + ¾.⅛
j=i

which is formally identical to the standard VAR model, with the only difference
that the distribution of the error term, which is usually assumed to be Gaussian,
takes here into account the binary nature of V.i∙⁸ The total number of parame-
ters is further reduced to 4£ + 3. It is important to explore the precise meaning
of these restrictions on the dynamics of the process, and find other, more inter-
pretable ways to set up a priori restrictions of the parameter space of the VAP
models. Notice however that the problems related to using overparameterized
models are here mitigated, with respect to the standard Vector AutoRegressive
literature, by the availability of individual data, since the usable data points are
here N × T, rather than T (see also Section 4).

For the unrestricted VAP(£) model, the Granger поп-causality conditions
are formally identical to (11) and (12), but the restriction matrices are defined
as follows:

I times

In this case, the restrictions matrices exclude all the regressors involving yj_-1
from Pr {yj_t I y_itt-ι, ■ ■ .,y_itt-e}, and all the regressors involving y}~₁,...,
yj__£ from Pr{y?£ I yt_lt-ι, ∙ ∙ ∙, yi,t-e}∙ Again, the simultaneous independence
condition (13) remains unchanged. The restriction matrices for the restricted
versions of the VAP, as well as those needed for the VAPX may be obtained
accordingly.

4 Estimation and Testing

The purpose of this Section is to discuss the properties of the parameters es-
timates in model (7), as well as the properties of the tests for the hypotheses
(11)-(13). Some hints will also be given about the generalizations illustrated in
Section 3. We will discuss the asymptotic properties of Maximum Likelihood es-
timates and LR tests, although several other standard procedures for estimating
and testing may be used. Some finite sample results will be also illustrated.

We assume that each individual i, i = 1,... N, is observed at each time t
during a period of known length t = 1,..., T; the extension to the case in which
every individual i is observed for a length T^,_i is straightforward if we suppose

⁸Notice also that, in this model, dropping the constant would impose the following restric-
tion, probably uninteresting in most applications:

Pr {y⅛ = 1 I Y_i,_t-₁ = ... = Y_itt-_l = (0,0)'} = 0.5

Pr{y⅛ = 1 I y_i,_t.₁ = ... = Y_iιt._t = (0,0)'} = 0.5

11

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