Non-causality in Bivariate Binary Panel Data

the usual asymptotic properties. Possible problems in finite samples are also
illustrated. Section 5 shows how the proposed analysis does specialize when one
is interested in a specific four states Markov chain, corresponding to discrete
time bivariate survival data. Sections 6 and 7 illustrate the proposed methodol-
ogy, using respectively data about marriage and fertility timing in a sample of
266 American women and about the adoption of two interrelated technologies
by 552 Italian metalworking plants. Section 8 concludes.

2 A Markov Dynamic Bivariate Probit Model
for Homogeneous Population

Essentially, the type of data set where we want to check for поп-causality con-
sists in observations on the choices of N individuals facing two interacting bi-
nary choices in discrete time. To do this, it seems then natural to use, as a
statistical model, a dynamic version of a bivariate discrete choice model. The
static univariate and multivariate probit model, viewed in a latent regression
perspective, is briefly reviewed in Appendix A. In this Section a simple dynamic
version of the bivariate probit model is presented,⁵ derived under the following
assumptions:

• the population is homogeneous (no covariates are introduced);

• the process is Markov (all the information from the history of the process
which is relevant for the transition probabilities in t is embodied in the
state of the process in (t — 1)).

Both these simplifying assumptions will be relaxed in Section 3. Notice
that the derivation of the dynamic model requires the events to take place
only in discrete time. In fact, qualitative variable models as logit and probit
are sensitive to the length of grouping: when analyzing events taking place
in continuous time subject to grouping, the estimates obtained using discrete
choice models can be subject to a severe bias. The phenomenon is much more
prominent when analyzing multi-state models, in which interactions between
events are under study.

In order to use the bivariate probit setting for representing the distribution
of Yi_it =            conditional on the state of the system in (t — 1), let us

introduce the following notation:

Z₁ 1       2       1      2    √

Si,t-1 (ɪɪ        yi,t-l ? Vift — lVift — l)

D_yit = 2diag(₂∕_i,_i)-I₂

It is worth to point out that s⅛-ι is an invertible linear transformation of

s_i*-₁ = [(1 - ₂∕_i¹-r) (1 - ⅜²-r) ,⅜¹-₁ (1 - 2/Lr) , (1 - ⅜¹-₁) ₂∕_i²-₁,₂∕_i¹-₁¾²-₁]

⁶Dynamic versions of the univariate probit model are discussed, for example, in Heckman
(1981).

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