Non-causality in Bivariate Binary Panel Data



which involves four mutually exclusive dummies representing the four states of
the process in (t — 1); in fact St-ι = Qs't~l, with

Illl
0 1 0 1
y- OOll
_ ° ° ° 1 _

The reason for using st-ι instead of sjLl to describe the state of the system in
(t — 1) is that, doing so, the поп-causality restrictions are more easily written
and interpreted.

As shown in Appendix A, the joint distribution of Yitt conditional on the
state of the system in (t — 1) can be written as follows:

Pr{2∕i,i I 2∕i,i.1} = Φ2 f¾,t∕3'4i-15O,¾,t [ ɪ pi't 1 D1yi }     (7)

∖                    L Pi,t 1J ’ /

where p⅛ι⅛ is given by

= 2 exp (τ'⅜⅜-ι)
Pl,t l + exp(7'si,i-ι)

β = [β1,β2∖ and 7 are parameter matrices of dimension 4 × 2 and 4 × 1 re-
spectively, while Φ2 (∙; μ, R) denotes the integrated bivariate normal with mean
μ and correlation matrix R. The logit-type functional form in (8) is chosen
so as to bound the correlation coefficient between —1 and 1: other choices are
possible. As a whole, the distribution depends on 12 parameters freely varying
in R12, and it is easily shown that the transition probabilities are a bijective
transformation of β and 7. Notice that the marginal distribution of Yi1t and Y?t
(given yi,t-ι) is given by

Pr {y[t I yi,t~1} = Φ1 ((2y[t - 1) β'1si,t-1; 0,1)              (9)

Pr {ylt I yi,t~1} = Φ1 ((2^ - 1) ∕¾si,i-ι; 0,1)            (10)

It may be useful to give an interpretation of the model in terms of latent re-
gression model. Each individual i has to make two binary choices at time t, i.e.
to choose the value of the binary bivariate vector Yiit. The latent regression
approach assumes that the individual will choose Yi1t = 1 when a latent contin-
uous random variable Yi1t crosses a threshold, which, in the current framework,
is assumed to depend on the choice made in (t — 1). The same holds for Y?t.
The latent regression here is:

⅛=Mm+⅛

2∕⅝ = ∕⅛4i~1+⅛

where:

⅛√⅜)~≡Qo]κ. d)



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