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 sjL_l 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 Y_itt 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 D¹_yi }     (7)

∖                    L Pi,t ¹J ’ /

where p⅛_ι⅛ is given by

₌ 2 exp (τ'⅜⅜-ι)
^Pl,t l + exp(7's_i,_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 Y_i¹_t and Y?_t
(given y_i,_t-ι) is given by

Pr {y[_t I y_i,_t~₁} = Φ₁ ((2y[_t - 1) β'₁s_i,_t-₁; 0,1)              (9)

Pr {yl_t I y_i,_t~₁} = Φ₁ ((2^ - 1) ∕¾s_i,_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 Y_iit. The latent regression
approach assumes that the individual will choose Y_i¹_t = 1 when a latent contin-
uous random variable Y_i¹_t 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 = ∕⅛_4i~₁+⅛

where:

⅛√⅜)~≡Qo]κ. d)

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