Non-causality in Bivariate Binary Panel Data

or, in compact vectorial form:

Y* = Bx + ε = V + ε

^Y = [⅛∙>o}]

where Y*,Y,υ and ε are (n × 1) vectors of elements Y_j*,Yj, βjX and e_j∙, whereas
B = [β₁, ∙ ∙ ∙ , β_n∖'. Notice that no loss of generality is involved in assuming that
the same vector x enters all equations, since some of the coefficients in B may
be zero.

The probability of Yj can be easily calculated by adapting the results of the
previous section:

^Pr (^yj = Уз I <≈) = Φ ((2 ∙ y_j - 1) ∙ β_jχ∙, 0, σ²)

i.e., the marginal distribution of Yj is a simple univariate probit model; this
result can be very useful in applications, since it shows that the distribution of
Yj depends only on the parameters of the corresponding latent regression Y.*.

However, in this case, it is more interesting to express the probability of a
vector у = [τ∕ι,... ,y_n]' whose elements are only O’s or l’s; in order to do so,
define the diagonal matrix D_y

D_y = 2 ∙ diag (y) - I_n

Then the probability of у is

Pr (У = у I x) = Pr (D_y (Bx + ɛ) > 0)

= Pr (D_yBx + D_yε > 0)

= Pr (—D_yε < D_yBx)

Now suppose that ε is a random Gaussian vector, so that

ε ~ N (0, Σ)

Then -D_yε is itself a random Gaussian vector

-D_yε ~ N (θ,D_yΣD'_y)

and the probability of observing у is:

Pr (У = у I x) = Φ_n (p_yBx; 0, D_yΣD'_y)

This formula allows to express the probability of у in compact form, involving
only an «-dimensional integral parameterized as a function of y. Here, D_yYD¹_y
is a positive definite matrix: in fact, consider d_y, a column vector equal to the
main diagonal of D_y; then, by Equation (2.11) in Styan (1973)

DyVD¹_y = (d_yd!_y) * Σ
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