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 Yj*,Yj, βjX and ej∙, whereas
B = [β1, ∙ ∙ ∙ , βn∖'. 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 ∙ yj - 1) ∙ βjχ∙, 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,... ,yn]' whose elements are only O’s or l’s; in order to do so,
define the diagonal matrix Dy
Dy = 2 ∙ diag (y) - In
Then the probability of у is
Pr (У = у I x) = Pr (Dy (Bx + ɛ) > 0)
= Pr (DyBx + Dyε > 0)
= Pr (—Dyε < DyBx)
Now suppose that ε is a random Gaussian vector, so that
ε ~ N (0, Σ)
Then -Dyε is itself a random Gaussian vector
-Dyε ~ N (θ,DyΣD'y)
and the probability of observing у is:
Pr (У = у I x) = Φn (pyBx; 0, DyΣD'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, DyYD1y
is a positive definite matrix: in fact, consider dy, a column vector equal to the
main diagonal of Dy; then, by Equation (2.11) in Styan (1973)
DyVD1y = (dyd!y) * Σ
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