where * is the Hadamard product. But Σ and {dyd'y) are both positive definite,
and so is their Hadamard product (see Theorem 3.1 in Styan (1973)).
Alternatively, we can write the probabilities Pr (У = y ∣ x) as
Pr (У = y I x) = Pr (Dy (Bx + ε) > 0)
= Pr (DyBx + Dyε > 0)
where DyBx + Dyε is a random Gaussian vector
DyBx + Dyε ~ N (DyBx,DyΣD'y)
Then the probability of observing y is:
Pr (У = y I æ) = Pr (DyBx + Dyε > 0)
+ ∞
= f φn (pyBx, DyΣD'y) dς
= ʃ ' Φ∏ (DyBx, DyΣDy) dς
= (⅛∈r+})
where R" is the non-negative orthant of the «-dimensional Euclidean space.
Similar to the univariate case, it is simple to show that this model suffers
from an identification problem, insofar as B and Σ cannot be recovered unam-
biguously. As in the univariate case, the problem is solved by standardizing,
which in the multivariate setting amounts at replacing the variance covariance
matrix Σ with the correlation matrix R.
B Variables of the Application
This Appendix simply reports the variables used in the application of Section
6. For every year and every variable, we have reproduced the field of the PSID
database corresponding to the couple: as an example V30020 is the field corre-
sponding to the variable INTERVIEW in 1969. In parentheses, the selection rule
we applied is reported: since for V30020 a recorded value of 0 corresponds to
a missing value, we have selected only the individuals having a recorded value
V30020 greater than 0.
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