Bayesian Inference in the Seemingly Unrelated Regressions Model

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⎛
U U U
y ⎞ ⎛
() t X ⎞ ⎛
() t e ⎞
() t
⎟=⎜ ⎟β +⎜ ⎜ O O O
y ⎟ ⎜
() t X ⎟ ⎜
() t e ⎟
⎝ ⎠ ⎝ ⎠ ⎝ () t ⎠
(58)
where we write
⎡
UU UO
Σ Σ ⎤
E⎡ ⎣
e() t e()
′
t
⎤
⎦
= ⎢ ⎥
OU OO
⎢⎣Σ
Σ ⎥⎦
(59)
U
t
The conditional posterior pdf f( y() | βΣ , , y()
) is a multivariate normal distribution
with mean
O
t
U O O UO OO−1
O O
() t () t () t () t () t
E( y | β, Σ , y ) = X β +Σ Σ ( y −X
β )
(60)
and covariance matrix
U O UU UO OO−1
OU
() t y()
t
V( y | β, Σ , ) =Σ −Σ Σ Σ (61)
U
t
O
t
Furthermore, ( y() | βΣ , , y()
), t = 1 ,2,…,
T are independent. Thus, for generating y()
t
within the Gibbs sampler, we use the conditional normal distributions defined by
equations (60) and (61) for all observations where an unobserved component is
present.
U
Suppose, now, that the unobserved components represent negative values of a
Tobit-type latent variable. In this case we have the additional posterior information
U
that the elements of y () t are negative. The conditional posterior pdf for
U O
() t y()
t
( y | βΣ , , ) becomes a truncated (multivariate) normal distribution with a
U
truncation that forces y () t to be negative. Its location vector and scale matrix (no
longer the mean and covariance matrix) are given in equations (60) and (61). A