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