For a conditional simulation, this distribution of
must be conditioned on the values of the CONDITION variables. The relevant general result concerning conditional distributions of multivariate normal random variables is the following. Let
, where
and where
is
,
is
,
is
,
is
, and
is
, with
. The full vector
has simply been partitioned into two subvectors,
and
, and
has been similarly partitioned into covariances and cross covariances.
With this notation, the distribution of
conditioned on
is
, with
and
See Searle (1971, pp. 46–47) for details.
Using the SIMNORMAL procedure corresponds with the conditional simulation as follows. Let
be the VAR variables as before (
is the number of variables in the VAR list). Let the mean vector for
be denoted by
. Let the CONDITION variables be denoted by
(where
is the number of variables in the COND list). Let the mean vector for
be denoted by
and the conditioning values be denoted by
Then stacking
the variance of
is
where
,
, and
. By using the preceeding general result, the relevant covariance matrix is
and the mean is
By using
and
, simulating
now proceeds as in the unconditional case.