st: re: std error covariances between equations

["Kit Baum, Faculty Micro Resource Center" <fmrc@bc.edu>]

I believe that Alan's problem, in a two-equation context, could be viewed as
looking for the covariance term
cov [x1 0] V [0 x2'] = x1 V12 x2'
using V12 to refer to the NE block of the
covariance matrix. Alternatively, we could be looking for
cov [0 x2] V [x1' 0] = x2 V21 x1'
where V21 is the SW block of the covariance
matrix. Since x1 and x2 may have different numbers of columns, these 
off-diagonal blocks are not symmetric.

The predict option stddp calculates the standard error of the difference in
linear predictions. Using the same algebra, and the formula given in [r] 
predict,
this is x1 V11 x1' - x2 V21 x1' - x1 V12 x2' + x2 V22 x2'
where the first and last terms are related to the variances of the two 
individual equations. It would seem like they could be calculated (via stdp 
for each equation) and added, and the result negated. Without working it 
out in detail, I am not sure how - [- x2 V21 x1' - x1 V12 x2' ] relates to 
what Alan seeks, but it seems like you ought to be able to get it out of 
this.

Since Stata will calculate this stddp for any pair of equations, it would
be quite straightforward to have an ado do all the necessary predicts (for
both the stdp and stddp) and crank out the desired quantities, even for >2 
eqns,
if they can be derived from the stddp magnitudes.

Kit
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Kit Baum, Faculty Micro Resource Center                  fmrc@bc.edu
Academic Technology Services, Boston College   http://www.bc.edu/ats
http://fmwww.bc.edu/FMRC/                http://fmwww.bc.edu/GStat/
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