Thanks for the suggestion Paul but I am not sure this works, maybe you (or
someone else) can clarify. I knew I could bootstrap, as you suggested,
however, I think there is a problem you omit. The joint inequalities imply
that the frontier of the acceptance region is not continuous, which is a
problem for hypothesis testing (deriving the distribution for the test). To
see this, notice that my original hypothesis a > 0 and a + b < 0 can be
re-written as (a+b)^(sign of (-a)) < 0. However, even if bs would give me a
test result if I bootstrapped (a+b)^(sign of (-a)) < 0 I do not think that
result is valid. The composite hypothesis makes it quite difficult I
believe. Redefine a + b = -g, then the region we are looking for, in a (a,g)
graph, is the positive quadrant. Think about estimates such that you are
really close to (0,0) but in the region where a is negative, now draw
different sized circles around that point to represent different critical
values, as you can see, there's no easy formula to represent the area of
these circles that falls into the positive quadrant. I think this is more or
less a way to represent the difficulty of this problem. Is there something I
am missing about bootstrapping that solves this problem? Thanks again Paul,
since nobody was answering I was thinking that this might just be so trivial
that nobody wanted to bother answering it.
This can be bootstrapped, without any programming, using the -bs- command.
Using the auto data set (& changing the hypothesis slightly),
the following will do it:
bs "xtprobit foreign weight, re i(rep78)"
I trust that your application makes more sense than this one.
Adapt as necessary. No spaces are permitted when describing
the hypotheses to be tested. A reps option is needed.
See help for the command -bs- for more details, inclusing the meanings of
the difference confidence intervals.