-_diparm- is a very useful utility for displaying ancillary
parameters. These parameters are often estimated in a transformed
metric, but we are interested in the estimate in the original metric.
Handily, -_diparm- includes several standard transformations as options.
My questions are:
1. In the example below, why does the value for the standard error
of the ancillary parameter "rho" that gets pumped out by -_diparm- with
the tanh option differ from the value that I calculated by hand.
2. More generally, why does -_diparm- /not/ produce the "z" and "P > z"
values? ... and yet it produces a c.i.? [I, for one, would like to
see "z" and "P > z" reported as well.]
Motivation:
In a model with a correlation ("rho") as ancillary parameter,
a model is typically estimated in terms of
t = atanh(rho) = .5*ln((1+rho)/(1-rho)),
to ensure that rho lies in the range (-1,1). A bivariate
probit (-biprobit-) is an example of such a model.
The following output illustrates the basis of my questions:
. * Use same data set as for -biprobit- in Ref [A-G], p. 142
. use http://www.stata-press.com/data/r7/school.dta, clear
. biprobit (private=years logptax loginc) (vote=years logptax loginc), nolog
Seemingly unrelated bivariate probit Number of obs = 95
Wald chi2(6) = 9.59
Log likelihood = -89.254028 Prob > chi2 = 0.1431
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
private |
years | -.0118884 .0256778 -0.46 0.643 -.0622159 .0384391
logptax | -.1066962 .6669782 -0.16 0.873 -1.413949 1.200557
loginc | .3762037 .5306484 0.71 0.478 -.663848 1.416255
_cons | -4.184694 4.837817 -0.86 0.387 -13.66664 5.297253
-------------+----------------------------------------------------------------
vote |
years | -.0168561 .0147834 -1.14 0.254 -.0458309 .0121188
logptax | -1.288707 .5752266 -2.24 0.025 -2.416131 -.1612839
loginc | .998286 .4403565 2.27 0.023 .1352031 1.861369
_cons | -.5360573 4.068509 -0.13 0.895 -8.510188 7.438073
-------------+----------------------------------------------------------------
/athrho | -.2764525 .2412099 -1.15 0.252 -.7492153 .1963102
-------------+----------------------------------------------------------------
rho | -.2696186 .2236753 -.6346806 .1938267
------------------------------------------------------------------------------
Likelihood ratio test of rho=0: chi2(1) = 1.38444 Prob > chi2 = 0.2393
. local t = [athrho]_b[_cons]
. local set = [athrho]_se[_cons]
. /* rho = (exp(2*t)-1) / (exp(2*t)+1) = g(t), where t = [athrho]_b[_cons]
> By delta method:
> std error of rho = g'(t)*(std. error of t) where ( ... I think that ...)
> g'(t) = 2*sqrt(exp(2*t))/((exp(2*t)+1)
>
> -biprobit- produces rho and se(rho) using the "tanh" option in a call to _diparm
> (see ..\ado\base\b\biprobit.ado)
> */
.
. di "rho = " (exp(2*`t')-1) / (exp(2*`t')+1)
rho = -.26961864
. di " std error of rho = " `set'*2*sqrt(exp(2*`t'))/(exp(2*`t')+1)
std error of rho = .23227723
Observe that
(1) the hand-calculated value for s.e.(rho) is close to, but definitely
not the same as, the value printed out in the -biprobit- output. Why?
(2) -_diparm- (called by -biprobit-) did not produce "z" and "P > z"
statistics for rho.
Stephen
----------------------
Professor Stephen P. Jenkins <[email protected]>
Institute for Social and Economic Research (ISER)
University of Essex, Colchester, CO4 3SQ, UK
Tel: +44 (0)1206 873374. Fax: +44 (0)1206 873151.
http://www.iser.essex.ac.uk
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