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st: Standard error for correlation coefficient in "biprobit"


From   kubo kensuke <kuboken@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   st: Standard error for correlation coefficient in "biprobit"
Date   Mon, 18 Oct 2010 11:03:36 +0900

I am using the "biprobit" command for bivariate probit regression, and
would like to know how Stata computes the standard error for the
correlation coefficient ("rho") between the two error terms.

I have read the manual and understand that Stata indirectly estimates
"rho", by estimating its arc-hyperbolic tangent ("\athrho"), and
transforming it back to the original parameter using the inverse
function.  I also understand how the confidence interval of "rho" is
calculated from the confidence interval of "\athrho".  However,
nowhere can I find any explanation on how the reported standard error
of "rho" is calculated.  I tried a delta-method calculation to obtain
the standard error of "rho" from the standard error of "\athrho" and
the analytical derivative of the hyperbolic tangent function, but the
result was clearly different from what was reported by Stata.

Hoping for an explanation, I purchased Professor Cox's Stata Journal
article ("Speaking Stata: Correlation with confidence, or Fisher’s z
revisited", 2008), but could not find a direct answer to this specific
question.

All I need is the formula that Stata uses to obtain the standard error
of "rho", based on its direct estimation of the arc-hyperbolic
tangent.   Any suggestions on this topic would be greatly appreciated.

With thanks,
Ken Kubo
Institute of Developing Economies, Japan

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