# st: Inverse hyperbolic sine function

 From smerryman@kc.rr.com To statalist@hsphsun2.harvard.edu Subject st: Inverse hyperbolic sine function Date Thu, 31 Mar 2005 10:22:20 -0600

```Thanks a lot, Scott.  Just two follow-up questions.

1) If my dependent variable is net worth (say the var name is "wealth"),
do I simply use log(wealth+sqrt(wealth^wealth+1)) as the dependent
variable in regression?
2) How to interpret the coefficients for control variables (eg. gender,
edu, logged wages)? Same as the log transformation?

Ying
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First, as Roger noted, there was a typo in my reply.  The function should be

IHS = log(z + sqrt(z^2 + 1))

1.  Yes, but more generally, the ihs fuctions is ln(theta*z + sqrt((theta^2 * z^2
+ 1))/theta; where theta is scale parameter that could be estimated.

2.  If z is large relative to 1/theta, the ihs function approximates log(z) for positive values and -log(z) for negative values.  The graph looks like:

clear

set obs 800

egen z = fill(-20 -19.95)

gen ihs_z = ln(z + sqrt(z^2 +1))

twoway line ihs z, xline(0) yscale(noline) yline(0) xscale(noline)

The marginal effects of wealth with respect to a change in x would be:

ihs(wealth) = ln(wealth + sqrt(wealth^2 +1)) = b_0 + b_1*x + e

d(w)/d(x) = d(ihs(w))/d(x) * d(w)/d(ihs(w)) where the first term is the estimated coefficient on x (b_1) and the second term is sqrt(w^2 +1).

Hope this helps,
Scott

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