Richard Sperling wrote:
I would like to use Stata (9.2 SE) to estimate the following
inherently non-linear function by IRLS:
C = (a + b*X^g)*e,
where e is multiplicatively normal with mean 1 and standard deviation
0.15. I cannot use -nl- to estimate this function because it assumes
that the error term enters additively. One suggestion I have received
is to take logs of both sides:
ln(C) = ln(a+b*X^g) + ln(e).
Is there an alternative way of estimating this equation?
--------------------------------------------------------------------------------
Take a look at the -lnlsq()- option of -nl-. I've illustrated its
application to your problem in the do-file below. (Pharmacokineticists have
a variety of so-called error models or variance models. Yours is the
"constant-coefficient-of-variation" or "proportional" variance model.)
You can also use -ml- in lieu of -nl- for nonlinear regression where you
want to model the variance explicitly. It's illustrated in the FAQ section
of StataCorp's website at www.stata.com/support/faqs/stat/nl_ml.html .
(It's called "extended least squares" in the pharmacokinetics literature.)
Joseph Coveney
clear
set more off
set seed `=date("2007-04-07", "ymd")'
set obs 500
// C = (a + b*X^g)*e
local a 2
local b 3
local g 0.75
generate double X = uniform()
generate double e = 0.15 * invnormal(uniform())
generate double C = (`a' + `b' * X^`g') * (1 + e)
nl (C = {a} + {b} * X^{g}), lnlsq(-1) initial(a `a' b `b' g `g')
exit
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