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# st: Re: standard errors after nonlinear function of ...

 From rgutierrez@stata.com (Roberto G. Gutierrez, StataCorp.) To statalist@hsphsun2.harvard.edu Subject st: Re: standard errors after nonlinear function of ... Date Mon, 14 Oct 2002 18:50:38 -0500

```Dean Yang <dyang@fas.harvard.edu> asks:

>> Is there a non-linear analog to the -lincom- command? I'd like to calculate
>> a nonlinear function of some regression coefficients, and the associated
>> standard error.

>> I know we can test nonlinear hypotheses using -testnl-, but the command only
>> saves chi or F-statistics and degrees of freedom. It doesn't give you the
>> standard error on the estimate.

and John Gibson <jkgibson@mngt.waikato.ac.nz> replied:

> From testnl, if you use the
> ...   ,g(matname)
> option you can save the Jacobian, and then plug that into the delta method
> formula.

To elaborate on what John suggests, I'll give an example using the auto data.

We begin by fitting a linear regression

. regress price weight length mpg

[some output omitted]

------------------------------------------------------------------------------
price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
weight |   4.364798   1.167455     3.74   0.000     2.036383    6.693213
length |  -104.8682   39.72154    -2.64   0.010    -184.0903   -25.64607
mpg |  -86.78928   83.94335    -1.03   0.305     -254.209    80.63046
_cons |   14542.43   5890.632     2.47   0.016      2793.94    26290.93
------------------------------------------------------------------------------

Suppose then that our nonlinear combinations of parameters is

f(b) = _b[length]/_b[mpg]

that is, the ratio of the coefficients on -length- and on -mpg-.

We call -testnl- with a test of f(b) = 0, and retrieve the matrix of first
derivatives (the Jacobian), G.

. testnl _b[length]/_b[mpg] = 0, g(G)

(1)  _b[length]/_b[mpg] = 0

F(1, 70) =        1.05
Prob > F =        0.3099

. mat list G

G[1,4]
c1          c2          c3          c4
r1           0  -.01152216   .01392232           0

We note that the derivatives of f(b) with respect to the first and last
element of e(b), namely _b[weight] and _b[_cons], are zero.  That's comforting
considering that f(b) is not a function of these two parameters.

Applying the delta method,

. mat A = G*e(V)*G'

. mat list A

symmetric A[1,1]
r1
r1  1.3953568

gives the estimated variance.

--Bobby
rgutierrez@stata.com
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