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st: How to implement Wald estimator (for IV that is a ratio of coefficients)?


From   Misha Spisok <misha.spisok@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   st: How to implement Wald estimator (for IV that is a ratio of coefficients)?
Date   Fri, 9 Oct 2009 18:21:39 -0700

Hello, Statalist!

In short, does -ivregress- (or -reg3-) include what I think is called
the Wald estimator?  If so, how can I implement it for a problem like
the one below?  I've searched for a command for the Wald estimator,
but can only find references to Wald _tests_.

I am considering a model similar to Ashenfelter and Greenstone (2004)
with two reduced-form equations, the estimates of which are used to
find an instrumental variable estimator in a third equation, the one
of primary interest.

My question is, how can I do this in Stata in one fell swoop?

The two equations are

F = lambda_F*VMT + PI_F*1(65mph limit in force) + epsilon
H = lambda_H*VMT + PI_H*1(65mph limit in force) + epsilon'

where 1(.) is an indicator variable which I'll call "65mph" below.

The equation of interest is

H = beta*VMT + theta*F + nu

The parameter of interest is theta.  From the estimate of the reduced
form equations the IV for theta, theta_IV, is

theta_IV = (PI_H)/(PI_F)

Given estimates of PI_H and PI_F (as presented in the paper), one can
form the corresponding theta_IV.  It seems that the authors use a
formula for the standard error of theta_IV like the following:

se_theta = theta_IV*sqrt((se_PI_H/PI_H)^2 + (se_PI_F/PI_F)^2)

I tried doing this in the following ways, but the results are not the
same.  I wouldn't expect them to be, but I can't find a reference for
Wald estimator in Stata, so I thought I'd try it.

Method 1:
. reg F VMT 65mph
. reg H VMT 65mph
Calculate theta_IV from coefficients on 65mph in the above equations.

Method 2:
. ivregress 2sls H VMT (F 65mph)
Hope that theta_IV would be the coefficient on F.

Method 3:
. reg3 (F VMT 65mph) (H VMT F)
Hope that theta_IV would be the coefficient on F in the equation for H.

What is the correct way to get this ratio of coefficients (theta_IV =
(PI_H)/(PI_F)) and its standard error all at once in Stata?

Thanks,

Misha
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