Re: st: How to obtain standard errors for the 2SLAD model (qreg):Theory behind...

["Brian P. Poi" <bpoi@stata.com>]

On Tue, 13 Jun 2006, Alexandros Zagg wrote:

> Dear STATA-listers,
>
> Follow the instructions given by Statacorp's Brian P. Poi in the
> statalist dated 26 Sep 2003 (look below) I was wondering how
> bootstrapping will actually correct the standard errors? Could anybody
> briefly outline the theory or direct me to any references?
> Also, is it possible to perform tests such as the exogeneity test of
> instruments or a test of over-identification assumption?
> Many thanks for your time!
>
> Alexandros
Alexandros,

The bootstrap is a very general method of obtaining standard errors based 
on the principle of resampling.  For estimators such as the OLS estimator 
in linear regression, deriving a mathematical formula for the covariance 
matrix is straightforward and relatively easy to program.  For two-step 
estimators, several different techniques are available for deriving an 
asymptotically-justified covariance matrix, and Amemiya does this for the 
2SLAD estimator in a 1982 paper in Econometrica.

Bootstrapping, on the other hand, is based on the idea that the standard 
error of an estimate tells you the variance of the sampling distribution 
you would obtain if you were able to go out, draw a sample from a 
population, apply your estimator, draw a new sample from the population, 
apply your estimator again, and so on, many times.  In practice we cannot 
go out and draw new samples, so the bootstrap pretends your original 
sample (i.e. dataset) is representative of the population and draws (with 
replacement) samples from your original sample.

Deriving the covariance matrix for a two-step estimator is often a 
challenge, but writing a program that implements the estimator for a 
single sample, then making use of Stata's bootstrapping commands is 
straightforward.

Cameron and Trivedi's 2005 textbook "Microeconometrics" has a very good 
chapter on the bootstrap, and Davidson and MacKinnon's 2004 textbook 
"Econometric Theory and Methods" introduces the bootstrap relatively early 
in the text and makes use of it throughout the remainder of the book.

Two "From the Helpdesk" Stata Journal articles also discuss the bootstrap: 
Weihua Guan wrote an article in 2003 (volume 3, no. 1), and I wrote an 
article on the bootstrap in 2004 (volume 4, no. 3) that included a handy 
program for determining the number of bootstrap replications to use.

Regarding tests of exogeneity and overidentifying restrictions as 
they apply specifically to the 2SLAD estimator, I have to pass, as I do 
not have a definitive answer.

I hope this helps.

  -- Brian Poi
  -- bpoi@stata.com
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