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st: Re: Generating predicted values for OLS with transformed dependent variables


From   "Rodrigo A. Alfaro" <ralfaro76@hotmail.com>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: Re: Generating predicted values for OLS with transformed dependent variables
Date   Wed, 12 Apr 2006 09:23:11 -0400

Daniel,

Let me simplify the problem. Considere u~normal(0,s_u^2) and g(y)=u. You want E(y)... right? Sometimes you can find the distribution of y using the jacobian transformation. Suppose that h() is the inverse of g() then y=h(u)... then you need to find the distribution of y. This is a change of variable, you have to evaluate the normal with h() and multiply by the jacobian.

Confuse? take g() =ln()... then ln(y)=u which is a simplification of the regression with log in the dependent variable. Note that the inverse of g() is known then h()=exp() and finally y=exp(u). You need to know the distribution of y, which is lognormal!!! (http://www.xycoon.com/logn_relationships1.htm). Using this distribution we can get the expected value of y E(y) = exp(0+0.5*s_u^2) (http://www.xycoon.com/logn_expectedvalue.htm). If ln(y)=bx+u, you can find that for nonstochastic x E(y)=exp(bx+0.5*s_u^2)=exp(bx)*exp(0.5*s_u^2), the second term is the "adjustment".

In your problem you have to find the distribution of y for y=u^4 and u~normal. I understand that you cannot use the jacobian transformation, but the proof of "the square of a standard normal is a chi-square 1" is a useful source to solve your problem.

Good luck,
Rodrigo.

----- Original Message ----- From: "Daniel Schneider" <daniel.schneider@stanford.edu>
To: <statalist@hsphsun2.harvard.edu>
Sent: Monday, April 10, 2006 5:47 PM
Subject: st: Generating predicted values for OLS with transformed dependent variables



I have a question on statistics and a possible implementation in Stata:

I know that generating predicted values from an OLS with a
log-transformed dependent variable needs some adjustment to generate
consistent and unbiased values (van Garderen 2001 or Wooldridges
textbook explain this).

Does anyone know if a similar logic applies to other transformations,
for example a "root-root" (2x square root) transformation? If yes, does
anyone know how that adjustment would look like and how it would be
implemented in Stata?


Daniel
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