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


From   "Nick Cox" <n.j.cox@durham.ac.uk>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: RE: Re: Generating predicted values for OLS with transformed dependent variables
Date   Wed, 12 Apr 2006 14:31:58 +0100

As posted earlier, -glm- offers the back-to-basics 
Jacobin solution as an alternative to this use
of Jacobians. 

Nick 
n.j.cox@durham.ac.uk 

Rodrigo A. Alfaro
 
> 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.
> 

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