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RE: st: help log linear

From   Stephen Schmidt <[email protected]>
To   [email protected]
Subject   RE: st: help log linear
Date   Sun, 22 Feb 2004 22:18:02 -0500 (EST)

On Sun, 22 Feb 2004, Vincenzo Verardi wrote:
> 1) run the regression lny=xb
> the I would type the command
> predict lnyhat, xb
> 2) I would then exponentiate my result generating a new variable
> gen ypred=exp(lnyhat)
> I would then have the predicted y.

This is not correct, because the desired quantity is E(y), not
exp(E(logyhat)). Because

  logy = logyhat + err

one can exponentiate both sides to get:

  exp(logy) = exp(logyhat + err)

and can then take expectations:

  E(y) = exp(Elogy)) = E(exp(logyhat + err))

It is not, however, valid to break up the right side as

  E(y) = E(exp(logyhat)) + E(exp(err))

because exp is not a linear function. One needs to integrate
the expected value explicitly. I have a memory, probably
incorrect, that E(y) = exp(logyhat) + 1/2*sigma-squared,
where sigma-squared is the variance of the error term, so
that E(y) is a little higher than exp(logyhat). But that
memory dates back to graduate school and isn't relaible
at this time....

Steve Schmidt

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