Sorry if this is a straightforward question, but I have tried digging into econometrics book and cannot find anything about it. I worked on a model with log(wage) = experience + experience^2 + experience^3 and i have the results for that now. However, my results are in the form of E(log(wage)). Instead, now, I wanna move to E(wage).
Is there a transformation I should perform? Or does anyone have literature or terms I should google for?
Hopefully someone can help me
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shadowtalker
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4By Jensen's inequality, $\mathbb E[\log(X)]\le\log(\mathbb E[X])$ assuming both expectations are defined. – Xi'an Apr 20 '22 at 17:10
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Okay, and is there a way to make a transformation such that E(log(x)) transfers to log(E(x))? – notest Apr 20 '22 at 17:15
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3Maybe asking about Duan's smearing? – Andy W Apr 20 '22 at 17:21
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3Is this a duplicate of Transforming back after a log transformation? – shadowtalker Apr 20 '22 at 17:28
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2If you want $\mathbb{E}[\mathrm{wage}]$ instead, you should use a different model, perhaps with a logarithmic link function. How did you fit your model for $\log(\mathrm{wage})$? – Ben Apr 20 '22 at 17:28
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The relationship between $E[\ln(x)]$ and $E[x]$ will in general depend on the distribution of $\ln(x)$.
If you fit $\ln(x)$ with an ordinary least squares (OLS) regression, then by assumption $\ln(x)$ is normally distributed around its expectation value with some standard deviation $\sigma_{\log}$, which should have been reported in by your OLS fitting software. Therefore, $x$ is log-normal distributed. Conveniently, the log-normal distribution is parameterized by the mean and standard deviation of the log, $\mu_{\log}$ and $\sigma_{\log}$. In terms of these parameters, the expectation value of $x$ is $\exp\left(\mu_{\log} + \frac{\sigma^2_{\log}}{2}\right)$. (The median, btw, is independent of $\sigma_{\log}$ in this case; it's just $\exp(\mu_{\log})$).
If you did something other than an OLS regression (e.g., you fit a GLM with a log link function), then the assumed conditional distribution for $\ln(x)$ is different, and therefore the relationship between the expectations of the log and the variable will be different. In particular, for a Poisson regression, $E[x]$ is just $\exp(E[\ln(x)])$. For other families you may have to look up the relationship.
Also, note that all of these formulae ignore the uncertainty in your estimate of $E[\ln(x)]$, which may or may not be a reasonable thing to do in your application.
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