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I assume a simple model $\log(y_i) \sim \mathcal{N}(\mu_i,\sigma)$ with $\mu_i=\alpha + \beta x_i$ . Now if I have the estimates $\hat{\alpha}$ and $\hat{\beta}$, how do I calculate the fitted values $\hat{y}$? (Using $\hat{y}= \exp(\hat{\alpha} + \hat{\beta} x_i)$ seems to work; this would be the median of the log-normal distribution.)

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  • $\begingroup$ You seem to have answered your own question as to how to calculate fitted values; what to call them (apart from 'fitted values'), or how to describe them unambiguously in words rather than equations, would be different questions. Conditional geometric means? Geometric conditional means? $\endgroup$
    – onestop
    Mar 15, 2011 at 14:28

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The conditional expectation of $y_i$ at the value $x_i$ can be estimated as $\exp(\hat{\alpha} + \hat{\beta}x_i + s^2/2)$ where $s^2$ is the mean squared error in your regression (estimating $\sigma^2$). Don't trust this until you have closely looked for evidence of heteroscedasticity (i.e., changes in the true value of $\sigma$ for different values of $x$) and found nothing of importance.

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