# Fitted values for a log-normal model

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.)

• 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? Mar 15, 2011 at 14:28

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.