Regarding log-normal specification

Question

I'm working with a log-normal regression model. However, some of the dependent variable equal zero (not missing). Can I use an alternative specification like $log(y+1)$ ~ $X$ (most $y$s are really large)? Or should I just omit those observations? Do I have to do balance check every time I drop some observations and report all the result in appendix?

Answer 1

The answer to this question really depends on the meaning of $y=0$ in your application. You've said most $y$ are really large -- this means that those $y$'s that are zero are quite different from the other $y$'s. Are they really likely to be generated by the same process that generates the large $y$'s?

If $y=0$ is truly comparable with $y\gg0$ then you could use the approach you suggest of taking $\log(y+1)$. This would equate to a functional form of

$$ y_i = e ^ {\beta_0} e^ {\beta_1 x_i,1} ... e^{\beta_n x_i,n}e^{\epsilon_i}+1 $$

Note that the errors $\epsilon_i$ are exponential, not additive using this transformation.

If most $y$'s are really large, then $\log(y+1)$ will also be much larger when $y\neq0$, so including those observations for which $y=0$ will increase the error in your regression significantly. The distribution of error terms will be non-normal (and may effectively be a mixture of two different distributions). Coefficient estimates will potentially be biased if the process generating the $y=0$ terms is different to that generating the $y>0$ terms.

You could ignore the cases where $y=0$ entirely, if you considered that the aim of your model is to understand the relationship between the $y$'s and the $x$'s conditional of $y>0$.

Alternatively, and perhaps more satisfactorily if $y=0$ has meaning in your context, you could explicitly model the probability of $y=0$ vs $y>0$ using a binary model (ie a logitistic regression)and then have a sub-model that only considered the case where $y>0$. This would avoid omitting any data points, at the cost of having to consider what explanatory variables are suitable for explaining the probability of $y=0$.

Rob Hyndman's blog has some further discussion on transforming data with zeroes in it.