I would like to fit a model of the form
$z = k x^\alpha y^\beta$
to some data I have (it's a spatial gravity model). Now I know you could take logs of both sides and fit a linear regression
$\log z = \log k + \alpha \log x + \beta \log y$
But there are two problems:
Does this violate assumptions of OLS regression if $\log x$ and $\log y$ don't have a normal distribution?
Some of my $z$s are zero - how to deal with that?
In response to whuber's question. This is a model of how many people from each of a number of districts travel to an event. $x$ is district population and $y$ is (mean) distance (from each district) - thus I expect $\alpha \approx 1$ and $\beta < 0$. $z$ is sampled from a survey of under 1% of over 100k visitors. I am interested in obtaining estimates for where the remaining 99% of visitors came from, for which I do not expect $z=0$ for any district, but in the limited sample data $z$ is sometimes zero because the survey people didn't happen to meet anybody from that district.
If anyone has comments about how the above modifies expectations about the error distribution on $z$ and hence the best way to fit the model I would be interested to hear it. As I was expecting $a \approx 1$, I tried nonlinear least squares with $a$ fixed at 1, but that led to extremely bad variance on $k$ ($\sigma_k > k$). I have since fitted GLM with a poisson link function which gives sensible $k$ and $\beta$ but $\alpha \approx 0.85$ - not great, but it's only a rough model so I can live with it.