This may be an obvious question, but I'm flummoxed.
Let's say I've got a nested logit model with a degenerate nest—say, a healthcare choice model where the branches are Home vs. Hospital. The Home branch has only one twig, called NoTreatment, while the Hospital branch has six twigs, representing each of the possible hospitals.
I have data from a study showing coefficients on each of the alternative and individual characteristics within the Hospital branch, as well as the inclusive value, so I can easily construct
$$U_{nj|k} = \beta'X_{j|k} + \varepsilon_{j|k}$$
where $n$ represents the individual, $j$ represents the hospital, and $k$ represents the branch (ie, that the individual chose to be treated at a hospital in the first place).
My understanding is that the probability that individual $n$ chooses provider $j$ is:
$$P_{nj} = \frac{e^{\frac{V_j}{\sigma}}\left[\sum_{m\ne j}e^{\frac{V_m}{\sigma}}\right]^{\sigma -1}}{\left[\sum_{m}e^{\frac{V_m}{\sigma}}\right]^{\sigma }}$$
where $\sigma-1$ is the inclusive value.
Assuming that $U_{home} = 0+\varepsilon$, what is the probability that someone chooses to be home? Is it simply:
$$P_{n0} = \frac{\left[\sum_{m\ne home}e^{\frac{V_m}{\sigma}}\right]^{\sigma -1}}{\left[\sum_{m}e^{\frac{V_m}{\sigma}}\right]^{\sigma }}$$
or do I need to deal with $\sigma$ differently?
EDIT: grammar