[Limdep Nlogit List] Generating a gamma distribution from the random effect ?alpha? estimate

[Bjorn Lardner]

Dear LIMDEP users,
We?ve estimated a panel Poisson model with a gamma distributed random 
effect, and look for advice on how to visualize the random panel effect. 
Our confusion stems from the fact that a gamma distribution has two 
essential variables: a shape variable (a) and a scale variable (b), while 
the LIMDEP output only gives one variable pertaining to that error 
distribution; ?alpha?. The key to the issue appears to be (LIMDEP v. 9.0 
manual, p. E26-3),
??ui is a random effect for the ith group such that exp(ui) has a gamma 
distribution with parameters (?, ?). Thus, E[exp(ui)] has mean 1 and 
variance 1/? = [alpha] .?
Despite these leads, we suspect that our attempts to produce a gamma 
distribution may be flawed. We eventually want to illustrate the gamma 
distribution around a predicted model count - a count produced by entering 
certain character states and/or covariates into the model, using the 
variable coefficients given in the LIMDEP output. Perhaps there is even 
specific LIMDEP code that we can use to generate the random effect 
estimates for each panel?
All our variables are either effect coded and balanced, or continuous 
variables transformed so that each variable estimated has a mean of 0 
(zero). Hence, if we make a prediction at the mean of the variable 
states/values, all that really affects the model prediction is the 
intercept, and the predicted count will be Exp[intercept]. How then do we 
use the ?alpha? value given by LIMDEP to calculate/illustrate the gamma 
distribution around that Exp[intercept] value?
If it may be of help in illustrating the solution to our problem, let us 
ASSUME that LIMDEP produces a model intercept coefficient =0.1 and an 
alpha = 0.4. Because E[exp(ui)] has a variance of 1/? = alpha, it should 
come that ? = 1/alpha = 2.5. Also, because the coefficient of the 
intercept is 0.1 the predicted count from the model at the variables means 
(ignoring the random effect) is 1.105. However, we suspect that the 
predicted count may not be part of this step but that we should first 
visualize the distribution of the intercept?s coefficient; then transform 
that to the model predicted count distribution ? right? If anybody could 
show us the calculation steps necessary to arrive at the shape (a) and 
scale (b) variables that are required to plot the gamma distribution (e.g. 
by the Excel GAMMADIST function), we would be most grateful!
Sincerely,
Bjorn Lardner / ecologist