gmnl in WTP space: distribution of the price coefficient

[Simone Angioloni]

Hi,

I am using the gmnl Stata command to estimate a generalised multinational model with random parameters in willingness to pay space:

Fitting the Generalized Multinomial Logit Model in Stata - Yuanyuan Gu, Arne Risa Hole, Stephanie Knox, 2013 (sagepub.com)

I would like to run a Poe test of the distribution of the price coefficient as in:

Computational Methods for Measuring the Difference of Empirical Distributions - Poe - 2005 - American Journal of Agricultural Economics - Wiley Online Library

I do not have any problem to do this for all the other random coefficients of the model. However, I am not sure about the distribution of the price coefficient. Below, the model I estimated and the distributions generated (start matrix of initial values omitted):

constraint 1 [Mean]mprice = 1
gmnl choice mprice, rand(x1 x2t) group(cs) id(id) het(const) constraint(1) gamma(0) from(start, copy) nrep(500)
matrix b = e(b)
matrix V = e(V)
drawnorm mprice mean_x1 mean_x2 het sd_x1 sd_x2 tau, mean(b) cov(V) n(1000) seed(183831)

The variable 'mprice' in the line above is just an expedient to match the structure of e(b) and e(V) since just a column of 1 is generated. Given that there is a constant of normalisation set to -0.5*tau^2 as explained in slides 5 and 6 here:

DCMD.pdf (sheffield.ac.uk)

I am wondering if the corresponding distribution of the price coefficient is just -exp (-0.5*tau^2 +theta*Z) or just -exp(theta*Z).

Any help would be greatly appreciated.

Simone

[Hong Il Yoo]

The price coefficient is given by -1 * exp(-0.5*tau^2 + theta*Z + tau*v) where v is a standard normally distributed random variable. In other words, the intercept term for the natural log of -1 times your price coefficient is equal to -0.5*tau^2 + the coefficient on the constant regressor, -const-.

If you're interested in the WTP space model and your model specification does not account for observed heterogeneity in the scale parameter, perhaps you can use Arne Risa Hole's -mixlogitwtp- command instead: To download, -ssc install mixlogitwtp- as usual. With this command, you don't have to deal with -0.5^tau^2 or generate the constant regressor manually: Stata will produce a coefficient on -mprice- which corresponds to -0.5*tau^2 + the coefficient on the constant regressor in your -gmnl- specification.

[Simone Angioloni]

Thanks Hong, this is clear and useful.

[Simone Angioloni]

Dear Hong,

Apologies in advance if the question below is redundant, but I would like to double check two aspects about generating the distribution of the price coefficient with gmnl.

Given the definition of the price coefficient as -1 * exp(-0.5*tau^2 + theta*Z + tau*v), once estimated the model, I just need to take a random sample of the standard normal error v and plug it into the definition above with the estimated parameters?

Besides, in case theta*Z does not just include the constant term but a covariate for observed heterogeneity in the scale parameter such as Z1, I am wondering if using the mean value of this regressor is appropriate when generating the distribution of the price coefficient. I mean this: say v1 is the generated standard normal random error, the distribution of the price coefficient would be:

-1 * exp(-0.5*tau^2 + theta0*constant + theta1*mean(Z1) + tau*v1)

Thank you very much,

Simone