ApolloChoiceModelling forum

Hi,

I have been estimating a mixed logit discrete choice model in Apollo and I was receiving errors indicating positivity in the hessian and some NaN's for standard errors while using the maxLik package to estimate the hessian. If I use the numDeriv package the model has stabilised (okay hessian and all se's estimated). Should I be concerned? I have estimated the model using other software and output/model is stable (converging okay, negative hessian and se's okay). Parameter estimates are consistent for all versions. I have been estimating at 6000 sobol draws (3030 obs, 11 random parameters and 11 fixed parameters).

Interested to hear your thoughts on the impact of using these different packages to estimate the hessian in Apollo.

Cheers

Hi Jane,

What version of Apollo are you using? You can see it when you load the library.

Thanks
David

Currently I am running Apollo 0.2.1 and the comments above are from working with this version, but I was using 0.0.9 and actually (if I remember correctly i had specified the maxLik package) in this version I didn't have the warnings about positive hessian and nan's for se.

Cheers,
Alaya

Hi Alaya,

Newer versions of Apollo include more information on the Hessian than older versions. For example, the biggest eigenvalue was not reported prior to v0.1.0. So it may be that the issue was present in v0.0.9, but it simply wasn't reported. You can check this by manually calculating the eigenvalues for the model$hessian object.

Since v0.2.0, Apollo supports three ways to calculate the Hessian:

• By numerically differentiating the analytical gradient of the model (only available for the most common models)

• Full numerical differentiation using the numDeriv package

• Full numerical differentiation using the maxLik package

In theory, they should all render the same results, but in practice there may be differences due to numerical issues. By default, Apollo will try the methods in the described order, e.g. jumping to the second if the first one fails. When available, we always recommend using the analytical gradient, as it is faster to compute and relies less on numerical approximations. The numDeriv package uses more precise approximations than maxLik does, but in our experience is slightly less robust.

In summary, if Apollo does manage to calculate the Hessian (and the corresponding s.e.) I wouldn't worry too much about the method used to obtain it, as they are all consistent among them. If you get the warning about positive eigenvalues (which happens more often in complicated models with multiple local maximum), you might be in a saddle point, so I would recommend trying different starting values in the hope of finding a better solution. You can do this automatically with the apollo_searchStart function.

Cheers
David