Triple hurdle model for sequential binary decisions

[Vikram SP]

Dear All,

I am working on a data set which has sequential binary decisions and am wondering whether triple hurdle model can be run.
The model is something like this-
1. Whether to participate or not, Y1= Probit
2. If participate, propose less than market price or more than market price, Y2=probit
3. If propose less than, whether buy or Not, Y3= Probit.

It would be really nice if you could please place your suggestions on this and preferably syntax for running the model.
Thank you in advance!
Vikram

[Stephen Jenkins]

There is a large literature, mainly by sociologists, that has looked at these sorts of models. Google on "Mare model" which looks at sequential binary choices in education transition contexts. Also look at the 'sequential logit' contributions by Statalist member Maarten Buis.
Applications of these models usually ignore the possibility that the errors in the equation at each stage may be correlated. Some economists (including Heckman and collaborators, see e.g. http://https://www.jstor.org/stable/10.1086/250010) have noted that this may mean that an important aspect of 'selection' is being ignored, and so they propose shifting to probit (rather than logit) and formulating the cross-equation error correlations using the multivariate normal distribution. Such models may be hard to identify (in the statistical sense). More generally, one is dealing with one problem (selection) at a potential cost (functional form assumptions).
The Heckman type approach could be programmed using the approach and software provided by Cappellari & Jenkins, Stata Journal, 2006, 6(2). Free download from SJ website. The tricky parts include getting the expressions for the likelihood function correct ... and then adapting our code appropriately. But there should be expressions "out there" that you could find.

Example reference: https://pure.sfi.dk/ws/files/233504/WP_11_2008.pdf . Subsequently published as Anders Holm and Mads M. Jæger, (2011), ‘Dealing with Selection Bias in Educational Transition Models: The Bivariate Probit Selection Model’, Research in Social Stratification and Mobility. 29(3), 239-350.

[Vikram SP]

Thank you very much Stephen for the wonderful information.
I will explore this.