Micro-econometrics: discrete choice models

In this post, I want to discuss some statistical models from [1]. I'll implement these models in GAMS. First of all to emphasize these are all (nonlinear) optimization problems. Instead of using canned routines using a statistical package, this can help to get a better understanding of what is really going on. At least for me, not using a black-box routine, forces me to understand the underlying optimization models. Another application can be to have this part of a larger GAMS model. Some mathematical programming models just need some estimation code before the real model can be attacked. If the rest of the model is in GAMS, it may be a little bit easier to also use GAMS in the estimation tasks, instead of using a statistical package. This actually happens quite a lot. Finally, it may be easier to add constraints using a GAMS formulation compared to a canned routine in a statistical package.

In this case, the first argument was the reason for using GAMS. Reproducing results is for me a good tool to help me understand a dense text.  

Median, quantiles and quantile regression as linear programming problems

Quantile regression is a bit of an exotic type of regression [1,2,3]. It can be seen as a generalization of \(\ell_1\) or LAD regression [4], and just as LAD regression we can formulate and solve it as an LP.

First I want to discuss some preliminaries: how to find the median and the quantiles of a data vector \(\color{darkblue}y\). That will give us the tools to formulate the quantile regression problem as an LP. The reason for adding these preliminary steps is to develop some intuition about how Quantile Regression problems are defined. I found that most papers just "define" the underlying optimization problem, without much justification. I hope to show with these small auxiliary models how we arrive at the Quantile Regression model. Along the way, we encounter some interesting titbits. I'll discuss a few details that papers typically glance over or even skip, but I find fascinating.