Matching Markets 

 Rich's  post  on instruments the other day reminded me of a conversation that I've been having with a faculty member; although the connection may not be particularly clear, at least at first. 

 The setup is that there are many markets in which buyers and sellers are distinct types of actors, for example, the market for spouses has, until recently, been such a market (although I make no claim as to which side of the market is buying and which is selling).  This market, in the form of college applications, was analyzed by Gale and Shapley in a famous 1962  paper  in which they proved that there was a solution to this type of matching problem. 

  
 


 Another example, that motivated my interest in the question, is the market for medical residents (see  here ).  Shortly before graduation medical students apply for positions with various residency programs across the country by submitting rank order lists to a central clearinghouse; residency programs enter into a similar process of ranking medical students.  The clearinghouse then produces an assignment that is optimal in an economic sense. 

 Unfortunately, this setup does not permit the applied researcher (or poor grad student) much traction for identifying the effect of being assigned to a particular residency program. 

 One solution comes from some  work  by Morten Sorensen on matching in venture capital.  His idea is to model the decision process leading to investments by venture capitalists in early stage companies and at the same time to model his outcome of interest (company goes public) thus allowing for correlations between the respective error terms of the attractiveness / matching model and the outcome equation.  Sorensen makes the point that this method makes use of the characteristics of other investors and investments in the market as instruments in order to address the fact that "better" investors may invest in "better" companies. 

 While in principle this method is attractive, it is computationally difficult and does not convince everyone--the faculty member I was talking to agreed that in principle this method is attractive, but that the results would be more credible with an instrumental variable that affects the probability of being assigned to a particular program.  However, he also pointed out the value of these structural models--they provide estimates that may be valid over a broader range of values and can be used to do policy experiments that one may not be able to do with a model identified by instrumental variables.