Empirical researchers who use matching models to study matching markets such as marriage are often faced with the difficulty that they can observe the results of the market--e.g. who marries whom--but not the intermediate choices that produced these matches (such as who courted whom, who proposed and was rejected, etc.). One approach is to look for particular markets in which such additional data can be found (as in these studies in India, the U.S., and Korea). Another is to develop statistical tools to infer the missing data, e.g. about the preferences of men and women, from the readily observable outcomes.
A NBER working paper that takes this latter approach is Identification in Matching Games, by Jeremy T. Fox - http://papers.nber.org/papers/W15092.
Here's the first paragraph from the introduction:
"Matching games are a new and important area of empirical interest. Consider the classic example of marriage. A researcher may have data on a set of marriages in each of a set of independent matching markets, say a set of towns. The researcher observes characteristics of each man and each woman in each town, as well as the sets of marriages that occurred. The researcher observes equilibrium outcomes, here marriages, and not choice sets, so identification in this type of model will not be able to rely trivially on the analysis of single-agent demand models. What type of parameters can be identified from these data?"
And here's the formal abstract:
Abstract: I study a many-to-many, two-sided, transferable-utility matching game. Consider data on matches or relationships between agents but not on the choice set of each agent. I investigate what economic parameters can be learned from data on equilibrium matches and agent characteristics. Features of a production function, which gives the surplus from a match, are nonparametrically identified. In particular, the ratios of complementarities from multiple pairs of inputs are identified. Also, the ordering of production levels is identified.