ACCIÓN TÉRMICA.

A core motivation for the thesis has been the assertion that, in studying sorting and com- petition in education systems, it is vital to understand family preferences and decision- making with regards to school choice. Faced with a lack of data on choices, and an abundance of data on allocations, we sought a way to reveal information about the pref- erences underlying allocations. In order to achieve this, we investigated the branch of game theory dealing with two-sided matchings, known as stable matching theory, and the current state of the art regarding structural estimation of these models. We found that current structural models were either too restrictive, requiring strong restrictions on model form that could not be met in our context, or prohibitively computationally demanding. This motivated a re-writing of the incomplete likelihood model for stability, leading to an explicit functional form for the likelihood and more flexible options for estimation.

Apart from the immediate benefit of being able to implement the model to answer our research questions, the investigation led to informal insights about the properties of this class of models, which it is hoped will aid the intuitive understanding of the field for applied researchers. Using simulation evidence it was conjectured that two-sided models possess good large-sample properties for bias and identification in one-to-many matching markets (such as school markets). Such arguments, alongside the large sample sizes we have been able to achieve by combining datasets, lead to confidence in the validity of using two-sided models in the context of the thesis.

10.2.1 Assumptions impacting upon the validity of results

However, these arguments assume that the model is complete and perfectly specified. In developing the model, Chapter 5 described a number of simplifying assumptions that had to be made in the specification of both pupil and school utility functions. The most important assumptions are discussed here.

The first assumption, which motivated the entire empirical approach, was that the ob- served allocations constitute a stable matching. This assumption was justified on the grounds that each local authority’s admissions system uses stable matching algorithms

to compute the allocation. However, it is possible that a small number of pupils admit- ted during the year – i.e. outside the normal admissions process – might create blocking pairs, as a school might not usually prefer those pupils to other pupils the school has rejected. There has been little work on the sensitivity of two-sided models to small devi- ations from stability; this should be a priority for further work on empirical matchings. It is also assumed that parents express their true preferences on the application form. Although expressing one’s true preferences is a weakly optimal strategy for stable match- ing markets, Chapter 8 discusses possible conditions under which this assumption might fail.

Of the two sides of the market, the school side was most limited in terms of available explanatory variables. We lacked information about siblings, religious observance, special educational needs, looked after children, feeder schools, and catchment areas, all of which have some bearing on probability of admission to a school. The sparsity of the school-side model risks introducing omitted-variables bias into estimates of preference parameters on the pupil side. Another direction for future research on empirical matching models is the extent to which each side of the model is sensitive to omission of variables on the other side of the model.

It is also not known how sensitive the model is to the capacity of each school. In the absence of data on capacities we have assumed that each school’s capacity is equal to its intake, implying that all schools are full. In future work it would be advisable to obtain data about school capacities, and generalise the model to allow for schools with empty places.

On the pupil side of the model the dataset was sufficiently detailed to allow us to investi- gate rich interactions between pupil and school characteristics. However, there were two restrictions in the pupil utility function that may have limited the ability of the model to predict aggregate demand. First, there were no fixed effects for schools in the model. Since these terms ensure that predicted aggregate demand is equal to observed aggregate demand in the sample, their omission might be expected to affect estimates of aggregate demand.

The other main restriction of pupil utility functions is the assumption that the unobserved portion of utilityas is uncorrelated across alternatives for each pupil. Whilst it is not

uncommon for discrete choice studies to make this assumption, it is well-known that assuming uncorrelated residuals limits the ability of the model to capture substitution patterns in aggregate demand. However, it is not clear whether a model with correlated residuals would even be identifiable in the context of a two-sided model. In Chapter 5 it is argued that the abundance of terms interacting pupil and school characteristics creates rich substitution patterns in aggregate demand, so that the omission of substitution-on- unobservables is less serious a concern.

10.2.2 Other applications of two-sided matching models

The two-sided matching paradigm has wide applicability in social science and Economics, and there are several possible areas in which two-sided empirical models may be useful. Two-sided empirical models have already been used to study topics such as the matching of teachers to schools (Boyd et al., 2013), or industrial organisation (Sørensen, 2007). Although we have argued that the stability likelihood method works best in many-to-one markets where there are a large number of individuals for each institution, it is possible that further advances could be made in applications with one-to-one markets. This would enable the model to be used in diverse applications such as job matching, University admissions, the matching of directors to boards, or even the market for sportspeople in team sports.

It is even possible that stable matching models could be applied to one-sided matching markets. Analogous theories of stability and theoretical results exist for one-sided match- ing markets, in which individuals match with others of the same type (the so-calledStable Roommates Problem). If the empirical models we have investigated could be adapted, this may potentially provide a new avenue for the study of social networks. Although statistical models for the probability of forming an edge between two people already ex- ist in social networks research, these models currently do not allow the participants in the match to have separate utility functions for the match; therefore they are not struc- tural. However, there would be much work to be done to ascertain the identification and computational feasibility of such models.