This chapter discusses the mathematical structure of the set of stable matchings, including some computational algorithms. Some readers may prefer to simply skim this chapter, at least on first reading.

The core of a game

One of the most important "solution concepts" in cooperative game theory is the core of a cooperative game. In this section we will see that the set of stable matchings in a marriage problem is equal to the core of the game. (In subsequent chapters we will see that in some of the more complex two-sided matching markets we will consider, the set of stable matchings may be only a subset of the core, and in the case of many-to-many matching, [pairwise] stable matchings need not even be in the core at all.)

As discussed earlier, when we formally model various kinds of games, we will generally want to specify a set of players, a set of feasible outcomes, preferences of players over outcomes, and rules, which determine how the game is played. In how much detail we will want and need to specify the rules will depend on what kind of phenomena we are describing, and what kind of theory we are trying to construct. It is often sufficient to summarize the rules of the game by specifying which coalitions (i.e., subsets) of players are empowered by the rules of the game to enforce which outcomes. (Thus in our analysis of the marriage market in Chapter 2, we concentrated on the fact that in order for a marriage to take place, it is necessary and sufficient for the man and woman involved to agree.)

Introduction

Misspecification of the model may result in inconsistency of maximum likelihood estimates and of estimators of their standard errors. We have seen examples of this in the last chapter. In this chapter we shall discuss some numerical and graphical methods for detecting such misspecification, whose implementation ought to be a routine part of the econometric analysis of transition data. Of course, the main way in which an econometrician detects whether his model is wrong is when the estimates do not make economic sense. But the purely statistical tests we describe in this chapter are important supplements to the economist's judgment and are particularly valuable when economics does not provide a clear guide to what are sensible results.

When an econometrician proposes to fit a model f(t | x) to data drawn from N people with regressor vectors x he is tentatively asserting that the data distribution, conditional on x, is in the family t - which may be specified fully or semi-parametrically - and that this is true for all people. There is both an assumption about functional forms and an assumption about homogeneity conditional on x. Much of this book has been about data that are not homogeneous and this is a natural preoccupation of a social scientist. The diversity of people is surely greater than can be accounted for by the values of five or ten regressor variables. The emphasis in this chapter will therefore again be on the problem of heterogeneity. But we shall also point out the form taken by some tests for incorrect functional form.

Introduction and Overview

To construct the likelihood functions described in chapter 8 requires that an investigator assume a form for the joint probability density function of the data up to a finite set of unknown parameters. That assumption constitutes information supplied by the investigator, and since it gives the complete joint data distribution it is appropriate to call it full information. It is possible to carry out an econometric analysis using assumptions that amount to less than such a full specification. It seems useful to call such analyses limited information by analogy with the econometric simultaneous equations model in which limited information analysi requires less than a full specification of all equations in the system. These methods are also called semi-parametric.

The situation in which economic theory suggests only part of the model for data is rather common in econometrics. It corresponds to the idea that there may be components of a model specification that are, from the economic point of view, at best uncertain and at worst entirely arbitrary. We have already considered at some length the case in which theory might indicate a specification of the hazard function of the form ϴ = νμ(x1β) given the regressor vector x and the unobserved scalar ν. A complete model then requires us to specify the density function of V given x, and economic theory offers little guidance on this. In this case, theory does not lead us to a complete model and the question then arises as to whether we can devise a likelihood function that will enable us to make inferences about β while allowing us to avoid specifying the distribution of V.

This chapter looks at models of many-to-one matching between firms and workers that generalize the college admissions model in two important ways, by allowing firms to have a larger class of preferences over groups of workers, and by explicitly putting money into the model, so salaries are determined as part of the outcome of the game, rather than specified in the model as part of the job description. Of course, when we look at a model that does both these things together, we will also have to specify how the preferences of firms and workers deal with different combinations of job assignments and wages. Section 6.2 considers a version of a model proposed by Kelso and Crawford, with both complex preferences and negotiated wages. (We model wages here as a discrete variable, which is a natural modeling assumption since, for example, contracts cannot specify wages more closely than to the nearest penny. The next chapters model wages as a continuous variable, which also has some advantages.) But first we construct a simpler model in which we can examine complex preferences while continuing to treat salaries as an implicit part of the job description. Throughout this chapter we continue to make the simplifying assumption that workers are indifferent to which other workers are employed by the same firm.

Introduction and Overview

The process of movement from state to state generates a sequence of points on the time axis - the times at which transitions are made. Since movements are probabilistic the passage of a person over time is a realisation of a stochastic, point process. There is a very large literature dealing with the mathematical properties of such processes. Much of this literature deals with the long-run or equilibrium properties when the transition probabilities are constant over time and the process is stationary. In econometrics we usually have to model processes observed over rather short periods of time and which are not stationary, so results from the theory of point processes are not directly relevant. Nevertheless some knowledge of the basic stochastic processes is helpful in thinking about the properties of econometric data as is illustrated by the following example.

Many government statistical services collect and publish information about the duration of unemployment. They obtain this by sampling the population of registered unemployed people and asking them how long they have been unemployed and then they collect the answers in a grouped frequency distribution. Table 5.1 below gives such a distribution for the UK in 1984.

Introduction and Overview

The theoretical basis for a specification of a hazard function is a model of optimal choice by the agents whose transitions are to be studied. Such a model can be specified in great detail leading to a hazard function that is determined by economic theory up to some set of unknown parameters. Or the investigator might abstain from such a detailed specification, choosing instead only to let economic theory suggest the relevant regressor variables and the probable directions of their effects. Both approaches have been used in the econometric literature. Sometimes the former is called a structural approach and the latter, rather disparagingly, a reduced form approach. There is, however, no clear distinction between them but rather only a difference of degree. In chapter 6 we shall give an account of some structural models whereas in this chapter and the next we shall be concerned with families of models whose functional form is not, precisely, dictated by economic-theoretical considerations, but which are convenient vehicles for an econometric investigation. Such families need to allow for the following facts:

1. The duration distributions of different people differ because, for example, they face different prices, have different wealth and income, and have differing stocks and types of human capital.

2. These sources of difference can be represented by a regressor vector, x, for each person, where x may have components which should, according to our economic theory, have been measured, but were not.

3. The regressor vector may have elements that are functions either of calendar time, for example, regressor values changing over the business cycle, or of duration, which is time measured from the date of entry to the state. An example is the stream of unemployment benefits, which in the British system used to vary both with the duration of unemployment and among persons. Thus we have, in general, x = x (t, s) where, say, t is duration and s calendar time. Though in our notation we shall sometimes suppress the possible time dependence of x, it must be borne in mind.

We close with a few open questions and suggestions of possible directions for further research. In keeping with the emphasis of the book on both analysis and modeling, some of the questions and directions are of each type; that is, some call for the statement and proof of theorems, whereas progress on others will (first) involve the construction of new models.

1. Since many entry level labor markets and other two-sided matching situations don't employ centralized matching procedures, and yet aren't observed to experience the kinds of market failure that seem to be associated with unstable matching, we can conjecture that at least some of these markets reach stable outcomes by means of decentralized decision making. So one of the chief modeling problems that will arise in studying such markets will be to develop decentralized models of stable matching. (We noted that a consequence of Theorem 2.33 is that a random process that begins from an arbitrary matching and continues by satisfying a randomly selected blocking pair must eventually converge with probability one to a stable matching, provided each blocking pair has a probability of being selected that is bounded away from zero. Perhaps this kind of result will provide the building blocks for models of decentralized matching.)

2. …

This chapter presents one of the generalizations of the assignment game, in which agents' preferences may be represented by nonlinear utility functions. So in this model agents are allowed to make somewhat more complex tradeoffs than in the assignment model between whom they are matched with and how much money they receive. Nevertheless, each of the principal results we proved for the marriage market has a close parallel in the present model.

This model is a variant of a model introduced by Demange and Gale (1985). The only difference between the model presented here and their model is in the definition of feasible outcomes, which here allow monetary transfers to be made not only among matched pairs of agents, but also among arbitrary coalitions of agents, as in the assignment model and the one-seller model explored in the previous two chapters. Aside from making this model a generalization of these other models, this change allows us not to rule out a priori the kinds of strategic opportunities available to coalitions of bidders, for example, that we discussed in Sections 1.2 and 7.2.1. However most of the results from Demange and Gale's model carry over unchanged to the case when monetary transfers are allowed between unmatched agents. The reason is that as in the assignment game, no such transfers are made at stable outcomes. That is, we will see that the only monetary transfers that occur at stable outcomes are between agents who are matched to each other.

The formal model

There are clear similarities between the hospital intern market and the simple model of a marriage market studied in the previous chapters. There are two kinds of agents, hospitals and medical students, and the function of the market is to match them. (Strictly speaking, we should speak of hospital programs rather than hospitals, because different internship programs within a hospital are separately administered, and students apply to specific programs.) Because interns' salaries are part of the job description of each position, and not negotiated as part of the agreement between each hospital and intern, salaries will not play an explicit role in our model, but will simply be one of the factors that determine the preferences that students have over the hospitals. Similarly, we will assume that hospitals have preferences over students - that is, they are able to rank order the students who have applied to them for positions, as they are asked to do by the National Resident Matching Program. The major difference from the marriage problem is that each hospital program may employ more than one student, although each student can take only one position. (All the positions offered by a given hospital program are identical, since hospitals offering different kinds of positions must divide them into different programs.) The rules of the market are that any student and hospital may sign an employment contract with each other if they both agree, any hospital may choose to keep one or more of its positions unfilled, and any student may remain unmatched if he or she wishes (and seek employment later in a secondary market).

Fully parametric inference is where the investigator specifies the joint distribution of the data completely apart from a fixed, finite number of unknown parameters. This distribution provides the likelihood function whose study is the basis of inference both about the unknown parameters and about the adequacy of that distribution as a model for the process generating the data. The joint distribution depends upon two factors. The first is the specification of the probability law governing the passage of individuals from state to state. For a Markov or semi-Markov process this amounts to specifying the transition intensities - how they depend upon the date, upon the elapsed duration, upon both constant and time-varying regressors, possibly including unmeasured person-specific heterogeneity. The second is the sampling scheme, in particular whether we have sampled, for example, the population of entrants to a state, the population of people regardless of their state, or the population of members of a particular state. Thus we can identify four stages in fully parametric inference.

As in any game-theoretic analysis, it will be important in what follows to keep clearly in mind the “rules of the game” by which men and women may become married to one another, as these will influence every aspect of the analysis. (If, for example, our imaginary village were located in a country in which a young woman required the consent of her father before she could marry, then the fathers of eligible women would have a prominent role to play in the model.) We will suppose the general rules governing marriage are these: Any man and woman who both consent to marry one another may proceed to do so, and any man or woman is free to withhold his or her consent and remain single. We will consider more detailed descriptions of possible rules (concerning, e.g., how proposals are made, or whether a marriage broker plays a role) at various points in the discussion.

The formal (cooperative) model

The elements of the formal model are as follows. There are two finite and disjoint sets M and W: M = {mi m2, …,mn} isthe set of men, and W= {w1, w2,..., wp} is the set of women. Each man has preferences over the women, and each woman has preferences over the men. These preferences may be such that, say, a man m would prefer to remain single rather than be married to some woman w he doesn't care for.

In these chapters we will examine in detail the two-sided matching market without money that arises when each agent may be matched with (at most) one agent of the opposite set. For obvious reasons this model is, somewhat playfully, often called a "marriage market," with the two sets of agents being referred to as "men" and "women" instead of students and colleges, firms and workers, or physicians and hospitals. We will follow this practice here. In this whimsical vein, it may be helpful to think of the men and women as being the eligible marriage candidates in some small and isolated village.

The marriage market will be simpler to describe and investigate than a labor market in which a firm may employ many workers. And we will see in Part II that many (although not all) of the conclusions reached about this model will also apply to the hospital intern market, in which a hospital, of course, typically employs many interns. The marriage market will therefore be a good model with which to begin the mathematical investigation. In some of the discussion that follows, it will nevertheless be helpful to remember that much of our interest in this problem is motivated by labor markets, rather than by marriage in its full human complexity. (Thus we will sometimes speak about courtship, but never about dependent children or mid-life crises.)

Chapters 5 and 6 will explore models of many-to-one matching. In terms of the economic phenomena that motivate the models in this book, many to- one matching in two-sided markets is perhaps the most typical case, where one side of the market consists of institutions and the other side of individuals. Thus colleges admit many students, firms hire many workers, and hospitals employ many interns, all at the same time. But students typically attend only one college (at least at any given time), and so forth.

A central issue in formulating a model of many-to-one matching will be how to model the preferences of the institutions, since these involve comparisons of different groups of students, workers, and so on. No comparable question arose in the marriage model, where preferences over individuals were sufficient to determine preferences over matchings. We will see that there are both important differences and striking similarities between one-to-one and many-to-one matching. The results presented in Chapter 5 will also complete the explanation begun in Section 1.1 of the phenomena described there concerning the labor market for medical interns. And some further empirical observations in related markets will briefly be described.

Introduction

An Overview of the Book

This book is about the movement of individuals among a set of states, and a transition is a movement from one state to another. Transition data record the sequence of states that were occupied and the times at which movements between them occurred. The states will be finite in number and may be defined in any way that the economics of the problem suggests is useful. All that is required is that they be clearly defined and that we have a rule for telling which state a person is in at each moment of time. Some simple examples of states might be

1. Unemployed,

2. Employed,

3. Out of the labour force;

or

1. Married,

2. Unmarried;

or

1. Employed,

2. Retired.

Our concern will be with the passage of people among such sets of states. We shall give an account of the building of models for such movement, of the fitting of such models to data, and of the use of models and data to test economic and statistical hypotheses. In the first part of the book - chapters 1 through 6 - we shall describe ways of building models for fitting to transition data. In the rest of this introductory chapter we shall give an account of the two fundamental tools for the study of duration data, the hazard function and the Exponential distribution, and we shall examine some duration data.

The last part of this book examines how econometricians formalised and generalised their ideas on the relationship between economic theory and economic data. We see that they developed three formal models of the way in which observed statistical relationships might correspond to their expected theoretical relationships. Each model provided an explanation of the data-theory gap and a rationalisation of the approximate, rather than exact, fit found in measured economic laws. One model explained these approximations in terms of measurement errors: the errors-in-variables model. Another explanation rested on variables omitted from the measurement equation: the errors-inequations model. The third model provided a more general explanation of the relationship between empirical results and economic theory by treating the theoretical relationship as probabilistic. Each of the three models was associated with an appropriate statistical analysis.

So far, this history has concentrated on the development of econometrics within its applied context. Econometricians have been portrayed as responding (not always successfully) to problems thrown up in their applied work. These last chapters provide a more integrated history of the field. They draw both on the applied work of the period (discussed in the earlier chapters) and on the theoretical discussions of econometricians, which began in earnest only in the 1930s. In particular, Chapter 7 offers a synthetic reconstruction of the development of the ideas involved in formal models. It concentrates on the development of the errors-in-equations and errors-in-variables models and brings in the probability model only briefly.

It is no accident that the standard economic model used in textbooks to illustrate the identification problem in econometrics is the simple market demand and supply model, for it was while attempting to estimate demand elasticities that pioneering econometricians first discovered the identification problem and related correspondence problems.

In a general sense identification is concerned with the correspondences between economic activity, the data that activity generates, the theoretical economic model and the estimated relationship. In early econometric work on demand, these correspondence problems were attacked in various ways. In some cases, investigators examined their data to see which relationships could be estimated and then sought to interpret these relationships in terms of economic theory. This involved two ideas of identification: one of identifying as ‘interpreting’ an estimated relationship in terms of economic theory, and secondly, one of identifying as ‘locating’ a relationship in the data. This notion of locating a relationship in the data has its closest modern counterpart in economic time-series analysis, which is concerned with finding a model which characterises the data given certain criteria. In other early work, investigators started with economic theory models and examined the circumstances under which these models could be estimated. This latter approach is nearest to the present-day scope of ‘the identification problem’, which deals with the question of whether the parameters of a model can be uniquely determined, given that the model is known (that is, assumed to be the ‘true’ model).

Wiliam Stanley Jevons was one of the first economists to break away from the casual tradition of applied work which prevailed in the nineteenth century and combine theory with statistical data on many events to produce a general account of the business cycle. Then, in the early twentieth century when statistical work was still uncommon, Henry Ludwell Moore adopted more sophisticated techniques to develop an alternative theory of the cycle. This chapter relates how Jevons and Moore set about building their theories. Both of them relied heavily on statistical regularities in the formation of their hypotheses, which featured periodic economic cycles caused by exogenous changes in heavenly bodies. The development of the econometric approach to business cycle analysis between the 1870s and the 1920s is shown in their work, and in its changing reception from reviewers. Jevons' and Moore's cycle work deserves to be taken seriously as pioneering econometrics, yet the periodic cycle programme they initiated did not prove to be very influential. The last section of this chapter explores why this was so.

Jevons' sunspot theory

Jevons' sunspot theory of the cycle has always been the object of mirth to his fellow economists, despite the fact that by the time he began to work on the subject in the 1870s he was already well known and distinguished for his contributions to mainstream economics. He was also renowned for statistical analyses of various problems, particularly the problem of index numbers.