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1、1 Unobserved Heterogeneity in Matching Games Jeremy T. Fox University of Michigan and NBER Chenyu Yang University of Michigan July 2012 Abstract Agents in two-sided matching games vary in characteristics that are unobservable in typical data on matching markets. We investigate the identification of
2、the distribution of these unobserved characteristics using data on who matches with whom. The distribution of match-specific unob- servables cannot be fully recovered without information on unmatched agents, but the distribution of a combination of unobservables, which we call unobserved complementa
3、rities, can be identified. Knowledge of the unobserved complementarities is sufficient to construct certain counterfactuals. The distribution of agent-specific unobservables is identified under different conditions. Thanks to Stephan Lauermann, Ariel Pakes, and seminar participants at various confer
4、ences and univer- sities for helpful suggestions. Our email addresses are jtfoxumich.edu and chnyyangumich.edu. 2 1 Introduction Matching games model the sorting of agents to each other. Men sort to women in marriage based on characteristics such as income, schooling, personality and physical appear
5、ance, with more desirable men typically matching to more desirable women. Upstream firms sort to downstream firms based on the product qualities and capacities of each of the firms. This paper is partially motivated by such applications in industrial organization, where downstream firms pay upstream
6、 firms money, and thus it is reasonable to work with transferable utility matching games (Koopmans and Beckmann, 1957; Becker, 1973; Shapley and Shubik, 1972). There has been recent interest in the structural estimation of (both transferable utility and non- transferable utility) matching games.1 Th
7、e papers we cite are unified in estimating some aspect of the preferences of agents in a matching game from data on who matches with whom as well as the observed characteristics of agents or of matches. The sorting patterns in the data combined with assumptions about equilibrium inform the researche
8、r about the structural primitives in the market, namely some function that transforms an agents own characteristics and his potential partners characteristics into some notion of utility or output. These papers are related to but not special cases of papers estimating discrete, non-cooperative (Nash
9、) games, like the entry literature in industrial organization (Bresnahan and Reiss, 1991; Berry, 1992; Mazzeo, 2002; Tamer, 2003; Seim, 2006; Bajari et al., 2010) and the discrete outcomes peer effects literature (Brock and Durlauf, 2007; de Paula and Tang, 2012). Matching games typically use the co
10、operative solution concept of pairwise stability.2 The empirical literature cited previously structurally estimates how various structural or equilib- rium objects, such as payoffs or preferences, are functions of the characteristics of agents observed in the data. For example, Choo and Siow (2006)
11、study the marriage market in the United States and estimate how the equilibrium payoffs of men for women vary by the ages of the man and the woman. Srensen (2007) studies the matching of venture capitalists to entrepreneurs as a function of observed venture capitalist experience. Fox (2010a) studies
12、 matching between automotive assemblers (downstream firms) and car parts suppliers (upstream firms) and asks how observed specialization measures in the portfolios of car parts sourced or supplied contribute to agent profit functions. The above papers all use data on a relatively limited set of agen
13、t characteristics. In Choo and Siow, personality and physical attractiveness are not measured, even though those characteristics are likely important in determining the equilibrium pattern of marriages. Similarly, in Fox each firms product quality is not directly measured and is only indirectly infe
14、rred. In Srensen, the unobserved ability 1See, among others: Dagsvik (2000); Boyd et al. (2003); Choo and Siow (2006); Srensen (2007); Fox (2010a); Gordon and Knight (2009); Chen (2009); Ho (2009); Park (2008); Yang et al. (2009); Logan et al. (2008); Levine (2009); Baccara et al. (forthcoming); Sio
15、w (2009); Galichon and Salanie (2010); Chiappori et al. (2010b); Crawford and Yurokoglu (forthcoming); Weese (2010); Christakis et al. (2010); Echenique et al. (2011); Menzel (2011). 2Transferable utility matching games can be seen as special cases of models of hedonic equilibrium (Brown and Rosen,
16、1982; Ekeland et al., 2004; Heckman et al., 2010; Chiappori et al., 2010a). Unlike the empirical literature on hedonic equilibrium, the estimation approaches in most matching papers do not rely on data on equilibrium prices or transfers. Compared to the current work, the hedonic papers do not allow
17、for unobserved characteristics. 3 of each venture capitalist is not measured. If matching based on observed characteristics is found to be important, it is a reasonable conjecture that matching based on unobserved characteristics is also important. Ackerberg and Botticini (2002) provide empirical ev
18、idence that farmers and landlords sort on unobservables such as risk aversion and monitoring ability, without formally estimating a matching game or the distribution of these unobservables. Our discussion of the empirical applications cited above shows that unobserved characteristics are potentially
19、 important. As the consistency of estimation procedures for matching games depends on assumptions on the unobservables, empirical conclusions might be more robust if the estimated matching games allow richly specified distributions of unobserved agent heterogeneity. This paper investigates what data
20、 on the sorting patterns between agents can tell us about the distributions of unobserved agent characteristics relevant for sorting. In particular, we study the nonparametric identification of distributions of unobserved agent heterogeneity in two-sided matching games. With the distribution of unob
21、servables, the researcher can explain sorting and construct counterfactual predictions about market assignments. This paper allows for this empirically relevant heterogeneity in partner preferences using data on only observed matches (who matches with whom), not data from, say, an online dating site
22、 on rejected profiles (Hitsch et al., 2010) or on equilibrium transfers, such as wages in a labor market (Eeckhout and Kircher, 2011). Transfers are often confidential data in firm contracts (Fox, 2010a) and are rarely observed in marriage data (Becker, 1973). In the following specific sense, this p
23、aper on identification is ahead of the empirical matching litera- ture because no empirical papers have parametrically estimated distributions of unobserved character- istics in matching games. Thus, this paper seeks to introduce a new topic for economic investigation, rather than to simply loosen p
24、arametric restrictions in an existing empirical literature. This paper contributes to the literature on the nonparametric (allowing infinite dimensional objects) identifica- tion of transferable utility matching games (Fox, 2010b; Graham, 2010). Our paper is distinguished because of its focus on ide
25、ntifying distributions of unobservables, rather than mostly deterministic functions of observables. Our focus on using data on many markets with finite numbers of agents in each (transferable utility) market follows Fox (2010a,b).3 We first consider a baseline model, which is stripped down to focus
26、on the key problem of identifying distributions of heterogeneity from sorting data. In our baseline transferable utility matching game, 3In addition to our study of identifying distributions of unobservables, there are many modeling differences between our paper and the literature on transferable ut
27、ility matching games following the approach of Choo and Siow (2006), including Galichon and Salanie (2010) and Chiappori et al. (2010b). We use data on many markets with finite numbers of players and different realizations of observables and unobservables in each market; the Choo and Siow approach a
28、pplies to one large market with an infinity of agents. We require at least one continuous, observable characteristic per match or per agent; the Choo and Siow literature allows only a finite number of observable characteristic values. Unobservables in the Choo and Siow literature are typically i.i.d
29、. shocks for the finite observable types rather than than unobserved agent characteristics or unobserved preferences on observed, ordered characteristics, such as random coefficients. This so-called “separability” assumption in the Choo and Siow literature has the key implication that the unobservab
30、le, economically endogenous transfers in the Choo and Siow literature are independently distributed (within the single market they consider) from the unobservable, economically exogenous econometric errors. The unobserved transfers in our model are not independently distributed from the unobserved a
31、gent characteristics, for example. 4 2 the primitive that governs sorting is the matrix that collects the production values for each potential match in a matching market. The production level of each match is additively separable in observable and unobservable terms. The observable term is captured
32、by a match-specific regressor. The unknown primitive is therefore the distribution (representing randomness across markets) of the matrix that collects the unobservable terms in the production of each match in a market. We call this distribution the distribution of match-specific unobservables. Matc
33、h-specific unobservables nest many special cases, such as agent-specific unobservables. We provide three main results, and some extensions. Our first main result states that the distribu- tion of match-specific unobservables is not identified in a one-to-one matching game with data on who matches wi
34、th whom but without data on unmatched or single agents. Our second main result states that the distribution of a change of variables of the unobservables, the distribution of what we call unobserved complementarities, is identified. We precisely define unobserved complementarities below. Our identif
35、ication proof works by tracing the joint (across possible matches in a market) cumulative distribution function of these unobserved complementarities using the match-specific observables. We also show that knowledge of the distribution of unobserved complementarities is sufficient for com- puting as
36、signment probabilities. Our third main result says that the distribution of the primitively specified, match-specific unobservables is actually identified when unmatched agents are observed in the data. Our three main results can be intuitively understood by reference to a classic result in Becker (
37、1973). He studies sorting in two-sided, transferable utility matching games where agents have scalar characteristics (types). He shows that high-type agents match to high-type agents if the types of agents are complements in the production of matches. Many production functions for match output exhib
38、it complementarities. Say in Beckers model male and female types are xm and xw , respectively. A production function with horizontal preferences, such as (xm xw ) , and one with vertical pref- erences, such as 2xmxw , can both have the same cross-partial derivative, here 2. Beckers result that compl
39、ementarities alone drive sorting means that data on sorting cannot tell these two production functions apart. In our more general class of matching games, our first main result is that we cannot identify the distribution of match-specific unobservables. Our second main result is that we can iden- ti
40、fy the distribution of our notion of unobserved complementarities. These two results are analogous to Beckers results for a more general class of matching games. Our third main result uses data on unmatched agents. In a matching game, agents can unilaterally decide to be single or not. If all other
41、agents are single and hence available to match, the fact that one particular agent is single can only be explained by the production of all matches involving that agent being less than the production from being single. This type of direct comparison between the production of being single and the pro
42、duction of being matched is analogous to the way identification proceeds in discrete Nash games, where the payoff of a players observed (in the data) strategy must be higher than strategies not chosen, given the strategies of rivals. Thus, the availability of data on 5 unmatched agents introduces an
43、 element of individual rationality that maps directly into the data and is therefore useful for identification of the primitive distribution of match-specific characteristics. Many empirical researchers might be tempted to specify a parametric distribution of match-specific unobservables. Our three
44、results together suggest that estimating a matching model with a parametric distribution of match-specific unobservables will not lead to credible estimates without using data on unmatched agents, as a more general nonparametrically specified distribution is not identified. One could instead impose
45、a parametric distribution for unobserved complementarities. We examine several extensions to the baseline model that add more empirical realism. Our baseline model imposes additive separability between unobservables and observables in the production of a match. We examine an extension where addition
46、al observed characteristics enter match production and these characteristics may, for example, have random coefficients on them, reflecting the random preferences of agents for partner characteristics. For example, observationally identical men are often observed to marry observationally distinct wo
47、men. One important hypothesis is that these men have heterogeneous preferences for the observable characteristics of women. In a model with random preferences, we identify the distribution of match production conditional on the characteristics of agents and matches other than the match-specific char
48、acteristics used in the baseline model. This object of identification follows identification work using special regressors in the multinomial choice literature (Lewbel, 2000; Matzkin, 2007; Berry and Haile, 2010). In another extension, we identify fixed-across-markets but heterogeneous-within-a-mark
49、et coeffi- cients on the the match-specific characteristics used in the baseline model. This relaxes the assumption that the match-specific characteristics enter the production of each match in the same manner. An- other extension considers models where key observables vary at the agent and not the match level. We can achieve identification of the primitive distribution of match-specific unobservables without relying on data on unmatched agents, but by imposing a perhaps stronger functional form for match production. Our results on one-to-one, tw