Abstract
This note studies the empirical content of a simple marriage matching model with transferable utility, based on Becker (J Polit Econ 81:813–846, 1973). Under Becker’s conditions, the equilibrium matching is unique and assortative. However, this note shows that when the researcher only observes a subset of relevant characteristics, the unique assortative matching does not uniquely determine a distribution of observed characteristics. This precludes standard approaches to point estimation of the underlying model parameters. We propose a solution to this problem, based on the idea of “random matching.”
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Notes
The assumptions are similar to those of the transferable utility model studied in Chiappori et al. (2012), although we do not assume that X and \(\epsilon \) (Y and \(\eta \)) are separable in the single index functions.
In Becker’s original formulation, these single indices are interpreted as the time inputs that a husband or wife contributes toward household production. See also Roth and Sotomayor (1990).
As such, the fact that \(\epsilon \) and \(\eta \) are unobserved has nothing to do with the underlying indeterminacy problem. Even if \(\epsilon \) and \(\eta \) were observed by the researcher, the index structure of the problem still causes the joint distribution of (X, Y) indeterminate in equilibrium.
This is the joint distribution of (U, V) having marginal distributions \(F_U\) and \(F_V\) with maximal positive correlation between U and V; see, for example, Joe (1997).
In the assumption and the rest of this section, we ignore the parameter \(\theta \) for notational simplicity.
See Gourieroux and Monfort (1997), for example.
We also tried Nelder–Mead, but the performance is poor because it tends to get stuck in local minima.
In a sense, this is not a fair comparison because the canonical correlation method is not shown to be consistent when the coefficients are random or when the covariates are non-normal.
We also estimated Model 3, although the result is harder to interpret because of the different way of normalization. The results are available upon request.
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We thank Alejandro Robinson-Cortes for research assistance.
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Cao, J., Shi, X. & Shum, M. On the empirical content of the Beckerian marriage model. Econ Theory 67, 349–362 (2019). https://doi.org/10.1007/s00199-018-1106-z
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DOI: https://doi.org/10.1007/s00199-018-1106-z