cupid_matching¶

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A Python package to solve, simulate and estimate separable matching models

Free software: MIT license
Documentation: https://bsalanie.github.io/cupid_matching
See also: An interactive Streamlit app

Installation¶

1	`pip install [-U] cupid_matching`

Importing functions from the package¶

For instance:

from cupid_matching.min_distance import estimate_semilinear_mde

How it works¶

The following only describes the general ideas. See here for more background and the API reference on this site for the technical documentation.

The cupid_matching package has code

to solve for the stable matching using our Iterative Projection Fitting Procedure (IPFP) in variants of the model of bipartite, one-to-one matching with perfectly transferable utility. It has IPFP solvers for variants of the Choo and Siow 2006 model with or without singles, homoskedastic and heteroskedastic; and also for a class of nested logit models
to estimate the parameters of separable models with semilinear surplus and entropy using a minimum distance estimator
to estimate the parameters of semilinear Choo and Siow models using a Poisson GLM estimator
for a Streamlit interactive app that demonstrates solving and estimating the Choo and Siow model using the cupid_matching package. You can try it here.

Incidentally, my ipfp_R Github repository contains R code to solve for equilibrium in (only) the basic version of the Choo and Siow model.

The package builds on the pioneering work of Choo and Siow JPE 2006 and on my work with Alfred Galichon, especially our REStud 2022 paper and this working paper.

At this stage, it only deals with bipartite models. As the heterosexual marriage market is a leading example, I will refer to the two sides as men and women. Each man $m$ (resp. each women $w$) has an observed, discrete-valued type $x$ (resp. $y$), along with unobserved heterogeneity.

The primitives¶

The primitives of this class of matching models are

the margins: the numbers $n_x$ of men of type $x=1,\ldots,X$ and the numbers $m_y$ of women of type $y=1,\ldots,Y$
the joint surplus created by the match of a man $m$ of type $x$ and a woman $w$ of type $y$. We assume separability: this joint surplus takes the form $$ \Phi_{xy}+\varepsilon_{my} +\eta_{xw}. $$ I will only describe here the case when the data also has singles, i.e., when $x=y=0$ is a possible type. The fuller documentation explains how to deal with the case when singles are not observed.

A single man has utility $\varepsilon_{m0}$ and a single woman has utility $\eta_{0w}$.

The modeler chooses the distributions of the vectors $(\varepsilon_{m0},\varepsilon_{m1},\ldots, \varepsilon_{mY})$ and $(\eta_{0w},\eta_{1w},\ldots,\eta_{Xw})$.

The solution: the stable matching¶

We denote $\mu_{xy}$ the number of matches between men of type $x$ and women of type $y$, $\mu_{x0}$ the number of single men of type $x$, and $\mu_{0y}$ the number of single women of type $y$.

The total numbers must add up to the margins: $$ \sum_{y=1}^Y \mu_{xy}+\mu_{x0}=n_x \; \text{ and } \; \sum_{x=1}^X \mu_{xy}+\mu_{0y}=m_y. $$ The total number of individuals is $N_i=\sum_{x=1}^X n_x+ \sum_{y=1}^Y m_y$ and the total number of households is $N_h = N_i - \sum_{x=1}^X \sum_{y=1}^Y \mu_{xy}$.

Galichon-Salanié (REStud 2022) shows that in large markets, if the vectors $\varepsilon$ and $\eta$ have full support, the stable matching is the unique solution to the strictly convex program $$ \max_{\mu} \left(\sum_{x=1}^X \sum_{y=1}^Y \mu_{xy}\Phi_{xy} + \mathcal{E}(\mu; n, m)\right) $$ where $\mathcal{E}$, the generalized entropy function, depends on the assumed distributions of the $\varepsilon$ and $\eta$ random vectors.

The files choo_siow.py, choo_siow_gender_heteroskedastic, choo_siow_heteroskedastic, and nested_logit provide EntropyFunctions objects that compute the generalized entropy and at least its first derivative for, respectively,

the original Choo and Siow 2006 model, in which the $\varepsilon$ and $\eta$ terms are iid draws from a type I extreme value distribution
the same model, without singles (to be used when only couples are observed)
an extension of 1. that allows for a scale parameter $\tau$ for the distribution of $\eta$
an extension of 3. that has type-dependent scale parameters $\sigma_x$ and $\tau_y$ (with $\sigma_1=1$
a two-layer nested logit model in which singles (type 0) are in their own nest and the user chooses the structure of the other nests.

Users of the package are welcome to code EntropyFunctions objects for different distributions of the unobserved heterogeneity terms.

Solving for the stable matching¶

Given any joint surplus matrix $\Phi$; margins $n$ and $m$; and a generalized entropy $\mathcal{E}$, one would like to compute the stable matching patterns $\mu$.

For the five classes of models above, this can be done efficiently using the IPFP algorithm in Galichon-Salanié (REStud 2022) , which is coded in ipfp_solvers.py for the four Choo and Siow variants and in model_classes.py for the nested logit.

Here is an example, given a Numpy array $\Phi$ that is an $(X,Y)$ matrix and a number $\tau>0$:

import numpy as np

from cupid_matching.ipfp_solvers import ipfp_gender_homoskedastic solver

solution = ipfp_gender_heteroskedastic_solver(Phi, n, m, tau)
mus, error_x, error_y = solution
muxy = mus.muxy

The mus above is an instance of a Matching object (defined in matching_utils.py). mus.muxy has the number of couples by $(x,y)$ cell at the stable matching; the vectors mus.mux0 and mus.mu0y contain the numbers of single men and women of each type.

The vectors error_x and error_y are estimates of the precision of the solution (see the code in ipfp_solvers.py).

Estimating the joint surplus¶

Given observed matching patterns $\mu$; a class of generalized entropy functions $(\mathcal{E}^{\alpha})$; and a class of joint surplus functions $(\Phi^{\beta})$, one would like to estimate the parameter vectors $\alpha$ and $\beta$.

The package provides two estimators, which are described extensively in this paper:

the minimum distance estimator in min_distance.py
the Poisson estimator in poisson_glm.py, which only applies to the Choo and Siow homoskedastic model.

At this stage cupid_matching only allows for linear models of the joint surplus:

\[ \Phi^{\beta}_{xy} = \sum_{k=1}^K \phi_{xy}^k \beta_k \]

where the basis functions $(\phi^1,\ldots,\phi^K)$ are imposed by the analyst.

The minimum distance estimator works as follows, given

an observed matching stored in a Matching object mus
an EntropyFunction object entropy_model that allows for p parameters in $\alpha$
an $(X,Y,K)$ Numpy array of basis functions phi_bases:

mde_results = estimate_semilinear_mde(
    mus, phi_bases, entropy_model)
mde_results.print_results(n_alpha=p)

The mde_results object contains the estimated $\alpha$ and $\beta$, their estimated variance-covariance, and a specification test.

The Poisson-GLM estimator of the Choo and Siow homoskedastic model takes as input the obserrved matching and the basis functions, and returns the estimated $\beta$ and its estimated variance-covariance.

poisson_results = choo_siow_poisson_glm(mus_sim, phi_bases)
    _, mux0_sim, mu0y_sim, n_sim, m_sim = mus_sim.unpack()
poisson_results.print_results()

The poisson_results object contains the estimated $\beta$, the expected utilities $u_x$ and $v_y$, and their estimated variance-covariance.

Examples¶

The following can be found in the Github repository and in the source code on PyPI:

example_choosiow.py shows how to run minimum distance and Poisson estimators on a Choo and Siow homoskedastic model.
example_choosiow_no_singles.py does the same for a model without singles.
example_nested_logit.py shows how to run minimum distance estimators on a two-layer nested logit model.

Warnings¶

many of these models (including all variants of Choo and Siow) rely heavily on logarithms and exponentials. It is easy to generate examples where numeric instability sets in.
as a consequence, the numeric versions of the minimum distance estimator (which use numerical derivatives) are not recommended.
the bias-corrected minimum distance estimator (corrected) may have a larger mean-squared error and/or introduce numerical instabilities.
the estimated variance of the estimators assumes that the observed matching was sampled at the household level, and that sampling weights are all equal.