Solves, simulates, and estimates separable matching TU models
Project description
cupid_matching
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
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. The full documentation is here.
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, onetoone 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, myy 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, discretevalued 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 $$ \Phi_{xy}+\varepsilon_{my} +\eta_{xw}. $$ 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 $N_h = N_i  \sum_{x=1}^X \sum_{y=1}^Y \mu_{xy}$.
GalichonSalanié (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 typedependent scale parameters $\sigma_x$ and $\tau_y$ (with $\sigma_1=1$
 a twolayer 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 GalichonSalanié (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 file example_choosiow.py
has a demo of minimum distance and Poisson estimators on the Choo and Siow 2006 model. For other models, the minimum distance estimator works as follows, given
 an observed matching stored in a
Matching
objectmus
 an
EntropyFunction
objectentropy_model
that allows forp
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 variancecovariance, and a specification test.
Examples
example_choosiow.py
shows how to run minimum distance and Poisson estimators on a Choo and Siow homoskedastic model.example_nestedlogit.py
shows how to run minimum distance estimators on a twolayer 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 biascorrected minimum distance estimator (
corrected
) may have a larger meansquared 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.
Release notes
version 1.1.1
 improved documentation
 the package now relies on my utilities package
bs_python_utils
. TheVarianceMatching
class inmatching_utils,py
is new; this should be transparent for the user.
version 1.0.8
 deleted spurious print statement.
version 1.0.7
 fixed error in biascorrection term.
version 1.0.6
 corrected typo.
version 1.0.5
 simplified the biascorrection for the minimum distance estimator in the Choo and Siow homoskedastic model.
version 1.0.4
 added an optional biascorrection for the minimum distance estimator in the Choo and Siow homoskedastic model, to help with cases when the matching patterns vary a lot across cells.
 added two complete examples:
example_choosiow.py
andexample_nestedlogit.py
.
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