# Monte Carlo Inference on Two-Sided Matching Models

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## Abstract

**:**

## 1. Introduction

## 2. Monte Carlo Inference: Overview

#### 2.1. Monte Carlo Inference

#### 2.2. Subvector Inference

## 3. A Two-Sided Random Matching Model

#### 3.1. A College Admissions Model

**Definition**

**1.**

- (i) A matching $\mu :{N}_{s}^{\prime}\to {N}_{c}^{\prime}$ is stable with respect to $\mathit{\pi}=({\mathit{\pi}}_{s},{\mathit{\pi}}_{c})\in \mathbf{\Pi}$, if the following two conditions are satisfied.
- (a) There is no $i\in {N}_{s}$ such that $0{\succ}_{{\pi}_{i}}\mu (i)$ and no $j\in {N}_{c}$ such that $0{\succ}_{{\pi}_{j}}{i}^{\prime}$ for some ${i}^{\prime}\in {\mu}^{-1}(j)$.
- (b) There is no pair $(i,j)\in {N}_{s}\times {N}_{c}$ such that both $j{\succ}_{{\pi}_{i}}\mu (i)$ and $i{\succ}_{{\pi}_{j}}{i}^{\prime}$ for some ${i}^{\prime}\in {\mu}^{-1}(j)$.

- (ii) A matching mechanism $\mu :{N}_{s}^{\prime}\times \mathbf{\Pi}\to {N}_{c}^{\prime}$ is stable if $\mu (\xb7;\mathit{\pi})$ is stable with respect to each $\mathit{\pi}\in \mathbf{\Pi}.$

#### 3.2. Random Preferences

#### 3.2.1. Students’ Heterogeneous Preferences

**Definition**

**2.**

#### 3.2.2. Colleges’ Homogeneous Preferences

#### 3.3. The Joint Distribution of a Large Observed Matching

**Assumption**

**1.**

**Theorem**

**1.**

## 4. Monte Carlo Inference

#### 4.1. Test Statistics, Critical Values and Confidence Sets

#### 4.2. Constructing Test Statistics

## 5. Monte Carlo Simulation Studies

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Serial Dictatorship Algorithm

`x_s`,`x_c`: ${n}_{s}\times 1$ vector of student characteristics and ${n}_{c}\times 1$ vector of college characteristics.`theta_s`,`theta_c`: scalar student and college preference parameters.`N_s`,`N_c`: number of students and colleges.`student_score`: ${n}_{s}\times 1$ vector of college preferences over students, based on`theta_s`,`x_s`and standard normal random variables.`coll_rank`: The indices associated with the preferred students of colleges. For example,`coll_rank(1)`is the index of the most preferred student according to colleges.`val_ji`: ${n}_{c}\times {n}_{s}$ matrix whose $(j,i)$ element is associated with the value that student i places on college j based on`theta_c`,`x_c`and standard normal random variables.`pos_vec`: ${n}_{s}\times 1$ vector whose ith element says the college associated with the ith college position. For example, if`pos_vec`$={[1,1,2,2,3]}^{\prime}$, it means that there are three colleges, where college 1 and 2 each have two positions and college 3 has 1.`val_ji_pos`is an ${n}_{s}\times {n}_{s}$ matrix whose $(j,i)$ element is the value that student i has for position $j.$`val_fv`: ${n}_{s}\times {n}_{s}$ matrix whose $(j,r)$ value is the value that the student ranked r highest according to coll_rank has for college position j.`matching`: ${n}_{s}\times 1$ vector, where`matching(i)`$=j$ means student $i\in \{1,\dots ,{n}_{s}\}$ is matched with college position $j\in \{1,\dots ,{n}_{s}\}$.

`serial_dictatorship`returns

`matching`given

`theta_s`

`theta_c`,

`x_c`,

`x_s`,

`N_s`,

`N_c`, and

`pos_vec`when the colleges’ ranking of students is based on students’ score generated from a model with additive normal errors. One can change this specification in the code for alternative specification of the way colleges rank students.

`function matching =serial_dictatorship(theta_s, theta_c, x_c, x_s, N_s, N_c,pos_vec)`

`student_score = theta_s*x_s + normrnd(0,1,N_s,1);`

`[~,coll_rank] = sort(student_score,’descend’);`

`val_ji=repmat((theta_c*x_c),[1 N_s]) + normrnd(0,1,N_c,N_s);`

`val_ji_pos=val_ji(pos_vec,:);`

`val_fv=val_ji_pos(:,coll_rank);`

`inverse_matching=zeros(N_s,1);`

`matching=inverse_matching;`

`id=1:N_s;`

`for i=1:N_s`

`[~,index]=max(val_fv(:,i));`

`inverse_matching(coll_rank(i))=index;`

`val_fv(index,:)=-inf;`

`end`

`matching(inverse_matching)=id;`

`end`

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1 | This assumption of homogeneity in preferences of one side is certainly restrictive, and yet this asymmetry of preference heterogeneity between the two sides reflects various many-to-one matching markets in practice. For example, colleges mostly agree on who the best students are whereas many students face tradeoff between the distance from their homes to a college and the college’s quality. This assumption of homogeneous preference on one side is not unprecedented in the literature either. See for example Agarwal (2015) who used this assumption in the analysis of medical residents’ matching market. |

2 | Canen et al. (2018) for an empirical model of linear interactions over a large network. Using a set of behavioral assumptions, they produce best responses that exhibit local dependence, and permit partial observation of the players by the econometrician for inference. |

3 | Schwartz (2018) used the Monte Carlo subvector inference approach following this paper’s proposal. However, his setting permits using standard inference on part of the parameter vector as a first step, applying Monte Carlo inference for the remaining parameters. This two-step approach no longer ensures finite sample validity. Nevertheless, it sharpens the inference results and reduces the computational costs. |

4 | By the definition of a matching mechanism as a map on ${N}_{s}^{\prime}\times \Pi $, it is anonymous in the sense that the matching mechanism remains invariant to the relabeling of the agents’ indices. |

5 | A proper development will require defining preferences over sets of students by colleges, and defining stability of a matching in terms of these preferences. When the preferences are so-called responsive, the group stability is equivalent to pairwise stability. As we make use of pairwise stability for econometric inference, we refer the reader to Chapter 5 of Roth and Sotomayor (1990) for further details. |

6 | Adding an additive term whose covariate depends only on ${x}_{s,i}$ instead of the interaction term ${x}_{c,j}$ is superfluous for “identification”, because variations in ${x}_{s,i}$ do not change the ranking of the colleges by the student i. |

7 | Our choice of notation for matching as $\mathbf{Y}$ is to emphasize that matching is an endogenous outcome. |

8 | In the literature of mechanism design, this mechanism is called a serial dictatorship mechanism. (See Satterthwaite and Sonnenschein (1981)). |

9 | Using the notation $[i]$ instead of i is meant as a reminder that the quantity depends on the “history” $[i]=(1,2,\dots ,i-1,i)$, rather the current agent index i. |

${\mathit{\theta}}_{\mathit{s}}$ | ${\mathit{\theta}}_{\mathit{c}}$ | $\mathit{K}\mathbf{=}\mathbf{5}$ | $\mathit{K}\mathbf{=}\mathbf{10}$ | $\mathit{K}\mathbf{=}\mathbf{20}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|

0.5 | 1.0 | 1.5 | 0.5 | 1.0 | 1.5 | 0.5 | 1.0 | 1.5 | ||

0.5 | $n=200$ | 0.9870 | 0.8720 | 0.8180 | 0.9700 | 0.8340 | 0.7620 | 0.9470 | 0.7840 | 0.6680 |

$n=400$ | 1.0000 | 0.9990 | 0.9920 | 1.0000 | 1.0000 | 0.9960 | 1.0000 | 0.9980 | 0.9810 | |

$n=600$ | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | |

1.0 | $n=200$ | 0.2320 | 0.0640 | 0.1040 | 0.2430 | 0.0620 | 0.0830 | 0.2500 | 0.0470 | 0.1150 |

$n=400$ | 0.4170 | 0.0340 | 0.0770 | 0.4360 | 0.0450 | 0.0970 | 0.5100 | 0.0500 | 0.1010 | |

$n=600$ | 0.6170 | 0.0580 | 0.0910 | 0.6290 | 0.0460 | 0.1040 | 0.6590 | 0.0510 | 0.1110 | |

1.5 | $n=200$ | 0.0700 | 0.6820 | 0.9340 | 0.0790 | 0.6380 | 0.9210 | 0.0840 | 0.5890 | 0.9100 |

$n=400$ | 0.1040 | 0.9660 | 0.9980 | 0.1350 | 0.9520 | 1.0000 | 0.1570 | 0.9150 | 0.9960 | |

$n=600$ | 0.1590 | 0.9990 | 1.0000 | 0.1810 | 0.9960 | 1.0000 | 0.2300 | 0.9940 | 1.0000 |

${\mathit{\theta}}_{\mathit{s}}$ | ${\mathit{\theta}}_{\mathit{c}}$ | $\mathit{K}\mathbf{=}\mathbf{5}$ | $\mathit{K}\mathbf{=}\mathbf{10}$ | $\mathit{K}\mathbf{=}\mathbf{20}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|

0.5 | 1.0 | 1.5 | 0.5 | 1.0 | 1.5 | 0.5 | 1.0 | 1.5 | ||

0.5 | $n=200$ | 0.9860 | 0.8850 | 0.8300 | 0.9760 | 0.8370 | 0.7470 | 0.9520 | 0.7820 | 0.6670 |

$n=400$ | 1.0000 | 0.9990 | 0.9960 | 1.0000 | 0.9990 | 0.9960 | 1.0000 | 0.9980 | 0.9860 | |

$n=600$ | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | |

1.0 | $n=200$ | 0.2280 | 0.0610 | 0.0870 | 0.2530 | 0.0510 | 0.0910 | 0.2400 | 0.0490 | 0.1110 |

$n=400$ | 0.4230 | 0.0310 | 0.0690 | 0.4470 | 0.0530 | 0.0960 | 0.4940 | 0.0510 | 0.0920 | |

$n=600$ | 0.6280 | 0.0570 | 0.0880 | 0.6250 | 0.0470 | 0.0960 | 0.6640 | 0.0450 | 0.1110 | |

1.5 | $n=200$ | 0.0650 | 0.6820 | 0.9510 | 0.0710 | 0.6500 | 0.9210 | 0.0910 | 0.5990 | 0.9200 |

$n=400$ | 0.1010 | 0.9650 | 0.9980 | 0.1260 | 0.9520 | 1.0000 | 0.1590 | 0.9150 | 0.9960 | |

$n=600$ | 0.1540 | 0.9990 | 1.0000 | 0.1710 | 0.9950 | 1.0000 | 0.2100 | 0.9930 | 1.0000 |

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**MDPI and ACS Style**

Kim, T.; Schwartz, J.; Song, K.; Whang, Y.-J.
Monte Carlo Inference on Two-Sided Matching Models. *Econometrics* **2019**, *7*, 16.
https://doi.org/10.3390/econometrics7010016

**AMA Style**

Kim T, Schwartz J, Song K, Whang Y-J.
Monte Carlo Inference on Two-Sided Matching Models. *Econometrics*. 2019; 7(1):16.
https://doi.org/10.3390/econometrics7010016

**Chicago/Turabian Style**

Kim, Taehoon, Jacob Schwartz, Kyungchul Song, and Yoon-Jae Whang.
2019. "Monte Carlo Inference on Two-Sided Matching Models" *Econometrics* 7, no. 1: 16.
https://doi.org/10.3390/econometrics7010016