tcb@breakable

#  Individual Rationality Conditions of Identifying Matching Costs in
Transferable Utility Matching Games††thanks: I thank my advisor Jeremy Fox for
his valuable advice.

Suguru Otani<EMAIL_ADDRESS>Rice University, Department of Economics. Kraft
Hall, Room 443, 6100 Main St. Houston, TX 77005, United States.

###### Abstract

As the widely applied method for measuring matching assortativeness in a
transferable utility matching game, a matching maximum score estimation is
proposed by Fox (2010). This article reveals that combining unmatched agents,
transfers, and individual rationality conditions with sufficiently large
penalty terms makes it possible to identify the coefficient parameter of a
single common constant, i.e., matching costs in the market.

Keywords: Matching, Matching cost, Individual rationality, Identification
JEL Codes: C78, C57

##  1 Introduction

One of the most well-known methods for measuring matching assortativeness was
developed by Fox (2010). Although the method allows researchers to flexibly
incorporate additional inequalities from equilibrium conditions, the
additional information that helps identification is unknown. This article
focuses on pairwise stability in a transferable utility (TU) matching model
and investigates the identification by adding unmatched agents, transfer data,
and individual rationality (IR) conditions, which are available in many
empirical applications.111Examples using transfer data are Fox and Bajari
(2013), Akkus et al. (2015), and Pan (2017). Examples using unmatched data are
Fox et al. (2018) and Otani (2021).

##  2 Model

###  2.1 Baseline matching model

I consider a two-sided one-to-one TU merger matching game in a single market.
Let $\mathcal{N}_{b}$ and $\mathcal{N}_{s}$ be the sets of potential finite
buyers and sellers respectively. Let $b=1,\cdots,|\mathcal{N}_{b}|$ be buyer
firms and let $s=1,\cdots,|\mathcal{N}_{s}|$ be seller firms where $|\cdot|$
is cardinality. Let $\mathcal{N}_{b}^{m}$ denote the set of ex-post matched
buyers and $\mathcal{N}_{b}^{u}$ denote that of ex-post unmatched buyers such
that $\mathcal{N}_{b}=\mathcal{N}_{b}^{m}\cup\mathcal{N}_{b}^{u}$ and
$\mathcal{N}_{b}^{m}\cap\mathcal{N}_{b}^{u}=\emptyset$. For the seller side,
define $\mathcal{N}_{s}^{u}$ and $\mathcal{N}_{s}^{m}$ as the set of ex-post
matched and unmatched sellers such that
$\mathcal{N}_{s}=\mathcal{N}_{s}^{m}\cup\mathcal{N}_{s}^{u}$ and
$\mathcal{N}_{s}^{m}\cap\mathcal{N}_{s}^{u}=\emptyset$. Let $\mathcal{M}^{m}$
be the sets of all ex-post matched pairs
$(b,s)\in\mathcal{N}_{b}^{m}\times\mathcal{N}_{s}^{m}$. Let $\mathcal{M}$
denote the set of all ex-post matched pairs $(b,s)\in\mathcal{M}^{m}$ and
unmatched pairs $(\tilde{b},\emptyset)$ and $(\emptyset,\tilde{s})$ for all
$\tilde{b}\in\mathcal{N}_{b}^{u}$ and $\tilde{s}\in\mathcal{N}_{s}^{u}$ where
$\emptyset$ means a null agent generating unmatched payoff.

Each firm can match at most one agent on the other side, so
$|\mathcal{N}_{b}^{m}|=|\mathcal{N}_{s}^{m}|$. The matching joint production
function is defined as $f(b,s)=V_{b}(b,s)+V_{s}(b,s)$ where
$V_{b}:\mathcal{M}\rightarrow\mathbb{R}$ and
$V_{s}:\mathcal{M}\rightarrow\mathbb{R}$. The net matching values for buyer
$b$ and seller $s$ are defined as $V_{b}(b,s)=f(b,s)-p_{b,s}$ and
$V_{s}(b,s)+p_{b,s}$, where $p_{b,s}\in\mathbb{R}_{+}$ is the equilibrium
merger price paid to seller firm $s$ by buyer firm $b$ and
$p_{b\emptyset}=p_{\emptyset s}=0$. For scale normalization, I assume
$V_{b}(b,\emptyset)=0$ and $V_{s}(\emptyset,s)=0$ for all
$b\in\mathcal{N}_{b}$ and $s\in\mathcal{N}_{s}$. Each buyer maximizes
$V_{b}(b,s)$ across seller firms, whereas each seller maximizes $V_{s}(b,s)$
across buyer firms.

The stability conditions for buyer firm $b\in\mathcal{N}_{b}$ and seller firm
$s\in\mathcal{N}_{s}$ are as follows:

$\displaystyle V_{b}(b,s)$ $\displaystyle\geq V_{b}(b,s^{\prime})\quad\forall
s^{\prime}\in\mathcal{N}_{s}\cup\emptyset,s^{\prime}\neq s,$ (1)
$\displaystyle V_{s}(b,s)$ $\displaystyle\geq V_{s}(b^{\prime},s)\quad\forall
b^{\prime}\in\mathcal{N}_{b}\cup\emptyset,b^{\prime}\neq b.$

Based on Equation (1) and equilibrium price conditions $p_{b^{\prime},s}\leq
p_{b,s}$ and $p_{b,s^{\prime}}\leq p_{b^{\prime},s^{\prime}}$ in Akkus et al.
(2015), I construct the inequalities for matches $(b,s)\in\mathcal{M}$ and
$(b^{\prime},s^{\prime})\in\mathcal{M},(b^{\prime},s^{\prime})\neq(b,s)$ as
follows:

$\displaystyle f(b,s)-f(b,s^{\prime})$ $\displaystyle\geq
p_{b,s}-p_{b,s^{\prime}}\geq p_{b,s}-p_{b^{\prime},s^{\prime}},$ (2)
$\displaystyle f(b^{\prime},s^{\prime})-f(b^{\prime},s)$ $\displaystyle\geq
p_{b^{\prime},s^{\prime}}-p_{b^{\prime},s}\geq
p_{b^{\prime},s^{\prime}}-p_{b,s},$ $\displaystyle
V_{s}(b,s)-V_{s}(b^{\prime},s)$ $\displaystyle\geq 0,$ $\displaystyle
V_{s^{\prime}}(b^{\prime},s^{\prime})-V_{s}(b,s^{\prime})$ $\displaystyle\geq
0,$

where $p_{b^{\prime},s}$ and $p_{b,s^{\prime}}$ are unrealized equilibrium
merger prices that cannot be observed in the data. The last two inequalities
cannot be derived from the data because the researchers cannot observe how the
total matching value $f(b,s)$ is shared between buyer $b$ and seller $s$.

###  2.2 Transfer and unmatched data.

In many empirical applications, researchers do not obtain data on transfers
and unmatched firms. The minimum number of possible inequalities is discussed
in Result 1.

######  Result 1.

Suppose that researchers obtain (1) transfer data $p_{bs}$ for matched buyer
firm $b$ and seller firm $s$ for all $b\in\mathcal{N}_{b}^{m}$ and
$s\in\mathcal{N}_{s}^{m}$, and (2) unmatched data about $\mathcal{N}_{b}^{u}$
and $\mathcal{N}_{s}^{u}$ with their observed characteristics. Let
${}_{n}C_{k}$ be the number of combinations of $k$ objects from $n$ objects,
and ${}_{n}P_{k}$ be the number of permutations, representing the different
ordered arrangements of a $k$-element subset of an $n$-set.222For reference,
${}_{n}C_{k}=\frac{n!}{k!(n-k)!}$ and ${}_{n}P_{k}=\frac{n!}{(n-k)!}$ where
$!$ is a factorial function. Then, the followings hold:

  1. (i)

Researchers can construct the following inequalities for matches
$(b,s)\in\mathcal{M}$ and
$(b^{\prime},s^{\prime})\in\mathcal{M},(b^{\prime},s^{\prime})\neq(b,s)$:

$\displaystyle f(b,s)-f(b,s^{\prime})$ $\displaystyle\geq
p_{b,s}-p_{b^{\prime},s^{\prime}}.$ (3)

The minimum number of total inequalities is
${}_{|\mathcal{M}^{m}|+|\mathcal{N}_{b}^{u}|+|\mathcal{N}_{s}^{u}|}P_{2}$.

  2. (ii)

If the data does not contain (1), researchers can construct the following
inequalities for matches $(b,s)\in\mathcal{M}$ and
$(b^{\prime},s^{\prime})\in\mathcal{M},(b^{\prime},s^{\prime})\neq(b,s)$:

$\displaystyle f(b,s)+f(b^{\prime},s^{\prime})$ $\displaystyle\geq
f(b,s^{\prime})+f(b^{\prime},s).$ (4)

The minimum number of total inequalities is
${}_{|\mathcal{M}^{m}|+|\mathcal{N}_{b}^{u}|+|\mathcal{N}_{s}^{u}|}C_{2}$.

  3. (iii)

If the data does not contain (2), researchers can construct Inequality (4) for
matches $(b,s)\in\mathcal{M}^{m}$ and
$(b^{\prime},s^{\prime})\in\mathcal{M}^{m},(b^{\prime},s^{\prime})\neq(b,s)$.
The minimum number of total inequalities is ${}_{|\mathcal{M}^{m}|}P_{2}$.

  4. (iv)

If the data does not contain (1) and (2), researchers can construct Inequality
(3) for matches $(b,s)\in\mathcal{M}^{m}$ and
$(b^{\prime},s^{\prime})\in\mathcal{M}^{m},(b^{\prime},s^{\prime})\neq(b,s)$.
The minimum number of total inequalities is ${}_{|\mathcal{M}^{m}|}C_{2}$.

###  2.3 Individual rationality conditions

IR conditions are implicitly included in Inequality (3) when transfer data is
available because $\mathcal{M}$ restores unmatched buyer $\tilde{b}$ and
seller $\tilde{s}$ as $(\tilde{b},\emptyset)$ and $(\emptyset,\tilde{s})$,
respectively. For matched pair $(b,s)\in\mathcal{M}$ and unmatched buyer
$(\tilde{b},\emptyset)\in\mathcal{M}$, Inequality (3) gives
$f(b,s)-f(b,\emptyset)=f(b,s)\geq p_{b,s}-p_{\tilde{b},\emptyset}\geq
p_{b,s}$, i.e.,

$\displaystyle f(b,s)-p_{b,s}$ $\displaystyle\geq 0.$ (5)

Importantly, unlike pairwise inequalities, IR condition holds for each matched
firm on each side regardless of the availability of transfer data. To
distinguish IR conditions from pairwise inequalities, let
$IR(\mathcal{M}^{m})=2|\mathcal{M}^{m}|=|\mathcal{M}_{b}^{m}|+|\mathcal{M}_{s}^{m}|$
be the number of inequalities from IR conditions.

Table 1 summarizes the above results and illustrates a useful configuration of
pairwise inequalities. Note that
${}_{|\mathcal{M}^{m}|+|\mathcal{N}_{b}^{u}|+|\mathcal{N}_{s}^{u}|}P_{2}$ and
${}_{|\mathcal{M}^{m}|}P_{2}$ include the same inequalities in
$IR(\mathcal{M}^{m})$. Section 3 uses the result to demonstrate a necessary
modification for identifying matching costs.

Table 1: The minimum number of total inequalities. |  | Unmatched
---|---|---
|  | Available | Unavailable
Transfer | Available | ${}_{|\mathcal{M}^{m}|+|\mathcal{N}_{b}^{u}|+|\mathcal{N}_{s}^{u}|}P_{2}$ | ${}_{|\mathcal{M}^{m}|}P_{2}$
| Unavailable | ${}_{|\mathcal{M}^{m}|+|\mathcal{N}_{b}^{u}|+|\mathcal{N}_{s}^{u}|}C_{2}+IR(\mathcal{M}^{m})$ | ${}_{|\mathcal{M}^{m}|}C_{2}+IR(\mathcal{M}^{m})$

###  2.4 Matching maximum score estimator

Fox (2010) proposes a maximum score estimator using Inequality (3) or (4). The
maximum score estimator is consistent if the model satisfies a rank order
property, i.e., the probability of observing matched pairs is larger than the
probability of observing swapped matched pairs. I specify $f(b,s)$ as a
parametric form $f(b,s|X,\beta)$ where $X$ is a vector of observed
characteristics of all buyers and sellers and $\beta$ is a vector of
parameters. Given $X$, one can estimate $\beta$ without IR conditions by
maximizing the following objective function:

$\displaystyle
Q(\beta)=\begin{cases}\sum_{(b,s)\in\mathcal{M}}\sum_{(b^{\prime},s^{\prime})\in\mathcal{M},(b^{\prime},s^{\prime})\neq(b,s)}\mathbbm{1}[f(b,s|X,\beta)-f(b,s^{\prime}|X,\beta)\geq
p_{b,s}-p_{b^{\prime},s^{\prime}}]\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text{if transfer data is
available},\\\
\sum_{(b,s)\in\mathcal{M}}\sum_{(b^{\prime},s^{\prime})\in\mathcal{M},(b^{\prime},s^{\prime})\neq(b,s)}\mathbbm{1}[f(b,s|X,\beta)+f(b^{\prime},s^{\prime}|X,\beta)\geq
f(b,s^{\prime}|X,\beta)+f(b^{\prime},s|X,\beta)]\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text{otherwise},\end{cases}$
(6)

where $\mathbbm{1}[\cdot]$ is an indicator function. If unmatched data is
unavailable, $\mathcal{M}$ is replaced with $\mathcal{M}^{m}$. If IR
conditions are included, $Q(\beta)$ is modified to the following:

$\displaystyle\tilde{Q}(\beta)=Q(\beta)+\lambda\cdot\sum_{(b,s)\in\mathcal{M}^{m}}\mathbbm{1}[f(b,s|X,\beta)\geq
0],\quad\lambda\geq 1,$ (7)

where $\lambda$ is the importance weight of IR conditions. If $\lambda$ is
larger, the importance of the IR condition term is larger for the evaluation
of $\tilde{Q}(\beta)$. If the transfer data are available, the correction term
is redundant as in Table 1, but it does not affect the search for the
maximizer of $\tilde{Q}(\beta)$. Section 3 investigates the importance of
$\lambda$.

##  3 IR conditions identify the coefficient parameter of a constant

The estimation of fixed costs is one of the fundamental tasks in structural
empirical studies. Suppose that researchers want to estimate an additive
separable matching cost in a single market. Let $c$ be the matching cost and
assume $c<0$ for exposition.333If researchers expect $\beta X_{b}X_{s}$ to be
negative, i.e., the market to generate negative assortative matchings and want
to investigate a subsidy effect inducing matchings, they can assume $c>0$ as
the subsidy effect. The logic is the same in the main text. Following the
literature, I specify $f(b,s)$ as follows:

$\displaystyle f(b,s)=\beta X_{b}X_{s}+c\cdot\mathbbm{1}[b\neq\emptyset\text{
or }s\neq\emptyset],$ (8)

where $X_{b}$ and $X_{s}$ are vectors of continuous observable characteristics
for buyer $b$ and seller $s$ and $\beta$ is a vector of parameters. Note that
hypothetical matching cost $c$ exists for unmatched pairs. Then, Result 2
holds.

######  Result 2.

Suppose the matching joint production function is specified as (8). Then,

  1. (i)

by using pairwise inequalities based only on matched pairs, $c$ cannot be
identified via $Q(\cdot)$;

  2. (ii)

in addition to (i), even if transfer data is available, $c$ cannot be
identified via $Q(\cdot)$;

  3. (iii)

in addition to (i), even if unmatched data is available, the lower bound of
$c$ cannot be identified without IR conditions;

  4. (iv)

in addition to (iii), $c$ can be identified only when IR conditions are used
with a sufficiently large importance weight via $\tilde{Q}(\beta)$, whether
the transfer data is included or not.

Note that $(iv)$ of Result 2 does not reveal how large the importance weight
$\lambda$ should be. The appropriate weight of $\lambda$ depends on the sample
sizes of matched and unmatched agents as a tuning parameter. The working paper
version provides detailed numerical experiments to show the necessity of the
weight.

##  4 Conclusion

This article investigates the identification of a common fixed cost of
matching in a TU matching game. For identification, IR conditions with a
sufficiently large importance weight are necessary unless you have data of
both unmatched agents and transfers.

##  Appendix A Proof of results (For Online Publication)

##### Proof of Result 1

###### Proof.

I demonstrate $(i)$ because other parts are analogously proven. $\mathcal{M}$
restores unmatched buyer $\tilde{b}$ and seller $\tilde{s}$ as
$(\tilde{b},\emptyset)$ and $(\emptyset,\tilde{s})$, so the size of the set of
unmatched pairs is $(|\mathcal{N}_{b}^{u}|+|\mathcal{N}_{s}^{u}|)$. The number
of possible combinations of $(b,s)\in\mathcal{M}$ and
$(b^{\prime},s^{\prime})\in\mathcal{M},(b^{\prime},s^{\prime})\neq(b,s)$ is
${}_{|\mathcal{M}^{m}|+|\mathcal{N}_{b}^{u}|+|\mathcal{N}_{s}^{u}|}C_{2}$ and
each combination gives two inequalities as in Inequality (3). This gives

$\displaystyle
2\cdot_{|\mathcal{M}^{m}|+|\mathcal{N}_{b}^{u}|+|\mathcal{N}_{s}^{u}|}C_{2}=2\frac{(|\mathcal{M}^{m}|+|\mathcal{N}_{b}^{u}|+|\mathcal{N}_{s}^{u}|)(|\mathcal{M}^{m}|+|\mathcal{N}_{b}^{u}|+|\mathcal{N}_{s}^{u}|-1)}{2!}=_{|\mathcal{M}^{m}|+|\mathcal{N}_{b}^{u}|+|\mathcal{N}_{s}^{u}|}P_{2}.$
(9)

∎

##### Proof of Result 2

###### Proof.

To prove $(i)$, substituting (8) into (3) gives

$\displaystyle\beta X_{b}X_{s}+c\cdot\mathbbm{1}[b\neq\emptyset\text{ or
}s\neq\emptyset]+\beta
X_{b^{\prime}}X_{s^{\prime}}+c\cdot\mathbbm{1}[b^{\prime}\neq\emptyset\text{
or }s^{\prime}\neq\emptyset]$ $\displaystyle\quad\geq\beta
X_{b}X_{s^{\prime}}+c\cdot\mathbbm{1}[b\neq\emptyset\text{ or
}s^{\prime}\neq\emptyset]+\beta
X_{b^{\prime}}X_{s}+c\cdot\mathbbm{1}[b^{\prime}\neq\emptyset\text{ or
}s\neq\emptyset],$

where the indicator functions must be 1 for matched pair $(b,s)$ and
unrealized matched pair $(b,s^{\prime})$ so that matching cost $c$ is canceled
out. Similarly, $(ii)$ is also proved in the same way.

To prove $(iii)$, it is sufficient to consider pairwise inequalities based on
an unmatched pair as the pairwise inequalities of matched pairs are shown
above. With transfer data, by substituting (8) into (3), the pairwise
inequalities based on unmatched pairs denoted by $(\tilde{b},\emptyset)$ and
$(\emptyset,\tilde{s})$ can be reduced to a single inequality as follows:

$\displaystyle 0$ $\displaystyle\geq\beta
X_{\tilde{b}}X_{\tilde{s}}+c-p_{\tilde{b},\tilde{s}},$

where $p_{\tilde{b},\tilde{s}}=0$. This only provides the upper bound of $c$
as $-\beta X_{b}X_{s}\geq c$. Without transfer data, the same inequality is
derived using (4).

To prove $(iv)$, when transfer data is available, IR condition (5) for matched
pair $(b,s)$ gives

$\displaystyle\beta X_{b}X_{s}+c-p_{b,s}\geq 0,$

whereas, when transfer data is not available, IR condition (5) for matched
pair $(b,s)$ gives

$\displaystyle\beta X_{b}X_{s}+c\geq 0.$

Thus, IR condition (5) provides the lower bound of $c$ as $c\geq-\beta
X_{b}X_{s}+p_{b,s}$ or $c\geq-\beta X_{b}X_{s}$.

Finally, I demonstrate the necessity of $\lambda$. In the modified objective
function in (7), the additional term

$\displaystyle\lambda\cdot\sum_{(b,s)\in\mathcal{M}^{m}}1[\beta
X_{b}X_{s}+c\geq 0],$

takes up to $\lambda\cdot|\mathcal{M}^{m}|$. However, $Q(\beta)$ of (7) can
take up to either of the numbers in Table 1 which are much larger than
$|\mathcal{M}^{m}|$. In finite samples, the part does not need to achieve its
perfect score, so some fractions of pairwise inequalities are not satisfied
even at the maximizer of the objective function. This implies that, if
$\lambda$ and $|\mathcal{M}^{m}|$ are small, the evaluation of IR conditions
is dominated by the number of unsatisfied pairwise inequalities, which can be
larger than $|\mathcal{M}^{m}|$. Thus, $\lambda$ must be large enough,
corresponding to $|\mathcal{M}^{m}|$. ∎

##  Appendix B Numerical experiments (For Online Publication)

In Section B, I examine the improvement of estimation accuracy via each type
of inequality through the objective function of the matching maximum score
estimator in a finite sample.

###  B.1 Computation of a matching equilibrium

Let $F(b,s)=f(b,s)+\varepsilon_{b,s}$ be the match value function of match
$(b,s)$ where $\varepsilon_{b,s}$ is drawn from zero median distribution. The
equilibrium one-to-one matching allocation
$\\{m(b,s)\\}_{b\in\mathcal{N}_{b},s\in\mathcal{N}_{s}}$ is calculated by the
following linear programming problem proposed by Shapley and Shubik
(1971):444See Galichon (2018) for the computational properties and recent
results.

$\displaystyle\max_{\\{m(b,s)\\}_{b\in\mathcal{N}_{b},s\in\mathcal{N}_{s}}}$
$\displaystyle F(b,s)\cdot m(b,s)$ $\displaystyle\text{s.t. }0$
$\displaystyle\leq\sum_{b\in\mathcal{N}_{b}}m(b,s)\leq 1$ $\displaystyle 0$
$\displaystyle\leq\sum_{s\in\mathcal{N}_{s}}m(b,s)\leq 1$ $\displaystyle 0$
$\displaystyle\leq m(b,s)\quad\forall b\in\mathcal{N}_{b},\forall
s\in\mathcal{N}_{s}$

where $F(b,s)$ is specified as some parametric form of observed
characteristics of buyer $b$ and seller $s$ by researchers. The dual of this
linear programming problem also gives equilibrium prices. I assume that
unobservable hypothetical equilibrium prices for matchings with unmatched
buyers and sellers equal to zero.

###  B.2 Numerical results

We are interested in the improvement of identification and estimation accuracy
by adding unmatched firms, transfer data, and IR conditions. Also, I
investigate possibly negative assortative matching which may generate
unmatched agents via binding IR conditions, which is not investigated by the
previous study. In Section B.2, I compare the finite sample performances of
the four models in Table 1. For the exposition, I list the models below.

  * •

Model 1: with unmatched data and transfer data, denoted by $(U,T)$

  * •

Model 2: without unmatched data but with transfer data, denoted by $(T)$

  * •

Model 3: with unmatched data but without transfer data, denoted by $(U)$

  * •

Model 4: without unmatched data and transfer data, denoted by $(None)$

and I compare each model with and without IR conditions.

For simplicity, I assume $|\mathcal{N}_{b}|=|\mathcal{N}_{s}|$ in a single
matching market. I specify the market size as one of $\\{10,20,30,50,100\\}$.
The relative sizes of unmatched agents, i.e., $|\mathcal{N}^{u}_{b}|$ and
$|\mathcal{N}^{u}_{s}|$ depend on values of parameters. The error term
$\varepsilon_{b,s}$ is i.i.d. drawn from $N(0,1)$. I examine the following two
cases:

  * •

Case 1: I specify the joint production function as
$f(b,s)=\beta_{0}X_{b0}X_{s0}+\beta_{1}\cdot X_{b1}X_{s1}+\beta_{2}\cdot
X_{b2}X_{s2}$ where $\beta_{0}=1$ for normalization, $\beta_{1}=0.5$, and
$\beta_{2}\in\\{-3-2,-1,0,1\\}$ as true parameters. I assume that for
$i\in\\{b,s\\}$

$\displaystyle\left(\begin{array}[]{c}X_{i0}\\\ X_{i1}\\\
X_{i2}\end{array}\right)\sim_{iid}N\left(\left(\begin{array}[]{c}3\\\ 3\\\
3\end{array}\right),\left(\begin{array}[]{ccc}1&0.25&0.25\\\ 0.25&1&0.25\\\
0.25&0.25&1\end{array}\right)\right).$

For example, positive $\beta_{2}$ induces positive assortative matchings,
whereas negative $\beta_{2}$ induces negative assortative matchings which may
generate unmatched agents via binding IR conditions.

  * •

Case 2: The model is the same as Case 1 except that I specify the joint
production function as $f(b,s)=\beta_{0}X_{b0}X_{s0}+\beta_{1}\cdot
X_{b1}X_{s1}+\beta_{2}\cdot\kappa\cdot\mathbbm{1}[b\neq\emptyset\text{ or
}s\neq\emptyset]$ where $\kappa$ is a positive constant adjusting the level.
For example, if $\beta_{2}$ is negative, the term captures matching fixed
costs common in the market. I assume that $\kappa=8$ for generating unmatched
agents in simulations.555The value of positive constant $\kappa$ only affect
matching outcomes through the data generating process. For example, in Case 2,
observed interaction term $\beta_{0}X_{b0}X_{s0}+\beta_{1}X_{b1}X_{s1}$ is
$13.5(=1\times 3\times 3+0.5\times 3\times 3)$ on average. Then, to create
unmatched firms, $13.5-\kappa\beta_{1}+\varepsilon_{bs}$ needs to be at least
negative for some matching pair $(b,s)$. Thus, researchers expect composite
estimand $\kappa\beta_{1}$ to be large than $13.5$ in practical situations
since $\kappa$ is unknown.

For each scenario, I simulate 100 datasets and estimate parameters for each
dataset under the same importance weight of IR conditions. I fix the weight
$\lambda=100$.666In Appendix C.2, I investigate how the importance weight of
IR conditions affects identification of parameters in detail.

####  B.2.1 Case 1: including negative assortativeness.

Figure 1 illustrates contour maps of the objective function over $\beta_{1}$
and $\beta_{2}$ under the same single simulated dataset across different
models with and without IR conditions (left and right). Note that it captures
only how the objective function is shaped and does not capture the identified
set for set-identified cases, so that parameters are point estimated. Models
1, 2, 3, and 4 are listed from top to bottom in order. I find that transfer
data in itself is helpful for estimation regardless of including IR conditions
and unmatched firms. It also shows that if researchers do not have transfer
data, IR conditions trace out the lower bound of $\beta_{2}$ so that the
objective function is sharpened. However, the estimated maximizer of the
objective function can be slightly biased.

Tables 2 and 3 show the bias and Root Mean Squared Error (henceforth, RMSE)
across four models with and without IR conditions. I find that for Case 1,
including IR conditions decreases the bias and RMSE of $\beta_{2}$ in almost
all specifications. In contrast, $\beta_{1}$ estimated with IR conditions is
upward biased significantly for all models. Thus, for Case 1, there is no
strong recommendation about whether IR conditions are included or not.

Figure 1: Contour maps of the objective function of each model for Case 1
($\lambda=100$).

  * a

Note: I use the same simulated single dataset with
$\beta_{0}=+1,\beta_{1}=0.5,\beta_{2}=-2$ for all specifications. The only
difference is the specification of the objective function. The left figures do
not include IR condition, whereas the right figures do include this. Model 1,
2, 3, and 4 are listed from top to bottom in order. The maxima of the
objective function are plotted as red points.

####  B.2.2 Case 2: including matching fixed costs.

Figure 2 illustrates contour maps of the objective function over $\beta_{1}$
and $\beta_{2}$ under the same single simulated dataset across different
models with and without IR conditions. The only difference from Figure 1 is
the specification of the post-merger matching joint production function
$f(b,s)$ and generated data under the specification. First, the second and
fourth left panels show that without IR conditions and unmatched data,
$\beta_{2}$ cannot be identified completely. This is because pairwise
inequalities based only on matched pairs cancel out the constant term
regarding $\beta_{2}$. Also, the third left panel shows that even if unmatched
data is available, without IR conditions, the lower bound of $\beta_{2}$
cannot be identified. This implies that pairwise inequalities based on
unmatched firms and matched firms investigate only why unmatched firms do not
match any other firms. The case is satisfied when the matching cost is
infinitely large. In other words, pairwise inequalities based on unmatched
firms and matched firms do not answer why matched firms can match their
sellers. IR conditions answer the question.

The right panels in Figure 2 show how IR conditions trace out the lower bound
of $\beta_{2}$. In particular, the first and third right panels illustrate
that if the data of unmatched firms is available, IR conditions help to
identify the bound of $\beta_{2}$ regardless of the availability of transfer
data. Since collecting data of unmatched agents is easier than collecting
transfer data for each matching pair given the specified market definition,
this finding is practically important.

Tables 4 and 5 confirm that a combination of IR conditions and unmatched data
gives less biased estimates of $\beta_{1}$ and $\beta_{2}$ for almost all
specifications. However, if less than 20 percent of the samples are unmatched
as in the case of $\beta_{2}=-1$, the estimated $\beta_{2}$ is heavily biased
upwardly because the information of unmatched choices is too less. Table 7 in
Appendix C.2 suggests that for such a case, $\lambda=1$ is an adequate weight.
Table 6 in Appendix C.1 suggests that whether 30 percent of the full samples
are unmatched or not is the practical threshold about whether researchers
should use sufficiently high $\lambda$ or $\lambda=1$. The issue of the choice
of $\lambda$ can be resolvable before implementing estimations given the data.

Figure 2: Contour maps of the objective function of each model for Case 2
($\lambda=100$).

  * a

Note: I use the same simulated single dataset with
$\beta_{0}=+1,\beta_{1}=0.5,\beta_{2}=-2$ for all specifications. The only
difference is the specification of the objective function. The left figures do
not include IR condition, whereas the right figures do include this. Model 1,
2, 3, and 4 are listed from top to bottom in order. The maxima of the
objective function are plotted as red points

Table 2: Results of monte carlo simulations for Case 1 without
IR.($\lambda=100$)

| Num of agents |  | 10 | 20 | 30 | 50 | 100 |  | 10 | 20 | 30 | 50 | 100
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\beta_{1}$ |  |  |  |  |  |  |  | $\beta_{2}$ |  |  |  |  |
0.5 | unmatched | Mean Num | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| U,T | Bias | 0.08 | 0.08 | 0.28 | 0.19 | 0.13 |  | 0.21 | 0.1 | 0.66 | 0.38 | 0.3
|  | RMSE | (0.5) | (0.35) | (0.68) | (0.39) | (0.27) |  | (0.58) | (0.39) | (1.39) | (0.72) | (0.63)
| T | Bias | 0.17 | 0.08 | 0.28 | 0.21 | 0.13 |  | 0.25 | 0.09 | 0.66 | 0.42 | 0.32
|  | RMSE | (0.96) | (0.34) | (0.68) | (0.43) | (0.28) |  | (0.71) | (0.37) | (1.39) | (0.78) | (0.65)
| U | Bias | 0.23 | 0.09 | 0.28 | 0.21 | 0.13 |  | 0.25 | 0.11 | 0.66 | 0.42 | 0.33
|  | RMSE | (1.11) | (0.36) | (0.68) | (0.43) | (0.28) |  | (0.7) | (0.4) | (1.39) | (0.78) | (0.65)
| Non | Bias | 0.16 | 0.09 | 0.28 | 0.21 | 0.13 |  | 0.25 | 0.1 | 0.66 | 0.42 | 0.33
|  | RMSE | (0.96) | (0.35) | (0.68) | (0.43) | (0.28) |  | (0.71) | (0.39) | (1.39) | (0.78) | (0.65)
0.5 | unmatched | Mean Num | 0.01 | 0.0 | 0.0 | 0.0 | 0.03 | 0.0 | 0.01 | 0.0 | 0.0 | 0.0 | 0.03
| U,T | Bias | 1.19 | 0.46 | 0.46 | 0.33 | 0.38 |  | 0.21 | 0.04 | -0.0 | 0.02 | -0.01
|  | RMSE | (2.76) | (0.96) | (0.87) | (0.59) | (0.55) |  | (2.14) | (0.44) | (0.31) | (0.21) | (0.17)
| T | Bias | 1.26 | 0.43 | 0.46 | 0.35 | 0.38 |  | 0.19 | 0.04 | -0.01 | 0.01 | -0.0
|  | RMSE | (2.82) | (0.91) | (0.87) | (0.6) | (0.55) |  | (2.17) | (0.45) | (0.31) | (0.21) | (0.17)
| U | Bias | 1.23 | 0.43 | 0.46 | 0.36 | 0.39 |  | 0.19 | 0.03 | -0.01 | 0.01 | -0.01
|  | RMSE | (2.8) | (0.91) | (0.87) | (0.61) | (0.55) |  | (2.17) | (0.45) | (0.31) | (0.21) | (0.17)
| Non | Bias | 1.26 | 0.43 | 0.46 | 0.36 | 0.39 |  | 0.2 | 0.03 | -0.01 | 0.01 | -0.01
|  | RMSE | (2.82) | (0.91) | (0.87) | (0.61) | (0.55) |  | (2.17) | (0.45) | (0.31) | (0.21) | (0.17)
0.5 | unmatched | Mean Num | 1.69 | 3.08 | 4.21 | 6.77 | 13.29 | -1.0 | 1.69 | 3.08 | 4.21 | 6.77 | 13.29
| U,T | Bias | 0.81 | 0.34 | 0.2 | 0.21 | 0.15 |  | -1.28 | -0.5 | -0.26 | -0.34 | -0.27
|  | RMSE | (1.75) | (0.65) | (0.43) | (0.38) | (0.31) |  | (2.04) | (0.79) | (0.49) | (0.58) | (0.44)
| T | Bias | 0.57 | 0.38 | 0.17 | 0.22 | 0.14 |  | -1.17 | -0.57 | -0.26 | -0.3 | -0.25
|  | RMSE | (1.54) | (1.04) | (0.42) | (0.39) | (0.3) |  | (1.98) | (1.15) | (0.61) | (0.58) | (0.42)
| U | Bias | 0.87 | 0.35 | 0.19 | 0.19 | 0.15 |  | -1.34 | -0.53 | -0.25 | -0.31 | -0.27
|  | RMSE | (1.86) | (0.67) | (0.44) | (0.37) | (0.3) |  | (2.18) | (0.84) | (0.5) | (0.56) | (0.45)
| Non | Bias | 0.6 | 0.38 | 0.18 | 0.22 | 0.14 |  | -1.21 | -0.58 | -0.26 | -0.31 | -0.27
|  | RMSE | (1.56) | (1.04) | (0.44) | (0.39) | (0.3) |  | (2.03) | (1.15) | (0.6) | (0.59) | (0.44)
0.5 | unmatched | Mean Num | 5.68 | 10.65 | 15.73 | 25.47 | 50.12 | -2.0 | 5.68 | 10.65 | 15.73 | 25.47 | 50.12
| U,T | Bias | 0.31 | 0.15 | 0.12 | 0.14 | 0.11 |  | -1.43 | -0.63 | -0.5 | -0.37 | -0.27
|  | RMSE | (1.42) | (0.41) | (0.33) | (0.3) | (0.3) |  | (2.48) | (0.99) | (0.8) | (0.64) | (0.53)
| T | Bias | 0.49 | 0.44 | 0.31 | 0.27 | 0.09 |  | -1.52 | -0.92 | -0.71 | -0.67 | -0.21
|  | RMSE | (2.41) | (1.48) | (1.03) | (0.67) | (0.24) |  | (2.74) | (1.92) | (1.58) | (1.56) | (0.6)
| U | Bias | 0.28 | 0.12 | 0.09 | 0.12 | 0.08 |  | -1.38 | -0.58 | -0.48 | -0.36 | -0.24
|  | RMSE | (1.41) | (0.38) | (0.28) | (0.27) | (0.22) |  | (2.38) | (0.98) | (0.75) | (0.64) | (0.47)
| Non | Bias | 0.35 | 0.46 | 0.31 | 0.27 | 0.09 |  | -1.27 | -0.91 | -0.71 | -0.69 | -0.22
|  | RMSE | (2.34) | (1.49) | (1.03) | (0.67) | (0.24) |  | (2.83) | (1.92) | (1.58) | (1.56) | (0.63)
0.5 | unmatched | Mean Num | 7.67 | 14.72 | 21.95 | 36.08 | 71.83 | -3.0 | 7.67 | 14.72 | 21.95 | 36.08 | 71.83
| U,T | Bias | -0.08 | 0.19 | 0.11 | 0.08 | 0.09 |  | -1.8 | -0.88 | -0.62 | -0.43 | -0.36
|  | RMSE | (2.04) | (0.51) | (0.35) | (0.29) | (0.2) |  | (2.76) | (1.64) | (1.27) | (0.82) | (0.64)
| T | Bias | -0.26 | 0.45 | 0.25 | 0.07 | 0.21 |  | -0.48 | -0.91 | -0.88 | -0.17 | -0.79
|  | RMSE | (3.42) | (2.15) | (1.04) | (0.38) | (0.47) |  | (4.18) | (2.42) | (2.35) | (0.57) | (1.68)
| U | Bias | -0.09 | 0.17 | 0.09 | 0.06 | 0.08 |  | -1.82 | -0.93 | -0.59 | -0.42 | -0.39
|  | RMSE | (2.04) | (0.51) | (0.32) | (0.27) | (0.2) |  | (2.76) | (1.71) | (1.22) | (0.82) | (0.63)
| Non | Bias | -0.05 | 0.53 | 0.3 | 0.08 | 0.21 |  | -0.55 | -0.9 | -0.93 | -0.17 | -0.79
|  | RMSE | (3.58) | (2.35) | (1.2) | (0.37) | (0.47) |  | (4.11) | (2.41) | (2.42) | (0.57) | (1.68)

  * Note:The objective function was numerically maximized using differential evolution (DE) algorithm in BlackBoxOptim.jl package. For DE algorithm, I require setting the domain of parameters and the number of population seeds so that I fix the former to $[-10,10]$ for parameters and the latter to 400. The reported point estimates are the best-found maxima. I omit the parameter $\beta_{0}$ that can take on only a finite number of values (here +1) converge at an arbitrarily fast rate, they are superconsistent.

Table 3: Results of monte carlo simulations for Case 1 with IR.($\lambda=100$)

| Num of agents |  | 10 | 20 | 30 | 50 | 100 |  | 10 | 20 | 30 | 50 | 100
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\beta_{1}$ |  |  |  |  |  |  |  | $\beta_{2}$ |  |  |  |  |
0.5 | unmatched | Mean Num | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| U,T | Bias | 0.18 | 0.07 | 0.3 | 0.21 | 0.12 |  | 0.35 | 0.16 | 0.72 | 0.43 | 0.31
|  | RMSE | (0.63) | (0.26) | (0.66) | (0.44) | (0.26) |  | (1.05) | (0.53) | (1.32) | (0.82) | (0.58)
| T | Bias | 0.18 | 0.08 | 0.3 | 0.23 | 0.13 |  | 0.35 | 0.14 | 0.71 | 0.48 | 0.34
|  | RMSE | (0.63) | (0.27) | (0.65) | (0.47) | (0.28) |  | (1.05) | (0.49) | (1.32) | (0.88) | (0.61)
| U | Bias | 0.18 | 0.08 | 0.3 | 0.23 | 0.12 |  | 0.34 | 0.17 | 0.72 | 0.48 | 0.33
|  | RMSE | (0.63) | (0.29) | (0.65) | (0.47) | (0.28) |  | (1.05) | (0.58) | (1.32) | (0.88) | (0.62)
| Non | Bias | 0.18 | 0.08 | 0.3 | 0.23 | 0.12 |  | 0.34 | 0.17 | 0.72 | 0.48 | 0.33
|  | RMSE | (0.63) | (0.29) | (0.65) | (0.47) | (0.28) |  | (1.05) | (0.58) | (1.32) | (0.88) | (0.62)
0.5 | unmatched | Mean Num | 0.01 | 0.0 | 0.0 | 0.0 | 0.02 | 0.0 | 0.01 | 0.0 | 0.0 | 0.0 | 0.02
| U,T | Bias | 0.62 | 0.57 | 0.64 | 0.45 | 0.41 |  | 0.27 | 0.13 | 0.0 | 0.01 | 0.03
|  | RMSE | (1.68) | (1.17) | (1.12) | (0.66) | (0.58) |  | (1.16) | (0.56) | (0.27) | (0.21) | (0.19)
| T | Bias | 0.59 | 0.59 | 0.66 | 0.45 | 0.42 |  | 0.27 | 0.15 | 0.0 | 0.0 | 0.04
|  | RMSE | (1.67) | (1.18) | (1.13) | (0.67) | (0.59) |  | (1.16) | (0.56) | (0.28) | (0.22) | (0.19)
| U | Bias | 0.61 | 0.59 | 0.66 | 0.45 | 0.41 |  | 0.27 | 0.15 | 0.0 | -0.01 | 0.03
|  | RMSE | (1.69) | (1.18) | (1.13) | (0.67) | (0.59) |  | (1.16) | (0.56) | (0.28) | (0.22) | (0.19)
| Non | Bias | 0.61 | 0.6 | 0.66 | 0.45 | 0.41 |  | 0.27 | 0.14 | 0.0 | -0.01 | 0.03
|  | RMSE | (1.69) | (1.18) | (1.13) | (0.67) | (0.59) |  | (1.16) | (0.56) | (0.28) | (0.22) | (0.19)
0.5 | unmatched | Mean Num | 1.72 | 3.17 | 4.33 | 6.67 | 12.68 | -1.0 | 1.72 | 3.17 | 4.33 | 6.67 | 12.68
| U,T | Bias | 1.2 | 0.8 | 0.59 | 0.49 | 0.45 |  | -0.54 | -0.2 | -0.08 | 0.06 | 0.06
|  | RMSE | (2.13) | (1.25) | (0.85) | (0.72) | (0.61) |  | (1.15) | (0.58) | (0.41) | (0.3) | (0.24)
| T | Bias | 1.31 | 0.71 | 0.62 | 0.5 | 0.49 |  | -0.57 | -0.14 | -0.03 | 0.1 | 0.08
|  | RMSE | (2.42) | (1.16) | (0.91) | (0.73) | (0.67) |  | (1.31) | (0.54) | (0.43) | (0.34) | (0.26)
| U | Bias | 1.21 | 0.77 | 0.57 | 0.5 | 0.48 |  | -0.55 | -0.19 | -0.08 | 0.05 | 0.05
|  | RMSE | (2.13) | (1.23) | (0.85) | (0.75) | (0.66) |  | (1.16) | (0.57) | (0.4) | (0.32) | (0.24)
| Non | Bias | 1.22 | 0.71 | 0.62 | 0.5 | 0.48 |  | -0.52 | -0.15 | -0.03 | 0.09 | 0.08
|  | RMSE | (2.21) | (1.16) | (0.91) | (0.73) | (0.66) |  | (1.25) | (0.55) | (0.42) | (0.35) | (0.26)
0.5 | unmatched | Mean Num | 5.6 | 10.6 | 15.9 | 25.38 | 50.44 | -2.0 | 5.6 | 10.6 | 15.9 | 25.38 | 50.44
| U,T | Bias | 0.82 | 0.52 | 0.38 | 0.38 | 0.36 |  | -1.16 | -0.32 | -0.12 | -0.05 | 0.02
|  | RMSE | (2.0) | (0.89) | (0.7) | (0.61) | (0.58) |  | (2.28) | (0.9) | (0.59) | (0.53) | (0.57)
| T | Bias | 1.21 | 0.98 | 0.51 | 0.53 | 0.4 |  | -0.72 | -0.37 | 0.08 | 0.07 | 0.33
|  | RMSE | (2.66) | (1.82) | (0.94) | (0.85) | (0.62) |  | (2.55) | (1.63) | (0.61) | (0.67) | (0.55)
| U | Bias | 0.76 | 0.51 | 0.38 | 0.4 | 0.33 |  | -1.07 | -0.3 | -0.13 | -0.05 | 0.07
|  | RMSE | (1.95) | (0.87) | (0.65) | (0.62) | (0.51) |  | (2.21) | (0.87) | (0.57) | (0.54) | (0.44)
| Non | Bias | 1.09 | 0.98 | 0.51 | 0.52 | 0.38 |  | -0.56 | -0.37 | 0.1 | 0.07 | 0.33
|  | RMSE | (2.58) | (1.82) | (0.94) | (0.85) | (0.6) |  | (2.62) | (1.63) | (0.63) | (0.67) | (0.55)
0.5 | unmatched | Mean Num | 7.62 | 15.01 | 22.0 | 36.08 | 71.75 | -3.0 | 7.62 | 15.01 | 22.0 | 36.08 | 71.75
| U,T | Bias | 0.45 | 0.35 | 0.33 | 0.28 | 0.26 |  | -1.71 | -0.48 | -0.28 | -0.01 | 0.05
|  | RMSE | (2.24) | (0.6) | (0.63) | (0.53) | (0.53) |  | (2.83) | (1.25) | (1.11) | (0.72) | (0.83)
| T | Bias | 1.22 | 1.17 | 0.85 | 0.64 | 0.36 |  | 0.23 | 0.24 | -0.33 | 0.15 | 0.36
|  | RMSE | (3.15) | (2.29) | (1.47) | (1.09) | (0.61) |  | (4.12) | (2.93) | (1.76) | (1.42) | (0.81)
| U | Bias | 0.4 | 0.33 | 0.29 | 0.26 | 0.21 |  | -1.64 | -0.46 | -0.23 | 0.01 | 0.13
|  | RMSE | (2.23) | (0.61) | (0.51) | (0.52) | (0.38) |  | (2.77) | (1.27) | (0.91) | (0.7) | (0.6)
| Non | Bias | 1.41 | 1.21 | 0.87 | 0.65 | 0.34 |  | 0.26 | 0.23 | -0.35 | 0.14 | 0.38
|  | RMSE | (3.33) | (2.4) | (1.49) | (1.12) | (0.61) |  | (4.09) | (2.95) | (1.77) | (1.42) | (0.81)

  * Note:The objective function was numerically maximized using differential evolution (DE) algorithm in BlackBoxOptim.jl package. For DE algorithm, I require setting the domain of parameters and the number of population seeds so that I fix the former to $[-10,10]$ for parameters and the latter to 400. The reported point estimates are the best-found maxima. I omit the parameter $\beta_{0}$ that can take on only a finite number of values (here +1) converge at an arbitrarily fast rate, they are superconsistent.

Table 4: Results of monte carlo simulations for Case 2 without
IR.($\lambda=100$)

| Num of agents |  | 10 | 20 | 30 | 50 | 100 |  | 10 | 20 | 30 | 50 | 100
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\beta_{1}$ |  |  |  |  |  |  |  | $\beta_{2}$ |  |  |  |  |
0.5 | unmatched | Mean Num | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| U,T | Bias | 0.31 | 0.28 | 0.14 | 0.04 | 0.01 |  | -0.53 | -1.12 | -0.69 | -1.79 | -1.59
|  | RMSE | (1.64) | (0.84) | (0.46) | (0.29) | (0.16) |  | (5.73) | (5.12) | (5.76) | (5.88) | (5.88)
| T | Bias | 0.41 | 0.31 | 0.14 | 0.04 | 0.0 |  | -0.49 | -1.17 | -0.75 | -1.71 | -1.82
|  | RMSE | (1.77) | (0.89) | (0.47) | (0.29) | (0.16) |  | (5.6) | (5.16) | (5.77) | (5.85) | (6.0)
| U | Bias | 0.41 | 0.32 | 0.14 | 0.04 | 0.0 |  | -0.51 | -1.29 | -0.74 | -1.7 | -1.58
|  | RMSE | (1.77) | (0.89) | (0.47) | (0.29) | (0.16) |  | (5.57) | (5.18) | (5.75) | (5.86) | (5.98)
| Non | Bias | 0.41 | 0.32 | 0.14 | 0.04 | 0.0 |  | -0.51 | -1.26 | -0.74 | -1.7 | -1.58
|  | RMSE | (1.77) | (0.89) | (0.47) | (0.29) | (0.16) |  | (5.57) | (5.14) | (5.75) | (5.86) | (5.98)
0.5 | unmatched | Mean Num | 0.01 | 0.0 | 0.0 | 0.0 | 0.02 | 0.0 | 0.01 | 0.0 | 0.0 | 0.0 | 0.02
| U,T | Bias | 0.51 | 0.24 | 0.14 | 0.04 | 0.01 |  | 0.97 | 0.2 | 0.3 | -0.86 | -0.94
|  | RMSE | (1.99) | (0.74) | (0.46) | (0.29) | (0.16) |  | (5.55) | (5.0) | (5.71) | (5.66) | (5.8)
| T | Bias | 0.57 | 0.24 | 0.14 | 0.04 | 0.0 |  | 0.92 | 0.2 | 0.25 | -0.83 | -0.78
|  | RMSE | (2.06) | (0.74) | (0.47) | (0.29) | (0.16) |  | (5.5) | (5.0) | (5.73) | (5.68) | (5.81)
| U | Bias | 0.6 | 0.24 | 0.14 | 0.04 | 0.0 |  | 1.01 | 0.12 | 0.26 | -0.71 | -0.56
|  | RMSE | (2.1) | (0.74) | (0.47) | (0.29) | (0.16) |  | (5.42) | (4.96) | (5.71) | (5.65) | (5.81)
| Non | Bias | 0.51 | 0.24 | 0.14 | 0.04 | 0.0 |  | 0.94 | 0.12 | 0.26 | -0.71 | -0.56
|  | RMSE | (1.99) | (0.74) | (0.47) | (0.29) | (0.16) |  | (5.52) | (4.96) | (5.71) | (5.65) | (5.81)
0.5 | unmatched | Mean Num | 1.56 | 2.87 | 3.81 | 5.61 | 10.47 | -1.0 | 1.56 | 2.87 | 3.81 | 5.61 | 10.47
| U,T | Bias | 0.28 | 0.33 | 0.12 | 0.03 | 0.02 |  | -3.72 | -4.97 | -4.81 | -4.24 | -4.71
|  | RMSE | (1.45) | (1.07) | (0.43) | (0.21) | (0.15) |  | (5.35) | (5.64) | (5.57) | (5.05) | (5.53)
| T | Bias | 0.28 | 0.38 | 0.14 | 0.04 | 0.03 |  | -3.08 | -3.14 | -2.92 | -2.55 | -2.81
|  | RMSE | (1.43) | (1.25) | (0.41) | (0.25) | (0.16) |  | (5.45) | (5.96) | (5.42) | (5.32) | (5.26)
| U | Bias | 0.29 | 0.3 | 0.12 | 0.04 | 0.01 |  | -3.78 | -5.04 | -5.08 | -4.52 | -4.98
|  | RMSE | (1.45) | (1.06) | (0.44) | (0.22) | (0.15) |  | (5.36) | (5.66) | (5.77) | (5.23) | (5.68)
| Non | Bias | 0.28 | 0.38 | 0.15 | 0.05 | 0.02 |  | -3.08 | -3.01 | -2.84 | -2.53 | -2.86
|  | RMSE | (1.43) | (1.25) | (0.48) | (0.25) | (0.15) |  | (5.45) | (5.98) | (5.56) | (5.31) | (5.32)
0.5 | unmatched | Mean Num | 5.57 | 11.0 | 16.11 | 26.31 | 51.08 | -2.0 | 5.57 | 11.0 | 16.11 | 26.31 | 51.08
| U,T | Bias | 0.26 | 0.07 | 0.05 | 0.04 | 0.02 |  | -4.29 | -4.56 | -4.45 | -4.37 | -4.34
|  | RMSE | (0.86) | (0.31) | (0.24) | (0.16) | (0.11) |  | (4.89) | (5.07) | (5.04) | (5.02) | (4.9)
| T | Bias | 0.25 | 0.13 | 0.43 | 0.35 | 0.07 |  | -0.99 | 0.11 | -0.51 | 0.88 | 0.81
|  | RMSE | (2.0) | (1.47) | (1.66) | (0.94) | (0.28) |  | (5.73) | (5.32) | (5.64) | (5.84) | (5.36)
| U | Bias | 0.28 | 0.05 | 0.03 | 0.03 | 0.02 |  | -4.46 | -4.69 | -4.63 | -4.61 | -4.52
|  | RMSE | (0.88) | (0.29) | (0.2) | (0.15) | (0.1) |  | (5.0) | (5.13) | (5.12) | (5.16) | (4.99)
| Non | Bias | 0.25 | 0.14 | 0.43 | 0.35 | 0.07 |  | -0.99 | 0.06 | -0.61 | 0.72 | 0.8
|  | RMSE | (2.0) | (1.47) | (1.66) | (0.94) | (0.28) |  | (5.73) | (5.35) | (5.6) | (5.76) | (5.37)
0.5 | unmatched | Mean Num | 8.88 | 17.4 | 25.72 | 42.52 | 84.13 | -3.0 | 8.88 | 17.4 | 25.72 | 42.52 | 84.13
| U,T | Bias | -0.38 | 0.1 | 0.08 | 0.01 | 0.04 |  | -2.78 | -3.87 | -3.32 | -3.72 | -3.6
|  | RMSE | (2.38) | (0.55) | (0.4) | (0.19) | (0.18) |  | (4.1) | (4.37) | (3.91) | (4.28) | (4.08)
| T | Bias | -0.04 | 0.3 | 0.23 | 0.08 | 0.91 |  | 0.41 | 2.7 | 1.65 | 3.34 | 2.82
|  | RMSE | (4.16) | (3.19) | (2.98) | (0.88) | (2.33) |  | (5.6) | (6.82) | (5.84) | (6.3) | (6.16)
| U | Bias | -0.39 | 0.09 | 0.06 | 0.0 | 0.01 |  | -2.8 | -4.08 | -3.5 | -3.98 | -3.86
|  | RMSE | (2.37) | (0.54) | (0.36) | (0.18) | (0.11) |  | (4.12) | (4.52) | (4.01) | (4.42) | (4.27)
| Non | Bias | -0.13 | 0.29 | 0.22 | 0.1 | 0.93 |  | 0.6 | 2.52 | 1.54 | 3.24 | 2.97
|  | RMSE | (4.24) | (3.19) | (2.98) | (0.88) | (2.33) |  | (5.6) | (6.74) | (5.71) | (6.3) | (6.17)

  * Note:The objective function was numerically maximized using differential evolution (DE) algorithm in BlackBoxOptim.jl package. For DE algorithm, I require setting the domain of parameters and the number of population seeds so that I fix the former to $[-10,10]$ for parameters and the latter to 400. The reported point estimates are the best-found maxima. I omit the parameter $\beta_{0}$ that can take on only a finite number of values (here +1) converge at an arbitrarily fast rate, they are superconsistent.

Table 5: Results of monte carlo simulations for Case 2 with IR.
($\lambda=100$)

| Num of agents |  | 10 | 20 | 30 | 50 | 100 |  | 10 | 20 | 30 | 50 | 100
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\beta_{1}$ |  |  |  |  |  |  |  | $\beta_{2}$ |  |  |  |  |
0.5 | unmatched | Mean Num | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| U,T | Bias | 0.17 | 0.25 | 0.16 | 0.05 | 0.01 |  | 3.87 | 4.25 | 4.13 | 4.28 | 4.67
|  | RMSE | (0.96) | (0.78) | (0.49) | (0.28) | (0.16) |  | (4.84) | (5.06) | (5.07) | (5.16) | (5.43)
| T | Bias | 0.17 | 0.26 | 0.16 | 0.05 | 0.01 |  | 3.87 | 4.32 | 4.19 | 4.29 | 4.66
|  | RMSE | (0.96) | (0.78) | (0.49) | (0.28) | (0.17) |  | (4.84) | (5.12) | (5.14) | (5.14) | (5.41)
| U | Bias | 0.17 | 0.29 | 0.15 | 0.05 | 0.01 |  | 3.87 | 4.24 | 4.22 | 4.46 | 4.62
|  | RMSE | (0.96) | (0.86) | (0.49) | (0.28) | (0.17) |  | (4.84) | (5.05) | (5.15) | (5.29) | (5.38)
| Non | Bias | 0.17 | 0.29 | 0.15 | 0.05 | 0.01 |  | 3.87 | 4.24 | 4.22 | 4.46 | 4.62
|  | RMSE | (0.96) | (0.86) | (0.49) | (0.28) | (0.17) |  | (4.84) | (5.05) | (5.15) | (5.29) | (5.38)
0.5 | unmatched | Mean Num | 0.01 | 0.0 | 0.0 | 0.0 | 0.02 | 0.0 | 0.01 | 0.0 | 0.0 | 0.0 | 0.02
| U,T | Bias | 0.63 | 0.26 | 0.16 | 0.05 | 0.01 |  | 4.79 | 5.33 | 5.13 | 5.41 | 5.56
|  | RMSE | (1.98) | (0.78) | (0.49) | (0.28) | (0.16) |  | (5.68) | (5.99) | (5.91) | (6.13) | (6.21)
| T | Bias | 0.59 | 0.26 | 0.16 | 0.05 | 0.01 |  | 4.86 | 5.33 | 5.19 | 5.41 | 5.55
|  | RMSE | (1.95) | (0.78) | (0.49) | (0.28) | (0.16) |  | (5.72) | (5.99) | (5.98) | (6.14) | (6.21)
| U | Bias | 0.63 | 0.29 | 0.15 | 0.05 | 0.01 |  | 4.84 | 5.24 | 5.22 | 5.46 | 5.55
|  | RMSE | (1.98) | (0.86) | (0.49) | (0.28) | (0.17) |  | (5.7) | (5.91) | (6.0) | (6.16) | (6.18)
| Non | Bias | 0.63 | 0.29 | 0.15 | 0.05 | 0.01 |  | 4.84 | 5.24 | 5.22 | 5.46 | 5.55
|  | RMSE | (1.98) | (0.86) | (0.49) | (0.28) | (0.17) |  | (5.7) | (5.91) | (6.0) | (6.16) | (6.18)
0.5 | unmatched | Mean Num | 1.52 | 2.97 | 3.83 | 5.78 | 10.73 | -1.0 | 1.52 | 2.97 | 3.83 | 5.78 | 10.73
| U,T | Bias | 0.59 | 0.44 | 0.24 | 0.19 | 0.2 |  | 2.59 | 1.61 | 1.94 | 1.67 | 1.88
|  | RMSE | (1.59) | (1.16) | (0.59) | (0.44) | (0.38) |  | (4.73) | (3.42) | (3.76) | (3.54) | (3.72)
| T | Bias | 0.63 | 0.41 | 0.23 | 0.14 | 0.15 |  | 2.94 | 2.83 | 2.9 | 3.52 | 3.15
|  | RMSE | (1.69) | (1.18) | (0.57) | (0.4) | (0.34) |  | (4.98) | (4.6) | (4.49) | (5.25) | (4.81)
| U | Bias | 0.59 | 0.41 | 0.23 | 0.2 | 0.2 |  | 2.57 | 1.49 | 1.74 | 1.45 | 1.79
|  | RMSE | (1.59) | (1.18) | (0.58) | (0.44) | (0.39) |  | (4.7) | (3.22) | (3.46) | (3.19) | (3.61)
| Non | Bias | 0.67 | 0.43 | 0.22 | 0.17 | 0.15 |  | 2.92 | 2.5 | 2.61 | 3.21 | 3.25
|  | RMSE | (1.73) | (1.24) | (0.57) | (0.41) | (0.34) |  | (5.05) | (4.32) | (4.23) | (4.94) | (4.98)
0.5 | unmatched | Mean Num | 5.5 | 11.15 | 16.41 | 26.41 | 51.16 | -2.0 | 5.5 | 11.15 | 16.41 | 26.41 | 51.16
| U,T | Bias | 0.55 | 0.33 | 0.24 | 0.29 | 0.35 |  | -0.44 | -0.08 | 0.1 | 0.05 | 0.01
|  | RMSE | (1.17) | (0.68) | (0.47) | (0.56) | (0.71) |  | (1.34) | (0.68) | (0.52) | (0.5) | (0.7)
| T | Bias | 0.86 | 0.46 | 0.6 | 0.44 | 0.12 |  | 1.63 | 3.91 | 4.15 | 5.24 | 5.92
|  | RMSE | (2.18) | (1.71) | (1.83) | (1.02) | (0.37) |  | (4.3) | (5.93) | (5.63) | (6.6) | (7.07)
| U | Bias | 0.55 | 0.29 | 0.21 | 0.29 | 0.28 |  | -0.45 | -0.04 | 0.13 | 0.07 | 0.1
|  | RMSE | (1.17) | (0.63) | (0.45) | (0.54) | (0.49) |  | (1.34) | (0.65) | (0.53) | (0.46) | (0.41)
| Non | Bias | 0.82 | 0.46 | 0.65 | 0.43 | 0.13 |  | 1.74 | 3.94 | 4.14 | 5.4 | 5.86
|  | RMSE | (2.07) | (1.71) | (1.9) | (1.02) | (0.37) |  | (4.29) | (5.94) | (5.62) | (6.74) | (6.99)
0.5 | unmatched | Mean Num | 8.81 | 17.45 | 25.71 | 42.59 | 84.21 | -3.0 | 8.81 | 17.45 | 25.71 | 42.59 | 84.21
| U,T | Bias | -0.19 | 0.33 | 0.38 | 0.18 | 0.21 |  | -1.48 | -0.58 | -0.56 | -0.05 | -0.0
|  | RMSE | (2.38) | (0.82) | (0.74) | (0.43) | (0.44) |  | (2.72) | (1.63) | (1.32) | (0.68) | (0.68)
| T | Bias | 1.26 | 1.28 | 0.42 | 0.13 | 1.4 |  | 1.42 | 4.63 | 5.38 | 6.08 | 5.82
|  | RMSE | (3.9) | (3.62) | (1.92) | (1.0) | (2.96) |  | (5.36) | (6.96) | (7.02) | (7.09) | (7.12)
| U | Bias | -0.19 | 0.33 | 0.35 | 0.18 | 0.18 |  | -1.48 | -0.56 | -0.49 | -0.05 | 0.05
|  | RMSE | (2.38) | (0.87) | (0.72) | (0.43) | (0.37) |  | (2.72) | (1.69) | (1.24) | (0.69) | (0.57)
| Non | Bias | 1.28 | 1.37 | 0.41 | 0.12 | 1.42 |  | 1.5 | 4.74 | 5.31 | 6.1 | 5.78
|  | RMSE | (3.87) | (3.67) | (1.92) | (1.0) | (2.97) |  | (5.4) | (7.02) | (6.93) | (7.1) | (7.08)

  * Note:The objective function was numerically maximized using differential evolution (DE) algorithm in BlackBoxOptim.jl package. For DE algorithm, I require setting the domain of parameters and the number of population seeds so that I fix the former to $[-10,10]$ for parameters and the latter to 400. The reported point estimates are the best-found maxima. I omit the parameter $\beta_{0}$ that can take on only a finite number of values (here +1) converge at an arbitrarily fast rate, they are superconsistent.

##  Appendix C Other experiments (For Online Publication)

###  C.1 How many unmatched agents should we include?

I investigate the safe levels of the proportion of the number of unmatched
agents to that of matched agents. In particular, I focus on Model 3, i.e., the
case when unmatched data is available. Table 6 shows that at least 30 % of the
full sample ($N=100$) should be unmatched agents. For example, if
$\beta_{2}=-1.6$, then 33 % of the full samples are not matched and the bias
is 0.19. Table 6 also suggests that if the number of unmatched agents is
relatively small, identification of $\beta_{2}$ may fail and the estimate will
be heavily biased upward. For example, if $\beta_{2}=-1.2$ and less than 20 %
of the full samples are unmatched, then the bias is 0.76. The case approaches
to the case where unmatched data is not available, like Model 2 and 4.

Table 6: How many unmatched agents should we include?(Case 2 with IR,
$\lambda=100$).

| Num of agents |  | 10 | 20 | 30 | 50 | 100 |  | 10 | 20 | 30 | 50 | 100
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\beta_{1}$ |  |  |  |  |  |  |  | $\beta_{2}$ |  |  |  |  |
0.5 | unmatched | Mean Num | 1.5 | 2.81 | 3.8 | 5.78 | 10.76 | -1.0 | 1.5 | 2.81 | 3.8 | 5.78 | 10.76
| U | Bias | 0.59 | 0.46 | 0.25 | 0.2 | 0.17 |  | 2.63 | 2.29 | 2.26 | 2.43 | 2.38
|  | RMSE | (1.63) | (1.22) | (0.56) | (0.47) | (0.32) |  | (4.49) | (4.15) | (4.03) | (4.23) | (3.93)
0.5 | unmatched | Mean Num | 1.93 | 3.38 | 4.87 | 7.38 | 13.71 | -1.1 | 1.93 | 3.38 | 4.87 | 7.38 | 13.71
| U | Bias | 0.6 | 0.44 | 0.33 | 0.22 | 0.19 |  | 2.25 | 1.93 | 1.57 | 2.66 | 1.98
|  | RMSE | (1.46) | (0.87) | (0.62) | (0.44) | (0.36) |  | (4.44) | (4.01) | (3.35) | (4.46) | (3.84)
0.5 | unmatched | Mean Num | 2.28 | 4.12 | 5.89 | 9.11 | 17.19 | -1.2 | 2.28 | 4.12 | 5.89 | 9.11 | 17.19
| U | Bias | 0.49 | 0.52 | 0.33 | 0.33 | 0.36 |  | 1.85 | 1.11 | 1.22 | 1.67 | 0.76
|  | RMSE | (1.59) | (1.2) | (0.55) | (0.52) | (0.52) |  | (4.08) | (3.0) | (2.93) | (3.46) | (2.24)
0.5 | unmatched | Mean Num | 2.52 | 4.96 | 7.02 | 11.03 | 20.59 | -1.3 | 2.52 | 4.96 | 7.02 | 11.03 | 20.59
| U | Bias | 0.6 | 0.49 | 0.33 | 0.35 | 0.29 |  | 1.5 | 1.06 | 0.77 | 1.29 | 0.53
|  | RMSE | (1.44) | (0.85) | (0.58) | (0.6) | (0.44) |  | (3.62) | (3.03) | (2.28) | (2.99) | (1.32)
0.5 | unmatched | Mean Num | 2.96 | 5.75 | 8.2 | 13.0 | 24.6 | -1.4 | 2.96 | 5.75 | 8.2 | 13.0 | 24.6
| U | Bias | 0.67 | 0.41 | 0.4 | 0.44 | 0.34 |  | 0.86 | 0.93 | 0.5 | 0.48 | 0.3
|  | RMSE | (1.49) | (0.82) | (0.72) | (0.64) | (0.52) |  | (2.92) | (2.68) | (1.61) | (1.58) | (0.42)
0.5 | unmatched | Mean Num | 3.32 | 6.57 | 9.51 | 15.18 | 28.66 | -1.5 | 3.32 | 6.57 | 9.51 | 15.18 | 28.66
| U | Bias | 0.84 | 0.49 | 0.43 | 0.35 | 0.31 |  | 0.05 | 0.59 | 0.51 | 0.33 | 0.28
|  | RMSE | (1.69) | (0.93) | (0.65) | (0.5) | (0.5) |  | (2.14) | (2.28) | (1.8) | (1.02) | (0.43)
0.5 | unmatched | Mean Num | 3.79 | 7.32 | 10.71 | 17.6 | 33.29 | -1.6 | 3.79 | 7.32 | 10.71 | 17.6 | 33.29
| U | Bias | 0.78 | 0.62 | 0.4 | 0.37 | 0.38 |  | -0.18 | 0.17 | 0.16 | 0.26 | 0.19
|  | RMSE | (1.39) | (0.96) | (0.69) | (0.61) | (0.54) |  | (1.82) | (1.58) | (0.7) | (0.99) | (0.37)
0.5 | unmatched | Mean Num | 4.24 | 8.23 | 12.1 | 19.67 | 37.79 | -1.7 | 4.24 | 8.23 | 12.1 | 19.67 | 37.79
| U | Bias | 0.66 | 0.53 | 0.46 | 0.39 | 0.39 |  | -0.39 | -0.03 | 0.05 | 0.21 | 0.14
|  | RMSE | (1.27) | (0.89) | (0.77) | (0.6) | (0.62) |  | (1.41) | (0.76) | (0.56) | (0.45) | (0.44)
0.5 | unmatched | Mean Num | 4.75 | 9.2 | 13.47 | 22.06 | 42.32 | -1.8 | 4.75 | 9.2 | 13.47 | 22.06 | 42.32
| U | Bias | 0.57 | 0.58 | 0.43 | 0.46 | 0.36 |  | -0.23 | -0.03 | 0.13 | 0.05 | 0.15
|  | RMSE | (0.97) | (0.91) | (0.77) | (0.71) | (0.54) |  | (1.54) | (1.39) | (0.91) | (0.52) | (0.42)
0.5 | unmatched | Mean Num | 5.12 | 10.2 | 14.87 | 24.3 | 46.62 | -1.9 | 5.12 | 10.2 | 14.87 | 24.3 | 46.62
| U | Bias | 0.7 | 0.57 | 0.47 | 0.43 | 0.39 |  | -0.45 | -0.14 | 0.05 | 0.03 | 0.07
|  | RMSE | (1.32) | (0.9) | (0.78) | (0.68) | (0.58) |  | (1.49) | (0.73) | (0.56) | (0.55) | (0.4)
0.5 | unmatched | Mean Num | 5.55 | 11.08 | 16.12 | 26.46 | 51.1 | -2.0 | 5.55 | 11.08 | 16.12 | 26.46 | 51.1
| U | Bias | 0.58 | 0.47 | 0.45 | 0.41 | 0.44 |  | -0.37 | -0.08 | 0.04 | 0.08 | -0.01
|  | RMSE | (1.04) | (0.75) | (0.75) | (0.6) | (0.65) |  | (1.22) | (0.64) | (0.59) | (0.45) | (0.45)
0.5 | unmatched | Mean Num | 6.05 | 11.86 | 17.49 | 28.55 | 55.25 | -2.1 | 6.05 | 11.86 | 17.49 | 28.55 | 55.25
| U | Bias | 0.72 | 0.55 | 0.51 | 0.33 | 0.31 |  | -0.66 | -0.27 | -0.11 | 0.1 | 0.1
|  | RMSE | (1.47) | (0.92) | (0.83) | (0.5) | (0.54) |  | (1.88) | (1.01) | (0.72) | (0.41) | (0.52)
0.5 | unmatched | Mean Num | 6.44 | 12.58 | 18.69 | 30.6 | 59.37 | -2.2 | 6.44 | 12.58 | 18.69 | 30.6 | 59.37
| U | Bias | 0.64 | 0.49 | 0.43 | 0.36 | 0.32 |  | -0.59 | -0.2 | -0.03 | 0.06 | 0.04
|  | RMSE | (1.35) | (0.84) | (0.71) | (0.56) | (0.5) |  | (1.81) | (0.91) | (0.65) | (0.56) | (0.48)
0.5 | unmatched | Mean Num | 6.91 | 13.35 | 19.78 | 32.49 | 63.34 | -2.3 | 6.91 | 13.35 | 19.78 | 32.49 | 63.34
| U | Bias | 0.71 | 0.38 | 0.47 | 0.38 | 0.32 |  | -0.79 | -0.09 | -0.13 | -0.01 | 0.04
|  | RMSE | (1.46) | (0.66) | (0.71) | (0.6) | (0.53) |  | (1.99) | (0.75) | (0.68) | (0.58) | (0.56)
0.5 | unmatched | Mean Num | 7.2 | 14.1 | 20.79 | 34.17 | 67.23 | -2.4 | 7.2 | 14.1 | 20.79 | 34.17 | 67.23
| U | Bias | 0.63 | 0.45 | 0.35 | 0.3 | 0.31 |  | -0.74 | -0.26 | -0.05 | 0.08 | 0.03
|  | RMSE | (1.4) | (0.84) | (0.61) | (0.57) | (0.53) |  | (2.04) | (1.14) | (0.71) | (0.63) | (0.61)
0.5 | unmatched | Mean Num | 7.46 | 14.79 | 21.7 | 35.69 | 70.88 | -2.5 | 7.46 | 14.79 | 21.7 | 35.69 | 70.88
| U | Bias | 0.49 | 0.47 | 0.33 | 0.33 | 0.24 |  | -0.6 | -0.34 | -0.03 | 0.03 | 0.09
|  | RMSE | (1.37) | (0.94) | (0.62) | (0.56) | (0.42) |  | (1.98) | (1.37) | (0.69) | (0.6) | (0.5)
0.5 | unmatched | Mean Num | 7.82 | 15.33 | 22.67 | 37.37 | 74.26 | -2.6 | 7.82 | 15.33 | 22.67 | 37.37 | 74.26
| U | Bias | 0.32 | 0.37 | 0.29 | 0.29 | 0.31 |  | -0.75 | -0.29 | -0.02 | 0.01 | -0.03
|  | RMSE | (1.31) | (0.79) | (0.59) | (0.51) | (0.52) |  | (1.98) | (1.27) | (0.71) | (0.59) | (0.62)
0.5 | unmatched | Mean Num | 8.11 | 15.99 | 23.57 | 38.96 | 77.06 | -2.7 | 8.11 | 15.99 | 23.57 | 38.96 | 77.06
| U | Bias | 0.34 | 0.45 | 0.27 | 0.3 | 0.19 |  | -0.8 | -0.37 | -0.02 | -0.03 | 0.17
|  | RMSE | (1.18) | (0.76) | (0.61) | (0.49) | (0.36) |  | (1.82) | (1.08) | (0.83) | (0.57) | (0.48)
0.5 | unmatched | Mean Num | 8.38 | 16.54 | 24.43 | 40.21 | 79.72 | -2.8 | 8.38 | 16.54 | 24.43 | 40.21 | 79.72
| U | Bias | 0.26 | 0.35 | 0.39 | 0.29 | 0.25 |  | -0.97 | -0.33 | -0.27 | 0.01 | 0.04
|  | RMSE | (1.32) | (0.67) | (0.68) | (0.49) | (0.44) |  | (2.1) | (1.09) | (0.94) | (0.58) | (0.59)
0.5 | unmatched | Mean Num | 8.63 | 17.01 | 25.25 | 41.41 | 82.31 | -2.9 | 8.63 | 17.01 | 25.25 | 41.41 | 82.31
| U | Bias | 0.26 | 0.43 | 0.37 | 0.31 | 0.32 |  | -1.43 | -0.58 | -0.3 | -0.06 | -0.08
|  | RMSE | (1.63) | (0.88) | (0.61) | (0.52) | (0.54) |  | (2.73) | (1.63) | (0.88) | (0.67) | (0.73)
0.5 | unmatched | Mean Num | 8.81 | 17.42 | 25.84 | 42.61 | 84.38 | -3.0 | 8.81 | 17.42 | 25.84 | 42.61 | 84.38
| U | Bias | -0.22 | 0.48 | 0.35 | 0.26 | 0.21 |  | -1.45 | -0.73 | -0.32 | -0.03 | 0.05
|  | RMSE | (2.38) | (1.01) | (0.64) | (0.49) | (0.39) |  | (2.72) | (1.81) | (1.0) | (0.69) | (0.54)

  * Note:The objective function was numerically maximized using differential evolution (DE) algorithm in BlackBoxOptim.jl package. For DE algorithm, I require setting the domain of parameters and the number of population seeds so that I fix the former to $[-10,10]$ for parameters and the latter to 400. The reported point estimates are the best-found maxima. I omit the parameter $\beta_{0}$ that can take on only a finite number of values (here +1) converge at an arbitrarily fast rate, they are superconsistent.

###  C.2 Sensitivity of $\lambda$, the importance weight of IR conditions.

Figure 3 illustrates how the importance weight of IR conditions sharpens the
objective function of $\beta_{2}$ for Model 3, that is, the case when
unmatched data are available. The first left panel for $\beta_{2}=-2$ shows
that even the lowest importance weight ($\lambda=1$) is helpful to obtain a
maximizer point. Furthermore, the larger the importance weight is, the tighter
the lower bound of $\beta_{2}$ is. This implies that DE algorithm is easier to
find the maximizer under the larger importance weight. On the contrary, the
first right panel of $\beta_{2}=-1$ shows that the lowest importance weight
($\lambda=1$) is enough for correction, which is consistent with Table 7.
Then, large $\lambda$ may loosen the upper bound of $\beta_{2}$. It should be
noted that incorporating IR conditions with importance weight is not
significantly harmful for estimations of all other parameters. Therefore, the
researchers who want to estimate a matching cost must include IR conditions
with the recommended importance weight in any case.

Tables 7, 8, 9, 10, 11 correspond to Table 5 with different importance weight
$\lambda$. These suggest an adequate weight of $\lambda$. Concretely, if
$\beta_{2}$ is too low and more than half of the samples are not matched, then
small $\lambda$ induces the downward bias of $\beta_{2}$. This is because the
correction term (7) is ignorable and the lower bound is loose when
$|\mathcal{M}^{m}|$ is small. On the other hand, if $\beta_{2}$ is not low and
more than 90 % of sample are matched, then $\lambda=1$ gives unbiased
estimates rather than large $\lambda$ because $|\mathcal{M}^{m}|$ is
sufficiently large. As a numerical finding, whether 30 % of the full samples
are unmatched or not is the practical threshold about whether researchers
should use sufficiently high $\lambda$ or $\lambda=1$.

Figure 3: Contour maps of the objective function of Model 3 for Case 2 under
different $\lambda$ ($\beta_{2}=-2$ (left) and $\beta_{2}=-1$ (right)).

  * a

Note: I use Model 3 (i.e., $(U)$) under the same simulated single dataset with
$\beta_{0}=+1,\beta_{1}=0.5$ and $\beta_{2}=-2$ (left) and $-1$ (right). The
only difference is $\lambda$, the importance weight of IR conditions. I pick
up $\lambda$ from $\\{1,10,25,50\\}$. The maximum point of the objective
function is plotted. The maxima of the objective function are plotted as red
points

Table 7: Case 2 with IR ($\lambda=1$).

| Num of agents |  | 10 | 20 | 30 | 50 | 100 |  | 10 | 20 | 30 | 50 | 100
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\beta_{1}$ |  |  |  |  |  |  |  | $\beta_{2}$ |  |  |  |  |
0.5 | unmatched | Mean Num | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| U,T | Bias | 0.42 | 0.17 | 0.15 | 0.05 | 0.01 |  | 3.71 | 4.09 | 4.3 | 3.48 | 4.15
|  | RMSE | (1.65) | (0.5) | (0.5) | (0.28) | (0.15) |  | (4.65) | (5.09) | (5.17) | (4.69) | (5.12)
| T | Bias | 0.46 | 0.17 | 0.15 | 0.05 | 0.01 |  | 3.65 | 4.09 | 4.3 | 3.51 | 4.19
|  | RMSE | (1.69) | (0.5) | (0.5) | (0.28) | (0.16) |  | (4.64) | (5.09) | (5.17) | (4.72) | (5.14)
| U | Bias | 0.46 | 0.17 | 0.15 | 0.05 | 0.01 |  | 3.63 | 4.12 | 4.43 | 3.56 | 4.23
|  | RMSE | (1.69) | (0.5) | (0.5) | (0.29) | (0.16) |  | (4.62) | (5.1) | (5.25) | (4.73) | (5.14)
| Non | Bias | 0.46 | 0.17 | 0.15 | 0.05 | 0.01 |  | 3.63 | 4.12 | 4.43 | 3.56 | 4.23
|  | RMSE | (1.69) | (0.5) | (0.5) | (0.29) | (0.16) |  | (4.62) | (5.1) | (5.25) | (4.73) | (5.14)
0.5 | unmatched | Mean Num | 0.01 | 0.0 | 0.0 | 0.0 | 0.02 | 0.0 | 0.01 | 0.0 | 0.0 | 0.0 | 0.02
| U,T | Bias | 0.62 | 0.13 | 0.15 | 0.05 | 0.01 |  | 4.56 | 4.87 | 5.3 | 4.39 | 5.14
|  | RMSE | (1.99) | (0.44) | (0.5) | (0.28) | (0.15) |  | (5.52) | (5.74) | (6.03) | (5.4) | (5.93)
| T | Bias | 0.62 | 0.13 | 0.15 | 0.05 | 0.01 |  | 4.53 | 4.87 | 5.3 | 4.46 | 5.16
|  | RMSE | (1.99) | (0.44) | (0.5) | (0.28) | (0.16) |  | (5.5) | (5.74) | (6.03) | (5.45) | (5.95)
| U | Bias | 0.62 | 0.13 | 0.15 | 0.05 | 0.01 |  | 4.53 | 4.9 | 5.43 | 4.56 | 5.22
|  | RMSE | (1.99) | (0.44) | (0.5) | (0.29) | (0.16) |  | (5.5) | (5.75) | (6.12) | (5.52) | (5.98)
| Non | Bias | 0.62 | 0.13 | 0.15 | 0.05 | 0.01 |  | 4.53 | 4.9 | 5.43 | 4.56 | 5.22
|  | RMSE | (1.99) | (0.44) | (0.5) | (0.29) | (0.16) |  | (5.5) | (5.75) | (6.12) | (5.52) | (5.98)
0.5 | unmatched | Mean Num | 1.6 | 2.83 | 3.91 | 5.74 | 10.73 | -1.0 | 1.6 | 2.83 | 3.91 | 5.74 | 10.73
| U,T | Bias | 0.49 | 0.41 | 0.21 | 0.12 | 0.07 |  | 1.98 | 0.6 | 0.34 | 0.17 | -0.07
|  | RMSE | (1.54) | (1.13) | (0.52) | (0.3) | (0.23) |  | (4.4) | (2.75) | (1.75) | (1.22) | (0.31)
| T | Bias | 0.6 | 0.38 | 0.21 | 0.11 | 0.07 |  | 2.18 | 2.15 | 1.86 | 2.07 | 1.81
|  | RMSE | (1.77) | (1.17) | (0.51) | (0.32) | (0.22) |  | (4.34) | (4.27) | (3.59) | (4.05) | (3.81)
| U | Bias | 0.49 | 0.41 | 0.21 | 0.13 | 0.07 |  | 1.95 | 0.52 | 0.35 | 0.12 | -0.04
|  | RMSE | (1.54) | (1.13) | (0.52) | (0.31) | (0.22) |  | (4.35) | (2.64) | (1.75) | (1.16) | (0.27)
| Non | Bias | 0.6 | 0.37 | 0.21 | 0.12 | 0.07 |  | 2.15 | 2.2 | 1.82 | 1.95 | 1.44
|  | RMSE | (1.77) | (1.17) | (0.51) | (0.33) | (0.22) |  | (4.3) | (4.34) | (3.56) | (3.91) | (3.3)
0.5 | unmatched | Mean Num | 5.66 | 11.08 | 16.08 | 26.43 | 50.88 | -2.0 | 5.66 | 11.08 | 16.08 | 26.43 | 50.88
| U,T | Bias | 0.35 | 0.16 | 0.14 | 0.1 | 0.04 |  | -0.59 | -0.53 | -0.41 | -0.3 | -0.59
|  | RMSE | (0.97) | (0.4) | (0.32) | (0.23) | (0.14) |  | (1.46) | (1.42) | (1.08) | (0.61) | (1.48)
| T | Bias | 0.66 | 0.67 | 0.35 | 0.42 | 0.09 |  | 1.83 | 3.45 | 4.74 | 5.11 | 5.31
|  | RMSE | (1.8) | (2.18) | (1.51) | (1.06) | (0.31) |  | (4.21) | (5.55) | (6.27) | (6.45) | (6.6)
| U | Bias | 0.35 | 0.15 | 0.13 | 0.08 | 0.03 |  | -0.58 | -0.52 | -0.39 | -0.28 | -0.61
|  | RMSE | (0.98) | (0.38) | (0.3) | (0.2) | (0.14) |  | (1.45) | (1.41) | (1.07) | (0.59) | (1.51)
| Non | Bias | 0.68 | 0.67 | 0.4 | 0.42 | 0.09 |  | 1.9 | 3.38 | 4.81 | 5.09 | 5.43
|  | RMSE | (1.8) | (2.18) | (1.58) | (1.06) | (0.32) |  | (4.27) | (5.5) | (6.35) | (6.41) | (6.71)
0.5 | unmatched | Mean Num | 8.85 | 17.29 | 25.92 | 42.62 | 84.41 | -3.0 | 8.85 | 17.29 | 25.92 | 42.62 | 84.41
| U,T | Bias | -0.24 | 0.28 | 0.14 | 0.03 | 0.03 |  | -1.57 | -1.32 | -0.82 | -1.0 | -0.99
|  | RMSE | (2.41) | (0.83) | (0.46) | (0.19) | (0.17) |  | (2.86) | (2.47) | (1.74) | (2.09) | (2.13)
| T | Bias | 1.41 | 1.49 | 1.85 | 0.84 | 1.36 |  | 1.49 | 4.68 | 5.08 | 6.1 | 5.88
|  | RMSE | (4.14) | (3.58) | (3.99) | (2.47) | (2.95) |  | (5.47) | (6.86) | (6.96) | (7.43) | (7.09)
| U | Bias | -0.25 | 0.27 | 0.13 | 0.04 | 0.02 |  | -1.56 | -1.29 | -0.8 | -1.07 | -1.04
|  | RMSE | (2.41) | (0.82) | (0.44) | (0.19) | (0.14) |  | (2.86) | (2.46) | (1.71) | (2.16) | (2.15)
| Non | Bias | 1.45 | 1.49 | 1.84 | 0.83 | 1.37 |  | 1.52 | 4.68 | 5.06 | 6.1 | 5.85
|  | RMSE | (4.17) | (3.58) | (3.99) | (2.47) | (2.95) |  | (5.48) | (6.86) | (6.94) | (7.43) | (7.07)

  * a

Note:The objective function was numerically maximized using differential
evolution (DE) algorithm in BlackBoxOptim.jl package. For DE algorithm, I
require setting the domain of parameters and the number of population seeds so
that I fix the former to $[-10,10]$ for parameters and the latter to 400. The
reported point estimates are the best-found maxima. I omit the parameter
$\beta_{0}$ that can take on only a finite number of values (here +1) converge
at an arbitrarily fast rate, they are superconsistent.

Table 8: Case 2 with IR ($\lambda=2$).

| Num of agents |  | 10 | 20 | 30 | 50 | 100 |  | 10 | 20 | 30 | 50 | 100
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\beta_{1}$ |  |  |  |  |  |  |  | $\beta_{2}$ |  |  |  |  |
0.5 | unmatched | Mean Num | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| U,T | Bias | 0.04 | 0.09 | 0.14 | 0.06 | 0.02 |  | 4.3 | 4.04 | 3.96 | 3.7 | 4.45
|  | RMSE | (0.22) | (0.31) | (0.48) | (0.29) | (0.16) |  | (5.13) | (4.96) | (4.93) | (4.59) | (5.44)
| T | Bias | 0.04 | 0.08 | 0.14 | 0.06 | 0.01 |  | 4.3 | 4.03 | 3.9 | 3.59 | 4.46
|  | RMSE | (0.22) | (0.29) | (0.48) | (0.29) | (0.17) |  | (5.13) | (4.94) | (4.86) | (4.48) | (5.45)
| U | Bias | 0.07 | 0.09 | 0.14 | 0.06 | 0.01 |  | 4.3 | 4.04 | 3.88 | 3.64 | 4.46
|  | RMSE | (0.41) | (0.31) | (0.48) | (0.29) | (0.17) |  | (5.12) | (4.96) | (4.85) | (4.54) | (5.44)
| Non | Bias | 0.04 | 0.08 | 0.14 | 0.06 | 0.01 |  | 4.33 | 4.04 | 3.93 | 3.57 | 4.46
|  | RMSE | (0.22) | (0.29) | (0.48) | (0.29) | (0.17) |  | (5.13) | (4.95) | (4.89) | (4.47) | (5.44)
0.5 | unmatched | Mean Num | 0.02 | 0.0 | 0.0 | 0.0 | 0.02 | 0.0 | 0.02 | 0.0 | 0.0 | 0.0 | 0.02
| U,T | Bias | 0.23 | 0.07 | 0.14 | 0.05 | 0.01 |  | 5.15 | 5.19 | 4.95 | 4.82 | 5.42
|  | RMSE | (1.3) | (0.25) | (0.48) | (0.29) | (0.16) |  | (5.91) | (5.97) | (5.72) | (5.58) | (6.25)
| T | Bias | 0.23 | 0.08 | 0.14 | 0.06 | 0.02 |  | 5.15 | 5.21 | 4.99 | 4.7 | 5.42
|  | RMSE | (1.3) | (0.27) | (0.48) | (0.29) | (0.16) |  | (5.91) | (6.0) | (5.75) | (5.45) | (6.24)
| U | Bias | 0.26 | 0.07 | 0.14 | 0.06 | 0.01 |  | 5.16 | 5.2 | 4.97 | 4.71 | 5.42
|  | RMSE | (1.34) | (0.25) | (0.48) | (0.29) | (0.17) |  | (5.91) | (5.98) | (5.74) | (5.46) | (6.25)
| Non | Bias | 0.26 | 0.08 | 0.14 | 0.06 | 0.01 |  | 5.16 | 5.22 | 5.03 | 4.64 | 5.42
|  | RMSE | (1.34) | (0.27) | (0.48) | (0.29) | (0.17) |  | (5.91) | (6.01) | (5.78) | (5.38) | (6.25)
0.5 | unmatched | Mean Num | 1.53 | 2.87 | 3.9 | 5.85 | 10.63 | -1.0 | 1.53 | 2.87 | 3.9 | 5.85 | 10.63
| U,T | Bias | 0.52 | 0.33 | 0.22 | 0.1 | 0.06 |  | 1.86 | 0.75 | 1.36 | 0.56 | 0.03
|  | RMSE | (1.56) | (1.03) | (0.5) | (0.3) | (0.21) |  | (4.44) | (2.75) | (3.59) | (1.93) | (0.28)
| T | Bias | 0.54 | 0.25 | 0.16 | 0.1 | 0.06 |  | 2.21 | 2.05 | 2.24 | 2.09 | 1.58
|  | RMSE | (1.57) | (0.96) | (0.44) | (0.31) | (0.21) |  | (4.74) | (4.12) | (4.19) | (3.82) | (3.32)
| U | Bias | 0.53 | 0.33 | 0.21 | 0.11 | 0.06 |  | 1.83 | 0.8 | 1.2 | 0.55 | 0.04
|  | RMSE | (1.56) | (1.03) | (0.5) | (0.31) | (0.22) |  | (4.4) | (2.85) | (3.36) | (1.93) | (0.27)
| Non | Bias | 0.47 | 0.23 | 0.16 | 0.11 | 0.06 |  | 2.39 | 2.16 | 2.24 | 1.99 | 1.55
|  | RMSE | (1.48) | (0.96) | (0.44) | (0.32) | (0.21) |  | (4.83) | (4.21) | (4.19) | (3.73) | (3.36)
0.5 | unmatched | Mean Num | 5.64 | 11.15 | 16.06 | 26.17 | 50.89 | -2.0 | 5.64 | 11.15 | 16.06 | 26.17 | 50.89
| U,T | Bias | 0.36 | 0.18 | 0.16 | 0.1 | 0.07 |  | -0.57 | -0.32 | -0.2 | -0.15 | -0.22
|  | RMSE | (0.98) | (0.42) | (0.33) | (0.28) | (0.19) |  | (1.42) | (0.85) | (0.52) | (0.45) | (0.57)
| T | Bias | 0.42 | 0.1 | 0.08 | 0.21 | 0.05 |  | 1.82 | 4.31 | 4.51 | 5.35 | 5.64
|  | RMSE | (1.5) | (0.61) | (0.22) | (0.65) | (0.23) |  | (4.23) | (6.08) | (6.02) | (6.58) | (6.88)
| U | Bias | 0.36 | 0.17 | 0.15 | 0.08 | 0.07 |  | -0.56 | -0.3 | -0.18 | -0.13 | -0.26
|  | RMSE | (0.98) | (0.41) | (0.32) | (0.25) | (0.2) |  | (1.42) | (0.85) | (0.49) | (0.4) | (0.65)
| Non | Bias | 0.47 | 0.12 | 0.13 | 0.21 | 0.05 |  | 1.92 | 4.19 | 4.49 | 5.31 | 5.57
|  | RMSE | (1.78) | (0.63) | (0.54) | (0.65) | (0.23) |  | (4.39) | (6.01) | (5.98) | (6.53) | (6.82)
0.5 | unmatched | Mean Num | 8.81 | 17.34 | 25.76 | 42.53 | 84.41 | -3.0 | 8.81 | 17.34 | 25.76 | 42.53 | 84.41
| U,T | Bias | -0.26 | 0.31 | 0.18 | 0.08 | 0.07 |  | -1.58 | -1.13 | -0.63 | -0.41 | -0.51
|  | RMSE | (2.36) | (0.82) | (0.44) | (0.24) | (0.19) |  | (2.82) | (2.19) | (1.4) | (1.04) | (1.38)
| T | Bias | 1.29 | 0.89 | 0.37 | 0.11 | 0.08 |  | 1.4 | 5.0 | 5.23 | 5.97 | 6.1
|  | RMSE | (3.94) | (3.01) | (1.9) | (0.98) | (0.32) |  | (5.39) | (7.26) | (6.77) | (7.09) | (7.32)
| U | Bias | -0.25 | 0.29 | 0.17 | 0.08 | 0.06 |  | -1.58 | -1.1 | -0.62 | -0.45 | -0.55
|  | RMSE | (2.36) | (0.8) | (0.44) | (0.24) | (0.18) |  | (2.82) | (2.17) | (1.39) | (1.08) | (1.41)
| Non | Bias | 1.32 | 1.08 | 0.35 | 0.19 | 0.1 |  | 1.53 | 5.05 | 5.23 | 5.96 | 6.18
|  | RMSE | (3.91) | (3.18) | (1.89) | (1.34) | (0.37) |  | (5.44) | (7.39) | (6.72) | (7.11) | (7.38)

  * a

Note:The objective function was numerically maximized using differential
evolution (DE) algorithm in BlackBoxOptim.jl package. For DE algorithm, I
require setting the domain of parameters and the number of population seeds so
that I fix the former to $[-10,10]$ for parameters and the latter to 400. The
reported point estimates are the best-found maxima. I omit the parameter
$\beta_{0}$ that can take on only a finite number of values (here +1) converge
at an arbitrarily fast rate, they are superconsistent.

Table 9: Case 2 with IR ($\lambda=5$).

| Num of agents |  | 10 | 20 | 30 | 50 | 100 |  | 10 | 20 | 30 | 50 | 100
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\beta_{1}$ |  |  |  |  |  |  |  | $\beta_{2}$ |  |  |  |  |
0.5 | unmatched | Mean Num | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| U,T | Bias | 0.05 | 0.09 | 0.14 | 0.04 | 0.01 |  | 3.79 | 4.15 | 4.17 | 4.42 | 4.12
|  | RMSE | (0.18) | (0.28) | (0.44) | (0.27) | (0.16) |  | (4.73) | (5.01) | (5.15) | (5.27) | (4.96)
| T | Bias | 0.05 | 0.08 | 0.14 | 0.04 | 0.0 |  | 3.79 | 4.28 | 4.18 | 4.43 | 4.1
|  | RMSE | (0.18) | (0.26) | (0.44) | (0.27) | (0.16) |  | (4.73) | (5.06) | (5.15) | (5.28) | (4.95)
| U | Bias | 0.08 | 0.08 | 0.14 | 0.04 | 0.0 |  | 3.78 | 4.22 | 4.14 | 4.48 | 4.11
|  | RMSE | (0.39) | (0.26) | (0.44) | (0.27) | (0.16) |  | (4.65) | (5.04) | (5.13) | (5.33) | (4.95)
| Non | Bias | 0.08 | 0.08 | 0.14 | 0.05 | 0.0 |  | 3.78 | 4.3 | 4.16 | 4.44 | 4.1
|  | RMSE | (0.39) | (0.26) | (0.44) | (0.28) | (0.16) |  | (4.65) | (5.09) | (5.14) | (5.29) | (4.97)
0.5 | unmatched | Mean Num | 0.01 | 0.0 | 0.0 | 0.0 | 0.02 | 0.0 | 0.01 | 0.0 | 0.0 | 0.0 | 0.02
| U,T | Bias | 0.04 | 0.07 | 0.06 | 0.04 | 0.01 |  | 4.94 | 5.1 | 5.15 | 5.4 | 4.99
|  | RMSE | (0.2) | (0.23) | (0.33) | (0.27) | (0.16) |  | (5.69) | (5.72) | (5.91) | (6.14) | (5.7)
| T | Bias | 0.04 | 0.07 | 0.06 | 0.04 | 0.01 |  | 4.94 | 5.01 | 5.15 | 5.43 | 5.07
|  | RMSE | (0.2) | (0.24) | (0.33) | (0.27) | (0.16) |  | (5.69) | (5.66) | (5.91) | (6.15) | (5.78)
| U | Bias | 0.07 | 0.07 | 0.05 | 0.04 | 0.0 |  | 4.92 | 5.12 | 5.14 | 5.41 | 5.07
|  | RMSE | (0.4) | (0.23) | (0.34) | (0.27) | (0.16) |  | (5.62) | (5.74) | (5.9) | (6.15) | (5.77)
| Non | Bias | 0.04 | 0.07 | 0.05 | 0.05 | 0.0 |  | 4.99 | 5.03 | 5.13 | 5.44 | 5.1
|  | RMSE | (0.2) | (0.24) | (0.34) | (0.28) | (0.16) |  | (5.71) | (5.68) | (5.89) | (6.16) | (5.82)
0.5 | unmatched | Mean Num | 1.68 | 2.9 | 3.85 | 5.83 | 10.72 | -1.0 | 1.68 | 2.9 | 3.85 | 5.83 | 10.72
| U,T | Bias | 0.24 | 0.4 | 0.21 | 0.18 | 0.16 |  | 3.13 | 1.4 | 0.92 | 0.82 | 0.35
|  | RMSE | (0.6) | (1.16) | (0.49) | (0.42) | (0.3) |  | (4.85) | (3.41) | (2.31) | (2.36) | (1.59)
| T | Bias | 0.23 | 0.33 | 0.18 | 0.15 | 0.11 |  | 3.59 | 2.81 | 2.38 | 2.47 | 1.96
|  | RMSE | (0.58) | (1.09) | (0.47) | (0.4) | (0.26) |  | (5.25) | (4.85) | (4.22) | (4.17) | (3.78)
| U | Bias | 0.24 | 0.4 | 0.2 | 0.19 | 0.16 |  | 3.13 | 1.36 | 0.85 | 0.81 | 0.36
|  | RMSE | (0.6) | (1.16) | (0.49) | (0.43) | (0.31) |  | (4.85) | (3.38) | (2.21) | (2.36) | (1.59)
| Non | Bias | 0.24 | 0.32 | 0.18 | 0.15 | 0.12 |  | 3.72 | 2.75 | 2.38 | 2.43 | 1.82
|  | RMSE | (0.59) | (1.09) | (0.47) | (0.4) | (0.27) |  | (5.36) | (4.8) | (4.22) | (4.15) | (3.57)
0.5 | unmatched | Mean Num | 5.6 | 11.14 | 16.01 | 26.32 | 50.83 | -2.0 | 5.6 | 11.14 | 16.01 | 26.32 | 50.83
| U,T | Bias | 0.48 | 0.17 | 0.17 | 0.14 | 0.12 |  | -0.56 | -0.1 | -0.1 | -0.09 | -0.14
|  | RMSE | (1.2) | (0.45) | (0.35) | (0.3) | (0.23) |  | (1.56) | (0.61) | (0.46) | (0.39) | (0.45)
| T | Bias | 0.49 | 0.1 | 0.05 | 0.08 | 0.08 |  | 1.76 | 4.73 | 4.47 | 5.65 | 5.71
|  | RMSE | (1.58) | (0.64) | (0.25) | (0.24) | (0.31) |  | (4.08) | (6.39) | (5.95) | (6.97) | (6.69)
| U | Bias | 0.48 | 0.16 | 0.16 | 0.15 | 0.12 |  | -0.56 | -0.09 | -0.08 | -0.1 | -0.15
|  | RMSE | (1.2) | (0.43) | (0.33) | (0.3) | (0.24) |  | (1.56) | (0.59) | (0.45) | (0.39) | (0.44)
| Non | Bias | 0.57 | 0.1 | 0.1 | 0.08 | 0.08 |  | 1.88 | 4.72 | 4.53 | 5.65 | 5.69
|  | RMSE | (1.8) | (0.65) | (0.55) | (0.24) | (0.31) |  | (4.22) | (6.41) | (6.02) | (6.97) | (6.68)
0.5 | unmatched | Mean Num | 8.93 | 17.36 | 25.72 | 42.63 | 84.3 | -3.0 | 8.93 | 17.36 | 25.72 | 42.63 | 84.3
| U,T | Bias | -0.26 | 0.3 | 0.17 | 0.12 | 0.07 |  | -1.44 | -0.84 | -0.47 | -0.25 | -0.16
|  | RMSE | (2.4) | (0.84) | (0.5) | (0.28) | (0.18) |  | (2.77) | (1.85) | (1.37) | (0.84) | (0.71)
| T | Bias | 1.27 | 0.93 | 0.38 | 0.11 | 0.08 |  | 1.45 | 5.17 | 5.36 | 5.94 | 6.43
|  | RMSE | (3.95) | (3.08) | (1.9) | (0.98) | (0.33) |  | (5.37) | (7.15) | (6.92) | (6.96) | (7.43)
| U | Bias | -0.26 | 0.3 | 0.18 | 0.12 | 0.08 |  | -1.45 | -0.83 | -0.48 | -0.25 | -0.18
|  | RMSE | (2.4) | (0.84) | (0.5) | (0.28) | (0.19) |  | (2.78) | (1.85) | (1.37) | (0.84) | (0.72)
| Non | Bias | 1.3 | 1.04 | 0.37 | 0.2 | 0.1 |  | 1.58 | 5.28 | 5.39 | 6.0 | 6.4
|  | RMSE | (3.92) | (3.16) | (1.9) | (1.34) | (0.38) |  | (5.43) | (7.25) | (6.89) | (7.06) | (7.41)

  * a

Note:The objective function was numerically maximized using differential
evolution (DE) algorithm in BlackBoxOptim.jl package. For DE algorithm, I
require setting the domain of parameters and the number of population seeds so
that I fix the former to $[-10,10]$ for parameters and the latter to 400. The
reported point estimates are the best-found maxima. I omit the parameter
$\beta_{0}$ that can take on only a finite number of values (here +1) converge
at an arbitrarily fast rate, they are superconsistent.

Table 10: Case 2 with IR ($\lambda=10$).

| Num of agents |  | 10 | 20 | 30 | 50 | 100 |  | 10 | 20 | 30 | 50 | 100
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\beta_{1}$ |  |  |  |  |  |  |  | $\beta_{2}$ |  |  |  |  |
0.5 | unmatched | Mean Num | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| U,T | Bias | 0.04 | 0.13 | 0.16 | 0.05 | 0.0 |  | 3.61 | 4.73 | 4.25 | 4.16 | 4.49
|  | RMSE | (0.18) | (0.53) | (0.48) | (0.25) | (0.16) |  | (4.68) | (5.4) | (5.09) | (5.01) | (5.34)
| T | Bias | 0.04 | 0.13 | 0.16 | 0.05 | 0.01 |  | 3.61 | 4.73 | 4.27 | 4.24 | 4.5
|  | RMSE | (0.18) | (0.53) | (0.48) | (0.26) | (0.16) |  | (4.68) | (5.4) | (5.12) | (5.09) | (5.34)
| U | Bias | 0.07 | 0.13 | 0.15 | 0.06 | 0.01 |  | 3.63 | 4.66 | 4.3 | 4.2 | 4.52
|  | RMSE | (0.39) | (0.53) | (0.48) | (0.26) | (0.16) |  | (4.65) | (5.35) | (5.13) | (5.05) | (5.36)
| Non | Bias | 0.07 | 0.13 | 0.15 | 0.06 | 0.01 |  | 3.63 | 4.66 | 4.22 | 4.2 | 4.52
|  | RMSE | (0.39) | (0.53) | (0.48) | (0.26) | (0.16) |  | (4.65) | (5.35) | (5.07) | (5.05) | (5.36)
0.5 | unmatched | Mean Num | 0.01 | 0.0 | 0.0 | 0.0 | 0.02 | 0.0 | 0.01 | 0.0 | 0.0 | 0.0 | 0.02
| U,T | Bias | 0.05 | 0.1 | 0.16 | 0.06 | 0.01 |  | 4.84 | 5.69 | 5.28 | 5.07 | 5.61
|  | RMSE | (0.19) | (0.37) | (0.48) | (0.28) | (0.16) |  | (5.69) | (6.32) | (5.97) | (5.86) | (6.28)
| T | Bias | 0.05 | 0.1 | 0.16 | 0.06 | 0.01 |  | 4.84 | 5.69 | 5.28 | 5.09 | 5.61
|  | RMSE | (0.19) | (0.37) | (0.48) | (0.28) | (0.16) |  | (5.69) | (6.32) | (5.97) | (5.85) | (6.28)
| U | Bias | 0.05 | 0.1 | 0.15 | 0.06 | 0.01 |  | 4.88 | 5.62 | 5.22 | 5.08 | 5.6
|  | RMSE | (0.19) | (0.37) | (0.48) | (0.28) | (0.16) |  | (5.7) | (6.27) | (5.93) | (5.86) | (6.29)
| Non | Bias | 0.09 | 0.1 | 0.15 | 0.06 | 0.01 |  | 4.88 | 5.62 | 5.31 | 5.05 | 5.6
|  | RMSE | (0.39) | (0.37) | (0.48) | (0.29) | (0.16) |  | (5.7) | (6.27) | (5.99) | (5.82) | (6.29)
0.5 | unmatched | Mean Num | 1.44 | 2.89 | 3.84 | 5.72 | 10.67 | -1.0 | 1.44 | 2.89 | 3.84 | 5.72 | 10.67
| U,T | Bias | 0.48 | 0.3 | 0.21 | 0.19 | 0.14 |  | 3.64 | 2.26 | 1.52 | 1.1 | 1.17
|  | RMSE | (1.49) | (0.69) | (0.53) | (0.41) | (0.32) |  | (5.6) | (4.1) | (3.22) | (2.56) | (2.82)
| T | Bias | 0.47 | 0.24 | 0.22 | 0.2 | 0.11 |  | 3.8 | 3.28 | 2.74 | 2.51 | 2.46
|  | RMSE | (1.49) | (0.59) | (0.54) | (0.44) | (0.29) |  | (5.72) | (4.9) | (4.43) | (4.26) | (4.16)
| U | Bias | 0.48 | 0.3 | 0.2 | 0.2 | 0.14 |  | 3.64 | 2.18 | 1.47 | 1.12 | 1.18
|  | RMSE | (1.49) | (0.7) | (0.53) | (0.42) | (0.32) |  | (5.6) | (4.02) | (3.19) | (2.63) | (2.82)
| Non | Bias | 0.4 | 0.22 | 0.22 | 0.2 | 0.11 |  | 3.96 | 3.26 | 2.66 | 2.45 | 2.55
|  | RMSE | (1.22) | (0.59) | (0.54) | (0.44) | (0.29) |  | (5.78) | (4.88) | (4.34) | (4.11) | (4.26)
0.5 | unmatched | Mean Num | 5.54 | 11.09 | 15.98 | 26.51 | 51.44 | -2.0 | 5.54 | 11.09 | 15.98 | 26.51 | 51.44
| U,T | Bias | 0.65 | 0.24 | 0.2 | 0.23 | 0.15 |  | -0.68 | -0.11 | -0.06 | -0.13 | -0.07
|  | RMSE | (1.22) | (0.52) | (0.45) | (0.41) | (0.3) |  | (1.61) | (0.7) | (0.55) | (0.51) | (0.39)
| T | Bias | 0.55 | 0.11 | 0.2 | 0.19 | 0.13 |  | 1.91 | 4.6 | 4.79 | 5.31 | 5.74
|  | RMSE | (1.51) | (0.66) | (0.97) | (0.51) | (0.35) |  | (4.24) | (6.21) | (6.16) | (6.64) | (7.03)
| U | Bias | 0.65 | 0.25 | 0.2 | 0.21 | 0.15 |  | -0.68 | -0.13 | -0.07 | -0.11 | -0.06
|  | RMSE | (1.22) | (0.57) | (0.45) | (0.39) | (0.3) |  | (1.61) | (0.74) | (0.56) | (0.49) | (0.38)
| Non | Bias | 0.61 | 0.12 | 0.25 | 0.19 | 0.13 |  | 1.86 | 4.65 | 4.85 | 5.33 | 5.77
|  | RMSE | (1.77) | (0.68) | (1.09) | (0.51) | (0.35) |  | (4.22) | (6.23) | (6.24) | (6.65) | (7.06)
0.5 | unmatched | Mean Num | 8.76 | 17.34 | 25.8 | 42.51 | 84.2 | -3.0 | 8.76 | 17.34 | 25.8 | 42.51 | 84.2
| U,T | Bias | -0.23 | 0.37 | 0.24 | 0.14 | 0.09 |  | -1.54 | -0.76 | -0.41 | -0.12 | -0.07
|  | RMSE | (2.39) | (0.79) | (0.59) | (0.3) | (0.25) |  | (2.85) | (1.7) | (1.27) | (0.57) | (0.51)
| T | Bias | 1.29 | 1.01 | 0.42 | 0.1 | 0.08 |  | 1.41 | 5.31 | 5.42 | 6.19 | 6.76
|  | RMSE | (3.94) | (3.22) | (1.91) | (0.99) | (0.34) |  | (5.41) | (7.22) | (7.04) | (7.16) | (7.72)
| U | Bias | -0.23 | 0.36 | 0.2 | 0.14 | 0.08 |  | -1.54 | -0.73 | -0.33 | -0.13 | -0.04
|  | RMSE | (2.39) | (0.79) | (0.52) | (0.3) | (0.22) |  | (2.85) | (1.69) | (1.08) | (0.56) | (0.46)
| Non | Bias | 1.34 | 1.05 | 0.4 | 0.1 | 0.11 |  | 1.52 | 5.33 | 5.48 | 6.18 | 6.75
|  | RMSE | (3.92) | (3.22) | (1.9) | (0.99) | (0.42) |  | (5.47) | (7.22) | (7.01) | (7.15) | (7.72)

  * a

Note:The objective function was numerically maximized using differential
evolution (DE) algorithm in BlackBoxOptim.jl package. For DE algorithm, I
require setting the domain of parameters and the number of population seeds so
that I fix the former to $[-10,10]$ for parameters and the latter to 400. The
reported point estimates are the best-found maxima. I omit the parameter
$\beta_{0}$ that can take on only a finite number of values (here +1) converge
at an arbitrarily fast rate, they are superconsistent.

Table 11: Case 2 with IR ($\lambda=20$).

| Num of agents |  | 10 | 20 | 30 | 50 | 100 |  | 10 | 20 | 30 | 50 | 100
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\beta_{1}$ |  |  |  |  |  |  |  | $\beta_{2}$ |  |  |  |  |
0.5 | unmatched | Mean Num | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| U,T | Bias | 0.12 | 0.05 | 0.04 | 0.05 | 0.01 |  | 3.82 | 4.07 | 4.7 | 4.44 | 4.54
|  | RMSE | (0.89) | (0.22) | (0.29) | (0.27) | (0.17) |  | (4.8) | (4.92) | (5.43) | (5.3) | (5.35)
| T | Bias | 0.12 | 0.05 | 0.04 | 0.05 | 0.01 |  | 3.85 | 4.07 | 4.64 | 4.42 | 4.54
|  | RMSE | (0.89) | (0.22) | (0.29) | (0.27) | (0.17) |  | (4.84) | (4.92) | (5.38) | (5.3) | (5.35)
| U | Bias | 0.2 | 0.05 | 0.03 | 0.05 | 0.01 |  | 3.8 | 4.06 | 4.67 | 4.37 | 4.51
|  | RMSE | (1.07) | (0.22) | (0.29) | (0.27) | (0.17) |  | (4.81) | (4.89) | (5.39) | (5.24) | (5.32)
| Non | Bias | 0.12 | 0.05 | 0.04 | 0.05 | 0.01 |  | 3.9 | 4.06 | 4.64 | 4.37 | 4.51
|  | RMSE | (0.89) | (0.22) | (0.29) | (0.27) | (0.17) |  | (4.85) | (4.89) | (5.38) | (5.24) | (5.32)
0.5 | unmatched | Mean Num | 0.01 | 0.0 | 0.0 | 0.0 | 0.02 | 0.0 | 0.01 | 0.0 | 0.0 | 0.0 | 0.02
| U,T | Bias | 0.04 | 0.11 | 0.12 | 0.05 | 0.02 |  | 5.02 | 5.55 | 5.31 | 5.5 | 5.64
|  | RMSE | (0.18) | (0.4) | (0.43) | (0.27) | (0.17) |  | (5.84) | (6.16) | (6.05) | (6.23) | (6.31)
| T | Bias | 0.04 | 0.11 | 0.12 | 0.05 | 0.02 |  | 5.02 | 5.55 | 5.26 | 5.38 | 5.64
|  | RMSE | (0.18) | (0.4) | (0.44) | (0.28) | (0.17) |  | (5.84) | (6.16) | (6.0) | (6.11) | (6.31)
| U | Bias | 0.07 | 0.11 | 0.12 | 0.05 | 0.01 |  | 4.97 | 5.53 | 5.28 | 5.33 | 5.6
|  | RMSE | (0.39) | (0.4) | (0.44) | (0.28) | (0.17) |  | (5.79) | (6.13) | (6.02) | (6.06) | (6.26)
| Non | Bias | 0.07 | 0.11 | 0.12 | 0.05 | 0.01 |  | 5.0 | 5.53 | 5.28 | 5.33 | 5.6
|  | RMSE | (0.39) | (0.4) | (0.44) | (0.28) | (0.17) |  | (5.83) | (6.13) | (6.02) | (6.06) | (6.26)
0.5 | unmatched | Mean Num | 1.63 | 2.83 | 3.69 | 5.7 | 10.62 | -1.0 | 1.63 | 2.83 | 3.69 | 5.7 | 10.62
| U,T | Bias | 0.58 | 0.32 | 0.17 | 0.18 | 0.21 |  | 3.79 | 1.84 | 1.56 | 1.37 | 1.26
|  | RMSE | (1.58) | (0.79) | (0.52) | (0.39) | (0.38) |  | (5.58) | (3.67) | (3.06) | (3.01) | (3.04)
| T | Bias | 0.53 | 0.24 | 0.11 | 0.15 | 0.15 |  | 4.08 | 2.82 | 2.65 | 2.39 | 2.61
|  | RMSE | (1.54) | (0.59) | (0.43) | (0.39) | (0.33) |  | (5.76) | (4.39) | (4.26) | (3.91) | (4.36)
| U | Bias | 0.58 | 0.32 | 0.17 | 0.18 | 0.2 |  | 3.71 | 1.84 | 1.43 | 1.34 | 1.25
|  | RMSE | (1.58) | (0.79) | (0.52) | (0.39) | (0.38) |  | (5.53) | (3.67) | (2.89) | (3.04) | (3.04)
| Non | Bias | 0.54 | 0.22 | 0.11 | 0.15 | 0.16 |  | 4.26 | 2.81 | 2.57 | 2.4 | 2.61
|  | RMSE | (1.54) | (0.59) | (0.43) | (0.39) | (0.33) |  | (5.92) | (4.36) | (4.17) | (3.94) | (4.33)
0.5 | unmatched | Mean Num | 5.43 | 11.28 | 16.02 | 26.5 | 50.92 | -2.0 | 5.43 | 11.28 | 16.02 | 26.5 | 50.92
| U,T | Bias | 0.64 | 0.31 | 0.26 | 0.18 | 0.23 |  | -0.53 | -0.12 | -0.05 | -0.01 | -0.09
|  | RMSE | (1.42) | (0.57) | (0.47) | (0.36) | (0.42) |  | (1.97) | (0.6) | (0.47) | (0.42) | (0.5)
| T | Bias | 0.52 | 0.13 | 0.11 | 0.19 | 0.11 |  | 2.07 | 4.83 | 5.08 | 5.23 | 5.94
|  | RMSE | (1.57) | (0.69) | (0.28) | (0.56) | (0.3) |  | (4.38) | (6.38) | (6.53) | (6.49) | (7.23)
| U | Bias | 0.63 | 0.3 | 0.26 | 0.18 | 0.22 |  | -0.52 | -0.11 | -0.05 | -0.01 | -0.06
|  | RMSE | (1.42) | (0.56) | (0.47) | (0.37) | (0.41) |  | (1.97) | (0.59) | (0.47) | (0.42) | (0.47)
| Non | Bias | 0.61 | 0.13 | 0.15 | 0.19 | 0.11 |  | 2.08 | 4.82 | 5.15 | 5.28 | 5.88
|  | RMSE | (1.85) | (0.69) | (0.57) | (0.56) | (0.3) |  | (4.39) | (6.37) | (6.6) | (6.55) | (7.18)
0.5 | unmatched | Mean Num | 8.8 | 17.38 | 25.7 | 42.57 | 84.31 | -3.0 | 8.8 | 17.38 | 25.7 | 42.57 | 84.31
| U,T | Bias | -0.26 | 0.43 | 0.19 | 0.19 | 0.13 |  | -1.54 | -0.73 | -0.21 | -0.19 | -0.05
|  | RMSE | (2.43) | (0.91) | (0.52) | (0.37) | (0.33) |  | (2.94) | (1.73) | (0.99) | (0.64) | (0.55)
| T | Bias | 1.35 | 1.02 | 0.39 | 0.13 | 0.15 |  | 1.46 | 5.33 | 5.56 | 6.18 | 6.32
|  | RMSE | (4.01) | (3.23) | (1.9) | (0.99) | (0.84) |  | (5.45) | (7.22) | (7.07) | (7.17) | (7.38)
| U | Bias | -0.26 | 0.42 | 0.18 | 0.21 | 0.12 |  | -1.54 | -0.72 | -0.2 | -0.24 | -0.03
|  | RMSE | (2.43) | (0.91) | (0.5) | (0.4) | (0.31) |  | (2.94) | (1.72) | (0.92) | (0.7) | (0.54)
| Non | Bias | 1.34 | 1.14 | 0.38 | 0.12 | 0.17 |  | 1.6 | 5.43 | 5.48 | 6.2 | 6.29
|  | RMSE | (3.94) | (3.31) | (1.9) | (0.99) | (0.86) |  | (5.47) | (7.33) | (6.97) | (7.18) | (7.36)

  * a

Note:The objective function was numerically maximized using differential
evolution (DE) algorithm in BlackBoxOptim.jl package. For DE algorithm, I
require setting the domain of parameters and the number of population seeds so
that I fix the former to $[-10,10]$ for parameters and the latter to 400. The
reported point estimates are the best-found maxima. I omit the parameter
$\beta_{0}$ that can take on only a finite number of values (here +1) converge
at an arbitrarily fast rate, they are superconsistent.

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