normal, or centered at zero. We argue that this flexibility enables bias
reduction for the plug-in estimator, particularly in cases where first-stage
sampling error induces significant bias in the second stage. We introduce a
debiased plug-in estimator and demonstrate that its limiting distribution has
a zero mean. Finite sample performance confirms the finite sample performance
of our debiased estimator.

We leverage the proposed approach to study peer effects on adolescents’ fast-
food consumption habits. We avoid the issue of weak instruments that can occur
in this model by expanding the pool of instruments. This approach induces
distortion in the estimates because the number of weak instruments is large.
Our method is designed to correct this bias and provide a valid inference.

This paper contributes to the emerging literature on inference methods when
standard regularity conditions are violated. The proposed approach is
straightforward to implement. We do not impose a particular class of estimator
in the first stage. For instance, it may be a Bayesian estimator, which is not
necessarily normally distributed. Our method is also suitable for complex
models. Unlike resampling methods, we eliminate the need for multiple
computations of both the first stage. We hope that the simplicity and
flexibility of our approach will promote its use to infer multiple-stage
estimators.

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##  Appendix A Appendix – Proofs

###  A.1 Proof of Theorem 4.1

Equation (5) is given by

$\textstyle\Delta_{n}=\sqrt{n}(\boldsymbol{\hat{\theta}}_{n}-\boldsymbol{\theta}_{0})=\mathbf{A}_{n}^{-1}\boldsymbol{\dot{q}}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\,.$

Let
$\boldsymbol{\Sigma}_{n}=\mathbb{V}(\boldsymbol{\dot{q}}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n}))$ and
$\boldsymbol{\Sigma}_{0}=\lim\boldsymbol{\Sigma}_{n}$. The existence of
$\boldsymbol{\Sigma}_{n}$ and $\boldsymbol{\Sigma}_{0}$ is guaranteed by
Assumption 4.1. As
$\mathbb{E}(\lVert\boldsymbol{\dot{q}}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\rVert^{\nu})<\infty$ for some $\nu>2$, it follows that
the limiting distribution of
$\boldsymbol{\dot{q}}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})$ has variance $\boldsymbol{\Sigma}_{0}$ (see Chung,
2001, Theorem 4.5.2). Moreover, Assumption 4.2 implies that
$\operatorname{plim}\mathbf{A}_{n}=\mathbf{A}_{0}$. Therefore,
$\mathbb{V}(\textstyle\Delta_{0})=\mathbf{A}_{0}^{-1}\boldsymbol{\Sigma}_{0}\mathbf{A}_{0}^{-1}$.
We have

$\textstyle\boldsymbol{\Sigma}_{n}^{\kappa}=\mathbf{V}_{n}+\textstyle\frac{1}{\kappa-1}\sum_{s=1}^{\kappa}(\mathcal{E}_{n,s}-\boldsymbol{\Omega}_{n}^{\kappa})(\mathcal{E}_{n,s}-\boldsymbol{\Omega}_{n}^{\kappa})^{\prime},$
(12)

where
$\boldsymbol{\Omega}_{n}^{\kappa}=\frac{1}{\kappa}\sum_{s=1}^{\kappa}\mathcal{E}_{n,s}$.
The second term of the RHS of Equation (12) is the sampling variance of
$\mathcal{E}_{n,s}$. The variables $\mathcal{E}_{n,1},\leavevmode\nobreak\
\dots,\leavevmode\nobreak\ \mathcal{E}_{n,\kappa}$ are independent and
identically distributed as $\mathcal{E}_{n}$. By the LLN,
$\textstyle\frac{1}{\kappa-1}\sum_{s=1}^{\kappa}(\mathcal{E}_{n,s}-\boldsymbol{\Omega}_{n}^{\kappa})(\mathcal{E}_{n,s}-\boldsymbol{\Omega}_{n}^{\kappa})^{\prime}$
converges in probability to $\mathbb{V}(\mathcal{E}_{n})$ as $\kappa$ grows to
infinity; that is,
$\operatorname{plim}_{\kappa}\boldsymbol{\Sigma}_{n}^{\kappa}=\mathbf{V}_{n}+\mathbb{V}(\mathcal{E}_{n})$,
where $\operatorname{plim}_{k}$ is the limit in probability as $\kappa$ grows
to infinity ($n$ set fixed).

As
$\mathbb{E}(\lVert\mathcal{E}_{n}\rVert^{\nu})\leq\mathbb{E}(\lVert\boldsymbol{\dot{q}}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\rVert^{\nu})<\infty$ for some $\nu>2$, we have
$\lim\mathbb{V}(\mathcal{E}_{n})=\mathbb{V}(\mathcal{E}_{0})$. Additionally,
$\operatorname{plim}\mathbf{V}_{n}=\mathbf{V}_{0}$ by Condition (ii) of
Assumption 4.1. As a result,
$\textstyle\operatorname{plim}_{n,\kappa}\boldsymbol{\Sigma}_{n}^{\kappa}=\mathbf{V}_{0}+\mathbb{V}(\mathcal{E}_{0})=\boldsymbol{\Sigma}_{0}$
and
$\mathbb{V}(\textstyle\Delta_{0})=\mathbf{A}_{0}^{-1}(\operatorname{plim}_{n,\kappa}\boldsymbol{\Sigma}_{n}^{\kappa})\mathbf{A}_{0}^{-1}$.

###  A.2 Proof of Theorem 4.2

Let
$F_{u,n}(\mathbf{t})=\mathbb{P}(\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t})$ for all
$\boldsymbol{t}\in\mathbb{R}^{K_{\theta}}$. We first state and show the
following lemma.

######  Lemma A.1 (Unconditional Asymptotic Normality).

Under Assumption 4.4, the unconditional distribution of
$\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\ \mathbf{\hat{B}}_{n})$
is asymptotically normal, in the sense that $\lim
F_{u,n}(\mathbf{t})=\Phi(\boldsymbol{t})$ for each $\boldsymbol{t}$.

###### Proof.

We have $\displaystyle
F_{u,n}(\mathbf{t})=\mathbb{E}[\mathbb{P}(\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}|\mathbf{\hat{B}}_{n})]$, where the
expectation is taken with respect to $\mathbf{\hat{B}}_{n}$. Thus,
$\displaystyle\lim
F_{u,n}(\mathbf{t})=\lim\mathbb{E}\\{\mathbb{P}(\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}|\mathbf{\hat{B}}_{n})\\}$. Given
that $\lvert\mathbb{P}(\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}|\mathbf{\hat{B}}_{n})\rvert\leq 1$,
we can interchange the expectation and the limit in the previous equation (see
Lebesgue–Vitali theorem in Bogachev and Ruas, 2007, Theorem 4.5.4). It follows
that $\displaystyle\lim
F_{u,n}(\mathbf{t})=\mathbb{E}\\{\lim\mathbb{P}(\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}|\mathbf{\hat{B}}_{n})\\}$. By
Assumption 4.4,
$\lim\mathbb{P}(\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}|\mathbf{\hat{B}}_{n})=\Phi(\boldsymbol{t})$.
Hence, $\lim F_{u,n}(\mathbf{t})=\Phi(\boldsymbol{t})$. ∎

We now show Theorem 4.2. By substituting
$\boldsymbol{\dot{q}}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})=\mathbf{V}_{n}^{1/2}\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})+\mathcal{E}_{n}$ in Equation (5) we obtain
$\Delta_{n}=\mathbf{A}_{n}^{-1}\mathbf{V}_{n}^{1/2}\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})+\mathbf{A}_{n}^{-1}\mathcal{E}_{n}$. By Lemma A.1,
$\boldsymbol{u}_{n}(\mathbf{y}_{n},\mathbf{\hat{B}}_{n})$ converges in
distribution to a standard normal distribution, independently of
$\mathbf{\hat{B}}_{n}$. This convergence also holds independently of
$\mathcal{E}_{n}$ since the latter is random only because of
$\mathbf{\hat{B}}_{n}$. Recall also that
$\operatorname{plim}\mathbf{A}_{n}=\mathbf{A}_{0}$ and
$\operatorname{plim}\mathbf{V}_{n}=\mathbf{V}_{0}$, where $\mathbf{A}_{0}$ and
$\mathbf{V}_{0}$ are nonstochastic. Therefore, by the Slutsky theorem, the
conditional distribution of $\Delta_{n}$, given $\mathcal{E}_{n}$, and the
conditional distribution of
$\boldsymbol{\psi}_{n}=\mathbf{A}_{0}^{-1}\mathbf{V}_{n}^{1/2}\boldsymbol{\zeta}+\mathbf{A}_{0}^{-1}\mathcal{E}_{n}$,
given $\mathcal{E}_{n}$, have the same limit, where $\boldsymbol{\zeta}\sim
N(0,\boldsymbol{I}_{K_{\theta}})$. Put differently,
$\operatorname{plim}\mathbb{P}(\Delta_{n}\preceq\boldsymbol{t}|\mathcal{E}_{n})=\operatorname{plim}\mathbb{P}(\boldsymbol{\psi}_{n}\preceq\boldsymbol{t}|\mathcal{E}_{n})$.
As this is true for almost all $\mathcal{E}_{n}$, it follows that
$\mathbb{E}\big{(}\operatorname{plim}\mathbb{P}(\Delta_{n}\preceq\boldsymbol{t}|\mathcal{E}_{n})\big{)}=\mathbb{E}\big{(}\operatorname{plim}\mathbb{P}(\boldsymbol{\psi}_{n}\preceq\boldsymbol{t}|\mathcal{E}_{n})\big{)}$.
As is the case in the proof of Lemma A.1, we can interchange the expectation
and the limit because
$\lvert\mathbb{P}(\Delta_{n}\preceq\boldsymbol{t}|\mathcal{E}_{n})\rvert\leq
1$ and
$\lvert\mathbb{P}(\boldsymbol{\psi}_{n}\preceq\boldsymbol{t}|\mathcal{E}_{n})\rvert\leq
1$. Thus,
$\lim\mathbb{E}\big{(}\mathbb{P}(\Delta_{n}\preceq\boldsymbol{t}|\mathcal{E}_{n})\big{)}=\lim\mathbb{E}\big{(}\mathbb{P}(\boldsymbol{\psi}_{n}\preceq\boldsymbol{t}|\mathcal{E}_{n})\big{)}$.
As a result,
$\lim\mathbb{P}(\Delta_{n}\preceq\boldsymbol{t})=\lim\mathbb{P}(\boldsymbol{\psi}_{n}\preceq\boldsymbol{t})$.
This completes the proof of Theorem 4.2.

###  A.3 Uniform Convergence of the Distribution of the Plug-in Estimator

We establish a more general result than Theorem 4.2 in terms of uniform
convergence. To achieve this result, we consider a stronger version of
Assumption 4.4 using uniform convergence.

######  Assumption A.1 (Conditional Asymptotic Normality).

The conditional distribution of the standardized influence function
$\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})$, given $\mathbf{\hat{B}}_{n}$, uniformly converges in
distribution to $N(0,\leavevmode\nobreak\ \boldsymbol{I}_{K_{\theta}})$,
almost surely; that is,
$\sup_{\boldsymbol{t}\in\mathbb{R}^{K_{\theta}}}\lvert\mathbb{P}(\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}|\mathbf{\hat{B}}_{n})-\Phi(\boldsymbol{t})\rvert=o_{p}(1)$.

Assumption A.1 can be implied by a strong version of the CLT to
$\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\ \mathbf{\hat{B}}_{n})$
conditional on $\mathbf{\hat{B}}_{n}$. This variant of the CLT, often referred
to as the uniform CLT, is an improvement of the standard CLT, as introduced by
Berry and Esseen (see Jirak, 2016; Raič, 2019). Given $\mathbf{\hat{B}}_{n}$
set fixed as a predetermined sequence, the uniform CLT can be applied as in
the case of a single-step approach. As in Lemma A.1, Assumption A.1 also
implies the uniform convergence of the unconditional distribution.

######  Lemma A.2 (Unconditional Asymptotic Normality).

Under Assumption A.1, $\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})$ uniformly converges to a standard normal distribution,
in the sense that
$\sup_{\boldsymbol{t}\in\mathbb{R}^{K_{\theta}}}\lvert\mathbb{P}(\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t})-\Phi(\boldsymbol{t})\rvert=o(1)$.

###### Proof.

Let
$m_{\boldsymbol{t}}(\mathbf{\hat{B}}_{n})=\mathbb{P}(\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}|\mathbf{\hat{B}}_{n})-\Phi(\boldsymbol{t})$.
We have
$\mathbb{E}(m_{\boldsymbol{t}}(\mathbf{\hat{B}}_{n}))=\mathbb{P}(\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t})-\Phi(\boldsymbol{t})$, where the
expectation is taken with respect to $\mathbf{\hat{B}}_{n}$. As
$\sup_{\boldsymbol{t}\in\mathbb{R}^{K_{\theta}}}\lvert\mathbb{E}(m_{\boldsymbol{t}}(\mathbf{\hat{B}}_{n}))\rvert\leq\mathbb{E}(\sup_{\boldsymbol{t}\in\mathbb{R}^{K_{\theta}}}\lvert
m_{\boldsymbol{t}}(\mathbf{\hat{B}}_{n})\rvert)$, it turns out that
$\sup_{\boldsymbol{t}\in\mathbb{R}^{K_{\theta}}}\lvert\mathbb{P}(\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t})-\Phi(\boldsymbol{t})\rvert\leq\mathbb{E}(\sup_{\boldsymbol{t}\in\mathbb{R}^{K_{\theta}}}\lvert
m_{\boldsymbol{t}}(\mathbf{\hat{B}}_{n})\rvert)$. Since
$\sup_{\boldsymbol{t}\in\mathbb{R}^{K_{\theta}}}\lvert
m_{\boldsymbol{t}}(\mathbf{\hat{B}}_{n})\rvert$ is bounded by one and is
$o_{p}(1)$, then
$\lim\mathbb{E}(\sup_{\boldsymbol{t}\in\mathbb{R}^{K_{\theta}}}\lvert
m_{\boldsymbol{t}}(\mathbf{\hat{B}}_{n})\rvert)=0$. This completes the proof.
∎

The following theorem establishes the uniform convergence of the distribution
$\Delta_{n}$.

######  Theorem A.1 (Aymptotic Distribution).

Assumptions 2.1–4.2, and A.1 hold. Let
$\boldsymbol{\chi}_{n}=\boldsymbol{\zeta}+\mathbf{V}_{n}^{-1/2}\mathcal{E}_{n}$,
where $\boldsymbol{\zeta}\sim N(0,\leavevmode\nobreak\
\boldsymbol{I}_{K_{\theta}})$. Let
$G(\boldsymbol{t})=\lim\mathbb{P}(\boldsymbol{\chi}_{n}\preceq\boldsymbol{t})$
for $\boldsymbol{t}\in\mathbb{R}^{K_{\theta}}$ be the limiting distribution
function of $\boldsymbol{\chi}_{n}$. We have
$\sup_{\boldsymbol{t}\in\mathbb{R}^{K_{\theta}}}\lvert\mathbb{P}(\sqrt{n}\mathbf{V}_{n}^{-1/2}\mathbf{A}_{0}(\boldsymbol{\hat{\theta}}_{n}-\boldsymbol{\theta}_{0})\preceq\boldsymbol{t})-G(\boldsymbol{t})\rvert=o(1)$.

###### Proof.

As
$\\{\boldsymbol{\chi}_{n}\preceq\boldsymbol{t}\\}=\\{\boldsymbol{\zeta}\preceq\boldsymbol{t}-\mathbf{V}_{n}^{-1/2}\mathcal{E}_{n}\\}$,
we have
$G(\boldsymbol{t})=\lim\mathbb{E}\big{\\{}\mathbb{P}(\boldsymbol{\zeta}\preceq\boldsymbol{t}-\mathbf{V}_{n}^{-1/2}\mathcal{E}_{n}|\mathcal{E}_{n})\big{\\}}$.
Thus,

$G(\boldsymbol{t})=\lim\mathbb{E}\\{\Phi(\boldsymbol{t}-\mathbf{V}_{n}^{-1/2}\mathcal{E}_{n})\\},$
(13)

because $\operatorname{plim}\mathbf{V}_{n}$ is nonstochastic and
$\boldsymbol{\zeta}\sim N(0,\leavevmode\nobreak\
\boldsymbol{I}_{K_{\theta}})$. Moreover, as
$\Delta_{n}=\mathbf{A}_{n}^{-1}\mathbf{V}_{n}^{1/2}\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})+\mathbf{A}_{n}^{-1}\mathcal{E}_{n}$, then
$\\{\mathbf{V}_{n}^{-1/2}\mathbf{A}_{n}\Delta_{n}\preceq\boldsymbol{t}\\}=\\{\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}-\mathbf{V}_{n}^{-1/2}\mathcal{E}_{n}\\}$.
This translates to
$\lim\mathbb{P}(\mathbf{V}_{n}^{-1/2}\mathbf{A}_{0}\Delta_{n}\preceq\boldsymbol{t})=\lim\mathbb{P}\\{\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}-\mathbf{V}_{n}^{-1/2}\mathcal{E}_{n}\\}$
because $\operatorname{plim}\mathbf{A}_{n}=\mathbf{A}_{0}$ is nonstochastic.
Since $\mathbb{P}\\{\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}-\mathbf{V}_{n}^{-1/2}\mathcal{E}_{n}\\}=\mathbb{E}\\{\mathbb{P}(\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}-\mathbf{V}_{n}^{-1/2}\mathcal{E}_{n}|\mathcal{E}_{n})\\}$,
we thus have

$\lim\mathbb{P}(\mathbf{V}_{n}^{-1/2}\mathbf{A}_{n}\Delta_{n}\preceq\boldsymbol{t})=\lim\mathbb{E}\big{\\{}\mathbb{P}\\{\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}-\mathbf{V}_{n}^{-1/2}\mathcal{E}_{n}|\mathcal{E}_{n}\\}\big{\\}}.$
(14)

From (13) and (14),
$\lim\lvert\mathbb{P}(\mathbf{V}_{n}^{-1/2}\mathbf{A}_{0}\Delta_{n}\preceq\boldsymbol{t})-G(\boldsymbol{t})\rvert=\textstyle\lim\lvert\mathbb{E}\big{\\{}\mathbb{P}\\{\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}-\mathbf{V}_{n}^{-1/2}\mathcal{E}_{n}|\mathcal{E}_{n}\\}-\Phi(\boldsymbol{t}-\mathbf{V}_{n}^{-1/2}\mathcal{E}_{n})\big{\\}}\rvert$.
In addition, For any $\boldsymbol{t}$ and $\mathcal{E}_{n}$, we have
$\mathbb{P}\\{\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}-\mathbf{V}_{n}^{-1/2}\mathcal{E}_{n}|\mathcal{E}_{n}\\}-\Phi(\boldsymbol{t}-\mathbf{V}_{n}^{-1/2}\mathcal{E}_{n})\leq\sup_{\boldsymbol{t}\in\mathbb{R}^{K_{\theta}}}\lvert\mathbb{P}\\{\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}|\mathcal{E}_{n}\\}-\Phi(\boldsymbol{t})\rvert$.
Thus,
$\lim\lvert\mathbb{P}(\mathbf{V}_{n}^{-1/2}\mathbf{A}_{0}\Delta_{n}\preceq\boldsymbol{t})-G(\boldsymbol{t})\rvert\textstyle\leq\lim\mathbb{E}\big{\\{}\sup_{\boldsymbol{t}\in\mathbb{R}^{K_{\theta}}}\lvert\mathbb{P}\\{\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}|\mathcal{E}_{n}\\}-\Phi(\boldsymbol{t})\rvert\big{\\}}$.

By Lemma A.2,
$\sup_{\boldsymbol{t}\in\mathbb{R}^{K_{\theta}}}\lvert\mathbb{P}\\{\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}|\mathcal{E}_{n}\\}-\Phi(\boldsymbol{t})\rvert=o_{p}(1)$
because conditioning on $\mathcal{E}_{n}$ involves conditioning on
$\mathbf{\hat{B}}_{n}$. As
$\sup_{\boldsymbol{t}\in\mathbb{R}^{K_{\theta}}}\lvert\mathbb{P}\\{\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}|\mathcal{E}_{n}\\}-\Phi(\boldsymbol{t})\rvert$
is bounded, this implies that
$\lim\mathbb{E}\\{\sup_{\boldsymbol{t}\in\mathbb{R}^{K_{\theta}}}\lvert\mathbb{P}(\boldsymbol{u}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\preceq\boldsymbol{t}|\mathcal{E}_{n})-\Phi(\boldsymbol{t})\rvert\\}=0$.
As a result,
$\lim\sup_{\boldsymbol{t}\in\mathbb{R}^{K_{\theta}}}\lvert\mathbb{P}(\mathbf{V}_{n}^{-1/2}\mathbf{A}_{0}\Delta_{n}\preceq\boldsymbol{t})-G(\boldsymbol{t})\rvert=0$.
This completes the proof. ∎

###  A.4 Proof of Corollary 4.1

By the definition of $\Delta_{n}$, we have
$\boldsymbol{\theta}_{0}(\iota)=\boldsymbol{\hat{\theta}}_{n}(\iota)-\Delta_{n}/\sqrt{n}$.
Thus ,
$\mathbb{P}(\boldsymbol{\theta}_{0}(\iota)\in[T_{\frac{\alpha}{2}},\leavevmode\nobreak\
T_{1-\frac{\alpha}{2}}])=\mathbb{P}(\Delta_{n}(\iota)\in[\sqrt{n}(\boldsymbol{\hat{\theta}}_{n}(\iota)-T_{1-\frac{\alpha}{2}}),\leavevmode\nobreak\
\sqrt{n}(\boldsymbol{\hat{\theta}}_{n}(\iota)-T_{\frac{\alpha}{2}})])$, with
$\sqrt{n}(\boldsymbol{\hat{\theta}}_{n}(\iota)-T_{1-\frac{\alpha}{2}})$ and
$\sqrt{n}(\boldsymbol{\hat{\theta}}_{n}(\iota)-T_{\frac{\alpha}{2}})$ being
the empirical quantiles of the sample
$\\{\boldsymbol{\psi}_{n,s}(\iota),\leavevmode\nobreak\
s=1,\leavevmode\nobreak\ \dots,\leavevmode\nobreak\ \kappa\\}$ at
$\frac{\alpha}{2}$ and $1-\frac{\alpha}{2}$, respectively. Since
$\Delta_{n}(\iota)$ and $\boldsymbol{\psi}_{n,s}(\iota)$ have the same
limiting distribution when $n$ and $\kappa$ grow to infinity, these quantiles
have the same limit as the theoretical quantiles of $\Delta_{n}(\iota)$ at
$\frac{\alpha}{2}$ and $1-\frac{\alpha}{2}$. This completes the proof.

###  A.5 Proof of Corollary 4.2

By Condition (ii) of Theorem 4.1, $\\{\mathbf{V}_{n},\leavevmode\nobreak\
\mathcal{E}_{n}\\}$ converges in distribution to
$\\{\mathbf{V}_{0},\leavevmode\nobreak\ \mathcal{E}_{0}\\}$. As
$\\{\mathbf{V}_{n},\leavevmode\nobreak\ \mathcal{E}_{n}\\}$ is independent of
$\boldsymbol{\zeta}$, Theorem 4.2 implies that
$\sqrt{n}(\boldsymbol{\hat{\theta}}_{n}-\boldsymbol{\theta}_{0})$ converges in
distribution to
$\boldsymbol{\psi}_{0}:=\mathbf{A}_{0}^{-1}\mathbf{V}_{0}^{1/2}\boldsymbol{\zeta}+\mathbf{A}_{0}^{-1}\mathcal{E}_{0}$,
which is the sum of two independent variables. As a result,
$\boldsymbol{\psi}_{0}$ is normally distributed with mean
$\mathbb{E}(\boldsymbol{\psi}_{0})=\mathbf{A}_{0}^{-1}\mathbb{E}(\mathcal{E}_{0})$
and variance
$\mathbb{V}(\boldsymbol{\psi}_{0})=\mathbf{A}_{0}^{-1}(\mathbf{V}_{0}+\mathbb{V}(\mathcal{E}_{0}))\mathbf{A}_{0}^{-1}$.

###  A.6 Proof of Theorem 4.3

We have
$\sqrt{n}(\boldsymbol{\theta}_{n,\kappa}^{\ast}-\boldsymbol{\theta}_{0})=\Delta_{n}-\mathbf{\hat{A}}_{n}^{-1}\boldsymbol{\hat{\Omega}}_{n}^{\kappa}$.
As $\Delta_{n}=O_{p}(1)$,
$\operatorname{plim}\mathbf{\hat{A}}_{n}=\mathbf{A}_{0}$, and
$\operatorname{plim}_{\kappa,n}\boldsymbol{\hat{\Omega}}_{n}^{\kappa}=\mathbb{E}(\mathcal{E}_{0})$,
it follows that
$\sqrt{n}(\boldsymbol{\theta}_{n,\kappa}^{\ast}-\boldsymbol{\theta}_{0})=O_{p}(1)$,
that is, $\boldsymbol{\theta}_{n,\kappa}^{\ast}$ is a $\sqrt{n}$-consistent
estimator of $\boldsymbol{\theta}_{0}$. This completes the proof of Statement
(i).

$\Delta_{n}$ has the same asymptotic distribution as
$\boldsymbol{\psi}_{n}=\mathbf{A}_{0}^{-1}\mathbf{V}_{n}^{1/2}\boldsymbol{\zeta}+\mathbf{A}_{0}^{-1}\mathcal{E}_{n}$
and $\mathbf{\hat{A}}_{n}^{-1}\boldsymbol{\hat{\Omega}}_{n}^{\kappa}$
converges in probability to the constant
$\mathbf{A}_{0}^{-1}\mathbb{E}(\mathcal{E}_{0})$, as $\kappa$ and $n$ grow to
infinity. Thus, by the Slutsky theorem,
$\sqrt{n}(\boldsymbol{\theta}_{n,\kappa}^{\ast}-\boldsymbol{\theta}_{0})=\Delta_{n}-\mathbf{\hat{A}}_{n}^{-1}\boldsymbol{\hat{\Omega}}_{n}^{\kappa}$
has the same asymptotic distribution as
$\boldsymbol{\psi}_{n}-\mathbf{A}_{0}^{-1}\mathbb{E}(\mathcal{E}_{0})=\mathbf{A}_{0}^{-1}\mathbf{V}_{n}^{1/2}\boldsymbol{\zeta}+\mathbf{A}_{0}^{-1}(\mathcal{E}_{n}-\mathbb{E}(\mathcal{E}_{0}))$.
As a result $\boldsymbol{\psi}_{n}^{\ast}$ and
$\sqrt{n}(\boldsymbol{\theta}_{n,\kappa}^{\ast}-\boldsymbol{\theta}_{0})$ have
the same distribution, have $\kappa$ and $n$ grow to infinity. This completes
the proof of Statement (ii).

Statement (iii) is a direct implication of Statement (ii). The variable
$\boldsymbol{\psi}_{n}^{\ast}$ converges in distribution to
$\boldsymbol{\psi}_{0}^{\ast}:=\mathbf{A}_{0}^{-1}\mathbf{V}_{0}^{1/2}\boldsymbol{\zeta}+\mathbf{A}_{0}^{-1}(\mathcal{E}_{0}-\mathbb{E}(\mathcal{E}_{0}))$,
with $\mathbb{E}(\boldsymbol{\psi}_{0}^{\ast})=0$ and
$\mathbb{V}(\boldsymbol{\psi}_{0}^{\ast})=\mathbf{A}_{0}^{-1}(\mathbf{V}_{0}+\mathbb{V}(\mathcal{E}_{0}))\mathbf{A}_{0}^{-1}$.
As $\boldsymbol{\psi}_{n}^{\ast}$ and
$\sqrt{n}(\boldsymbol{\theta}_{n,\kappa}^{\ast}-\boldsymbol{\theta}_{0})$ have
the same distribution, the result follows.

##  Appendix B Online Appendix

Supplementary material related to this paper can be found online at
https://ahoundetoungan.com/files/Papers/InferenceTSE_OA.pdf

Supplemental Appendix for "Inference for Two-Stage Extremum Estimators"

This supplemental appendix includes additional results and technical details
omitted from the main text. Section S.1 discusses primitive conditions of some
high-level assumptions introduced in the paper. Section S.2 provides technical
details regarding our simulation study, including a description of how we
approximate the asymptotic distributions. Section S.3 presents detailed
results of our empirical application with network data.

##  Appendix S.1 Online Appendix–Proofs

###  S.1.1 Consistency of Plug-in Estimators

We impose lower-level assumptions that result in a consistent plug-in
estimator. For notational ease, we omit $\mathbf{y}_{n}$ and $\mathbf{X}_{n}$
in $Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{y}_{n},\leavevmode\nobreak\ \mathbf{X}_{n},\leavevmode\nobreak\
\mathbf{B})$ and we simply write
$Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\ \mathbf{B})$. We also write
$q_{i}(\boldsymbol{\theta},\leavevmode\nobreak\ \boldsymbol{\beta}_{i})$
instead of $q(\boldsymbol{\theta},\leavevmode\nobreak\
y_{i},\leavevmode\nobreak\ \boldsymbol{x}_{i},\leavevmode\nobreak\
\boldsymbol{\beta}_{i})$. For any $\boldsymbol{\theta}\in\boldsymbol{\Theta}$
and $a$, $\tau\leavevmode\nobreak\ >0$, we define

$\textstyle\mathcal{H}_{i}(a,\tau)=\sup_{\lVert\boldsymbol{\beta}_{i}-\boldsymbol{\beta}_{0,i}\rVert<\tau}\dfrac{\lvert
q_{i}(\boldsymbol{\theta},\leavevmode\nobreak\
\boldsymbol{\beta}_{i})-q_{i}(\boldsymbol{\theta},\leavevmode\nobreak\
\boldsymbol{\beta}_{0,i})\rvert}{\lVert\boldsymbol{\beta}_{i}-\boldsymbol{\beta}_{0,i}\rVert^{a}},$

where $\lVert.\rVert$ is the $\ell^{2}$-norm.

######  Assumption S.1 (Primitive Conditions for Assumption 2.3).

(i) $\boldsymbol{\Theta}$is a compact subset and $\boldsymbol{\theta}_{0}$ is
an interior point of $\boldsymbol{\Theta}$.
(ii) The function $Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{B}_{0})$ converges uniformly in probability (across
$\boldsymbol{\theta}\in\boldsymbol{\Theta}$) to a nonstochastic function
$Q_{0}(\boldsymbol{\theta})$ that is maximized only at
$\boldsymbol{\theta}_{0}$.
(iii) For any $\boldsymbol{\theta}\in\boldsymbol{\Theta}$, there exists
constants $a(\boldsymbol{\theta})$,
$\tau(\boldsymbol{\theta})\leavevmode\nobreak\ >0$, such that,
$\max_{i}\mathcal{H}_{i}(a(\boldsymbol{\theta}),\tau(\boldsymbol{\theta}))=O_{p}(1)$.
(iv) There exists a sequence of neighborhoods
$\mathcal{O}(\boldsymbol{\beta}_{0,1}),\leavevmode\nobreak\
\dots,\leavevmode\nobreak\ \mathcal{O}(\boldsymbol{\beta}_{0,n})$, such that
$\partial_{\boldsymbol{\theta}}q_{i}(\boldsymbol{\theta},\leavevmode\nobreak\
\boldsymbol{\beta}_{i})$ is $O_{p}(1)$, uniformly across
$\boldsymbol{\theta}\in\boldsymbol{\Theta}$,
$\boldsymbol{\beta}_{i}\in\mathcal{O}(\boldsymbol{\beta}_{0,i})$ and $i$, that
is, $\max_{\boldsymbol{\theta}\in\boldsymbol{\Theta},\leavevmode\nobreak\
\boldsymbol{\beta}_{i}\in\mathcal{O}(\boldsymbol{\beta}_{0,i}),\leavevmode\nobreak\
i\leq
n}\textstyle\lVert\partial_{\boldsymbol{\theta}}q_{i}(\boldsymbol{\theta},\leavevmode\nobreak\
\boldsymbol{\beta}_{i})\rVert=O_{p}(1)$.

The compactness restriction in Condition (i) allows for the plug-in estimator
to converge in its support. Condition (ii) is a classical identification
condition also required for a standard M-estimator. Importantly, this
condition does not involve the estimator $\mathbf{\hat{B}}_{n}$. It is an
identification condition that is set at the true $\mathbf{B}_{0}$. Condition
(iii) implies that $\lvert q_{i}(\boldsymbol{\theta},\leavevmode\nobreak\
\boldsymbol{\beta}_{i})-q_{i}(\boldsymbol{\theta},\leavevmode\nobreak\
\boldsymbol{\beta}_{0,i})\rvert\leq\lVert\boldsymbol{\beta}_{i}-\boldsymbol{\beta}_{0,i}\rVert^{a(\boldsymbol{\theta})}O_{p}(1)$
for all $i$ and
$\lVert\boldsymbol{\beta}_{i}-\boldsymbol{\beta}_{0,i}\rVert<\tau(\boldsymbol{\theta})$.
A similar assumption is also imposed by Cattaneo et al. (2019) and requires
$q_{i}(\boldsymbol{\theta},\leavevmode\nobreak\ \boldsymbol{\beta}_{i})$ to be
smooth in $\boldsymbol{\beta}_{i}$. We use this condition and the uniform
convergence of $\boldsymbol{\hat{\beta}}_{n,i}$ (Assumption 2.1) to show that
$Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})-Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{B}_{0})$ converges in probability to $0$ for each
$\boldsymbol{\theta}$. Condition (iv) allows us to generalize this point-wise
convergence to a uniform convergence.111Note that
$\partial_{\boldsymbol{\theta}}q_{i}(\boldsymbol{\theta},\leavevmode\nobreak\
\boldsymbol{\beta}_{i})$ is be bounded in probability for each $i$ as it must
have finite moments. Consequently, one can find a neighborhood
$\mathcal{O}(\boldsymbol{\beta}_{0,i})$ such
$\max_{\boldsymbol{\theta}\in\boldsymbol{\Theta},\leavevmode\nobreak\
\boldsymbol{\beta}_{i}\in\mathcal{O}(\boldsymbol{\beta}_{0,i})}\textstyle\lVert\partial_{\boldsymbol{\theta}}q_{i}(\boldsymbol{\theta},\leavevmode\nobreak\
\boldsymbol{\beta}_{i})\rVert$ is bounded in probability. The additional
requirement of Condition (iv) is that the bound is uniform across $i$ as $n$
grows to infinity.

######  Proposition S.1.

Under Assumptions 2.1, 2.2, and S.1, the estimator
$\boldsymbol{\hat{\theta}}_{n}$ converges in probability to
$\boldsymbol{\theta}_{0}$.

###### Proof.

The proof is performed in two steps.

Step 1: We show that $Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})$ converges uniformly in probability to
$Q_{0}(\boldsymbol{\theta})$.
For any $\boldsymbol{\theta}$, we have $\lvert
Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})-Q_{0}(\boldsymbol{\theta})\rvert\leq\lvert
Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})-Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{B}_{0})\rvert+\lvert Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{B}_{0})-Q_{0}(\boldsymbol{\theta})\rvert$. Since
$Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{B}_{0})-Q_{0}(\boldsymbol{\theta})$ converges uniformly in probability
in $\boldsymbol{\theta}$ to $0$ (Condition (ii) of Assumption S.1), it is
sufficient to show that $Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})-Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{B}_{0})$ also converges uniformly in probability in
$\boldsymbol{\theta}$ to $0$.

By Assumption 2.1, for $n$ large enough,
$\displaystyle\max_{i}\textstyle\lVert\boldsymbol{\hat{\beta}}_{n,i}-\boldsymbol{\beta}_{0,i}\rVert<\tau(\boldsymbol{\theta})$
with probability approaching one. Thus, by Condition (iii) of Assumption S.1,

$\lvert q_{i}(\boldsymbol{\theta},\leavevmode\nobreak\
\boldsymbol{\hat{\beta}}_{n,i})-q_{i}(\boldsymbol{\theta},\leavevmode\nobreak\
\boldsymbol{\beta}_{0,i})\rvert\leq\lVert\boldsymbol{\hat{\beta}}_{n,i}-\boldsymbol{\beta}_{0,i}\rVert^{a(\boldsymbol{\theta})}\max_{i}\mathcal{H}_{i}(a(\boldsymbol{\theta}),\tau(\boldsymbol{\theta}))\quad\text{for
all $i$}$

with probability approaching one. As $\lvert
Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})-Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{B}_{0})\rvert\leq\max_{i}\lvert
q_{i}(\boldsymbol{\theta},\leavevmode\nobreak\
\boldsymbol{\hat{\beta}}_{n,i})-q_{i}(\boldsymbol{\theta},\leavevmode\nobreak\
\boldsymbol{\beta}_{0,i})\rvert$, this implies that

$\lvert Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})-Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{B}_{0})\rvert\leq\max_{i}\lVert\boldsymbol{\beta}_{i}-\boldsymbol{\beta}_{0,i}\rVert^{a(\boldsymbol{\theta})}\max_{i}\mathcal{H}_{i}(a(\boldsymbol{\theta}),\tau(\boldsymbol{\theta}))$

with probability approaching one. As a result
$Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})-Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{B}_{0})$ converges in probability to zero because
$\max_{i}\lVert\boldsymbol{\beta}_{i}-\boldsymbol{\beta}_{0,i}\rVert^{a(\boldsymbol{\theta})}=o_{p}(1)$
and
$\max_{i}\mathcal{H}_{i}(a(\boldsymbol{\theta}),\tau(\boldsymbol{\theta}))=O_{p}(1)$.

To show that the convergence is uniform, we apply the mean value theorem to
$Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})-Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{B}_{0})$ with respect to $\boldsymbol{\theta}$. For any
$\boldsymbol{\tilde{\theta}}\in\boldsymbol{\Theta}$, we have

$Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})-Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{B}_{0})-\big{(}Q_{n}(\boldsymbol{\tilde{\theta}},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})-Q_{n}(\boldsymbol{\tilde{\theta}},\leavevmode\nobreak\
\mathbf{B}_{0})\big{)}=(\boldsymbol{\theta}-\boldsymbol{\tilde{\theta}})^{\prime}\boldsymbol{\hat{Q}}_{n},$

where
$\boldsymbol{\hat{Q}}_{n}=\frac{1}{n}\sum_{i=1}^{n}\partial_{\boldsymbol{\theta}}\big{(}q_{i}(\boldsymbol{\theta}^{+}_{n},\leavevmode\nobreak\
\boldsymbol{\hat{\beta}}_{n,i})-q_{i}(\boldsymbol{\theta}^{+}_{n},\leavevmode\nobreak\
\boldsymbol{\beta}_{0,i})\big{)}$, for some $\boldsymbol{\theta}^{+}_{n}$ that
lies between $\boldsymbol{\theta}$ and $\boldsymbol{\tilde{\theta}}$. Thus,

$\lvert Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})-Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{B}_{0})-\big{(}Q_{n}(\boldsymbol{\tilde{\theta}},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})-Q_{n}(\boldsymbol{\tilde{\theta}},\leavevmode\nobreak\
\mathbf{B}_{0})\big{)}\rvert\leq\lVert\boldsymbol{\hat{Q}}_{n}\rVert\lVert\boldsymbol{\theta}-\boldsymbol{\tilde{\theta}}\rVert.$

As $\boldsymbol{\hat{Q}}_{n}=O_{p}(1)$ (Condition (iv) of Assumption S.1) and
$\boldsymbol{\Theta}$ is compact, it follows from Lemma 2.9 of Newey and
McFadden (1994) that ${Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})-Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{B}_{0})}$ converges uniformly in probability to $0$.

Step 2: We establish the consistency of the estimator
$\boldsymbol{\hat{\theta}}_{n}$.

Let $\mathcal{O}(\boldsymbol{\theta}_{0})^{c}$ be the complement of
$\mathcal{O}(\boldsymbol{\theta}_{0})$ in $\boldsymbol{\Theta}$. Note that
$\mathcal{O}(\boldsymbol{\theta}_{0})^{c}$ is nonempty and compact and
$\max_{\boldsymbol{\theta}\in\mathcal{O}(\boldsymbol{\theta}_{0})^{c}}Q_{0}(\boldsymbol{\theta})$
exists. Let
$\delta=Q_{0}(\boldsymbol{\theta}_{0})-\max_{\boldsymbol{\theta}\in\mathcal{O}(\boldsymbol{\theta}_{0})^{c}}Q_{0}(\boldsymbol{\theta})$
and $J_{n}=\big{\\{}|Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})-Q_{0}(\boldsymbol{\theta})|<\delta/2,\,\text{ for all
}\boldsymbol{\theta}\big{\\}}$. We know that $Q_{0}(\boldsymbol{\theta})$ is
uniquely maximized at $\boldsymbol{\theta}_{0}$ and that
$\boldsymbol{\theta}_{0}\notin\mathcal{O}(\boldsymbol{\theta}_{0})^{c}$. Thus
$\delta>0$. Moreover, since $Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})$ converges uniformly in probability to
$Q_{0}(\boldsymbol{\theta})$, we have ${\lim\mathbb{P}(J_{n})=1}$.

$J_{n}\implies\big{\\{}Q_{0}(\boldsymbol{\hat{\theta}}_{n})>Q_{n}(\boldsymbol{\hat{\theta}}_{n},\mathbf{\hat{B}}_{n})-\delta/2\big{\\}}\cap\big{\\{}Q_{n}(\boldsymbol{\theta}_{0},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})>Q_{0}(\boldsymbol{\theta}_{0})-\delta/2\big{\\}}$ (S.1)

As
$\boldsymbol{\hat{\theta}}_{n}=\operatorname{argmax}_{\boldsymbol{\theta}}Q_{n}(\boldsymbol{\theta},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})$, we also have
$Q_{n}(\boldsymbol{\hat{\theta}}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})\geq Q_{n}(\boldsymbol{\theta}_{0},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})$. Thus,
$\big{\\{}Q_{0}(\boldsymbol{\hat{\theta}}_{n})>Q_{n}(\boldsymbol{\hat{\theta}}_{n},\mathbf{\hat{B}}_{n})-\delta/2\big{\\}}$
implies
$\big{\\{}Q_{0}(\boldsymbol{\hat{\theta}}_{n})>Q_{n}(\boldsymbol{\theta}_{0},\mathbf{\hat{B}}_{n})-\delta/2\big{\\}}$.
It follows from (S.1) that

$\displaystyle J_{n}$
$\displaystyle\implies\big{\\{}Q_{0}(\boldsymbol{\hat{\theta}}_{n})>Q_{n}(\boldsymbol{\theta}_{0},\mathbf{\hat{B}}_{n})-\delta/2\big{\\}}\cap\big{\\{}Q_{n}(\boldsymbol{\theta}_{0},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})>Q_{0}(\boldsymbol{\theta}_{0})-\delta/2\big{\\}},$
$\displaystyle J_{n}$
$\displaystyle\implies\big{\\{}Q_{0}(\boldsymbol{\hat{\theta}}_{n})>Q_{0}(\boldsymbol{\theta}_{0})-\delta\big{\\}}.$
(S.2)

As
$\delta=Q_{0}(\boldsymbol{\theta}_{0})-\max_{\boldsymbol{\theta}\in\mathcal{O}(\boldsymbol{\theta}_{0})^{c}}Q_{0}(\boldsymbol{\theta})$,
it turns out from (S.2) that

$\displaystyle J_{n}$
$\displaystyle\implies\big{\\{}Q_{0}(\boldsymbol{\hat{\theta}}_{n})>\max_{\boldsymbol{\theta}\in\mathcal{O}(\boldsymbol{\theta}_{0})^{c}}Q_{0}(\boldsymbol{\theta})\big{\\}},$
$\displaystyle J_{n}$
$\displaystyle\implies\boldsymbol{\hat{\theta}}_{n}\in\mathcal{O}(\boldsymbol{\theta}_{0}).$
(S.3)

As ${\lim\mathbb{P}(J_{n})=1}$, then (S.3) implies that
${\lim\mathbb{P}(\boldsymbol{\hat{\theta}}_{n}\in\mathcal{O}(\boldsymbol{\theta}_{0}))=1}$.
This is true for any open subset $\mathcal{O}(\boldsymbol{\theta}_{0})$ that
contains $\boldsymbol{\theta}_{0}$. As a result,
$\boldsymbol{\hat{\theta}}_{n}$ converges in probability to
$\boldsymbol{\theta}_{0}$. ∎

###  S.1.2 Primitive Conditions for Assumption 4.2

For notational ease, let
$\ddot{q}_{i}(\boldsymbol{\theta},\leavevmode\nobreak\
\boldsymbol{\beta}_{i})=\partial_{\boldsymbol{\theta}}\partial_{\boldsymbol{\theta}^{\prime}}q(\boldsymbol{\theta},\leavevmode\nobreak\
y_{i},\leavevmode\nobreak\ \boldsymbol{x}_{i},\leavevmode\nobreak\
\boldsymbol{\beta}_{i})$. For any $\boldsymbol{\theta}\in\boldsymbol{\Theta}$
and $a$, $\tau\leavevmode\nobreak\ >0$, we define

$\textstyle\ddot{\mathcal{H}}_{i}(a,\tau)=\sup_{(\lVert\boldsymbol{\theta}-\boldsymbol{\theta}_{0}\rVert+\lVert\boldsymbol{\beta}_{i}-\boldsymbol{\beta}_{0,i}\rVert)^{a}<\tau}\dfrac{\lVert\ddot{q}_{i}(\boldsymbol{\theta},\leavevmode\nobreak\
\boldsymbol{\beta}_{i})-\ddot{q}_{i}(\boldsymbol{\theta}_{0},\leavevmode\nobreak\
\boldsymbol{\beta}_{0,i})\rVert}{(\lVert\boldsymbol{\theta}-\boldsymbol{\theta}_{0}\rVert+\lVert\boldsymbol{\beta}_{i}-\boldsymbol{\beta}_{0,i}\rVert)^{a}}\,.$

We impose lower-level conditions that imply Assumption 4.2.

######  Assumption S.2 (Primitive Conditions for Assumption 4.2).

(i) The matrix
$\frac{1}{n}\sum_{i=1}^{n}\ddot{q}_{i}(\boldsymbol{\theta}_{0},\leavevmode\nobreak\
\boldsymbol{\beta}_{0,i})$ converges in probability to a finite nonsingular
matrix $\mathbf{A}_{0}$ defined by
$\lim\mathbb{E}\big{(}\frac{1}{n}\sum_{i=1}^{n}\ddot{q}_{i}(\boldsymbol{\theta}_{0},\leavevmode\nobreak\
\boldsymbol{\beta}_{0,i})\big{)}$.
(ii) There exists constants $a^{\ast},\leavevmode\nobreak\ \tau^{\ast}>0$ such
that $\max_{i}\ddot{\mathcal{H}}_{i}(a^{\ast},\tau^{\ast})=O_{p}(1)$.

Condition (i) imposes that
$\frac{1}{n}\sum_{i}^{n}\ddot{q}_{i}(\boldsymbol{\theta}_{0},\leavevmode\nobreak\
\boldsymbol{\beta}_{0,i})$ converges in to
$\lim\mathbb{E}\big{(}\frac{1}{n}\sum_{i=1}^{n}\ddot{q}_{i}(\boldsymbol{\theta}_{0},\leavevmode\nobreak\
\boldsymbol{\beta}_{0,i})\big{)}$. This condition is classical as in the case
of a single-step estimator. It does not involve any estimator and can be
implied by the weak law of large numbers (WLLN). For
$\ddot{q}_{i}(\boldsymbol{\theta}_{0},\leavevmode\nobreak\
\boldsymbol{\beta}_{0,i})$’s dependent across $i$, WLLN for dependent
processes can be used. Condition (ii) is similar to Condition (iii) of
Assumption S.1. It requires
$\ddot{q}_{i}(\boldsymbol{\theta},\leavevmode\nobreak\
\boldsymbol{\beta}_{i})$ to be smooth in both $\boldsymbol{\beta}_{i}$ and
$\boldsymbol{\theta}$, uniformly in $i$.

######  Proposition S.2.

Under Assumptions 2.1, 2.2, 2.3, S.1 and S.2, the Hessian of the objective
function evaluated at any consistent estimator $\boldsymbol{\theta}_{n}^{+}$,
given by
$\frac{1}{n}\sum_{i=1}^{n}\ddot{q}_{i}(\boldsymbol{\theta}_{n}^{+},\leavevmode\nobreak\
\boldsymbol{\hat{\beta}}_{n,i})$, converges in probability to a finite
nonsingular matrix
$\mathbf{A}_{0}=\lim\mathbb{E}\big{(}\frac{1}{n}\sum_{i=1}^{n}\ddot{q}_{i}(\boldsymbol{\theta}_{0},\leavevmode\nobreak\
\boldsymbol{\beta}_{0,i})\big{)}$.

###### Proof.

By Assumption 2.1, for $n$ large enough,
$\lVert\boldsymbol{\theta}_{n}^{+}-\boldsymbol{\theta}_{0}\rVert+\displaystyle\max_{i}\textstyle\lVert\boldsymbol{\hat{\beta}}_{n,i}-\boldsymbol{\beta}_{0,i}\rVert<\tau^{\ast}$
with probability approaching one. Thus,
$\lVert\ddot{q}_{i}(\boldsymbol{\theta}_{n}^{+},\leavevmode\nobreak\
\boldsymbol{\hat{\beta}}_{n,i})-\ddot{q}_{i}(\boldsymbol{\theta}_{0},\leavevmode\nobreak\
\boldsymbol{\beta}_{0,i})\rVert\leq(\lVert\boldsymbol{\theta}_{n}^{+}-\boldsymbol{\theta}_{0}\rVert+\lVert\boldsymbol{\hat{\beta}}_{n,i}-\boldsymbol{\beta}_{0,i}\rVert)^{a^{\ast}}\max_{i}\ddot{\mathcal{H}}_{i}(a^{\ast},\tau^{\ast})$,
for all $i$, with probability approaching one.
As
$\lVert\frac{1}{n}\sum_{i=1}^{n}\ddot{q}_{i}(\boldsymbol{\theta}_{n}^{+},\leavevmode\nobreak\
\boldsymbol{\hat{\beta}}_{n,i})-\frac{1}{n}\sum_{i=1}^{n}\ddot{q}_{i}(\boldsymbol{\theta}_{0},\leavevmode\nobreak\
\boldsymbol{\beta}_{0,i})\rVert\leq\max_{i}\lVert\ddot{q}_{i}(\boldsymbol{\theta}_{n}^{+},\leavevmode\nobreak\
\boldsymbol{\hat{\beta}}_{n,i})-\ddot{q}_{i}(\boldsymbol{\theta}_{0},\leavevmode\nobreak\
\boldsymbol{\beta}_{0,i})\rVert$, it follows that
$\lVert\frac{1}{n}\sum_{i=1}^{n}\ddot{q}_{i}(\boldsymbol{\theta}_{n}^{+},\leavevmode\nobreak\
\boldsymbol{\hat{\beta}}_{n,i})-\frac{1}{n}\sum_{i=1}^{n}\ddot{q}_{i}(\boldsymbol{\theta}_{0},\leavevmode\nobreak\
\boldsymbol{\beta}_{0,i})\rVert\leq(\lVert\boldsymbol{\theta}_{n}^{+}-\boldsymbol{\theta}_{0}\rVert+\max_{i}\lVert\boldsymbol{\hat{\beta}}_{n,i}-\boldsymbol{\beta}_{0,i}\rVert)^{a^{\ast}}\max_{i}\ddot{\mathcal{H}}_{i}(a^{\ast},\tau^{\ast})$
with probability approaching one, where
$\lVert\boldsymbol{\theta}_{n}^{+}-\boldsymbol{\theta}_{0}\rVert+\max_{i}\lVert\boldsymbol{\beta}_{i}-\boldsymbol{\beta}_{0,i}\rVert^{a(\boldsymbol{\theta})}=o_{p}(1)$
and $\max_{i}\ddot{\mathcal{H}}_{i}(a^{\ast},\tau^{\ast})=O_{p}(1)$. As a
result,
$\frac{1}{n}\sum_{i=1}^{n}\ddot{q}_{i}(\boldsymbol{\theta}_{n}^{+},\leavevmode\nobreak\
\boldsymbol{\hat{\beta}}_{n,i})-\frac{1}{n}\sum_{i=1}^{n}\ddot{q}_{i}(\boldsymbol{\theta}_{0},\leavevmode\nobreak\
\boldsymbol{\beta}_{0,i})=o_{p}(1)$. Given that
$\frac{1}{n}\sum_{i=1}^{n}\ddot{q}_{i}(\boldsymbol{\theta}_{0},\leavevmode\nobreak\
\boldsymbol{\beta}_{0,i})$ converges in probability to a finite nonsingular
matrix $\mathbf{A}_{0}$, the result follows. ∎

##  Appendix S.2 Supplementary Materials on the Simulation Study

This section provides a detailed explanation of the method used to estimate
the asymptotic variance and the asymptotic cumulative distribution function
(CDF) of the plug-in estimators in the simulation study. Our replication code
available at https://github.com/ahoundetoungan/InferenceTSE implements this
method. We also present the estimates of the asymptotic CDF of the debiased
estimators.

###  S.2.1 Asymptotic Variance and Asymptotic CDF

##### DGPs A and B

DGP A is a treatment effect model with endogeneity. The model is defined as
follows:

$\displaystyle y_{i}=\theta_{0}d_{i}+\varepsilon_{i},\quad
d_{i}=\mathbbm{1}\\{z_{i}>0.5(\varepsilon_{i}+1.2)\\},\quad
z_{i}\sim\text{Uniform}[0,\leavevmode\nobreak\
1],\quad\varepsilon_{i}\sim\text{Uniform}[-1,1],$

where $d_{i}$ is a treatment status indicator, $z_{i}$ is an instrument for
the treatment and $\theta_{0}=1$. In the first stage, we perform two OLS
regressions: a regression of $y_{i}$ on
$\boldsymbol{z}_{i}=(1,z_{i})^{\prime}$ and another regression of $d_{i}$ on
$\boldsymbol{z}_{i}$. For DGP B, the vector of regressors in the first stage
is $\boldsymbol{z}_{i}=(1,\leavevmode\nobreak\ z_{1,i},\leavevmode\nobreak\
\dots,z_{k_{n},i})^{\prime}$.

Let $\boldsymbol{\hat{\gamma}}_{n}^{(y)}$ and
$\boldsymbol{\hat{\gamma}}_{n}^{(d)}$ be the respective OLS estimators and let
$\boldsymbol{\hat{\gamma}}_{n}=(\boldsymbol{\hat{\gamma}}_{n}^{(y)\prime},\leavevmode\nobreak\
\boldsymbol{\hat{\gamma}}_{n}^{(d)\prime})^{\prime}$ be the joint first-stage
estimator. Let also $\hat{\nu}_{i}^{(y)}$ and $\hat{\nu}_{i}^{(d)}$ be the
residuals of the regressions; that is,
$\hat{\nu}_{i}^{(y)}=y_{i}-\boldsymbol{z}_{i}^{\prime}\boldsymbol{\hat{\gamma}}_{n}^{(y)}$
and
$\hat{\nu}_{i}^{(d)}=d_{i}-\boldsymbol{z}_{i}^{\prime}\boldsymbol{\hat{\gamma}}_{n}^{(d)}$.
We define
$\boldsymbol{z}_{i}^{(\nu)}=(\hat{\nu}_{i}^{(y)}\boldsymbol{z}_{i}^{\prime},\leavevmode\nobreak\
\hat{\nu}_{i}^{(d)}\boldsymbol{z}_{i}^{\prime})^{\prime}$. The estimator of
the asymptotic distribution of $\boldsymbol{\hat{\gamma}}_{n}$ is a normal
distribution with mean $\boldsymbol{\hat{\gamma}}_{n}$ and covariance matrix
$\hat{\mathbb{V}}(\boldsymbol{\hat{\gamma}}_{n})=\frac{1}{n}\mathbf{\hat{H}}_{n}^{-1}\mathbf{\hat{J}}_{n}\mathbf{\hat{H}}_{n}^{-1}$,
where

$\textstyle\mathbf{\hat{H}}_{n}=\frac{1}{n}\sum_{i=1}^{n}\operatorname{diag}(\sum_{i=1}^{n}\boldsymbol{z}_{i}\boldsymbol{z}_{i}^{\prime},\leavevmode\nobreak\
\sum_{i=1}^{n}\boldsymbol{z}_{i}\boldsymbol{z}_{i}^{\prime})\quad\text{and}\quad\textstyle\mathbf{\hat{J}}_{n}=\frac{1}{n}\sum_{i=1}^{n}\boldsymbol{z}_{i}^{(\nu)}\boldsymbol{z}_{i}^{(\nu)\prime}.$

The notation $\operatorname{diag}$ stands for the bloc diagonal matrix
operator. In the second stage, the objective function to be maximized is
$\textstyle Q_{n}(\theta,\leavevmode\nobreak\
\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})=-\frac{1}{n}\sum_{i=1}^{n}(\boldsymbol{z}_{i}^{\prime}\boldsymbol{\hat{\gamma}}_{n}^{(y)}-\theta\boldsymbol{z}_{i}^{\prime}\boldsymbol{\hat{\gamma}}_{n}^{(d)})^{2}$,
where
$\mathbf{\hat{B}}_{n}=(\boldsymbol{\hat{\beta}}_{n,1},\leavevmode\nobreak\
\dots,\leavevmode\nobreak\ \boldsymbol{\hat{\beta}}_{n,n})^{\prime}$ and
$\boldsymbol{\hat{\beta}}_{n,i}=(\boldsymbol{z}_{i}^{\prime}\boldsymbol{\hat{\gamma}}_{n}^{(y)},\leavevmode\nobreak\
\boldsymbol{z}_{i}^{\prime}\boldsymbol{\hat{\gamma}}_{n}^{(d)})^{\prime}$.
This implies that
$\boldsymbol{\dot{q}}_{n}(\textbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})=\frac{2}{\sqrt{n}}\sum_{i=1}^{n}\boldsymbol{z}_{i}^{\prime}\boldsymbol{\hat{\gamma}}_{n}^{(d)}(\boldsymbol{z}_{i}^{\prime}\boldsymbol{\hat{\gamma}}_{n}^{(y)}-\theta_{0}\boldsymbol{z}_{i}^{\prime}\boldsymbol{\hat{\gamma}}_{n}^{(d)})$.
We define the following expressions:

$\textstyle\hat{A}_{n}=\frac{1}{n}\sum_{i=1}^{n}(\boldsymbol{z}_{i}^{\prime}\boldsymbol{\hat{\gamma}}_{n}^{(d)})^{2}\quad\text{and}\quad\hat{\mathcal{E}}_{n,s}=\frac{2}{\sqrt{n}}\sum_{i=1}^{n}\boldsymbol{z}_{i}^{\prime}\boldsymbol{\hat{\gamma}}_{n}^{(d,s)}(\boldsymbol{z}_{i}^{\prime}\boldsymbol{\hat{\gamma}}_{n}^{(y,s)}-\hat{\theta}_{n}\boldsymbol{z}_{i}^{\prime}\boldsymbol{\hat{\gamma}}_{n}^{(d,s)}),$

where $(\boldsymbol{\hat{\gamma}}_{n}^{(y,1)\prime},\leavevmode\nobreak\
\boldsymbol{\hat{\gamma}}_{n}^{(d,1)\prime})^{\prime},\leavevmode\nobreak\
\dots,\leavevmode\nobreak\
(\boldsymbol{\hat{\gamma}}_{n}^{(y,\kappa)\prime},\leavevmode\nobreak\
\boldsymbol{\hat{\gamma}}_{n}^{(d,\kappa)\prime})^{\prime}\overset{i.i.d}{\sim}N(\boldsymbol{\hat{\gamma}}_{n},\leavevmode\nobreak\
\hat{\mathbb{V}}(\boldsymbol{\hat{\gamma}}_{n}))$. Let
$\hat{\psi}_{n,s}=\frac{\hat{\mathcal{E}}_{n,s}}{\hat{A}_{n}}$. The asymptotic
CDF of $\sqrt{n}(\hat{\theta}_{n}-\theta_{0})$ can be estimated by the
empirical CDF of the sample: $\\{\hat{\psi}_{n,s},\leavevmode\nobreak\
s=1,\leavevmode\nobreak\ \dots,\leavevmode\nobreak\ \kappa\\}$. The estimator
of the asymptotic variance of $\hat{\theta}_{n}$ is

$\textstyle\hat{\mathbb{V}}(\hat{\theta}_{n})=\dfrac{\sum_{s=1}^{\kappa}(\hat{\mathcal{E}}_{n,s}-\hat{\mathbb{E}}(\mathcal{E}_{n}))^{2}}{n(\kappa-1)\hat{A}_{n}^{2}},$

where
$\hat{\mathbb{E}}(\mathcal{E}_{n})=\frac{1}{\kappa}\sum_{s=1}^{\kappa}\hat{\mathcal{E}}_{n,s}$.
The debiased estimator is given by

$\theta_{n,\kappa}^{\ast}=\hat{\theta}_{n}-\hat{\mathbb{E}}(\mathcal{E}_{n})/(\sqrt{n}\hat{A}_{n}).$

Let
$\hat{\psi}_{n,s}^{\ast}=\frac{\hat{\mathcal{E}}_{n,s}^{\ast}-\hat{\mathbb{E}}(\mathcal{E}_{n}^{\ast})}{\hat{A}_{n}^{\ast}}$,
where $\hat{A}_{n}^{\ast}$, $\hat{\mathcal{E}}_{n,s}^{\ast}$, and
$\hat{\mathbb{E}}(\mathcal{E}_{n}^{\ast})$ are defined as $\hat{A}_{n}$,
$\hat{\mathcal{E}}_{n,s}$, and $\hat{\mathbb{E}}(\mathcal{E}_{n})$,
respectively, with the difference that they are computed using
$\theta_{n,\kappa}^{\ast}$ and not $\hat{\theta}_{n}$. We can estimate the
asymptotic CDF of $\sqrt{n}(\theta_{n,\kappa}^{\ast}-\theta_{0})$ by the
empirical CDF of the sample: $\\{\hat{\psi}_{n,s}^{\ast},\leavevmode\nobreak\
s=1,\leavevmode\nobreak\ \dots,\leavevmode\nobreak\ \kappa\\}$.

##### DGP C

DGP C is a Poisson model with a latent covariate that is defined as:

$y_{i}\sim\text{Poisson}(\exp(\theta_{0,1}+\theta_{0,2}p_{i})),\quad
p_{i}=\sin^{2}(\pi z_{i}),\quad z_{i}\sim\text{Uniform}[0,\leavevmode\nobreak\
10],\quad d_{i}\sim\text{Bernoulli}(p_{i}),$

where $p_{i}$ is an unobserved probability and
$\boldsymbol{\theta}_{0}=(\theta_{0,1},\leavevmode\nobreak\
\theta_{0,2})^{\prime}=(0.5,\leavevmode\nobreak\ 2)^{\prime}$. The
practitioner observes the pairs $(y_{i},\leavevmode\nobreak\ z_{i})$ for all
$i$ but only observes $d_{i}$ for a representative subsample of size
$n^{\ast}\leq n$. In the first stage, $p_{i}=\mathbb{E}(d_{i}|z_{i})$ is
estimated using a generalized additive model (GAM) of $d_{i}$ on $z_{i}$ in
the subsample of size $n^{\ast}$ where $d_{i}$ is observed. The GAM involves
approximating $p_{i}$ by piecewise polynomial functions of $z_{i}$.222See
HASTIE, T. J. (2017): ”Generalized additive models,” in Statistical models in
S, Routledge, 249–307. We consider cubic polynomial functions on the intervals
$[0,\leavevmode\nobreak\ 0.5],\leavevmode\nobreak\ \dots,\leavevmode\nobreak\
[9.5,\leavevmode\nobreak\ 10]$. This approach can thus be regarded as an OLS
regression of $d_{i}$ on numerous explanatory variables called bases, which
are computed from $z_{i}$. We can write
$\hat{p}_{i}=h(z_{i},\leavevmode\nobreak\ \boldsymbol{\hat{\gamma}}_{n})$,
where $h$ is a piecewise cubic polynomial function and
$\boldsymbol{\hat{\gamma}}_{n}$ is the OLS estimator. The regression results
can be used to compute $\hat{p}_{i}$ for any $i$ in the full sample because we
observe $z_{i}$ of all $i$. The estimator of the asymptotic distribution of
$\boldsymbol{\hat{\gamma}}_{n}$ is a normal distribution with mean
$\boldsymbol{\hat{\gamma}}_{n}$ and the standard OLS variance denoted
$\hat{\mathbb{V}}(\hat{\theta}_{n})$.

In the second stage, we perform a maximum likelihood (ML) estimation by
assuming that $y_{i}$ follows a Poisson distribution with mean
$\exp(\boldsymbol{\hat{\beta}}_{n,i}^{\prime}\boldsymbol{\theta})$, where
$\boldsymbol{\hat{\beta}}_{n,i}=(1,\leavevmode\nobreak\ \hat{p}_{i})$. The
objective function is thus given by $Q_{n}(\theta_{0},\leavevmode\nobreak\
\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})=\frac{1}{n}\sum_{i=1}^{n}\big{(}y_{i}\boldsymbol{\hat{\beta}}_{n,i}^{\prime}\boldsymbol{\theta}-\exp(\boldsymbol{\hat{\beta}}_{n,i}^{\prime}\boldsymbol{\theta})\big{)}$
and $\boldsymbol{\dot{q}}_{n}(\mathbf{y}_{n},\leavevmode\nobreak\
\mathbf{\hat{B}}_{n})=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\big{(}y_{i}-\exp(\boldsymbol{\hat{\beta}}_{n,i}^{\prime}\boldsymbol{\theta}_{0})\big{)}\boldsymbol{\hat{\beta}}_{n,i}$.
Therefore,

$\displaystyle\textstyle\mathbf{\hat{A}}_{n}=\frac{1}{n}\sum_{i=1}^{n}\exp\big{(}\boldsymbol{\hat{\beta}}_{n,i}^{\prime}\boldsymbol{\hat{\theta}}_{n}\big{)}\boldsymbol{\hat{\beta}}_{n,i}\boldsymbol{\hat{\beta}}_{n,i}^{\prime}\,,\quad\hat{\mathbf{V}}_{n}=\frac{1}{n}\sum_{i=1}^{n}\exp\big{(}\boldsymbol{\hat{\beta}}_{n,i}^{\prime}\boldsymbol{\hat{\theta}}_{n}\big{)}\boldsymbol{\hat{\beta}}_{n,i}\boldsymbol{\hat{\beta}}_{n,i}^{\prime},\quad\text{and}$
$\displaystyle\textstyle\hat{\mathcal{E}}_{n,s}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\big{(}\exp(\boldsymbol{\hat{\beta}}_{n,i}^{(s)^{\prime}}\boldsymbol{\hat{\theta}}_{n})-\exp(\boldsymbol{\hat{\beta}}_{n,i}^{(s)\prime}\boldsymbol{\hat{\theta}}_{n})\big{)}\boldsymbol{\hat{\beta}}_{n,i}^{(s)},$

where $\boldsymbol{\hat{\beta}}_{n,i}^{(s)}=h(z_{i},\leavevmode\nobreak\
\boldsymbol{\hat{\gamma}}_{n}^{(s)})$, for $s=1,\leavevmode\nobreak\
\dots,\leavevmode\nobreak\ \kappa$, and
$\boldsymbol{\hat{\gamma}}_{n}^{(1)}\leavevmode\nobreak\
\dots,\leavevmode\nobreak\
\boldsymbol{\hat{\gamma}}_{n}^{(\kappa)}\overset{i.i.d}{\sim}N(\boldsymbol{\hat{\gamma}}_{n},\leavevmode\nobreak\
\hat{\mathbb{V}}(\boldsymbol{\hat{\gamma}}_{n}))$.

Let
$\boldsymbol{\hat{\psi}}_{n,s}=\mathbf{\hat{A}}_{n}^{-1}\hat{\mathbf{V}}_{n}^{1/2}\boldsymbol{\zeta}_{s}+\mathbf{\hat{A}}_{n}^{-1}\hat{\mathcal{E}}_{n,s}$,
where $\boldsymbol{\zeta}_{1}$, …,
$\boldsymbol{\zeta}_{\kappa}\overset{i.i.d}{\sim}N(0,\leavevmode\nobreak\
\boldsymbol{I}_{K_{\theta}})$. The asymptotic CDF of
$\sqrt{n}(\boldsymbol{\hat{\theta}}_{n}-\boldsymbol{\theta}_{0})$ can be
estimated by the empirical CDF of the sample:
$\\{\boldsymbol{\hat{\psi}}_{n,s},\leavevmode\nobreak\
s=1,\leavevmode\nobreak\ \dots,\leavevmode\nobreak\ \kappa\\}$. The asymptotic
variance of $\boldsymbol{\hat{\theta}}_{n}$ is estimated by

$\hat{\mathbb{V}}(\boldsymbol{\hat{\theta}}_{n})=\displaystyle\frac{\mathbf{\hat{A}}_{n}^{-1}\boldsymbol{\hat{\Sigma}}_{n}^{\kappa}\mathbf{\hat{A}}_{n}^{-1}}{n},$

where
$\boldsymbol{\hat{\Sigma}}_{n}^{\kappa}=\hat{\mathbf{V}}_{n}+\frac{1}{\kappa-1}\sum_{s=1}^{\kappa}(\hat{\mathcal{E}}_{n,s}-\boldsymbol{\hat{\Omega}}_{n}^{\kappa})(\hat{\mathcal{E}}_{n,s}-\boldsymbol{\hat{\Omega}}_{n}^{\kappa})^{\prime}$
and
$\boldsymbol{\hat{\Omega}}_{n}^{\kappa}=\frac{1}{\kappa}\sum_{s=1}^{\kappa}\hat{\mathcal{E}}_{n,s}$.
The debiased estimator is given by

$\boldsymbol{\theta}_{n,\kappa}^{\ast}=\boldsymbol{\hat{\theta}}_{n}-\mathbf{\hat{A}}_{n}^{-1}\boldsymbol{\hat{\Omega}}_{n}^{\kappa}/\sqrt{n}.$

Let
$\boldsymbol{\hat{\psi}}_{n,s}^{\ast}=(\mathbf{\hat{A}}_{n}^{\ast})^{-1}(\mathbf{\hat{V}}_{n}^{\ast})^{1/2}\boldsymbol{\zeta}_{s}+(\mathbf{\hat{A}}_{n}^{\ast})^{-1}(\hat{\mathcal{E}}_{n,s}^{\ast}-\boldsymbol{\hat{\Omega}}_{n}^{\ast\kappa})$,
where $\mathbf{\hat{A}}_{n}^{\ast}$, $\mathbf{\hat{V}}_{n}^{\ast}$,
$\hat{\mathcal{E}}_{n,s}^{\ast}$, and
$\boldsymbol{\hat{\Omega}}_{n}^{\ast\kappa}$ are respectively defined as
$\mathbf{\hat{A}}_{n}$, $\mathbf{\hat{V}}_{n}$, $\hat{\mathcal{E}}_{n,s}$, and
$\boldsymbol{\hat{\Omega}}_{n}^{\kappa}$, with the difference that they are
computed using $\boldsymbol{\theta}_{n,\kappa}^{\ast}$ and not
$\boldsymbol{\hat{\theta}}_{n}$. We can estimate the asymptotic CDF of
$\sqrt{n}(\boldsymbol{\hat{\theta}}_{n}^{\ast}-\boldsymbol{\theta}_{0})$ by
the empirical CDF of the sample:
$\\{\boldsymbol{\hat{\psi}}_{n,s}^{\ast},\leavevmode\nobreak\
s=1,\leavevmode\nobreak\ \dots,\leavevmode\nobreak\ \kappa\\}$.

##### DGP D

DGP D is a copula-based multivariate time-series model. We consider $k_{n}$
returns $y_{1,i}$, …, $y_{k_{n},i}$, where $i$ is time and $k_{n}\geq 2$. Each
$y_{p,i}$, for $p=2,\leavevmode\nobreak\ \dots,\leavevmode\nobreak\ k_{n}$,
follows an AR(1)-GARCH(1, 1) model, such that

$y_{p,i}=\phi_{p,0}+\phi_{p,1}y_{p,i-1}+\sigma_{p,i}\varepsilon_{p,i},\quad\sigma_{p,i}^{2}=\beta_{p,0}+\beta_{p,1}\sigma_{p,i-1}^{2}\varepsilon_{p,i-1}^{2}+\beta_{p,2}\sigma_{p,i-1}^{2},$

where $\phi_{p,0}=0$, $\phi_{p,i-1}=0.4$, $\beta_{p,0}=0.05$,
$\beta_{p,1}=0.05$, $\beta_{p,2}=0.9$, and $\varepsilon_{p,i}$ follows a
standardized Student distribution of degree-of-freedom $\nu_{p}=6$. We account
for the correlation between the returns using the Clayton copula. The joint
density function of $y_{i}=(y_{1,i},\leavevmode\nobreak\
\dots,\leavevmode\nobreak\ y_{p,i})^{\prime}$ conditional on
$\mathcal{F}^{i-1}$ is given by
$c_{i}(G_{1,i}(\boldsymbol{\beta}_{0,1}),\leavevmode\nobreak\
\dots,\leavevmode\nobreak\
G_{k_{n},i}(\boldsymbol{\beta}_{0,k_{n}}),\leavevmode\nobreak\ \theta_{0})$,
where $\boldsymbol{\beta}_{0,p}=(\phi_{p,0},\leavevmode\nobreak\
\phi_{p,1},\leavevmode\nobreak\ \beta_{p,0},\leavevmode\nobreak\
\beta_{p,1},\leavevmode\nobreak\ \beta_{p,2},\leavevmode\nobreak\
\nu_{p})^{\prime}$, $G_{p,i}(\boldsymbol{\beta}_{0,p})$ is the CDF of
$y_{p,i}$ conditional on $\mathcal{F}^{i-1}$, and $c_{i}$ is the probability
density function (PDF) of $k_{n}$-dimensional Clayton copula of parameter
$\theta_{0}=4$. A multi-stage estimation strategy can be used to estimate
$\theta_{0}$. In the first $k_{n}$ stages, we separately estimate each
$\boldsymbol{\beta}_{0,p}$ by applying an AR(1)-GARCH(1, 1) model to the
sample $y_{p,1},\leavevmode\nobreak\ \dots,\leavevmode\nobreak\ y_{p,n}$.

$\textstyle\boldsymbol{\hat{\beta}}_{n,p}=\arg\max_{\boldsymbol{\beta}_{p}}\ell_{p}:=\frac{1}{n}\sum_{i=1}^{n}\underbrace{\log
g_{p,i}(\boldsymbol{\beta}_{p})}_{\ell_{p,i}}\,,\,\,\text{ for
}\,p=1,\,\dots,\,k_{n}\,,$

where $g_{p,i}(\boldsymbol{\beta}_{0,p})$ is the PDF of $y_{p,i}$ conditional
on $\mathcal{F}^{i-1}$. Let $\boldsymbol{\hat{\beta}}_{n}$ be the estimator
$\boldsymbol{\beta}_{0}:=(\boldsymbol{\beta}_{0,1}^{\prime},\leavevmode\nobreak\
\dots,\leavevmode\nobreak\ \boldsymbol{\beta}_{0,k_{n}}^{\prime})^{\prime}$.
The estimator of the asymptotic distribution of $\boldsymbol{\hat{\beta}}_{n}$
is a normal distribution with mean $\boldsymbol{\hat{\beta}}_{n}$ and variance
given by
$\hat{\mathbb{V}}(\boldsymbol{\hat{\beta}}_{n})=\frac{1}{n}\mathbf{\hat{H}}_{n}^{-1}\mathbf{\hat{J}}_{n}\mathbf{\hat{H}}_{n}^{-1}$,
where

$\textstyle\mathbf{\hat{H}}_{n}=\frac{1}{n}\sum_{i=1}^{n}\operatorname{diag}(\frac{\partial^{2}}{\partial\boldsymbol{\beta}^{\prime}\partial\boldsymbol{\beta}}\ell_{1,i},\,\dots,\,\frac{\partial^{2}}{\partial\boldsymbol{\beta}^{\prime}\partial\boldsymbol{\beta}}\ell_{k_{n},i})\quad\text{and}\quad\textstyle\mathbf{\hat{J}}_{n}=\mathbb{V}_{\text{HAC}}(\frac{1}{\sqrt{n}}\sum_{i=1}^{n}(\partial_{\boldsymbol{\beta}^{\prime}}\ell_{1,i},\,\dots,\,\partial_{\boldsymbol{\beta}^{\prime}}\ell_{k_{n},i})^{\prime})\,.$

The notation $\mathbb{V}_{\text{HAC}}$ is the heteroskedasticity and
autocorrelation consistent (HAC) covariance matrix to account for the serial
correlation.333See ANDREWS, D. W. (1991): ”Heteroskedasticity and
autocorrelation consistent covariance matrix estimation,” Econometrica:
Journal of the Econometric Society, 59, 817–858. In the HAC approach, we use
the quadratic spectral kernel and set the bandwidth to $\frac{3}{4}n^{1/3}$.
The gradient and the Hessian of the likelihood $\ell_{p,i}$ do not have a
closed form. Fortunately, they can be approximated numerically in most
statistical software.

In the last stage, we estimate $\theta_{0}$ by ML after replacing
$\boldsymbol{\beta}_{0}$ in the density function of $y_{i}$ with
$\boldsymbol{\hat{\beta}}_{n}$. Let $q_{n,i}(\theta,\leavevmode\nobreak\
\boldsymbol{\hat{\beta}}_{n})=\log\big{(}c_{i}(G_{1,i}(\boldsymbol{\hat{\beta}}_{n,1}),\,\dots,\,G_{k_{n},i}(\boldsymbol{\hat{\beta}}_{n,k_{n}}),\leavevmode\nobreak\
\theta)\big{)}$, where $\log c_{i}(u_{1},\leavevmode\nobreak\
\dots,\leavevmode\nobreak\
u_{k_{n}},\theta)=\sum_{p=1}^{k_{n}-1}\log(p\theta+1)-(\theta+1)\sum_{p=1}^{k_{n}}\log
u_{p}-(k_{n}+\tfrac{1}{\theta})\log(\sum_{p=1}^{k_{n}}u_{p}^{-\theta}-k_{n}+1)$
. The objective function is $Q_{n}(\theta,\leavevmode\nobreak\
\boldsymbol{\hat{\beta}}_{n})=\frac{1}{n}\sum_{i=1}^{n}q_{n,i}(\theta,\leavevmode\nobreak\
\boldsymbol{\hat{\beta}}_{n})$. To compute
$\partial_{\theta}q_{n,i}(\theta_{0},\leavevmode\nobreak\
\boldsymbol{\hat{\beta}}_{n})$ and
$\frac{\partial^{2}}{\partial\theta^{2}}q_{n,i}(\theta_{0},\leavevmode\nobreak\
\boldsymbol{\hat{\beta}}_{n})$, we need the first and second derivatives of
$\log c_{i}(u_{1},\leavevmode\nobreak\ \dots,\leavevmode\nobreak\
u_{k_{n}},\theta)$ that can be expressed as follows:

$\displaystyle\textstyle\partial_{\theta}\log c_{i}(u_{1},\leavevmode\nobreak\
\dots,\leavevmode\nobreak\
u_{k_{n}},\theta)=\sum_{p=1}^{k_{n}-1}\frac{p}{p\theta+1}-\sum_{p=1}^{k_{n}}\log
u_{p}+\frac{\log\left(\sum_{p=1}^{k_{n}}u_{p}^{-\theta}-k_{n}+1\right)}{\theta^{2}}+\frac{(k_{n}+\tfrac{1}{\theta})\sum_{p=1}^{k_{n}}u_{p}^{-\theta}\log
u_{p}}{\sum_{p=1}^{k_{n}}u_{p}^{-\theta}-k_{n}+1}$
$\displaystyle\textstyle\frac{\partial^{2}}{\partial\theta^{2}}\log
c_{i}(u_{1},\leavevmode\nobreak\ \dots,\leavevmode\nobreak\
u_{k_{n}},\theta)=\frac{(k_{n}+\tfrac{1}{\theta})\left(\sum_{p=1}^{k_{n}}u_{p}^{-\theta}\log
u_{p}\right)^{2}}{\left(\sum_{p=1}^{k_{n}}u_{p}^{-\theta}-k_{n}+1\right)^{2}}-\frac{(k_{n}+\tfrac{1}{\theta})\sum_{p=1}^{k_{n}}u_{p}^{-\theta}\left(\log
u_{p}\right)^{2}}{\sum_{p=1}^{k_{n}}u_{p}^{-\theta}-k_{n}+1}-\sum_{p=1}^{k_{n}}\left(\frac{p}{p\theta+1}\right)^{2}-$
$\displaystyle\quad\quad\textstyle\frac{2\sum_{p=1}^{k_{n}}u_{p}^{-\theta}\log
u_{p}}{\theta^{2}\left(\sum_{p=1}^{k_{n}}u_{p}^{-\theta}-k_{n}+1\right)}-\frac{2\log\left(\sum_{p=1}^{k_{n}}u_{p}^{-\theta}-k_{n}+1\right)}{\theta^{3}}\,.$

We define the following expressions:
$\hat{A}_{n}=\frac{1}{n}\sum_{i=1}^{n}\frac{\partial^{2}}{\partial\theta}q_{n,i}(\hat{\theta}_{n},\leavevmode\nobreak\
\boldsymbol{\hat{\beta}}_{n})$,
$\hat{V}_{n}=\mathbb{V}_{\text{HAC}}(\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\partial_{\theta}q_{n,i}(\hat{\theta}_{n},\boldsymbol{\hat{\beta}}_{n}))$,
and
$\quad\hat{\mathcal{E}}_{n,s}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\partial_{\theta}q_{n,i}(\hat{\theta}_{n},\boldsymbol{\hat{\beta}}_{n}^{(s)})$,
where $\boldsymbol{\hat{\beta}}_{n}^{(1)},\leavevmode\nobreak\
\dots,\leavevmode\nobreak\
\boldsymbol{\hat{\beta}}_{n}^{(\kappa)}\overset{i.i.d}{\sim}N(\boldsymbol{\hat{\beta}}_{n},\leavevmode\nobreak\
\hat{\mathbb{V}}(\boldsymbol{\hat{\beta}}_{n}))$. Let
$\hat{\psi}_{n,s}=\frac{\sqrt{\hat{V}_{n}}\zeta_{s}+\hat{\mathcal{E}}_{n,s}}{\hat{A}_{n}}$,
where $\zeta_{1}$, …, $\zeta_{\kappa}$ are independent variables from
$N(0,1)$. The asymptotic CDF of $\sqrt{n}(\hat{\theta}_{n}-\theta_{0})$ can be
estimated by the empirical CDF of the sample:
$\\{\hat{\psi}_{n,s},\leavevmode\nobreak\ s=1,\leavevmode\nobreak\
\dots,\leavevmode\nobreak\ \kappa\\}$. The estimator of the asymptotic
variance of $\hat{\theta}_{n}$ is

$\hat{\mathbb{V}}(\hat{\theta}_{n})=\textstyle\frac{1}{n\hat{A}_{n}^{2}}(\hat{V}_{n}+\frac{1}{\kappa-1}\sum_{s=1}^{\kappa}(\hat{\mathcal{E}}_{n,s}-\hat{\mathbb{E}}(\mathcal{E}_{n}))^{2}),$

where
$\hat{\mathbb{E}}(\mathcal{E}_{n})=\frac{1}{\kappa}\sum_{s=1}^{\kappa}\hat{\mathcal{E}}_{n,s}$.
The debiased estimator is given by

$\theta_{n,\kappa}^{\ast}=\hat{\theta}_{n}-\hat{\mathbb{E}}(\mathcal{E}_{n})/(\sqrt{n}\hat{A}_{n}).$

Let
$\hat{\psi}_{n,s}^{\ast}=\frac{\sqrt{\hat{V}_{n}^{\ast}}\zeta_{s}+\hat{\mathcal{E}}_{n,s}^{\ast}-\hat{\mathbb{E}}(\mathcal{E}_{n}^{\ast})}{\hat{A}_{n}^{\ast}}$,
where $\hat{A}_{n}^{\ast}$, $\hat{V}_{n}^{\ast}$,
$\hat{\mathcal{E}}_{n,s}^{\ast}$, and
$\hat{\mathbb{E}}(\mathcal{E}_{n}^{\ast})$ are defined as $\hat{A}_{n}$,
$\hat{V}_{n}$, $\hat{\mathcal{E}}_{n,s}$, and
$\hat{\mathbb{E}}(\mathcal{E}_{n})$, respectively, with the difference that
they are computed using $\theta_{n,\kappa}^{\ast}$ and not $\hat{\theta}_{n}$.
We can estimate the asymptotic CDF of
$\sqrt{n}(\theta_{n,\kappa}^{\ast}-\theta_{0})$ by the empirical CDF of the
sample: $\\{\hat{\psi}_{n,s},\leavevmode\nobreak\ s=1,\leavevmode\nobreak\
\dots,\leavevmode\nobreak\ \kappa\\}$.

###  S.2.2 Estimates of the Asymptotic Distribution of the Debiased Estimators

This section presents the estimates of the asymptotic CDF of
$\Delta_{n,\kappa}^{\ast}:=\sqrt{n}(\boldsymbol{\theta}_{n,\kappa}^{\ast}-\theta_{0})$,
where $\boldsymbol{\theta}_{n,\kappa}^{\ast}$ is the debiased estimator. In
contrast to the case of the classical plug-in estimator, the true sampling
CDFs are asymptotically centered at zero because
$\mathbb{E}(\Delta_{n,\kappa}^{\ast})$ converges to zero asymptotically.
Overall, the results demonstrate that the estimator of the CDF of
$\Delta_{n,\kappa}^{\ast}$, as outlined in Theorem 4.3, performs well.

DGP A

DGP B

Figure S.5: Monte Carlo Simulations: Estimates of asymptotic CDFs (DGPs A and
B)

This figure displays average estimates of the asymptotic CDF of
$\sqrt{n}(\boldsymbol{\theta}_{n,\kappa}^{\ast}-\theta_{0})$ for DGPs A and B.
$F_{0}^{\ast}$ represents the true sampling CDF whereas $\hat{F}_{n}^{\ast}$
corresponds to the average estimate of the CDF using our simulation approach.
The $L_{1}$-Wasserstein distance between each estimated CDF and $F_{0}^{\ast}$
is enclosed in parentheses.

DGP C

DGP D: Mean Correction

DGP D: Median Correction

Figure S.6: Monte Carlo Simulations: Estimates of asymptotic CDFs (DGPs C and
D)

This figure displays average estimates of the asymptotic CDF of
$\sqrt{n}(\boldsymbol{\theta}_{n,\kappa}^{\ast}-\theta_{0})$ for DGPs A and B.
$F_{0}^{\ast}$ represents the true sampling CDF whereas $\hat{F}_{n}^{\ast}$
corresponds to the average estimate of the CDF using our simulation approach.
The $L_{1}$-Wasserstein distance between each estimated CDF and $F_{0}^{\ast}$
is enclosed in parentheses.

##  Appendix S.3 Supplementary Materials on the Application

###  S.3.1 Data Summary

Our dependent variable is the weekly fast-food consumption frequency, measured
by the reported frequency (in days) of fast-food restaurant visits in the past
week. We control for 25 observable characteristics in $\mathbf{X}_{r}$ such as
students’ gender, grade, race, weekly allowance, and parent’s education and
occupation. On average, students report consuming fast food 2.35 days per
week. (see Table S.1).

Table S.1: Data Summary

Statistic | Mean | St. Dev. | Min | Max
---|---|---|---|---
Fast food consumption | $2.353$ | $1.762$ | $0$ | $7$
Female | $0.501$ | $0.500$ | $0$ | $1$
Age | $16.628$ | $1.554$ | $12$ | $21$
Hispanic | $0.200$ | $0.400$ | $0$ | $1$
Grade 7–8 | $0.100$ | $0.300$ | $0$ | $1$
Grade 9–10 | $0.230$ | $0.421$ | $0$ | $1$
Grade 11–12 | $0.533$ | $0.499$ | $0$ | $1$
Race (White) |  |  |  |
| Black | $0.142$ | $0.349$ | $0$ | $1$
| Asian | $0.138$ | $0.345$ | $0$ | $1$
| Other | $0.117$ | $0.321$ | $0$ | $1$
With parents | $0.675$ | $0.468$ | $0$ | $1$
Allowance per week | $7.893$ | $11.609$ | $0$ | $95$
Mother Education (High school) |  |  |  |
|  $<$ High school | $0.146$ | $0.353$ | $0$ | $1$
|  $>$ High school and not graduated | $0.173$ | $0.378$ | $0$ | $1$
|  $>$ High school and graduated | $0.226$ | $0.418$ | $0$ | $1$
| Missing | $0.124$ | $0.330$ | $0$ | $1$
Father Education (High school) |  |  |  |
|  $<$ High school | $0.124$ | $0.329$ | $0$ | $1$
|  $>$ High school and not graduated | $0.137$ | $0.344$ | $0$ | $1$
|  $>$ High school and graduated | $0.202$ | $0.402$ | $0$ | $1$
| Missing | $0.284$ | $0.451$ | $0$ | $1$
Mother Job (None) |  |  |  |
| Professional | $0.157$ | $0.364$ | $0$ | $1$
| Other | $0.623$ | $0.485$ | $0$ | $1$
| Missing | $0.088$ | $0.283$ | $0$ | $1$
Father Job (None) |  |  |  |
| Professional | $0.053$ | $0.223$ | $0$ | $1$
| Other | $0.663$ | $0.473$ | $0$ | $1$
| Missing | $0.240$ | $0.427$ | $0$ | $1$

  * This table presents the mean, standard deviation (St. Dev.), minimum, and maximum of the variables used in the empirical application. For the categorical explanatory variables, the level in parentheses is set as the reference level. "With parents" is a dummy variable taking 1 if the student lives with their mother and father.

###  S.3.2 Estimation and Inference

The following table displays the full results of estimations of the peer
effect model.

Table S.2: Estimation Results: OLS approach

Fixed Effects No Yes Coef Sd. Err Coef Sd. Err Peer effects: $\theta_{0,1}$
$0.192$ $0.031$ $0.150$ $0.032$ Individual characteristics:
$\boldsymbol{\theta}_{0,2}$ Female $-0.158$ $0.075$ $-0.161$ $0.074$ Age
$0.109$ $0.040$ $0.083$ $0.040$ Hispanic $0.289$ $0.130$ $0.103$ $0.147$ Grade
7–8 $-0.033$ $0.257$ $0.102$ $0.260$ Grade 9–10 $0.020$ $0.168$ $0.020$
$0.169$ Grade 11–12 $0.258$ $0.116$ $0.205$ $0.117$ Race (White) Black $0.075$
$0.145$ $-0.085$ $0.167$ Asian $0.302$ $0.144$ $0.134$ $0.167$ Other $-0.118$
$0.138$ $-0.145$ $0.139$ With parents $-0.001$ $0.134$ $0.002$ $0.134$
Allowance per week $0.008$ $0.003$ $0.006$ $0.003$ Mother Education (High
school) $<$ High school $0.101$ $0.119$ $0.084$ $0.119$ $>$ High school and
non graduated $0.054$ $0.100$ $0.023$ $0.100$ $>$ High school and graduated
$0.132$ $0.106$ $0.086$ $0.106$ Missing $-0.035$ $0.191$ $-0.103$ $0.190$
Father Education (High school) $<$ High school $-0.251$ $0.128$ $-0.244$
$0.127$ $>$ High school and non graduated $-0.100$ $0.112$ $-0.112$ $0.112$
$>$ High school and graduated $-0.005$ $0.109$ $-0.038$ $0.110$ Missing
$0.016$ $0.186$ $-0.024$ $0.186$ Mother Job (None) Professional $0.015$
$0.133$ $-0.007$ $0.133$ Other $0.065$ $0.101$ $0.061$ $0.101$ Missing $0.376$
$0.227$ $0.441$ $0.226$ Father Job (None) Professional $-0.247$ $0.224$
$-0.281$ $0.223$ Other $-0.230$ $0.165$ $-0.234$ $0.164$ Missing $-0.252$
$0.250$ $-0.228$ $0.250$ Contextual peer effects: $\boldsymbol{\theta}_{0,3}$
Female $0.044$ $0.120$ $-0.001$ $0.121$ Age $-0.030$ $0.022$ $-0.008$ $0.023$
Hispanic $-0.091$ $0.195$ $-0.189$ $0.203$ Grade 7–8 $-0.268$ $0.274$ $-0.281$
$0.277$ Grade 9–10 $-0.189$ $0.209$ $-0.107$ $0.209$ Grade 11–12 $-0.028$
$0.189$ $0.028$ $0.188$ Race (White) Black $0.165$ $0.198$ $0.129$ $0.206$
Asian $-0.118$ $0.187$ $-0.122$ $0.196$ Other $-0.511$ $0.228$ $-0.470$
$0.228$ With parents $-0.199$ $0.216$ $-0.230$ $0.218$ Allowance per week
$0.006$ $0.005$ $0.003$ $0.005$ Mother Education (High school) $<$ High school
$0.417$ $0.196$ $0.359$ $0.196$ $>$ High school and non graduated $-0.163$
$0.170$ $-0.226$ $0.171$ $>$ High school and graduated $-0.065$ $0.178$
$-0.189$ $0.180$ Missing $-0.288$ $0.373$ $-0.386$ $0.372$ Father Education
(High school) $<$ High school $-0.090$ $0.216$ $-0.111$ $0.217$ $>$ High
school and non graduated $0.140$ $0.176$ $0.084$ $0.179$ $>$ High school and
graduated $0.093$ $0.177$ $0.052$ $0.182$ Missing $0.289$ $0.324$ $0.278$
$0.324$ Mother Job (None) Professional $-0.217$ $0.223$ $-0.246$ $0.224$ Other
$-0.249$ $0.171$ $-0.272$ $0.172$ Missing $-0.193$ $0.427$ $-0.105$ $0.428$
Father Job (None) Professional $0.484$ $0.370$ $0.111$ $0.376$ Other $0.368$
$0.268$ $0.238$ $0.274$ Missing $0.044$ $0.424$ $-0.116$ $0.430$ For the
categorical variables, the level in parentheses is set as the reference level.

Table S.3: Estimation Results: Classical and Optimal GMM approaches

Model CIV CIV OIV OIV Fixed Effects No Yes No Yes Coef Sd. Err Coef Sd. Err
Coef Sd. Err Coef Sd. Err Peer effects: $\theta_{0,1}$ $0.149$ $0.160$ $0.081$
$0.169$ $-0.065$ $0.287$ $0.016$ $0.208$ Individual characteristics:
$\boldsymbol{\theta}_{0,2}$ Female $-0.156$ $0.075$ $-0.158$ $0.075$ $-0.145$
$0.077$ $-0.154$ $0.075$ Age $0.111$ $0.040$ $0.085$ $0.040$ $0.121$ $0.042$
$0.087$ $0.041$ Hispanic $0.288$ $0.130$ $0.095$ $0.148$ $0.284$ $0.132$
$0.088$ $0.149$ Grade 7–8 $-0.033$ $0.258$ $0.105$ $0.260$ $-0.035$ $0.261$
$0.108$ $0.261$ Grade 9–10 $0.019$ $0.168$ $0.018$ $0.169$ $0.017$ $0.171$
$0.016$ $0.169$ Grade 11–12 $0.258$ $0.116$ $0.203$ $0.117$ $0.261$ $0.118$
$0.201$ $0.117$ Race (White) Black $0.073$ $0.145$ $-0.091$ $0.168$ $0.066$
$0.147$ $-0.097$ $0.168$ Asian $0.305$ $0.144$ $0.133$ $0.167$ $0.318$ $0.147$
$0.132$ $0.167$ Other $-0.123$ $0.139$ $-0.153$ $0.141$ $-0.144$ $0.142$
$-0.160$ $0.142$ With parents $-0.008$ $0.136$ $-0.006$ $0.135$ $-0.042$
$0.143$ $-0.015$ $0.137$ Allowance per week $0.008$ $0.003$ $0.006$ $0.003$
$0.008$ $0.003$ $0.006$ $0.003$ Mother Education (High school) $<$ High school
$0.102$ $0.120$ $0.085$ $0.119$ $0.110$ $0.121$ $0.087$ $0.119$ $>$ High
school and not graduated $0.057$ $0.100$ $0.025$ $0.100$ $0.070$ $0.103$
$0.026$ $0.101$ $>$ High school and graduated $0.131$ $0.106$ $0.082$ $0.107$
$0.128$ $0.108$ $0.079$ $0.107$ Missing $-0.045$ $0.194$ $-0.121$ $0.195$
$-0.094$ $0.204$ $-0.138$ $0.198$ Father Education (High school) $<$ High
school $-0.252$ $0.128$ $-0.246$ $0.127$ $-0.260$ $0.130$ $-0.247$ $0.128$ $>$
High school and not graduated $-0.100$ $0.112$ $-0.113$ $0.112$ $-0.101$
$0.113$ $-0.114$ $0.112$ $>$ High school and graduated $-0.006$ $0.109$
$-0.040$ $0.111$ $-0.011$ $0.111$ $-0.042$ $0.111$ Missing $0.019$ $0.187$
$-0.020$ $0.186$ $0.037$ $0.190$ $-0.017$ $0.186$ Mother Job (None)
Professional $0.011$ $0.134$ $-0.014$ $0.134$ $-0.009$ $0.137$ $-0.021$
$0.135$ Other $0.066$ $0.101$ $0.061$ $0.101$ $0.069$ $0.103$ $0.061$ $0.102$
Missing $0.381$ $0.228$ $0.452$ $0.228$ $0.404$ $0.232$ $0.463$ $0.229$ Father
Job (None) Professional $-0.242$ $0.224$ $-0.278$ $0.224$ $-0.219$ $0.228$
$-0.274$ $0.224$ Other $-0.225$ $0.165$ $-0.228$ $0.165$ $-0.203$ $0.169$
$-0.222$ $0.166$ Missing $-0.255$ $0.251$ $-0.230$ $0.250$ $-0.267$ $0.254$
$-0.231$ $0.251$ Contextual peer effects: $\boldsymbol{\theta}_{0,3}$ Female
$0.037$ $0.123$ $-0.013$ $0.124$ $0.003$ $0.130$ $-0.024$ $0.126$ Age $-0.025$
$0.029$ $0.001$ $0.031$ $0.000$ $0.040$ $0.009$ $0.034$ Hispanic $-0.075$
$0.203$ $-0.169$ $0.209$ $0.002$ $0.222$ $-0.150$ $0.213$ Grade 7–8 $-0.264$
$0.274$ $-0.273$ $0.278$ $-0.246$ $0.278$ $-0.266$ $0.279$ Grade 9–10 $-0.186$
$0.209$ $-0.099$ $0.210$ $-0.174$ $0.212$ $-0.092$ $0.211$ Grade 11–12
$-0.011$ $0.199$ $0.056$ $0.201$ $0.073$ $0.222$ $0.083$ $0.207$ Race (White)
Black $0.172$ $0.199$ $0.141$ $0.208$ $0.205$ $0.205$ $0.153$ $0.209$ Asian
$-0.106$ $0.192$ $-0.101$ $0.202$ $-0.047$ $0.205$ $-0.082$ $0.206$ Other
$-0.519$ $0.230$ $-0.483$ $0.230$ $-0.560$ $0.237$ $-0.495$ $0.232$ With
parents $-0.212$ $0.221$ $-0.247$ $0.222$ $-0.276$ $0.235$ $-0.264$ $0.224$
Allowance per week $0.007$ $0.006$ $0.004$ $0.005$ $0.009$ $0.006$ $0.004$
$0.006$ Mother Education (High school) $<$ High school $0.420$ $0.197$ $0.359$
$0.196$ $0.438$ $0.200$ $0.359$ $0.197$ $>$ High school and not graduated
$-0.159$ $0.170$ $-0.223$ $0.172$ $-0.139$ $0.174$ $-0.220$ $0.172$ $>$ High
school and graduated $-0.053$ $0.184$ $-0.178$ $0.182$ $0.010$ $0.199$
$-0.168$ $0.184$ Missing $-0.264$ $0.383$ $-0.351$ $0.382$ $-0.146$ $0.409$
$-0.319$ $0.387$ Father Education (High school) $<$ High school $-0.112$
$0.231$ $-0.144$ $0.231$ $-0.223$ $0.264$ $-0.175$ $0.239$ $>$ High school and
not graduated $0.124$ $0.186$ $0.054$ $0.193$ $0.043$ $0.209$ $0.025$ $0.201$
$>$ High school and graduated $0.082$ $0.181$ $0.035$ $0.186$ $0.029$ $0.192$
$0.020$ $0.189$ Missing $0.282$ $0.325$ $0.265$ $0.326$ $0.251$ $0.331$
$0.254$ $0.327$ Mother Job (None) Professional $-0.225$ $0.225$ $-0.262$
$0.227$ $-0.265$ $0.232$ $-0.276$ $0.229$ Other $-0.245$ $0.172$ $-0.268$
$0.173$ $-0.227$ $0.175$ $-0.265$ $0.173$ Missing $-0.209$ $0.431$ $-0.125$
$0.431$ $-0.286$ $0.445$ $-0.144$ $0.433$ Father Job (None) Professional
$0.501$ $0.375$ $0.119$ $0.377$ $0.587$ $0.392$ $0.125$ $0.378$ Other $0.382$
$0.273$ $0.256$ $0.277$ $0.452$ $0.287$ $0.273$ $0.280$ Missing $0.061$
$0.429$ $-0.091$ $0.434$ $0.146$ $0.444$ $-0.067$ $0.438$ For the categorical
variables, the level in parentheses is set as the reference level.

Table S.4: Estimation Results: Many Instrument Approaches

Model IV-MI IV-MI DIV-MI DIV-MI Fixed Effects No Yes No Yes Coef Sd. Err Coef
Sd. Err Coef Sd. Err Coef Sd. Err Peer effects: $\theta_{0,1}$ $0.276$ $0.063$
$0.208$ $0.067$ $0.300$ $0.063$ $0.218$ $0.067$ Individual characteristics:
$\boldsymbol{\theta}_{0,2}$ Female $-0.162$ $0.072$ $-0.164$ $0.070$ $-0.163$
$0.072$ $-0.165$ $0.070$ Age $0.105$ $0.038$ $0.082$ $0.038$ $0.104$ $0.038$
$0.082$ $0.038$ Hispanic $0.290$ $0.126$ $0.109$ $0.135$ $0.290$ $0.126$
$0.110$ $0.135$ Grade 7–8 $-0.032$ $0.252$ $0.100$ $0.241$ $-0.030$ $0.252$
$0.102$ $0.241$ Grade 9–10 $0.021$ $0.157$ $0.021$ $0.154$ $0.020$ $0.157$
$0.023$ $0.154$ Grade 11–12 $0.257$ $0.118$ $0.207$ $0.119$ $0.255$ $0.118$
$0.209$ $0.119$ Race (White) Black $0.078$ $0.147$ $-0.080$ $0.158$ $0.080$
$0.147$ $-0.078$ $0.158$ Asian $0.297$ $0.134$ $0.135$ $0.151$ $0.296$ $0.134$
$0.136$ $0.151$ Other $-0.110$ $0.129$ $-0.138$ $0.128$ $-0.108$ $0.129$
$-0.135$ $0.128$ With parents $0.012$ $0.128$ $0.010$ $0.127$ $0.016$ $0.129$
$0.011$ $0.128$ Allowance per week $0.008$ $0.003$ $0.006$ $0.003$ $0.008$
$0.003$ $0.006$ $0.003$ Mother Education (High school) $<$ High school $0.098$
$0.113$ $0.082$ $0.114$ $0.097$ $0.114$ $0.082$ $0.114$ $>$ High school and
not graduated $0.048$ $0.093$ $0.022$ $0.093$ $0.047$ $0.094$ $0.021$ $0.093$
$>$ High school and graduated $0.133$ $0.095$ $0.089$ $0.094$ $0.134$ $0.095$
$0.089$ $0.094$ Missing $-0.015$ $0.179$ $-0.088$ $0.178$ $-0.008$ $0.179$
$-0.085$ $0.178$ Father Education (High school) $<$ High school $-0.248$
$0.114$ $-0.243$ $0.112$ $-0.247$ $0.114$ $-0.244$ $0.112$ $>$ High school and
not graduated $-0.100$ $0.102$ $-0.111$ $0.102$ $-0.100$ $0.102$ $-0.111$
$0.102$ $>$ High school and graduated $-0.003$ $0.103$ $-0.036$ $0.105$
$-0.004$ $0.103$ $-0.035$ $0.105$ Missing $0.009$ $0.180$ $-0.027$ $0.179$
$0.007$ $0.180$ $-0.025$ $0.179$ Mother Job (None) Professional $0.023$
$0.122$ $-0.001$ $0.124$ $0.024$ $0.122$ $0.002$ $0.124$ Other $0.064$ $0.093$
$0.060$ $0.095$ $0.063$ $0.093$ $0.062$ $0.095$ Missing $0.366$ $0.216$
$0.432$ $0.213$ $0.362$ $0.216$ $0.429$ $0.213$ Father Job (None) Professional
$-0.256$ $0.192$ $-0.284$ $0.196$ $-0.261$ $0.192$ $-0.286$ $0.196$ Other
$-0.238$ $0.136$ $-0.239$ $0.139$ $-0.241$ $0.137$ $-0.239$ $0.139$ Missing
$-0.248$ $0.230$ $-0.227$ $0.231$ $-0.246$ $0.230$ $-0.226$ $0.231$ Contextual
peer effects: $\boldsymbol{\theta}_{0,3}$ Female $0.058$ $0.111$ $0.010$
$0.112$ $0.062$ $0.111$ $0.013$ $0.112$ Age $-0.040$ $0.020$ $-0.015$ $0.021$
$-0.043$ $0.020$ $-0.016$ $0.021$ Hispanic $-0.121$ $0.167$ $-0.206$ $0.180$
$-0.129$ $0.167$ $-0.210$ $0.180$ Grade 7–8 $-0.275$ $0.241$ $-0.287$ $0.241$
$-0.279$ $0.241$ $-0.286$ $0.242$ Grade 9–10 $-0.194$ $0.179$ $-0.114$ $0.180$
$-0.193$ $0.179$ $-0.112$ $0.180$ Grade 11–12 $-0.061$ $0.171$ $0.003$ $0.173$
$-0.067$ $0.172$ $0.003$ $0.173$ Race (White) Black $0.152$ $0.190$ $0.119$
$0.197$ $0.147$ $0.190$ $0.113$ $0.197$ Asian $-0.141$ $0.173$ $-0.139$
$0.181$ $-0.149$ $0.173$ $-0.145$ $0.181$ Other $-0.495$ $0.177$ $-0.459$
$0.180$ $-0.489$ $0.177$ $-0.456$ $0.180$ With parents $-0.174$ $0.194$
$-0.215$ $0.198$ $-0.165$ $0.194$ $-0.215$ $0.198$ Allowance per week $0.005$
$0.005$ $0.003$ $0.005$ $0.005$ $0.005$ $0.003$ $0.005$ Mother Education (High
school) $<$ High school $0.410$ $0.180$ $0.359$ $0.177$ $0.409$ $0.180$
$0.361$ $0.177$ $>$ High school and not graduated $-0.171$ $0.147$ $-0.229$
$0.146$ $-0.173$ $0.147$ $-0.228$ $0.146$ $>$ High school and graduated
$-0.090$ $0.161$ $-0.199$ $0.166$ $-0.096$ $0.162$ $-0.199$ $0.167$ Missing
$-0.335$ $0.351$ $-0.415$ $0.354$ $-0.345$ $0.352$ $-0.419$ $0.354$ Father
Education (High school) $<$ High school $-0.046$ $0.182$ $-0.083$ $0.180$
$-0.031$ $0.183$ $-0.082$ $0.180$ $>$ High school and not graduated $0.172$
$0.155$ $0.110$ $0.158$ $0.184$ $0.156$ $0.115$ $0.158$ $>$ High school and
graduated $0.113$ $0.157$ $0.065$ $0.163$ $0.122$ $0.158$ $0.065$ $0.163$
Missing $0.301$ $0.241$ $0.288$ $0.243$ $0.308$ $0.241$ $0.289$ $0.242$ Mother
Job (None) Professional $-0.202$ $0.200$ $-0.233$ $0.201$ $-0.196$ $0.200$
$-0.231$ $0.201$ Other $-0.256$ $0.150$ $-0.276$ $0.153$ $-0.254$ $0.150$
$-0.275$ $0.153$ Missing $-0.163$ $0.385$ $-0.088$ $0.384$ $-0.150$ $0.386$
$-0.082$ $0.384$ Father Job (None) Professional $0.450$ $0.289$ $0.105$
$0.298$ $0.438$ $0.289$ $0.106$ $0.298$ Other $0.341$ $0.187$ $0.223$ $0.199$
$0.332$ $0.187$ $0.219$ $0.198$ Missing $0.010$ $0.322$ $-0.137$ $0.332$
$0.000$ $0.322$ $-0.142$ $0.331$ For the categorical variables, the level in
parentheses is set as the reference level.