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\\
\textbf{Xi Qu}\\
{Department of Economics, Antai College of Economics and Management, Shanghai Jiao Tong University, 1954 Huashan Road, Shanghai, 200030, China PRC. Email: [email removed].}
\\
\textbf{Jiajun Zhang}\\{International Business School, Shanghai University of International Business and Economics, 201620, China PRC. E-mail: [email removed].}
\end{minipage}
\end{center}
\setstretch{1.2}
\section{Additional lemmas}
\begin{lemma}\label{lem:normleq}
Let $A$ and $B$ be $n \times n$ real matrices, then $\|AB\|\leq\|A\|_{\mathrm{sp}}\|B\|$ or $\|AB\|\leq\|B\|_{\mathrm{sp}}\|A\|$.
\end{lemma}
\begin{proof}
This is a standard matrix norm inequality. For a formal discussion and proof, see Section 5.6 of \cite{horn2012matrix}.
\end{proof}
\begin{lemma}\label{lem:boundB}
(\romannumeral1) For any time-varying square matrix $\mathcal{B}_{t}$, if
$\sup\limits_{t}\|\mathcal{B}_{t}\|_{1}=\sup\limits_{t}\|\mathcal{B}_{t}\|_{\infty}=O(1)$, then $\sup\limits_{t}\|\mathcal{B}_{t}\|_{\mathrm{sp}}=O(1)$.
\\
(\romannumeral2) For any time-varying square random matrix $\mathcal{B}_{t}$, depending on an index $n$. If
$\sup_{t}\|\mathcal{B}_{t}\|_{1}=O_p(1)$
and
$\sup_{t} \|\mathcal{B}_{t}\|_{\infty} = O_p(1)$,
then $\sup_{t} \|\mathcal{B}_{t}\|_{\mathrm{sp}} = O_p(1)$
\end{lemma}
\begin{proof}
(\romannumeral1)
We use the standard matrix norm inequality $\|\mathcal{B}_{t}\|_{\mathrm{sp}} \leq \sqrt{\|\mathcal{B}_{t}\|_{1} \|\mathcal{B}_{t}\|_{\infty}}$. Taking the supremum over $t$ yields:
$\sup_{t} \|\mathcal{B}_{t}\|_{\mathrm{sp}} \leq \sup_{t} \sqrt{\|\mathcal{B}_{t}\|_{1} \|\mathcal{B}_{t}\|_{\infty}} \leq \sqrt{\left( \sup_{t} \|\mathcal{B}_{t}\|_{1} \right) \left( \sup_{t} \|\mathcal{B}_{t}\|_{\infty} \right)}=O(1).$
\\
(\romannumeral2)
According to the proof of (\romannumeral1),
let $X = \sup_{t} \|\mathcal{B}_{t}\|_{1}$ and $Y = \sup_{t} \|\mathcal{B}_{t}\|_{\infty}$. Then, $Z = XY = O_p(1)$. Since norms are non-negative, $Z \ge 0$. Therefore, $\sqrt{Z} = \sqrt{XY} = O_p(1)$. Hence, $0 \leq \sup_{t} \|\mathcal{B}_{t}\|_{\mathrm{sp}} \leq \sqrt{XY}= O_p(1)$.
\end{proof}
\begin{lemma}\label{lem:e-I}
Let the $n \times n$ matrix $A$ with its typical element $a_{ij}$ satisfy
$\sup_{i,j}|a_{ij}|=O_{p}(d^{-\nu})$ and
$\sup_{i}\sum_{j=1}^{n}|a_{ij}|=O_{p}(d^{-\nu})$ for some $\nu > 0$. Suppose that $n^{1/2}d^{-\nu} \to 0$ as $n \to\infty$, then:
$$
\|e^A - I\|=O_{p}(\|A\|)=O_{p}(n^{1/2}d^{-\nu}).
$$
\end{lemma}
\begin{proof}\sloppy
First, it is easy to conclude that $\|A\|=O_{p}(n^{1/2}d^{-\nu})$ since $\sum_{j=1}^{n}|a_{ij}|^2 \leq \left(\sup_{k}|a_{ik}| \right) \left(\sum_{j=1}^{n}|a_{ij}|\right)\leqO_{p}(d^{-2\nu})$. Therefore, $\|A\|^2=O_{p}(n d^{-2\nu})$.
Second,
Using the triangle inequality for the Frobenius norm and the submultiplicative property, we get:
$\|e^A - I\|\leq\|A\|+\frac{\|A^2\|}{2!}+\frac{\|A^3\|}{3!}+\cdots$.
Since $\|A^k\| \leq \|A\|^k$, this becomes:
$\|e^A-I\|\leq\|A\|+\frac{\|A\|^2}{2!}+\frac{\|A\|^3}{3!}+\cdots=e^{\|A\|}-1$.
Since $\|A^k\| \leq \|A\|^k$, this becomes:
$\|e^A - I\|\leq\|A\|+\frac{\|A\|^2}{2!} + \frac{\|A\|^3}{3!}+\cdots=e^{\|A\|}-1$.
Let $X_p=\|A\|$,
we have $e^{X_p} - 1 = X_p + o_{p}(1)$.
Combining above all, we conclude that $\|e^A - I\|=O_{p}(\|A\|)$.
\end{proof}
Lemma \ref{lem:e-I} shows that the MESS has the same asymptotic behavior as that of the SAR.
\begin{lemma}\label{lem:hTt}
When $T \to\infty$,
$\sum_{t=1}^{T-1}h_{Tt}^2=O(T)$ and $\sum_{t=1}^{T-1}(1-h_{Tt}^2)=O(\ln{T})$, where $h_{Tt}=\sqrt{\frac{T-t}{T-t+1}}$.
\end{lemma}
\begin{proof}
Note that $h_{Tt}^2=1-\frac{1}{T-t+1}$, and we have $\sum_{t=1}^{T-1}h_{Tt}^2=O(T)-O(\ln{T})=O(T)$ since $\sum_{t=1}^{T-1}\frac{1}{T-t+1}=O(\ln{T})$.
\end{proof}
\begin{lemma}\label{lem:Rr}
Under Assumption \ref{ass:appro_G} and $\ell_{k}^{-\varsigma_{k}}+n^{1/2}\ell_{k}^{-\varsigma_{k}} \to 0$ as $n \to\infty$:\\
(\romannumeral1) $e^{\Delta_{k}}-I_{n}$ satisfies
\hyperref[propzero]{\ref*{propzero} {$O_{p}(\ell_{k}^{-\varsigma_{k}})$}}
, where $\Delta_{k}$ is defined in Notation \ref{notation:sup}(\romannumeral2). Also, $R_{k}$ and $S_{k}R_{k}$ satisfy
\hyperref[propzero]{\ref*{propzero} {$O_{p}(\ell_{k}^{-\varsigma_{k}})$}}
, and $B_{k}$ and $S_{k}$ satisfy \ref{propUB};
\\
(\romannumeral2)
$H_k$ satisfies
\hyperref[propzero]{\ref*{propzero} {$O_{p}(\ell_{3}^{-\varsigma_{3}})$}}
, where $H_{k}=R_{k}B_{k}^{-1}$;
\\
(\romannumeral3)
$\operatorname{E}\|r_{kt}^{*}\|=O(n^{1/2}\ell_{k}^{-\varsigma_{k}}h_{Tt})$ where $h_{Tt}=\sqrt{\frac{T-t}{T-t+1}}$, $r_{kt}^{*}=h_{Tt}(r_{kt}-\frac{1}{T-t}\sum_{h=t+1}^{T}r_{kt})$ and $r_{kt}$ is defined in equation \eqref{eq:Vt}.
\end{lemma}
\begin{proof}
For (\romannumeral1), first, from Lemma \ref{lem:e-I}, we can derive that $\|e^{\Delta_{k}}-I_{n}\|=O_{p}(\|\Delta_{k}\|)=O_{p}(n^{1/2}\ell_{k}^{-\varsigma_{k}})$.
Second, $\Vert S_{k}\Vert_{\mathrm{rc}}\leq e^{\sum_{p_{k}=1}^{\ell_{k}}|\lambda_{p_{k}}|\Vert\varPhi_{kp_{k}}\Vert_{\mathrm{rc}}}=O_{p}(1)$ from Assumption \ref{ass:appro_G} (\romannumeral1) and (\romannumeral2) and thus $\|S_{k}\|_{\mathrm{sp}}=O_{p}(1)$.
Third, since
$R_{k}=S_{k}(e^{\Delta_{k}}-I_{n})$, by Lemmas \ref{lem:normleq} and \ref{lem:e-I}, we have
\[
\|R_{g_{k}}\|\leq\|S_{k}\|_{\mathrm{sp}}\|e^{\Delta_{k}}-I_{n}\|=O_{p}(1)O_{p}(\|\Delta_{k}\|)=O_{p}(n^{1/2}\ell_{k}^{-\varsigma_{k}}).
\]
The results of $\|R_{k}\|_{\mathrm{rc}}=O_{p}(\ell_{k}^{-\varsigma_{k}})$ since $\|e^{\Delta_{k}}-I_{n}\|_{\mathrm{rc}}=O_{p}(\|\Delta_{k}\|_{\mathrm{rc}})=O_{p}(\ell_{k}^{-\varsigma_{k}})$.
Fourth, we have
\[
\|S_{k}R_{g_{k}}\|\leq\|S_{k}\|_{\mathrm{sp}}\|R_{g_{k}}\|=O_{p}(n^{1/2}\ell_{k}^{-\varsigma_{k}}).
\]
For (\romannumeral2), first, combining \eqref{eq:sdpd_np} and \eqref{eq:matrix function}, we have
\begin{flalign}\label{eq:Ut, Et and RS}
U_{t}=B_{3}^{-1}E_{t}=(R_{3}+S_{3})^{-1}E_{t}
\end{flalign}
When $k=3$, we have $r_{3t}=R_{3}U_{t}=H_{3}E_{t}$, where $H_{3}=R_{3}B_{3}^{-1}$.
Since $B_{k}$, $S_{k}$ and $R_{k}$ are invertible, we have
$B_{3}^{-1}=(I_{n}+S_{3}^{-1}R_{3})^{-1}S_{3}^{-1}$.
Then, we have $\|S_{k}^{-1}R_{k}\|_{\mathrm{sp}}=O_{p}(n^{1/2}\ell_{k}^{-\varsigma_{k}})$ which implies $\ell_{k}^{\varsigma_{k}}=o(n^{1/2})$. Thus, by the Neumann series, we have $\|(I_{n}+S_{3}^{-1}R_{3})^{-1}\|_{\mathrm{sp}}=O(1)$, $\|B_{3}^{-1}\|_{\mathrm{sp}}=O(1)$ and $\|H_{3}\|_{\mathrm{sp}}=O_{p}(\ell_{k}^{-\varsigma_{3}})$.
For (\romannumeral3), consider $k=3$, and $r_{3t}^{*}=h_{Tt}(r_{3t}-\frac{1}{T-t}\sum_{h=t+1}^{T}r_{3t}).$
Note that
\begin{flalign*}
\begin{split}
\operatorname{E}\|r_{3t}\|^2&
=\operatorname{E}\sum_{i=1}^{n}r_{3t,i}^2=\operatorname{E}\sum_{i=1}^{n}(\sum_{j=1}^{n}h_{3,ij}\epsilon_{jt})^2
=\operatorname{E}\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{k=1}^{n}h_{3,ij}h_{3,ik}\epsilon_{jt}\epsilon_{kt}
\\
&=\operatorname{E}\sum_{i=1}^{n}\sum_{j=1}^{n}h_{3,ij}^2\epsilon_{jt}^2
\leq\sqrt{\operatorname{E}\left[\sum_{i=1}^{n}(\sum_{j=1}^{n}h_{3,ij}^2)^2\right]}\sqrt{\operatorname{E}\left[\sum_{j=1}^{n}\epsilon_{j}^4\right]}\\
&=O(\sqrt{n}\ell_{3}^{-2\varsigma_{3}})\cdot O(\sqrt{n})
=O(n\ell_{3}^{-2\varsigma_{3}}),
\end{split}
\end{flalign*}
by Cauchy-Schwartz inequality and the fact that $\sum_{j=1}^{n}h_{3,ij}^2\leq(\sum_{j=1}^{n}|h_{ij}|)^2\leq \|H_{3}\|_{\infty}^2=O_{p}(\ell_{3}^{-2\varsigma_{3}})$.
Thus,
$\operatorname{E}\|r_{3t}\|\leq\sqrt{\operatorname{E}\|r_{3t}\|^2}=O(n^{1/2}\ell_{k}^{-\varsigma_{3}})$. So we have
$\operatorname{E}\|\frac{1}{T-t}\sum_{h=t+1}^{T}r_{3t}\|\leq\frac{1}{T-t}\sum_{h=t+1}^{T}\operatorname{E}\|r_{3t}\|=O(n^{1/2}\ell_{k}^{-\varsigma_{3}}).$
Finally, $\operatorname{E}\|r_{3t}^{*}\|=O(n^{1/2}\ell_{3}^{-\varsigma_{3}}h_{Tt})$. Similarly, we have
$\operatorname{E}\|r_{1t}^{*}\|=O(n^{1/2}\ell_{1}^{-\varsigma_{1}}h_{Tt})$ and $\operatorname{E}\|r_{2t}^{*}\|=O(n^{1/2}\ell_{2}^{-\varsigma_{2}}h_{Tt})$ as $\rho(A)<1$.
\end{proof}
\begin{lemma}\label{lem:EPE}
Under Assumption \ref{ass:sieve-basis}(\romannumeral1), the covariance of $m_{N}^{\mathtt{line}}(\theta_{0})$ and $m_{N}^{\mathtt{quad}}(\theta_{0})$ is asymptotically zero as $(n,T)\to\infty$.
\end{lemma}
\begin{proof}
Similar to the Proof of Theorem \ref{thm:gm_pi}.
\end{proof}
\begin{lemma}\label{lem:norm of M}
Under Assumption \ref{ass:IV}, $\|M_{t}\|_{\mathrm{rc}}=O_{p}\left(\ell_n\right)$, and $\|M_{t}\|=O_{p}\left(\sqrt{\ell_n}\right)$.
\end{lemma}
\begin{proof}
Note that the entries of $M_{t}$ are
\[
m_{ijt}=\tfrac{1}{n}\bigl[J_{n}Q_{t}\bigr]_{i}'\bigl(Q_{nt}'J_{n}\Sigma_{t}J_{n}Q_{nt}\bigr)^{-1}\bigl[J_{n}Q_{t}\bigr]_{i}
\]
and thus
\[
|m_{ijt}|=O_{p}\Bigl(\tfrac{1}{n}\Big\Vert\bigl[J_{n}Q_{t}\bigr]_{i}\Big\Vert\Big\Vert\bigl[J_{n}Q_{t}\bigr]_{j}\Big\Vert\Bigr)
=O_{p}\Bigl(\frac{\ell_{n}}{n}\Bigr)
\]
uniformly in $i,j$ for each $t$.
Similarly, we also observe that
\[
\sum_{j=1}^{n}m_{ijt}^2=O_{p}\Bigl(\frac{\ell_{n}}{n}\Bigr).
\]
uniformly in $i$ for each $t$.
So we have
$\|M_{t}\|_{\mathrm{rc}}=O_{p}(\ell_{n})$, $\|M_{t}\|_{\mathrm{sp}}=O_{p}(\ell_{n})$
and $\|M_{t}\|^2=O_{p}(\ell_{n})$.
\end{proof}
Lemma \ref{lem:norm of M} indicates that the linear moments may be subject to various moment issues due to the cross-sectional dimension, which can be viewed as an extension of those presented in \cite{lee2014efficient}.
\begin{lemma}\label{lem:VBV}
For $n\times n$ time-varying matrix $\mathcal{B}_{t}$, suppose $\mathcal{B}_{t}$ satisfies
\hyperref[propzero]{\ref*{propzero} {$O_{p}(b)$}}
and $Q_{jt}$ is the $j$-th column components of the IV $Q_{t}$, under Assumptions \ref{ass:nT}, \ref{ass:appro_G} and \ref{ass:IV}:\\
(\romannumeral1) $\operatorname{E}\|r_{kt}^{*}\|^2=O(n^2h_{Tt}^4\ell_{k}^{-4\varsigma_{k}})$;
\\
(\romannumeral2)
$\operatorname{E}|r_{kt}^{*\prime}{\mathcal{B}_{t}}r_{kt}^{*}|=O\left(nh_{Tt}^2\ell_{k}^{-2\varsigma_{k}}b\right)$;\\
(\romannumeral3)
$\operatorname{E}|r_{kt}^{*\prime}\mathcal{B}_{t} E_{t}^{*}|=O(nh_{Tt}^2\ell_{k}^{-\varsigma_{k}}b)$.\\
(\romannumeral4)
$\operatorname{E}|Q_{jt}'\mathcal{B}_{t}(r_{kt}^{*}+E_{t}^{*})|=O(\sqrt{n}h_{Tt}\ell_{k}^{-\varsigma_{k}}b)$.
\end{lemma}
\begin{proof}
For (\romannumeral1)
$\operatorname{E}\|r_{kt}\|^2=\operatorname{E}\left[\sum_{i}r_{kt,i}^2\right]^2=\operatorname{E}\left[\sum_{i}(\sum_{j}h_{3,ij}\epsilon_{j})^2\right]^2=\operatorname{E}(E_{t}'H_{3}'H_{3}E_{t})^2$. Note that $H_{3}'H_{3}$ satisfies
\hyperref[propzero]{\ref*{propzero} {$O_{p}(h^2)$}}
with $h=\ell_{3}^{-\varsigma_{3}}$, and we have $\operatorname{E}\|r_{kt}\|^2=O(n^2h_{Tt}^4\ell_{k}^{-4\varsigma_{k}})$.
For (\romannumeral2), taking $k=3$ as an example, according to Lemmas \ref{lem:normleq} and \ref{lem:Rr}, we know that the dominant term is $h_{Tt}\operatorname{E}|r_{3t}^{\prime}{\mathcal{B}_{t}}r_{3t}|$. According to the Cauchy-Schwarz inequality,
$\operatorname{E}|r_{kt}^{\prime}{\mathcal{B}_{t}}r_{3t}|\leq\sqrt{\operatorname{E}\|r_{kt}^{\prime}{\mathcal{B}_{t}}r_{3t}\|^2}=\sqrt{\operatorname{E}\|E_{t}^{\prime}H_{3}'\mathcal{B}_{t} H_{3}E_{t}\|^2}$, also, $H_{3}'\mathcal{B}_{t} H_{3}$ satisfies
\hyperref[propzero]{\ref*{propzero} {$O_{p}(b\ell_{3}^{-2\varsigma_{3}})$}}
. Thus, $\operatorname{E}\|r_{3t}^{\prime}{\mathcal{B}_{t}}r_{3t}\|^2=O\left(n^2\ell_{3}^{-4\varsigma_{3}}b^2\right)$.
The result of (\romannumeral3) can be proved similarly to (\romannumeral2). For (\romannumeral4), we know that $Q_{t}$ can be expressed as a linear combination of $Y_{t-1}^{(*,-1)}$ and $X_{t}^{*}$. Since that $Y_{t-1}^{(*,-1)}$ and $X_{t}^{*}$ are both uncorrelated with $E_{t}^{*}$, the desired result follows immediately by arguments analogous to those used for (\romannumeral2) and (\romannumeral3).
\end{proof}
\begin{lemma}\label{lem:bgmm_trans}
Under Assumption \ref{ass:nT}, $\operatorname{Var}(
\ifstrequal{t}{N}{
\pmb{M}({\bm{\Sigma}_{t}})
}{
{M}({\Sigma_{t}})
}
\Sigma_{t}^{-1/2}E_{t}^{*})=
\ifstrequal{t}{N}{
\pmb{M}({\bm{\Sigma}_{t}})
}{
{M}({\Sigma_{t}})
}
+o(1)$.
\end{lemma}
\begin{proof}
For the case of \ref{var:homo} and \ref{var:hetei}, we have $\operatorname{Var}(
\ifstrequal{t}{N}{
\pmb{M}({\bm{\Sigma}_{t}})
}{
{M}({\Sigma_{t}})
}
\Sigma_{t}^{-1/2}E_{t}^{*})=
\ifstrequal{t}{N}{
\pmb{M}({\bm{\Sigma}_{t}})
}{
{M}({\Sigma_{t}})
}
$. Hence, we only need to investigate the case of \ref{var:hetet}. Denote $H_{t}=
\ifstrequal{t}{N}{
\pmb{M}({\bm{\Sigma}_{t}})
}{
{M}({\Sigma_{t}})
}
\Sigma_{t}^{-1/2}E_{t}^{*}$.
To prove that $\operatorname{Var}(H_t)=
\ifstrequal{t}{N}{
\pmb{M}({\bm{\Sigma}_{t}})
}{
{M}({\Sigma_{t}})
}
+o(1)$ under the case of \ref{var:hetet} where $\Sigma_t=\sigma_t^2 I_n$, we first observe that the scaling matrix simplifies to $\Sigma_t^{-1/2}=\sigma_t^{-1} I_n$ and the projection matrix reduces to the standard centering matrix $M(\Sigma_t)=I_n- (\sigma_t^{-1}l_n)(n \sigma_t^{-2})^{-1}(\sigma_t^{-1}l_n)'=J_n$.
Since the variance of the forward orthogonal deviation error $E_t^*$ is dominated by the current period variance $\sigma_t^2 I_n$ for large $T$, specifically, $\operatorname{Var}(E_t^*) = \sigma_t^2 I_n + O((T-t)^{-1})$, the variance of the transformed error becomes
$\operatorname{Var}(H_t)=J_n(\sigma_t^{-1} I_n)[\sigma_t^2 I_n + o(1)](\sigma_t^{-1} I_n)J_n
=J_n(I_n+o(1))J_n=J_n+o(1)$. Given that $M(\Sigma_t)=J_n$ in this specification, it follows that $\operatorname{Var}(H_t)=
\ifstrequal{t}{N}{
\pmb{M}({\bm{\Sigma}_{t}})
}{
{M}({\Sigma_{t}})
}
+o_{p}(1)$ as $T \to \infty$.
\end{proof}
\begin{lemma}\label{lem:norm of the moments}
Under Assumptions \ref{ass:disterbance}-\ref{ass:IV}, $\operatorname{E}\|m_{N}(\theta_{0})\|=o(1)$.
\end{lemma}
\begin{proof}
Observe that
\begin{flalign*}
\begin{split}
\operatorname{E}\|m_{N}(\theta_{0})\|&
\leq\frac{1}{n(T-1)}\Bigl(\operatorname{E}\|\sum_{j=1}^{\ell_{p}}\mathbf{V}_{N}^{*\prime}\mathbf{J}_{N}\mathbf{P}_{Nj}\mathbf{J}_{N}\mathbf{V}_{N}^{*}\|+\operatorname{E}\|\sum_{j=1}^{\ell_{q}}\mathbf{Q}_{Nj}'\mathbf{J}_{N}\mathbf{V}_{N}^{*}\|\Bigr)
\\&
=O\left((\ell_{n}^{-2\underline{\varsigma}}+\ell_{n}^{-\underline{\varsigma}})\cdot\ell_{n}^{1/2}\right)=o(1)
\end{split}
\end{flalign*}
where the last two relations follow by the same argument used to bound $\mathscr{A}_{1N}$ and $\mathscr{A}_{2N}$ in the proof of Lemma \ref{lem:CLT}.
\end{proof}
\begin{lemma}\label{lem:foc of mess}
(\romannumeral1) For the MESS, when $j=1,...,\ell_{k}$ and $k=1,2,3$, under Assumption \ref{ass:sieve-basis} and $S_{k}$'s satisfy \ref{propUB}, we have
\begin{flalign*}
\|\tfrac{\partial S_{k}}{\partial\lambda_{kj_0}}-\varPhi_{kj}S_{k}\|_{\mathrm{sp}}=O(\|\varPhi_{kj}\|_{\mathrm{sp}})=O(1).
\end{flalign*}
(\romannumeral2) $\|\frac{\partial S_{k}}{\partial\lambda_{k_0}}\|_{\mathrm{sp}}=O(\sqrt{\ell_{n}})$, where $\frac{\partial S_{k}}{\partial\lambda_{k_0}}=\left[\frac{\partial S_{k}}{\partial\lambda_{k_{10}}},...,\frac{\partial S_{k}}{\partial\lambda_{{k\ell_{k}}_0}}\right]$ for the true estimand $\lambda_{kj_0}$.
\end{lemma}
\begin{proof}
Since $\varPhi_{kj}$ does not commute with $S_{k}$, the Fr\'{e}chet derivative of the matrix exponential gives
\begin{flalign*}
\tfrac{\partial S_{k}}{\partial\lambda_{kj_0}}=\int_0^1 e^{(1-t) \Xi_k}\varPhi_{kj} e^{t\Xi_k} d s
\end{flalign*}
Because the norm of an integral is less than the integral of the norm, we have
\begin{flalign*}
\begin{split}
\|\tfrac{\partial S_{k}}{\partial\lambda_{kj_0}}\|_{\mathrm{sp}}&\leq \left(\int_0^1\|e^{(1-t) \Xi_k}\varPhi_{kj} e^{t\Xi_k}\|_{\mathrm{sp}}dt\right)\leq \left(\int_0^1\|e^{(1-t)\Xi_k}\|_{\mathrm{sp}} \|\varPhi_{kj}\|_{\mathrm{sp}}\|e^{t\Xi_k}\|_{\mathrm{sp}}dt\right)
\\
&\leq
\|\varPhi_{kj}\|_{\mathrm{sp}}\left(\int_0^1e^{(1-t)\|\Xi_k\|_{\mathrm{sp}}} e^{t\|\Xi_k\|_{\mathrm{sp}}}dt\right)\leq \|\varPhi_{kj}\|_{\mathrm{sp}}\cdot e^{\|\Xi_k\|_{\mathrm{sp}}}=O(\|\varPhi_{kj}\|_{\mathrm{sp}})=O(1).
\end{split}
\end{flalign*}
from Assumption \ref{ass:sieve-basis}.
Thus, we have
\begin{flalign*}
\|\tfrac{\partial S_{k}}{\partial\lambda_{kj_0}}-\varPhi_{kj}S_{k}\|_{\mathrm{sp}}\leq\|\tfrac{\partial S_{k}}{\partial\lambda_{kj_0}}\|_{\mathrm{sp}}+\|\varPhi_{kj}S_{k}\|_{\mathrm{sp}}=O(\|\varPhi_{kj}\|_{\mathrm{sp}}).
\end{flalign*}
\end{proof}
\section{Estimation of heteroskedastic variances}\label{sec:estimate_variance}
First, we explain the transformation operator $
\ifstrequal{t}{N}{
\pmb{J}({\bm{\Sigma}_{t}})
}{
{J}({\Sigma_{t}})
}
$ derived in BGMME, which can be motivated from the approximated log-likelihood function
\begin{flalign*}
L_{nT}(\theta,\mathbf{c}_{n},\bm\alpha_{T},\Sigma_{t})=-\frac{nT}{2}\ln(2\pi)-\frac{nT}{2}\ln|\Sigma_{t}(\theta)|+T\bigl(\ln|(S_{1}(\lambda)|+\ln|S_{3}(\lambda)|\bigr)-\sum_{t=1}^{T}V_{t}^{c\prime}(\theta)\Sigma_{t}^{-1}(\theta)V_{t}^{c}(\theta).
\end{flalign*}
The use of the approximate likelihood relies on the negligibility of $r_{t}$, which in turn permits the replacement of the true $g_{k0}$ with asymptotically negligible cost.
Concentrating out $\bm\alpha_{T}$ by the first-order condition, we have
\begin{flalign}
L_{nT}(\theta,\mathbf{c}_{n},\Sigma_{t})=-\frac{nT}{2}\ln(2\pi)-\frac{nT}{2}\ln|\Sigma_{t}(\theta)|+T\bigl(\ln|(S_{1}(\lambda)|+\ln|S_{3}(\lambda)|\bigr)-\sum_{t=1}^{T}V_{t}^{c\prime}(\theta)
\ifstrequal{t}{N}{
\pmb{J}({\bm{\Sigma}_{t}})
}{
{J}({\Sigma_{t}})
}
V_{t}^{c}(\theta),
\end{flalign}
where $V_{t}^{c}(\theta)=S_{3}(\lambda)S_{1}(\lambda)Y_{t}-S_{3}\bigl((\gamma I_{n}+S_{2})Y_{t-1}+X_{t}\beta+\mathbf{c}_{n}\bigr)$ and $
\ifstrequal{t}{N}{
\pmb{J}({\bm{\Sigma}_{t}})
}{
{J}({\Sigma_{t}})
}
=\Sigma_{t}^{-1}-\Sigma_{t}^{-1}l_{n}(l_{n}'\Sigma_{t}^{-1}l_{n})^{-1}l_{n}'\Sigma_{t}^{-1}$. Thus, $
\ifstrequal{t}{N}{
\pmb{J}({\bm{\Sigma}_{t}})
}{
{J}({\Sigma_{t}})
}
$ enables the best moment conditions to mimic the score of the likelihood function, and thus it can provide the moment conditions more efficiently than the counterparts relying on the operator $J_{n}$.
The BGMME has two main advantages over MLE. First, when the model is SAR, it avoids evaluating the Jacobian determinant. Second, the BGMME is subject only to approximation bias from sieves, whereas MLE can suffer additional bias from the incidental-parameters problem, as documented in the dynamic panel data literature.
\section{Additional results}\label{sec:mmc}
In this section, we present comprehensive results from our Monte Carlo experiments. While the main text focused on MESS specifications, here we provide the corresponding finite-sample performance for SAR models.
Tables are presented on the following pages.
\newpage
\begin{table}[htbp]
\centering \footnotesize \setlength{\abovecaptionskip}{0cm} \setlength{\tabcolsep}{0.8mm}
\caption{Finite sample performance of $\pi$ for the MESS, $\operatorname{Var}(\epsilon_{it})=\sigma_i^2$. Robustness check.}
\begin{spacing}{1}
\begin{tabular}{lcccccclcccccc}
\toprule
\multicolumn{7}{c}{$(n,T,\ell_{n})=(100,10,2)$} & \multicolumn{7}{c}{$(n,T,\ell_{n})=(100,10,[n^{1/5}]+2)$} \\
\midrule
\multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} \\
\cmidrule{2-7}\cmidrule{9-14} & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ \\
\midrule
Bias & -0.0539 & 0.0062 & -0.0544 & 0.0049 & -0.0562 & 0.0044 & Bias & -0.0508 & 0.0047 & -0.0517 & 0.0036 & -0.0524 & -0.0016 \\
ESD & 0.0393 & 0.0381 & 0.0388 & 0.0345 & 0.0435 & 0.0386 & ESD & 0.0381 & 0.0350 & 0.0374 & 0.0339 & 0.0398 & 0.0375 \\
RMSE & 0.0667 & 0.0386 & 0.0668 & 0.0348 & 0.0711 & 0.0388 & RMSE & 0.0635 & 0.0353 & 0.0638 & 0.0341 & 0.0658 & 0.0375 \\
CP & 0.7800 & 0.9190 & 0.7770 & 0.9240 & 0.7260 & 0.8750 & CP & 0.7890 & 0.8950 & 0.7940 & 0.9180 & 0.7220 & 0.8840 \\
\midrule
\multicolumn{7}{c}{$(n,T,\ell_{n})=(100,25,2)$} & \multicolumn{7}{c}{$(n,T,\ell_{n})=(100,25,[n^{1/5}]+2)$} \\
\midrule
& \multicolumn{1}{c}{2SLS} & & \multicolumn{1}{c}{OGMM} & & \multicolumn{1}{c}{BGMM} & & \multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} \\
\cmidrule{2-7}\cmidrule{9-14} &$\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ \\
\midrule
Bias & -0.0445 & 0.0095 & -0.0434 & 0.0080 & -0.0429 & 0.0077 & Bias & -0.0441 & 0.0074 & -0.0433 & 0.0060 & -0.0428 & 0.0008 \\
ESD & 0.0246 & 0.0243 & 0.0223 & 0.0240 & 0.0211 & 0.0214 & ESD & 0.0231 & 0.0237 & 0.0204 & 0.0219 & 0.0201 & 0.0205 \\
RMSE & 0.0508 & 0.0261 & 0.0488 & 0.0253 & 0.0478 & 0.0227 & RMSE & 0.0498 & 0.0248 & 0.0479 & 0.0227 & 0.0473 & 0.0205 \\
CP & 0.8850 & 0.8820 & 0.8540 & 0.9020 & 0.8770 & 0.8520 & CP & 0.8930 & 0.8950 & 0.8840 & 0.9060 & 0.8820 & 0.9030 \\
\midrule
\multicolumn{7}{c}{$(n,T,\ell_{n})=(200,10,2)$} & \multicolumn{7}{c}{$(n,T,\ell_{n})=(200,10,[n^{1/5}]+2)$} \\
\midrule
\multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} \\
\cmidrule{2-7}\cmidrule{9-14} & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ \\
\midrule
Bias & -0.0396 & 0.0071 & -0.0372 & 0.0067 & -0.0369 & 0.0066 & Bias & -0.0349 & 0.0050 & -0.0343 & 0.0035 & -0.0339 & -0.0050 \\
ESD & 0.0320 & 0.0282 & 0.0260 & 0.0246 & 0.0256 & 0.0231 & ESD & 0.0299 & 0.0278 & 0.0249 & 0.0232 & 0.0249 & 0.0222 \\
RMSE & 0.0509 & 0.0291 & 0.0454 & 0.0255 & 0.0449 & 0.0240 & RMSE & 0.0460 & 0.0282 & 0.0424 & 0.0235 & 0.0421 & 0.0228 \\
CP & 0.8970 & 0.9090 & 0.9000 & 0.9270 & 0.9070 & 0.8340 & CP & 0.8960 & 0.9010 & 0.9010 & 0.9140 & 0.8990 & 0.9060 \\
\midrule
\multicolumn{7}{c}{$(n,T,\ell_{n})=(200,25,2)$} & \multicolumn{7}{c}{$(n,T,\ell_{n})=(200,25,[n^{1/4}])$}
\\
\midrule
\multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} \\
\cmidrule{2-7}\cmidrule{9-14} & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ \\
\midrule
Bias & -0.0199 & 0.0079 & -0.0124 & 0.0087 & -0.0123 & 0.0063 & Bias & -0.0155 & 0.0062 & -0.0109 & 0.0055 & -0.0103 & -0.0048 \\
ESD & 0.0138 & 0.0145 & 0.0135 & 0.0130 & 0.0132 & 0.0128 & ESD & 0.0135 & 0.0135 & 0.0133 & 0.0128 & 0.0130 & 0.0126 \\
RMSE & 0.0242 & 0.0165 & 0.0183 & 0.0156 & 0.0180 & 0.0143 & RMSE & 0.0206 & 0.0149 & 0.0172 & 0.0139 & 0.0166 & 0.0135 \\
CP & 0.9140 & 0.8990 & 0.9160 & 0.9210 & 0.9250 & 0.9320 & CP & 0.9310 & 0.9610 & 0.9680 & 0.9210 & 0.9700 & 0.9440 \\
\midrule
\multicolumn{7}{c}{$(n,T,\ell_{n})=(400,10,2)$} & \multicolumn{7}{c}{$(n,T,\ell_{n})=(400,10,[n^{1/5}]+2)$} \\
\midrule
\multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} \\
\cmidrule{2-7}\cmidrule{9-14} & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ \\
\midrule
Bias & -0.0173 & -0.0055 & -0.0161 & 0.0031 & -0.0153 & 0.0010 & Bias & -0.0151 & 0.0023 & -0.0131 & 0.0002 & -0.0126 & -0.0010 \\
ESD & 0.0241 & 0.0233 & 0.0177 & 0.0158 & 0.0177 & 0.0157 & ESD & 0.0241 & 0.0226 & 0.0170 & 0.0155 & 0.0170 & 0.0154 \\
RMSE & 0.0297 & 0.0239 & 0.0239 & 0.0161 & 0.0234 & 0.0157 & RMSE & 0.0284 & 0.0227 & 0.0215 & 0.0155 & 0.0212 & 0.0154 \\
CP & 0.9470 & 0.9140 & 0.9610 & 0.9180 & 0.9520 & 0.9230 & TSD & 0.9260 & 0.9200 & 0.9450 & 0.9240 & 0.9180 & 0.9420 \\
\midrule
\multicolumn{7}{c}{$(n,T,\ell_{n})=(400,25,2)$} & \multicolumn{7}{c}{$(n,T,\ell_{n})=(400,25,[n^{1/5}]+2)$}
\\
\midrule
\multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} \\
\cmidrule{2-7}\cmidrule{9-14} & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ \\
\midrule
Bias & -0.0142 & -0.0095 & -0.0125 & 0.0031 & -0.0122 & 0.0010 & Bias & -0.0133 & 0.0029 & -0.0113 & 0.0010 & -0.0071 & 0.0008 \\
ESD & 0.0152 & 0.0155 & 0.0095 & 0.0100 & 0.0095 & 0.0099 & ESD & 0.0140 & 0.0121 & 0.0091 & 0.0091 & 0.0091 & 0.0090 \\
RMSE & 0.0208 & 0.0182 & 0.0157 & 0.0105 & 0.0155 & 0.0100 & RMSE & 0.0193 & 0.0124 & 0.0145 & 0.0092 & 0.0115 & 0.0090 \\
CP & 0.9430 & 0.9510 & 0.9530 & 0.9180 & 0.9430 & 0.9640 & CP & 0.9570 & 0.9490 & 0.9310 & 0.9520 & 0.9420 & 0.9400 \\
\bottomrule
\end{tabular}
\hspace*{-1cm}
\begin{tablenotes}
\footnotesize
\item \textbf{Note:} The true parameters are set to $\pi_0=(0.4,-0.7)'$, and $\bar{d}_0=15\%$. The results are based on 1,000 Monte Carlo replications. Bias denotes the mean bias of the estimates, ESD denotes the standard deviation, RMSE denotes the root mean squared error, and CP denotes the 95\% coverage probability. The 2SLS refers to the two-stage least square estimator, OGMM refers to the feasible optimal GMM estimator, and BGMM refers to the feasible best GMM estimator.
\end{tablenotes}
\end{spacing}
\label{sim:hei_robust}
\end{table}
\begin{sidewaystable}[htbp]
\centering\footnotesize\setlength{\abovecaptionskip}{0cm} \setlength{\tabcolsep}{0.8mm} \vspace*{-5mm}
\caption{Finite sample performance of $G_{1}, G_{2}$ and $G_{3}$ for the MESS, $\operatorname{Var}(\epsilon_{it})=\sigma_i^2$. Robustness check.}
\begin{spacing}{0.5}\hspace*{-5mm}
\begin{tabular}{lrrrrrrrrrlrrrrrrrrr}
\toprule
\multirow{4}[3]{*}{} & \multicolumn{9}{c}{$(n,T,\ell_{n})=(100,10,2)$} & & \multicolumn{9}{c}{$(n,T,\ell_{n})=(100,10,[n^{1/5}]+2)$} \\
\cmidrule{2-20}
\multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} \\
\cmidrule{2-10}\cmidrule{12-20} & \multicolumn{1}{c}{$\tilde{g}_1$} & \multicolumn{1}{c}{$\tilde{g}_2$} & \multicolumn{1}{c}{$\tilde{g}_3$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} \\
\midrule
MAE & 0.0339 & 0.0352 & 0.0530 & 0.0338 & 0.0351 & 0.0514 & 0.0340 & 0.0352 & 0.0505 & MAE & 0.0323 & 0.0342 & 0.0509 & 0.0322 & 0.0341 & 0.0499 & 0.0327 & 0.0343 & 0.0489 \\
Bias & -0.0204 & -0.0257 & -0.0464 & -0.0201 & -0.0259 & -0.0457 & -0.0197 & -0.0258 & -0.0451 & Bias & -0.0187 & -0.0255 & -0.0450 & -0.0185 & -0.0257 & -0.0446 & -0.0180 & -0.0255 & -0.0441 \\
RMSE & 0.0228 & 0.0288 & 0.0531 & 0.0225 & 0.0289 & 0.0517 & 0.0224 & 0.0289 & 0.0506 & RMSE & 0.0209 & 0.0284 & 0.0509 & 0.0207 & 0.0285 & 0.0500 & 0.0204 & 0.0284 & 0.0491 \\
\midrule
& \multicolumn{9}{c}{$(n,T,\ell_{n})=(100,25,2)$} & & \multicolumn{9}{c}{$(n,T,\ell_{n})=(100,25,[n^{1/5}]+2)$} \\
\cmidrule{2-20}
\multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} \\
\cmidrule{2-10}\cmidrule{12-20} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} \\
\midrule
MAE & 0.0329 & 0.0342 & 0.0509 & 0.0329 & 0.0341 & 0.0499 & 0.0331 & 0.0343 & 0.0489 & MAE & 0.0323 & 0.0329 & 0.0488 & 0.0322 & 0.0328 & 0.0478 & 0.0327 & 0.0330 & 0.0467 \\
Bias & -0.0190 & -0.0255 & -0.0475 & -0.0185 & -0.0257 & -0.0467 & -0.0180 & -0.0255 & -0.0457 & Bias & -0.0187 & -0.0240 & -0.0450 & -0.0185 & -0.0243 & -0.0446 & -0.0175 & -0.0247 & -0.0441 \\
RMSE & 0.0216 & 0.0287 & 0.0541 & 0.0211 & 0.0288 & 0.0526 & 0.0209 & 0.0287 & 0.0512 & RMSE & 0.0196 & 0.0250 & 0.0472 & 0.0194 & 0.0252 & 0.0466 & 0.0186 & 0.0257 & 0.0459 \\
\midrule
& \multicolumn{9}{c}{$(n,T,\ell_{n})=(200,10,2)$} & & \multicolumn{9}{c}{$(n,T,\ell_{n})=(200,10,[n^{1/5}]+2)$} \\
\cmidrule{2-20}
\multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} \\
\cmidrule{2-10}\cmidrule{12-20} & \multicolumn{1}{c}{$\tilde{g}_1$} & \multicolumn{1}{c}{$\tilde{g}_2$} & \multicolumn{1}{c}{$\tilde{g}_3$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} \\
\midrule
MAE & 0.0199 & 0.0209 & 0.0375 & 0.0199 & 0.0209 & 0.0368 & 0.0198 & 0.0209 & 0.0361 & MAE & 0.0194 & 0.0209 & 0.0322 & 0.0192 & 0.0208 & 0.0317 & 0.0192 & 0.0208 & 0.0311 \\
Bias & -0.0159 & -0.0180 & -0.0352 & -0.0156 & -0.0181 & -0.0346 & -0.0153 & -0.0180 & -0.0342 & Bias & -0.0128 & -0.0162 & -0.0305 & -0.0125 & -0.0163 & -0.0302 & -0.0121 & -0.0162 & -0.0298 \\
RMSE & 0.0164 & 0.0191 & 0.0370 & 0.0160 & 0.0191 & 0.0362 & 0.0158 & 0.0191 & 0.0357 & RMSE & 0.0137 & 0.0175 & 0.0326 & 0.0133 & 0.0175 & 0.0321 & 0.0130 & 0.0175 & 0.0316 \\
\midrule
& \multicolumn{9}{c}{$(n,T,\ell_{n})=(200,25,2)$} & & \multicolumn{9}{c}{$(n,T,\ell_{n})=(200,25,[n^{1/5}]+2)$} \\
\cmidrule{2-20}
\multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} \\
\cmidrule{2-10}\cmidrule{12-20} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} \\
\midrule
MAE & 0.0195 & 0.0200 & 0.0367 & 0.0195 & 0.0200 & 0.0360 & 0.0195 & 0.0200 & 0.0354 & MAE & 0.0186 & 0.0196 & 0.0314 & 0.0184 & 0.0196 & 0.0309 & 0.0183 & 0.0197 & 0.0302 \\
Bias & -0.0158 & -0.0180 & -0.0357 & -0.0153 & -0.0181 & -0.0350 & -0.0148 & -0.0182 & -0.0344 & Bias & -0.0126 & -0.0159 & -0.0313 & -0.0122 & -0.0160 & -0.0308 & -0.0115 & -0.0163 & -0.0301 \\
RMSE & 0.0160 & 0.0183 & 0.0363 & 0.0154 & 0.0184 & 0.0355 & 0.0149 & 0.0186 & 0.0349 & RMSE & 0.0129 & 0.0163 & 0.0320 & 0.0125 & 0.0164 & 0.0314 & 0.0119 & 0.0167 & 0.0306 \\
\midrule
& \multicolumn{9}{c}{$(n,T,\ell_{n})=(400,10,2)$} & & \multicolumn{9}{c}{$(n,T,\ell_{n})=(400,10,[n^{1/5}]+2)$} \\
\cmidrule{2-20}
\multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} \\
\cmidrule{2-10}\cmidrule{12-20} & \multicolumn{1}{c}{$\tilde{g}_1$} & \multicolumn{1}{c}{$\tilde{g}_2$} & \multicolumn{1}{c}{$\tilde{g}_3$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} \\
\midrule
MAE & 0.0094 & 0.0099 & 0.0198 & 0.0094 & 0.0099 & 0.0198 & 0.0093 & 0.0099 & 0.0194 & MAE & 0.0092 & 0.0098 & 0.0166 & 0.0091 & 0.0098 & 0.0166 & 0.0092 & 0.0099 & 0.0164 \\
Bias & -0.0075 & -0.0084 & -0.0181 & -0.0074 & -0.0083 & -0.0181 & -0.0074 & -0.0083 & -0.0180 & Bias & -0.0062 & -0.0076 & -0.0159 & -0.0062 & -0.0075 & -0.0159 & -0.0061 & -0.0075 & -0.0158 \\
RMSE & 0.0077 & 0.0090 & 0.0190 & 0.0076 & 0.0089 & 0.0190 & 0.0076 & 0.0089 & 0.0189 & RMSE & 0.0068 & 0.0083 & 0.0173 & 0.0068 & 0.0082 & 0.0173 & 0.0067 & 0.0083 & 0.0170 \\
\midrule
& \multicolumn{9}{c}{$(n,T,\ell_{n})=(400,25,2)$} & & \multicolumn{9}{c}{$(n,T,\ell_{n})=(400,25,[n^{1/5}]+2)$} \\
\cmidrule{2-20}
\multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} \\
\cmidrule{2-10}\cmidrule{12-20} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} \\
\midrule
MAE & 0.0092 & 0.0095 & 0.0188 & 0.0092 & 0.0095 & 0.0187 & 0.0092 & 0.0096 & 0.0184 & MAE & 0.0088 & 0.0095 & 0.0169 & 0.0088 & 0.0095 & 0.0169 & 0.0088 & 0.0096 & 0.0164 \\
Bias & -0.0075 & -0.0087 & -0.0181 & -0.0074 & -0.0086 & -0.0180 & -0.0073 & -0.0087 & -0.0178 & Bias & -0.0064 & -0.0078 & -0.0169 & -0.0064 & -0.0078 & -0.0168 & -0.0060 & -0.0079 & -0.0164 \\
RMSE & 0.0076 & 0.0089 & 0.0183 & 0.0075 & 0.0088 & 0.0182 & 0.0074 & 0.0089 & 0.0180 & RMSE & 0.0066 & 0.0081 & 0.0173 & 0.0066 & 0.0081 & 0.0172 & 0.0062 & 0.0082 & 0.0167 \\
\bottomrule
\end{tabular}
\hspace*{-1cm}
\begin{tablenotes}
\footnotesize
\item \textbf{Note:} The $\tilde{g}_{k}$ extracts the column vectors composed of non-zero elements from the upper triangular submatrix of $G_{k}$. The $\bar{d}_0=5\%$ in DGP and $\bar{d}_0=15\%$ in estimation. The results are based on 1,000 Monte Carlo replications. MAE denotes the mean absolute error. Bias denotes the mean bias of the estimates, and RMSE denotes the root mean squared error. The 2SLS refers to the two-stage least square estimator, OGMM refers to the feasible optimal GMM estimator, and BGMM refers to the feasible best GMM estimator.
\end{tablenotes}
\end{spacing}
\label{sim:heteG_d0}
\end{sidewaystable}
\begin{table}[htbp]
\centering \footnotesize \setlength{\abovecaptionskip}{0cm} \setlength{\tabcolsep}{0.8mm}
\caption{Finite sample performance of $\pi$ for the SAR, $\operatorname{Var}(\epsilon_{it})=\sigma_i^2$.}
\begin{spacing}{1}
\begin{tabular}{lcccccclcccccc}
\toprule
\multicolumn{7}{c}{$(n,T,\ell_{n})=(100,10,2)$} & \multicolumn{7}{c}{$(n,T,\ell_{n})=(100,10,[n^{1/5}]+2)$} \\
\midrule
\multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} \\
\cmidrule{2-7}\cmidrule{9-14} & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ \\
\midrule
Bias & 0.0021 & -0.0024 & 0.0108 & -0.0015 & 0.0110 & -0.0018 & Bias & -0.0020 & -0.0041 & 0.0089 & -0.0029 & 0.0087 & -0.0027 \\
ESD & 0.0361 & 0.0292 & 0.0406 & 0.0287 & 0.0416 & 0.0294 & ESD & 0.0378 & 0.0288 & 0.0424 & 0.0283 & 0.0423 & 0.0292 \\
RMSE & 0.0362 & 0.0293 & 0.0420 & 0.0288 & 0.0430 & 0.0295 & RMSE & 0.0378 & 0.0291 & 0.0433 & 0.0285 & 0.0432 & 0.0294 \\
CP & 0.7970 & 0.9070 & 0.7950 & 0.9280 & 0.7850 & 0.9250 & CP & 0.8930 & 0.9030 & 0.8830 & 0.9100 & 0.8720 & 0.8990 \\
\midrule
\multicolumn{7}{c}{$(n,T,\ell_{n})=(100,25,2)$} & \multicolumn{7}{c}{$(n,T,\ell_{n})=(100,25,[n^{1/5}]+2)$} \\
\midrule
& \multicolumn{1}{c}{2SLS} & & \multicolumn{1}{c}{OGMM} & & \multicolumn{1}{c}{BGMM} & & \multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} \\
\cmidrule{2-7}\cmidrule{9-14} &$\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ \\
\midrule
Bias & 0.0163 & -0.0002 & 0.0163 & -0.0002 & 0.0149 & 0.0000 & Bias & 0.0152 & 0.0012 & 0.0152 & 0.0012 & 0.0125 & 0.0002 \\
ESD & 0.0221 & 0.0175 & 0.0221 & 0.0163 & 0.0212 & 0.0163 & ESD & 0.0202 & 0.0173 & 0.0202 & 0.0165 & 0.0196 & 0.0164 \\
RMSE & 0.0275 & 0.0175 & 0.0275 & 0.0163 & 0.0259 & 0.0163 & RMSE & 0.0253 & 0.0173 & 0.0253 & 0.0165 & 0.0232 & 0.0164 \\
CP & 0.8000 & 0.8990 & 0.8060 & 0.9260 & 0.7960 & 0.9150 & CP & 0.8430 & 0.8880 & 0.8140 & 0.9200 & 0.8040 & 0.9090 \\
\midrule
\multicolumn{7}{c}{$(n,T,\ell_{n})=(200,10,2)$} & \multicolumn{7}{c}{$(n,T,\ell_{n})=(200,10,[n^{1/5}]+2)$} \\
\midrule
\multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} \\
\cmidrule{2-7}\cmidrule{9-14} & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ \\
\midrule
Bias & 0.0088 & -0.0017 & 0.0086 & -0.0007 & 0.0035 & -0.0006 & Bias & 0.0057 & -0.0002 & 0.0054 & -0.0003 & -0.0017 & -0.0023 \\
ESD & 0.0286 & 0.0206 & 0.0283 & 0.0203 & 0.0270 & 0.0200 & ESD & 0.0270 & 0.0211 & 0.0266 & 0.0208 & 0.0252 & 0.0206 \\
RMSE & 0.0299 & 0.0207 & 0.0296 & 0.0203 & 0.0272 & 0.0200 & RMSE & 0.0276 & 0.0211 & 0.0271 & 0.0208 & 0.0253 & 0.0207 \\
CP & 0.8960 & 0.9070 & 0.8840 & 0.9270 & 0.8730 & 0.9160 & CP & 0.8980 & 0.9140 & 0.8990 & 0.9180 & 0.8880 & 0.9070 \\
\midrule
\multicolumn{7}{c}{$(n,T,\ell_{n})=(200,25,2)$} & \multicolumn{7}{c}{$(n,T,\ell_{n})=(200,25,[n^{1/4}])$}
\\
\midrule
\multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} \\
\cmidrule{2-7}\cmidrule{9-14} & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ \\
\midrule
Bias & 0.0123 & 0.0019 & 0.0123 & 0.0019 & 0.0109 & -0.0002 & Bias & 0.0089 & 0.0017 & 0.0089 & 0.0017 & 0.0063 & -0.0006 \\
ESD & 0.0150 & 0.0118 & 0.0150 & 0.0115 & 0.0147 & 0.0115 & ESD & 0.0141 & 0.0120 & 0.0139 & 0.0115 & 0.0139 & 0.0115 \\
RMSE & 0.0194 & 0.0120 & 0.0194 & 0.0117 & 0.0183 & 0.0115 & RMSE & 0.0167 & 0.0121 & 0.0165 & 0.0116 & 0.0153 & 0.0115 \\
CP & 0.9120 & 0.9220 & 0.9270 & 0.9300 & 0.9190 & 0.9320 & CP & 0.9130 & 0.9300 & 0.9320 & 0.9360 & 0.9240 & 0.9330 \\
\midrule
\multicolumn{7}{c}{$(n,T,\ell_{n})=(400,10,2)$} & \multicolumn{7}{c}{$(n,T,\ell_{n})=(400,10,[n^{1/5}]+2)$} \\
\midrule
\multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} \\
\cmidrule{2-7}\cmidrule{9-14} & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ \\
\midrule
Bias & 0.0063 & 0.0011 & 0.0062 & 0.0011 & 0.0033 & -0.0004 & Bias & 0.0050 & 0.0018 & 0.0049 & 0.0015 & 0.0009 & 0.0000 \\
ESD & 0.0198 & 0.0145 & 0.0198 & 0.0144 & 0.0193 & 0.0144 & ESD & 0.0189 & 0.0146 & 0.0185 & 0.0145 & 0.0181 & 0.0144 \\
RMSE & 0.0208 & 0.0145 & 0.0207 & 0.0144 & 0.0196 & 0.0144 & RMSE & 0.0196 & 0.0147 & 0.0191 & 0.0146 & 0.0181 & 0.0144 \\
CP & 0.9260 & 0.9400 & 0.9360 & 0.9330 & 0.9250 & 0.9220 & CP & 0.9400 & 0.9350 & 0.9420 & 0.9420 & 0.9350 & 0.9400 \\
\midrule
\multicolumn{7}{c}{$(n,T,\ell_{n})=(400,25,2)$} & \multicolumn{7}{c}{$(n,T,\ell_{n})=(400,25,[n^{1/5}]+2)$}
\\
\midrule
\multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{2}{c}{2SLS} & \multicolumn{2}{c}{OGMM} & \multicolumn{2}{c}{BGMM} \\
\cmidrule{2-7}\cmidrule{9-14} & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ & $\gamma$ & $\beta$ \\
\midrule
Bias & 0.0073 & 0.0019 & 0.0073 & 0.0019 & 0.0070 & 0.0001 & Bias & 0.0060 & 0.0018 & 0.0060 & 0.0018 & 0.0054 & -0.0007 \\
ESD & 0.0102 & 0.0081 & 0.0102 & 0.0081 & 0.0100 & 0.0081 & ESD & 0.0095 & 0.0084 & 0.0095 & 0.0084 & 0.0095 & 0.0083 \\
RMSE & 0.0125 & 0.0083 & 0.0125 & 0.0083 & 0.0122 & 0.0081 & RMSE & 0.0112 & 0.0086 & 0.0112 & 0.0086 & 0.0109 & 0.0083 \\
CP & 0.9250 & 0.9360 & 0.9320 & 0.9380 & 0.9250 & 0.9360 & CP & 0.9340 & 0.9420 & 0.9400 & 0.9450 & 0.9410 & 0.9420 \\
\bottomrule
\end{tabular}
\hspace*{-1cm}
\begin{tablenotes}
\footnotesize
\item \textbf{Note:} The true parameters are set to $\pi_0=(0.4,0.3)'$, and $\bar{d}_0=10\%$. The results are based on 1,000 Monte Carlo replications. Bias denotes the mean bias of the estimates, ESD denotes the standard deviation, RMSE denotes the root mean squared error, and CP denotes the 95\% coverage probability. The 2SLS refers to the two-stage least square estimator, OGMM refers to the feasible optimal GMM estimator, and BGMM refers to the feasible best GMM estimator.
\end{tablenotes}
\end{spacing}
\label{sim:hei_sar}
\end{table}
\begin{table}[htbp]
\centering \footnotesize \setlength{\abovecaptionskip}{0cm} \setlength{\tabcolsep}{0.8mm}
\caption{Estimated $\rho(\hat{A})$ for the SAR, $\operatorname{Var}(\epsilon_{it})=\sigma_i^2$.}
\begin{spacing}{1}\hspace*{-0.8cm}
\begin{tabular}{lcccccclcccccc}
\toprule
\multicolumn{7}{c}{$(n,T)=(100,10)$} & \multicolumn{7}{c}{$(n,T,\ell_{n})=(100,25)$} \\
\midrule
\multirow{2}[4]{*}{} & \multicolumn{3}{c}{$\ell_{n}=2$} & \multicolumn{3}{c}{$\ell_{n}=[n^{1/5}]+2$} & \multirow{2}[4]{*}{} & \multicolumn{3}{c}{$\ell_{n}=2$} & \multicolumn{3}{c}{$\ell_{n}=[n^{1/5}]+2$} \\
\cmidrule{2-7}\cmidrule{9-14} & 2SLS &OGMM & BGMM & 2SLS &OGMM & BGMM & & 2SLS &OGMM & BGMM & 2SLS &OGMM & BGMM \\
\midrule
Mean & 0.5407 & 0.5404 & 0.5397 & 0.5597 & 0.5618 & 0.5621 & Mean & 0.5819 & 0.5456 & 0.5461 & 0.5837 & 0.5550 & 0.5558 \\
Bias & -0.1927 & -0.1929 & -0.1937 & -0.1737 & -0.1715 & -0.1712 & Bias & -0.1514 & -0.1878 & -0.1872 & -0.1496 & -0.1784 & -0.1775 \\
ESD & 0.1353 & 0.1420 & 0.1408 & 0.1327 & 0.1365 & 0.1355 & ESD & 0.0918 & 0.0827 & 0.0827 & 0.0898 & 0.0859 & 0.0862 \\
RMSE & 0.2355 & 0.2395 & 0.2394 & 0.2186 & 0.2192 & 0.2184 & RMSE & 0.1771 & 0.2052 & 0.2047 & 0.1745 & 0.1980 & 0.1974 \\
\midrule
\multicolumn{7}{c}{$(n,T)=(200,10)$} & \multicolumn{7}{c}{$(n,T,\ell_{n})=(200,25)$} \\
\midrule
\multirow{2}[4]{*}{} & \multicolumn{3}{c}{$\ell_{n}=2$} & \multicolumn{3}{c}{$\ell_{n}=[n^{1/5}]+2$} & \multirow{2}[4]{*}{} & \multicolumn{3}{c}{$\ell_{n}=2$} & \multicolumn{3}{c}{$\ell_{n}=[n^{1/5}]+2$} \\
\cmidrule{2-7}\cmidrule{9-14} & 2SLS &OGMM & BGMM & 2SLS &OGMM & BGMM & & 2SLS &OGMM & BGMM & 2SLS &OGMM & BGMM \\
\midrule
Mean & 0.6485 & 0.6274 & 0.6282 & 0.6566 & 0.6464 & 0.6454 & Mean & 0.6818 & 0.6535 & 0.6535 & 0.6901 & 0.6619 & 0.6606 \\
Bias & -0.0849 & -0.1060 & -0.1051 & -0.0768 & -0.0870 & -0.0879 & Bias & -0.0515 & -0.0799 & -0.0798 & -0.0432 & -0.0714 & -0.0728 \\
ESD & 0.1520 & 0.1542 & 0.1535 & 0.1508 & 0.1557 & 0.1535 & ESD & 0.0855 & 0.0835 & 0.0835 & 0.0803 & 0.0806 & 0.0812 \\
RMSE & 0.1741 & 0.1871 & 0.1860 & 0.1692 & 0.1783 & 0.1769 & RMSE & 0.0999 & 0.1156 & 0.1155 & 0.0912 & 0.1077 & 0.1090 \\
\midrule
\multicolumn{7}{c}{$(n,T)=(400,10)$} & \multicolumn{7}{c}{$(n,T,\ell_{n})=(400,25)$} \\
\midrule
\multirow{2}[4]{*}{} & \multicolumn{3}{c}{$\ell_{n}=2$} & \multicolumn{3}{c}{$\ell_{n}=[n^{1/5}]+2$} & \multirow{2}[4]{*}{} & \multicolumn{3}{c}{$\ell_{n}=2$} & \multicolumn{3}{c}{$\ell_{n}=[n^{1/5}]+2$} \\
\cmidrule{2-7}\cmidrule{9-14} & 2SLS &OGMM & BGMM & 2SLS &OGMM & BGMM & & 2SLS &OGMM & BGMM & 2SLS &OGMM & BGMM \\
\midrule
Mean & 0.6885 & 0.6740 & 0.6741 & 0.7008 & 0.6927 & 0.6925 & Mean & 0.7490 & 0.7131 & 0.7131 & 0.7533 & 0.7353 & 0.7355 \\
Bias & -0.0448 & -0.0593 & -0.0592 & -0.0326 & -0.0407 & -0.0409 & Bias & 0.0156 & -0.0202 & -0.0202 & 0.0200 & 0.0020 & 0.0021 \\
ESD & 0.1834 & 0.1786 & 0.1785 & 0.1724 & 0.1742 & 0.1735 & ESD & 0.1237 & 0.1162 & 0.1162 & 0.1205 & 0.1160 & 0.1161 \\
RMSE & 0.1888 & 0.1882 & 0.1881 & 0.1755 & 0.1789 & 0.1783 & RMSE & 0.1247 & 0.1179 & 0.1179 & 0.1221 & 0.1161 & 0.1161 \\
\bottomrule
\end{tabular}
\hspace*{-1cm}
\begin{tablenotes}
\footnotesize
\item \textbf{Note:} $\bar{d}_0=10\%$. The results are based on 1,000 Monte Carlo replications. Mean denotes the mean of the estimates. Bias denotes the mean bias of the estimates, ESD denotes the standard deviation, and RMSE denotes the root mean squared error. The 2SLS refers to the two-stage least square estimator, OGMM refers to the feasible optimal GMM estimator, and BGMM refers to the feasible best GMM estimator.
\end{tablenotes}
\end{spacing}
\label{sim:rhoG_sar}
\end{table}
\begin{sidewaystable}[htbp]
\centering\footnotesize\setlength{\abovecaptionskip}{0cm} \setlength{\tabcolsep}{0.8mm} \vspace*{-5mm}
\caption{Finite sample performance of $G_{1}, G_{2}$ and $G_{3}$ for the SAR, $\operatorname{Var}(\epsilon_{it})=\sigma_i^2$.}
\begin{spacing}{0.5}\hspace*{-5mm}
\begin{tabular}{lrrrrrrrrrlrrrrrrrrr}
\toprule
\multirow{4}[3]{*}{} & \multicolumn{9}{c}{$(n,T,\ell_{n})=(100,10,2)$} & & \multicolumn{9}{c}{$(n,T,\ell_{n})=(100,10,[n^{1/5}]+2)$} \\
\cmidrule{2-20}
\multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} \\
\cmidrule{2-10}\cmidrule{12-20} & \multicolumn{1}{c}{$\tilde{g}_1$} & \multicolumn{1}{c}{$\tilde{g}_2$} & \multicolumn{1}{c}{$\tilde{g}_3$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} \\
\midrule
MAE & 0.0158 & 0.0178 & - & 0.0158 & 0.0190 & 0.0193 & 0.0159 & 0.0190 & 0.0195 & MAE & 0.0119 & 0.0123 & - & 0.0117 & 0.0124 & 0.0122 & 0.0116 & 0.0124 & 0.0123 \\
Bias & 0.0017 & -0.0039 & - & -0.0060 & -0.0022 & 0.0016 & -0.0056 & -0.0023 & 0.0017 & Bias & 0.0013 & -0.0012 & - & -0.0070 & -0.0005 & 0.0012 & -0.0069 & -0.0005 & 0.0013 \\
RMSE & 0.0157 & 0.0193 & - & 0.0165 & 0.0213 & 0.0206 & 0.0167 & 0.0212 & 0.0208 & RMSE & 0.0081 & 0.0098 & - & 0.0102 & 0.0099 & 0.0084 & 0.0101 & 0.0099 & 0.0084 \\
\midrule
& \multicolumn{9}{c}{$(n,T,\ell_{n})=(100,25,2)$} & & \multicolumn{9}{c}{$(n,T,\ell_{n})=(100,25,[n^{1/5}]+2)$} \\
\cmidrule{2-20}
\multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} \\
\cmidrule{2-10}\cmidrule{12-20} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} \\
\midrule
MAE & 0.0124 & 0.0124 & - & 0.0117 & 0.0129 & 0.0122 & 0.0117 & 0.0129 & 0.0122 & MAE & 0.0119 & 0.0123 & - & 0.0117 & 0.0124 & 0.0122 & 0.0116 & 0.0124 & 0.0123 \\
Bias & 0.0026 & -0.0014 & - & -0.0056 & -0.0001 & 0.0018 & -0.0054 & -0.0001 & 0.0019 & Bias & 0.0013 & -0.0012 & - & -0.0070 & -0.0005 & 0.0012 & -0.0069 & -0.0005 & 0.0013 \\
RMSE & 0.0099 & 0.0110 & - & 0.0103 & 0.0115 & 0.0097 & 0.0103 & 0.0115 & 0.0097 & RMSE & 0.0081 & 0.0098 & - & 0.0102 & 0.0099 & 0.0084 & 0.0101 & 0.0099 & 0.0084 \\
\midrule
& \multicolumn{9}{c}{$(n,T,\ell_{n})=(200,10,2)$} & & \multicolumn{9}{c}{$(n,T,\ell_{n})=(200,10,[n^{1/5}]+2)$} \\
\cmidrule{2-20}
\multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} \\
\cmidrule{2-10}\cmidrule{12-20} & \multicolumn{1}{c}{$\tilde{g}_1$} & \multicolumn{1}{c}{$\tilde{g}_2$} & \multicolumn{1}{c}{$\tilde{g}_3$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} \\
\midrule
MAE & 0.0082 & 0.0090 & - & 0.0077 & 0.0094 & 0.0093 & 0.0078 & 0.0094 & 0.0092 & MAE & 0.0074 & 0.0079 & - & 0.0069 & 0.0083 & 0.0071 & 0.0069 & 0.0083 & 0.0071 \\
Bias & 0.0022 & -0.0026 & - & -0.0015 & -0.0013 & 0.0001 & -0.0015 & -0.0013 & 0.0000 & Bias & 0.0011 & -0.0025 & - & -0.0030 & -0.0013 & -0.0004 & -0.0029 & -0.0013 & -0.0004 \\
RMSE & 0.0083 & 0.0100 & - & 0.0081 & 0.0105 & 0.0100 & 0.0083 & 0.0105 & 0.0099 & RMSE & 0.0061 & 0.0079 & - & 0.0065 & 0.0084 & 0.0061 & 0.0065 & 0.0083 & 0.0061 \\
\midrule
& \multicolumn{9}{c}{$(n,T,\ell_{n})=(200,25,2)$} & & \multicolumn{9}{c}{$(n,T,\ell_{n})=(200,25,[n^{1/5}]+2)$} \\
\cmidrule{2-20}
\multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} \\
\cmidrule{2-10}\cmidrule{12-20} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} \\
\midrule
MAE & 0.0064 & 0.0060 & - & 0.0057 & 0.0059 & 0.0058 & 0.0057 & 0.0059 & 0.0058 & MAE & 0.0063 & 0.0058 & - & 0.0056 & 0.0059 & 0.0056 & 0.0056 & 0.0059 & 0.0056 \\
Bias & 0.0029 & -0.0018 & - & -0.0004 & -0.0009 & 0.0000 & -0.0005 & -0.0009 & -0.0001 & Bias & 0.0017 & -0.0017 & - & -0.0011 & -0.0010 & -0.0008 & -0.0011 & -0.0010 & -0.0008 \\
RMSE & 0.0054 & 0.0055 & - & 0.0047 & 0.0053 & 0.0048 & 0.0048 & 0.0053 & 0.0047 & RMSE & 0.0039 & 0.0045 & - & 0.0036 & 0.0044 & 0.0037 & 0.0036 & 0.0044 & 0.0037 \\
\midrule
& \multicolumn{9}{c}{$(n,T,\ell_{n})=(400,10,2)$} & & \multicolumn{9}{c}{$(n,T,\ell_{n})=(400,10,[n^{1/5}]+2)$} \\
\cmidrule{2-20}
\multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} \\
\cmidrule{2-10}\cmidrule{12-20} & \multicolumn{1}{c}{$\tilde{g}_1$} & \multicolumn{1}{c}{$\tilde{g}_2$} & \multicolumn{1}{c}{$\tilde{g}_3$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} \\
\midrule
MAE & 0.0042 & 0.0046 & - & 0.0039 & 0.0048 & 0.0040 & 0.0039 & 0.0048 & 0.0040 & MAE & 0.0039 & 0.0043 & - & 0.0035 & 0.0044 & 0.0037 & 0.0035 & 0.0044 & 0.0037 \\
Bias & 0.0016 & -0.0004 & - & 0.0003 & 0.0002 & 0.0007 & 0.0003 & 0.0002 & 0.0008 & Bias & 0.0013 & -0.0005 & - & -0.0001 & 0.0001 & 0.0004 & -0.0001 & 0.0001 & 0.0005 \\
RMSE & 0.0044 & 0.0052 & - & 0.0040 & 0.0054 & 0.0045 & 0.0041 & 0.0054 & 0.0044 & RMSE & 0.0035 & 0.0045 & - & 0.0031 & 0.0046 & 0.0033 & 0.0031 & 0.0046 & 0.0033 \\
\midrule
& \multicolumn{9}{c}{$(n,T,\ell_{n})=(400,25,2)$} & & \multicolumn{9}{c}{$(n,T,\ell_{n})=(400,25,[n^{1/5}]+2)$} \\
\cmidrule{2-20}
\multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} & \multirow{2}[4]{*}{} & \multicolumn{3}{c}{2SLS} & \multicolumn{3}{c}{OGMM} & \multicolumn{3}{c}{BGMM} \\
\cmidrule{2-10}\cmidrule{12-20} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} & \multicolumn{1}{c}{$\tilde{g}_{1}$} & \multicolumn{1}{c}{$\tilde{g}_{2}$} & \multicolumn{1}{c}{$\tilde{g}_{3}$} \\
\midrule
MAE & 0.0035 & 0.0031 & - & 0.0032 & 0.0031 & 0.0027 & 0.0032 & 0.0031 & 0.0027 & MAE & 0.0034 & 0.0032 & - & 0.0029 & 0.0031 & 0.0025 & 0.0029 & 0.0031 & 0.0025 \\
Bias & 0.0022 & 0.0001 & - & 0.0016 & 0.0002 & 0.0005 & 0.0016 & 0.0002 & 0.0005 & Bias & 0.0017 & 0.0003 & - & 0.0008 & 0.0003 & 0.0000 & 0.0008 & 0.0003 & 0.0000 \\
RMSE & 0.0033 & 0.0030 & - & 0.0028 & 0.0030 & 0.0022 & 0.0028 & 0.0030 & 0.0022 & RMSE & 0.0026 & 0.0026 & - & 0.0017 & 0.0026 & 0.0015 & 0.0017 & 0.0026 & 0.0015 \\
\bottomrule
\end{tabular}
\hspace*{-1cm}
\begin{tablenotes}
\footnotesize
\item \textbf{Note:} The $\tilde{g}_{k}$ extracts the column vectors composed of non-zero elements from the upper triangular submatrix of $G_{k}$, and $\bar{d}_0=10\%$. The results are based on 1,000 Monte Carlo replications. MAE denotes the mean absolute error. Bias denotes the mean bias of the estimates, and RMSE denotes the root mean squared error. The 2SLS refers to the two-stage least square estimator, OGMM refers to the feasible optimal GMM estimator, and BGMM refers to the feasible best GMM estimator.
\end{tablenotes}
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\label{sim:heteG_sar}
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