Semi-nonparametric estimation of spatial dynamic panel data models with nonparametric spatial weights

lemmaLet $A$ and $B$ be $n \times n$ real matrices, then $\|AB\|\leq\|A\|_{\mathrm{sp}}\|B\|$ or $\|AB\|\leq\|B\|_{\mathrm{sp}}\|A\|$.

proofThis is a standard matrix norm inequality. For a formal discussion and proof, see Section 5.6 of horn2012matrix.

lemma(\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 $\sup_{t} \|\mathcal{B}_{t}\|_{\mathrm{sp}} = O_p(1)$

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, $0 \leq \sup_{t} \|\mathcal{B}_{t}\|_{\mathrm{sp}} \leq \sqrt{XY}= O_p(1)$.

lemmaLet 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: $$ \|e^A - I\|=O_{p}(\|A\|)=O_{p}(n^{1/2}d^{-\nu}). $$

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 $\|e^A - I\|=O_{p}(\|A\|)$.

lemmaWhen $T \to\infty$, $\sum_{t=1}^{T-1}h_{Tt}^2=O(T)$ and $\sum_{t=1}^{T-1}(1-h_{Tt}^2)=O(\ln{T})$, where $h_{Tt}=\sqrt{\frac{T-t}{T-t+1}}$.

proofNote 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 $\sum_{t=1}^{T-1}\frac{1}{T-t+1}=O(\ln{T})$.

lemmaUnder Assumption (ref) 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)(\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); \\ (\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 (ref).

proofFor (\romannumeral1), first, from Lemma (ref), 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) (\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) and (ref), 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 (ref) and (ref), we have \begin{flalign} 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 $\rho(A)<1$.

lemmaUnder Assumption (ref)(\romannumeral1), the covariance of $m_{N}^{\mathtt{line}}(\theta_{0})$ and $m_{N}^{\mathtt{quad}}(\theta_{0})$ is asymptotically zero as $(n,T)\to\infty$.

proofSimilar to the Proof of Theorem (ref).

lemmaUnder Assumption (ref), $\|M_{t}\|_{\mathrm{rc}}=O_{p}\left(\ell_n\right)$, and $\|M_{t}\|=O_{p}\left(\sqrt{\ell_n}\right)$.

proofNote 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 $\|M_{t}\|^2=O_{p}(\ell_{n})$.

lemmaFor $n\times n$ 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), (ref) and (ref):\\ (\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) $\operatorname{E}|Q_{jt}'\mathcal{B}_{t}(r_{kt}^{*}+E_{t}^{*})|=O(\sqrt{n}h_{Tt}\ell_{k}^{-\varsigma_{k}}b)$.

proofFor (\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) and (ref), 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).

lemmaUnder Assumption (ref), $\operatorname{Var}( \ifstrequal{t}{N}{ \pmb{M}({\bm{\Sigma}_{t}}) }{ {M}({\Sigma_{t}}) } \Sigma_{t}^{-1/2}E_{t}^{*})= \ifstrequal{t}{N}{ \pmb{M}({\bm{\Sigma}_{t}}) }{ {M}({\Sigma_{t}}) } +o(1)$.

proofFor the case of (ref) and (ref), 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). 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) 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 $T \to \infty$.

lemmaUnder Assumptions (ref)-(ref), $\operatorname{E}\|m_{N}(\theta_{0})\|=o(1)$.

proofObserve 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\varsigma}+\ell_{n}^{-\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).

lemma(\romannumeral1) For the MESS, when $j=1,...,\ell_{k}$ and $k=1,2,3$, under Assumption (ref) and $S_{k}$'s satisfy (ref), 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 $\lambda_{kj_0}$.

proofSince $\varPhi_{kj}$ 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). 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*}

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).

flalignL_{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),

sidewaystable[htbp] {0cm} {0.8mm} \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} \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} \begin{tablenotes} • 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}

sidewaystable[htbp] {0cm} {0.8mm} \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} \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} \begin{tablenotes} • 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} \end{spacing}

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