O_{p}\left(\varsigma(r)\max\left\\{\sqrt{d_{\tau}/n},\sqrt{\sum_{\ell=1}^{d_{\zeta}}r_{\ell}^{-2\kappa_{\ell}}}\right\\}\right),$

uniformly on $\mathcal{Z}$. Thus we have

$\left\|\Sigma\left(\widehat{\tau}\right)^{-1}-\Sigma^{-1}\right\|\leq\left\|\Sigma\left(\widehat{\tau}\right)^{-1}\right\|\left\|\Sigma\left(\widehat{\tau}\right)-\Sigma\right\|\left\|\Sigma^{-1}\right\|=O_{p}\left(\varsigma(r)\max\left\\{\sqrt{d_{\tau}/n},\sqrt{\sum_{\ell=1}^{d_{\zeta}}r_{\ell}^{-2\kappa_{\ell}}}\right\\}\right).$

And similarly,

$\displaystyle\left\|\left(\frac{1}{n}\Psi^{\prime}\Sigma\left(\widehat{\tau}\right)^{-1}\Psi\right)^{-1}-\left(\frac{1}{n}\Psi^{\prime}\Sigma^{-1}\Psi\right)^{-1}\right\|$
$\displaystyle\leq$
$\displaystyle\left\|\left(\frac{1}{n}\Psi^{\prime}\Sigma\left(\widehat{\tau}\right)^{-1}\Psi\right)^{-1}\right\|\left\|\frac{1}{n}\Psi^{\prime}\left(\Sigma\left(\widehat{\tau}\right)^{-1}-\Sigma^{-1}\right)\Psi\right\|\left\|\left(\frac{1}{n}\Psi^{\prime}\Sigma^{-1}\Psi\right)^{-1}\right\|$
$\displaystyle=$ $\displaystyle
O_{p}\left(\left\|\Sigma\left(\widehat{\tau}\right)^{-1}-\Sigma^{-1}\right\|\right)=O_{p}\left(\varsigma(r)\max\left\\{\sqrt{d_{\tau}/n},\sqrt{\sum_{\ell=1}^{d_{\zeta}}r_{\ell}^{-2\kappa_{\ell}}}\right\\}\right).$

As in the proof of Theorem 4.2,
$n\widehat{m}_{n}=\widehat{\sigma}^{-2}{u}^{\prime}\Sigma\left(\widehat{\tau}\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\widehat{\tau}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\tau}\right)^{-1}{u}+\widehat{\sigma}^{-2}\sum_{k=1}^{4}A_{k},$
where $\gamma$ in the parametric setting is changed to $\tau$ in this
nonparametric setting. Then, by the MVT,

$\displaystyle\left|u^{\prime}\left(\Sigma\left(\widehat{\tau}\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\widehat{\tau}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\tau}\right)^{-1}-\Sigma^{-1}\Psi[\Psi^{\prime}\Sigma^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma^{-1}\right)u\right|$
$\displaystyle\leq$ $\displaystyle
2\left(\sup_{t}\left\|\frac{1}{\sqrt{n}}u^{\prime}\Sigma\left(t\right)^{-1}\Psi\right\|\left\|\left(\frac{1}{n}\Psi^{\prime}\Sigma\left(t\right)^{-1}\Psi\right)^{-1}\right\|\right)\sum_{j=1}^{d_{\tau}}\left\|\frac{1}{\sqrt{n}}\Psi^{\prime}\left(\Sigma\left(\widetilde{\tau}\right)^{-1}\Sigma_{j}\left(\widetilde{\tau}\right)\Sigma\left(\widetilde{\tau}\right)^{-1}\right)u\right\|$
$\displaystyle\times$
$\displaystyle\left|\widetilde{\tau}_{j}-\tau_{j0}\right|+2\sup_{t}\left\|\frac{1}{\sqrt{n}}u^{\prime}\Sigma\left(t\right)^{-1}\Psi\right\|\left\|\left(\frac{1}{n}\Psi^{\prime}\Sigma\left(t\right)^{-1}\Psi\right)^{-1}\right\|\left\|\frac{1}{\sqrt{n}}\Psi^{\prime}\left(\Sigma_{0}-\Sigma\right)u\right\|$
$\displaystyle+\left\|\frac{1}{\sqrt{n}}u^{\prime}\Sigma^{-1}\Psi\right\|^{2}\left\|\left(\frac{1}{n}\Psi^{\prime}\Sigma\left(\widehat{\tau}\right)^{-1}\Psi\right)^{-1}-\left(\frac{1}{n}\Psi^{\prime}\Sigma^{-1}\Psi\right)^{-1}\right\|$
$\displaystyle=$ $\displaystyle
O_{p}(\sqrt{p})O_{p}(d_{\tau}\sqrt{p}\varsigma(r)/\sqrt{n})+O_{p}(\sqrt{p})O_{p}\left(\sqrt{p}\varsigma(r)\sqrt{\sum_{\ell=1}^{d_{\zeta}}r_{\ell}^{-2\kappa_{\ell}}}\right)$
$\displaystyle+$ $\displaystyle
O_{p}(p)O_{p}\left(\varsigma(r)\max\left\\{\sqrt{d_{\tau}/n},\sqrt{\sum_{\ell=1}^{d_{\zeta}}r_{\ell}^{-2\kappa_{\ell}}}\right\\}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(p\varsigma(r)\max\left\\{d_{\tau}/\sqrt{n},\sqrt{\sum_{\ell=1}^{d_{\zeta}}r_{\ell}^{-2\kappa_{\ell}}}\right\\}\right)=o_{p}(\sqrt{p}),$

where the last equality holds under the conditions of the theorem. Next, it
remains to show $A_{k}=o_{p}(p^{1/2}),k=1,\ldots,4$. The order of $A_{k}$,
$k\leq 3$, is the same as the parametric case:

$\displaystyle\left|A_{1}\right|$ $\displaystyle=$
$\displaystyle\left|{u}^{\prime}\Sigma\left(\widehat{\tau}\right)^{-1}\left({\theta}_{0}-\widehat{{f}}\right)\right|\leq\sup_{\alpha,t}\left\|u^{\prime}\Sigma\left(t\right)^{-1}\frac{\partial{f}(x,{\alpha})}{\partial\alpha_{j}}\right\|\left|\alpha_{j}^{\ast}-\widetilde{\alpha}_{j}\right|+\frac{p^{1/4}}{n^{1/2}}\sup_{t}\left\|u^{\prime}\Sigma\left(t\right)^{-1}h\right\|$
$\displaystyle=$ $\displaystyle
O_{p}(\sqrt{n})O_{p}(\frac{1}{\sqrt{n}})+O(\frac{p^{1/4}}{n^{1/2}})O_{p}(\sqrt{n})=O_{p}(p^{1/4})=o_{p}(p^{1/2}),$
$\displaystyle|A_{2}|$ $\displaystyle=$
$\displaystyle\left|(u\mathbf{+}\theta_{0}-\widehat{f})^{\prime}\left(\Sigma\left(\widehat{\tau}\right)^{-1}-\Sigma\left(\widehat{\tau}\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\widehat{\tau}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\tau}\right)^{-1}\right)e\right|$
$\displaystyle\leq$
$\displaystyle\sup_{t}|u^{\prime}\Sigma\left(t\right)^{-1}e|+\sup_{t}\left|u^{\prime}\Sigma\left(t\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(t\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(t\right)^{-1}e\right|$
$\displaystyle+\left\|{\theta}_{0}-\widehat{{f}}\right\|\sup_{t}\left(\left\|\Sigma\left(t\right)^{-1}\right\|+\left\|\Sigma\left(t\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(t\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(t\right)^{-1}\right\|\right)\left\|e\right\|$
$\displaystyle=$ $\displaystyle
O_{p}(p^{-\mu}n^{1/2})+O_{p}(p^{-\mu+1/4}n^{1/2})=O_{p}(p^{-\mu+1/4}n^{1/2})=o_{p}(\sqrt{p}),$
$\displaystyle\left|A_{3}\right|$ $\displaystyle=$
$\displaystyle\left|{u}^{\prime}\Sigma\left(\widehat{\tau}\right)^{-1}\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\tau}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\tau}\right)^{-1}({\theta}_{0}-\widehat{{f}})\right|$
$\displaystyle\leq$
$\displaystyle\sup_{\alpha,t}\sum_{j=1}^{d_{\alpha}}\left\|u^{\prime}\Sigma\left(t\right)^{-1}\Psi\left(\Psi^{\prime}\Sigma\left(t\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(t\right)^{-1}\frac{\partial{f}(x,{\alpha})}{\partial\alpha_{j}}\right\|\left|\alpha_{j}^{\ast}-\widetilde{\alpha}_{j}\right|$
$\displaystyle+\frac{p^{1/4}}{n^{1/2}}\sup_{t}\left\|u^{\prime}\Sigma\left(t\right)^{-1}\Psi\left(\Psi^{\prime}\Sigma\left(t\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(t\right)^{-1}h\right\|$
$\displaystyle=$ $\displaystyle
O_{p}(1)+O_{p}(p^{1/4})=O_{p}(p^{1/4})=o_{p}(p^{1/2}).$

However, $A_{4}$ has a different order. Under $H_{\ell}$,

$\displaystyle A_{4}$ $\displaystyle=$
$\displaystyle\left({\theta}_{0}-\widehat{{f}}\right)^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\left({\theta}_{0}-\widehat{{f}}\right)$
$\displaystyle=$
$\displaystyle\left({\theta}_{0}-\widehat{{f}}\right)^{\prime}\Sigma_{0}^{-1}\left({\theta}_{0}-\widehat{{f}}\right)+\left({\theta}_{0}-\widehat{{f}}\right)^{\prime}\left(\Sigma\left(\widehat{\tau}\right)^{-1}-\Sigma^{-1}\right)\left({\theta}_{0}-\widehat{{f}}\right)$
$\displaystyle=$
$\displaystyle\frac{p^{1/2}}{n}h^{\prime}\Sigma_{0}^{-1}h+o_{p}(1)+O_{p}\left(p^{1/2}\right)O_{p}\left(\varsigma(r)\max\left\\{\sqrt{d_{\tau}/n},\sqrt{\sum_{\ell=1}^{d_{\zeta}}r_{\ell}^{-2\kappa_{\ell}}}\right\\}\right)$
$\displaystyle=$
$\displaystyle\frac{p^{1/2}}{n}h^{\prime}\Sigma_{0}^{-1}h+o_{p}(\sqrt{p}),$

where the last equality holds under the conditions of the theorem. Combining
these together, we have
$n\widehat{m}_{n}=\widehat{\sigma}^{-2}\widehat{{v}}^{\prime}\Sigma\left(\widehat{\tau}\right)^{-1}\widehat{{u}}={\sigma_{0}^{-2}}\varepsilon^{\prime}\mathscr{V}\varepsilon+\left({p^{1/2}}/{n}\right){h}^{\prime}\Sigma_{0}^{-1}{h}+o_{p}(\sqrt{p}),$
under $H_{\ell}$ and the same expression holds with $h=0$ under $H_{0}$. ∎

###### Proof of Theorem 6.3.

Omitted as it is similar to the proof of Theorem 4.4. ∎

Supplementary online appendix to ‘Consistent specification testing under
spatial dependence’

Abhimanyu Gupta and Xi Qu

##  Appendix S.A Additional simulation results: Unboundedly supported
regressors and asymptotic critical values

This section provides additional simulation results using the same design as
in Section 8 of the main body of the paper. Recall that the paper reports only
bootstrap results for the compactly supported regressors case. Here we include
results using asymptotic critical values for both the compactly and unbounded
supported regressor cases, as well as bootstrap results for the latter,
focusing on the SARARMA(0,1,0) model. The results are in Tables OT.1-OT.4 and
our findings match those in the main text, with the bootstrap typically
offering better size control.

##  Appendix S.B Proofs of Theorems 4.2 and 4.4

###### Proof of Theorem 4.2.

From Corollary 4.1 and Lemma LS.2,
$\left\|\Sigma\left(\widehat{\gamma}\right)-\Sigma\right\|=O_{p}\left(\left\|\widehat{\gamma}-\gamma_{0}\right\|\right)=\sqrt{d_{\gamma}/n}$,
so we have, from Assumption R.3,

$\left\|\Sigma\left(\widehat{\gamma}\right)^{-1}-\Sigma^{-1}\right\|\leq\left\|\Sigma\left(\widehat{\gamma}\right)^{-1}\right\|\left\|\Sigma\left(\widehat{\gamma}\right)-\Sigma\right\|\left\|\Sigma^{-1}\right\|=O_{p}\left(\left\|\widehat{\gamma}-\gamma_{0}\right\|\right)=\sqrt{d_{\gamma}/n}.$
(S.B.1)

Similarly,

$\displaystyle\left\|\left(\frac{1}{n}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}-\left(\frac{1}{n}\Psi^{\prime}\Sigma^{-1}\Psi\right)^{-1}\right\|$
$\displaystyle\leq$
$\displaystyle\left\|\left(\frac{1}{n}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\right\|\left\|\frac{1}{n}\Psi^{\prime}\left(\Sigma\left(\widehat{\gamma}\right)^{-1}-\Sigma^{-1}\right)\Psi\right\|\left\|\left(\frac{1}{n}\Psi^{\prime}\Sigma^{-1}\Psi\right)^{-1}\right\|$
$\displaystyle\leq$
$\displaystyle\sup_{\gamma\in\Gamma}\left\|\left(\frac{1}{n}\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi\right)^{-1}\right\|\left\|\Sigma\left(\widehat{\gamma}\right)^{-1}-\Sigma^{-1}\right\|\left\|\frac{1}{\sqrt{n}}\Psi\right\|^{2}=O_{p}\left(\left\|\widehat{\gamma}-\gamma_{0}\right\|\right)=\sqrt{d_{\gamma}/n}.$

By Assumption R.2, we have $\widehat{\alpha}-\alpha^{\ast}=O_{p}(1/\sqrt{n})$.
Denote by $\theta^{\ast}(x)=\psi(x)^{\prime}\beta^{\ast}$, where
$\beta^{\ast}=\operatorname*{arg\,min}_{\beta}\mathcal{E}[y_{i}-\psi(x_{i})^{\prime}\beta)]^{2}$,
and set $\theta_{ni}=\theta(x_{i})$, $\theta_{0i}=\theta_{0}(x_{i})$,
$\widehat{\theta}_{i}=\psi_{i}^{\prime}\widehat{\beta}$,
$\widehat{f}_{i}=f(x_{i},\widehat{\alpha})$,
$f_{i}^{\ast}=f(x_{i},\alpha^{\ast})$. Then
$\widehat{u}_{i}=y_{i}-f(x_{i},\widehat{\alpha})=u_{i}+\theta_{0i}-\widehat{f}_{i}$.
Let
${\theta_{0}}=(\theta_{0}\left(x_{1}\right),\ldots,\theta_{0}\left(x_{n}\right))^{\prime}$
as before, with similar component-wise notation for the $n$-dimensional
vectors ${\theta^{\ast}}$, $\widehat{f}$, and $u$. As the approximation error
is ${e}={\theta}_{0}-{\theta}^{\ast}={\theta}_{0}-\Psi\beta^{\ast}$,

$\displaystyle\widehat{{\theta}}-{\theta}^{\ast}$ $\displaystyle=$
$\displaystyle\Psi(\widehat{\beta}-\beta^{\ast})=\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}({u+\theta}_{0}-\Psi\beta^{\ast})$
$\displaystyle=$
$\displaystyle\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}({u+e}),$

so that

$\displaystyle n\widehat{m}_{n}$ $\displaystyle=$
$\displaystyle\widehat{\sigma}^{-2}\widehat{{v}}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\widehat{{u}}=\widehat{\sigma}^{-2}\left(\widehat{{\theta}}-\widehat{f}\right)^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\left(y-\widehat{f}\right)$
$\displaystyle=$
$\displaystyle\widehat{\sigma}^{-2}\left(\widehat{{\theta}}-{\theta}^{\ast}+{\theta}^{\ast}-{\theta}_{0}+{\theta}_{0}-\widehat{{f}}\right)^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\left({u+\theta}_{0}-\widehat{{f}}\right)$
$\displaystyle=$
$\displaystyle\widehat{\sigma}^{-2}\left[\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}({u+e}{)}-{e}+{\theta}_{0}-\widehat{{f}}\right]^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\left({u+\theta}_{0}-\widehat{{f}}\right)$
$\displaystyle=$
$\displaystyle\widehat{\sigma}^{-2}{u}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}{u}+\widehat{\sigma}^{-2}{{u}^{\prime}}\Sigma\left(\widehat{\gamma}\right)^{-1}{\left({\theta}_{0}-\widehat{{f}}\right)}$
$\displaystyle{-}$
$\displaystyle\widehat{\sigma}^{-2}\left({u+\theta}_{0}-\widehat{{f}}\right)^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\left(I-\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\right){e}$
$\displaystyle+$
$\displaystyle\widehat{\sigma}^{-2}\left({\theta}_{0}-\widehat{{f}}\right)^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}{u}$
$\displaystyle+$
$\displaystyle\widehat{\sigma}^{-2}({\theta}_{0}-\widehat{{f}})^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}({\theta}_{0}-\widehat{{f}})$
$\displaystyle=$
$\displaystyle\widehat{\sigma}^{-2}{u}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}u+\widehat{\sigma}^{-2}\left(A_{1}+A_{2}+A_{3}+A_{4}\right),$

say. First, for any vector $g$ comprising of conditioned random variables,

$\mathcal{E}\left[\left(u^{\prime}\Sigma(\gamma)^{-1}{g}\right)^{2}\right]=g^{\prime}\Sigma(\gamma)^{-1}\Sigma\Sigma(\gamma)^{-1}{g}\leq\sup_{\gamma\in\Gamma}\left\|\Sigma(\gamma)^{-1}\right\|^{2}\left\|\Sigma\right\|\left\|g\right\|^{2}=O_{p}\left(\left\|g\right\|^{2}\right),$

uniformly in $\gamma\in\Gamma$, where the expectation is taken conditional on
$g$. Similarly,

$\displaystyle\mathcal{E}\left[\left(u^{\prime}\Sigma(\gamma)^{-1}\Psi\left(\Psi^{\prime}\Sigma(\gamma)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma(\gamma)^{-1}{g}\right)^{2}\right]$
$\displaystyle=$ $\displaystyle
g^{\prime}\Sigma(\gamma)^{-1}\Psi\left(\Psi^{\prime}\Sigma(\gamma)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma(\gamma)^{-1}\Sigma\Sigma(\gamma)^{-1}\Psi\left(\Psi^{\prime}\Sigma(\gamma)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma(\gamma)^{-1}{g}$
$\displaystyle\leq$
$\displaystyle\sup_{\gamma\in\Gamma}\left\|\Sigma(\gamma)^{-1}\right\|^{4}\left\|\Sigma\right\|\left\|\frac{1}{n}\Psi\left(\frac{1}{n}\Psi^{\prime}\Sigma(\gamma)^{-1}\Psi\right)^{-1}\Psi^{\prime}\right\|^{2}\left\|g\right\|^{2}=O_{p}\left(\left\|g\right\|^{2}\right),$

uniformly and, for any $j=1$, …, $d_{\gamma}$,

$\displaystyle\mathcal{E}\left[\left(u^{\prime}\Sigma(\gamma)^{-1}\Sigma_{j}\left(\gamma\right)\Sigma\left(\gamma\right)^{-1}{g}\right)^{2}\right]$
$\displaystyle=$ $\displaystyle
g^{\prime}\Sigma(\gamma)^{-1}\Sigma_{j}\left(\gamma\right)\Sigma\left(\gamma\right)^{-1}\Sigma\Sigma(\gamma)^{-1}\Sigma_{j}\left(\gamma\right)\Sigma\left(\gamma\right)^{-1}{g}$
$\displaystyle\leq$
$\displaystyle\sup_{\gamma\in\Gamma}\left\|\Sigma(\gamma)^{-1}\right\|^{4}\left\|\Sigma_{j}\left(\gamma\right)\right\|^{2}\left\|\Sigma\right\|\left\|g\right\|^{2}=O_{p}\left(\left\|g\right\|^{2}\right).$

Let $\Psi_{k}$ be the $k$-th column of $\Psi$, $k=1,\ldots,p$. Then, we have
$\left\|\Psi_{k}/\sqrt{n}\right\|=O_{p}(1)$ and for any $\gamma\in\Gamma$,

$\displaystyle\mathcal{E}\left\|\frac{1}{\sqrt{n}}u^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi\right\|^{2}$
$\displaystyle\leq$
$\displaystyle{\sum_{k=1}^{p}\mathcal{E}\left(u^{\prime}\Sigma\left(\gamma\right)^{-1}\frac{1}{\sqrt{n}}\Psi_{k}\right)^{2}}=O_{p}\left({p}\right),$
$\displaystyle\mathcal{E}\left\|\frac{1}{\sqrt{n}}u^{\prime}\Sigma\left(\gamma\right)^{-1}\Sigma_{j}\left(\gamma\right)\Sigma\left(\gamma\right)^{-1}\Psi\right\|^{2}$
$\displaystyle\leq$
$\displaystyle{\sum_{k=1}^{p}\mathcal{E}\left(u^{\prime}\Sigma\left(\gamma\right)^{-1}\Sigma_{j}\left(\gamma\right)\Sigma\left(\gamma\right)^{-1}\frac{1}{\sqrt{n}}\Psi_{k}\right)^{2}}=O({p}).$

Therefore, by Chebyshev’s inequality,

$\sup_{\gamma\in\Gamma}\left\|\frac{1}{\sqrt{n}}u^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi\right\|=O_{p}(\sqrt{p})\text{
\ \ and \ \
}\sup_{\gamma\in\Gamma}\left\|\frac{1}{\sqrt{n}}u^{\prime}\Sigma\left(\gamma\right)^{-1}\Sigma_{j}\left(\gamma\right)\Sigma\left(\gamma\right)^{-1}\Psi\right\|=O_{p}(\sqrt{p}).$

By the decomposition

$\displaystyle
u^{\prime}\left(\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}-\Sigma^{-1}\Psi[\Psi^{\prime}\Sigma^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma^{-1}\right)u$
$\displaystyle=$ $\displaystyle
u^{\prime}\left(\Sigma\left(\widehat{\gamma}\right)^{-1}+\Sigma^{-1}\right)\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\left(\sum_{i=1}^{n}e_{in}e_{in}^{\prime}\right)\left(\Sigma\left(\widehat{\gamma}\right)^{-1}-\Sigma^{-1}\right)u$
$\displaystyle+u^{\prime}\Sigma^{-1}\Psi\left([\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}-[\Psi^{\prime}\Sigma^{-1}\Psi]^{-1}\right)\Psi^{\prime}\Sigma^{-1}u$
$\displaystyle=$ $\displaystyle
u^{\prime}\left(\Sigma\left(\widehat{\gamma}\right)^{-1}+\Sigma^{-1}\right)\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\left(\sum_{i=1}^{n}e_{in}e_{in}^{\prime}\right)\sum_{j=1}^{d_{\gamma}}\left(\Sigma\left(\widetilde{\gamma}\right)^{-1}\Sigma_{j}\left(\widetilde{\gamma}\right)\Sigma\left(\widetilde{\gamma}\right)^{-1}\right)$
$\displaystyle\times$ $\displaystyle
u(\widetilde{\gamma}_{j}-\gamma_{j0})+u^{\prime}\Sigma^{-1}\Psi\left([\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}-[\Psi^{\prime}\Sigma^{-1}\Psi]^{-1}\right)\Psi^{\prime}\Sigma^{-1}u,$

where $e_{in}$ is an $n\times 1$ vector with $i$-th entry one and zeros
elsewhere, so $\sum_{i=1}^{n}e_{in}e_{in}^{\prime}=I_{n}$, and

$\displaystyle
e_{in}^{\prime}\left(\Sigma\left(\widehat{\gamma}\right)^{-1}-\Sigma^{-1}\right)u$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{d_{\gamma}}e_{in}^{\prime}\left(\Sigma\left(\widetilde{\gamma}\right)^{-1}\Sigma_{j}\left(\widetilde{\gamma}\right)\Sigma\left(\widetilde{\gamma}\right)^{-1}\right)u(\widetilde{\gamma}_{j}-\gamma_{j0})$
$\displaystyle=$ $\displaystyle
e_{in}^{\prime}\sum_{j=1}^{d_{\gamma}}\left(\Sigma\left(\widetilde{\gamma}\right)^{-1}\Sigma_{j}\left(\widetilde{\gamma}\right)\Sigma\left(\widetilde{\gamma}\right)^{-1}\right)u(\widetilde{\gamma}_{j}-\gamma_{j0})$

where $\widetilde{\gamma}$ is a value between $\widehat{\gamma}$ and
$\gamma_{0}$ due to the mean value theorem. We have

$\displaystyle\left|u^{\prime}\left(\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}-\Sigma^{-1}\Psi[\Psi^{\prime}\Sigma^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma^{-1}\right)u\right|$
$\displaystyle\leq$ $\displaystyle
2\sup_{\gamma\in\Gamma}\left\|\frac{1}{\sqrt{n}}u^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi\right\|\left\|\left(\frac{1}{n}\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi\right)^{-1}\right\|\sum_{j=1}^{d_{\gamma}}\left\|\frac{1}{\sqrt{n}}\Psi^{\prime}\left(\Sigma\left(\gamma\right)^{-1}\Sigma_{j}\left(\gamma\right)\Sigma\left(\gamma\right)^{-1}\right)u\right\|$
$\displaystyle\times$
$\displaystyle\left|\widetilde{\gamma}_{j}-\gamma_{j0}\right|+\left\|\frac{1}{\sqrt{n}}u^{\prime}\Sigma^{-1}\Psi\right\|^{2}\left\|\left(\frac{1}{n}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}-\left(\frac{1}{n}\Psi^{\prime}\Sigma^{-1}\Psi\right)^{-1}\right\|$
$\displaystyle=$ $\displaystyle
O_{p}(\sqrt{p})O_{p}(d_{\gamma}\sqrt{p}/\sqrt{n})+O_{p}(p)O_{p}(\sqrt{d_{\gamma}}/\sqrt{n})=O_{p}(d_{\gamma}p/\sqrt{n})=o_{p}(\sqrt{p}),$

where the last equality holds under the conditions of the theorem.

It remains to show that

$A_{i}=o_{p}\left({p^{1/2}}\right),i=1,\ldots,4.$ (S.B.2)

It is convenient to perform the calculations under $H_{\ell}$, which covers
$H_{0}$ as a particular case. Using the mean value theorem and either $H_{0}$
or $H_{\ell}$, we can express

${\theta}_{0i}-\widehat{{f}}_{i}={f}_{i}^{\ast}-\widehat{{f}}_{i}-(p^{1/4}/n^{1/2}){h_{i}}=\sum_{j=1}^{d_{\alpha}}\frac{\partial{f}(x_{i},\widetilde{\alpha})}{\partial\alpha_{j}}(\alpha_{j}^{\ast}-\widetilde{\alpha}_{j})-\frac{p^{1/4}}{n^{1/2}}{h_{i},}$
(S.B.3)

where $\widetilde{\alpha}_{j}$ is a value between $\alpha_{j}^{\ast}$ and
$\widehat{\alpha}_{j}$. Then, for any $j=1,\ldots,d_{\alpha}$,
$\left|\alpha_{j}^{\ast}-\widetilde{\alpha}_{j}\right|{=}O_{p}(1/\sqrt{n})$.
Based on

$\sup_{\gamma\in\Gamma}\left|u^{\prime}\Sigma(\gamma)^{-1}\Psi\left(\Psi^{\prime}\Sigma(\gamma)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma(\gamma)^{-1}{g}\right|{=}O_{p}\left(\left\|g\right\|\right)\text{
and
}\sup_{\gamma\in\Gamma}\left|u^{\prime}\Sigma(\gamma)^{-1}g\right|=O_{p}\left(\left\|g\right\|\right)$

for any $\gamma\in\Gamma$ and any conditioned vector $g$, if we take
$g={\partial{f}(x,{\alpha})}/{\partial\alpha_{j}}$ or $g=h$, then both satisfy
$O_{p}\left(\left\|g\right\|\right)=O_{p}\left(\sqrt{n}\right)$ and it follows
that

$\displaystyle\left|A_{1}\right|$ $\displaystyle=$
$\displaystyle\left|{u}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\left({\theta}_{0}-\widehat{{f}}\right)\right|\leq\sup_{\gamma,\alpha}\sum_{j=1}^{d_{\alpha}}\left\|u^{\prime}\Sigma(\gamma)^{-1}\frac{\partial{f}(x,{\alpha})}{\partial\alpha_{j}}\right\|\left|\alpha_{j}^{\ast}-\widetilde{\alpha}_{j}\right|+\frac{p^{1/4}}{n^{1/2}}\sup_{\gamma}\left\|u^{\prime}\Sigma(\gamma)^{-1}h\right\|$
$\displaystyle=$ $\displaystyle
O_{p}(\sqrt{n})O_{p}\left(\frac{1}{\sqrt{n}}\right)+O\left(\frac{p^{1/4}}{n^{1/2}}\right)O_{p}(\sqrt{n})=O_{p}(p^{1/4})=o_{p}(p^{1/2}).$

Similarly,

$\displaystyle\left|A_{3}\right|$ $\displaystyle=$
$\displaystyle\left|{u}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}({\theta}_{0}-\widehat{{f}})\right|$
$\displaystyle\leq$
$\displaystyle\sup_{\gamma,\alpha}\sum_{j=1}^{d_{\alpha}}\left\|u^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\frac{\partial{f}(x,{\alpha})}{\partial\alpha_{j}}\right\|\left|\alpha_{j}^{\ast}-\widetilde{\alpha}_{j}\right|$
$\displaystyle+\frac{p^{1/4}}{n^{1/2}}\sup_{\gamma}\left\|u^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}h\right\|$
$\displaystyle=$ $\displaystyle
O_{p}(1)+O_{p}(p^{1/4})=O_{p}(p^{1/4})=o_{p}(p^{1/2}).$

Also, by Assumptions R.2 and R.10, we have

$\left\|{\theta}_{0}-\widehat{{f}}\right\|\leq\sup_{\alpha}\sum_{j=1}^{d_{\alpha}}\left\|\frac{\partial{f}(x,{\alpha})}{\partial\alpha_{j}}\right\|\left|\alpha_{j}^{\ast}-\widetilde{\alpha}_{j}\right|+\left\|h\right\|\frac{p^{1/4}}{n^{1/2}}=O_{p}(p^{1/4}).$
(S.B.4)

By (3.2), we have $\left\|e\right\|=O(p^{-\mu}n^{1/2})$ and

$\displaystyle|A_{2}|$ $\displaystyle=$
$\displaystyle\left|(u\mathbf{+}\theta_{0}-\widehat{f})^{\prime}\left(\Sigma\left(\widehat{\gamma}\right)^{-1}-\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\right)e\right|$
$\displaystyle\leq$
$\displaystyle\sup_{\gamma}|u^{\prime}\Sigma\left(\gamma\right)^{-1}e|+\sup_{\gamma}\left|u^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}e\right|$
$\displaystyle+\left\|{\theta}_{0}-\widehat{{f}}\right\|\sup_{\gamma}\left(\left\|\Sigma\left(\gamma\right)^{-1}\right\|+\left\|\Sigma\left(\gamma\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}\right\|\right)\left\|e\right\|$
$\displaystyle=$ $\displaystyle
O_{p}(p^{-\mu}n^{1/2})+O_{p}(p^{-\mu+1/4}n^{1/2})=O_{p}(p^{-\mu+1/4}n^{1/2})=o_{p}(\sqrt{p}).$

where the last equality holds under the conditions of the theorem. Finally,
under $H_{\ell}$,

$\displaystyle A_{4}$ $\displaystyle=$
$\displaystyle\left({\theta}_{0}-\widehat{{f}}\right)^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\left({\theta}_{0}-\widehat{{f}}\right)$
$\displaystyle=$
$\displaystyle\left({\theta}_{0}-\widehat{{f}}\right)^{\prime}\Sigma^{-1}\left({\theta}_{0}-\widehat{{f}}\right)+\left({\theta}_{0}-\widehat{{f}}\right)^{\prime}\left(\Sigma\left(\widehat{\gamma}\right)^{-1}-\Sigma^{-1}\right)\left({\theta}_{0}-\widehat{{f}}\right)$
$\displaystyle=$
$\displaystyle\frac{p^{1/2}}{n}h^{\prime}\Sigma^{-1}h+o_{p}(1)+O_{p}\left(p^{1/2}d_{\gamma}^{1/2}/n^{1/2}\right)=\frac{p^{1/2}}{n}h^{\prime}\Sigma^{-1}h+o_{p}(\sqrt{p}).$

Combining these together, we have

$n\widehat{m}_{n}=\widehat{\sigma}^{-2}\widehat{{v}}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\widehat{{u}}=\frac{1}{\sigma_{0}^{2}}\varepsilon^{\prime}\mathscr{V}\varepsilon{+}\frac{p^{1/2}}{n}{h}^{\prime}\Sigma^{-1}{h}+o_{p}(\sqrt{p}),$

under $H_{\ell}$ and the same expression holds with $h=0$ under $H_{0}$.

∎

###### Proof of Theorem 4.4.

(1) Follows from Theorems 4.2 and 4.3. (2) Following reasoning analogous to
the proofs of Theorems 4.2 and 4.3, it can be shown that under $H_{1}$,
$\widehat{m}_{n}=n^{-1}{\sigma}^{*-2}(\theta_{0}-f^{\ast})^{\prime}\Sigma\left(\gamma^{\ast}\right)^{-1}(\theta_{0}-f^{\ast})+o_{p}(1).$
Then,

$\mathscr{T}_{n}=\left(n\widehat{m}_{n}-p\right)/{\sqrt{2p}}=\left({n}/{\sqrt{p}}\right){(\theta_{0}-f^{\ast})^{\prime}\Sigma\left(\gamma^{\ast}\right)^{-1}(\theta_{0}-f^{\ast})}/\left({\sqrt{2}n\sigma^{\ast
2}}\right)+o_{p}\left({n}/{\sqrt{p}}\right)$

and for any nonstochastic sequence $\\{C_{n}\\}$, $C_{n}=o(n/p^{1/2})$,
$P(\mathscr{T}_{n}>C_{n})\rightarrow 1,$ so that consistency follows. (3)
Follows from Theorems 4.2 and 4.3. ∎

##  Appendix S.C Proof of Theorem 5.1

###### Proof.

We prove the result under $H_{1}$, which is the more challenging case as it
involves nonparametric estimation. The proof under $H_{0}$ is similar. We will
show $\widehat{\phi}\overset{p}{\rightarrow}\phi_{0}$, whence
$\widehat{\beta}\overset{p}{\rightarrow}\beta_{0}$ and
$\widehat{\sigma}^{2}\overset{p}{\rightarrow}\sigma^{2}_{0}$ follow from (5.3)
and (5.4) respectively. First note that

$\mathcal{L}\left(\phi\right)-\mathcal{L}=\log\overline{\sigma}^{2}\left(\phi\right)/\overline{\sigma}^{2}-n^{-1}\log\left|T^{\prime}(\lambda)\Sigma(\gamma)^{-1}T(\lambda)\Sigma\right|=\log\overline{\sigma}^{2}\left(\phi\right)/\sigma^{2}\left(\phi\right)-\log\overline{\sigma}^{2}/\sigma_{0}^{2}+\log
r(\phi),$ (S.C.1)

where recall that
$\sigma^{2}\left(\phi\right)=n^{-1}\sigma_{0}^{2}tr\left(T^{\prime}(\lambda)\Sigma(\gamma)^{-1}T(\lambda)\Sigma\right),\text{
}\overline{\sigma}^{2}=\overline{\sigma}^{2}\left(\phi_{0}\right)=n^{-1}u^{\prime}E^{\prime}MEu,$
using (5.4) and also
$r(\phi)=n^{-1}tr\left(T^{\prime}(\lambda)\Sigma(\gamma)^{-1}T(\lambda)\Sigma\right)/\left|T^{\prime}(\lambda)\Sigma(\gamma)^{-1}T(\lambda)\Sigma\right|^{1/n}$.

We have
$\overline{\sigma}^{2}\left(\phi\right)=n^{-1}\left\\{S^{-1^{\prime}}\left(\Psi\beta_{0}+u\right)\right\\}^{\prime}S^{\prime}(\lambda)E(\gamma)^{\prime}M\left(\gamma\right)E(\gamma)S(\lambda)S^{-1}\left(\Psi\beta_{0}+u\right)=c_{1}\left(\phi\right)+c_{2}\left(\phi\right)+c_{3}\left(\phi\right)$,
where

$\displaystyle c_{1}\left(\phi\right)$ $\displaystyle=$ $\displaystyle
n^{-1}\beta_{0}^{\prime}\Psi^{\prime}T^{\prime}(\lambda)E(\gamma)^{\prime}M\left(\gamma\right)E(\gamma)T(\lambda)\Psi\beta_{0},$
$\displaystyle\text{\ }c_{2}\left(\phi\right)$ $\displaystyle=$ $\displaystyle
n^{-1}\sigma_{0}^{2}tr\left(T^{\prime}(\lambda)E(\gamma)^{\prime}M\left(\gamma\right)E(\gamma)T(\lambda)\Sigma\right),$
$\displaystyle c_{3}\left(\phi\right)$ $\displaystyle=$ $\displaystyle
n^{-1}tr\left(T^{\prime}(\lambda)E(\gamma)^{\prime}M\left(\gamma\right)E(\gamma)T(\lambda)\left(uu^{\prime}-\sigma_{0}^{2}\Sigma\right)\right)$
$\displaystyle+$ $\displaystyle
2n^{-1}\beta_{0}^{\prime}\Psi^{\prime}T^{\prime}(\lambda)E(\gamma)^{\prime}M\left(\gamma\right)E(\gamma)T(\lambda)u.$

Note that in the particular cases of Theorems 4.1 and 6.1, where
$T(\lambda)=I_{n}$, the $c_{1}$ term vanishes because
$M\left(\gamma\right)E(\gamma)\Psi=0$ and $M\left(\tau\right)E(\tau)\Psi=0$.
Proceeding with the current, more general proof

$\displaystyle\log\frac{\overline{\sigma}^{2}\left(\phi\right)}{\sigma^{2}\left(\phi\right)}$
$\displaystyle=$
$\displaystyle\log\frac{\overline{\sigma}^{2}\left(\phi\right)}{\left(c_{1}\left(\phi\right)+c_{2}\left(\phi\right)\right)}+\log\frac{c_{1}\left(\phi\right)+c_{2}\left(\phi\right)}{\sigma^{2}\left(\phi\right)}$
$\displaystyle=$
$\displaystyle\log\left(1+\frac{c_{3}\left(\phi\right)}{c_{1}\left(\phi\right)+c_{2}\left(\phi\right)}\right)+\log\left(1+\frac{c_{1}\left(\phi\right)-f\left(\phi\right)}{\sigma^{2}\left(\phi\right)}\right),$

where
$f\left(\phi\right)=n^{-1}\sigma_{0}^{2}tr\left(E^{\prime-1}T^{\prime}(\lambda)E(\gamma)^{\prime}\left(I_{n}-M\left(\gamma\right)\right)E(\gamma)T(\lambda)E^{-1}\right).$
Then (S.C.1) implies

$\displaystyle
P\left(\left\|\widehat{\phi}-\phi_{0}\right\|\in\overline{\mathcal{N}}^{\;\phi}\left(\eta\right)\right)$
$\displaystyle=$ $\displaystyle P\left(\inf_{\phi\in\text{
}\overline{\mathcal{N}}^{\;\phi}\left(\eta\right)}\mathcal{L}\left(\phi\right)-\mathcal{L}\leq
0\right)$ $\displaystyle\leq$ $\displaystyle
P\left(\log\left(1+\underset{\phi\in\text{
}\overline{\mathcal{N}}^{\;\phi}\left(\eta\right)}{\sup}\left|\frac{c_{3}\left(\phi\right)}{c_{1}\left(\phi\right)+c_{2}\left(\phi\right)}\right|\right)+\left|\log\left(\overline{\sigma}^{2}/\sigma_{0}^{2}\right)\right|\right.$
$\displaystyle\left.\geq\inf_{\phi\in\text{
}\overline{\mathcal{N}}^{\;\phi}\left(\eta\right)}\left(\log\left(1+\frac{c_{1}\left(\phi\right)-f\left(\phi\right)}{\sigma^{2}\left(\phi\right)}\right)+\log
r(\phi)\right)\right),$

where recall that
$\overline{\mathcal{N}}^{\;\phi}\left(\eta\right)=\Phi\backslash\mathcal{N}^{\phi}\left(\eta\right),$
$\mathcal{N}^{\phi}\left(\eta\right)=\left\\{\phi:\left\|\phi-\phi_{0}\right\|<\eta\right\\}\cap\Phi.$
Because $\overline{\sigma}^{2}/\sigma_{0}^{2}\overset{p}{\rightarrow}1,$ the
property $\log\left(1+x\right)=x+o\left(x\right)$ as $x\rightarrow 0$ implies
that it is sufficient to show that

$\displaystyle\underset{\phi\in\text{
}\overline{\mathcal{N}}^{\;\phi}\left(\eta\right)}{\sup}\left|\frac{c_{3}\left(\phi\right)}{c_{1}\left(\phi\right)+c_{2}\left(\phi\right)}\right|$
$\displaystyle\overset{p}{\longrightarrow}$ $\displaystyle\text{ }0,$ (S.C.2)
$\displaystyle\underset{\phi\in\text{
}\overline{\mathcal{N}}^{\;\phi}\left(\eta\right)}{\sup}\left|\frac{f\left(\phi\right)}{\sigma^{2}\left(\phi\right)}\right|$
$\displaystyle\overset{p}{\longrightarrow}$ $\displaystyle\text{ }0,$ (S.C.3)
$\displaystyle P\left(\inf_{\phi\in\text{
}\overline{\mathcal{N}}^{\;\phi}\left(\eta\right)}\left\\{\frac{c_{1}\left(\phi\right)}{\sigma^{2}\left(\phi\right)}+\log
r(\phi)\right\\}>0\right)$ $\displaystyle\longrightarrow$ $\displaystyle\text{
}1.$ (S.C.4)

Because
$\overline{\mathcal{N}}^{\;\phi}\left(\eta\right)\subseteq\left\\{\Lambda\times\overline{\mathcal{N}}^{\;\gamma}\left(\eta/2\right)\right\\}\cup\left\\{\overline{\mathcal{N}}^{\;\lambda}\left(\eta/2\right)\times\Gamma\right\\}$,
we have

$\displaystyle P\left(\inf_{\phi\in\text{
}\overline{\mathcal{N}}^{\;\phi}\left(\eta\right)}\left\\{\frac{c_{1}\left(\phi\right)}{\sigma^{2}\left(\phi\right)}+\log
r(\phi)\right\\}>0\right)$ $\displaystyle\geq$ $\displaystyle
P\left(\min\left\\{\underset{\Lambda\times\overline{\mathcal{N}}^{\;\gamma}\left(\eta/2\right)}{\inf}\frac{c_{1}\left(\phi\right)}{\sigma^{2}\left(\phi\right)},\underset{\overline{\mathcal{N}}^{\;\lambda}\left(\eta/2\right)}{\inf}\log
r(\phi)\right\\}>0\right)$ $\displaystyle\geq$ $\displaystyle
P\left(\min\left\\{\underset{\Lambda\times\overline{\mathcal{N}}^{\;\gamma}\left(\eta/2\right)}{\inf}\frac{c_{1}\left(\phi\right)}{C},\underset{\overline{\mathcal{N}}^{\lambda}\left(\eta/2\right)}{\inf}\log
r(\phi)\right\\}>0\right),$

from Assumption SAR.2, whence Assumptions SAR.3 and SAR.4 imply (S.C.4). Again
using Assumption SAR.2, uniformly in $\phi$,
$\left|f\left(\phi\right)/\sigma^{2}\left(\phi\right)\right|=O_{p}\left(\left|f\left(\phi\right)\right|\right)$
and

$\displaystyle\left|f\left(\phi\right)\right|$ $\displaystyle=$ $\displaystyle
O_{p}\left(tr\left(E^{\prime-1}T^{\prime}(\lambda)\Sigma(\gamma)^{-1}\Psi\left(\Psi^{\prime}\Sigma(\gamma)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma(\gamma)^{-1}T(\lambda)E^{-1}\right)/n\right)$
(S.C.5) $\displaystyle=$ $\displaystyle
O_{p}\left(tr\left(E^{\prime-1}T^{\prime}(\lambda)\Sigma(\gamma)^{-1}\Psi\Psi^{\prime}\Sigma(\gamma)^{-1}T(\lambda)E^{-1}\right)/n^{2}\right)=O_{p}\left(\left\|\Psi^{\prime}\Sigma(\gamma)^{-1}T(\lambda)E^{-1}/n\right\|_{F}^{2}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(\left\|\Psi/n\right\|_{F}^{2}\overline{\varphi}^{2}\left(\Sigma(\gamma)^{-1}\right)\left\|T(\lambda)\right\|^{2}\left\|E^{-1}\right\|^{2}\right)=O_{p}\left(\left\|\Psi/n\right\|_{F}^{2}\left\|T(\lambda)\right\|^{2}\overline{\varphi}\left(\Sigma\right)/\underline{\varphi}^{2}\left(\Sigma(\gamma)\right)\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(\left\|T(\lambda)\right\|^{2}/n\right),$

where we have twice made use of the inequality

$\left\|AB\right\|_{F}\leq\left\|A\right\|_{F}\left\|B\right\|$ (S.C.6)

for generic multiplication compatible matrices $A$ and $B$. (S.C.3) now
follows by Assumption SAR.1 and compactness of $\Lambda$ because
$T(\lambda)=I_{n}+\sum_{j=1}^{d_{\lambda}}\left(\lambda_{0j}-\lambda_{j}\right)G_{j}$.
Finally consider (S.C.2). We first prove pointwise convergence. For any fixed
$\phi\in\overline{\mathcal{N}}^{\;\phi}\left(\eta\right)$ and large enough
$n$, Assumptions SAR.2 and SAR.4 imply

$\displaystyle\left\\{c_{1}\left(\phi\right)\right\\}^{-1}$ $\displaystyle=$
$\displaystyle O_{p}\left(\left\|\beta_{0}\right\|^{-2}\right)=O_{p}(1)$
(S.C.7) $\displaystyle\left\\{c_{2}\left(\phi\right)\right\\}^{-1}$
$\displaystyle=$ $\displaystyle O_{p}(1),$ (S.C.8)

because
$\left\\{n^{-1}\sigma_{0}^{2}tr\left(T^{\prime}(\lambda)\Sigma(\gamma)^{-1}T(\lambda)E^{-1}\right)\right\\}^{-1}=O_{p}(1)$
and, proceeding like in the bound for $\left|f(\phi)\right|$,
$tE^{\prime-1}r\left(E^{\prime-1}T^{\prime}(\lambda)E(\gamma)^{\prime}\left(I-M\left(\gamma\right)\right)E(\gamma)T(\lambda)E^{-1}\right)=O_{p}\left(\left\|T(\lambda)\right\|^{2}/n\right)=O_{p}\left(1/n\right)$.
In fact it is worth noting for the equicontinuity argument presented later
that Assumptions SAR.2 and SAR.4 actually imply that (S.C.7) and (S.C.8) hold
uniformly over $\overline{\mathcal{N}}^{\phi}(\eta)$, a property not needed
for the present pointwise arguments. Thus
$c_{3}\left(\phi\right)/\left(c_{1}\left(\phi\right)+c_{2}\left(\phi\right)\right)=O_{p}\left(\left|c_{3}\left(\phi\right)\right|\right)$
where, writing
$\mathfrak{B}(\phi)=T^{\prime}(\lambda)E(\gamma)^{\prime}M\left(\gamma\right)E(\gamma)T(\lambda)$
with typical element $\mathfrak{b}_{rs}(\phi)$, $r,s=1,\ldots,n$,
$c_{3}\left(\phi\right)$ has mean $0$ and variance

$O_{p}\left(\frac{\left\|\mathfrak{B}(\phi)\Sigma\right\|_{F}^{2}}{n^{2}}+\frac{\sum_{r,s,t,v=1}^{n}\mathfrak{b}_{rs}(\phi)\mathfrak{b}_{tv}(\phi)\kappa_{rstv}}{n^{2}}+\frac{\left\|\beta_{0}^{\prime}\Psi^{\prime}\mathfrak{B}(\phi)E^{-1}\right\|^{2}}{n^{2}}\right),$
(S.C.9)

with $\kappa_{rstv}$ denoting the fourth cumulant of
$u_{r},u_{s},u_{t},u_{v}$, $r,s,t,v=1,\ldots,n$. Under the linear process
assumed in Assumption R.4 it is known that

$\sum_{r,s,t,v=1}^{n}\kappa^{2}_{rstv}=O(n).$ (S.C.10)

Using (S.C.6) and Assumptions SAR.1 and R.3, the first term in parentheses in
(S.C.9) is

$\displaystyle
O_{p}\left(\left\|\mathfrak{B}(\phi)\right\|_{F}^{2}\overline{\varphi}^{2}\left(\Sigma\right)/n^{2}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(\left\|T(\lambda)\right\|_{F}^{2}\left\|E(\gamma)\right\|^{4}\left\|M(\gamma)\right\|^{2}\left\|T(\lambda)\right\|^{2}/n^{2}\right)$
(S.C.11) $\displaystyle=$ $\displaystyle
O_{p}\left(\left\|T(\lambda)\right\|^{4}/n\underline{\varphi}^{2}\left(\Sigma(\gamma)\right)\right)=O_{p}\left(\left\|T(\lambda)\right\|^{4}/n\right),$

while the second is similarly

$O_{p}\left\\{\left(\left\|\mathfrak{B}(\phi)\right\|_{F}^{2}/n\right)\left(\sum_{r,s,t,v=1}^{n}\kappa^{2}_{rstv}/n^{2}\right)^{\frac{1}{2}}\right\\}=o_{p}\left(\left\|T(\lambda)\right\|^{4}\right),$
(S.C.12)

using (S.C.10). Finally, the third term in parentheses in (S.C.9) is

$O_{p}\left(\left\|\mathfrak{B}(\phi)\right\|^{2}/n\right)=O_{p}\left(\left\|T(\lambda)\right\|^{4}/n\right).$
(S.C.13)

By compactness of $\Lambda$ and Assumption SAR.1, (S.C.11), (S.C.12) and
(S.C.13) are negligible, thus pointwise convergence is established.

Uniform convergence will follow from an equicontinuity argument. First, for
arbitrary $\varepsilon>0$ we can find points
$\phi_{*}=\left(\lambda^{\prime}_{*},\gamma^{\prime}_{*}\right)^{\prime}$,
possibly infinitely many, such that the neighborhoods
$\left\|\phi-\phi^{*}\right\|<\varepsilon$ form an open cover of
$\overline{\mathcal{N}}^{\phi}(\eta)$. Since $\Phi$ is compact any open cover
has a finite subcover and thus we may in fact choose finitely many
$\phi_{*}=\left(\lambda^{\prime}_{*},\gamma^{\prime}_{*}\right)^{\prime}$,
whence it suffices to prove

$\underset{\left\|\phi-\phi_{{}_{\ast}}\right\|<\varepsilon}{\sup}\left|\frac{c_{3}\left(\phi\right)}{c_{1}\left(\phi\right)+c_{2}\left(\phi\right)}-\frac{c_{3}\left(\phi_{\ast}\right)}{c_{1}\left(\phi_{\ast}\right)+c_{2}\left(\phi_{\ast}\right)}\right|\overset{p}{\longrightarrow}0.$

Proceeding as in Gupta and Robinson (2018), we denote the two components of
$c_{3}\left(\phi\right)$ by $c_{31}\left(\phi\right),$
$c_{32}\left(\phi\right),$ and are left with establishing the negligibility of

$\displaystyle\frac{\left|c_{31}\left(\phi\right)-c_{31}\left(\phi_{\ast}\right)\right|}{c_{2}\left(\phi\right)}+\frac{\left|c_{32}\left(\phi\right)-c_{32}\left(\phi_{\ast}\right)\right|}{c_{1}\left(\phi\right)}+\frac{\left|c_{3}\left(\phi_{\ast}\right)\right|}{c_{1}\left(\phi\right)c_{1}\left(\phi_{\ast}\right)}\left|c_{1}\left(\phi_{\ast}\right)-c_{1}\left(\phi\right)\right|$
(S.C.14) $\displaystyle+$
$\displaystyle\frac{\left|c_{3}\left(\phi_{\ast}\right)\right|}{c_{2}\left(\phi\right)c_{2}\left(\phi_{\ast}\right)}\left|c_{2}\left(\phi_{\ast}\right)-c_{2}\left(\phi\right)\right|,$

uniformly on $\left\|\phi-\phi_{{}_{\ast}}\right\|<\varepsilon$. By the fact
that (S.C.7) and (S.C.8) hold uniformly over $\Phi$, we first consider only
the numerators in the first two terms in (S.C.14). As in the proof of Theorem
1 of Delgado and Robinson (2015), (S.C.6) implies that
$\mathcal{E}\left(\sup_{\left\|\phi-\phi_{{}_{\ast}}\right\|<\varepsilon}\left|c_{31}\left(\phi\right)-c_{31}\left(\phi_{\ast}\right)\right|\right)$
is bounded by

$n^{-1}\left(\mathcal{E}\left\|u\right\|^{2}+\sigma_{0}^{2}tr\Sigma\right)\sup_{\left\|\phi-\phi_{{}_{\ast}}\right\|<\varepsilon}\left\|\mathfrak{B}(\phi)-\mathfrak{B}(\phi_{*})\right\|=O_{p}\left(\sup_{\left\|\phi-\phi_{{}_{\ast}}\right\|<\varepsilon}\left\|\mathfrak{B}(\phi)-\mathfrak{B}(\phi_{*})\right\|\right),$

because $\mathcal{E}\left\|u\right\|^{2}=O(n)$ and $tr\Sigma=O(n)$.
$\mathfrak{B}(\phi)-\mathfrak{B}(\phi_{*})$ can be written as

$\displaystyle\left(T(\lambda)-T(\lambda_{*})\right)^{\prime}E(\gamma)^{\prime}M(\gamma)E(\gamma)T(\lambda)+T(\lambda_{*})^{\prime}\Sigma^{\prime}(\gamma_{*})M(\gamma_{*})E(\gamma_{*})\left(T(\lambda)-T(\lambda_{*})\right)$
(S.C.15) $\displaystyle+$ $\displaystyle
T^{\prime}(\lambda_{*})\left(E(\gamma)^{\prime}M(\gamma)E(\gamma)-E(\gamma_{*})^{\prime}M(\gamma_{*})E(\gamma_{*})\right)T(\lambda),$

which, by the triangle inequality, has spectral norm bounded by

$\displaystyle\left\|T(\lambda)-T(\lambda_{*})\right\|\left(\left\|E(\gamma)\right\|^{2}\left\|T(\lambda)\right\|+\left\|E(\gamma_{*})\right\|^{2}\left\|T(\lambda_{*})\right\|\right)$
$\displaystyle+$
$\displaystyle\left\|T(\lambda_{*})\right\|\left\|E(\gamma)^{\prime}M(\gamma)E(\gamma)-E(\gamma_{*})^{\prime}M(\gamma_{*})E(\gamma_{*})\right\|\left\|T(\lambda)\right\|$
$\displaystyle=$ $\displaystyle
O_{p}\left(\left\|T(\lambda)-T(\lambda_{*})\right\|+\left\|E(\gamma)^{\prime}M(\gamma)E(\gamma)-E(\gamma_{*})^{\prime}M(\gamma_{*})E(\gamma_{*})\right\|\right).$

By Assumption SAR.1 the first term in parentheses on the right side of
(LABEL:cons7) is bounded uniformly on
$\left\|\phi-\phi_{*}\right\|<\varepsilon$ by

$\sum_{j=1}^{d_{\lambda}}\left|\lambda_{j}-\lambda_{*j}\right|\left\|G_{j}\right\|\leq\max_{j=1,\ldots,d_{\lambda}}\left\|G_{j}\right\|\left\|\lambda-\lambda_{*}\right\|=O_{p}(\varepsilon),$
(S.C.17)

while because
$E(\gamma)^{\prime}M(\gamma)E(\gamma)=n^{-1}\Sigma(\gamma)^{-1}\Psi\left(n^{-1}\Psi^{\prime}\Sigma(\gamma)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma(\gamma)^{-1}$
for any $\gamma\in\Gamma$, the second one can be decomposed into terms with
bounds typified by

$\displaystyle
n^{-1}\left\|\Sigma(\gamma)^{-1}-\Sigma(\gamma_{*})^{-1}\right\|\left\|\Psi\right\|^{2}\left\|\left(n^{-1}\Psi^{\prime}\Sigma(\gamma)^{-1}\Psi\right)^{-1}\right\|\left\|\Sigma(\gamma)^{-1}\right\|^{2}$
$\displaystyle\leq$ $\displaystyle
n^{-1}\left\|\Sigma(\gamma)-\Sigma(\gamma_{*})\right\|\left\|\Psi\right\|^{2}\left\|\left(n^{-1}\Psi^{\prime}\Sigma(\gamma)^{-1}\Psi\right)^{-1}\right\|\left\|\Sigma(\gamma)^{-1}\right\|^{3}\left\|\Sigma(\gamma_{*})^{-1}\right\|$
$\displaystyle=$ $\displaystyle
O_{p}\left(\left\|\Sigma(\gamma)-\Sigma(\gamma_{*})\right\|\right)=O_{p}(\varepsilon),$

uniformly on $\left\|\phi-\phi_{*}\right\|<\varepsilon$, by Assumptions R.3
and R.8, Proposition 4.1 and the inequality
$\left\|A\right\|\leq\left\|A\right\|_{F}$ for a generic matrix $A$, so that

$\sup_{\left\|\phi-\phi_{*}\right\|<\varepsilon}\left\|\mathfrak{B}(\phi)-\mathfrak{B}(\phi_{*})\right\|=O_{p}(\varepsilon).$
(S.C.18)

Thus equicontinuity of the first term in (S.C.14) follows because
$\varepsilon$ is arbitrary. The equicontinuity of the second term in (S.C.14)
follows in much the same way. Indeed
$\sup_{\left\|\phi-\phi_{*}\right\|<\varepsilon}c_{32}\left(\phi\right)-c_{32}\left(\phi_{\ast}\right)=2n^{-1}\beta_{0}^{\prime}\Psi^{\prime}\sup_{\left\|\phi-\phi_{*}\right\|<\varepsilon}\left(\mathfrak{B}(\phi)-\mathfrak{B}(\phi_{*})\right)u=O_{p}\left(\sup_{\left\|\phi-\phi_{*}\right\|<\varepsilon}\left\|\mathfrak{B}(\phi)-\mathfrak{B}(\phi_{*})\right\|\right)=O_{p}(\varepsilon)$,
using earlier arguments and (S.C.18). Because $c_{1}(\phi)$ is bounded and
bounded away from zero in probability (see S.C.7) for sufficiently large $n$
and all $\phi\in\overline{\mathcal{N}}^{\phi}(\eta)$, the third term in
(S.C.14) may be bounded by
${\left|c_{3}(\phi_{*})\right|}/{c_{1}(\phi_{*})}\left(1+{c_{1}(\phi_{*})}/{c_{1}(\phi)}\right)\overset{p}{\longrightarrow}0,$
convergence being uniform on $\left\|\phi-\phi_{*}\right\|<\varepsilon$ by
pointwise convergence of $c_{3}(\phi)/\left(c_{1}(\phi)+c_{2}(\phi)\right)$,
cf. Gupta and Robinson (2018). The uniform convergence to zero of the fourth
term in (S.C.14) follows in identical fashion, because $c_{2}(\phi)$ is
bounded and bounded away from zero (see (S.C.8)) in probability for
sufficiently large $n$ and all $\phi\in\overline{\mathcal{N}}^{\phi}(\eta)$.
This concludes the proof. ∎

##  Appendix S.D Lemmas

######  Lemma LS.1.

Under the conditions of Theorem 4.1,
$c_{1}(\gamma)=n^{-1}\beta^{\prime}\Psi^{\prime}E^{\prime}(\gamma)M(\gamma)E(\gamma)\Psi\beta+o_{p}(1).$

###### Proof.

First,

$c_{1}(\gamma)=n^{-1}\beta^{\prime}\Psi^{\prime}E^{\prime}(\gamma)M(\gamma)E(\gamma)\Psi\beta+c_{12}(\gamma)+c_{13}(\gamma),$

with
$c_{12}(\gamma)=2n^{-1}{e}^{\prime}E^{\prime}(\gamma)M(\gamma)E(\gamma)\Psi\beta$
and
$c_{13}(\gamma)=n^{-1}{e}^{\prime}E^{\prime}(\gamma)M(\gamma)E(\gamma){e}$. It
is readily seen that $c_{12}(\gamma)$ and $c_{13}(\gamma)$ are negligible. ∎

######  Lemma LS.2.

Under the conditions of Theorem 4.2 or Theorem 5.2,
$\left\|\widehat{\gamma}-\gamma_{0}\right\|=O_{p}\left(\sqrt{d_{\gamma}/n}\right).$

###### Proof.

We show the details for the setting of Theorem 4.2 and omit the details for
the setting of Theorem 5.2. Write $l=\partial
L(\beta_{0},\gamma_{0})/\partial\gamma$. By Robinson (1988), we have
$\left\|\widehat{\gamma}-\gamma_{0}\right\|=O_{p}\left(\left\|l\right\|\right)$.
Now $l=\left(l_{1},\ldots,l_{d_{\gamma}}\right)^{\prime}$, with
$l_{j}=n^{-1}tr\left(\Sigma^{-1}\Sigma_{j}\right)-n^{-1}\sigma_{0}^{-2}u^{\prime}\Sigma^{-1}\Sigma_{j}\Sigma^{-1}u$.
Next,
$\mathcal{E}\left\|l\right\|^{2}=\sum_{j=1}^{d_{\gamma}}\mathcal{E}\left(l_{j}^{2}\right)$
and

$\mathcal{E}\left(l_{j}^{2}\right)=\frac{1}{n^{2}\sigma_{0}^{4}}var\left(u^{\prime}\Sigma^{-1}\Sigma_{j}\Sigma^{-1}u\right)=\frac{1}{n^{2}\sigma_{0}^{4}}var\left(\varepsilon^{\prime}B^{\prime}\Sigma^{-1}\Sigma_{j}\Sigma^{-1}B\varepsilon\right)=\frac{1}{n^{2}\sigma_{0}^{4}}var\left(\varepsilon^{\prime}D_{j}\varepsilon\right),$
(S.D.1)

say. But, writing $d_{j,st}$ for a typical element of the infinite dimensional
matrix $D_{j}$, we have

$var\left(\varepsilon^{\prime}D_{j}\varepsilon\right)=\left(\mu_{4}-3\sigma_{0}^{4}\right)\sum_{s=1}^{\infty}d_{j,ss}^{2}+2\sigma_{0}^{4}tr\left(D_{j}^{2}\right)=\left(\mu_{4}-3\sigma_{0}^{4}\right)\sum_{s=1}^{\infty}d_{j,ss}^{2}+2\sigma_{0}^{4}\sum_{s,t=1}^{\infty}d_{j,st}^{2}.$
(S.D.2)

Next, by Assumptions R.4, R.3 and R.9

$\sum_{s=1}^{\infty}d_{j,ss}^{2}=\sum_{s=1}^{\infty}\left(b_{s}^{\prime}\Sigma^{-1}\Sigma_{j}\Sigma^{-1}b_{s}\right)^{2}\leq\left(\sum_{s=1}^{\infty}\left\|b_{s}\right\|^{2}\right)\left\|\Sigma^{-1}\right\|^{2}\left\|\Sigma_{j}\right\|=O\left(\sum_{j=1}^{n}\sum_{s=1}^{\infty}b^{*2}_{js}\right)=O(n).$
(S.D.3)

Similarly,

$\sum_{s,t=1}^{\infty}d_{j,st}^{2}=\sum_{s=1}^{\infty}b_{s}^{\prime}\Sigma^{-1}\Sigma_{j}\Sigma^{-1}\left(\sum_{t=1}^{\infty}b_{t}b_{t}^{\prime}\right)\Sigma^{-1}\Sigma_{j}\Sigma^{-1}b_{s}=\sum_{s=1}^{\infty}b_{s}^{\prime}\Sigma^{-1}\Sigma_{j}\Sigma^{-1}\Sigma_{j}\Sigma^{-1}b_{s}=O(n).$
(S.D.4)

Using (S.D.3) and (S.D.4) in (S.D.2) implies that
$\mathcal{E}\left(l_{j}^{2}\right)=O\left(n^{-1}\right)$, by (S.D.1). Thus we
have $\mathcal{E}\left\|l\right\|^{2}=O\left(d_{\gamma}/n\right)$, and thus
$\left\|l\right\|=O_{p}\left(\sqrt{d_{\gamma}/n}\right)$, by Markov’s
inequality, proving the lemma. ∎

######  Lemma LS.3.

Under the conditions of Theorem 4.3,
$\mathcal{E}\left({\sigma_{0}^{-2}}\varepsilon^{\prime}\mathscr{V}\varepsilon\right)=p$
and
$Var\left({\sigma_{0}^{-2}}\varepsilon^{\prime}\mathscr{V}\varepsilon\right)/2p\rightarrow
1$.

###### Proof.

As
$\mathcal{E}\left({\sigma_{0}^{-2}}\varepsilon^{\prime}\mathscr{V}\varepsilon\right)=tr\left(\mathcal{E}[B^{\prime}\Sigma^{-1}\Psi(\Psi^{\prime}\Sigma^{-1}\Psi)^{-1}\Psi^{\prime}\Sigma^{-1}B]\right)=p,$
and

$Var\left(\frac{1}{\sigma_{0}^{2}}\varepsilon^{\prime}\mathscr{V}\varepsilon\right)=\left(\frac{\mu_{4}}{\sigma_{0}^{4}}-3\right)\sum_{s=1}^{\infty}\mathcal{E}(v_{ss}^{2})+\mathcal{E}[tr(\mathscr{V}\mathscr{V}^{\prime})+tr(\mathscr{V}^{2})]=\left(\frac{\mu_{4}}{\sigma_{0}^{4}}-3\right)\sum_{s=1}^{\infty}v_{ss}^{2}+2p,$
(S.D.5)

it suffices to show that

$(2p)^{-1}\sum_{s=1}^{\infty}v_{ss}^{2}\overset{p}{\rightarrow}0.$ (S.D.6)

Because $v_{ss}=b_{s}^{\prime}\mathscr{M}b_{s}$, we have
$v_{ss}^{2}=\left(\sum_{i,j=1}^{n}b_{is}b_{js}m_{ij}\right)^{2}$. Thus, using
Assumption R.4 and (A.5), we have

$\displaystyle\sum_{s=1}^{\infty}v_{ss}^{2}$ $\displaystyle\leq$
$\displaystyle\left(\sup_{i,j}\left|m_{ij}\right|\right)^{2}\sum_{s=1}^{\infty}\left(\sum_{i,j=1}^{n}\left|b^{*}_{is}\right|\left|b^{*}_{js}\right|\right)^{2}=O_{p}\left(p^{2}n^{-2}\left(\sup_{s}\sum_{i=1}^{n}\left|b^{*}_{is}\right|\right)^{3}\sum_{i=1}^{n}\sum_{s=1}^{\infty}\left|b^{*}_{is}\right|\right)$
(S.D.7) $\displaystyle=$ $\displaystyle O_{p}\left(p^{2}n^{-1}\right),$

establishing (S.D.6) because $p^{2}/n\rightarrow 0$. ∎

######  Lemma LS.4.

Under the conditions of Theorem 6.2,
$\left\|\widehat{\tau}-\tau_{0}\right\|=O_{p}\left(\sqrt{d_{\tau}/n}\right).$

###### Proof.

The proof is similar to that of Lemma LS.2 and is omitted. ∎

Denote $H(\gamma)=I_{n}+\sum_{j=m_{1}+1}^{m_{1}+m_{2}}\gamma_{j}W_{j}$ and
$K(\gamma)=I_{n}-\sum_{j=1}^{m_{1}}\gamma_{j}W_{j}$. Let
$G_{j}(\gamma)=W_{j}K^{-1}(\gamma)$, $j=1,\ldots,m_{1}$,
$T_{j}=H^{-1}(\gamma)W_{j}$, $j=m_{1}+1,\ldots,m_{1}+m_{2}$ and, for a generic
matrix $A$, denote $\overline{A}=A+A^{\prime}$. Our final conditions may
differ according to whether the $W_{j}$ are of general form or have ‘single
nonzero diagonal block structure’, see e.g Gupta and Robinson (2015). To
define these, denote by $V$ an $n\times n$ block diagonal matrix with $i$-th
block $V_{i}$, a $s_{i}\times s_{i}$ matrix, where
$\sum_{i=1}^{m_{1}+m_{2}}s_{i}=n$, and for $i=1,...,m_{1}+m_{2}$ obtain
$W_{j}$ from $V$ by replacing each $V_{j}$, $j\neq i$, by a matrix of zeros.
Thus $V=\sum_{i=1}^{m_{1}+m_{2}}W_{j}$.

######  Lemma LS.5.

For the spatial error model with SARMA$(p,q)$ errors, if

$\sup_{\gamma\in\Gamma^{o}}\left(\left\|K^{-1}(\gamma)\right\|+\left\|K^{\prime-1}(\gamma)\right\|+\left\|H^{-1}(\gamma)\right\|+\left\|H^{\prime-1}(\gamma)\right\|\right)+\max_{j=1,\ldots,m_{1}+m_{2}}\left\|W_{j}\right\|<C,$
(S.D.8)

then

$\left(D\Sigma(\gamma)\right)\left(\gamma^{\dagger}\right)=A^{-1}(\gamma)\left(\sum_{j=1}^{m_{1}}\gamma^{\dagger}_{j}\overline{H^{-1}(\gamma)G_{j}(\gamma)}+\sum_{j=m_{1}+1}^{m_{1}+m_{2}}\gamma^{\dagger}_{j}\overline{T_{j}(\gamma)}\right)A^{\prime-1}(\gamma).$

###### Proof.

We first show that $D\Sigma\in\mathscr{L}\left(\Gamma^{o},\mathcal{M}^{n\times
n}\right)$. Clearly, $D\Sigma$ is a linear map and (S.D.8)

$\left\|\left(D\Sigma(\gamma)\right)\left(\gamma^{\dagger}\right)\right\|\leq
C\left\|\gamma^{\dagger}\right\|_{1},$

in the general case and

$\left\|\left(D\Sigma(\gamma)\right)\left(\gamma^{\dagger}\right)\right\|\leq
C\max_{j=1,\ldots,m_{1}+m_{2}}\left|\gamma_{j}^{\dagger}\right|,$

in the ‘single nonzero diagonal block’ case. Thus $D\Sigma$ is a bounded
linear operator between two normed linear spaces, i.e. it is a continuous
linear operator.

With $A(\gamma)=H^{-1}(\gamma)K(\gamma)$, we now show that

$\frac{\left\|A^{-1}\left(\gamma+\gamma^{\dagger}\right)A^{\prime-1}\left(\gamma+\gamma^{\dagger}\right)-A^{-1}\left(\gamma\right)A^{\prime-1}\left(\gamma\right)-\left(D\Sigma(\gamma)\right)\left(\gamma^{\dagger}\right)\right\|}{\left\|\gamma^{\dagger}\right\|_{g}}\rightarrow
0,\text{ as }\left\|\gamma^{\dagger}\right\|_{g}\rightarrow 0,$ (S.D.9)

where $\left\|\cdot\right\|_{g}$ is either the 1-norm or the max norm on
$\Gamma$. First, note that

$\displaystyle
A^{-1}\left(\gamma+\gamma^{\dagger}\right)A^{\prime-1}\left(\gamma+\gamma^{\dagger}\right)-A^{-1}(\gamma)A^{\prime-1}(\gamma)$
(S.D.10) $\displaystyle=$ $\displaystyle
A^{-1}\left(\gamma+\gamma^{\dagger}\right)\left(A^{-1}\left(\gamma+\gamma^{\dagger}\right)-A^{-1}(\gamma)\right)^{\prime}+\left(A^{-1}\left(\gamma+\gamma^{\dagger}\right)-A^{-1}(\gamma)\right)A^{-1}(\gamma)$
$\displaystyle=$
$\displaystyle-A^{-1}\left(\gamma+\gamma^{\dagger}\right)A^{\prime-1}\left(\gamma+\gamma^{\dagger}\right)\left(A\left(\gamma+\gamma^{\dagger}\right)-A(\gamma)\right)^{\prime}A^{\prime-1}(\gamma)$
$\displaystyle-$ $\displaystyle
A^{-1}\left(\gamma+\gamma^{\dagger}\right)\left(A\left(\gamma+\gamma^{\dagger}\right)-A(\gamma)\right)A^{-1}(\gamma)A^{\prime-1}(\gamma).$

Next,

$\displaystyle A\left(\gamma+\gamma^{\dagger}\right)-A(\gamma)$
$\displaystyle=$ $\displaystyle
H^{-1}\left(\gamma+\gamma^{\dagger}\right)K\left(\gamma+\gamma^{\dagger}\right)-H^{-1}\left(\gamma\right)K\left(\gamma\right)$
$\displaystyle=$ $\displaystyle
H^{-1}\left(\gamma+\gamma^{\dagger}\right)\left(K\left(\gamma+\gamma^{\dagger}\right)-K(\gamma)\right)$
$\displaystyle+$ $\displaystyle
H^{-1}\left(\gamma+\gamma^{\dagger}\right)\left(H\left(\gamma\right)-H\left(\gamma+\gamma^{\dagger}\right)\right)H^{-1}\left(\gamma\right)K\left(\gamma\right)$
$\displaystyle=$
$\displaystyle-H^{-1}\left(\gamma+\gamma^{\dagger}\right)\left(\sum_{j=1}^{m_{1}}\gamma_{j}^{\dagger}W_{j}+\sum_{j=m_{1}+1}^{m_{1}+m_{2}}\gamma_{j}^{\dagger}W_{j}H^{-1}(\gamma)K(\gamma)\right).$

Substituting (LABEL:sem_SARMA1) in (S.D.10) implies that

$A^{-1}\left(\gamma+\gamma^{\dagger}\right)A^{\prime-1}\left(\gamma+\gamma^{\dagger}\right)-A^{-1}(\gamma)A^{\prime-1}(\gamma)=\Delta_{1}\left(\gamma,\gamma^{\dagger}\right)+\Delta_{2}\left(\gamma,\gamma^{\dagger}\right)=\Delta\left(\gamma,\gamma^{\dagger}\right),$
(S.D.12)

say, where

$\displaystyle\Delta_{1}\left(\gamma,\gamma^{\dagger}\right)$ $\displaystyle=$
$\displaystyle
A^{-1}\left(\gamma+\gamma^{\dagger}\right)A^{\prime-1}\left(\gamma+\gamma^{\dagger}\right)\left(\sum_{j=1}^{m_{1}}\gamma_{j}^{\dagger}W^{\prime}_{j}+K^{\prime}(\gamma)H^{\prime-1}(\gamma)\sum_{j=m_{1}+1}^{m_{1}+m_{2}}\gamma_{j}^{\dagger}W^{\prime}_{j}\right)$
$\displaystyle\times$ $\displaystyle
H^{\prime-1}\left(\gamma+\gamma^{\dagger}\right)A^{\prime-1}(\gamma),$
$\displaystyle\Delta_{2}\left(\gamma,\gamma^{\dagger}\right)$ $\displaystyle=$
$\displaystyle
A^{-1}\left(\gamma+\gamma^{\dagger}\right)H^{-1}\left(\gamma+\gamma^{\dagger}\right)\left(\sum_{j=1}^{m_{1}}\gamma_{j}^{\dagger}W_{j}+\sum_{j=m_{1}+1}^{m_{1}+m_{2}}\gamma_{j}^{\dagger}W_{j}H^{-1}(\gamma)K(\gamma)\right)$
$\displaystyle\times$ $\displaystyle A^{-1}(\gamma)A^{\prime-1}(\gamma).$

From the definitions above and recalling that
$A(\gamma)=H^{-1}(\gamma)K(\gamma)$, we can write

$\Delta\left(\gamma,\gamma^{\dagger}\right)=A^{-1}\left(\gamma+\gamma^{\dagger}\right)\Upsilon\left(\gamma,\gamma^{\dagger}\right)A^{\prime-1}\left(\gamma\right),$
(S.D.13)

with

$\displaystyle\Upsilon\left(\gamma,\gamma^{\dagger}\right)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{m_{1}}\gamma_{j}^{\dagger}G^{\prime}_{j}\left(\gamma+\gamma^{\dagger}\right)H^{\prime-1}\left(\gamma+\gamma^{\dagger}\right)+A^{\prime-1}\left(\gamma+\gamma^{\dagger}\right)A^{\prime}(\gamma)\sum_{j=m_{1}+1}^{m_{1}+m_{2}}\gamma_{j}^{\dagger}T^{\prime}_{j}\left(\gamma+\gamma^{\dagger}\right)$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{m_{1}}\gamma_{j}^{\dagger}H^{-1}\left(\gamma+\gamma^{\dagger}\right)G_{j}\left(\gamma\right)+\sum_{j=m_{1}+1}^{m_{1}+m_{2}}\gamma_{j}^{\dagger}T_{j}\left(\gamma+\gamma^{\dagger}\right).$

Then (S.D.12) implies that

$\displaystyle
A^{-1}\left(\gamma+\gamma^{\dagger}\right)A^{\prime-1}\left(\gamma+\gamma^{\dagger}\right)-A^{-1}(\gamma)A^{\prime-1}(\gamma)-\left(D\Sigma(\gamma)\right)\left(\gamma^{\dagger}\right)$
(S.D.14) $\displaystyle=$ $\displaystyle
A^{-1}\left(\gamma+\gamma^{\dagger}\right)A^{\prime-1}\left(\gamma+\gamma^{\dagger}\right)-A^{-1}(\gamma)A^{\prime-1}(\gamma)-\Delta\left(\gamma,\gamma^{\dagger}\right)-\left(D\Sigma(\gamma)\right)\left(\gamma^{\dagger}\right)+\Delta\left(\gamma,\gamma^{\dagger}\right)$
$\displaystyle=$
$\displaystyle\Delta\left(\gamma,\gamma^{\dagger}\right)-\left(D\Sigma(\gamma)\right)\left(\gamma^{\dagger}\right),$

so to prove (S.D.9) it is sufficient to show that

$\frac{\left\|\Delta\left(\gamma,\gamma^{\dagger}\right)-\left(D\Sigma(\gamma)\right)\left(\gamma^{\dagger}\right)\right\|}{\left\|\gamma^{\dagger}\right\|_{g}}\rightarrow
0\text{ as }\left\|\gamma^{\dagger}\right\|_{g}\rightarrow 0.$ (S.D.15)

The numerator in (S.D.15) can be written as
$\sum_{i=1}^{7}\Pi_{i}\left(\gamma,\gamma^{\dagger}\right)A^{\prime-1}(\gamma)$
by adding, subtracting and grouping terms, where (omitting the argument
$\left(\gamma,\gamma^{\dagger}\right)$)

$\displaystyle\Pi_{1}$ $\displaystyle=$ $\displaystyle
A^{-1}\left(\gamma+\gamma^{\dagger}\right)\sum_{j=1}^{m_{1}}\gamma_{j}^{\dagger}G^{\prime}_{j}\left(\gamma+\gamma^{\dagger}\right)H^{\prime-1}(\gamma)\left(H(\gamma)-H\left(\gamma+\gamma^{\dagger}\right)\right)^{\prime}H^{\prime-1}\left(\gamma+\gamma^{\dagger}\right),$
$\displaystyle\Pi_{2}$ $\displaystyle=$ $\displaystyle
A^{-1}\left(\gamma+\gamma^{\dagger}\right)\sum_{j=1}^{m_{1}}\gamma_{j}^{\dagger}H^{-1}\left(\gamma+\gamma^{\dagger}\right)\left(H(\gamma)-H\left(\gamma+\gamma^{\dagger}\right)\right)H^{-1}(\gamma)G_{j}\left(\gamma\right),$
$\displaystyle\Pi_{3}$ $\displaystyle=$ $\displaystyle
A^{-1}\left(\gamma+\gamma^{\dagger}\right)\sum_{j=m_{1}+1}^{m_{1}+m_{2}}\gamma_{j}^{\dagger}\left(A^{-1}\left(\gamma+\gamma^{\dagger}\right)-A^{-1}\left(\gamma\right)\right)T^{\prime}_{j}\left(\gamma+\gamma^{\dagger}\right),$
$\displaystyle\Pi_{4}$ $\displaystyle=$
$\displaystyle\left(A^{-1}\left(\gamma+\gamma^{\dagger}\right)-A^{-1}\left(\gamma\right)\right)\sum_{j=m_{1}+1}^{m_{1}+m_{2}}\gamma_{j}^{\dagger}\overline{T_{j}\left(\gamma+\gamma^{\dagger}\right)},$
$\displaystyle\Pi_{5}$ $\displaystyle=$ $\displaystyle
A^{-1}(\gamma)\sum_{j=m_{1}+1}^{m_{1}+m_{2}}\gamma_{j}^{\dagger}\overline{H^{-1}\left(\gamma+\gamma^{\dagger}\right)\left(H(\gamma)-H\left(\gamma+\gamma^{\dagger}\right)\right)H^{-1}(\gamma)W_{j}},$
$\displaystyle\Pi_{6}$ $\displaystyle=$
$\displaystyle\Delta\left(\gamma,\gamma^{\dagger}\right)\sum_{j=1}^{m_{1}}\gamma_{j}^{\dagger}W_{j}^{\prime}H^{\prime-1}(\gamma),$
$\displaystyle\Pi_{7}$ $\displaystyle=$
$\displaystyle\left(A^{-1}\left(\gamma+\gamma^{\dagger}\right)-A^{-1}\left(\gamma\right)\right)\sum_{j=1}^{m_{1}}\gamma_{j}^{\dagger}H^{-1}(\gamma)G_{j}(\gamma).$

By (S.D.8), (S.D.13) and replication of earlier techniques, we have

$\max_{i=1,\ldots,7}\sup_{\gamma\in\Gamma^{o}}\left\|\Pi_{i}\left(\gamma,\gamma^{\dagger}\right)A^{-1}(\gamma)\right\|\leq
C\left\|\gamma^{\dagger}\right\|^{2}_{g},$ (S.D.16)

where the norm used on the RHS of (S.D.16) depends on whether we are
considering the general case or the ‘single nonzero diagonal block’ case. Thus

$\frac{\left\|\Delta\left(\gamma,\gamma^{\dagger}\right)-\left(D\Sigma(\gamma)\right)\left(\gamma^{\dagger}\right)\right\|}{\left\|\gamma^{\dagger}\right\|_{g}}\leq
C\left\|\gamma^{\dagger}\right\|_{g}\rightarrow 0\text{ as
}\left\|\gamma^{\dagger}\right\|_{g}\rightarrow 0,$

proving (S.D.15) and thus (S.D.9). ∎

######  Corollary CS.1.

For the spatial error model with SAR$(m_{1})$ errors,

$\left(D\Sigma(\gamma)\right)\left(\gamma^{\dagger}\right)=K^{-1}(\gamma)\sum_{j=1}^{m_{1}}\gamma^{\dagger}_{j}\overline{G_{j}(\gamma)}K^{\prime-1}(\gamma).$

###### Proof.

Taking $m_{2}=0$ in Lemma LS.5, the elements involving sums from $m_{1}+1$ to
$m_{1}+m_{2}$ do not arise and $H(\gamma)=I_{n}$, proving the claim. ∎

######  Corollary CS.2.

For the spatial error model with SMA$(m_{2})$ errors,

$\left(D\Sigma(\gamma)\right)\left(\gamma^{\dagger}\right)=H(\gamma)\sum_{j=1}^{m_{2}}\gamma^{\dagger}_{j}\overline{T_{j}(\gamma)}H^{\prime}(\gamma).$

###### Proof.

Taking $m_{1}=0$ in Lemma LS.5, the elements involving sums from $1$ to
$m_{1}$ do not arise and $K(\gamma)=I_{n}$, proving the claim. ∎

######  Lemma LS.6.

For the spatial error model with MESS$(m_{1})$ errors, if

$\max_{j=1,\ldots,m_{1}}\left(\left\|W_{j}\right\|+\left\|W^{\prime}_{j}\right\|\right)<1,$
(S.D.17)

then

$\left(D\Sigma(\gamma)\right)\left(\gamma^{\dagger}\right)=\exp\left(\sum_{j=1}^{m_{1}}\gamma_{j}\left(W_{j}+W_{j}^{\prime}\right)\right)\sum_{j=1}^{m_{1}}\gamma_{j}^{\dagger}\left(W_{j}+W_{j}^{\prime}\right).$

###### Proof.

Clearly $D\Sigma\in\mathscr{L}\left(\Gamma^{o},\mathcal{M}^{n\times
n}\right)$. Next,

$\displaystyle\left\|A^{-1}\left(\gamma+\gamma^{\dagger}\right)A^{\prime-1}\left(\gamma+\gamma^{\dagger}\right)-A^{-1}(\gamma)A^{\prime-1}(\gamma)-\left(D\Sigma(\gamma)\right)\left(\gamma^{\dagger}\right)\right\|$
(S.D.18) $\displaystyle=$
$\displaystyle\left\|\exp\left(\sum_{j=1}^{m_{1}}\left(\gamma_{j}+\gamma^{\dagger}_{j}\right)\left(W_{j}+W_{j}^{\prime}\right)\right)-\exp\left(\sum_{j=1}^{m_{1}}\gamma_{j}\left(W_{j}+W_{j}^{\prime}\right)\right)-\left(D\Sigma(\gamma)\right)\left(\gamma^{\dagger}\right)\right\|$
$\displaystyle=$
$\displaystyle\left\|\exp\left(\sum_{j=1}^{m_{1}}\gamma_{j}\left(W_{j}+W_{j}^{\prime}\right)\right)\left(\exp\left(\sum_{j=1}^{m_{1}}\gamma^{\dagger}_{j}\left(W_{j}+W_{j}^{\prime}\right)\right)-I_{n}-\sum_{j=1}^{m_{1}}\gamma^{\dagger}_{j}\left(W_{j}+W_{j}^{\prime}\right)\right)\right\|$
$\displaystyle\leq$
$\displaystyle\left\|\exp\left(\sum_{j=1}^{m_{1}}\gamma_{j}\left(W_{j}+W_{j}^{\prime}\right)\right)\right\|\left\|\exp\left(\sum_{j=1}^{m_{1}}\gamma^{\dagger}_{j}\left(W_{j}+W_{j}^{\prime}\right)\right)-I_{n}-\sum_{j=1}^{m_{1}}\gamma^{\dagger}_{j}\left(W_{j}+W_{j}^{\prime}\right)\right\|$
$\displaystyle\leq$ $\displaystyle
C\left\|I_{n}+\sum_{j=1}^{p}\gamma^{\dagger}_{j}\left(W_{j}+W_{j}^{\prime}\right)+\sum_{k=2}^{\infty}\left\\{\sum_{j=1}^{m_{1}}\gamma^{\dagger}_{j}\left(W_{j}+W_{j}^{\prime}\right)\right\\}^{k}-I_{n}-\sum_{j=1}^{m_{1}}\gamma^{\dagger}_{j}\left(W_{j}+W_{j}^{\prime}\right)\right\|$
$\displaystyle\leq$ $\displaystyle
C\left\|\sum_{k=2}^{\infty}\left\\{\sum_{j=1}^{m_{1}}\gamma^{\dagger}_{j}\left(W_{j}+W_{j}^{\prime}\right)\right\\}^{k}\right\|\leq
C\sum_{k=2}^{\infty}\sum_{j=1}^{m_{1}}\left|\gamma^{\dagger}_{j}\right|\left\|\left(W_{j}+W_{j}^{\prime}\right)\right\|^{k}$
$\displaystyle\leq$ $\displaystyle
C\sum_{k=2}^{\infty}\left\|\gamma^{\dagger}\right\|^{k}_{g},$

by (S.D.17), without loss of generality, and again the norm used in (S.D.18)
depending on whether we are in the general or the ‘single nonzero diagonal
block’ case. Thus

$\frac{\left\|A^{-1}\left(\gamma+\gamma^{\dagger}\right)A^{\prime-1}\left(\gamma+\gamma^{\dagger}\right)-A^{-1}(\gamma)A^{\prime-1}(\gamma)-\left(D\Sigma(\gamma)\right)\left(\gamma^{\dagger}\right)\right\|}{\left\|\gamma^{\dagger}\right\|_{g}}\leq
C\sum_{k=2}^{\infty}\left\|\gamma^{\dagger}\right\|^{k-1}_{g}\rightarrow 0,$

as $\left\|\gamma^{\dagger}\right\|_{g}\rightarrow 0$, proving the claim. ∎

######  Theorem TS.1.

Under the conditions of Theorem 4.4 or 5.3,
$\mathscr{T}_{n}-\mathscr{T}_{n}^{a}=o_{p}(1)$ as $n\rightarrow\infty$.

###### Proof.

It suffices to show that
$n\widetilde{m}_{n}=n\widehat{m}_{n}+o_{p}(\sqrt{p})$. As
$\widehat{\eta}=y-\widehat{{\theta}},$ $\widehat{u}=y-\widehat{f}$, and
$\widehat{v}=\widehat{\theta}-\widehat{f}$, we have
$\widehat{u}=\widehat{\eta}+\widehat{v}$ and

$\displaystyle n\widetilde{m}_{n}$ $\displaystyle=$
$\displaystyle\widehat{\sigma}^{-2}\left(\widehat{u}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\widehat{u}-\widehat{\eta}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\widehat{\eta}\right)=\widehat{\sigma}^{-2}\left(2\widehat{u}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\widehat{v}-\widehat{v}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\widehat{v}\right)$
(S.D.19) $\displaystyle=$ $\displaystyle
2n\widehat{m}_{n}-\widehat{\sigma}^{-2}\left[\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}({u+e}{)}-{e}+{\theta}_{0}-\widehat{{f}}\right]^{\prime}$
$\displaystyle\Sigma\left(\widehat{\gamma}\right)^{-1}\left[\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}({u+e}{)}-{e}+{\theta}_{0}-\widehat{{f}}\right]$
$\displaystyle=$ $\displaystyle
2n\widehat{m}_{n}-\widehat{\sigma}^{-2}u^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}u\mathbf{-}\widehat{\sigma}^{-2}\left({\theta}_{0}-\widehat{{f}}\right)^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\left({\theta}_{0}-\widehat{{f}}\right)$
$\displaystyle\mathbf{+}\widehat{\sigma}^{-2}\left(2({\theta}_{0}-\widehat{{f}})-e\right)^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\left(I-\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\right)e$
$\displaystyle-2\widehat{\sigma}^{-2}\left({\theta}_{0}-\widehat{{f}}\right)^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}u$
$\displaystyle=$ $\displaystyle
2n\widehat{m}_{n}-\left(n\widehat{m}_{n}-\widehat{\sigma}^{-2}\left(A_{1}+A_{2}+A_{3}+A_{4}\right)\right)-\widehat{\sigma}^{-2}A_{4}$
$\displaystyle\mathbf{+}\widehat{\sigma}^{-2}\left(2({\theta}_{0}-\widehat{{f}})-e\right)^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\left(I-\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\right)e-2\widehat{\sigma}^{-2}A_{3}$
$\displaystyle=$ $\displaystyle
n\widehat{m}_{n}+\widehat{\sigma}^{-2}\left(A_{1}+A_{2}-A_{3}\right)$
$\displaystyle+\widehat{\sigma}^{-2}\left(2({\theta}_{0}-\widehat{{f}})-e\right)^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\left(I-\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\right)e.$

In the proof of Theorem 4.2, we have shown that

$\left|\left({\theta}_{0}-\widehat{{f}}\right)^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\left(I-\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\right)e\right|=o_{p}(\sqrt{p})$

in the process of proving $|A_{2}|=o_{p}(\sqrt{p})$. Along with

$\displaystyle\left|e^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\left(I-\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\right)e\right|$
$\displaystyle\leq$
$\displaystyle\left|e^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}e\right|+\left|e^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}e\right|$
$\displaystyle\leq$
$\displaystyle\left\|e\right\|^{2}\sup_{\gamma\in\Gamma}\left\|\Sigma\left(\gamma\right)^{-1}\right\|+\left\|e\right\|^{2}\sup_{\gamma\in\Gamma}\left\|\Sigma\left(\gamma\right)^{-1}\right\|^{2}\left\|\frac{1}{n}\Psi\left(\frac{1}{n}\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\right\|$
$\displaystyle=$ $\displaystyle
O_{p}\left(\left\|e\right\|^{2}\right)=O_{p}\left(p^{-2\mu}n\right)=o_{p}(\sqrt{p}),$

we complete the proof that
$n\widetilde{m}_{n}=n\widehat{m}_{n}+o_{p}(\sqrt{p}).$ In the SAR setting of
Section 5,

$\displaystyle n\widetilde{m}_{n}$ $\displaystyle=$
$\displaystyle\widehat{\sigma}^{-2}\left(\widehat{u}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\widehat{u}-\widehat{\eta}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\widehat{\eta}\right)=\widehat{\sigma}^{-2}\left(2\widehat{u}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\widehat{v}-\widehat{v}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\widehat{v}\right)$
$\displaystyle=$ $\displaystyle
2n\widehat{m}_{n}-\widehat{\sigma}^{-2}\left[\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\left(u+e+\sum_{j=1}^{d_{\lambda}}(\lambda_{j_{0}}-\widehat{\lambda}_{j})W_{j}y\right)-e+\theta_{0}-\widehat{f}\right]^{\prime}$
$\displaystyle\Sigma\left(\widehat{\gamma}\right)^{-1}\left[\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\left(u+e+\sum_{j=1}^{d_{\lambda}}(\lambda_{j_{0}}-\widehat{\lambda}_{j})W_{j}y\right)-e+\theta_{0}-\widehat{f}\right].$

Compared to the expression in (S.D.19), we have the additional terms

$-\widehat{\sigma}^{-2}\sum_{j=1}^{d_{\lambda}}(\lambda_{j_{0}}-\widehat{\lambda}_{j})W_{j}y^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\sum_{j=1}^{d_{\lambda}}(\lambda_{j_{0}}-\widehat{\lambda}_{j})W_{j}y$

and

$-2\widehat{\sigma}^{-2}\sum_{j=1}^{d_{\lambda}}(\lambda_{j_{0}}-\widehat{\lambda}_{j})W_{j}y^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\left(u+\theta_{0}-\widehat{f}\right).$

Both terms are $o_{p}(\sqrt{p})$ from the orders of $A_{5}$ and $A_{6}$ in the
proof of Theorem 5.2. Hence, in the SAR setting,
$n\widetilde{m}_{n}=n\widehat{m}_{n}+o_{p}(\sqrt{p})$ also holds.

We now present similar calculations that justify the validity of our bootstrap
test for the SARARMA($m_{1}$,$m_{2},m_{3}$) model. The bootstrapped test
statistic is constructed with

$n\widehat{m}_{n}^{\ast}=\widehat{{v}}^{\ast\prime}\Sigma\left(\widehat{\gamma}^{\ast}\right)^{-1}\widehat{{u}}^{\ast}=(\widehat{{\theta}}_{n}^{\ast}-{f}(x,\widehat{\alpha}_{n}^{\ast}))^{\prime}\Sigma\left(\widehat{\gamma}^{\ast}\right)^{-1}\left((I_{n}-\sum_{k=1}^{m_{1}}\widehat{\lambda}_{k}^{\ast}W_{1k})y^{\ast}-{f}(x,\widehat{\alpha}_{n}^{\ast})\right).$

Let $J_{n}=(I_{n}-\frac{1}{n}l_{n}l_{n}^{\prime})$. As
$y=S(\lambda)^{-1}(\theta(x)+R(\gamma)\xi)$, we have

$\displaystyle\widetilde{\mathbf{\xi}}$ $\displaystyle=$ $\displaystyle
J_{n}\widehat{\mathbf{\xi}}$ $\displaystyle=$ $\displaystyle
J_{n}\left(\left(\sum_{l=1}^{m_{3}}\gamma_{3l}W_{3l}+I_{n}\right)^{-1}+\left(\sum_{l=1}^{m_{3}}\gamma_{3l}W_{3l}+I_{n}\right)^{-1}\sum_{l=1}^{m_{3}}(\gamma_{3l}-\widehat{\gamma}_{3l})W_{3l}\left(\sum_{l=1}^{m_{3}}\widehat{\gamma}_{3l}W_{3l}+I_{n}\right)^{-1}\right)$
$\displaystyle\times\left(I_{n}-\sum_{l=1}^{m_{2}}\gamma_{2l}W_{2l}+\sum_{l=1}^{m_{2}}(\gamma_{2l}-\widehat{\gamma}_{2l})W_{2l}\right)\left(S(\lambda)y-\theta(x)+\sum_{k=1}^{m_{1}}(\lambda_{k}-\widehat{\lambda}_{k})W_{1k}y+\theta(x)-\psi^{\prime}\widehat{\beta}\right)$
$\displaystyle=$
$\displaystyle\xi-\frac{1}{n}l_{n}l_{n}^{\prime}\xi+J_{n}\left(\sum_{l=1}^{m_{3}}\gamma_{3l}W_{3l}+I_{n}\right)^{-1}\left(I_{n}-\sum_{l=1}^{m_{2}}\gamma_{2l}W_{2l}\right)\left(\sum_{k=1}^{m_{1}}(\lambda_{k}-\widehat{\lambda}_{k})W_{1k}y+\theta(x)-\psi^{\prime}\widehat{\beta}\right)$
$\displaystyle+J_{n}\left(\sum_{l=1}^{m_{3}}\gamma_{3l}W_{3l}+I_{n}\right)^{-1}\sum_{l=1}^{m_{2}}(\gamma_{2l}-\widehat{\gamma}_{2l})W_{2l}\left(S(\lambda)y-\theta(x)+\sum_{k=1}^{m_{1}}(\lambda_{k}-\widehat{\lambda}_{k})W_{1k}y+\theta(x)-\psi^{\prime}\widehat{\beta}\right)$
$\displaystyle+J_{n}\left(\sum_{l=1}^{m_{3}}\gamma_{3l}W_{3l}+I_{n}\right)^{-1}\sum_{l=1}^{m_{3}}(\gamma_{3l}-\widehat{\gamma}_{3l})W_{3l}\left(\sum_{l=1}^{m_{3}}\widehat{\gamma}_{3l}W_{3l}+I_{n}\right)^{-1}$
$\displaystyle\times\left(I_{n}-\sum_{l=1}^{m_{2}}\gamma_{2l}W_{2l}+\sum_{l=1}^{m_{2}}(\gamma_{2l}-\widehat{\gamma}_{2l})W_{2l}\right)\left(S(\lambda)y-\theta(x)+\sum_{k=1}^{m_{1}}(\lambda_{k}-\widehat{\lambda}_{k})W_{1k}y+\theta(x)-\psi^{\prime}\widehat{\beta}\right),$

which can be written as

$\widetilde{\mathbf{\xi}}=\xi+\sum_{j=1}^{r}\zeta_{1n,j}p_{nj}+\sum_{j=1}^{s}\zeta_{2n,j}Q_{nj}\xi,$

where $p_{nj}$ is an $n$-dimensional vector with bounded elements,
$Q_{nj}=[q_{nj,i}]$ is an $n\times n$ matrix with bounded row and column sum
norms, and $\zeta_{1n,j}$ and $\zeta_{2n,j}$’s are equal to
$l_{n}^{\prime}\xi/n$, elements of $\lambda_{k}-\widehat{\lambda}_{k}$,
$\gamma_{2l}-\widehat{\gamma}_{2l}$, $\theta(x)-\psi^{\prime}\widehat{\beta}$
or their products. This differs from the proof of Lemma 2 in Jin and Lee
(2015) in the term $\theta(x)-\psi^{\prime}\widehat{\beta}$ and potentially
increasing order of $d_{\gamma}$. Then,
$\zeta_{1n,j}=O_{p}(\sqrt{p^{1/2}/n}\vee\sqrt{d_{\gamma}/n})$ and
$\zeta_{2n,j}=O_{p}(\sqrt{p^{1/2}/n}\vee\sqrt{d_{\gamma}/n})$, instead of
$O_{p}(\sqrt{1/n})$ as in Jin and Lee (2015). Based on this result, the
assumptions in Theorem 4 of Su and Qu (2017) hold, so the validility of our
bootstrap test directly follows.

∎

|  | PS |  |  |  | Trig |  |  |  | B-s |
---|---|---|---|---|---|---|---|---|---|---|---
| 0.01 | 0.05 | 0.10 |  | 0.01 | 0.05 | 0.10 |  | 0.01 | 0.05 | 0.10
$n=60$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.01}$ | ${\small 0.032}$ | ${\small 0.05}$ |  | ${\small 0.01}$ | ${\small 0.028}$ | ${\small 0.054}$ |  | ${\small 0.02}$ | ${\small 0.042}$ | ${\small 0.064}$
| ${\small 0.036}$ | ${\small 0.084}$ | ${\small 0.122}$ |  | ${\small 0.02}$ | ${\small 0.056}$ | ${\small 0.084}$ |  | ${\small 0.044}$ | ${\small 0.008}$ | ${\small 0.11}$
${\small c=3}$ | ${\small 0.07}$ | ${\small 0.156}$ | ${\small 0.194}$ |  | ${\small 0.166}$ | ${\small 0.248}$ | ${\small 0.296}$ |  | ${\small 0.208}$ | ${\small 0.302}$ | ${\small 0.372}$
| ${\small 0.454}$ | ${\small 0.58}$ | ${\small 0.658}$ |  | ${\small 0.172}$ | ${\small 0.29}$ | ${\small 0.358}$ |  | ${\small 0.166}$ | ${\small 0.274}$ | ${\small 0.346}$
${\small c=6}$ | ${\small 0.37}$ | ${\small 0.532}$ | ${\small 0.644}$ |  | ${\small 0.688}$ | ${\small 0.806}$ | ${\small 0.854}$ |  | ${\small 0.688}$ | ${\small 0.82}$ | ${\small 0.884}$
| ${\small 0.998}$ | ${\small 1}$ | ${\small 1}$ |  | ${\small 0.676}$ | ${\small 0.822}$ | ${\small 0.866}$ |  | ${\small 0.576}$ | ${\small 0.726}$ | ${\small 0.81}$
$n=100$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.008}$ | ${\small 0.03}$ | ${\small 0.044}$ |  | ${\small 0.006}$ | ${\small 0.012}$ | ${\small 0.028}$ |  | ${\small 0.016}$ | ${\small 0.028}$ | ${\small 0.042}$
| ${\small 0.022}$ | ${\small 0.052}$ | ${\small 0.068}$ |  | ${\small 0.004}$ | ${\small 0.028}$ | ${\small 0.05}$ |  | ${\small 0.018}$ | ${\small 0.048}$ | ${\small 0.062}$
${\small c=3}$ | ${\small 0.352}$ | ${\small 0.478}$ | ${\small 0.574}$ |  | ${\small 0.27}$ | ${\small 0.39}$ | ${\small 0.484}$ |  | ${\small 0.376}$ | ${\small 0.518}$ | ${\small 0.614}$
| ${\small 0.54}$ | ${\small 0.666}$ | ${\small 0.744}$ |  | ${\small 0.288}$ | ${\small 0.412}$ | ${\small 0.508}$ |  | ${\small 0.316}$ | ${\small 0.462}$ | ${\small 0.544}$
${\small c=6}$ | ${\small 0.984}$ | ${\small 0.99}$ | ${\small 0.99}$ |  | ${\small 0.956}$ | ${\small 0.986}$ | ${\small 0.992}$ |  | ${\small 0.98}$ | ${\small 0.992}$ | ${\small 0.994}$
| ${\small 0.998}$ | ${\small 0.998}$ | ${\small 0.998}$ |  | ${\small 0.948}$ | ${\small 0.99}$ | ${\small 0.992}$ |  | ${\small 0.956}$ | ${\small 0.99}$ | ${\small 0.996}$
$n=200$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.002}$ | ${\small 0.016}$ | ${\small 0.034}$ |  | ${\small 0.002}$ | ${\small 0.014}$ | ${\small 0.034}$ |  | ${\small 0.038}$ | ${\small 0.074}$ | ${\small 0.102}$
| ${\small 0.008}$ | ${\small 0.026}$ | ${\small 0.048}$ |  | ${\small 0.012}$ | ${\small 0.028}$ | ${\small 0.036}$ |  | ${\small 0.01}$ | ${\small 0.036}$ | ${\small 0.074}$
${\small c=3}$ | ${\small 0.176}$ | ${\small 0.29}$ | ${\small 0.356}$ |  | ${\small 0.164}$ | ${\small 0.256}$ | ${\small 0.312}$ |  | ${\small 0.388}$ | ${\small 0.354}$ | ${\small 0.606}$
| ${\small 0.34}$ | ${\small 0.496}$ | ${\small 0.582}$ |  | ${\small 0.144}$ | ${\small 0.274}$ | ${\small 0.356}$ |  | ${\small 0.168}$ | ${\small 0.282}$ | ${\small 0.376}$
${\small c=6}$ | ${\small 0.888}$ | ${\small 0.942}$ | ${\small 0.96}$ |  | ${\small 0.818}$ | ${\small 0.898}$ | ${\small 0.934}$ |  | ${\small 0.944}$ | ${\small 0.974}$ | ${\small 0.986}$
| ${\small 0.99}$ | ${\small 0.998}$ | ${\small 1}$ |  | ${\small 0.816}$ | ${\small 0.904}$ | ${\small 0.944}$ |  | ${\small 0.862}$ | ${\small 0.932}$ | ${\small 0.954}$
Table OT.1: Rejection probabilities of SARARMA(0,1,0) using asymptotic test
${\mathscr{T}_{n}}$ at 1, 5, 10% levels, power series (PS), trigonometric
(Trig) and B-spline (B-s) bases. Compactly supported regressors.

|  | PS |  |  |  | Trig |  |  |  | B-s |
---|---|---|---|---|---|---|---|---|---|---|---
| 0.01 | 0.05 | 0.10 |  | 0.01 | 0.05 | 0.10 |  | 0.01 | 0.05 | 0.10
$n=60$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.01}$ | ${\small 0.032}$ | ${\small 0.05}$ |  | ${\small 0.01}$ | ${\small 0.028}$ | ${\small 0.054}$ |  | ${\small 0.06}$ | ${\small 0.01}$ | ${\small 0.016}$
| ${\small 0.036}$ | ${\small 0.084}$ | ${\small 0.122}$ |  | ${\small 0.02}$ | ${\small 0.056}$ | ${\small 0.084}$ |  | ${\small 0.044}$ | ${\small 0.008}$ | ${\small 0.116}$
${\small c=3}$ | ${\small 0.07}$ | ${\small 0.156}$ | ${\small 0.194}$ |  | ${\small 0.16}$ | ${\small 0.252}$ | ${\small 0.292}$ |  | ${\small 0.09}$ | ${\small 0.138}$ | ${\small 0.186}$
| ${\small 0.454}$ | ${\small 0.58}$ | ${\small 0.658}$ |  | ${\small 0.174}$ | ${\small 0.29}$ | ${\small 0.358}$ |  | ${\small 0.166}$ | ${\small 0.272}$ | ${\small 0.34}$
${\small c=6}$ | ${\small 0.37}$ | ${\small 0.532}$ | ${\small 0.644}$ |  | ${\small 0.682}$ | ${\small 0.798}$ | ${\small 0.85}$ |  | ${\small 0.514}$ | ${\small 0.644}$ | ${\small 0.714}$
| ${\small 0.998}$ | ${\small 1}$ | ${\small 1}$ |  | ${\small 0.676}$ | ${\small 0.822}$ | ${\small 0.866}$ |  | ${\small 0.572}$ | ${\small 0.714}$ | ${\small 0.8}$
$n=100$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.008}$ | ${\small 0.03}$ | ${\small 0.044}$ |  | ${\small 0.006}$ | ${\small 0.012}$ | ${\small 0.026}$ |  | ${\small 0}$ | ${\small 0.004}$ | ${\small 0.006}$
| ${\small 0.022}$ | ${\small 0.052}$ | ${\small 0.068}$ |  | ${\small 0.006}$ | ${\small 0.028}$ | ${\small 0.05}$ |  | ${\small 0.018}$ | ${\small 0.05}$ | ${\small 0.062}$
${\small c=3}$ | ${\small 0.352}$ | ${\small 0.478}$ | ${\small 0.574}$ |  | ${\small 0.268}$ | ${\small 0.396}$ | ${\small 0.486}$ |  | ${\small 0.158}$ | ${\small 0.23}$ | ${\small 0.288}$
| ${\small 0.54}$ | ${\small 0.666}$ | ${\small 0.744}$ |  | ${\small 0.288}$ | ${\small 0.412}$ | ${\small 0.508}$ |  | ${\small 0.322}$ | ${\small 0.466}$ | ${\small 0.55}$
${\small c=6}$ | ${\small 0.984}$ | ${\small 0.99}$ | ${\small 0.99}$ |  | ${\small 0.958}$ | ${\small 0.986}$ | ${\small 0.992}$ |  | ${\small 0.918}$ | ${\small 0.97}$ | ${\small 0.98}$
| ${\small 0.998}$ | ${\small 0.998}$ | ${\small 0.998}$ |  | ${\small 0.952}$ | ${\small 0.99}$ | ${\small 0.992}$ |  | ${\small 0.96}$ | ${\small 0.99}$ | ${\small 0.998}$
$n=200$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.002}$ | ${\small 0.016}$ | ${\small 0.034}$ |  | ${\small 0.002}$ | ${\small 0.018}$ | ${\small 0.038}$ |  | ${\small 0}$ | ${\small 0}$ | ${\small 0}$
| ${\small 0.008}$ | ${\small 0.026}$ | ${\small 0.048}$ |  | ${\small 0.012}$ | ${\small 0.028}$ | ${\small 0.032}$ |  | ${\small 0.01}$ | ${\small 0.036}$ | ${\small 0.064}$
${\small c=3}$ | ${\small 0.176}$ | ${\small 0.29}$ | ${\small 0.356}$ |  | ${\small 0.156}$ | ${\small 0.258}$ | ${\small 0.312}$ |  | ${\small 0.022}$ | ${\small 0.03}$ | ${\small 0.044}$
| ${\small 0.34}$ | ${\small 0.496}$ | ${\small 0.582}$ |  | ${\small 0.144}$ | ${\small 0.272}$ | ${\small 0.352}$ |  | ${\small 0.154}$ | ${\small 0.266}$ | ${\small 0.352}$
${\small c=6}$ | ${\small 0.888}$ | ${\small 0.942}$ | ${\small 0.96}$ |  | ${\small 0.816}$ | ${\small 0.908}$ | ${\small 0.936}$ |  | ${\small 0.43}$ | ${\small 0.522}$ | ${\small 0.554}$
| ${\small 0.99}$ | ${\small 0.998}$ | ${\small 1}$ |  | ${\small 0.816}$ | ${\small 0.904}$ | ${\small 0.944}$ |  | ${\small 0.856}$ | ${\small 0.924}$ | ${\small 0.944}$
Table OT.2: Rejection probabilities of SARARMA(0,1,0) using asymptotic test
${\mathscr{T}_{n}}^{a}$ at 1, 5, 10% levels, power series (PS), trigonometric
(Trig) and B-spline (B-s) bases. Compactly supported regressors.

|  | PS | ${\mathscr{T}_{n}}=\mathscr{T}_{n}^{a}$ |  |  | Trig | $\mathscr{T}_{n}$ |  |  | Trig | $\mathscr{T}_{n}^{a}$
---|---|---|---|---|---|---|---|---|---|---|---
| 0.01 | 0.05 | 0.10 |  | 0.01 | 0.05 | 0.10 |  | 0.01 | 0.05 | 0.10
$n=60$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.02}$ | ${\small 0.05}$ | ${\small 0.072}$ |  | ${\small 0.016}$ | ${\small 0.038}$ | ${\small 0.052}$ |  | ${\small 0.016}$ | ${\small 0.038}$ | ${\small 0.052}$
| ${\small 0.038}$ | ${\small 0.082}$ | ${\small 0.11}$ |  | ${\small 0.038}$ | ${\small 0.06}$ | ${\small 0.08}$ |  | ${\small 0.038}$ | ${\small 0.06}$ | ${\small 0.08}$
${\small c=3}$ | ${\small 0.106}$ | ${\small 0.158}$ | ${\small 0.224}$ |  | ${\small 0.062}$ | ${\small 0.11}$ | ${\small 0.146}$ |  | ${\small 0.062}$ | ${\small 0.11}$ | ${\small 0.146}$
| ${\small 0.152}$ | ${\small 0.25}$ | ${\small 0.31}$ |  | ${\small 0.09}$ | ${\small 0.158}$ | ${\small 0.204}$ |  | ${\small 0.09}$ | ${\small 0.158}$ | ${\small 0.204}$
${\small c=6}$ | ${\small 0.552}$ | ${\small 0.686}$ | ${\small 0.73}$ |  | ${\small 0.234}$ | ${\small 0.352}$ | ${\small 0.482}$ |  | ${\small 0.236}$ | ${\small 0.354}$ | ${\small 0.43}$
| ${\small 0.634}$ | ${\small 0.774}$ | ${\small 0.82}$ |  | ${\small 0.404}$ | ${\small 0.542}$ | ${\small 0.642}$ |  | ${\small 0.404}$ | ${\small 0.542}$ | ${\small 0.642}$
$n=100$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.008}$ | ${\small 0.024}$ | ${\small 0.036}$ |  | ${\small 0.002}$ | ${\small 0.018}$ | ${\small 0.036}$ |  | ${\small 0.002}$ | ${\small 0.018}$ | ${\small 0.036}$
| ${\small 0.024}$ | ${\small 0.05}$ | ${\small 0.068}$ |  | ${\small 0.012}$ | ${\small 0.026}$ | ${\small 0.052}$ |  | ${\small 0.012}$ | ${\small 0.026}$ | ${\small 0.052}$
${\small c=3}$ | ${\small 0.162}$ | ${\small 0.262}$ | ${\small 0.342}$ |  | ${\small 0.142}$ | ${\small 0.22}$ | ${\small 0.286}$ |  | ${\small 0.142}$ | ${\small 0.22}$ | ${\small 0.286}$
| ${\small 0.216}$ | ${\small 0.332}$ | ${\small 0.408}$ |  | ${\small 0.164}$ | ${\small 0.274}$ | ${\small 0.35}$ |  | ${\small 0.164}$ | ${\small 0.274}$ | ${\small 0.35}$
${\small c=6}$ | ${\small 0.824}$ | ${\small 0.894}$ | ${\small 0.926}$ |  | ${\small 0.79}$ | ${\small 0.868}$ | ${\small 0.892}$ |  | ${\small 0.79}$ | ${\small 0.866}$ | ${\small 0.894}$
| ${\small 0.888}$ | ${\small 0.944}$ | ${\small 0.952}$ |  | ${\small 0.862}$ | ${\small 0.896}$ | ${\small 0.928}$ |  | ${\small 0.862}$ | ${\small 0.896}$ | ${\small 0.928}$
$n=200$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.006}$ | ${\small 0.018}$ | ${\small 0.032}$ |  | ${\small 0.008}$ | ${\small 0.022}$ | ${\small 0.032}$ |  | ${\small 0.008}$ | ${\small 0.022}$ | ${\small 0.032}$
| ${\small 0.012}$ | ${\small 0.032}$ | ${\small 0.068}$ |  | ${\small 0.01}$ | ${\small 0.026}$ | ${\small 0.046}$ |  | ${\small 0.01}$ | ${\small 0.026}$ | ${\small 0.046}$
${\small c=3}$ | ${\small 0.096}$ | ${\small 0.182}$ | ${\small 0.258}$ |  | ${\small 0.076}$ | ${\small 0.152}$ | ${\small 0.212}$ |  | ${\small 0.078}$ | ${\small 0.15}$ | ${\small 0.208}$
| ${\small 0.126}$ | ${\small 0.24}$ | ${\small 0.33}$ |  | ${\small 0.098}$ | ${\small 0.184}$ | ${\small 0.26}$ |  | ${\small 0.098}$ | ${\small 0.184}$ | ${\small 0.26}$
${\small c=6}$ | ${\small 0.754}$ | ${\small 0.858}$ | ${\small 0.892}$ |  | ${\small 0.596}$ | ${\small 0.728}$ | ${\small 0.794}$ |  | ${\small 0.596}$ | ${\small 0.724}$ | ${\small 0.79}$
| ${\small 0.84}$ | ${\small 0.918}$ | ${\small 0.944}$ |  | ${\small 0.684}$ | ${\small 0.794}$ | ${\small 0.866}$ |  | ${\small 0.684}$ | ${\small 0.792}$ | ${\small 0.866}$
Table OT.3: Rejection probabilities of SARARMA(0,1,0) using asymptotic tests
${\mathscr{T}_{n}},{\mathscr{T}_{n}}^{a}$ at 1, 5, 10% levels, power series
(PS) and trigonometric (Trig) bases. Unboundedly supported regressors.

|  | PS | ${\mathscr{T}_{n}}^{\ast}=\mathscr{T}_{n}^{a\ast}$ |  |  | Trig | $\mathscr{T}_{n}^{\ast}$ |  |  | Trig | $\mathscr{T}_{n}^{a\ast}$
---|---|---|---|---|---|---|---|---|---|---|---
| 0.01 | 0.05 | 0.10 |  | 0.01 | 0.05 | 0.10 |  | 0.01 | 0.05 | 0.10
$n=60$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.008}$ | ${\small 0.058}$ | ${\small 0.108}$ |  | ${\small 0.01}$ | ${\small 0.046}$ | ${\small 0.124}$ |  | ${\small 0.01}$ | ${\small 0.046}$ | ${\small 0.124}$
| ${\small 0.008}$ | ${\small 0.042}$ | ${\small 0.094}$ |  | ${\small 0.006}$ | ${\small 0.044}$ | ${\small 0.102}$ |  | ${\small 0.006}$ | ${\small 0.044}$ | ${\small 0.102}$
${\small c=3}$ | ${\small 0.052}$ | ${\small 0.17}$ | ${\small 0.318}$ |  | ${\small 0.036}$ | ${\small 0.14}$ | ${\small 0.21}$ |  | ${\small 0.036}$ | ${\small 0.14}$ | ${\small 0.21}$
| ${\small 0.034}$ | ${\small 0.16}$ | ${\small 0.184}$ |  | ${\small 0.034}$ | ${\small 0.132}$ | ${\small 0.234}$ |  | ${\small 0.034}$ | ${\small 0.132}$ | ${\small 0.234}$
${\small c=6}$ | ${\small 0.35}$ | ${\small 0.67}$ | ${\small 0.808}$ |  | ${\small 0.16}$ | ${\small 0.392}$ | ${\small 0.556}$ |  | ${\small 0.16}$ | ${\small 0.392}$ | ${\small 0.558}$
| ${\small 0.262}$ | ${\small 0.656}$ | ${\small 0.794}$ |  | ${\small 0.204}$ | ${\small 0.468}$ | ${\small 0.66}$ |  | ${\small 0.204}$ | ${\small 0.468}$ | ${\small 0.66}$
$n=100$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.006}$ | ${\small 0.05}$ | ${\small 0.102}$ |  | ${\small 0.006}$ | ${\small 0.05}$ | ${\small 0.11}$ |  | ${\small 0.004}$ | ${\small 0.05}$ | ${\small 0.112}$
| ${\small 0.012}$ | ${\small 0.054}$ | ${\small 0.128}$ |  | ${\small 0.004}$ | ${\small 0.044}$ | ${\small 0.112}$ |  | ${\small 0.004}$ | ${\small 0.044}$ | ${\small 0.112}$
${\small c=3}$ | ${\small 0.13}$ | ${\small 0.342}$ | ${\small 0.516}$ |  | ${\small 0.128}$ | ${\small 0.324}$ | ${\small 0.488}$ |  | ${\small 0.126}$ | ${\small 0.32}$ | ${\small 0.488}$
| ${\small 0.122}$ | ${\small 0.326}$ | ${\small 0.498}$ |  | ${\small 0.114}$ | ${\small 0.298}$ | ${\small 0.474}$ |  | ${\small 0.114}$ | ${\small 0.298}$ | ${\small 0.474}$
${\small c=6}$ | ${\small 0.766}$ | ${\small 0.932}$ | ${\small 0.974}$ |  | ${\small 0.728}$ | ${\small 0.92}$ | ${\small 0.974}$ |  | ${\small 0.728}$ | ${\small 0.92}$ | ${\small 0.972}$
| ${\small 0.774}$ | ${\small 0.934}$ | ${\small 0.968}$ |  | ${\small 0.732}$ | ${\small 0.898}$ | ${\small 0.952}$ |  | ${\small 0.732}$ | ${\small 0.898}$ | ${\small 0.952}$
$n=200$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.03}$ | ${\small 0.056}$ | ${\small 0.088}$ |  | ${\small 0.028}$ | ${\small 0.06}$ | ${\small 0.098}$ |  | ${\small 0.028}$ | ${\small 0.06}$ | ${\small 0.098}$
| ${\small 0.028}$ | ${\small 0.084}$ | ${\small 0.128}$ |  | ${\small 0.022}$ | ${\small 0.068}$ | ${\small 0.118}$ |  | ${\small 0.022}$ | ${\small 0.068}$ | ${\small 0.118}$
${\small c=3}$ | ${\small 0.17}$ | ${\small 0.346}$ | ${\small 0.49}$ |  | ${\small 0.132}$ | ${\small 0.286}$ | ${\small 0.384}$ |  | ${\small 0.13}$ | ${\small 0.288}$ | ${\small 0.38}$
| ${\small 0.178}$ | ${\small 0.34}$ | ${\small 0.488}$ |  | ${\small 0.128}$ | ${\small 0.274}$ | ${\small 0.416}$ |  | ${\small 0.128}$ | ${\small 0.274}$ | ${\small 0.416}$
${\small c=6}$ | ${\small 0.794}$ | ${\small 0.92}$ | ${\small 0.966}$ |  | ${\small 0.682}$ | ${\small 0.866}$ | ${\small 0.93}$ |  | ${\small 0.678}$ | ${\small 0.864}$ | ${\small 0.93}$
| ${\small 0.84}$ | ${\small 0.936}$ | ${\small 0.976}$ |  | ${\small 0.698}$ | ${\small 0.888}$ | ${\small 0.93}$ |  | ${\small 0.698}$ | ${\small 0.888}$ | ${\small 0.93}$
Table OT.4: Rejection probabilities of SARARMA(0,1,0) using bootstrap tests
${\mathscr{T}_{n}}^{\ast},{\mathscr{T}_{n}}^{a\ast}$ at 1, 5, 10% levels,
power series (PS) and trigonometric (Trig) bases. Unboundedly supported
regressors.

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