\subsection{Proof of the other direction in Theorem \ref{th: pop_sparsity gnar}.b}
We proceed to prove the other direction of the second part ($\implies$).
\begin{proof}
Consider reparameterizing the process for some $s \leq r^*$ such that $\beta_{kr} \equiv 0$ if and only if $r > s$. 
By Proposition \ref{prop: var constrains} the new linear filters given by (\ref{eq: opt filter}) satisfy that if $(i, j) \notin \mathcal{A}_s,$ then
$\cov (u_{i, t}^{(ij)|s}, u_{j, t + k}^{(ij)|s}) \equiv 0,$
where $u_{i, t}^{(ij)|s}$ denotes the optimal conditional residuals when restricting the GNAR process to maximum active $\rstage$ depth equal to $s$.
\par
Following the ideas above, for $r \in \{1, \dots, r^*\}$ restricting the length of $\rstage$ neighbourhood regression to a value smaller than $r^*$ results in the optimal filters
\begin{equation*}
     u_{i, t}^{(ij)|r} = X_{i, t} - a_{i} - \sum_{k = 1}^{p} \sum_{l \in \mathcal{A}_r^{(j)}} 
        \gamma_{k}^{il} X_{l, t - k},
\end{equation*}
where $\mathcal{A}_r^{(j)} = \mathcal{A}^{(ij)} \cap \mathcal{A}_r$. Hence, using Theorem \ref{th: inv spec cross indentity}, \eqref{eq: end first part} and \eqref{eq: r stage inv matrix}, we have that
\begin{align} \label{eq: second part end equivalence neigh sets r}
    (i, j ) \in \mathcal{P}_r &\implies (i, j) \in \mathcal{A}_{r}, \nonumber \\
    &\iff |\cov (u_{i, t}^{(ij)|r}, u_{j, t + h}^{(ij)|r} )| \not\equiv 0, \nonumber \\
    &\iff |[\bmat{g}^{(r)} (\omega)]_{ij}| \not\equiv 0,
\end{align}
\par
and that the reparameterized GNAR process for distances no longer than $(r - 1)$ satisfies
\begin{align} \label{eq: second part end equivalence neigh sets r - 1}
    (i, j) \in \mathcal{P}_{r} &\implies (i, j) \notin \mathcal{A}_{r - 1} \nonumber \\
    &\iff \cov (u_{i, t}^{(ij)|r - 1}, u_{j, t + h}^{(ij)|r - 1}) \equiv 0 \nonumber \\
    &\iff [\bmat{g}^{(r - 1)} (\omega)]_{ij} \equiv 0,
\end{align}
where $\bmat{g}^{(r)} (\omega)$ and $\bmat{g}^{(r - 1)} (\omega)$ are the inverse spectral matrices of the constrained GNAR processes. Also, note that since
$(i, j) \in \mathcal{A}_{r^*}$ by Proposition \ref{prop: var constrains} we have that 
$[\bmat{S} (\omega)]_{i j} \not\equiv 0, $
which holds in conjunction with \eqref{eq: second part end equivalence neigh sets r} and \eqref{eq: second part end equivalence neigh sets r - 1}. Hence,
$$0 \equiv |[\bmat{g}^{(r - 1)} (\omega)]_{ij}| < |[\bmat{g}^{(r)} (\omega)]_{ij}| \leq |[\bmat{S} (\omega)]_{ij}| . $$
\par
Thus, for each $r \in \{1, \dots, r^*\}$ we can find two threshold values $\xi^{(r)}, \xi^{(r - 1)}$ such that $\xi^{(r)} < \xi^{(r - 1)}$ and
\begin{align*}
        \sft \big( \bmat{S} (\omega) + \xi^{(r)} \diag\{e^{\imath \phi_{ii}}\};  \xi^{(r)} \big) &= \bmat{g}^{(r)} (\omega), \\
        \sft \big( \bmat{S} (\omega) + \xi^{(r - 1)} \diag\{e^{\imath \phi_{ii}}\};  \xi^{(r - 1)} \big) &= \bmat{g}^{(r - 1)} (\omega),
\end{align*}
which gives $\xi^{(r)} \leq |[\bmat{S}(\omega)]_{ij}| < \xi^{(r - 1)}$, and we have that
\begin{equation} \label{eq: end second side b}
    (i, j) \in \mathcal{P}_{r} \implies \xi^{(r)} \leq |[\bmat{S}(\omega)]_{ij}| < \xi^{(r - 1)}, \forall \omega \in [-\pi, \pi].
\end{equation}
%Proof ends here
\end{proof}
Finally, combining \eqref{eq: end first part}, \eqref{eq: end first side part b}, \eqref{eq: end second side b} and Proposition \ref{prop: distance for conditional independence} finishes the proof of Theorem \ref{th: pop_sparsity gnar}.