\section{Model Predictions}
We perform one-step prediction ten times to have a more robust estimate of out-of-sample prediction error, the summarised results are shown below.
\begin{table}[H]
    \centering
    \begin{tabular}{ccccc}
        Model & MSE(sd) & Mean No. Parameters \\
        \hline
         GNAR(1, [1]) &  7.414(2.977) & 2 \\
         Sparse VAR(1) & 11.307(2.768) & 2962 \\
         Res. VAR(1) & 10.614(2.482) & 3767 \\
         VAR(1) & 12.275(3.184) & 19600 \\
         AR(1) & 87.402(5.146) & 140 \\
         GNAR(1, [1])* & 7.313(2.900) & 141
    \end{tabular}
    \caption{One-step predictive performance and average number of parameters for the COVID-19 (network) time series for five different
        time series models. The MSE standard deviation is shown within parenthesis. GNAR(1, [1])* uses a different $\alpha_{id}$ for every NHS trust.}
    \label{tab: mode comp summary}
\end{table}
One-step prediction error (i.e., $||\hat{\boldsymbol{X}}_t - \boldsymbol{X}_t ||^2$) comparison between model predictions and the known test values. We perform the experiment ten times for $t = 443, \dots, 452$, and show the standard deviation within parenthesis next to each MSE value. The third column indicates the average number of active (i.e., non-zero coefficients) each model has throughout the experiment.
\par
Interestingly, GNAR parsinomity not only results in the smallest one-step prediction error in this case, it also has the smallest variance and standard deviation. Moreover, it is possible to fit more complex GNAR models since the number of unknown parameters is drastically smaller than for the other models, most notably, fitting a GNAR(1, [1]) requires estimating two parameters, whereas sparse VAR(1) requires estimating, on average, 3061 coefficients. 
\par
Remarkably, GNAR(1, [1]) accomplishes a smaller squared prediction error than sparse VAR(1) does with much fewer parameters.
\subsection{Model Comparison for one-lag differenced series}
\begin{table}[H]
    \centering
    \begin{tabular}{ccccc}
        Model & MSE & Mean Active Parameters \\
        \hline
         GNAR(1, 1) &  7.847(2.908) & 2 \\
         Sparse VAR(1) & 7.686(2.940) & 451 \\
         Res. VAR(1) & 9.149(3.068) & 1840 \\
         VAR(1) & 11.371(3.525) & 19600 \\
         AR(1) & 7.816(2.931) & 140 \\
    \end{tabular}
    \caption{Mean number of parameters and one-step predictive performance for the differenced COVID-19 (network) time series for five different
        time series models. The MSE standard deviation is shown within parenthesis.}
    \label{tab: mode comp summary differenceed}
\end{table}
\subsection{Prediction Error Tables for the Ten Experiments}
Below are the prediction error tables for each of the test points $\boldsymbol{X}_t$, where $t = 443, \dots, 452$, and  GNAR(1, [1])* uses a different $\alpha_{id}$ for every NHS trust. 
\begin{table}[H]
    \centering
    \begin{tabular}{ccc}
 Model & Active Parameters & One-Step SPE \\ 
\hline
GNAR(1, [1]) & 2 & 5.441 \\
 VAR(1) & $140^2$ & 11.613 \\
 Res. VAR(1) & 3821 & 9.902 \\
 Sparse VAR(1) & 2980 & 10.212 \\
 AR(1) & $140$ & 82.055 \\
 GNAR(1, [1])* & 141 & 5.324 
\end{tabular}
    \caption{One-step prediction error, $\hat{\boldsymbol{X}}_{443}$ is predicted using the previous 442 observations.}
    \label{tab: pred 443}
\end{table}

\begin{table}[H]
    \centering
    \begin{tabular}{ccc}
 Model & Active Parameters & One-Step SPE \\ 
\hline
GNAR(1, [1]) & 2 & 7.86 \\
 VAR(1) & $140^2$ & 12.804 \\
 Res. VAR(1) & 3712 & 11.504 \\
 Sparse VAR(1) & 2992 & 11.533 \\
 AR(1) & $140$ & 78.748 \\
 GNAR(1, [1])* & 141 & 7.833 
\end{tabular}
    \caption{One-step prediction error, $\hat{\boldsymbol{X}}_{444}$ is predicted using the previous 443 observations.}
    \label{tab: pred 444}
\end{table}

\begin{table}[H]
    \centering
    \begin{tabular}{ccc}
 Model & Active Parameters & One-Step SPE \\ 
\hline
GNAR(1, [1]) & 2 & 10.286 \\
 VAR(1) & $140^2$ & 12.52 \\
 Res. VAR(1) & 3730 & 11.605 \\
 Sparse VAR(1) & 2785 & 12.986 \\
 AR(1) & $140$ & 82.571 \\
 GNAR(1, [1])* & 141 & 10.173 
\end{tabular}
    \caption{One-step prediction error, $\hat{\boldsymbol{X}}_{445}$ is predicted using the previous 444 observations.}
    \label{tab: pred 445}
\end{table}

\begin{table}[H]
    \centering
    \begin{tabular}{ccc}
 Model & Active Parameters & One-Step SPE \\ 
\hline
GNAR(1, [1]) & 2 & 12.953 \\
 VAR(1) & $140^2$ & 17.762 \\
 Res. VAR(1) & 3812 & 14.472 \\
 Sparse VAR(1) & 3283 & 15.057 \\
 AR(1) & $140$ & 86.781 \\
 GNAR(1, [1])* & 141 & 12.676 
\end{tabular}
    \caption{One-step prediction error, $\hat{\boldsymbol{X}}_{446}$ is predicted using the previous 455 observations.}
    \label{tab: pred 446}
\end{table}

\begin{table}[H]
    \centering
    \begin{tabular}{ccc}
 Model & Active Parameters & One-Step SPE \\ 
\hline
GNAR(1, [1]) & 2 & 7.525 \\
 VAR(1) & $140^2$ & 13.82 \\
 Res. VAR(1) & 3761 & 10.815 \\
 Sparse VAR(1) & 3315 & 11.995 \\
 AR(1) & $140$ & 89.097 \\
 GNAR(1, [1])* & 141 & 7.431 
\end{tabular}
    \caption{One-step prediction error, $\hat{\boldsymbol{X}}_{447}$ is predicted using the previous 446 observations.}
    \label{tab: pred 447}
\end{table}

\begin{table}[H]
    \centering
    \begin{tabular}{ccc}
 Model & Active Parameters & One-Step SPE \\ 
\hline
GNAR(1, [1]) & 2 & 5.088 \\
 VAR(1) & $140^2$ & 12.504 \\
 Res. VAR(1) & 3830 & 9.976 \\
 Sparse VAR(1) & 3609 & 10.169 \\
 AR(1) & $140$ & 88.181 \\
 GNAR(1, [1])* & 141 & 4.989 
\end{tabular}
    \caption{One-step prediction error, $\hat{\boldsymbol{X}}_{448}$ is predicted using the previous 447 observations.}
    \label{tab: pred 448}
\end{table}

\begin{table}[H]
    \centering
    \begin{tabular}{ccc}
 Model & Active Parameters & One-Step SPE \\ 
\hline
GNAR(1, [1]) & 2 & 4.44 \\
 VAR(1) & $140^2$ & 7.978 \\
 Res. VAR(1) & 3738 & 7.716 \\
 Sparse VAR(1) & 3030 & 7.918 \\
 AR(1) & $140$ & 91.004 \\
 GNAR(1, [1])* & 141 & 4.466 
\end{tabular}
    \caption{One-step prediction error, $\hat{\boldsymbol{X}}_{449}$ is predicted using the previous 448 observations.}
    \label{tab: pred 449}
\end{table}

\begin{table}[H]
    \centering
    \begin{tabular}{ccc}
 Model & Active Parameters & One-Step SPE \\ 
\hline
GNAR(1, [1]) & 2 & 3.061 \\
 VAR(1) & $140^2$ & 7.196 \\
 Res. VAR(1) & 3759 & 5.888 \\
 Sparse VAR(1) & 3017 & 6.842 \\
 AR(1) & $140$ & 88.066 \\
 GNAR(1, [1])* & 141 & 3.089 
\end{tabular}
    \caption{One-step prediction error, $\hat{\boldsymbol{X}}_{450}$ is predicted using the previous 449 observations.}
    \label{tab: pred 450}
\end{table}

\begin{table}[H]
    \centering
    \begin{tabular}{ccc}
 Model & Active Parameters & One-Step SPE \\ 
\hline
GNAR(1, [1]) & 2 & 8.868 \\
 VAR(1) & $140^2$ & 10.869 \\
 Res. VAR(1) & 3772 & 10.999 \\
 Sparse VAR(1) & 3016 & 11.68 \\
 AR(1) & $140$ & 91.151 \\
 GNAR(1, [1])* & 141 & 8.607 
\end{tabular}
    \caption{One-step prediction error, $\hat{\boldsymbol{X}}_{451}$ is predicted using the previous 450 observations.}
    \label{tab: pred 451}
\end{table}

\begin{table}[H]
    \centering
    \begin{tabular}{ccc}
 Model & Active Parameters & One-Step SPE \\ 
\hline
GNAR(1, [1]) & 2 & 8.623 \\
 VAR(1) & $140^2$ & 15.693 \\
 Res. VAR(1) & 3794 & 13.258 \\
 Sparse VAR(1) & 3299 & 15.715 \\
 AR(1) & $140$ & 96.361 \\
 GNAR(1, [1])* & 141 & 8.546 
\end{tabular}
    \caption{One-step prediction error, $\hat{\boldsymbol{X}}_{452}$ is predicted using the previous 451 observations.}
    \label{tab: pred 452}
\end{table}

