## 2 Materials and Methods
### Methodologies
#### 2.2.1 Granger Causality  
The first technique we have used was the Granger causality testing methodology, which is recognized as a kind of statistical hypothesis testing that determines if an association can be established between two time-series. It is based on the idea that a time series \(X\) Granger-causes another time series \(Y\), if past values of \(X\) predicts the future values of \(Y\) better than using only past values of \(Y\). This particular concept of causality is based on some epistemological assumptions. The reader interested in those issues can refer to References [32,33]. Instead, from a mathematical viewpoint, this Granger causality testing method requires that the time series under investigation are stationary. This is a \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**County**} & \multicolumn{4}{c}{**PM\({}_{2.5}\)**} & \multicolumn{4}{c}{**\# of Days with PM\({}_{2.5}\) \textgreater{} 10**} \\ \cline{2-6}  & **19/02–08/03** & **19/02–31/07** & **Increase** & **Percentage** & **19/02–08/03** & **19/02–31/07** \\ \hline New York & 7.48 & 6.63 & +0.85 & +12.8\% & 4 & 10 \\ Kings & 6.27 & 5.28 & +0.99 & +18.8\% & 2 & 8 \\ Bronx & 7.62 & 6.47 & +1.15 & +17.8\% & 5 & 15 \\ Queens & 6.87 & 5.79 & +1.08 & +18.7\% & 4 & 14 \\ Richmond & 6.86 & 5.63 & +1.23 & +21.8\% & 5 & 12 \\ Nassau & 6.52 & 5.18 & +1.34 & +25.9\% & 3 & 7 \\ Westchester & 6.08 & 4.77 & +1.31 & +27.5\% & 2 & 4 \\ Rockland & 6.14 & 4.82 & +1.32 & +27.4\% & 3 & 6 \\ Suffolk & 6.35 & 5.4 & +0.95 & +17.6\% & 2 & 9 \\ Monroe & 7.11 & 6,15 & +0.96 & +15.6\% & 3 & 13 \\ Onondaga & 8.51 & 6.38 & +2.13 & +33.4\% & 5 & 11 \\ Oneida & 6.63 & 4.38 & +2.25 & +51.4\% & 3 & 6 \\  & & & Average & 24.04\% & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The PM\({}_{2.5}\) (short and extended periods) per county, increase, number of days with exceedances (short and extended periods). condition to be checked prior to developing any experiment. Hence, before proceeding with the Granger causality methodology, we have verified that our time series (infections and pollution in NYC) met this stationary condition, using the augmented Dickey-Fuller method [34]. Finally, from a statistical viewpoint, a Granger causality test requires to define both a null hypothesis and an alternative hypothesis. The null hypothesis corresponds to the fact that the time series \(X\) does not Granger-causes the time series \(Y\). The alternative hypothesis is that the time series \(X\) Granger-causes the time series \(Y\). Coming to our specific case, the null hypothesis is that the series of the PM\({}_{2.5}\) values does not Granger-cause the time series of the new daily infections in NYC. Consequently, we could decide that the time series of that particulate Granger-causes the time series of the infections, only if the aforementioned null hypothesis should be rejected. To better understand, now, how a Granger causality hypothesis testing procedure works, we can start from our time series \(X\) and \(Y\) (i.e., PM\({}_{2.5}\) and COVID-19 infections) that can be modeled with the following Granger causality Equation (1) below:  
\[Y_{t}=\ \sum_{i=1}^{l}a_{i}\ Y_{t-i}+\ \sum_{i=1}^{l}b_{i}\ X_{t-14+i}+\ c_{t}. \tag{1}\]  
\(Y_{t}\) and \(X_{t}\) are the single values of the two series \(Y\) and \(X\), in our case corresponding to the values of the PM\({}_{2.5}\) and of the infections, recorded daily, yet staggered by 14 days, due to the maximum delay it may take for COVID-19 to manifest symptoms. This said, the above formula computes the current values of \(Y\), based on previous values of both \(X\) and \(Y\). How far back one can go with past values of \(X\) and \(Y\), to get the current values of \(Y\), is given by the value of \(l\), the so _lag_. In our case, we have chosen this _lag_ value equal to 5, as 5.2 is the mean incubation period for COVID-19, as estimated in Reference [27]. \(c_{t}\), instead, is a white-noise-random vector. The above formula, works based on the role played by the \(b\) coefficients.