## 2 Materials and Methods
### Methodologies
#### 2.2.1 Granger Causality
Essentially, \(X\) Granger-causes \(Y\), only if the \(b\) coefficients are different from zero. In fact, if the \(b\) coefficients are equal to zero, only the previous values of \(Y\) have an influence on the future values of \(Y\). It is now clear that modeling a causal relationship with the Granger formula above is equivalent to performing a statistical hypothesis test, where the null hypothesis is that all the \(b\) coefficients are zero, based on the following Formula (2):  
\[H_{0}:\ b_{1}=\ b_{2}=\ \ldots=\ b_{l}=0 \tag{2}\]  
Instead, the alternative hypothesis is that at least one of the \(b\) coefficients is not zero. At this point, assigned all the real values to \(Y\) and \(Y\), a vector autoregressive procedure must be run to get the \(b\) coefficients. Once that \(b\) coefficients are computed, a final \(F\)_test_ procedure is to be performed to verify if those values fit with the all zero distribution of the null hypothesis. This statistical test will return a \(p\)-value. The higher is this \(p\)-value, the more certain is the null hypothesis. With a lower \(p\)-value, instead, the alternative hypothesis comes confirmed (that is, \(X\) Granger-causes \(Y\)). In all the Granger experiments we have conducted, a level of significance equal to 5% was used. Said simply, in each experiment, we have rejected the null hypothesis only if the returned \(p\)-value was less than 0.05. More important is, now, to explain why and how we used this specific Granger procedure. To respond to the first question (why? ), suppose that one wants to decide if a relationship exists between the number of infections registered in a given day (e.g., 15 March) and the amount of PM\({}_{2.5}\) circulating in the air in the previous days. To do that, with other alternative approaches, one would have considered only the measurement of the values of interest, taken on just two days: The day of the registered infections (15 March) and the day when the pollutant circulating in the air might have favored an airborne transmission of this virus, with the result of the consequent infections. Assume that this specific day could be the one set 14 days before 15 March, that is 1 March. It is easy to understand that this would be an extremely reductive analysis, based on the role played by the amount of particulate circulating in the air on just one day (1 March), without the possibility of taking into account all the remaining days between 1 March to 15 March. With the approach based on the Granger formula, instead, we can take into simultaneous consideration multiple days, each with its amount of measured particulate. This is by virtue of the lag factor (i.e., the \(l\) index in the sum of the Granger formula) that allows one to go back as many days as one wants in the computation. And this, in turn, brings us to the answer to the second of the two questions we have posed before (how?). In particular, for each of the 12 counties of interest, we took into account three different pairs of time series. As to the (three) time series relative to the COVID-19 infections, all of them started in coincidence with the day of the beginning of the pandemic in NYC, that is 4 March; while the first, the second, and the third of those series ended, respectively, on the following days--20 March, 21 March, and 22 March. The idea was that using three different series, in our Granger analysis, would have corroborated our study, yielding more experiments/results for each of the 12 counties of interest. Finally, given the time lag of 14 days between a COVID-19 infection and its registered symptoms, the (three) time series relative to the PM\({}_{2.5}\) particulate all started on 19 February (14 days prior to 4 March) and ended, respectively, on the following days--6 March, 7 March, and 8 March. This is due to the fact that, to carry out a Granger analysis, the length of the compared series must be equal.