**2. Methodology**

We aim to build a monitoring model which can provide support to policy makers engaged in contrasting the spread of the COVID-19, and their economical consequences. To this aim, we propose a statistical model that can estimate when the peak of contagion is reached, so that preventive measures (such as mobility restrictions) can be applied and/or relaxed.

To be built the model requires, for each country (or region), the daily count of new infections. In the study of epidemics, it is usually assumed that infection counts follow an exponential growth, driven by the reproduction number *R* (see, e.g., Biggerstaff et al. 2014). The latter can be estimated by the ratio between the new cases arising in consecutive days: a short-term dependence. This procedure, however, may not be adequate: incubation time is quite variable among individuals and data occurrence and measurement is not uniform across different countries (and, sometimes, along time): these aspects induce a long-term dependence.

From the previous considerations, it follows that it would be ideal to model newly infected counts as a function of both a short-term and a long-term component. A model of this kind has been recently proposed by Agosto et al. (2016), in the context of financial contagion. We propose to adapt this model to the COVID-19 contagion.

Formally, resorting to the log-linear version of Poisson autoregression, introduced by Fokianos and Tjøstheim (2011), we assume that the statistical distribution of new cases at time (day) *t*, conditional on the information up to *t* − 1, is Poisson, with a log-linear autoregressive intensity, as follows:

$$y\_t | \mathcal{F}\_{t-1} \sim Poisson(\lambda\_t)$$

$$\log(\lambda\_t) = \omega + \alpha \log(1 + y\_{t-1}) + \beta \log(\lambda\_{t-1}).$$

where F*<sup>t</sup>*−<sup>1</sup> denotes the *σ*-field generated by {*y*0, ..., *yt*}, *yt* ∈ N, *ω* ∈ R, *α* ∈ R, *β* ∈ R. Note that the inclusion of log(1 + *yt*−<sup>1</sup>), rather than log(*yt*−<sup>1</sup>), allows to deal with zero values.

In the model, *ω* is the intercept term, whereas *α* and *β* express the dependence of the expected number of new infections, *λ<sup>t</sup>*, on the past counts of new infections. Specifically, the *α* component represents the short-term dependence on the previous time point. The *β* component represents a trend component, that is, the long-term dependence on all past values of the observed process. The inclusion of the *β* component is analogous to moving from an ARCH (Engle 1982) to a GARCH (Engle and Bollerslev 1986) model in Gaussian processes, and allows to capture long memory effects. The advantage of a log-linear intensity specification, rather than the linear one known as integer-valued GARCH (see, e.g., Ferland et al. 2006), is that it allows for negative dependence. From an inferential viewpoint, Fokianos and Tjøstheim (2011) show that the model can be estimated by a maximum likelihood method.
