The diffusion of COVID-19 across Italian provinces: a spatial dynamic panel data approach with common factors

ABSTRACT This study adopts a spatial dynamic panel data model with common factors and a connectivity matrix based on cross-province population flows to help explain the spread of COVID-19 infections across Italian provinces during the period 2020–21. We find that an increase in the infections in a province has a positive and statistically significant effect on neighbours’ infections, which highlights the relevance of spatial spillover effects. This finding is robust to several robustness checks. Furthermore, we investigate cross-provincial transmission heterogeneity using a heterogeneous spatial dynamic panel, which provides novel insights into the diffusion patterns of the disease.


INTRODUCTION
Since the outbreak was first reported in China in December 2019, the coronavirus disease (COVID-19) pandemic produced by the SARS-CoV-2 pathogen has spread to over 200 countries and territories, causing 629 million infections and 6.58 million deaths worldwide by the end of October 2022.
The rapid diffusion of this infectious disease all over the world contrasts with the few studies analysing the role played by spillovers and diffusion effects in shaping its spatial distribution.To the best of our knowledge, the only studies analysing the evolution of COVID-19 within a framework that allows for infection spillover effects across units are Guliyev (2020) and Han et al. (2021) for Chinese provinces and cities, Paez et al. (2021) for Spanish provinces, Laroze et al. (2021) for English districts, Bourdin et al. (2021) for Italian provinces, Ramirez et al. (2022) for Organisation for Economic Co-operation and Development (OECD) regions and Krisztin et al. (2020) for a global sample of countries.One finding that stands out from these spatial econometric studies is the existence of a positive spatial correlation in the number of infections among sample units.
While these studies represent substantial progress in our understanding of the spread of COVID-19, unobserved common factors affecting the evolution of the disease have been neglected in the modelling process.This makes it difficult to disentangle 'true spillover effects' from other potential sources of cross-sectional dependence (CSD), such as common trends/shocks that affect different units differently (Bailey et al., 2016a(Bailey et al., , 2016b)).This issue is relevant for the correct identification of spillovers in regional epidemic modelling, given that the emergence of genetic variants affecting the basal speed of viral transmission and the implementation of policy restrictions taking place in many geographical units at the same time, produced synchronized accelerations and slowdowns in the spread of the disease.These events resulted in spatially correlated infection rates, which can be easily confounded with a substantive process of cross-regional transmission.A second point that has been overlooked by the existing literature is the heterogeneity in disease transmission potential among units, given that infection spillovers have been assumed to be homogeneous.Nevertheless, even if all units in a geographical or mobility network have the exact same number of connections, it can be the case that not all units are equally important or influential (Aquaro et al., 2021).
Against this background, the present paper delves into the role played by spillovers and diffusion effects in shaping the evolution of COVID-19 infections in a sample of 107 Italian provinces during the period 26 February 2020-17 October 2021.To that end, we adopt a spatial dynamic panel data (SDPD) model with fixed-effects, common factors and a connectivity matrix based on population flows to explore the effect on infections produced by changes in time-varying factors such as human mobility, air pollution, climate factors and vaccination rates.
This study distinguishes itself from earlier papers on the determinants of COVID-19 diffusion in three main aspects.First, the identification of infection spillover effects relies on the distinction between weak and strong CSD (Pesaran, 2015;Pesaran et al., 2004).If the data exhibit 'strong' CSD, conventional spatial maximum likelihood, Bayesian and generalized method of moments (GMM) estimators will produce biased and spurious estimates of the spillover effects (Elhorst et al., 2020a(Elhorst et al., , 2020b)).As explained by Elhorst et al. (2020b), in such circumstances the use of common factors such as cross-sectional averages or principal components is required.We take this issue into account by controlling for the cross-sectional average of infections using the bias-corrected quasi-maximum likelihood estimator for SDPD developed by Yu et al. (2008).
Second, to investigate the spatiotemporal transmission mechanisms of COVID-19 infections we consider (1) information on large-scale cross-provincial population flows datasets provided by Enel-X/HERE Technologies and Facebook Data for Good Program, (2) a dataset with information on Facebook social network friendship links, and (3) conventional geographical proximity measures.In addition, rather than assuming a specific spatial model, we carry out a spatial Bayesian model selection exercise that allows a joint analysis of the model probabilities of different spatial models and connectivity matrices while accounting for the presence of strong CSD in the data.This allows us to draw inferences about the mechanism underlying infection diffusion across provinces, as different specifications of the connectivity matrix and dynamic spatial models imply different transmission and spillover processes at work (Elhorst, 2014;Ezcurra & Rios, 2020;Rios, 2017).
Third, and more innovative, our analysis accounts for the possible influence of heterogeneity in the spread of the epidemic disease across provinces and over time.This point is relevant as provinces are heterogeneous in important characteristics (e.g., size, location, accessibility), which causes one to suspect that spillover heterogeneity could be present in this setting.Thus, we relax the assumption of a homogeneous spatial coefficient and estimate a dynamic spatial heterogeneous parameter using the quasi-maximum likelihood (QML) algorithm developed by Aquaro et al. (2021).Following LeSage and Chih (2018), we also provide a decomposition of the spillover effects produced by COVID-19 infection shocks, disentangling the 'spill-in' effects from the 'spill-out' effects.This decomposition helps to identify which spatial units exert a stronger effect in the dissemination of infections and those that are more sensitive and vulnerable to changes in the infection levels elsewhere.Finally, we employ Bayesian model averaging (BMA) techniques to find the robust set of time-invariant socio-economic factors driving observed spatial spillover heterogeneity and the basal level of infections per capita.
The remainder of the paper is structured as follows.Section 2 reviews the literature on the drivers of COVID-19 severity differentials.Section 3 describes the spatiotemporal patterns on COVID-19 infections at the provincial level in Italy and provides a statistical analysis of the properties of the data.The econometric approach is discussed in section 4, while section 5 presents the result of the empirical analysis.Section 6 addresses the existence of spatial heterogeneity, whereas section 7 offers the main conclusions of the paper.

THE DETERMINANTS OF COVID-19 INFECTIONS: THEORETICAL CONSIDERATIONS
Epidemics can be affected by numerous socio-economic processes and environmental factors, which makes it difficult to identify the relevant determinants underlying their geographical spread (Bourdin et al., 2021;Ramirez et al., 2022;Rios & Gianmoena, 2021).This section reviews the key factors high-lightened by the literature analysing the time-varying drivers of COVID-19 infections.

Spillover effects and neighbours' infections
Cities, regions and countries around the world constitute a large network, with not only geographical structure but also a social structure shaped by domestic travel, friendships, international flights, shared borders and trade links (Han et al., 2021).Thus, in the context of an epidemic caused by a communicable human-to-human disease such as COVID-19, regional health outcomes will be interdependent.Infection spillovers across geographical units will occur with a certain probability if humans carrying the virus interact with other susceptible humans travelling later to other regions.Therefore, for a given level of disease prevalence, the more people from a region i who interact with people from another region j, the more likely that infected people from region j will end up spreading the infection to the population of region i, and vice versa (Laroze et al., 2021).
In this regard, the spatial studies of Laroze et al. ( 2021) for the districts of England and Han et al. (2021) for Chinese cities find positive and significant spillovers by specifying the connectivity between units with commuter flow data.Others, such as Guliyev (2020), Paez et al. (2021) and Bourdin et al. (2021) who focus on the spatial diffusion of COVID-19 across Chinese, Spanish and Italian provinces, respectively, find similar results using geographically based interaction matrices.Related contributions from the literature on complex systems and networks such as those of Fritz et al. (2022) and Celani and Giudici (2022) provide similar insights showing that epidemic models including the information on social media friendship networks and/or commuting flows outperform models that neglect regional interdependence.These studies explain the geographical spread of the disease via 'short-jump mode spillovers' (along relatively short transport routes) and 'local mode spillovers' (through neighbourhoods).On the other hand, Krisztin et al. (2020) highlight the role played by international flights, providing evidence of the existence of time-varying tunnel effects or 'long-jump mode spillover effects', which allowed the pathogen to travel long distances through international travel networks.
Although these studies differ substantially in their methods, their definition of neighbours and their sample composition, they all provide strong evidence suggesting that the stage of the epidemic in a region is likely to have an impact in the evolution of the disease in neighbouring regions.

Environmental factors
Climatic factors are likely to explain the spatial distribution and the dynamics of epidemics.The intuition is that disease agents and their vectors have specific environments that are optimal for growth, survival, transport or dissemination; and climatic factors such as precipitation, temperature, humidity, air circulation and solar radiation intensity ultimately define such an environment.From a review of 27 studies on the role played by climate factors in the spread of COVID-19 and their own empirical analysis, Rios and Gianmoena (2021) conclude that temperatures are one of the most robust drivers of the epidemic in Italy during the first wave, whereas the existing linkages between infections and other climate factors such as air humidity or solar radiation are not robust to model uncertainty.The theoretical intuition for the negative link between temperatures and infections is that lower temperatures (1) decrease defence barriers in human hosts (Makinen et al., 2009), (2) increase the effectiveness of the SARS-CoV-2 replications and provided that aerosol transmission is the dominant route of disease spread (Zhang et al., 2020), and (3) it can increase the frequency of social interactions that take place in indoor environments with poorer ventilation, where long-lasting airborne suspension of viral particles is higher.
On the other hand, studies investigating the spatial disparities of COVID-19 severity suggest that particulate matter pollution increases contagions and the subsequent health damages (Becchetti et al., 2022).Two hypotheses explain these findings: (1) the 'prolonged exposure' hypothesis and (2) the 'carrier-effect' one.The first hypothesis suggests that chronic air pollution exposure can exacerbate the vulnerability of populations to respiratory virus infections such as COVID-19, whereas the carrier conjecture suggests that airborne pollution particles may have been able to serve as a carrier for the pathogen.

Human factors
The idea that human movement spreads COVID-19 inspired lock-downs and stay-at-home orders adopted across the world in 2020 (Glaeser et al., 2020).Due to the airborne nature of COVID-19 transmission (Miller et al., 2021), the link between mobility and infections is expected to be strong, as large transit hubs enable super-spreading events.More important, higher levels of human mobility ultimately reflect increased volumes of social interactions and possibilities of contagion.In this regard, a vast amount of evidence suggests that mobility has a positive effect on infections and deaths (Hu et al., 2021a;Kraemer et al., 2020;Spelta & Pagnottoni, 2021;Zhou et al., 2020).As regarding public policy, lockdown restrictions and facemask mandates have been the most common health policy intervention during the early stage of the COVID-19 pandemic (Chernozhukov et al., 2020;Mitze et al., 2020), whereas the main pharmaceutical intervention to curve the spread of the disease has been the massive vaccination campaign initiated in late December 2020.The evidence in this regard suggests that although vaccination does not fully stop transmission, it reduces the risk of infection and accelerates viral clearance (Singanayagam et al., 2021).Thus, we expect a negative link between vaccination and infections.

Spatio-temporal patterns of COVID-19 infections
Our research uses data on the COVID-19 infections in Italian provinces provided by the Civil Protection Ministry (MPC).The key dependent variable through the empirical analysis is the natural logarithm of the average number of daily new COVID-19 infections per 100,000 inhabitants, calculated over windows of seven consecutive days (Y it ), from 26 February 2020 until 17 October 2021.Thus, the cross-sectional dimension of our panel data consists of N = 107 provinces, whereas the time sample consists of T = 85 periods (weeks).We adopt this measure to minimize administrative noisy signals produced by changes in reporting intensity and weekend reporting delays, which are present in raw daily data.
Figure 1 illustrates the evolution of infections over time since the beginning of the pandemic and the cross-sectional sample average.It emerges that infections per 100,000 inhabitants had the tendency to move together in the various provinces, but that the degree of co-movement differs from one province to another.
Figure 2 illustrates, on a common scale based on the percentiles of the pooled dataset, the spatial distribution of the new infections for selected time intervals.As observed in Figure 2(a), the epicentre of the COVID-19 outbreak in Italy was the province of Lodi, in the northern region of Lombardy.Neighbouring provinces such as Cremona and Bergamo were also severely affected with more than 1.5% of the population getting infected during the first epidemic wave according to official logs.Overall, the provinces in the centre and south of the country remained largely unaffected during the first wave of the pandemic.As evidenced in Figure 2(b), the north-south divide continued during the summer and most of the fall of 2020.Indeed, during the period between August and mid-November 2020, which covers the high peak of the second epidemic wave, provinces that experienced a higher growth in the number of infections per capita were Bolzano, Aosta, Varese and Milano; all located in the north of the country.
As shown in Figure 2(c), between November 2020 and February 2021 the spread of the disease across space became much more generalized.It was especially severe The diffusion of COVID-19 across Italian provinces: a spatial dynamic panel data approach with common factors 287 in the north-eastern provinces of Belluno, Bolzano, Treviso and Udine, where approximately 5% of the population got infected in those three months.However, provinces in the south also experienced marked increases of infections in that period.Between February and July 2021, the disease spread all over the country.Figure 2(d) shows that the only group of provinces that experienced relatively moderate infection rates in those months were those in the islands of Sicily and Sardinia.Nonetheless, from July to October 2021, which covers the summer season and the achievement of high vaccination rates, the picture reversed and the higher infection rates took place in the two islands (Figure 2e).These figures reveal three interesting stylized facts: (1) infections tend to be strongly correlated over time (the average time correlation, r = 0.796, p = 0.00).Therefore, the level of infections in period t − 1 is a good predictor of the level of infections in period t; (2) the level of infections at the provincial level is in parallel with the cross-sectional average; and (3) the levels of infections at the provincial level are locally correlated across space.Based on the local clustering of provinces in Figure 2, the levels of infections in the neighbours of a particular province are a good predictor of the level of infections in that province itself.

Assessing cross-sectional dependence (CSD) and serial dependence
We begin our empirical analysis by investigating the degree of CSD in our sample of provinces and testing if our panel data is stationary, as these two properties of the data have important implications for the modelling exercise.
We first analyse CSD following the two-step procedure suggested by Bailey et al. (2016b) to distinguish between weak and strong CSD, which is based on the cross-section dependence (CD) test developed by Pesaran et al. (2004) and Pesaran (2015), and the a-exponent estimator developed by Bailey et al. (2016a).The CD test verifies the degree of CSD in terms of the rate at which the average pairwise correlation in the cross-section of provinces varies with N .It is defined as: where rij denotes the sample correlation coefficient between Y it and Y jt of two units i and j observed over time.Bailey et al. (2016b) show that the average correlation coefficient has the following property: where a is a parameter that can take values on the [0, 1] interval.A value of a , 0.5 points to weak dependence since r N tends to go to zero very quickly.When 0.5 , a , 0.75, the data exhibit moderate dependence, where cross-unit connectivity is denser but decays sufficiently fast to employ a spatial econometric model.Finally, when 0.75 , a , 1, the data are said to exhibit strong dependence because of r N tends to zero at a slower rate than (N ) .In this latter case, common factors need to be accounted for.In turn, a can be 288 Lisa Gianmoena and Vicente Rios REGIONAL STUDIES calculated as: x t and where the u 2 v and c N are small sample bias-correction terms obtained by running separate regressions of x it on a constant and x t for each unit i, such that each of these regressions is based on T observations.These key statistics when calculated in our data on provincial infections are shown in Table 1.
As observed, the degree of CSD in the data on COVID-19 infections is strong.r N = 0.849 and the CD test statistic is 491.29, which is substantially higher than the 1.96 critical value at the 5% level.With the null hypothesis of weak CSD soundly rejected, we then estimated the a exponent of CSD due to Bailey et al. (2016b).We obtained â = 1.001(0.0271).COVID-19 infections are strongly correlated across provinces, and therefore before estimating any spatial model it would be necessary to first purge Y it of the common sources of their dependence, or alternatively to control for common factors within a spatial model framework.
A second issue is to verify whether the dependent variable is stationary or non-stationary, in which case it does have a unit root.The general model we employ for testing the existence of unit roots reads as: where DY it is the growth rate of infections of province i at time t, Y t and DY t denote level and growth rate cross-sectional averages, Trendt is a linear trend and u it is the error term.We follow Ciccarelli and Elhorst (2018) and apply the cross-sectionally augmented Dickey-Fuller test (CADF) to check if τ is equal to 0 (unit root) or smaller than 0 (stationary).Note that instead of common coefficients across all units, coefficients are assumed to be unit specific.By estimating the coefficients of the N unit-  The diffusion of COVID-19 across Italian provinces: a spatial dynamic panel data approach with common factors 289 specific regressions by ordinary least squares (OLS), a series of N t-values is obtained for t, one for each spatial unit.We use six alternative specifications of the CADF test and report the results in Table 2.In five out of six of the specifications of the CADF we find that it is possible to reject the null of no stationarity and that the percentage of provinces exhibiting a non-stationary behaviour is small, especially when cross-sectional averages Y t and DY t are included in the model. 1

A spatial dynamic panel data (SDPD) model with common factors
The results of our previous data analysis suggest that when modelling COVID-19 infections it is necessary to control for common factors to achieve a manageable degree of CSD and stationarity.Hence, in this empirical analysis, we adopt the following dynamic spatial lag model (DSLM) with common factors: where Y t denotes an N × 1 column vector that consists of one observation of the dependent variable for every unit (i = 1, . . ., N ) in the sample at time t (t = 1, . . ., T ), which for this study is the logarithm of the weekly average number of new infections per 100,000 inhabitants in the N = 107 Italian provinces over the period from 26 February 2020 to 17 October 2021 (T = 85 weeks).Y t−1 and WY t represent, respectively, the temporal and spatial lag, and WY t−1 the spatiotemporal lag of Y t , while f, r and h are the corresponding response parameters of these variables.The N × N matrix W is a non-negative matrix of known constants describing the connectivity among the provinces in the sample.The specification of this matrix will be further discussed in section 4.3.X t represents an N × K matrix of independent variables that involve time-varying environmental and human-related predictors.In equation ( 5) observable predictors are assumed to have homogeneous response parameters.The common factor term meant to cover potential strong CSD is the cross-sectional average of the dependent variable at time Y it , which has unit-specific coefficients given by the ′ is a vector of province fixed effects, which control for all region-specific time invariant variables whose omission could bias the estimates (i.e., income per capita, population density, etc.).Finally, the N × 1 vector e t = (e 1t , . . ., e Nt ) ′ consists of i.i.d.disturbances whose elements have zero mean and finite variance s 2 .
The control variables included in X have been selected based on available data and our literature review on the groups of determinants of COVID-19 infections in section (2).Within the set of 'human-related variables' in X , we control for (1) the evolution of human mobility taken from the COVID-19 Community Mobility Reports provided by Google.Our mobility index is calculated as an arithmetic average of different components in the Google mobility reports: where the L components considered are: (1) workplaces, (2) transit stations, (3) retail and recreation, and (4) grocery and pharmacy.This index captures aggregate movement trends by province with respect to a baseline prepandemic period.However, one limitation of Google's data is that it is not informative on the directionality of population movements. 3Our second control is (2) the share of population vaccinated with at least one dose, which is obtained from the Health Ministry of Italy.
As regards the set of environmental variables, we control for the changes in the (3) air quality of the province, which we measure using satellite and high-resolution gridded data of average concentration of particles matter between 0 and 50 m above the surface with a diameter less than 2.5 µm (PM 2.5 ).This data are provided by the Copernicus Atmospheric Monitoring Service (CAMS).Finally, we also consider the role of climate factors by including the (4) total precipitation and (5) the average temperature measured at 2 m from the surface, both of which are taken from the NASA-POWER v8 geographical information system (GIS) database. 4

Inference
To estimate equation ( 5) we employ the bias-corrected quasi-maximum likelihood (BCQML) estimator for dynamic spatial panel data (DSPD) models developed by Yu et al. (2008).This estimator will yield consistent parameter estimates provided that the model is stable and does not suffer from space-time cointegration (i.e., f + r + h , 1).Inference on the effect produced by changes in our regressors in the context of the DSLM is based on a partial derivative interpretation and the computation of direct, indirect and total effects (Elhorst, 2014; Rios , 2017).The matrix of short-run effects with respect to a change in a regressor X (k) is given by: On the other hand, the long-run effects are given by: In this context, direct effects (diagonal terms in equations 6 and 7) capture the effect on infections in i caused by a unit change in an exogenous variable X k in i. Indirect effects (offdiagonal terms) can be interpreted as the effect of a change in X k in all other provinces j = i on the infections in i.
Finally, the total effect is the sum of the direct and indirect impacts.

Spatial dynamic panel model selection
One relevant assumption in our baseline specification is that a change in any of our predictors X (k) i affects infections in another province Yj by influencing first infection rates in the population of origin.For example, if Y i increases, and the population of j interacts with that of i, then Y j may also increase by importing infections from i.
We now verify the probability of this causal mechanism of diffusion by considering alternative spatial processes using spatial dynamic Bayesian model selection techniques (Ezcurra & Rios, 2020;LeSage, 2014;Rios, 2017).Given that Bayesian estimation shares its properties with QML/ ML estimators for DSPD, and these algorithms require a weakly cross-correlated dataset, we control for strong CSD by including Y t in the various specifications and allowing for unit-specific coefficients.
We calculate posterior model z probabilities p(M z |D) as follows: where z = 1, . . ., Z is the specific model under consideration, p(M z ) is the prior model probability and the term p(D|M z ) is the marginal likelihood given by p(D|M z ) = p(D|Q z , M z )p(Q z |M z )dQ z where Q is the vector of parameters of the model.To avoid situations where the conclusions depend heavily on subjective prior information, we rely on diffuse prior distributions.In order to make each model M z equally likely a priori, the same prior probability p(M z ) = 1/Z is assigned to each model under consideration.Equation ( 8) allows us to compare our baseline DSLM with a variety of dynamic spatial panel models.We consider (1) the dynamic spatial Durbin model (DSDM), (2) a restricted version of the dynamic spatial Durbin model (rDSDM), (3) the dynamic spatial error model (DSEM), (4) the dynamic spatial Durbin error model (DSDEM) and ( 5) the dynamic spatial lag of X model (DSLXM) (for a taxonomy, see Elhorst (2014)).However, the estimation of these spatial models requires defining a connectivity matrix W .We consider three groups of connectivity matrices to model the potential mechanisms underlying the geographical transmission of the disease among provinces: (1) population flows through the travel network, (2) social network connectedness and (3) geographical proximity.
We first construct W matrices where entries w ij reflect the relative intensity of population movement flows (m ij ) between provinces . The reason is that higher cross-province movements increase the probability of spreading infections from i to j.Our first source of data on cross-province mobility comes from Enel X/Here Technologies, which provides with information on the weekly per capita flows among all Italian provinces from mid-January to mid-February 2020 using anonymized and aggregated data from connected vehicles, maps and navigation systems.The sample covers the 5-10% of the total population depending on the specific province.Our second source of data on population movements is the Movement between Administrative Regions dataset developed within the context of the Facebook Data for Good Program.These data provide measurements at 8h intervals from a sample of people sharing location history data with the Facebook app, and account for approximately the 6.5% of the population.The large scale of users considered in both makes them good candidate approximations of the movement patterns underlying the diffusion of an epidemic disease such as COVID-19.Nonetheless, with the aim of increasing the robustness of our measurement of between-province flows against distortions, we define a hybrid W to compensate their potential idiosyncratic biases as: . Because of the counting procedure and minimum movements thresholds to comply with privacy requirements, the Facebook dataset is likely to underestimate the size of cross-province movements in small provinces.On the other hand, Enel-X/Here Technologies may overestimate it by counting the same person crossing different regions in more than one entry of the origin-destination matrix, even if it was for a small amount of time. 5 Our second group of W matrices consists of (2) social connectedness matrices based on friendships in the social network of Facebook, which has been shown to perform well analysing COVID-19 diffusion over space (Fritz et al., 2022;Kuchler et al., 2021).This measure of the social network distance between two provinces i and j is based on the social connectedness index (SCI) introduced by Bailey et al. (2018) which measures the relative probability of a Facebook friendship link between a given Facebook user in location i and a given Facebook user in location j using a sample of 28.5 million users (i.e., 48.3% of the country's population).The index is given by: The rationale for using social network links is that areas with closer social friendship ties between themselves are more likely to generate close physical The diffusion of COVID-19 across Italian provinces: a spatial dynamic panel data approach with common factors 291 interactions between region's residents providing opportunities for the spread of the disease.Our third group of W matrices consists of geographical based matrices as in Bourdin et al. (2021) or Paez et al. (2021).
For each group of matrices, we compute weighted knearest neighbours' row-normalized matrices with k = 5, 7, 10 to capture different levels of interconnectedness for each type of W .Nevertheless, because of the effect of geographical distance might not decay linearly, we also consider more flexible specifications of the decay in mutual influence as distance increases.To that end we build a power-distance decay specification that takes the form of a gravity matrix where: and a negative exponential decay matrix: with u = 0.01, both with cut-offs at the first quartile of the distribution of distances (i.e., w ij = 0 unless Columns (1) to ( 6) in Table 3 report the probability of each spatial model (SM) and W matrix given the data (D), (P(SM, W |D)).We find the DSLM based on the 10-nearest neighbours' population flow's matrix appears to be the preferred DSPD specification, as it concentrates the 48.6% of the probability.The second best specification is the DSLM based on the 10-nearest geographical neighbours with a 38.9% of probability.Overall, the DSLM concentrates the 92.8% of the probability, whereas the DSDM accounts for the remaining 7.1%.Across W 's, the group of models considering population flow's matrices obtains a 54%, whereas geographically based specifications obtain a 46%.Thus, this spatial model selection exercise gives some important messages.The first is that the transmission channel of infections that better captures weak CSD is that of population flows, given that these W 's tend to concentrate most of the probability.The second takeaway message is that first-order interactions are more likely to occur with the 10 provinces to which one is more closely connected in terms of population movements, which suggests a mixture of short-mode jumps and local neighbourhood effects has been at work in the spread of the disease.Nevertheless, because of the spread of COVID-19 infections is better explained by a global spillover process instead of other alternative processes, a change in the infections in one province can set in motion a sequence of adjustments in potentially all units in the sample (through higher order interactions), such that a new long-run equilibrium of infections might arise.This finding is in line with the results in the field of spatial epidemics (Roques et al., 2020).Our results also show that the extension of the model with the WX terms seems unnecessary as both the rDSDM and the DSDM specifications are less consistent with the data.

Baseline results
Table 4 presents the results obtained using the DSLM including province fixed-effects and cross-sectional averages, estimated by the BCQML method proposed by Yu et al. (2008), and the 10-nearest neighbours hybrid W matrix based on population flows.Column 1 reports coefficient estimates.The estimates of cross-province transmission (WY t ) and the within-province transmission (Yt −1) are positive and significant, whereas the lagged value of the dependent variable in space and time (WY t−1 ) is negative and significant.These results confirms that the dynamic spatial panel data modelling framework used in this analysis is suitable for studying the evolution of COVID-19 infections.To find out whether the model under consideration is stable, the sum of the parameters f + r + h is calculated and a two-sided Wald test is carried out to investigate the null hypothesis of f + r + h = 1.Table 4 reports the parameter sum and the corresponding p-values of the F test.Importantly, the model is stable and does not suffer from spatial cointegration (i.e., t + r + h = 0.748, F = 542.7 with p ¼ 0.00).Regarding the degree of CSD, we find that controlling for common factors in our SDPD reduces substantially the CD test statistic (from 491.2 to 34.6) and the a-exponent of CSD (from 1.00 to 0.66).This viewpoint is supported by the estimates of the spatial and time-lagged effects reported in column (1), which satisfy the stability conditions associated with weakly cross-correlated processes.
Focusing on the goal of the paper, we find that the estimated coefficient of the spatial lag of the dependent variable is positive and statistically significant at the 1% level, indicating the existence of infection spillovers across Italian provinces.This means that infections in one province depend positively on the infections in provinces with which there is a higher degree of population flow connectivity.The spatial lag parameter reveals that an increase of 1% in the infections per capita in neighbouring provinces −i is expected to increase infections per capita in province i by 0.5%.Our estimates underline the importance of cross-province transmission in shaping the geographical distribution of infections in Italy.In substantive terms, the spillover coefficient from Table 4 reveals that if the neighbour's level of infections were at the peak of their infection curve, in just one week 1.7% of the total population in the province i would get infected.This shows that the effect of neighbouring provinces is quantitatively important. 7 To investigate the role played by our time-varying predictors we resort to the computation of direct effects, indirect effects and total effects in both the short and long runs.These are reported in columns (2) to (4).We find that all the variables in the analysis generate the expected results and display qualitatively similar effects in both, the short term and in the long term.Focusing on the direct effects of changes in environmental factors, we find that an increase in the air pollution in province i increases infections in i, whereas an increase in the temperatures reduces them.Second, with respect to human related factors, we observe that mobility exerts a positive and significant impact at the 1% level, whereas the effect of higher vaccination rates is to decrease infection rates.Short-run indirect effects are significant at the 5% level for four out of the five variables and in all cases amplify direct impacts.The indirect effects estimates show that the spatial amplification is particularly pronounced, as they account for the 48% of the total effect.The interpretation of this result is that if all provinces j = 1, . . ., N other than i experience a change in X k , this will have a similar effect in i that if only i experiments a change in X k , which can be explained by the fact that individual provinces are very small in size relative to the whole system of interacting units.Finally, we find that the total effects are significant at the 1% level for four out of the five variables.Vaccination and temperatures reduce transmission whereas mobility and air pollution increase it.
The differences in size between the long-run and the short-run effects imply that apart from the 'first period', where interaction effects are mainly pure spatial feedback effects, spatiotemporal feedbacks passing from one province to another seem to be relevant in order to explain the evolution of infections.Furthermore, the fact that on average short run effects only account for the 51.1% of the total long run effect, suggests that the diffusion of epidemic shocks takes time to unravel. 8 One final issue that is worth discussing (1) is the heterogeneous behaviour of provinces in response to the changing state of the epidemic at the cross-sectional level ( Y t ) captured by Ĝ 1 − r and (2) the differences in the fixed component of infections mi .In Figure 3, we plot the spatial distribution of the province fixed-effects and the unit-specific response to the common factor captured by the cross-sectional averages.As observed in Figure 3(a), the basal level of infections measured by the province-fixed effects tends to be higher in the north of the country.This result, coupled with the fact that most of the northern provinces do not seem to be very responsive to the national trend and southern ones are highly sensitive to it (Figure 3b), suggests that northern provinces have behaved like trend-setters in the diffusion of the epidemic, whereas provinces in the south have acted like trend-followers.

Robustness checks
In this section we investigate the robustness of spillover effects to changes in the modelling set up.In Table 5 we present a summary of the results of 19 different variations in our research design.To save space, we present only the   6) show the posterior model probability of each model of each pair of dynamic spatial model specifications (SM) and W's.All DSPD processes include spatial fixed effects and unit-specific coefficients for the cross-sectional average of the dependent variable.The Bayesian estimation of the models is based on 11,000 Markov chain Monte Carlo draws with a burn-in sample of 1000 draws.To avoid situations where the conclusions depend heavily on subjective prior information, we rely on diffuse prior distributions.In order to make each model equally likely a priori, the same prior probability p(M z ) = 1/Z is assigned to each model under consideration.We set a normal-gamma conjugate prior for F = (b, u, t, h, m, G) (i.e., p(F) N(c, T) and σ p 1 s 2 G(d, n), and a beta prior for r (in the DSLM/rDSDM/DSDM specifications) and l (in the DSEM/DSDEM specifications) such that p(r), p(l) Beta(a 0 , a 0 ) where we set c = 0 and T = 1E + 12, d = 0, n = 0 and a 0 = 1.1.
The diffusion of COVID-19 across Italian provinces: a spatial dynamic panel data approach with common factors 293 REGIONAL STUDIES results of the cross-provincial transmission effect (r), the stability condition of the model f + r + h, the a-exponent of CSD of the residuals, and the short-and long-run total effects of each predictor.However, the complete information of each of these robustness checks is presented in Tables B1-B19 in Appendix B in the supplemental data online.We begin by examining whether our results depend on the connectivity matrix used to capture the linkages between the sample provinces.Hence, we repeat the analysis using connectivity matrices based on cross-province population flows but allowing a varying number of connections.In addition, we use geographical and social connectedness matrices.We find that the coefficient r continues to be positive and statistically significant in all cases.
In a second check, we investigate if the results are affected by the way we modelled strong CSD.To that end, we employ an SDPD including fixed and time-period fixed effects and the principal components (PC) estimator for SDPD developed by Shi and Lee (2017) including two common factors.Although the size of the spillover decreases in these specifications, the spatial lag coefficient remains positive and significant.
We also explore to what extent our results depend on the specific spatial model used to investigate the relationship between own and neighbouring infections.We consider the DSDM, the rDSDM, the dynamic generalized nesting spatial model (DGNSM) and the dynamic spatial autoregressive with autoregressive error structure (DSARAR).These checks reveal that our main finding still holds.
We further examine if the results hold when looking at different time samples.Specifically, we consider (1) the period that covers the first wave from February 2020 to the end of the second wave in April 2021, (2) the period ranging from the beginning of the second wave in October 2020 to the end of the third wave in October 2021, and (3) 294 Lisa Gianmoena and Vicente Rios

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the period that covers the vaccination campaign from January to October 2021.We still find a significant effect of neighbouring provinces in all cases.Finally, we analyse the sensitivity of our results to the frequency of the data.We check if our results hold with two alternative frequencies of the time dimension: 10and 14-day windows.We find that these changes do not modify our results.
Summing up, the effect of the cross-province disease transmission parameter is positive and significant irrespective of the matrix, the spatial model specification, the procedure to capture strong CSD, the frequency of the data or the time sample considered.As refers to the effects of our predictors, we find that the most consistent variables explaining COVID-19 infections are the aggregate level of human mobility, vaccines and air pollution, which are significant at the 5% level in 18 out of the 19 checks.Temperatures tend to be negative and significant, although the effect is less robust and depends on the specification.Finally, precipitations are mostly insignificant.

The role of time-invariant socio-economic characteristics
The analysis performed so far has accounted for the role played by time-varying drivers within a specification that includes province-fixed effects and common factors.However, the inclusion of fixed effects has precluded us from exploring the role played by slowly varying socio-economic dimensions, given that the later do not change over the study's time frame.We now investigate if fixed socioeconomic characteristics explain the geographical variability of the basal provincial infection rates measured by the fixed effects estimates ( mi ) reported in Figure 3.
We begin by considering economic factors and the role played by the sectoral composition on infections (Ascani et al., 2021;Bourdin et al., 2021).To capture the impact of differentials in the level of development and the effect of sectoral specialization we consider (1) the logarithm of GDP per capita, (2) the employment in the public administration, (3) the employment in wholesale and retail trade, transport, accommodation, and food service activities and (4) the employment in manufacturing.
We also account for differences in socio-demographic characteristics, as these factors are among the top determinants associated with the disproportional impact of the COVID-19 pandemic across space (Caselli et al., 2022;Hu et al., 2021b).Among them, the age structure and the composition of the population have received a lot of attention in the literature.Thus, we consider (5) the share of the population below 25 years old, (6) the share of the population above 65 years old and (7) the share of immigrants.
In addition, other social aspects such as (8) the share of health spending with respect to the GDP, which helps to proxy health system capacity, might have been relevant.Similarly, (9) social capital might be of importance, given that greater interpersonal trust within a community, could endow individuals with greater concern for others, more respect for social distancing and higher compliance with governmental policy measures (Makridis & Wu, 2021;Rodríguez-Pose & Burlina, 2021).Furthermore, we consider an indicator of (10) multimodal accessibility that measures provincial accessibility by road, train, and air.Finally, we investigate the effects of agglomeration and density by considering (11) the logarithm of the total population and ( 12) the logarithm of the population density.Compared with low-density areas, a higher population density could favour the interaction and personal contacts reinforcing the transmission of the virus (Sy et al., 2021).The dependent variable is in all cases the logarithm of the average new infections per 100,000 inhabitants of the various provinces calculated over windows of seven days.*Significant at the 10% level, **significant at the 5% level, ***significant at the 1% level.The results are obtained using a 10-nearest neighbours connectivity matrix based on the average of cross-province population flows data drawn from Enel-X/Here Technologies and the Movement between Administrative Regions Dataset of Facebook Data for Good Program unless otherwise specified.Inferences regarding the statistical significance of these effects are based on the variation of 1000 simulated parameter combinations drawn from the variance-covariance matrix implied by the BCQML estimator of Yu et al. (2008) when employing cross-sectional averages to capture strong dependence or the PC-BCQML estimator of Shi and Lee (2017).
The diffusion of COVID-19 across Italian provinces: a spatial dynamic panel data approach with common factors

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To analyse the effect of these factors we combine the outputs of a Bayesian heteroskedastic model, BMA techniques and relative importance metrics analysis.The BMA approach solves the question of estimating variable importance from a probabilistic perspective and can be used to assess statistical significance in the context of a regression model with low degrees of freedom (Steel, 2020).The relative importance decomposition of the R 2 of the model is also useful, as it enables a detailed analysis of the contribution of each variable when explaining the geographical variability of infections by considering all potential model compositions given our set of explanatory variables (Gromping, 2007;Rios & Gianmoena, 2021).The results of employing these procedures are shown in Table 6.
Column (1) shows the results of a Bayesian heteroskedastic linear regression model where we investigate the role of all factors at once.In columns (2) to (4) we complement these results with those of the BMA.The key metrics to make inference in the BMA analysis are the (1) posterior inclusion probability (PIP) for each predictor, (2) the standardized posterior mean and standard deviation of the effects, and (3) the posterior sign certainty, which measures the frequency of models in which the sign of the corresponding coefficient is positive.In column (5), we provide the averaged decomposition of the R 2 of the model. 9 In column (1) we find positive and statistically significant effects for (1) the specialization in services, (2) the GDP per capita, (3) the share of immigrants and (4) the share of the young.The predictors that exert a negative and significant impact are (5) the social capital and (6) the regional health spending.Out of these six factors, five are robust to model uncertainty and obtain PIPs above the 50% in the BMA.These are the GDP per capita, the share of young, the specialization in services and health spending and the social capital.Importantly, the higher probabilistic importance attributed to these factors by the BMA is shared by the weights in the R 2 decomposition analysis, suggesting that these factors are robust drivers of the structural level of infections in Italy.
GDP per capita explains 29% of provincial disparities and obtains the highest PIP (89%).Overall, the finding of the positive impact of GDP per capita is in line with studies documenting that viruses hit harder in dynamic, and highly internationally connected places (Ramirez et al., 2022).In the second order of importance, we find the share of the population below 25 years.This variable displays a PIP of 69% and explains 14% of the geographical disparities in basal infections per capita.The link between younger populations and infections can be explained by the lower perception of risk and individual distancing efforts (Aubert & Augeraud-Veron, 2021).Another explanation for this result is that the mobility of people aged 65 and above in Italy, most of whom are retirees, is structurally lower than that of younger cohorts, which means lower social interactions and probabilities of infections (Caselli et al., 2022).Third, we find that health spending displays PIPs of the 60% and explains 7% of provincial variation in infections.The observed negative effect suggests that provinces with low health system capacity suffered of more infections per capita, which means that failing to devote enough resources to the health system decreased the ability of authorities to identify potential carriers, isolate them and reduce higher order transmission chains (Atkeson et al., 2020).The share of employment in restauration and other services activities explains the 11.54% of the variability of fixed effects and obtains a PIP of 53%.Because of restaurants and bars promote gatherings of humans in closed spaces, which greatly favours the spread of airborne diseases such as COVID-19, provinces with a large share of their economy specialized in these activities seem to have experienced higher infection rates.Finally, we find that social capital exerts a negative robust effect on the basal level of infections, although it explains a lower share of the model's fit (4.2%).The negative link between social capital and infections suggests that public's trust in government and a shared social contract among citizens also matters for health emergencies.

Estimating spillover heterogeneity
To increase our understanding of the spatial diffusion patterns of COVID-19 infections, we now investigate the possibility that the size of the spillover effects differs across Italian provinces.To that end, we consider the following heterogeneous dynamic spatial lag model (HDSLM): for i = 1, . . ., N and t = 1, . . ., T .To capture strong dependence, the model incorporates fixed effects a i and the cross-sectional averages of Y it .It displays full heterogeneity in the spatial and temporal autoregressive coefficients of infections (r i , f i , h i ), as well as the slope coefficients for the exogenous predictors b ik , the cross-sectional average of infections k i and a time trend g i .Innovations are assumed to be distributed as eIID(0, s 2 i,e ).The connectivity matrix w + ij is a doubly stochastic version of our preferred 10-nearest neighbour population flows' specification.Nonetheless, a difference between this W + and the W employed in equation ( 5) is that rows and columns sum up to 1.We transform the original 10-nearest row-standardized matrix into a doubly stochastic one using the Sinkhorn-Knopp algorithm developed by Knight (2008).The property of doubly stochasticity is needed to be able to compute spill-in and spill-out effects (LeSage & Chih, 2018).To estimate the model in equation ( 9), we use the algorithm of QML for heterogeneous SDPD models developed by Aquaro et al. (2021). 10Note: The dependent variable is in all cases the fixed-effects estimated component of the logarithm of the seven day average of the new infections per 100,000 inhabitants of the various provinces.*Significant at the 10% level, **significant at the 5% level, ***significant at the 1% level.Column (1) reports the estimation of a Bayesian heteroskedastic linear model with unit-specific error variances V = diag(v 1 , . . ., v n ).We set independent prior distributions p(v i ) for the relative variance parameters taking the form p(v −1 i |r) IDx 2 (r).The hyper-parameter r controlling the prior variance scalars v 1 , . . ., v n is set to r = 4.The Bayesian estimation of the model parameters is based on 2500 Markov chain Monte Carlo draws with a burn-in sample of 500 draws.Results in columns (2) to (4) reflect Bayesian model averaged estimates corresponding to the estimation of all the models in the model space including any combination of the 12 variables.We use a fixed BRIC g-prior on the parameters and a Binomial model prior, with prior mean model size equal to 6 in all cases.Column (2) reports the posterior inclusion probability of the predictor.Column (3) reflects the standardized conditional posterior mean for the linear marginal effects of each variable.Column (4) is the sign certainty probability, a measure of our posterior confidence in the sign of the coefficient.Column (5) reports the averaged share of the model's R 2 attributed to each variable when calculating the LMG, CAR, GEN and proportional marginal variance decomposition (PMVD) relative importance metrics decompositions (for details, see Gromping (2007).Standard deviations are shown in parentheses.
The diffusion of COVID-19 across Italian provinces: a spatial dynamic panel data approach with common factors

REGIONAL STUDIES
In Figure 4, we report individual parameter estimates of the contemporaneous cross-province transmission (r i ), the within-province transmission ( f i ) and the diffusion effect ( ĥi ).Regarding the cross-province transmission, we find positive and statistically significant estimates at the 5% level in 62 out of 107 provinces (57.9%), with only 17 estimates being negative but insignificant.These positive ripple effects are moderated after one week, with the coefficients of the diffusion effect ( ĥi ) being largely negative (i.e., 60 provinces exhibit a negative and statistically significant parameter).Similarly, Figure 4 displays the estimates of the within-province transmission, which are generally positive and highly significant.
The most striking result that emerges from the geographical display of the cross-province transmission estimates in Figure 4 is that the strength of the spillovers changes as we move from more economically developed and vibrant transport hubs in the north of the country towards the south.While the cluster of provinces in the north of the country displays a positive and significant spillover effect, the disease transmission among provinces in the south has been lower or insignificant in many cases.Overall, this finding points to stronger feedback effects among provinces in the north and can explain why these provinces have suffered from higher infection rates and a more sustained epidemic.
We estimate the short-run response to a 1% shock to each province to further investigate the directionality of cross-province transmission and the potential of the various Italian provinces to disseminate the disease and their degree of vulnerability.These are calculated using equation (10): 11 where d(.) denotes a N × N diagonal matrix. 12We follow LeSage and Chih (2018) who suggest separating indirect effects between spill-in and spill-out effects.Spill-in effects are row-specific sums of the off-diagonal elements in equation ( 10) and represent the sensitivity (vulnerability) of the infections in province i to a shock in all other provinces.Conversely, spill-out effects are column specific sums of the off-diagonal elements and represent the impact of a shock in unit i in the COVID-19 infections in all other units.We find that spill-out effects are significant in nearly 87% of the provinces whereas spill-in effects are significant in 54% of them.Figure 5 presents graphically this information.As shown, spill-in and spill-out effects are much stronger in the northern provinces, and specially within the region of Lombardy.In fact, eight out of the top 10 provinces in terms of spill-in and 10 out of the top 20 in spill-out effects belong this region.As refers the south of the country, it presents significant spill-out effects in Naples or Rome among other provinces, but the tendency to find significant spill-in effects is lower than the one of norther provinces.Hence, provinces located in the north have been more vulnerable but also more prone to spread the disease than those in the south.Taken together, these results suggest that the spread of the disease has mostly followed a north-north pattern, and to a lower extent, a south-north directionality.Nevertheless, spillovers directed from northern provinces to southern ones have been less relevant in this context.

The drivers of spatial heterogeneity in vulnerability and diffusion
Figure 5 reveals that spill-in and spill-out effects are heterogeneous over the geographical space.One candidate explanation for the spatial disparities observed in Figure 5 is that some structural social, economic, or demographic characteristics, which do not vary quickly over time, may enhance or reduce both outward diffusivity and vulnerability to exogenous infections shocks.Thus, to increase our understanding of the deep causes of the spatial heterogeneity in this context, we now investigate the role played  Lisa Gianmoena and Vicente Rios by different candidate predictors driving differences in (1) spill-in effects and (2) spill-out effects. 13We use BMA to produce a probabilistic ranking of the importance played by the set of factors considered in section 5.3 Columns (1) to (3) of Table 7 report the results regarding the role played by our set of candidate predictors on the size of the spill-in effects, while columns (4) to ( 6) report the results when the estimated spill-out effects are taken as our dependent variable.As observed, the disparities in the degree of vulnerability of i to an increase in the level of infections in neighbouring provinces j ≠ i, can be mainly explained by the degree of accessibility of the province of i (62%).The estimated posterior effect is positive (0.412) and significant at the 1% level.The intuition is that higher accessibility increases the propensity to import infections, given that provinces with large transport nodes with a massive influx of travellers should have a bias to attract epidemic outbreaks (Han et al., 2021).Another relevant factor reducing the sensitivity of i to the evolution of infections in neighbouring provinces, is public spending on health (50.6%).The lower sensitivity of provinces that belong to regions with higher health spending could be explained by the fact that greater medical resources may have allowed them to implement tests, intensify contact tracing and locate cases more efficiently, thereby curving the spread of the disease (Atkeson et al., 2020).As refers to the outward diffusivity, which is measured by the spill-out effects, we find that the most important explanatory variable with a PIP of 81% is the accessibility of the province.This result is not surprising, given that provinces with better communication networks and connections also have the greatest potential to transmit infections to other areas.
The local productive structure, although correlated with our inward and outward diffusivity estimates, receives lower PIPs.This means that after controlling for the variability in the degree of accessibility, the local productive structure is unlikely to be a driver of the variation in spillover effects.Therefore, the provincial heterogeneity displayed in Figure 5 is mainly driven by differences in health system capacity and accessibility.

CONCLUSIONS
This study employs a dynamic spatial panel data approach with common factors to investigate the role played by spillover effects in shaping the evolution of the COVID-19 pandemic in a sample of 107 Italian provinces during the period ranging from 26 February 2020 to 17 October 2021.
To investigate the transmission mechanism of the epidemic, we consider interprovincial connectivity matrices based on large-scale population flow databases, friendships in social networks, and geographical distance.We also consider spatial models with diverse interpretations and implications regarding the causal mechanism underlying the diffusion of the epidemic.Applying spatial Bayesian model selection techniques over patterns of connectivity and spatial models, we find that the CSD observed in the data of infections is best explained by the population flows among the 10 provinces with which there is greater connectivity and that the most plausible spatial process implies the existence of global spillovers.
Our baseline results show that infection rates in neighbouring provinces exert a positive and statistically significant effect on one province's infections, confirming the relevance of cross-province population movements, even when the influence of national (common) factors is filtered out.This finding is robust to a variety of changes in the research design.In a second step, we augment our baseline model to allow for heterogeneous spillovers.We find larger spill-in and spill-out effects in the north of the country.This suggests that provinces located in the north of the country have been more vulnerable but also more prone to spread the disease, which can help to explain the sustained infection rates and higher severity of the disease in these provinces during the period of the study.The diffusion of COVID-19 across Italian provinces: a spatial dynamic panel data approach with common factors Our analysis has important policy implications and provides some lessons that can be used in the social management of future epidemic outbreaks.
The first key lesson is that restrictions and containment measures implemented in situations of health emergency should take into account regional-level heterogeneity given that some spatial units are more likely to produce cascade effects into the rest of the territory than others.For the case of the COVID-19 epidemic in Italy, we find that heterogeneities in both, cascading potential and vulnerability across provinces are mainly explained by their accessibility.However, in future epidemic outbreaks, other features might be of higher importance.Therefore, health authorities should try to understand how these heterogeneous vulnerabilities and propensities to spill-out are built into the system, as this knowledge can be employed to minimize health and economic damages.
Second, there are slowly varying structural characteristics that make some places more likely to experience high and sustained infection rates in the context of epidemic outbreaks.We find that factors such as GDP per capita, the sectoral specialization, the age composition and the social capital have played a role in this regard.The impact of social capital shows that success in public health implementation is likely to be dependent on both the public's trust in government and in a shared social contract among citizens.This means that policymakers can reduce the risks of negative health outcomes by rebuilding trust in government, improving the perceptions of citizens and boosting the capabilities of government.Furthermore,  3) is the estimated spill-in effect in each province, whereas the dependent variable in columns (4) to ( 6) is the estimated spill-out effect.The results reported correspond to the estimation of all the models in the model space including any combination of the 12 variables.We use a fixed BRIC g-prior on the parameters and a binomial model prior with prior mean model size equal to 6 in all cases, which ensures a priori inclusion probabilities of the 50%.Columns (1) and ( 4) reflect the posterior inclusion probabilities, columns (2) and ( 5) the conditional posterior standardized marginal effects of each variable, and columns (3) and ( 6) the sign certainty probability respectively.*Significant at the 10% level; **significant at the 5% level; ***significant at the 1% level.Standard deviations are shown in parentheses.
302 Lisa Gianmoena and Vicente Rios REGIONAL STUDIES improving health system capacity with a proper endowment of economic resources appears as a key factor to minimize vulnerability to infectious diseases.Hence, if the goal is to cope with systemic threats in the future, public health policy could be oriented towards resilience rather than to efficiency.Third, as refers to time-varying predictors, our results show that on average, an increase in mobility and in air pollution raised infections per capita, whereas higher vaccination rates and higher temperatures reduced them.These findings suggest that stringent mobility and environmental policies can be used to slow down the spread of infectious respiratory diseases such as COVID-19.On the other hand, the ability of vaccines to curve infections, highlights the need to strengthen vaccine manufacturing capacity and distribution, as these two capabilities can ensure rapid access for the greatest number of people.

Figure 3 .
Figure 3.Fixed effects and responsiveness to the cross-sectional average.

Table 3 .
Dynamic spatial Bayesian model selection.
Yu et al. (2008)ent variable is in all cases the logarithm of the average new infections per 100,000 inhabitants of the various provinces calculated over windows of seven days.*Significant at the 10% level, **significant at the 5% level, ***significant at the 1% level.t-Statisticsareshown in parentheses.The results are obtained using a 10-nearest neighbour connectivity matrix based on cross-province population flows data drawn from Enel-X/Here Technologies and the Movement between Administrative Regions Dataset of Facebook Data for Good Program.Inferences regarding the statistical significance of these effects are based on the variation of 1000 simulated parameter combinations drawn from the variance-covariance matrix implied by the BCQML estimator ofYu et al. (2008)using equation (5).

Table 6 .
Time-invariant determinants of infections.

Table 7 .
Drivers of spatial diffusion heterogeneity.
Note: The dependent variable in columns (1) to (