There are many alternatives to model the time trends and dynamic properties of the data, including quadratic trends as in Li and Linton (2020) [

5]. We follow Yue et al. (2020) in considering autoregressions for the growth rates [

6]. Let

${Y}_{it}$ be the number of newly infected people in country

$i,i=1,\dots ,N$, at time

t,

$t=1,\dots ,T$. We model growth rates

${y}_{it}:=\mathrm{l}\mathrm{o}\mathrm{g}\left({Y}_{it}+1\right)-\mathrm{l}\mathrm{o}\mathrm{g}\left({Y}_{i,t-1}+1\right)$ via spatial autoregressive processes of order

p, i.e.,

where

${w}_{ij}$ are spatial weights with

${w}_{ii}=0$,

i = 1…,

N, and

${\mu}_{i},{\alpha}_{ij}$ are coefficients. The weights could, for example, be inversely related to the distance

${d}_{ij}$ between countries

i and

j. Beenstock and Felsenstein (2007) suggest to modulate inverse distances with the relative population size of both countries [

18], that is

where

${Z}_{i}$ is the population of country

i. Hence, bigger countries receive more weight than smaller countries. The growth rates of the nine countries with highest number of cases until end of April, 2020, are depicted in

Figure 1.

The error term can be written in vector form, stacking it into an (

N × 1) vector

${\epsilon}_{t}$, as

where

W is the spatial weight matrix

W with

${w}_{ii}=0$,

i = 1…,

N. An advantage of the spatial error model is that, once

$\rho $ is estimated, the structural shocks

${u}_{t}$ can readily be obtained by the transformation

$\left({I}_{N}-\rho W\right){\epsilon}_{t}$. This does not require decomposing a variance–covariance matrix as in classical VAR models, for which different methods are often a matter of debate in the empirical literature, see e.g., the discussion in Lütkepohl (2005) [

21].

The dynamics of

${y}_{t}$ can be generalized to a vector autoregressive model (VAR) of the form

where

${\epsilon}_{t}$ is a spatial error term as in Equation (4), and

${A}_{j}$ are

N ×

N parameter matrices. For large

N, restrictions are necessary for

${A}_{j}$ as otherwise there would be too many parameters to estimate for reasonable sample sizes. Indeed, in our case, the number of countries is larger than the number of days in the sample, so that methods are required that penalize model complexity.

To gain insight into the dynamics of infection growth rates, we will first look at univariate autoregressive models because it reveals some interesting dynamic features of the data. We will then consider spatial vector autoregressions to model the spatial and temporal relationships between countries. Of the more than 200 countries that have reported Covid-19 infections, we have selected the 100 countries with the highest number of infections as of end of April 2020. Thus, our cross-section dimension is given by N = 100. The reason to exclude countries with smaller number of cases is the highly erratic behavior of corresponding growth rates that could potentially bias the estimation of spatial correlation patterns.

#### 2.1. Univariate Analysis

We fit autoregressive models of order

$p$ to the growth rate of infections for each country, where the lag order

$p$ is selected by the Akaike information criterion (AIC). The maximum lag chosen is 10, but for no country the selected

$p$ is larger than 2. Results are reported in

Table A1 in the

Appendix.

The empirical results suggest that:

First order serial correlations of growth rates are negative.

For the countries with AR(2) dynamics, there are stochastic cycles in growth rates with an average length of about three days.

The spatial autocorrelation coefficient $\rho $ is strongly significant.

It is quite remarkable that except for one country (Andorra), all first order autocorrelation coefficients are negative with an average of −0.46. The second order autocorrelation coefficients are closer to zero, but still quite significant for many.

It turns out that among the 100 countries with the highest number of reported cases, 72 have growth rates with AR(2) dynamics, and of these all but one (Denmark) have roots of the characteristic equations that are complex, indicating the presence of stochastic cycles. The average length of these cycles is calculated as

All average cycle durations happen to be between 2 and 4 days, with an average of 2.8 days across countries. We do not have an explanation for the presence of such cycles, and whether they are genuine to the flux of new infections within a country, or artificially generated by the reporting practice. However, given that almost all autocorrelations of order larger than two tend to be small and often negligible, we can exclude seasonal or weekend effects as potential explanation. We next move on to the multivariate analysis in order to understand potential spillovers between countries.

#### 2.2. Multivariate Analysis

For the specification of the VAR model in Equation (5), we first choose the lag order of

p = 2, which corresponds to the maximum of the selected lag orders found for the univariate models. We then estimate the model by minimizing equation by equation the following criterion

with respect to the coefficients

${A}_{ijk}$, where

${\lambda}_{i}$ is chosen by ten-fold cross-validation. This criterion is known in statistics as the least absolute shrinkage and selection operator (LASSO), see e.g., Tibshirani (1996) [

22].

Figure 2 and

Figure 3 show the parameter estimates for the 50 × 50 subset of countries with most infections. The number of non-zero coefficients in

${A}_{1}$ by country ranges from 1 to 32 (out of 100), while the total number is 1741 (out of 10,000). Similarly, the number of non-zero coefficients in

${A}_{2}$ ranges from 6 to 35, with a total number of 1614 non-zero coefficients.

In a second step, the coefficient

$\rho $ of the spatial error component model in Equation (4) is estimated by Gaussian maximum likelihood, see e.g., Bivand et al. (2013) [

23]. The model obviously depends on the specification of the spatial weights matrix

W, for which we used inverse distances modulated by the relative population sizes, as in Equation (3). Distances are measured as the geographic distance in kilometers on the WGS ellipsoid between the centroids of two countries. This specification led to the highest likelihood among a set of alternative specifications including unweighted inverse distances,

k-nearest neighbor and contiguity matrices. We obtain an estimate of 0.2805 with associated standard error of 0.0518, which suggests that spatial correlation is highly significant and important to be included in the subsequent analysis. After estimation, the residuals

${u}_{t}$ have been tested for remaining spatial autocorrelation using the Baltagi et al. (2007) spatial autocorrelation test for panel models [

24], which is an extension of the classical Moran test. The

p-value of the test is 0.157, which suggests that spatial autocorrelation has been sufficiently captured by the model.

In the following, we study the implications of the estimated parameters for the decompositions of the variances of forecast errors, in the spirit of Diebold and Yilmaz (2014) [

20]. This allows quantifying the network relationships between countries. Starting from the reduced form VAR (2) model in Equation (5) with spatial error terms

${\epsilon}_{t}$ in Equation (4), we may obtain the infinite order vector moving average (VMA) representation,

with coefficient matrices

${\mathsf{\Phi}}_{j}$ that, for our VAR (2) model, can be obtained recursively as

${\mathsf{\Phi}}_{0}={I}_{N}$,

${\mathsf{\Phi}}_{1}={A}_{1}$,

${\mathsf{\Phi}}_{2}={\mathsf{\Phi}}_{1}{A}_{1}+{A}_{2},\dots ,{\mathsf{\Phi}}_{j}={\mathsf{\Phi}}_{j-1}{A}_{1}+{\mathsf{\Phi}}_{j-2}{A}_{2},\dots $, see e.g., Lütkepohl (2005) [

21].

From the spatial error model in Equation (4) we then obtain orthogonalized shocks as

${u}_{t}=\left({I}_{N}-\rho W\right){\epsilon}_{t}$ and rewrite the model as

where

${\mathsf{\Theta}}_{j}:={\mathsf{\Phi}}_{j}{\left({I}_{N}-\rho W\right)}^{-1}$. Thus, having orthogonalized the error terms, we can perform structural analysis, and this is independent of a particular decomposition of a variance-covariance matrix, which is a common problem in classical VAR models. The proportion of the

h-step forecast error variance of country

i accounted for by innovations in country

j is then given by

where

${\theta}_{ij,k}$ is the

ij-th element of the parameter matrix

${\mathsf{\Theta}}_{k}$. It is important to emphasize that

${\omega}_{ij,h}$ depends on both the dynamics of the VAR system via the parameter matrices

${\mathsf{\Phi}}_{j}$, and the spatial lag matrix,

$\left({I}_{N}-\rho W\right)$. Hence, forecast variance decompositions are determined by spatial and temporal dynamics jointly, which corroborates the importance of a joint spatio-temporal modeling approach.

The proportions

${\omega}_{ij,h}$ are reported graphically in

Figure 4 for the 50 countries having the highest number of infections. We chose a forecast horizon of

h = 10, which for our parameter estimates is almost identical to long-run forecasts (

$h=\infty $) as

${\omega}_{ij,h}$ converges quickly to a constant as

h increases. Not surprisingly, most countries’ forecast errors are explained to a good extent by their own history, i.e., the diagonal elements of

$\omega $ tend to be larger than the off-diagonal elements. The average of the diagonal elements is 0.47 with a maximum of 0.96 and a minimum of 0.1. This also implies that there are many countries for which the sum of the contributions of other countries is larger than that of their own history.

This analysis can be refined following Diebold and Yilmaz (2014) by viewing variance decompositions as weighted directed networks [

20]. They define two directional connectedness measures, “to” and “from”, for each country. The “to” measure is defined as

where we suppress the

h-index for simplicity. The

${\omega}_{\cdot j}$ measures the sum of the contributions of country

j to all other countries’ forecast errors, and it can be viewed as a “to”-degree of a node (i.e., a country) of the network. The support of the univariate distribution of

${\omega}_{\cdot j}$ is

$\left[0,N\right]$. The countries’ “to”-degrees are visualized in

Figure 5, and the corresponding distribution in

Figure 6. The distribution of “to”-degrees has a bi-modal structure, where the majority of “to”-degrees is close to zero, and a small group of countries having “to”-degrees of 2 or larger. The countries with highest “to”-degrees are Ghana, United Arab Emirates, Philippines, Ecuador, Oman, Belarus, Singapore, Iraq, Spain, and Egypt. Note that each continent has at least one country with a high “to”-degree, i.e., a country whose innovations in infection rates help to predict infection rates of other countries.

Likewise, the “from” directional connectedness is defined as

And measures the sum of the contributions of all other countries

j to explain the forecast error variance of country

i. The support of the univariate distribution of

${\omega}_{i\cdot}$ is

$\left[0,1\right]$. These measures are visualized in

Figure 7, and the corresponding distribution in

Figure 8. Note that, by construction, the means of the “to” and “from” distributions are the same, 0.5222, but that the “from” distribution is more concentrated around values close to one, meaning that there are many countries whose forecast variances can to a large proportion be explained by the innovations of other countries. The countries with highest “from”-degrees are Canada, Italy, USA, China, Belgium, Bulgaria, Mexico, Germany, Great Britain, and Denmark. Note that these countries tend to have high absolute number of infection cases. Note also that the proportion of developing and emerging countries among those with high “to”-degrees is larger than among those with high “from”-degrees.

Finally, the total connectedness measure for the network of countries is given by $\omega :={{\displaystyle \sum}}_{i=1}^{N}{\omega}_{i\cdot}={{\displaystyle \sum}}_{j=1}^{N}{\omega}_{\cdot j}$, which corresponds to the mean degree of the network. We obtain a mean degree of 52.22.