Andr ́ee between COVID-19 incidence and climatic conditions. Some COVID-19 related climatic zones are mapped by the Copernicus Earth Observation Programme, see (Copernicus Climate Change Service, 2020), and these put all but a select few municipalities analyzed in this study in the same zone. For this reason, one should expect that if the relationship between pollution and COVID-19 incidence varies regionally, it does so with a reasonable smoothness.

Additional results below are obtained using non-parametric penalized kernel regression following (Hainmueller and Hazlett, 2014; Andr ́ee et al., 2019). The estimates provide observation-level marginal coefficients that allow for nonlinearity conditional on levels in the data. Longitude and latitude have been added as additional controls, which allows the model to capture spatial trends in line with a spatial residual component. However, this time it also allows the model’s parameters to vary across spatial gradients in unobserved components, such as related to weather. The model nests a linear model, specifically, higher levels of regularization result in linearized relationships. Evidence that the relationship with air pollution is nonlinear is strengthened by using Model (2) and re-estimating it after applying a third-order Taylor approximation to the PM2 _._ 5 measurements. Calculating an auxiliary test statistic for the significance of the second and third terms overwhelmingly supports nonlinearity, a Likelihood Ratio obtains a _p_ -value below 0 _._ 001 (statistic of 15 _._ 18 on 2 degrees of freedom).

The fit of the nonparametric model is tuned using standard cross-validation procedures and out-of-sample prediction performance was estimated using 10-fold, repeated twice, cross-validation. To keep the flexibility of the models at a manageable level given the small number of observations, only a few predictors are used. In particular, the significant predictors from the final model (5) are taken, the share of unmarried is dropped as the model can now estimate nonlinear dependence on the share of single households, the share of long-term illness and population density are added back in because they remain of particular interest. The share of non-Western immigrants was dropped because it was insignificant and dropping it did not negatively impact the cross-validation results. Finally, the share of population in the 25 to 44 years group was dropped because estimating nonlinearly on only the share of population above 65 resulted in better fit.

Table 2: Dependent Variable: Confirmed COVID-19 cases <u>per</u> 100,000 inhabitants.

|**Variable**|**Avg.**|**Pr(**_|_**Avg.**_| _|_>_0**)**<br>**q.25**|**q.50**|**q.75**|
|---|---|---|---|---|---|
|Population density|-3.75|0.24|-8.40|-1.84|1.62|
|Share above 65|-0.41|0.44|-1.55|0.00|1.42|
|Share of single households|-1.57|0.00|-2.56|-1.01|0.19|
|Share with long-term illness|0.39|0.59|-1.70|-0.06|1.94|
|Mean PM2_._5|5.98|0.02|-1.24|3.16|11.45|

Insample R2: 0.50, CV R2 0.35, Longitude and Latitude used as additional controls.

Table 2 presents the estimation results. In it, Avg. takes the average across all the marginal coefficients and q.25-q.75 give the quantiles as an indication of parameter heterogeneity. To understand the shape of the nonlinearity, figure 3 plots conditional expectations across the range of values in the covariates. The values are produced by fixing all covariates, 10