Epidemic–economic complexity of COVID-19 policies across skill groups and geographies

ABSTRACT The COVID-19 pandemic has threatened public health and socio-economic activities across societal groups and geographies. We analyse the complex interplay between epidemic and economic factors using a structural panel vector autoregressive (PVAR) approach for Danish municipalities. Findings indicate that the pandemic shock and associated public health interventions led to significant increases in unemployment rates. Wage compensations reduce regional unemployment through both a direct local effect and indirect spatial spillovers. Decomposing the unemployment rate by skill, we find that the response to an increase in wage compensations is only significant for low-skilled persons and that it is larger in urban compared with rural settings.


INTRODUCTION
When the COVID-19 pandemic hit the world in early 2020, two types of policy responses were central in the public and academic debate: first, in the absence of a pharmaceutical cure, non-pharmaceutical public health policy interventions were introduced to slow the SARS-CoV-2 infectious spread through social distancing and reduced mobility rates.A large body of empirical research has evaluated the effectiveness of these policies (e.g., Alfano & Ercolano, 2020;Bourdin et al., 2021;Dall Schmidt & Mitze, 2022;Kosfeld et al., 2021;McCann et al., 2022).
A second type of public policies intended to cushion the detrimental socio-economic effects of (1) the pandemic development itself and (2) non-pharmaceutical public health interventions including mobility restrictions, lockdowns of businesses, etc.While firms were supported through lump-sum payments, tax reductions or specific loans, the main instruments to counter adverse labour market outcomes were the extension of sick leave payments, the extension of short-time work linked to wage compensation schemes and more generous unemployment benefits (Eichhorst et al., 2020; Organisation for Economic Co-operation and Development (OECD), 2021a).Several contributions have studied the trade-offs between public health interventions and socio-economic outcomes (e.g., Alexander & Karger, 2020;Bonaccorsi et al., 2020), but only few studies have evaluated the effects of socioeconomic policies on economic outcomes during the pandemic (exceptions include, for example, Meinen et al., 2021;and Milani, 2021).Virtually no evidence is available that models the complex interplay between SARS-CoV-2 infection dynamics, public health policy interventions, socio-economic policies and, ultimately, economic outcomes at the local area level.This is undertaken here.
We develop a structural panel vector autoregressive (PVAR) model for Danish municipalities (local administrative units (LAU) level 1) and study the response of the local unemployment rate to (1) a SAR-CoV-2 infection shock, (2) mobility reduction resulting from social distancing interventions in public health policy and (3) socio-economic policies running through a generous wage compensation scheme subsiding wage costs of firms to preserve jobs.A particular focus is on distinguishing to what extent (i) unemployment effects arrive from local vis-à-vis global (spatially indirect) variations in factors (1) to (3), (ii) effects differ by skill groups and (iii) vary across different geographies in Denmark.We argue that a systems approach to this epidemic-economic complexity allows policymakers to better understand the working of different types of interventions during a major crisis.It is essential input for well-informed policy decisions to save human lives while minimizing negative socio-economic effects in a future public health crisis.Our results also contribute to the literature on wage subsidies as an economic policy instrument in times of crisis.
Our methodical choice of a short-run structural PVAR approach that incorporates spatial spillovers is generally supported by the nature of the COVID-19 crisis, where 'shocks' and associated effects arrive in the short run as labour market behaviour and outcomes adapt quickly to changing conditions from SARS-CoV-2 infections and public health policy interventions.Denmark further represents an interesting case to investigate these linkages as public health interventions and socio-economic policies in the form of wage compensations subsidizing wage costs of firms have been implemented in a highly flexible manner.For instance, the Danish wage compensation scheme covered more than 250,000 jobs at its peak in April 2020 and returned to nearly zero just a few months later.One key difference to more 'static' wage compensation and furloughing policy approaches in European neighbouring countries was that the Danish socio-economic policy design closely followed SARS-CoV-2 infections and the implemented public health interventions over time (see Appendix A in the supplemental data online for details).Furthermore, entitlement of businesses to receive the nationally defined wage compensation policy support varied over the geography of Denmark depending on the local SARS-CoV-2 infection dynamics, lockdowns and social distancing measures imposed on specific sectors and then affecting the local business climate differently.All in all, our data (including policy instruments) thus exhibit significant space and time variations that are needed to properly identify effects.
Turning to the results, our structural PVAR approach confirms earlier findings of a significant reduction in mobility rates when a region is shocked by rising SARS-CoV-2 infections.We further find that regional unemployment rates increase both with (1) rising SARS-CoV-2 incidences rates and (2) lower mobility levels underlining the trade-off between public health interventions and economic outcomes.As a novel result, our PVAR approach point to the mitigating effect of increasing the number of persons in wage compensation jobs which helps to reduce the local unemployment rate.The effect is equally driven by a (1) local (direct) component and (2) indirect spatial spillovers.
When decomposing the unemployment rate by skill groups, we find that the mitigating effect from wage compensations is significant only for low-skilled unemployed.Lastly, effects are present in urban and rural municipalities, but the magnitude of the effects (particularly its spatial indirect component) is larger in an urban environment.We conclude that an effective policy mix to combat COVID-19 must consider the complex interplay between epidemic and socio-economic developments while accounting for spatial spillovers and heterogeneity across societal groups and geographies.
The paper proceeds as follows.Section 2 outlines our modelling approach.Section 3 reviews prior evidence on the role of wage compensation schemes for labour market outcomes during times of crises and provides institutional details about the Danish wage compensation scheme during COVID-19.Section 4 presents the data used for the empirical analysis and section 5 outlines the PVAR estimation approach.Section 6 reports the baseline empirical results and detailed results on heterogeneity in effects across skill groups and geographies.Section 7 offers a policy discussion beyond the COVID-19 crisis.

COMPLEXITIES IN POLICY RESPONSES DURING THE COVID-19 CRISIS
In Denmark, social distancing measures including strict lockdowns for large parts of the society and businesses to curb the SARS-CoV-2 infectious spread had been implemented throughout 2020 and 2021 (OECD, 2021b;Olagnier & Mogensen, 2020). 1 One side effect of lockdowns is that employed persons become vulnerable to being fired, which could lead to social and economic consequences in terms of unemployment spells.These mechanics of interdependencies between the health shock from infections, public health interventions and local socio-economic outcomes point to the complexity of assessing the policies undertaken during a public health crisis.
Our focus is on investigating the local labour market effects of socio-economic policies in terms of wage compensations, while controlling for local effects of the pandemic shock and public health policies associated with non-pharmaceutical interventions.The combination of mobility restrictions due to restrictive lockdowns, on the one hand, and wage compensation policies to remedy negative local social and economic consequences, on the other hand, is not unique to Denmark.But the Danish policy design features a high degree of flexibility over time, which makes it particularly well-suited for studying the geographical complexity between infectious spread, economic policies, and local labour markets responses in a dynamic set-up.
We apply a shock-response-feedback approach that particularly considers the response in socio-economic outcomes (unemployment) to the introduction of wage compensation schemes across different geographies, as indicated by the arrows in Figure 1.Immediate causal links are in this assessment running from changes in local incidence rates (box 1 in Figure 1) to public policy interventions changing mobility (box 2) and then to socio-economic policies (box 3) to counter detrimental socio-economic outcomes in a region (box 4).Local SARS-CoV-2 incidence rates are considered the most exogenous factor in this transmission process as they result from a global pandemic shock.The next step in the chain of causal events is the implementation of public health policies to curb infections by reducing local workplace mobility and social contacts (e.g., Cartenì et al., 2020;Tokay, 2021), upon which socio-economic policies were introduced to counter adverse local economic (side) effects.These mechanisms are likely to affect overall unemployment, as well as unemployment in different geographies and across skill groups.We particularly focus on links from socio-economic policies (box 3) to local socioeconomic outcomes (box 4) within this complex epidemic-economic system.One could consider other causal orders of immediate effects, but the given order running from infections, over public health interventions and socio-economic policies to local labour market outcomes mimics the policy decisions observed during the COVID-19 crisis.
This causal order of impulses and (policy) responses can be characterized as resulting from immediate effects of the pandemic shock and induced effects and feedbacks in the local epidemic-economic system.First, within the same region (blue circle in Figure 1) there will be feedbacks among the different variables considered.For example, a mobility reduction will affect infection rates as less social contact from mobility should curb infectious spread (e.g., Kosfeld et al., 2021).Immediate and induced effects between socio-economic policies and local labour market outcomes are indicated by solid lines with twosided arrow, that is, box 3 and 4 in Figure 1.Second, another type of induced effects relates to spatial dependence as shown by dashed arrows in Figure 1.More persons in a wage compensation scheme in one region may, for instance, reduce labour supply in neighbouring regions, which would tend to affect unemployment in the neighbouring region.Relatedly, interregional links between wage compensation jobs and unemployed persons may mirror underlying commuting patterns as wage compensations are paid out by workplace location while unemployment is registered at the location of residence.
The visualization of impulse-responses as shown in Figure 1 helps us to approach different key aspects in the epidemic-economic complexity of COVID-19.As a plausibility check, we first test the link between SARS-CoV-2 incidence rates (box 1 in Figure 1) and mobility rates (box 2), which has been extensively covered in the literature.We next turn to the core focus of this paper, that is, the question: Do socio-economic outcomes in terms of local unemployment rates (box 4) depend on local epidemic development (box 1), change in mobility mirroring public health policy interventions (box 2) and socio-economic policy in terms of wage compensations (box 3)?This is our second test which shall provide novel evidence on the importance of different policy mixes during a public health crisis.A third test focuses on whether effect transmission predominantly arrives from the region's own epidemic-economic system or arrives from spatial spillovers from local epidemic-economic systems in neighbouring areas.Finally, heterogeneity of effects across skill levels and geographical structures is considered.Heterogeneity across geographical structures is measured by comparing urban and rural areas.These investigations constitute our fourth and fifth test.

WAGE SUBSIDIES IN TIMES OF CRISIS
Prior evidence on the effects on unemployment of wage subsidies to firms remains inconclusive: Katz (1996) finds that wage subsidies to employers may have modest positive effects for disadvantaged groups.Other contributions that have considered the issue of targeted wage subsidy policies (e.g., Dubin & Rivers, 1993;Huttunen et al., 2013;Jaenichen & Stephan, 2011) point to effects on employment and re-employment in some cases but not all.
Wage compensation schemes running through public wage subsidies have been widely used during the global financial crisis of 2007/08 and even more during the COVID-19 pandemic (Ebbinghaus & Lehner, 2022;Vireck et al., 2022).During the global financial crisis, costs were incurred by firms mainly from loss of firm-specific human capital as result of dismissals and adjoin search, hiring and training workers in the following recovery.This was particularly relevant for specialized workers (Möller, 2010).Short-time work arrangements acting as wage subsidies further led to temporary labour hoarding in firms.Stressing this type of internal flexibility relative to external flexibility to firms, Möller (2010) sees such labour market responses as exemplary for crisis situations.Hamilton (2020) argues that the COVID-19 crisis affected demand unevenly across the economy, why broad cash stimuli may only affect some businesses, whereas wage subsidies can be targeted to businesses impacted more by COVID-19. For Germany, Christl et al. (2021) show that short-time working schemes combined with wage compensations were effective during the COVID-19 crisis in alleviating income loss for lowincome persons and households.Considering the period from March to June 2020 in Denmark, Borgensgaard (2022) finds that wage compensations reduced job losses especially for shorter tenures.
Different designs of wage subsidy policies may affect search behaviour, labour hoarding and relative costs of training and rehiring staff in different sectors and across different types of labour.How such policies unfold will, firstly, depend on regional structures in business, income and/or skill structure. 2Secondly, they will also depend on the channels through which an exogenous shock arrives, and the policy mix applied.It is noticeable that the design in Denmark differed from that in other countries (see Appendix A in the supplemental data online for details).
The Danish design is based on an employer-side wage subsidy (wage compensation) for wage costs incurred by employers.Wage subsidies were, though, only allocated under a set of specific conditions concerning the reduction in turnover and the share of employed subject to being sent home, while wage compensation percentages relative to the nominal wage level were quite generous.Eligibility to such wage compensations for employers may therefore vary considerably over regional contexts and business climates.These are for Denmark detailed in Table A1 of Appendix A in the supplemental data online.The wage subsidy scheme in Denmark was used in a very flexible manner limited to periods of high infectious spread and lockdowns, where job search was difficult.It was therefore less susceptible to moral hazard problems.As infections reduced and lockdowns were lifted, wage subsidy schemes were cut back.

DATA AND STYLIZED FACTS
A challenge for a comprehensive analysis along the lines of Figure 1 is establishing a database comprising epidemiological, socio-economic and policy information during the COVID-19 crisis.We have gathered data for 98 municipalities (LAU-1) between February 2020 and May 2021 in the following way: . Daily data on reported SARS-CoV-2 infection cases in the human population for each municipality have been retrieved from the official COVID-19 Dashboard governed by Statens Serum Institut (SSI). 3The data have been aggregated (summed) to a monthly frequency to match the frequency of available economic data, that is, labour market data.This aggregation also reduces the problem of outliers typically present in epidemiological data with higher time frequency. .Weekly data on mobility levels have been gathered from the Google COVID-19 Community Mobility Reports.We focus on workplace mobility, which maps the frequency of visits to workplace locations within a municipality (categorized by Google).Values are deviations from a baseline period of weekly visits (median) to workplace locations between 3 January and 6 February 2020. 4Data have been aggregated to a monthly frequency and are measured as mobility reductions (%) relative to the baseline level, that is, larger values indicate lower mobility rates. .Monthly data on the number of persons in wage compensation jobs in Danish municipalities have been obtained from the Danish Business Authorities under the Danish Freedom of Information Act. .Monthly data on the number of unemployed and the (seasonally adjusted) unemployment rate, that is, the number of unemployed persons in the labour force, per municipality have been obtained from Statistics Denmark. 5 .Skill-specific unemployment rates by month and municipality have been calculated on the basis of data on unemployed persons covered by different unemployment insurance funds as percentage of the total labour force. 6.A classification of urban and rural types of municipalities, as shown in Figure B1 in Appendix B in the supplemental data online, is taken from Kristensen et al. (2006).Based on this geo information, binary dummy variables haven been constructed that measure the geographical context of a municipality.
Epidemic-economic complexity of COVID-19 policies across skill groups and geographies 325 REGIONAL STUDIES . Data on geographical borders and municipal centroids have been obtained from the Danish data and map supply (Dataforsyningen). 7This information is used to construct a spatial weighting matrix.
Several aspects on the calculation of the skill-specific unemployment rate are worth mentioning.Since monthly skill-specific unemployment rates are not published by Statistics Denmark, we argue that data by unemployment insurance funds are a feasible approximation for skill groups.However, three potential limitations need to be considered.First, membership in an unemployment insurance fund is voluntary, why unemployed that are not members of an insurance funds are excluded and skill-specific unemployment rates may not add up to the total unemployment rate.Second, compared with rates calculated based on the skill-specific local labour force (for which data are unfortunately not available) this measure may be sensitive to compositional changes in the local labour force by skill levels, for example, through retirement, entry from/to the local labour force or geographical mobility.Given our relatively short sample period, we, though, argue that larger compositional changes from mobility in and out of the local labour force are likely absent.Moreover, typical mobility patterns in times of crisis, such as retiring after being fired, should have been alleviated by the Danish wage compensation scheme in place during COVID-19.Also, the inclusion of region-fixed effects in our regression approach accounts for overall (time-constant) differences in the skill composition of the local labour force across municipalities, which should render our measure of skill-specific unemployed out of the total local labour force insensitive to such between variation in the data.Third, while the categorization of insurance funds is based on education and job segment characteristics, the link to skill levels may remain somewhat fuzzy.Hence, we only distinguish two broad categories of high-and low-skilled unemployment, as defined in Table B1 of Appendix B in the supplemental data online.
Further information and summary statistics for the different variables collected are provided in Table 1.
Differences in sectoral structures, a varying dependency on labour market mobility and heterogeneous local SARS-CoV-2 infection dynamics (reflecting population densities and social structures) have affected the extent to which wage compensation schemes apply for a given geography.Panel A of Figure 2 shows that the number of persons in wage compensation jobs in the labour force varies considerably across Danish municipalities. 8Urban areas furthermore tend to be more dependent in relative terms on such policies.Some peripheral areas, though, also show considerable dependence on wage compensation policies during the first spike of SARS-CoV-2 infections in April 2020.
This begs the question if there is temporal persistence in the dependency on wage compensation schemes over time for a municipality.Panel B of Figure 2 reports the percentage change in the number of persons in wage compensation jobs by municipality between the first and second infection spike in April 2020 and January 2021, respectively.There is no clear evidence of persistence.For example, the municipality to the very southwest of Denmark shows a high dependency in panel A of Figure 2 but sees a sizeable reduction in panel B of Figure 2. The observed space and time variations in local dependencies on wage compensation schemes require a modelling approach that captures such dynamics.This is what we address next.

Baseline specification
We use a structural vector autoregression model for panel data (PVAR) to investigate the complex interplay between epidemic and socio-economic variables as shown in Figure 1.PVAR models are a flexible tool for dynamic analyses in a multivariate context (e.g., Holtz-Eakin et al., 1988).A particular strength is their ability to identify adjustment paths taking place after (transitory) shocks to the underlying system of variables (Mäki-Arvela, 2003).Extensions of the PVAR approach allow for a proper handling of spatial dependence and the identification of cross-sectional spillovers (Beenstock & Felsenstein, 2007;Di Giacinto, 2010;Elhorst et al., 2021;Mitze et al., 2018).
As highlighted in section 2, both temporal and spatial dynamics are considered important in our empirical application.We approach the issue in a sequential manner: Our baseline specification starts from a non-spatial PVAR model that stresses the time dynamics within and between variables.For a dynamic system of M equations with y m,it denoting the outcome variable of the m-th equation (with ), the baseline model can be written as: where m m,i is a vector of municipal-specific time-constant effects included in the m-th equation, a m,m denotes the regression coefficient indexed by equation/variable for a lag structure of one month and 1 m,it is an i.i.d.error term for the m-th equation with zero mean and finite variance.We also include K exogenous factors z that cover linear, quadratic, and cubic time trends and binary dummies for local lockdowns, for example, related to sharply rising infection numbers in some parts of Denmark (Dall Schmidt & Mitze, 2022).These factors capture non-linear trends in epidemiological data.The coefficient b k measures the strength of the correlation between the

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Torben Dall Schmidt and Timo Mitze REGIONAL STUDIES Epidemic-economic complexity of COVID-19 policies across skill groups and geographies 327 REGIONAL STUDIES k-th factor and each of the four endogenous variables of our epidemic-economic system.More compactly, we can express the above M-equation system stacked over crosssections and equations as: where y t =[y 1,1t ,...,y 1,Nt ,...,y M,1t ,...,y M,Nt ] ′ .Analogously, mis an NM ×1 vector of municipality-fixed effects; A 1 and A 2 are coefficient matrices, for example, A 1 =I N ⊗a with I N being an identity matrix of dimension N, ⊗ is the Kronecker product and a denotes an M ×M matrix of regression coefficients [a m,m ] M×M ; 1 t is the NM ×1 vector of residuals with E(1 t )=0, E(1 t 1 ′ t )=S and E(1 t 1 ′ t−h )=0 (for h=1,2,...), where S is a NM ×NM positive definite variance-covariance matrix.
We estimate the PVAR as a four-equation system with ir, mobr, rwagecomp, runemp as endogenous outcome variables (Table 1).All variables enter the model as logarithmic transformations denoted by the prefix l, for example, lir is the log-transformed SARS-CoV-2 incidence rate.The PVAR system is estimated in an equation-byequation manner as a dynamic fixed effects (DFE) specification using ordinary least squares.In addition, we also apply bias corrected DFE estimation with bootstrapped standard errors as discussed in Everaert and Pozzi (2007).

Spatially extended specification
Dynamic panel data models can be extended spatially in different ways by adding contemporaneous spatial lags and space-time lags of the endogenous and exogenous variables to each equation of the PVAR system (Elhorst, 2012).While adding more spatialized variables may reduce estimation problems associated with an omitted variable bias, prior research has also shown that it can lead to an under-identification problem.Considering this trade off, we follow the approach advocated in Mitze et al. (2018) and limit the spatial coverage to a spatial autoregressive (SAR) specification.Different from more general spatial econometric alternatives such as the spatial Durbin model (SDM), the SAR approach accounts for spatial dependence through the inclusion of a spatial lag of the dependent variable (W y t ) in each equation of the PVAR system in equation ( 2), where the matrix W = I M ⊗ W N is composed of an identity matrix of dimension M and an N × N spatial weighting matrix W N .The latter has elements w ij (for i,j ¼ 1, … , N and i = j) which take values of 1 if the centroids of two municipalities lie within a 50-km distance radius, and zero otherwise.The matrix W N is further row standardized so that values of the spatial lag term have the same dimension as y t and can be interpreted as the average value in the spatial neighbourhood of municipality i.
In the case of spatially extended models, estimation and effect interpretation is more complex than in nonspatial models.Regarding estimation, it needs to be considered that the inclusion of a spatial lag of the endogenous variable is correlated with 1 t .Also, as for the case of nonspatial estimators, the endogeneity of the included time lagged dependent variable needs to be considered.To account for both sources of endogeneity we apply the bias-corrected quasi-maximum likelihood (QML) estimator for dynamic spatial panel models proposed by Yu et al. (2008).In terms of effect identification, the application of a SAR model enables us to capture global (but not local) spatial spillovers running through the m-th equation's dependent variable.We argue that for our epidemic-economic system the presence of global spillovers is, in fact, a more likely outcome than the mere presence of local spillovers, which limits spatial effect transmission to firstorder neighbours.In comparison, global spillovers have an effect on all other regions in the geographical system.
The strength of direct and spatial indirect (spillover) effects can be computed based on the regression coefficients in the spatially extended PVAR model of equation ( 1), which includes the regression coefficient r m of the added spatial lag term in the m-th equation.If we take the regression equation for y 1 in equation ( 1) as an example, then the direct short-term effect of y 2 on y 1 can be computed as [ I N a 1,2 (I N − r 1 W ) −1 ] d , where the term (I N − r 1 W ) −1 is referred to as the global spatial multiplier and the superscript d denotes the operator that calculates the mean diagonal elements of a matrix over the cross-sectional dimension N × N .Similarly, the spatially indirect short-term effect, which measures the average effect of a change in y 2 for municipality j on y 1 for municipality i can be calculated as [(I N a 1,2 ) (I N − r 1 W ) −1 ] rsum , where rsum is the operator calculating the mean row sum of the non-diagonal elements.The sum of the direct and spatially indirect effects is the total short-term effect of y 2 on y 1 .To provide a measure for the statistical significance of the average effects, we calculate simulated standard errors as, for example, proposed by LeSage and Pace (2009) and Elhorst (2014).

Impulse-response functions and causal ordering
From the estimated direct (and spatially indirect) effects we compute impulse-response functions (IRFs), which allow us to graphically assess the system's short-term response to different shocks.While IRFs are a meaningful way to describe the response of one variable to innovations (shocks) in another variable of the system when holding all other shocks equal to zero, two aspects need to be considered: First, as pointed out by Di Giacinto (2010), causal interpretations of IRFs are limited for VAR models in their unrestricted reduced-form presentation.We address this limitation by imposing a structural causal ordering of variables in period t (motivated by Figure 1) as: where arrows indicate the direction of (Wold) causality.
Variables more to the left (e.g., the SARS-CoV-2 incidence rate) contemporaneously affect variables to the right (at time t), while feedback effects from the latter 328 Torben Dall Schmidt and Timo Mitze

REGIONAL STUDIES
to the former variables only take place with a time lag (t + 1).Hence, for example, incidence rates immediately affect mobility patterns and local labour market outcomes (directly and through indirect spatial spillovers).Feedback effects from changes in the unemployment rate to mobility patterns and SARS-CoV-2 incidences rates, among others, are restricted to period t + 1 onwards.Technically this is done by computing orthogonal IRFs based on a Cholesky decomposition of the model's variance-covariance matrix.Second, the graphical IRF analysis does not provide insights on the statistical significance of effects.We address this by plotting IRFs together with 95% confidence intervals calculated from Monte Carlo simulations (Love & Zicchino, 2006).

Baseline results
As a first plausibility check for the mechanisms at play in our estimated PVAR system, we focus on the link between SARS-CoV-2 infection dynamics and changes in mobility rates (a link which is well documented in the literature).As seen in the upper panel of Figure 3, a transitory, one unit shock to the incidence rate (measured as a 1 SD (standard deviation) shock for standardized variables, that is, the structural errors) results in a statistically significant mobility reduction, which peaks about 1-2 months after the shock.Responses of the outcome variable (%) are shown on the y-axis.The estimated mobility decline transmits through (1) higher levels of social distancing to avoid infection and (2) reduced socio-economic activity from public health policy interventions, for example, lockdowns.This effect is transitionary in nature and phases out after approx.six months.The flip side to this impulse-response pattern is that of a one unit shock to the mobility rate as shown in the lower panel of Figure 3: a shock to the mobility rate reduces SARS-CoV-2 incidence rates.An increase in mobility reduction (due to more restrictive public health interventions) implies less workplace mobility and less social contacts at workplaces, which then reduce incidence rates.In general, these results are in line with theoretical considerations and consistent with previous findings in the empirical literature (e.g., Caselli et al., 2022;Kosfeld et al., 2021).After this initial plausibility test of our PVAR approach to epidemic-economic complexities, we turn to our main focal points, that is, the dependencies of local socio-economic outcomes to changes in epidemic developments and  socio-economic policies including wage compensation schemes.These effects are summarized in Figure 4.
As revealed in the upper panel of Figure 4, a one unit shock to the SARS-CoV-2 incidence rate drives up the local unemployment rate.Again, this is consistent with theoretical expectations.As incidence rates increase, this restricts the opportunities of businesses to pursue normal operations and, hence, endangers the employment level as some businesses are forced to fire employees.Similarly, with increasing mobility reductions as shown in the mid panel of Figure 4, the local unemployment rate soars.With less mobility to workplaces due to non-pharmaceutical interventions, for example, lockdowns, this associates to higher unemployment rates.Finally, the lower panel of Figure 4 attends the question whether wage compensations can mitigate effects on the unemployment rate or whether they just render moral hazards in labour market behaviour.A one unit shock (increase) to the share of wage compensation jobs in the local labour force clearly reduces the overall local unemployment in a municipality.The effect is significant for about two months.Approximately, the effect size measured is a 1 percentage point reduction in the unemployment rate for 1 SD increase in the share of wage compensation jobs in the labour force (which corresponds to about 4% in our sample).This lends credence to the effectiveness of socio-economic policies to curb local unemployment rates in times of crisis.

Spatial effects
Ignoring spatial dependence may bias our estimates as the baseline results shown in Figures 3 and 4 only measure local direct effects.Figure 5 therefore compares the direction and magnitude of IRFs computed based on the non-spatial effects from DFE estimation with those for a spatially augmented SAR model.IRFs for direct and spatial indirect effects in the latter model are computed as described in section 5.For ease of presentation, no confidence intervals for the IRFs are shown, but a full set of IRFs for each estimation set-up is provided in Appendix C in the supplemental data online.Figure 5 clearly reveals that spatial dependence is of importance.The non-spatial effects from DFE estimates and the total effects reported by the SAR model largely coinciding in the upper two panels of Figure 4. Furthermore, the spatially direct and spatially indirect effects are of the same sign and similar magnitude.(linear, quadratic and cubic), binary dummies for regional lockdowns and municipal fixed effects.For ease, only point estimates are displayed, but associated 95% confidence intervals are shown in the Appendix in the supplemental data online as well as the full set of IRFs across the different estimators.
It is noticeable, however, that the non-spatial effects from DFE estimates for the response of the unemployment rate to a shock in wage compensations as shown in the lower panel of Figure 5 are sizably smaller than the results based on a SAR approach (by a factor of about 0.4).The non-spatial effects from DFE estimates are in terms of size and curvature close to the direct spatial effects of the SAR model.Not considering spatial dependence in shocks from wage compensations would accordingly render too low (conservative) effect estimates of socio-economic policies on local unemployment.
Spatial dependence in effects of wage compensations may reflect different aspects.If neighbouring municipalities have similar business structures and climates with a similarly high eligibility to receive wage compensations, this would ease the pressure towards increasing unemployment in each of the municipalities.While Figure 4 already pointed to the importance of wage compensation policy to address adverse COVID-19 crisis effects on local unemployment, adding spatially indirect effects in Figure 5 emphasizes this even more.Further, spatial indirect effects likely mirror latent commuter relationships among municipalities.Persons in wage compensation jobs are measured at the workplace location while unemployment is registered at the location of residence, why significant spatial spillovers may point to this transmission channel.Ignoring these linkages would systematically underestimate the effect of wage compensation schemes on the local unemployment rate.

Effect channels: skill groups
One of the consequences of the spread of SARS-CoV-2 infections in society was the public health policy response of restricting on-site access to workplaces through, for example, lockdowns.In comparison with low-skilled workers, high-skilled workers may be less reliant on onsite activities at the workplace and be more able to work remote.Skill groups may accordingly react differently to shocks.Figure 6 shows the response of the highly skilled unemployment rate (upper left) and low-skilled unemployment rate (upper right) to a one unit shock in the share of wage compensation jobs in the local labour force.A policy shock that increases the number of wage compensation jobs does not render statistically significant effects on the unemployment rate of high-skilled workers, but it leads to a clear drop in the low-skilled unemployment rate.Low-skilled workers are likely more dependent on on-site activities at the workplace and then more dependent on being furloughed instead of being fired.For high-skilled workers, this effect is absent, which one may conjecture is caused by much easier access to remote work and less dependence on on-site work at the workplace.
The strong mitigating effect of wage compensations on unemployment rates of low-skilled workers arrives from both direct spatial effects (left-most lower panel of Figure 6) and indirect spatial effects (right-most lower panel of Figure 6).This taps into the issue of commuting-based spatial spillovers, as an increase in the share of wage compensation jobs in neighbouring municipalities reduces labour supply of low-skilled workers in a municipality and consecutively low-skilled unemployment rates.This is not the case for high-skilled unemployed.A scarce literature has considered remote work and COVID-19 (e.g., Althoff et al., 2022;Brynjolfsson et al., 2020) often based on early data or simulations.We do not measure remote work directly, but the findings by skill groups are indicative of how socio-economic policies interact with job types by skill levels in times of a public health crisis likely correlating with varying abilities to undertake remote work.

Effect channels: geographies
Figure 7 reports the results for the SAR model based on interaction terms between the included regressors and binary dummies for urban and rural municipalities.The interaction term approach (different from sample split regressions) allows us to incorporate the full set of Danish municipalities in the calculation of spatial spillovers running through W y t .A one unit shock (increase) to the share of wage compensation jobs in the labour force has the clear immediate effect of driving down local unemployment rates in both rural and urban areas.As such, this policy would appear important for both types of regions.But effects are only significant and negative for a longer period in urban areas, while effects phase out faster in rural areas.
Hotels, restaurants, other private services and cultural activities are to a higher extent concentrated in urban areas.These were very vulnerable to the pandemic crisis following public health interventions closing down such businesses.Dependencies on wage compensation schemes in urban areas may be longer lasting than in rural areas, as crisis effects and public health interventions tend to be more structurally determined with respect to, for instance, sectoral dependence.What particularly trigger the difference in size and curvatures between urban and rural areas are the spatially indirect effects shown in the lower right panel of Figure 7: the rapid decay in rural areas but more persistence in urban areas.This may reflect that urban areas, through agglomeration economies, are more interconnected in production structures across neighbouring municipalities than rural areas.Wage compensation schemes may therefore help businesses seeing problems in their operations in one municipality, which may spillover more strongly to businesses in other municipalities within broader urban areas.This effect subsides more rapidly in rural areas, with less tightly knitted business structures over neighbouring municipalities.

DISCUSSION
The COVID-19 pandemic has led to several policy responses to curb SARS-CoV-2 infection risks.One policy measure comprised public health interventions, for example, lockdowns.Socio-economic policies in terms of wage compensation schemes were instigated to counterbalance adverse local economic effects.Our focus is to assess the effectiveness of these latter compensation policies on local economic outcomes while taking the epidemic-economic complexity into account.Denmark can be seen as a relevant case study here due to the tradition of flexibly adaptable labour market policies, which resulted in considerable space-time variations in the implementation of wage compensation schemes during the COVID-19 crisis at the local municipal level.
We use rich data to track the epidemiological and socio-economic development in Danish municipalities from February 2020 until May 2021 covering the two first waves of the SARS-CoV-2 pandemic.Our specified epidemic-economic system using non-spatial and spatially extended PVAR models considers dynamic space-time interdependencies between SARS-CoV-2 incidence rates, public health interventions affecting local mobility rates, wage compensations and the (skill specific) unemployment rate at the municipal level.
Four important policy findings emerge: (1) wage compensations can alleviate detrimental local labour market outcomes during the COVID-19 crisis; (2) spatial dependence matters for the unemployment rate response to changes in wage compensations; (3) particularly lowskilled unemployed are likely more dependent on on-site work modes and thus benefit from wage compensation policies; and (4) supportive effects of wage compensation schemes to local labour market outcomes are more visible in urban areas.We interpret these results as reflecting the importance of considering intersections of low-skilled jobs, wage compensation policies and agglomeration effects in urban areas.Our findings inform on the importance of policy mixes in future public health and similar crises resulting from 'force majeure' global events for different geographies.We also notice that a shock in the SARS-CoV-2 incidence rate significantly reduces mobility and that an increase in the mobility reduction translates into lower incidence rates, which is plausible in the light of earlier findings (e.g., Caselli et al., 2022;Kosfeld et al., 2021).
More pathways for future research should be considered: While Denmark provides considerable space and time variation in policies designs, questions to be addresses are: Does a higher permanency in (wage compensation) policy implementation introduce stronger issues of moral hazard and will generosity in compensation, hence, matter for policy effectiveness?Answers to these questions call for cross-country comparisons.Providing harmonized and comparable data over a series of countries on elaborate epidemic-economic systems at local levels is challenging but may provide an important extension to the results presented here.From a policy perspective, this is highly relevant as future crises may call for similar actions in terms of re-introducing wage compensations and related shorttime work schemes for mass use (Weber & Yilmaz, 2022).One example of such a policy scenario is the evolving European energy crisis in the wake of the Russian-Ukrainian war.Similar to the COVID-19 pandemic, the energy crisis has heterogeneous effects across sectors (Hutter & Weber, 2022) and European regions, largely reflecting their specific energy dependence, and may thus call for tailored fiscal policy and labour market solutions, as the crisis lingers on.8. Stylized facts of wage compensation developments are also provided in Figures B2 and B3 in Appendix B in the supplemental data online.

Figure 2 .
Figure 2. Spatial and temporal variation in the regional importance of wage compensations: (a) wage compensation jobs as share of labour force (April 2020, %); and (b) change in wage compensation jobs (April 2020-January 2021, %).Source: Data on persons in wage compensation jobs delivered according to the Danish Freedom of Information Act from the Danish Business Authorities.

Figure 3 .
Figure 3. Impulse-response functions (IRFs) for the mobility response to a one unit shock to the municipal SARS-CoV-2 incidence rate (upper) and the response of the SARS-CoV-2 incidence rate to a one unit shock to mobility rates (lower).Note: IRFs are plotted for monthly data.Underlying estimates are based on dynamic fixed effects (DFE) model including time trends (linear, quadratic and cubic), binary dummies for regional lockdowns and municipal fixed effects.95% Confidence intervals are calculated based on Monte Carlo simulations with 200 repetitions.

Figure 4 .
Figure 4. Impulse-response functions (IRFs) for the unemployment rate response to a one unit shock to the municipal SARS-CoV-2 incidence rate (upper), a mobility rate reduction (middle) and the share of wage compensation jobs in the labour force (lower).Note: IRFs are plotted for monthly data.Underlying estimates are based on dynamic fixed effects (DFE) model including time trends (linear, quadratic and cubic), binary dummies for regional lockdowns and municipal fixed effects.95% Confidence intervals are calculated based on Monte Carlo simulations with 200 repetitions.

Figure 5 .
Figure 5.Comparison of impulse-response functions (IRFs) for the unemployment rate response to a one unit shock to the remaining variables of the panel vector autoregressive (PVAR) system for (1) the non-spatial DFE baseline model and (2) the augmented spatial autoregressive (SAR) model.Spatial IRFs are additionally decomposed into a direct and spatially indirect component.Note: See the main text for a definition of the direct and indirect effects in the SAR model.All non-spatial and spatial estimates further include time trends(linear, quadratic and  cubic), binary dummies for regional lockdowns and municipal fixed effects.For ease, only point estimates are displayed, but associated 95% confidence intervals are shown in the Appendix in the supplemental data online as well as the full set of IRFs across the different estimators.

Figure 6 .
Figure 6.Impulse-response functions (IRFs) for the skill-specific unemployment rate response to a one unit shock to wage compensations: (upper left) the unemployment rate response for highly skilled persons; and (upper right) for low-skilled persons.Note: Underlying estimates are based on the spatial autoregressive (SAR) model including time trends (linear, quadratic and cubic), binary dummies for regional lockdowns and municipal fixed effects.The lower panels decompose the skill-specific unemployment rate response into a direct (lower left) and indirect (lower right) component.IRFs are plotted for monthly data.95% Confidence intervals (upper left and right) are calculated based on Monte Carlo simulations with 200 repetitions.For confidence intervals of direct and indirect IRFs, see the Appendix in the supplemental data online.

Table 1 .
Descriptive statistics for epidemic and economic variables in the empirical analysis.