The novel corona virus (SARS -CoV-2), hereafter referred to as pandemic has spread exponentially across the world claiming thousands of lives every day. With no readily available vaccines or pharmaceutical therapies, precautionary measures like lockdowns, maintaining at least six feet of distance between two individuals and restricting respiratory contacts through masks have become the commonly resorted interventions to arrest the infection growth. However, given the highly infectious nature of the virus, the benefits of an individual adhering to the protocols would be negligible if others in the community are reluctant to adopt these protective measures. It follows therefore that ensuring compliance is identical to the problem of collective action where coordinated among communities would add to the stock of public health. In particular, public agencies and health bodies in India initiated widespread informational campaigns on COVID-19 during the pandemic. Community based information networks are likely to have taken cognizance of this information to promote voluntary compliance behavior. The realization of cooperative equilibrium, however, could be particularly challenging in the context of developing countries that are populous and are commonly characterized by fragmentation along social, religious and economic lines.
In such a setting, could an individual residing in a diverse neighborhood be more susceptible to COVID-19 infections as against someone in a more homogeneous setting? What is the nature of the heterogeneities that matter? Are social, religious and economic differences equally important or do some heterogeneities matter more than the others? We pose these questions in the context of India, which remains among the worst hit country in Asia with number of infected cases only second to the United States of America (USA) and deaths just below USA and Brazil. At the point of drafting the paper, close to 8.6 million cases have been reported with a fatality of more than 127,000 (As of November 10, 2020).
Particularly, in this paper, we study the implications of social, religious and economic diversity on the growth in the number of reported infections starting from the beginning of the nationwide lockdown (March 25, 2020) till July 26, 2020. Thereafter this lockdown was implemented in four phases panning for about two months: Phase 1: 25 March-April 14; Phase 2: April 15 to May 3; Phase 3: May 4 to May 17 and Phase 4: May 18 till May 31. Thereafter, the economic unlocking process was initiated and accordingly we consider four additional phases for about two months, Phase 5: June 1 -June 14, Phase 6:
June 15-June 28, Phase 7: June 29 -July 12, Phase 8: July 13 to July 26 for analysis. This allows us to examine the effects on unlocking over the growth in the number of cases and the associated implications of diversity.
The question on diversity and its influence in determining the efficacy of community action against COVID-19 is pertinent due to strong presence of caste and religious stratification in the social architecture in India. Dating back to 1500-500 BCE the caste system comprises of over 4000 social categories making India one of the most genetically diverse countries in the world (Munshi, 2019) . In this paper we define heterogeneity in terms of four broad caste groups. This includes the Upper Caste (Others), the Scheduled Caste, SC (formerly untouchable caste), the ethnic and the tribal group belonging to Scheduled Tribes (ST) and Other Backward Classes (OBC). As (Munshi, 2019) describes, these groups are delineated with distinct spatial segregation that extends over the clusters in terms of information sharing and social identity and are likely to play an important role in protective effort against the spread of infection.
Such segregation is not restricted to caste and social group affinity alone. Ever since the independence of India in 1947 that entailed a violent partition along religious lines, religious conflicts, especially the ones between Hindus and Muslims have also been viewed as a source of social cohesion or fragmentation (Field, Levinson, Pande, & Visaria, 2008) ; (Susewind, 2017) ; (Mitra & Ray, 2014) . While the existing literature stuides the effects of social fragmentation on public good provisioning, the impact of caste and religion based diversity on infection spread remains relatively less explored . Thus, in a country like India where caste and religion are deeply embedded in societal affairs, exploring the impact of diversity in these dimensions on COVID-19 spread could provide important policy pointers for accelerating medical interventions. Notably, in addition to caste and religion heterogeneity which are largely social in nature, we consider economic diversity manifested through inequality in household consumption expenditure and explore its relationship with the spread of the infection. Higher economic inequality is likely to reduce group solidarity and act as a deterrent to collective provisioning (Wade, 1994) ; (Dayton-Johnson, 2000) . In other words, we posit that notions of trust and coordinated community action can go beyond social and religious affiliations and be aligned along economic lines with respect to adherence to the COVID-19 compliance protocols.
The idea that diversity may undermine voluntary cooperation to public good, that in our case is adherence to compliance protocols has been a persistent theme across developing countries (Banerjee, Iyer, & Somanathan, 2005) ; (Habyarimana, Humphreys, Posner, & Weinstein, 2007) .The underlying mechanism that drives the relationship is understood through a number of channels: first is through the preference link where individuals of a particular group care more for in-group members rather than outsiders (Tajfel, Billig, Bundy, & Flament, 1971) ; (Vigdor, 2004) ; (Alesina & Ferrara, Ethnic diversity and economic performance, 2005) . Secondly, homogeneous societies, by virtue of higher social interactions often possess greater levels of social capital, that influences the expected reciprocity from others reflecting higher level of interpersonal trust. Accordingly, higher social ties, and networks ensure homogeneous communities to monitor the violations more efficiently and hence able to impose social sanctions effectively, which can potentially lead to improved collective action (Banerjee, Iyer, & Somanathan, 2005) ; (Miguel & Gugerty, 2005) . Finally, greater diversity is also associated with divergence in preferences and varied risk perception that may lead to sub-optimal level of compliance (Alesina, Baqir, & Easterly, 1999) ; (Munshi & Rosenzweig, 2018) . However, given the highly infectious nature of the virus, it may also be plausible that there exists a possibility of lower inter-personal interaction in diverse communities as a result of which the chances of spread of the virus may be low in more heterogeneous communities (Bosancianu, et al., 2020) . This countervailing possibility turns the potential relationship between diversity and spread of the infection to be an ambiguous one and thus necessitate empirical validation.
To study the relationships of interest, we integrate data from multiple sources. The district-wise data on COVID-19 cases (outcome) for the period of the study, as described previously comes from the Development Data Lab's (DDL) COVID India database (Asher, Tobias, Matsuura, & Novosad, 2020) .
We also construct a range of relevant controls at the district level using multiple recent rounds of the National Sample Survey Organization (NSSO), which arguably are among the most credible official surveys for India. In particular, we examine the changes in the effect size for social, religious and economic homogeneity at the district level for the time period starting from March 25 to July 26, 2020 dividing, this window it into 8 periods, the first four capturing the phases of the nation-wide lockdown and the latter corresponding to the four periods of unlocking.
While estimating the relationship of interest, we can negate the possibility of bias from reverse causation that entails residential self-selection because of the exogenous and unprecedented nature of the pandemic. Instead central challenge in ascertaining a causal relationship comes from the possibility of endogeneity that may stem from Omitted Variable Bias (OVB). 1 Despite controlling for various confounders, it is possible that we are unable to account for unobservables that are simultaneously correlated with both the variable of interest and the outcomes, thereby inducing bias in our findings. To account for such possibilities, we undertake the following measures. First, we control for the COVID-19 cases reported in a district before the first phase of the lockdown (25 th March 2020). This is critical to address the selection on unobservables as it subsumes the cumulative influence of initial hotspots, superior international connectivity, and integration of certain districts into the global supply chain among others, all of which are likely to be associated with residential patterns that may shape measures of 1 Given that COVID-19 was an exogenous shock, endogeneity via reverse causality is less of a concern. heterogeneities at the district level. For example, note that the COVID-19 virus did not originate in India 2 , but entered the country through districts/cities that facilitate international travel and handle large volumes of passengers, especially from China, Europe and the Middle East. 3 These districts are also likely to be better integrated with global supply chains and thereby be favored employment destinations across social, religious and economic lines. Such factors may result in higher or lower heterogeneity depending on socio-economic or socio-religious mobility. Such an effect would confound with the relationship of interest thereby leading to biased estimates. Second, to account for the dynamic supply side responses for each phase of the lockdown, we control for number of deaths in the district for the previous period in all the regressions, in addition to the district level physical and human health infrastructure. Here, a plausible assumption is that the district and other levels of administration are likely to be more sensitive to the number of COVID-19 deaths in the previous period when formulating or reviewing a policy response.
Third, for all the regressions, we take intra-state, region fixed effects to account for any region level unobserved characteristics that might be correlated with both the variables of interest and the outcome.
Specifically, this variable along with the time/ period dummies is likely to control for the variation in COVID-19 testing that is likely to be similar within but vary across these regions with time. Finally, we use the strategy by (Altonji, Elder, & Taber, 2005) and then (Oster, 2019) to examine the bounds of our coefficients with a sizeable and standard assumption of selection on unobservables. These steps ensure that the influence of unobservable confounders are accounted for and the coefficients of homogeneity derived from the main regression estimations can be viewed from a robust causal lens, which aids devising important policy instruments.
Our findings indicate, ceteris paribus, districts with higher caste-group homogeneity are likely to experience lower growth in number of COVID-19 infections on average. The relationship becomes stronger through extended phases of the lockdown but weakens with the beginning of the unlocking process. The overall relationship remains statistically significant for the first month of unlocking and dissipates thereafter. We also find a modest but positive effect of economic equality on spread of infection in the first three phases of the lockdown thereby underscoring the importance of social ties based on economic lines. We find no evidence of a significant relationship with religious homogeneity in a similar context. These results are statistically valid for a range of robustness checks. We also examine the implications of the unlocking procedure on the relationships of interest. Our findings indicate a significant influence of the unlocking process, with the gains of caste-group homogeneity attenuating 2 The virus first appeared in Wuhan, China with the Chinese authorities confirming the possibility of human to human transmission in late January https://www.nytimes.com/2020/01/20/world/asia/coronavirus-chinasymptoms.html 3 The first few cases in India were recorded in the state of Kerala along with the cities of Delhi and Mumbai among others with high international passengers and transnational migrants once the lockdown was lifted. Interestingly, for the districts that were predominantly rural, we find that these gains increased in the unlocking phase. This heterogeneous finding potentially underscores the importance of social cooperation and ties which is likely to be much stronger in rural areas in comparison to the more urbanized ones.
Our main contribution through the paper is two-fold. One, we contribute to the growing literature on the relationship between social fragmentation and provisioning of public goods. Importantly, we evidence this relationship in context of a unique public good that is tied to public health and compliance with COVID-19 protocols. This has far reaching implications for preserving both lives and livelihoods in context of post pandemic world order. Second, from a policy perspective, our findings would imply identifying socially heterogeneous geographical pockets as potential hotspots and ensuring early administrative allocation of scarce medical facilities and services in these pockets, until the supply of these can be scaled up. Also, our findings underscore the importance of social protection and economic support, especially during lockdowns to ensure that the gains made in arresting the growth of infection during lockdown remains intact even after the initiation of unlocking the economy. Importantly, our research underlines the importance of social cohesion and community ties and builds a case against divisive political agendas that deepens the existing fissures across communities. Instead, our work outlines the potential gains from promoting harmony across different social groups, especially in the fragmented pockets.
The structure of this paper is as follows. Section 2 introduces the data used and presents summary statistics. Section 3 discusses the empirical framework by outlining the estimation and the identification strategy. Section 4 discusses the results alongside some robustness checks. Section 5 concludes with a summary of findings, limitations of the study and its policy implications.
District-day-level data on COVID-19 cases and deaths comes from the Development Data Lab's (DDL) COVID India database, which is available for 719 of 733 (98%) of the total districts in India. 4 The period of analysis ranges from 25 th March 2020 to 26 th July 2020 and covers the period of nation-wide lockdown, which ended on the 31 st of May 2020, as well as the subsequent two month period of unlocking. 5 Figure 1 which provides a spatial view of spread of COVID-19 infections in India through 4 The Socioeconomic High-resolution Rural-Urban Geographic Platform for India (SHRUG) is an open source platform to facilitate data sharing between researchers working on India. This may be accessed at: http://www.devdatalab.org/shrug. These numbers are taken from covidindia.com, a crowd sourced project. 5 GoI announced nationwide lockdown in four continuous phases starting from the 25 th of March that lasted until the end of 31 st May with detailed guidelines issued on measures such as social distancing and hygiene practices. subsequent phases of the lockdown indicates shrinking of the green area, which represents low infection band. Post lockdown we observe a rising number of reported cases as indicated in Table 1 . Thereafter, the respective state governments were given more freedom to decide on further restrictions outside a countrywide negative list that for example included international travel and opening of educational institutions https://timesofindia.indiatimes.com/india/lockdown-4-0-after-may-31-states-to-have-own-curbslists/articleshow/76024524.cms To assess district level social, religious and economic diversity, we use the Herfindahl Hirschman Index (HHI) 6 (Rhoades, 1993) and the Gini Index (GI), respectively. As discussed previously, for caste diversity, we use the broad caste groups that includes the historically disadvantaged groups of SCs, STs and OBCs in addition to the privileged castes aggregated under "Others". 7 For religion, representation across 8 major religious groups is accounted for. 8 The HHI varies on a scale of 0 to 1 with a higher value implying increased homogeneity or concentration. GI is based on monthly household consumption expenditure, which is also computed on a scale of 0 to 1 with a higher value implying higher inequality. 9 Figure 2 gives a spatial overview of the social group, religion and economic diversity across India districts. For analysis, the variables of interest are taken as standardized Z-scores, which are standard deviation units of difference from all-India means. 10 Data for this comes from the 76 th round of the National Sample Survey Organization (NSSO) on drinking water, sanitation, hygiene, and housing 6 The HHI index is a measure of size of firm with respect to the industry and is typically used as a measure of competition with a higher value implying higher monopolization. 7 The social networks within the Indian caste system are splintered into various sub-caste or jatis. In this paper, we restrict ourselves to the broad caste groupings as district level representative data is only available for this level for period as recent as 2018. 8 These include Hindu, Muslim, Christian, Sikh, Jain, Buddhist, Zoroastrian and Others. 9 We assume that intra-household as against inter-household coordination and cooperation is relatively much easier to achieve. Thus, in this paper we compute the variables of interest (HHI and GI) using data at the household level, which is then aggregated at the district level using weights. We do however check for the robustness of our findings using individual level variables which are qualitatively similar.
conditions collected between July-December 2018. 11 This is the latest available data, to the best of our knowledge that would provide representative estimates of the socio-economic profile at the district level by using appropriate sampling weights. Notably, data from NSSO is often used to estimate prevalence of poverty, and unemployment by the Government of India along with generating estimates on educational, agricultural and health indicators among others, which gives credence to our main variables of interest as well as the control ones.
We also control for a range of socio-economic, demographic and mobility controls from the 76 th round of the NSSO survey. This includes share of households with access to basic amenities such as improved latrines and water sources as per the WHO guidelines. Access to these amenities is likely to facilitate reduced movement and exposure to infections from others in the vicinity. As proximity to other households is critical in achieving compliance with social distancing measures, we control for density of population 12 as well as share of households in the district living in slum or squatter settlements. Given the importance of hygiene, especially handwashing habits in preventing the spread of the COVID-19 infection, we account for share of households with human faeces spotted around premises during the survey, as well as share of households whose members wash hands regularly with soap after defecating. 13 Emerging literature suggests that the COVID-19 pandemic has an inherent urban character with a critical role of cities in in disease transmission (Mishra, Gayen, & Haque, 2020) . To account for this, share of households in the district that reside in urban areas is controlled for. Given the vulnerability of the elderly population (>=60 years) to infections (Daoust, 2020) , share of elderly population is also accounted for.
For economic profile, share of households with at least a member with a graduate degree and regular employment are considered. We control for mobility in various ways, given it is a critical channel for transmission: firstly we account for the family mobility trend by including the share of households in the district with current place of residence being at least 10 years. Secondly, we check for the high frequency mobility pattern by including the share of households where someone travels for more than 15 kilometres (km) for work.
Additionally, to account for health profile of the district, information on share of population in a district with chronic ailment is taken from the 75 th NSSO round on Health Consumption. 14 For availability of health infrastructure, both physical and human, number of hospitals as well as number of doctors per million households in a district is considered. This is taken from the DDL COVID India database that sources this information from the Population Census (2011) and the Socio-Economic Caste Census (2012) (Asher, Tobias, Matsuura, & Novosad, 2020) . The above list of controls is available for 670 of the 733 (91%) districts of India and we are able to map these variables with information on COVID-19 cases and deaths for 661 of the 719 (92%) districts.
To explore the relationship between diversity and spread of COVID-19 infection, we first present a scatter of social group homogeneity and the natural logarithm of cases across the eight periods considered for the analysis. These periods comprise of the four phases each of lockdown and unlocking, respectively. We also use the administrative data from Government of India (GoI) notification dated 30 th April 2020, issued by the Ministry of Health and Family Welfare (MoHFW). As per the notification, districts were classified into red, orange and green zones (dangerous, mild, no apparent danger respectively) for the week following the end of second phase of lockdown (3 rd May 2020) using a multifactorial approach. 15 Here, the movement of the districts coded as green across the distribution of HHI for different phases suggests that the districts at the lower end of the distribution (lower social group homogeneity) moved further to the right as compared to the green districts at the upper end of the distribution, which were relatively more homogeneous (Figure 3 ).The trend is similar for consumption inequality with cases appearing to rise faster for green coded districts with higher economic inequality. 16 15 This approach accounted for number of cases, extent of testing, surveillance feedback and doubling rates in respective districts. As per this criterion, a district was classified under green zone if there were no confirmed cases of COVID-19 in the past 21 days. No subsequent notifications with such categorizations were disseminated by the Ministry to the best of our knowledge. 16 The scatter plots for GI from household consumption inequality are available on request.
Note: Author's calculations We now turn to a discussion on descriptive statistics and provide summary measures of outcomes, variables of interest and other control variables. It must be noted that across the districts of India, close to 85% of the districts had no COVID-19 cases reported until the beginning of the lockdown (25 th March 2020). For each of the four phases of the lockdown, which were considered to be among the most stringent across the world, 17 the share of districts with no cases reduced to 47%, 31%, 26% and 15 % respectively. The average number of cases (deaths) per district in this duration increased from about 4 (0) before the first phase of the lockdown to 2669 (80) for the period ending 31 st of May 2020. For the last period of analysis ending 26 th July 2020, the average cases and deaths were as high as 21,279 and 537 respectively (Table 1) .
Among the variables of interest, given the vast majority of the country is Hindu, the degree of religious homogeneity is significantly larger than social group homogeneity, on average. Religious homogeneity is found to be the highest in districts of Odisha, Uttarakhand, and Gujarat with 11, six and five districts featuring in the top decile of the distribution. For social group homogeneity, the states of To further understand how the outcome of interest varies with measures of homogeneity, we use the kernel weighted local polynomial smoothened plots with 95% confidence interval that makes no restrictive distributional assumptions on the error term (Fan & Gijbels, 1996) . In Figure 4 , for the duration of the analysis, we find that the rate of growth of COVID-19 cases falls as social group homogeneity rises along the x-axis. The growth of infections also appears to be rising with higher income inequality as measured by higher values of the Gini coefficient along the x-axis. As this analysis is devoid of potential confounders, we now turn to a regression-based approach to validate the relationships of interest. For this, we first present the empirical framework and then present the corresponding findings over the next two sections. Next as discussed, we divide the time since the commencement of the first lockdown (March 25, 2020) till July 26, 2020 into 8 periods each consisting of about 15 days. 19 The first four periods correspond to the four lockdown phases and the next four represent the unlocking phases. To examine the implications of homogeneity across these time periods, we estimate the following pooled OLS regression equation:
18 The National Sample Survey Organization divides all the districts into a number of mutually exclusive and exhaustive administrative regions, which are used as fixed effects in our regressions. 19 Initial phases (1 and 2) were longer as lockdown phase lasted more than 2 weeks in the beginning but later was extended by intervals of 2 weeks.
Here denotes the periods dummies. ̅ represents 1 added to the number of reported cases in period, and ̅ is the reported number of deaths till the lagged period, in district, d. To assess the relationship between homogeneity and the changes in infections over time, we examine the coefficients, .
Our central hypothesis links caste and religious homogeneity, alongside lower economic inequality with a potentially slower growth of infection in a spatial unit. The underlying pathway for this, as argued is that of within group cooperation and trust, which facilitates coordinated community action to tackle the spread of a pandemic. To establish causality of homogeneity or inequality to the number of reported infection, we need to account for the potential endogeneity, the two main concerns of which are reverse causality and the Omitted Variable Bias (OVB).
In the context of our study, reverse causality is of negligible concern. Given that COVID-19 induced lockdown in India was an unanticipated shock and introduced with just a four-hour notice, the chances of residential clustering on social, religious and economic lines across districts as a response to the outbreak would be improbable. This sounds especially true when studies have documented the low levels of inter-district or inter-state migration in India even during the non-COVID times (Munshi and Rosenzweig 2009; Munshi 2016 ; Rowchowdhury 2019). Post lockdown, although mobility increased, it is still likely that this was s considerably lower than that during the pre-COVID period. Hence, the chances of reverse causality restricting us to view the estimates from causal lens are almost negligible.
To control for the OVB, we introduce a host of control variables that range from district level economic conditions and urbanization to sanitation and hygiene practice along with health indicators and availability of physical infrastructure and health workers among many others. Note that the initial hotspots where the chances of the spread of infection remain high have been controlled for through the number of cases reported across districts before the initiation of the lockdown. This in our view is critical to address the issue of endogeneity as it controls for various characteristics of districts that became epicenters of COVID-19 epidemic in India. For example, the major metropolitan cities of Delhi, Mumbai, Kolkata and Chennai accounted for more than half the total tally of cases 20 and are also major centers for urban jobs in India. The time variant potential health measures in response to COVID-19 deaths have been controlled through the district level number of deaths in the lagged period. Given the administrative structure of India where each district is under the stewardship of a bureaucrat (district collector) that directly reports to the Chief Minister of a state, allocation of scarce administrative resources in a pandemic is likely to be most sensitive to COVID-19 related casualties. In addition, the region fixed effect are introduced to control for all the time invariant characteristics that may be common to all the districts lying within the respective regions.
Despite these elaborate set of control variables, one may still argue some potential confounders getting omitted from the regressions, which can yield biased estimates. To ensure this is not the case, we examine the potential variation in the estimates after accounting for the OVB through a strategy developed by (Oster, 2019) based on (Altonji, Elder, & Taber, 2005 )that rests on the assumption that selection on unobservable variables can be measured through the extent of selection on observables. For further elaboration, consider the following regression equation:
Here is the dependent variable, denotes the primary variable of interest, is the vector of control variables and is the vector of all unobserved components that was omitted from the regression. Our aim is to examine the corresponding changes in or whether it changes its sign when is considerably high.
Here the primary assumption is:
The above equation indicates that the relationship between and the unobserved component, is proportional to the correlation between and the observed component ( and the degree of proportionality is given by . With this, (Oster, 2019) show that it is possible to derive a possible range of through two parameters: and . is the R-squared value of a hypothetical regression that incorporates all the observable along with the unobserved components and hence is the maximum value of R-square it can achieve. While there can be various options to compute the bound on , using data from a wide sample of randomized experiments, (Oster, 2019) 
We first present the estimates of the regression of the natural logarithmic value of the reported cases on the three different indices of homogeneity as outlined in equation (1) in Table 2 . To check the level of stability of the coefficient of interest, we use four different specifications. In the first specification, apart from the variables of interest, we incorporate natural log of cases reported prior to the initiation of lockdown in the district and number of deaths in the previous period alongside regional specific fixed effects. To this, in specification 2, we add indicators on access to amenities such as improved toilet and primary source of water. Further, we also add variables on share of households (HH) in the district that reside in slums/squatter settlements and general hygiene and cleanliness practices around their premises.
In specification 3, we subsequently control for the economic and demographic variables. In this specification, in addition we account for share of individuals in the district reporting chronic ailments. In the last specification, we control for the mobility indicators as specified previously alongside average density of population in the district as well as the existing physical and health infrastructure of the district.
Because it controls for a host of observable covariates spanning from economic to demographic characteristics along with sanitary and health indicators and also the supply side response to the outbreak, this is our preferred specification, which is used in the subsequent regressions as well.
The results appear to be quite stable across specifications with findings from the final specification (column 4) suggesting that a one standard deviation increase in homogeneity across caste groups, ceteris-paribus, is associated with a decrease in the logarithmic value of the reported cases by about 0.18 on an average. The results for income GINI are qualitatively similar in the sense that an increase in inequality is related to higher number of cases albeit with a lower magnitude with an average effect size of about 0.13 for every standard deviation increase, thereby underscoring the importance of equity. Notably, the bearing on religious homogeneity remains statistically indistinguishable from zero.
The findings from other set of covariates indicate expected results as we observe that the number of reported cases and deaths in the pre-lockdown phase to be significantly associated with the current number of reported cases. Next, we run a pooled OLS regression using the number of district level reported cases at the end of each of the 8 periods we considered through equation (2) Note: The coefficients come from the full control scheme, as described in model 2 above with confidence bands of 95%. The vertical line at x=4 signifies end of nationwide lockdown (31 st May 2020). To ensure that districts with no cases in any period do not drop out of the analysis, unity was added to cases before taking natural log. Results robust to adding a smaller positive number. Results robust with three-and seven-day rolling averages replacing daily reported cases for each period. Detailed tables available on request.
How effective is our variables of interest in arresting the growth in the reported cases of COVID-19 infection? Does diversity in terms of caste groups or in economic lines lead to higher number of cases?
As argued earlier, we include a host of control covariates in our regression that account for most of the variations in the number of cases across districts. The information about the number of tests carried out at the district level is unavailable but the region fixed effects along with the period dummies have been incorporated in the model. To the extent that there are no systematic changes in the testing across time for districts within a region, the fixed effects would account for large part of the effect through differential testing numbers. Note that the associated R-square value from regressions corresponding with figure 5 is observed to be 0.75, which indicates 75% of the variation in the dependent variable has already been accounted for through the set of control variables used in the regression. To that extent, we argue that we can view the inferences drawn through causal lens.
Nevertheless, because one may still argue about some omitted variables biasing our estimations, we use strategy developed by (Oster, 2019) based on (Altonji, Elder, & Taber, 2005 )that estimates the bounds of the main coefficient of interest assuming that the extent of selection on unobservables can be measured through the selection on observables as explained earlier. The results from this exercise are presented in Table 3 . For interaction coefficients which are found to be statistically significant at 10% level or less, the needs to be well over 1 to bring them to zero (Oster, 2019) . This, as argued earlier is unlikely especially since we are already controlling for a host of covariates, which potential accounts for many unobservable factors as well. However, even if we assume the extent of unobservable characteristics confounding the dependent variables is proportional to that which is explained through the observed covariates, the significant effects of homogeneity with regards to caste and also that related to economic inequality through Gini are preserved and do not wither away. 218 Note: Authors' calculations. *** p-value < 0.01, ** p-value < 0.05, * p-value <0.10. The uncontrolled coefficient is from regression of outcome exclusively on variable of interest. Controlled coefficients are from the hypothetical regression with and where is from the final regression with all observed variables. Results obtained via user written Stata command psacalc (Oster, 2016) .
The findings, we argue have considerable implications in terms of identifying the target group for scaling up compliance inducing actions. Among all measures of social cohesion, our findings show, caste heterogeneity remains persistently negative predictor of infection spread during lockdowns relative to religion and income. Further, impact of higher caste homogeneity seems to get multiplied with each subsequent phase of the lockdown. 22 Its only after the lockdown was lifted did the gains from caste group cohesion start declining. Here one may argue that because of the continuous stretch of nationwide lockdowns spanning for more than two months, despite collective benefits, the associated private costs due to disruptions in livelihood and declining economic well-being led to weakening of the social ties among people of similar caste groups (Ahmed et al. 2020; Lancet, 2020) . Incidentally, studies have indicated a decline in private compliance to COVID-9 non-pharmaceutical protocols like maintaining social distancing and wearing facemasks over time possibly due to fatigue and economic costs (Briscese et. al., 2020; Das et al., 2020) .To complement our findings using period-wise data, we also run pooled OLS regressions taking daily data on the number of infection cases. Here instead of period dummies, we incorporate daily dummies and examine the coefficients on the daily interactions with our main variables 22 According to the Blavatnik School of Governance study, India is found to have the most stringent lockdown among all the major countries in the world (https://www.bsg.ox.ac.uk/research/research-projects/coronavirusgovernment-response-tracker) (accessed on September 30, 2020) of interest. Figure 6 presents the estimations from the regression. The findings indicate similar inferences that we obtained earlier. The gains from caste homogeneity seems to increase over each day of the lockdown on an average, but post lockdown the effect size is found to get reduced over time. Economic equality is also found to bear some discernible gains in the initial phases of the lockdown but the effect seems to be withered with time. 
We run a battery of robustness checks to ensure our inferences are robust.
First, we consider Shannon's entropy measure (SEM) of diversity instead of the HHI indices for caste group and religious homogeneity. This value of this measure ranges from zero to infinity with a higher value implying greater diversity (Frenken, 2004) . Mathematically, this can be presented as ∑
where is the share of each group in overall population which can be divided into mutually exclusive groups. The results for this are presented in Figure 7 . Unlike the HHI that ranges between 0 and 1 with higher value implying higher homogeneity, SEM ranges from 0 to infinity with higher values representing greater diversity. Thus, although the HHI and SEM are inversely related, the coefficients are not directly comparable. However, the results suggest the relationships of interest are robust to the measure of diversity. Note that in some of the districts, many of the groups may not be present which would make their share in the population to be zero. As a result, these districts drop out of the analysis. To validate that this is not driving our findings, we re-estimate the model using HHI of heterogeneity for the sub-set of districts for which the SEM could be computed. We are able to ascertain that the relationship of interest is robust and the findings can be provided on request. Note: The coefficients come from the full control scheme, as described in model 2 with confidence bands of 95%. The vertical line at x=4 signifies end of nationwide lockdown (31 st May 2020). To ensure that districts with no cases in any period do not drop out of the analysis, unity was added to cases before taking natural log. Results are robust to adding a smaller positive number. Results are robust with three-and seven-day rolling averages replacing daily reported cases for each period. Detailed tables are available on request.
Data indicates that the bigger cities are clusters with higher economic activity and larger in-flow of migrants. Thus, they could potentially drive our results. Though we adequately controlled for these characteristics in our specifications, as robustness check, we remove the districts that contain the capital of a state and re-estimate the model with non-capital districts across all the Indian states. This is because it is likely that these districts with the state capitals are likely to be the major hotspots within a state. Figure 8 which presents the findings indicate that we are able to replicate our results with such this subsample of districts as well. Note: The coefficients come from the full control scheme, as described in model 2 with confidence bands of 95%. The vertical line at x=4 signifies end of nationwide lockdown (31 st May 2020). To ensure that districts with no cases in any period do not drop out of the analysis, unity was added to cases before taking natural log. Results are robust to adding a smaller positive number. Results are robust with three-and seven-day rolling averages replacing daily reported cases for each period. Detailed tables are available on request.
We argue social cohesion on caste lines and to a lesser extent economic equality, among other factors can be the key to arrest growth in COVID-19 infection cases over time. This is evident especially during the lockdown phases as we find increasing gains of homogeneity across caste over time and higher returns of economic equality at least during the early phases of the government mandated nationwide lockdown. As the unlocking commenced after May 31, 2020, figure 4 indicate that no discernible gain from economic equality or even if there is a gain, which is statistically distinguishable from zero, the effect size is lower.
So, is it the case government unlocking of the economy after the mandated lockdown played a role in altering the influence of homogeneity on caste lines or that of economic equality? Did the returns from lower diversity erode with unlocking of the economy?
We examine this using the following regression model:
̅ ̅ This looks same as equation (2) with the only difference being introduction of an unlock dummy instead of period dummies. The "unlock" dummy takes the value of 1 for period 5 to 8 and 0 for the periods 1 to 4. Note that as in the earlier case, this regression controls for the number of fatalities in the lagged period along with the other confounders as included earlier.
The findings from the regression, presented in table 4 suggest a positive and significant rise in the cases post unlocking in areas with higher caste homogeneity. In other words, the marginal gains of higher caste homogeneity through lower infection cases seem to get diluted once the unlocking process of the economy started and it led to about a 0.61 increase in the logarithmic value of the number of reported cases with each additional standard deviation increase in caste homogeneity. Please note that the relationship between this homogeneity and the reported cases remains negative and statistically significant even during the initial phases of the unlocking period. The positive sign on the interaction term just indicates that the influence of the social cohesion working through caste get reduced significantly after the initial of the unlock process from the government.
Interestingly, this relationship does not seem to hold for religious homogeneity or economic inequality as we observe their corresponding interaction coefficients to be statistically insignificant. The larger implication of the findings as indicated earlier indicates that the social cohesion for caste seem to have worked with a mandatory deterrence measure of nationwide stringent lockdown. Nevertheless with the unlocking process, this cohesion possibly weakened potentially in the wake of massive economic disruptions which led a rise in cases even in areas with higher homogeneity, though this would need further research to pin point the exact reasons Nonetheless, the implications of caste homogeneity in areas which are more likely to be associated with higher social ties and coordination with stronger inter-personal relationship are likely to offer some insights into isolating the economic considerations from social cohesion. Since rural India is more likely to be characterized by these features, we examine if the gains through caste homogeneity hold in the less urbanized districts in the post unlocking period (Banerjee B. , 1986) . To give evidence on this, we look at the differential effects of unlocking in areas with varying levels of urbanization. To understand this, we create a dummy variable, , which takes the value of 1 for districts with less than percent population living in urban areas and 0 for those with or more than k percent urban population. We use this in our regression, given by the following:
Here is our coefficient of interests. hypothesized. In areas with low urbanization, the marginal gains from higher caste homogeneity seem to not only remain intact even after the unlock process but also show positive effects in reducing the growth in reported COVID-19 cases, even after lockdown. Interestingly, for districts with more than 90% and even 80% rural population, the implications of caste homogeneity on the unlocking initiation is found to bear a significant and negative relationship with the number of reported cases with the effect size being larger for the former than the latter. This underscores the importance of higher social cohesion and interpersonal ties. Coordination action through strong caste networks is likely to play an important role in ensuring low growth in infections. This is observed to be true even during the period when the unlocking process was initiated and the potential mobility increases along with the likelihood of complying with the pandemic protocols like social distancing among others. Note: The coefficients come from the full control scheme, as described in model 4 with confidence bands of 95%. Unlocking dummy takes value 0 if data from period before end of nationwide lockdown and takes value 1 if after unlocking.
The key findings of the paper can be summarized via three important results. First, we find that castegroup homogeneity has a significant influence in curtailing the spread of COVID-19 spread in India.
Religious fragmentation, however has a limited role. In addition, we find that income inequality exerts a negative influence on growth of COVID-19 cases, which tend to rise faster in more unequal districts.
Accordingly, our results emphasize the importance of strong community cohesion mainly working through caste-groups and to a lesser through economic affinity in ensuring socially beneficial compliance.
A major challenge to sustain the community norms arises from the economic hardship due to subsequent lockdown phases. These are likely to intensify economic needs to violate compliance norms. Evidently, the gains through cohesion seem to attenuate after easing of these phases of mandatory lockdown. In this context, firstly the paper emphasizes the importance of robust social protection and redistributive measures, which are a vital link that needs to strengthened during the pandemic, not only as a step to mitigate the economic hardship but also to arrest the spread of the infection through sustained community coordination.
Finally, the effect from caste-group homogeneity is found to be more prominent in areas that are less urbanized and that this effect continues to operate even after mobility restrictions are withdrawn. It is our surmise that the result is suggestive of the fact that relative importance of different dimension of social fragmentation varies with the degree of urbanization. In districts that are less urbanized, community cohesion is likely to be dominated by social identity. People in rural areas have a relatively restricted social network but this network entails strong ties relative to urbanized households that have more contacts but weaker social ties (Sato & Zenou, 2015) . In other words, urban interactions are more frequent but ties are voluntary in nature, which in rural areas are mostly based on social norm and kinship (White & Guest, 2003) . Studies have also found that people living in urban areas are less likely to participate in collective action (Bovaird, Van Ryzin, Loeffler, & Parrado, 2014) . We believe among other factors, the nature of interaction and community structure in the rural areas are the enabling reasons for a stronger relationship between homogeneity and spread of infections through better social cohesion.
Importantly we still observe that community effect continues to display significant resilience in rural areas even in the unlock period when movement restrictions are partially lifted albeit its effect is found to be disappear in more urbanized areas. Given the fact, that COVID-19 cases continue to spread in India these possibilities open up a number of policy avenues for health interventions.
First, government should prioritize socially heterogeneous areas relative to homogeneous ones as they might be more vulnerable and possess lower social capital to tackle infection growth. Our findings provide a basis for identifying and zoning of local areas into socially homogenous and heterogeneous blocks. The heterogeneous blocks can be prioritized for medical intervention as well as income support to compensate for the already weak community cohesion as a short and medium response to pandemic.
Secondly, in the relatively homogenous areas, as existing social ties are getting weakened with onset of gradual opening up of the economy strategies to strengthen community networks might be undertaken.
One possible channel could be decentralizing health interventions through community channels involving greater community participation. Research indicates that positive perception about the community is found to enhance private compliance behavior with respect to the pandemic (Das et al. 2020 ). Here targeted intervention and educational messages catering to the relevance of the community through workers belonging from the same community can prove to be important. The effectiveness of these policies, if implemented would then be an important area of future research.
Curtailing growth in infection requires strong social cohesion and ties within a community, something that might be difficult in a diverse and heterogenous context. Such a setting places obstacles in realization of trust and cooperation that is required for coordinated community action during pandemics. Using high frequency data on COVID-19 infection at the district level, we find that social homogeneity through caste to play a significant role in ensuring lower growth in COVID-19 cases in the Indian context. During the nationwide lockdown, social cohesion played an important role in arresting the growth of infection. However, the marginal gains seem to dampen after the initiation of the unlocking process that potentially led to higher mobility. Importantly community effect continues to display significant resilience in rural areas even in the unlocked period albeit its effect is found to disappear in more urbanized areas. As we have accounted for a wide range of potential confounders, we have reasons to believe that our estimates reveal the impact of community cohesion on infection spread via coordinated compliance. This becomes even more robust after we account for potential selection on unobservables, which might nullify the positive effects of caste-group homogeneity. Our analysis suggest that caste-wise heterogeneous areas and to some extent economically unequal areas are most likely to be vulnerable to infection growth and government must prioritize a response to such areas while expanding its emergency capacity. In areas which are relatively homogenous, we stress on community strengthening efforts.
Despite our robust findings, the exact mechanisms through which caste-based heterogeneity becomes important in the context of in India remain unexplored in our paper. We propose this as an important agenda for future research.