Detailed regression results can be found in Table 1 and Supplementary Table 2; Figures 3, 4 and 5 present the main estimates graphically.

Highly correlated unobservable variables influence both government policy and confirmed case and death growth rates The estimated covariances (and associated correlations) between the national-level latent variables in Eqs. 1, 2, and 4 were all positive and significantly different from zero at the 0.001% level: (rho _{lambda ,12})=0.98 [95% CI: 0.95,1.00], (rho _{lambda ,14})=0.66 [0.54,0.79], (rho _{lambda ,24})=0.68 [0.56,0.80]. This validates the hypothesis that significant unobservable variables may simultaneously influence case and death growth rates and the severity of government policies. Ignoring these relationships by estimating a single regression of case or death growth rates, as is commonly done in the literature, would have led to biased estimates.

Adaptive expectations of the risk of infection impact nonresidential mobility Citizens reacted to increases in the seven-day lag in growth rates of confirmed cases by reducing mobility. This is consistent with a theory of citizens forming adaptive expectations about the severity of the pandemic at any point in time and the risk of contraction based on the data. The impact of the daily growth in confirmed cases is (hat{gamma }_{2}=-0.214) [-0.255,-0.174]. Putting the effect size into perspective, an increase in the case rate of 1 percentage point would result on average in a relatively weak reduction in mobility of 0.21 percentage points compared to the baseline.

The indirect links between NPIs and confirmed case/death growth rates The indirect links between NPI mandates and confirmed case/death growth rates consist of two components: for cases, ((SIrightarrow M)) and ((Mrightarrow dot{C})), and for deaths, ((SIrightarrow M)) and ((Mrightarrow dot{D})). Below, I will present evidence that nonresidential mobility mediated the relationship between NPI stringency and case/death growth rates. I begin by reporting the evidence regarding the (SIrightarrow M) component, which is common to both instances.

Higher NPI severity restricts nonresidential mobility Nonresidential mobility is clearly impacted by the SI (see the estimates of (gamma _{l,t}) in Table 1 and Figure 3), as the null hypothesis that all SI dummy variable coefficients are equal to zero was rejected ((chi ^{2}left( 10right) =428.38,p<0.0001)). Furthermore, the absolute value of the effect size was monotonically increasing in the 10 SI ranges, i.e., higher stringency led to reduced mobility.

Lower nonresidential mobility increases case and death growth rates The theoretical motivation behind restricting mobility through lockdown measures is that restricting it will reduce the number of social interactions, thereby reducing transmission and infection. However, a reduction in nonresidential mobility necessarily entails an increase in the time spent within residences—indeed, these two effects are highly negatively correlated in the Google mobility data, where the median correlation within panels is -0.89. Consequently, reductions in nonresidential mobility may have a detrimental effect on case/death growth rates; although transmission outside the home may be reduced, within-household transmission may be enhanced as people spend increasingly more time with cohabitants in the confines of a closed space [41]. Since the two effects work in opposite directions, the question of whether restricting nonresidential mobility reduces case/death growth rates must be resolved empirically. The finding that the impact of (M_{i,t-7}) was significantly negative for cases, (hat{beta }_{1}^{c}=-0.0417) (left( left[ -0.0578,-0.0256right] ,p<0.001right)), and of (M_{i,t-21}) was significantly negative for deaths (hat{beta }_{1}^{d}=-0.0162) (left( left[ -0.03,-0.002right] ,p=0.025right)), supports the hypothesis that the benefits of reduced nonresidential mobility are more than outweighed by the detrimental effects of increased within-household transmission (conditioned on the stringency of government policies). The effect size is moderate, as a 10 percentage point decrease in mobility leads to an increase of 0.4 percentage points in the case growth rate.

The direct link between SI and case/death growth rates The null hypothesis that NPI mandates had no direct impact on case growth rates was rejected, as the set of (beta _{2,l}^{c}) estimates are all equal not significantly different from zero ((chi ^{2}left( 10right) =137.28,p<0.0001)). Furthermore, there was evidence of a nonlinear relationship between NPI stringency and case/death growth rates, with increasingly stringent restrictions offering strongly decreasing returns (see Table 1; the conclusions are similar for death growth rates).

The total effect of NPI stringency on case/death growth rates and the optimal level of stringency The total effect of the stringency of NPI mandates on case growth rates can be computed by adding the direct path (beta _{2,l}) and the indirect path (gamma _{1,l}times beta _{1}) for each stringency level of l. This combined effect of stringency on (dot{C}_{i,t}) and (dot{D}_{i,t}) is documented in Supplementary Tables 2 and 4, and presented graphically in Figure 4.

Total effect of the Stringency Index level

The maximum impact of NPI stringency on case growth is observed for the SI range of 61–70. I tested the difference in effectiveness of all other levels against the most effective range, 61–70, correcting for multiple comparisons using the Sidak correction.^{Footnote 2} There was no difference in effectiveness for the ranges 51–60 and 71–80 compared to 61–70 (see Supplementary Table 3 for the test statistics). The ranges 81–90 and 91–100 were significantly less effective than the range 61–70. Consequently, there are no further gains to be achieved beyond the SI range of 51–60. The socially optimal SI range, however, must account not only for the positive effects of NPIs, but also for the significant impact on physical and mental health [42,43,44,45,46,47,48] and economic costs that result from restrictions (for example, see [5,6,7]). While this would require a full cost–benefit analysis [49,50,51,52] that is beyond the scope of this paper, it is possible to derive the approximate upper bound of the socially optimal SI level with a single assumption about the cost profiles of different SI levels: that the costs are monotonically increasing in the SI level. Consequently, without the need to quantify costs, I conclude that the upper bound of the socially optimal SI level, SI*, is 51–60—that is, the minimum SI range that is not significantly different from the maximum effect at 61–70.

Quantifying the exact costs of NPIs is an important endeavour, but one that is fraught with difficulties, such as converting non-economic outcomes into monetary terms. Furthermore, doing so would require multiple assumptions about highly uncertain possible effects, many in the distant future. By contrast, this upper bound on SI* is extremely robust, albeit less informative. Although the derivation of SI* is based on statistical significance, it is possible that lower SI levels may be statistically significantly different from SI*, but that the difference in the effect size is practically of little importance. Indeed, the ratio of the effect sizes of an SI range of 31–40 relative to the average of the 51–80 range, is 91% [(85%,97%)], that is, the former is 91% as effective as the latter. The relative effectiveness of the even laxer 21–30 range is 72% [(57%,86%].) If the costs of NPIs increase quickly with NPI stringency, it is conceivable that moderately severe policy responses in the 31–40 SI range (and possibly even 21–30) may in fact be close to the socially optimal SI* (arising from a full cost–benefit analysis), as it achieves 91% effectiveness without accounting for costs.

Similarly, the maximum effectiveness on death growth rates was also observed in the SI range of 61–70. I tested the difference in effectiveness of all other levels against this range and found no significant increase in effectiveness for levels beyond 41–50 (see Supplementary Table 5). An SI range of 31–40 achieved 93% [(87%,98%)] of the effectiveness of the average of the 41–90 range; the laxer 21–30 range achieved 73% [(58%,88%].)

Finally, while the SI aggregates individual NPI policies, examining the median values of the individual policies in the dataset for each SI range can also be informative (see Table 2). Note that in the 31–40 SI range, which achieves at least 90% of the maximum impact, the median policies do not include any restrictions on transport and internal movement, nor do they include stay-home restrictions. Furthermore, closing schools and workplaces and cancelling public events were recommended but not mandatory. The only stringent individual policies typically arising in the 31–40 SI range were quarantining high-risk cases from international travel and restrictions on gatherings of 100–1,000 people; however, these two policies are not reflective of citizens’ everyday behavior. Consequently, voluntary behavioral changes appear to be more important drivers of the impact of NPIs on case and death growth rates than are mandatory measures.^{Footnote 3} This is consistent with other studies that also conclude that the flattening of NPI effectiveness with increasing stringency reflects a relatively stronger voluntary (vs. mandatory) component to behavioral changes (e.g., [17,18,19, 23]).

What causes this voluntary behavioral change? As I have shown, it is partly due to citizens’ expectations of the risk of infection and severity as captured by (hat{gamma }_{2}) in Eq. 3—the effect size, however, was found to be relatively small. The majority of voluntary behavioral change, is likely due to the signalling value of policy decisions. Citizens can use information about the stringency of government measures to infer the severity of the pandemic. This implies that recommendations by governments, for SI ranges of up to 40, were heeded by citizens, who significantly changed their behavior in ways beyond those captured by mobility in Eq. 3, such as by implementing preventative measures including diligent hand washing, wearing masks, and self-isolating when infected.

Extensive public testing significantly reduces case and death growth rates, contact tracing does not Figure 5 presents the estimated coefficients and associated 95% confidence intervals regarding contact tracing and public testing (see Table 1 for detailed regression results). All three levels of the testing regime were jointly significantly different from zero ((chi ^{2}left( 3right) =66.03,p<0.0001)), leading to progressively greater declines in case growth rates as testing became more extensive (robust to multiple comparison Sidak corrections).^{Footnote 4} Note that the most extensive testing policy had an impact of -7.39 [-9.926,-4.854], which is 51% [27%,76%] of that of the most impactful SI range (61–70). Similarly, the implementation of testing policies significantly reduced the death growth rate ((chi ^{2}left( 3right) =42.81,p<0.0001)). Unlike lockdowns, the only costs that testing policies incur are financial; mental health, for example, is generally not affected. Coupled with its significant impact on COVID-19 dynamics, this renders extensive testing a desirable tool.

Estimated coefficients across levels of testing policy (TPl) and contact tracing (CTl), where l=level

Contact tracing of any level did not have a significant impact on confirmed case growth rate ((chi ^{2}left( 2right) =1.93,p=0.38)) or death growth rate ((chi ^{2}left( 2right) =1.31,p=0.52)). However, contact tracing may still be effective if the number of daily new cases is small, when efficient tracing is more manageable. There remain important challenges to scaling contact tracing [53,54,55] that could hinder its effectiveness during significant outbreaks.

The proportion of the population tested daily does not significantly affect case and death growth rates While both estimates were negative, as expected, increasing the proportion of daily tests did not significantly reduce the case growth rate, (hat{beta }_{5}=-0.18) [(-0.403,-0.035,p=0.1)], nor did it significantly reduce the death growth rate, -0.126 [(-0.315,0.063,p=0.191)]. Including different levels of the ordinal testing policy variable may partially capture this effect, as they will be correlated to some degree with the proportion tested—that is, extensive testing as coded in the ordinal variable would likely be associated with more testing. Finally, this may be the result of implicitly assuming exogeneity; testing may also be endogenously determined as governments are likely to step up testing during phases with higher transmission.

Vaccination reduces confirmed case and death growth rates Despite few datapoints where vaccination was already well underway, each 1 percentage point increase in the cumulative vaccination % reduced the case growth rate by -0.017 percentage points ([-0.032,-0.002,p=0.022]) and the death growth rate by -0.0187 percentage points [(-0.035,-0.002,p=0.026)]. Note that the median (nonzero) cumulative vaccination % across countries was only 2.6% and the 10th and 90th percentiles ware 0.07% and 19.8%, respectively. Consequently, these relatively low estimates should not be assumed to extrapolate for higher vaccination levels, especially since this is likely to be a nonlinear relationship in reality—that is, the effect of vaccinations may increase at an accelerating rate as the population approaches herd immunity.

Government policy is endogenous and exhibits hysteresis Government policy is strongly endogenous, in contrast to the common implicit assumption of exogeneity. The seven-day lagged confirmed growth rate coefficient (hat{delta }_{1}=0.048) (left[ 0.035,0.061,p<0.0001right]) was positively related to NPI severity. Furthermore, I found significant hysteresis in the de-escalation of NPIs. For the same case growth rate, NPIs were significantly more stringent if the case growth rate had recently been falling rather than rising: (hat{delta }_{2}=0.531 [0.453,0.609,p<0.0001]).