Did lockdowns have an influence on the growth rate of COVID-19 cases or was the decay in growth rates already well under way? How should we think about this question and where do epidemiologists go wrong?
Statistical Model
We start with a purely statistical model described as intervention analysis by econometricians. Assume daily case growth (x) can be described as an AR(1) process with a lockdown intervention dummy (l). This dummy takes the value zero before lockdown measures have been introduced and one afterwards.
From this model we can derive the effect of lockdown measures on the cases in period (t+j):
Taking limits (j goes to infinity), the long run impact of lockdown intervention becomes
which leads to a new long run case growth
versus the old long run growth (before Intervention)
We can apply our model to Italian cases
The Intervention date is given by the 9th of March 2020. We estimate a highly significant ar(1) coefficient of 0.74 (t-value of 11.6) and a highly significant Intervention dummy of -5.08 (t-value of -2.9) with an intercept of 5.98 (t-value of 3). From this, the impact of Italian intervention amounts to a reduction in case growth by 19.6% points
dropping from pre Intervention growth of 23% to 3.4% (all values are rounded). Intervention was highly successful and statistically significant. We can of course improve this model by more carefully modelling the strength of intervention (transfer function modelling) or estimating a joint VAR (with case series and intervention variables).
Economics versus Statistics versus Epidimology
While the model confirms a statistically significant impact, it has many weaknesses as an economist would quickly point out. Most importantly it assumes (as epidemiological models do) constant coefficients, i.e. rational agents do not change their behaviour. This is of course blatantly wrong (as in all reduced form models) and explains the instability of the virus’ observed reproduction rate. In epidemiological models the reproduction rate only changes over time as the virus will face an increasing number of already infected (immune). In real life, rational agents will change their behaviour irrespective of government intervention. This equally applies when lockdown measures are relaxed by politicians with higher ambitions. Rational agents are our only chance against the virus and incompetent politicians.
Without a model of how rational agents respond (deciding on the involved tradeoffs) we can not deduce from models like the one above, that it was government intervention that came to the rescue. This should also come as a warning to data scientists who generally operate without theoretical models.