Given limited supply of approved vaccines and constrained medical resources, design of a vaccination strategy to control a pandemic is an economic problem. We use time-series and panel methods with real-world country-level data to estimate effects on COVID-19 cases and deaths of two key elements of mass vaccination - time between doses and vaccine type. We find that new infections and deaths are both significantly negatively associated with the fraction of the population vaccinated with at least one dose. Conditional on first-dose coverage, an increased fraction with two doses appears to offer no further reductions in new cases and deaths. For vaccines from China, however, we find significant effects on both health outcomes only after two doses. Our results support a policy of extending the interval between first and second doses of vaccines developed in Europe and the US. As vaccination progresses, population mobility increases, which partially offsets the direct effects of vaccination. This suggests that non-pharmaceutical interventions remain important to contain transmission as vaccination is rolled out.