Accurate counts of cases and deaths are critical for devising an optimal pandemic response. Yet, as the COVID-19 pandemic has progressed, so too has the recognition that cases and deaths have been underreported, perhaps vastly so. Here, we present an econometric strategy to estimate the true number of COVID-19 cases and deaths for roughly 60 countries from January 1 through November 3, 2020. Specifically, we estimate a 'structural' model based on the workhorse SIR epidemiological model extended to incorporate measurement error. The results indicate significant under-reporting by magnitudes that align with existing research and conjectures by public health experts. While our approach requires some strong assumptions, these assumptions are very different from the equally strong assumptions required by other approaches addressing under-reporting in the assessment of the extent of the pandemic. Thus, we view our approach as a complement to existing methods.