The oxidative capacity of past atmospheres is highly uncertain. We present here a new climate-biosphere-chemistry modeling framework to determine oxidant levels in the present and past troposphere. We use the GEOS-Chem chemical transport model driven by meteorological fields from the NASA Goddard Institute of Space Studies (GISS) ModelE, with land cover and fire emissions from dynamic global vegetation models. We present time-slice simulations for the present day, late preindustrial era (AD 1770), and the Last Glacial Maximum (LGM, 19-23 ka), and we test the sensitivity of model results to uncertainty in lightning and fire emissions. We find that most preindustrial and paleo climate simulations yield reduced oxidant levels relative to the present day. Contrary to prior studies, tropospheric mean OH in our ensemble shows little change at the LGM relative to the preindustrial era (0.5 +/- 12 %), despite large reductions in methane concentrations. We find a simple linear relationship between tropospheric mean ozone photolysis rates, water vapor, and total emissions of NOx and reactive carbon that explains 72% of the variability in global mean OH in 11 different simulations across the last glacial-interglacial time interval and the industrial era. Key parameters controlling the tropospheric oxidative capacity over glacial-interglacial periods include overhead stratospheric ozone, tropospheric water vapor, and lightning NOx emissions. Variability in global mean OH since the LGM is insensitive to fire emissions. Our simulations are broadly consistent with ice-core records of Delta O-17 in sulfate and nitrate at the LGM, and CO, HCHO, and H2O2 in the preindustrial era. Our results imply that the glacial-interglacial changes in atmospheric methane observed in ice cores are predominantly driven by changes in its sources as opposed to its sink with OH.