The hydroxyl radical (OH) is the main atmospheric oxidant and the primary sink of the greenhouse gas CH4. In an attempt to constrain atmospheric levels of OH, two recent studies combined a tropospheric two-box model with hemispheric-mean observations of methyl chloroform (MCF) and CH4. These studies reached different conclusions concerning the most likely explanation of the renewed CH4 growth rate, which reflects the uncertain and underdetermined nature of the problem. Here, we investigated how the use of a tropospheric two-box model can affect the derived constraints on OH due to simplifying assumptions inherent to a two-box model. To this end, we derived species- A nd timedependent quantities from a full 3-D transport model to drive two-box model simulations. Furthermore, we quantified differences between the 3-D simulated tropospheric burden and the burden seen by the surface measurement network of the National Oceanic and Atmospheric Administration (NOAA). Compared to commonly used parameters in two-box models, we found significant deviations in the magnitude and timedependence of the interhemispheric exchange rate, exposure to OH, and stratospheric loss rate. For MCF these deviations can be large due to changes in the balance of its sources and sinks over time. We also found that changes in the yearly averaged tropospheric burden of CH4 and MCF can be obtained within 0.96 ppb yr-1 and 0.14%yr-1 by the NOAA surface network, but that substantial systematic biases exist in the interhemispheric mixing ratio gradients that are input to two-box model inversions. To investigate the impact of the identified biases on constraints on OH, we accounted for these biases in a two-box model inversion of MCF and CH4. We found that the sensitivity of interannual OH anomalies to the biases is modest (1 %-2 %), relative to the uncertainties on derived OH (3 %-4 %). However, in an inversion where we implemented all four bias corrections simultaneously, we found a shift to a positive trend in OH concentrations over the 1994-2015 period, compared to the standard inversion. Moreover, the absolute magnitude of derived global mean OH, and by extent, that of global CH4 emissions, was affected much more strongly by the bias corrections than their anomalies (∼ 10 %). Through our analysis, we identified and quantified limitations in the two-box model approach as well as an opportunity for full 3-D simulations to address these limitations. However, we also found that this derivation is an extensive and species-dependent exercise and that the biases were not always entirely resolvable. In future attempts to improve constraints on the atmospheric oxidative capacity through the use of simple models, a crucial first step is to consider and account for biases similar to those we have identified for the two-box model.