Abstract A central issue for understanding past climates involves the use of sparse time-integrated data to recover the physical properties of the coupled climate system. This issue is explored in a simple model of the midlatitude climate system that has attributes consistent with the observed climate. A quasigeostrophic (QG) model thermally coupled to a slab ocean is used to approximate midlatitude coupled variability, and a variant of the ensemble Kalman filter is used to assimilate time-averaged observations. The dependence of reconstruction skill on coupling and thermal inertia is explored. Results from this model are compared with those for an even simpler two-variable linear stochastic model of midlatitude air–sea interaction, for which the assimilation problem can be solved semianalytically. Results for the QG model show that skill decreases as the length of time over which observations are averaged increases in both the atmosphere and ocean when normalized against the time-averaged climatological variance. Skill in the ocean increases with slab depth, as expected from thermal inertia arguments, but skill in the atmosphere decreases. An explanation of this counterintuitive result derives from an analytical expression for the forecast error covariance in the two-variable stochastic model, which shows that the ratio of noise to total error increases with slab ocean depth. Essentially, noise becomes trapped in the atmosphere by a thermally stiffer ocean, which dominates the decrease in initial condition error owing to improved skill in the ocean. Increasing coupling strength in the QG model yields higher skill in the atmosphere and lower skill in the ocean, as the atmosphere accesses the longer ocean memory and the ocean accesses more atmospheric high-frequency “noise.” The two-variable stochastic model fails to capture this effect, showing decreasing skill in both the atmosphere and ocean for increased coupling strength, due to an increase in the ratio of noise to the forecast error variance. Implications for the potential for data assimilation to improve climate reconstructions are discussed.