The estimation of allocative and technical inefficiency has grown to an enormous body of literature, both theoretical and empirical. Ideally, one would estimate time-varying firm and input-specific parameters describing allocative inefficiency in order to minimize aggregation bias. However, this has never been previously accomplished. Typically, only industry-wide allocative efficiency parameters have been empirically identified. Our proposed solution is to employ Gibbs sampling to approximate posterior distributions from a Bayesian limited information model, embedding GMM moment conditions imposed via an instrumental variables step to obtain plant-specific parameters estimates that vary flexibly over time. For a panel of Chilean hydroelectric power plants, posterior distributions of these estimates display substantial differences in location and precision. By contrast, the standard GMM approach which produces industry-wide, time-varying allocative inefficiency parameters, not only fails to reveal the inter-plant differences by construction, but does not even produce posterior medians that approximate a weighted average of the plant-specific posterior medians.