A popular approach to perform inference on a target parameter in the presence of nuisance parameters is to construct estimating equations that are orthogonal to the nuisance parameters, in the sense that their expected first derivative is zero. Such first-order orthogonalization may, however, not suffice when the nuisance parameters are very imprecisely estimated. Leading examples where this is the case are models for panel and network data that feature fixed effects. In this paper, we show how, in the conditional-likelihood setting, estimating equations can be constructed that are orthogonal to any chosen order. Combining these equations with sample splitting yields higher-order bias-corrected estimators of target parameters. In an empirical application we apply our method to a fixed-effect model of team production and obtain estimates of complementarity in production and impacts of counterfactual re-allocations.