This paper focuses on the computational complexity of computing empirical plug-in estimates for causal effect queries. Given a causal graph and observational data, any identifiable causal query can be estimated from an expression over the observed variables, called the estimand. The estimand can then be evaluated by plugging in probabilities computed empirically from data. In contrast to conventional wisdom which assumes that high dimensional probabilistic functions will lead to exponential evaluation time, we show that estimand evaluation can be done efficiently, potentially in time linear in the data size, depending on the estimand's hypergraph. In particular, we show that both the treewidth and hypertree width of the estimand's structure bound the evaluation complexity, analogous to their role in bounding the complexity of inference in probabilistic graphical models. In settings with high dimensional functions, the hypertree width often provides a more effective bound, since the empirical distributions are sparse.