The Honest Truth About Causal Trees: Accuracy Limits for Heterogeneous Treatment Effect Estimation

Recursive decision trees are widely used to estimate heterogeneous causal treatment effects in experimental and observational studies. These methods are typically implemented using CART-type recursive partitioning and are often viewed as adaptive procedures capable of discovering treatment effect heterogeneity in high-dimensional settings. We study causal tree estimators based on adaptive recursive partitioning and establish lower bounds on their estimation accuracy. Under basic conditions, we show that causal trees constructed via standard CART-type splitting rules cannot achieve polynomial-in-$n$ convergence rates in the uniform norm (where $n$ denotes the sample size). The underlying mechanism is that greedy recursive partitioning selects highly imbalanced splits with non-vanishing probability, producing terminal nodes containing very few observations and leading to large estimation variance. We further show that sample splitting (``honesty'') yields at most negligible improvements in convergence rates. As a consequence, causal tree estimators may converge arbitrarily slowly and can even be inconsistent in some settings. Our results also clarify the role of balanced partition assumptions in existing theoretical guarantees for causal forests and related ensemble methods. The analysis develops new probabilistic tools for studying adaptive recursive partitioning procedures, including non-asymptotic approximations for suprema of partial sums and Gaussian processes. As a technical by-product, we also identify and correct an error in Eicker (1979).