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Computationally efficient approximation for the double bootstrap mean bias correction

We propose a computationally efficient approximation for the double bootstrap bias adjustment factor without using the inner bootstrap loop. The approximation converges in probability to the population bias correction factor. We study the finite sample properties of the approximation in the context of a linear instrumental variable model. In identified versions of the model considered in our Monte Carlo experiments, the proposed approximation leads to estimators with lower variance than those based on the double bootstrap and, lower adjusted mean-squared error than estimators based on the single bootstrap. Evidence from the experiments we consider suggests that the bootstrap is less effective in reducing the bias when the instrumental variable is weak and endogeneity is strong.