This paper considers the problem of choosing the number of bootstrap repetitions B for bootstrap standard errors, confidence intervals, confidence regions, hypothesis tests, p-values, and bias correction. For each of these problems, the paper provides a three-step method for choosing B to achieve a desired level of accuracy. Accuracy is measured by the percentage deviation of the bootstrap standard error estimate, confidence interval length, test’s critical value, test’s p-value, or bias-corrected estimate based on B bootstrap simulations from the corresponding ideal bootstrap quantities for which B��. The results apply quite generally to parametric, semiparametric, and nonparametric models with independent and dependent data. The results apply to the standard nonparametric iid bootstrap, moving block bootstraps for time series data, parametric and semiparametric bootstraps, and bootstraps for regression models based on bootstrapping residuals. Monte Carlo simulations show that the proposed methods work very well.

This paper studies the extent to which qualitative features of Walrasian equilibria are refutable given a nite data set. In particular, we consider the hypothesis that the observed data are Walrasian equilibria in which each price vector is locally stable under t^atonnement. Our main result shows that a nite set of observations of prices, individual incomes and aggregate consumption vectors is rationalizable in an economy with smooth characteristics if and only if it is rationalizable in an economy in which each observed price vector is locally unique and stable under t^atonnement. Moreover, the equilibrium correspondence is locally monotone in a neighborhood of each observed equilibrium in these economies. Thus the hypotheses that equilibria are locally stable under t^atonnement, equilibrium prices are locally unique and equilibrium comparative statics are locally monotone are not refutable with a nite data set. 1