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Suppose I have an estimate (say an OLS coefficient), I can obtain its standard error using the standard OLS formula. I can also use nonparametric bootstrap and compute the standard error. My question is: should these two ways always give (almost) the same answer? If not, what are possible reasons for the difference?

(Note that there could be heteroscedasticity or autocorrelation in the regression error term but I am ignoring them both in the OLS estimation or in performing bootstrap).

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The confidence level of 'standard' t-tests in OLS depends on normality assumptions or asymptotic arguments. If the residuals or dependent variable are (surely) normally distributed, the t statistic is surely t-distributed, hence you have perfect $\alpha$ level control in the test. If the variates are not normally distributed, the argument still holds asymptotically in large samples as long as the dependent variable is continuous and residuals are independent and identically distributed.

Bootstrapped test statistics are always then useful if we do not trust the distributional assumptions underlying standard test procedures or if the sample size is too small to allow an asymptotic argument. The big advantage of the bootstrap is that it then leads to conclusions with accurate confidence, while standard tests may fail (in the sense of rejecting too liberal or conservative).

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