In a classic paper on conducting bootstrapped hypothesis tests, Hall & Wilson (1991) present the following guideline, which I am trying to understand:
On first read, this seemed obvious: $\widehat{\theta}^{*}$ are being resampled under $H_0$, so are consistent for $\theta_0$, whereas $\widehat{\theta}$ is consistent for $\theta$. Thus, $\widehat{\theta}^{*} - \widehat{\theta}$ is consistent for $\theta_0 - \theta$, which makes sense. In contrast, $\widehat{\theta}^{*} - \theta_0$ is consistent for $\theta_0 - \theta_0 = 0$, which is not useful.
But then I realized, on rereading the paragraph above the First Guideline and a subsequent example, that Hall & Wilson are computing $\widehat{\theta}^{*}$ by resampling directly from the sample, rather than under the null. In that case, I have no idea why they recommend resampling $\widehat{\theta}^{*} - \widehat{\theta}$, which is consistent for 0!
References
Hall, P., & Wilson, S. R. (1991). Two guidelines for bootstrap hypothesis testing. Biometrics, 757-762.