TABLE 3.

Algorithm of wild bootstrap inference based on martingale representation of the MI estimator

Step WB-1.	Sample uk, for k = 1,…, (1 + m)n, that satisfy $\mathbb{E}\left(u_k\mid O_{1:n}\right)=0,\mathbb{E}\left(u_k^2\mid O_{1:n}\right)=1$ and $\mathbb{E}\left(u_k^4\mid O_{1:n}\right)<\infty$.
Step WB-2.	Compute the bootstrap replicate as $W_L^{*}=n^{-1/2}{\sum }_{k=1}^{(1+m)n}{\widehat{\xi }}_{n,k}{u}_k$, where ${\widehat{\xi }}_{n,k}$ is the empirical version of ξn,k by replacing the unknown quantities with their estimators and the one-dimensional integrals by the numerical integration.
Step WB-3.	Repeat Steps 1 and 2 B times, and estimate the variance of ${\widehat{\Delta }}_{\tau ,\text{mi}}$ by the sample variance of these copies of $W_L^{*}$.