Does block-boostrapping clean the autocorrelation of residuals?

Question

Based on two of my previous questions

How to compute the likelihood's Jacobian of a bootstrapped spatial autoregressive specification?

How to bootstrap a (spatial) lag operator?

I decided to finally ML-block-bootstrap-estimate $\hat{\rho}$ -- involved in a spatial autoregressive model -- where $\hat{\rho} = \max_{\rho} \mathcal{B}\left(\mathcal{L}(\rho)\right)$ and

$$ \mathcal{L}(\rho)= \left(\frac{2\pi e}{n}\boldsymbol{u}^{'}\boldsymbol{u}\right)^{-\frac{n}{2}} \begin{vmatrix} \boldsymbol{S}\left(\rho\right) \end{vmatrix} $$

with

$$ \boldsymbol{u} = \mathcal{B}(\boldsymbol{y}) - \rho \mathcal{B}(\mathcal{B}(\boldsymbol{W})')'\mathcal{B}(\boldsymbol{y}) - \mathcal{B}(\boldsymbol{X})\boldsymbol{\beta}_{\rho} $$

and where $\mathcal{B}(.)$ stands for the rows-bootstrapping resampling process operator. Note that the transpose operator ($'$) is used so as to bootstrap over columns as well.

And what follows are the convergence and histogram charts of $\rho$

I did $1000$ iterations, at which the block size is randomly chosen between $2$ and $n/2$. Blocks are allowed to overlap. "Initial value" stands for the estimation coming from the original MLE,( with no resampling).

Does block-boostrapping clean the autocorrelation of (my) residuals or what ?

I must mention that all other coefficients converge as well.

Answer 1

Actually what we are shown in the above convergence and distribution charts is not related with bootstrapping cleaning autocorrelation, but rather to a mis-bootstrapped lag-operator $\boldsymbol{W}$.

In my case, $\boldsymbol{W}$ was a (very sparse) Boolean matrix ($k$-nn with $k=1$), which once boostrapped led (boot-sample-)$\rho$ to simply multiply almost... nothing.

As may have been outlined (I guess) by Adam in comment, one way to do this better is to recompute the lag operator for each resampling, so as to consider the newly bootstrapped geography. The two charts below are what I get now:

Also, note that the notation I use for the auto-regressive coefficient $\rho$ can be confusing: it would have been better denoted by $\bar{\rho}$, i.e., the bootstrap mean, which is to $\hat{\rho}$, what $\hat{\rho}$ itself is to $\rho$.

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Some references about spatial bootstrap which are clearly better than my homemade solution:

Lin, K., Long, Z., & Mei, W. (2007). Bootstrap Test Statistics for Spatial Econometric Models.

Nordman, D. J., Lahiri, S. N., & Fridley, B. L. (2007). Optimal block size of variance estimation by a spatial block bootstrap method. Sankhya: The Indian Journal of Statistics, 69(3), 468–493.