# Does block-boostrapping clean the autocorrelation of residuals?

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.

• Is time measured with the index of the series or with a separate series of values? For instance, if I have a block {a, b, c} at times 1, 2, and 3 and I bootstrap this block to obtain {c, c, c}, is the times of these entries 1, 2, and 3 using the index, or all at 3, 3, 3 (coincident) using the time values? Commented Jun 29, 2017 at 16:06
• @Adam Thx. Time in the time-series's sense ? I have no time dimension in the above estimation. Only cross-section spatial data. Thus, I would say that all my entries are coincident. Did I understand you well ? The only "time" I have is the virutal one of my iterations. Commented Jun 29, 2017 at 16:15
• I assumed time, but for space my question would be: when you bootstrap rows of data, do these rows contain variables indicating the latitude/longitude or appropriate space components? Commented Jun 29, 2017 at 16:34
• @Adam Thx. Are you asking whether, for a given iteration, the bootstrap operator transforms $\boldsymbol{y}$, $\boldsymbol{X}$ and $\boldsymbol{W}$ so as to preserve the observed spatial structure of data ? If so, yes it does. Noting that for the square matrix (the lag operator) $\boldsymbol{W}$, it does so over rows first, and then over columns, whence $\mathcal{B}(\mathcal{B}(\boldsymbol{W})')'$. Did I understand you well ? But actually these rows do not explicitly contain variables indicating the latitude/longitude nor appropriate space components. Commented Jun 29, 2017 at 16:55
• @Adam. Did you mean that I should bootstrap over coordinates as well and recompute a distance-related matrix of weights for each sampling ? Commented Jul 3, 2017 at 19:57

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 $\rho$.

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.