Kernel choice for HAC standard errors

[JosiahParry]

Hi all, I am reviewing the literature around estimating standard errors in spatial lag SAR models (defined as $y = \rho Wy + X\beta + u$). The use of HAC standard errors is suggested when there is still residual spatial autocorrelation—though I'm struggling to follow some of the logic in the use of HAC standard errors.

I'd love other more experienced insight into this.

I have two primary questions regarding the use of HAC standard errors.

1. Which kernel should be used and why?

{sphet} R package requires you to specify which kernel to be used. The options are epanechnikov, triangular, bisquare, parzen, th, rectangular, qs. In libpysal's spreg when robust = 'hac' the gwk argument must be provided which should be kernel weights.

In Anselin & Rey (2014), spreg, and sphet, there is no recommendation as to which kernel to use. How do we choose which kernel? Just go for it and compare outputs and see which one feels better?

2. How is the neighborhood for the HAC identified and why is it different than the neighborhood used to calculate the spatial lag?

In Anselin and Rey (2014) they cite Kelejian and Prucha (2007, pdf attached) saying that the minimum number of neighbors for HAC standard errors is the cubed root of the number of observations (n^1/3).
kelejian-prucha-2007.pdf I cannot find this reference—granted there is a lot of math that I cannot keep up with.

It feels odd that one would use a different neighborhood than the one that is used in creating spatial lags.

[martinfleis]

cc @pedrovma as I'm not sure you're following this space.

[JosiahParry]

Also going to ping @rsbivand and @gpiras as they are both listed as authors on {sphet}

[pedrovma]

The choice of Kernel follows the same procedure as for a spatial weights matrix: it depends on how you assume the spatial process you are modelling to be.
For example, if you assume all neighbours have the same influence, regardless of their distance, a uniform kernel should be used. If you assume distance decay, you should use other types. If you assume that much more weight should be given to closer neighbours, triangular is the most usual (also for its simplicity with linear decay). If you assume that distant neighbours are relevant, Gaussian is better. And so on... (see the figure to compare the different distance decays). You need to reflect on the spatial process you are modelling to make your choice.

By Brian Amberg - CC BY-SA 3.0, Link

Standard spatial weights matrix and kernel weights are different in their purpose. The former is generally used to create a spatial average of any given variable in a parametric setting. In contrast, the latter is generally used for spatial smoothing in a non-parametric setting.

Both libpysal and spreg have evolved much since Anselin and Rey (2014), so the formulas there may no longer apply. But the intuition remains.

[rsbivand]

Chapter 9, pp. 185-208, in Kelejian & Piras (2017) https://www.sciencedirect.com/book/9780128133873/spatial-econometrics gives more of the bases for the HAC approach. It asserts that while the weights matrix is fixed and used in the parametric part of the model, the HAC part operates on the error term and adjusts the standard errors of the parametric part based on the non-parametric HAC estimates. These in turn came by analogy with the use of similar kernel adjustment in time-series models.