Disaggregation of spatial autocorrelation parameter

Question

I have data aggregated at state level. When I estimate a spatial autoregressive model such as $$y = \rho W y + X\beta + \epsilon$$ on this data, I see that the autoregressive parameter $\rho$ is moderately significant ($\rho = 0.2$, $p$-val$= 0.06$).

But I am preparing to apply this estimated model to data aggregated at county level; to be precise, I have ${X, W}$ at both state and county levels, but $y$ only at the state. As I understand spatial autoregression, there should be a significantly higher correlation as I move down in geography. A priori I would expect somewhere around 0.8, based on my prior experience.

This project is for a practical model rather than research, so a precise mathematical solution is not necessary (and I imagine doesn't exist). I am interested if others have run into this problem or if what I am doing is recklessly irresponsible...

Answer 1

I'm going to give an answer to a different question! Because I am unsure if the original one is tractable, but it is how I would frame the problem - so I hope it is helpful.

Let's just start with the simpler task of decomposing a set of covariances between two spatial weights matrices for the same units of analysis. Lets say that $W = W^a + W^b$, and we want to know how $\text{Cov}(y,Wy)$ relates to the covariances of $\text{Cov}(y,W^ay)$ and $\text{Cov}(y,W^by)$.

So assuming a mean centered $y$, the sample covariance can be written as:

$$\text{Cov}(y,Wy) = \frac{N}{\sum_i \sum_j w_{ij}} \frac{\sum_i \sum_j w_{ij}(y_i \cdot y_j)}{\sum_i y_i^2}$$

Where $N$ is the total number of observations and $w_{ij}$ are the weights between units $i$ and $j$. Because I specified that $W$ is the sum of two separate weight matrices we can decompose the numerator in the above equation to the covariance contributed to by $W^a$ and the contribution of $W^b$:

$$\sum_i \sum_j w_{ij}(y_i \cdot y_j) = \sum_i \sum_j (w_{ij}^a + w_{ij}^b)(y_i \cdot y_j)$$

So if you formulated the problem like this, we have a whole to part relationship. One slightly different way to do this with the county level model is to have you binary contiguity matrix be $W$, but then have $W^a$ be the within-state weights and $W^b$ be the between state weights. Then you can estimate a model of the form

\begin{align} y &= \rho_a W^a y + \rho_b W^b y + X \beta \\ y &= \rho W y + X \beta \end{align}

I'm not sure if $\rho_a + \rho_b = \rho$, but hopefully this is a good start.

So there are a few difficulties/moving parts with the aggregation that make me think that a direct decomposition is not going to happen:

• If you consider contiguity matrices for the state level compared to the county level, the weights do not overlap nicely like they do in this exposition. For example, at the state level Maryland is a neighbor of Pennsylvania. At the county level, any county in northern PA is not going to be first order neighbors of any county in Maryland.
• The variance of the aggregated state level series is not the same as the variance of the county level. In most social science lit. when you aggregate areas up the correlations tend to be higher. I would expect the spatial autocorrelation with the smaller units to actually be smaller. In applied social science research a spatial auto-correlation of 0.2 is about the highest I've seen.
• The variance of the aggregated series is partially a function of the within state correlation. So I don't believe you can decompose the within state spatial auto-correlation from the aggregate between state spatial auto-correlation as stated.

I would like to see it if I am wrong though!