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...

  • $\begingroup$ Because a p-value of $0.6$ is about as non-significant as you can get, I wonder whether you mistyped this number. I don't completely follow your question, either: are you implying you believe an estimated autocorrelation of $0.2$ at the state level would be incompatible with one of $0.8$ at the county level? If so, why? And how would the $W$ matrices for the state and county datasets be related exactly? $\endgroup$
    – whuber
    Jul 18 '14 at 17:27
  • $\begingroup$ Indeed, I have corrected the $p$ value to be 0.06. $\endgroup$ Jul 18 '14 at 17:30
  • $\begingroup$ Two states that are adjacent to each other likely share some characteristics, leading to correlation in their lm error terms that is corrected by including the $\rho W y$ term. Two counties next to each other likely share far more characteristics, leading to much higher correlation. In either case, I specify $W$ exogenously as adjacent shapes. $\endgroup$ Jul 18 '14 at 17:32
  • $\begingroup$ Although I suppose I could specify 2nd- or 3rd- order adjacency for the county $W$; this would perhaps allow me to use the $\rho$ estimated from the state model. $\endgroup$ Jul 18 '14 at 17:33

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!

  • $\begingroup$ I really like this answer, even though it answered a slightly different question. As to the correlation decreasing when moving upwards, this doesn't make much sense to me; I know that residuals get smaller at higher levels, but shouldn't autocorrelation increase? Your county in northern PA is marked as a neighbor to the county in MD at the state level, so it will weaken the autoregression; at the county level it will be paired with counties around it an in New York. Wouldn't this increase autocorrelation? $\endgroup$ Jul 18 '14 at 19:16
  • $\begingroup$ My previous experience with spatial models has been at the urban level, there I regularly see $\rho$ approaching 0.95. Perhaps my expectations are skewed... $\endgroup$ Jul 18 '14 at 19:17
  • $\begingroup$ Crime data with which I am more familiar spatial auto-correlation is much smaller (at every unit of analysis). I am less familiar with other demographic info. What is the data you are working with and what are the "urban" units of analysis? I agree in principal with your logic about the spatial auto-correlation, smaller places should be more related to the other smaller places, because they are closer (Tobler's law). My experience in practice though is the opposite, smaller units of analysis have a smaller spatial auto-correlation for nearby units. $\endgroup$
    – Andy W
    Jul 18 '14 at 19:24
  • $\begingroup$ My current dataset is freight production by commodity in US states; I'm trying to disaggregate this to county level where I have employment statistics by industry. My previous applications have been in home prices, where I had values and characteristics for almost all homes (points) in a large metropolitan area. $\endgroup$ Jul 18 '14 at 19:27
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    $\begingroup$ Basically the covariance of the aggregated units is the sum of all the covariances of the micro level units - so when you aggregate more you are adding up more spatial covariance terms. I go through a similar exposition about how aggregation bias occurs in Chapter 2 of my prospectus. That is about the best hand-wavy reasoning I can give to the spatial auto-correlation off-the-cuff though. (I doubt I understand the implications fully.) $\endgroup$
    – Andy W
    Jul 18 '14 at 19:27

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