MAUP: correlation increases with aggregation, would error decrease? I know that, due to MAUP, as you coarsen the scale of analysis, you can expect the correlation coefficient to increase, variance to decrease (?), and regression coefficients to fluctuate.  Is there any predictable effect on the errors from a regression?  
 A: The Modifiable Areal Unit Problem (MAUP) is a particular case of aggregation bias, and the seminal papers on the MAUP build off of early investigations into such bias. Your question has, to my mind, a few misconceptions in it. First, it is not the case that correlation necessarily increases with aggregation; correlation can go either way. For example, I have seen real data where depressive symptoms are associated with BMI in roughly a 'U' shape. However, if one examines the association with the aggregated categories 'underweight', 'normal weight', or 'overweight' the association diminishes to almost nil, since the sloped part of the non-aggregated relationships do not correspond to the numerical cut-offs for the definitions of 'overweight' and 'underweight'.
Second, your question leaves out a critically important issue: it is not merely scale that affects correlations, autocorrelations, variance measures, and inferences, but categorization. That is for a given scale (i.e. a given number of units to be aggregated to) the function that determines which individual (or lower level unit) ends up in which higher level unit also affects all these statistical measures.
The critical insight to take away from the MAUP literature is not the idea that there is a bias to be corrected, but rather that aggregation requires substantive theoretical justification. For example, if interested in school-children's vehicular-pedestrian collisions, school districts (which implement crossing-guard policies) would be a more appropriate definition of neighborhood than, say, police precincts. 
Example of categorization
In the picture below, the circle represents a city, and the blue dots represent affluent individuals, while the yellow triangles represent impoverished individuals (the thick black horizontal line might represent the tracks for which there is a "wrong side of" :). The red lines divide the city into three neighborhoods; notice that there is an economic gradient with the wealthier neighborhood up north, the middling neighborhood in the center, and the impoverished neighborhood in the south.

Now notice in this second image, that the city is identical with respect to the residents and their individual distributions of affluence/impoverishment, except that the red lines have been rearranged so that no there is no economic gradient, in other words the between neighborhood variance is now zero (also, for some reason the tracks are a different color, but that's irrelevant :). The scale (3 neighborhoods) remains the same, but the categorization function (who ends up in which neighborhood) is different.

References
Dudley, G. (1991). Scale, aggregation, and the modifiable areal unit problem. The Operational Geographer, 9(3):28–33.
Gehlke, C. E. and Biehl, K. (1934). Certain Effects of Grouping Upon the Size of the Correlation Coefficient in Census Tract Material. Journal of the American Statistical Association, 29(185):169–170. [One of the first ever papers to address aggregation bias.]
Lee, H. T. K. and Kemp, Z. (2000). Hierarchical reasoning and on-line analytical processing of spatial and temporal data. In Proceedings of the 9th International Symposium on Spatial Data Handling, Beijing, P.R. China. International Geographic Union. [This is one of the best papers I have read on the subject, but I think you have to track one of the authors down to obtain a copy.]
Openshaw, S. (1983). The modifiable areal unit problem. Concepts and Techniques in Modern Geography. Geo Books, Norwich, UK.
Openshaw, S. and Taylor, P. J. (1979). A million or so correlation coefficients: Three experiments on the modifiable area unit problem. In Wrigley, N. (ed.) Statistical Applications in the Spatial Sciences, pages 127–144. Pion, London, UK.
