  Yule (1926) identified the issue of "nonsense correlations" in time series
data, where dependence within each of two random vectors causes overdispersion
-- i.e. variance inflation -- for measures of dependence between the two.
During the near century since then, much has been written about nonsense
correlations -- but nearly all of it confined to the time series literature. In
this paper we provide the first, to our knowledge, rigorous study of this
phenomenon for more general forms of (positive) dependence, specifically for
Markov random fields on lattices and graphs. We consider both binary and
continuous random vectors and three different measures of association:
correlation, covariance, and the ordinary least squares coefficient that
results from projecting one random vector onto the other. In some settings we
find variance inflation consistent with Yule's nonsense correlation. However,
surprisingly, we also find variance deflation in some settings, and in others
the variance is unchanged under dependence. Perhaps most notably, we find
general conditions under which OLS inference that ignores dependence is valid
despite positive dependence in the regression errors, contradicting the
presentation of OLS in countless textbooks and courses.
