  It is well known that the dependence structure for jointly Gaussian variables
can be fully captured using correlations, and that the conditional dependence
structure in the same way can be described using partial correlations. The
partial orrelation does not, however, characterize conditional dependence in
many non-Gaussian populations. This paper introduces the local Gaussian partial
correlation (LGPC), a new measure of conditional dependence. It is a local
version of the partial correlation coefficient that characterizes conditional
dependence in a large class of populations. It has some useful and novel
properties besides: The LGPC reduces to the ordinary partial correlation for
jointly normal variables, and it distinguishes between positive and negative
conditional dependence. Furthermore, the LGPC can be used to study departures
from conditional independence in specific parts of the distribution. We provide
several examples of this, both simulated and real, and derive estimation theory
under a local likelihood framework. Finally, we indicate how the LGPC can be
used to construct a powerful test for conditional independence, which, again,
can be used to detect Granger causality in time series.
