  It has been recently shown in Jaworski, P., Jelito, D. and Pitera, M. (2024),
'A note on the equivalence between the conditional uncorrelation and the
independence of random variables', Electronic Journal of Statistics 18(1), that
one can characterise the independence of random variables via the family of
conditional correlations on quantile-induced sets. This effectively shows that
the localized linear measure of dependence is able to detect any form of
nonlinear dependence for appropriately chosen conditioning sets. In this paper,
we expand this concept, focusing on the statistical properties of conditional
correlation estimators and their potential usage in serial dependence
identification. In particular, we show how to estimate conditional correlations
in generic and serial dependence setups, discuss key properties of the related
estimators, define the conditional equivalent of the autocorrelation function,
and provide a series of examples which prove that the proposed framework could
be efficiently used in many practical econometric applications.
