  It\^{o} processes are the most common form of continuous semimartingales, and
include diffusion processes. This paper is concerned with the nonparametric
regression relationship between two such It\^{o} processes. We are interested
in the quadratic variation (integrated volatility) of the residual in this
regression, over a unit of time (such as a day). A main conceptual finding is
that this quadratic variation can be estimated almost as if the residual
process were observed, the difference being that there is also a bias which is
of the same asymptotic order as the mixed normal error term. The proposed
methodology, ``ANOVA for diffusions and It\^{o} processes,'' can be used to
measure the statistical quality of a parametric model and, nonparametrically,
the appropriateness of a one-regressor model in general. On the other hand, it
also helps quantify and characterize the trading (hedging) error in the case of
financial applications.
