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

ANOVA for diffusions and Itô processes