Optimum thresholding using mean and conditional mean squared error

We consider a univariate semimartingale model for (the logarithm of) an asset price, containing jumps having possibly infinite activity. The nonparametric threshold estimator IV̂n of the integrated variance IV≔∫0Tσs2ds proposed in Mancini (2009) is constructed using observations on a discrete time grid, and precisely it sums up the squared increments of the process when they are below a threshold, which depends on the observation time step and, sometimes, model parameters or latent variables, that need to be estimated. All the threshold functions satisfying given conditions allow asymptotically consistent estimates of IV, however the finite sample properties of IV̂n can depend on the specific choice of the threshold. We aim here at optimally selecting the threshold by minimizing either the estimation mean squared error (MSE) or the conditional mean squared error (cMSE). The last criterion allows to reach a threshold which is optimal not in mean but for the specific volatility and jumps paths at hand.