  The paper concerns the problem of pointwise adaptive estimation in regression
when the noise is heteroscedastic and incorrectly known. The use of the local
approximation method, which includes the local polynomial smoothing as a
particular case, leads to a finite family of estimators corresponding to
different degrees of smoothing. Data-driven choice of localization degree in
this case can be understood as the problem of selection from this family. This
task can be performed by a suggested in Katkovnik and Spokoiny (2008) FLL
technique based on Lepski's method. An important issue with this type of
procedures - the choice of certain tuning parameters - was addressed in
Spokoiny and Vial (2009). The authors called their approach to the parameter
calibration "propagation". In the present paper the propagation approach is
developed and justified for the heteroscedastic case in presence of the noise
misspecification. Our analysis shows that the adaptive procedure allows a
misspecification of the covariance matrix with a relative error of order
1/log(n), where n is the sample size.
