Given a stationary process, let us predict it using a first-order pred
ictor whose single coefficient is adapted to the current observations
using a constant gain identification algorithm. We investigate the pre
diction error variance as a function of the adaptation gain i.e., the
length of the memory (the number of observations) of the identificatio
n scheme. An infinite-memory corresponds to the asymptotically constan
t optimal predictor and a finite memory to a locally adaptive time var
ying predictor. We show that, in some specified situations, the predic
tion error variance associated with the finite memory adaptation schem
e is smaller that the optimal variance. This of course can only occur
if the model is misspecified i.e., the structure of the optimal predic
tor too simple.