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Forecasting unstable processes
Previous analysis on forecasting theory either assume knowing the true
parameters or assume the stationarity of the series. Not much are known on the
forecasting theory for nonstationary process with estimated parameters. This
paper investigates the recursive least square forecast for stationary and
nonstationary processes with unit roots. We first prove that the accumulated
forecast mean square error can be decomposed into two components, one of which
arises from estimation uncertainty and the other from the disturbance term. The
former, of the order of , is of second order importance to the latter
term, of the order T. However, since the latter is common for all predictors,
it is the former that determines the property of each predictor. Our theorem
implies that the improvement of forecasting precision is of the order of
when existence of unit root is properly detected and taken into
account. Also, our theorem leads to a new proof of strong consistency of
predictive least squares in model selection and a new test of unit root where
no regression is needed. The simulation results confirm our theoretical
findings. In addition, we find that while mis-specification of AR order and
under-specification of the number of unit root have marginal impact on
forecasting precision, over-specification of the number of unit root strongly
deteriorates the quality of long term forecast. As for the empirical study
using Taiwanese data, the results are mixed. Adaptive forecast and imposing
unit root improve forecast precision for some cases but deteriorate forecasting
precision for other cases.Comment: Published at http://dx.doi.org/10.1214/074921706000000969 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
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