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 $\log(T)$, 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 $\log(T)$ 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

Observation of Galactic and Solar Cosmic Rays from October 13, 1959 to February 17, 1961 with Explorer VII (Satellite 1959 Iota)

This paper gives a comprehensive summary of cosmic-ray intensity observations at high latitudes over North America and over Australia in the altitude range 550 to 1100 kilometers by means of Geiger tubes in Explorer VII (Earth satellite 1959 Iota). The time period covered is October 13, 1959 to February 17, 1961. Of special interest are the observational data on some 20 solar cosmic-ray events including major events of early April 1960, early September 1960, and of mid-November 1960. Detailed study of the latitude dependence of solar cosmic ray intensity will be presented in a later companion paper