Time series — information and prediction

Abstract

A time series Y _t can be transformed into another time series V _t by means of a linear transformation. Should the matrix of that transformation have an inverse, the pair (Y _t, V _t) is called invertible. Based on the decomposition procedure for stationary time series introduced in a previous paper it is shown that a sufficient condition for the invertibility of the pair (Y _t, V _t) is that V _t be the first component of Y _t, i.e. V _t = V ¹ _t . By the invertibility property V ¹ _t can be used for forecasting, that is, predictions are made on V ¹ _t which is then transformed into Y _t. This is accomplished by means of a special kind of predictor permitting to make one-step-ahead forecasts in a straightforward way. Since the first component depends on a parameter α i.e. V ¹ _t = V ¹ _t (α), a procedure is proposed that allows us to find the optimal parameter value, α = α₀. Thus, it is shown that better forecasting accuracy may result by fitting a simple autoregression to the first component V ¹ _t (α₀) than if the process Y _t were described by a more elaborate model. Model building is therefore no longer a prerequisite in forecasting. The forecasting procedure is then extended so as to cope with the homogeneous nonstationary case, and examples are given to illustrate the forecasting accuracy as compared to customary model-based approaches. In the light of these results the problem of the information conveyed by the values of the series is discussed in terms of the spreading rate concept, thus highlighting the role of the current time value, as well as that of the remote values of the series, in forecasting stationary and nonstationary time series.