Differential geometrical structures related to forecasting error variance ratios

Differential geometrical structures (Riemannian metrics, pairs of dual affine connections, divergences and yokes) related to multi-step forecasting error variance ratios are introduced to a manifold of stochastic linear systems. They are generalized to nonstationary cases. The problem of approximating a given time series by a specific model is discussed. As examples, we use the established scheme to discuss the AR (1) approximations and the exponential smoothing of ARMA series for multi-step forecasting purpose. In the process, some interesting results about spectral density functions are derived and applied.