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Forecasting the evolution of complex systems is one of the grand challengesof modern data science. The fundamental difficulty lies in understanding thestructure of the observed stochastic process. In this paper, we show that everyuniformly-positive-definite-in-covariance and sufficiently short-rangedependent non-stationary and nonlinear time series can be well approximatedglobally by an auto-regressive process of slowly diverging order. When linearprediction with ${\cal L}^2$ loss is concerned, the latter result facilitates aunified globally-optimal short-term forecasting theory for a wide class oflocally stationary time series asymptotically. A nonparametric sieve method isproposed to globally and adaptively estimate the optimal forecastingcoefficient functions and the associated mean squared error of forecast. Anadaptive stability test is proposed to check whether the optimal forecastingcoefficients are time-varying, a frequently-encountered question forpractitioners and researchers of time series. Furthermore, partialauto-correlation functions (PACF) of general non-stationary time series arestudied and used as a visual tool to explore the linear dependence structure ofsuch series. We use extensive numerical simulations and two real data examplesto illustrate the usefulness of our results.

Globally Optimal And Adaptive Short-Term Forecast of Locally Stationary Time Series And A Test for Its Stability