  Forecasting the evolution of complex systems is one of the grand challenges
of modern data science. The fundamental difficulty lies in understanding the
structure of the observed stochastic process. In this paper, we show that every
uniformly-positive-definite-in-covariance and sufficiently short-range
dependent non-stationary and nonlinear time series can be well approximated
globally by an auto-regressive process of slowly diverging order. When linear
prediction with ${\cal L}^2$ loss is concerned, the latter result facilitates a
unified globally-optimal short-term forecasting theory for a wide class of
locally stationary time series asymptotically. A nonparametric sieve method is
proposed to globally and adaptively estimate the optimal forecasting
coefficient functions and the associated mean squared error of forecast. An
adaptive stability test is proposed to check whether the optimal forecasting
coefficients are time-varying, a frequently-encountered question for
practitioners and researchers of time series. Furthermore, partial
auto-correlation functions (PACF) of general non-stationary time series are
studied and used as a visual tool to explore the linear dependence structure of
such series. We use extensive numerical simulations and two real data examples
to illustrate the usefulness of our results.
