Short Time Series Forecasting: Polynomial Extrapolation and Arima Methodology

Introduction. Forecasting from short time series is difficult because model identification, stationarity testing and parameter estimation are performed with limited information. ARIMA remains a central statistical benchmark, but for short records the stationarization stage is often more consequential than the subsequent selection of autoregressive and moving-average orders. This study establishes an explicit link between finite differencing and local polynomial extrapolation. Materials and Methods. The non-seasonal ARIMA construction logic is summarized with emphasis on stationarization, order selection, estimation, diagnostics and recursive forecasting. For equally spaced observations, polynomial one-step extrapolation is represented through binomial coefficients and compared algebraically with finite-difference operators. On this basis, a forecasting method that averages a polynomial prediction sequence and its increment-based form are formulated. Results. It is proved that the m-th finite difference at the forecast point is exactly the one-step error of a polynomial extrapolator of degree m−1 built from the preceding m observations. Consequently, differencing in ARIMA admits an interpretation as a sequence of local polynomial forecast residuals. A compact averaging estimator is obtained for short-series forecasting. Conclusions. The derived identity provides a direct mathematical bridge between ARIMA stationarization and polynomial extrapolation. It also motivates forecast combination across polynomial orders when the available history is short and the appropriate local trend order is uncertain.