Time series modeling and forecasting by mathematical programming

We present an innovative approach to time-series analysis motivated by a desire to automate autoregressive integrated moving average (ARIMA) modeling and by extension, the process of building a reliable forecast. Our approach relies on optimization modeling and the decomposition of a time series into fixed seasonal effects, autoregressive (AR) terms, moving average (MA) terms, white noise, and any free variation outside of the selected model's framework. The simultaneous selection of appropriately-lagged terms and estimation of their coefficients is achieved via mathematical programming to minimize normed errors subject to constraints that include a limit on the number of combined AR and MA terms to achieve desired parsimony. An estimator for white noise serves as a stationary reference signal for determining the MA coefficients and is modeled using wavelets, which facilitate the formulation of integer linear constraints. We report the results of numerical testing of this methodology on several real-world datasets and discuss the main implications for research and practice. We find that our models (i) achieve one step-ahead forecasting accuracy that outperforms other, widely-used interpretable methodologies, including random forests and gradient boosting models, (ii) offer interpretable results, and (iii) are computationally tractable.