A Novel Theoretical Framework for Exponential Smoothing

Abstract

Simple Exponential Smoothing is a classical technique used for smoothing time series data by assigning exponentially decreasing weights to past observations through a recursive equation; it is sometimes presented as a rule of thumb procedure. We introduce a novel theoretical perspective where the recursive equation that defines simple exponential smoothing occurs naturally as a stochastic gradient ascent scheme to optimize a sequence of Gaussian log-likelihood functions. Under this lens of analysis, our main theorem shows that -- in a general setting -- simple exponential smoothing converges to a neighborhood of the trend of a trend-stationary stochastic process. This offers a novel theoretical assurance that the exponential smoothing procedure yields reliable estimators of the underlying trend shedding light on long-standing observations in the literature regarding the robustness of simple exponential smoothing.