An Analysis of Linear Time Series Forecasting Models

TL;DR

This paper analyzes various linear time series forecasting models, showing their equivalence to standard linear regression and demonstrating that simpler models often outperform more complex variants in many settings.

Contribution

It characterizes the function spaces of linear models, proves their equivalence to linear regression, and shows simpler models often yield better forecasts.

Findings

Linear variants are equivalent to standard linear regression.

Models can be reinterpreted as linear regression over augmented features.

Simpler closed-form models outperform complex variants in 72% of tests.

Abstract

Despite their simplicity, linear models perform well at time series forecasting, even when pitted against deeper and more expensive models. A number of variations to the linear model have been proposed, often including some form of feature normalisation that improves model generalisation. In this paper we analyse the sets of functions expressible using these linear model architectures. In so doing we show that several popular variants of linear models for time series forecasting are equivalent and functionally indistinguishable from standard, unconstrained linear regression. We characterise the model classes for each linear variant. We demonstrate that each model can be reinterpreted as unconstrained linear regression over a suitably augmented feature set, and therefore admit closed-form solutions when using a mean-squared loss function. We provide experimental evidence that the models under inspection learn nearly identical solutions, and finally demonstrate that the simpler closed form solutions are superior forecasters across 72% of test settings.