  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.
