How Good Can Linear Models Be
for Time-Series Forecasting?

The adaptive context story

One forecaster, eight datasets, eight different optima.

Time-series forecasting research has been moving steadily toward larger architectures, on the assumption that capacity is what unlocks accuracy. Most of that gap closes with cheaper preprocessing tuning, not bigger models. Ridge regression makes the test concrete: a closed-form solution and interpretable weights let the tuned hyperparameters be read off the search directly. The widget below shows what the search finds. Drag the horizon slider, switch between datasets, and watch the optimal lookback span grow, plateau, or shrink — set by each dataset's stationarity, not by $H$.

Pick a dataset above; the colored span shows $L^\star(H)$ for that dataset and horizon, anchored at the prediction-start line. The numbers are the full searched optima per dataset.

Live playground

Try the search yourself.

Pick a dataset, series, and forecast horizon, then drag the lookback $L$ and regularization $\alpha$ sliders to refit closed-form Ridge in your browser. The remaining hyperparameters (scaler scope and method, augmentation, local-norm fraction $r$) stay pinned to the searched optimum for the current horizon; autotune snaps $L$ and $\alpha$ there too. Bundled datasets: ETTh1, ETTh2, and Exchange.

Method

Closed-form Ridge over a searched preprocessing space.

The pipeline is small: sliding windows → scaling → optional augmentation → closed-form Ridge. Optuna searches the preprocessing stages while the model class stays fixed. Ridge has a closed form, $\boldsymbol{\beta} = (X^\top X + \alpha I)^{-1} X^\top \mathbf{y}$, so each trial is a few-millisecond Cholesky solve.

Two grouping levers regularize the search: $g_h$ groups consecutive horizons to share one HP setting (default 48; barely costs accuracy), and $g_s$ groups series similarly (the optimal value is dataset-specific, ranging from fully shared on ETTh1 to fully per-series on Weather).

Three diagnostic patterns

What the search reveals about the data.

Efficiency / accuracy tradeoffs

How cheap can the search get?

Forecasts

What the tuned model predicts.

The same locality and seasonality the search recovers in the hyperparameters are visible directly in the trained weights. See Figure 4 in the paper for the full lag-by-horizon weight heatmap across the eight datasets.

Benchmark results

Long-term multivariate forecasting (MSE).

Per-horizon test MSE on the eight standard benchmarks. Within the linear class, tuned Ridge takes the best average on seven of eight datasets and ties FITS on ETTh2. Against nonlinear baselines, it wins the average on six of eight benchmarks. Bold marks the best per row; underline marks the second-best. OLS, FITS, and DLinear are reproduced with instance normalization from Toner et al. (2024); nonlinear baselines come from their original publications.

Limitations

Scope notes.

The eight benchmarks are smooth sensor and energy series; the lookback-scaling and locality patterns we report are best read as findings on this regime rather than universal claims. We follow prior work in reporting point-estimate MSE; very small margins should be treated as ties. Nonlinear baselines are quoted from their original publications, so comparisons reflect their published configurations rather than a jointly retuned protocol. Per-series search scales with channel count; finer-grained search is an open direction. The study fixes the model class to Ridge. Applying the same preprocessing search to deeper models is the obvious next step.

Citation

BibTeX

Copy@article{huang2026linear,
  title   = {How Good Can Linear Models Be for Time-Series Forecasting?},
  author  = {Huang, Lang and Xu, Jinglue and Darlow, Luke},
  journal = {arXiv preprint arXiv:2606.27282},
  year    = {2026}
}