Scalable estimation of VARMA models
2026-08-06 • Machine Learning
Machine Learning
AI summaryⓘ
The authors developed a new method to estimate VARMA models, which are complex time series models useful for capturing patterns with fewer parameters but have been too difficult to compute for large datasets. Their approach speeds up calculations so each step does not depend on the length of the data, making it much more practical for bigger problems. By reparametrizing the model and using clever math, they ensure the model stays stable and accurate without bias even as it scales. They show theoretically and empirically that their estimators perform well and are better than many existing methods on real-world datasets.
VARMAvector autoregressive moving-averagetime seriesstationarityinvertibilitymaximum a posterioripartial autocorrelationregularized least squaresParseval identityforecasting
Authors
Daniel Paulin, Victor Elvira
Abstract
Vector autoregressive moving-average (VARMA) models have long been considered impractical beyond moderate dimensions: the likelihood is non-convex, the parametrization is identified only up to equivalence, and every evaluation costs a pass over the entire series. Yet their moving-average term captures with a few parameters what a pure autoregression matches only with many lags. We introduce an estimation framework that removes this computational barrier: each optimization iteration is independent of the series length $T$. The framework combines a partial-autocorrelation reparametrization that guarantees stationarity and invertibility by construction, Gaussian priors on the reparametrized coefficients with separate scales for diagonal and off-diagonal entries, and losses that depend on the data only through fixed-size sufficient statistics, evaluated by a Parseval (Fourier) identity at near-linear cost in the truncation length. This yields two point estimators: a regularized least-squares fit and a covariance-marginalized maximum-a-posteriori estimator. We prove that both recover the infinite-autoregressive representation of the true process at a near-parametric rate in fixed dimension, so the truncation introduces no asymptotic bias. The same machinery extends, at the same leading cost, to seasonal dynamics, exogenous regressors (VARMAX), and rolling-window refits. Empirically, the estimators stay close to the oracle forecast error from $d=10$ to $d=40$ (where classical conditional MLE returns non-invertible fits whose forecasts diverge) and match or beat VAR, Bayesian-VAR, component-wise ARMA, and sparse-VARMA baselines on retail-demand, meteorological, and air-quality data. This brings likelihood-based VARMA estimation, at a per-iteration cost independent of the series length, to the problem sizes where practitioners have so far relied on VAR models.