Paulin & Elvira's VARMA framework scales to d=40, beats VAR baselines

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 the moving-average term captures with a few parameters what a pure autoregression matches only with many lags. In a new paper, statisticians Daniel Paulin (University of Edinburgh) and Victor Elvira (University of Edinburgh) introduce an estimation framework that removes this computational barrier, making per-iteration cost independent of series length T.

The framework combines a partial-autocorrelation reparametrization that guarantees stationarity and invertibility by construction, Gaussian priors on coefficients with separate scales for diagonal and off-diagonal entries, and losses depending on data only through fixed-size sufficient statistics, evaluated via a Parseval (Fourier) identity at near-linear cost in truncation length. This yields two point estimators: a regularized least-squares fit and a covariance-marginalized maximum-a-posteriori estimator. The authors prove both recover the infinite-autoregressive representation at near-parametric rates in fixed dimension, so truncation introduces no asymptotic bias. The machinery extends to seasonal dynamics, exogenous regressors (VARMAX), and rolling-window refits at the same leading cost.

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 to problem sizes where practitioners have so far relied on simpler VAR models, opening the door to more expressive time-series modeling in production forecasting systems.