Scalable estimation of VARMA models

2026-08-06

AI Summary

This work addresses the practical limitations of high-dimensional VARMA models, which suffer from non-convex likelihoods, parameter non-identifiability, and computational complexity that scales with time series length. The authors propose the first scalable maximum likelihood estimation framework for VARMA whose per-iteration cost is independent of sequence length. Their approach employs a partial autocorrelation reparameterization to guarantee stationarity and invertibility, leverages Parseval’s identity to accelerate Fourier-domain computations, and introduces a Gaussian prior with diagonal/off-diagonal scale separation, enabling optimization using only fixed-size sufficient statistics. The method accommodates seasonality, exogenous variables, and rolling updates without asymptotic bias. In dimensions d = 10–40, it achieves near-oracle prediction error and substantially outperforms VAR, Bayesian VAR, component-wise ARMA, and sparse VARMA, as demonstrated on retail, meteorological, and air quality datasets.

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.

Problem

• VARMA
• scalable estimation
• high-dimensional time series
• computational barrier
• non-convex likelihood

Innovation

• VARMA
• scalable estimation
• reparameterization
• Fourier-based sufficient statistics
• invertibility