Sparse Vector Autoregressive Modeling

The vector autoregressive (VAR) model has been widely used for modeling temporal dependence in a multivariate time series. For large (and even moderate) dimensions, the number of AR coefficients can be prohibitively large, resulting in noisy estimates, unstable predictions and difficult-to-interpret temporal dependence. To overcome such drawbacks, we propose a 2-stage approach for fitting sparse VAR (sVAR) models in which many of the AR coefficients are zero. The first stage selects non-zero AR coefficients based on an estimate of the partial spectral coherence (PSC) together with the use of BIC. The PSC is useful for quantifying the conditional relationship between marginal series in a multivariate process. A refinement second stage is then applied to further reduce the number of parameters. The performance of this 2-stage approach is illustrated with simulation results. The 2-stage approach is also applied to two real data examples: the first is the Google Flu Trends data and the second is a time series of concentration levels of air pollutants.

💡 Research Summary

The paper addresses the well‑known “parameter explosion” problem in vector autoregressive (VAR) modeling of multivariate time series when the number of series (K) and the autoregressive order (p) are moderate to large. A fully parameterized VAR(p) contains K²·p coefficients, which leads to noisy estimates, unstable forecasts, and difficult interpretation. To obtain a parsimonious “sparse VAR” (sVAR) model, the authors propose a two‑stage procedure that exploits the partial spectral coherence (PSC) – a frequency‑domain measure of conditional dependence between two series after removing the linear effect of all other series.

Stage 1 – PSC‑based group selection.
For each unordered pair (i, j) of series, the PSC is defined as the normalized cross‑spectral density of the residual series obtained after optimal linear filtering out the remaining K‑2 series. Using the inverse spectral density matrix gY(ω)=fY(ω)⁻¹, PSCij(ω)=−gYij(ω)/√