Stable Reduced-Rank VAR Identification

Abstract

The vector autoregression (VAR) has been widely used in system identification, econometrics, natural science, and many other areas. However, when the state dimension becomes large the parameter dimension explodes. So rank reduced modelling is attractive and is well developed. But a fundamental requirement in almost all applications is stability of the fitted model. And this has not been addressed in the rank reduced case. Here, we develop, for the first time, a closed-form formula for an estimator of a rank reduced transition matrix which is guaranteed to be stable. We show that our estimator is consistent and asymptotically statistically efficient and illustrate it in comparative simulations.

Figures (6)

• Figure 1: Pole locations of the first $50$ repeats.
• Figure 2: Pole magnitude histograms of the $1000$ repeats: $*$ are the true poles. Unstable poles for LS for each $T$: $25.9\%, 6.9\%, 0.2\%$.
• Figure 3: Histograms of estimation errors $e_{ {{RLS}}},e_{ {{RFB}}}$ and prediction errors $\epsilon_{ {{RLS}}},\epsilon_{ {{RFB}}}$: the red '$*$' marks and the numbers at the right upper corners are the medians and the blue '$|$' marks are the upper and lower quantiles.
• Figure 4: Computational times $t_{comp}$ in log scale: $t_{comp}$ for LS and FB are almost identical and $\log(t_{comp})$ is almost linear with $\log n$. The average computational time is about $49.6$s for model order $n=3072$.
• Figure 5: Box plots of estimation errors $e_{RLS},e_{RFB}$ plotted against $k=\log_2(n/6)$: The errors of both estimators distribute very similarly and their medians converge to constants as the order $n$ grows with $T$ being a multiple of $n$.