Optimal Break-Point Estimation Rate in Grouped Time-Varying Network VAR

§ Problem Statement

Setup

Let $\{X_t\}_{t=1}^T$ be an $N$ -dimensional time series, $X_t=(X_{1t},\dots,X_{Nt})^\top\in\mathbb R^N$ , generated by a single-break grouped time-varying VAR(1):

X_t=A\!\left(\frac tT\right)X_{t-1}+\varepsilon_t,\qquad t=1,\dots,T,

where $\{\varepsilon_t\}$ is a zero-mean innovation sequence (for example, a martingale difference array with uniformly bounded sub-Gaussian tails), and the coefficient matrix is piecewise smooth with one unknown break fraction $\tau_0\in(0,1)$ :

A(u)= \begin{cases} A^{(1)}(u), & u\le \tau_0,\\ A^{(2)}(u), & u>\tau_0, \end{cases} \qquad t_0:=\lfloor T\tau_0\rfloor .

Assume $A^{(1)}$ and $A^{(2)}$ satisfy the regularity conditions used for local-linear estimation away from the break (in particular, smoothness on each side and stability of the VAR), and that the jump at $\tau_0$ is nonzero.

The grouped structure means there is an unknown partition of nodes into latent groups; following the paper's notation for the Stage-1 rate, let $\bar n=\bar n_{NT}$ denote the network-size quantity entering that bound. Let $h^{\ddagger}$ and $\bar h^{\ddagger}$ be the bandwidth sequences used in the Stage-1 break-detection procedure, with $h^{\ddagger}\to0$ and $\bar h^{\ddagger}\to0$ as $N,T\to\infty$ .

This setup follows Li et al. (2023).

For the Stage-1 break estimator $\hat t$ , it is known that

\left|\frac{\hat t-t_0}{T}\right|=O_P(r_{NT}),\qquad r_{NT}:=\sqrt{\frac{\bar n\,\bar h^{\ddagger}\log(N\vee T)}{T}}+(h^{\ddagger})^2,

where $N\vee T:=\max\{N,T\}$ .

An

Unsolved Problem

Construct a break-location estimator $\tilde t$ (for example, based on one-sided local-linear smoothing near the candidate break) and prove a strictly faster rate

\left|\frac{\tilde t-t_0}{T}\right|=O_P(a_{NT}),\qquad a_{NT}=o(r_{NT}),

uniformly over an appropriate class of single-break grouped time-varying network VAR data-generating processes. A further open objective is to determine the optimal achievable rate $b_{NT}$ (equivalently, a minimax rate) via matching upper and lower bounds:

\inf_{\breve t}\sup_{P\in\mathcal P}\Pr_P\!\left(\left|\frac{\breve t-t_0}{T}\right|>C\,b_{NT}\right)\to0\ \text{for some }C>0,

and

\inf_{\breve t}\sup_{P\in\mathcal P}\Pr_P\!\left(\left|\frac{\breve t-t_0}{T}\right|>c\,b_{NT}\right)\not\to0\ \text{for some }c>0,

where $\mathcal P$ is the model class above. Identifying a normalization $s_{NT}$ and a nondegenerate limit law $F$ for $s_{NT}(\tilde t-t_0)$ is also an open objective. These minimax/lower-bound/limit-distribution targets are research goals rather than established results in the cited paper. See Li et al. (2023) for context.

§ Discussion

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§ Significance & Implications

Li et al. (2023) notes that the proved Stage-1 break-rate is conservative and may be improvable. Sharper break estimation would directly affect downstream accuracy of estimated group numbers, memberships, and coefficient functions before/after the break.

§ Known Partial Results

Theorem 5.1(i) proves consistency and the conservative Stage-1 rate above for the scaled break estimator. Remark 5.2(i) states that one-sided local-linear smoothing may improve the approximation rate, but the paper does not provide a sharper proved rate, minimax lower bound, or limit law for the break-location estimator.

§ Tags

change-point convergence-rate time-varying-var network-var latent-groups