Inference on Dynamic Spatial Autoregressive Models with Change Point Detection

TL;DR

This work develops a dynamic spatial autoregressive framework that allows the spillover structure to be a time-varying, sparse linear combination of multiple pre-specified weight matrices. By embedding instrumental-variable–based adaptive LASSO in a profiled regression setup, the authors establish oracle properties for variable selection, derive convergence rates, and prove the consistency of the time-varying spatial weight estimators. The framework further extends to change-point settings, enabling consistent detection of thresholds and structural shifts in both coefficient regimes and spatial weights, with substantial computational advantages. Through simulations and real-data applications (notably stock returns), the approach demonstrates strong statistical performance and practical utility for uncovering evolving spatial spillovers while mitigating model-uncertainty in weight matrices. Overall, the methodology provides a principled, scalable tool for data-driven spatial structure learning and change-point analysis in panel-like spatio-temporal contexts.

Abstract

We analyze a varying-coefficient dynamic spatial autoregressive model with spatial fixed effects. One salient feature of the model is the incorporation of multiple spatial weight matrices through their linear combinations with varying coefficients, which help solve the problem of choosing the most ``correct'' one for applied econometricians who often face the availability of multiple expert spatial weight matrices. We estimate and make inferences on the model coefficients and coefficients in basis expansions of the varying coefficients through penalized estimations, establishing the oracle properties of the estimators and the consistency of the overall estimated spatial weight matrix, which can be time-dependent. We further consider two applications of our model in change point detections in dynamic spatial autoregressive models, providing theoretical justifications in consistent change point locations estimation and practical implementations. Simulation experiments demonstrate the performance of our proposed methodology, and real data analyses are also carried out.

Figures (6)

• Figure 1: Histogram of $T^{1/2} (\widehat{\mathbf{R}}_{\widehat{H}} \widehat{\mathbf{S}}_{\boldsymbol{\gamma}} \widehat{\mathbf{R}}_{\beta} \widehat{\boldsymbol{\Sigma}}_{\beta} \widehat{\mathbf{R}}_{\beta}^\top \widehat{\mathbf{S}}_{\boldsymbol{\gamma}}^\top \widehat{\mathbf{R}}_{\widehat{H}}^\top)^{-1/2} (\widehat{\bm{\phi}}_{\widehat{H}} -\bm{\phi}_{\widehat{H}}^\ast)$ for $(T,d) =(200,50)$, shown for the first coordinate (left panel) and the second coordinate (right panel). The red curves are the empirical density, and the black dotted curves are the density for $\mathcal{N}(0,1)$.
• Figure S1: Estimator performance for model \ref{['eqn: sim_dgp_endo']} with $m$ varying from $0$ to $100$, under the setting $(T,d)=(50,25)$. The left panel shows the MSE for $\widehat{\bm{\phi}}$ (pink), $\widehat{\bm{\beta}}$ (teal), and $\widehat{\boldsymbol{\mu}}$ (purple); the right panel shows the specificity (red) and sensitivity (blue) of $\widehat{\bm{\phi}}$. Each value is averaged over 500 repetitions.
• Figure S3: Histograms of estimated change locations under true models \ref{['eqn: sim_dgp_remark1']} (left panel) and \ref{['eqn: sim_dgp_remark1_nochange']} (right panel). Both experiments are repeated 500 times.
• Figure S4: Illustration of the total profits of industrial enterprises within considered Chinese provinces and direct-administered municipalities, in 100 million yuans.
• Figure S5: Illustration of $\widehat{\mathbf{W}}_t$ of Model 2 in \ref{['eqn: realdata_NBS_alternative']} among Beijing, Shanghai and Guangdong, from August 2016 to August 2024.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 1
• Corollary 2
• Corollary 3
• Remark 1
• Corollary 4
• Lemma 1
• Lemma 2