Selected papers

Background

Spatial autoregressive (SAR) and related models require a spatial weighting matrix \(W\). In practice, \(W\) is often assumed or selected from a small candidate set, which risks misspecification. This work proposes a data-driven method to estimate the full matrix \(W\) while also allowing for an unknown number of location-specific structural breaks in the mean.

Key takeaways

• Joint problem: Spatial dependence and mean breaks can be easily confounded. The method estimates both simultaneously.
• Two-stage adaptive LASSO: Stage 1 finds candidate change points per location; Stage 2 estimates \(W\) and local mean paths with constraints.
• Model-based selection of \(\lambda\): A criterion aligns sample and model-implied spatial autocorrelation to choose the penalty parameter.
• Output: A sparse, interpretable estimate of \(W\) (nonnegative, row sums \(\le 1\)) and location-wise mean trajectories over time.
• Use cases: Panels with possible regime shifts (policy, markets) where spatial link strength and directionality are unknown.

Intuition

If means jump at different times across locations, those jumps can mimic spatial spillovers; conversely, strong spatial spillovers can look like simultaneous breaks. The estimator separates these by first over-detecting plausible change points in each local series, then letting the full spatiotemporal model retain only those needed while estimating \(W\).

Model

Observations at time \(t=1,\dots,T\) and locations \(s_1,\dots,s_n\):

\[ Y_t = W Y_t + a_t + \varepsilon_t,\quad \varepsilon_t \sim \mathcal{N}(0,\sigma^2 I_n),\qquad \text{with } \rho(W) < 1,\; \mathrm{diag}(W)=0,\; W\ge 0. \]

The vector \(a_t\) collects location-specific mean levels that may change at unknown times (structural breaks, not necessarily synchronized across locations). Reduced form:

\[ Y_t = (I - W)^{-1} a_t + (I - W)^{-1}\varepsilon_t. \]

Estimation (two-stage adaptive LASSO)

1. Stage 1: change-point candidates (per location). For each location \(i\), fit a lower-triangular “step” design with adaptive LASSO to detect level changes in the overall mean trajectory \(\alpha_{t,i}\) (which already contains spillovers). Collect candidate change times \(\mathcal{T}^{(1)}_i\).
2. Stage 2: joint estimation of \(W\) and local means. Stack all times/locations and solve a constrained LASSO: \[ \min_{\tilde a,\;\xi}\;\|Z\xi + \Phi \tilde a - y\|_2^2 + \lambda_b \|\xi\|_1, \] subject to \( \tilde a_{t,i}=0\) for \(t\notin \mathcal{T}^{(1)}_i\), \(w_{ii}=0\), \(w_{ij}\ge 0\), and \(\sum_j w_{ij}\le 1\) for all \(i\). Here \( \xi=\mathrm{vec}(W)\), \( \Phi\) maps change increments to levels, and \(Z = I_n \otimes Y\).

Penalty selection

Standard CV on prediction error can neglect spatial error correlation. The paper proposes selecting \(\lambda_b\) by matching the sample spatial autocorrelation (from mean-adjusted data) with the model-implied spatial autocorrelation (from \((I-W)^{-1}\varepsilon\)), e.g., via an MAE between correlation matrices. This targets the correct strength and pattern of \(W\).

Practical guidance

• Diagnostics: If residual maps suggest level shifts at different times across locations, and SAR estimates depend heavily on the chosen \(W\), consider this estimator.
• Weights interpretation: Estimated links need not be purely geographic; they may reflect network or functional proximity. Inspect on a map or network graph.
• Directionality: In short panels with few breaks, one-way links can be partially mirrored (weak reverse links). Focus on the stronger edge.
• Constraints: Nonnegativity and row-sum \(\le 1\) stabilize estimation and ensure \((I-W)\) invertibility in practice.

Application

The method is illustrated with Berlin condominium prices (1995–2014), where both spatial spillovers and structural breaks are evident. It detected location-specific regime shifts (e.g. around the 2008 financial crisis) and recovered a spatial weights matrix \(W\) that highlighted heterogeneous connections between districts. Importantly, the analysis shows that intra-city dependence often extends beyond simple geographical proximity, reflecting similarities in culture, lifestyle, or socio-economic conditions. This provides a more realistic picture of market dynamics than standard SAR models with pre-specified weights and is particularly valuable for real-estate pricing, where overlooking such hidden dependencies can lead to biased valuations and misguided investment or policy decisions. The approach is broadly relevant for urban economics, finance, and regional studies, where simultaneous spatial spillovers and regime changes are common.