Dynamic Spatial Durbin Models

• Dynamic Spatial Durbin Models (DSDM) is an econometric framework that captures both temporal lags and spatial spillovers in panel data.
• It differentiates between direct treatment effects and indirect network spillovers, enabling nuanced analysis of technology adoption in financial systems.
• DSDM employs rigorous estimation methods such as MLE, QMLE, and Bayesian MCMC to address simultaneity, feedback, and systemic risk.

A Dynamic Spatial Durbin Model (DSDM) is an econometric framework for panel data that incorporates both dynamic temporal dependencies and cross-sectional (spatial or network-driven) interactions in outcomes and treatments. In the context of empirical banking research, DSDM enables discrimination between direct effects of a covariate (e.g., technology adoption) on an entity’s outcome and indirect effects propagated through a defined inter-entity network. This structure captures simultaneity, feedback, and lagged contagion within a spatially connected system, making it salient for studying network spillovers and systemic risk in applications such as the diffusion of Generative AI in financial institutions (Kikuchi, 2 Feb 2026).

1. Model Structure and Mathematical Formulation

The DSDM augments a standard panel fixed effects (FE) specification for $N$ units over $T$ periods by including temporal lags, contemporaneous spatial lags, spatial–temporal lags, and lagged treatment network effects. Let $Y_{it}$ denote the outcome for unit $i$ at time $t$ (e.g., bank productivity), $D^{AI}_{it}$ a binary treatment indicator (e.g., Generative AI adoption), $X_{it}$ a conformable set of exogenous controls, and $W$ the $N \times N$ spatial weight matrix ($w_{ii}=0$). The scalar DSDM is:

$$\begin{aligned} Y_{it} &= \tau Y_{i,t-1} + \rho \sum_{j=1}^N w_{ij} Y_{jt} + \eta \sum_{j=1}^N w_{ij} Y_{j,t-1}\ &+ \beta D^{AI}_{it} + \theta \sum_{j=1}^N w_{ij} D^{AI}_{jt} + \gamma' X_{it} + \mu_i + \delta_t + \varepsilon_{it} \end{aligned}$$

Here, $\tau$ captures own temporal persistence, $\rho$ is the coefficient on the contemporaneous spatial lag, $\eta$ governs spatial–temporal lag (i.e., lagged outcome in peers), $\beta$ measures the direct treatment effect, and $\theta$ captures network spillovers in the treatment (Kikuchi, 2 Feb 2026).

Matrix notation for $Y_t \in \mathbb{R}^{N}$ at time $t$ is:

$$Y_t = \tau Y_{t-1} + \rho W Y_t + \eta W Y_{t-1} + \beta D^{AI}_t + \theta W D^{AI}_t + \Gamma X_t + \mu + \delta_t \iota_N + \varepsilon_t$$

2. Variable Specification and Parameter Definitions

• Dependent Variables:
  • $$ROA_{it} = 100 \times \frac{\text{Net Income}_{it}}{\text{Total Assets}_{it}}$$
  • $$ROE_{it} = 100 \times \frac{\text{Net Income}_{it}}{\text{Total Equity}_{it}}$$
• Treatment Variable:
  • $D^{AI}_{it}=1$ if GenAI keywords are detected in a bank’s 10-Q at $t$; $0$ otherwise.
• Controls ($X_{it}$):
  • $\ln(\text{Assets}_{it})$ (size), Tier-1 capital ratio, Digitalization index, CEO age.
• Parameters:
  • $\tau$: AR(1) serial persistence
  • $\rho$: Contemporaneous spatial autoregression
  • $\eta$: Spatial–temporal lag
  • $\beta$: Direct (own-adoption) effect
  • $\theta$: Network (spillover) effect
  • $\Gamma$: Control slopes
  • $\mu_i$, $\delta_t$: Unit and time effects
  • $\varepsilon_{it}$: Idiosyncratic error

3. Construction and Normalization of the Spatial Weight Matrix

The spatial weight matrix $W$ operationalizes peer influence. Each $w_{ij}$ encodes the relationship between units $i$ and $j$, normalized so $\sum_{j \neq i} w_{ij} = 1$ and $w_{ii} = 0$. Two principal constructions are employed:

• Network (Asset-Similarity) Weights:

  $$w_{ij}^{\mathrm{net,raw}} = \exp\left(-\frac{(\ln \bar{A}_i-\ln \bar{A}_j)^2}{2h^2}\right)$$

  Here, $\bar{A}_i$ is the time-average assets; $h$ is the standard deviation across $\ln \bar{A}_i$.

• Geographic (Headquarters Proximity) Weights:

  $$w_{ij}^{\mathrm{geo,raw}} = \exp\left(-\frac{d_{ij}}{d_{\text{median}}}\right)$$

  $d_{ij}$ is the Haversine distance; $d_{\text{median}}$ is the median pairwise distance.

Both forms are row-normalized: $$w_{ij} = \frac{w_{ij}^{\mathrm{raw}}}{\sum_{k \neq i} w_{ik}^{\mathrm{raw}}}$$

4. Estimation Approach and Identification

Simultaneity and feedback in $W Y_t$ induce endogeneity. Three estimation strategies are implemented:

• Maximum Likelihood (MLE):

  Maximizes the concentrated Gaussian log-likelihood, incorporating the Jacobian term $\ln\lvert I-\rho W\rvert$.

• Quasi-Maximum Likelihood (QMLE):

  Uses sandwich-robust standard errors for heteroskedastic or non-Gaussian $\varepsilon_{it}$.

• Bayesian MCMC:

  Applies weakly informative priors and a Gibbs–Metropolis sampler (10,000 draws, 5,000 burn-in), reporting posterior means and credible intervals.

Key identification rests on correct $W$, exogeneity of $X_{it}$, controlling for heterogeneity via $\mu_i$, and absorbing macro shocks via $\delta_t$ (Kikuchi, 2 Feb 2026).

5. Interpretation of Dynamic and Spatial Parameters

• Direct Effect $(\beta)$:

  Average productivity difference associated with own AI adoption; predominantly captures high-performing (frontier) firm selection.

• Spillover Effect $(\theta)$:

  Marginal effect of peer adoption (spatially weighted) on own productivity. Positive values indicate strategic complementarity and knowledge diffusion; negative values would indicate competitive displacement.

• Spatial Autoregression $(\rho)$:

  Immediate cross-sectional dependence—high values reflect tight synchronous outcome linkages.

• Temporal Persistence $(\tau)$ and Spatio-Temporal Lag $(\eta)$:

  AR(1) self-persistence and lagged neighbor effects; empirically, $\eta<0$ implies that lagged network shocks partially wash out.

Spatial feedback necessitates decomposition of treatment effects (LeSage & Pace, 2009):

$$\frac{\partial Y}{\partial D^{AI}} = S[\beta I_N + \theta W],\quad S=(I_N-\rho W)^{-1}$$

with

$$\begin{aligned} \text{Direct} &= \frac{1}{N} \mathrm{tr}[S\beta I_N] \ \text{Indirect} &= \frac{1}{N} \iota_N'(S\theta W)\iota_N \ \text{Total} &= \text{Direct} + \text{Indirect} \end{aligned}$$

6. Model Diagnostics and Robustness Checks

Model adequacy is validated via:

• Spatial autocorrelation:
  • Moran’s $I$ on residuals, Lagrange multiplier tests (LM-lag, LM-error) for omitted spatial structure.
• Serial correlation:
  • Wooldridge’s AR(1) panel test on residuals.
• Robustness to $W$ choice:
  • Re-estimation with loan-profile, Fed-district, and geographic matrices; $\theta$ and $\rho$ remain significant and positive.
• Placebo and falsification:
  • Pre-treatment assignments (pre-ChatGPT period) yield null $\theta$, and random adopter permutation nullifies spillovers.
• Sensitivity to control set:
  • Lagged controls, non-linear size, and industry-trend interactions do not overturn coefficient sign/magnitude.

7. Empirical Results: Interbank Spillovers and Systemic Coupling

Under the preferred network specification ($W_{\mathrm{network}}$), key posterior means (Bayesian MCMC) for 2018–2025 U.S. bank panel data (Kikuchi, 2 Feb 2026):

Parameter	ROA	ROE
Direct effect $(\beta)$	$0.0373$ (0.0139)	$0.4199$ (0.0927)
Spillover $(\theta)$	$0.1606$ (0.0508)	$0.6787$ (0.3361)
Spatial AR $(\rho)$	$0.6166$ (0.0170)	$0.7453$ (0.0132)
Temporal persistence $(\tau)$	$0.6363$ (0.0047)	$0.6967$ (0.0044)
Spatio-temporal lag $(\eta)$	$-0.2898$ (0.0168)	$-0.4653$ (0.0134)

• Spillover decomposition: Over 80% of total productivity gain from AI adoption arises indirectly through network effects rather than directly via own adoption.
  • ROA: Direct = 0.0923, Indirect = 0.4289 $\rightarrow$ Total = 0.5212
  • ROE: Direct = 0.7634, Indirect = 3.8941 $\rightarrow$ Total = 4.6575
• Heterogeneity: For large (top-25%) banks, $\theta_{ROE}=3.13$ (p < 0.01); a one-SD increase in peer adoption raises own ROE by 313 basis points. Small banks display negligible spillovers.

These results indicate strong “strategic complementarity” in GenAI adoption. Constructive synchronization within the system—termed “algorithmic coupling”—enhances knowledge diffusion but introduces a new channel for systemic risk. Technical faults, vendor outages, or model drift in widely deployed AI solutions can generate correlated shocks, amplifying systemic vulnerability. Elevated $\rho$ and $\theta$ serve as quantitative early-warning indicators, suggesting that regulatory focus is warranted on vendor concentration, AI model diversity, and coordinated stress-testing.

Summary

Dynamic Spatial Durbin Models provide a rigorous, quantitatively interpretable framework to analyze the intertwined temporal and network spillovers of technology adoption in financial systems. The DSDM captures both the productivity synergies and systemic risks engendered by the increasing algorithmic coupling of U.S. banks during the GenAI adoption era (Kikuchi, 2 Feb 2026).