All proximities matter: A Bayesian combination approach to inter-firm spillovers and economic performance

Technical Summary: A Bayesian Combination Approach to Inter-firm Spillovers and Economic Performance

Problem Statement
Firm performance, particularly in knowledge-intensive sectors, is driven not only by internal capabilities but also by proximity to other firms, which facilitates knowledge spillovers. While traditional literature acknowledges that proximity extends beyond geography to include technological, cognitive, and industrial dimensions, empirical approaches often rely on synthetic network indices (e.g., centrality measures) or single-matrix spatial models. These methods frequently overlook the multifaceted nature of inter-firm connections and the complex interdependencies between different proximity dimensions. Furthermore, existing maximum-likelihood approaches for combining multiple proximity matrices are often limited to a few dimensions, and standard spatial models may fail to capture the joint contribution of multiple proximity types or the uncertainty regarding which specific dimensions drive spillovers.

Methodology
The authors propose a unified framework that integrates multiple proximity networks into a Spatial Durbin Model (SDM) using a Bayesian convex combination approach.

1. Semantic Proximity Construction:

   • Instead of relying on static industry codes or patent classes, the study constructs six proximity matrices derived from the textual content of corporate websites.
   • Using Latent Dirichlet Allocation (LDA) and natural language processing (NLP), firms are mapped onto common topics to extract vectors representing their technologies, knowledge bases, competencies, and specializations.
   • Pairwise cosine similarities are transformed into an inverted-U function (following Boschma, 2005; Nooteboom et al., 2007) to construct proximity matrices. This transformation posits that moderate similarity maximizes spillovers, while extreme similarity or dissimilarity hinders them.
   • The six matrices include: Geographical, General, Technological, Cognitive, Competence-based, and Industrial.

2. Bayesian Convex Combination in SDM:

   • The study extends the convex combination framework (Debarsy and LeSage, 2022) from the Spatial Autoregressive (SAR) model to the Spatial Durbin Model (SDM).
   • A combined weight matrix $W_c(\Gamma)$ is defined as a weighted sum of the $L$ individual proximity matrices: $W_c(\Gamma) = \sum_{\ell=1}^{L} \gamma_\ell W_\ell$, where $\gamma_\ell$ are weights summing to 1.
   • Model Specification: The authors propose a specific formulation where the same combined weight matrix $W_c(\Gamma)$ governs both the spatial lag of the dependent variable and the spatial lags of the covariates: $y = \rho W_c(\Gamma)y + X\beta + W_c(\Gamma)X\theta + \epsilon$. This differs from previous formulations that applied the combination only to the dependent variable's lag, offering better interpretability of the $\gamma$ parameters and greater parsimony.
   • Estimation: The model is estimated using Markov Chain Monte Carlo (MCMC) methods. The approach employs flat priors and approximates the Jacobian term using a Monte Carlo trace estimator to handle the varying weight matrices across draws.
   • Model Uncertainty: Bayesian Model Averaging (BMA) is utilized to address model uncertainty. The authors employ a backward selection procedure to identify the optimal subset of proximity matrices, comparing models based on log-marginal likelihoods and posterior probabilities.

3. Impact Calculation:

   • Direct, indirect (spillover), and total impacts are derived from the partial derivatives of the reduced-form SDM.
   • To manage computational complexity, the authors use a trace-based approximation strategy (expanding the inverse matrix into an infinite series) to calculate scalar summary measures of impacts from the posterior distributions.

Empirical Application and Results
The framework is applied to a dataset of 505 firms within Milan's tech ecosystem, using sales growth (2022–2023) as the dependent variable.

• Model Selection: Through backward elimination and BMA, the study identifies a specification with four proximity matrices (Geographical, Technological, Cognitive, and Industrial) as the best fit, achieving a posterior probability of over 75%. This specification significantly outperforms both the full six-matrix model and all SAR specifications (which received zero posterior probability).
• Proximity Weights: In the best specification, Technological proximity carries the highest weight ($\gamma \approx 0.32$), followed by Cognitive proximity ($\gamma \approx 0.29$) and Industrial proximity ($\gamma \approx 0.25$). Geographical proximity remains significant but secondary ($\gamma \approx 0.13$). General and competence-based proximities were found to be redundant and excluded from the optimal model.
• Spillover Effects:
  • The spatial autoregressive parameter ($\rho$) is high ($\approx 0.76$), indicating strong overall dependence.
  • Firm Size (Sales): Shows a negative direct effect but a positive indirect effect, suggesting compensatory spillovers from neighbors' scale.
  • Knowledge Base (Employees): Exhibits strong positive direct and indirect effects, confirming that human capital expansion drives both internal growth and external spillovers.
  • Network Centrality: When using the full proximity matrix, the significance of individual network centrality indices (e.g., technological centrality) diminishes or changes, suggesting that the proximity matrix captures the underlying relational structure more effectively than synthetic centrality measures.

Key Contributions and Significance
The paper claims three primary contributions to the literature:

1. Methodological Extension: It is the first application of the Bayesian convex combination approach to the Spatial Durbin Model at the firm level. The authors argue this formulation is superior to previous SAR-based or alternative SDM specifications because it ensures the same spatial structure governs both endogenous interactions and exogenous spillovers, improving parameter interpretability and reducing the number of estimated parameters.
2. Data Innovation: By deriving proximity measures from corporate website text rather than static industry codes or patent classes, the study captures the dynamic, self-reported nature of firm activities and knowledge profiles. This allows for a more up-to-date and granular understanding of inter-firm similarity.
3. Empirical Insights: The study demonstrates that multidimensional proximity networks drive firm performance through specific channels. It finds that while geographical proximity matters, technological and cognitive alignments are the dominant drivers of spillovers in Milan's tech ecosystem. The results suggest that policy interventions should prioritize semantic alignments (e.g., technology matching, knowledge transfer) over simple physical co-location.

Conclusion
The paper concludes that preserving the full topology of inter-firm connections via adjacency matrices, rather than relying on synthetic centrality measures, provides a more robust understanding of spillover mechanisms. The Bayesian framework successfully handles model uncertainty, identifying that a subset of four proximity dimensions (Geographical, Technological, Cognitive, Industrial) is sufficient to explain the complex dependence structure in the Milan tech ecosystem. These findings offer concrete guidance for place-based innovation policies, suggesting that fostering technological and cognitive linkages may be more effective than generic clustering incentives.