Estimating Firm-level Production Functions with Spatial Dependence in Output, Input, and Productivity

This paper proposes a three-stage efficient GMM estimation algorithm for estimating firm-level production functions given spatial dependence across firms due to supplier/customer relationships, sharing of input markets, or knowledge spillover. The procedure builds on Ackerberg, Caves and Frazer (2015) and Wooldridge (2009), but in addition, allows the productivity process to depend on the lagged output levels and lagged input usages of related firms, and potential spatially correlated productivity shocks across firms, where the set of related firms can differ across the three dimensions of spatial dependence. We establish the asymptotic properties of the proposed estimator, and conduct Monte Carlo simulations to validate these properties. The estimator is consistent under DGPs with or without spatial dependence, and with strong/weak or positive/negative spatial dependence. In contrast, the conventional estimators lead to biased estimates of the production function parameters when the underlying DGPs have spatial dependence structure, and the magnitudes of the bias increase with the strength of spatial dependence in the underlying DGPs. We apply the proposed estimation algorithm to a Japanese firm-to-firm dataset for the period 2009–2018. We find significant and positive spatial coefficients in the Japanese firm-level productivity process via all three channels proposed above.