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

This paper proposes a three-stage GMM estimation procedure for estimating firm-level productivity in the presence of potential spatial dependence across firms via the product market, the input market, and the supply chain. 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 input usages of related firms, and to accommodate spatially correlated productivity shocks across firms. The procedure provides the estimates of the production function parameters (the capital and labor shares in value-added, and the degree of serial correlation in the productivity process), and the spatial dependence parameters (of productivity on related firms’ past outputs and inputs, and current innovation shocks), 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. In particular, our proposed estimator is consistent under DGPs with or without spatial dependence. In contrast, the conventional estimators are biased when the true DGPs are indeed characterized by spatial dependence.