Production Function Estimation Robust to Flexible Timing of Labor Input

Control function approaches to estimating production functions rely on a proxy that is monotone in an unobserved scalar productivity conditioning on other state variables. Ackerberg, Caves, and Frazer (2015) point out a potential functional dependence problem when conditioning only on capital. They provide a simple solution by conditioning on both capital and labor. This approach allows flexible timing of labor input for when firms learn all or part of their productivity. However, we demonstrate that ACF's moment condition may suffer from weak identification, and its significance depends on the timing of labor input. We then propose easy-to-implement modified procedures that remedy these issues, and provide Monte Carlo evidence. Estimation of production functions with flexible timing assumption of inputs is important because it can allow for unobserved, serially correlated, firm-specific wage shocks, labor being chosen prior to other variable inputs, or labor having dynamic implications. Moreover, since the exact timing of input choices is unknown and even may differ across firms in practice, our proposal is valuable as it fully incorporates the flexible framework but avoids weak identification