Production function estimation using nonparametric shape constrained methods can be computationally challenging. For the standard additive error model, y = f(x) + ε, with monotonicity and concavity constraints imposed at each observation, the associated programming problem has a quadratic objective function and linear inequality constraints and thus is easy from a computational complexity point of view. However, the number of constraints grows at the rate n2. Thus, estimation on a grid, as described in Preciado Arreola et al. (2016) and Yagi et al. (2018), can reduce the computational difficulty significantly. In contrast, the multiplicative error model, y = f(x) eε, which models heteroskedasticity in the error term with monotonicity and concavity constraints has a quasi-convex objective function requiring generally non-linear optimization methods. We discuss these and other computational issues related to nonparametric shape constrained methods.
