Census of manufacturing data is gathered by statistical agency around the world and can be used to benchmark performance within industries or summarize production behavior within industries. In this talk, I present results our laboratory have found for data from the U.S., Japan and Chile. I present some of the modeling challenges and limitations of the data. I focus on the use of nonparametric shape constrained production function estimators and describe some of the benefits that result from their use. I close with a variety of policy insights that can be gained from analyzing Census of manufacturing data.

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) e^ε, 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.

Assumptions motivated by either logical or existing theory can be imposed during model estimation to restrict the feasible region of the parameters. The restrictions, implemented as shape constraints, may not provide any benefits in an asymptotic analysis, but will improve the estimator’s finite sample performance. This paper briefly reviews an illustrative set of research on shape constrained estimation in the economics and operations research literature. We highlight the methodological innovations and applications, with a particular emphasis on utility functions, production economics, and sequential decision making applications.

We develop a new approach to estimate a production function based on the economic axioms of the Regular Ultra Passum law and convex non-homothetic input isoquants. Central to the development of our estimator is stating the axioms as shape constraints and using shape constrained nonparametric regression methods.
We implement this approach using data from the Japanese corrugated cardboard industry from 1997-2007. Using this new approach, we find most productive scale size is a function of the capital-to-labor ratio and the largest firms operate close to the largest most productive scale size associated with a high capital-to-labor ratio. We also measure the productivity growth across the panel periods based on the residuals from our axiomatic model. We find that capital-to-labor ratio is negatively correlated with the productivity growth due to the over-investment in capital.

We examine a novel way of imposing shape constraints on a local polynomial kernel estimator. The proposed approach is referred to as Shape Constrained Kernel-weighted Least Squares (SCKLS). We prove uniform consistency of the SCKLS estimator with monotonicity and convexity/concavity constraints and establish its convergence rate.
In addition, we propose a test to validate whether shape constraints are correctly specified.
The competitiveness of SCKLS is shown in a comprehensive simulation study. Finally, we analyze Chilean manufacturing data using the SCKLS estimator and quantify production in the plastics and wood industries. The results show that exporting firms have significantly higher productivity.

We examine a novel way of imposing shape constraints on a local polynomial kernel estimator. The proposed approach is referred to as Shape Constrained Kernel-weighted Least Squares (SCKLS). We prove uniform consistency of the SCKLS estimator with monotonicity and convexity/concavity constraints and establish its convergence rate.
In addition, we propose a test to validate whether shape constraints are correctly specified.
The competitiveness of SCKLS is shown in a comprehensive simulation study. Finally, we analyze Chilean manufacturing data using the SCKLS estimator and quantify production in the plastics and wood industries. The results show that exporting firms have significantly higher productivity.

We examine a novel way of imposing shape constraints on a local polynomial kernel estimator. The proposed approach is referred to as Shape Constrained Kernel-weighted Least Squares (SCKLS). We prove uniform consistency of the SCKLS estimator with monotonicity and convexity/concavity constraints and establish its convergence rate.
In addition, we propose a test to validate whether shape constraints are correctly specified.
The competitiveness of SCKLS is shown in a comprehensive simulation study. Finally, we analyze Chilean manufacturing data using the SCKLS estimator and quantify production in the plastics and wood industries. The results show that exporting firms have significantly higher productivity.

We develop a new approach to estimate a production function based on the economic axioms of the Regular Ultra Passum law and convex non-homothetic input isoquants. Central to the development of our estimator is stating the axioms as shape constraints and using shape constrained nonparametric regression methods.
We implement this approach using data from the Japanese corrugated cardboard industry from 1997-2007. Using this new approach, we find most productive scale size is a function of the capital-to-labor ratio and the largest firms operate close to the largest most productive scale size associated with a high capital-to-labor ratio. We also measure the productivity growth across the panel periods based on the residuals from our axiomatic model. We find that capital-to-labor ratio is negatively correlated with the productivity growth due to the over-investment in capital.