Shape-Constrained Kernel-Weighted Least Squares: Estimating Production Functions for Chilean Manufacturing Industries

ABSTRACT

In this article, 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.

Key words:

• Kernel estimation
• Local polynomials
• Multivariate convex regression
• Nonparametric regression
• Shape constraints.

ACKNOWLEDGMENTS

We thank two anonymous reviewers and the Associate Editor for providing useful suggestions that helped improve this article. We also thank Chris Parmeter, Jeff Racine, and Qi Li for their helpful comments.

Notes

1 A variable bandwidth method allows the bandwidth associated with a particular regressor to vary with the density of the data.

2 Since we underestimate the level of the errors in Step 1 by a factor of roughly n^{− 2/(4 + d)}, for the theoretical development, we address this bias issue by modifying the p-value to be $p_n=\frac{1}{B}{\sum }_{k=1}^B{\mathbf{1}}_{\{T_n\le T_{nk}+{\Delta }_n\}}$, where Δ_n = O(n^{− 2/(4 + d)}log n). Note that if we fix m and pick h = O(n^{− η}) for $\eta \in (\frac{1}{4+d},\frac{1}{d})$, then Δ_n/T_nk = o_p(1) as n → ∞, that is, this correction has a negligible effect. Indeed, our experience suggests that this modification offers little improvement in terms of finite sample performance in our simulation study.

3 The CNLS estimates include the second stage linear programming estimation procedure described by Kuosmanen and Kortelainen (2012) to find the minimum extrapolated production function.

4 Note that firms’ decisions, that is, selecting labor and capital levels with considerations for productivity levels or whether to export, are potentially endogenous. Solutions to this issue are to instrument or build a structural model based on timing assumptions. Our estimator can be embedded within the estimation procedures such as those described in Ackerberg, Caves, and Frazer (2015) to address this issue.

5 The data are available at http://www.ine.cl/estadisticas/economicas/manufactura .

6 The definition of Labor includes full-time, part-time, and outsourced labors. Capital is defined as a sum of the fixed assets balance such as buildings, machines, vehicles, furniture, and technical software. Value added is computed by subtracting the cost of raw materials and intermediate consumption from the total amount produced. Further details are available at http://www.ine.cl/estadisticas/economicas/manufactura .

7 We apply a Cobb–Douglas OLS to the second stage data {X_j, y_j − Z′_jγ}ⁿ_{j = 1} which removes the effect of contextual variables from observed output. See Appendix F for details.