CNLS
): An alternative formulation¶Hildreth (1954) was the first to consider nonparametric regression subject to monotonicity and concavity constraints in the case of a single input variable $x$ (see also Hanson and Pledger 1976). Kuosmanen (2008) extended Hidreth’s approach to the multivariate setting with a vector-valued $\bf{x}$, and coined the term convex nonparametric least squares (CNLS
) for this method. CNLS builds upon the assumption that the true but unknown production function f belongs to the set of continuous, monotonic increasing and globally concave functions, $F_2$, imposing exactly the same production axioms as standard DEA.
The CNLS
estimator of function $f$ is obtained as the optimal solution to the infinite dimensional least squares problem
where $\alpha_i$ and $\beta_i$ define the intercept and slope parameters of tangent hyperplanes that characterize the estimated piece-wise linear frontier. $\varepsilon_i$ denotes the CNLS residuals.
To estimate the above CNLS
model on Python, we have to resort to an alternative CNLS
formulation.
Proof 1: The constraints in the two formulations are equivalent.
Therefore, the second constraint is redefined as:
\begin{align}
& \alpha_i + \beta_iX_i \le \alpha_j + \beta_jX_i \quad \forall i, j\\
& \hat{y}_i \le \hat{y}_j + \beta_j^{'} (X_i-X_j) \\
& \hat{y}_j - \hat{y}_i \ge \beta_j^{'} (X_j-X_i) \\
& \hat{y}_i - \hat{y}_j \ge \beta_i^{'} (X_i-X_j) \quad \blacksquare \\
\end{align}
Proof 2: The objective function is a standard quadratic problem (QP).
The standard QP
formulation:
\begin{align}
\text{min} f(x) = \frac{1}{2}x^TAx + q^Tx
\end{align}
The objective function in the alternative CNLS
formulation:
\begin{align}
& \text{min} \frac{1}{2} \mid\mid y-\hat{y}\mid\mid_2^2 \\
& = \frac{1}{2} (\hat{y} - y)^T(\hat{y} - y) \\
& = \frac{1}{2}(\hat{y}^T\hat{y} -y^T\hat{y} -\hat{y}^Ty+ y^Ty) \\
& = \frac{1}{2}\hat{y}^T\hat{y} - y^T\hat{y} + \frac{1}{2}y^Ty \\
\end{align}
since $\frac{1}{2}y^Ty$ is a contant term, it is sufficient to solve the QP-form objective function. Note that A is a identity matrix, q= -y. $\blacksquare$
Now, we can estimate the alternative CNLS
model based on convex quadratic problem solver in Python, eg., CVXOPT.
[1] Johnson, A. L. and Kuosmanen, T. (2015) An Introduction to CNLS and StoNED Methods for Efficiency Analysis: Economic Insights and Computational Aspects, in Ray, S. C., Kumbhakar, S. C., and Dua, P. (eds) Benchmarking for Performance Evaluation: A Production Frontier Approach. Springer, pp. 117–186.
[2] Kuosmanen, T., Johnson, A. and Saastamoinen, A. (2015) Stochastic Nonparametric Approach to Efficiency Analysis: A unified Framework, in Zhu, J. (ed.) Data Envelopment Analysis. Springer, pp. 191–244.