Isotonic CNLS

This section introduces a variant of CNLS estimator, Isotonic Convex Nonparametric Least squares (ICNLS), where ICNLS only relies on the monotonic assumption. To relax the concavity assumption in CNLS estimation (i.e., estimating a production function), we have to rephrase the Afriat inequality constraint in Problem （2.2). To do that, we define a binary matrix \(P=[p_{ij}]_{n x n}\) to represent the isotonicity (Keshvari and Kuosmanen, 2013).

Define the binary matrix \(P=[p_{ij}]_{n x n}\) as follows

\begin{align*} p_{ij} = \begin{cases} 1 & \text{if } x_i \preccurlyeq x_j \\ 0 & \text{otherwise} \end{cases} \end{align*}

Applying enumeration method to define the elements of matrix \(P\), we solving the following QP problem

\begin{align*} & \underset{\alpha, \beta, \varepsilon} {min} \sum_{i=1}^n\varepsilon_i^2 \\ & \text{s.t.} \\ & y_i = \alpha_i + \beta_i^{'}X_i + \varepsilon_i \quad \forall i \\ & p_{ij}(\alpha_i + \beta_i^{'}X_i) \le p_{ij}(\alpha_j + \beta_j^{'}X_i) \quad \forall i, j\\ & \beta_i \ge 0 \quad \forall i \\ \end{align*}

Note that the concavity constraints between units \(i\) and \(j\) is relaxed by the matrix \(P_{ij}=0\). If the \(P_{ij}=1\) for all \(i\) and \(j\), then the above QP problem (i.e., ICNLS problem) reduces to CNLS problem.

Example: Isotonic CNLS(ICNLS) [.ipynb]

In the following code, we estimate an additive production function using ICNLS approach.

# import packages
from pystoned import ICNLS
from pystoned.constant import CET_ADDI, FUN_PROD, OPT_LOCAL, RTS_VRS
from pystoned.dataset import load_Finnish_electricity_firm

# import Finnish electricity distribution firms data
data = load_Finnish_electricity_firm(x_select=['OPEX', 'CAPEX'], y_select=['Energy'])

# define and solve the ICNLS model
model = ICNLS.ICNLS(y=data.y, x=data.x, z=None, cet = CET_ADDI, fun = FUN_PROD, rts = RTS_VRS)
model.optimize(OPT_LOCAL)

# display residuals
model.display_residual()