FronPy

Python package for stochastic frontier analysis

This package was built was to facilitate direct maximum likelihood estimation of the normal-gamma stochastic frontier model using a closed-form expression for the log-likelihood function in terms of the parabolic cylinder function. Efficiency predictors for the model are also available in closed-form using the parabolic cylinder function. To our knowledge, this is the first such implementation of the normal-gamma model using direct maximum likelihood estimation; other implementations use various approaches to approximate the log-likelihood - see the working paper cited below. The package produces parameter estimates, standard errors, log-likelihoods, efficiency predictors, and more in a convenient format.

The package however also includes options to estimate other stochastic frontier specifications, and may be of more general use to anyone who wishes to use Python for stochastic frontier analysis. Currently the package is limited to models of the form $$y_i=\boldsymbol{x'_i\beta}+E_i, \qquad E_i=V_i-sU_i, \qquad V_i\sim N\left(0,\sigma_V^2\right),$$ where $U_i$ may be either $N^+\left(0,\sigma_U^2\right)$ (half normal), $N^+\left(\mu,\sigma_U^2\right)$ (truncated normal), $\mathrm{Gamma}\left(\mu,\sigma_U\right)$, or $\mathrm{Exponential}\left(\sigma_U\right)$; N.B. in the latter two cases $\sigma_U$ is a scale parameter. The package may be extended in future in order to accomodate additional specifications.

Installation

Provided you have the git package installed, you can install FronPy by entering:

pip install git+https://github.com/AlexStead/FronPy.git

Getting started

The code block below estimates the normal-gamma stochastic frontier model, and demonstrates the package's basic syntax:

import fronpy
import numpy as np
electricity = fronpy.utils.dataset('electricity.csv')

nexpmodel = fronpy.estimate(electricity,model='nexp',cost=True)
ngmodel = fronpy.estimate(electricity,model='ng',startingvalues=np.append(nexpmodel.theta,0),cost=True)
ngmodel

In the final line, ngmodel produces rather a lot of output, but is useful to see all of the outputs produced by the fronpy.estimate. For example, we can see that ngmodel.lnlikelihood would give us the log-likelihood, ngmodel.beta would give us the estimated frontier parameters, ngmodel.eff_bc would give us $\mathbb{E}[e^{-U_i}|E_i=\varepsilon_i]$ for each observation, and so on. Note that currently, fronpy.estimate requires a numpy.ndarray with rows representing observations and columns representing variables, where:

• all data are numeric.
• the first column contains the dependent variable.
• columns 2,...,n contain the independent variables, including a constant if desired.
• there are no missing, NaN, inf, -inf or similarly invalid values. The package then assumes that the frontier is linear in its parameters, e.g. Cobb-Douglas or translog; note that any interactions must be included as columns in the numpy.ndarray. The package may be generalised in future to allow for arbitrary functional forms.

Linked publications

• Stead AD. 2023. Maximum likelihood estimation of normal-gamma and normal-Nakagami stochastic frontier models. Working paper, University of Leeds.