Stochastic Frontier Analysis

A Python package for estimating cross-sectional and panel data stochastic frontier models using maximum likelihood and Bayesian methods.

Features

The package currently supports:

• classical normal linear regression;
• cross-sectional normal-exponential stochastic frontier models;
• cross-sectional normal-half-normal stochastic frontier models;
• panel normal-exponential stochastic frontier models;
• panel normal-half-normal stochastic frontier models;
• panel random-effects models;
• maximum likelihood estimation;
• Bayesian estimation;
• marginal data density calculations (aka integrated likelihood);
• inefficiency and efficiency estimation.

The current release focuses on estimating individual stochastic frontier models. Model search and Bayesian model averaging procedures will be added in future versions.

Installation

Install the package from PyPI with:

pip install stochastic-frontier-analysis

For local development, clone the repository and install it in editable mode from the project root:

python -m pip install -e .

To install the optional dependencies used by the examples:

python -m pip install -e ".[examples]"

Although the distribution name is stochastic-frontier-analysis, the Python import package is NAMED short: sfa.

Package Structure

The main public modules are:

• sf_model — model estimation and presentation of estimation results;
• inefficiency — estimation of inefficiency and efficiency scores.

They can be imported with:

from sfa import sf_model, inefficiency

Model and efficiency estimation is kept separate. This allows users to estimate and compare alternative SF specifications before computing efficiency scores for the preferred model, which may be quite demanding.

Available Models

The argument sfa_opt selects the model specification:

Value	Model
0	Classical normal linear regression
1	Cross-sectional normal-exponential stochastic frontier
2	Cross-sectional normal-half-normal stochastic frontier
3	Panel normal-exponential stochastic frontier
4	Panel normal-half-normal stochastic frontier
5	Panel Gaussian random-effects model

Model Estimation

The main estimation function is:

model = sf_model.fit(X, y, n, T, sfa_opt, dec_crit, if_mdd)

It returns a results object containing parameter estimates, standard errors, model fit measures, and other diagnostics.

Depending on the selected options, the returned object may include:

• maximum likelihood estimates;
• Bayesian posterior estimates;
• Akaike information criterion;
• Bayesian information criterion;
• maximized log-likelihood;
• maximized log-posterior;
• marginal data density;
• confidence or credible intervals;
• additional summary statistics.

Display the estimation results with:

model.summary()

Because summary() prints the results directly, it should not be wrapped in print().

Function Arguments

X

The regressor matrix.

As the intercept is required, X must already contain a column of ones. It is recommended to place the intercept column as the first column:

X = np.column_stack((np.ones(len(y)), regressors))

y

The dependent variable. Its observations must be ordered consistently with the rows of X.

n

The number of cross-sectional units.

For a cross-sectional model, this usually equals the total number of observations. For a panel model, it is the number of units.

T

The number of time periods.

For cross-sectional models:

T = 1

For panel models, observations should be ordered by unit and then by time:

unit 1: period 1, ..., period T
unit 2: period 1, ..., period T
...
unit n: period 1, ..., period T

dec_crit

The principal model-fit criterion:

dec_crit = 1

uses the marginal data density, whereas:

dec_crit = 0

uses the Bayesian information criterion.

The marginal data density provides a fully Bayesian measure of model fit, while BIC is faster to compute.

if_mdd

Controls whether Bayesian calculations and the marginal data density are computed:

if_mdd = 1

enables Bayesian estimates and marginal data density calculations.

if_mdd = 0

uses maximum likelihood estimation without calculating the marginal data density.

For a fully Bayesian analysis:

dec_crit = 1
if_mdd = 1

For a faster estimation based on BIC:

dec_crit = 0
if_mdd = 0

Basic Example

import numpy as np

from sfa import sf_model, inefficiency

# Dependent variable
y = np.asarray(y, dtype=float).reshape(-1)

# Regressor matrix with the intercept in the first column
X = np.column_stack((np.ones(len(y)), x1, x2, x3))

# Cross-sectional normal-exponential stochastic frontier
n = len(y)
T = 1
sfa_opt = 1

# Fully Bayesian model evaluation
dec_crit = 1
if_mdd = 1

model = sf_model.fit(X, y, n, T, sfa_opt, dec_crit, if_mdd)
model.summary()

# Efficiency scores based on Bayesian estimates
bayes = 1
u, te = inefficiency.jondrow(model, X, y, bayes)

Set:

bayes = 0

to estimate scores using the maximum likelihood.

Example data

Example scripts and datasets are stored in the examples directory on GitHub. Just copy-paste them (with the data folder) into your working directory and run there.

A portable way to load an Excel file from examples/data is:

import os
import pandas as pd

my_dir = os.path.dirname(os.path.abspath(__file__))
data_path = os.path.join(my_dir, 'data', 'dane.xlsx')
data = pd.read_excel(data_path, header=None, sheet_name=sf_label)

This works on Windows, macOS, and Linux.

Complete Examples

The repository on GitHub includes complete examples based on simulated and empirical data, including data preparation, model estimation, result summaries, and (in)efficiency scores estimation.

See the files in:

examples/

Project Background and development status

This Python implementation is based on code developed for a broader MATLAB project on Bayesian model averaging in SFA. The MATLAB implementation includes the model search and Bayesian model averaging procedures, whereas this Python package currently focuses on a single-model estimation. The porject is under development.

MATLAB package:

https://github.com/KamilMakiela/SF-BMA

Python package:

https://github.com/KamilMakiela/Python-SFA

References

Jondrow, J., Lovell, C. A. K., Materov, I. S., and Schmidt, P. (1982). "On the estimation of technical inefficiency in the stochastic frontier production function model". Journal of Econometrics, 19(2–3), 233–238. https://doi.org/10.1016/0304-4076(82)90004-5

Makieła, K. (2026). "Model uncertainty under non-Gaussian errors: Bayesian model averaging and selection in stochastic frontier models". Available on arXiv.