stochastic-frontier-analysis 1.0.0

Estimation of cross-sectional and panel stochastic frontier models

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 Gaussian random-effects models;
• maximum-likelihood estimation;
• Bayesian posterior estimation;
• marginal data density calculations;
• technical inefficiency and efficiency estimation.

The current release focuses on estimating individual stochastic frontier models. Model-search and Bayesian model-averaging procedures may 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 technical-efficiency scores.

They can be imported with:

from sfa import sf_model, inefficiency

The estimation and efficiency calculations are kept separate. This allows users to estimate and compare alternative model specifications before computing efficiency scores for the preferred model.

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

Estimating a Model

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.

When an intercept is required, X must already contain a column of ones. It is recommended to place the intercept in 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 inference 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 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 calculate scores using the maximum-likelihood estimates.

Example Data

Example scripts and datasets are stored in the examples directory. Just copy-paste them (with data folder) into your working directory and run there, or run them directly from the package directory.

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 includes complete examples based on simulated and empirical data, including data preparation, model estimation, result summaries, and efficiency-score calculations.

See the files in:

examples/

Project Background

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

MATLAB package:

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

Python package:

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

Development Status

The package is under active development. The current version supports individual cross-sectional and panel stochastic frontier models. Additional model-search and Bayesian model-averaging procedures may be introduced in future releases.

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.