help xtkumblienhard

Title

Performs Estimations of Generalized Four-Component Panel Data Stochastic Frontier Models

Syntax

xtkumblienhard
depvar
indepvars
[if] [in] [weight]
, stub(string) [options]

* stub(string) designates a string name from which new variable names will be created

fe use the fixed-effects estimator instead of the default random-effects estimator

vce(vcetype) vcetype may be robust,
cluster clustvar

level(#) set confidence level; default is level(95)

frontier_options in addition to the options listed above, all options of the command [R] frontier can be used

* stub() is required.

You must tsset or xtset your data before using xtkumblienhard;
see [TS] tsset and [XT] xtset.

depvar, indepvars may contain time-series operators; see tsvarlist.

indepvars may contain factor variables; see fvvarlist.

fweights and pweights are allowed; see weight.

Description

xtkumblienhard is a powerful and user-friendly Stata command designed to estimate Generalized Four-Component
Panel Data Stochastic Frontier Models, all in a single line of code. This package streamlines the estimation of
a sophisticated stochastic frontier framework where the composite error term is decomposed into four distinct
components: unobserved individual heterogeneity, persistent inefficiency, transitory inefficiency and random
noise. By clearly separating these sources of variation, xtkumblienhard enables researchers to gain deeper
insights into performance dynamics across panel data settings. This model builds on the influential and excellent
works of Kumbhakar, Lien and Hardaker (Journal of Productivity Analysis, 2014), Kumbhakar, Wang and Horncastle
(Cambridge University Press, 2015), and Nguyen, Sickles and Zelenyuk (Springer, 2022), who
originally implemented it through multi-step procedures. The innovation of xtkumblienhard lies in its
simplicity: it brings the full power of this four-component framework to Stata users with native ADO language
syntax, eliminating the need for complex coding or external scripts. Rather than replacing these foundational and
outstanding contributions, xtkumblienhard democratizes access to advanced stochastic frontier analysis, making
it faster, easier, and more intuitive for applied researchers, analysts, and students alike. Whether you are studying
firm-level productivity, benchmarking efficiency, or exploring heterogeneity in performance, xtkumblienhard is
your go-to tool for robust and elegant estimation.

Econometric Model

The generalized four-component panel data stochastic frontier model is specified as:

y_it = beta_0 + x'_it*beta + c_i - eta_i + v_it - u_it (1)

where the distributional assumptions are:

c_i ~ iid N(0, sigma_c^2), eta_i ~ iid N+(0, sigma_eta^2)

v_it ~ iid N(0, sigma_v^2), u_it ~ iid N+(0, sigma_u^2)

In Equation (1):

y_it is the dependent variable

x'_it are the regressors

beta_0 and beta are the parameters of interest

c_i captures unobserved individual heterogeneity

eta_i denotes persistent inefficiency

u_it denotes transitory inefficiency

v_it is the random disturbance

As shown by Kumbhakar et al. (2014), the model in Equation (1) can be estimated through a multi-step
procedure (three steps in total). To facilitate estimation, the model can be rewritten as:

y_it = beta_0* + x'_it*beta + alpha_i + epsilon_it (2)

with the following definitions:

beta_0* = beta_0 – E[eta_i] – E[u_it] (3)

alpha_i = c_i – eta_i + E[eta_i] (4)

epsilon_it = v_it – u_it + E[u_it] (5)

In Equations (3)–(5), E[.] denotes the expectation operator. The reformulated model
in Equation (2) is equivalent to a standard panel data specification and can be estimated using
conventional panel data methods. Once Equation (2) is estimated, predicted values
of alpha_i and epsilon_it (alpha_i_hat and epsilon_it_hat) are
obtained. These predictions are then used to recover the persistent and transitory inefficiency
components by applying standard stochastic frontier techniques to Equations (4)
and (5), substituting alpha_i_hat and epsilon_it_hat for alpha_i
and epsilon_it, respectively.

Options

stub(string) designates a string name from which new variable names will be
created. To form this option, you put inside the parentheses a string name (without the
double quotes). Then new variable names will be created from this string. You must
specify this option in order to get a result. Hence this option is required.

fe requests the fixed-effects (within) regression estimator. If you specify this
option, you request to use the fixed-effects estimator instead of the default random-effects estimator.

vce(vcetype) specifies the type of standard error reported, which
includes types that are robust to some kinds of misspecification (robust), that allow
for intragroup correlation (cluster clustvar).

vce(robust) is equivalent to specifying
vce(cluster panelvar).

vce(cluster clustvar) specifies that standard errors
allow for intragroup correlation within groups defined by one
variable in clustvar, relaxing the usual requirement that the
observations be independent. For example, vce(cluster clustvar)
produces cluster-robust standard errors that allow for observations
that are independent across groups defined by clustvar but not
necessarily independent within groups.

For your information, the Defaults Variance Estimators are: conventional for
the first step, and oim for the second and third steps.

level(#) specifies the confidence level, as a percentage, for confidence
intervals. The default is level(95) or as set by set level.
See [R] Estimation options for more information.

frontier_options in addition to the options listed above, all options
of the command [R] frontier can be used. You can form these options in exactly
the same way as you would do with the command frontier. Simply enter them as if you were
using the command frontier. See [R] frontier for more details.

Return values for xtkumblienhard

The command xtkumblienhard stores its results in e(), following the three-step estimation
procedure described above. Each step produces its own set of return values, which can be accessed and
inspected individually. This design allows users to verify intermediate results, replicate calculations, and
better understand how the estimates are obtained.

To access the saved results, you need to restore the corresponding estimation set and
then list the stored values. The procedure is as follows:

To get the first-step results, you type:

estimates restore step_1_stub

ereturn list

To get the second-step results, you type:

estimates restore step_2_stub

ereturn list

To get the third-step results, you type:

estimates restore step_3_stub

ereturn list

In these commands, the placeholder stub refers to the string name you specified in
your estimation call. This stub is used to generate new variable names and to organize the stored results
consistently across the three steps. Choosing a clear and memorable stub name will make it easier to
track your outputs.

For additional details on how to manage and inspect stored estimation
results, see [R] estimates and [P] ereturn. These references explain
how Stata handles estimation sets and return values, and they provide guidance on
integrating xtkumblienhard outputs into your broader workflow.

Examples

Before beginning the estimations, we use the set more off instruction to tell
Stata not to pause when displaying the output.

set more off

We illustrate the use of the command xtkumblienhard with the dataset xtkumblienharddata.dta. This
dataset contains a sample of panel data for developing countries in the World. It contains 7 periods of
4 non overlapping years from 1996-1999, 2000-2003 to 2020-2023.

use http://fmwww.bc.edu/repec/bocode/x/xtkumblienharddata.dta, clear

Next we describe the dataset to see the definition of each variable.

We observe that the dataset begins with the World Bank country codes, country names, and time periods. It
also includes qualitative variables that define subsamples within the database, allowing users to distinguish
groups of interest. The panel structure has already been declared with xtset, so the data are ready for
panel estimation. Following this, the dataset provides the main quantitative variables of interest. All
quantitative variables are expressed in logarithmic form, consistent with the estimation of stochastic
frontier production functions presented in this Examples section.

describe

We begin the regressions by estimating a random-effects Cobb–Douglas stochastic frontier
production function. To do so, we indicate the name of the command xtkumblienhard, followed by the dependent
variable lgnetodaidrec. We then list the explanatory variables:
lgmilitexppcgdp, lgextdebtst, lgpoptotal, and lgevitotal. After specifying the model, we
include the relevant options. In particular, we provide the string sfadj1, without quotation marks, to
the option stub(), which determines the suffix used when generating new variables during the estimation
steps.

xtkumblienhard lgnetodaidrec lgmilitexppcgdp lgextdebtst lgpoptotal lgevitotal, stub(sfadj1)

The output produced by the command xtkumblienhard is composed of three parts, each corresponding
to each step of the multi-step estimation procedure. The first part,
titled STEP 1: Random-Effects Panel Data Regression displays
the Random-effects GLS regression which contains the estimated parameters of interest. The second
part, titled STEP 2: SFA Estimation to Obtain the Persistent (In)Efficiency displays
the Stochastic frontier normal/half-normal model that allows to calculate the Persistent Inefficiency
and Efficiency Scores. The third part, titled STEP 3: SFA Estimation to Obtain the Transitory (In)Efficiency
displays the Stochastic frontier normal/half-normal model that allows to calculate the Transitory Inefficiency
and Efficiency Scores.

Let us interpret the results we just found in this estimation.
In this Examples section, our objective is to estimate a country's Aid Absorption Capacity by
regressing actual foreign aid on key economic and social characteristics. Each variable has an expected sign
grounded in standard aid-allocation patterns. Higher military spending is often associated with greater aid
inflows, as donors may support countries viewed as strategic or important for regional stability. External debt
is also expected to have a positive sign, since highly indebted countries frequently receive assistance to ease
financial pressures. By contrast, larger populations tend to reduce aid per capita, implying a negative
effect. Countries exposed to external economic shocks - such as commodity dependence or vulnerability to
natural disasters - typically receive more aid, so a positive sign is expected. The first-step results confirm these
expectations: all coefficients are statistically
significant, display the anticipated signs, and have economically meaningful magnitudes. These findings
indicate that the model behaves as expected and provides a consistent basis for the inefficiency and efficiency
analyses.

Next, we use describe to display all previously generated variables along with their
labels, allowing us to verify that each variable has been created and documented correctly.

describe Alpha_sfadj1 Epsilon_sfadj1 Ineff_Pers_sfadj1 Eff_Pers_sfadj1 Ineff_Trans_sfadj1 Eff_Trans_sfadj1 Overall_TE_sfadj1 Overall_Ineff_sfadj1

We now summarize these variables to obtain a quick overview of their distributions and basic
descriptive statistics.

summarize Alpha_sfadj1 Epsilon_sfadj1 Ineff_Pers_sfadj1 Eff_Pers_sfadj1 Ineff_Trans_sfadj1 Eff_Trans_sfadj1 Overall_TE_sfadj1 Overall_Ineff_sfadj1, sep(0)

The summary statistics show that the average of the estimated composed random individual-specific effects
is very small, as expected. The mean of the composed error term is approximately zero, which is consistent with its
definition. The estimated average Persistent Inefficiency is 175.65%, while the corresponding Persistent Efficiency
averages 31%. The estimated average Transitory Inefficiency is 37.72%, with a Transitory Efficiency
of 71.21%. Finally, the estimated Overall Technical Efficiency averages 22.40%, and Overall Inefficiency
averages 213.38%.

We now list the World Bank country codes, the time periods, and all previously generated variables
to inspect them in detail and verify that each has been created correctly.

list pbm period Alpha_sfadj1 Epsilon_sfadj1 Ineff_Pers_sfadj1 Eff_Pers_sfadj1 Ineff_Trans_sfadj1 Eff_Trans_sfadj1 Overall_TE_sfadj1 Overall_Ineff_sfadj1, sep(7)

Let us plot a histogram of the Overall Technical Efficiency scores.

histogram Overall_TE_sfadj1, bin(100) normal

We notice that, we have a right-skewed or positively-skewed distribution for the overall efficiency
scores. Hence, the mean efficiency is greater than the median efficiency scores.

To display the results from the first step, we type:

estimates restore step_1_sfadj1

ereturn list

To display the results from the second step, we type:

estimates restore step_2_sfadj1

ereturn list

To display the results from the third step, we type:

estimates restore step_3_sfadj1

ereturn list

For more information on working with stored estimation results, see [R] estimates
and [P] ereturn.

We now illustrate how to use xtkumblienhard in combination with the predict command.

We begin by restoring the estimation results from the first step.

estimates restore step_1_sfadj1

Then, we compute the linear prediction based on the first-step model.

predict double lgnetodaidrechat, xb

We describe the previously created variable to see its label.

describe lgnetodaidrechat

We summarize this variable.

summarize lgnetodaidrechat

We now illustrate how to tabulate the estimation results produced by xtkumblienhard. We
begin by running a new regression and supplying a different string, sfadj2, to the option stub() so
that the results from this estimation are stored separately from the previous one.

xtkumblienhard lgnetodaidrec lggdppcapcstd lgmilitexppcgdp lggovernanceidx lggoodpolicyidx, stub(sfadj2)

The expected coefficient signs follow well-established patterns in the aid allocation literature. Higher GDP
per capita generally reduces aid inflows, as wealthier countries are viewed as less dependent on external
assistance; a negative sign is therefore expected. Military spending often attracts more aid because
donors may support countries seen as strategic partners or contributors to regional stability, implying a
positive sign. Stronger governance - reflected in better institutions, lower corruption, and more effective
public administration - tends to increase donor confidence and thus aid receipts, again suggesting a positive
sign. Similarly, countries with sound macroeconomic policies are typically rewarded with higher aid, as donors
prefer environments where funds are more likely to be used effectively. The first-step results confirm these
expectations: all coefficients are statistically significant, display the anticipated signs, and have economically
meaningful magnitudes.
These findings confirm that the estimation behaves as anticipated and
provides a reliable basis for our study.

We can now tabulate the first-step results from both regressions using estimates table.

estimates table step_1_sfadj1 step_1_sfadj2, b(%7.4f) p(%7.4f) stats(N r2_o)

The same comparison can be produced with [R] etable.

etable, estimates(step_1_sfadj1 step_1_sfadj2) cstat(_r_b) cstat(_r_p, nformat(%7.4f)) mstat(N) mstat(r2_o)

Readers/Users may also utilize outreg (if installed), outreg2
(if installed), estout (if installed), or any other Stata command designed to tabulate
stored estimation results.

Next, we illustrate how to use some frontier_options. In some cases, the second and third
steps of the estimation may encounter convergence difficulties. When this occurs, these options can help improve
numerical stability and guide the optimizer toward a solution. In the example below, we
apply difficult, technique(dfp), iterate(20000), and nrtolerance(0.005) to demonstrate how
such options can be incorporated when needed.

xtkumblienhard lgnetodaidrec lggdppcapcstd lgmilitexppcgdp lggovernanceidx lggoodpolicyidx, stub(sfadj3) difficult technique(dfp) iterate(20000) nrtolerance(0.005)

Up to this point, the first-step estimation has relied on the default random-effects estimator. The
following example shows how to instead use the fixed-effects estimator in the first step.

xtkumblienhard lgnetodaidrec lgmilitexppcgdp lgextdebtst lgevitotal, stub(sfadj4) fe

The results indicate that all regressors remain statistically significant, retain their expected
signs, and exhibit economically meaningful absolute values. This confirms that the first-step estimates of our study are
robust to the choice between random-effects and fixed-effects specifications.

We now illustrate how to use xtkumblienhard with the [U] if qualifier, and simultaneously
show how to apply the vce(robust) option. We begin by estimating a first model for Low-Income Countries only:

`"xtkumblienhard lgnetodaidrec lggdppcapcstd lggovernanceidx lggoodpolicyidx if incomegrpwb == "Low income", stub(sfadj5) vce(robust)"'

Next, we estimate a second model for Lower Middle-Income and Upper Middle-Income Countries combined:

`"xtkumblienhard lgnetodaidrec lgmilitexppcgdp lgextdebtst lgevitotal if (incomegrpwb == "Lower middle income" | incomegrpwb == "Upper middle income"), stub(sfadj6) vce(robust)"'

In both subsample estimations, all regressors remain statistically significant, preserve their expected
signs, and display economically meaningful magnitudes. These results confirm that our findings are robust when the
analysis is conducted on income-group subsamples.

We now illustrate how to include a time trend when using xtkumblienhard. To do so, we add the
variable period to our standard Cobb-Douglas specification. At the same time, we show how to apply the
option vce(cluster clustvar) to obtain cluster-robust standard errors.

xtkumblienhard lgnetodaidrec period lggdppcapcstd lgpoptotal, stub(sfadj7) vce(cluster id)

The estimated coefficient on period (0.060353) suggests a positive time trend in aid absorption
capacity. Interpreted in growth-rate terms, this implies that total factor productivity associated with Net ODA and
Aid received as a percentage of GDP increased by roughly 6.0% every four-year period, on average, across all
countries in the sample from 1996 to 2023. This indicates a gradual improvement in countries' ability to absorb
aid over time.

We next illustrate how to use the level(#) option to specify the confidence level reported in the
estimation output. This option is useful when users require wider or narrower confidence intervals for
inference. In the example below, we estimate the model using a 99% confidence interval.

xtkumblienhard lgnetodaidrec period lggdppcapcstd lgpoptotal, stub(sfadj8) vce(robust) level(99)

Specifying level(99) instructs Stata to compute 99% confidence intervals for all estimated
coefficients, which is appropriate when a more conservative inference threshold is desired.

To conclude this Examples section, we now switch from the Cobb-Douglas specification used so
far to a more flexible Translog production function. This also provides opportunities to demonstrate the use
of the [R] margins command for computing marginal effects
and the implementation of factor variables (see fvvarlist). We begin by estimating a Translog specification.

xtkumblienhard lgnetodaidrec per lggdppcapcstd lgpoptotal c.per#c.per c.lggdppcapcstd#c.lggdppcapcstd c.lgpoptotal#c.lgpoptotal c.per#c.lggdppcapcstd c.per#c.lgpoptotal c.lggdppcapcstd#c.lgpoptotal, stub(tl) vce(r)

We then restore the first-step estimation results.

estimates restore step_1_tl

Next, we use [R] margins to compute the average marginal effects of all independent
variables. The option dydx(*) requests marginal effects for every regressor, while post stores the
results in e() for possible tabulation. We also specify nochainrule because xtkumblienhard is
a community-contributed command, and margins must avoid applying the chain rule automatically.

margins, dydx(*) post nochainrule

The resulting marginal effects indicate that all regressors remain statistically significant, retain
their expected signs, and exhibit economically meaningful magnitudes. These findings confirm that our results
are robust to adopting a more flexible functional form.

EPILOGUE

In preparing this Examples section, we pursued two main objectives. First, we aimed to demonstrate
how to use xtkumblienhard effectively through simple, transparent, and reproducible examples. Second, we
sought to conduct an original empirical exercise on countries' Aid Absorption Capacity and Efficiency using
stochastic frontier models for panel data, thereby offering a modest contribution to this line of
research. Of course, these examples only scratch the surface of what can be achieved
with xtkumblienhard, the accompanying dataset, and the many possibilities that arise when combining
this command with other Stata tools. The flexibility of the estimator, the richness of the data, and the
breadth of Stata's ecosystem open numerous avenues for further exploration. We leave these extensions to
the reader/user, who is encouraged to adapt, expand, and refine the analyses according to her or his own research
interests.

References

Kumbhakar Subal C., Lien Gudbrand and Hardaker J. Brian: 2014,
"Technical Efficiency in Competing Panel Data Models: A Study of Norwegian Grain Farming",
Journal of Productivity Analysis 41(2), 321–337, April.

Kumbhakar Subal C., Wang Hung-Jen and Horncastle Alan P.: 2015,
"A Practitioner's Guide to Stochastic Frontier Analysis Using Stata",
Cambridge University Press, Cambridge, ISBN 9781107029514.

Nguyen Bao Hoang, Sickles Robin C. and Zelenyuk Valentin: 2022,
"Efficiency Analysis with Stochastic Frontier Models Using Popular Statistical Softwares",
in: Duangkamon Chotikapanich, Alicia N. Rambaldi and Nicholas Rohde (eds.),
Advances in Economic Measurement, Chapter 3, pp. 129–171, Springer.

Citation and Donation

The command xtkumblienhard is not an Official Stata command. Like a paper, it is a free contribution to
the research community. If you find the command xtkumblienhard and its accompanying dataset useful and
utilize them in your
works, please cite them like a paper as it is explained in the Suggested Citation section of
the IDEAS/RePEc webpage of the command. Please, also cite Kumbhakar, Lien and Hardaker (2014),
Kumbhakar, Wang and Horncastle (2015), and
Nguyen, Sickles and Zelenyuk (2022) in your
works.
Thank you infinitely, in advance, for doing all these gestures! Please, note that citing this
command xtkumblienhard and these references are a good way to disseminate their use and their
discovery by other researchers and analysts. Doing these actions, could also, potentially, help us, as a community, to solve
challenging current problems and those that lie ahead in the future.

I would also like to ask you about one more thing, please! I hope you are finding my Stata Packages useful
and insightful. If you have appreciated the work I do and would like to support me financially in continuing
to develop these resources, I would be incredibly grateful. You can help fund my work
through My Patreon Page (LINK HERE)
or through My PayPal Page (LINK HERE),
which will allow me to dedicate more time and resources to creating even better
tools and updates. Any contribution, no matter how small, is greatly appreciated
and will go directly towards furthering my work.
Thank you so much in advance for your valuable support ! Best and Kind Regards !

Acknowledgements

The command xtkumblienhard is a Stata ADO File Language implementation inspired by the
original Do-file program developed by Nguyen, Sickles, and Zelenyuk (2022). The name and theoretical
foundation of this command are based on the influential work of Kumbhakar, Lien and Hardaker (2014), whose
contributions to the field have been instrumental in shaping this package. I am deeply grateful to
Bao Hoang Nguyen, Robin C. Sickles, and Valentin Zelenyuk for their pioneering implementation, and
to Subal C. Kumbhakar, Gudbrand Lien, J. Brian Hardaker, Hung-Jen Wang, and Alan P. Horncastle for their
extensive research and methodological advancements. I also thank StataCorp LLC for making their
software, documentation, and resources widely accessible through both official and commercial
channels. This Stata package is built upon and inspired by the collective work of these scholars
and company. I extend my sincere appreciation to all of them. As always, any remaining errors or shortcomings
are entirely my own. Constructive feedback is warmly welcomed!

Thank you, you the reader/user, for downloading and exploring this Stata package. Your time, curiosity, and
commitment to rigorous research mean the world to me. I am truly honored to be part of your analytical
journey, and I hope this tool empowers your work with clarity, precision, and purpose. As an economist and
data scientist, I offer consulting services to individuals, institutions, and companies across the
globe. If you are working on a project that could benefit from collaboration, or if you would like to
explore how we might work together, please feel free to reach out. I also welcome financial contributions
to support future research, publications, and the continued development of open-access tools like
this one. My contact details are included above and below for your convenience. I would like to close
with a personal note of profound gratitude. To my father, my mother, my family, all Prophets, Messengers
and their Companions, and to the Great Allah, thank you for your unwavering love, support, and faith throughout
this long and challenging journey. Your strength has been my foundation, and your encouragement has carried me
through every step of this work. I profoundly and sincerely thank you all. I am also deeply grateful to
my Patreon and PayPal supporters: Aissata Coulibaly, Djedje Hermann Yohou, and Yeo Nibontenin, whose generosity
and belief in my mission have been both humbling and motivating. Your support has helped bring this project
to life, and I offer you my heartfelt thanks. To all readers/users of this package: may it serve as a
launchpad for bold ideas, impactful research, and meaningful change! Keep pushing boundaries, keep asking
questions, and above all - enjoy the journey! With sincere appreciation!

Author

Diallo Ibrahima Amadou

FERDI (Fondation pour les Etudes et Recherches sur le Developpement International)

63 Boulevard Francois Mitterrand

63000 Clermont-Ferrand

France

E-Mail: zavren@gmail.com

Diallo Ibrahima Amadou

Zavren Consulting and Publishing

E-Mail: zavren@gmail.com

Also see

Online: help for [XT] xtreg, [R] frontier, [XT] xtfrontier,
[R] estimates,
[P] ereturn, [R] etable, [R] margins,
outreg (if installed), outreg2 (if installed), estout (if installed),
sfcross (if installed), sfpanel (if installed), xtnondynthreshsfa (if installed),
frontierhtail (if installed), sfkk (if installed), xtsfkk (if installed)