SFB 
823 
Simar and Wilson two-stage 
efficiency analysis for Stata 
D
iscussion P
aper 
 
Oleg Badunenko, Harald Tauchmann  
 
 
 
Nr. 11/2018 
 
 
 
 
 
 
 
 
 
Simar and Wilson two-stage efficiency analysis
for Stata
Oleg Badunenko
University of Portsmouth
Portsmouth, UK
oleg.badunenko@port.ac.uk
Harald Tauchmann
Universita¨t Erlangen-Nu¨rnberg,
and RWI – Leibniz Institut fu¨r Wirtschaftsforschung,
and CINCH – Health Economics Research Center
Findelgasse 7/9, 90402 Nu¨rnberg, Germany
harald.tauchmann@fau.de
May, 2018
Abstract. When analyzing what determines the efficiency of production, regressing
efficiency scores estimated by DEA on explanatory variables has much intuitive
appeal. Simar and Wilson (2007) show that this na¨ıve two-stage estimation
procedure suffers from severe flaws, that render its results, and in particular
statistical inference based on them, questionable. At the same time they propose
a statistically grounded bootstrap based two-stage estimator that eliminates the
above mentioned weaknesses of its na¨ıve predecessors and comes in two variants.
This article introduces the new Stata command simarwilson that implements
either variant of the suggested estimator in Stata. The command allows for various
options, and extends the original procedure in some respects. For instance, it
allows for analyzing both, output- and input-oriented efficiency. To demonstrate
the capabilities of the new command simarwilson we use data from the Penn
World Tables and the Global Competitiveness Report by the World Economic
Forum to perform a cross-country empirical study about the importance of quality
of governance of a country for its efficiency of output production.
Keywords: DEA, two-stage estimation, truncated regression, bootstrap, efficiency,
bias correction, environmental variables.
1 Introduction
Analyzing the technical efficiency of production/decision making units (DMUs) has
developed into a major field in empirical economics and management science.1 From a
methodological perspective, two major strands of the literature can be distinguished: (i)
analyses that rest on parametric, regression based methods, namely stochastic frontier
analysis (SFA, Aigner et al. 1977), and (ii) analyses that use non-parametric methods,
namely data envelopment analysis (DEA, Charnes et al. 1978). The pros and cons of
either approach have been discussed extensively (e.g. Hjalmarsson et al. 1996; Murillo-
Zamorano 2004).
One of the advantages of the parametric approaches, namely the truncated-normal
stochastic frontier model, is that it does not only allow for measuring inefficiency but also
incorporates a model of the determinants of inefficiency.2 In contrast, non-parametric
approaches are primarily concerned with estimating a production-possibility frontier (or
an input requirement frontier) and with measuring the distance of observed input-output
combinations to this frontier. Yet, shedding light on what determines the magnitude of
this distance is out of the narrow3 scope of non-parametric approaches such as DEA.
For many research questions, however, identifying determinants of inefficiency is of
much greater relevance than determining its magnitude for specific DMUs. For this
reason, in the domain of non-parametric efficiency analysis, semi-parametric two-stage
approaches that combine efficiency measurement by DEA with a regression analysis that
uses DEA estimated efficiency as dependent variables have become popular. Simar and
Wilson (2007) list almost fifty published articles and mention hundreds of unpublished
papers that employ such two-stage procedures. In these (early) applications the second
stage is typically a censored (tobit like) regression to account for the bounded nature of
DEA efficiency scores, or just simply OLS (Simar and Wilson 2007).
Despite their popularity and their intuitive appeal, such na¨ıve two-stage estimators
are criticized by Simar and Wilson (2007) mainly for two reasons. Firstly, they stress
the absence of a clear theory of the underlying data generating process, that would
justify the na¨ıve two-stage approach.4 Secondly, they criticize the conventional inference
1. In a supplement to their recent survey article Emrouznejad and Yang (2018) list more than 10 000
publications, only considering the non-parametric strand of this literature.
2. The Stata command frontier includes the option cm() that allows for specifying the conditional
mean of the truncated normal distribution, from which the distance to the frontier is assumed to be
drawn, as linear function of exogenous variables.
3. Some DEA models were proposed that directly include environmental variables in the DEA linear
programming problem (cf. Coelli et al. 2005). Yet, these models suffer from several shortcomings.
For instance, they allow only for continuous environmental variables. More recently, smoothing
based, fully non-parametric methods for estimating conditional frontiers, which substantially extend
the familiar DEA framework, have been suggested (Daraio and Simar 2005, 2007). Unlike two-stage
approaches, these model allow for environmental variables affecting the shape of the frontier. Yet,
they are not considered as determinants of the distance from the frontier.
4. In a closely related article (Simar and Wilson 2011), Simar and Wilson discuss further contributions
(Hoff 2007; McDonald 2009; Ramalho et al. 2010; Banker and Natarajan 2008) to the literature
on two-stage estimators. These are not ‘na¨ıve’ in the sense that they did not make an attempt of
justifying the proposed procedures. They rather provide some kind of rationale and/or statistical
justification. Simar and Wilson (2011, p. 206), however, deny the former three a decent basis
2
that is pursued in most of the two-stage applications, for ignoring that estimated DEA
efficiency scores are calculated from a common sample of data. Treating them as if
they were independent observations is not appropriate since the problems related to
invalid inference due to serial correlation arise. Simar and Wilson (2007) develop a
two-stage procedure that takes the above mentioned issues into account. They construct
an underlying data generating process that is consistent with a two-stage estimation
procedure, which – as the most obvious difference to the earlier na¨ıve approach – implies
a truncated rather than censored regression model. This reflects that the substantial
share of fully efficient DMUs typically found in DEA is an artifact of the finite sample
bias inherent in DEA, but does not represent a feature of the true underlying data
generating process. Moreover, they develop a parametric bootstrap procedure that is
consistent with the assumed data generating process and addresses the second issue. It
yields estimated standard errors and confidence intervals that do not suffer from bias
due to estimated efficiency scores being correlated.
The Simar and Wilson (2007) procedure has become a workhorse of empirical
efficiency analysis with hundreds of applications from various fields of economics.5 This
popular, yet technically involved estimator has not yet been available to Stata users,
unless they developed their own code. The present paper introduces the new Stata
command simarwilson that allows for applying this estimator in Stata.6 In doing this,
it greatly benefits from the recently published user written Stata command teradial
(Badunenko and Mozharovskyi 2016), which is required for running simarwilson. For
the first time, teradial enables fast estimation of DEA in Stata even for large samples.
This is essential for practical applications of the Simar and Wilson (2007) estimator,
because it involves bootstrapping the DEA estimator.7
The remainder of this paper is organized as follows. The following section gives
a brief summary of the Simar and Wilson (2007) two-stage estimator. The syntax of
simarwilson is described in section 3. Section 4 presents an application to cross country
data. Section 5 concludes.
2 The Simar and Wilson (2007) estimator in brief
2.1 Some essential ideas
Simar and Wilson (2007) consider a setting in which a researcher observes three types of
variables xi, yi, and zi, for a sample of i = 1, . . . , N DMUs. xi denotes a vector of P
in statistical theory. With respect to the latter they argue that the claimed desirable statistical
properties rely on very restrictive assumptions.
5. Google Scholar (2018, March 27) lists more than 2200 citations (for instance Fragkiadakis et al.
2016; Pe´rez Urdiales et al. 2016; Glass et al. 2015; Chortareas et al. 2013, just to mention few
recent applications). Interestingly, the inventors of this popular method dissociate themselves from
advocating two-stage approaches (Simar and Wilson 2011, p. 216).
6. An earlier, less powerful version of the ado-file that accompanies this article has been made available
through ssc in 2016.
7. This applies to algorithm #2 but not to algorithm #1; see section 2. Prior to teradial being
available, algorithm #2 was hence effectively out of reach for Stata users.
3
inputs to production. yi is a vector of Q outputs from production. zi denotes a row
vector of K environmental variables that may affect the ability of DMU i to efficiently
combine the consumed inputs to the produced outputs. The effect of zi on efficiency
is in the focus of the empirical analysis. The production technology is assumed to be
homogeneous across DMUs. That is, a common production-possibility frontier – the
boundary of the convex production-possibility set – represents all combinations (y∗j , x
∗
j )
that are fully efficient in the sense that no output can be increased without decreasing
at least one other output or increasing at least one input. A crucial assumption is that
the shape of the production-possibility frontier does not depend on zi, which is referred
to as separability in Simar and Wilson (2007).
The output-input set (yi, xi) observed for DMU i, will regularly fail in realizing a
point at the frontier. This deviation is necessarily directional, i.e. i produces less output
than technically feasible or it consumes more input than technically feasible. The widely
used output oriented Farrell (1957) distance measure quantifies the deviation from the
frontier as the relative radial distance in output direction θi. That is θi denotes the
factor by with output generation yi of DMU i has to be proportionally increased in
order to project (yi, xi) onto the frontier. θi is hence a measure of inefficiency that is
bounded to the [1,∞) interval. Alternatively, one may measure the Farrell distance in
input direction as ϑi, that is the factor by which input consumption xi of DMU i has to
be proportionally reduced in order to project (yi, xi) onto the frontier. ϑi is hence a
measure of efficiency that is bounded to the (0, 1] interval.8 Yet, in Simar and Wilson
(2007) the focus is on θi.
9
The key idea in Simar and Wilson (2007) about the data generating process is that
efficiency θi linearly depends on zi
θi = ziβ + εi (1)
where β denotes a column vector of coefficients, estimating which is the ultimate objective
of the empirical analysis. The disturbances εi are assumed to be – conditionally on zi
– indpendently10, truncated normally distributed, with parameters μ = 0 and σ, and
left-truncation at 1− ziβ.11 This assumption guarantees that θi cannot be smaller than
unity, irrespective of the values the variables in zi may take. Though full efficiency
(θi = 1) is in principle possible, it occurs with zero probability. Conditional on θi, DMU
i chooses a set of outputs and inputs (yi, xi) as (1/θiy
∗
i ,x
∗
i ), with (y
∗
i ,x
∗
i ) denoting some
point at the production-possibility frontier.12
8. An alternative to the Farrell (1957) distance measure is one introduced by Shephard (1970), which
is just the reciprocal of θi and ϑi, respectively. simarwilson accommodates the Shephard measure
by the option invert; see sections 2.3 and 3.
9. Simar and Wilson (2007) do not explicitly consider input oriented efficiency, i.e. ϑi. simarwilson
straightforwardly extends the original Simar and Wilson (2007) to accommodating input oriented
efficiency too; see sections 2.3 and 3.
10. This implies that simarwilson is meant for being used with standard cross-sectional data but not
with panel- or other types of clustered data.
11. The first and the second moment of the conditional error distribution are E(εi|zi) = σλi and
Var(εi|zi) = σ2
(
1 + ((1−ziβ)/σ)(λi − λ2i )
)
, with λi denoting the inverse Mills ratio
φ((1−ziβ)/σ)
1−Φ((1−ziβ)/σ) .
12. The data generating process can hence be described as sampling from the joint distribution f (x,y, z),
4
It is key for understanding the shortcomings of na¨ıve two-stage approaches that θi is
genuinely unobservable. In consequence the estimated efficiency score θ̂i one obtains
from running a DEA13 is not θi. In other words, θ̂i is not the distance of (yi, xi) to the
true production-possibility frontier but the distances to an estimate of the latter. Due
to the boundary estimation framework of DEA, this estimate suffers from finite sample
bias and in turn θ̂i is biased towards the value of one. That means that (1) cannot be
estimated straightforwardly and θi has to be replaced in (1) by the biased estimate θ̂i
in order to formulate an operational regression equation. As pointed out in Simar and
Wilson (2007), this generates two major problems for na¨ıve two step approaches. Firstly,
although the errors εi are assumed to be statistically independent across DMUs, the
operational errors in a regression of θ̂i on zi are not, since the θ̂i are estimated from a
common sample of data. Secondly, in any application of DEA some – usually numerous –
θ̂i take the value of one, though according to (1) θi takes this value with zero probability.
In the procedure14 suggested in Simar and Wilson (2007), the former issue is addressed
by estimating standard errors and confidence intervals for β̂ by the means of a parametric
bootstrap procedure, in which artificial pseudo errors are independently drawn from
the truncated normal distribution with left-truncation at 1 − ziβ̂. The latter issue is
addressed in Simar and Wilson (2007) in two different ways, which leads to two different
suggested estimation procedures (algorithm #1 and algorithm #2). Algorithm #1
simply excludes those DMUs from the regression analysis, for which DEA yields scores
θ̂i that equal one. These are obviously artifacts of finite sample bias. The remaining
M (with M < N) DEA scores enter a truncated regression model (left-truncation at
1) as left-hand-side variable. Estimating this model yields β̂, which, together with
the estimate for the variance parameter σ̂, enters the bootstrap procedure mentioned
above. The second suggested approach (algorithm #2) is more involved and rests on bias
corrected DEA scores θ̂bci as left-hand-side variable. Since θ̂
bc
i > 1 holds for i = 1 . . . N ,
unlike algorithm #1, all DMUs are considered in the truncated regression analysis and
the subsequent bootstrap procedure. The bias correction itself rests on a bootstrap
procedure that incorporates the assumptions regarding the process that generates θi,
i.e. equation (1). For this reason, it is computationally simpler and more parametric
than alternative bias correction procedures that have been suggested in the literature
(Simar and Wilson 2000; Kneip et al. 2008) and have recently been made available to
Stata users by the user written command teradialbc (Badunenko and Mozharovskyi
2016).
Figure 1 graphically illustrates the concepts of true, DEA estimated, and bias
which can be written as f (x,y|θ, z) · f (θ|z) · f (z), (Simar and Wilson 2007, p. 35). The focus of
the empirical analysis is on f (θ|z).
13. Discussing the linear programming problem that has to be solved to obtain θ̂i is out of the scope of
this article. Readers not familiar with DEA are referred to the seminal article Charnes et al. (1978)
and standard textbooks such as Coelli et al. (2005) and Cooper et al. (2007). A Stata oriented,
brief, and intuitive introduction is also provided in Badunenko and Mozharovskyi (2016).
14. The present article focuses on the intuition behind the Simar and Wilson (2007) estimator, and its
practical implementation in Stata. For this reason, its statistical properties are not discussed in
depth. Readers, who are interested in more theory oriented discussion of the estimator, are referred
to the original article (Simar and Wilson 2007).
5
AB
yA
yA*
yA
DEA*
yA
bc*
ou
tp
ut
 (y
)
input (x)
observed DMUs true frontier
DEA estimated frontier bias corrected est. frontier
Figure 1: Graphical illustration of true and estimated inefficiency. Considering
DMU A, true inefficiency θA is y
∗
A/yA, (uncorrected) DEA estimated inefficiency θ̂A
is y
DEA∗
A /yA, and bias corrected estimated inefficiency θ̂bcA is y
bc∗
A /yA. In this finite and
small (N = 20) artificial sample, DEA systematically underestimates true inefficiency.
Bias correction adjusts estimated inefficiency upwards. DMU B for instance, which is
seemingly fully efficient according to conventional DEA, is inefficient according to the bias
corrected estimated frontier. Indeed, the inefficiency of B is even overestimated by θ̂bcB .
Unlike for conventional DEA, with bias correction the estimated production-possibility
set is not convex, the estimated frontier is not even monotone. Notes: Input quantities x
randomly drawn from continuous uniform U(0, 2) distribution; true frontier (production
function) y = x
1
4 ; inefficiency generated according to (1), with β = 0 and σ = 3; variable
returns to scale assumed in the DEA; bias correction follows steps 1–4, algorithm #2
(Simar and Wilson 2007, see below). Source: Own calculations.
corrected estimated inefficiency, using randomly generated data and considering a simple
single-input–single-output production technology.
2.2 The procedures suggested in Simar and Wilson (2007)
This subsection in detail describes the suggested procedures algorithm #1 and algorithm
#2, mentioned above. In doing this, it almost one-to-one reproduces what is found at
pages 41-43 in Simar and Wilson (2007). This, in particular, applies to the subsequent
6
paragraphs that describe the steps of the estimation procedure(s) in almost exactly the
same way as they are described in the key reference.
Algorithm #1 consists of the following steps:
1. Compute θ̂i for all DMUs i = 1, . . . , N using DEA.
2. Use those M (with M < N) DMUs, for which θ̂i > 1 holds, in a truncated
regression (left-truncation at 1) of θ̂i on zi to obtain coefficient estimates β̂ and an
estimate for variance parameter σ̂ by maximum likelihood.
3. Loop over the following steps 3.1–3.3 B times, in order to obtain a set of B
bootstrap estimates (β̂b, σ̂b), with b = 1, . . . , B.
3.1 For each DMU i = 1, . . . ,M , draw an artificial error ε˜i from the truncated
N(0, σ̂) distribution with left-truncation at 1− ziβ̂.
3.2 Calculate artificial efficiency scores θ˜i as ziβ̂ + ε˜i for each DMU i = 1, . . . ,M .
3.3 Run a truncated regression (left-truncation at 1) of θ˜i on zi to obtain maximum
likelihood, bootstrap estimates β̂b and σ̂b.
4. Calculate confidence intervals and standard errors for β̂ and σ̂ from the bootstrap
distribution of β̂b and σ̂b.
The more involved algorithm #2 consists of the following steps:
1. Compute θ̂i for all DMUs i = 1, . . . , N using DEA.
2. Use those M (M < N) DMUs, for which θ̂i > 1 holds, in a truncated regression
(left-truncation at 1) of θ̂i on zi to obtain coefficient estimates β̂ and an estimate
for variance parameter σ̂ by maximum likelihood.
3. Loop over the following steps 3.1–3.4 B1 times, in order to obtain a set ob B1
bootstrap estimates θ̂bi for each DMU i = 1, . . . , N , with b = 1, . . . , B1.
3.1 For each DMU i = 1, . . . , N , draw an artificial error ε˜i from the truncated
N(0, σ̂) distribution with left-truncation at 1− ziβ̂.
3.2 Calculate artificial efficiency scores θ˜i as ziβ̂ + ε˜i for each DMU i = 1, . . . , N .
3.3 Generate i = 1, . . . , N artificial DMUs with input quantities x˜i = xi and
output quantities y˜i = (
̂θi/˜θi)yi.
3.4 Use the N artificial DMUs, generated in step 3.3, as reference set in a DEA
that yields θ̂bi for each original DMU i = 1, . . . , N .
4. For each DMU i = 1, . . . , N , calculate a bias corrected efficiency score θ̂bci as
θ̂i −
(
1
B1
∑B1
b=1 θ̂
b
i − θ̂i
)
.
7
5. Run a truncated regression (left-truncation at 1) of θ̂bci on zi to obtain coefficient
estimates
̂̂
β and an estimate for variance parameter ̂̂σ by maximum likelihood.
6. Loop over the following steps 6.1–6.3 B2 times, in order to obtain a set ob B2
bootstrap estimates (
̂̂
β
b
, ̂̂σb), with b = 1, . . . , B2.
6.1 For each DMU i = 1, . . . , N , draw an artificial error ˜˜εi from the truncated
N(0, ̂̂σ) distribution with left-truncation at 1− zî̂β.
6.2 Calculate artificial efficiency scores
˜˜
θi as zi
̂̂
β + ˜˜εi for each DMU i = 1, . . . , N .
6.3 Run a truncated regression (left-truncation at 1) of
˜˜
θi on zi to obtain bootstrap
estimates
̂̂
β
b
and ̂̂σb by maximum likelihood.
7. Calculate confidence intervals and standard errors for
̂̂
β and ̂̂σ from the bootstrap
distribution of
̂̂
β
b
and ̂̂σb.
simarwilson uses the inverse transform method for generating pseudo truncated
normal random variates.15 Choosing sufficiently large values for B1 and B2 – the
latter corresponds to B in algorithm #1 – is crucial for the bias correction and the
estimation of percentile based confidence intervals yielding meaningful results. For B1
and B2 simarwilson uses the default of 100 and 1000 bootstrap repetitions, respectively.
The former default value is suggested in Simar and Wilson (2007), yet depending on
the data used, choosing a substantially larger number for B1 may be advisable. If
normal-approximated confidence intervals (option cinormal) are preferred, one may
choose a much smaller number than the default for B1, and B2 respectively. Running
simarwilson, in particular algorithm #2, requires a substantial amount of computing
time, which rapidly increases in the number of observations. For small samples, looping
over truncated regression takes the lion’s share in computing time. If the sample is large,
looping over DEA consumes relatively more time.16
15. In rare cases, for which the linear prediction ziβ̂ takes extreme values, generating pseudo truncated
normal random variates may fail (Chopin 2011). Therefore simarwilson stops and issues an error
message, if the initial truncated regression (step 2 or step 5) yields abs((1−zi ̂β)/σ̂) > 37.5 for at
least one observation. This provides strong indication for the model being ill-specified or the data
suffering from a sever outlier problem.
16. In the application (N = 131, M = 113, Q = 1, P = 3, K = 7), presented in section 4, with
B = 2000, run time for algorithm #1 is 162 seconds, while it is 170 seconds for algorithm #2,
with B1 = 1000 and B2 = 2000 (Stata/SE 15.1, Windows 10 Enterprise, Intel
R© CoreTM i7-3520M
2.90 GHz, 8 GB RAM). Using the default values for B, B1, and B2 reduces run time to 79 and
81 seconds, respectively. If the the sample is expanded by the factors 2, 5, 10, and 25, run time
for algorithm #1 is increased by the factors 1.1, 1.3, 1.6, and 2.7, respectively. The corresponding
factors for algorithm #2 are 1.2, 1.7, 2.4, and 5.0.
8
2.3 Some minor extensions
The new Stata command simarwilson is meant to implement the above procedures
one-to-one in Stata. It only deviates from what is suggested in Simar and Wilson (2007)
by allowing for some settings and features that are not explicitly considered there.
• simarwilson allows for analyzing input oriented efficiency, while Simar and Wilson
(2007) only consider the output oriented counterpart. This requires estimating an
input oriented efficiency measure ϑ̂i in step 1 (algorithm #1) and steps 1 and 3.4
(algorithm #2) and interchanging the roles of inputs and outputs in the step 3.3
(algorithm #2). Beyond this only two minor changes are required: (i) all truncated
regressions, by default, consider two-sided truncation (at 0 from the left and at 1
from the right) rather than one-sided truncation. (ii) rather than sampling from
a one-sided truncated normal distribution the artificial errors are drawn from a
two-sided truncated normal distribution with left-truncation at −ziβ̂ and right-
truncation at 1− ziβ̂ (algorithm #2, step 6.1 −zî̂β and 1− zî̂β, respectively).17
By this, it is taken into account that the Farrell input oriented efficiency measure
is bounded to the unit interval. Specifying the option base(input) invokes these
deviations from the default procedure. One may optionally (option notwosided)
stick to one-sided truncation and only consider truncation from the right when
analyzing input oriented efficiency. Using option notwosided seems questionable
insofar as it rests on simulating a data generating process that is inconsistent with
the non-negative nature of θi. In particular, notwosided is not recommendable
with algorithm #2.18
• One may opt for the Shephard rather than the Farrell distance measure (option
invert). This simply means that all (internally; see below) estimated scores are
inverted through all steps of the estimation procedure. If constant returns to scale
are assumed for the production technology, this is equivalent to switching from
output to input oriented efficiency. For variable and non-increasing returns, this
one-to-one correspondence does not hold. For option invert being specified with
output oriented efficiency, the same changes to the estimation procedure apply
as described above with respect to option base(input) (without option invert).
Considering the input oriented Farrell or the output oriented Shephard efficiency
measure, which are both bounded to the unit interval, may lead to counterintuitive
results when performing the bias correction in algorithm #2. More precisely, it
may happen that the bias corrected scores are negative for some DMUs. Negative
scores do not enter the truncated regression analysis, unless option notwosided
is specified. If negative efficiency measures occur, simarwilson issues a warning
and recommends switching to Farrell output oriented or Shephard input oriented
17. We would like to thank Ramon Christen for suggesting implementing two-sided truncation in
simarwilson.
18. Sampling from the right-truncated normal distribution may result in ϑ˜i < 0 (step 3.2) and, in
consequence, may make the bias correction fail (x˜i = (
̂ϑi/˜ϑi)xi < 0; step 3.3, alg. #2). For this
reason, notwosided is ignored in step 3.1 of algorithm #2.
9
efficiency, for which bias correction cannot result in negative scores. Yet, ultimately,
the decision how to respond to this problem is up to the user.
• One may assume a data generating process that deviates from (1) by considering
log-(in)efficiency as left-hand-side variable (option logscore), that is
ln(θi) = ziβ + i (2)
Here i is assumed to be truncated normally distributed, with left-truncation at
−ziβ.19 If ln(ϑi) is considered as left-hand-side variable, truncation at −ziβ is
from the right. If all θ̂i are close to unity, specifying the option logscore will
make little difference. Yet, if the data include DMUs that according to the DEA
are very inefficient, specifying logscore may result in a model specification that
is more easily estimated in the truncated regressions.
• simarwilson allows for restricting the reference set for the DEA to a subset of
the considered DMUs (option reference()); cf. Figure 4. Unlike teradial, it
does not allow for considering DMUs as elements of the reference set for which
no efficiency scores are estimated.20 Restricting the reference set to a sub-sample
of the considered DUMs will regularly result in some irregular (super-efficient)
estimated scores. Such DMUs are ignored in the truncated regressions. In general,
restricting the reference set makes the DEA model substantially deviate from what
is considered in Simar and Wilson (2007). Users should, hence, carefully think
about whether using the option reference() really makes sense.
• simarwilson allows for using efficiency scores which were beforehand estimated by
some estimation procedure, using Stata21 or any other software. This effectively
means that step one in algorithm #1 is skipped. If externally estimated, bias
corrected scores are available, one may in principle also skip steps 1–4 in algorithm
#2. However, the bias correction procedure suggested above is specific and
incorporates the assumptions on which the subsequent steps are based. Appropriate
bias corrected scores will, hence, rarely be available. The scores calculated by
teradialbc, though similar in some respect (cf. Simar and Wilson 2007), deviate
from what is computed in steps 1–4 of algorithm #2. Since using any kind of
numeric, non-negative variable as externally estimated score is technically feasible,
it is the user’s responsibility to make sure that this variable is a radial measure of
technical efficiency.
• simarwilson allows for weighted estimation (only pweights and iweights are
allowed). It is important to note that weights are immaterial for the DEA steps
19. Note that (2) is not equivalent to ln(θi − 1) = ziβ + ζi, with ζi ∼ N
(
0, σ2
)
, which might
– erroneously – be regarded as an obvious choice. Unlike (2), this process not only assumes
θi = 1 to occur with zero probability, but regards full efficiency as genuinely impossible. This
conflicts with the production-possibility set including its boundary. Moreover, unlike (2), the above
process assumes that for any DMU reaching some neighbourhood of θi = 1 is relatively unlikely,
i.e. Pr (1 ≤ θi < 1 + τ) < Pr (1 + τ ≤ θi < 1 + 2τ)∀i if τ → 0.
20. This is technically infeasible since in steps 3.2 and 3.3 (algorithm #2) an estimated score is required
for any DMU that contributes to the artificial reference set.
21. Besides teradial the user written command dea (Ji and Lee 2010) allows for this.
10
within simarwilson, but only affect truncated regression estimation. Zero weights
can hence be used for excluding some DMUs from the truncated regression analysis
that are considered in the DEA.
3 The simarwilson command
simarwilson requires Stata 12 or higher. Unless externally estimated efficiency scores
are used, simarwilson requires the user written ado teradial including the associated
plugin (st0444). With internal DEA the number of observations is limited by the value
of [R] matsize that is 11 000 at the maximum. The prefix commands by and svy are
not allowed. The prefix command bootstrap is technically allowed with externally
estimated scores, however using it is entirely counterproductive. pweights (default) and
iweights are allowed, aweights and fweights are not allowed; see [U] 11.1.6 weight.
Weights only affect the truncated regression steps within simarwilson but not the DEA
steps. If iweights are used, (regression) numbers of observations are expressed in terms
of rounded sums of weights.
3.1 Syntax for simarwilson
The syntax for simarwilson reads as follows:
simarwilson
[
(outputs = inputs)
] [
depvar
]
indepvars
[
if
] [
in
] [
,
algorithm(1|2) notwosided logscore nounit rts(crs|nirs|vrs)
base(output|input) reference(varname) invert tename(newvar)
tebc(newvar) biaste(newvar) reps(#) bcreps(#) saveall(name)
bcsaveall(name) dots cinormal bbootstrap level(#) noomitted
baselevels noprint nodeaprint trnoisily
]
outputs is the list of outputs from the production process, while inputs is the corresponding
list of inputs. Either varlist may only include numeric, non-negative variables. Factor
variables and times-series operators are not allowed. The number of output and input
variables must not exceed the number of considered DMUs.
depvar specifies an existing variable that contains an externally estimated efficiency
measure (score), meant to enter the regression model as dependent variable. Specifying
depvar is only possible, if (outputs = inputs) is not specified. That means, with (outputs
= inputs) specified, any variable in the following varlist is interpreted as element of
indepvars. simarwilson expects depvar to be a radial efficiency measure that is either
bounded to the (0, 1] interval or to the [1,∞) interval. This implies that depvar must
not be measured in percent. If some values of depvar are smaller than one while others
exceed one, simarwilson issues a warning and ignores observations, depending on how
the option nounit is specified. This may happen if the preceding efficiency analysis
was carried out using a reference set that does not include all observations for which
11
efficiency scores are estimated. Note that Simar and Wilson (2007) do not consider this
case. Only numeric and strictly positive values are allowed for depvar.
indepvars denotes the list of explanatory variables. Unlike outputs and inputs, factor
variables are allowed in indepvars; see [U] 1.4.3 Factor variables. Time-series operators
such as L. and F. are not allowed.
3.2 Options for simarwilson
algorithm(1|2) specifies whether algorithm #1 or #2 is applied. algorithm(2) re-
quires (outputs = inputs). algorithm(1) is the default. If external DEA scores are
used as depvar, one has to opt for algorithm(1) even if the externally estimated
scores are bias corrected.22
notwosided makes simarwilson apply a one-sided truncated regression model, irre-
spective of whether (regular) efficiency scores are bounded to the (0, 1] interval or
to the [1,∞) interval. For (regular) scores within (0, 1] the default (twosided) is
to use a two-sided truncated regression model and to sample from the two-sided
truncated normal distribution. With twosided, the procedure hence takes into
account that input oriented (Farrell) efficiency scores are not only less than or equal
to 1 but are also strictly positive. The latter is ignored with notwosided. Hence,
with notwosided, simarwilson mirror-inverted applies the procedure suggested in
Simar and Wilson (2007) – that only consider scores within [1,∞) – to efficiency
scores within (0, 1]. For (regular) efficiency scores ≥ 1, specifying notwosided has
no effect. notwosided is not recommended in conjunction with algorithm(2).
logscore makes simarwilson use the natural logarithm of the efficiency score as left-
hand-side variable in the truncated regressions. With logscore specified, truncation
is at 0 rather than at 1 and is always one-sided. If externally estimated scores are
used, one must not take the logarithm beforehand. One rather has to specify the
original untransformed score as depvar.
nounit specifies whether inefficiency is indicated by efficiency score < 1 (unit) or by
efficiency score > 1 (nounit). Specifying this option will rarely be necessary. If the
DEA is carried out internally, simarwilson internally sets nounit depending on
how the options base() and invert are specified. If externally estimated scores are
used and all observations of depvar are either in the (0, 1] or in the [1,∞) interval,
specifying the nounit option is also not required, since simarwilson recognizes
which DMUs are inefficient and which are efficient. Only if external scores are used
that are neither bounded to the (0, 1] interval nor to the [1,∞) interval, nounit is
required to specify which observation of depvar are regular (inefficient) ones and
which are irregular (super-efficient) ones. Note that Simar and Wilson (2007) do not
consider irregular (super-efficient) DMUs.
rts(crs|nirs|vrs) specifies under which assumption regarding the returns to scale of
22. With bias corrected (externally estimated) scores in hand, steps 2–4 of algorithm #1 are fully
equivalent to steps 5–7 of algorithm #2; cf. section 2.2.
12
the considered production process, the measure of technical efficiency is estimated.
crs requests constant returns to scale, nirs requests non-increasing returns to scale,
and vrs requests variable returns to scale. rts(crs) is the default. rts() is passed
through to teradial. If externally estimated scores are used, specifying rts() has
no effect.
base(output|input) specifies orientation/base of the radial measure of technical effi-
ciency. output requests output orientation while input requests input orientation.
base(output) is the default. base() is passed through to teradial and has no
effect if externally estimated scores are used.
reference(varname) specifies the indicator variable that defines which data points of
outputs and inputs (DMUs) form the technology reference set. varname needs to be
binary (numeric or string), with the (alphanumerically) larger value indicating being
part of the reference set. Since for each reference DMU an efficiency score is required
when running simarwilson, the full set of DMUs or a subset of DMUs may serve
as reference set. Yet, the reference set may not include any observations for which
technical efficiency is not estimated. This precludes the specification (ref outputs
= ref inputs), which is allowed in teradial. Specifying a subset of observation as
reference set will frequently result in irregular efficiency estimates (super-efficient
DMUs). Note that Simar and Wilson (2007) consider the full set of observations as
reference set. Specifying a subset as reference, hence, results in a DEA model that
substantially deviates from what is assumed in Simar and Wilson (2007).
invert makes simarwilson calculate and use the Shephard instead of the Farrell
(default) efficiency measure. That is all estimated efficiency scores are inverted,
unless they where externally estimated. invert is redundant for base(crs) since
for constant returns to scale input oriented efficiency is just the reciprocal of output
oriented efficiency. Hence rather than specifying invert one can just switch the
base. Yet, this does not hold for base(nirs) and base(vrs). With externally
estimated scores, specifying invert has no effect. One rather has to manually invert
the externally estimated scores prior to running simarwilson, if one wants to switch
between the Farrell and the Shephard measure.
tename(newvar) creates the new variable newvar that contains estimates of radial
technical efficiency (DEA scores).
tebc(newvar) creates the new variable newvar that contains bias corrected estimates
of radial technical efficiency (bias corrected DEA scores). tebc(newvar) requires
algorithm(2).
biaste(newvar) creates the new variable newvar that contains bootstrap bias esti-
mate for original radial measures of technical efficiency. biaste(newvar) requires
algorithm(2).
reps(#) specifies the number of bootstrap replications for estimating confidence intervals
and standard errors for the regression coefficients. The default is 1000 replications.
bcreps(#) specifies the number of bootstrap replications for the bias correction of DEA
13
scores. The default is 100 replications as suggested in Simar and Wilson (2007).
saveall(name) makes simarwilson save all bootstrap estimates of the regression
coefficients to the (reps×K + 1) mata matrix name. Any existing mata matrix
name is replaced.
bcsaveall(name) makes simarwilson save all bootstrap efficiency scores that are
estimated in the bias correction procedure to the (bcreps× N dea) mata matrix name.
Any existing mata matrix name is replaced. Depending on bcreps(#) and the
number of considered DMUs, the saved mata matrix may be huge.
dots makes simarwilson display one dot character for each bootstrap replication.
cinormal makes simarwilson display normal-approximated confidence intervals rather
than percentile based bootstrap confidence intervals for the regression coefficients.
One may change the reported type of confidence intervals by retyping simarwilson
without arguments and only specifying the option cinormal.
bbootstrap makes simarwilson display mean bootstrap coefficients rather than the
original coefficients from estimating the truncated regression model. One may
change the type of the reported coefficient vector by retyping simarwilson without
arguments and only specifying the option bbootstrap.
level(#); see [R] level estimation options. One may change the reported confidence level
by retyping simarwilson without arguments and only specifying the option level(#).
For percentile based confidence intervals this requires the option saveall(name).
noomitted specifies that variables that were omitted because of collinearity not to be
displayed. The default is to include in the results table any variables omitted because
of collinearity and to label them as omitted by the o. prefix.
baselevels makes simarwilson display base categories of factor variables in the table
of results and label them as base by the #b. prefix.
noprint prevents simarwilson from displaying warnings. Error messages are displayed
irrespective of whether or not noprint is specified.
nodeaprint prevents simarwilson from displaying DEA output.
trnoisily makes simarwilson display genuine output of truncreg for the initial
truncated regression(s) (not for truncated regressions within bootstrap procedures).
Specifying this option might be useful if simarwilson issues the error message
‘truncated regression failed’ or ‘convergence not achieved in truncated regression’ and
the accompanying return code is inconclusive about what makes truncreg fail.
In addition, simarwilson allows for all maximization options that are allowed with
truncreg, which are simply passed through; see [R] maximize. Moreover, one may specify
the truncreg options noconstant, offset(varname), and constraints(constraints),
which are also passed through; see [R] truncreg.
14
3.3 Saved results for simarwilson
simarwilson saves the following results to e():
Scalars
e(N) # of observations
(inefficient DMUs)
e(chi2) model chi-squared
e(N lim) # of limit observations
(efficient DMUs)
e(p) model significance, p-value
e(N irreg) # of irregular observations
(super-efficient DMUs)
e(N reps) # of completed bootstrap reps
e(N all) overall # of observations
(all DMUs with eff. score)
e(N misreps) # of failed bootstrap reps
e(wgtsum) sum of weights
(if weights are specified)
e(level) confidence level
e(sigma) estimate of sigma e(algorithm) algorithm used (1 or 2)
e(ll) pseudo log-likelihood
(initial truncated reg.)
e(noutps) # of output variables
e(ic) # of iterations
(initial truncated reg.)
e(ninps) # of input variables
e(converged) 1 converged, 0 otherwise
(initial truncated reg.)
e(N dea) # of DMUs for which efficiency
scores are estimated
e(rc) return code e(N dearef) # of DMUs in reference set
e(k eq) # of equations in e(b) e(N deaneg) # of negative bias corrected
scores
e(df m) model degrees of freedom e(N bc) # of completed bootstrap
reps (bias correction)
Macros
e(title) Simar & Wilson (2007)
two-stage efficiency analysis
e(marginsdefault) default predict()
specification for margins
e(shorttitle) Simar & Wilson (2007)
eff. analysis
e(predict) program used to
implement predict
e(cmdline) command as typed e(cinormal) cinormal (if option cinormal is
specified)
e(cmd) simarwilson e(bbootstrap) bbootstrap (if option
bbootstrap is specified)
e(unit) either unit or nounit e(scoretype) either score or bcscore
e(truncation) either twosided or onesided e(invert) either Farrell or Shephard
e(logscore) logscore if option logscore
is specified
e(biaste) varname of estimated bias (if op-
tion biaste is specified)
e(wtype) either pweight or iweight
(if weights are specified)
e(tebc) varname of estimated bias cor-
rected efficiency (if option tebc
is specified)
e(wexp) exp
(if weights are specified)
e(tename) varname of estimated uncor-
rected efficiency (if option tename
is specified)
e(depvarname) name of lhs variable e(rts) returns to scale (CRS or NIRS or
VRS) (if DEA is internal)
e(depvar) either efficiency or inefficiency e(base) base/orientation (output or
input) (if DEA is internal)
e(saveall) name if option
saveall(name) is specified
e(outputs) varlist of outputs (if DEA is in-
ternal)
e(bcsaveall) name if option
bcsaveall(name) is specified
e(inputs) varlist of inputs (if DEA is inter-
nal)
e(marginsok) predictions allowed by margins e(properties) b V
15
Matrices
e(b) vector of estimated coefficients e(b bstr) bootstrap estimates of coeffi-
cients
e(V) estimated coefficient
variance-covariance matrix
e(bias bstr) bootstrap estimated biases
e(Cns) constraints matrix
(if constraints are specified)
e(ci percentile) bootstrap percentile confi-
dence intervals
Functions
e(sample) marks estimation sample
Note that e(sample) and e(N) refer to those observations that enter the truncated
regression analysis.
3.4 simarwilson postestimation
The postestimation commands that are available after simarwilson are almost the
same as for truncreg; see [R] truncreg postestimation. Among others, these
are test, testnl, lincom, nlcom, predict, predictnl, and [R] margins. margins,
dydx(indepvars) appears to be particularly valuable. After simarwilson, margins
behaves slightly differently than it behaves after truncreg. The default is to estimate
marginal effects on expected (in)efficiency that is on E (θi|θi > 1, zi) (Farrell output ori-
ented) and E (ϑi|0 < ϑi < 1, zi) (Farrell input oriented), respectively.23 That is margins,
by default, internally sets the options predict(e(1,.)) and predict(e(0,1)), respec-
tively.24 If one wants to estimate marginal effects on the linear index, specifying the option
predict(xb) is required. The options predict(ystar(a,b)) and predict(pr(a,b))
are not allowed with margins after simarwilson. They make margins consider a
censored outcome, which makes little sense with simarwilson.
Users should, in general, be careful in interpreting the results one obtains from postes-
timation commands, such as predict, used after simarwilson. The postestimation
commands treat the results of simarwilson as if they where generated by truncreg.
One should, however, be aware that in terms of the underlying model both are not
the same. Besides the estimated variance-covariance matrix, the key difference is that
truncreg usually assumes that the left-hand-side variable of the data generating process
is observed for not-truncated observation and may in principle also be observable for
truncated observations. In contrast simarwilson rests on the assumption that the true
outcome variable is genuinely unobservable. Moreover, while in many applications of
truncreg truncation originates from missing information, for simarwilson truncation
is a genuine feature of the data generating process; see section 2.
23. Since the data generating process (1) assumes a truncated distribution for εi, E (θi|θi > 1, zi)
coincides with E (θi|zi). I.e. θi < 1 is not only not observed, it is rather impossible according
to (1). The same line of argument applies to E (ϑi|0 < ϑi < 1, zi) as well as to the Shephard
measures. This argument would not hold for standard applications of truncreg, which illustrates
that the truncreg postestimation commands should be used with some caution after simarwilson.
With the options nounit and notwosided simultaneously specified, margins by default considers
E (ϑi|ϑi < 1, zi) that is the option predict(e(.,1)) is internally set.
24. With option logscore the default is predict(e(0,.)) and predict(e(.,0)), respectively. That is
margins, by default, yields semi-elasticities.
16
4 An application of simarwilson
4.1 Comparison of estimation methods
In order to illustrate how simarwilson can be used in applied work, in this section
we use the command for empirically addressing the question of whether the quality of
governance, including quality of the judicial system, at the national level matters for
the efficiency of gross domestic product (GDP) generation. The analysis is based on
cross country data that is provided through the Penn World Table data base, version 9
(Feenstra et al. 2015) and the World Economic Forum, Global Competitiveness Report,
version 2018-02-26 (World Economic Forum 2018; Schwab 2017). Though both data
bases are publicly available on the internet, only the Penn World Table allows for being
straightforwardly used with Stata. For this reason, this article is accompanied by the user
written ado-file gciget.ado that facilitates the retrieval of the Global Competitiveness
Index data using Stata. See subsection 4.3 for a more detailed description of gciget.
The Stata log below is from using gciget to load three selected variables (EOSQ048,
EOSQ051, EOSQ144) of the Global Competitiveness Index into Stata and merging them
to the Penn World Table data.
. gciget EOSQ048 EOSQ051 EOSQ144
DISCLAIMER: The World Economic Forum is the provider of the Global
Competitiveness Index 2017-2018, a framework and a corresponding set of
indicators for 137 economies. The software gciget.ado provides a practical
way to read the indicators into Stata (R). The responsibility of complying
with the terms and conditions of use under which the owner of the data
grants access to the indicators is entirely with the user but not with the
authors of the software gciget.ado. Any user of gciget.ado is responsible
for making him or herself familiar with the terms of use under which she or
he is allowed to work with the data of the Global Competitiveness Index. For
more information and methodology, please see http://wef.ch/gcr17. In no
event will the authors, owners, and creators of gciget.ado, or their
employers or any other party who may modify and/or redistribute this
software, accept liability for any loss or damage suffered as a result of
using the gciget.ado software.
Downloading the GCI_Dataset_2007-2017.xlsx file
Importing the GCI_Dataset_2007-2017.xlsx file
Processing EOSQ048: 1.09 Burden of government regulation, 1-7 (best)
Processing EOSQ051: 1.01 Property rights, 1-7 (best)
Processing EOSQ144: 1.06 Judicial independence, 1-7 (best)
. tempfile data_gci_selected
. quietly save `data_gci_selected´
. use "https://www.rug.nl/ggdc/docs/pwt90.dta", clear
. quietly merge 1:1 countrycode year using "`data_gci_selected´"
We consider a national-level production process that generates the single output real
GDP (rgdpo) by using three inputs: capital stock (ck), number of persons engaged (emp),
and human capital (hc). We assume variable returns to scale and consider the output
oriented Farrell efficiency measure. As key explanatory variables we consider the ‘burden
of government regulation’ (EOSQ048), ‘property rights protection’ (EOSQ051), and ‘judicial
independence’ (EOSQ144). While all the rest of the data used are from the Penn World
17
Table, the latter three variables are provided through the Global Competitiveness Report.
These indices are measured on a continuous scale ranging from 1 to 7 and originate from
answers to the following questions in the World Economic Forum, Executive Opinion
Survey (see Schwab 2017, Appendix C for details): “In your country, how burdensome
is it for companies to comply with public administrations requirements (e.g., permits,
regulations, reporting)? [1 = extremely burdensome; 7 = not burdensome at all]”; “In
your country, to what extent are property rights, including financial assets, protected?
[1 = not at all; 7 = to a great extent]”; “In your country, how independent is the
judicial system from influences of the government, individuals, or companies? [1 = not
independent at all; 7 = entirely independent]” (World Economic Forum 2018). In order
to address possible endogeneity concerns regarding these regressors, we let them enter
the model as lagged values. In addition to the three explanatory variables of primary
interest, we include lagged log-population (lpop) as control. To allow for country-size
related heterogeneity in the link between governance quality and national efficiency, we
interact the governance quality indices with lpop in the regression models.
After loading the working data into Stata’s memory, we generate the explanatory
variables that we actually need in the empirical analysis and give them more telling
names. Since simarwilson does not allow for times-series operators, we generate lagged
values ‘by hand’. To make the code easier to read, we place the governance quality
variables in the global macro g list and define the global macro z list that contains
the comprehensive list of explanatory variables. Since the sample size is relatively small,
we opt for a rather generous level of significance by setting the confidence level to 90%.
Moreover, we set a new seed for Stata’s random number generator.25 To facilitate the
replication of results, the random number generator is reset to this state every time
simarwilson runs in the application. To preserve the spirit of a randomness, this should
be avoided in own applications.
. qui gen regu = EOSQ048[_n-1] if countrycode == countrycode[_n-1]
. qui gen prop = EOSQ051[_n-1] if countrycode == countrycode[_n-1]
. qui gen judi = EOSQ144[_n-1] if countrycode == countrycode[_n-1]
. qui gen lpop = ln(pop[_n-1]) if countrycode == countrycode[_n-1]
. global g_list "regu prop judi"
. global z_list "regu prop judi lpop c.regu#c.lpop c.prop#c.lpop c.judi#c.lpop"
. set level 90
. set seed 341566575
Second, we use teradial to generate externally estimated DEA efficiency scores
(te vrs o) using the most recent year that is available in the data, that is 2014. We
restrict the DEA to countries for which information on all right-hand-side variables is
available.26 Since we do not define a reference set that deviates from the sample for
which efficiency measures are estimated, the option base(o) makes te vrs o taking
25. The default random number generator (mt64) of Stata 15 is used.
26. This is makes the DEA steps in algorithms #1 and #2 using the same sample. Only for the latter,
the right-hand-side variables are required for estimating (bias corrected) efficiency scores; cf. step 3,
alg. #2, p. 7.
18
vales equal or larger than one.27 Then we let Stata report descriptive statistics for
the variables used in the subsequent regressions. Due to missing information in some
variables only 131 countries out of 182 covered by the Penn World Table can be used for
estimation.
. teradial rgdpo = ck emp hc if year == 2014 & regu <. & prop <. & judi <. & lp
> op <., te(te_vrs_o) rts(v) base(o) noprint
. sum te_vrs_o regu prop judi lpop if e(sample)
Variable Obs Mean Std. Dev. Min Max
te_vrs_o 131 1.699949 .6236905 1 5.513838
regu 131 3.435143 .6711715 1.846199 5.42263
prop 131 4.304648 1.030568 1.610298 6.378975
judi 131 3.897085 1.315987 1.113236 6.678279
lpop 131 2.566502 1.586448 -1.264066 7.217087
As the next step of the analysis, we use four empirical models to explain (in)efficiency
in GDP generation. Besides simarwilson, algorithm(1) and algorithm(2) we also
consider tobit and truncreg as references. Since the model coefficients themselves do
not allow for being straightforwardly interpreted in quantitative terms, we use margins,
dydx() to estimate average marginal effects of the governance quality indices on national
GDP efficiency.
We start with tobit estimation which – according to Simar and Wilson (2007) –
erroneously regards full efficiency (te vrs o = 1) as outcome of the underlying data
generating process rather than an artifact of finite sample bias.28 Consistent with this
misinterpretation we use the option predict(ystar(1,.)) with margins. Estimated
marginal effects are not displayed but stored with estimates store for later comparison.
The output from tobit reveals that, according to DEA, 18 countries are fully efficient
while 113 are found to be inefficient. With judical independence being the only exception,
the governance variables are individually significant at the 10% level and bear the
expected negative signs. However, because of the model including several interactions
with log-population, making any statement about the link between governance quality
and GDP efficiency is hardly possible without examining marginal effects. At least, the
sings of coefficients attached to the interaction variables seem to indicate that possible
efficiency gains trough less business regulation and better protection of property rights
are first of all a matter of small countries.
27. Unless option invert is used, positive coefficient of a variable in simarwilson would imply it has
a negative effect on efficiency. Conversely, negative coefficient would mean that the variable has
positive effect on efficiency.
28. The options nolstretch and vsquish are just for making the displayed output fit on a printed page.
19
. tobit te_vrs_o $z_list, ll(1) nolstretch vsquish
Refining starting values:
Grid node 0: log likelihood = -130.85914
Fitting full model:
Iteration 0: log likelihood = -130.85914
Iteration 1: log likelihood = -128.73992
Iteration 2: log likelihood = -128.71027
Iteration 3: log likelihood = -128.7102
Iteration 4: log likelihood = -128.7102
Tobit regression Number of obs = 131
Uncensored = 113
Limits: lower = 1 Left-censored = 18
upper = +inf Right-censored = 0
LR chi2(7) = 20.43
Prob > chi2 = 0.0047
Log likelihood = -128.7102 Pseudo R2 = 0.0735
te_vrs_o Coef. Std. Err. t P>|t| [90% Conf. Interval]
regu -.3925008 .2014823 -1.95 0.054 -.7264042 -.0585973
prop -.5199721 .2574393 -2.02 0.046 -.9466096 -.0933347
judi .2488415 .1888903 1.32 0.190 -.064194 .5618771
lpop -.8211409 .2667289 -3.08 0.003 -1.263173 -.3791084
c.regu#
c.lpop .1484147 .0687327 2.16 0.033 .0345084 .2623209
c.prop#
c.lpop .1251518 .0871451 1.44 0.153 -.0192682 .2695717
c.judi#
c.lpop -.0858924 .0693701 -1.24 0.218 -.2008549 .0290701
_cons 4.589835 .7747277 5.92 0.000 3.305929 5.873741
var(e.te_v~o) .4098449 .0562185 .3265083 .514452
. qui margins, dydx($g_list) predict(ystar(1,.)) post
. estimates store tobit
Then we turn to the truncated regression by using truncreg. Unlike tobit, this
approach does not consider observations for which te vrs o = 1 holds. For this reason,
we use the option predict(e(1,.)) when estimating marginal effects. The estimated
coefficients look quite different compared to their counterparts from tobit. Yet in terms
of the signs, the results are similar to their counterparts from tobit. According to the
results from truncreg judical independence seem to matter for efficiency, since both
judi and its interaction with lpop are statistically significant at the 10% level. This
points to judical independence being negatively associated with efficiency, at least in
small countries. However, following the argument of Simar and Wilson (2007), this result
might be an artifact of incorrectly estimated standard errors.
20
. truncreg te_vrs_o $z_list, ll(1) nolstretch vsquish
(note: 18 obs. truncated)
Fitting full model:
Iteration 0: log likelihood = -76.432745
Iteration 1: log likelihood = -68.518139
Iteration 2: log likelihood = -67.617016
Iteration 3: log likelihood = -67.606346
Iteration 4: log likelihood = -67.606307
Iteration 5: log likelihood = -67.606307
Truncated regression
Limit: lower = 1 Number of obs = 113
upper = +inf Wald chi2(7) = 18.90
Log likelihood = -67.606307 Prob > chi2 = 0.0085
te_vrs_o Coef. Std. Err. z P>|z| [90% Conf. Interval]
regu -.9258069 .4299484 -2.15 0.031 -1.633009 -.2186048
prop -1.243902 .4991533 -2.49 0.013 -2.064936 -.4228676
judi .7784162 .3780368 2.06 0.039 .156601 1.400231
lpop -1.739993 .5952224 -2.92 0.003 -2.719046 -.7609389
c.regu#
c.lpop .4253728 .1720618 2.47 0.013 .1423563 .7083894
c.prop#
c.lpop .2581352 .1794841 1.44 0.150 -.0370899 .5533604
c.judi#
c.lpop -.2592945 .1497392 -1.73 0.083 -.5055935 -.0129955
_cons 7.447817 1.629842 4.57 0.000 4.766965 10.12867
/sigma .7222912 .0925133 7.81 0.000 .5701204 .8744621
. qui margins, dydx($g_list) predict(e(1,.)) post
. estimates store truncreg
Hence, in the next step, we turn to simarwilson, algorithm(1). Since externally
estimated efficiency scores are already available, we do not rerun the DEA within
simarwilson but use te vrs o as dependent variable. Using the (rgdpo = ck emp hc)
syntax instead, and specifying the options rts(v) and base(o) would have generated
identical results. Since we report percentile confidence intervals for the coefficients,
we request a large number (2000) for the bootstrap replications. This choice results
in a substantial computing time of 162 seconds (Stata/SE 15.1).29 Specifying the
option predict() is not required for margins, since the appropriate specification is
set internally. As a practical matter we advise to use 1 processor in the MP version
of Stata by typing set processors 1 before executing simarwilson. The estimated
coefficient necessarily coincide with what we got from truncreg, since simarwilson,
algorithm(1) only affects the estimated standard errors and confidence intervals. Yet,
even with respect to the latter two, the deviation from their na¨ıve counterparts from
truncreg is rather moderate. This is in line with what is frequently found in applications
of algorithm #1.
29. Carrying out the DEA internally affects computing time just marginally.
21
. simarwilson te_vrs_o $z_list, reps(2000)
Simar & Wilson (2007) eff. analysis Number of obs = 113
(algorithm #1) Number of efficient DMUs = 18
Number of bootstr. reps = 2000
Wald chi2(7) = 21.63
inefficient if te_vrs_o > 1 Prob > chi2(7) = 0.0029
Data Envelopment Analysis: externally estimated scores
Observed Bootstrap Percentile
inefficiency Coef. Std. Err. z P>|z| [90% Conf. Interval]
te_vrs_o
regu -.9258069 .4021808 -2.30 0.021 -1.61701 -.2838173
prop -1.243902 .4715584 -2.64 0.008 -2.042034 -.5066595
judi .7784162 .356048 2.19 0.029 .1985666 1.371619
lpop -1.739993 .5688841 -3.06 0.002 -2.670876 -.8296476
c.regu#c.l~p .4253728 .1611459 2.64 0.008 .1649802 .6971241
c.prop#c.l~p .2581352 .1692766 1.52 0.127 -.013744 .5414581
c.judi#c.l~p -.2592945 .1400455 -1.85 0.064 -.4829779 -.0310317
_cons 7.447817 1.557957 4.78 0.000 4.988534 9.974977
/sigma .7222912 .0877174 8.23 0.000 .5537709 .8368937
. qui margins, dydx($g_list) post
. estimates store alg_1
Then we turn to algorithm(2). In this procedure tailored, bias corrected efficiency
scores enter the regression model at the left-hand-side. Hence, we cannot use exter-
nally estimated scores but let simarwilson carry out the bias correction internally.
This requires the (rgdpo = ck emp hc) syntax along with the options rts(v) and
base(o). The latter two determine the DEA model used. By specifying the option
tebc(tebc vrs o) we save the estimated, bias corrected efficiency scores for possible
later use. We opt for 1000 replications in the bias correction bootstrap, which is well
above the default suggested in Simar and Wilson (2007). Estimating this model takes
170 seconds. Due to the relatively small sample size, using algorithm(2) increases
computing time by only 5%; cf. footnote 16. Since we de not use externally estimated
scores as left-hand-side variable, but let simarwilson run the DEA internally, the re-
ported output also involves comprehensive information about the DEA model used.30
In the present application, using bias corrected instead of uncorrected scores has just a
moderate impact on the estimated coefficients and the associated estimated confidence
intervals, see the output below.
30. The option nodeaprint suppresses displaying the DEA related information.
22
. simarwilson (rgdpo = ck emp hc) $z_list if year == 2014, alg(2) rts(v) base(o
> ) reps(2000) bcreps(1000) tebc(tebc_vrs_o)
Simar & Wilson (2007) eff. analysis Number of obs = 131
(algorithm #2) Number of efficient DMUs = 0
Number of bootstr. reps = 2000
Wald chi2(7) = 21.10
inefficient if tebc_vrs_o > 1 Prob > chi2(7) = 0.0036
Data Envelopment Analysis: Number of DMUs = 131
Number of ref. DMUs = 131
output oriented (Farrell) Number of outputs = 1
variable returns to scale Number of inputs = 3
bias corrected efficiency measure Number of reps (bc) = 1000
Observed Bootstrap Percentile
inefficiency Coef. Std. Err. z P>|z| [90% Conf. Interval]
tebc_vrs_o
regu -.9161716 .3876275 -2.36 0.018 -1.547635 -.2845122
prop -1.209787 .5058764 -2.39 0.017 -2.062226 -.4182389
judi .6717764 .3638639 1.85 0.065 .0861624 1.271846
lpop -1.796833 .5587313 -3.22 0.001 -2.725979 -.8933189
c.regu#c.l~p .4237473 .1520378 2.79 0.005 .1775733 .6736762
c.prop#c.l~p .2335061 .1755503 1.33 0.183 -.0346361 .53319
c.judi#c.l~p -.2302095 .1372156 -1.68 0.093 -.45712 -.0115268
_cons 7.887194 1.604777 4.91 0.000 5.281076 10.45693
/sigma .8735555 .1007678 8.67 0.000 .6807801 1.010889
. estimates store alg_2_raw
. qui margins, dydx($g_list) post
. estimates store alg_2
sum gives us descriptive statistics for estimated, bias corrected inefficiency. Comparing
them to the descriptives for te vrs o shows that the bias correction adjusts the estimated
scores away from unity, ruling out (seemingly) fully efficient countries.
. sum tebc_vrs_o
Variable Obs Mean Std. Dev. Min Max
tebc_vrs_o 131 1.956169 .7083735 1.068883 6.482759
In order to allow for interpreting the results in qualitative terms we examine the
estimated mean marginal effects. This yields a rather clear picture. While, on average,
the regulatory burden and judical independence appear to be immaterial for the efficiency
of GDP generation, the protection of property rights matters. Except for tobit, the
estimated marginal effect is clearly significant and amounts to roughly 1/3 (Farrell, output-
oriented) units, by which inefficiency is reduced in response to an one unit increase in
property rights protection. This appears to be a strong effect that corresponds to a shift
from the median to the 27th percentile of the sample distribution of tebc vrs o.
23
. estimates table tobit truncreg alg_1 alg_2, title(Estimated Mean Marginal Eff
> ects) p
Estimated Mean Marginal Effects
Variable tobit truncreg alg_1 alg_2
regu -.02001409 .04003719 .04003719 .04118827
0.8110 0.6720 0.6607 0.6814
prop -.17286701 -.33398801 -.33398801 -.33925269
0.1211 0.0049 0.0042 0.0108
judi .02948397 .08804266 .08804266 .06698891
0.7278 0.3449 0.3325 0.5203
legend: b/p
Measuring effects in terms of Farrell (output oriented) efficiency units appears not to
be particularly telling. One may, hence, prefer a scaled efficiency measure, that allows for
interpreting marginal effects in terms of percentage points. This calls for switching from
the Farrell to the Shephard efficiency measure. Switching from output to input oriented
efficiency, which would also yield efficiency scores within the unit interval, does not have
much appeal for the present application. It would imply the thought experiment of
reducing input consumption, which appears rather odd given that the national capital
stock and human capital are among the inputs variables.
While switching to the Shephard measure was straightforward for algorithm(1) –
one just has to use the reciprocal of te vrs o as dependent variable – in the present
application it causes difficulties with algorithm(2). As indicated by a warning issued by
simarwilson (see below), the bias correction yields some negative scores; cf. subsection
2.3. These are not used in the truncated regressions. Thus only 127, not 131, countries
enter the regression analysis. In qualitative terms, using the Shephard measure as
left-hand-side variable does not change the general pattern of results. As expected (see
footnote 27), the signs of all coefficients are just reversed and all coefficients remain
statistically significant.
24
. simarwilson (rgdpo = ck emp hc) $z_list if year == 2014, alg(2) rts(v) base(o
> ) reps(2000) bcreps(1000) invert
warning: bias-correction yields at least one negative score; consider
dropping opt. invert or switching to base(input)
Simar & Wilson (2007) eff. analysis Number of obs = 127
(algorithm #2) Number of efficient DMUs = 0
Number of bootstr. reps = 2000
inefficient if bcscore < 1 Wald chi2(7) = 93.44
twosided truncation Prob > chi2(7) = 0.0000
Data Envelopment Analysis: Number of DMUs = 131
Number of ref. DMUs = 131
output oriented (Shephard) Number of outputs = 1
variable returns to scale Number of inputs = 3
bias corrected efficiency measure Number of reps (bc) = 1000
Observed Bootstrap Percentile
efficiency Coef. Std. Err. z P>|z| [90% Conf. Interval]
bcscore
regu .0691237 .0376769 1.83 0.067 .006694 .1316281
prop .2030272 .049161 4.13 0.000 .123413 .2827933
judi -.1056821 .0366765 -2.88 0.004 -.1656055 -.0459
lpop .2506157 .0484025 5.18 0.000 .1726308 .3321083
c.regu#c.l~p -.0408009 .013037 -3.13 0.002 -.062014 -.0190109
c.prop#c.l~p -.0632933 .0165594 -3.82 0.000 -.0897817 -.0356286
c.judi#c.l~p .0520734 .013219 3.94 0.000 .0304517 .074041
_cons -.3667276 .1422998 -2.58 0.010 -.605271 -.1369738
/sigma .1190544 .0076651 15.53 0.000 .1025166 .1276049
. qui margins, dydx($g_list) post
. estimates store alg_2_inv
One may force simarwilson to use negative bias corrected scores in the regression
analysis by combining invert with the option notwosided. By doing this one accepts,
however, two inconsistencies. Besides allowing for negative efficiency scores, which
arguably makes little sense, one makes simarwilson apply different truncation rules in
different steps of the estimation procedure; see footnote 18. As can be seen from the
output below, simarwilson points the user to this issue. Indeed, forcing simarwilson
to consider the few observations with negative scores has noticeable impact on the
estimated coefficients.
25
. simarwilson (rgdpo = ck emp hc) $z_list if year == 2014, alg(2) rts(v) base(o
> ) reps(2000) bcreps(1000) invert notwosided
warning: opt. notwosided not recommendable with alg. #2; in step 3.1
(alg. #2) sampling is from the twosided-truncated normal distribution
warning: bias-correction yields at least one negative score; consider
dropping opt. invert or switching to base(input)
Simar & Wilson (2007) eff. analysis Number of obs = 131
(algorithm #2) Number of efficient DMUs = 0
Number of bootstr. reps = 2000
inefficient if bcscore < 1 Wald chi2(7) = 98.54
onesided truncation Prob > chi2(7) = 0.0000
Data Envelopment Analysis: Number of DMUs = 131
Number of ref. DMUs = 131
output oriented (Shephard) Number of outputs = 1
variable returns to scale Number of inputs = 3
bias corrected efficiency measure Number of reps (bc) = 1000
Observed Bootstrap Percentile
efficiency Coef. Std. Err. z P>|z| [90% Conf. Interval]
bcscore
regu .0400735 .0429141 0.93 0.350 -.031693 .1113316
prop .2704966 .0544725 4.97 0.000 .1828421 .3623146
judi -.1504577 .0401147 -3.75 0.000 -.216313 -.0863225
lpop .2538996 .056574 4.49 0.000 .1615896 .3499556
c.regu#c.l~p -.0301825 .0149123 -2.02 0.043 -.0546047 -.0048266
c.prop#c.l~p -.0856331 .0186131 -4.60 0.000 -.1159246 -.0555597
c.judi#c.l~p .0696979 .0147094 4.74 0.000 .0452582 .0940363
_cons -.4241697 .1654795 -2.56 0.010 -.7034528 -.1535988
/sigma .1404817 .0088438 15.88 0.000 .1210069 .1504123
. qui margins, dydx($g_list) post
. estimates store alg_2_notwo
In order to specify a model which renders interpreting estimation results in quan-
titative terms more convenient, using the option logscore is a possible alternative to
invert. By considering log-inefficiency as left-hand-side variable, marginal effects can
be interpreted as percentage reductions in inefficiency. Hence we rerun or preferred
model (algorithm #2, Farrell output-oriented efficiency) using the option logscore. The
statistical significance and the signs of the estimated coefficients are equivalent to those
from the specification of reference.
26
. simarwilson (rgdpo = ck emp hc) $z_list if year == 2014, alg(2) rts(v) base(o
> ) reps(2000) bcreps(1000) logscore
Simar & Wilson (2007) eff. analysis Number of obs = 131
(algorithm #2) Number of efficient DMUs = 0
Number of bootstr. reps = 2000
Wald chi2(7) = 37.48
inefficient if ln(bcscore) > 0 Prob > chi2(7) = 0.0000
Data Envelopment Analysis: Number of DMUs = 131
Number of ref. DMUs = 131
output oriented (Farrell) Number of outputs = 1
variable returns to scale Number of inputs = 3
bias corrected efficiency measure Number of reps (bc) = 1000
Observed Bootstrap Percentile
inefficiency Coef. Std. Err. z P>|z| [90% Conf. Interval]
ln(bcscore)
regu -.2212588 .0942768 -2.35 0.019 -.3813767 -.0688785
prop -.3127128 .1199882 -2.61 0.009 -.5092462 -.1177127
judi .1617935 .0877577 1.84 0.065 .0224931 .3082684
lpop -.4737086 .1279737 -3.70 0.000 -.6940146 -.2691337
c.regu#c.l~p .0927766 .0340445 2.73 0.006 .038239 .1526358
c.prop#c.l~p .0777964 .0414616 1.88 0.061 .0125089 .1486909
c.judi#c.l~p -.0644932 .0323986 -1.99 0.047 -.1192701 -.0134126
_cons 2.312936 .3706691 6.24 0.000 1.701108 2.91863
/sigma .2882745 .020727 13.91 0.000 .2448331 .3138336
. qui margins, dydx($g_list) post
. estimates store alg_2_log
One may not feel comfortable with using a (bias corrected) efficiency measure that
conflicts with convexity of the production-possibility set; cf. Figure 1. One way of
addressing this issue, is to once again envelope the non-convex bias corrected estimated
frontier by a convex hull and to use the distance to this convexified bias corrected frontier
as dependent variable in the regression analysis (cf. Badunenko et al. 2013, and Figure
5). The (ref outputs = ref inputs) specification of teradial allows for straightforwardly
implementing this procedure; see the below Stata log and Badunenko and Mozharovskyi
(2016). Compared to its direct counterpart (simarwilson, algorithm(2) without
invert and logscore), using this once more adjusted efficiency measure changes the
estimated coefficients markedly. Yet, qualitatively, the pattern of estimates remains the
same.
27
. qui gen rgdpo_front = tebc_vrs_o*rgdpo
. teradial rgdpo = ck emp hc (rgdpo_front = ck emp hc) if year == 2014 & regu <
> . & prop <. & judi <. & lpop <., te(tebc_vrs_o_convex) rts(v) base(o) noprint
. simarwilson tebc_vrs_o_convex $z_list if year == 2014, reps(2000)
Simar & Wilson (2007) eff. analysis Number of obs = 131
(algorithm #1) Number of efficient DMUs = 0
Number of bootstr. reps = 2000
Wald chi2(7) = 24.89
inefficient if tebc_vrs_o_convex > 1 Prob > chi2(7) = 0.0008
Data Envelopment Analysis: externally estimated scores
Observed Bootstrap Percentile
inefficiency Coef. Std. Err. z P>|z| [90% Conf. Interval]
tebc_vrs_o~x
regu -.8525049 .351338 -2.43 0.015 -1.436492 -.2731832
prop -1.104847 .4583591 -2.41 0.016 -1.884511 -.3711119
judi .5759623 .3329715 1.73 0.084 .0401372 1.122519
lpop -1.748744 .5217818 -3.35 0.001 -2.625194 -.919087
c.regu#c.l~p .3752556 .1349932 2.78 0.005 .1569001 .5928116
c.prop#c.l~p .2194959 .1578936 1.39 0.164 -.0314325 .4908592
c.judi#c.l~p -.1856799 .1233833 -1.50 0.132 -.3866211 .0144541
_cons 7.956469 1.475931 5.39 0.000 5.576847 10.362
/sigma .8986468 .0911453 9.86 0.000 .7240245 1.017877
. qui margins, dydx($g_list) post
. estimates store bc_convex
Finally, we compare the marginal effects for all specifications of simarwilson that
we have estimated. Somewhat surprisingly, unlike the specification of reference, the
specifications using the Shephard measure argue for more regulatory interference im-
proving efficiency (p-values 0.039 and 0.096, respectively). One may, hence, speculate
that regu not only captures detrimental but also beneficial facets of business regulation.
In terms of the point estimates, all model specification yield a positive association
of property right protection and GDP efficiency. Only for the Shephard measure as
left-hand-side variable (without option notwosided) the average marginal effect of prop
turns statistically insignificant at the 10% level.
Using the Shephard measure (option invert) or the option logscore makes inter-
preting the estimated marginal effect easier. According to the specification using the
Shephard measure (without notwosided) a one unit increase in prop on average improves
efficiency by 3.6 percentage points. According to the specification using log-inefficiency
at the left-hand-side, the mean effect is a 10.7 percent reduction in inefficiency. With
respect to judi, the estimated marginal effects are throughout statistically insignificant.
In terms of estimated average marginal effects, basing the analysis on a convex estimated
hull has almost no effect as compared to using the non-convex, bias corrected estimated
frontier.
28
. estimates table alg_1 alg_2 alg_2_inv alg_2_notwo alg_2_log bc_convex, title(
> Estimated Mean Marginal Effects) p b(%5.4f) p(%4.3f)
Estimated Mean Marginal Effects
Variable alg_1 alg_2 alg_2~v alg_2~o alg_2~g bc_co~x
regu 0.0400 0.0412 -0.0378 -0.0358 0.0074 0.0210
0.661 0.681 0.039 0.096 0.861 0.848
prop -0.3340 -0.3393 0.0359 0.0532 -0.1067 -0.3541
0.004 0.011 0.139 0.063 0.056 0.015
judi 0.0880 0.0670 0.0310 0.0256 0.0020 0.0828
0.332 0.520 0.104 0.253 0.964 0.455
legend: b/p
4.2 Effect heterogeneity
We complete our application by analyzing possible heterogeneity in the efficiency effects
of ‘burden of government regulation’, ‘property rights protection’, and ‘judicial indepen-
dence’. In doing this, we focus on simarwilson, algorithm(2) without invert and
logscore as our preferred estimation method. The estimated mean marginal effects
from this model suggest that only the protection of property rights matters for efficiency.
However, this result might just be an artifact of averaging heterogenous effects. We
graphically examine possible effect heterogeneity using [R] marginsplot command; see
[R] marginsplot and the Stata log below. We consider two dimensions of heterogeneity:
heterogeneity with respect to country size measured by lpop (Fig. 2, right panel), and
heterogeneity with respect to the respective considered dimension of governance quality
(Fig. 2, left panel).
. estimates restore alg_2_raw
(results alg_2_raw are active now)
. local h_list "$g_list lpop"
. foreach h of varlist `h_list´ {
2. qui sum `h´ if e(sample)
3. local mymin = r(min)*0.98
4. local myxmin = ceil(`mymin´)
5. local mymax = r(max)*1.02
6. local myxmax = floor(`mymax´)
7. local mystep = (`mymax´-`mymin´)/25
8. foreach g of varlist `h_list´ {
9. local r_list : list h_list - h
10. qui margins if e(sample), dydx(`g´) at(`h´=(`mymin´(`myste
> p´)`mymax´) (asobserved) `r_list´)
11. qui marginsplot, xlabel(`myxmin´(1)`myxmax´) recast(line)
> recastci(rarea) scheme(s2manual)
12. *qui graph export "${figures}marginsplot_aso_`g´_`h´_${dat
> e}.eps", as(eps) preview(off) replace fontface(Times)
. qui graph export "marginsplot_aso_`g´_`h´_${date}.eps", as(ep
> s) preview(off) replace fontface(Times)
13. }
14. }
29
−
.2
0
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E(
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2 3 4 5
burden of government regulation (regu)
Average Marginal Effects of regu with 90% CIs
−
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Average Marginal Effects of regu with 90% CIs
−
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property rights protection (prop)
Average Marginal Effects of prop with 90% CIs
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Average Marginal Effects of prop with 90% CIs
−
.2
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.2
.4
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judicial independence (judi)
Average Marginal Effects of judi with 90% CIs
−
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−1 0 1 2 3 4 5 6 7
log population (lpop)
Average Marginal Effects of judi with 90% CIs
Figure 2: Estimated marginal effects of governance quality indices on inefficiency
by country size (right column) and its respective own value (left column). Notes: Farrell
output oriented efficiency as dependent variable; Simar and Wilson (2007), algorithm
#2 used for estimation; 90% confidence bands indicated by shaded areas. Source:
Own calculations based on Penn World Table and World Economic Forum Global
Competitiveness Report data.
30
The left panel of Fig. 2 does not suggest that effect heterogeneity with respect to the
respective category of governance quality is a big issue, at least qualitatively. The effects
of both, ‘burden of government regulation’ and ‘judicial independence’ on inefficiency
are statistically insignificant at any level of regu and judi. This is perfectly in line with
the small and statistically insignificant estimated mean effects. Yet, if one focuses on
the point estimates of the marginal effect of regu and for a moment ignores statistical
significance, then Fig. 2 points to relaxing government regulation being beneficial, if the
regulator burden is high but exerting a negative effect on efficiency, if it is already small.
This pattern arguably makes much sense. The pattern for ‘property rights protection’
does also not conflict with what we found for the mean effect. Here we find a significant
inefficiency reducing effect of better property rights protection over the entire range
of prop. Yet, the effect seems to be much stronger for low levels of property rights
protection, though the estimated marginal effect gets increasingly noisy for small values
of prop.
The overall picture is somewhat different for heterogeneity with respect to country
size (Fig. 2, right panel). There, the marginal effect of all three governance indicators
exhibits substantial heterogeneity. While focusing on mean marginal effects suggested
that the level of regulation was immaterial for national efficiency, considering effect
heterogeneity challenges this finding. More specifically, Fig. 2 suggests that relaxing
government regulation reduces inefficiency in small countries. Yet in big countries, it
seems to exerts a negative effect on national efficiency. This pattern corroborates our
earlier hypothesis of ‘regulatory burden’ being an ambiguous concept, since in certain
circumstances some regulation may be well required for efficient production. A similar
pattern of heterogeneity is found for the effect of prop. In small countries, improving the
protection of property rights is clearly beneficial for efficiency of GDP production, while
for big counties such effect is not found in the data. The reverse pattern of heterogeneity
is found with respect to judicial independence. While the effect of judi on efficiency is
statistically insignificant for a wide range of values of lpop, Fig. 2 suggests a statistically
significant, efficiency reducing effect for very small countries. This somewhat surprising
finding has, however, to be interpreted with caution. Near collinearity might be a
technical explanation for the mirror inverted patterns found for prop and judi. Both
variables are strongly correlated (0.903) in the estimation sample, while their respective
correlations with regu (0.506 and 0.448) are much weaker.
4.3 The gciget command
As mentioned in Section 4.1 importing the indices from the Global Competitiveness
Report that we used in our empirical study is not straightforward. We have developed
the new Stata command gciget to get the indices directly into the memory of Stata
from the World Economic Forum’s Global Competitiveness Report.
gciget proceeds in three steps. First it downloads the excel file GCI Dataset 2007-
2017.xlsx from the The Global Competitiveness Report section (http://wef.ch/gcr17)
of the World Economic Forum website. The user can optionally indicate the path to
the excel file GCI Dataset 2007-2017.xlsx stored locally. Second, gciget imports the
31
excel file. See [D] import excel regarding the requirement for the version of Stata.
Third, gciget processes the variables that use has specified after gciget. The resulting
data are in a long format and are by default declared to be panel data, see [XT] xtset.
The syntax for gciget reads as follows:
gciget
[
varlist
] [
, options
]
The user can optionally specify the varlist from the list of indices in the Global
Competitiveness Report (see the excel file GCI Dataset 2007-2017.xlsx for the possible
names). If no valid name of the index is specified, all indices will be processed.
The following options are available:
clear replace data in memory
noxtset do not declare the loaded data to be panel data
noquery suppress summary calculations by xtset
panelvar(newvarname) generate numeric panelvar newvarname
url(filename) download link
sheet(‘‘sheetname’’) excel worksheet to load
cellrange([start][:end]) excel cell range to load
nowarnings do not display warnings
gciget only helps user get the data from the World Economic Forum into Stata.
Thus any liability for the data or its usage is disclaimed. That the data comes from the
World Economic Forum, also puts restriction on the data availability and the terms and
conditions under which the data can be used. As of this writing the data are available
for 2007–2017. The following code illustrate a simple import of four indices and plotting
GCI index for four countries.
. gciget EOSQ048 EOSQ051 GCI GCI.A.02.01, clear
DISCLAIMER: The World Economic Forum is the provider of the Global
Competitiveness Index 2017-2018, a framework and a corresponding set of
indicators for 137 economies. The software gciget.ado provides a practical
way to read the indicators into Stata (R). The responsibility of complying
with the terms and conditions of use under which the owner of the data
grants access to the indicators is entirely with the user but not with the
authors of the software gciget.ado. Any user of gciget.ado is responsible
for making him or herself familiar with the terms of use under which she or
he is allowed to work with the data of the Global Competitiveness Index. For
more information and methodology, please see http://wef.ch/gcr17. In no
event will the authors, owners, and creators of gciget.ado, or their
employers or any other party who may modify and/or redistribute this
software, accept liability for any loss or damage suffered as a result of
using the gciget.ado software.
Downloading the GCI_Dataset_2007-2017.xlsx file
Importing the GCI_Dataset_2007-2017.xlsx file
Processing EOSQ048: 1.09 Burden of government regulation, 1-7 (best)
32
Processing EOSQ051: 1.01 Property rights, 1-7 (best)
Processing GCI: Global Competitiveness Index
Processing GCI_A_02_01: A. Transport infrastructure
. xtline GCI if countrycode == "USA" | countrycode == "DEU" | countrycode == "F
> RA" | countrycode == "GBR", overlay i(country) t(year) scheme(sj) xlabel(2007
> (2)2017)
. qui graph export "GCI_four_cns.eps", as(eps) preview(off) replace fontface(Ti
> mes)
5
5.
2
5.
4
5.
6
5.
8
G
lo
ba
l C
om
pe
tit
iv
en
es
s I
nd
ex
2007 2009 2011 2013 2015 2017
Year
country = France country = Germany
country = United Kingdom country = United States
Figure 3: Global Competitiveness Index for Fracne, Germany, the UK and the USA.
Source: World Economic Forum’s Global Competitiveness Report.
5 Summary and conclusions
In this article, the new user written Stata command simarwilson was introduced that
implements Simar and Wilson (2007) two-stage efficiency analysis. This estimator has
substantial value for applied efficiency analysis as it puts regression analysis of DEA scores
on firm statistical ground. The new Stata command extends the originally proposed
procedure in some (minor) respects, which increases its applicability in applied empirical
work. simarwilson complements the contributions of Ji and Lee (2010), Tauchmann
(2012), and in particular Badunenko and Mozharovskyi (2016), who have already made
related methods of non-parametric efficiency analysis available to Stata users.
33
6 Acknowledgements
This work has been supported in part by the Collaborative Research Center “Statistical
Modelling of Nonlinear Dynamic Processes” (SFB 823) of the German Research Founda-
tion (DFG). The authors are grateful to Ramon Christen, Rita Maria Ribeiro Bastiao,
Akash Issar, Ana Claudia Sant’Anna, Jarmila Curtiss, Meir Jose´ Behar Mayerstain, Erik
Alda, Annika Herr, Hendrik Schmitz, Franziska Valder, Franz Josef Zorzi, Christian
Merkl, and participants of the 2015 German Stata Users Group meeting for many
valuable comments.
7 Supplementary Figures
The figures below graphically illustrate the concepts of a ‘restricted reference set’ (Figure
4) and a ‘convexified frontier’ (Figure 5) that were referred to in this article, using the
same artificial data that was used to illustrate DEA in Figure 1.
A
ByB
yB
DEA*
yB
bc*
ou
tp
ut
 (y
)
input (x)
reference DMUs non−reference DMUs
true frontier DEA est. frontier
bias corrected est. frontier
Figure 4: Estimated inefficiency for sub-sample of DMUs used as reference.
Considering only as sub-sample of DMUs as reference set renders DMU B seemingly
super-efficient, both according to conventional DEA (θ̂B = y
DEA∗
B /yB < 1) and according
to bias corrected DEA (θ̂bcB = y
bc∗
B /yB < 1). DMU A, is still estimated to be inefficient,
cf. Figure 1. Yet, the magnitude of estimated inefficiency is somewhat smaller. Note:
Artificial data generated in the same way as for Figure 1. Source: Own calculations.
34
AB
yA
yA*
yA
DEA*
yA
convex*
ou
tp
ut
 (y
)
input (x)
observed DMUs true frontier
DEA est. frontier bias cor. est. frontier
convexified bias cor. est. frontier
Figure 5: Convexified bias corrected estimated frontier. Measuring inefficiency
relative to the convexified bias corrected frontier either does not affect estimated bias
corrected inefficiency (e.g. DMU B) or increases estimated bias corrected inefficiency
(e.g. DMU A). Note: Artificial data generated in the same way as for Figure 1. Source:
Own calculations.
8 Software availability
The ado-files simarwilson.ado and gciget.ado and the accompanying help-files are
available from ssc. Type
ssc install simarwilson
and
ssc install gciget
to install the ado-files on your machine.
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