Munich Personal RePEc Archive

Regional Efficiency Dispersion,
Convergence, and Efficiency Clusters:
Evidence from the Provinces of Indonesia
1990-2010
Mendez, Carlos
Nagoya University

11 September 2019

Online at https://mpra.ub.uni-muenchen.de/95972/
MPRA Paper No. 95972, posted 11 Sep 2019 07:21 UTC

Regional Efficiency Dispersion, Convergence,and Efficiency Clusters: Evidence from theProvinces of Indonesia 1990-2010
Carlos Mendez*
Nagoya University
Graduate School of International Development(Preliminary Version 1.0)

Abstract
This paper studies efficiency convergence across provinces in Indonesia overthe 1990-2010 period. Through the lens of both classical and distributional convergence frameworks, the dispersion dynamics of pure technical efficiency andscale efficiency are contrasted. The results suggest that—on average—there isregional convergence in both measures of efficiency. However, results from the
distributional framework indicate the existence of two separate convergenceclusters within the pure technical efficiency distribution. Thus, since scale efficiency is characterized by only one convergence cluster, the two clusters ofpure technical efficiency appear to be driving the overall efficiency dynamics ofIndonesia.

Keywords:
Efficiency, Convergence, Distribution-based clustering, Nonparametric distribution, Indonesia* I am grateful to all participants of the Economic Development Policy and Management
seminar at Nagoya University for the insightful comments and suggestions.
Contact Email: carlos@gsid.nagoya-u.ac.jp
Homepage: https://carlos-mendez.rbind.io

2

1 Introduction
Large per-capita income differences across provinces is a well known issue ofthe Indonesian economy (Esmara, 1975). Regional income differences seem topersist despite considerable policy efforts that aim to promote a more balanced
growth path. Although a series of five-year development plans, fiscal transfer programs, internal migration policies, and integrated economic zones initiatives aimed to reduce regional disparities, the outcomes still remain below
their original targets (Akita, 1988; Garcia and Soelistianingsih 1998; Kataoka,2012).
In the economic growth literature, per-capita income differences are commonly accounted for by two proximate sources: the accumulation of factor inputs and the efficiency with which those inputs are used (Caselli, 2005). Factor
inputs typically include measures of both physical capital and human capital.
Efficiency, on the other hand, can be further decomposed into different typesof efficiencies. The Data Envelopment Analysis (DEA) framework of Charnes
et al. (1978), for instance, decomposes overall efficiency into pure technicalefficiency and scale efficiency.
Both the accumulation of factor inputs and the efficiency with which they
are used are two key sources for understanding the large income differences
across the Indonesian provinces. However, as noted by Kataoka (2018), relatively few studies have examined the role of efficiency in determining regional
income disparities in Indonesia. The work of Kataoka (2018) is one those studies; by estimating the three measures of efficiency of the DEA framework, theresults of his study indicate that efficiency differences across provinces havedecreased over the 1990-2010 period. In other words, provinces in Indonesia
have tended to converge in terms of their relative efficiency levels.
In an attempt to contribute and extend the convergence findings of
Kataoka (2018), this paper studies the dynamics of regional efficiency disparities through the lens of both the classical convergence approach of Barroand Sala-i-Martin (1992) and the distributional convergence approach of Quah
(1997). Results from the classical approach suggest that—on average—there
is regional convergence in the three measures of efficiency of the DEA framework. Results from the distributional approach, which emphasizes tendencies

3
beyond average behavior, suggest that there are two separate clusters of convergence in overall efficiency (OE) and in pure technical efficiency (PE). In contrast, scale efficiency (SE) is the only variable where the Indonesian provinces
appear to be converging to a unique equilibrium. These results imply that theoverall dynamics of efficiency appear to be mostly driven by the dynamics ofpure technical efficiency and not necessarily scale efficiency.
From a methodological standpoint, the contribution of this paper is two
fold. First, it extends and complements the original framework of Quah (1997)
by integrating the distribution-based clustering algorithm of Menardi and Azzalini (2014). Second, it applies the conditional density estimator of Hyndman
et. al (1996), which is shown to have better statistical properties than the original estimator of Quah (1997).
The rest of this paper is organized as follows. Section 2 describes thedata and the empirical frameworks. Section 3 presents the results of both classical and distributional convergence analyses. Finally, Section 4 offers someconcluding remarks and points out some avenues for further research.

2 Data and Methods
2.1

Efficiency Measurements and Data

This paper uses efficiency data for 26 contiguous provinces of Indonesia overthe 1990-2010 period. More specifically, the data is from the recent work of
Kataoka (2018). In this paper, this author uses the Data Envelope Analysis(DEA) framework to quantify the relative efficiency with which inputs are used
to produce output at the provincial level in Indonesia. Provincial output ismeasured by the Gross Regional Domestic Product, and inputs include bothmeasures of human capital (labor force adjusted by educational attainment)
and physical capital (measured in in monetary units).
DEA is a nonparametric framework that is used to measure the relative efficiency of decision-making units (Charnes et al., 1978). Originally, thisframework has been mainly used for micro-level production units1 . However,1

The DEA literature usually refers to this production units as decision-making units(DMUs). They are responsible for turning inputs into output(s). Note that for the purposes

4this framework has become increasingly more popular for the analysis of mesolevel production units, such as cities, districts, and provinces (Charnes et al.,
1989; Halkos and Tzeremes, 2010; Schaffer et al., 2011) .
One particularly appealing feature of the DEA framework is that the measured overall efficiency (OE) can be further decomposed into pure technicalefficiency (PE) and scale efficiency (SE). Equation 1 summarizes this decomposition:
Overall efficiency = Pure efficiency ∗ Scale efficiency.
(1)
At a basic level, pure (technical) efficiency refers to the managerial capability ofusing inputs and scale efficiency refers to the exploitation of economies of scaleby operating at an optimal size. Kataoka (2018, Table 2) provides estimates forthese three efficiency measurements for a sample of 26 Indonesian provinces
over the 1990-2010 period.

2.2

Classical Convergence Framework

2.2.1 Beta Convergence
The classical convergence framework of Barro and Sala-i-Martin (1991,1992)
for testing the income convergence the macro level has been increasingly applied to the regional-level context (Magrini, 2004). It has also been applied toother variables beyond income. In this framework, the growth rate of a variable, gy , is a function of its initial level in log terms, log(y0 ). If, in a regression
setting, there is an inverse relationship between these two variables, then aprocess of regional convergence is taking place. This notion of convergence iscommonly referred as beta convergence in the economic growth literature. In
the context of the current paper, the existence of beta convergence would indicate that the less efficient provinces of Indonesia are catching up with mostefficient ones.
For econometric estimation purposes, Barro and Sala-i-Martin(1991,1992) suggest the following specification for studying convergenceacross the regions of a country:of the current paper, the terms decision-making units and production units are used interchangeably.

5gy = γ −

1 − e−βt

log y0 + ut ,(2)twhere gy is the average growth rate of the variable y, γ is a constant, t is a timeindex, β is the speed of convergence, y0 is the initial level of the variable, and
ut is a random disturbance. After estimating the speed of convergence, β, asecond parameter of interest, known as the “half-life” measure of convergence,
can be computed as:
log2
half -lif e =.
(3)βThis measure indicates the time (usually measured in years) that the averageregion would need to halve the distance between its initial efficiency level andits final convergence equilibrium.
2.2.2 Sigma Convergence
There is also a much more general notion of convergence, which is known assigma convergence. This type of convergence refers to the secular decrease in
the cross-regional dispersion of a variable over time. The cross-regional dispersion is then typically measured by the standard deviation of the logarithm ofa variable at different time periods. For the purposes of the current paper, thecross-regional dispersion, σt , is measured as follows:
v
u
u
σ =t
t

N 
2
1 X
log (yi,t ) − log(yt ) ,N − 1 i=1

(4)where N is the number of regions (provinces) in the sample, yi,t is the efficiency
score of region i at time t, and log(yt ) is the average of the natural logarithm ofefficiency at time t.

2.3

Distributional Convergence Framework

The classical convergence framework is sometimes criticized by its excess focus
on average behavior and the existence of a unique convergence equilibrium. It
ignores possible heterogenous patterns of convergence and the formation of
local convergence clusters. Motivated by these limitations, Quah (1993; 1996;

6
Figure 1. Distribution dynamics framework: A graphical summary

(a)

(a)

(b)

(c)

(c)

(b)

(d)

(d)

t

t+s

Time

Note: y/ȳ indicates that the value of the variable y is expressed in relative terms. ȳ indicates the convergence benchmark or the sample average.
Source: Adapted from Quah (1993).

1997) introduced the distributional convergence approach. In contrast to theclassical framework, the distributional framework focuses on the dynamics ofthe entire cross-sectional distribution.2
Figure 1 shows a graphical summary of the distributional convergenceframework. Two essential features should be noted. First, the external-shape
dynamics of the distribution indicate the formation of two local convergenceclusters. Second, the intra-distributional dynamics indicate the sources of theclustering process. For instance, some regions (such as a and d) may show large
and persistent differences over time. Others (such as c), may move forward andcatch-up with the upper cluster. And others (such as b), may move backward
and catch-down with the bottom cluster.
To formalize the intuition of Figure 1 in the context of the variables of the2

See Epstein et al. (2003), Magrini (1999; 2004), Bianco (2016) or Mendez-Guerra (2018) for
more comprehensive presentations of the distributional convergence framework

7
current paper, let us first define ft (x) as the cross-regional efficiency distribution at time t and ft+s (y) as the distribution at time t + s. Next, the evolution
from time t to time t + s is commonly modeled as a first-order autoregressive
process of a time-homogeneous Markov chain. That is,
ft+s(y)| {z }=

F uture Distribution

ˆ

ft(x)

ft+s|Zt=x
|

{z

dx,

(5)

| {z }}

Stochastic Kernel Initial Distribution

where the transformation of the initial distribution, ft(x), into the future distribution, ft+s(y), depends on the stochastic kernel, ft+s|Zt=x(x), which is a transitional operator commonly estimated as a conditional distribution function.
That is,ft+s|Zt =x (x) =ft,t+s(y, x),
ft (x)(6)

where ft,t+s (y, x) is an unconditional joint distribution and ft (x) is the marginal
distribution of x. Lastly, let us define the unconditional joint distribution as:nx − xiz − zi
1 X
Kx,
ft,t+s (y, x) =Kz
nhz hx i=1
hz
hx









(7)

where Ky and Kx denote Gaussian kernel functions, and hz and hx denote kernel bandwidths for y and x, respectively.3
The stochastic kernel provides valuable information about intradistributional dynamics and the formation of local convergence clusters. As
shown in Equation 6, it depends on two components: the unconditional joint
distribution, ft,t+s (y, x), and the initial marginal distribution, ft (x). To evaluate the effect of the former, we will follow the approach of Azzalini et. al (2007)
and Menardi and Azzalini (2014). To evaluate the effect of the latter, we will
follow the approach of Hyndman et. al (1996).

3

Optimal flexible bandwidhts are derived from the the minimization of the asymptotic mean
integrated square error (AMISE).

8

3 Results and Discussion
3.1

Classical Convergence

Table 1 presents the results of the beta convergence framework. The overall finding is that the less efficient provinces are catching up with the mostefficient ones. This convergence process is statistically significant, consistentacross the three efficiency scores, and considerably fast. In particular, convergence in pure (technical) efficiency (PE) appears to be the fastest. In this score,
the provinces of Indonesia are expected to halve its efficiency differences in thenext 5.6 years. Interestingly, the speed of convergence of both pure efficiency(PE) and scale efficiency (SE) appears to be very similar.
Table 1
Beta convergence across Indonesian Provinces

Dependent Variable
Growth of Overall Growth of Pure Growth of Scale
Efficiency (OE)
Efficiency (PE) Efficiency (SE)(1990-2010)(1990-2010)(1990-2010)
Beta coefficient- 0.84-0.91- 0.91
Speed of convergence
0.09
0.12
0.12
Half-life time (in years)
7.6
5.6
5.7
R-squared
0.84
0.89
0.82
P-value of beta
0.00
0.00
0.00
Note: All regressions include a constant term, which is not presented in the table.
Source: Author’s calculations using data from Kataoka (2018).

Table 2 presents the results of the sigma convergence framework. Consistent with the beta-convergence finding, there is a reduction in the efficiency
dispersion over time. This process is both statistically significant and consistent across the three efficiency scores. When contrasting the dispersion reduction in pure efficiency with that of scale efficiency, the former shows a larger
absolute change. Thus, pure efficiency accounts for more of the variation in
overall efficiency.

9
Table 2Sigma convergence across Indonesian Provinces

Dispersion in 1990
Dispersion in 2010

Dispersion Ratio σσ1990
2010
F-Statistic
P-value

Overall
Pure
Scale
Efficiency (OE) Efficiency (PE) Efficiency (SE)
0.63
0.54
0.46
0.25
0.18
0.20
2.53
3.03
2.31
6.40
9.20
5.30
0.00
0.00
0.00

Note: Following the convention of the literature, the dispersion is measured by the standard deviation of the natural
logarithm of each variable.
Source: Author’s calculations using data from Kataoka (2018).

3.2

Distributional Convergence3.2.1 Overall Efficiency
Figure 2 shows the distributional convergence results for the overall efficiency
score (OE). As shown in Panel (a), the most noticeable feature of the regional
dynamics is the existence of two separate clusters. A relatively small and highefficiency cluster is located at around 1 in each year. The regions belonging tothis cluster are those high-efficiency provinces of 1990 that remained highly efficient in 2010. A relatively large and low-efficiency cluster is located at around
0.8 of the efficiency score of 2010. The regions belonging to this cluster used
to have an efficiency score below 0.6 in 1990. By the year 2010, however, theylargely improved relative to their initial position and started process of convergence both among themselves and with the regions of the efficiency frontier. However, this second convergence process is not yet complete and there
remains a significant distance between the high-efficiency cluster and the lowefficiency one.
Based on the clustering framework of Azzalini and Torelli (2007), Panel(c) refines the cluster composition by identifying the “core” members in eachcluster. Through the lens of this framework, regions are further classified into

10
3 groups: the core cluster 1, conformed by the low-efficiency regions; the corecluster 2, conformed by the high-efficiency regions; and a remaining group oftransitional outliers (marked with the number 0). This latter group is mostly
composed by those regions that drastically changed their relative position between the 1990-2010 period.

1.0

50
75

0.8

50

75

0.6

25

1

0.4

0.8
0.6

0.4

0.6

0.8

0
0

1
0
1

0.4

1

1

1.2
1.0

2

1

0.8

1

●

●

●

●

●

●

●
●

●

●

●

●

●

●

●

0.6

0.8

111
1

0.6

oe10

0

1
1
1

0.2

0.4

2
2

0

0.2

(b) Conditional distribution

Overall Efficiency in 2010

1.0(a) Unconditional distribution

0

1

0.8
0.6
Overall Ef
0.4
ficiency
in 2010

Relative Overall Efficiency in 1990

0

0.4

1.2

1.0

in 19900.2

Overall Efficie
ncy

Relative Overall Efficiency in 2010

Figure 2. Distributional convergence and clusters in overall
efficiency

0

0.2

0.4

0.6

0.8

oe90

(c) Distribution-based clusters

1.0

0.2

0.4

0.6

0.8

1.0

Overall Efficiency in 1990(d) Conditional distribution: Modal boxplots

Notes: Panel A: The uncondtional bivarate distribution is estimated using the nonparametric methods described in
Menardi and Azzalini (2014). Panel B: The uncondtional bivarate distribution is estimated using the nonparametricmethods described in Hyndman et. al (1996). Panel C: The distribution-based clusters are indentified using algorithmof Menardi and Azzalini (2014). Panel D: The modal boxplots are also described in Hyndman et. al (1996). Darkershaded stripes indicate higher density areas, and the mode for each conditional density is shown as a bullet point.
Source: Author’s calculations using data from Kataoka (2018).

Based on the distributional convergence framework of Quah (1996; 1997)
and the stochastic kernel estimator of Hyndman, Bashtannyk and Grunwald(1996), Panels (b) and (d) show evolution of the efficiency distribution condi-

11tional on its initial level in 1990. On the one hand, the regions with an efficiency score between 0.2 and 0.6 in 1990 tended to convergence to an efficiency
score of 0.8 in 2010. On the other, regions with an efficiency score between 0.7
and 1 in 1990 tended to convergence to an efficiency score just under 1 in 2010.
These results are consistent with those of Panels (a) and (c) in the sense thatthe dynamics of the overall efficiency distribution suggest the formation of two
different convergence clubs.

50
25

0.9

75

50

0.7

25

1

75

0.5

0.8
0.6

0.4

0.6

0.8

1.2

1.0

Relative Pure Efficiency in 1990

2

0.9
0.8

1

1

0

1
1

1

11

1

0.7

pe10

0

0

1
0

0
0

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

0.4

0.5

0.6

0

1.0

2
0

0.8

0

0.6

0

0.4

(b) Conditional distribution

Pure Efficiency in 2010

1.0(a) Unconditional distribution

0.4

1
0.8
0.6
Pure Effic
iency in2010

in 1990

0.2

Pure Efficiency

Relative Pure Efficiency in 2010

Figure 3. Distributional convergence and clusters in pure efficiency

0

0.4

0.6

0.8

pe90

(c) Distribution-based clusters

1.0

0.2

0.4

0.6

0.8

1.0

Pure Efficiency in 1990(d) Conditional distribution: Modal boxplots

Notes: Panel A: The uncondtional bivarate distribution is estimated using the nonparametric methods described in
Menardi and Azzalini (2014). Panel B: The uncondtional bivarate distribution is estimated using the nonparametricmethods described in Hyndman et. al (1996). Panel C: The distribution-based clusters are indentified using algorithmof Menardi and Azzalini (2014). Panel D: The modal boxplots are also described in Hyndman et. al (1996). Darkershaded stripes indicate higher density areas, and the mode for each conditional density is shown as a bullet point.
Source: Author’s calculations using data from Kataoka (2018).

12

1.0
0.8

25
50
75

0.6

1
0.8

0.4

0.6

0.4

0.6

0.8

1.2

1.0

0.8
0.6
Scale Efficiency
in 2010

Relative Scale Efficiency in 1990

1
1
1
1
1

0.2

0.4

0.6

0.8

se90

(c) Distribution-based clusters

1.0

1.0

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

0.8

11 11 1
1
11
1

0.6

1

1

0.4

1 1
11

1

1

0.2

0.4

(b) Conditional distribution

Scale Efficiency in 2010

0.5 0.6 0.7 0.8 0.9 1.0

se10(a) Unconditional distribution

1

0.4
1

in 1990

0.2

Scale Efficiency

Relative Scale Efficiency in 2010

Figure 4. Distributional convergence and clusters in scale efficiency

0.2

0.4

0.6

0.8

1.0

Scale Efficiency in 1990(d) Conditional distribution: Modal boxplots

Notes: Panel A: The uncondtional bivarate distribution is estimated using the nonparametric methods described in
Menardi and Azzalini (2014). Panel B: The uncondtional bivarate distribution is estimated using the nonparametricmethods described in Hyndman et. al (1996). Panel C: The distribution-based clusters are indentified using algorithmof Menardi and Azzalini (2014). Panel D: The modal boxplots are also described in Hyndman et. al (1996). Darkershaded stripes indicate higher density areas, and the mode for each conditional density is shown as a bullet point.
Source: Author’s calculations using data from Kataoka (2018).

3.2.2 Pure (Technical) Efficiency vs Scale Efficiency
Figure 3 and Figure 4 show the distributional convergence results for the twosources of overall efficiency (OE): pure technical)efficiency (PE) and scale efficiency (SE). On the one hand, the convergence dynamics of PE are largely
similar to those of OE. First, the unconditional distribution shows two separate clusters (Panel a). A relatively small and high-efficiency cluster is locatedat around 1 and a relatively large and low-efficiency cluster is located at around
0.8 of the efficiency score of 2010. Next, the distribution-based clustering algo-

13rithm of Azzalini and Torelli (2007), indicates the existence of two core clusters
and a group of transitional outliers (Panel c). Finally, the conditional distribution approach of Quah (1996; 1997) suggests that the regions with an efficiency
score between 0.2 and 0.6 in 1990 tended to convergence to an efficiency scoreof 0.8 in 2010 and the regions with an efficiency score between 0.7 and 1 in1990 tended to convergence to an efficiency score just under 1 in 2010. Taken
together, all these results suggest that the convergence dynamics of PE tend to
mirror those of the OE.
On the other hand, the dynamics of SE are largely different from thoseof OE (Figure 4). There is only one cluster of regions with no transitional
outliers (Panel a and c). Furthermore, the conditional distribution (Panel band d) indicates that most regions tended to converge towards the efficiency
frontier. Thus, taken together, these results suggest that, at least in termsof scale efficiency, the provinces of Indonesia have been converging towards aunique equilibrium.

4 Concluding Remarks
Large and persistent per-capita income differences exist across the provincesof Indonesia. Standard economic growth theory suggests that, in a proximate
sense, those differences can be driven by differences in capital accumulationand efficiency. Few studies about Indonesia have evaluated the potential roleof efficiency. Among them, Kataoka (2018) finds that differences in overall efficiency and its two determinants (pure technical efficiency and scale efficiency)
have decreased over the 1990-2010 period.
In this context, this paper evaluates the evolution of provincial efficiency
differences through the lens of both the classical convergence approach of Barroand Sala-i-Martin (1992) and the distributional convergence approach of Quah
(1997). Results from the classical approach support the previous finding of
Kataoka (2018). That is,—on average—there is regional convergence in thethree measures of efficiency. However, results from the distributional approachsuggest significant heterogeneous patterns of convergence, which go beyond
the performance of the average province. In particular, results suggest theexistence of two local convergence clusters in both overall efficiency and pure

14
technical efficiency. Scale efficiency is the only variable where the Indonesian
provinces appear to be converging to a unique equilibrium. Taken together,
these results imply that in order to reduce even further the overall efficiency
differences across provinces, additional policy efforts should aim to reduce thedistance between the two clusters of pure technical efficiency.
Finally, further research on efficiency convergence in Indonesia seems
promising in at least Three immediate fronts. First, the robustness of the classical convergence analysis could be tested even further. For instance, Cole and
Neumayer (2003) show that the results of beta convergence (or divergence)
could be reversed when sample weights are added to the analysis. For Indonesia, those weights could be the number of districts, industries, or firms of eachprovince. In the case of the sigma convergence analysis, Dalgaard (2001) andRam (2018) show that different measures of dispersion, such as the coefficientof variation, could indicate not only different magnitudes of convergence, butalso a different convergence direction. In other words, the dispersion dynamics
identified by the coefficient of variation could contradict those of the standard
deviation. Second, the distribution dynamics analysis could be extended andinclude an evaluation of convergence of the long-run (ergodic) distribution. The
continuous state space approach of Johnson (2005) may prove useful for thisendeavor. Finally, alternative clustering frameworks could be considered. Of
particular relevance are those that identify clusters based on probability distribution functions. Among them, the finite mixture approach of Pittau andZelli (2016) is a recent alternative that could be helpful for evaluating the robustness of the convergence clusters.
5

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