LAJED No 30 Noviembre 2018 - Abril 2019 87 - 115
ISSN: 2074 - 4706

Beta, Sigma and Distributional
Convergence in Human
Development? Evidence from theMetropolitan Regions of Bolivia¿Convergencia beta, sigma y distribucionalen desarrollo humano? Evidencia delas regiones metropolitanas de Bolivia

Carlos Mendez-Guerra*

Abstract**
Almost half of the population of Bolivia lives in the metropolitan regions of La Paz, Santa
Cruz, and Cochabamba. Motivated by the development potential of these regions, this paper
evaluates the process of regional convergence in human development through the lens of
three frameworks: beta, sigma, and distributional convergence. The overall result highlights
an increase in the speed of convergence that is driven by both relative forward mobility of
the less developed regions and relative backward mobility of the more developed regions.
Additionally, the distributional convergence framework indicates that the formation ofmultiple convergence clusters is a salient feature of inequality reduction. In the long-run,convergence appears to be characterized by the transformation of a trimodal distribution into
a left–skewed unimodal distribution. This last result implies that the least developed regions
are still relatively far from achieving complete convergence in human development.
Keywords: Convergence, distribution dynamics, human development, Bolivia*

Lecturer of Economics at the Faculty of Arts and Science, Kyushu University.
Contact: mendez.carlos@artsci.kyushu-u.ac.jp** This paper is an extended version of the working paper entitled “On the Distribution Dynamics of Human
Development: Evidence from the Metropolitan Regions of Bolivia”, MPRA Working Paper No 87571.

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Beta, Sigma and Distributional Convergence in Human Development?

ResumenMás de la mitad de la población de Bolivia vive en las regiones metropolitanas de La Paz,
Santa Cruz y Cochabamba. Motivado por el potencial de desarrollo de estas regiones, este
artículo evalúa el proceso de convergencia regional en desarrollo humano a través de tres
análisis de convergencia: beta, sigma y convergencia distribucional. El resultado general
resalta un incremento en la velocidad de convergencia que está determinado conjuntamente
por la movilidad relativa hacia delante de las regiones menos desarrolladas y por la
movilidad relativa hacia atrás de las regiones más desarrolladas. Adicionalmente, el análisis
de convergencia distribucional indica que la formación de múltiples clusters de convergencia
es una característica saliente del proceso de reducción de desigualdades. En el largo plazo, laconvergencia parece estar caracterizada por la transformación de una distribución trimodala una distribución unimodal sesgada a la izquierda. Este último resultado implica que lasregiones menos desarrolladas están todavía relativamente lejos de alcanzar una convergencia
completa en desarrollo humano.
Palabras clave: Convergencia, distribución dinámica, desarrollo humano, Bolivia
Classification/Clasificación JEL: O15, O40, 047

1. Introduction
Since the mid-1980s, Bolivia has experienced large political, social, and economictransformations. Among its social and demographic transformations, there has been acontinuous movement of population toward the most urban and metropolitan areas of
the country. By the year 2013, 46% of the total population are concentrated in the main
metropolitan regions of Bolivia (UNDP, 2016).
Given the notion that metropolitan regions within a country are more likely to share acommon technological and institutional environment,1 the neoclassical growth model
would predict that these regions are expected to converge in terms of their living standards.
Motivated by this prediction and the observed socioeconomic progress of the metropolitan
regions of Bolivia, this paper documents the evolution of human development disparities
and convergence patterns over the 1992-2013 period. In particular, using the United1

For instance, compared to urban and rural differences within a country or high-income and low-income differences
across countries, metropolitan regions within a country are expected to have a higher degree of homogeneity.

88Carlos Mendez-Guerra

Nations’ human development index at the municipal level, this paper evaluates the processof regional convergence through the lens of three frameworks: beta, sigma, and distributional
convergence.
Due to the lack of systematic datasets at the municipal level, most convergence studies
about Bolivia have focused on the regional per-capita income differences among the nine
departments of the country (Evia et al., 1999; Sandoval, 2003); Soruco, 2012; among others)2.
The work of Machicado et al. (2012) is a notable exception that uses municipal-level data to
analyze the spatial determinants of human development, which is measured by the United
Nations’ human development index. In addition, most studies primarily focus on classicalsummary measures of convergence such as beta and sigma convergence3.
Besides an analysis of beta and sigma convergence, the main contribution of this article
is the analysis of distributional convergence using a representative sample4 of metropolitan
regions from Bolivia. The key hypothesis is that even in a scenario of fast beta and sigma
convergence, the presence of highly asymmetric distribution (in the long run) still suggests
the persistence of an incomplete process of convergence. As argued by Quah (1993, 1996 and1997), an important limitation of summary measures such as beta and sigma convergenceis the assumption that all regions converge at the same rate and to the same steady state. In
contrast, the distributional convergence approach allows for heterogeneous transitional
dynamics and potentially multiple long-run equilibriums. In other words, it helps identify
patterns that would remain hidden under the lens of classic summary measures of convergence.
Overall, the results suggest that there is a tendency toward regional convergence thatis largely driven by both forward mobility of the less developed regions and backward
mobility of the more developed regions. Moreover, increasing rates of both sigma and beta
convergence suggest that the dynamics of the 2001-2013 subperiod largely explain theprocess of inequality reduction observed in the entire 1992-2013 period. Results from the
distributional convergence analysis provide further insights and suggest that the formationof multiple clusters of convergence is a salient feature of inequality reduction. The 1992-2001
period appears to be characterized by three separate convergence clusters. The 2001-2013
2

More recent references include Kuscevic-Montero and Rivera-del-Rio (2013), Caballero-Claure and CaballeroMartinez (2016), and Mendez-Guerra (2017).

3

The work of Mendez-Guerra (2017) is an exception that studies convergence using a nonparametric distributional
approach.

4

By “representative sample” I only mean that 46% of the total population of Bolivia are concentrated in thismetropolitan regions.

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Beta, Sigma and Distributional Convergence in Human Development?

period, on the other hand, highlights the merge between the central cluster and the highdevelopment cluster identified in the previous decade. Given these clustering dynamics, thelong-run equilibrium distribution appears to be characterized by the transformation of atrimodal distribution into a left–skewed unimodal distribution.
The rest of the paper is organized as follows. Section 2 briefly describes the three
convergence frameworks and the data of the study. Section 3 presents the results for eachframework, with particular emphasis on the increasing rates of convergence over time. Finally,
Section 4 offers some concluding remarks.

2. Methodology and Data
2.1. Two Classical Measures of Convergence: Beta and Sigma
The main prediction of the standard neoclassical growth model is conditional
convergence in the sense that the growth rate of an economy is higher when there is a large
distance between this economy’s level of output and its own steady state. When economies
or regions share a common technological and institutional environment —such as the caseof the municipal regions within a country— they are likely to converge to the same steadystate. Thus, given this common setting and the existence of diminishing marginal returns tocapital accumulation, the poor regions would tend to grow faster than the rich ones. In theeconomic growth literature, this inverse relationship between the initial level of output and its
subsequent growth rate is typically referred as beta convergence.
Although the neoclassical growth model and its conditional convergence prediction
have been originally proposed and econometrically evaluated in the context of per-capita
GDP differences, this article evaluates convergence in the context of human development
differences. This does not imply that per-capita GDP differences and human development
differences are treated as equivalent at a conceptual level. This paper only implements somestatistical tools and tests that were initially applied in economic growth literature. Thesestatistical tests of convergence have already been applied in several contexts beyond GDP,and they have provided valuable new insights regarding the evolution of human and social
differences across regions5.
5See, for instance, Marchante and Ortega (2006), Petrakos and Saratsis (2000), Royuela, and García (2015) and thereferences therein.

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Carlos Mendez-Guerra

In what follows, I briefly6 describe the derivation of the beta convergence test in thecontext of the variables of this article. In addition, I explain the concept of sigma convergenceand point out how it is related to beta convergence.
2.1.1. Beta Convergence
The empirical analysis of beta convergence has been documented in a vast collection of
papers7. In particular, the analysis of Barro and Sala-i-Martin (1991, 1992a and 1992b) has
been largely influential. Their analysis typically starts by considering the average growth rateof output per worker (or alternatively, for the present paper, the human development index)
over an interval of t time periods, starting at 0. That is,



1 e
y 
1
log  t    tt
 y0 

t

 log y  log A  1  e  log y ,


t
t

*

0

0

(1)

where β is the speed with which a representative economy converges toward its steady state,
y0 is the initial level of output, (1/t)log(yt/y0) is the average growth rate between time 0 andtime t, φ is the exogenous growth of technology, A0 is the initial level of technology, and y * isthe steady-state value of output per effective worker. When a group of economies (or regions)share a common technological and institutional environment, Equation 1 becomes



1 e
y 
1
log  t   tt
 y0 

t

 log y ,0and the rate of beta convergence (β) can be computed by estimating the following univariate
regression



1 e
y 
1
log  t    tt
 y0 

t

 log y  u ,0t

6

See Barro and Sala-i-Martin (2004) for a more complete presentation.

7

See De-la-Fuente (1997, 2000), Islam (2003) and Abreu et al. (2005) for a survey of this literature.

91

(2)

Beta, Sigma and Distributional Convergence in Human Development?

where γ is a constant and ut is a random (white noise) disturbance that reflects unexpectedchanges in production conditions, preferences, technologies, or institutions.
In addition to the rate of beta convergence (β), a second parameter of interest, known as
the “half-life” measure of convergence, can be computed as

half  life 

log 2
.


(3)

This second parameter is particularly informative because it measures the time that a
representative economy needs to halve the gap between its initial and steady-state level.
2.1.2. Sigma Convergence
Sigma convergence is a more general notion of convergence that is complementary to
the analysis of beta convergence. It describes the decline of the cross-sectional variation ofa variable (such as the GDP per worker or the human development index) over time. Beta
convergence is one determinant of sigma convergence, but as argued by Quah (1992), Salai-Martin (1996), and other authors, beta convergence is not a sufficient condition for sigma
convergence.
The analytical justification for this argument is as follows. First, let us restate Equation 2 ina two-period context.

 y log  t     1  e   log yt 1  ut .
 yt 1 





2

(4)

Next, let σ t represent the cross-sectional variation of log yt at time t. Equation 4 and2
the assumed white noise properties of ut imply that σ t evolves over time in accordance withthe following first-order difference equation

 t2  e 2   t21   ut2 .

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Carlos Mendez-GuerraWhen the variance of the of the random disturbance is constant over time  ut2   u2the solution to this equation is

 2
 u2
 u2  2  t   0 e ,1  e 2  
1  e 2  
2
t



(5)

where σ 0 is the variance of log y0.
2





2
2
2
2
Equation 5 implies that as σ t approaches its steady-state value  t   t 1   ,the cross-sectional variation becomes

2 

 u2
,
1  e 2 

(6)2

where σ2 declines with an increase in β or with a decline in σ u . In other words, beta
convergence (an increase in β) is not a sufficient condition for sigma convergence (a declinein σ2), since a reduction of the cross-sectional variation (σ2) also depends on the variation of2
the random disturbance ( σ u ). Thus, this observation led Sala-i-Martin (1996) and otherauthors to suggest that the study of beta convergence should be complemented by a study ofsigma convergence.
2.2. Distributional Convergence
Classical summary measures of convergence, such as beta and sigma convergence,characterize the behavior of a representative economy that approaches smoothly to a uniquesteady state. Thus, cross-sectional differences in the speed of convergence are not taken into
account. It could be possible that at a point of time some economies are converging fasteror slower than others. Largely motivated by this limitation, Quah (1993, 1996, 1997)
introduces the distributional convergence framework, which describes the evolution of theentire cross-section of economies. At its core, this framework characterizes the dynamics ofa system in terms of the transitional dynamics and long-run equilibrium of a non-parametric
distribution function. Transitional dynamics are modeled via an estimated stochastic kernel,
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Beta, Sigma and Distributional Convergence in Human Development?

which is a continuous state-space representation of a Markovian transition matrix. The longrun equilibrium is modeled via an estimated ergodic distribution, which is a continuous
representation of a Markov chain equilibrium. In what follows, I briefly8 describe more themain components of the distributional convergence framework in the context of the variablesof this paper.
2.2.1. Transitional Dynamics
Denote pt(x) as the initial cross-sectional distribution of human development9 at timet. Likewise, pt+s(y) is the human development distribution at some future time t + s. To
model the evolution from time t to time t + s, the literature typically assumes a first-order
autoregressive process of a time-homogeneous Markov chain. That is,pt  s  y 

 


Future Distribution







P( y | x)pt  x dx.

Transitional Operator Initial Distribution(7)

where the transition between the initial distribution, pt(x), and the future distribution, pt+s(y),is mapped by a transitional probability operator, P ( y | x) , which is commonly referred in
the literature as the stochastic kernel.
To estimate this stochastic kernel, most studies exploit the advantages of nonparametricmethods (Wand and Jones, 1995; Li and Racini, 2007; Henderson and Parmeter, 2015). The
first step in the estimation is to define the stochastic kernel as a conditional distribution

P( y | x) 

pt ,t  s  y, x pt  x ,(8)where pt,t+s(y, x) is an unconditional joint distribution. The next step is to specify this joint
distribution in terms of two kernel functions and a pair of smoothing parameters. A common
candidate is8

See Epstein et al. (2003), Magrini (2004 and 2009) or Bianco (2016) for a more complete presentation.

9

For the rest of this analysis, the human development level of each region is expressed in relative terms. That is, thereported HDI level of each region is normalized by the cross-sectional average of the sample.

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Carlos Mendez-Guerra

pt ,t  s  y, x   y  yi   x  xi 
1 nKy  Kx 
,nhy hx i 1  hy   hx where y and x denote (relative) human development in each region at time t andt + s, respectively, Ky and Kx denote Gaussian kernel functions, and hy and hx denote thesmoothing parameters for y and x respectively. Following Magrini (1999, 2009) and Kar,Jha, and Kateja (2011), the optimal selection of the smoothing parameters is based on theminimization of the asymptotic mean integrated square error (AMISE). The final step is the
specification of the marginal distributions (pt+s(y) and pt(x)) , which are estimated using asingle Gaussian kernel function and a single smoothing parameter10.
2.2.2. Long-run Equilibrium
To estimate the ergodic distribution, the approach of Johnson (2000 and 2005) is
implemented. Considering the dynamics described in Equation 7, the long-run equilibriumof the system is given by the solution to the following problem:p  y   P( y | x) p  x  dx  p  x 

(9)

If a solution exists, then the shape of the ergodic distribution, p  y  , provides valuable
information regarding the long-run convergence patterns of the economic system. To
compute this solution, this paper uses the MATLAB functions developed by Magrini (2009).

3. Data and Some Stylized Facts
The dataset comes from the 2016 Human Development Report for Bolivia. The United
Nations Development Program (UNDP, 2016) constructed a municipal-level Human
Development Index (HDI) that covers 20 municipalities from the metropolitan regions of
La Paz, Cochabamba, and Santa Cruz. The temporal dimension of this dataset comprises
four years: 1992, 2001, 2005 and 2013. The construction of this dataset required census data,
household surveys, and administrative records of public services.
10 The smoothing parameter for each marginal distribution is also derived from the minimization of the asymptotic
mean integrated square error (AMISE).

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Beta, Sigma and Distributional Convergence in Human Development?

Figure 1: Human Development Progress: Metropolitan Regions and National Average

Source: Authors’ calculations using data from the United Nations Development Program (UNDP, 2016).

To illustrate some of the basic patterns associated with this dataset, Figure 1 compares
the human development progress of the metropolitan regions and the national average. As
expected, the level of development is higher in the metropolitan regions. More recently,
however, growth at the national level has increased and, as a result, the average municipalityof Bolivia has been catching up with the level of development of the average metropolitan
municipality.
Particularly for the analysis of distributional convergence, it is important to control for
aggregate shocks that might affect all regions alike. Hence, a relative (ratio) measure of thehuman development index is used as the main unit of analysis. More specifically, the index ofeach municipality is rescaled by the cross-sectional mean of each year. Finally, to facilitate theinterpretation of the results, the relative human development index of each municipality ispresented in natural-log transformed values. This transformation simply rescales the index in a
way that the sample average now takes a value of zero at each point in time.

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Carlos Mendez-Guerra

Figure 2: Forward and Backward Mobility in Human Development, 1992-2013 Period

Notes: Relative HDI means that the reported human development index of each municipality
is normalized by the cross-sectional mean of each year. Then, by using a logarithmictransformation, the cross-sectional mean is equal to zero in each year.
Source: Authors’ calculations using data from the United Nations Development Program (UNDP, 2016)

Using this rescaled version of the index, Figure 2 illustrates the dynamics of regional
differences in human development during the 1992-2013 period. By simply looking at thedispersion of the points along each axis, it is clear that there is a noticeable reduction in humandevelopment differences over time. For instance, relative to the sample mean of the year 1992,
human development differences ranged between 17% below the mean (the case of Palca)
and 20% above the mean (the case of Santa Cruz). By the year 2013, this range has noticeably
decreased. Human development differences ranged between 12% below the mean (the caseof Laja) and 7% above the mean (the case of Santa Cruz).
Figure 2 also shows that while some regions moved forward (those located above thedashed 45-degree line) relatively to their initial position in the year 1992, other regions moved
backward (those located below he dashed 45-degree line). The slope of the fitted regression
line summarizes these dynamics and suggests that regions with relatively lower levels ofhuman development appear to be moving forward, whereas the regions with relatively higher
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Beta, Sigma and Distributional Convergence in Human Development?

levels of human development appear to be moving backward11. Naturally, the outcome ofthese dynamics is a process of regional convergence.
Some extreme examples of mobility dynamics may be illustrative. The region of Tiquipaya
improved its relative position from 3% below the mean in 1992 to 7% above the mean in 2013.
In contrast, the region of El Alto deteriorated its position from 7% above the mean in 1992 to
7% below the mean in 2013.
When focusing on shorter time periods, Figure 3 suggests that the speed of regional
convergence differs over time. Panel A of Figure 3 shows relatively little regional mobility inthe 1992-2001 subperiod. In contrast, the relatively large regional mobility of the 2001-2013
subperiod appears to be driving the convergence process observed in the entire 1992-2013
period.
Although the fitted regression lines of Figures 2 and 3 summarize —to some extent—
the overall convergence pattern across metropolitan regions, there are some key aspects ofthe convergence process that require further investigation. A more formal analysis usingclassical measures of convergence (that is, sigma and beta convergence) can be helpful forevaluating the robustness of previous results. More interestingly perhaps, the distributional
convergence framework can be helpful for evaluating nonlinear dynamics and the formationof convergence clusters. Finally, a more complete dynamic analysis should also include bothnotions of transition and long-run equilibrium. All these important aspects of the convergenceprocess are presented in the next section.

11 Note that a backward movement in relative terms does not imply a backward movement in absolute terms.

98Carlos Mendez-Guerra

Figure 3: Mobility in Human Development, 1992-2001 and 2001-2013 Subperiods
(a)

(b)

Notes: Relative HDI means that the reported human development index of each municipality
is normalized by the cross-sectional mean of each year. Then, by using a logarithmictransformation, the cross-sectional mean is equal to zero in each year.
Source: Authors’ calculations using data from the United Nations Development Program (UNDP, 2016)

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Beta, Sigma and Distributional Convergence in Human Development?

4. Results
4.1. Two Classical Measures of Convergence4.1.1. Beta Convergence
Figure 4 shows the inverse relationship between the initial level of human development
and its subsequent growth rate. This result implies that, on average, the less developed regions
are growing faster than the more developed ones. Hence, a process of beta convergenceappears to be occurring among the metropolitan regions of Bolivia.
Figure 4: Beta Convergence in Human Development, 1992-2013 Period

Source: Authors’ calculations using data from the United Nations Development Program (UNDP, 2016)

Table 1 shows the rates of beta convergence and the “half-life” indicator for the entire
1992-2013 period and its two subperiods. For the 1992-2001 subperiod, the estimated valueof the rate of convergence is 2.23% and the implied half-life indicator is just over 31 years.
In contrast, for the 2001-2013 subperiod, the estimated value of the rate of convergence is
7.21% and the implied half-life indicator is almost 10 years. Thus, these results re-emphasize

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Carlos Mendez-Guerra

the observation that the higher speed of convergence of the 2001-2013 subperiod appears tobe driving the speed of convergence of the entire 1992-2013 period.
Table 1
Speed of Beta Convergence in Human Development over time
1992-2013 Period
Variablelog(y0)/T

Coefficientt-statistic

p-value-0.7099

-8.07

0.00

Coefficientt-statistic

p-value-0.1816

-2.54

0.02

Coefficientt-statistic

p-value-0.5789

-7.22

0.00

R2

0.78

Speed of convergence (β)

5.89%

Half-life (periods)

11.76

1992-2001 Subperiod
Variablelog(y0)/T
R2

0.26

Speed of convergence (β)

2.23%

Half-life (periods)

31.12

2001-2013 Subperiod
Variablelog(y0)/T
R2

0.73

Speed of convergence (β)

7.21%

Half-life (periods)

9.62

Notes: For presentation purposes, the regression constant is omitted from the table.
Source: Authors’ calculations using data from the United Nations Development Program (UNDP, 2016)

4.1.2. Sigma Convergence
Figure 5 shows the evolution of regional disparities over time. Two measures of crosssectional dispersion, the standard deviation and the interquartile range, are calculated
in the years for which data is available. The overall result is a clear reduction in the humandevelopment disparities over time. Hence, also a process of sigma convergence appears to beoccurring among the metropolitan regions of Bolivia.

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Beta, Sigma and Distributional Convergence in Human Development?

Figure 5: Sigma Convergence in Human Development, 1992-2013 Period

Source: Authors’ calculations using data from the United Nations Development Program (UNDP, 2016)

A common concern in the analysis of dispersion has to do with the sensitivity of thestandard deviation to extreme values. Since the computation of the standard deviation requires
the estimation of the sample mean, which in turn tends to be sensitive to outliers, it is plausiblethat the sigma convergence finding may not be robust. To alleviate this concern, the declining
tendency of the interquartile range, which is an indicator less sensitive to outliers, reassuresthat the human development differences across regions appear indeed to be decreasing overtime12. Figure 5 also suggests that the rate of change of sigma convergence has changed overtime. In particular, over the 2001-2013 subperiod, differences in human development havedeclined at a faster speed. This phenomenon is particularly more notorious when using theinterquartile range.

12 See Mendez-Guerra (2017) for an example in which the standard deviation and the interquartile range move in
opposite directions in the context of the per-capita income disparities among the nine administrative regions of
Bolivia.

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Carlos Mendez-Guerra

4.2. Distributional Convergence4.2.1. Transitional Dynamics
Figures 6, 7 and 8 show the transitional dynamics of convergence through the lens of theestimated stochastic kernel. Convergence implications derive from the shape of the threedimensional plot in Panel (a) or from its corresponding contour plot in Panel (b). Note that inthese figures, both panels illustrate the same transitional dynamics, but from different angles.
One of the main features of the stochastic kernel is the graphical identification of patternsof stagnation, transition, and clustering. For instance, if most of the density mass of thestochastic kernel is concentrated along the 45-degree line, then the elements in the crosssectional distribution remain where they started and a pattern of stagnation characterizes the
dynamics of the system under study. In contrast, if most of the density appears to be rotating
counter-clockwise or clockwise from the 45-degree line, then the elements in the crosssectional distribution moved from where they started and a pattern of transition characterizes
the system. More specifically, a tendency towards convergence would be signaled by theconcentration of the density mass around the zero-value of the time t + s axis and parallelto the time t axis. Finally, the existence of separate regions of high density (multiple modes)
signals the formation of different clusters of convergence.
The overall finding associated with Figures 6, 7 and 8 is that the formation of clusters ofconvergence characterizes the observed inequality reduction in human development acrossthe metropolitan regions of Bolivia. Moreover, these clustering dynamics are largely different
across the two decades of the analysis. Most notoriously, the merging of clusters during the
2001-2013 subperiod appears to be driving the convergence process observed in the entire
1992-2013 period.

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Beta, Sigma and Distributional Convergence in Human Development?

Figure 6: Transitional Dynamics in Human Development, 1992-2013 Period
(a)

(b)

Notes: Relative HDI means that the reported human development index of each municipality is normalized
by the cross-sectional mean of each year. Then, by using a logarithmic scale, the cross-sectional mean isequal to zero in each year. For a detailed list of the regions and their respective cluster, see the Annex.
Source: Authors’ calculations using data from the United Nations Development Program (UNDP, 2016).

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Carlos Mendez-Guerra

Figure 7: Transitional Dynamics in Human Development, 1992-2001 Subperiod
(a)

(b)

Notes: Relative HDI means that the reported human development index of each municipality is normalized
by the cross-sectional mean of each year. Then, by using a logarithmic scale, the cross-sectional mean isequal to zero in each year. For a detailed list of the regions and their respective cluster, see the Annex.
Source: Authors’ calculations using data from the United Nations Development Program (UNDP, 2016).

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Beta, Sigma and Distributional Convergence in Human Development?

Figure 8: Transitional Dynamics in Human Development, 2001-2013 Subperiod
(a)

(b)

Notes: Relative HDI means that the reported human development index of each municipality is normalized
by the cross-sectional mean of each year. Then, by using a logarithmic scale, the cross-sectional mean isequal to zero in each year. For a detailed list of the regions and their respective cluster, see the Annex.
Source: Authors’ calculations using data from the United Nations Development Program (UNDP, 2016).

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Carlos Mendez-Guerra

Figure 7 shows the transitional dynamics for the 1992-2001 subperiod. The estimatedstochastic kernel clearly points to three separate clusters of high density. Relative to thecentral cluster, located around the mean human development level of the year 2001, there isa low human development cluster located at about 10% below the mean. On the other sideof the distribution, there is a high human development cluster located at about 14% above
the cross-sectional mean of the year 2001. In addition, note that in Panel (b), the low humandevelopment cluster is mostly located above the 45-degree line (that is, forward mobility)
and the high human development cluster is mostly located below the 45-degree line (that is,backward mobility). Thus, over time, these two clusters are transitioning toward the centralcluster.
Figure 8 shows the transitional dynamics for the 2001-2013 subperiod. Most notoriously
in this case, the stochastic kernel highlights the merging between the central cluster and thehigh-development cluster identified in the previous decade. The newly merged cluster is
now located at about 3% above the mean human development level of the year 2013. The
relatively low development cluster, on the other hand, is located at about 4% below the crosssectional mean of the same year. Taken together, these transitional dynamics suggest that theconvergence process arising from the bottom the distribution is slower compared to that
arising from the top of the distribution.
4.2.2. Long-Run Equilibrium
Figures 9 and 10 compare the features of the estimated long-run (ergodic) distribution to
those of the year 1992, 2001 and 2013. Note that the main purpose of an ergodic distribution
analysis is just to clarify and magnify the effects of the observed transitional dynamics13.
Similar to the stochastic kernel, the study of the shape of the ergodic distribution allows forthe identification of patterns of divergence, unimodal convergence, and convergence clusters.
For instance, if the ergodic distribution shows a tendency toward a unique point of mass
with relatively symmetric and thin tails, then there is a pattern of unimodal convergence. In
contrast, if the ergodic distribution displays a tendency toward multimodality, then one caninterpret this density stratification phenomenon as a manifestation of convergence clusters.
The overall finding associated with Figures 9 and 10 is that the process of convergencecharacterized by the evolution of a trimodal distribution (year 1992) into a left–skewed
13 Quah (1997) emphasizes that the estimation of a long-run distribution should not be considered as a forecast ofwhat will happen in the future.

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Beta, Sigma and Distributional Convergence in Human Development?

unimodal distribution (ergodic distribution). Moreover, similar to the transitional dynamics
findings, the two periods of analysis imply different convergence dynamics in the long run.
Most notoriously, the long-run reduction in human development inequality in the entire
1992-2013 period largely depends on the continuation of the convergence dynamics of the
2001-2013 subperiod.
Figure 9: Equilibrium Distribution of Human Development, 1992-2013 Period

Notes: Relative HDI means that the reported human development index of eachmunicipality is normalized by the cross-sectional mean of each year. Then, by using alogarithmic scale, the cross-sectional mean is equal to zero in each year.
Source: Authors’ calculations using data from the United Nations Development Program (UNDP, 2016).

Panel (a) of Figure 10 shows the marginal distributions for the years 1992 and 2001,
and the long-run (ergodic) distribution associated to that time span. As expected, given thepreviously described transitional dynamics, human development differences are smaller inthe long run. However, the asymmetric and bumpy shape of the ergodic distribution may stillsuggest the existence of two convergence clubs for two reasons. First, the human development
distribution clearly shows two density peaks in the year 2001. Second, consistent with thisbimodality, the ergodic distribution still shows two areas of high probability: one density
bump located at about 12% below the mean and the main peak at about 2% above the mean.
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Carlos Mendez-Guerra

Figure 10: Equilibrium Distribution of Human Development, 1992-2001 and 2001-2013
(a)

(b)

Notes: Relative HDI means that the reported human development index of eachmunicipality is normalized by the cross-sectional mean of each year. Then, by using alogarithmic scale, the cross-sectional mean is equal to zero in each year.
Source: Authors’ calculations using data from the United Nations Development Program (UNDP, 2016).

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Beta, Sigma and Distributional Convergence in Human Development?

Panel (b) of Figure 10 shows the long-run (ergodic) distribution given the transitional
dynamics of the 2001-2013 subperiod. Although there are no multiple modes or clear bumps
in the long run, the shape of the ergodic distribution is still largely asymmetric. Indeed, thedistance between the left tail and the mode of the distribution suggests that the least developed
regions of the sample are still relatively14 far from achieving convergence in the long run.

5. Concluding Remarks
This article has documented the reduction of human development disparities (as measured
by the United Nations’ human development index) across the metropolitan regions of Bolivia
over the 1992-2013 period. In particular, through the lens of three convergence frameworks,
the process of inequality reduction across regions has been characterized in terms of its variable
rates of convergence (both beta and sigma), transitional dynamics, and long-run equilibrium.
Overall, there is a tendency toward regional convergence that is largely driven by both forward
mobility of less developed regions and the backward mobility of the more developed regions.
Moreover, increasing rates of both sigma and beta convergence suggest that the dynamics
of the 2001-2013 subperiod largely explain the process of inequality reduction in the entire
1992-2013 period.
In addition, the transitional dynamics analysis, via the estimated stochastic kernel, suggeststhat the formation of different convergence clusters is a salient feature of inequality reduction
in human development. In line with the increasing rates of sigma and beta convergence, theseclustering dynamics largely differ across the two decades of the analysis. While the 1992-2001
subperiod appears to be characterized by the formation of three separate clusters, the 20012013 subperiod highlights the merging between the central cluster and the high-development
cluster identified in the previous decade.
Although a detailed evaluation of the specific factors that are driving the formation ofthese convergence clusters is beyond the scope of this article, the economic history of Bolivia
highlights the role of strong cyclical fluctuations. Between 1992 and 1998, Bolivia experienced
a low-growth cycle, leading to a financial crisis between 1998 and 2002. In contrast, since2003, the country entered a high-growth cycle that is mostly driven by the fast growth of its
exports (mainly natural gas)15. Thus, at least in the Bolivian case, it appears that decade-length
14 Relative to the level and speed of convergence experienced by the most developed regions in the sample.
15 The author gratefully acknowledges this valuable observation from an anonymous referee.

110Carlos Mendez-Guerra

cycles of low (high) GDP growth at the national level are associated with slow (fast) rates
of convergence in human development at the regional level. In this context, further researchis needed not only to quantitatively asses this claim, but also to evaluate the sensitivity andasymmetries of each of the components of the human development index.
The long-run equilibrium analysis, via the estimated ergodic distribution and the observed
marginal distributions, suggests that the process of regional convergence is characterized
by the transformation of a trimodal distribution (year 1992) into a left-skewed unimodal
distribution (ergodic estimation). This unimodal transformation, however, largely depends
on the continuation of the human development dynamics observed in the 2001-2013
period. If, for instance, the dynamics of the 1992-2001 period are taken as a more realistic
determinant of the long run, then the human development distribution is more likely to be
characterized by two convergence clubs. In any of these cases, it appears to be clear that thehuman development distribution is quite sticky at the bottom, and thus the least developed
regions are still relatively far from achieving complete convergence in the long run.
Finally, from a methodological standpoint, further research on regional convergence in
Bolivia seems promising in at least three fronts. First, adjusting for spatial correlation may
not only accelerate or retard the rates of beta and sigma convergence, but also change thecomposition and size of the convergence clubs. In this line of research, the work of Royuela
and Garcia (2015) is an interesting illustration of how this adjustment can be implemented in
the analysis of beta and sigma convergence, and the work of Gerolimetto and Magrini (2015)
suggests an innovative spatial extension to be implemented in the distributional convergenceframework. Second, alternative clustering frameworks can help evaluate the robustnessof the convergence clusters identified in this paper. In particular, the convergence test andclustering algorithm develop by Phillips and Sul (2007, 2009) appears to be a compelling
alternative. Third, since the higher rates of convergence documented in this paper occurredin periods of high economic growth at the national level, it seems promising to have a moreformal evaluation of regional convergence responses to fluctuations in the business cycle.
Making progress on this front, Shibamoto et al. (2016) suggests using the panel analysis of
nonstationarity in idiosyncratic and common components developed by Bai and Ng (2004).
Fecha de recepción: 2 de julio de 2018
Fecha de aceptación: 20 de septiembre de 2018
Manejado por la A.B.C.E./IISEC
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Beta, Sigma and Distributional Convergence in Human Development?

References
1.

Abreu, M., H. De Groot and R. Florax. 2005. “A meta-analysis of beta-convergence: The
legendary 2 percent”. Journal of Economic Surveys, 19(3):389-420.

2.

Bai, J. and S. Ng. 2004. “A panic attack on unit roots and cointegration”. Econometrica,
72(4):1127-1177.

3.

Barro, R. J. and X. Sala-I-Martin. 1991. “Convergence across states and regions”. Brookings
Papers on Economic Activity, 1991(1):107-182.

4.

---------- 1992a. “Convergence”. Journal of Political Economy, 100(2):223-251.

5.

---------- 1992b. “Regional growth and migration: A Japan-United States comparison”.
Journal of the Japanese and International Economies, 6(4):312-346.

6.

---------- 2004. Economic growth. MIT Press, Cambridge, Mass.

7.

Bianco, S. D. 2016. “Going clubbing in the eighties: convergence in manufacturingsectors at a glance”. Empirical Economics, 50(2):623-659.

8.

Caballero-Claure, B. and R. Caballero-Martínez. 2016. “Sigma convergencia,
convergencia beta y condicional en Bolivia, 1990-2011”. Economía coyuntural, 1(1):2559.

9.

De la Fuente, A. 1997. “The empirics of growth and convergence: A selective review”.
Journal of Economic Dynamics and Control, 21(1):23–73.

10.

---------- 2000. “Convergence across countries and regions: theory and empirics”. EIB
papers, 5(2):25–45.

11.

Epstein, P., P. Howlett and M.S. Schulze. 2003. “Distribution dynamics: stratification,
polarization, and convergence among OECD economies, 1870–1992”. Explorations in
Economic History, 40(1):78-97.

12.

Evia, J. L., O. Nina, M. Urquiola, L. Andersen and E. Antelo. 1999. “Geography anddevelopment in Bolivia: Migration, urban and industrial concentration, welfare, andconvergence: 1950-1992”. Inter-American Development Bank, Working Paper 3085.

13.

Gerolimetto, M. and S. Magrini. 2015. “Spatial distribution dynamics”. European
Regional Science Association, Conference Paper 1172.

14.

Henderson, D. J. and C. F. Parmeter. 2015. Applied Nonparametric Econometrics.
Cambridge University Press, Cambridge.
112Carlos Mendez-Guerra

15.

Islam, N. 2003. “What have we learnt from the convergence debate?” Journal of economic
surveys, 17(3):309-362.

16.

Johnson, P. A. 2000. “A nonparametric analysis of income convergence across the us
states”. Economics Letters, 69(2):219-223.

17.

Johnson, P. A. 2005. “A continuous state space approach to convergence by parts”.
Economics Letters, 86(3):317-321.

18.

Kar, S., D. Jha and A. Kateja. 2011. “Club-convergence and polarization of states: A
nonparametric analysis of post-reform India”. Indian Growth and Development Review,
4(1):53-72.

19.

Kuscevic-Montero, C. and M. Rivera-del Rio. 2013. “Convergencia en Bolivia: Un
enfoque espacial con datos de panel dinámicos”. Revista de Economía del Rosario,
16(2):233-256.

20.

Li, Q. and J. S. Racine. 2007. Nonparametric econometrics: Theory and practice. Princeton
University Press, New Yersey.

21.

Machicado, C. G., N. B. Nina and L. C. Jemio. 2012. Factores que inciden en el crecimiento
y el desarrollo en Bolivia: Análisis nacional y regional (1989-2009). INESAD-PIEB, La Paz.

22.

Magrini, S. 1999. “The evolution of income disparities among the regions of theEuropean Union”. Regional Science and Urban Economics, 29(2):257-281.

23.

---------- 2004. “Regional (di)convergence”. In: J.V. Henderson and J.F. Thisse (ed.)
Handbook of regional and urban economics, 4: 2741-2796, Elsevier.

24.

---------- 2009. “Why should we analyse convergence using the distribution dynamics
approach?” Scienze Regionali, 8(1):5-34.

25.

Marchante, A. J., & B. Ortega. 2006. “Quality of life and economic convergence acrossSpanish regions, 1980-2001”. Regional Studies, 40(5), 471-483.

26.

Mendez-Guerra, C. 2017. “Heterogeneous Growth and Regional (Di)Convergence in
Bolivia: A Distribution Dynamics Approach”. Economía coyuntural, 2(4):81-108.

27.

Petrakos, G., & Y. Saratsis. 2000. “Regional inequalities in Greece”. Papers in Regional
Science, 79(1), 57-74.

28.

Phillips, P. C. and D. Sul. 2007. “Transition modeling and econometric convergencetests”. Econometrica, 75(6):1771-1855.

113

Beta, Sigma and Distributional Convergence in Human Development?

29.

---------- 2009. “Economic transition and growth”. Journal of Applied Econometrics,
24(7):1153-1185.

30.

Quah, D. 1993. “Galton’s fallacy and tests of the convergence hypothesis”. The
Scandinavian Journal of Economics, 95(4), 427-443.

31.

---------- 1996. “Twin peaks: growth and convergence in models of distribution
dynamics”. Economic Journal, 106(437), 1045-1055.

32.

---------- 1997. “Empirics for growth and distribution: Stratification, polarization, andconvergence clubs”. Journal of Economic Growth, 2(1):27-59.

33.

Royuela, V. and G. A. García. 2015. “Economic and social convergence in Colombia”.
Regional Studies, 49(2):219-239.

34.

Sala-i Martin, X. 1996. “The classical approach to convergence analysis”. Economic
Journal, pages, 106(437), 1019-1036.

35.

Sandoval, F. 2003. “Situación, tendencias y perspectivas de la convergencia regional en
Bolivia 1980-1997”. Working Paper, Banco Central de Bolivia.

36.

Shibamoto, M., Y. Tsutsui and C. Yamane. 2016. “Understanding regional growth
dynamics in Japan: Panel co-integration approach utilizing the panic method”. Journalof the Japanese and International Economies, 40:17-30.

37.

Soruco, C. 2012. “Espacio, convergencia y crecimiento regional en Bolivia: 1990-2010”.
Working Paper, Banco Central de Bolivia, 01/2012.

38.

UNDP. 2016. El nuevo rostro de Bolivia: transformación social y metropolización. United
Nations Development Program, Human Development Report for Bolivia, Bolivia, La
Paz.

39.

Wand, M. P. and M. C. Jones. 1995. Kernel Smoothing. Chapman and Hall, London.

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Annex
List of Regions and Cluster Identification
RegionClusters
Period 1992-2001

Clusters
Period 2001-2013

Clusters
Period 1992-2013

La Paz

High

High

High

Cochabamba

High

High

High

Santa Cruz

High

High

High

El Torno

Low-Middle

High

High

Tiquipaya

Middle

High

High

Vinto

MiddleHigh

High

SacabaMiddleHigh

High

CotocaMiddleHighHigh

La Guardia

MiddleHigh

High

WarnesMiddleHigh

High

Quillacollo

Middle-High

High

High

Colcapirhua

Middle-High

High

High
Low-High

ViachaLow-Middle

Low-High

Mecapaca

Low

Low

Low

Achocalla

Low

Low

Low

Laja

Low

Low

Low

Sipesipe

Low

Low

Low

Porongo

Low

Low

Low

Palca

MiddleLow

Low

El AltoMiddleLow

Low

Notes: Low-Middle, Middle-High, and Low-High classifications indicate that a region is between clusters.
Source: Authors’ calculations using data from the United Nations Development Program (UNDP, 2016).

115