Munich Personal RePEc Archive

Beta, Sigma and Distributional
Convergence in Human Development?
Evidence from the Metropolitan Regionsof Bolivia
Mendez-Guerra, Carlos

28 June 2018

Online at https://mpra.ub.uni-muenchen.de/87627/
MPRA Paper No. 87627, posted 02 Jul 2018 04:26 UTC

Beta, Sigma and Distributional Convergence in Human Development?
Evidence from the Metropolitan Regions of Bolivia
Carlos Mendez-Guerra
Kyushu University

Abstract
Almost half of the population of Bolivia currently lives in the metropolitan regions of La Paz, Santa Cruz, and Cochabamba. Motivated by the potential for growth and development of these regions, this article documents
the evolution of their human development differences over the 1992-2013
period. In particular, using the United Nations’ human development index
at the municipal level, this paper evaluates the process of regional convergence through the lens of three frameworks: beta, sigma, and distributional
convergence. The overall result is an increasing tendency toward convergence that is driven by both slower forward mobility of the less developed
regions and faster backward mobility of the more developed regions. The
distributional convergence framework provides further insights and suggests that the formation and merging of multiple convergence clusters isa salient feature of inequality reduction. Furthermore, in the long-run,convergence appears to be characterized by the transformation of a trimodal distribution into a left–skewed unimodal distribution. The article
concludes emphasizing that the cross-regional distribution of human development in Bolivia is quite sticky at its left tail and, as a result, the least
developed regions are still relatively far from achieving complete convergence in the long run.

JEL Codes: O15, O40, 047
Keywords: convergence, distribution dynamics, human development, Bolivia

1

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CARLOS MENDEZ-GUERRA

1. Introduction
Since the mid-1980s, Bolivia has experienced large political, social, and economic transformations. Among its social and demographic transformations,
there has been a continuous movement of population toward the most urban
and metropolitan areas of the country. By the year 2013, forty six percent of thetotal 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 a common technological and institutional environment,1 the neoclassical growth model would predict that these regions are expected to converge interms of their living standards. Motivated by this prediction and the observedsocioeconomic 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 United Nations’ humandevelopment index at the municipal level, this paper evaluates the process ofregional convergence through the lens of three frameworks: beta, sigma, anddistributional 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 others2 ). 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
1

For instance, compared to urban and rural differences within a country or high-income andlow-income differences across countries, metropolitan regions within a country are expected tohave a higher degree of homogeneity.
2
More recent references include Kuscevic-Montero and Rivera-del-Rio (2013), CaballeroClaure and Caballero-Martinez (2016), and Mendez-Guerra (2017).

CONVERGENCE IN HUMAN DEVELOPMENT?

3classical summary measures of convergence such as beta and sigma convergence.3 As argued by Quah (1993, 1996, 1997), an important limitation of thesesummary measures is that they assume that all regions converge at the samerate and to the same steady state. In this context, this paper provides not only
an evaluation of the human development differences across metropolitan municipalities, but also implements a more comprehensive and flexible analysisof the process of convergence. That is, besides using the summary measures ofbeta and sigma convergence, it evaluates the transitional dynamics and longrun equilibrium of the entire cross-sectional distribution of human development.
Overall, the results suggest that there is a tendency toward regional convergence that is 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 the process of inequality reduction
observed in the entire 1992-2013 period. Results from the distributional convergence analysis suggest that the formation (and merge) of multiple clusters ofconvergence is a salient feature of inequality reduction in human development.
The 1992-2001 period appears to be characterized by three separate convergence clusters. The 2001-2013 period, on the other hand, highlights the merge
between the central cluster and the high-development cluster identified in theprevious decade. Given these clustering dynamics, the long-run equilibrium
distribution appears to be characterized by the transformation of a trimodal
distribution into a left–skewed unimodal distribution.
The rest of the paper is organized as follows. Section 2 briefly describes thethree convergence frameworks and the data of the study. Section 3 presents theresults for each framework, with particular emphasis on the increasing rates of
convergence over time. Finally, Section 4 offers some concluding remarks.
3

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

4

CARLOS MENDEZ-GUERRA

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 development and its
own steady state. When economies or regions share a common technological and institutional environment—such as the case of the municipal regions
within a country—they are likely to converge to the same steady state. Thus,
given this common setting and the existence of diminishing marginal returns
to capital accumulation, less developed regions would tend to grow faster thanmore developed ones. In the growth and development literature, this inverse relationship between the initial level of development and its subsequent growthrate is typically referred as beta convergence. In what follows, I briefly4 describe
the main components of the beta convergence framework in the context of thevariables of this paper. In addition, I explain the concept of sigma convergenceand point out how it is related to the beta convergence measure.

Beta Convergence
The empirical analysis of beta convergence has been documented in a vast collection of papers.5 In particular, the analysis of Barro and Sala-i-Martin (1991,
1992a, 1992b) has been largely influential. Their analysis typically starts by considering the average growth rate of output per worker (or alternatively, for thepresent paper, the human development index) over an interval of t time periods, starting at 0. That is,
1
log
t
4



yt
y0





1 − e−βt1 − e−βt∗=ϕ+(log ỹ + log A0 ) −log y0 ,tt(1)

See Barro and Sala-i-Martin (2004) for a more complete presentation.
See De-la-Fuente (1997, 2000), Islam (2003) and Abreu et al. (2005) for a survey of this literature.
5

CONVERGENCE IN HUMAN DEVELOPMENT?

5

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 and time t, ϕ is the exogenous growth of technology,
A0 is the initial level of technology, and ỹ ∗ is the 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
log
t



yt
y0




1 − e−βt=−log y0 ,t

and the rate of beta convergence (β) can be computed by estimating the following univariate regression
1
log
t



yt
y0




1 − e−βt=γ−log y0 + ut ,t(2)

where γ is a constant and ut is a random (white noise) disturbance that reflects
unexpected changes 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 -lif e =

log2
.
β

(3)

This second parameter is particularly informative because it measures the timethat a representative economy needs to halve the gap between its initial andsteady-state level.
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 thecross-sectional variation of a variable (such as the GDP per worker or the human development index ) over time. Beta convergence is one determinant of6

CARLOS MENDEZ-GUERRA

sigma convergence, but as argued by Quah (1992), Sala-i-Martin (1996), andother 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 in a two-period context.
log



yt
yt−1




= γ − 1 − e−β log yt−1 + ut .

(4)

Next, let σt2 represent the cross-sectional variation of log yt at time t. Equation 4
and the assumed white noise properties of ut imply that σt2 evolves over time inaccordance with the following first-order difference equation22
σt2 = e−2β σt−1
+ σut
.

2
When the variance of the of the random disturbance is constant over time (σut
= σu2 ),

the solution to this equation is
σt2 =



σu2
σu2
2
e−2βt ,
+ σ0 −
−2β
−2β
1−e
1−e

(5)

where σ02 is the variance of log y0 .
2Equation 5 implies that as σt2 approaches its steady-state value σt2 = σt−1
= σ2 ,

the cross-sectional variation becomes
σ2 =

σu2
,
1 − e−2β(6)where σ 2 declines with an increase in β or with a decline in σu2 . In other words,
beta convergence (an increase in β) is not a sufficient condition for sigma convergence (a decline in σ 2 ), since a reduction of the cross-sectional variation (σ 2) also depends on the variation of the random disturbance (σu2 ). Thus, this observation led Sala-i-Martin (1996) and other authors to suggest that the study ofbeta convergence should be complemented by a study of sigma convergence.

CONVERGENCE IN HUMAN DEVELOPMENT?

7

2.2. Distributional Convergence
Classical summary measures of convergence, such as beta and sigma convergence, characterize the behavior of a representative economy that approachessmoothly to a unique steady state. Thus, cross-sectional differences in the speedof convergence are not taken into account. It could be possible that at a point oftime some economies are converging faster or slower than others. Largely motivated by this limitation, Quah (1993, 1996, 1997) introduces the distributional
convergence framework, which describes the evolution of the entire cross-sectionof economies. At its core, this framework characterizes the dynamics of a system in terms of the transitional dynamics and long-run equilibrium of a nonparametric distribution function. Transitional dynamics are modeled via an estimated stochastic kernel, which is a continuous state-space representation ofa Markovian transition matrix. The long-run equilibrium is modeled via an estimated ergodic distribution, which is a continuous representation of a Markov
chain equilibrium. In what follows, I briefly6 describe more the main components of the distributional convergence framework in the context of the variables of this paper.

Transitional Dynamics
Denote pt (x) as the initial cross-sectional distribution of human development7
at time t. 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-homogeneous6

See Epstein et al. (2003), Magrini (2004, 2009) or Bianco (2016) for a more complete presentation.
7
For the rest of this analysis, the human development level of each region is expressed inrelative terms. That is, the reported HDI level of each region is normalized by the cross-sectional
average of the sample.

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CARLOS MENDEZ-GUERRA

Markov chain. That is,
pt+s (y)| {z }

F uture Distribution

=

ˆ ∞
0

P (y | x)| {z }

pt (x)| {z }dx.

(7)

T ransitional Operator Initial Distribution

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
nonparametric methods (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 is




n
1 X
x − xiy − yipt,t+s (y, x) =Kx,
Kynhy hx i=1
hyhxwhere y and x denote (relative) human development in each region at time t
and t + s respectively, Ky and Kx denote Gaussian kernel functions, and hy andhx denote the smoothing parameters for y and x respectively. Following Magrini (1999, 2009) and Kar, Jha, and Kateja (2011), the optimal selection of thesmoothing parameters is based on the minimization 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 a single Gaussian
kernel function and a single smoothing parameter.88The smoothing parameter for each marginal distribution is also derived from the the minimization of the asymptotic mean integrated square error (AMISE).

CONVERGENCE IN HUMAN DEVELOPMENT?

9Long-run Equilibrium
To estimate the ergodic distribution, the approach of Johnson (2000, 2005) is
implemented. Considering the dynamics described in Equation 7, the long-run
equilibrium of the system is given by the solution to the following problem:p∞ (y) =ˆ ∞P (y | x)p∞ (x)dx = p∞ (x).

(9)

0

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

2.3. Data and Some Stylized Facts
The dataset comes from the 2016 Human Development Report for Bolivia.9 The
United Nations Development Program (UNDP, 2016) constructed a municipallevel Human Development Index (HDI) that covers 20 municipalities from themetropolitan regions of La Paz, Cochabamba, and Santa Cruz. The temporaldimension 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.
To illustrate some of the basic patterns associated with this dataset, Figure
1 compares the human development progress of the metropolitan regions andthe national average. As expected, the level of development is higher in themetropolitan regions. More recently, however, growth at the national level has
increased and, as a result, the average municipality of Bolivia has been catching
up with the level of development of the average metropolitan municipality.
9The
reportcanbedownloadedfromthefollowing
website:http://www.bo.undp.org/content/dam/bolivia/docs/
undp_bo_IDH2016.pdf . Table 1 of the appendix is used to construct the dataset of this study.

10

CARLOS MENDEZ-GUERRA

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

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

Particularly for the analysis of distributional convergence, it is important tocontrol for aggregate shocks that might affect all regions alike. Hence, a relative(ratio) measure of the human development index is used as the main unit ofanalysis. More specifically, the index of each municipality is rescaled by thecross-sectional mean of each year. Finally, to facilitate the interpretation of
the results, the relative human development index of each municipality is presented in natural-log transformed values. This transformation simply rescales
the index in a way that the sample average now takes a vale of zero at each point
in time.
Using this rescaled version of the index, Figure 2 illustrates the dynamics ofregional differences in human development during the 1992-2013 period. By
simply looking at the dispersion of the points along each axis, it is clear that
there is a noticeable reduction in human development differences over time.
For instance, relative to the sample mean of the year 1992, human development

CONVERGENCE IN HUMAN DEVELOPMENT?

11

differences ranged between 17 percent below the mean (the case of Palca) and
20 percent above the mean (the case of Santa Cruz). By the year 2013, this range
has noticeably decreased. Human development differences ranged between 12
percent below the mean (the case of Laja) and 7 percent above the mean (thecase of Santa Cruz).
Figure 2: Forward and Backward Mobility in Human Development, 1992-2013
PeriodSource: Authors’ calculations using data from the United Nations Development Program(UNDP, 2016)

Figure 2 also shows that while some regions moved forward (those locatedabove the dashed 45-degree line) relatively to their initial position in the year
1992, other regions moved backward (those located below he dashed 45-degreeline). The slope of the fitted regression line summarizes these dynamics andsuggests that regions with relatively lower levels of human development appear
to be moving forward, whereas the regions with relatively higher levels of human development appear to be moving backward.10 Naturally, the outcome of
10

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

12

CARLOS MENDEZ-GUERRA

these dynamics is a process of regional convergence.
Figure 3: Forward and Backward Mobility in Human Development, 1992-2001
and 2001-2013 Subperiods

CONVERGENCE IN HUMAN DEVELOPMENT?

13

Some extreme examples of mobility dynamics may be illustrative. The region of Tiquipaya improved its relative position from 3 percent below the meanin 1992 to 7 percent above the mean in 2013. In contrast, the region of El Alto
deteriorated its position from 7 percent above the mean in 1992 to 7 percent
below the mean in 2013.
When focusing on shorter time periods, Figure 3 suggests that the speed ofregional convergence differs over time. Panel A of Figure 3 shows relatively little
regional mobility in the 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 someextent—the overall convergence pattern across metropolitan regions, there aresome key aspects of the convergence process that require further investigation.
A more formal analysis using classical measures of convergence (that is, sigma
and beta convergence) can be helpful for evaluating the robustness of previous
results. More interestingly perhaps, the distributional convergence framework
can be helpful for evaluating nonlinear dynamics and the formation of convergence clusters. Finally, a more complete dynamic analysis should also include
both notions of transition and long-run equilibrium. All these important aspects of the convergence process are presented in the next section.

3. Results
3.1. Two Classical Measures of Convergence
Beta Convergence
Figure 4 shows the inverse relationship between the initial level of human development and its subsequent growth rate. This results implies that, on average, the less developed regions are growing faster than the more developed
ones. Hence, a process of beta convergence appears to be occurring among the

14

CARLOS MENDEZ-GUERRA

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 forthe entire 1992-2013 period and its two subperiods. For the 1992-2001 subperiod, the estimated value of the rate of convergence is 2.23 percent 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 percent and theimplied half-life indicator is almost 10 years. Thus, these results re-emphasize
the observation that the higher speed of convergence of the 2001-2013 subperiod appears to be driving the speed of convergence of the entire 1992-2013
period.

15

CONVERGENCE IN HUMAN DEVELOPMENT?

Table 1: Speed of Beta Convergence in Human Development over time

1992-2013 Period
Variable

Coefficient t-statistic p-valuelog(y0)/T-0.7099

R2

0.78

-8.07

0.00

Speed of convergence (β) 5.89%
Half-life (years)

11.76

1992-2001 Subperiod
Variable

Coefficient t-statistic p-valuelog(y0)/T-0.1816

R2

0.26

-2.54

0.02

Speed of convergence (β) 2.23%
Half-life (years)

31.12

2001-2013 Subperiod
Variable

Coefficient t-statistic p-valuelog(y0)/T-0.5789

R2

0.73

-7.22

0.00

Speed of convergence (β) 7.21%
Half-life (years)

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)

16

CARLOS MENDEZ-GUERRA

Sigma Convergence
Figure 5 shows the evolution of regional disparities over time. Two measures ofcross-sectional dispersion, the standard deviation and the interquartile range,
are calculated in the years for which data is available. The overall result is aclear reduction in the human development disparities over time. Hence, also a
process of sigma convergence appears to be occurring among the metropolitan
regions of Bolivia.
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 due with the sensitivity of the standard deviation to extreme values. Since the computation of
the standard deviation requires the estimation of the sample mean, which inturn tends to be sensitive to outliers, it is plausible that 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

CONVERGENCE IN HUMAN DEVELOPMENT?

17

decreasing over time.11
Figure 5 also suggests that the rate of change of sigma convergence has changed
over time. In particular, over the 2001-2013 subperiod, differences in humandevelopment have declined at a faster speed. This phenomenon is particularly
more notorious when using the interquartile range.

3.2. Distributional Convergence
Transitional Dynamics
Figures 6, 7 and 8 show the transitional dynamics of convergence through thelens of the estimated stochastic kernel. Convergence implications derive fromthe shape of the three-dimensional plot in Panel (a) or from its corresponding
contour plot in Panel (b). Note that in these figures, both panels illustrate thesame transitional dynamics, but from different angles.
One of the main features of the stochastic kernel is the graphical identification of patterns of stagnation, transition, and clustering. For instance, if mostof the density mass of the stochastic kernel is concentrated along the 45-degreeline, then the elements in the cross-sectional 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 counterclockwise 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 the tendency towards convergence would be signaled by the concentration of the density mass aroundthe zero-value of the time t + s axis and parallel to the time t axis. Finally, theexistence of separate regions of high density (multiple modes) signals the formation of different clusters of convergence.
11

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.

18

CARLOS MENDEZ-GUERRA

Figure 6: Transitional Dynamics 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 logarithmic scale, thecross-sectional mean is equal to zero in each year.
Source: Authors’ calculations using data from the United Nations Development Program(UNDP, 2016)

CONVERGENCE IN HUMAN DEVELOPMENT?

19

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

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, thecross-sectional mean is equal to zero in each year.
Source: Authors’ calculations using data from the United Nations Development Program(UNDP, 2016)

20

CARLOS MENDEZ-GUERRA

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

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, thecross-sectional mean is equal to zero in each year.
Source: Authors’ calculations using data from the United Nations Development Program(UNDP, 2016)

CONVERGENCE IN HUMAN DEVELOPMENT?

21

The overall finding associated with Figures 6, 7 and 8 is that the formation ofclusters of convergence characterizes the observed inequality reduction in human development across the metropolitan regions of Bolivia. Moreover, theseclustering 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.
Figure 7 shows the transitional dynamics for the 1992-2001 subperiod. The
estimated stochastic kernel clearly points to three separate clusters of high density. Relative to the central cluster, located around the mean human development level of the year 2001, there is a low human development cluster located atabout 10 percent below the mean. On the other side of the distribution, there isa high human development cluster located at about 14 percent above the crosssectional mean of the year 2001. In addition, note that in Panel (b), the low
human development cluster is mostly located above the 45-degree line (that is,
forward mobility) and the high human development cluster is mostly locatedbelow the 45-degree line (that is, backward mobility). Thus, over time, thesetwo clusters are transitioning toward the central cluster.
Figure 8 shows the transitional dynamics for the 1992-2001 subperiod. Most
notoriously in this case, the stochastic kernel highlights the merging between
the central cluster and the high-development cluster identified in the previous
decade. The newly merged cluster is now located at about 3 percent above themean human development level of the year 2013. The relatively low development cluster, on the other hand, is located at about 4 percent below the crosssectional mean of the same year. Taken together, these transitional dynamics
suggest that the convergence process arising from the bottom the distribution
is slower compared to that arising from the top of the distribution.

22

CARLOS MENDEZ-GUERRA

Figure 9: Equilibrium Distribution of 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 logarithmic scale, thecross-sectional mean is equal to zero in each year.
Source: Authors’ calculations using data from the United Nations Development Program(UNDP, 2016)

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 purposeof an ergodic distribution analysis is just to clarify and magnify the effects of theobserved transitional dynamics.12 Similar to the stochastic kernel, the study ofthe shape of the ergodic distribution allows for the identification of patterns ofdivergence, unimodal convergence, and convergence clusters. For instance, if
the ergodic distribution shows a tendency toward a unique point of mass with
12

Quah (1997) emphasizes that the estimation of a long-run distribution should not be considered as a forecast of what will happen in the future.

CONVERGENCE IN HUMAN DEVELOPMENT?

23

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 can interpret this density stratification phenomenon as amanifestation of convergence clusters.
The overall finding associated with Figures 9 and 10 is that the process of
convergence characterized by the evolution of a trimodal distribution (year 1992)
into a left–skewed 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 longrun reduction in human development inequality in the entire 1992-2013 period
largely depends on the continuation of the convergence dynamics of the 20012013 subperiod
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 the previously described transitional dynamics, human development differences are smaller in the long run. However, the asymmetricand bumpy shape of the ergodic distribution may still suggest 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 withthis bimodality, the ergodic distribution still shows two areas of high probability: one density bump located at about 12 percent below the mean and the themain peak at about about 2 percent above the mean.
Panel (b) of Figure 10 shows the long-run (ergodic) distribution given thetransitional 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, the distance between the left tail and themode of the distribution suggests that the least developed regions of the sample
are still relatively13 far from achieving convergence in the long run.
13

Relative to the level and speed of convergence experienced by the most developed regionsin the sample.

24

CARLOS MENDEZ-GUERRA

Figure 10: Equilibrium Distribution of Human Development, 1992-2001 and2001-2013 Subperiods

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, thecross-sectional mean is equal to zero in each year.

CONVERGENCE IN HUMAN DEVELOPMENT?

25

4. Concluding Remarks

This article has documented the reduction of human development disparities(as measured by the United Nations’ human development index) across themetropolitan regions of Bolivia over the 1992-2013 period. In particular, throughthe 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 mobilityof the more developed regions. Moreover, increasing rates of both sigma andbeta 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, suggests that the formation of different convergence clusters is a salientfeature of inequality reduction in human development. In line with the increasing rates of sigma and beta convergence, these clustering 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, the2001-2013 subperiod highlights the merging between the central cluster andthe high-development cluster identified in the previous decade.
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 thehuman development dynamics observed in the 2001-2013 period. If, for instance, the dynamics of the 1992-2001 period are taken as a more realistic input
for 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

26

CARLOS MENDEZ-GUERRA

clear that the human 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, further research on regional convergence at the municipal level 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 the composition 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
convergence framework. Second, alternative clustering frameworks can help
evaluate the robustness of the convergence clusters identified in this paper. In
particular, the convergence test and clustering algorithm develop by Phillips
and Sul (2007, 2009) appears to be a compelling alternative. Third, since thehigher rates of convergence documented in this paper occurred in periods ofhigh 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).

Acknowledgement

This working paper is an extended version of the working paper entitled “On theDistribution Dynamics of Human Development: Evidence from the Metropolitan Regions of Bolivia”, MPRA Working Paper No. 87571

CONVERGENCE IN HUMAN DEVELOPMENT?

27

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