In the literature on convergence, the simple Markov chain model indicates evolution towards a twin-peaked world. Although cleansing the ergodic distribution of income across countries of short-run noise reinforces its twin-peaked shape, these twin peaks are not statistically significant. Moreover, the specific type high immobility reflected by the data on income renders the estimated transition matrix particularly prone to the generation of twin-peaked ergodic distributions.

Gisele Hites

English

Research Papers in Economics

1
On the Robustness of the Twin-peaked
Ergodic Distribution of Income Across
Countries
Gisèle Hites
*
ECARES, Université Libre de Bruxelles
Abstract
In the literature on convergence, the simple Markov chain model
indicates evolution towards a twin-peaked world. Although cleansing the
ergodic distribution of income across countries of short-run noise reinforces
its twin-peaked shape, these twin peaks are not statistically significant.
Keywords: Ergodic distribution; Filters; Income distribution; Markov chains; Twin peaks.
JEL classification: C23; O57.
                                                       
* Address: ECARE-CP 139, ULB, 50 Av Roosevelt, 1050 Brussels, Belgium. Tel:
32-2-650-3860. Fax: 32-2-650-4012. E-mail: ghites@ulb.ac.be.2
1. Introduction
The simple Markov chain model has been widely used in the social
sciences to study the phenomenon of mobility. Empirical applications have
included geographic, labor, and social mobility. The prime attraction in this
approach lies in the simplicity of the characterization of the steady state. One
active area of research in which the long run properties of a panel of data are
of particular interest is the study of the distribution of income across
countries. In this body of literature, Quah’s (1993a,b) application of the
simple Markov chain model reveals evidence of a world in which the rich and
the poor are diverging to form ‘twin peaks’. In what follows, the robustness of
this conclusion is assessed. It is shown that although cleansing the ergodic
distribution of income across countries of short-run noise reinforces its twin
peaked shape, these twin peaks are not statistically significant. Moreover, the
specific type of high immobility reflected by the data on income renders the
estimated transition matrix particularly prone to the generation of twin-peaked
ergodic distributions.
2.  Filter
First, let us establish notation. There are a finite number of states m
) ,..., 1 ( m i =  and transitions between these states are observed at regular
intervals for a finite length of time T  ) ,..., 1 ( T t = . Let  ) (t N  be the matrix of
observed transitions at time t where the ijth element is  ) (t nij  (the number of
transitions from state i to state j observed at time t), and let  ) (t n  be the
distribution of the observations across the states at time t where the ith
element is  ) (t ni  (the number of observations in state i at time t). We assume
that the observed transitions are generated by a simple (i.e. of order one) time-
homogenous Markov chain according to the matrix of transition probabilities
P where the ijth element is pij (the probability of transiting from state i to state
j). The maximum likelihood estimator of P is denoted P ˆ , where the ijth
element is  ( ) ( ) ￿ ￿ ￿ = = = =
m
j
T
t ij
T
t ij ij t n t n p
1 1 1 / ˆ . The model can then be
summarized by the following expression:  ( ) ( )
1 ) 0 ( 1
+ ￿ = ￿ = +
t P n P t n t n . In
this paper, we are interested in the long-run tendencies of the distribution of
the observations, i.e. in the ergodic distribution  P n n ￿ ¥ = ¥ ) ( ) ( .
Second, let us review Quah's application of the simple Markov chain
model. The data used is the Lapeyres index of annual real per capita income
from the Summers and Heston (1991) Penn World Tables for 118 countries
(relative to the world average) for the 1962-84 time period. Five possible
states are defined by discretizing the set of possible values of relative incomes
into intervals at 1/4, 1/2, 1, and 2. The results are presented below.3
27 . 0 16 . 0 16 . 0 18 . 0 24 . 0 ) (
99 . 0 01 . 0 0 0 0
02 . 0 94 . 0 04 . 0 0 0
0 04 . 0 92 . 0 04 . 0 0
0 0 04 . 0 92 . 0 05 . 0
0 0 0 03 . 0 97 . 0
ˆ
= ¥
=
Q
Q
n
P
(1)
The estimated ergodic distribution clearly indicates an evolution towards a
bipolar world of haves and have-nots.
Finally, let us turn to the issue of filtering. In the simple Markov
chain model, the estimated transition probability matrix is used to extract
information concerning the mobility of countries within the distribution of
incomes. This information is camouflaged by two sources of noise. The first
is generated by inaccuracies in the data and will not be discussed here. The
second results from using continuous data to estimate a categorical model (i.e.
from translating continuous data into categorical data by defining income
class frontiers) and will be discussed below. Whereas the first source of noise
affects the inference in ways that are unknown to us, the second source of
noise can be observed directly.
In the simple Markov chain model, transitions represent mobility. In
reality, however, transitions can occur for two reasons. Transitions can result
from higher (or lower) than average world growth in a country. This is what
we call mobility and this is what we would like to measure. Transitions can
also result from business cycle type variations in a country's income when the
level of income is situated very close to one defining a frontier between
classes. This is clearly not mobility and since such transitions are included in
our calculation of mobility, it is necessary to purge the data of such noise. In
sum, we need to correct for the bias that short run fluctuations in income
introduce into the calculation of long run tendencies in the distribution of
world incomes.
1
In order to cleanse the data, it is necessary to tighten the conditions
under which a transition is considered to represent mobility. Here this is
achieved by requiring a transition to last a minimum number of periods in
order for it to be counted as mobility, this minimum number of periods being
defined as just over the average number of periods spanned by a business
cycle. Four increasingly fine filters are applied to the original data.
2 The first
                                                       
1 This is an estimation problem arising from the bias introduced into the transition
matrix by fitting a categorical  to a finite sample of continuous data. In theory, the
ergodic distribution is independent of short run noise.
2 The data used is the same as that used by Quah (1993a,b) with the following
adjustments: the data comes from a more recent version of the Penn World Tables
(i.e. Mark 5.6). and the sample is composed of the 111 countries for which there is4
filter counts transitions as mobility if they last for at least one year, that is if
no transition is observed during the year following the initial transition. The
second, third and fourth filters do the same for two, three and four year spans,
respectively. The results obtained from application of these four filters are
presented below:
1967 . 0 0685 . 0 0894 . 0 1355 . 0 5098 . 0 ) (
1892 . 0 0659 . 0 0983 . 0 1274 . 0 5192 . 0 ) (
2562 . 0 0893 . 0 1226 . 0 1240 . 0 4079 . 0 ) (
1893 . 0 0788 . 0 1449 . 0 1650 . 0 4221 . 0 ) (
4
3
2
1
= ¥
= ¥
= ¥
= ¥
H
H
H
H
n
n
n
n
(2)
Two observations are interesting to note. First, of the 127 transitions
initially counted, almost half are due to short-term fluctuations in income and
not the long-term tendencies that we are trying to measure. Indeed, of the 127
transitions initially counted, 29 (15, 12 and 4, respectively) lasted less than
two (three, four and five, respectively) years. Second, the application of
increasingly fine filters reinforces the twin peaked shape of the ergodic
distribution. In very rough terms, comparing the ergodic distribution
calculated from the most finely filtered data to the ergodic distribution
calculated from the unfiltered data, the part of the distribution falling into the
poorest class (left hand peak) increases by 20% to reach 50%, the part of the
distribution falling into the richest class (right hand peak) remains at 20%,
and the part of the distribution falling into the middle classes decreases
uniformly.
In sum, when the data is purged of short run noise, the evidence for
twin peaks is reinforced. But just how robust is this evidence? In the next
section, statistical inference is carried out.
3.  Statistical inference
Anderson and Goodman (1957) study the asymptotic properties of
first-order Markov chains and show that, for each state i, under the null
hypothesis  ij ij p p ~ ˆ =   ) ,..., 1 , ( m j i = :
)) 1 ( ( ~ ~
) ~ ˆ (
2
1 1
2
* -
-
￿￿
= =
m m
p
p p
n
m
i
m
j ij
ij ij
i c
(3)
where  ￿ = =
T
t i i t n n
0
* ) (  and  ij p ~  are the transition probabilities under the null
hypothesis. However, this result is developed under the assumption that every
                                                                                                                                 
continuously available data for the period 1960-89, that is the 118 countries used by
Quah minus Afghanistan, Sudan, Ethiopia, Liberia, Nepal, Iraq, and Tanzania.5
0 > ij p  and this is clearly not the case here.
3 Nevertheless, we can
circumvent this problem by simply neglecting the zeros in each row. Indeed,
we are not interested in testing whether these elements are truly zero; rather,
we are interested in testing whether the positive elements are truly the
estimated values. This is equivalent to treating the first and fifth rows of the
estimated transition matrix as the results from the estimation of a two-state
Markov chain model (i.e.  2 5 1 = = m m ), and the second to fourth rows as
the results from the estimation of a three-state Markov chain (i.e.
3 4 3 2 = = = m m m ). Each row has a 
2 c  distribution with  1 - i m  degrees
of freedom and since these rows are asymptotically independent, they can be
added to obtain a 
2 c  distribution with  ) 1 ( - i m m  degrees of freedom. So, m
varies from row to row, and Equation becomes:
) ( ~ ~
) ~ ˆ (
1
2
1 1
2
* ￿ ￿￿
= = =
- m
i
i
m
i
m
j ij
ij ij
i m
p
p p
n
i
c
(4)
Using the test presented above, different matrices generating different
ergodic distributions were tested under the null. These matrices were created
by taking the estimated transition matrix resulting from application of the
fourth filter and perturbing the off-diagonal elements in a strategic manner.
The following three hypotheses were amongst those not rejected. In the first
case, decreasing  45 ˆ p  by 1%, causes the ergodic distribution to become
strongly unimodal in state 1:
0269 . 0 0825 . 0 1078 . 0 1640 . 0 6188 . 0 ) ( 1 = ¥ n               (5)
In the second case, increasing  12 ˆ p  by 1% and decreasing  21 ˆ p  by 2%, causes
the ergodic distribution to become bimodal in states 2 and 5:
3215 . 0 1132 . 0 1474 . 0 2235 . 0 1945 . 0 ) ( 2 = ¥ n (6)
In the third case, increasing  12 ˆ p  by 1%, decreasing  21 ˆ p  by 2%, and
decreasing  45 ˆ p  by 1% causes the ergodic distribution to become unimodal in
state 2:
0517 . 0 1577 . 0 2059 . 0 3126 . 0 2722 . 0 ) ( 3 = ¥ n              (7)
These results show that the confidence region for the estimated transition
probabilities is big enough to contain matrices generating a whole range of
very different ergodic distributions.
Notice that the nonrobustness of the twin peaks result does not
originate in the particularly large size of the confidence region, but rather in
                                                       
3 Anderson and Goodman’s test statistic is an asymptotic result, valid only for
sufficiently large samples (i.e. those for which the expected frequency in each of the
cells of the transition matrix is superior to 5).6
the extreme sensitivity of the ergodic distribution to perturbations to the
estimated transition matrix. Analysis of the elasticities of the elements of the
ergodic distribution with respect to the elements of a triple diagonal transition
matrix provides an explanation of this fragility.
4 First, the elasticities of the
elements of the ergodic distribution with respect to the transition probabilities
tend to increase as the off-diagonal elements of the triple diagonal transition
matrix decrease. Second, given a ‘balanced’ triple diagonal transition matrix
(i.e. one generating a uniform ergodic distribution), the elasticities of states 1
and 5 of the ergodic distribution with respect to the transition probabilities are
higher than those of states 2, 3 and 4 of the ergodic distribution. Intuitively,
the relatively high elasticity of the endpoints of the ergodic distribution can be
explained by the fact that whereas the three middle states all have two exits
(i.e. one to a lower state and one to a higher state), the two endpoints only
have one exit. So, whereas blocking one of these single exits necessarily leads
to a pile up in one of the endpoints in the long run, blocking one of the other
exits leads to a pile up that could potentially be distributed amongst all the
states behind the blocked exit via the unblocked exit. So, whenever the
estimated transition matrix presents a triple diagonal structure, it is easier to
generate modes in states 1 and 5 than in any of the other states, especially if
the off-diagonal elements of the transition matrix are particularly small.
5.  Concluding remarks
Two lessons emerge from this paper. First, when Markov chain
models are fitted to continuous data, a bias towards excess mobility is
introduced into the estimated transition matrix. Second, when the estimated
transition matrix of a simple Markov chain model presents a certain type of
high immobility, the corresponding ergodic distribution is characterized by an
extreme fragility of a very particular sort.
Acknowledgements
I would like to thank Marco Lippi and Lucrezia Reichlin for helpful
comments on previous versions of this paper.
References
Anderson, T.W., and L.A. Goodman, 1957, Statistical inference about
Markov chains, Annals of Mathematical Statistics 28, 89-110.
                                                       
4 For the general expressions of the elasticities of the elements of the ergodic
distribution with respect to the transitions probabilities along with their analytical
derivation, please contact the author.7
Quah, D., 1993a, Empirical cross-section dynamics in economic growth,
European Economic Review 37, 426-434.
Quah, D., 1993b, Galton’s fallacy and tests of the convergence hypothesis,
Scandinavian Journal of Economics 95, 427-443.
Quah, D., 1996, Empirics for economic growth and convergence, European
Economic Review 40, 1353-1375.
Summers, R. and A. Heston, 1991, The Penn World Table (Mark 5): an
expanded set of international comparisons, 1950-1988, The Quarterly Journal
of Economics 106, 327-368.

1993b, Galton’s fallacy and tests of the convergence hypothesis,

Empirics for economic growth and convergence,

Statistical inference about Markov chains,

The Penn World Table (Mark 5): an expanded set of international comparisons,

On the Robustness of the Twin-Peaked Ergodic Distribution of Income Across Countries

http://fmwww.bc.edu/RePEc/es2000/1369.pdf

On the Robustness of the Twin-Peaked Ergodic Distribution of Income Across Countries

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