Re-Examining the Convergence of the World Economy:
Markov Chain Model with Population Weighting
Alvaro Domínguez
Asian Growth Research Institute (AGI)
Hiroshi Sakamoto
Asian Growth Research Institute (AGI)

Working Paper Series Vol. 2022-02
February 2022

The view expressed in this publication are those of the author(s) and do
not necessarily reflect those of the Institute.

No part of this article may be used or reproduced in any mannerwhatsoever without written permission except in the case of brief
quotations embodied in articles and reviews. For information, pleasewrite to the Institute.

Asian Growth Research InstituteRe-Examining the Convergence of the World Economy:
Markov Chain Model with Population Weighting
Alvaro Domínguez 1
Asian Growth Research Institute (AGI)
11-4 Otemachi, Kokurakita, Kitakyushu, 803-0814 JAPAN

Hiroshi Sakamoto 2
Asian Growth Research Institute (AGI)
11-4 Otemachi, Kokurakita, Kitakyushu, 803-0814 JAPAN

Abstract
We study the convergence hypothesis of economic growth relying on recent data. To fix the
population size of each country for rank changes, we re-estimate the transition probability
matrix by applying a population weight to changes in the income rank of each country. We thenfind that with no population weighting, the world economy can be divided into two peaks as
before. Nevertheless, the population-weighted probability model yields more optimistic
results: We divide the world economy into several regions, estimate similar probability models,and calculate the convergence distribution. We then divide the world into the optimistic andpessimistic region. The optimistic region, with high income, is composed of East Asia andEurope; The pessimistic region, with low income, is composed of Sub-Saharan Africa. These
two extremes cause the observed twin peaks. The transition information of China and India has
a significant impact when considering population weighting. These two countries show rapid
economic growth, which produces optimistic results in our population-weighted model.
JEL classification: C49, D39, O50, R11
Keywords: Convergence, World Economy, Markov Chain.

1E-mail address: alvdom@agi.or.jp

2E-mail address: sakamoto@agi.or.jp

1. Introduction
We study the long-term growth trends of the world economy since the 1950s from theperspective of convergence. We analyze the hypothesis of whether economically poor countries(regions) can catch up with rich countries (regions) in the context of economics. The possibilityof convergence increases if we assume a production function with diminishing returns
represented à la Solow (1956). This is due to productivity being higher when there are low
inputs and gradually decreasing as input increases. We thus link this to empirical research byreplacing “low input” with “poor country (region)”.
There is large literature analyzing the trends of convergence at the world scale (Barro andSala-i-martin, 2004; Islam, 2003). Traditionally, this has been analyzed mainly througheconometric methods. In this study, instead, we analyze it relying on the stochastic method
proposed by Quah (1993, 1996a, 1996b). This unique analytical method, classified as a
“distribution approach” (Islam 2003), produces distinctive results compared to the econometricmethodology. It divides the logarithmic value of relative income, which compares eachcountry's GDP per capita with the average, into several ranks in descending order of income.
Then, the time-series changes of the rank (income states) in each sample are totaled, and these
are represented by probabilities. The Markov transition matrix obtained by such a procedure
has an eigenvalue (the dominant eigenvalue), whose absolute value is 1. Therefore, theconvergence distribution (ergodic distribution) can be estimated by obtaining the eigenvector
associated to the dominant eigenvalue. With this approach, Quah argues that the world
economy divides into two extremes with twin peaks: a rich economy and a poor economy.
Up to this point, the changes in the income state have been calculated using a single country(region) as a sample, but it is not easy for the income state to change in countries such as China
or India, which have large populations. Taking this into consideration, in this study we multiply
each sample by the population weight and we estimate the Markov transition matrix from theaggregated results to obtain the convergence distribution. For the population weightthe ratio
was obtained from the total of these worlds and was multiplied by the income state movement.
This is expected to make the stochastic model more precise, but the aggregation work becomes
more complex because the population ratio is multiplied for all states changes.
The rest of this article is organized in the following way. Section 2 describes the stochastic
model using Markov chains. Section 3 explains the data used. The world economy is divided
into several regions. This aims at finding out the characteristics between regions by conducting
similar analyses per region. Section 4 details the results of the regional model without
population weights while section 5 does so for the case including population weights. Finally,
section 6 concludes.

12. The model
The distribution approach of Quah (1993, 1996a, b) relies on a Markov transition matrix toshow the change in the distribution from a period to the next. In this section, we briefly explain
the essentials of this methodology.
Let Ft be the income distribution state in period t expressed as a 1×N vector (number ofstates). The Markov process is a mathematical expression of the situation in which the income
distribution situation Ft+1 in the next term depends on the income distribution states Ft in thecurrent term. In other words, the fluctuation of the income distribution states between the twotime points is defined as follows.
𝐹𝐹𝑡𝑡+1 = 𝐹𝐹𝑡𝑡 ∙ 𝑀𝑀𝑡𝑡

(1)

Where Mt is the transition matrix. Since the income distribution is ranked for this transition,
the transition matrix is estimated by aggregating the changes in the rank. The estimation
method is as follows, with P as the number of changes in rank.
𝑀𝑀𝑡𝑡,𝑗𝑗𝑗𝑗 = 𝑃𝑃𝑡𝑡,𝑗𝑗𝑗𝑗 ⁄∑𝑛𝑛𝑘𝑘=1 𝑃𝑃𝑡𝑡,𝑗𝑗𝑗𝑗(2)This shows the probability of how many of the total number of changes (total number ofincome state movements from state k), including the number staying in the same state, thatmoved to state j. Then, we can take advantage of the ergodic property that is characteristic of
Markov chains, and find the convergence distribution (ergodic distribution).
F＝F･M (3)The transition matrix obtained through this procedure usually has an eigenvalue with anabsolute value of 1. Relying on the associated eigenvector, the estimation of the convergence
distribution (ergodic distribution) can be obtained using the solver in Excel or through anyother method to compute it. In this case, eq (3) holds when M is given, and the model isconstrained so that the sum of F equal to 1 is solved.
The convergence distribution we analyze is a fixed distribution that is reached when looking
at the future, where this model is maintained for a long term in a stochastic model composedof samples from 1953 to 2017. The convergence hypothesis examines whether indicators such
as GDP per capita in the world economy, or a specific economic group settle at a certain value.
In the verification of the convergence hypothesis by the stochastic model, the shape of theconvergence distribution is a concern. Therefore, given that there is convergence to a certain
value, it is desirable that the distribution is mountain-shaped, centered on the middle-income
group. In the case of other distributions, it cannot be confirmed that the convergence hypothesis
is universally established.
2

3. Data
In this study we rely on the Penn World Table (PWT), 3 which provides internationallycomparable data (Feenstra, Inklaar and Timmer, 2015). The PWT provides macroeconomic
indicators such as GDP, for the case of some countries, presenting estimates going back to 1950.
However, since the data for China is only available from 1952 to 2017, we set the period ofanalysis for that period. We show that the number of countries (regions) change during thisperiod. Figure 1 shows the change in the number of samples in the analysis, with a maximum
value of 182. As we can see in the figure, it is possible to classify and analyze the world
economy by dividing it into several regions. Table 1 shows the applicable countries whenclassified by region. Based on this table, we separate it into the following seven regions:
East Asia & Pacific (ea, es, as)Europe & Central Asia (eu)
Latin America & Caribbean (la, nl)
Middle East & North Africa (me, ms, as)North America (na, nl)South Asia (sa, es, ms, as)
Sub-Saharan Africa (af)This regional classification is based on the WDI (World Development Indicators)
classification. In addition, for the convenience of the analysis, we combine and analyze theadjacent regions into a larger area (mentioned in parenthesis). For example, (ea) in East Asia
& Pacific is simply referred to as “East Asia & Pacific”, (es) alludes to East Asia & Pacific andSouth Asia, and (as) references to East Asia & Pacific, Middle East & North Africa and SouthAsia, all combined.
We next use real GDP and population data. The PWT estimates real GDP from productionand expenditure, with some slight differences in the estimated values. In this study, we adopt
real GDP (“rgdpo” in the PWT notation) from the production side. The relative income percapita is calculated from this data, and is defined as the logarithm when the average income(per year) is 1 (𝑙𝑙𝑙𝑙(𝑦𝑦⁄𝑛𝑛⁄∑ 𝑦𝑦⁄∑ 𝑛𝑛), with y being real GDP, and n the population).
In this paper we analyze the convergence, in fluctuations, of the relative income of eachcountry and region through the lens of the method proposed by Quah (1993, 1996a, 1996b). In
this methodology, the logarithmic value of the relative income previously mentioned is ranked
in ascending order of income, the change in the rank is followed in chronological order, andthe aggregated value is indicated through a probability. Nevertheless, since this ranking isarbitrary, the logarithmic value of the relative income for ranking is fixed with the following
numbers based on an average of 0.

3

Penn World Table version 9.1, https://www.rug.nl/ggdc/productivity/pwt/
3

5 states: (lo, ml, mi, mh, hi or 1, 2, 3, 4, 5).
-1.386 (1/4), -0.693 (1/2), 0.000 (1), 0.693 (2)7 states: (ll, lo, ml, mi, mh, hi, hh or 1, 2, 3, 4, 5, 6, 7).
-1.500, -1.000, -0.500, 0.000, 0.500, 1.000
The numbers in parenthesis from the five states show the values for 1/4, 1/2 and 2 times theaverage income. This allows for changes in the number of samples.
Tables 2 and 4 display the hierarchical distribution by region. In table 2 we see that North
America (na) only has the highest income group (5) whereas South Asia (sa) possesses all but
the highest income group. In the case of table 4, na only has positive values in states 6 and 7,
while sa has positive values in all states except in 6 and 7. In regards to the convergence
distribution, World (wd) is evenly distributed whilst other regions are biased. Among theseregions, East Asia & Pacific (ea), Europe & Central Asia (eu), and Middle East & North Africa(me) are concentrated in the high-income group, and Latin America & Caribbean (la) is in the
middle-income group. Finally, Sub-Saharan Africa (af) is concentrated in the low-income
group.
Tables 3 and 5 show the changes in the hierarchy of each member of ea and sa, at a given
time. We see that in the seventh state of table 5, Japan falls from a value of 7 to 6. In the caseof Australia and Brunei, they remain in the highest income class in every state. In the otherextreme, for each state, Nepal persists in the lowest income class. Countries with large
populations like China have, in recent years, risen from 2 to 4, while India has lately jumped
to 3 after a period of falling from 2 to 1. We would expect that countries with big populations
lift their income groups as they grow, although we don’t observe this with low-income
countries such as Cambodia, Bangladesh, Nepal, and Pakistan.
4. Regional Model of Markov Chain without Population Weight
We first estimate the stochastic model proposed by Quah (1993, 1996a, 1996b) and find its
convergence distribution. The stochastic model estimates models by region as well as the entire
world economy and compares their convergence distributions.
Tables 6 and 7 show the transition probability matrices for the world (wd) and East Asia &Pacific (ea). A five states model (5×5 matrix) and a seven states model (7×7 matrix) aredisplayed, respectively (see tables 1 and 2 in the Appendix for results of all areas analyzed,
including these). Each matrix shows only the elements in which the income state movements
are seen. The blank elements are the places where the state movements are not observed, andthe movement probability is zero. In addition, since the values are displayed with 4 digits after
the decimal point, even if state movements are spotted, they are displayed as 0.0000 if theprobability is extremely low. For the convenience of computation, the movement probability iszero at this point as well.
We can observe from table 6 that there are areas where the stochastic model does not hold.
4For example, ea has a probability of moving from hi (highest income group) to hi 1.0000. Since
there is a probability of moving from mh to hi in the next income group (0.0561), this showsthat once the highest income group is reached, it will not fall to the lower level. Therefore, it isexpected that the convergence distribution will be concentrated at one point in hi.
Based on these results, the ergodic convergence distribution is calculated in tables 8 and 9.
From the above explanation, there are areas where the transition probability matrix does nothold, thus blank parts in the convergence distribution of these areas appear. Additionally, in thetable, the income states with the largest distribution are colored yellow, the states with thesecond largest distribution with blue, and the states with the smallest distribution with orange.
Based on this, the tendency of the ergodic convergence distribution in each region is divided
as follows. First, only wd is divided into two extremes, the highest-income group and thelowest-income group, and the rest is a distribution with a concentration in one peak. Amongthem, there are many regions that are concentrated in the highest income class, and the regional
divisions of ea, eu, me, na, and es fall under this category. The regional divisions of la, sa, nl,
and ms are concentrated in the middle-income group, and only af (Sub-Saharan Africa) isconcentrated in the lowest-income group. This difference in the convergence distribution ofeach region seems to indicate the “twin peak” that Quah claims, that is, the tendency of theworld economy to differentiate into two peaks.
5. Regional Model of Markov Chain with Population Weight
Tables 10 and 11 show the transition probability matrix for the world (wd) and East Asia &Pacific (ea). Since the income state movement itself is irrelevant with or without population
weighting, the positions of the elements in which the probabilities are described are the same.
By considering the population weight, only the value of the probability is different (see tables
3 and 4 in the Appendix for results for all the regions analyzed, including these). When wecompare the values of wd in table 6 with those in table 10, the former’s values are 0.9624,
0.9100, 0.9110, 0.9233, 0.9833, whereas the latter’s values are 0.9058, 0.9125, 0.9441, 0.9395,
and 0.9976.
Based on this, tables 12 and 13 present the ergodic distributions. There is a clear distinctionin each distribution. Initially, the distribution is convergent throughout wd, showing a bipolar
distribution without population weighting. When we add population weighting, the distribution
becomes concentrated in the high-income group. In addition, there are many regions where the
distribution is concentrated in the high-income class and, besides wd, there are regional
divisions corresponding to ea, eu, na, nl, es, and as. The areas concentrated in the middleincome group are divided into two areas, la and me, while af concentrates in the low-income
group even when population weighting is taken into consideration. As a result of adding
population weight, there is a slight difference in the tendency of the convergence distribution
between the five and seven states. Therefore, it becomes difficult to judge the two regional
divisions of sa and ms.
5

In tables 6 and 10, we compare again this difference in convergence distribution. As we
described above, in the elements of the diagonal matrix the probability of lo decreases, while
the other probabilities increase. Next, the probability of moving from lo to ml increases from
0.0370 without population weighting to 0.0942 with population weighting. From this, we cansay that the population weighted model is more likely to get out of the low-income class.
Furthermore, the probability that the income class will rise from ml to mi, mi to mh, and mh to
hi is higher without population weight, whereas the probability that the income class will fallfrom hi to mh, mh to mi, and mi to ml is higher with population weight. It is likely that the
distribution will be concentrated in the high-income group, since no alternative is higher. So,
we can consider that the discrepancy in the result of the convergence distribution is due to sucha difference in probability.
6. Conclusion
We analyzed the convergence distribution of the world economy by region using a stochastic
model based on a Quah-style Markov chain. As a feature of the study, we compared a normal
stochastic model without population weighting with a model that considers it. We found thatthe model without population weighting still shows the tendency of the world economy tobecome polarized (Twin Peaks). One of the factors for this is the characteristics of each region.
Sub-Saharan Africa has a convergent distribution of low incomes, but many regions have highor medium incomes.
On the other hand, when we considered population weighting, the distribution tended to beconcentrated in the high-income group, indicating that the possibility of twin peaks is excluded.
However, it is still regional and sub-Saharan Africa continues being a low-income region. The
convergence hypothesis does not hold without population weighting, but it does so withpopulation weighting. Nevertheless, the fact that income is concentrated in high-income groups
does not suggest that income inequality will disappear in the foreseeable future.

References
Barro, Robert J. and Xavier Sala-i-Martin. (2004) Economic growth (Second edition),
Cambridge: MIT Press
Feenstra R., Inklaar. R, and Timmer. M, (2015) “The Next Generation of the Penn World Table,”
American Economic Review, 105(10), pp. 3150-3182.
Islam, Nazrul. (2003) “What Have We Learnt from the Convergence Debate? A Review of theConvergence Literature,” Journal of Economic Surveys, 17(3), pp. 309-362.
Quah, Danny. (1993) “Empirical Cross-Section Dynamics in Economic Growth,” European
Economic Review, 37, pp. 426-434.
Quah, Danny. (1996a) “Empirics for Economic Growth and Convergence,” European
6Economic Review, 40, pp. 1353-1375.
Quah, Danny. (1996b) “Twin Peaks: Growth and Convergence in Model of Distribution
Dynamics,” Economic Journal, 106, pp. 1045-1055.
Solow, Robert M. (1956) “A Contribution to the Theory of Economic Growth,” Quarterly
Journal of Economics, 70, pp. 65-94.

7

Table 1 Classification of Region
East Asia & PacificMiddle East & North Africa

North America

South Asia

Australia
Brunei Darussalam
China
Fiji
China, Hong Kong SAR
Indonesia
Japan
Cambodia
Republic of Korea
Lao People's DR
China, Macao SAR
MyanmarMongolia
Malaysia
New ZealandPhilippines
Singapore
ThailandTaiwanViet Nam

United Arab Emirates
BahrainDjibouti
Algeria
Egypt
Iran (Islamic Republic of)
Iraq
IsraelJordanKuwaitLebanonMoroccoMalta
Oman
State of Palestine
Qatar
Saudi Arabia
Syrian Arab Republic
Tunisia
Yemen

BermudaCanadaUnited States

Bangladesh
BhutanIndia
Sri Lanka
Maldives
Nepal
Pakistan

Figure 1 Number of Countries
200
180
160
140
120
100
80
60
40
20
1952
1954
1956
1958
1960
1962
1964
1966
1968
1970
1972
1974
1976
1978
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
2012
2014
2016

0

East Asia & Pacific

Europe & Central Asia

Latin America & Caribbean

Middle East & North Africa

North America

South Asia

Sub-Saharan Africa

8

Continues
Europe & Central Asia

Latin America & Caribbean

Sub-Saharan Africa

Albania
Armenia
Austria
Azerbaijan

Italy
Kazakhstan
Kyrgyzstan
Lithuania

MexicoMontserrat
Nicaragua
PanamaAngolaBurundiBenin
Burkina Faso

Mozambique
Mauritania
Mauritius
Malawi

Belgium
Bulgaria

Luxembourg
LatviaAruba
Anguilla
Argentina
Antigua andBarbudaBahamas
BelizePeru
Paraguay

Namibia
Niger

Bosnia andHerzegovina

Republic ofMoldova

El Salvador

Belarus
Switzerland

North Macedonia
Montenegro

Bolivia(Plurinational
State of)BrazilBarbados

Botswana
Central African
Republic
Côte d'Ivoire

Netherlands

Chile

Cameroon
D.R. of the
Congo
Congo

RwandaSudan

CyprusCzech Republic

NorwayColombia

Comoros

Sierra Leone

Germany

PolandCosta Rica

Suriname
Sint Maarten(Dutch part)
Turks and Caicos
Islands
Trinidad andTobagoUruguay

Cabo Verde

Denmark

PortugalCuraçao

Ethiopia

Spain

Romania

Cayman Islands

Gabon

Seychelles

Estonia

Dominica

Ghana

Chad

Finland

RussianFederation
SerbiaSt. Vincent and
the Grenadines
Venezuela(Bolivarian
Republic of)British Virgin
Islands

Sao Tome andPrincipe
Eswatini

GuineaTogo

FranceSlovakia

Dominican
Republic
EcuadorGambiaUnited Kingdom
Georgia

Slovenia
SwedenGrenadaGuatemalaGreeceCroatia
Hungary
Ireland

Tajikistan
Turkmenistan
TurkeyUkraineIcelandUzbekistan

Honduras
Haiti
JamaicaSaint Kitts and
Nevis
Saint Lucia

Guinea-Bissau
Equatorial
GuineaKenya
Liberia
LesothoMadagascar

U.R. of Tanzania:
Mainland
UgandaSouth Africa

Mali

9

Nigeria

SenegalZambiaZimbabwe

Table 2 Distribution of Income State among Region (Five States)
All wdEast Asia & Pacific
Europe & Central Asia
Latin America & Caribbean
Middle East & North Africa
North America
South Asia
Sub-Saharan Africa

1

2

3

4

5

1,820
195
43
50
91
0
151
1,290

1,869
248
118
320
223
0
172
788

2,088
209
314
881
268
0
77
339

1,774
109
647
631
183
0
11
193

2,319
405
1,092
299
307
180
0
36

Table 3 Income State Changes (Five States, East Asia & Pacific, South Asia)
Australia
Brunei Darussalam
China
Fiji
China, Hong Kong SAR
Indonesia
Japan
Cambodia
Republic of Korea
Lao People's DR
China, Macao SAR
MyanmarMongolia
Malaysia
New ZealandPhilippines
Singapore
ThailandTaiwanViet Nam
Bangladesh
BhutanIndia
Sri Lanka
Maldives
Nepal
Pakistan

1952
5

1960
5

2

2
3
4
2
4

3

1970
5
5
2
3
4
1
5
2
2
1
4
1
1
3
5
2
4
2
3
1
2
2
1
3
2
1
2

2

5
2
2
2

3
5
2
3
2
3
2

1
3

2
3

2

1
2

10

1980
5
5
2
3
5
2
5
1
3
1
5
1
2
3
5
2
5
2
4
1
1
2
1
2
3
1
2

1990
5
5
2
3
5
2
5
1
4
1
5
1
2
4
5
2
5
3
5
1
1
2
1
2
3
1
2

2000
5
5
2
3
5
2
5
1
5
1
5
1
2
4
5
2
5
3
5
1
1
2
1
2
3
1
2

2010
5
5
3
3
5
3
5
1
5
2
5
2
3
4
5
2
5
3
5
2
1
3
2
3
3
1
2

2017
5
5
3
3
5
3
5
1
5
2
5
2
3
4
5
3
5
4
5
2
1
3
2
3
3
1
2

Table 4 Distribution of Income State among Region (Seven States)
All wdEast Asia & Pacific
Europe & Central Asia
Latin America & Caribbean
Middle East & North Africa
North America
South Asia
Sub-Saharan Africa

1

2

3

4

5

6

7

1,584
163
28
39
78
0
129
1,147

1,230
154
69
110
104
0
138
655

1,459
189
126
447
214
0
90
393

1,504
146
252
655
186
0
43
222

1,427
90
507
528
128
0
11
163

1,058
117
503
227
145
5
0
61

1,608
307
729
175
217
175
0
5

2010
7
7
4
3
7
3
7
1
6
2
7
2
3
5
6
3
7
4
7
2
1
3
2
4
4
1
2

2017
7
7
4
3
6
4
6
1
6
3
7
3
4
5
6
3
7
5
7
3
2
3
3
4
4
1
2

Table 5 Income State Changes (Seven States, East Asia & Pacific, South Asia)
Australia
Brunei Darussalam
China
Fiji
China, Hong Kong SAR
Indonesia
Japan
Cambodia
Republic of Korea
Lao People's DR
China, Macao SAR
MyanmarMongolia
Malaysia
New ZealandPhilippines
Singapore
ThailandTaiwanViet Nam
Bangladesh
BhutanIndia
Sri Lanka
Maldives
Nepal
Pakistan

1952
7

1960
7

2

2
4
5
2
5

4

1970
7
7
2
4
5
2
6
2
3
1
5
1
2
4
7
3
5
3
4
1
2
2
2
3
3
1
2

2

7
3
2
3

3
7
3
4
2
3
2

2
4

2
4

2

1
2

11

1980
7
7
2
4
6
2
7
1
4
1
6
1
2
4
6
3
7
3
5
1
1
2
1
2
3
1
2

1990
7
7
2
4
7
3
7
1
5
1
7
1
3
5
6
3
7
4
6
1
1
2
1
3
4
1
2

2000
7
7
3
4
7
3
7
1
6
1
7
1
2
5
7
3
7
4
7
2
1
3
1
3
4
1
2

Table 6 Five States Markov Transition Matrix without Population Weight (wd and ea)
lo

ml

mi

mh

hi

wd

lo

0.96240.03700.0006wd

ml

0.04280.91000.0472wd

mi0.00150.04010.91100.0474wd

mh

0.00060.05020.92330.0260wd

hi

0.00040.00040.01580.9833ealo0.93300.0670ea

ml

0.04080.91020.0490ea

mi

0.02450.93140.0441eamh

0.02800.9159eahi

0.05611.0000

Table 7 Seven States Markov Transition Matrix without Population Weight (wd and ea)
ll

lo

ml

mi

mh

hi

hh

wd

ll

0.96200.03680.0006wd

lo0.05840.87740.06260.0016wd

ml

0.00070.05230.88850.0578wd

mi0.00140.00070.05430.87840.06450.0007wd

mh

0.00070.00070.06080.89330.04370.0007wd

hi

0.00100.00100.05050.88730.0603wd

hh

0.0006

0.03780.9616eall

0.93830.0617ealo0.05190.87010.0779ea

ml

0.02170.93480.0435ea

mi0.01400.9231

0.0629eamh0.03410.8977eahi

0.9027

0.0973eahh

0.02980.9702120.00060.00070.0682

Table 8 Five States Ergodic Distribution without Population Weight
lo
wd

ml

mi

0.20150.1678mh
0.1879hi
0.17310.2697ea

1.0000eu

0.01130.03830.10030.23830.6118la0.03670.17730.40220.2882

0.0957me

0.06110.17440.23820.18240.3439na

1.0000sa

0.3446

0.39320.22820.0339af

0.63350.23400.08000.04330.0092nl

0.03470.16780.38050.27270.1443ms

0.12460.22900.23750.14730.2616as

0.06850.11920.14620.12350.5426es

1.0000

Table 9 Seven States Ergodic Distribution without Population Weight
ll
wd

lo
0.1820ml
0.1118mi
0.1307mh
0.1334hi
0.1375eahh0.11750.18700.23440.7656eu

0.00580.01760.0356

0.07380.17350.22550.4682la0.03400.06990.22700.29540.24200.07970.0519me

0.04370.07320.17140.17490.13440.19220.21020.03350.9665sa

0.23670.25150.27130.18960.0509af

0.60610.2166

0.09730.03920.02750.01220.0010nl

0.03230.06650.21620.28140.23060.07740.0956

0.23440.7656ms

0.09010.12110.18870.17120.11430.15020.1643as

0.03650.04470.08730.09790.09200.22170.4199naes

13

Table 10 Five States Markov Transition Matrix with Population Weight (wd and ea)
lo

ml

mi

mh

hi

wd

lo

0.90580.09420.0000wd

ml

0.06320.91250.0243wd

mi0.00220.01950.94410.0342wd

mh

0.00000.03980.93950.0207wd

hi

0.00000.00000.0024

0.9976ealo0.71340.2866ea

ml

0.06360.91060.0258ea

mi0.0136

0.96530.0210eamh

0.03300.9006eahi

0.06641.0000

Table 11 Seven States Markov Transition Matrix with Population Weight (wd and ea)
ll

lo

ml

mi

mh

hi

hh

wd

ll

0.92310.07690.0000wd

lo

0.03850.91530.04620.0000wd

ml

0.00000.04840.90460.0470wd

mi0.00320.0001

0.02130.92630.04920.0000wd

mh

0.00000.00020.04810.90410.04750.0001wd

hi

0.00000.00000.04030.90490.0548wd

hh

0.00000.01640.9836eall

0.69880.3012ealo0.03230.9111

0.0566ea

ml

0.05050.9066

0.0430ea

mi

0.0001

0.96330.0366eamh

0.04090.8747eahi

0.91880.0812eahh

0.04600.954014

0.00000.00000.0844

Table 12 Five States Ergodic Distribution with Population Weight
lo
wd

ml

mi

0.05720.0820mh
0.0920hi
0.07910.6896ea

1.0000eu

0.00090.00580.02270.10060.8700la

0.00930.05420.54530.38770.0036me

0.09010.18120.37510.26470.0889na

1.0000sa

0.56050.42810.01140.0000af

0.68650.22140.06660.02540.0001nl

0.00480.02790.2806

0.19950.4872ms

0.44970.3713

0.09820.06050.0203as

0.06200.10540.11540.06580.6514es

1.0000

Table 13 Seven States Ergodic Distribution with Population Weight
ll
wd

lo
0.0404ml
0.0711

mi
0.0602mh
0.1144hi
0.1166eahh
0.13780.45950.36130.6387eu

0.00060.00190.01440.03200.12970.26070.5608la0.01810.02410.1436

0.43360.34020.03950.0009me

0.0566

0.06340.20570.28850.27830.06680.04090.00001.0000na
sa

0.11420.16070.6726

0.05210.0004af

0.68680.19980.07800.02480.01060.00010.0000nl

0.01070.01430.08520.25750.20200.02350.40670.36130.6387es
ms

0.16390.2132

0.23600.17520.1526

0.03660.0224as

0.00920.02280.0206

0.07670.0631

0.32270.484915

Appendix Table 1 Five States Markov Transition Matrix without Population Weight
lo

ml

mi

mh

hi

wd

lo

0.96240.0370wd

ml

0.04280.91000.0472wd

mi0.00150.04010.91100.0474wd

mh

0.00060.05020.92330.0260wd

hi

0.00040.01580.9833ealo0.93300.0670ea

ml

0.04080.91020.0490ea

mi

0.02450.93140.0441eamh

0.02800.9159

ea

hi

eu

lo

0.88370.1163eu

ml0.03450.8534

eu

mi

eu

mh

eu

hi

la

lo

0.91670.0625laml

0.01280.92950.0577lami0.00120.02420.93080.0438lamh

0.00160.06000.92060.0178lahi

0.00340.00340.04700.9463melo0.85560.1444me

ml

0.04550.88180.0727memi0.00380.04940.91250.0342memh

0.04470.90500.0503mehi

0.0267

0.9733

na

lo

na

ml

na

mi

na

mh

na

hi

0.00040.0006

0.05611.00000.04280.11210.88160.07570.03180.94900.01910.0074

0.99260.02081.00001.00001.00001.00001.000016

Continues
lo

ml

mi

mh

hi

sa

lo

0.97990.0201sa

ml

0.0176

0.93530.0471sa

mi

0.08110.89190.0270samh0.18180.8182

sa

hi

af

lo

0.97690.0231af

ml

0.06140.91300.0256af

mi0.00300.07190.87720.0479af

mh

0.08850.87500.0365afhi

0.17140.8286nl

lo0.91670.0625nl

ml0.01280.9295nl

mi0.0012

nl

mh

nl

hi

0.0021eslo0.95340.0466es

ml

0.03130.92050.0482es

mi0.03960.92090.0396esmh

0.04240.9068

es

hi

ms

lo

0.9331

0.0669ms

ml

0.03330.9051

0.0615ms

mi0.00300.05640.90800.0326ms

mh

0.05260.90000.0474ms

hi

0.0267

0.9733aslo0.93300.0670asml

0.03620.9071

0.0567asmi0.00180.04440.91680.0370as

mh

0.04380.90570.0505ashi

0.01150.98851.00000.02080.05770.02420.93080.04380.00160.06000.92060.01780.00210.02950.96630.05081.000017

Appendix Table 2 Seven States Markov Transition Matrix without Population Weight
ll

lo

ml

mi

mh

hi

hh

wd

ll

0.96200.03680.00060.0006wd

lo0.05840.87740.06260.0016wd

ml

0.00070.05230.88850.0578wd

mi0.00140.00070.05430.87840.06450.0007wd

mh

0.00070.00070.06080.89330.0437wd

hi

0.00100.00100.0505wd

hh

0.0006eall

0.93830.0617ealo0.05190.87010.0779ea

ml

0.02170.93480.0435ea

mi0.01400.9231

0.0629eamh0.03410.89770.00070.00070.88730.06030.03780.9616

0.0682eahi

0.9027

0.0973eahh

0.02980.9702eull

0.82140.1786eulo0.05880.79410.1471eu

ml

0.06400.83200.1040eu

mi

0.0041

0.04130.85950.0950eu

mh

0.00200.03840.90910.0505euhi0.03890.91210.0491euhh0.0236

0.9764lall

0.91890.0541lalo

0.02780.90740.05560.0093laml

0.00230.01830.91280.0642lami

0.05260.89010.05570.0015lamh

0.06980.89730.0310lahi

0.0045

0.09380.83040.0670lahh

0.0057

0.09710.8971mell

0.89610.09090.0130melo

0.04900.82350.1275meml

0.05210.8531

0.0948memi

0.08740.85790.0492memh

0.06400.86400.06400.00800.05040.89930.05040.0512

0.9488

me

hi

me

hh

na

ll

na

lo

na

ml

na

mi

na

mh

0.02700.00190.00450.00550.00231.00001.00001.00001.00001.0000na

hi

0.50000.5000na

hh

0.01730.982718

Continues
ll

lo

ml

mi

mh

hi

sa

ll

0.97660.0234sa

lo0.0221

0.94120.0368sa

ml

0.03410.93180.0341sa

mi

0.04880.90240.0488samh0.18180.8182

sa

hi

sa

hh

af

ll

0.97320.0268aflo0.07420.87790.04640.0015af

ml0.10490.86700.0281af

mi0.07370.84790.0737af

mh

0.10490.8580hh

1.00001.00000.0046

af

hi

af

hh

nl

ll

0.91890.0541nl

lo0.02780.90740.05560.0093nl

ml

0.00230.01830.91280.0642nl

mi

0.05260.89010.05570.0015nl

mh

0.06980.89730.0310nl

hi

0.00440.09210.82460.0746nl

hh

0.00290.05750.9397esll

0.95520.0448eslo0.03790.90340.0586es

ml

0.02570.93380.0404es

mi0.02170.91850.0598esmh0.05050.8889eshi

0.9027

0.0973eshh0.02980.9702ms

ll

0.94630.04880.0049ms

lo0.03360.89080.0756ms

ml

0.04680.87630.0769ms

mi0.08040.86610.0491ms

mh

0.07350.86030.05880.0074ms

hi

0.05040.89930.0504ms

hh

0.0512

0.9488asll

0.94280.05450.0027aslo0.04080.88270.0765asml

0.03730.8986

0.0642asmi

0.05450.88830.0545asmh

0.05800.87500.06250.00450.02780.90080.07140.03870.9613

as

hi

as

hh

0.08330.03700.00440.00270.05000.40000.02700.00190.00450.86670.600019

0.00230.0606

Appendix Table 3 Five States Markov Transition Matrix with Population Weight
lo

ml

mi

mh

hi

wd

lo

0.90580.09420.0000wd

ml

0.06320.91250.0243wd

mi0.00220.01950.94410.0342wd

mh

0.00000.03980.93950.0207wd

hi

0.00000.00000.0024

0.9976ealo0.71340.2866ea

ml

0.06360.91060.0258ea

mi0.0136

0.96530.0210eamh

0.03300.9006

ea

hi

eu

lo

0.88570.1143eu

ml0.01750.85940.1232eu

mi0.03140.85700.1116eu

mh

0.0251

0.94730.0276euhi

0.00320.9968lalo0.95670.0433laml

0.00740.90770.0849lami

0.00000.00840.95920.0324lamh

0.00000.0456

0.95400.0005lahi

0.00030.00030.04900.9504melo

0.92390.0761me

ml

0.03060.88570.0837memi0.00350.03690.9226

0.0370memh

0.05240.93580.0117mehi

0.03500.9650

na

lo

na

ml

na

mi

na

mh

na

hi

0.06641.00000.00011.00001.00001.00001.00001.000020

Continues
lo

ml

mi

mh

hi

sa

lo

0.94030.0597sa

ml

0.07810.91990.0020sa

mi

0.07370.92550.0007samh0.19030.8097

sa

hi

af

lo

0.98070.0193af

ml0.05380.92680.0194af

mi0.01970.0446

0.89330.0423afmh

0.11090.88770.0014afhi

0.50000.5000nl

lo0.95670.0433nl

ml

0.00740.90770.0849nl

mi0.00000.00840.95920.0324nl

mh

0.00000.0456

0.95400.0005nl

hi

0.00000.00000.00020.9998eslo0.88500.1150es

ml

0.06750.91310.0194es

mi0.01550.96410.0204esmh0.03310.9005

es

hi

ms

lo

0.93950.0605ms

ml

0.07240.91580.0118ms

mi0.0031

0.0416

0.92300.0324ms

mh

0.0526

0.93570.0117ms

hi

0.03500.9650aslo

0.88640.1136asml

0.06620.9121

0.0217asmi

0.00060.01920.95690.0233asmh

0.04090.91470.0444ashi

0.00450.99551.00000.0001

0.06631.000021

Appendix Table 4 Seven States Markov Transition Matrix with Population Weight
ll

lo

ml

mi

mh

hi

hh

wd

ll

0.92310.07690.0000wd

lo

0.03850.91530.04620.0000wd

ml

0.00000.04840.90460.0470wd

mi0.00320.0001

0.02130.92630.04920.0000wd

mh

0.00000.00020.04810.90410.04750.0001wd

hi

0.00000.00000.04030.90490.0548wd

hh

0.00000.01640.9836eall

0.69880.3012ealo0.03230.91110.05050.00000.00000.0566

ea

ml

ea

mi

0.9066

0.04300.0001

0.96330.0366eamh

0.04090.8747

ea

hi

ea

hh

eu

ll

0.84340.1566eu

lo0.04670.76110.1923eu

ml

0.02220.89030.0875eu

mi0.00130.03610.82590.1367eu

mh

0.00050.03330.89540.0709euhi

0.03530.9136

0.05110.02380.97620.08440.91880.08120.04600.9540

eu

hh

la

ll

0.97130.0287lalo0.02150.9271

0.05130.0001laml

0.00000.0086

0.8992

0.0922lami

0.03050.92930.04010.0000lamh

0.0511

0.92290.0260lahi

0.00000.22340.77110.0055lahh

0.00070.23320.7660mell

0.95080.04900.0003melo0.0201

0.83240.1475meml

0.03820.89070.0711memi

0.04550.89400.0553memh

0.05730.92700.01490.0008mehi

0.06560.88490.0495mehh

0.08670.9133

na

ll

na

lo

na

ml

na

mi

na

mh

na

hi

0.4722

0.5278nahh

0.00001.00000.00010.00000.00000.00520.00001.00001.00001.00001.00001.000022

Continues
ll

lo

ml

mi

mh

hi

sa

ll

0.93700.0630sa

lo0.04480.93180.0234sa

ml

0.00560.99000.0044sa

mi

0.0567

0.94180.0015samh0.19030.8097

sa

hi

sa

hh

af

ll

0.98150.0185aflo0.06090.8733hh

1.00001.00000.16140.06570.0001

af

ml

af

mi

af

mh

af

hi

af

hh

nl

ll

0.97130.0287nl

lo0.02150.9271

0.05130.0001nl

ml

0.00000.0086

0.8992

0.0922nl

mi0.03050.92930.04010.0000nl

mh

0.0511

0.92290.0260nl

hi

0.00000.22340.77100.0056nl

hh

0.00000.00030.9997esll

0.89570.1043eslo0.03750.91980.0427es

ml0.04590.91500.0391es

mi0.0016

0.96270.0357esmh

0.04110.8746

es

hi

es

hh

ms

ll

0.93790.06210.0000ms

lo0.04380.92790.0283ms

ml

0.02210.93980.0381ms

mi

0.0466

0.89860.0501ms

mh

0.0575

ms

hi

ms

hh

as

ll

0.89880.10120.0000as

lo

0.03720.91830.0444asml

0.04520.91280.0421asmi

0.01030.94910.0396asmh

0.0481

as

hi

as

hh

0.02230.82300.01560.02740.90250.04780.11160.88710.00130.13390.7991

0.06700.6486

0.35140.00010.00000.00000.00470.001023

0.00000.08430.91880.08120.04600.95400.92680.01490.00080.06560.88490.04950.08670.91330.89730.05430.00040.01070.91320.07610.05070.9493