Irregular Economic Growth in the World Economy: Fluctuations ofErgodic Distributions through a Markov Chain Model

Alvaro Domínguez
Asian Growth Research Institute (AGI)
Hiroshi Sakamoto
Asian Growth Research Institute (AGI)

Working Paper Series Vol. 2022-03
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Irregular Economic Growth in the World Economy:
Fluctuations of Ergodic Distributions through a Markov Chain
Model

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 reexamine the convergence hypothesis of economic growth. Traditionally, it was analyzed
using econometric methods, although estimating long-term economic fluctuations with a linear
model is not always ideal. We thus employ a Markov chain stochastic model that divides thelogarithmic value of relative income, comparing each country's GDP per capita with theaverage, into several ranks in descending order of income. Using the most recent data, we total
the time-series changes of the income states in each sample, and represent them throughprobabilities. We observe the changing ergodic distribution and show that the world economy
is not growing monotonously, and proceed to correct the population size of each country forrank changes. The transition probability matrix is re-estimated by applying population weights
to changes in the income states of each country. When there is no population weighting, themodel shows that the world economy may be divided into two peaks as before. However, whenusing population weights, the model yields more optimistic results.

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 analyze the long-term growth trends of the world economy from the 1950s, from theperspective of convergence. We study whether economically poor countries (regions) can catch
up with rich ones. Economic theory tells us that if we assume production functions withdiminishing returns à la Solow (1956), the possibility of convergence increases. This is due to
productivity being high at the stage of low input and gradually decreasing as inputs increase.
By replacing “low input” with “poor country (region)”, we can then link it to empirical research.
There is a large literature related to whether the world economy is converging (Barroand Sala-i-martin, 2004; Islam, 2003). One finding is that there is a lack of convergence due tothe economic growth of each country being non-monotonous. Various economic growthmodels such as Solow (1956) emphasize monotony as a characteristic. However, the economic
growth of each country does not behave exactly as in a model.
This discussion has traditionally been analyzed mainly through econometric methods.
In this study, however, we rely on the stochastic method proposed by Quah (1993, 1996a,
1996b). This unique analytical method, classified by Islam (2003) as a “distribution approach,”produces distinctive results compared to econometric methods. It divides the logarithm ofrelative income, which compares each country'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 as probabilities. The Markov transition
matrix obtained by such a procedure has an eigenvalue (the “dominant eigenvalue”) whose
absolute value is 1. Therefore, the convergence distribution (ergodic distribution) can be
estimated by obtaining the associated eigenvector of the eigenvalue taking a value of 1. With
this approach, Quah argues that the world economy is divided into two extremes or “TwinPeaks”: a rich economy and a poor economy.
In this paper, we analyze how the ergodic distribution obtained by the above procedure
varies over time. This shows a well-known tendency in econometrics that changing the sample
changes the result, but conversely, the data does not change monotonously. In other words,
trends in the world economy change in a complex way.
In previous studies, the changes in income state were calculated using a country(region) as a sample. This has the problem that it is not easy for the income state to change incountries with large populations such as China and India. Hence, in this work, we multiply
each sample by the population weight and calculate the Markov transition matrix from theaggregated results to obtain the convergence distribution. In the case of the population weight,
the average of the year in which the sample as the number of populations appears and the year
after it was calculated, the ratio is obtained from the total of these worlds, and is multiplied by
the income states movement. This makes the stochastic model more precise, although theaggregation work becomes more complex because the population ratio has to be multiplied forall state changes.
This work is organized as follows. In section 2 we describe the stochastic model using
1Markov chains. Section 3 presents the data we use. We show the results of the convergence
distribution without population weight in section 4 and with population weight in section 5.
Section 6 concludes.

2. The Stochastic Model
The “stochastic model” using a Markov transition matrix is defined as follows. Let Ft be theincome distribution state in period t, expressed as a 1×N vector, representing the number ofstates. The Markov process is an expression of the situation in which the income distribution
situation Ft+1 in the next state depends on the income distribution state Ft in the current state.
In other words, the fluctuation of the income distribution states between the two time points isdefined as follows.
𝐹𝐹𝑡𝑡+1 = 𝐹𝐹𝑡𝑡 ∙ 𝑀𝑀𝑡𝑡

(1)

Where Mt is a transition matrix. Now, since the income distribution is ranked for thistransition, the transition matrix is estimated by aggregating the changes of income states. The
estimation method is as follows, with P defined as the number of changes of income states.
𝑀𝑀𝑡𝑡,𝑗𝑗𝑗𝑗 = 𝑃𝑃𝑡𝑡,𝑗𝑗𝑗𝑗 ⁄∑𝑛𝑛𝑘𝑘=1 𝑃𝑃𝑡𝑡,𝑗𝑗𝑗𝑗(2)

Next, when population weighting is applied to each transition, the transition probability
matrix is estimated after totaling the population weighting W that differs in each country(region).
𝑛𝑛
𝑚𝑚
𝑀𝑀𝑡𝑡,𝑗𝑗𝑗𝑗 = ∑𝑚𝑚
𝑟𝑟=1 𝑊𝑊𝑡𝑡,𝑗𝑗𝑗𝑗,𝑟𝑟 ⁄∑𝑘𝑘=1 ∑𝑟𝑟=1 𝑊𝑊𝑡𝑡,𝑗𝑗𝑗𝑗,𝑟𝑟(2)’This shows the probability of how much of the total number of changes (total numberof income states movements from state k), including the number staying in the same state has
moved to state j. Then, we take advantage of the ergodicity which is a characteristic of Markov
chains, and find the convergence distribution (ergodic distribution).
F＝F･M (3)The transition matrix obtained through such a procedure usually has an eigenvalue with
an absolute value of 1. Therefore, by finding the eigenvector associated to this eigenvalue, theestimation of the convergence or ergodic distribution can be done by using a solver such as theone found in Excel. However, in this study, we make use of GAMS (General Algebraic
Modeling System) to calculate various convergence distributions. In this case, equation 3 holds
2when M is given, and the model is constrained so that the sum of F equal to 1 is solved.
The convergence distribution analyzed in this study consists of a fixed distribution thatis reached through a stochastic model composed of samples from 1953 to 2017. The
convergence hypothesis examines whether indicators such as GDP per capita for the world
economy or a specific economic group settle into a certain value. When verifying whether there
is convergence through this model, the shape of the convergence distribution is a concern.
Hence, assuming that there is convergence to a certain value, it is desirable that the distribution
is concentrated in some income states. Particularly, it is expected that the distribution will bemountain-shaped, centered on the middle-income group. For other distributions instead,convergence may not be achieved.

3. Data
For the data in this study, we rely on the PWT (Penn World Table version 9.1,
https://www.rug.nl/ggdc/productivity/pwt/), which provides internationally comparable data
(Feenstra, Inklaar and Timmer, 2015). The PWT provides only macroeconomic indicators such
as GDP, but for some countries, it is superior because it provides estimates from 1950. However,since the data for China is available only from 1952 onwards, we analyze from 1952 to 2017
(the last obtainable year). The number of country (region) changes during the period, is set toa maximum of 182.
Next, we use real GDP and population. The PWT estimates real GDP from two sides,
production and expenditure, with the estimated values being slightly different. In this work, weutilize real GDP (“rgdpo” in the PWT notation) from the production side. Then, from this data
we calculate the relative income per capita. The relative income is defined as the logarithm
when the average income (of each year) is 1 (𝑙𝑙𝑙𝑙(𝑦𝑦⁄𝑛𝑛⁄∑ 𝑦𝑦⁄∑ 𝑛𝑛)), with y representing the realGDP, and n being the population.
The analysis of convergence in this paper shows fluctuations in the relative income of eachcountry (region) through a stochastic model. Specifically, we rely on the method proposed by
Quah (1993, 1996a, 1996b). In this method, the logarithmic value of the relative income
mentioned above is ranked in ascending order of income, with the change in rank being
followed in chronological order, and the aggregated value being expressed as a probability.
However, since this ranking is arbitrary, we fix the logarithmic value of the relative income forthe ranking, with the following numbers based on the average of 0:
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

3The numbers in parentheses in the five states indicate 1/4, 1/2, and 2 times the average
income. This makes it possible to respond to changes in the number of samples.

4. Time Series Change of Convergence Distribution without Population Weight
Up to this point, we have observed the convergence distribution through the stochastic modelby region and confirmed the distribution of the income groups that make up this model. Next,
we analyze the changes in the time series with respect to the convergence of the world economy.
In this study, the purpose of fixing the relative income (i.e., the standard of the income class),is that it is possible to deal with changes in the number of samples. This allows us to analyze
the difference in the convergence distribution when the measurement period is changed. Thus,
we start by calculating the convergence distribution using the sample going from 1953 to 1972
for each income state. We then extend the convergence distribution period, on a yearly basis(e.g., 1973, 1974, and so on), by computing the cumulative type samples. Additionally, we
compute the convergence distribution through the moving average type samples for 20 years(e.g., 1954-73, 1955-74, etc.), and we then compare it to the cumulative type samples obtained
previously.
Figure 1 shows the convergence distribution of five income states from the cumulative
sample in chronological order. From this figure, we see that the convergence distribution of the
1953-72 sample converges to the high-income class, and then it gets polarized over time. In
table 1, the distribution of the high-income group is the largest throughout the entire period,
but the second-largest group changes to the low-income group. This indicates that theconvergence distribution changes depending on the sample used, and that it does not become auniversal event for future trends in the world economy.
Figure 2 shows the convergence distribution of the five income states of the cumulative
moving average sample in chronological order. First, in the early stages, the distribution isconcentrated in the high-income group, and then from around the 1980s, it tends to becomepolarized. From around 1994, it becomes a bipolar differentiation concentrated in the lowincome group. Then, approximately from 2010, it becomes a distribution concentrated in the
middle-income group. From this result, the convergence distribution tends to change
significantly when the period is divided.
The results of these calculations, in seven income states, are shown in Figure 3(cumulative sample) and Figure 4 (moving average sample). There is a slight difference in theyear of change depending on the moving average type of the sample, but the tendency is notmuch different from the case of five income states. There is no universality in the future trendsof the world economy, and it becomes clear that the convergence problem has not yet beensolved.

4

5. Time Series Change of Convergence Distribution with Population Weight
Figure 5 shows the convergence distribution of five income states through the cumulative
sample in chronological order. We can see from this figure that the overwhelming concentrationof distribution in the high-income group tends to gradually disappear. However, from 2000 to
2002 the tendency to concentrate once again returns, and since then it gradually eases.
Nevertheless, it shows a high ratio of 60% or more throughout the measurement period,
indicating that the world economy is likely to be concentrated in the high-income group. The
group with the lowest ratio is the middle-income group until around 2005, but after 2006 it
becomes the low-income group. Accordingly, up to 2005 we can say that the differentiation isvery weak, even though it is concentrated in the high-income group.
Figure 6 shows the convergence distribution of five income states from the moving averagetype of the cumulative sample in chronological order. In the figure, we can observe the color
changes. Instead, in table 6 the concentration on the high-income group continues until 1990.
It should be noted though, that the ratio drops significantly. Additionally, from 1991 thehierarchy of the highest ratio changes rapidly. In section 4 we pointed out that the convergence
distribution tends to change significantly when the period is divided, but it is safe to say that asimilar tendency occurs even if the population weight is considered.
The results of these calculations into seven income states are shown in Figure 7 (cumulative
sample) and Figure 8 (moving average sample). We can see that the highest-income group has
the highest ratio in the cumulative sample, although the ratio gradually decreases. There is aweak bipolar differentiation between 1992 and 2008, but it is concentrated in the second lowest
income group (lo) rather than the lowest income one (ll). For the moving average type ofsample, the same concentration on the highest income group (as in the five states) continues
until 1989, whereas in the concentration on the lowest income group persists from 1990 up to
2001. From then on, with some exceptions, the distribution concentrates in the middle-income
class. Although there are some differences, these do not considerably differ with those from thetendency in the five states. In addition, the possibility of bipolar differentiation disappears when
we consider population weighting. We can confirm that the “Twin Peaks” claimed by Quah are
not found.
Finally, regarding the convergence hypothesis, the concentration of the distribution in thehigh-income states can initially be seen as a confirmation of the hypothesis. However, in thisstudy it is difficult to say that it is achieved in two different ways. In the first, if strictly attained,
the income distribution should be more average, in which case the distribution ratio of theaverage income group would be the highest. In the second, although the distribution isconcentrated in the high-income group, the ratio declines. Due to this, the absolute convergence
hypothesis cannot be obtained, and we conclude that instead conditional convergence occurs.
This is similar to the case of many studies relying on econometric methods.

5

6. Summary
We considered the traditional convergence hypothesis discussed in economics as a way toanalyze the inequality distribution affecting various countries. We then verified this relying ondata from the Penn World Table. After introducing a Quah-style Markov chain model, we
presented results while considering populations with and without weights. We showed that
there is a tendency for a bipolar differentiation of the world economy, whereas the convergence
hypothesis does not hold. Meanwhile, the distribution tends to be concentrated in the higher
income group when considering population weighting. Possibly, this tendency differs
depending on the time under analysis, and does not necessarily lead to the establishment of the
hypothesis. Nonetheless, since the growth of the world economy is not considered to beuniform, it is an expected result.

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
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.

6

Figure 1 Five States Ergodic Distribution without Population Weight (Accumulation from
1972)
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%

lo

ml

mi

mh

hi

Figure 2 Five States Ergodic Distribution without Population Weight (20 Years Moving
Average)
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%

lo

ml

mi

7

mh

hi

Figure 3 Seven States Ergodic Distribution without Population Weight (Accumulation from
1972)
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%

ll

lo

ml

mi

mh

hi

hh

Figure 4 Seven States Ergodic Distribution without Population Weight (20 Years Moving
Average)
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%

ll

lo

ml

mi

8

mh

hi

hh

Figure 5 Five States Ergodic Distribution with Population Weight (Accumulation from 1972)
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%

lo

ml

mi

mh

hi

Figure 6 Five States Ergodic Distribution with Population Weight (20 Years Moving Average)
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%

lo

ml

mi

9

mh

hi

Figure 7 Seven States Ergodic Distribution with Population Weight (Accumulation from 1972)
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%

ll

lo

ml

mi

mh

hi

hh

Figure 8 Five States Ergodic Distribution with Population Weight (20 Years Moving Average)
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%

ll

lo

ml

mi

10

mh

hi

hh

Appendix Table 1 Five States Ergodic Distribution without Population Weight (Accumulationfrom 1972)
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017

lo

0.05750.06250.0581
0.05120.08100.08090.0827
0.09780.0892
0.09650.08960.0866
0.09850.09990.11870.12150.12830.12850.13800.15240.16930.16410.18200.17180.17080.17920.19020.18850.19130.20240.20150.20240.19220.18450.17150.18380.17640.17290.18320.18380.18390.18520.20000.19700.19600.2015ml

0.08250.08900.07800.06820.10270.11050.11290.13340.12200.12870.11930.11450.12590.13460.14010.13810.14160.14490.15050.16070.17400.17200.17500.16900.17350.17390.17910.17320.17170.17360.16830.16850.16690.16250.15500.15600.15540.15110.16140.16040.16140.16120.16660.16770.16440.1678mi

0.07440.08030.07480.07650.10400.10230.10850.11470.11220.11850.12400.12310.12470.13690.13210.12400.12670.12380.12520.13720.13770.13980.13790.13940.14480.14670.15280.14970.14540.14790.15430.15400.15730.15420.15880.15480.15510.16620.17560.17830.18110.18020.18060.18510.18460.1879mh

0.12690.12170.12270.11490.15620.14510.14110.14820.13240.14550.14600.15820.15120.15450.15650.15040.14580.14420.13900.14910.14530.13930.12720.12960.13350.12980.12950.12540.12220.12310.13300.13210.13900.13760.14870.14340.14530.15780.16290.16960.17240.17660.17010.17300.17320.173111hi

0.65870.64650.66630.68910.55610.56110.55480.50590.54420.51080.52100.51770.49970.47410.4526
0.46600.45760.45860.44740.4006
0.37380.38480.37790.39020.37740.37040.34850.36320.36930.3531
0.34280.34310.34470.36120.36590.36210.36780.35200.31700.30790.30120.29680.28270.27710.28190.2697

Appendix Table 2 Five States Ergodic Distribution without Population Weight (20 Years
Moving Average)
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017

lo

0.05750.05970.07120.0511
0.09290.09350.10670.13200.13500.12940.12100.11750.11750.13470.15920.16460.16920.16950.20130.23710.24600.23190.27630.26950.24800.27830.30540.29550.32610.3556
0.36720.36940.31250.30600.23100.26350.23230.21720.21500.20230.17450.17870.18550.18840.19780.1868ml

0.08250.0921
0.09140.07100.11510.13180.14270.17560.16910.16940.14360.13500.13220.14670.14590.14680.14700.15410.16480.19570.2052
0.19880.21050.21390.21500.20320.20700.18790.19540.19160.18540.19150.18250.17090.1586
0.16270.16020.14060.14570.13950.12610.12190.12810.13440.12250.1223mi

0.07440.08700.08070.08440.11380.11550.12930.13850.13800.13940.13790.13630.12620.14170.13040.12830.12960.13270.13280.15400.15320.15450.15170.15040.16070.16280.1636
0.16520.15550.15350.15140.15300.16990.15620.18690.19180.19260.2233
0.23650.23280.24770.23770.24420.25250.2426
0.2455mh

0.12690.12030.11640.11100.14440.13730.14280.14820.13150.14850.13570.14850.12980.12920.13300.12710.12310.12820.11640.12850.12620.12240.09850.10180.10450.10200.09840.0945
0.09360.0862
0.09590.08870.10920.10850.13230.12910.14080.16480.17530.18590.20600.23470.24680.24580.24270.248412hi

0.65870.64100.64030.68250.53380.52180.47840.4056
0.42640.41330.46180.46270.49430.44770.4316
0.43310.43110.41560.38470.28470.26930.29240.26290.26440.27190.25370.22570.25680.22930.21310.20010.19740.22600.25840.29120.25290.27420.25420.22750.23940.24560.22710.19540.17880.19450.1970Appendix Table 3 Seven States Ergodic Distribution without Population Weight (Accumulationfrom 1972)
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017

ll

0.03850.06250.04090.03690.04300.04890.05480.05030.06810.0576
0.05560.04580.06220.07850.0887
0.09130.10180.11530.13260.15410.16800.17910.19560.20110.19330.19330.20510.20720.2055
0.19830.18150.20310.18490.17320.16470.17220.17050.16930.17320.16550.17070.17300.18360.18150.17770.1820lo

0.04430.05440.03660.04090.04380.04590.05270.04970.06230.05700.05510.04530.05920.07200.07420.06900.07800.08290.09150.09950.11030.11540.12100.12130.12280.11870.12820.12280.11590.11620.11550.12480.12090.1156
0.1094
0.10850.10510.10440.10640.10430.10510.10780.10900.10880.10960.1118ml

0.07780.07030.0592
0.06390.06880.06880.0781
0.06680.07440.06730.06430.05780.07300.0911
0.09670.08560.08760.0846
0.09720.10140.11050.12110.12320.12170.12130.12350.13010.12520.12020.1206
0.11790.1226
0.12430.12350.11650.11740.11220.11290.11860.11630.11740.11910.12380.12590.12740.1307

mi

0.06390.06450.05240.05620.05780.05520.05800.04930.0587
0.05370.05480.04840.06090.07610.08290.07690.07700.07500.08190.08840.0907
0.0955
0.0966
0.09650.0942
0.09550.0996
0.09950.09930.10220.10090.10120.10570.10810.10470.10440.10560.11140.11690.11820.12050.12190.12550.12840.13060.133413mh

0.12780.11560.10150.09910.10230.09250.08880.07510.08100.07680.07470.07140.0847
0.09810.11200.10650.10090.09910.10260.10780.10670.10580.09920.0997
0.09680.09440.0945
0.0936
0.09380.0954
0.0973
0.09670.10480.10880.1106
0.1096
0.11250.12010.12340.12770.13020.13530.13330.13500.13750.1375hi

0.20170.19460.22460.22020.22790.23250.20920.20590.19390.19480.20000.20240.21250.19980.17130.17270.16190.15840.15310.14380.12640.12180.11690.11310.1126
0.11250.10420.10470.10780.10610.11080.10430.10830.11270.11570.11410.11820.11620.11500.12100.11910.12140.11480.11660.11890.1175hh

0.44590.4381
0.48480.48280.45640.45610.45840.50300.46170.49290.49550.52900.44740.38440.37430.3982
0.39280.38480.34120.30520.28730.26140.24740.2466
0.25900.26220.23820.24710.25740.26110.27610.24720.25110.25810.27840.27380.27600.26580.24650.24700.23700.22130.21000.2037
0.19840.1870Appendix Table 4 Seven States Ergodic Distribution without Population Weight (20 Years
Moving Average)
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017

ll

0.03850.05430.0481
0.04110.05550.07250.07690.07000.10190.08290.07680.05340.07250.10890.11870.13510.14270.16170.22340.25000.27650.25420.33060.42010.3711
0.36150.39000.44180.40040.41820.3546
0.40380.32220.27530.25010.26000.24390.20710.19490.15330.14210.13170.13160.12070.11110.1157lo

0.04430.04930.04440.04950.05780.06540.07090.0661
0.09180.08300.0696
0.05140.0693
0.09360.08940.08370.08980.09550.1106
0.11480.13370.13950.16130.1426
0.15320.1466
0.15550.14070.13190.13190.14480.15930.15160.13940.13950.14680.12480.11270.1081
0.09820.08330.07880.07180.06920.0626
0.0670ml

0.07780.06300.07070.07520.08750.08940.10090.0812
0.09890.08650.06930.05780.0782
0.09730.09970.08850.08700.0825
0.0936
0.09670.10820.13390.13310.11220.11820.12400.12420.11540.11900.11960.12980.12690.13440.12910.12260.13850.12760.13580.13260.13130.11900.10220.10570.11440.11080.1080mi

0.06390.06420.06480.07170.07530.06920.0706
0.05340.0706
0.06450.04970.04500.06070.08140.08780.08440.08120.07920.08430.09240.09120.10120.10100.08640.0906
0.09780.10180.10200.10720.11170.11520.11160.12620.12780.12450.12890.13840.16200.16990.17270.18520.17630.18400.19340.20810.213714mh

0.12780.10580.11120.11190.11450.09950.09480.06970.08120.08270.05850.05820.07380.08720.1042
0.09710.08870.08710.08290.08510.08050.08470.06930.06170.06230.06480.06500.06220.06950.06770.07880.07120.09040.09900.1001
0.09910.11680.13750.14580.16010.17890.20000.21280.21110.2311
0.2394hi

0.20170.17670.21220.18290.18370.1836
0.16620.1746
0.15540.16970.19300.19970.20810.18150.15660.15260.13930.13680.11410.10650.08790.0871
0.06240.0531
0.05620.05540.04730.04040.04950.04290.04920.03870.04860.06050.07250.06530.07740.07930.08390.09940.11770.13280.12550.13450.14420.1415hh

0.44590.48670.44870.46770.42570.42030.41970.48500.40020.43080.48320.53460.43740.35010.3436
0.35870.37120.35730.29120.25440.22200.19940.14230.12380.14840.14990.1162
0.09750.12250.10800.12760.08860.12650.16890.19050.16140.17100.16570.16480.18510.17370.17800.1686
0.15670.13220.1147

Appendix Table 5 Five States Ergodic Distribution with Population Weight (Accumulationfrom 1972)
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017

lo

0.00790.00730.00600.00620.00890.01000.01120.01640.02300.01940.01620.03940.05720.05590.06650.06500.0656
0.06950.07340.08280.08720.08300.0871
0.0661
0.06180.06670.10800.12570.12680.12750.0711
0.06580.06510.05780.0546
0.0556
0.05500.0502
0.05340.0546
0.05280.05190.05500.05890.05670.0572ml

0.01130.01060.00870.00900.01040.01090.01430.02110.02960.02500.02090.05130.07420.07320.08510.08270.08300.0881
0.09280.10420.10970.10400.10840.08170.07600.08110.13100.15170.15200.15210.08350.07620.07460.07510.07170.07370.07370.06780.07310.07540.07350.07260.07740.08360.08080.0820mi

0.00620.00660.00550.00580.00580.00570.00740.01030.01160.01130.01180.02940.03020.03040.02890.02730.02750.02770.02900.03220.03290.03600.03700.03470.03400.03630.04470.04920.04840.04880.0551
0.05350.05500.05720.05710.0597
0.06130.06400.06880.07380.07360.07440.07940.08740.08880.0920mh

0.0276
0.02690.02650.02620.02600.02580.02550.02550.02470.02470.0246
0.06230.05910.05930.05680.0537
0.05330.05240.05150.05470.05530.0526
0.05120.05430.05520.0546
0.05730.05370.05200.05160.06040.0611
0.06130.0616
0.0626
0.06220.06230.06390.07300.07730.07830.07920.07860.07780.07760.079115hi

0.9471
0.9486
0.95340.95290.94880.9476
0.9416
0.92660.91110.91970.92650.81770.77940.7811
0.76270.77140.7706
0.76220.75340.72610.71500.72450.71640.76320.77300.76120.65900.61970.62070.61990.72990.74340.74400.74840.75400.74870.74780.75410.7316
0.71890.72170.72190.70960.69240.69610.6897

Appendix Table 6 Five States Ergodic Distribution with Population Weight (20 Years Moving
Average)
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017

lo

0.00790.00690.00800.00650.00980.01590.02610.03870.1341
0.06510.00410.07550.13040.12460.13090.12570.07610.1462
0.23730.31970.3046
0.29690.32810.26510.08110.08680.39370.41720.40580.42500.29950.20800.18080.04960.02290.02700.02150.01660.01710.01120.00970.01120.01170.01320.01320.0094ml

0.01130.01010.00920.00890.01000.01500.02950.04350.12050.07200.00460.08590.11070.12290.13010.12300.1276
0.26470.26150.34900.35720.31180.33900.26230.28720.30170.29210.30130.28470.29440.18470.10850.09000.10090.08850.11600.1142
0.09700.1071
0.09950.08980.10800.11800.15100.14160.1329mi

0.00620.00660.00560.00580.00580.00810.01550.02190.0281
0.02200.00210.0396
0.03790.04190.03570.03480.0371
0.06540.05990.07590.07560.07770.0837
0.09480.12070.12880.08630.08590.08600.08310.1393
0.09330.10420.12800.14820.21060.22170.24320.27400.28380.28200.34730.40220.48060.48540.4879mh

0.0276
0.02680.02590.02540.02490.03490.03550.03530.02790.03160.00280.05440.04780.0466
0.04390.03990.04090.06880.05830.06200.06590.05850.04450.0674
0.09380.09000.05240.04610.04790.04230.08050.05740.06170.07020.08130.10420.10510.11210.15300.15850.16190.24120.25330.19660.17610.186216hi

0.9471
0.9496
0.95130.95340.94940.92620.89330.86050.68940.80920.98640.7446
0.6732
0.66400.6594
0.67670.71820.45490.38290.19350.19670.25520.20470.31050.4172
0.39270.17550.14940.1756
0.15520.29600.53290.56330.65130.6591
0.54240.53750.53120.44880.44690.45670.29230.21470.15860.18370.1836

Appendix Table 7 Seven States Ergodic Distribution with Population Weight (Accumulationfrom 1972)
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017

ll

0.00030.00050.00140.00110.00160.00180.00390.00430.00740.00640.00640.00420.00900.00880.01030.01050.01690.04240.04830.05350.05990.0655
0.0611
0.06600.06720.07400.0822
0.0934
0.0936
0.09250.06980.0736
0.06820.06530.06000.05940.05740.0386
0.03760.04020.03890.0396
0.03490.04080.04120.0404lo

0.00280.00400.00810.00800.00880.00890.01700.01730.02750.02260.02140.01330.02700.02510.02450.02340.0372
0.09250.10050.10690.11660.12140.10790.11200.10990.11650.12780.13790.13490.12880.12440.13150.12540.12110.11130.10980.10550.06870.06700.07150.06930.07080.06050.0706
0.07100.0711ml

0.00110.00110.00250.00190.00210.00200.00380.00330.00490.00430.00420.00400.00830.00800.01800.01930.02000.02020.02260.02410.02630.04350.04170.04630.04850.0551
0.06410.07300.07520.07550.07620.07560.07830.07810.07380.07700.07190.04640.04530.04780.04580.04650.04990.05740.05880.0602mi

0.00190.00200.00450.00370.00390.00360.00590.00510.00700.00640.00660.00620.01260.01190.02290.02200.02170.02050.02250.02380.02440.03970.03910.04020.03830.03960.04400.04890.04930.04900.04850.0476
0.04820.0491
0.04680.04780.04700.06140.06310.0756
0.07630.08100.09080.10490.11020.114417mh

0.00840.0081
0.02190.01730.01790.01680.02080.01310.01520.01450.01400.01340.02510.02350.04500.04360.04260.03910.04050.04070.04100.0576
0.05420.06290.0632
0.06170.05980.05690.05650.0552
0.05670.06020.06200.06340.06350.0636
0.06290.08230.08850.10320.1036
0.10830.1111
0.11370.11570.1166hi

0.06850.06720.19110.15230.15650.14680.14100.1032
0.0977
0.09550.09240.09060.16870.15850.14540.14210.13600.12290.11950.11890.11330.10410.10600.10120.0996
0.09540.08970.08410.08310.08090.08310.08740.08750.08900.08720.08580.09330.09980.11640.13530.13790.13910.14040.13380.13710.1378hh

0.91710.91720.77050.81550.80910.82010.80770.85360.84030.85030.85500.86830.74930.76410.73390.73900.72550.66250.6461
0.63210.6186
0.56820.58990.57130.5732
0.55770.53240.50590.50740.5181
0.5412
0.52410.53050.53410.55750.55670.56210.60290.5822
0.5264
0.52830.51470.51230.47890.46600.4595

Appendix Table 8 Seven States Ergodic Distribution with Population Weight (20 Years Moving
Average)
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017

ll

0.00030.00040.00320.00190.00300.00350.01090.01070.03190.01380.02400.01370.03000.03190.05750.07110.11990.19620.31500.33470.35980.36670.24340.74810.42690.4384
0.33610.44850.38550.45740.12200.12700.09580.09620.06900.06370.05740.01480.01180.00340.00220.00450.00180.00240.00130.0006lo

0.00280.00320.01720.01310.01470.01460.03910.03360.08850.05450.05010.02600.05150.04980.0702
0.0664
0.10980.18120.22680.21940.22090.21730.21210.07290.14530.14470.15520.14500.14040.11710.17290.18230.15090.15190.16940.18150.16030.02420.02050.01550.01030.02180.00900.01280.00750.0038ml

0.00110.00090.00540.00320.00360.00330.00930.00670.01670.01020.00810.00740.01580.01590.05780.06680.08500.05690.07270.07350.07670.13380.15480.06290.14980.17400.21420.23220.25990.24920.42720.41390.38440.40880.29730.31370.28230.12070.10290.09710.07030.04710.04990.05590.03170.0157mi0.00190.00160.00950.00620.00640.00560.01300.00930.02100.01450.01120.01160.02450.02500.07650.07670.0676
0.05720.04730.04670.04300.06110.08090.02470.05290.05230.06820.07020.08290.07210.10470.1043
0.09300.11020.10810.11400.11970.23250.23510.29900.31610.25340.31720.39780.45330.462518mh

0.00840.00670.04490.02820.02950.02660.03290.01660.02280.02220.01540.01650.03150.02790.09190.08760.07590.0636
0.04800.03950.03760.0451
0.04270.01720.0416
0.03850.0456
0.04170.0516
0.04140.06890.0806
0.0763
0.09330.10010.10200.10880.21350.24820.30480.31870.29790.32170.2536
0.24290.2476hi

0.06850.05720.28100.17550.18170.16410.15730.10780.0956
0.0997
0.0952
0.09410.17840.15910.12600.11650.09440.07420.04540.04400.03810.02540.02800.00860.01960.01670.01910.01120.01430.01000.01680.01670.01490.01730.02480.02170.03800.05530.08980.10030.11810.16710.14370.14120.14760.1584hh

0.91710.93000.63870.77170.7611
0.78220.73740.81530.72350.78510.79610.83070.66840.69040.52030.51490.44730.3706
0.24480.24210.22390.15060.2381
0.06570.16380.13540.1616
0.05120.06540.05280.08760.07530.18470.12230.23150.20340.2336
0.33890.29170.17990.16420.20820.15670.13630.11570.1113