Empirics on Convergence

8/6/2019 Empirics on Convergence

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Empirics on Convergence Name: Seunghoon Ko, ID:3379941

Introduction

The purpose of this paper is to examine the empirics on convergence of the economy,

that is the convergence of Real GDP per capita across countries. While there are many factors

that determine growth rate, this paper looks at the factors that play a key role in determining

growth rate or factors that are considered as important. Such factors could be found in

macroeconomic theory, for example Solow, RCK, Barro, and Romer model looks at

Technology, Capital (both Human and Physical), Labour, Savings rate. Focus was on the

convergence (conditional convergence) of the growth rate where the finding was that poor

countries grow faster than rich countries hence the growth rate also depends on the initial

state as well as steady state of the country, theoretically under the assumption that all the

production function of each countries are the same. While review of the literature of above

mentioned theories could be found Section 1. It is worthwhile noting here that assumptions

made in the macroeconomic theory are questionable, for example unrealistic assumption of

same production function across countries.

One of the examples of the questionable assumptions as stated above is that the

countries have the identical production function, if one assumes this is true then to find the

empirical convergence of Real GDP per capita one would use the single cross country

regression however if production functions differ across countries then we would have an

omitted variable bias (omitted regressor in disturbance term is correlated with included

regressors), even if the production functions across countries are the same there would be

other unobservable factors that determine the growth rate which differ across countries and

hence again causing omitted variable bias. Such bias could be fixed to certain degree by


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using Instrumental variable or panel data approach. However instrumental variable method is

very hard to implement as one need to find instruments that satisfy exogeneity (to

disturbances) and relevance (to regressors that are being instrumented). Hence in this paper

Panel Data approach is taken to find growth empirics however to see the effectiveness of

panel data approach, problems in cross sectional regression is also looked at. My contribution

would be to use latest data available as all the literatures are very old using data before 1980s

and usage of the different method which is panel data method.

This paper is organized as follows;

1. Literature review2. Panel data approach to convergence (my contribution)3. Conclusion and Limitations


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Literature review

To measure growth rate from the macroeconomic theory which focuses on the fact

that at the steady state GDP per capita grows at a constant rate which could not be confirmed

because we would never know whether countries are at their steady state or not. However

from Solow model this is intuitively right because of the property of diminishing marginal

returns(DMR) (of inputs), that is if capital has a characteristic of DMR then economy

converges hence if we can find that property of DMR exists then we could conclude that the

economy converges as existence of DMR of capital means DMR of income per capita (or

equivalently output) as growth rate of income per capita isfunction of capital per capita.

Hence if negative correlation between initial levels of income and subsequent growth rates

could be seen then one can say that economy converges.

Baumol(1986) reported finding convergence. But this is under assumption that there

is no country-specific-effect even if it is true that capital has a property of DMR, this does

not mean output converges unless there exists no country specific effects that determine

output along with capital differently across countries.

The model used by Baumol(1986) and findings were;

Which implies almost perfect convergence.

I have replicated his model using current data (using data ranging from 1970 to 2005)

and obtained similar result, but more I look at this method more I think it makes no sense in

econometric terms, first of all in order to obtain growth rate that is of I neededto do and this is a dependent variable of our regression but the


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regressor is which is already included in the dependent variable hence there is omittedvariable bias. Despite this main finding was that the higher a countrys initial productivity

level the slower the growth rate. Below are the graph and regression results of selected

successful countries (as regards to growth) using up to date dataset. In this literature I have

not explained the variables in detail as it will be done in my empirical work;

. reg lngrowth ratep, vce(robust)

Linear regression Number of obs = 7F( 1, 5) = 8.26Prob > F = 0.0348R-squared = 0.6184Root MSE = .12356

------------------------------------------------------------------------------

| Robustlngrowth | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------ratep | -.3576176 .1244166 -2.87 0.035 -.6774408 -.0377945_cons | 3.852694 .2537628 15.18 0.000 3.200376 4.505012

------------------------------------------------------------------------------

As could be seen the regression using different time periods produce different

coefficient. Rebelo (1991) who used different time periods even found that coefficient of

above regression to be zero or even positive.

0

0.5

1

1.5

2

2.5

0 10 20 30 40

Growth rate %

Initial GDP per working hour (000)

Growth rate and initial GDP

Australia

France

Germany

Italy

Japan

Sweden


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These discrepancies arise from assuming no country specific effects that is in

econometric terms no fixed effects or random effects and also omitting the very important

variable called initial human capital among others. Also another flaw in this model is that the

country is selected in such a way to acquire desired outcome, in econometric terms there is a

selection bias.

To account for human capital, Barro (1989) has controlled for these omitted variable

bias by including initial human capital and found negative correlation. Also Barro has

eliminated selection bias by including all countries (except for ones which does not have

adequate dataset), below is the regression (somewhat simplified) of Barros but using latest

available data.

. reg gr7005 lny0 human, vce(robust)


------------------------------------------------------------------------------| Robust

gr7005 | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------

lny0 | -.004833 .0027742 -1.74 0.085 -.0103361 .0006702human | .0628622 .0197107 3.19 0.002 .0237615 .1019629_cons | .0467984 .0215834 2.17 0.032 .0039827 .0896141

------------------------------------------------------------------------------

As could be seen the human capital is very significant. But there are apparently 124

unobservable regressors that significantly explain GDP growth rate. However in this paper

we are interested in the validity of macroeconomic theory, which is we want to use

theoretically proven factors that could explain the existence of convergence of GDP as

opposed to what factors explain the growth itself. However the omitted variable in the error

term is nevertheless a problem such problem could be explicitly seen in MRK empirical work.


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Findings of Barro(1989) implied that Solow and RCK implied not absolute

convergence but conditional convergence where each country reach their steady states which

are different from each other in our simplified form of replication of Barros work.

(Intuitively Solow models assumption of similar attributes between countries implies

adjustment for the difference across countries have been taken into account but in Barros

empirical work this adjustment has not yet made.)

The bottom line is that all of the above cross sectional regression requires significant

factors influencing growth to be included in the model however it is not possible to do so as

many factors are not observable. One could argue that IV could be used but as mentioned

before instrument is very hard to find. Hence findings from Barro, Rebelo, and Baumol

cannot be justified in econometric sense.

Such problem could be seen explicitly in MRW empirical work ignoring human

capital for a moment or consider human capital as contained in unobservable . LetsAssume that Technology is the same across the countries and this seems reasonable to

assume if we view technology as public good, that is if one country invents production

increasing technology another country would buy or at least find a matching technology to

remain competitive. (Note however this does not mean that

is the same across countries because it encompasseslabour augmenting factors other than technology such as the culture, weather, and many

country specific production augmenting factors which does not vary over time at least not

significantly.)

Below is the derivation of Macroeconomic theory in such a way to see the problem in

implementing econometric methods;


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Starting from Labour augmenting technology progress (Macroeconomic assumption

of production process of a country Solows work);

Then assuming countries are at their steady states we could substitute

into the

equation above and taking a log of substituted equation we get;

Since different countries have different savings rate, and growth of labour force. We

could write above equation into more econometric friendly terms;


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Note this is not a panel data because of the assumption that we are at the steady state,

the saving rate and population growth rate is constant over time and only differ across

countries. However we could see from the dataset that it is not constant hence we average

saving and population growth rate over time so that we could do the regression which is close

to the theory, this regression is mainly to show why the method used by Baumol, and Barro

is arguably wrong, mainly because of the behaviour of unobservable term which variesover individuals, and leaving more sounding panel data approach to later sections.

Note, A(0) are unobservable and if it were constant across countries, that is no

individual effects, it would not have mattered but intuitively it is not constant as, mentioned

above, it reflects not only technology but also resource endowment and things that augment

labour which are different across countries hence we would have heteroskedastic error terms

which could be accounted for by robust standard error or transform the variable so ultimately

doing GLS but what is troublesome is the fact that some of these unobservable are correlated

with regressors (s and n).

Note is a constant term in cross sectional model, that is for a given t, for examplesuch given t in MRW is average of 2 time periods so variables were averaged over time, we

regress log growth rate of income per capita for each country on savings and labour growth

rate of that country.

Results were allegedly quite successful in explaining a large fraction of the cross

country variations in income though , capital elasticity of output, is unrealistically high. But

in my opinionthe model is completely wrong ( not independent or at least correlated with


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regressors) findings of parameters in a misspecified model are most likely, if not always,

would be wrong. Another problematic assumption is that countries are at their steady states

which MRW account for by log linearizing around the steady state which means we will be

assuming that we are near the steady state as opposed to at the steady state.

As reader would have noticed the dependent variable is not a growth rate which we

would want to regress on the initial GDP per capita to see whether the coefficient is negative

(which implies convergence). Above MRW approach is to show that omitted variable bias

exists. That is the regression done by Baumol, and Barro has the form;

What if we assume that we are not at the steady state and let variables to vary across

time and individuals then we could use panel data approach to eliminate country specific

effect, but if we assume that we would not have been able to obtain the above model as

assumption of, that is the steady state value of capital where variable is not varying overtime, is necessary to obtain above model. But Mankiw, Romer, and Weil (MRW) found a

way to account for the fact that the economy may not be in the steady state and relaxes this

assumption by looking at the behaviour at the vicinity of steady state as opposed to at the

steady state. Below is the work of MRW using current available data.

MRW first looked at the behaviour of the economy in vicinity of steady state, i.e.

linearly approximating income per effective labour around steady state (Macroeconomic

methods), where , where we get;


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Hence;

Subtracting from both sides we get;

Substituting in we get; (note s and n differ across countries)

Similarly if we include human capital that is then production function would look like;

And log linearizing around the steady state i.e. following the steps employed above

we would end up with;


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We should note that above specification is per effective labour terms that is . But in MRWs work, they have used output per labour due to unobservable A(t).

This method I believe is wrong due to measurement error in variables and the reason

for linearizing around steady state is to account for unobservable A(0) which caused omitted

variable bias as mentioned in previous example. But here A(0) is still causing the problem

now through measurement errors in variables which ultimately causes similar problem as

omitted variable bias.

Despite this MRW regressed using heteroskedastic robust method. Below is the

replication of their work using latest available data.

Description of variables ;

1) Lny7005 : ln(GDP per capita 2005)ln(GDP per capita 1970)2) Lny_0 : ln(GDP per capita 1970)3) Lnngd : ln(average population growth rate from 1970 to 2005 + g + d), where g+d

= Technology growth rate + depreciation rate = assumed to be .05

4) Lns7005: ln(average saving rate from 1970 to 2005)5) Lnhuman: attendance % of secondary school of working population.


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Regression results;

. reg lny7005 lny_0, vce(robust)


------------------------------------------------------------------------------| Robust

lny7005 | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------

lny_0 | .0279914 .0744417 0.38 0.708 -.1196633 .1756461

_cons | .3073197 .6427392 0.48 0.634 -.9675504 1.58219------------------------------------------------------------------------------

. reg lny7005 lnngd lny_0 lns7005, vce(robust)


------------------------------------------------------------------------------| Robust

lny7005 | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------

lnngd | -1.387302 .4469528 -3.10 0.002 -2.274044 -.5005607lny_0 | -.2231771 .0680494 -3.28 0.001 -.3581852 -.0881691

lns7005 | .5875024 .1499033 3.92 0.000 .2900985 .8849063_cons | -.3852019 1.26639 -0.30 0.762 -2.897683 2.12728

------------------------------------------------------------------------------

. reg lny7005 lnhuman lnngd lny_0 lns7005, vce(robust)


------------------------------------------------------------------------------

| Robustlny7005 | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------lnhuman | .289698 .1190042 2.43 0.017 .0535678 .5258281lnngd | -.9610455 .4708914 -2.04 0.044 -1.895396 -.0266949lny_0 | -.3361484 .0778304 -4.32 0.000 -.4905808 -.1817159

lns7005 | .4776744 .1630971 2.93 0.004 .1540545 .8012944_cons | .72775 1.309137 0.56 0.580 -1.869863 3.325363

------------------------------------------------------------------------------


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MRW reported that when we only regress growth rate on initial value we see no

convergence (first regression result) but in second regression results we see that it converges

when we adjust for savings, and population growth rate. And the third regression which

includes human capital additional to savings and population growth rate shows even more

significant convergence.

In summary in this literature review we see that the econometric methods used in past

literatures seem problematic mainly due to omitted variable bias and measurement errors. To

account for these problems one would need to find other ways to empirically prove

convergence. Other possible ways would be to;

1) Find all significant regressors. (100s of them)2) Find instrumental variables. (I dont see how)3) Reparameterize the equation so that we could use panel data approach where we

could eliminate or account for country specific factors.

In the next section of this paper we look at the third option which turns out we could

eliminate problems by using Panel Data approach.


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Panel Data approach.

This part is my formal contribution to this paper which is similar to Islam (1995) but

focusing on convergence and a bit of alterations.

Here we build on MRW empirical work where the specification was;

The problem was that the variable are in per effective labour termswhich we could not measure and instead just used per labour terms that is instead of;

We have just used;

This caused measurement error bias.

We can actually get by;


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Then using we get;

Substituting this into MRW framework we get;

In our MRW framework we have used cross sectional data by letting . We alter this framework by using five year time intervals, that isobtain the data in 5 year intervals ranging from 1970 to 2000. Ive chose to use 5 year

intervals because if 1 year interval is used then short term disturbances may loom large in

such brief time spans [Islam (1995)]. Thus saving and population growth rates (which are

explanatory variables) are averaged over 5 years. And as a consequence the error terms are

now five years apart hence may be thought to be less influenced by business cycle

fluctuations and less likely to be serially correlated than they would be in a yearly data setup.

We ending up with 7 time points then we could rewrite the above cross sectional framework

into the panel data framework and obtain;


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In econometric friendly terms;

Where;

Datasets of above variables are obtained from;

Penn world table

Barro- Lee website

World bank


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Now that the model is specified we could do rigorous panel data procedures to

account for country specific effects.

We will be looking at;

1) Pooled OLS estimation2) Fixed Effect estimation3) Random Effect estimation4) Hausman Test

1) Pooled OLSIn Pooled OLS we are just doing OLS ignoring the panel feature. We are basically

assuming that there is no individual effect that is We should note that POLS is consistent if is uncorrelated with regressors. And

efficient if

for all i and

is white noise. Also that If

is nonzero but uncorrelated

with regressors, RE is better.

Below is the result of POLS;

. reg lng_it y_initial lnh_it lnn_itgd lns_it, vce(robust)


------------------------------------------------------------------------------| Robust

lng_it | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------

y_initial | -.0233163 .0156138 -1.49 0.136 -.05397 .0073374

lnh_it | .0286706 .0122637 2.34 0.020 .0045939 .0527473

lnn_itgd | .0138115 .0395443 0.35 0.727 -.0638238 .0914468lns_it | .0654242 .0150738 4.34 0.000 .0358305 .0950178_cons | .3486285 .1310444 2.66 0.008 .0913554 .6059016

------------------------------------------------------------------------------


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We should note that the results does not show, at least statistically at 10% level,

conditional convergence

Clearly this POLS is not a method we would like to use as we know from the

macroeconomic theory that there exists individual effects which means POLS above is

inconsistent if individual effects is Fixed effect, and consistent but has wrong standard error

and inefficient if the individual effect is Random effect, either way POLS is the least

preferred method hence we will be working with assumption of RE or/and FE rather than

POLS.

2) Fixed effects estimation.In within group estimation we assume the country specific effect is fixed effect.

In fixed effects estimation can be correlated with regressors. This is because weeliminate (together with any time invariant regressors) by the within-group transformation.For example;

where y_i is the average over time, etc.

Hence this estimator is consistent whether or not is correlated with as termis eliminated.


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Below is the result of fixed effects estimation;

. xtreg lng_it y_initial lnh_it lnn_itgd lns_it, fe

Fixed-effects (within) regression Number of obs = 728Group variable: id Number of groups = 104

R-sq: within = 0.1426 Obs per group: min = 7between = 0.1239 avg = 7.0overall = 0.0069 max = 7

F(4,620) = 25.79corr(u_i, Xb) = -0.9108 Prob > F = 0.0000

------------------------------------------------------------------------------lng_it | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------y_initial | -.2137626 .0222973 -9.59 0.000 -.25755 -.1699751

lnh_it | .0210071 .0140227 1.50 0.135 -.0065306 .0485449lnn_itgd | .0653934 .0337861 1.94 0.053 -.0009557 .1317425lns_it | -.0190566 .0219526 -0.87 0.386 -.062167 .0240538_cons | 1.994534 .2002841 9.96 0.000 1.601217 2.387851

-------------+----------------------------------------------------------------sigma_u | .27615923sigma_e | .16614842

rho | .73423024 (fraction of variance due to u_i)------------------------------------------------------------------------------F test that all u_i=0: F(103, 620) = 2.81 Prob > F = 0.0000

The result shows clear convergence significant at 1% level.

This proves that there exists individual effect.

3) Random effects estimation.In random effects estimation we assume that is uncorrelated with regressors. And

we do feasible GLS (efficient estimation) based on the variance formula for . Herewe assume that individual effect is like randomly distributed within the cross-section ofthe individual population. If we observe the entire population, a priori the fixed effect

assumption is valid however we do not observe the entire population mainly due to

unavailability of datasets. Random effects estimation is consistent and efficient if isuncorrelated with regressors and is iid across i and t and inconsistent if is correlatedwith regressors.


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Below is the result of the Random effects estimation;

. xtreg lng_it y_initial lnh_it lnn_itgd lns_it, re

Random-effects GLS regression Number of obs = 728Group variable: id Number of groups = 104

R-sq: within = 0.0140 Obs per group: min = 7between = 0.1779 avg = 7.0overall = 0.0546 max = 7

Random effects u_i ~ Gaussian Wald chi2(4) = 29.86corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000

------------------------------------------------------------------------------lng_it | Coef. Std. Err. z P>|z| [95% Conf. Interval]

-------------+----------------------------------------------------------------y_initial | -.0332879 .0109639 -3.04 0.002 -.0547766 -.0117991

lnh_it | .0330096 .0109213 3.02 0.003 .0116042 .054415lnn_itgd | .0297319 .0316838 0.94 0.348 -.0323672 .0918309lns_it | .0606473 .0145949 4.16 0.000 .0320418 .0892528_cons | .4574249 .1172364 3.90 0.000 .2276459 .687204

-------------+----------------------------------------------------------------sigma_u | .05477756sigma_e | .16614842

rho | .09803938 (fraction of variance due to u_i)------------------------------------------------------------------------------

Result shows that there exists convergence but at a lower rate than that of FE

estimation.

The question is which estimation method is better. That is, is FE estimation method

better than RE estimation method? Intuitively we would think FE estimation method is more

consistent with macroeconomic theory.

We can test this by using Hausman test which we do below;


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4) Hausman testIn Hausman test we compare FE estimator and RE estimator. We use the fact that if

RE is true then both are consistent but if FE is true then RE estimator is not consistent.

So by setting the hypothesis;

Then if we could reject this hypothesis this would mean that RE model is not valid.

Below is the result of Hausman test;

. hausman FE RE, sigmamore

---- Coefficients ----| (b) (B) (b-B) sqrt(diag(V_b-V_B))| FE RE Difference S.E.

-------------+----------------------------------------------------------------y_initial | -.2137626 -.0332879 -.1804747 .0211786

lnh_it | .0210071 .0330096 -.0120025 .0102794lnn_itgd | .0653934 .0297319 .0356615 .017377

lns_it | -.0190566 .0606473 -.079704 .0183923

------------------------------------------------------------------------------b = consistent under Ho and Ha; obtained from xtregB = inconsistent under Ha, efficient under Ho; obtained from xtreg

Test: Ho: difference in coefficients not systematic

chi2(4) = (b-B)'[(V_b-V_B)^(-1)](b-B)= 94.48

Prob>chi2 = 0.0000

As could be seen from the result we reject null hypothesis which means that RE

model is not valid.

Hence the best method I believe out of all the models weve seen including literature

reviews to prove convergence using Solow macroeconomic model is to use FE model.


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Conclusions and Limitations.

The results of FE estimation shows slower rate of convergence than that of MRW

work. And since I believe this FE estimation is better than MRW we could conclude that

countries do converge conditional on saving, population and human capital level but at a

slower rate than previously seen in literatures.

The limitation of FE estimation as well as all other estimations was that we are only

using few of many variables that determine convergence and also we only take Solow

seriously and disregard any other theories on convergence for example endogenous theory

which was looked at by Romer. Therefore all the empirical work could be thought of as IF

Solow model is perfect representation of the production procedure of the country then

conditional on saving rates, population growth rates, and human capital levels that is

countries with same saving, population growth rate, and human capital levels would converge

to the same income per capita value in the long run regardless of the initial level of output per

capita.

This limitation in econometric terms could be seen from the behaviour of whichwe assumed in our model to be white noise i.e. not correlated between individuals and

through time is questionable. But all the econometric methods used assumes this regardless,

which could cause problems.

Hence the further studies could be done on empirics on convergence where one could

use Panel data with Instrumental variable approach, which in effect would account for

problems we have with .


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Reference

Nazrul Islam Growth Empirics: A Panel Data Approach The MIT Press, Vol. 110.

No.4 (1995)

Barro, Robert J., Economic Growth in a Cross Section of Countries, The Quarterly

Journal of Economics, Vol. 106, No.2 (1991)

Baumol, William J., Productivity Growth, Convergence and Welfare: What the

Long-Run Data Show?American Economic Review, LXXVI (1986), 1072-85.

Solow, Robert M., A Contribution to the theory of Economic Growth, Quarterly

Journal of Economics, LXX (1956), 65-94

Rebelo, Sergio, Long-Run Policy Analysis and Long Run Growth,Journal of

Political Economy, XCIX(1991), 500-21

Mankiw, N. Gregory, David Romer, and David Weil, A Contribution to the Empirics

of Economic Growth, Quarterly Journal of Economics, CVII(1992), 407-37

ECON 711 Notes, The University of Auckland, 2010




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Empirics on Convergence