Growth and Inequalities across Countries: A Data Analysisfabcol.free.fr/pdf/macro3-2.pdf · Growth...

Growth and Inequalities across Countries: A Data Analysis

Growth and Inequalities across Countries:A Data Analysis

F. Collard


Objective

Objective

Very simple question:

“Why are the riches rich and the poors poor?”

Seems to be a very simple question!

Very complicated issue: Economists have been struggling withthis issue since Smith [1776]!!


Objective

Objective

Various ways of approaching the problem

Understand inequalities across individuals within a countryand make international comparisons (largely dealt with in theliterature, ex: Sala–i–Martin)

Understand cross–countries inequalities in growthperformances (less studied but Quah [1997], Jones [1997])


Basic Observations

Basic ObservationsWhat data?

Data: output per worker available in the World–Penn Table6.0 over the period 1960 to 1998.

Output: real GDP at constant world prices =⇒ Avoid priceand exchange rate effects;Labor: # of individuals aged between 15 and 64.

Build log of output per worker.

Restrict to Non Sub–Saharan African countries: Why?

Assess inequalities in output: yi − yus (does this matter?)


Basic Observations

Basic ObservationsHow to account for inequalities?

Think of simple descriptive statistics

Standard deviation:

σt =

√√√√ 1

N

N∑i=1

(yit − y t)2

with y t = 1N

∑Ni=1 yit .

Plot of these quantities


Basic Observations


Standard Deviation

1960 1965 1970 1975 1980 1985 1990 1995 20000.75

0.76

0.77

0.78

0.79

0.8

0.81

Years


Basic Observations


Lorenz Curve

Cumulative % of Total Population

Cum

ula

tive

%ofTota

lO

utp

ut

A

B

Gini Index=B/(A+B)


Basic Observations


Gini Index

1960 1965 1970 1975 1980 1985 1990 1995 20000.33

0.34

0.35

0.36

0.37

0.38

0.39

Years


Basic Observations


Problem: Does not locate where the phenomenon takes place

Why is it important?

In the lower tail: the poors get poorer =⇒ Essentially aproblem of development economicsAround the mean: The gap occurs among developed countries=⇒ Essentially a problem of growth theory

We would like to plot the whole distribution


Basic Observations


How to estimate it? Non parametric methods

Kernel density estimator: Rosenblatt–Parzen kernel densityestimator

f (y) =1

Nh

N∑i=1

K

(y − yi

h

)where the kernel, K (.), is a continuous, bounded, symmetricreal–valued function that integrates to 1.

Gaussian kernel: K (ω) = exp(−ω2/2)/√

2π,

Bandwidth parameter: h = 1.0592σN−1/5 where σ is thestandard deviation of the data and N is the number ofobservations.


Basic Observations


Kernel Estimates of Distribution

−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5196019781998


Basic Observations


Locating the breaking point

Think of more sophisticated (but simple!) descriptive statistics

Percentile differences

Sort the data yiNi=1 and the cdf i/N

Interpolate to find the percentileCompute differences

Plot of the time series


Basic Observations


Percentile differences

1960 1965 1970 1975 1980 1985 1990 19950.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Years

20−8025−7530−70


Basic Observations

Basic ObservationsSummary

Drastic change in the shape of the distribution of (log–)output–per–worker between 1960 and 1998:

In 1960: clearly uni–modal and close to normal (Quah [1997],Jones [1997])Over time, the middle of distribution hollowed–out quitesubstantially;In 1998: Bi–modal distribution (Twin–Peaks)


Standard theory

Standard Theory

What theory to explain cross countries differences in growthperformances?

Standard growth theory!

Extremely powerful theory in general to account for thelong–run

Can it explain what we observed?


Standard theory

Standard Theory

What can the basic Solow growth model say about that?

Very standard and basic framework

Technology: Yi ,t = Kαi ,t(ΩiΓtLi ,t)

1−α with 0 6 α 6 1

Ωi : country specific labor augmenting knowledge,Γt : common time varying component (Γt+1 = (1 + γ)Γt ,Γ0 > 0 given).

Population: Li ,t+1 = (1 + ni )Li ,t , Li ,0 given.

Accumulation: Ki ,t+1 = siYi ,t + (1− δ)Ki ,t , Ki ,0 > 0 given.


Standard theory

Standard Theory

Denote yi ,t = Yi ,t/(ΓtLi ,t) and ki ,t = Ki ,t/(ΓtLi ,t), we have

yi ,t = kαi ,tΩ

1−αi

(1 + γ)(1 + ni )ki ,t+1 = siyi ,t + (1− δ)ki ,t


Standard theory

Standard Theory

Steady State

Along the steady state growth path, we have thatyi = log(yit)− log(y0t) satisfies the following relationship

yi =α

1− ανi

with νi = si(1+ni )(1+γ)−(1−δ) and νi = log(νi )− log(ν0).


Standard theory

Standard Theory

Change of Variable Formula

Let a random variable y be a function of another random variablex with distribution µx(x). y and x are related by the continuouslydifferentiable function f (·), such that y = f (x). Then

µy(y) =µx(x)

|f ′(x)|


Standard theory

Standard Theory

Distribution of Output

The steady state distribution of (log–) output–per–worker relativeto a reference economy (y0), is given by

µy(y) =1− α

αµν (ν)

where µν(·) denotes the distribution of ν.

Hence if ν is unimodal, so is output.


Standard theory

Standard Theory

Distribution of ν

−4 −2 0 20

0.2

0.4

0.6

0.8

1Investment to output ratio

19601978

−0.04 −0.02 0 0.02 0.040

10

20

30

40Population growth

19601978


Standard theory

Standard Theory

The distribution of ν is unimodal, so is that of output,

The Standard Model just Fails to Account for the Facts!

Need to go beyond the standard model

Will use Econometrics to put some structure on the data andgrasp some intuition to setup a model.


Structuring the Data

Putting some Structure on the Data

What can be learnt from Econometrics?

More than what you think!

Econometrics enables the Ceteris Paribus assumption.

Need to setup a strategy!




Methodology borrowed from the Labor literature:Juhn, Murphy and Pierce [JPE, 1993]

Where can changes come from?

Build counterfactuals

Recall that

y98 = y60 + 18×∆y60−78i + 20×∆y78−98

i

Understanding output level amounts to understand the growthprocess!




Split the sample in two periods and estimate a growthregression for each

∆y0i = x0

i β0 + ε0i (1)

∆y1i = x1

i β1 + ε1i (2)

Change in the distribution of variables:

∆yd

i = x0i β1 + ε1

i

Change in the magnitude of these mechanisms:

∆ym

i = x1i β0 + ε1

i

Change in the residual factors:

∆yε

i = x1i β1 + ε1

i

plot the distributions



Putting some Structure on the DataStrategy

1 Find a good way to approach the data (which model to use,need for theory)

2 Find a good representation of the data (which DGP to use)

3 Setup the counterfactual analysis



Putting some Structure on the DataStep 1: Which theory?

Standard Growth theory =⇒ Huge literature on the empiricsof growth;

Use β–convergence type of regressions




Basic idea: Use Solow growth model

k(t)

k(t)= sAk(t)α−1 − (n + γ + δ)

Steady state (k(t) = 0): k? =(

αAn+γ+δ

) 11−α

Log–linear representation:

k(t)

k(t)=s(α− 1)Ak?α−1(log(k(t))− log(k?))

=− s(1− α)(n + γ + δ)(log(k(t))− log(k?))

=− λ(log(k(t))− log(k?))




Solution

(log(k(t))− log(k?)) = e−λt(log(k(0))− log(k?))

orlog(k(t)) = e−λt log(k(0)) + (1− e−λt) log(k?))




Use the production function

log(y(t)) = log(A) + α(e−λt log(k(0)) + (1− e−λt) log(k?))

)= e−λt log(y(0)) + (1− e−λt) log(A) + α(1− e−λt) log(k?)

Finally, one gets

log(y(t))− log(y(0))

t=

(e−λt − 1)

tlog(y(0)) +

(1− e−λt)

t

log(A)

1− α

+α

1− α

(1− e−λt)

t[log(s)− log(n + γ + δ)]




Now Acknowledge heterogeneity

log(y(i , t))− log(y(i , 0))

t=

(e−λt − 1)

tlog(y(i , 0)) +

(1− e−λt)

t

log(Ai )

1− α

+α

1− α

(1− e−λt)

t[log(si )− log(ni + γ + δ)]

Suggest regression of the form

log(yi,t)− log(yi,0)

t= cst+β0 log(y(i , 0))+βi/y log(i/y)i,t+βn log(ni,t)+ui,t




All this work for . . . Nothing!Still need to estimate everything

Go to Step 2



Putting some Structure on the DataStep 2: Which DGP?

What tool?

Econometrics! (Basic Econometrics)

Split the sample into 2: 1960–1978 ; 1978–1998




OLS Estimates:

60− 78 : ∆yi = 0.185(0.025)

− 0.013(0.002)

yi0 − 0.137(0.164)

ni + 0.016(0.003)

(i

y

)i

78− 98 : ∆yi = 0.208(0.026)

− 0.014(0.002)

yi0 − 0.606(0.163)

ni + 0.027(0.004)

(i

y

)i

Test for global stability: 14.64, p–value: 0.002!

Test for convergence stability: 0.03, p–value: 0.855

Test for stability of n and i/y: 12.02, p–value: 0.002!




Should we be happy with that work?

Clearly: NO!

Several problems: weighting of countries, endogeneity biases,specification biases may account for the facts!

Need to work harder!



Putting some Structure on the DataStep 2: Which DGP?The weighting

Weighting scheme: All countries are assigned the sameweight! what about India and China!

Weighted regressions: β solves min(Y − Xβ)′W (Y − Xβ)where W is the weighting matrix.



Putting some Structure on the DataStep 2: Which DGP?The weighting

Weight: log(POP)

60− 78 : ∆yi = 0.161(0.023)

− 0.010(0.002)

yi0 − 0.118(0.173)

ni + 0.018(0.003)

(i

y

)i

78− 98 : ∆yi = 0.213(0.021)

− 0.015(0.002)

yi0 − 0.623(0.161)

ni + 0.025(0.004)

(i

y

)i

Weight: log(Y )

60− 78 : ∆yi = 0.180(0.024)

− 0.013(0.002)

yi0 − 0.133(0.162)

ni + 0.017(0.003)

(i

y

)i

78− 98 : ∆yi = 0.211(0.025)

− 0.014(0.002)

yi0 − 0.618(0.161)

ni + 0.026(0.004)

(i

y

)i



Putting some Structure on the DataStep 2: Which DGP?Endogeneity Biases

Both the investment–output ratio and population growthought to be endogenous

i/y : That’s what optimal growth models suggest!n: That’s what endogenous fertility models suggest!

Instrumental variables: Set of instruments, Z , such that Z isexogenous and highly correlated with he variable we want tocorrect

β = (XPZX ′)−1X ′PZY with PZ = Z (Z ′Z )−1Z ′



Putting some Structure on the DataStep 2: Which DGP?Endogeneity Biases

i/y : initial investment–output ratio

n: rate of growth over the 15 first periods of the sample

IV estimates:

60− 78 : ∆yi = 0.149(0.027)

− 0.011(0.002)

yi0 − 0.203(0.172)

ni + 0.009(0.004)

(i

y

)i

78− 98 : ∆yi = 0.204(0.028)

− 0.014(0.002)

yi0 − 0.638(0.166)

ni + 0.026(0.004)

(i

y

)i



Putting some Structure on the DataStep 2: Which DGP?Specification Biases

Essentially a problem of omitted variables!

Human CapitalNon–linear dynamicsInstitutions

Introduce each type of variable



Putting some Structure on the DataStep 2: Which DGP?Specification Biases: Education

Huge amount of work on that topic: Mankiw, Romer andWeil (1991), Barro and Lee (1993), Klenow andRodrıguez-Clare [1997], Durlauf and Quah (1999)

No clear conclusions! (measurement problem)

Enrollment rate

Use both OLS and IV estimates (education instrumented bythe stock (years))



Putting some Structure on the DataStep 2: Which DGP?Specification Biases: Education

OLS IV1 IV260–78 78–98 60–78 78–98 60–78 78–98

Const. 0.218 0.216 0.197 0.218 0.197 0.249( 0.032) ( 0.036) ( 0.033) ( 0.036) ( 0.036) ( 0.046)

y0 -0.017 -0.014 -0.015 -0.014 -0.015 -0.017( 0.003) ( 0.003) ( 0.003) ( 0.003) ( 0.003) ( 0.004)

n -0.082 -0.710 -0.117 -0.736 -0.117 -0.709( 0.164) ( 0.159) ( 0.171) ( 0.161) ( 0.171) ( 0.164)

i/y 0.014 0.029 0.006 0.030 0.006 0.028( 0.003) ( 0.004) ( 0.004) ( 0.005) ( 0.004) ( 0.005)

Educ. 0.010 -0.002 0.015 -0.004 0.015 0.008( 0.006) ( 0.009) ( 0.006) ( 0.009) ( 0.008) ( 0.014)

Q(Total) 24.275 [0.000] 28.970 [0.000] 28.844 [0.000]Q(y0) 0.342 [0.559] 0.053 [0.818] 0.152 [0.697]Q(n, i/y) 18.934 [0.000] 24.660 [0.000] 19.879 [0.000]Q(Educ.) 1.488 [0.223] 3.100 [0.078] 0.220 [0.639]



Putting some Structure on the DataStep 2: Which DGP?Specification Biases: Nonlinearities

Possibility of nonlinear convergence (Durlauf or Quah (1999))

Include powers of initial output per worker (up to degree 3)=⇒ non significant =⇒ No nonlinearities

Possibility of scale effects

Include powers of initial output and/or Total population=⇒ non significant =⇒ No scale effects



Putting some Structure on the DataStep 2: Which DGP?Specification Biases: Institutions

What is the role of institution?

Can it undo what we’ve found so far?

Include a measure of social infrastructure: Weighted averageof

An index of government anti–diversion policies created byPolitical Risk Services, a firm that specializes in riskassessments to international investors.An index of trade openness complied by Sachs and Warner(1995).

Non significant, does not affect the previous findings




Here, we have not exhausted all the candidates;

But it has been done

What to conclude?




Main Result

Differences in international output growth performances can beessentially accounted for by

The rate of investment in physical capital (or the savings rate)

The rate of population growth

=⇒ Traditional growth factors!

Go to Step 3



Putting some Structure on the DataStep 3: Counterfactual Analysis

Let us recall that we now have

∆y60−78i = β60−78

0 + β60−78y y60

i + X 60−78i β60−78

x + ε60−78i

∆y78−98i = β78−98

0 + β78−98y y78

i + X 78−98i β78−98

x + ε78−98i

It is then easy to build y98 as

y98i = y60

i + 18×∆y60−78i + 20×∆y78−98

i

Then build counterfactuals and associated distributions

This will help us to setup a model



Putting some Structure on the DataStep 3: Counterfactual Analysis: Interpreting the results

No effect of the change!

19601998corr.



Putting some Structure on the DataStep 3: Counterfactual Analysis: Interpreting the results

Big effect of the change!

19601998corr.




Control Convergence:

∆y60−78i = β60−78

0 + β60−78y y60

i + X 60−78i β60−78

x + ε60−78i

∆y78−98i = β78−98

0 + β60−78y y78

i + X 78−98i β78−98

x + ε78−98i

−4 −2 0 2 40

0.1

0.2

0.3

0.4

0.5

(a) Observables (y0)

19601998Corr.

−4 −2 0 2 40

0.1

0.2

0.3

0.4

0.5

(b) Coefficients (β78−98y

=0)




Control for changes in the residual factors:

∆y60−78i = β60−78

0 + β60−78y y60

i + X 60−78i β60−78

x + ε60−78i

∆y78−98i = β78−98

0 + β78−98y y78

i + X 78−98i β78−98

x + ε60−78i

−4 −2 0 2 40

0.1

0.2

0.3

0.4

0.5Unobservables




Control for changes in the distribution of variables:

∆y60−78i = β60−78

0 + β60−78y y60

i + X 60−78i β60−78

x + ε60−78i

∆y78−98i = β78−98

0 + β78−98y y78

i + X 60−78i β78−98

x + ε78−98i

−4 −2 0 2 40

0.1

0.2

0.3

0.4

0.5(c) Distribution (n,i/y)




Control for changes in the importance of variables:

∆y60−78i = β60−78

0 + β60−78y y60

i + X 60−78i β60−78

x + ε60−78i

∆y78−98i = β78−98

0 + β78−98y y78

i + X 78−98i β60−78

x + ε78−98i

−4 −2 0 2 40

0.2

0.4

0.6

0.8Coefficients (n,i/y)




Conclusion

Changes in the distribution of output per worker can be essentiallyaccounted for by an increase in the impact of

The rate of investment in physical capital (or the savings rate)

The rate of population growth

=⇒ Traditional growth factors!




Way more important than you may think!

the coefficient in front of (i/y) carries some information onthe returns to accumulation

Recall

log(y(t))− log(y(0))

t=

(e−λt − 1)

tlog(y(0)) +

(1− e−λt)

t

log(A)

1− α

+α

1− α

(1− e−λt)

t[log(s)− log(n + γ + δ)]



Putting some Structure on the DataUsing all this as a guide for a model

denote β ≡ (e−λt−1)t

∆y = β log(y(0))− βlog(A)

1− α− αβ

1− α[log(s)− log(n + γ + δ)]

implies that α ' 0.4 in the first sample, and α ' 0.65 in thesecond!

Suggest a model of technological change in which thetechnology became more capital intensive

Can we go a little bit further?

Yes!




Something else, very important, happened during that period

Opening of trade

Can it help understanding the data (I swear this is the lasttable!)




Total O60−78 <med(O60−78) O60−78 >med(O60−78)60–78 78–98 60–78 78–98 60–78 78–98

Const. 0.121 0.121 0.096 0.125 0.169 0.126( 0.021) ( 0.022) ( 0.023) ( 0.024) ( 0.035) ( 0.041)

y0 -0.011 -0.014 -0.009 -0.013 -0.015 -0.015( 0.002) ( 0.002) ( 0.003) ( 0.003) ( 0.004) ( 0.004)

K 0.009 0.029 0.017 0.018 0.006 0.041( 0.003) ( 0.004) ( 0.004) ( 0.004) ( 0.005) ( 0.006)

Stab.(Total) 17.846 [0.000] 1.565 [0.457] 48.966 [0.000]Stab.(y0) 0.834 [0.361] 1.453 [0.228] 0.001 [0.971]Stab.(K) 15.647 [0.000] 0.048 [0.827] 27.795 [0.000]

K is log(s/(n + γ + δ)).




Everything took place among open countries

Therefore suggests a model that features

Technological changeGlobalization

Growth and Inequalities across Countries: A Data Analysisfabcol.free.fr/pdf/macro3-2.pdf · Growth...

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