transition matrices matrix-based mobility measures … · transition matrices matrix-based mobility...

Transition matricesMatrix-based mobility measures

Other mobility measuresReferences

Measuring mobility

Austin Nichols

July 31 2014

Austin Nichols Measuring mobility



Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation

Topics

I transition matrices

I matrix-based mobility measures

I other mobility measures





Focus

We will examine various means of measuring mobility with a focus oneconomic mobility of individuals over time primarily due to changes in income

But most of these measures can be applied in many areas and are used forexample to measure changes in measured teacher quality in educationresearch or any changing states such as marital status or occupation disabilityor morbidity or changing prices or market shares

Many (but not all) of commonly used measures rely on an estimated transitionmatrix so letrsquos start there





Transition matrices

The usual setup for a transition matrix is to measure status s at time t minus 1 andagain at time t then to estimate the matrix M1 (the one denotes a one-perioddelta)

st = M1stminus1

in which case each column of M1 sums to one or sometimes its transpose

sTt = sTtminus1MT1

in which case each row of MT1 sums to one

s may measure for example which fifth (or half or hundredth) of the incomedistribution a panel survey respondent falls into in one year and then whichfifth they fall in the next year





Alternative transition matrices

One could also measure which fifth of the income distribution a respondentfalls into then which fifth their child appears in 30 years later(intergenerational mobility)

Or we could measure which fifth of a different economic status distribution arespondent falls into eg a measure of educational attainment then whichfifth of the income distribution their child appears in 30 years later (in whichcase we do not have s on both sides of that equation)

Or states defined by an absolute measure such as the US poverty line (unlikethe traditional European model for measuring poverty the US cutoff is definedby a theoretical lower bound budget adjusted only for measured inflation) Thiscan define multiple states as well we can look at who is poornonpoor in eachperiod but we can also look at below the poverty line [12) times the povertyline [23) times the poverty line etc





One (related) alternative to transition matrices

We could measure at which point in the overall distribution at time t + 1 (orgeneration t + 1) each conditional quantile reaches calculating quantilesconditional on starting points this is similar in spirit to a transition matrix

Can do this nonparametrically with a series of kernel-weighted quantileregressions as in Nichols and Favreault (2009)





Nichols and Favreault (2009)

02

46

81

0 20 40 60 80 100

Rank Sum Parents Years of Education

Avg earnings position in birth cohort1935-39

02

46

81

0 20 40 60 80 100



02

46

81

0 20 40 60 80 100



02

46

81

0 20 40 60 80 100







One alternative to transition matrices

That paper also critiques the common ldquointergenerational elasticityrdquo measurewhich regresses log income of the child on log income of the parent (note thatwe could do this for income of a person at two points in time as well tomeasure ldquointragenerational elasticityrdquo)

Making sure the nonparametric quantile regressions satisfy some basicadding-up constraints is no easy matter and is not even attempted in Nicholsand Favreault (2009)





Quintile transition matrices

The canonical example defines st so it measures fifths of an incomedistribution so M1 in

st = M1stminus1

is a quintile transition matrix (all the same theory applies to any quantiletransition matrix but 5 categories seems to be the optimal number for ourlimited attention) In that case s = (02 02 02 02 02)T in every period andM1 must be bistochastic ie rows and columns must sum to one

Now there is no information in s ie the ldquomiddle classrdquo cannot grow or shrinkif it is always 60 percent of the population and we can devote our attentionexclusively to the properties of M1





Displaying transition matrices

We donrsquot need to show the numbers in a table of course transition bar chartscommon in reports





Displaying comparisons of transition matrices





Markov matrices

Much of the appeal of transition matrices arises from the idea that we can usethem to describe longer-run dynamics In particular if the probability a personwinds up in a particular row of st depends only on which row of stminus1 they arein then M1 is a Markov matrix and we can describe the probability they windup in a particular row at time t + 1

st+1 = M1st = M1M1stminus1 = M21 stminus1

or time t + k

st+k = Mk1 st

so we donrsquot need to compute two-period transition matrix M2 or thethree-period transition matrix M3 or measure any longer transitions





Markov failures

Unfortunately it rarely is the case that the data satisfies the Markov assumptionthat history does not matter for transition rates even at the first comparisonpoint of the two-period transition matrix M2 (recall the subscript describes thedelta in time periods whereas the superscript denotes the power ie M2 is theproduct of M1 and M1 the square of single-period transition matrices)

st+1 = M2stminus1 6= M21 stminus1

but sometimes it is possible to expand the space over which the states in s aremeasured and get a transition matrix that comes close to satisfying the Markovassumption





Markov failures

For example instead of measuring poornonpoor in one period and trying topredict poornonpoor in the next period and all future periods we can measurepoornonpoor in two periods and try to predict the next two periods and allfuture periods (instead of two proportions in s now there are 4)

The meaning of the matrix changes in that case of course but it allowslong-run projections if Markov assumptions are satisfied





Transition estimation

Stata has a command xttrans for measuring transition rates but it does notrespect the panel structure properly tabulate works fine though and svytabulate allows weights cluster-robust standard errors and tests of hypotheses

webuse nlswork clear

keep if inlist(year707172737778)

egen m=median(ln_wage) by(year)

gen above=ln_wagegtm if ln_wagelt

gen lastyr=labove

bysort idcode (year) gen wrong=above[_n-1]

gen nextyr=fabove

tab above nextyr nofreq row

tab lastyr above nofreq row

tab wrong above nofreq row

xttrans above





xttrans


| above

lastyr | 0 1 | Total

-----------+----------------------+----------

0 | 8004 1996 | 10000

1 | 1285 8715 | 10000

-----------+----------------------+----------

Total | 4479 5521 | 10000


| above

wrong | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000

xttrans above

| above

above | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000





Quantile transition matrix estimation

Of course tabulate does not know that the resulting matrix is supposed to bebistochastic in the previous example and so it predicts 45 percent will be belowthe median next period and 55 percent above which cannot happen

This error can be due to a mass of people right at quantile breaks (if a lot ofpeople are right at the median wage) and tie-breaking rules or an unbalancedpanel






The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)

One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights

A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems





A quick and dirty adjustment




egen m2=median(ln_wage) if lln_wagelt by(year)


gen above2=ln_wagegtm2 if ln_wagelt

gen nextyr=fabove

gen nexty2=fabove2

tab above nextyr if year==70 nofreq row

tab above nexty2 if year==70 nofreq row





A quick and dirty adjustment cont


| nextyr

above | 0 1 | Total

-----------+----------------------+----------

0 | 7394 2606 | 10000

1 | 1341 8659 | 10000

-----------+----------------------+----------

Total | 4167 5833 | 10000


| nexty2

above | 0 1 | Total

-----------+----------------------+----------

0 | 8192 1808 | 10000

1 | 2240 7760 | 10000

-----------+----------------------+----------

Total | 5019 4981 | 10000






One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)




Transition matricesMobility measures

Matrix properties

Wersquove estimated a matrix now what Is the society highly mobile or not

Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions

Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo





Mobility statistics for transition matrices

Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation

Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix





Mobility measures

Letrsquos denote each of the commonly used measures with a single letter

I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)

I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)

I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)

I M Mean crossing measure the sum over i and j (from 1 to m) of Mij

times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)

All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix




Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading

Other mobility measures

Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R

Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M





Nichols (2008 2010) measure

The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality

The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data





Nichols measure cont

Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility

If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework

yit = ui + ri t + eit





Nichols (2008) graphic

Variation inGrowth Rates Variation in

Mean Incomes

Variation around trend0

1020

3040

5060

7080

Inco

me

-4 -3 -2 -1 0 1 2 3 4 5

TimeAustin Nichols Measuring mobility




Comparing across time and space

This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time






0

05

1

15

2

25

3

R

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year

United States CanadaGreat Britain Germany

R=Aggregate risk

0

05

1

15

2

25

3

I

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


I=Long-run inequality






0

01

02

03

V

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


V=Variability around trend

0

01

02

03

M

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


M=Mobility risk





Mobility measures

Adding to our list






I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)

I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)





Comparing measures

All of these measure different concepts of mobility and will rate mobility in thesame data differently

For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures

That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)





Comparing measures for 30 countries

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M

M

M

M

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T

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T

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T

T

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T

T

T

T

T

T

T

T

T

T

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D

D

D

D

D

D

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D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada

BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep

IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece

SlovakiaEstoniaPoland

United StatesLithuania

LatviaKorea

-3 -2 -1 0 1 2 3Standardized measure

M=Mobility risk T=Trace

D=Determinant R=Ratio

E=Eigenvalue C=Mean crossings





Pro-poor growth

There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)

FranceSwitzerland

ItalyAustria

DenmarkGermany

Great BritainUnited States

SpainAustralia

LuxembourgCanadaSwedenFinland

BelgiumGreeceEstonia

KoreaPortugal

IrelandIcelandNorway

-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes





See also

The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at

I httpppedorg

I httpwwwurbanorgeconomyEconomic-Insecuritycfm

I httpwwwurbanorginequality

I httpwwwurbanorgeconomicmobility




Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17

Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press

Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis

Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394

Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer

Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site




Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008

Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67

Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128

Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97

Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559

Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102




Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541

Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94

Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11

Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute

Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116




Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93

Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24

Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625

Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211

Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85


Transition matrices

Transition matrices

Alternatives

Quantile transition matrices

Markov matrices

Estimation

Matrix-based mobility measures

Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates

Comparing across measures

Further reading

References




Topics

I transition matrices

I matrix-based mobility measures

I other mobility measures





Focus








Transition matrices


st = M1stminus1


sTt = sTtminus1MT1























02

46

81

0 20 40 60 80 100



02

46

81

0 20 40 60 80 100



02

46

81

0 20 40 60 80 100



02

46

81

0 20 40 60 80 100
















st = M1stminus1


















Markov matrices



or time t + k

st+k = Mk1 st






Markov failures








Markov failures













gen lastyr=labove


gen nextyr=fabove




xttrans above





xttrans


| above


-----------+----------------------+----------

0 | 8004 1996 | 10000

1 | 1285 8715 | 10000

-----------+----------------------+----------

Total | 4479 5521 | 10000


| above

wrong | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000

xttrans above

| above

above | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000



























gen nextyr=fabove

gen nexty2=fabove2









| nextyr

above | 0 1 | Total

-----------+----------------------+----------

0 | 7394 2606 | 10000

1 | 1341 8659 | 10000

-----------+----------------------+----------

Total | 4167 5833 | 10000


| nexty2

above | 0 1 | Total

-----------+----------------------+----------

0 | 8192 1808 | 10000

1 | 2240 7760 | 10000

-----------+----------------------+----------

Total | 5019 4981 | 10000











Matrix properties















Mobility measures




































Mean Incomes


1020

3040

5060

7080

Inco

me

-4 -3 -2 -1 0 1 2 3 4 5












0

05

1

15

2

25

3

R

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


R=Aggregate risk

0

05

1

15

2

25

3

I

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year








0

01

02

03

V

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year



0

01

02

03

M

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


M=Mobility risk





Mobility measures

Adding to our list












Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References




Focus








Transition matrices


st = M1stminus1


sTt = sTtminus1MT1























02

46

81

0 20 40 60 80 100



02

46

81

0 20 40 60 80 100



02

46

81

0 20 40 60 80 100



02

46

81

0 20 40 60 80 100
















st = M1stminus1


















Markov matrices



or time t + k

st+k = Mk1 st






Markov failures








Markov failures













gen lastyr=labove


gen nextyr=fabove




xttrans above





xttrans


| above


-----------+----------------------+----------

0 | 8004 1996 | 10000

1 | 1285 8715 | 10000

-----------+----------------------+----------

Total | 4479 5521 | 10000


| above

wrong | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000

xttrans above

| above

above | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000



























gen nextyr=fabove

gen nexty2=fabove2









| nextyr

above | 0 1 | Total

-----------+----------------------+----------

0 | 7394 2606 | 10000

1 | 1341 8659 | 10000

-----------+----------------------+----------

Total | 4167 5833 | 10000


| nexty2

above | 0 1 | Total

-----------+----------------------+----------

0 | 8192 1808 | 10000

1 | 2240 7760 | 10000

-----------+----------------------+----------

Total | 5019 4981 | 10000











Matrix properties















Mobility measures




































Mean Incomes


1020

3040

5060

7080

Inco

me

-4 -3 -2 -1 0 1 2 3 4 5












0

05

1

15

2

25

3

R

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


R=Aggregate risk

0

05

1

15

2

25

3

I

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year








0

01

02

03

V

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year



0

01

02

03

M

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


M=Mobility risk





Mobility measures

Adding to our list












Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

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D

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E

E

E

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E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References




Transition matrices


st = M1stminus1


sTt = sTtminus1MT1























02

46

81

0 20 40 60 80 100



02

46

81

0 20 40 60 80 100



02

46

81

0 20 40 60 80 100



02

46

81

0 20 40 60 80 100
















st = M1stminus1


















Markov matrices



or time t + k

st+k = Mk1 st






Markov failures








Markov failures













gen lastyr=labove


gen nextyr=fabove




xttrans above





xttrans


| above


-----------+----------------------+----------

0 | 8004 1996 | 10000

1 | 1285 8715 | 10000

-----------+----------------------+----------

Total | 4479 5521 | 10000


| above

wrong | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000

xttrans above

| above

above | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000



























gen nextyr=fabove

gen nexty2=fabove2









| nextyr

above | 0 1 | Total

-----------+----------------------+----------

0 | 7394 2606 | 10000

1 | 1341 8659 | 10000

-----------+----------------------+----------

Total | 4167 5833 | 10000


| nexty2

above | 0 1 | Total

-----------+----------------------+----------

0 | 8192 1808 | 10000

1 | 2240 7760 | 10000

-----------+----------------------+----------

Total | 5019 4981 | 10000











Matrix properties















Mobility measures




































Mean Incomes


1020

3040

5060

7080

Inco

me

-4 -3 -2 -1 0 1 2 3 4 5












0

05

1

15

2

25

3

R

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


R=Aggregate risk

0

05

1

15

2

25

3

I

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year








0

01

02

03

V

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year



0

01

02

03

M

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


M=Mobility risk





Mobility measures

Adding to our list












Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References




















02

46

81

0 20 40 60 80 100



02

46

81

0 20 40 60 80 100



02

46

81

0 20 40 60 80 100



02

46

81

0 20 40 60 80 100
















st = M1stminus1


















Markov matrices



or time t + k

st+k = Mk1 st






Markov failures








Markov failures













gen lastyr=labove


gen nextyr=fabove




xttrans above





xttrans


| above


-----------+----------------------+----------

0 | 8004 1996 | 10000

1 | 1285 8715 | 10000

-----------+----------------------+----------

Total | 4479 5521 | 10000


| above

wrong | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000

xttrans above

| above

above | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000



























gen nextyr=fabove

gen nexty2=fabove2









| nextyr

above | 0 1 | Total

-----------+----------------------+----------

0 | 7394 2606 | 10000

1 | 1341 8659 | 10000

-----------+----------------------+----------

Total | 4167 5833 | 10000


| nexty2

above | 0 1 | Total

-----------+----------------------+----------

0 | 8192 1808 | 10000

1 | 2240 7760 | 10000

-----------+----------------------+----------

Total | 5019 4981 | 10000











Matrix properties















Mobility measures




































Mean Incomes


1020

3040

5060

7080

Inco

me

-4 -3 -2 -1 0 1 2 3 4 5












0

05

1

15

2

25

3

R

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


R=Aggregate risk

0

05

1

15

2

25

3

I

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year








0

01

02

03

V

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year



0

01

02

03

M

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


M=Mobility risk





Mobility measures

Adding to our list












Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References













st = M1stminus1


















Markov matrices



or time t + k

st+k = Mk1 st






Markov failures








Markov failures













gen lastyr=labove


gen nextyr=fabove




xttrans above





xttrans


| above


-----------+----------------------+----------

0 | 8004 1996 | 10000

1 | 1285 8715 | 10000

-----------+----------------------+----------

Total | 4479 5521 | 10000


| above

wrong | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000

xttrans above

| above

above | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000



























gen nextyr=fabove

gen nexty2=fabove2









| nextyr

above | 0 1 | Total

-----------+----------------------+----------

0 | 7394 2606 | 10000

1 | 1341 8659 | 10000

-----------+----------------------+----------

Total | 4167 5833 | 10000


| nexty2

above | 0 1 | Total

-----------+----------------------+----------

0 | 8192 1808 | 10000

1 | 2240 7760 | 10000

-----------+----------------------+----------

Total | 5019 4981 | 10000











Matrix properties















Mobility measures




































Mean Incomes


1020

3040

5060

7080

Inco

me

-4 -3 -2 -1 0 1 2 3 4 5












0

05

1

15

2

25

3

R

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


R=Aggregate risk

0

05

1

15

2

25

3

I

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year








0

01

02

03

V

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year



0

01

02

03

M

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


M=Mobility risk





Mobility measures

Adding to our list












Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References















Markov matrices



or time t + k

st+k = Mk1 st






Markov failures








Markov failures













gen lastyr=labove


gen nextyr=fabove




xttrans above





xttrans


| above


-----------+----------------------+----------

0 | 8004 1996 | 10000

1 | 1285 8715 | 10000

-----------+----------------------+----------

Total | 4479 5521 | 10000


| above

wrong | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000

xttrans above

| above

above | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000



























gen nextyr=fabove

gen nexty2=fabove2









| nextyr

above | 0 1 | Total

-----------+----------------------+----------

0 | 7394 2606 | 10000

1 | 1341 8659 | 10000

-----------+----------------------+----------

Total | 4167 5833 | 10000


| nexty2

above | 0 1 | Total

-----------+----------------------+----------

0 | 8192 1808 | 10000

1 | 2240 7760 | 10000

-----------+----------------------+----------

Total | 5019 4981 | 10000











Matrix properties















Mobility measures




































Mean Incomes


1020

3040

5060

7080

Inco

me

-4 -3 -2 -1 0 1 2 3 4 5












0

05

1

15

2

25

3

R

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


R=Aggregate risk

0

05

1

15

2

25

3

I

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year








0

01

02

03

V

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year



0

01

02

03

M

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


M=Mobility risk





Mobility measures

Adding to our list












Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References




Markov failures








Markov failures













gen lastyr=labove


gen nextyr=fabove




xttrans above





xttrans


| above


-----------+----------------------+----------

0 | 8004 1996 | 10000

1 | 1285 8715 | 10000

-----------+----------------------+----------

Total | 4479 5521 | 10000


| above

wrong | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000

xttrans above

| above

above | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000



























gen nextyr=fabove

gen nexty2=fabove2









| nextyr

above | 0 1 | Total

-----------+----------------------+----------

0 | 7394 2606 | 10000

1 | 1341 8659 | 10000

-----------+----------------------+----------

Total | 4167 5833 | 10000


| nexty2

above | 0 1 | Total

-----------+----------------------+----------

0 | 8192 1808 | 10000

1 | 2240 7760 | 10000

-----------+----------------------+----------

Total | 5019 4981 | 10000











Matrix properties















Mobility measures




































Mean Incomes


1020

3040

5060

7080

Inco

me

-4 -3 -2 -1 0 1 2 3 4 5












0

05

1

15

2

25

3

R

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


R=Aggregate risk

0

05

1

15

2

25

3

I

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year








0

01

02

03

V

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year



0

01

02

03

M

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


M=Mobility risk





Mobility measures

Adding to our list












Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References




Markov failures













gen lastyr=labove


gen nextyr=fabove




xttrans above





xttrans


| above


-----------+----------------------+----------

0 | 8004 1996 | 10000

1 | 1285 8715 | 10000

-----------+----------------------+----------

Total | 4479 5521 | 10000


| above

wrong | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000

xttrans above

| above

above | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000



























gen nextyr=fabove

gen nexty2=fabove2









| nextyr

above | 0 1 | Total

-----------+----------------------+----------

0 | 7394 2606 | 10000

1 | 1341 8659 | 10000

-----------+----------------------+----------

Total | 4167 5833 | 10000


| nexty2

above | 0 1 | Total

-----------+----------------------+----------

0 | 8192 1808 | 10000

1 | 2240 7760 | 10000

-----------+----------------------+----------

Total | 5019 4981 | 10000











Matrix properties















Mobility measures




































Mean Incomes


1020

3040

5060

7080

Inco

me

-4 -3 -2 -1 0 1 2 3 4 5












0

05

1

15

2

25

3

R

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


R=Aggregate risk

0

05

1

15

2

25

3

I

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year








0

01

02

03

V

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year



0

01

02

03

M

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


M=Mobility risk





Mobility measures

Adding to our list












Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References










gen lastyr=labove


gen nextyr=fabove




xttrans above





xttrans


| above


-----------+----------------------+----------

0 | 8004 1996 | 10000

1 | 1285 8715 | 10000

-----------+----------------------+----------

Total | 4479 5521 | 10000


| above

wrong | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000

xttrans above

| above

above | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000



























gen nextyr=fabove

gen nexty2=fabove2









| nextyr

above | 0 1 | Total

-----------+----------------------+----------

0 | 7394 2606 | 10000

1 | 1341 8659 | 10000

-----------+----------------------+----------

Total | 4167 5833 | 10000


| nexty2

above | 0 1 | Total

-----------+----------------------+----------

0 | 8192 1808 | 10000

1 | 2240 7760 | 10000

-----------+----------------------+----------

Total | 5019 4981 | 10000











Matrix properties















Mobility measures




































Mean Incomes


1020

3040

5060

7080

Inco

me

-4 -3 -2 -1 0 1 2 3 4 5












0

05

1

15

2

25

3

R

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


R=Aggregate risk

0

05

1

15

2

25

3

I

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year








0

01

02

03

V

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year



0

01

02

03

M

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


M=Mobility risk





Mobility measures

Adding to our list












Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References




xttrans


| above


-----------+----------------------+----------

0 | 8004 1996 | 10000

1 | 1285 8715 | 10000

-----------+----------------------+----------

Total | 4479 5521 | 10000


| above

wrong | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000

xttrans above

| above

above | 0 1 | Total

-----------+----------------------+----------

0 | 7833 2167 | 10000

1 | 1714 8286 | 10000

-----------+----------------------+----------

Total | 4741 5259 | 10000



























gen nextyr=fabove

gen nexty2=fabove2









| nextyr

above | 0 1 | Total

-----------+----------------------+----------

0 | 7394 2606 | 10000

1 | 1341 8659 | 10000

-----------+----------------------+----------

Total | 4167 5833 | 10000


| nexty2

above | 0 1 | Total

-----------+----------------------+----------

0 | 8192 1808 | 10000

1 | 2240 7760 | 10000

-----------+----------------------+----------

Total | 5019 4981 | 10000











Matrix properties















Mobility measures




































Mean Incomes


1020

3040

5060

7080

Inco

me

-4 -3 -2 -1 0 1 2 3 4 5












0

05

1

15

2

25

3

R

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


R=Aggregate risk

0

05

1

15

2

25

3

I

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year








0

01

02

03

V

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year



0

01

02

03

M

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


M=Mobility risk





Mobility measures

Adding to our list












Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References


























gen nextyr=fabove

gen nexty2=fabove2









| nextyr

above | 0 1 | Total

-----------+----------------------+----------

0 | 7394 2606 | 10000

1 | 1341 8659 | 10000

-----------+----------------------+----------

Total | 4167 5833 | 10000


| nexty2

above | 0 1 | Total

-----------+----------------------+----------

0 | 8192 1808 | 10000

1 | 2240 7760 | 10000

-----------+----------------------+----------

Total | 5019 4981 | 10000











Matrix properties















Mobility measures




































Mean Incomes


1020

3040

5060

7080

Inco

me

-4 -3 -2 -1 0 1 2 3 4 5












0

05

1

15

2

25

3

R

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


R=Aggregate risk

0

05

1

15

2

25

3

I

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year








0

01

02

03

V

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year



0

01

02

03

M

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


M=Mobility risk





Mobility measures

Adding to our list












Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References






| nextyr

above | 0 1 | Total

-----------+----------------------+----------

0 | 7394 2606 | 10000

1 | 1341 8659 | 10000

-----------+----------------------+----------

Total | 4167 5833 | 10000


| nexty2

above | 0 1 | Total

-----------+----------------------+----------

0 | 8192 1808 | 10000

1 | 2240 7760 | 10000

-----------+----------------------+----------

Total | 5019 4981 | 10000











Matrix properties















Mobility measures




































Mean Incomes


1020

3040

5060

7080

Inco

me

-4 -3 -2 -1 0 1 2 3 4 5












0

05

1

15

2

25

3

R

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


R=Aggregate risk

0

05

1

15

2

25

3

I

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year








0

01

02

03

V

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year



0

01

02

03

M

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


M=Mobility risk





Mobility measures

Adding to our list












Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References










Matrix properties















Mobility measures




































Mean Incomes


1020

3040

5060

7080

Inco

me

-4 -3 -2 -1 0 1 2 3 4 5












0

05

1

15

2

25

3

R

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


R=Aggregate risk

0

05

1

15

2

25

3

I

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year








0

01

02

03

V

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year



0

01

02

03

M

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


M=Mobility risk





Mobility measures

Adding to our list












Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References











Mobility measures




































Mean Incomes


1020

3040

5060

7080

Inco

me

-4 -3 -2 -1 0 1 2 3 4 5












0

05

1

15

2

25

3

R

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


R=Aggregate risk

0

05

1

15

2

25

3

I

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year








0

01

02

03

V

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year



0

01

02

03

M

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


M=Mobility risk





Mobility measures

Adding to our list












Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References




























Mean Incomes


1020

3040

5060

7080

Inco

me

-4 -3 -2 -1 0 1 2 3 4 5












0

05

1

15

2

25

3

R

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


R=Aggregate risk

0

05

1

15

2

25

3

I

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year








0

01

02

03

V

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year



0

01

02

03

M

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


M=Mobility risk





Mobility measures

Adding to our list












Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References











0

05

1

15

2

25

3

R

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


R=Aggregate risk

0

05

1

15

2

25

3

I

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year








0

01

02

03

V

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year



0

01

02

03

M

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


M=Mobility risk





Mobility measures

Adding to our list












Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References





0

01

02

03

V

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year



0

01

02

03

M

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Year


M=Mobility risk





Mobility measures

Adding to our list












Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References




Mobility measures

Adding to our list












Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References




Comparing measures









M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References





M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

SwedenDenmark

FinlandPortugal

LuxembourgNorway

GermanyCanada


IcelandHungary

AustriaAustralia

CyprusItaly

FranceIreland

SpainGreece



LatviaKorea









Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References




Pro-poor growth


FranceSwitzerland

ItalyAustria

DenmarkGermany


SpainAustralia



KoreaPortugal







See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References




See also


I httpppedorg







































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References



































Transition matrices

Transition matrices

Alternatives


Markov matrices

Estimation


Transition matrices

Mobility measures


Other measures

GE(2) measure

GE(2) estimates


Further reading

References

transition matrices matrix-based mobility measures … · transition matrices matrix-based mobility...

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