How fat is the top tail of the wealth distribution?
Philip Vermeulen
European Central Bank
DG-Research
Household wealth data and Public Policy, London
9/March/2015
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This slideshow should not be reported as representing the views of the European Central
Bank (ECB). The views expressed are those of the author and do not necessarily reflect
those of the ECB.
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Intro
Wealth distribution and its tail
30 percent of wealth is held by 1 percent of the population (Kennickell, 2009)
Piketty “Capital in the Twenty-First Century”
Stiglitz “The Price of Inequality: How Today’s Divided Society Endangers Our
Future”
analysis of tax and redistribution policies needs detailed distributional information
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Intro
Purpose of the research
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Intro
Purpose of the research
How much wealth is in the tail?
Philip Vermeulen
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9/March/2015
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Intro
Purpose of the research
How much wealth is in the tail?
Does survey data give a good account of tail wealth?
Philip Vermeulen
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9/March/2015
4 / 44
Intro
Purpose of the research
How much wealth is in the tail?
Does survey data give a good account of tail wealth?
Show the importance of differential unit non-response for tail wealth estimation
Philip Vermeulen
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Intro
Purpose of the research
How much wealth is in the tail?
Does survey data give a good account of tail wealth?
Show the importance of differential unit non-response for tail wealth estimation
Improve on the estimation of tail wealth in the presence of differential unit
non-response.
Philip Vermeulen
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9/March/2015
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Intro
Purpose of the research
How much wealth is in the tail?
Does survey data give a good account of tail wealth?
Show the importance of differential unit non-response for tail wealth estimation
Improve on the estimation of tail wealth in the presence of differential unit
non-response.
New estimates of tail wealth: US, UK, Germany, France, Spain, Italy, The
Netherlands, Belgium, Austria, Finland, Portugal
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Intro
Wealth distribution data
Household wealth surveys
US Survey of consumer finances
UK Wealth and Assets survey
European Household Finance and Consumption Survey
Complex survey designs: stratification, clustering, weighting
representative of population
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Intro
Literature: Tail wealth estimation
Wealth tax records: Roine and Waldenström (2009), Alvaredo and Saez (2009), Dell,
Piketty and Saez (2007)
estate tax records: Atkinson and Harrison (1978), Atkinson, Gordon and Harrison
(1989), Kopczuk and Saez (2004), Piketty, Postel-Vinay and Rosenthal (2006)
Capitalization capital income: Atkinson and Harrison (1978), Saez and Zucman
(2014)
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Intro
0
2.0e−06
Density
4.0e−06
6.0e−06
Part of the wealth distribution Germany
0
Philip Vermeulen
200000
400000
600000
wealtheuro
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800000
1000000
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Intro
0
1.0e−06
Density
2.0e−06 3.0e−06
4.0e−06
5.0e−06
Part of the wealth distribution France
0
Philip Vermeulen
200000
400000
600000
wealtheuro
How fat is the top tail
800000
1000000
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Intro
0
1.0e−06
Density
2.0e−06
3.0e−06
4.0e−06
Part of the wealth distribution Italy
0
Philip Vermeulen
200000
400000
600000
wealtheuro
How fat is the top tail
800000
1000000
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Intro
0
1.0e−06
Density
2.0e−06
3.0e−06
Part of the wealth distribution Spain
0
Philip Vermeulen
200000
400000
600000
wealtheuro
How fat is the top tail
800000
1000000
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Intro
The wealth distribution is special
Practically all familiar distributions: inflation, bond returns, stock returns, growth
rates,
“mean” and “variance” that makes sense (central tendency and fluctuations around
it)
wealth distribution in contrast:
an astronomically long tail: roughly twenty thousand times larger than the first 95
percent of the distribution
Has a “mean” and “variance” that don’t make sense.
NEED to sample especially from the tail to estimate its importance
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Intro
Household wealth survey data: two problems - one solution
PROBLEM I = Efficiency: a simple random sample yields an inefficient estimate of
the distribution of wealth
PROBLEM II = BIAS: survey weights that remain uncorrected for differential unit
non-response
SOLUTION : Oversample the wealthy
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Intro
Household wealth survey data: oversampling the rich
very different strategies to oversample wealthy households
US, UK: income tax files
Spain, France: wealth tax files
Finland: income register data
Germany, Belgium: income of regions
Portugal, Austria: metropolitan areas
Italy, Netherlands: No oversampling
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Intro
The effect of different oversampling strategies
Number of wealthy households in the samples > 2 million euro
Absolute Number vs Pct in sample
Sample size
Absolute number
Pct of sample
USA
6482
965
15
Spain
6197
544
9
UK
20165
949
5
France
15006
638
4
Germany
3565
85
2
Italy
7951
78
1
Netherlands
1301
2
0.15
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The data
The extreme tail
The forbes list (billion euro)
USA
Germany
UK
Italy
Spain
France
Austria
Netherlands
Portugal
Belgium
Finland
Philip Vermeulen
Individuals
396
52
37
14
12
11
5
3
2
1
1
Total wealth
978.6
183.3
84.8
46.6
28.3
60.1
13.0
4.8
4.1
1.9
1.0
How fat is the top tail
Pct country wealth
2.3
2.4
0.7
0.7
0.6
0.9
1.2
0.4
0.7
0.1
0.2
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The data
The Household data: the missing end of the tail
USA
Spain
France
UK
Finland
Germany
Belgium
Portugal
Austria
Italy
Netherlands
Philip Vermeulen
MIND THE GAP
Max wealth Survey vs Min wealth at Forbes
Million euros
Max wealth SCF/WAS/HFCS
Min wealth Forbes
806
737
409
780
153
810
92
780
15
958
76
818
8
1920
27
1110
22
1560
26
893
5
958
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The data
Some observations
large gap between the highest wealth household in HFCS sample and the poorest
household in Forbes
suggest combination of non-response and lack of effective oversampling
suggest to estimate the tail of the distribution survey + Forbes data
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Tail distributions
How to estimate tail wealth
Survey Sample only
Directly from the observations in the survey samples
Assume tail is Pareto:(pseudo) maximum likelihood
Assume tail is Pareto: Regression method on survey sample
Survey sample + Forbes
Assuming tail is Pareto: Regression method on survey sample+Forbes data
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Tail distributions
A candidate distribution for the tail
Pareto distribution (Power law)
complementary CDF or tail function
P (W > w) = (
wmin α
)
w
(1)
defined on the interval [wmin , ∞[ and α > 0
α tail index determines the fatness of the tail. The lower α, the fatter the tail.
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Tail distributions
Method 1: Pseudo maximum likelihood
Maximum likelihood is impossible, cannot write the likelihood function
Pseudo maximum likelihood
act ”as if the data were a census”
Q
L(α, wmin | W ) = f (wi )swi
Does not deal with the differential non-response problem
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Tail distributions
An interesting property of the Pareto
complementary CDF or tail function
P (W > w) = (
wmin α
)
w
(2)
n(wi ) the number of households that have wealth above wi (rank of the household)
wmin α
n(wi )
=(
)
N
wi
(3)
log(n(wi )) = log(N ) + αlog(wmin ) − αlog(wi )
log of the rank of the household and log of its wealth are on a straigth line.
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Tail distributions
Method 2: Regression method
log(n(wi )) = constant − αlog(wi )
log of the rank of the household and log of its wealth are on a straight line.
use survey data to construct n(wi )
also use Forbes data to construct n(wi )
αRH
ˆ : survey data only
αRHF
ˆ : survey data + Forbes data
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Tail distributions
Log rank of the household and log wealth (USA)
0
5
10
15
20
USA
10
15
20
25
lnwealtheuro
g_lnr
f_lnr
Graphs by country
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Tail distributions
0
5
10
15
Log rank of the household and log wealth (Germany)
14
16
18
20
lnwealtheuro
ln_rank_hfcs
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22
24
ln_rank_forbes
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Tail distributions
0
5
10
15
Log rank of the household and log wealth (UK)
14
16
18
20
22
24
lnwealtheuro
ln_rank_hfcs
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ln_rank_forbes
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Tail distributions
0
5
10
15
Log rank of the household and log wealth (France)
14
16
18
20
lnwealtheuro
ln_rank_hfcs
Philip Vermeulen
22
24
ln_rank_forbes
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Tail distributions
0
5
10
15
Log rank of the household and log wealth (Italy)
14
16
18
20
lnwealtheuro
ln_rank_hfcs
Philip Vermeulen
22
24
ln_rank_forbes
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Tail distributions
0
5
10
15
Log rank of the household and log wealth (Spain)
14
16
18
20
lnwealtheuro
ln_rank_hfcs
Philip Vermeulen
22
24
ln_rank_forbes
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Tail distributions
0
5
10
15
Log rank of the household and log wealth (Netherlands)
14
16
18
lnwealtheuro
ln_rank_hfcs
Philip Vermeulen
20
22
ln_rank_forbes
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Tail distributions
0
5
10
15
Log rank of the household and log wealth (Belgium)
14
16
18
lnwealtheuro
ln_rank_hfcs
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20
22
ln_rank_forbes
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Tail distributions
0
5
10
15
Log rank of the household and log wealth (Austria)
14
16
18
lnwealtheuro
ln_rank_hfcs
Philip Vermeulen
20
22
ln_rank_forbes
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Tail distributions
The effect of differential unit non-response: Monte Carlo
1 million households draw wealth from Pareto distribution with tail index α
rich list (households more than 740 million euro)
A survey sample is drawn for 750 households:
1) No oversampling: simple random sample
2) Oversampling (4 strata: 75, 150, 225, 300)
non-response probability follows 0.097+0.0365*ln(Wealth)
Estimate tail index using pseudo maximum likelihood, regression method survey,
regression method survey + rich list
twenty thousand Monte Carlo iterations
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Tail distributions
Pareto tail index estimates under differential unit non-response
Monte Carlo estimates of Pareto tail index
TRUE MAX REGR REGR Resp Rich list
α
LIK
+LIST
obs
obs
(1)
(2)
(3)
(4)
(5)
(6)
No Oversampling
1.50
1.61
1.61
1.49
280
50
1.60
1.50
1.60
0.10
0.13
0.03
13
7
1.71
1.71
1.60
281
26
0.10
0.14
0.04
13
5
Oversampling the rich
1.52
1.57
1.50
273
50
0.04
0.11
0.03
13
7
1.62
1.67
1.60
275
26
0.04
0.12
0.04
13
5
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Tail distributions
Tail wealth estimates under differential unit non-response
Monte Carlo estimates of tail wealth
as a proportion of true tail wealth
(1)
(2)
(3)
(4)
(5)
TRUE survey est. MAX REGR
REGR
α
LIK
+ LIST
No Oversampling
1.50
0.87
0.90
0.91
1.01
1.60
1.50
1.60
0.25
0.10
0.15
0.04
0.90
0.92
0.93
1.01
0.15
0.08
0.12
0.04
Oversampling the rich
0.91
0.97
0.94
1.01
0.23
0.05
0.14
0.04
0.93
0.98
0.95
1.00
0.12
0.04
0.11
0.04
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Tail distributions
TAIL WEALTH DISTRIBUTION (MONTE CARLO EXAMPLE)
Monte Carlo: Tail of the wealth distribution
0
10
−1
10
−2
10
−3
P(X>=x)
10
−4
10
Empirical ccdf (Sample)
Empirical ccdf (Rich list)
Regression (SAmple)
Regression (Sample and Rich list)
TRUE power law
−5
10
−6
10
−7
10
0
10
Philip Vermeulen
1
10
2
3
10
10
Wealth (in million euro)
How fat is the top tail
4
10
5
10
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Estimation results
TAIL WEALTH DISTRIBUTION (Germany)
Tail of the wealth distribution
0
10
−1
10
−2
10
−3
P(X>=x)
10
−4
10
Empirical ccdf (Survey)
Empirical ccdf (Forbes)
Regression (survey)
Regression (survey and Forbes)
Pseudo Maxlik(survey)
−5
10
−6
10
−7
10
0
10
Philip Vermeulen
1
10
2
3
10
10
Wealth (in million euro)
How fat is the top tail
4
10
5
10
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Estimation results
TAIL WEALTH DISTRIBUTION (USA)
Tail of the wealth distribution
0
10
−1
10
−2
10
−3
P(X>=x)
10
−4
10
Empirical ccdf (Survey)
Empirical ccdf (Forbes)
Regression (survey)
Regression (survey and Forbes)
Pseudo Maxlik(survey)
−5
10
−6
10
−7
10
0
10
Philip Vermeulen
1
10
2
3
10
10
Wealth (in million euro)
How fat is the top tail
4
10
5
10
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Estimation results
TAIL WEALTH DISTRIBUTION (USA)
Tail of the wealth distribution
0
10
−1
10
−2
10
−3
P(X>=x)
10
−4
10
Empirical ccdf (Survey)
Empirical ccdf (Forbes)
Regression (survey)
Regression (survey and Forbes)
−5
10
Pseudo Maxlik(survey)
−6
10
−7
10
1
10
Philip Vermeulen
2
10
3
10
Wealth (in million euro)
How fat is the top tail
4
10
5
10
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Estimation results
TAIL WEALTH DISTRIBUTION (UK)
Tail of the wealth distribution
0
10
−1
10
−2
10
−3
10
P(X>=x)
−4
10
−5
10
Empirical ccdf (Survey)
Empirical ccdf (Forbes)
Regression (survey)
Regression (survey and Forbes)
−6
10
−7
10
Pseudo Maxlik(survey)
−8
10
−9
10
0
10
Philip Vermeulen
1
10
2
3
10
10
Wealth (in million euro)
How fat is the top tail
4
10
5
10
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Estimation results
TAIL WEALTH DISTRIBUTION (UK)
Tail of the wealth distribution
0
10
−1
10
−2
P(X>=x)
10
−3
10
Empirical ccdf (Survey)
Empirical ccdf (Forbes)
Regression (survey)
Regression (survey and Forbes)
−4
10
Pseudo Maxlik(survey)
−5
10
1
10
Philip Vermeulen
2
10
3
10
Wealth (in million euro)
How fat is the top tail
4
10
5
10
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Estimation results
Average Pareto tail index
Regres
Regres
∆
excl Forbes incl Forbes
countries using tax/income records to oversample
USA
1.59
1.52
0.07
UK
2.05
1.74
0.31
France
1.76
1.62
0.14
Spain
1.77
1.69
0.07
Finland
2.11
1.88
0.23
other or no oversampling
Germany
1.68
1.39
0.29
Austria
1.65
1.46
0.20
Belgium
2.18
1.87
0.31
Portugal
1.45
1.47
-0.02
Italy
2.02
1.58
0.44
Netherlands
5.09
1.48
3.61
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Estimation results
Percentage wealth share of top 1 pct of households
Regres
∆
SURVEY incl Forbes
countries using tax/income records to oversample
USA
34
31-37
-3 to +3
UK
13
14-18
+1 to +5
France
18
19-21
+1 to +3
Spain
15
15-17
+0 to +2
Finland
12
13-15
+1 to +3
other or no oversampling
Germany
24
32-34
+8 to +10
Austria
23
31-32
+8 to +9
Belgium
12
15-16
+3 to +4
Portugal
21
23-27
+2 to +6
Italy
14
20-21
+6 to +7
Netherlands
9
10-19
+1 to +10
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Estimation results
Percentage wealth share of top 5 pct of households
Regres
∆
SURVEY incl Forbes
countries using tax/income records to oversample
USA
61
53-63
-8 to +2
UK
30
31-35
+1 to +5
France
37
38-39
+1 to +2
Spain
31
31-33
+0 to +2
Finland
31
32-33
+1 to +2
other or no oversampling
Germany
46
51-54
+5 to +8
Austria
48
52-54
+4 to +6
Belgium
31
33-34
+2 to +3
Portugal
41
42-45
+1 to +4
Italy
32
37-38
+5 to +6
Netherlands
26
27-36
+1 to +10
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Conclusion
Conclusion
unobserved differential non-response creates substantially downward biased estimates
of tail wealth
oversampling (using tax records) allows to observe and address (partially) differential
non-response
combining Forbes/rich lists tail wealth individuals with survey data improves tail
wealth estimates
Germany, Austria, The Netherlands, Italy have substantially higher tail wealth than
can be derived from the HFCS data only
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