Methods for summarizing and comparing wealth distributions

1
Methods for summarizing and comparing wealth
distributions

• Stephen P. Jenkins
• Institute for Social Economic Research
• University of Essex, UK
• and
• Markus Jäntti
• Åbo Akademi University
• Turku, FIN

2
The brief given to us

• Andrea Brandolini
• The main issue is to understand whether the
  standard tools for inequality measurement used
  for income work for wealth as well. For instance,
  the issue of negative values cannot be ignored as
  easily as with income. The Gini can deal with
  negative values, but are we happy?
• Tim Smeeding
• My preference is for a paper that opens
  doors and suggests approaches rather than being
  the last word on a topic.

3
The quick response

• Many of the tools commonly used to summarize
  income distributions can also be applied to
  wealth distributions,
• albeit adapted in order to account for the
  distinctive features of wealth distributions

4
Outline

• What is different about wealth distributions?
• Distribution and density functions
• Lorenz curves (relative, generalized, absolute)
• Inequality indices
• Fitting parametric size distributions
• Concluding remarks
• References
• Appendix 1. Wealth survey data for Finland
• Appendix 2. Practising what we preach Stata code
  to derive the estimates and draw the figures
• Tables 14
• Figures 19

Used for illustrations of methods
Much distributional analysis can be undertaken
using readily-available software
At the end of the paper!
5
Whats included whats left out

• We assume
• unit record (micro) data
• definitional issues resolved (focus on W G D)
• cross-sectional distribution(s)
• We do not consider
• methods for analysis of joint distributions, e.g.
  
• longitudinal dynamics
• income and wealth
• gross wealth and debt
• statistical inference

6
What is different about wealth distributions?

• Support
• G gt 0 D gt 0 W can take values lt 0, 0, gt 0
• Cf. income, for which distributional analysis
  tools developed, and for which typically assumed
  that Y gt 0
• Problems for applications to wealth
  distributions?
• Amiel, Cowell, Polovin (Economica, 1996) make
  persuasive case that assumptions of monotonicity
  and transfer principle should also apply when
  values of the variable of interest are negative
• (e.g.) mean-preserving spread reduces social
  welfare even if Wi lt 0, Wj lt 0, and µW lt 0
• e.g. non-crossing Lorenz curves have standard
  interpretation

7
What is different about wealth distributions?
(ctd.)

• Concentration of density mass, especially spike
  at zero (cf. income distributions), and
  relatively high concentration close to zero
• Right-skewed with long sparse tails (more so than
  income distributions)
• Related
• Non-trivial prevalence of extreme values
  either dirt (cf. Cowell Victoria-Feser), or
  high leverage
• estimates sensitive to treatment of extreme
  values?

8
Illustrations Finnish wealth survey data

• Unit household
• Gross wealth (G), Debt (D), Net wealth (W) are
  expressed in 2000 international
• Not equivalized
• Survey data for 1994 (n 5,210), 1998 (n
  3,893)
• All calculations used the sampling weights
• See Appendix 1 and Jäntti (1994) for more details
  and analysis. Focus here on net wealth, W

9
Table 1. Summary statistics, net wealth
10
Boxplots for net wealth
Extreme values are manifest!
Upper adjacent value p75 1.5IQR
p75 top of box p25 bottom
Lower adjacent value p25 ? 1.5IQR
Increase in mean raw data 28 trimming top
and bottom 1 24
11
Distribution function (Pens parade) and quantiles
Distribution function (CDF) and quantile function
are well defined

• Giants really are tall relative to dwarfs
• Upside down households at start of parade
• Positive heights only after p gt 10
• Anti-clockwise twist over time ? inequality rose?

Pens Parade graph of W against F(W) p See
Table 2 for quantile estimates.
12
Probability density functions

• PDF provides more detail about the wealth of the
  dwarfs comprising the majority of the
  population
• Histograms have well-known problems kernel
  density estimates often preferred nowadays
• Kernel density estimation methods to account for
  right skewness and sparse tail
• estimate in transformed metric (e.g. logs), and
  back transform problems with zero and negatives
  ? use other transformations e.g. inverse
  hyperbolic sine?
• Adaptive kernel density estimates, but issues of
  how to estimate spikes given smoothing

13
Histogram, 1000 bins
Both methods provide similar pictures about
Finnish net wealth distributions, as it happens
Adaptive kernel density estimates, Epanechnikov
kernel, 1000 data points
14
Lorenz curves and inequality

• Slope of LC at p F(W) is W/µW
• LC below horizontal axis (negative slope) for W lt
  0, given µW gt 0
• If µW lt 0, LC flipped vertically relative to
  usual case
• Finland single-crossing of LCs (later year from
  above) ? 1998 more unequal than 1994 for
  transfer-sensitive standard relative inequality
  measures iff CV1998 gt CV1994 (as happened)

Share of poorest 40 2.3 (1994)
2.9 (1998) Share of richest 10 35.5 (1994)
38.2 (1998)
15
Alternative representational devices

• If µW 0, LC is undefined if µW ? 0, numerical
  instability ?

• (a) Summarize social welfare rather than
  inequality generalized Lorenz curves
• slope at p F(W) is W (so may lie below
  horizontal axis)
• GL ordinate is µW when p 1 (may be negative)

GLCs for Finland cross no unambiguous ordering
according to SWFs satisfying monotonicity,
principle of transfers
16
Alternatives (b) absolute Lorenz curves

• (b) summarize inequality using absolute rather
  than relative measures absolute Lorenz curves
• ALC(p) p(µW(p) µW)
• cf. LC(p) pµW(p)/µW
• slope at p F(W) is W µW
• flipped vertically if µW ltlt 0
• Finland ALC1998 lies everywhere below ALC1994 ?
  1998 more unequal than 1994 for all standard
  absolute inequality measures

• NB both GLC and ALC are in units of W
• ? getting price deflator and PPP rate
• especially important

17
Inequality indices

• Many standard aggregative inequality measures
  are undefined for negative incomes, and a
  substantial class of these measures will not work
  even for zero incomes, in the sense that they are
  either undefined, or are unbounded, or attain
  their maximum value at any income distribution
  that has one or more zero incomes. (Amiel,
  Cowell, Polovin, 1996, p. S65.)
• Good news some standard relative inequality
  indices can be calculated
• Bad news many are particularly sensitive to
  extreme values! E.g. CV, GE(2)
• Gini well-defined (but may be gt 1)
• CV, Gini are negative if µW lt 0 undefined if µW
  0

18
Absolute inequality indices

• For example,
• absolute Gini (meanGini)
• Kolm indices K(?)
• Measures are not unit-free (price deflators, PPPs
  crucial)
• Kolm indices relatively unfamiliar ? more
  difficult to benchmark and interpret results for
  different ?
• Marginal social value of W is ?W, so if ?
  1/µW, then elasticity of marginal valuation of W
  equals 1 at the mean (Atkinson Brandolini,
  2004)
• but which mean should one use for comparisons?

19
Inequality of net wealth in Finland(Table 3,
extract)
Increase in inequality of net wealth for
the standard inequality indices, both
relative and absolute ? as expected from Lorenz
curve results
20
Index sensitivity regarding extreme values?
21
Fitting parametric size distributions

• Single-parameter Pareto distribution commonly
  fitted (characterises distribution above some
  lower bound W0 gt 0)
• straightforward to fit, especially given data
  available
• simple expressions for moments and inequality for
  distribution of W gt W0 in terms of shape
  parameter ?, and W0
• expressions for distribution as a whole rather
  more complicated, however (Atkinson Harrison
  1978),
• ... so why not fit models to the distribution as
  a whole?
• Parametric models for income distributions less
  relevant, given W may be zero or negative
• Few suitable candidates at present for wealth
  distributions?
• 3-parameter (displaced) lognormal is
  problematic to fit, and shape not necessarily
  appropriate
• finite mixture models much more promising

22
Dagum-III finite mixture model

• Characterization (Dagum 1990)
• Exponential distribution (one parameter) to
  characterise negative wealth values
• Discrete probability mass point at zero
• Dagum I (Burr 3) distribution for positive values
  (3 parameters scale plus 2 shape parameters)
• Table 4 estimates for Finland in 1994, 1998
• only estimates we know of, other than Dagums
  (1990) for Italy
• model fits well (though under-estimates µWW gt
  0)
• Could explore alternative characterizations
• Parameters can be made functions of covariates
  (Appendix 2)
• possibilities for cross-country or cross-time
  decompositions of wealth differences in
  differences in parameters and differences in
  distributions of characteristics (cf. DiNardo et
  al., using kernel density estimation)

23
Concluding remarks

• Many of the tools commonly used to summarize
  income distributions can also be applied to
  wealth distributions, albeit adapted in order to
  account for the distinctive features of wealth
  distributions
• Our illustrations with Finnish data have
  supported this case (and others can use our code)
• But remember the caveats concerning issues not
  addressed! For example,
• treatment of extreme values
• summarizing joint distributions
• statistical inference