Comparing register and survey wealth data

Fredrik Johansson and Anders Klevmarken

Department of EconomicsUppsala University

An ideal measure?

• The measure on which the decision maker acts!

• Is the survey response such a measure? – Probably not!

• What about the market value of an asset? – Yes, perhaps if there is a well defined market value.

• But, what is the market value of a house which is not put on the market?

Less of nonresponse and measurement errors in register data?

• Complete enumeration – but not always of the target population

• Register data are collected for administrative purposes and not for statistical purposes

• Self-interest in underreporting assets and over reporting liabilities

• But in Sweden: Banks, insurance companies, brokers and housing associations report to the tax authorities for each single individual.

• The market value of real estate is an estimate produced by Statistics Sweden

The estimates of market values of real property in register data

. ;ij j ij ij ijh T h T

Error=estimate – true market value

Table 2. Descriptive statistics for true and estimated market values by property, sales data 2003

Variable Median* Mean* Std dev* Skewness Kurtosis N

Home equity

Estimate962.2 1 240.5

963.9 1.99 5.87 54.25

3

True 942.4 1 222.6

950.9 1.88 5.57 54.25

3

Error27.0 17.9 316.8 -0.74 14.74 54.25

3

Corr(True, Error) -0.125 (<0.001)

What is a measurement error?

• Survey response – ”ideal” value• We will use the market value as of the last of

December 2003 as the ideal value of a financial asset assuming that register data have no measurement errors.

• For real property we recognize that there are measurement errors both in the survey and in the register measures. We will account for the measurement error in register data when ever possible.

Assets included

• Home equity (owner occupied house or condominium)

• Other real estate• Bank holdings• Bonds• Stocks and shares• Mutual funds• Debts

Data sources

• Register data:LINDA 2002 sample size approx. 1.1 million individuals

• Survey data:UU_RAND 2002, sample size 1431 individuals (households) aged 50+, response 893, subsample from LINDA

SHARE_SE 2003, sample size 4700 households, response 2208, at least one household member 50+

Table 3. Descriptive statistics for SHARE_SE and corresponding register data

Variable Median Mean Std Dev N

Home equity

Survey 885 981 1 166 7581 015 617 1,398

Register 808 750 1 022 380 922 035 1,398

Difference 42 943 144 377 771 502 1,398

Other real property

Survey 492 212 889 1401 452 201 645

Register 327 570 711 9101 542 788 645

Difference 98 442 177 2301 103 324 645

Bank

Survey 49 221 130 524 224 029 1,511

Register 49 773 130 198 223 480 1,511

Difference 984 326 139 202 1,511

Bonds

Survey 49 221 84 537 127 470 307

Register 26 025 74 517 145 690 307

Difference -166 10 020 91 416 307

Variable Median Mean Std Dev N

Stocks

Survey 49 221 191 236 344 467 685

Register 34 100 153 676 382 360 685

Difference 4 384 37 560 293 909 685

Mutual funds

Survey 98 442 181 519 266 968 934

Register 109 631 219 918 301 978 934

Difference -5 322 -38 399 229 176 934

Total debt

Survey 246 106 409 761 837 956 731

Register 300 000 440 240 731 472 731

Difference -21 746 -30 479 549 625 731

Total net worth

Survey 821 993 1 251 1621 530 660 1,515

Register 613 865 1 007 8391 432 051 1,515

Difference 70 404 243 3231 095 564 1,515

The very rich.6

.7.8

.91

1.1

Ra

tio

90 92 94 96 98 1002002

Reference Register_response

Survey_response Register_all_sampled

Total net worth

.4.6

.81

1.2

Ra

tio

90 92 94 96 98 1002003

Reference Register_response

Survey_response Register_all_sampled

Total net worth

UU-RAND compared to LINDA 2002 SHARE_SE compared to LINDA 2003

.6.8

11

.21

.4R

atio

90 92 94 96 98 1002003

Reference Register_response

Survey_response Register_all_sampled

Total net worth

Measurement errors and the variance (inequality) of wealth

Var(W)=Var(W*)+Var(u)+2Cov(W*,u)

* ; ( ) 0;

* ; ( ) 0;

uW W u E u

W W E

Asset Var(W) Var(W*) Var(u) ρ(W*,u) S(u)/S(W*)

Own home, corrected 1.03E+12 9.04E+11

4.95E+11 -0.275 0.740

Bank accounts 5.02E+10 4.99E+10

1.94E+10 -0.307 0.623

Bonds1.62E+10 2.12E+10

8.36E+09 -0.501 0.627

Stocks1.19E+11 1.46E+11

8.64E+10 -0.507 0.769

Mutual funds 7.13E+10 9.12E+10

5.25E+10 -0.523 0.759

Debts7.02E+11 5.35E+11

3.02E+11 -0.168 0.751

Table 4. The relative importance of measurement errors in estimating the variance of an asset, by type of asset

Wealth as a dependent variable

0 1

* ;* ;

* ; ( | ) 0;

uW W uW W

W X E X

*1 1

*1 1

1 1

;

;

0;

lim 0;

lim( ) lim( )

uX

X

X

X

uX

For financial assets

For real assets assume p

then p p

Estimated regression slopes with measurement errors in the dependent variable; independent

variable is age

1 1 uXb

Own home -12,654 -8,821 -3,833

Other real estate -5,323 -1,845 -3,478

Bank accounts 1,836 2,007 -171

Bonds 557 931 -374

Stocks 1,556 2,779 -1,223

Mutual funds 2,171 3,203 -1,031

Debts -15,877 -10,968 -4,910

Gross wealth 27,129 21,292 5,837

Net wealth 43,006 32,260 10,747

Estimated regression slopes with measurement errors in the dependent variable; independent

variable if ”healthy”

11

uXbOwn home 71591 105627 -34035

Other real estate 150175 124325 25850

Bank accounts 24508 25349 -841

Bonds 12991 5489 7501

Stocks -6888 -7140 253

Mutual funds 50412 50437 -24

Debts 4049 -41256 45305

Gross wealth 474173 188870 285303

Net wealth 470124 230126 239998

Wealth as explanatory variable

0 1

* ;

* ;

( | *) 0; ( ) 0;

uW W u

Y W

E W E u

2 2* * *

1 1 * 1 2 2 2*

*2*

1ˆ ;

2 ( *, )1 2

uW W W u uW W W

u W uuW

W

b S b S bb b

S S S Cov W ub

S

*1 1 2

*2*

1 limˆlim ;

lim1 2 lim

lim

uW

uuW

W

p bp

p Sp b

p S

2

*2*

1 1

lim0.5 lim 0.5

lim

lim ;

uuW

W

p SIf and p bp S

then p

* *1 1 2

*2*

1 lim limˆlim ;

lim1 2 lim

lim

uW W

uuW

W

p b p bp

p Sp b

p S

If Y is error prone with the additive error ν then,

Estimated regression slopes with measurement errors in the independent variable (gross wealth)

1̂1 *uW

b 2uS

/2

*WSDep. Var.

Own home

0.277(0.013)

0.310(0.011) -0.159

1.20E+12

3.32E+12 0,360

Other real estate

0.521(0.028)

0.634(0.018) -0.544

2.01E+12

5.72E+12 0,352

Bank accounts

0.045(0.003)

0.047(0.003) -0.002

9.70E+11

2.95E+12 0,328

Bonds 0.000(0.004)

0.002(0.004) -0.007

1.69E+12

6.80E+12 0,249

Stocks 0.039(0.006)

0.047(0.006) -0.028

1.76E+12

4.70E+12 0,376

Mutual funds

0.057(0.005)

0.055(0.005) 0.000

1.31E+12

3.89E+12 0,336

Debts 0.264(0.013)

0.254(0.011) -0.101

1.24E+12

4.50E+12 0,275

Health 1,55E-08(9,15E-

09)

1,68E-08(1,03E-

08)-1,19E-01 1,1E+12

2,14E+12 0,505

2

2*

u

W

SS

Multivariate regression with one error prone variable (gross wealth)

,*

*

210

XWY

uWW

.

2 2 2

* * * * *2 2 2* * *

2 2 2

* * * * *2 2 2* * *

ˆ ˆˆ

ˆ ˆ1

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2 ( )

. ;ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ1 2 ( )

YX yW WXyX W

WX XW

u u uyX yW W X yX uW yX W X yu uX yW yu

W W W

u u uXW W X uW Xu W X uX yW yu

W W W

S S S

S S S

S S S

S S S

Regressions of assets (y) on error prone gross wealth (W) and age (X)

Own home 0.268(0.012)

-4682(2271)

0.361(0.012)

-5161(2013)

Other real estate 0.431

(0.025)3012

(6070)0.636

(0.018)4203

(4259)

Bank accounts

0.021(0.002)

2439( 377)

0.023(0.002)

2329( 374)

Bonds 0.000(0.003)

1035( 580)

0.000(0.002)

1036( 578)

Stocks 0.051(0.007)

7172(1516)

0.063(0.007)

6685(1475)

Mutual funds

0.054(0.005)

4904( 949)

0.062(0.005)

4499(926)

Debts 0.056(0.005)

-8244(1157)

0.053(0.005)

-8698(1162)

.yW X .yX W*.yW X . *yX W

Conclusions

• With the exception of the top 1% SHARE_SE does not underestimate the average level of wealth. The survey has rather a tendency to over estimate wealth.

• At the top 1% the underestimate is due to selective nonresponse. Very, very rich people do not participate, while there is no tendency for those who participate to underreport.

• The main problem in the survey is the large error variance and the negative correlation between errors and true values.

• In our data the error variance ranges from almost 40% of the true variance (bank holdings) to almost 60% (stocks).

• The correlation ranges from -0.17 (debts) to -0.52 (mutual funds)

Conclusions cont.

The consequences are:1. No severe overestimates of inequality.2. In regressions with error prone gross wealth as an

explanatory variable the negative bias from the error variance is to a large extent compensated by the negative correlation between error and true value. The survey estimate of the marginal effect of gross wealth appears to have little bias.

3. In regressions with wealth as a dependent variable the correlation between the measurement errors and explanatory variables will bias the slope estimates. The sign of the bias depends on asset and explanatory variable.

Conclusions cont.

• Measurement errors in (our) wealth surveys do not have classical properties.

• Compensating error properties give decent estimates of the inequality of wealth and of the marginal effect of wealth,

• But approximately the right estimates for the wrong reason is a poor consolation!

• We need to learn more to be able to compensate for the effects of errors in survey wealth measures and if possible design surveys such that measurement errors are in controle.

.6.8

11

.21

.4R

atio

50 60 70 80 90 1002003

Reference Register_response

Survey_response Register_all_sampled

Total net worth

SHARE_SE compared to LINDA 2003