MEASURES OF GOODNESS OF FIT 

FOR EXTRAPOLATIONS: 

INITIAL RESULTS USING THE INDIVIDUAL 

TAX MODEL DATABASE 

Robert E. Gillette 

Office of Tax Analysis

U.S. Treasuiy Department 

OTA Paper 62 February 1989 

OTA Papers and Briefs are circulated so that the preliminary findings of tax research conductecl by

staff members and others associated with the Office of Tax Analysis may reach a wider audience. The 

views expressed are those of the authors. and do not reflect Treasury policy. Comments are invited,

but OTA Paper and Briefs should not be quoted without permission from the authors. Aclclitional 

copies of this publication may be purchased from the National Technical Information Service 5285 

Port Royal Road. Springfield. VA 22 I 161. Phone (703) 487-4660. 

Office of Tax Analysis 

U.S. Treasury Department. Room 4040 

Washington. DC 2022 
ABSTRACT 

Recent experiences during Tax Reform sug est that estimates of the impact of various tax proposals 

are quite sensitive to the extrapolation oP the tax data to future years. In this paper various 

aspects of the extrapolation process are examined. We discuss several alternative extrapolation

methods and describe the extrapolation procedure currently used by the Office of Tax Analysis. A 

set of statistics for evaluating the accuracy of an extrapolation are developed and then used to 

compare several extrapolations of the I98 I Individual Tax Model Database to 1983 levels. 
MEASURES OF GOODNESS OF FIT FOR EXTRAPOLATIONS: 

INITIAL RESULTS USING THE INDIVIDUAL TAX MODEL DATABASE 

Contents 

Page 

I. Introduction 1 

11. Methods of Extrapolation 2 

111. Measures of Goodness of Fit 4 

IV. Extrapolating the Tax Model Database 10 

V. Conclusions 19 

Endnotes 21 

References 22 

Tables: 

1. Percent’Error in the Prediction of the Total Value 12 

2. Inforination Gain in the Predicted Distribution 13 

Figures: 

1. Multiplicative Variance Decoinposition Variance 

Coinpoiient Attributed 15 to Extrapolation 

2. Multiplicative Variance Decoiiiposition Variance 

Component Attached 17 to Data Items 

3. Multiplicative Variance Decomposition Variance 

Coinpoiieiit Attached 18 to AGI Class 
MEASURES OF GOODNESS OF FIT 

FOR EXTRAPOLATIONS: 

INITIAL RESULTS USING THE INDIVIDUAL 

TAX MODEL DATABASE 

I. INTRODUCTION 

During the course of tax refoim. there was a great deal of interest in how revenue estimates for 

future tax years were obtained. The estimated tax impact of many of the tax provisions considered 

during tax reform were found to be sensitive to the extrapolation of current and prior year tax 

return data to reflect the years being considered in the reform proposal. Thus. the accuracy of the 

extrapolation process used in conjunction with the Treasury's Office of Tax Analysis (OTA) 

Individual Tax Model Database has become increasingly important (for a description of the Tax Model, 

see Wyscai-ver and Cilke [ 19871). 

The purpose of this paper is to present some initial results of a study of the extrapolation of 

databases used in microeconomic simulations. The eventual goal is to develop an improved 

extrapolation procedure that can generate a five year panel file for both analytic and revenue use. 

The initial results. however, are more limited. The results presented in this paper are based on a 

set of test extrapolations of the Treasury's 198 I Indiviclual Tax Model Database to I983 levels. The 

year 1983 was chosen as the target year for the initial extrapolation test since it was the most 

recent year for which extensive IRS Statistics of Income (SO]) information was available when the 

project began. 

The body of the paper is divided into four sections. In the first section of the paper. we review 

the various techniques that have been used to extrapolate tax databases. In the second section. we 

present some statistical measures that may be used to examine the success or 'fit' of an 

extrapolation. In the third section. we discuss the extrapolations macle to acljust or 'age' the 

Individual Tax Model Database to 1983 levels. Finally in the fourth section. we discuss some of the 

conclusions that can be drawn from these initial extypolation tests and suggest some areas for 

future research. 

-I-
-2-

11. METHODS OF EXTRAPOLATION 

In general. there are two approaches by which a tax database can be extrapolated. The database can 

be reweighted so that the individual variables on each tax return are adjusted together in order 

to achieve some set of aggregate and/or distributional targets. Reweighting changes the number of 

tax retums represented by each return in the database. AIternatively. the individual variables on 

the database can be adjusted separately to match a set of targets. Each variable can be multiplied 

by some constant adjustment factor for all the returns in the database. 

Most of OTA's extrapolation procedures. including the one used for most recent extrapolations 

(i.e.. the extrapolation creating the tax reform database in 1985) have emphasized reweighting each 

record as the best procedure for adjusting the database. For example, in the extrapolation from 

1981 to 1983. a single growth factor (the Consumer Price Index) was applied to all income items 

except the itemized deductions (which were adjusted separately). All of the other adjustments to 

the database were made by re-weighting the individual records. For a more complete description of 

the current extrapolation procedures used by OTA. see Wyscarver and Cilke [ 19871. 

Extrapolation procedures that emphasize the reweighting approach derive from a concern for 

preserving the actual information reported on a taxpayer's return. Maintaining the integrity of a 

retum is important since there exist significant correlations between the various items reported on 

a return that should be preserved. Therefore. each retum should be treated as a unit. rather than 

as a group of independent items. 

In order to maintain the relationships of the variahles within a given return. any change in the 

levels reported on a return should only be adjusted through the application of a uniform growth 

factor to all of the items on the return. Of course. the use of a uniform growth rate does not 

allow the composition (i-e., the relative mix) of the various items on the database as a whole to 

change. Adjustment of the composition of the database must be done through modification of the 

weights on the various returns found on the database. 

Lindsey ( I985), however, argues that extrapolation procedures which are based on the reweighting 

approach have two fundamental flaws. First. the use of a uniform growth factor for all income items 

does not take into account possible changes in the functional distribution of income (i.e.. changes 

in the relative mix of the various income components clue to changes in the economy. such as an 

increase in the rate of return). Second. making changes in the functional distribution of income 
-3-

throiigh altering the database's sample weights implicitly assiimes that all changes in the relative 

values of the components of income are caused by a change in the number of returns containing that 

component of income. rather than a change in the level of that income component reported on the 

returns (i.e.. the mean value of the income component remains constant). 

In order to correct for these problems. Lindsey developed an extrapolation process for the National 

Bureau of Economic Research (NBER) TAXSIM model's database that emphasizes indiviclual adjustments to the levels of the various items on the database.1 In particular Lindsey suggests that a 

better estimate of the change in the income distribution resulting from a change in a component of 

income (such as capital gains) would be obtained through an increase (or decrease as the case may 

be) of the level of that income component for the existing recipients rather than through an 

increase (decrease) in the number of the recipients. 

Lindsey's extrapolation method is quite similar to the extrapolation procedure used by OTA in the 

mid- 1970's. This earlier version of the OTA extrapolation process allowed for indivitlual arljustment 

factors to be applied to the various components of income prior to any reweighting of the file. 

Thus. the emphasis in the earlier OTA extrapolations seems to have been on individual adjustment of 

the inconie variables. rather than on reweighting records. In recent extrapolations, OTA choose to 

emphasize reweighting. and the multiple adjustment factors were eliminated. 2 

The differences between the reweighting approach and the individual acljustment approach derives from 

different views of portfolio adjustment. The assumptions behind the individual adjustment approach 3 

suggest that, in the short run at least, an individual's portfolio is inelastic with respect to 

changes in the relative rates of return for the various income components. In other words, by not 

allowing new recipients of a particular income component to appear on the extrapolated database, the 

individual adjustment approach assumes that an individual cannot or will not adjust his or her 

portfolio in response to changes in the rates of return on alternative assets. 

This assumption is in sharp contrast to the extremely responsive portfolio. behavior implied by the 

reweighting approach. If all aggregate changes on the tax datahase result from acl-justmentsin the 

number of taxpayers with different portfolios found on the clatahase. then no change in the relative 

rates of return for the various income components is assumed. This implies that any economic 
-4-

circumstance that might change the rate of retum of a given component of income will he compensated for through an adjustment in the taxpayers' portfolios. Thus, by not allowing the 

relationships between the various components of income on a record to change. the reweighting 

approach assumes that all taxpayer's portfolios are completely flexible. 

Currently. however. there is no means for choosing between the assumption of flexible portfolios and 

the assumption that, in the short run at least, portfolios are inflexible. In fact, it seems more 

reasonable to assume that. given the differences in the liquidity of assets composing the 

portfolios. the portfolios are in the 'short run' more flexible with respect to some components of 

income (e.g., capital gains and dividends) and less flexible with respect to other income coniponents 

(e.g.. wages and pensions). In addition it seems reasonable to assume that the flexibility of a 

taxpayer's portfolio will vary with the taxpayer's place in his (or her) life cycle. Therefore. an 

accurate extrapolation procedure would need to combine the use of indiviclual adjustment factors 

with the reweighting approach. 

OTA currently implements an extrapolation routine that combines the reweighting approach and the 

individual adjustment approach. Specifically. OTA's most recent extrapolation procedure operates in 

two stages. In the first stage, the levels of various components of income (such as wages and 

salaries. capital gains, interest. dividends. pensions. and business income) are independently 

acljusted to hit various macro-targets. In the second stage. the weights on the database are 

adjusted to achieve other aggregate targets (such as the two earner clecluction. the Foreign Tax 

Credit [ FTC). the Investment Tax Credit [ ITC]. itemized deductions. and the distribution of Adjusted 

Gross Income [AGI]. By using various combinations of first and second stage targets. it is possible 

create a large, if not infinite, number of extrapolations for a given set of targets. These 

extrapolations range from extrapolation based solely upon reweighting to extrapolation using only 

adjustment factors. 

The determination of the best mix of individual adjustment and reweighting. however. requires an 

examination of the accuracy of the different extrapolations with respect to the target values. This 

examination requires some means for comparing the fit of the different extrapoiations being 

considered. The next section describes some possible measures of the 'goodness of fit' of an 

extrapolation. 
-5 -

-

-

111. MEASURES OF GOODNESS OF FIT 

A key step in the development of some means for evaluating alternative extrapolations is 

constructing some measure of how well the ac@isted database fits the actual targets. In this 

initial study the targets are provided by the 1983 SOI. Since we were concerned with both the 

aggregate values of selected variables and the distribution of the variables by AGI class, it was 

necessary to have ‘goodness of fit’ measures that considered both the aggregate value as well as the 

distribution of the variables, Further, each extrapolation consists of many variables. so it was 

desirable to have a measure that could be used to compare both individual variables and the 

extrapolation as a whole (e.g., something like T-statistics and F-statistics in linear regression 

equations). 

Clearly. no one measure could satisfy all of these requirements. Therefore. three separate measures 

for the fit of an extrapolation were developed. Two of these measures examine the fit of indiviclunl 

variables: the percent error and the information gain (for the aggregate value and the distribution. 

respectively), The third measure examines the fit of the extrapolation as a whole: the multiplicative decomposition of the variance. 

A Measure of the Fit of a Value 

There are any number of possible mechanisms for measuring the fit of a value. One obvious measure 

of the error in the predicted value of a variable is the percent error: 

where f3 is the estimated value of some variable B and b is the true value. The percent error has 

several desirable properties. The most important of these properties is that the percent error is a 

relative measure of the deviation from the true value and so can be used to measure the relative 

accuracy of the various extrapolated variables. 

A Measure of the Fit of a Distribution 

The most commonly used non-parametric statistic for comparing two distributions is the 

Kolmogorov-Smimov two sample test. Unfortunately. the Kolmogorov-Smirnov test compares the 
-6-

empirical cumulative distribution functions of the two distributions being considered. Since we are 

interested in a variable's distribution by AGI class. the Kolmogorov-Smirnov test is inappropriate. 

Further. the standard alternative. the Chi-squared test. is not invariant to changes in the scale 

of measurement. Therefore an alternative measure. the information gain. was chosen (see Theil 

[ 19673). 

The information gain measure is derived from information theory which examines the relationship 

between probabilities and events. For example. suppose that some event Ei is expected to occur with 

some probability pi. If at some latter point in time a message is received that states that event 

Ei has occurred. then some amount of information has been received. Intuitively. the amount of 

information that is received from such a message is inversely related to the probability of E,'s 

occurrence (i.e., the more probable an event is. the less information that is obtained from that 

event's occurrence). To foimalize this relationship, let: 

where h(pi) is a measure of the amount of information generated by knowledge of the event and 

In(pi) is the natural log of probability pi 

Next. suppose there are N possible events. that collectively exhaust the outcome space (i.e.. one of 

the N events must occur). Then. prior to actually receiving a message the expected infomiation 

inherent in the message is: 

N 

(3) H(p) = c -piln(pi) i=l 

where 

N 

(4) c pi=l

i=l 

and H(p) is the expected information or entropy measure. Note that if the empirical probabilities. 

pi. are calculated as the share of the total value of a given variable (vj) that falls into a given 

AGI class (this limits the test to variables whose values have the same sign for each AGI class). 

then the distribution of a variable by AGI class can be \viewed as an algorithm for allocating 
-7-

expenclitrires into different classes and therefore can be summarized using the entropy measure. 

Two distributions can be compared by examining the expected change in information that results from 

moving from one distribution to the other. Let. 

(5) h(xi) - h(yi) = -In(xi/yi) xi .yi > 0 

. where xi is the prior probability (1 983 SOI's i-th AGI class share of vj) and yi is the posterior 

probability (the adjusted Tax Model Database's i-th AGI class share of vj). Then the expected 

information gain can be measured by 

N 

I(y:x) = C -yiln(xilyi)

i= 1 

where 

N N 

(7) E xi = E yi = 1 and xi.yi 2 0. 

i= I i= 1 

The information gain measure is a useful tool for examining the differences between the shapes of 

4

distributions. Since the calculations are all based on shares (technically, allocation probabilities) the measure is completely independent of the unit of measurement or the aggregate value of 

the variable. The only requirement is that the number of cells into which the distributions are 

divided be identical. Thus, the information gain is a very powerful tool for examining broad 

questions about distributions for it can be used to compare the distribution of any two variables 

(e.g.. one could examine how closely the distribution of dividends by AGI resembles the distribution 

5 of interest income). 

A Measure of the Overall Fit of an Extraoolation 

Clearly in addition to having measures of how well individual variables are predicted by an 

extrapolation. it is desirable to have a measure of the overall fit of an extrapolation. Most 

common measures of the fit of a predictor are based on the variance. Unfortunately none of standard 

measures are applicable. Therefore. we have modified a measure suggested by Theil (1 967). 
-8-

In developing a general measure of the fit of an extrapolation we shall make use of several Ivtsicdefinitions. Any given extrapolation t (or, niore generally, any predictor t where t = I,. .,,T) 

generates an IxJ matrix of variables. V,. where the ij-th element of V, represents the t-th 

extrapolation's prediction of the i-th variable's value in the j-th AGI class. Now, let X, be 

the natural log of V,. where the natural log is taken element by element. Then. assuming that 

where XT is a matrix of the natural logs of the true values, the error variance of X, can be written 

as: 

and the variance for each element of X, can be written as. 

Note that if E(xi jt -xi jT)=pijt $0. then estimates of the error variance based on the mean square 

error will overstate the true error variance and the efficiency of the extrapolation will be 

understated. In the absence of any information to the contrary, however, we assume that the 

predictors are unbiased. Unfortunately, even with the assumption of unbiasedness in the predictions 

of the extrapolation. this specification still has too many estimable parameters. Adclitional 

simplifying assumptions are necessary. 

Let us assume that the error variance can be decomposed in the following multiplicative manner: 

where aimeasures the inaccuracy corresponding to the i-th variahle. the inaccuracy corresponding 

to the j-th AGI class. and Y, the inaccuracy of the t-th extrapolation. The use of a multiplicative 

2 2 2

decomposition instead of the additive decomposition (i.e.. 6i +E +Ct ) generally used in variance 

analysis has a major benefit. With the multiplicative clecomposition. a change in one of the 

decomposition factors has the same percentage effect on all the variances. Thus if y, changes from 2 

1 to 1/2 this reduces all uijt by half, while if <, changes from I to 1/2 the percentage effect on 

2 

ui jt depends on the values of 6, and E 

2 2 

. 
-9-

The estimation of ai,e,, and Y, is relatively straight forward. By definition: 

Using this identity we can derive the following estimators: 

J r-1 

(W. (l/ni) C c (Xi,,-xij T)2 -

- a, 

2 

n, =J+r-I 2 2 

j=l t=l bj Ct 

I T-l 7 

( I3b) (I/nj) C c (Xij, 

2

-xi 

2 

jr IC = bj 2 nj =I +r-l 

ai ‘t i=l t=l 

I J 

2 2 2 2 2 2 where a, . bj , and c, are estimators for a, . 8, . and Y, respectively. Since equation I I has 

two multiplicative degrees of freedom (as one can multiply every aiby A,. every Bj by A,, and every 

y, by I /A, A, without changing the left hand side of the equation) the estimates of b, and c, must be 

normalized. 

These equations are nonlinear in the estimators and must be solved recursively. Although recursive 

estimation of equations 13a - 13c does not guarantee convergence. in all the cases considered the 

estimates of the decomposition factors converged within twenty iterations. 

In order to engage in statistical testing using the decomposition factors it is necessary to make 

two restrictive distributional assumptions: that the prediction errors are normally distrihuted. and 

that the predictors are uncorrelated. Given the nature of the extrapolation procedure. it is 

improbable that these two assumptions hold. Despite this. the decomposition factors can be used 

to rank the different variables. AGI classes. or extrapolations according to their relative 

effciengcies (e.g.. to compare the relative efficiency of different extrapolations simply by 

comparing their ranking by ct).6 
-- 

-- 

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IV. EXTRAPOLATING THE TAX MODEL DATABASE 

The initial test of OTA’s extrapolation process was a series of extrapolations of the Individual Tax 

Model Database based on the 198 I SO1 to the year 1983. Since the I983 SO1 was available, all of the 

targets were derived from the actual values found on the SOI.’ Thus. the fit of the different 

extrapolations reflected the mix of targeted variables chosen rather than the accuracy of the 

targets themselves. 

The key problem in extrapolating the Individual Tax Model Database to 1983 SO1 levels was the 

selection of the mix of targets to be used in the extrapolation process. It was decided that the 

initial group of targets should be chosen from the set of variables that are important in the new 

tax law but are in the Tax Model Database. These variahles were: AGI. capital gains. partnership 

income. rental income, pensions, dividends. interest. wages and salaries. the earned income credit. 

the investment credit. and the foreign tax credit. 

As was mentioned previously. the extrapolation process allows variables to be adjusted in two 

different procedures. A variable can be “blown up” (multiplied) by some constant adjustment factor 

for all the returns in the Tax Model Database this is the Stage One adjustment in the extrapolation process. Alternatively. a variable can be adjusted by changing the weights (the number of 

tax returns represented by each return in tile database) for some subset of the returns in the 

database not surprisingly. this is called the Stage Two adjustment in the extrapolation 

process. 

There are. of course, costs and benefits to using either the adjustment factor approach or the 

reweighting approach to extrapolate a database; To a large extent. the decision to use one of the 

extrapolation method depends on how the value of a variable is known or believed to have changed 

in the actual population. The modification of a variable only through the use of an adjustment 

factor assumes that the incidence of the variable in the popillation has not changed and that 

modification can be accounted for by an increase or decrease in the mean value of the variable per 

return. Changing the value of a variable using only the reweighting approach. on the other hand. 

assumes that the average value of the variable per retuin has not changed. and that the change can 

be explained by an increase or decrease in the incidence of the variable in the population. Using 

both adjustment factors and reweighting. allows the change to be caused by changes in both the 

incidence and the level of the variable. 
-11-

The first extrapolation attempt adjusted the Tax Model Database using only the Stage Two reweighting 

method. This led to major changes in the weights on the adjusted database. This implied a massive 

change in the types of returns filed by taxpayers, Since OTA believed that such a massive change 

was unlikely in a two year period. this initial extrapolation was rejected. 

In order to minimize the weight changes resulting from the extrapolation process all of the monetary 

targets were blown up by the inflation rate in a Stage One adjustment prior to the Stage Two 

reweighting. This significantly reduced both the size of the weight changes and the number of 

iterations the extrapolation routines required to meet the target levels. Hereafter. this 

extrapolation will be referred to as the initial extrapolation. 

Although this initial extrapolation was successful in achieving the required aggregate levels for 

the target variables. the distribution of the targets by AGI class did not resemble the target 

variables‘ 1983 distributions. Therefore. it was decided to explicitly directly target both capital 

gains and partnership income by AGI class in the Stage Two reweighting (AGI had already been 

targeted by AGI class). This. however. resulted in more targets for the Stage Two reweighting than 

the program was designed to handle. 

In order to achieve both the aggregate and distributional targets, the next extrapolation was 

divided into two subproblems: adjusting aggregate values and adjusting distributions. The first 

subproblem adjusted selected target values to the correct aggregate level. In addition. in the 

first subproblem. two distributional items were targeted: AGI by AGI class and the population by 

age strata. 

The second subproblem was intended to correctly distribute selected target variables by AGI class, 

Therefore, no further Stage One adjustments were made for this problem. The subproblem’s second 

stage was used to target the distribution of various targets. In aclclition, in order to correct for 

potential errors caused by reweighting, AGI was also targeted by class as were filing status and 

some income aggregates. 

Many attempts were made to adjust the Tax Model Datahase to the 1983 SO1 using this type of 

extrapolation. Although many different extrapolation runs were macle the overall results of the 

adjustment process can be summarized by extrapolations 1-4 presented in Tables I and 2. 

Specifically, Table I presents the percent error in the prediction of the aggregate value of 
-12-

Table 1: PERCENT ERROR IN THE PREDICTION OF THE TOTAL VALUE 

Data Extrapolation 1/

Items Initial 1 2 3 4 

Adjusted Gross 

Income 1.14 0.0 - 0.71 - 0.01 - 0.01 

Capital Gain -21.63 0.0 0.0 0.0 0.0 

Pensions - 6.47 0.0 0.0 0.0 0.0 

Dividends 4.38 - 0.64 - 2.78 - 5.56 - 4.17 

Interest 8.28 0.0 0.0 0.0 0.0 

Wages 0.98 - 0.61 - 0.78 - 1.54 - 1.19 

Partnership Gain - 1.68 0.0 0.0 0.0 0.0 

Rental Gain - 2.99 0.78 0.06 - 3.22 18.64 

Rental Loss -24.02 - 2.96 - 2.12 - 4.23 - 0.37 

Other Schedule E 

Gain -17 41 52.09 0.02 35.52 28.52 

Other Schedule E 

Loss - 1.45 - 6.44 0.0 - 7.87 10.12 

Earned Income 

Credit . - 8.8 -23.8 0.11. 0.0 0.0 

Investment Tax 

Credit 1.77 2.26 5.70 4.33 4.42 

Foreign Tax Credit 

Two Earner 

4.70 - 0.16 0.0 0.0 0.0 

Deduction 101e 5 102.1 - 7.09 -51.39 1.24 

Medical Deductions 0.50 38.09 30.78 27.13 -15.43 

Single Return 

Joint Return 

2.05 

- 0.57 

- 0.05 

0.08 

0.05 

0.08 

- 0.05 

0.08 

- 0.04 

0.08 

Married, Filing

Separately

Head of Household 

18.18 

- 5.02 

0.0 

- 0.47 

0.0 

- 0.47 

0.0 

- 0.47 

0.0 

- 0.47 

Partnership Loss - 9.46 - 0.1 0.0 - 0.01 0.0 

1 Each of the extrapolations are described in the text. 
-13-

Table 2: INFORMATION GAIN IN THE PREDICTED DISTRIBUTION 

(IN THOUSANDS) 

Data 

Items 

AdjustedlGross 

Income 

Capital Gain 

Pensions 

Dividends 

Interest 

Wages

Partnership Gain 

Partnership Loss 

Rental Gain 

Rental Loss 

Other Schedule E 

Gain 

Other Schedule E 

Loss 

Earned Income 

Credit 

Investment Tax 

Credit 

Extrapolation 1/

Initial 1 2 3 4 

N/C N/C N/C N/C N/C 

1.61 2.1 2.38 2.58 1.47 

6.97 10.7 8.25 8.48 26.1 

9.4 14.3 10.4 14.5 43.8 

9.99 9.44 6.78 8.37 17.9 

3.27 0.59 1.3 0.77 1.43 

9.84 8.98 8.34 10.4 63.7 

5.9 17.6 90.8 16.0 14.4 

14.2 9.45 11.0 9.84 26.9 

14.3 11.3 36.4 9.07 42.7 

6.94 83.2 44.6 108.0 147.0 

70.4 98.6 29.8 106.0 70.0 

N/C N/C N/C N/C N/C 

N/C N/C N/C N/C N/C 

Foreign Tax Credit2 N/C N/C N/C N/C N/C

Two Earner 1050.0 1080.0 19.5 1120.0 1310.0 

Medical Deduction2 N/C N/C N/C N/C N/C

Single Return 1.57 0.72 0.37 0.61 1.25 

Joint Return 5.20 1.69 2.01 1.87 2.83 

Married, Filing

Separately 81.0 91.1 87.0 87.3 104.0 

Head of Household 4.90 5.04 4.98 5.30 0.87 

Each of the extrapolations are described in the text. 

N/C = Not calculated due to zero or negative values in cell. 

1 
- l4-

selected varialdes for the initial and the four experimental extrapolations. Takle 2 presents the 

information gain in the predicted distributions for the four experimental extrapolations. The major 

adjustments and additions made to the targets for each extrapolation. as well as the results of the 

different target mixes. are summarized below. 

Extrapolation 1's relative efficiency proved to be superior to the initial extrapolation. The error 

variance attributable to the extrapolation (ct2) is 24.4% smaller than the variance attributable to 

the initial extrapolation (see Figure I), In examining the individual predictions of the 

extrapolation 1, however, it was decided that the extrapolation's major flaws lie in its prediction 

of the two eamer deduction and its prediction of the Other Schedule E income and loss. Therefore. 

the second extrapolation included five additional targets in the second pass' second stage to 

improve the extrapolation's prediction of the distribution of Other Schedule E income. In addition. 

the two eamer deduction's income share imputation was replaced with an imputation based on 1983 

data. 

The relative efficiency of extrapolation 2 was also far better than that of the initial 

extrapolation. As Figure 5 shows, the error variance attrihutahle to extrapolation 2 was 2.1% 

smaller than the error variance of extrapolation I. In addition. extrapolation 2 improved the 

prediction of Other Schedule E income (see Tables I and 2). Unfortunately. extrapolation 2 had a 

poorer prediction of the levels of wages and salaries and dividends than did extrapolation I. 

Further. extrapolation 2's prediction of the medical deduction was significantly worse than that of 

the initial extrapolation. 

For extrapolation 3. therefore, it was decided to add adjustment factors for the medical expense 

deduction to Stage One of the first subproblem and to add an aggregate target for this variable to 

Stage Two to improve the extrapolation's prediction of these variables. After an initial attempt. 

subproblem 1's blowups of Other Schedule E Gains and Losses were removed to improve the extrapolation's predictive power. These modifications to the subproblem 1 of the extrapolation did not 

improve the relative efficiency of the extrapolation. In fact. the error variance attributable to 

the third extrapolation was 14.4% larger than that of the second extrapolation and the predictions 

for many of the individual variables were much worse (see Tables I and 2. and Figure I). 

For extrapolation 4. an initial attempt was made to fix the third extrapolation by adding targets 

for the medical expenses deduction and the two earner clecluction to sukproblem 2's second stage. 

Under this target specification. however. the extrapolation proceclures did not converge. After some 
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-16-

experimentation. it was discovered that the key set of targets was tlie Other Schedule E distrihution. Removal of the Other Schedule E targets resulted in convergence of tlie extrapolation 

procedure. 

The results of extrapolation 4 are much worse than those of any of the previous extrapolations, as 

the error variance of the fourth extrapolation was 88.7% larger than the error variance attributable 

to the initial extrapolation. Although the predictions of the individual variables that were poorly 

predicted by the prior extrapolations were improved in the fourth extrapolation. other predictions 

of individual variables are worse than those in both the prior extrapolations and the initial 

extrapolation. 

The results of extrapolations 1 through 4 indicate that there are significant trade-offs to be made 

between individual variables in the extrapolation (see Figures 2 and 3) and it is impossible to 

choose between them without some external criterion. In fact, without some sort of lexicographic 

preference ordering about the fit of different variables the only reasonable selection criteria for 

choosing the best extrapolation procedure is the relative efficiency of the various extrapolations. 

Based on this criteria. the second extrapolation provides the best adjustment of the Tax Model 

Database to 1983 SO1 levels. 

V. CONCLUSIONS 

The use of extrapolation methods to acljust the OTA's Individual Tax Model Database has in some 

regards been only partially successful. Although it has proven possible to improve the efficiency 

of the Tax model Database as a predictor of the 1983 SOI. it has not been possible to totally 

eliminate the error variance of the database. In the course of this process, however, a great deal 

has been learned about the nature of the extrapolation procedures. First and foremost, it has 

proven impossible to improve the predictive power of the extrapolation by simply adcling more 

targets. As the number of targets grows. the extrapolation procedures produce an increasingly 

distorted database as seen in both the changes in the pattern of the sample weights and in the 

distribution of the examined variables. The addition of more targets to acljust for the increased 

distortion eventually leads to nonconvergence of the extrapolation routines. This is especially 

true when the targets are sources of income (e.g.. note the effect of targeting Other Schedule E 

income when most of the other income sources were already targeted). Perhaps the extrapolation 

proceclure needs some 'degrees of freedom' in order to successfully atljust a database. In particular 
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-19-

it appears that at least three soiirces of income must he untargeted (i.e.. three ‘degrees of 

0

freedom‘) if the extrapolation procedure is to converge. 

Second. the changes caused by the second pass indicate that Stage Two of the extrapolation process m 

does not preserve the distributions of either targeted or untargeted variables. Due to the nature 

of the criterion function (Le.. to minimize 

4 

(14) L = (w,/wo) + (w1/w0)-4- 2 

where w1 is the adjusted weight and wo is the initial weight) and the fact that variables are not 

identically distributed across AGI classes, the reweighting scheme cannot conserve the distribution 

of a variable by AGI class unless each class is targeted independently. 

Third. although there are no large (i.e.. larger than 0.1) covariances (as calculated using the 

Pearson covariance estimator, the nonparametric covariancesare somewhat higher) between the various 

sources of income for the entire population. there do seem to be such correlations within various 

subgroups of the population. This may explain some of the changes in the non-tat-geted variables 

that result from the extrapolation process. In addition. the ahsence of correlations for the whole 

datahase and their presence for subgroups suggests that it might be clesirahle to re-weight subgroups 

rather than the entire database. 

Together these three finclings suggest that it would be fi-uitful to continue research into 

improvements on the extrapolation process in two general directions. First. since it is impossible 

to target all variables in the database. a procedure must be developed that determines what are the 

key variables on the database to be used as targets. One possible source of these variables is to 

examine the highly correlated variables that are found for the subpopulations. Another possibility 

is to use artificial variables derived from principle component analysis as targets. 

A second area where the extrapolation process may be improved is in its clistributional effects. To 

the extent that some distortion of distributions is inevitable. it can he minimized through the 

proper selection of a criterion function. Perhaps a criterion function designed to minimize the 

change in the shape of the distribution of a group of varialiles will have less effect on the 

distribution than does a criterion function intended to minimize relative weight changes. 

Alternatively. perhaps the database ought to be extrapolated hy subgroups rather than as a whole. 

An additional area of long-term research is the mechanism for selecting targets. The research in 

this paper was based on using known SO1 values as targets. For most extrapolations. the targets 
-20-

will he forecasts rather than actual values. and the accuracy of the extrapolation will be dependent 

on the accuracy of the forecast. At this time. however. there has been little investigation of the 

properties needed in the forecast of extrapolation targets. 

To summarize. the recent use of the extrapolation process to adjust the Individual Tax Model 

Database to 1983 SO1 levels has resulted in a substantial increase in our knowledge of the behavior 

of the extrapolation procedure and has suggested several ways in which the extrapolation process 

might be improved. 
-21-

ENDNOTES 

1 It is important to note that there are. in fact. some items on the tax database that cannot be 

, adjusted through inflation factors. In general. such items are either discrete variables or items 

that are dependent on the value of other items on the record. Some examples of these are the 

number of deductions. the value of various income dependent credits. and the itemized clecluctions. 

Lindsey ( 1985) does not suggest that these items should be extrapolated using inflation factors: his 

arguments in favor of adjustment factors are limited to the components of income. 

'1 wish to thank John Wilkins for informing me of the existence of this earlier OTA 

extrapolation procedure. 

3 I am using the term 'portfolio adjustment' in a much broader sense than is common in the 

finance literature. Specifically, I am including all sources of income in an incliviclual's portfolio

rather than just the individual's capital assets. 

4 Unfortunately. traditional hypothesis testing cannot be used with the expected information 

statistic. There does exist a weaker concept -- the minimum discrimination information -- which is 

analogous to the Cramer-Rao inequality (Theil [ I97 I]. and Kullback [ 19591). Essentially. this 

concept determines the minimum value of I(y:x) which allows one to re-ject the null hypothesis that 

the two distributions are identical. For our purposes. the discriminating level for I(y:x) is zero 

(i.e. if I(y:x) > 0.0. then the two distributions are not the same). Note, however. that the 

discriminating concept does not provide a mechanism for calculating confidence intemals or for 

.comparing the fit of two distributions relative to a third: in these cases. analysis is limited to 

an ordinal ranking of the values of the expected information statistics. 

5 I am grateful to Roy Wyscaiver for pointing out some of the potential uses of the expected

information concept (e.g. in the imputation process when imputed variahles are being distributed 

using another variable's distribution). Another potential use of the expected information concept

is to examine changes in the distribution of variables over time. 

6 Two modifications were macle to the formula so that the sqriaretl logarithmic prediction errors 

could be calculated for all the cell values, First. since there were no cases where the predicted

cell value had a different sign than the true cell value all logs were taken using absolute values. 

Second, for those cells where either the true or the predicted value equaled zero, one was added to 

both cell values before the logs were calculated. 

7

An interesting set of comparison values for the extrapolations were the estimated errors 

resulting from the subsampling process used to generate the reduced SO1 file used in the Tax Model. 

Not surprisingly. the errors found for the reduced SOI were far smaller than those found for any of 

the extrapolations. 

6 Not surprisingly the requirement that there be approximately three degrees of freedom iniplies

that AGI and the various components of income are related in an extreniely nonlinear fashion. 

Unfortunately the exact nature of this nonlinear relationship is not clear. This relationship may

be clarified through future research. 
-22-

REFERENCES 

Cilke. James. and Roy Wyscarver (1 987) "The Treasuiy Individual Income Tax Simulation Model" in 

Compendium of Tax Research. U.S. Department of the Treasury. Washington. DC: Government 

Printing Office. 

Kullback. Solomon (1959) Information Theory and Statistics. New York: John Wiley and Sons. Inc. 

Lindsey, Lawrence ( 1985) 'Creating a Baseline Income Distribution for Tax Data'. Unpublished 

Manuscript. 

Theil. Henri (1967) Economicsand Information Theory. Amsterdam. The Netherlands: North-Holland 

Publishing Co. 

Theil. Henri (1 97 I ) Principles of Econometrics. New York: John Wiley and Sons. Inc.