Technical Paper Series

Congressional Budget Office

Washington, D.C.

FORECASTING CAPITAL

GAINS REALIZATIONS

Preston Miller

Federal Reserve Bank of Minneapolis and

Congressional Budget Office

Larry Ozanne

Congressional Budget Office

August 2000

2000-5

Technical papers in this series are preliminary and are circulated to stimulate

discussion and critical comment. They are not subject to CBO’s formal review and

editing processes. The analysis and conclusions expressed in them are those of the

authors and do not necessarily represent the position of the Congressional Budget

Office, the Federal Reserve Bank of Minneapolis, or the Federal Reserve System.

References in publications should be cleared with the authors. Papers in this series

can be obtained by sending an email to techpapers@cbo.gov.

For additional information about this paper, contact Larry Ozanne at (202) 226-2684

or by email at larry@cbo.gov. The authors wish to thank Nicholas Bull, Christopher

Sims, Richard M. Todd, and Christopher Williams for their comments on this paper.
ABSTRACT

As an input to its estimates of federal revenues, the Congressional Budget Office

(CBO) requires an estimate of capital gains realizations in the current calendar year

and forecasts of realizations for the next 10 years. The purpose of our study is to

improve the accuracy of the current-year estimates. As background, we describe the

models and methods that CBO now uses and briefly mention models developed at

other institutions. We then discuss in detail our method for constructing a new model

and compare the model’s performance with that of other models by determining how

well they would have estimated realizations had they been used in the past. Those

comparisons are conducted using increasingly realistic assumptions about the amount

of information available at the time the estimates are made. We find that if CBO had

used our new model throughout the 1990s, with only the information available when

the estimates of realizations were made, CBO’s estimates of current-year realizations

would have been more accurate. 
CONTENTS

I. INTRODUCTION 1

II. CURRENT PRACTICES 3

Current-Year Estimation 3

Out-Year Projections 6

III. FORMULATING A NEW MODEL 7

Model Formulation 8

Dependent Variable 9

Explanatory Variables 9

Tax Rates 9

Business Cycle 11

Equity Net Gains 11

Property Net Gains 12

Modeling the Gains in Current and Future Years Separately 13

IV. IN-SAMPLE EVALUATION AND COMPARISONS 15

V. OUT-OF-SAMPLE EVALUATION AND COMPARISONS 18

VI. REAL-TIME EVALUATION AND COMPARISONS 22
1. The Tax Analysis Division combines the estimates of realizations with estimates of wages and other income

sources and runs them through its microsimulation tax calculator to generate estimates of individual income

tax revenues. The contribution of capital gains to total income tax liability is not separately identified on

tax returns, and therefore it cannot be forecast directly by time-series or other methods.

I. INTRODUCTION

Each December, the Congressional Budget Office (CBO) completes its budget

estimates (known as the budget baseline) for the current and 10 succeeding fiscal

years. As an input to the revenue estimates, CBO’s Tax Analysis Division requires,

by early December, estimates of capital gains realizations for the current and 10

succeeding calendar years.1

At the time the estimates are made, last year’s and most of the current year’s

capital gains have been realized. CBO cannot observe those gains until people file

their tax returns and the Internal Revenue Service (IRS) processes them, a sequence

that takes over a year. Last year’s realizations can be closely approximated from

preliminary data from the IRS, but no such data are available for the current year.

However, other factors that indicate the amount of gains likely to be realized—such

as the year’s movements in stock prices, the strength of the economy, and any

changes in tax rates on capital gains—are largely known by the time of the budget

estimates. Consequently, CBO estimates the amount of gains realized in the current

year by using historically estimated equations that relate capital gains realizations to

those other aggregate factors.

Projecting realizations in future years is more difficult because the

macroeconomic factors used to estimate gains in the current year are unknown in the

future. CBO does forecast major macroeconomic variables, but stock prices and

other asset prices, which are important determinants of capital gains, are very

difficult to forecast. Consequently, CBO does not project future realizations using

the same equations that it uses to estimate current-year realizations. Instead, CBO

assumes that realizations gradually revert to their historical size relative to gross

domestic product (GDP), and it combines that assumption with its projection of GDP

to project gains.

The empirical research described in this paper was designed to improve

CBO’s estimates of current-year realizations. We originally intended to address

current-year estimation and future-year prediction with a single model, but we found

that the two had different enough problems to benefit from separate treatments. We

will discuss the problems of forecasting future gains in a second paper. This paper

describes CBO’s current practice for both current-year estimation and future-year

projection but discusses only our efforts to improve the former. It also briefly

mentions models developed at other institutions because they provide ideas for

refining CBO’s model of current-year realizations.
2

Our efforts had some success. We conclude that if CBO had used our new

model throughout the 1990s with only the information available when the estimates

were made, its estimates of current-year realizations would have been improved.

(Our new model, like CBO’s model, consists of four variants of a basic equation.)

Between 1991 and 1998, the CBO model had a root mean squared error of 15.1

percentage points. Our model has a root mean squared error of 11.7 percentage

points. But even those smaller errors can still lead to substantial errors in the dollar

level of realizations and revenues.

The new model has not yet withstood the test of time, however. Some of its

coefficient estimates were honed by extreme events, and there were not nearly

enough such events to have much confidence in the estimates. As a result, similar

extreme events in the future could well lead to large errors from our model as well

as major shifts in some of its coefficients. Nevertheless, a major advantage of the

model is that it affords the opportunity to refine the estimates of some coefficients

in the face of future extreme events, whereas most other models do not offer that

opportunity.
3

II. CURRENT PRACTICES

The measure of capital gains realizations that CBO estimates is the annual total of net

gains that are reported by taxpayers who have net gains. Taxpayers who have net

losses are excluded and their losses are estimated separately. Net losses are much

smaller than net gains and grow more regularly because of a $3,000 limit on losses

per return. 

Historically, realizations have tended to grow at the same rate as GDP but

with much greater year-to-year fluctuation. Thus, the ratio of gains to GDP changes

from year to year but shows little trend (see Figure 1). Its average from 1952 to 1998

was 2.7 percent. Gains reached their high-water mark of 7.4 percent of GDP in 1986,

when people rushed to realize gains ahead of a tax increase that was enacted that year

but did not take effect until 1987. In 1998, the most recent year for which tax return

processing is largely complete, realizations were at their second highest level, 5.1

percent. That peak was reached after an uncharacteristically steady climb beginning

in the early 1990s.

Although many assets generate capital gains, the two most important classes

of assets for tax revenues are equities and real estate (see Table 1). Equities

accounted for 30 percent to 40 percent of classifiable asset sales in the 1970s and

1980s, and the tremendous increases in the value of equity holdings in the 1990s have

undoubtedly made gains from equities even more important. Taxable gains on real

estate come from rental residential and commercial real estate. Some of those gains

are passed through partnerships and thus account for some of the gains attributed to

partnerships in Table 1. Few gains on owner-occupied homes are taxed, and all such

gains are excluded from Table 1.

Current-Year Estimation

The model that CBO uses to estimate current-year capital gains realizations consists

of single-equation regressions based on macroeconomic time series. CBO’s baseline

estimate of realizations is usually a central tendency of estimates from the group of

equations. 

Macroeconomic regressions that explain gains were developed in the 1980s

by analysts in the Treasury, CBO, and academia. The focus of their work was to

measure how realizations respond to changes in tax rates. But the basic idea behind

the equations—to explain realizations in terms of the outstanding pool of gains held

by taxpayers and the cost of realizing those gains—carries over to predicting

aggregate gains.
0

2

4

6

8

55 60 65 70 75 80 85 90 95

Years

Percent

Figure 1: Ratio of Capital Gains Realizations to GDP

SOURCES: Capital gains realizations are from the Department of the Treasury, Office of Tax

Analysis, and GDP is from the Department of Commerce, Bureau of Economic

Analysis.
TABLE 1: DISTRIBUTION OF CAPITAL GAINS BY ASSET TYPE

1977 1981 1985 1977 1981 1985

Total 53,066 97,057 194,689 100 100 100

Corporate stock, CGD 14,783 39,447 81,814 28 41 42

Other securities (bonds) 560 1,065 3,054 1 1 2

Options and futures contracts 1,689 3,683 3,406 3 4 2

Partnerships, S-corporations, trusts and estates 5,112 9,485 42,977 10 10 22

Residential rental property 4,596 8,229 18,748 9 8 10

Depreciable business personal property 2,256 3,576 1,335 4 4 1

Depreciable business real property 3,410 3,420 14,067 6 4 7

Other assets 20,660 28,152 29,290 39 29 15

SOURCE: Internal Revenue Service, Statistics of Income Bulletin (Winter 1985-86 and Spring 1999).

NOTES: Data are not fully comparable among years, especially 1977 versus 1981 and 1985.

 CGD is capital gains distributions from mutual funds.

a. Excludes all capital gains on personal residences.

Gross Capital Gains (Millions of dollars) Distribution of Gains (Percent)

a
2. The error-correction term is the lagged residual from an equation that explains the logarithm of the ratio

of gains to GDP as a function of the capital gains tax rate and the 1986 dummy variable. The residual

indicates whether gains differ from their expected long-run size relative to GDP.

4

Outstanding capital gains are those that have accrued during the current and

prior years less those that were realized in previous years or were exempted from tax

because their owner died. Accrued gains are estimated and reported in the Federal

Reserve Board’s flow-of-funds accounts as revaluations of corporate and

noncorporate equity held by the household sector. Those two revaluation measures

were rarely used in the early development of the equations. Instead, analysts

approximated accrued gains on stocks using the value of corporate equities held by

the household sector (also from the flow-of-funds accounts) or stock price indexes.

GDP was commonly used to approximate accrued gains on assets other than stocks.

No reliable and timely data on gains exempted at death are available, so that factor

was ignored in most equations.

The cost of realizing gains was represented by tax rates on capital gains. In

addition, equations developed after 1987 included a variable to capture the transitory

effects of the large increase in the tax rate on capital gains in 1987 that was passed

in 1986. CBO’s equations used a dummy variable to isolate those effects. Another

cost, that of selling an asset, has not been incorporated in the equations, although the

cost of trading stocks has been falling since the 1970s and could be affecting the

willingness to realize gains. Empirically, the stage of the business cycle has seemed

to affect people’s willingness to realize gains.

CBO began estimating capital gains at the end of 1986, but it did not adopt

an equation approach until 1988. From 1988 through 1990, CBO used regressions

that explain the logarithm of realized gains to estimate gains in the current year and

forecast gains over a five-year horizon. In 1991, it shifted to its current model, which

explains changes in the logarithm of gains, or, roughly speaking, the annual growth

rate of gains. In most years, CBO uses four variants of a basic equation, which differ

in their inclusion of multifamily housing starts and an error-correction term (see

Table 2).2

 The equations are estimated from 1955 through the last complete year and

then used to predict gains in the current year.

In most years, CBO averages the current-year estimates from the four

equations. In 1999, however, the regressions with error correction were omitted

because of their large errors the previous year. Also, in some years the estimates are

adjusted for special factors, such as the initial effect of the 1997 tax cut. Predictions

made by the four equations in early December 1999 are shown in Table 3. The

predictions have large standard errors, as indicated by their 95 percent confidence

intervals.
TABLE 2: CURRENT CBO EQUATIONS (Dependent variable is change in log of gains, estimated 1955-1998)

No Error Correction No Error Correction Error Correction Error Correction

 No Starts No Starts Starts

Explanatory Variables Coefficient t-stat Coefficient t-stat Coefficient t-stat Coefficient t-stat

Constant term -0.060 -0.8 -0.096 -1.6 -0.148 -2.3 -0.120 -2.0

Growth rate of prices 1.024 1.0 1.691 1.8 1.788 1.7 1.985 2.1

Real growth rate of household equity holdings 0.528 4.3 0.533 4.9 0.517 4.3 0.484 4.4

Growth rate of real GDP 2.562 1.9 2.921 3.1 4.335 4.5 3.304 3.5

Acceleration of real GDP 1.287 1.4

Growth rate of multifamily housing starts 0.243 3.0 0.223 2.8

Change in maximum tax rate -0.027 -3.5 -0.025 -3.6 -0.025 -3.4 -0.026 -3.7

Indicator: 1986 = 1, 1987 = -1 (0 otherwise) 0.544 5.7 0.523 6.0 0.581 6.1 0.555 6.3

Error-correction term -0.203 -2.0 -0.163 -1.7

Adjusted R-squared 0.725 0.767 0.737 0.778

Durbin Watson 2.072 2.018 1.856 1.888

SOURCE: CBO calculations.

 Starts
TABLE 3: FORECAST OF CHANGE IN LOGARITHM AND LEVEL OF CAPITAL GAINS FOR 1999

 Level of Gains (Billions of dollars)

Standard 95% Confidence Interval 95% Confidence Interval

Equations Mean Error Low High Mean Low High

0.142 0.129 -0.117 0.401 507 391 657

0.127 0.119 -0.111 0.365 500 394 634

Error correction, no starts 0.027 0.141 -0.255 0.309 452 341 600

0.028 0.129 -0.231 0.287 452 349 586

Average of change in logs 0.081 -0.179 0.340 477 368 618

SOURCE: CBO calculations.

NOTE: The estimated level of gains in 1999 is based on preliminary 1998 gains of $440 billion.

Error correction, starts

 Change in Log of Gains

 (Based on CBO forecasts of December 7 and 9, 1999)

No error correction, no starts

No error correction, starts
3. The models do not fully describe the process of forecasting gains at the four institutions. The institutions

have access to other models and consider factors outside any specific model.

5

In addition to the regressions developed at CBO, regression models of capital

gains realizations have been developed by analysts at other institutions to provide

input to their forecasts. Table 4 highlights some of the salient differences in the

selection of variables among four of those models. The four were developed by

Nicholas Bull and David Richardson of the Treasury Department; Randall Mariger,

formerly of the Federal Reserve Board; Prawpan Siwapradit of the New York State

Division of the Budget; and Thomas Stinson of the Minnesota Department of

Finance.3

For 1999, CBO estimated several equations adapted from the BullRichardson, Mariger, and Siwapradit models. The equations generated current-year

estimates similar to those of the CBO equations without error correction. CBO used

the estimates from those adapted equations in settling on its 1999 baseline estimate.

The accuracy of CBO’s current-year predictions can be evaluated for 1986

through 1998 (see Table 5). CBO was farthest off in 1986, when it underestimated

how much people would respond to the impending tax increase in 1987. Other large

errors occurred when gains were overestimated in 1989 and 1990 and underestimated

in 1996. The errors also show a cyclical pattern, overestimating growth in 1989

through 1991 and underestimating it in most years since then. 

The root mean squared error on predicted annual growth rates from 1986 to

1998 is 26 percent, compared with growth rates that ranged from an increase of 90

percent to a decrease of 56 percent. Looking just at the experience from 1991 to

1998, which excludes the unusual years of 1986 and 1987 and coincides with the use

of CBO’s current equations, the root mean squared error is 16 percent. During that

period, growth rates ranged from a 45 percent increase to a 10 percent decline. The

errors since 1991 are still large compared with the growth rates of gains, and they

reflect the substantial uncertainty in predicting capital gains even when other

macroeconomic variables are largely known. That uncertainty is reflected in the

standard errors of the equations themselves.

CBO’s record in estimating current-year realizations and some of the

statistical properties of its equations suggest there is room for improvement. First

and foremost, the current-year estimates are often far off the mark. Second, the

coefficients on some variables, such as the growth rates of inflation and real GDP,

vary considerably based on the set of other explanatory variables. Third, the

coefficients on the error-correction terms vary considerably over time. Finally, the

use of dummy variables for 1986 and 1987 prevents the equations from estimating

the transitory effects of a future large changes in tax rates, should one occur.
TABLE 4: OTHER MODELS OF CAPITAL GAINS REALIZATIONS

Model Dependent Variable Explanatory Variables

Bull-Richardson Dollar change in capital gains

realizations.

Current-year change in tax rate.

Next year’s change in tax rate if

positive; zero otherwise.

Accumulated monthly increases in

S&P 500 per year.

Accumulated monthly decreases in

S&P 500 per year.

Mariger Change in the ratio of capital gains

realizations to nominal potential

GDP.

Change in the ratio of actual GDP

to potential GDP.

Change in the ratio of equities held

by households to potential GDP. 

Dummy for 1986.

Siwapradit Change in the log of capital gains

realizations.

Change in the log of the value of

shares traded on the New York,

Nasdaq, and American stock

exchanges.

Tax rate combines federal, New

York State, and New York City

maximums. Dummy for 1986.

Stinson Change in the ratio of capital gains

realizations to the value of

household assets.

Growth in the value of household

assets and GDP. Dummy for 1986.
TABLE 5: Current-Year Forecasts and Actual Capital Gains CURRENT-YEAR FORECASTS AND ACTUAL CAPITAL GAINS

 Growth Rate (Percent)

Actual Forecast Error Actual Forecast Error

1985 171

1986 324 223 101 90.2 30.7 59.5

1987 144 144 0 -55.6 -55.6 0.1

1988 162 151 11 12.3 4.7 7.5

1989 154 224 -70 -5.2 38.4 -43.5

1990 124 170 -46 -19.4 11.7 -31.1

1991 112 132 -20 -9.8 10.0 -19.8

1992 127 119 8 13.5 10.2 3.3

1993 152 131 21 20.2 6.1 14.1

1994 153 150 3 0.3 3.4 -3.1

1995 180 177 3 17.9 14.9 2.9

1996 261 196 65 45.0 8.9 36.1

1997 365 382 -17 39.8 46.4 -6.6

1998 450 418 32 23.4 13.0 10.4

1999 500 13.6

Root mean squared error, 1986-1998 25.6

Root mean squared error, 1991-1998 16.1

SOURCE: CBO calculations.

a. Growth rate forecasts are from preliminary values for the prior year.

b. The December 1995 forecast was modified to $175 billion in the delayed baseline of March 1996.

c. Preliminary.

Level of Gains (Billions of dollars) a

b

c
6

Out-Year Projections

Because capital gains have shown little trend relative to GDP, CBO projects that in

the years after the current year, gains will move back to their expected size relative

to GDP. That expected size is the historical average adjusted by a regression

equation for differences between current and historical average tax rates on capital

gains. For example, at the end of 1999, CBO’s equations predicted that realizations

that year would be around $500 billion, or 5.5 percent of GDP. As noted earlier,

gains have averaged 2.7 percent of GDP historically, but because tax rates are

currently below their historical average, the expected ratio is about 3.1 percent.

Thus, CBO projects that the ratio of gains to GDP will fall toward 3.1 percent

starting in 2000.

The rate at which that ratio declines is based on the estimated coefficients of

the error-correction terms in CBO’s two equations with such terms. Those

coefficients suggest a rate of decline per year equal to about 20 percent of the gap

between the previous year’s ratio and expected long-run ratio. The 20 percent rate

was used in the December 1999 projection. Given the rate of decline, ratios of gains

to GDP can be calculated for each year in the projection period. Multiplying those

ratios by CBO’s forecast of GDP in each year of the projection period gives the outyear projections of capital gains. As can be seen in Table 6, the 1999 projection

shows gains declining from 1999 through 2004 and then growing back to about $500

billion in 2010.
TABLE 6: OUT-YEAR PROJECTIONS OF CAPITAL GAINS 

1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

GDP 8,301 8,760 9,235 9,692 10,154 10,610 11,069 11,544 12,054 12,589 13,148 13,734 14,362 15,024

 Past and current 365 440 500

 Projected 480 466 456 449 447 449 453 461 471 483 498

Ratio of gains to GDP

 Past and current 4.39 5.02 5.41

 Projected 4.96 4.59 4.29 4.06 3.87 3.72 3.60 3.50 3.43 3.37 3.32

Growth rate of gains 20.61 13.64 -3.94 -3.00 -2.19 -1.39 -0.54 0.36 1.06 1.65 2.16 2.70 3.07

Assumptions

500 = Equations' predictions for 1999

3.12% = target ratio of realizations to GDP

20% = approximate rebound rate in error correction equations

SOURCES: GDP in 1997 and 1998 are from the Bureau of Economic Analysis as of December 1999. GDP in 1999 through 2010 is from CBO's forecast of December 9, 1999.

Capital gains in 1997 are from the Statistics of Income Division, Internal Revenue Service. Other data are from CBO projections and calculations.

Capital gains

In Billions of Dollars

In Percent
7

III. FORMULATING A NEW MODEL

This section describes the steps taken to formulate our new model. The sections that

follow examine the model’s performance and compare it with alternative models.

We examined performance in three stages:

• In-Sample. For this stage, we estimate coefficients once over the 1952-1998

period using the data set as it existed at the end of 1999. We generate

estimates of realizations for any given year by applying those coefficients to

the actual values of the explanatory variables for that year.

• Out-of-Sample. This stage uses the same data set as the in-sample stage, but

we estimate coefficients and realizations on the basis of what is sometimes

called “recursive regression.” For instance, we estimate an equation over the

years 1952 through 1959 and apply those coefficients to the actual values of

explanatory variables in 1960 to estimate realizations in 1960. We then

estimate the model over the years 1952 through 1960 and apply those

coefficients to the actual values of explanatory variables in 1961 to estimate

realizations in 1961. We repeat that step until we estimate the model from

1952 through 1998 and use it to generate an estimate of realizations in 1999.

The set of coefficients estimated in this last step, using data from 1952

through 1998, is the same as the single set estimated for the in-sample stage.

• Real-Time. For this stage, we use recursive regression as we did in the

previous stage but with two important differences. First, at each step in the

recursive regression, the coefficients are based on the data set that actually

existed at the end of November in that year. Second, values for all

explanatory variables are forecast through the end of the current year on the

basis of the partial and preliminary data available at the end of November.

The ultimate question we are addressing is, Which model will generate the

most accurate estimates of capital gains realizations now and into the future? Our

methodology assumes that useful information on that question can be derived by

examining hypothetical past performance—in particular, by looking at which model

would have generated the most accurate estimates over past periods had it been

available to analysts. We examine that question in the out-of-sample stage and, with

more realism, in the real-time stage.

Arguably, examining actual or hypothetical past performance is not relevant

to determining the best model now. For instance, although a model may have

generated large errors before, perhaps it was “learning” along the way and may do

well from now on. We acknowledge that point of view by examining the in-sample

fit. However, we believe that looking at how models did, or would have done, in the
8

past also yields important information. In particular, analysts can have more

confidence in using a model in the future if, over a considerable period, its

coefficients have been stable and its errors in estimating gains have been small.

Model Formulation

Our search for an improved model was guided by several principles. One was to

choose a reasonable set of explanatory variables. By reasonable, we mean few in

number, with each variable having a clear economic interpretation. It also means that

each variable has a coefficient that is of a plausible magnitude and stable across

specifications and time. A second principle was to replace dummy variables with tax

variables, so that there would be a way in the future to calculate the effects of tax rate

changes and to learn from actual experience. In addition, that principle reflects our

philosophy that extreme events can be useful in refining coefficient estimates and so

should not be discarded as aberrations. A third principle was to allow for dynamic

effects. We wanted to account for both the delayed effects of past developments and

the effects of anticipated future developments. A fourth principle was to make the

model useful in real time; that is, it should estimate current-year realizations well

using the data that is available when the estimates are made. That principal requires

that explanatory variables be reported with little delay and be subject to little

revision.

Our set of explanatory variables was chosen to capture four influences:

• Capital gains tax rates,

• The business cycle,

• Net gains on equity, and

• Net gains on property.

Our original specification was determined by in-sample fit. Variables for

inclusion were evaluated according to the significance of their coefficients as

measured by t-statistics and their contribution to fit as measured by the adjusted Rsquareds of the regressions. The evaluations were done sequentially in the order of

the influences listed above. That is, we first selected tax rate variables based on the

t-statistics and adjusted R-squareds. Then we selected a business cycle variable

based on the same criteria but applied to the regression including our selected tax

variables. After the business cycle variable was chosen and evaluated in the

regression, we went on to select the variable representing equity net gains and finally

the variable representing property net gains.
4. See Alan Auerbach and Jonathan Siegel, “Capital-Gains Realizations of the Rich and Sophisticated,”

American Economic Review (May 2000), pp. 276-282.

9

Dependent Variable

For the dependent variable, we took the difference in the ratio of capital gains

realizations to nominal potential GDP, using the series constructed by CBO for the

denominator. Using potential GDP as the scale variable is equivalent to assuming

that gains change in proportion to potential GDP, controlling for other influences.

The idea of using a scale variable was also employed by Stinson and Mariger. We

followed Mariger’s choice of the scale variable partly because CBO uses the ratio of

gains to GDP in projecting realizations in future years, and the projection of GDP

follows the growth of potential GDP. Our initial thought in selecting this variable

was that one equation could improve both current-year estimation and future-year

forecasting.

We used the first difference in the ratio to increase the likelihood that our

dependent variable is stationary. The ratio itself had appeared to be stationary when

CBO’s current method was developed in 1991 using data from 1955 through 1990.

That appearance was the basis for the assumption used in the out-year projections

that gains would revert to their historical size relative to GDP. Tests on data from

1955 through 1993 also soundly rejected the hypothesis of a unit root in the ratio.

However, the ratio appears less stationary now because of two data changes in the

current analysis. First, we expanded the sample period by adding 1952 through 1954.

We added no years before 1952 to be able to test various lag lengths of explanatory

variables. Second, the ratio can now be observed through 1998. The newly added

early years are record lows in the ratio, and the newly available recent years are

unusual highs. A Dicky-Fuller test of the ratio of gains to GDP now barely rejects

the hypothesis of a unit root at the 5 percent probability level. Using the first

difference in the ratio of gains to GDP removes all evidence of a unit root, returning

us to a more clearly stationary dependent variable. 

We tested for the addition of lags in the dependent variable, but their

coefficients were insignificant, and they added nothing to the fit.

Explanatory Variables

Tax Rates. As our measure of capital gains tax rates, we use the rate faced by

taxpayers in the highest income tax bracket. And, as is done in microstudies, we

separate the steady-state and transitory effects of changes in that rate.4

 The transitory

effects are generally expected to be larger than the steady-state effects because the

former allow substitution of realizations between years, but the latter do not. Bull
5. The construction of our legislated tax variable and its values year by year are described in a memo by Larry

Ozanne, which is available on request.

10

and Richardson followed a similar strategy, although they specified their tax terms

differently.

For the steady-state tax rate variable, MTRNEXT, we use the rate legislated

as of the current year that is to be in effect in the next year.5

 The steady-state tax rate

enters the realizations regression linearly (which implies that revenues are quadratic

in the steady-state tax rate).

For our transitory tax rate variable, MTRTRANS, we take the difference

between this year’s rate and the steady-state rate, square that quantity, and preserve

the sign. The basic idea behind the nonlinear specification of the transitory tax rate

term is that large anticipated tax changes can be expected to have disproportionately

larger effects on realizations than small anticipated tax changes will. That idea can

be motivated by a transactions-cost theory of financial markets. Transactions costs

create thresholds for investors. For small changes in anticipated tax rates, few

investors would cross their thresholds and change their portfolios. However, more

and more would cross their thresholds and rearrange their portfolios as the

anticipated tax change grew larger. Transactions costs lessen in importance over the

long run, which is consistent with the linear form of the steady-state tax term.

More specifically, we assume that the level of gains (ignoring potential GDP

for the moment) is explained as follows:

GAINS = α*MTRNEXT + β*MTRTRANS + γ*X + μ 

where MTRTRANS = (MTR-MTRNEXT)2

 *sign(MTR-MTRNEXT)

and MTR is the current-year tax rate; X is other variables; α, β, and γ are coefficients;

and μ is the unexplained residual. Furthermore, because we transform our dependent

variable to first differences, our estimating equation becomes:

D(GAINS) = α*D(MTRNEXT) + β*D(MTRTRANS) + γ*D(X) + D(μ)

where D(.) indicates the change from the previous year.

To illustrate the results of this tax rate specification, consider the years

surrounding the Tax Reform Act of 1986, which raised the top tax rate on capital

gains to 28 percent starting in 1987. The actual top tax rate on gains, our MTR, was

.20 in 1985 and 1986 and .28 in 1987 and thereafter. But the legislated rate to be in

effect next year, our MTRNEXT, was .20 in 1985 and .28 in 1986 and thereafter.
11

Thus, D(MTRNEXT) has a value of 0 in 1985, .08 in 1986, and 0 again in 1987.

With the coefficient α negative, this pattern implies that taxpayers will permanently

reduce their realizations starting in 1986.

The transitory tax term, MTRTRANS, which depends on the difference

between MTR and MTRNEXT, is (.20-.20)2

 = 0 in 1985, -(.20-.28)2 = -.0064 in

1986, and (.28-.28)2

 = 0 in 1987. Thus, D(MTRTRANS) is 0 in 1985, -.0064 in

1986, and +.0064 in 1987. With the coefficient β negative, this pattern implies that

taxpayers will increase their realizations in 1986 to take advantage of the temporarily

low rate and then reverse that increase in 1987. The estimated magnitudes of

coefficients α and β should be such that the transitory effect of a big announced tax

change is larger than the permanent effect, thereby accounting for the increase in

gains in 1986 and the reduction in 1987.

Business Cycle. For our measure of the business cycle, we employed the ratio of

annual GDP to annual potential GDP, as used in Mariger’s model. We chose annual

averages so that the business cycle variable is measured consistently with the scale

variable used as the denominator of the dependent variable. Since the equation is in

first-difference form, the business cycle variable also enters as a first difference. That

formulation, which separates the scale and business cycle variables, fixes a problem

with CBO’s current equations. In them, GDP often plays both roles, which makes

its coefficient difficult to interpret and probably contributes to making its value vary

from equation to equation.

Equity Net Gains. For the variable representing net gains in equity, we chose the

difference in the logarithm of the Standard & Poor’s (S&P) 500 index from the fourth

quarter of the previous year to the fourth quarter of the current year. Although that

measure was not our first choice on conceptual grounds, it turned out to be more

practical than the alternatives. 

Initially, we considered three main contenders for this variable: one based on

flow-of-funds revaluations, one based on stock market volume, and one based on

stock prices. Conceptually, the first one, a revaluations variable, seemed most

closely related to our dependent variable, as discussed in Section I. And empirically,

with full-sample information, it was the best. In first-difference form and entering

with three lags, it provided the best fit—but only marginally better than that of the

current year’s growth in the S&P 500. Consideration of real-time problems,

however, tipped the balance in favor of the S&P 500. In real time, in early December

of any year, the S&P 500 is known up to the day the current-year estimate of

realizations is made. However, at that time, there is no figure for revaluations in that

year and only a preliminary figure for revaluations in the previous year, which is

likely to face major revisions. So if the revaluations measure were used as an

explanatory variable, its current-year value would have to be estimated using current
12

stock prices. When we experimented by using the current year’s growth in stock

prices and the lagged growth in revaluations, the lagged terms were insignificant.

Thus, we dropped revaluations from consideration.

We also did not choose the second option, the dollar-volume measure used

by Siwapradit. On conceptual grounds, we felt that its relationship to realizations has

been changing. One reason is that the share of stock market volume accounted for

by institutions not subject to the capital gains tax has been steadily rising since the

1950s (although that change would also affect the relationship of stock prices to

realizations). Another reason for discarding stock market volume is that the costs of

trading stocks have declined sharply in recent years, especially with trading over the

Internet. The result is almost certainly an increasing amount of sales with smaller

gains per trade. On empirical grounds, we found that the current year’s growth in the

S&P 500 fit better than the volume variable over the whole sample period.

Even choosing to use a stock price variable to represent equity net gains does

not necessarily suggest that the measure should be simply the current year’s growth

in the S&P 500. For example, we allowed for up to four lagged growth rates in the

index, but they did not significantly improve the fit. We also considered a broaderbased stock index, the Wilshire 5000, which includes all stocks on the New York,

Nasdaq, and American exchanges. Surprisingly, we found that in the years when

both the S&P 500 and Wilshire 5000 are available, they track each other very closely.

So even though Nasdaq and S&P 500 stock prices have behaved differently over the

past few years, the difference has been offset in the Wilshire 5000 with the behavior

of the prices of other stocks that are in neither the S&P nor Nasdaq. Since the S&P

500 is available over our whole sample period but the Wilshire 5000 is not, we opted

to use the S&P 500.

Property Net Gains. For the variable representing property net gains, we chose

multifamily housing starts. Although that is also the choice in the current CBO

equations, its inclusion seems, on first thought, to be strange. It won an empirical

process of elimination, however, and we have a hypothesis for why it contributes to

our model’s fit.

We were seeking a measure of accrued gains on real estate investments and

other unincorporated business assets held by households. Conceptually, the variable

closest to that desired measure is the revaluations on noncorporate equity in the flowof-funds accounts. Empirically, however, that variable is not measured well,

probably because prices of existing property are not measured well. Moreover, that

variable would present the same real-time problems of delayed availability and large

revisions that revaluations of corporate equity do. Further, when we included it in

our regression, it added nothing to fit.
13

Our next choice was an index of real estate prices, excluding single-family

homes. We tried measures of prices from the National Association of Real Estate

Investment Trusts, the National Council of Real Estate Investment Fiduciaries, the

National Real Estate Index (compiled by CB Real Estate), and the national income

and product accounts (NIPA). None of them helped—the first three suffer from short

histories, limited coverage of real estate, and failure to measure change in the market

price of the same basket of properties, and the NIPA measures reflect the cost of

investment rather than the price of existing properties. 

Our third choice was NIPA investment variables. But they, too, did not help.

Thus, we ended up using the series for multifamily housing starts, which does

contribute marginally to fit. Our hypothesis for that result is based on a Tobin’s Q

story for real estate investment. The story is that changes in demand or supply of real

estate first cause changes in the prices of existing properties, which change the ratio

of prices of new structures to prices of existing structures (Tobin’s Q). Changes in

Tobin’s Q then lead to changes in the rate of investment in new structures. Although

changes in multifamily housing starts lag behind changes in prices of existing

structures, the former are measured much more accurately. Meanwhile, starts lead

investment, which is just a distributed lag of past starts.

When we examine the contribution of starts to fit, it appears small over the

whole period. But in another sense, it is large, because virtually all of that

contribution comes during roughly 10 percent of the sample period (in the late 1980s

and early 1990s). During those years, realizations were much weaker than would be

expected on the basis of stock prices and the other explanatory variables in our

model. A reasonable explanation, supported by limited data from tax returns, is that

part of the shortfall during those years resulted from the collapse in real estate

markets. The multifamily housing starts variable picks up that effect.

Modeling the Gains in Current and Future Years Separately

A final exercise did not change our model, but it did change our research strategy in

important ways. We experimented with including an interest-rate-spread variable—

that is, the difference between a long-term and a short-term interest rate. We

reasoned that the spread is commonly found to be useful as a leading indicator and

as a variable in time-series forecasts. We found that the first lag of the spread

marginally contributed to fit, but its coefficient was unstable. All of its contribution

came early in the sample period, and it seemed to be capturing the process of

intermediation that occurred under the Federal Reserve’s Regulation Q. In the latter

part of the sample, its role changed to leading economic indicator, but the main

variables it was predicting were already largely known quantities on the right-hand
14

side of our equation, such as the business cycle and stock prices. As a result, we did

not include the interest rate spread in our model. 

Nevertheless, we changed our research strategy because of that experiment.

It led us to separate current-year estimation from out-year forecasting because of the

difference in the predictability of right-hand-side variables. For instance, for currentyear estimation, stock prices are largely known, and including them as an explanatory

variable essentially removes the usefulness of the previous year’s interest rate spread.

We suspect that for forecasting, even one year ahead, the reverse would be true: the

current year’s value of the interest rate spread would mostly be known, but stock

prices over the next year would have to be predicted (probably badly).

Thus, although we originally chose the ratio form for our model because we

felt it would be useful for out-year forecasting, we concluded that was no longer

necessary. We decided to approach current-year estimation and out-year forecasting

using different models. The criterion of the first would be to minimize a sum of

squared residuals, and the criterion of the second would be to minimize a sum of

squared forecast errors. The variables in the two models could be different, as could

the forms of the equations. That reasoning suggested that we need not state our

current-year estimation regression in ratio form; instead, we could choose whatever

form seemed to work best.
6. The specification of MTRTRANS also allows inferences about a pure transitory tax change such as would

occur if 1986 legislation had reduced the 1986 capital gains tax rate by 8 percentage points but left future

tax rates at 20 percent. The coefficient of MTRTRANS in Table 7 implies a pure transitory elasticity of

-2.2 for such a change.

A few caveats are important to keep in mind when comparing our elasticities with those in earlier studies.

In most equations that have been reported, the elasticity varies with the marginal tax rate and possibly other

factors, so any direct comparison of elasticities is flawed unless the conditions under which they were

evaluated are known. A second distinction is that our tax rate is the maximum capital gains rate, whereas

many behavioral responses attempt to measure responses to the average marginal capital gains tax rate.

Third, our measure of capital gains includes net short-term gains, whereas most behavioral estimates focus

on net long-term gains because those gains are directly affected by the capital gains tax rate. Two recent

studies reporting elasticities and references to earlier studies are Matthew Eichner and Todd Sinai, “Capital

Gains Tax Realizations and Tax Rates,” National Tax Journal (forthcoming in 2000); and Auerbach and

Siegel, “Capital-Gains Realizations of the Rich and Sophisticated.”

15

IV. IN-SAMPLE EVALUATION AND COMPARISONS

All the regressions we evaluated in this stage of our model’s development were

estimated by ordinary least squares over calendar years 1952 through 1998 based on

the full set of data as it existed at the end of 1999. In this section, we evaluate the fit

of two versions of our model—without and with multifamily housing starts—and

compare their fit to the fit of our adaptations of other models. Although our model

does relatively well compared with our versions of other models, the comparison

suggested some additional avenues that we pursued in the out-of-sample evaluation.

Our estimated regression without starts is displayed in Table 7. Its

coefficients seem plausible. With realized gains scaled to potential GDP, the

coefficient on the ratio of GDP to potential GDP indicates a strong business cycle

effect. The stock price effect is also strong and significant. The coefficients on the

tax terms are both of the expected sign and significant. The extraordinary

significance of the coefficient on D(MTRTRANS) arises from the variable

explaining the spike in realizations in 1986. The reasonableness of the coefficients

on the tax terms can be judged by the elasticities they imply: a permanent elasticity

of -.36 for a tax increase from 20 percent to 28 percent (which is near the lower end

of results found in other studies) and a transitory elasticity of 1.8 for realizations in

the current year from an announced tax increase next year from 20 percent to 28

percent (which is near the lower end of results found in studies of panels of

taxpayers).6

The coefficients in Table 7 also vary little across various specifications of

other variables. We checked for the stability over time of the coefficients by dividing

the sample period in half and estimating the regression separately over each half.

Although most coefficients were comparable in both subperiods, the coefficient on

the transitory tax term seemed to shift by a sizable magnitude. We reasoned that the

shift could result either from a fundamental instability or from the fact that the large

tax increase in 1987 that was legislated in 1986 was an extreme event that allowed
TABLE 7: OUR RATIO EQUATION WITHOUT MULTIFAMILY HOUSING STARTS

Dependent Variable: D(TOTGAIN2/GDPFE)

Method: Least Squares

Sample: 1952-1998

Variable Coefficient Standard Error t-Statistic Probability

D(MTRNEXT) -0.0802 0.0197 -4.0718 0.0002

D(MTRTRANS) -6.0706 0.3969 -15.2954 0.0000

D(GDP/GDPFE) 0.0955 0.0233 4.0908 0.0002

DLOG(SP500Q4) 0.0181 0.0029 6.1303 0.0000

R-squared 0.877522 Mean dependent variable 0.000695

Adjusted R-squared 0.868977 Standard deviation of dependent variable 0.009309

Standard error of regression 0.00337 Akaike info criterion -8.46671

Sum of squared residuals 0.000488 Schwarz criterion -8.30925

Log likelihood 202.9677 F-statistic 102.6949

Durbin-Watson statistic 1.700098 Probability (F-statistic) 0

Key:

D is the first difference operator: D[x(t)] = x(t) - x(t-1)

TOTGAIN2 is capital gains realizations

GDPFE is nominal potential GDP

MTRNEXT is our permanent tax rate term

MTRTRANS is our transitory tax rate term

SP500Q4 is the average value of the S&P 500 index in the fourth quarter

SOURCE: CBO calculations.
16

the coefficient to be estimated more precisely. The regression, split at the midpoint,

passes a formal Chow test for stability, which supports the second explanation for the

transitory tax term coefficient.

Our estimated regression with multifamily housing starts is displayed in Table

8. Including starts modestly lowers the coefficients on the business cycle and S&P

500 variables and increases the adjusted R-squared by only a little more than 1

percentage point. Although overall the improvement is small, it all comes in the

roughly 10 percent of the sample period from the late 1980s to the early 1990s. As

before, we split the sample period in two and estimated the regression including starts

over the two halves. This time, the coefficient on multifamily housing starts also

seems to shift by a sizable magnitude. Again, the regression passes a formal Chow

test, which suggests that the change in the value of the coefficient reflects the greater

precision afforded by the extreme event of the real estate collapse that began in the

late 1980s.

Neither equation suffers from serial correlation of its residuals, based on the

Durbin-Watson statistic and a correlogram of the residuals. Neither equation fails

the White test for heteroskedastic residuals.

We next compared the fit of the two versions of our model with our

adaptations of other models. To make the statistics of fit comparable among models

with different dependent variables, we judged the fit of all models in terms of how

well they explain the annual growth rate of realizations in-sample. Thus, we

converted the predicted output of each model to a predicted growth rate in each year

and compared it with actual growth rates. The comparison included the following

regressions:

• The four current CBO equations specified in Table 2 (with and without starts

and with and without error correction).

• Four equations with Siwapradit’s dollar-volume variable. Value of volume

and GDP enter nominally, and no price index is included; otherwise, the

equations match the CBO equations (with and without starts and with and

without error correction). These equations are referred to as the dollarvolume model. 

• Three equations (following those of Bull and Richardson) with separate

measures of stock price increases and decreases, which we refer to as the

SPUD (stock price up and down) model. The three equations differ in the

number of variables with lagged terms: none, two (the stock price changes),

and three (those two plus the dependent variable). The tax rate terms in those

equations are described in Table 4.
TABLE 8: OUR RATIO EQUATION WITH MULTIFAMILY HOUSING STARTS

Dependent Variable: D(TOTGAIN2/GDPFE)

Method: Least Squares

Sample: 1952-1998

Variable Coefficient Standard Error t-Statistic Probability 

D(MTRNEXT) -0.0837 0.0186 -4.4982 0.0001

D(MTRTRANS) -6.0210 0.3744 -16.0808 0.0000

D(GDP/GDPFE) 0.0704 0.0241 2.9165 0.0057

DLOG(SP500Q4) 0.0161 0.0029 5.5864 0.0000

DLOG(STARTS) 0.0055 0.0022 2.5391 0.0149

R-squared 0.893821 Mean dependent variable 0.000695

Adjusted R-squared 0.883709 0.009309

Standard error of regression 0.003175 Akaike info criterion -8.566962

Sum of squared residuals 0.000423 Schwarz criterion -8.370138

Log likelihood 206.3236 F-statistic 88.38997

Durbin-Watson statistic 1.65308 Probability (F-statistic) 0

Key:

STARTS is the number of dwelling units started in structures with two or more dwellings.

Other variables are defined in Table 7.

SOURCE: CBO calculations.

 Standard deviation of dependent variable
7. The Durbin-Watson statistic failed to find significant serial correlation of residuals in any equation.

17

• The two versions of our equation in difference-in-ratio form (with and

without starts).

The statistics of fit that we used to compare models, in terms of annual

growth rates, are R-squared, adjusted R-squared, and root mean squared error over

the entire sample period, and root mean squared error in the 1990s. Those statistics

are displayed in Table 9.7

Examining those statistics leads to a number of conclusions. First, our

equations seem to do about as well overall as the current CBO and dollar-volume

equations, even though the latter have two distinct advantages: they are specified in

growth-rate form and the criterion is in terms of growth rates; and they use dummy

variables for 1986 and 1987, whereas ours do not. Second, across different models,

multifamily housing starts helped somewhat. Third, using the dollar-volume variable

in a growth-rate specification helped in the 1990s. Finally, although the SPUD

equations do not fit the entire period well, they do fit the 1990s well.

That final conclusion led us to experiment with our method of estimating

coefficients. We observed that the dollar value of capital gains realizations has

grown over time. We reasoned that the SPUD equations—which minimize the sum

of squared errors in the dollar changes in capital gains realizations—are like

weighted regressions that give more weight to more recent observations. We

therefore experimented with our equations using both weighted regressions and

Kalman-based, time-varying coefficients. The improvement was marginal, however,

and we judged that it did not justify the added complexity.

Although our first conclusion about the relatively good fit of our new model

was encouraging, other conclusions relating to the success of different specifications

in the 1990s left us open to alternative specifications of the model. Thus, we

continued to experiment based on out-of-sample comparisons and, as a result,

changed the preferred specification of our model.
TABLE 9: COMPARING IN-SAMPLE ERRORS OF EQUATIONS

Adjusted Root Mean Squared Error

R-squared R-squared 1952-1998 1990-1998

 Current CBO equations

 No error correction, no starts 0.705 0.661 0.134 0.140

 No error correction, starts 0.767 0.733 0.119 0.126

 Error correction, no starts 0.702 0.657 0.135 0.143

 Error correction, starts 0.777 0.737 0.117 0.131

 Dollar-volume equations

 No error correction, no starts 0.676 0.646 0.141 0.122

 No error correction, starts 0.766 0.737 0.120 0.114

 Error correction, no starts 0.699 0.663 0.136 0.132

 Error correction, starts 0.776 0.743 0.117 0.120

 No lags -3.147 -3.542 0.504 0.157

 Lagged SPUD -2.348 -2.850 0.453 0.110

 Lagged SPUD and gains -2.663 -3.321 0.474 0.115

 Without starts or constant 0.704 0.683 0.135 0.122

 With starts, no constant 0.756 0.732 0.122 0.119

SOURCE: CBO calculations.

NOTE: SPUD = stock price up and down; GDPFE = potential nominal GDP.

Growth-Rate Equations [DLOG(GAINS)]

Change in Dollar Gains (SPUD)

Change in Ratio of Gains to GDPFE
8. Care must be exercised in interpreting the out-of-sample RMSEs because the uncertainty surrounding

estimates of realizations changes from year to year. For example, uncertainty tends to be larger in earlier

years because fewer observations are available to estimate an equation’s coefficients. Thus, it may be

appropriate to weight each error by the uncertainty surrounding the forecast at the time. Comparing root

mean squared errors by decade controls for some differences in uncertainty.

18

V. OUT-OF-SAMPLE EVALUATION AND COMPARISONS

The purpose of the out-of-sample exercise was to determine the performance of

different models when there was more limited knowledge of future events at the time

each estimate was made than in the in-sample exercise. Both the in-sample and outof-sample exercises used the data set as it existed at the end of 1999. And both

exercises assumed that values were known for right-hand-side variables in the current

year and for lagged dependent variables in the previous year. The difference in the

two exercises was in how the models were estimated. For the in-sample exercise, the

models were estimated over the entire sample, so the fitted values of the dependent

variable at each date built in knowledge of events that occurred at future dates. For

the out-of-sample exercise, that problem of building in knowledge of future events

was limited by using recursive regressions. For example, regressions were estimated

from 1952 through 1959, and the coefficients were used to estimate capital gains

realizations for 1960. Next, the regressions were estimated from 1952 through 1960,

and the new coefficients were used to generate realization estimates for 1961. The

process was repeated through 1999. That process parallels the annual updating of

equations that CBO actually uses, although CBO has not used the same equations for

so long a period.

The versions of models selected for the initial out-of-sample comparison were

the same as those for the in-sample comparison, with one exception. We included

only one regression with error correction and examined its performance only for the

1990s. That exception was made because recursive regression is more difficult when

there is an error-correction term and because we knew from experience that equations

with error-correction terms produced large errors in the late 1990s.

We judged the performance of models by their out-of-sample root mean

squared errors (RMSEs).8

 If, over the full estimation period, a model’s recursively

estimated coefficients were unchanging, its in-sample and out-of-sample RMSEs

would be the same. Thus, a big deterioration in a model’s performance in this stage

of testing compared with its in-sample performance indicates coefficient instability.

We also investigated coefficient instability more directly by applying a Chow test to

the model’s coefficients estimated over the first and second halves of the sample and

by examining plots of the coefficients and their associated standard errors computed

in each year of the recursive regressions.
19

Our general conclusions from comparing accuracy and stability across

equations were that our difference-in-ratio model and the dollar-volume model

seemed best, but neither was clearly superior. Both models had lower errors than the

current CBO and SPUD models over the whole period. In particular, error correction

hurt accuracy in the late 1990s, judging by the CBO equation in that form, because

realizations did not revert back to their historical norm relative to GDP. Meanwhile,

the difference-in-ratio and dollar-volume regressions had similar root mean squared

errors. Our regressions were more accurate in the 1980s and over the 1980-1998

period, but the dollar-volume regressions were more accurate in the 1970s and over

the entire 1960-1998 period. The out-of-sample RMSEs are compared in Table 10.

Our difference-in-ratio regressions have more stable coefficients over the

1980s and 1990s than do the SPUD regressions. Figures 2 and 3 compare the

recursive coefficients from versions of each model. (The model underlying Figure

2 is shown in Table 8, and the one underlying Figure 3 is shown in Table 11.)

Note that when coefficients change in our difference-in-ratio equation, the

new values are still well within the 95 percent confidence intervals of previous

coefficient estimates. In contrast, the changes in coefficients in the SPUD equation

following the 1987 tax increase are well beyond the confidence intervals of

previously estimated coefficients. The coefficients change because tax terms in the

SPUD equation must treat the fallback in realizations in 1987 proportionately to

responses to other tax changes. That leaves a large part of the fallback to be

explained by other variables. Nevertheless, both the SPUD equation and ours pass

a Chow test when the sample is divided in halves. (The stability of the current CBO

and dollar-volume regressions could not be tested over the full period because their

inclusion of dummy variables essentially assumes a structural break in 1986 and

1987.)

Based on those general conclusions, we experimented with changing the

functional form and variables of our model. Our aim was to see whether we could

improve the model by borrowing from some successes of other models, especially

the dollar-volume model. Tables 12 and 13 show the in-sample and out-of-sample

fits, respectively, of several alternative versions of our model. (Those tables show

results for the new versions as additions to the results shown in Tables 9 and 10.)

In terms of functional form, we compared the performance of our equations

in their original difference-in-ratio form with difference-in-logarithm and percentage

change forms. Each equation used the explanatory variables from our original

equations plus a constant term. (We added the constant term because it improved the

in-sample fit and out-of-sample accuracy, although it worsened the White test scores

for homoskedasticity.) All three forms have similar adjusted R-squared statistics

over the full sample, but the difference-in-logarithm form is slightly better than the
TABLE 10: COMPARING OUT-OF-SAMPLE ROOT MEAN SQUARED ERRORS 

(Errors are actual growth rates of gains less estimates)

 Addendum:

 Forecast of

 Growth Rate

1960-1998 1980-1998 1960-1969 1970-1979 1980-1989 1990-1998 for 1999 

 Current CBO equations

 No error correction, no starts 0.295 0.200 0.147 0.493 0.237 0.148 0.198

 No error correction, starts 0.312 0.198 0.174 0.525 0.243 0.133 0.171

 Error correction, no starts 0.157 0.105

 Error correction, starts 0.146 0.093

 Dollar-volume equations

 No error correction, no starts 0.198 0.219 0.138 0.208 0.276 0.128 0.268

 No error correction, starts 0.195 0.214 0.159 0.190 0.272 0.121 0.231

 No lags 0.246 0.286 0.106 0.266 0.355 0.179 0.125

 Lagged SPUD 0.229 0.257 0.143 0.242 0.318 0.165 0.212

 Lagged SPUD and gains 0.313 0.361 0.217 0.296 0.472 0.165 0.168

 Without starts or constant 0.221 0.189 0.137 0.322 0.232 0.126 0.138

 With starts, no constant 0.224 0.180 0.160 0.328 0.218 0.125 0.122

SOURCE: CBO calculations.

NOTE: Data used in the forecasts differs slightly from those used in the in-sample fits. SPUD = stock price up and down; GDPFE = 

 potential nominal GDP.

Growth-Rate Equations [DLOG(GAINS]

Change in Dollar Gains (SPUD)

Change in Ratio of Gains to GDPFE
-1.2

-0.8

-0.4

0.0

0.4

0.8

65 70 75 80 85 90 95

D(MTRNEXT) ± 2 S.E.

-80

-60

-40

-20

0

20

65 70 75 80 85 90 95

D(MTRTRANS) ± 2 S.E.

-0.05

0.00

0.05

0.10

0.15

0.20

65 70 75 80 85 90 95

D(GDP/GDPFE) ± 2 S.E.

0.005

0.010

0.015

0.020

0.025

0.030

0.035

65 70 75 80 85 90 95

DLOG(SP500Q4) ± 2 S.E.

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

65 70 75 80 85 90 95

DLOG(STARTS) ± 2 S.E.

Figure 2: Recursive Regression Coefficients (and Two Standard Errors Up or Down)

 of Difference-in-Ratio Equation with Starts, by Year

SOURCE: CBO calculations.

NOTE: For details of the ratio equation, see Table 8.
-15

-10

-5

0

5

10

15

20

65 70 75 80 85 90 95

Constant ± 2 S.E.

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

65 70 75 80 85 90 95

SPUP ± 2 S.E.

-0.4

0.0

0.4

0.8

1.2

1.6

65 70 75 80 85 90 95

SPDOWN ± 2 S.E.

-15

-10

-5

0

5

65 70 75 80 85 90 95

D(CBOMTR) ± 2 S.E.

-10

-5

0

5

10

15

20

65 70 75 80 85 90 95

CBOPOS(1) ± 2 S.E.

Figure 3: Recursive Regression Coefficients (and Two Standard Errors Up or Down)

 of Simple SPUD Equation with Starts, by Year

SOURCE: CBO calculations.

NOTE: For details of the simple SPUD equation, see Table 11.
TABLE 11: SIMPLE SPUD MODEL

Dependent Variable: D(TOTGAIN2)

Method: Least Squares

Sample: 1952-1998

Variable Coefficient Standard Error t-Statistic Probability 

Constant 0.286711 3.116311 0.092003 0.9271

SPUP 0.326442 0.053031 6.155634 0

SPDOWN 0.555197 0.141336 3.928194 0.0003

D(CBOMTR) -11.74883 1.309981 -8.968706 0

CBOPOS(1) 12.13001 1.448326 8.375189 0

R-squared 0.863226 Mean dependent variable 9.437381

Adjusted R-squared 0.8502 42.57696

Standard error of regression 16.47897 Akaike info criterion 8.542335

Sum of squared residuals 11405.37 Schwarz criterion 8.739159

Log likelihood -195.7449 F-statistic 66.2692

Durbin-Watson statistic 2.213039 Probability (F-statistic) 0

Key:

D(TOTGAIN2) is change in capital gains, in billions of dollars

SPUP is the sum of the monthly increases in the S&P 500 index for those months with a net

 increase over the previous month

SPDOWN is analogous to SPUP for decreases in the S&P 500

D(CBOMTR) is the change in an average tax rate on capital gains

CBOPOS(1) is the increase in CBOMTR in the coming year when the rate increases; 0 otherwise

SOURCE: CBO calculations.

NOTE: SPUD = stock price up and down.

 Standard deviation of dependent variable
TABLE 12: COMPARING IN-SAMPLE ERRORS OF MORE EQUATIONS

Adjusted Root Mean Squared Error 

R-squared R-squared 1952-1998 1990-1998

 Current CBO equations

 No error correction, no starts 0.705 0.661 0.134 0.140

 No error correction, starts 0.767 0.733 0.119 0.126

 Error correction, no starts 0.702 0.657 0.135 0.143

 Error correction, starts 0.777 0.737 0.117 0.131

 Dollar-volume equations

 No error correction, no starts 0.676 0.646 0.141 0.122

 No error correction, starts 0.766 0.737 0.120 0.114

 Error correction, no starts 0.699 0.663 0.136 0.132

 Error correction, starts 0.776 0.743 0.117 0.120

 No lags -3.147 -3.542 0.504 0.157

 Lagged SPUD -2.348 -2.850 0.453 0.110

 Lagged SPUD and gains -2.663 -3.321 0.474 0.115

 Without starts or constant 0.704 0.683 0.135 0.122

 With starts, no constant 0.756 0.732 0.122 0.119

 Without starts, with constant 0.720 0.693 0.131 0.115

 With starts and constant 0.770 0.741 0.119 0.113

 S&P 500, without starts 0.716 0.689 0.132 0.121

 S&P 500, with starts 0.783 0.757 0.115 0.114

 Dollar volume, without starts 0.696 0.667 0.136 0.109

 Dollar volume, with starts 0.760 0.731 0.121 0.110

 Constant, no starts 0.732 0.707 0.128 0.119

 Constant, starts 0.766 0.738 0.120 0.115

SOURCE: CBO calculations.

NOTE: SPUD = stock price up and down; GDPFE = potential nominal GDP. The scaled growth rate and 

 percentage growth rate equations include a constant term. 

Percentage Growth Rate of Gains/GDPFE

Growth-Rate Equations [DLOG(GAINS)]

Change in Dollar Gains (SPUD)

Change in Ratio of Gains to GDPFE

Scaled Growth Rate [DLOG(GAINS/GDPFE)]
TABLE 13: COMPARING OUT-OF-SAMPLE ROOT MEAN SQUARED ERRORS IN MORE EQUATIONS

(Errors are actual growth rates of gains less estimates)

Addendum:

Forecast of

Growth Rate

1960-1998 1980-1998 1960-1969 1970-1979 1980-1989 1990-1998 for 1999 

Current CBO equations

 No error correction, no starts 0.295 0.200 0.147 0.493 0.237 0.148 0.198

 No error correction, starts 0.312 0.198 0.174 0.525 0.243 0.133 0.171

 Error correction, no starts 0.157 0.105

 Error correction, starts 0.146 0.093

Dollar-volume equations

 No error correction, no starts 0.198 0.219 0.138 0.208 0.276 0.128 0.268

 No error correction, starts 0.195 0.214 0.159 0.190 0.272 0.121 0.231

No lags 0.246 0.286 0.106 0.266 0.355 0.179 0.125

Lagged SPUD 0.229 0.257 0.143 0.242 0.318 0.165 0.212

Lagged SPUD and gains 0.313 0.361 0.217 0.296 0.472 0.165 0.168

Without starts or constant 0.221 0.189 0.137 0.322 0.232 0.126 0.138

With starts, no constant 0.224 0.180 0.160 0.328 0.218 0.125 0.122

Without starts, with constant 0.216 0.184 0.145 0.311 0.226 0.121 0.129

With starts and constant 0.219 0.176 0.167 0.317 0.214 0.121 0.116

S&P 500, no starts 0.188 0.157 0.192 0.234 0.180 0.126 0.216

S&P 500, starts 0.184 0.139 0.208 0.227 0.155 0.120 0.183

Dollar volume, no starts 0.223 0.249 0.150 0.230 0.327 0.113 0.285

Dollar volume, starts 0.215 0.231 0.175 0.221 0.298 0.116 0.242

Constant, no starts 0.206 0.143 0.259 0.245 0.158 0.124

Constant, starts 0.209 0.150 0.277 0.228 0.171 0.122

SOURCE: CBO calculations.

NOTE: Data used in the forecasts differ slightly from those used in the in-sample fits. SPUD = stock price up and down; 

 GDPFE = potential nominal GDP. The scaled growth rate and percentage growth rate equations include a constant term.

Percentage Growth Rate of Gains/GDPFE

Growth-Rate Equations [DLOG(GAINS)]

Change in Dollar Gains (SPUD)

Change in Ratio of Gains to GDPFE

Scaled Growth Rate [DLOG(GAINS/GDPFE)]
20

others in terms of out-of-sample root mean squared errors. It tends to have relatively

smaller errors in years when all of the equations make large errors. A Dicky-Fuller

test could not reject the presence of a unit root in the logarithm of the ratio of gains

to potential GDP, but it could in the first difference, suggesting that the difference-inlogarithm variable is stationary. 

In terms of variables, beyond the constant term, we experimented in our

difference-in-logarithm equations with substituting the dollar-volume variable for the

S&P 500 variable. Replacing the S&P 500 variable with the dollar-volume variable

was not a clear success. The equations with the S&P 500 fit slightly better over the

whole sample period. Their out-of-sample errors are smaller over the 1960-1998 and

1980-1998 periods and during the 1980s. The equations with dollar volume have

slightly lower errors in the 1990s and much lower errors in the 1970s. Equations

with either variable do not have autocorrelated residuals. The S&P 500 equations

pass the Chow test but fail the White test. The dollar-volume equations pass the

White test, but the equation without starts fails the Chow test and the one with starts

barely passes it. Moreover, in the recursive regressions, the S&P 500 equations have

more stable coefficients on the variables representing the business cycle and equity

net gains.

Based on those results, our preferred equation uses the difference-in-log form

(see Table 14). It includes a constant; the difference in the permanent tax rate (the

legislated tax rate for next year); the difference in the transitory tax rate (the current

rate less the permanent rate, squared but with the sign preserved); the difference in

the log of the ratio of GDP to potential GDP; the difference in the log of the S&P

500; and the difference in the log of multifamily housing starts. It has the highest

adjusted R-squared. It is most accurate over the entire period and over 1980 through

1998, although it does not dominate in the 1990s. It fails the White test but passes

the Chow test, and its coefficients are more stable over the 1980s and 1990s than

those of the other equations.

The coefficients of our preferred specification also seem reasonable. The size

and significance of the coefficient on DLOG(GDP/GDPFE) confirm that the

excessive size of the coefficient on GDP in the current CBO equations (shown in

Table 2) resulted from the sensitivity of realizations to the business cycle. The

coefficient on DLOG(SP500Q4) indicates that a 1 percent increase in the S&P 500

leads to a 0.7 percent increase in the growth rate of gains relative to potential GDP.

Finally, the tax terms imply a permanent elasticity of -0.47 and a transitory elasticity

of 1.8 for changes in tax rates like those in 1986.

Despite our model’s apparent good performance, we believe some caveats are

necessary. First, there is good reason to suspect that the coefficients on the transitory

tax term and multifamily housing starts are more uncertain than their standard errors
TABLE 14: PREFERRED EQUATION FROM IN-SAMPLE AND OUT-OF-SAMPLE TESTS

Dependent Variable: DLOG(TOTGAIN2/GDPFE)

Method: Least Squares

Sample: 1952-1998

Variable Coefficient Standard Error t-Statistic Probability

Constant -0.042628 0.019472 -2.189185 0.0343

D(MTRNEXT) -2.585064 0.675268 -3.828206 0.0004

D(MTRTRANS) -115.4395 13.61065 -8.481552 0

DLOG(GDP/GDPFE) 2.358513 0.871639 2.705837 0.0099

DLOG(SP500Q4) 0.736638 0.121054 6.085178 0

DLOG(STARTS) 0.218479 0.079682 2.741875 0.009

R-squared 0.802356 Mean dependent variable 0.020822

Adjusted R-squared 0.778253 0.24483

Standard error of regression 0.11529 Akaike info criterion -1.36398

Sum of squared residuals 0.544968 Schwarz criterion -1.127791

Log likelihood 38.05353 F-statistic 33.2887

Durbin-Watson statistic 1.910765 Probability (F-statistic) 0

Key:

D is the first difference operator: D[x(t)] = x(t) - x(t-1)

LOG(x) is the logarithm of x

TOTGAIN2 is capital gains realizations

GDPFE is nominal potential GDP

MTRNEXT is our permanent tax rate term

MTRTRANS is our transitory tax rate term

SP500Q4 is the average value of the S&P 500 index in the fourth quarter

STARTS is the number of dwelling units started in structures with two or more dwellings

SOURCE: CBO calculations.

 Standard deviation of dependent variable
21

indicate. The coefficient on the transitory tax term is estimated primarily from one

event: the large tax increase legislated in 1986 that took effect in 1987. Similarly,

the coefficient on multifamily housing starts is estimated mainly from one event: the

real estate collapse that occurred from the late 1980s to the early 1990s. Although

those two coefficients are estimated reasonably based on historical information, we

suspect they could be subject to large revisions if significant changes occurred in the

future to anticipated capital gains tax rates or real estate market conditions.

Another caveat is that the superiority of our equation largely comes in the

1980s, when it had an advantage. None of the other models had a way to forecast

out-of-sample the effects on realizations of the anticipated 1987 tax increase. (The

SPUD model could anticipate the reaction in 1986 but only part of the reaction in

1987.) Thus, it is fair to argue that knowledge of what happened in 1986 and 1987

guided our specification. However, the specification still has an advantage over the

others. Should changes in tax rates be announced in the future, our equation can

estimate the effect on realizations, even if that estimate is imprecise. And the new

observation will allow the coefficient estimate to be refined. In contrast, equations

that use dummies for 1986 and 1987 cannot generate such estimates, and the SPUD

equation will have trouble estimating the effect of the tax change in the year

following the change. 
22

VI. REAL-TIME EVALUATION AND COMPARISONS

Out-of-sample comparisons are better than in-sample comparisons at indicating how

analysts would have fared using alternative models. The advantage of out-of-sample

comparisons is that they limit the degree to which unknown future developments can

affect the realization estimates of alternative models. But out-of-sample comparisons

do not go far enough. In this section, we examine how analysts would have fared in

estimating realizations using different models if they had only the information

available at the date the estimates were made.

Such real-time comparisons differ from out-of-sample comparisons in two

fundamental ways. First, out-of-sample model estimation is based on the data set as

it currently exists. However, that data set has been revised several times and can

differ greatly from the set that analysts had when they made their estimates of

realizations. In real time, by contrast, the models are estimated as of November 30

of each year based on the data the modeler would have had at that time. Second, the

realizations estimates from out-of-sample model estimation assume that the lagged

dependent variable and the right-hand-side variables are known with certainty. But,

of course, that is not actually the case. Realizations in the previous year have not

been fully tabulated by the Internal Revenue Service as of November 30, so that

figure must be extrapolated. Meanwhile, right-hand-side variables are known for

only part, if any, of the current year and must be projected through the end of the

year.

The main questions we posed in our real-time experiment were: How well

would CBO have estimated the current-year growth of capital gains in the 1990s had

it had access to the models we consider? Which of those models would have

performed best? And would that model have improved on the estimates that CBO

actually made? Thus, for the extrapolations of realizations in the previous year, we

used the ones CBO actually made in each December from 1991 through 1998. For

1990, we extrapolated the way that CBO would have, assuming it used the same

method it did in later years. For projections of right-hand-side variables for the year,

we used those actually produced by CBO’s Macroeconomic Analysis Division

(MAD) for the variables they project. For variables not projected by MAD, such as

stock prices or value of volume, we used simple time-series methods of calculation.

In particular, we assumed that each of those series follows a continuous-time random

walk with drift. Thus, for each series, we took the value at the end of November and

increased it by its historical average monthly gain (through that November) to get its

December value.

Because real-time considerations about the reliability of data and the

predictability of right-hand-side variables could change the ranking of equations in
23

terms of estimating accuracy, we included four versions of our difference-inlogarithm specification:

• With and without multifamily housing starts, and

• With an S&P 500 variable or a dollar-volume variable.

For comparison, we also included four of CBO’s current forecasting

equations, with and without starts and with and without error-correction terms.

Background information for the real-time comparisons appears in Table 15.

The first line shows the annual growth rate of realizations based on the data provided

by the IRS for complete years. Although the actual figures computed by the IRS are

available after a considerable delay (and would not be available in real time), they

still are the ones that CBO is trying to estimate. Thus, the accuracy of realizations

estimates is measured with respect to the figures in the first line. However, because

of the delay, an actual figure for 1999 is not yet available, and the figure for 1998

could change slightly.

The second line of Table 15 shows the baseline estimates of growth in

realizations that CBO made in early December of each year. In 1991 through 1999,

those estimates relied on input from the current CBO equations as well as on

judgment. In 1990, the estimate was based on a different model.

The next five lines show the estimates of the four current CBO equations and

their average. Those estimates were made in 1991 through 1999 and provided the

starting point for the forecast shown in the second line of the table. The average of

those equations differs noticeably from the baseline estimate in 1991, 1994, 1997,

and 1999, when additional considerations were incorporated.

The next five lines show revised estimates from the current CBO equations,

which we reestimated to make them more comparable with our new equations. The

current equations’ sample period was extended from 1955 back to 1952, the same as

for the new equations. Also, the value of corporate equities in the current year was

projected using the growth rate of the S&P 500, as we projected that index for

December in our new equations. Originally, corporate equities were updated using

the New York Stock Exchange composite index without any drift added for the

remainder of the year.

The last five lines of Table 15 parallel the previous five lines, but the

estimates are from our new equations. They predict much stronger growth in

realizations in 1999 than the current equations or CBO’s baseline estimate do. Only

time will tell which estimate is closer to the actual outcome.
TABLE 15: ACTUAL AND ESTIMATED GROWTH RATES OF CAPITAL GAINS REALIZATIONS

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999

Actual -0.196 -0.098 0.135 0.202 0.003 0.179 0.447 0.399 0.234 n.a.

CBO Baseline 0.117 0.100 0.102 0.061 0.034 0.150 0.089 0.464 0.130 0.136

Current CBO Equations

Dependent Variable = DLOG(GAINS)

 No error correction, no starts n.a. 0.052 0.111 0.061 0.041 0.140 0.105 0.534 0.201 0.152

 No error correction, starts n.a. -0.053 0.056 0.036 0.131 0.161 0.103 0.464 0.182 0.135

 Error correction, no starts n.a. 0.052 0.093 0.093 0.055 0.170 0.058 0.301 0.068 0.028

 Error correction, starts n.a. -0.029 0.102 0.077 0.152 0.175 0.073 0.294 0.073 0.028

 Average n.a. 0.005 0.090 0.067 0.095 0.162 0.085 0.394 0.129 0.084

Reestimated CBO Equations

Dependent Variable = DLOG(GAINS)

 No error correction, no starts -0.085 -0.019 0.083 0.055 0.058 0.188 0.144 0.460 0.284 0.121

 No error correction, starts -0.136 -0.127 0.004 0.022 0.141 0.203 0.141 0.404 0.255 0.099

 Error correction, no starts -0.136 -0.047 0.019 0.058 0.046 0.183 0.097 0.308 0.132 0.026

 Error correction, starts -0.152 -0.115 0.032 0.041 0.142 0.190 0.112 0.293 0.124 0.024

 Average -0.127 -0.077 0.035 0.044 0.097 0.191 0.124 0.366 0.199 0.067

New Equations

Dependent Variable = DLOG(GAINS/GDPFE)

 S&P 500, no starts -0.114 0.046 0.073 0.102 0.054 0.245 0.175 0.507 0.252 0.197

 S&P 500, starts -0.140 -0.052 0.076 0.073 0.148 0.243 0.180 0.472 0.239 0.168

 Dollar volume, no starts -0.081 0.000 0.081 0.200 0.090 0.186 0.148 0.380 0.213 0.269

 Dollar volume, starts -0.114 -0.091 0.082 0.155 0.183 0.190 0.156 0.362 0.204 0.229

 Average -0.112 -0.024 0.078 0.133 0.119 0.216 0.165 0.430 0.227 0.215

SOURCE: CBO calculations.

NOTE: GDPFE = potential nominal GDP; n.a. = not applicable.

a. The current CBO equations were first used in 1991.

a
9. The relative ranking of the equations remains unchanged when their target is growth from the preliminary

level of gains in the previous year to the actual level of gains reached in the current year. Recall that prioryear realizations are based on incomplete tax return information and therefore contain errors. Thus, even

if an equation accurately predicts the growth rate in gains as shown by final tax return data, it can still miss

the level of gains reached in the current year because its growth is from the wrong base. An alternative

target for comparing the equations is how well they grow from the estimated prior-year base to the correct

current-year gain. When the equations are judged by that alternative standard, their root mean squared

errors change slightly, but the rankings do not change. The root mean squared error for the average of the

new equations falls from .114 to .113, and that for the reestimated CBO equations rises from .132 to .136.

The changes in root mean squared errors are small because the errors in the prior-year level of gains are

small in most years. The relative rankings are also preserved because the errors in prior-year gains appear

to be unrelated to errors that the equations make in predicting current-year growth.

24

Table 16 compares the degree to which the estimates shown in Table 15 differ

from the actual growth rates. That comparison of errors yields three major

conclusions:

• The new equations perform best overall;

• The average of the four new equations slightly outperforms any one of the

new equations individually; and

• The reestimated CBO equations do better than the current CBO ones, which

in turn do better than CBO’s baseline estimates.9

Compared with the reestimated equations, the average improvement in

accuracy from the new equations is on the order of 15 percent. The improvement is

not uniform, however, since the current equation without multifamily housing starts

or error correction does only slightly worse than some of the new equations. For all

of the specifications, multifamily housing starts help only in our new equation with

the S&P 500. The average estimate of our new equations improves on the baseline

estimates that CBO made in the 1991-1998 period (when CBO was using the current

equations) by reducing the root mean squared error from .160 to .117, roughly a 27

percent improvement. Most of that improvement comes from reducing the root mean

squared error of the current-equation estimates from .151 to .117, roughly a 23

percent improvement.

The average estimate from the four new equations does better than any of the

new equations individually. The equation with dollar volume but not starts does the

best of the four, particularly from 1991 through 1998. The equation with the S&P

500 and starts, chosen as our preferred equation on the basis of its in-sample and outof-sample performance, comes in second place. The superiority of the dollar-volume

equations in the real-time comparisons is consistent with their superiority for the

1990s in both in-sample and out-of-sample comparisons. However, those equations

did not dominate out-of-sample in the period since the 1960s and did particularly

poorly in the 1980s. Moreover, the fact that the average did the best suggests that all
TABLE 16: ERRORS IN ESTIMATES OF GROWTH RATES OF CAPITAL GAINS REALIZATIONS

1990 1991 1992 1993 1994 1995 1996 1997 1998 1990-1998 1991-1998

CBO Baseline -0.314 -0.198 0.033 0.141 -0.031 0.029 0.358 -0.064 0.104 0.184 0.160

Current CBO Equations

Dependent Variable = DLOG(GAINS) 

 No error correction, no starts n.a. -0.150 0.024 0.141 -0.038 0.039 0.342 -0.135 0.032 n.a. 0.151

 No error correction, starts n.a. -0.046 0.080 0.165 -0.128 0.019 0.344 -0.064 0.051 n.a. 0.149

 Error correction, no starts n.a. -0.150 0.043 0.109 -0.052 0.009 0.389 0.099 0.166 n.a. 0.169

 Error correction, starts n.a. -0.069 0.033 0.125 -0.149 0.005 0.374 0.105 0.161 n.a. 0.166

 Average n.a. -0.104 0.045 0.135 -0.092 0.018 0.363 0.005 0.104 n.a. 0.151

Reestimated CBO Equations

Dependent Variable = DLOG(GAINS)

 No error correction, no starts -0.111 -0.080 0.052 0.147 -0.055 -0.009 0.303 -0.061 -0.050 0.127 0.128

 No error correction, starts -0.060 0.029 0.131 0.179 -0.138 -0.024 0.306 -0.004 -0.022 0.136 0.143

 Error correction, no starts -0.060 -0.052 0.116 0.144 -0.043 -0.003 0.350 0.091 0.102 0.143 0.150

 Error correction, starts -0.044 0.017 0.103 0.161 -0.139 -0.011 0.336 0.107 0.109 0.147 0.155

 Average -0.069 -0.021 0.101 0.158 -0.094 -0.012 0.324 0.033 0.035 0.132 0.138

New Equations

Dependent Variable = DLOG(GAINS/GDPFE)

 S&P 500, no starts -0.083 -0.144 0.062 0.100 -0.051 -0.066 0.273 -0.107 -0.018 0.122 0.126

 S&P 500, starts -0.057 -0.046 0.059 0.128 -0.145 -0.064 0.267 -0.072 -0.006 0.119 0.124

 Dollar volume, no starts -0.115 -0.099 0.054 0.002 -0.087 -0.006 0.299 0.019 0.020 0.117 0.118

 Dollar volume, starts -0.082 -0.007 0.053 0.047 -0.180 -0.011 0.291 0.038 0.029 0.121 0.125

 Average -0.084 -0.074 0.057 0.069 -0.116 -0.037 0.283 -0.031 0.006 0.114 0.117

SOURCE: CBO calculations.

NOTE: GDPFE = potential nominal GDP; n.a. = not applicable.

a. The current CBO equations were first used in 1991.

Root Mean Squared Error

a
25

of the new equations should be carried into the future until finer discriminations

among them are possible.

Finally, the reestimated CBO equations do better than the original baseline

for two reasons. First, the reestimated equations are more accurate than the ones

estimated at the time (the RMSE of the average falls from .151 to .138), apparently

because of the longer sample period and alternative projection of current-year equity

values. Second, adjustments made to the current-equation estimates to reflect

additional considerations worsened the estimates (those adjustments raised the

RMSE from .151 to .160). Adjustments made in 1991 and 1997 increased the error,

and one made in 1994 reduced it, but not by enough to offset the other two. It is too

early to know how the adjustment in 1999 affected accuracy.