MEASURING P m RESPONSES TO 

CAPITAL GAINS TAX CHANGES IN PANEL DATA* 

Leonard E. Burman and 

William C. Randolph 

Congressional Budget Office 

U.S. Congress 

OTA Paper 68 August 1994 

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

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 Papers and Briefs should not be quoted without permission from 

the authors. Additional copies of this publication may be purchased from the National Technical 

Information Service, 5285 Port Royal Road, Springfield, VA 22161 

Office of Tax Analysis 

U.S. Treasury Department, Room 1064 

Washington, DC 20220 

* Forthcoming, American Economic Review. This paper was written while Randolph 

was a staff member of the Office of Tax Analysis, U.S. Treasury Department. 
Measuring Permanent Responses to 

Capital Gains Tax Changes in Panel Data 

By LEONARD E.BURMANAND WILLIAM C. RANDOLPH.

This paper uses panel data and information about direrences in state tax rates to 

separate the efects of transitory and permanent tax rate changes on capital gains realizations behavior. The efect of permanent change isfound to be substantially smaller than the efect of transitory change. The estimated diference is even larger than past diferences between estimatesfrom carejkl micro data studies, which have 

primarily measured the transitory efect, and time series studies, which haveprimarily measured (at best) the permanent efect. Our results thus resolve a longstanding conflict between micro data and time series studies of how marginal tax rates afect capital gains realizations behavior. 

"Observe due measure, for right timing is in all things

the most important factor" --Hesiod (700 BC) 

For more than forty years, policy analysts and economists have debated about how capital

gains realizations respond to changes in capital gains tax rates. (See, e.g., Lawrence H. Seltzer,

1951) The issue has received attention in part because, if realizations of capital gains are 

responsive enough, the tax rate on capital gains could be cut at no cost to the Treasury. But it 

is also an important issue for tax reform because some argue that the welfare cost of the capital

gains tax could be large relative to the tax revenues collected if realizations are very sensitive 

to tax rates. (Patric H. Hendershott et al., 1991) 

The debate has been fueled by an array of disparate statistical estimates of the elasticity of 

capital gains realizations with respect to the marginal tax rate on capital gains. The evidence 

from time series appears entirely inconsistent with the evidence from individual tax return data. 

Time series studies have generally found that capital gains are relatively unresponsive to tax 

rates. Estimates based on micro data, however, generally suggest that realizations are highly

elastic. 

These empirical estimates are viewed with great skepticism by many who have studied the 

issue. Some authors of time series studies (Alan J. Auerbach, 1989; Jonathan Jones, 1989;

Robert Gillingham and John S.Greenlees, 1992) have discounted their findings because they are 

subject to intractable aggregation biases and are extremely sensitive to sample period and 

seemingly minor changes in specification. Estimates from micro data studieshave been even less 

robust. 

'Burman: Congressional Budget Office, U.S. Congress, Washington, DC 20515; Randolph: Office of Tax Analysis, 

U.S. Treasury, Washington, DC 20220. We are grateful to B.K. Atrostic, Jerry Auten, Charley Ballad, Joe Cordes, 

Glenn Hubbard, Jody Magliolo, Randy Mariger, Jim NUMS,Larry Ozanne, R.P. Trost, Jenny Wahl, and seminar 

participants at George Washington, Georgia State, Maryland, Michigan, and Northeastern, and three anonymous referees 

for helpful comments and suggestions. Jim Cilke and Gordon Wilson developed the tax calculators. Views expressed 

do not necessarily represent the views or policies of the Congressional Budget Office or the Department of the Treasury. 
More fundamentally, some authors of micro data studies have recognized that their 

estimates may systematically overstate the long-term response to tax changes. Indeed, the 

seminal empirical study of the effect of tax rates on realizations of capital gains raised this 

caveat: 

An individual whose tax rate varies substantially from year to year will 

tend to sell more when his rate is low. To the extent that low rates in 

1973are only temporarily low, our estimates will overstate the sensitivity of selling to the tax rate. We have no way of knowing how imponant this 

is. (Martin Feldstein et al., 1980, p. 785. Emphasis added.) 

Such timing behavior is very important. The Tax Reform Act of 1986 (TU)created a 

natural experiment to test the hypothesis that timing matters. TRA was passed by Congress at 

the end of September, 1986. It turned a "permanent" 20 percent maximum tax rate, in effect 

since 1981, into a temporary rate, to be replaced by a higher maximum capital gains rate of 28 

percent in 1987. In response, long-term capital gains on corporate stock in December 1986were 

nearly seven times their level in December 1985. (Burman et al., 1993) 

Timing behavior probably explains why micro data studies have produced such large

elasticity estimates.' As the transitory component of individuals' taxable income varies, it 

provides them with opportunities to time capital gains realizations in years when tax rates are 

relatively low. In a particular year, those with the lowest tax rates, other things constant, would 

be those with the largest capital gains realizations. As a result, a regression based on micro data 

is likely to measure a negative correlation between marginal tax rates and capital gains

realizations. But, without more information, it is impossible to determine how much of the 

measured correlation represents purely transitory timing behavior. Much of the policy debate,

however, has centered on the permanent or long-term response to statutory tax changes2 

To distinguish permanent from transitory tax effects, we define a "permanent tax rate" as 

the taxrate on long-term capital gains, purged of individual and aggregatetransitory effects. We 

estimate the relationship between capital gains and permanent tax rates in a panel of tax returns 

using an instrumental variables estimator that also accounts for the endogeneity of tax rates and 

self-selection. Our instrument for permanent tax rates is the maximum combined federal and 

state tax rate on long-term capital gains. This instrument, which only varies among states, 

removes individual transitory effects because it is uncorrelated with transitory variations in 

individuals' income. We remove aggregate transitory effects by using time dummies. 

Our estimates imply that the elasticity of capital gains realizations with respect to permanent 

tax changes is much smaller than the transitory response. Our point estimates of permanent tax 

effects are smaller in absolute value than most estimates from time series. Our estimates of 

transitory tax effects are larger than estimates from previous studies based on micro data. 

1

See Gerald E. Auten and Charles T. Clotfelter (1982), Joseph E. Stiglitz (1983), Jane G. Gravelle (1987, Auerbach 

(1988), Auten et al. (1989), Joel Slemrod and William Shobe (1990), and Donald W. Kiefer (1990) for more discussion 

on this point. 

2Auerbach (1989) and Gravelle (1991) have questioned the large elasticities found in most micro data studies on 

conceptual grounds. They argue that such large elasticitieswould imply that even modest changes in tax rates could cause 

realizations to exceed accruals. 

2 
I. The Decision to Realize Capital Gains 

There are two inherent problems in measuring taxpayer responses to capital gains tax 

changes: standard theoretical models do not explain why people realize significant amounts of 

taxable capital gains, and some variables that would enter almost any theoretical model are not 

observed in available data. Nonetheless, the typical empirical model may be interpreted as a 

reduced form, given data limitations, to test the most general implications of theory. Our 

analysis extends the basic empirical model to permit identification of a key policy parameter, the 

effect of permanent changes in tax rates. 

The capital gains tax is relatively easy to avoid. Tax on an asset's gain or loss is not due 

until the asset is sold, and may be avoided entirely if an asset is held until death or donated to 

charity. Stiglitz (1983) showed that, by borrowing, hedging, accelerating losses, and deferring

gains, capital gains taxes can be avoided altogether if capital markets are perfect and transactions 

are costless. He concluded that the existence of substantial taxable capital gains realizations 

implies that the underlying assumptions of his model must be violated in practice. 

George M. Constantinides (1984) showed that realizing gains on stocks as soon as they

qualify for preferential long-term tax rates may be optimal for very volatile stocks with low 

transaction costs. Yves Balcer and Kenneth L. Judd (1987) showed that, if borrowing and 

liquidity constraintsare binding and options markets do not exist, capital gains would be realized 

following a LIFO strategy to maximize the benefits of deferral. However, none of these models 

would explain the $100 to $200 billion in taxable gains reported in a typical year. 

Kiefer (1990) and Burman and Randolph (1992) developed models in which capital gains

realizations occur because capital markets are limited--there are liquidity constraints and no 

options markets--and individuals believe they can beat the market by trading. The latter study

also showed that transaction costs could be important. As well as characterizing a long-run

equilibrium in which significant amounts of capital gains could be realized, these analyses also 

shed light on the transition path from one steady state to another after tax laws change.

Conventional wisdom holds that the short-run response to a permanent cut in capital gains tax 

rates would be larger than the long-run response because of an immediate "unlocking" effect. 

Taxpayers holding assets to avoid capital gains tax suddenly flood the market with these assets 

when the tax rate is lowered because the tax cost of selling the assets is reduced. However, this 

conventional view ignores the fact that the cost of selling assets is also an increasing function of 

accrued gain as a fraction of asset value. On average, this fraction would be higher immediately

after a capital gains tax cut than it would be in the new steady state. The high level of accrued 

gains will initially increase the cost of asset sales relative to the steady state, and will therefore 

dampen adjustment in the short run. If the initial level of unrealized accruals is high enough,

the short-run increase in realizations of capital gains could actually be smaller than the long-run

increase. 

The response to temporary reductions in capital gains tax rates is clearer. A temporarily low capital gains tax rate provides taxpayers with an opportunity to gain from timing. A 

temporary tax cut reduces the tax cost of selling now, but leaves the tax cost of selling in the 

future unchanged. In contrast, a permanent tax cut reduces the tax cost of selling at any time. 

Thus, realizations of capital gains will be higher under a transitory rate cut than under a 

permanent cut, as illustrated by the response to TRA. (Paul J. Bolster et al., 1989; Burman et 

al., 1993) 

In micro data, variations in capital gains tax rates include both permanent and transitory 

components. The permanent component results from expected differences in earnings capacity, 

3 
sources of income and deductions, and because capital gains tax rates vary across states. It may

change when tax laws change. The transitory component results from tax planning and 

temporary changes in income and deductions. The tax law may also cause aggregate transitory changes if the statutory change is anticipated or if a new tax law is phased in over several years.

An empirical model should allow for the possibility that people respond differently to changes

in the permanent and transitory components. 

II. Empirical Model of Permanent and Transitory Tax Effects 

An individual decides whether to sell specific portfolio assets and, incidentally, whether 

to realize capital gains. Capital gains enter the decision because the tax price of selling an asset 

is the product of the capital gains tax rate and the share of asset value that is a capital gain.

Assets with relatively larger accumulated gains are more costly to sell than assets with smaller 

accumulated gains. Unfortunately, our panel of tax returns from 1979 to 1983only includes total 

capital gains and losses with no detail about sales of specific assets. Thus, like all previous

empirical studies of capital gains, we estimate a reduced form relationship between total longterm capital gains and factors that may affect the decision to sell assets with capital gains. 

For taxpayers who choose to realize capital gains, we model the relationship between 

capital gains and tax rates as follows: 

where g is realized capital gains by an individual at time t, X is a vector of predetermined and 

exogenous variables, 7, is the permanent tax rate, 2,is the current tax rate in year t, yo, yl,yz,

and y3, are fixed parameters, and e2is a random error term. This semi-log functional form has 

been used in most empirical capital gains research. It implies that the elasticity of capital gains

realizations with respect to the marginal tax rate is an approximately linear functionm3 

The decision of an individual to realize capital gains depends on the costs and benefits of 

realizing gains, the size and composition of the portfolio, and preferences. Taxes affect the costs 

and benefits of selling. The cost of selling depends on the effective marginal tax rate on capital

gains and on the size of the average accrued gain. Equation (1) separates the marginal tax rate 

into permanent and transitory components. The permanent tax rate is the tax rate purged of its 

individual and aggregate transitory components. It is the expected (normal) tax rate in a typical 

year given federal and state tax laws and normal levels of income for each individual. The 

remaining transitory component represents the tax cost of selling when the tax rate is unusually

high, or holding when the tax rate is unusually low. The lagged tax rate, is also included as a 

proxy for the unobservable size of accrued gains. For example, if the previous year's tax rate 

was unusually high, then accrued gains should be larger than usual because realizations would 

have been postponed. 

Other variables summarizing individual differences are included in X. The cost of selling

depends on transaction costs, so the composition of the portfolio is important. Lagged data from 

individual tax forms provide indirect informationabout whether the portfolio is likely to include 

real estate or business property, which is relatively costly to sell.4 Those data, as well as lagged 

3

In estimation, we also tested the assumption that the elasticity is approximately constant. This alternative does not 

affect the empirical results substantially. 

%e data are discussed in Section III. 

4 
data on sources of capital income, also allow us to create proxies for wealth, which represents the potential size of accrued unrealized gains, and for the share of wealth held as corporate stock. 

Sales and purchases of assets are a part of life-cycle consumption decisions. Thus, 

permanent and transitory income, age, marital status, and family size may be important

determinants. Panel data from tax returns allow us to estimate permanent and transitory income,

and age data are matched from social security records. In addition, regional dummies are 

included to control for regional differences in investment preferences. Time dummies are 

included to control for the aggregateeconomic factors that affect investment opportunities. These 

dummies also control for the average effect of tax law changes, as occurred in 1981. 

We account for the decision to realize a capital gain as well as the level of capital gains.

Our full empirical specification in Equations (2)-(4) represents (1) as a generalized tobit model,

and also accounts for the endogeneity of current marginal tax rates.' The tax terms are 

rearranged algebraically to simplify estimation. 

(3) lng= 

and 

where I' is a latent indicator of the decision to realize capital gains, the a and p terms are 

unknown parameters, and and E, are normally distributed error terms, uncorrelated with X, 

T ,or T,-~, such that E(q~.~?=cr~ for ij=1,2. The combined federal and state marginal tax rate 

&nction,f, is a known nonlinear function of capital gains and 2,a vector including income items 

from various sources, deductions, exemptions, transfers, carried over taxlosses and credits, and 

taxpayer filing status. 

A. Estimation Procedure 

We extend the instrumental variables procedure developed by Lung-Fei Lee et al. (1980) 

to allow for the presence of an unobserved variable, zP,and an endogenous variable, zt, in both 

the criterion function, (2), and the level equation, (3). The procedure is similar to the two-step

regression estimation method developed by James J. Heckman (1976), except that fitted values 

are used in place of T, and T~.The fitted value, zP,is created by regressing 2,on X,T~;,and T,,

the maximum combined federal and state tax rate in each individual's state. The fitted value, 

9,, is created by regressing T, on X,T~-~, T,, and T~, a "first-dollar" marginal tax rate on capital

'A separate appendix shows that correcting for endogeneity and sample selection is especially important in this type of 

model. Failure to properly account for these problems may explain much of the volatility in previous research on capital 

gains. The appendix is available upon request. 

'The yparameters in (1) have a simple relationship to the p's in (3): y,=~,+p,+~,; y2=p2+p3;y3=-ps. 

5 
gains. The first-dollar marginal tax rate is computed by setting g and the other sources of 

income and deductions that are jointly determined with g equal to zero.' 

The parameters of (2) are estimated by probit maximum likelihood with the fittedTp and T,

used in place of the actual values. The level equation, (3), is estimated by least squares using

the sample of realizers, including the estimated inverse Mills ratio as a regressor to control for 

sample selectivity.* For estimation of (3), values for zP and 7, are reestimated for the sample

of realizers including the inverse Mills ratio as an additional variable in the fitted equations. The standard errors are corrected using a formula derived by Lee et al.9 

B. Consistency of the IV Estimator 

Previous micro-data studies, which lacked appropriate instruments for T,, could only have 

produced consistent estimates of tax effects if transitory and permanent responses are the same. 

Under this condition, our estimation procedure would produce consistent estimates using almost 

any exogenous instruments for the permanent and transitory tax rates. However, if transitory

and permanent responses are different, then appropriate instruments for T, and 2,are essential to 

estimate permanent and transitory tax effects consistently. 

The estimation problem is unusual and interesting because we need to estimate the effects 

of two unobservable components of the tax rate. If T, is defined as 

where pt is transitory deviations in tax rates, then both T~ and pt enter (1) as explanatory

variables." This problem is similar in form to an errors-in-variable model. However, in the 

standard errors-in-variables model, only the systematic component, T,, would enter the model as 

an explanatory variable. To consistently estimate the effect of T,, we need an exogenous

instrument that is correlated with T,, but uncorrelated with pt, conditional on X and Tt-l. 

Although much of the variation in T, is related to the other exogenous variables, especially 

permanent income, wealth, and the portfolio mix, differences in state tax law provide an 

7

The firstdollar marginal taxrate is computed by setting long-term capital gains and other income and deduction items 

that are likely to be endogenous equal to zero and then computing the marginal tax rate on a defined long-term capital 

gain. This instrument retains a substantial amount of variation independent of the other explanatory variables because 

marginal tax rates are a known nonlinear function of numerous exogenous factors that do not directly affect capital gains, 

including consumer and mortgage interest deductions, contributions to pensions and IRAs, property taxes, certain health 

expenses, business and employment expenses, paid alimony, and many other deductions and adjustments to income. 

me inverse Mills ratio is computed based on the fitted values from the probit step. 

me standard error estimates may be understated because the formula does not account for the use of instrumental 

variables in the probit equation. To check the standard errors, we randomly split the sample into 10 parts and estimated 

the parameters for each subsample. The standard errors of the sample mean of the 10 estimates were very close to those 

produced by the formula. 

'we ignore sample selection in this discussion to focus on the key estimation problem. The lV results for the linear 

model, (l),are extended to the full model with truncation, (2)-(3), in a separate appendix. 

6 
exogenous source of variation that is easily measured.” Moreover, the variation in state taxes 

is closely related to an important policy question: how do realizations differ under different tax 

laws? Because state income taxes tend to be less graduated than the federal tax schedule, most 

gains are realized by taxpayers in the top state tax brackets. Thus, the top combined federal and 

state tax rate (z,) captures most of the important differencesin statutes, and does not vary amon 

individuals within a state. It is thus unlikely to be correlated with the transitory component, pt.i4 

To consistently estimate the transitory effect 0.3, we need a second exogenous instrument 

that is correlated with z, but uncorrelated with zp, conditional on X,z6], and z,. Our instrument 

has been used in various forms in most previous micro data studies of capital gains: the firstdollar tax rate (zo). Because marginal tax rates are a highly nonlinear function of many variables 

that do not directly affect capital gains (see footnote 7),this instrument captures much of the 

variation in zt, but is purged of its endogenous components. Further, zo is unlikely to be 

correlated with zp, conditional on zt-l,z, and the variables included in X. 

The standard errors-in-variables model assumes that the random component (pJ is 

uncorrelated with the X variables--an unwarranted assumption in our model. Transitory tax 

differences may well be correlated with such X variables as transitory income. As a result of 

this correlation and the nonstandard form of our estimator, the coefficients on the X variables 

may be inconsistent, reflecting a combination of the direct effect on gains (yo)and indirect effects 

through correlation with pt. While this may make interpretation of the effects of other variables 

more difficult, it does not affect the estimates of permanent and transitory tax effects. 

Under our assumptions, it canbe shown that the estimates of permanent and transitory tax 

coefficients in (1) will approach the following limits:I3 

and 

(7) 

11Others have used state variation to identify permanent effects of statutory changes in a panel or cross-section in 

different contexts. See Daniel Feenberg (1987) for an application to charitable contributions and David Neumark and 

William Washer (1991) for estimating the effects of minimum wages. William T. Bogart and William M.Gentry (1993) 

also use state data to estimate capital gains tax effects, but their state-level aggregate data averages out individual 

differences. 

12Theinstrument could be endogenous if the choice of state depends on capital gains tax rates. While we consider it 

unlikely that state tax rates on capital gains are very important to residential decisions, we test for this possible source 

of bias in our estimation. 

I3Probabilitylimits for all the parameters are derived in a separate appendix. The expression cov(x,y 1 z) is defined to 

be the partial covariance between x and y given z, Le., after the linear influence of z is removed from x and y. 

7 
Equations (6) and (7) show that the probability limits for the permanent and transitory coefficient 

estimates are weighted averages of the true values of the coefficients. If, as assumed, the state 

tax rate instrument, T,, is correlated with zP,and uncorrelated with p,, conditional on the other 

variables, (6) implies that the estimated permanent coefficient is consistent because 8, =O. 

Similarly, (7) implies that, if the first-dollar instrument, T,, is correlated with p,, and 

uncorrelated with zP,.conditional on the other variables, then the estimated transitory coefficient, 

yz, will also be consistent because 8, = O. 

Under the null hypothesis that the permanent and transitory tax coefficients are the same,

(6) and (7) imply that the estimates are consistent even if 8, and 8, are nonzero. Under the 

alternative hypothesis that y1#y2,the estimated difference between y, and y2 is biased toward 

zero if 8, or 8, is nonzero because both must lie between zero and one. Thus, even if the 

assumptions for consistency are violated, our estimates provide a conservative test of the 

hypothesis that y1=y2,which is the key assumption required for the validity of previous micro 

data studies. 

C,Altemative Estimators 

Two studies (Auten and Clotfelter, 1982, and U.S. Department of the Treasury, 1985)

attempted to measure the permanent tax rate directly from panel data by using three-year 

averages of marginal taxrates on capital gains. The fundamental problem with this approach is 

that a three-year average of federal income tax rates would be correlated with the transitory 

component of the tax rate. Thus, such a proxy cannot be used to estimate separately the effects 

of permanent and transitory tax rates since it is, itself, a combination of the two. 

Slemrod and Shobe (1990) used a fixed-effects model to control for differences in 

permanent tax rates and other unobservable fixed effects that may affect parameter estimates. 

This approachcan produce consistent estimates of the coefficientsof transitory tax rates and other 

non-fixed factors, but does not allow identification of the response of capital gains to permanent 

tax rates, as was recognized by the authors.14 

Bogart and Gentry (1993) use aggregate state data to estimate permanent capital gains tax 

effects. This approach mitigates the problem of limited sample size common to aggregate time 

series models, and the data set includes more years than our study. However, aggregate data 

precludes dealing with most of the econometric problems that we have found to be empirically

important and suffers from some of the same problems that affect aggregate time series studies. 

14Because the combined federal and state taxrates vary over time as well as among individuals, we could conceivably 

estimate an individual fixed effect in our model. We did not do this for two reasons. First, modelling fixed effects would 

make it difficult or impossible to control for sample selectivity, which Auten et al. (1989)found to cause substantial 

biases. Second, because only a minority of mostly small states changed their taxrates on capital gains between 1980and 

1983 (Bogart and Gentry, 1993),only about 3 percent of the independent variation in the state tax instrument remains 

after controlling for both time and individual fixed effects. The sources of this variation are individuals who moved 

between 1980and 1983,the 14 states that changed tax rates between 1980and 1983,and the interaction effect between 

the change in federal tax rates in 1981 and the net capital gains tax rate. Since the precision of instrumental variables 

estimates depends on the correlation between permanent tax rates and the instrument, removing almost all of the variation 

in the instrument would yield uninformative results. Moreover, to the extent that the remaining variation corresponds 

to movers, who may have reasons for realizing capital gains independent of tax effects, interpreting the estimated 

coefficient as primarily a permanent tax effect may be unwarranted. 

8 
III. Data 

The data are from a panel of individual income tax returns for about 11,OOO taxpayers for 

the years 1979 to 1983. (U.S. Department of the Treasury, 1979-1983) In addition to detailed 

tax return data, the panel includes the ages of taxpayers for each return. The panel sample was 

stratified to oversample high-income taxpayers; thus, a much larger proportion of the sample

(53.4percent) had capital gains than in the population at large (18.5 percent). The Treasury department edited the data for consistency and developed programs to calculate marginal tax 

rates. (James M. Cilke and Roy A. Wyscarver, 1987) Some observations were discarded 

because the data were internally inconsistent. 

Summary statistics for the data and instruments used in estimation are shown in Table 1. 

Weighted and unweighted statistics differ because the sample was stratified. We use unweighted

data for estimation,but test for the possibility that endogenous stratification biases the estimates.l5 

Our data were originally prepared by Auten et al. (1989), but we have made several 

improvements. We created the instrument for permanent tax variation (2,) by computing the 

combined federal and state marginal tax rate on capital gains for a taxpayer with $100 million 

of taxable income. We also modified the first-dollar tax rate instrument (T~)by setting several 

possibly endogenous components of income and deductions equal to zero. This was done for 

long- and short-termcapital gains and losses, capital loss carryovers, interest, dividends, business 

losses, charitable contributions, and the deduction for taxes paid and investment interest expense.

Auten et al. did not consider the deduction items other than charitable contributions to be 

endogenous. 

The sample period for estimation is 1980 to 1983 so that lagged values could be used. 

Observations on individuals were included in estimation whenever the current and lagged data 

were valid, which yielded a total of 42,406 observations. The dependent variable is net longterm capital gains before carryover of prior year losses as reported on Schedule D. The tax rate 

measure is the combined federal and state marginal rate, based on applicable tax law for each 

year and each taxpayer's income and deductions.'6 To smooth out kinks in the tax schedule and 

to represent the lumpiness of capital gains transactions, the marginal tax rate on capital gains was 

computed for a defined transaction, rather than for a dollar of capital gains. The capital gain on 

the defined transaction is the maximum of $l,OOO or the square root of imputed wealth.17 

'%e sample was stratified based on income, which includes capital gains realizations and other possibly endogenous 

variables. 

I6Because of the way the data were coded by the IRS, state of residence is available for all returns only in 1981. In 

other years, we used the actual state if it was available, or the state in 1981, otherwise. 

"As a sensitivity test, we also estimated the model using a marginal tax rate computed with a defined transaction of 

$1,OOO. This made almost no difference for the estimated effect of permanent tax rates, but increased slightly the 

estimated effect of transitory tax rates. 

9 
Population-Weighted 

Variable Description Mean Coefficient 

of Variation 

Net Long-Term Capital Gains 3.0 31.67 

Percentage with Net Positive Gains 18.5 ... Marginal Tax Rate on Capital Gains 11.9 0.54 

First-Dollar Tax Rate Instrument (T,,) 11.1 0.54 

Maximum Tax Rate Instrument (7,) 23.3 0.15 

Imputed Permanent Income 28.8 0.96 

Current Income (Exogenous Parts) 28.8 2.07 

Imputed Wealth (Gross Assets) 125.3 0.83 

Imputed Corporate Stock 11.3 4.81 

Business Losses Lagged 1.6 15.19 

Rent Losses Lagged 0.4 13.25 

Unweighted 

Mean Coefficient 

of Variation 

245.4 7.00 

53.4 ... 15.8 0.66 

13.1 0.73 

23.4 0.15 

125.7 1.64 

283.2 3.30 

286.5 6.70 

118.3 8.61 

101.8 3.96 

10.1 8.15 

Notes: Dollar amounts in thousands of 1981 dollars. Marginal tax rates are in percentages.

Statistics are for pooled years, 1980-1983. 

10 
The computed marginal taxrate is the change in tax liability divided by the amount of the defined 

capital gain.18 

Other regressors are discussed above in Section II, and summarized in Table 1. Wealth 

was imputed by using a tobit model to regress the logarithm of total wealth, as reported in a 1982 

sample of estate tax returns, on age and log capital income reported on 1981income tax returns. 

The estimated wealth regression was used to impute wealth for taxpayers in our panel sample for 

each year based on lagged values of the regressors. Corporate stock was imputed the same way.

Lagged business losses were computed as the sum of losses on rental property, losses reported 

on partnership returns, and losses reported by personal services corporations. 

Permanent income was imputed by using the panel sample to regress the logarithm of a 5-

year average (1979-1983) of real positive income on taxpayer characteristic^.'^ The regression

estimates were used to impute annual permanent income based on lagged values for the 

regressors. Current income, listed in Table 1, is defined as positive income excluding

endogenous sources such as capital gains.20 In the regression model, transitory income is the 

logarithm of the ratio of current to permanent income. 

Family size is the number of personal and dependent exemptions claimed on the tax return. 

Marital status is based on tax filing status. Age was derived from social security records. 

The sample period includes the Economic Recovery Tax Act (ERTA) of 1981, which 

reduced top tax rates on both ordinary income and capital gains by 29 percent and introduced 

many new tax preferences. The advantage of covering this period is that the change in tax law 

adds substantial variation to the statutory tax rates, The primary disadvantage is that the major 

tax change was far from a controlled experiment, and some of the response to the statutory

change in capital gains tax rates may be transitory, although the aggregate change was controlled 

for by time dummies. 

''Slemrod and Shobe (1990)argue that the marginal rate should be adjusted when the taxpayer has net capital losses 

to account for the fact that unused losses are at least partially deductible in future years. In our tax calculation, we 

consider only the current year, which implies a zero marginal tax rate on capital gains for taxpayers with nondeductible 

net losses. Given the small fraction of returns subject to the capital loss limitation, this difference is unlikely to affect 

empirical estimates. Moreover, since the capital losslimitationis essentiallytransitory, the estimatesof permanent effects 

are unlikely to depend on the treatment of losses. 

'Vositive income includes all positive components of income (including net positive capital gains). It is an 

approximation of economic income used by the IRS and in several earlier studies. 

%merit income and permanent income were scaled so that they had the same weighted population means. The 

unweighted mean for current income exceeds mean permanent income because the sample was stratified to oversample 

high-income taxpayers. Thus, people in the sample tend to have high transitory incomes. The wealth and stock variables 

were scaled to match the aggregates reported in the Survey of Consumer Finance for 1983,converted to 1981 dollars. 

11 
IV.Estimation Results 

Estimates for equations (2) and (3) are shown in Tables 2 and 3. Table 2 shows estimates 

of tax rate coefficients, the corresponding elasticities, and results for three restricted models. 

Table 3 shows the coefficient estimates for the non-tax variables. 

The first three columns in Table 2 show the marginal tax rate coefficients in the level 

equation (3) and the criterion function (2). To interpret these coefficient estimates, recall from 

footnote 6 that the total effect of changes in the permanent tax rate depends on the sum of the 

three taxrate coefficients, whereas the effect of transitory deviations depends only on coefficients 

of the current tax rate. Thus, the coefficient of the permanent tax rate in the full model is a 

measure of the difference in the effects of changes in the permanent and transitory tax rates. 

The estimated current tax rate coefficient in the full model is negative and statistically

significant at the 99 percent level in both the criterion function and level equation. The sign implies that individuals are more likely to realize capital gains (the criterion function) and realize 

more capital gains (the level equation) when they face temporarily low marginal tax rates. The 

size and significance of the coefficients imply that transitory changes in tax rates have a stroni 

effect on taxpayer decisions about whether to sell appreciated assets and realize capital gains. 

The lagged tax rate coefficient is small and insignificant, which implies that lagged tax rates 

do not affect current capital gains decisions, holding current and permanent tax rates and other 

included variables constant. This result is inconsistent with the conventional wisdom that the 

response in the first year to a capital gains tax change is larger than the long-run response. It 

is, however, consistent with Kiefer's simulations, discussed in Section I.22 

In contrast to the transitory effects, the estimated coefficient of the permanent tax rate is 

positive, nearly as large as the current tax rate coefficient, and significant at the 99 percent level 

in both the level equation and criterion function. This result implies that permanent changes in 

the taxrate have substantially smaller effects than transitory changes. These large and significant differences refute the basic assumption underlying the validity of previous micro data studies. 

21Although our model includes measures of permanent and transitory income, the transitory tax rate component may 

also proxy for variation in transitory income not controlled for by other variables. 

=Kiefer's simulations suggest a potentially larger "intermediate-term" effect, several years after a tax change. 

Unfortunately, there is not enough independent variation in tax rates in our data set to allow us to measure such effects 

with any precision. 

12 
TABLE 2-ESTIMATED COEFFICIENTS AND ELASTICITIES 

OF MARGINAL TAX RATE VARIABLES 

Estimated Model 

Full Model 

Level Equation 

(Equation 3) 

Criterion Function 

(Equation 2) 

Marginal Tax Rate Coefficient 

Current Lagged Permanent 

-0.145 0.0013 

(0.014) (0.011) 

-0.084 0.003 

(0.005) (0.006) 

0.116 

(0.036) 

0.088 

(0.016) 

-0.020 

(0.022) 

0.007 

(0.0 10) 

... 

... 

... 

... 

Permanent Transitory 

Elasticity Elasticity 

-0.18 -6.42 

(0.48) (0.34) 

-0.17 ... (0.42) 

... -6.10 

(0.33) 

... -4.19 

(0.22) 

Exclude Transitory and Lagged Tax Rates 

Level Equation ... ... 

Criterion Function ... ... 

Exclude Permanent Tax Rate 

Level Equation -0.144 0.039 

(0.014) (0.005) 

Criterion Function -0.083 0.036 

(0.005) (0.002) 

Exclude Permanent and Lagged __ Tax Rates

Level Equation -0.113 ... (0.0 10) 

Criterion Function -0.051 ... f0.0031 

Notes: Standard errors are in parentheses. Estimated coefficients of other variables 

included in the model are in Table 3. Elasticities are computed at an average tax rate of 

18.0 and an average lambda of 2.52. See equation 8. 

13 
The last two columns show the elasticities. The elasticity (e)measures the effect of a small 

change in the permanent tax rate: 

where h(h+ap)is the reciprocal of the Mills ratio evaluated at the mean of the systematic part

of the criterion function (h)plus the covariance between the error terms in the criterion function 

and the level equation (op). The transitory elasticity is given by a similar equation, excluding

the permanent and lagged tax rate coefficients. It is interpreted as the elasticity with respect to 

a change in the current tax rate, holding the permanent and lagged tax rates constant. 

The estimated permanent elasticity is -0.18, which implies that a 1 percent decrease in 

permanent tax rates would increase expected realized net long term capital gains by approximately 0.18 percent at average levels for all variables in 1983. However, the relatively

large standard error implies that we cannot reject the hypothesis that permanent changes in capital gains tax rates have no long-term effect on capital gains realizations.24 The standard error is 

also large enough that long-run elasticities of 0.0 and -1.0 are both included in a 95-percent confidence interval. 

The estimated transitory elasticity is -6.42, which is larger in absolute value than most 

previous elasticity estimates from micro data.25 The high transitory elasticity suggests that the 

response to a temporary taxchange would be extraordinary, with realizationsexpected to increase 

by more than six times the percentage change in the tax rate. This is consistent with the dramatic 

increase in realizations just after passage of the Tax Reform Act of 1986, as discussed in the 

introduction. 

The second panel of the table shows what happens to estimates of the permanent elasticity when the current and lagged tax rates are excluded from the estimated model. Assuming that 

the transitory component of the tax rate is uncorrelated with the permanent (state) tax rate 

instrument, the estimates of the permanent tax rate coefficient and elasticity are still consistent 

when the current and lagged tax rates are excluded. The permanent elasticity estimate changes 

very little, from -0.18 to -0.17, but the precision increases slightly. The transitory elasticity 

cannot be determined from this specification. 

The third panel shows the effect of excluding the permanent tax rate, but including the 

current and lagged tax rates, as in Auten et al. (1989). The current tax rate coefficients and 

implied transitory elasticity decrease slightly, and the lagged tax rate coefficients increase and 

become highly significant. This result makes sense because the average of tax rates over two 

years should be positively correlated with the omitted permanent tax rate. The omission would 

thus positively bias the current and lagged tax rate coefficientestimates. This result suggests that 

the lagged tax rate partially proxies for the omitted permanent tax rate. 

"Derivation is available from the authors on request. Permanent and transitory elasticities were computed for 1983 

means of the permanent tax rate and A, which were 18.0 and 2.52, respectively. 

"We refer to long-term changes because variation in the permanent tax rate instrument represents essentially crosssection variation in state marginal tax rates. While the combined state and federal marginal tax rates changed over time 

during our sample period, much of the possible influence of this source of variation was removed by including time 

dummy variables in our model. 

"See, e.&, Slemrod and Shobe (1991), Auten et al. (1989), and Gillingham et al. (1989) for recent estimates. 

14 
The fourth panel shows the effect of omitting both the lagged and permanent tax rates, as 

in Gillingham et al. (1989). The transitory elasticity estimatebecomes smaller, probably because 

the first-dollar tax rate instrumentis positively correlated with the permanent taxrate. A positive correlation would cause a positive bias in the transitory elasticity estimate, which may explain

why previous micro data studies have yielded smaller transitory elasticity estimates. This result 

is consistent with Slemrod and Shobe's (1990) finding that elasticity estimateswere biased toward 

zero by failure to control for unmeasured fixed effects, such as the permanent tax rate. 

The effects of other variables are summarized in Table 3, which reports estimated 

coefficients for the level equation, the criterion function, and the combined effects of implied

changes in the values of both functions on the expected value of capital gains realizations.26 For 

continuous variables, the estimates in Table 3 are reported as elasticities. For dummy variables,

i.e., those followed by (D) in the table, the effects are reported as percentage changes in 

expected capital gains realizations implied by changing each dummy variable from zero to one. 

The results seem generally consistent with life-cycle motives for saving and consumption,

modified somewhat by the incentive to hold assets with gains until death. Capital gains

realizations are significantly positively related to permanent income, but negatively related to 

transitory income, suggesting a consumption motive for realizations. Wealthier people are much 

more likely to realize capital gains, and realize larger gains than average. The composition of 

wealth also matters. A larger share of stocks in the portfolio--as measured by the stocwwealth 

variable--makes people significantly more likely to realize gains, but the average size of a gain

is smaller, ceteris paribw. This result may be a consequence of the lower transaction costs for 

stocks than for other kinds of assets, such as real estate. The positive and significant relationship

between gains and lagged business losses reflects the well-known relationship between tax 

shelters and capital gains, although rental losses (a subset of business losses) do not seem to have 

a very large independent effect on realizations. 

Holding wealth and other variables constant, the pattern of realizations follows the expected life-cycle profile except for the oldest cohort. The level of realizations declines steadily through

the peak earning years of 50-59, and then increases. The likelihood of realizing gains steadily

increases, perhaps reflecting the fact that older people are more likely to own assets that yield

capital gains. The percentage change in realizations is also U-shaped through age 69. However,

the oldest taxpayers realize less capital gains than the 60-69 cohort, and are slightly less likely 

to realize. Although this difference is statistically insignificant, it is consistent with older 

taxpayers avoiding realizations to take advantage of the step-up in basis at death. 

The Mills ratio coefficient equals the product of the standard error of the error term in the 

level equation, (3), and the correlation between the error terms in equations (2) and (3). The fact 

that the coefficient is nonzero implies that ignoring sample selectivity would lead to biased and 

inconsistent parameter estimates. The negative sign implies that the error terms are negatively

correlated. Thus, the tobit model used in some previous studies, which assumes a correlation 

of one, would be inappropriate. 

26Recall from Section II that these coefficients may not be estimated consistently. 

15 
18 

2

0

e 

& 

n 

-5 

2 M

.I0C

e n

zli I 0 

0 0 

1

\o

20l o 
Sensitivity tests for alternative specifications and segments of the data set are reported in 

Table 4. The results all confirm our basic finding that permanent elasticities are much smaller 

than transitory elasticities. The permanent elasticity is not significantly different from zero in 

any specification. 

The log-log model tests an approximately constant elasticity specification, which we view 

as inferior to the semi-log form. The results are similar to those under the semi-log form, but 

have higher standard errors. Weighted estimates are also consistent with our basic results. This 

result suggests that Joseph J. Minarik's (1981) finding that weighting could substantially alter 

elasticity estimates was a consequence of other estimation problems rather than endogenous

sample stratification. We excluded taxpayers from high- and low-tax states to test for the 

possibility of endogeneity bias in our state tax rate instrument. This experiment raises the 

standard errors significantly because much of the variation in the instrument is sacrificed, but 

does not alter the key conclusions. Results are similar when the sample is restricted to 1982 and 

1983 (after enactment of ERTA). Even when truncation is ignored and the model is estimated 

by two-stage least Squares, the elasticity estimates do not change much. Estimating the model 

by two-stage least squares based on a sample of realizers only, the transitory elasticity changed significantly, but the effect on the permanent elasticity estimate is small and insignificant. 

V. Conclusion 

It has long been suspected that differences between transitory and permanent responses to 

capital gains tax changes were at the heart of the conflicting empirical evidence from crosssection and time-series data. Using state tax rates to distinguish transitory from permanent tax 

effects, and correcting other econometric problems with previous studies, we find that the 

difference is large and statistically significant. The difference in estimated response is even 

larger than the differencesbetween past empirical results from careful micro-data studies, which 

measured a combination of permanent and transitory effects, and time-series studies, which are 

likely to have measured primarily permanent effects of changes in tax rates. 

Our analysis has some limitations. First, the capital gains realizations elasticity is only one 

of many factors that affect the proper taxation of capital gains. For example, our analysis

ignores the effects of capital gains taxes on the cost of capital and the allocation of capital among

kinds of investments, and it says nothing about arguments for taxing capital gains on equity

grounds. Second, this paper has followed all previous empirical research in estimating a reduced 

form model. Although this was necessitated by data limitations, it was also important to show 

that permanent and transitory tax effects could be estimated separately using a model otherwise 

similar to previous research. Any explicit structural model would require assumptions about the 

nature of preferences and individuals' optimization problems and the estimation method itself 

would be a radical departure from all prior research. That might lay open such an analysis to 

the criticism that the structure of the model was generating the results. The drawback of 

estimating a reduced form, however, is that the estimated parameters are functions of the tax law 

and macroeconomic environment and may thus change over time. 

The distinction between transitory and permanent tax effects may explain some other 

empirical anomalies. For example, the empirical evidence on the tax-sensitivity of charitable 

contributions seems to exhibit a similar divergencebetween time series and micro data estimates. 

The methodology developed here may help to resolve such disparities. 

17 
Y 

el 

W

W 0 

0, em 

r Iz \o

u Y.I 

c? c? 

Y.I 0 

0

0 

2.

a e

Lo

b E

.C. 

i *

..

i 

4 

V 7

U 

i 

c5 

v)

3 

c 

Y

.Ig

F 8 

.L 

c 

: 

Yi

> 

; 

.L 

c 

.L 

c a M V U 
Auerbach, Alan J., "Capital Gains Taxation in the United States: Realizations, Revenue and 

Rhetoric," Brookings Papers on Economic Activity, 1988, (2), 595-631. 

* -7 "Capital Gains Taxation and Tax Reform," National Tax Journal, September 1989, 42, 

39 1-401. 

. Auten, Gerald E., Burman, Leonard E. and Randolph, William C., "Estimation and 

Interpretation of Capital Gains Realization Behavior: Evidence From Panel Data,"

National Tax Journal, September 1989, 42, 353-374. 

-and Clotfelter, Charles T., "Permanent Versus Transitory Effects and the Realization of 

Capital Gains," Quarterly Journal of Economics, November 1982, 97,613-632. 

Balcer, Yves and Judd, Kenneth L., "Effectsof Capital Gains Taxation on Life-Cycle Investment 

and Portfolio Management," The Joumal of Finance, July 1987, 42, 743-758. 

Bogart, William T. and Gentry, William M., "Capital Gains Taxes and Realizations: Evidence 

from Interstate Comparisons,I' National Bureau of Economic Research (Cambridge, MA)

Working Paper 4254, January 1993. 

Bolster, Paul J., Lindsey, Lawrence B. and Mitrusi, Andrew, "Tax-Induced Trading: The 

Effect of the 1986 Tax Reform Act on Stock Market Activity," The Journal of Finance,

June 1989, 44, 327-344. 

Burman, Leonard E. and Randolph, William C., "TheoreticalDeterminants of Aggregate Capital Gains Realizations," mimeo, Congressional Budget Office and U.S.Treasury, September

1992. 

-9 Clausing, Kimberly and O'Hare, John, "Tax Reform and Realizations of Capital Gains in 

1986,'I mimeo, CongressionalBudget Office, Harvard University, and Joint Committee on 

Taxation, March 1993. 

Cilke, James M. and Wyscarver, Roy A., "The Individual Income Tax Simulation Model," in 

Compendium of TaxResearch 1987,Washington, DC: Government Printing Office, 1987. 

Constantinides,George M., "Optimal Stock Trading With Personal Taxes," Journal of Financial 

Economics, March 1984, 13, 65-89. 

Feenberg, Daniel, "Are Tax Price Models Really Identified? The Case of Charitable Giving," National Tax Journal, December 1987, 60, 629-633. 

Feldstein, Martin, Slemrod, Joel and Yitzhaki, Shlomo, "The Effects of Taxation on the Selling of Corporate Stock and the Realization of Capital Gains," Quarterly Journal of Economics,

June 1980, 95,777-791. 

Gillingham, Robert and Greenlees, John S.,"The Effect of Capital Gains Tax Rates on Capital Gains Tax Revenues: Another Look at the Evidence," National Tax Journal, June 1992, 

45, 167-178. 

19 
-7 -and Zieschang, Kim D., "New Estimates of Capital Gains Realization Behavior: 

Evidence from Pooled Cross-Section Data," Office of Tax Analysis Paper 66, U.S. 

Department of the Treasury, May 1989. 

Gravelle, Jane G., "A Proposal for Raising Revenue by Reducing Capital Gains Taxes,"

Congressional Research Service Report 87-562E, Library of Congress, June 30, 1987. 

-7 "Limits to Capital Gains Feedback Effects,'I Congressional Research ServiceReport 91-250 

RCO, Library of Congress, March 15, 1991. 

Heckman, James J., "The Common Structure of Statistical Models of Truncation, Sample

Selection and Limited Dependent Variables and a Simple Estimator for Such Models,"

Annals of Economic and Social Measurement, Fall 1976, 5, 475-492. 

Hendershott, Patric H., Toder, Eric and Won, Yunhi, "Effects of Capital Gains on Revenue and 

Economic Efficiency," National Tax Journal, March 1991, 44, 21-40. 

Jones, Jonathan D., "An Analysis of Aggregate Time Series Capital Gains Equations," Office 

of Tax Analysis Paper 65, U.S. Department of the Treasury, May 1989. 

Kiefer, Donald W., "Lock-In Effect Within a Simple Model of Corporate Stock Trading," National Tax Journal, March 1990, 43, 75-95. 

Lee, Lung-Fei, Maddala, G. S. and Trost, R. P., "Asymptotic Covariance Matrices of TwoStage Probit and Two-Stage Tobit Methods for Simultaneous Equations Models With 

Selectivity," Ecommetrica, March 1980, 48, 491-503. 

Minarik, Joseph J., "Capital Gains," in H.J. Aaron and J.H. Pechman, eds., How Taxes Afect 

Economic Behavior, Washington, DC: The Brookings Institution, 1981, 241-277. 

Neumark, David and Wascher, William, "Employment Effects of Minimum and Subminimum 

Wages: Panel Data on State Minimum Wage Laws," Industrial and Labor Relations 

Review, October 1992, 46, 55-81. 

Seltzer, Lawrence H.,me Nature and Tax Treatment of Capital Gains and Losses, New York: 

National Bureau of Economic Research, 1951. 

Slemrod, Joel and Shobe, William, "TheTax Elasticity of Capital Gains Realizations: Evidence 

from a Panel of Taxpayers," National Bureau of Economic Research (Cambridge, MA)

Working Paper 3237, January 1990. 

Stiglitz, Joseph E., "Some Aspects of the Taxation of Capital Gains," Joumal of Public 

Economics, June 1983, 21, 257-294. 

U.S. Department of the Treasury, Internal Revenue Service, Statistics of Income Division,

Special Panel of Tax Retums, 1979-1983. 

U.S. Department of the Treasury, Report to the Congress on the Capital Gains Tax Reductions 

of 1978,Washington, DC: U.S. Government Printing Office, September 1985. 

J 

20 
APPENDICES 

These appendices derive conditions for consistency of the IVestimator for permanent and transitory 

tax effects discussed in Section 11, derive a consistent estimator for the generalized tobit model with 

an endogenous regressor in both the criterion and level equations, discussed in Section 111, derive 

elasticities for the simultaneous selection model, examine the simultaneous equations bias induced by 

using actual tax rates or proxy variables as regressors to estimate tax effects, as has been done in several 

influential earlier studies and examine the variation in tax rates in the panel. 

Appendix A. Consistency of the N Estimator 

The linear capital gains model can be written as 

g = Hr+e 

where H =[Xi tpi p], I"=[yi i y1 i y2], p = t -tp,and it is assumed for simplicity that all variables are 

expressed as deviations from means. Note that the model has been expressed for convenience in terms 

of the permanent tax rate, tp,and the transitory component of the current marginal tax rate, p. It is 

also assumed that there are n i.i.d. observations, and that His of full column rank, conformable with 

r? 

The estimator, f,can be written as 

f = (AI@-'A'g, 

where &=[Xi +p i PI, +p=Wi(W:Wl)-'W[t,W,=[Xitl], P=i-tp,i=W&WJ-'W?/T, and 

W2=[W,i to]. Note that F is orthogonal to X and i,, from which it follows that To and are 

instrumental variables estimates, where i, is the instrument �or tPin equation (Al). As shown below, 

consistency of f oand fl depends on the covariance between Xand t, with p and E. Consistency of f2 

depends on the covariance of z0with t, and E, conditional on Xand ts. 

1

It is also assumed that the variables have finite moments up to at least the third order. In this appendix, the 

matrixxis not necessarily defined in the same way as in the main body of the paper. In the context of secondstage estimation of the generalized tobit model, for example, the estimated inverse Mill's ratio can be treated 

as a column of X. 
A. The Probability Limit of f 

By using the fact that F is orthogonal to X and tP,and applying well known rules �or a 

partitioned inverse, f can be rewritten as 

f, = (i$,$Jl$Mfi (444) 

f* = (P’P)Vg, (4 

where M, =[In-i,(Q:ip)-lQi] and M, = [I,-X(X’X)-’X’]. In the remainder of this appendix, equations 

(M),(A4), and (A5)are used to derive the probability limits of fo, fl, and f2 as n -00. The 

derivations that follow also use the fact that i,can be rewritten as 

i, = X(X/AyX’T+ MxT,(T:MxT,)-’T;Mxq 

and iican be rewritten as 

P =4v,T o( Tlk,z0) Tl$fWI T9 

where Mw1=[l,-W1(W~W,)-lW~]. 

1. Theprobabdity hit of fo 

By combining (Al) and (N),fo (the coefficient on X) can be written as 

30 = (X/h4&)-’X/Mp(m+�) = yo+(X’Mp-)-’[y~’M,fp+y~/Mp~+X~Mp�]. 

Because 5, =T -c, and MPz30,3, can be rewritten as 

fo = yo + (y*-y,)(X~Mp-)“X/M,p + (x’Mp-)-’x’M,c. 

22 
Under the assumptions that var(X) is nonsingular, vur(5, IX) >0, and by repeated application of 

Khintchine's theorem to the terms in equation (A6), it can be shown that as n-=, 

where k, and k, are known (messy) nonzero functions of the second order moments. Furthermore, 

using the expressions for k, and k,, it can be shown that if cov(rS,~1;T)=cov(r,,e1 X)=O, and 

cov(X,~)4,equation (A7) reduces to 

Plim(fo) = yo + (y,-y,)var(X)-'cov(X,p). (4 

(We define the expression cov(x,y I z) to be the partial covariance betweenx andy givenz, Le., after the 

linear influence of z is removed from x and y.) 

Thus,if rJ is uncorrelated with p and E, conditional on X,and X is uncorrelated with E, then 

the probability limit of fo differs from yo by the product of the difference between the transitory and 

permanent tax effects and a familiar term for omitted variable bias, where p is the omitted variable. 

2. The probability limit of f1 

By combining (Al) and (A4), fl (the coefficient of rp)can be written as 

Under the assumption that COV(T,,T Ix) $0,repeated application of Kintchine's theorem can be used 

to show that 

Thus, fl is a consistent estimate for y, if r, is uncorrelated with p and E conditional on X. 

Note that under the assumption, cov(s,,e (9=0,(A9)can be rewritten as a weighted average 

of y1 and yz: 

23 
3. The probability limit of f2 

By similar reasoning, under the assumption that cov(s0,rlX,rJ '0, it can be shown that the 

probability limit of f2 (the coefficient of p,) is 

Note that under the assumption, COV(T~,E~X,T,)'0, (A10) can be rewritten as a weighted 

average of y2 and yl: 

24 
Appendix B. Consistent Estimation of Generalized Tobit Model With Endogenous Regressors in 

Both the Probit and Level Equations 

The generalized tobit model in the paper is the following three equations c 

I' = Xq, + a,r, + g,r, + %r,-, + E, , 

and 

To simplify exposition, we begin by treating tpas if it is known and exogenous, and focus on 

the endogeneity of t, Assume that, for purposes of the first-stage probit estimates, the reduced form 

for t, is approximately linear in X,zp, T,-~,and zW2This constructed instrument, t,,, is a "first-dollar" 

marginal tax rate, computed with the endogenous sources of income, g and Y,set equal to 0: 

(recalling thatfis the tax �unction in equation (A15)). 

Let 2 be the matrix: 

2 qx i tp i r,,, i To]. 

Then the linearized reduced form for r, is 

t,=Zx +u, 

where x is a parameter vector and u is an error term that is assumed to be uncorrelated with 2. 

%e assumption of linearity is only for convenience. The IV procedure could produce consistent estimates 

even if the tax rates were highly non-linear. 

25 
Substituting the reduced form for rl in (A13) yields 

I' = XqJ + alTp + Or,(Zx) + $TI-, + (E1 + $U) . 

Following Lee, et al (1980), the conditional expectation of capital gains is 

E(g(l'>O) = xp, + PITp + p*E(r,(I*>o)+ + E(�,JI'>O) (KO) 

Using the standard formula for truncated means of normal random variates, the conditional 

expectations may be written as: 

E(r,II'>O) = zx + (UuI + QzuJ5,@ 

and 

where 6 and d are the standard normal density and distribution functions, respectively, evaluated at 

XqJ+%Tp +aZ(Zx>+%%,-I * 

The difference from the model of Lee, et al(1980), appears in the u,, term that appears in the 

conditional mean for rt. Thisterm is unambiguously positive, which implies that the conditional mean 

for T~is almost surely different from the unconditional mean, even if the covariances between different 

error terms are zero. This implies that correcting for selectivity is essential to finding consistent 

parameter estimates even if the error terms in equations (A13) and (A14) are uncorrelated, Le., even 

if ul2=O. 

A similar potential bias from ignoring selectivity would occur because xP is unobserved. The 

IVestimator described in section I1 replaces z Pwith Q,in both the probit and level equations. Thus, 

the preceding analysis applies, using Q,-T, -X(X'X)-'X'b in place of u and with a1substituted for 

aT The potential selectivity bias discussed above is thus compounded. Fortunately, the solution is the 

same in both cases. 

26 
Appendix C. Computing Permanent Elasticities in the Selection Model 

Let gi be net long-term capital gains for individual i, and let E be the expectation operator 

(conditional on ri,pand Xi, and ri,,=~i,,.l=~i,p). Then the long-term elasticity of gains with respect to 

permanent tax rates, e;, is 

For the semi-log model, it may be shown that the expression in (A23) is: 

where A( -)is the reciprocal of the Mills-ratio function (+/a),hi is the systematic part of the criterion 

function, equation (A13), and up is the covariance between the errors in the twoeq~ations.~The first 

part of (A24)is the response of the level of capital gain conditional on realizing a capital gain. The 

second part is the sum of the direct effect of tax rates on the probability of realizing and the indirect 

effect through the covariance in errors between the criterion and level equations. 

For the log-log model, the elasticity is:' 

The elasticity of aggregate realizations with respect to the permanent tax rate is the weighted 

sum of the individual elasticities, evaluated at the permanent tax rates? The correct weights are the 

sample weights multiplied by the amount of capital gains. In practice, the results are virtually identical 

3See Appendix D for the proof. 

4Recall that the tax terms have the form log(1+rLP)and that rip is measured in percents. 

'One might be tempted to compute the individual elasticities at the actual tax rates rather than the expected 

permanent tax rates. Thisprocedure would bias estimates because tax rates are endogenous and correlated with 

the level of capital gains. There is also a selectivity bias. 

27 
if (A24)and (A25)are evaluated at the gains-weighted means of permanent tax rates and A.6 Because 

the first part of each expression is conditional on realizing a capital gain, the appropriate average tax 

rate is the average estimated permanent tax rate for realizers, which is 19 percent in 1983. For the 

second part, we use the unconditional average permanent tax rate, which is 18percent. The weighted 

average values of A, which depend on estimated parameters, are 3.1 for the semi-log model and 2.9 �or P 

the log-log model. 

Appendix D. Derivation of Elasticity Formula 

The formula for elasticity is complicated somewhat because the dependent variable in the 

selection model is in logarithms. This appendix derives the general formula �or the log-selection model, 

which was applied in Appendix C. 

A. Expectation in a Generalized Tobir Model with Log Dependent Variable 

The model may be written in general form as: 

In Y,= f(x,)+ v1, if h (Xi) + 2 0 

=o otherwise , 

for i=1,...,A? Assume that, conditional on Xi, both vli and ut.are independent, identically distributed 

randomvariablessuchthat vli - N(O,u;), uZ - N(0,l),and E(v1,u2J = uI2.Thecorrelation between 

vli and % is p = u,Jul. We can rewrite the nonstandard normal random variable, vli, in terms of the 

standard normal,uli, as vlI ulul,. 

6

Computing the elasticities for the main specifications using micro-simulation changed only the third 

significant digit of the estimates. We report elasticities at the mean because they are easier to reproduce, 

requiring only a calculator and the parameter values reported below. 

28 
The expectation of Y,,conditional on X, is: 

(treating the value of EY, as 0 if h (x) +uzi <O). 

The complication in equation (A26)is the conditional expectation. The following lemma 

derives its value. 

Lemma: 

Pmfi 

Let u1and u2 be jointly standard normal with correlation p. Their joint density is: 

The conditional mean ofe'"' given u2 2 c is 

where Q is the univariate normal distribution function. 

29 
Finding the solution to (A29) involves integrating out ut and then recognizing that the 

remaining terms in u2are proportional to a normal density. The numerator of (A29)may rewritten as: 

.... 

where 

Rearrange q so that it is in the form 

4 (U1I4 = (u, - PI2 + W I 

where p does not depend on u, and r(uJ depends only on + Then 

4 (Ul,lC2) = - 2(1 +2) (u: - 2 ph4 - 241 -p2)u1+Z& 

30 
Substitute into (A30)and rearrange: 

The integrand inside the parentheses is (by design) the normal density for mean [pu, + u( 1-p? ] and 

variance 1 - p2. Thus, the integral is 1. 

N, therefore, simplifies to: 

Complete the square on the bracketed expression: 

= c~212[l - Q (c-up)] a 

Substituting this for the numerator of (A29)completes the proof. 8 

31 
Substituting (A27) into (A26),the unconditional expectation of Y may be rewritten as: 

B. Elasticity 

The elasticity is defined as 

', = 

The partial derivative, from (A31),is: 

Substituting into (A32)yields 

where Ai is the inverse Mills ratio: 

32 
Appendix E. Identification and Bias in a Simplified Model of Endogenous Taxable Income 

While virtually all past studies of capital gains have recognized the endogeneity of tax rates, the 

corrections for endogeneity have not followed standard econometric procedure. Thisappendix shows 

how these unconventional approaches are likely to bias estimates of capital gains responses.' 

A. Identification 

A simple model of capital gains realizations is used to illustrate problems with previous 

solutions to the endogeneity problem. The model ignores sample selection and the excluded 

permanent tax rate, which are discussed in the paper. Suppose that the marginal tax rate on capital 

gains is a non-linear function of the exogenous variables,X, but that it is approximately linear in capital 

gains, g,and other endogenous income, which is grouped together in one variable, Y. The system of 

equations may be written as 

g = BIT, + Wl + q1 = Z,P + 71 9 

y = YlT, + YJ2 + 12 = Z,Y + % 9 

where Zi*, q.], for i=1,2, and 

T, zf(X,Yg) = to(X) + & ' (g + q . 

Suppose that gand Yare always observable, and that permanent tax rates are observed and are 

part of the vectors X,and X, which are subsets of X. The variables in (A33)-(A35) are conformable 

data matrices with N observations, so g, Y,t,,ql, q2, and to are Nxl, X,is Nxk,, X2is Nxk, and Xis 

Nxk, where kkmax(k,,k2). The error terms have asymptotic covariance 

7

Technically, we are not examining bias but the difference between the probability limit of alternative 

estimators and the actual parameter values: the magnitude of the inconsistency. We use the terms "bias" or 

"asymptotic bias" as convenient shorthand. 

33 
uC = E(q.qj),for i=1,2, and j=1,2, and are uncorrelated with X. The parameter vectors have 

conforming dimensions. In addition, define a matrix, Z, as 

where 7,is the first-dollar marginal tax rate. 

Suppose also that the moments of the 2 matrices converge to positive definite matrices, i.e., 

z; zi plim -= Qi , for i = 0,1,2 . N- N 

Under these conditions, consistent estimation is straightforward. Because t,(X) is a known 

non-linear function and is correlated with t,,but not with q1or qa T,, is a good instrument for t,. All 

of the model's parameters are identifiable for two reasons. First, non-linearity of towould be sufficient 

by itself. Second, many of the elements of X that enter the tax calculation (i.e., exogenous factors that 

determine transitory tax rates) do not enter equations (A33) and (A34),so the parameters of the model 

could be identified even if tax rates were linear. 

The key coefficients, p1 and Pr can be estimated by two-stage least squares using 7, as an 

instrument for t, Thisestimator is analogous in this simple model to the estimator described in the 

Paper. 

B. Previous Studies 

Instead of two-stage least squares, many past studies have used a proxy variable in place of tr 

The proxy tax rate was either a "firstdollar" tax rate, which is a marginal tax rate computed by setting 

realized capital gains, but not other income, equal to zero, or a marginal tax rate computed by setting 

capital gains equal to an estimated value, conditional on exogenous taxpayer characteristics. 

The class of proxy variables used in previous studies may be written as 

When h(X) equals 0, 3, is the first-dollar tax rate used in previous studies. When h(X) is a function 

whose expected value equals actual capital gains, g, i, is a "fitted lastdollar tax rate." Because parts 

34 
of other income, Y,are endogenous, 3, is correlated with the error term, ql, in equation (A33). it 

follows that such proxies will produce inconsistent estimates regardless of the choice of h(X>.' 

1. Least squares bias 

The problems in using the proxy variables can be illustrated by examining the naive least 

squares estimator of the tax rate coefficient in the capital gains equation, p,. The asymptotic bias in 

this estimator, B1, is 

plim 8, - p1 = q1 plim

N- Nwhere q1is the top left element of el-'Thisscalar is positive because Q1is positive definite. 

The reduced form for T,is, from equations (A33)-(A35), 

T, = -

6 kro(X) + k(X1yl+X*yJ+ h,+ k% 9 

The constant factor, k,is positive because 6 is non-negative (marginal tax rates are assumed to be a 

nondecreasing function of income) and p1 and y1 are negative (endogenous income falls with tax 

rates). Substituting this reduced form into (A39) yields the asymptotic covariance between T, and q I: 

~~ 

bere are other problems with such proxies. First, the use of a proxy variable in place of actual tax rates 

causescoefficientestimates to be biased toward zero. Second, the use of firstdollar tax rates in place of actual 

tax rates would causecoefficient estimates to be biased away from zero. However, if estimation were otherwise 

appropriate (Le., the first-dollar rate were really exogenous, selectivity were properly accounted for, and the 

model were properly specified), then elasticity estimates based on first-dollar tax rates would be consistent 

(except for the proxy-variable bias just noted). 

35 
The bias is non-zero almost surely. IF the absolute value of the variance of q 1 exceeds the 

covariance between the two equations' errors (which seem likely), the bias will be positive (i.e., 

towards zero). 

Figure 1 illustrates this bias for the case of uI2=O (or capital gains is the only endogenous 

source of income). Suppose that two taxpayers were identical, except that they had different 

unobserved v 1values. On the figure, the taxpayer with the lower v 1corresponds to gain function, GG, 

and taxable income function 11. The taxpayer with the high q1has higher gains at every marginal tax 

rate, represented by the shift from GG to G'G'; taxable income is correspondingly higher represented 

by 1'1'. Because the tax schedule is progressive (marginal tax rates are upward sloping with respect 

to income), the equilibrium tax rate is higher for the taxpayer with higher q, than for the other. In this 

example, equilibrium gains actually increase from G to G' as equilibrium tax rates increase from 

T to TI. A regression line drawn through the two points would slope upward. It is clear that this 

positive correlation has nothing to do with the behavioral response of taxpayers to capital gains tax 

changes. In fact, gains are negatively related. Thus,the least squares bias reverses the sign of the 

relationship in this example. 

2. Estimates based on pmxy tax rate 

Most studies of capital gains realization behavior have used a proxy tax rate, such as i,defined 

in (A38). From (A35),the proxy tax rate is 

Substituting in the reduced form for Y, 3, may be written as 

36 
(A43) 

The asymptotic bias of an estimator based on proxies from its "true"' value is proportional to 

If u12is zero (or negative, but small), the asymptotic bias will be negative because y1 is assumed to be 

negative. Thus,using the first-dollar tax rate is likely to bias realization elasticity estimates away from 

zero causing estimated elasticities to be too large in absolute value. 

3. Average tar rate as measure ofpermanent tar rate 

As a proxy for the permanent marginal tax rate, Auten and Clotfelter used a 3-year moving 

average of actual capital gains tax rates, including the actual current year tax rate. This procedure 

results in biased and inconsistent estimates of permanent tax effects for the same reason that "naive" 

least squares, discussed above, results in inconsistent estimates of the transitory tax effect. Assuming, 

as above, that the other regressors are not correlated with the disturbance terms, the Auten-Clotfelter 

procedure is roughly equivalent to using tJ3, tJ3, and t,$ as regressors. Because the lagged tax 

rates are assumed to be predetermined, they would be part of XI,and the analysis of bias simply 

proceeds substituting zJ3 for t, in Equations (A33)-(A35). It can be shown that the resulting bias 

would be proportional to k(ull + u,&3. Thus,the bias would have the same sign as the "naive" least 

squares bias that results from ignoring endogeneity. Under plausible assumptions, the magnitude of 

the bias would be about a third of the bias in the naive model. Because the least squares bias can be 

severe, reducing the bias by two-thirds could still result in seriously flawed estimates, even if estimation 

were othewise appropriate. 

In a long enough panel, if tax law remained constant, the endogeneity bias could be limited or 

avoided-for example by excluding the current year fromtax rate averages. However, with currently 

9

The "true"parameter is not 0, from Equation (1) if a first-dollar tax rate is used (h(X)=O). In that case, the 

parameter would reflect the relationship between capital gains and the first dollar rate. However, this, by itself, 

does not bias elasticity estimates. (See footnote 8.) 

37 
available data sets, using average tax rates to proxy for permanent effects would result in potentially 

serious biases because of the endogeneity of current tax rates just discussed and because averages of 

a few years’ tax rates are insufficient to identify the separate effect of permanent from transitory tax 

changes. 

38 
Appendix F. Variation in Tax Rates in the Panel 

The central identification problem--estimation of the permanent effect-would disappear if 

there were no transitory variation in capital gains tax rates. In this case, all differences in tax rates 

would represent permanent tax effects and econometric evidence from micro dat'a would only reflect 

the effects of permanent tax changes. The effect of transitory tax changes could not be identified, but 

it would be irrelevant. 

The data, however, show that there is considerable transitoryvanation. Tax rates for individual 

taxpayers traced over five years (1979 to 1983)vary substantially, and the variance increases over time, 

even after controlling for the effect of the major tax legislation in 1981, as illustrated in Figure 2. 

To create Figure 2, taxpayers were divided into ten groups that correspond to deciles of the 

unconditional sample distribution of firstdollar capital gains tax rates in 1979." Each decile group was 

then followed through 1983 to examine how closely the group's conditional distribution in following 

years corresponded to the unconditional distribution of first-dollar capital gains tax rates for each year. 

In the figure, the distribution of each group is represented by its quartiles. A tendency for the 

conditional quartiles to approach the unconditional quartiles quickly would indicate a high degree of 

intertemporal variation in firstdollar tax rates for each taxpayer. However, if the conditional 

distributions remained relatively fied, this would suggest that most variation in tax rates represents 

permanent differences among taxpayers. 

For each of the ten decile groups, the figure displays three lines that show how the fit, second 

(median), and third quartiles of their conditional distributions are related to the percentiles of the 

unconditional distribution in each of the five years from 1979 through 1983. For example, consider 

taxpayers inthe first decile of the unconditional distribution in 1979,represented by the left-most panel 

of Figure2 By construction, the median for the first decile in 1979corresponds to percentile 5 of the 

unconditional distribution; the first and third quartiles of the conditional distribution correspond to 

percentiles 2 and 7of the unconditional distribution. However, after five years, the first, second, and 

third quartiles ofthe conditional distribution for the first group equaled percentiles 5, 13, and 28 of the 

unconditional distriiution. Similarly, the figure shows that the three conditional quartiles for taxpayers 

1

%e sample used to construct Figure 2 only includes taxpayers who realized net positive long-term capital 

gains at least once between 1979 and 1983. The firstdollar tax rate is the marginal tax rate on the first dollar 

of long-term capital gains, Le., computed with capital gains set to zero. 

39 
who were in the 5th decile in 1979equaled percentiles 42,45, and 47 of the unconditional distribution, 

but changed to equal percentiles 30, 50, and 63 of the unconditional distribution by 1983." 

The graph showsthat there was substantial intertemporal variation in first-dollar capital gains 

tax rates between 1979 and 1983. In all ten decile groups, the conditional distributions noticeably 

increased in dispersion over the five year period. Further, the tendency of conditional medians to drift 

toward the unconditional medians shows that taxpayers with low tax rates in 1979were likely to have 

had tax rates below their permanent levels; taxpayers with high rates in 1979were likely to have been 

experiencing unusually high rata in that year. 

While providing strong evidence of transitory volatility in tax rates, Figure 2 also illustrates that 

there are systematic differences in the permanent tax rates ofdifferent taxpayers. The distribution of 

the bottom five deciles remains below the population distribution for all fives years, and the top five 

deciles remain above the population distribution. Were tax rates purely random (transitory), the 

conditional distributions would have equalled the population distribution in 1980 through 1983. 

11By comparingquartiles of conditional distributions to percentiles of the unconditional distributions,we have 

deliberately abstracted from intertemporal variations that would have resulted from general shifts in the 

marginal tax-rate schedule due, for example, to statutory changes in marginal tax rates in 1981and 1982. As a 

result, Figure 2 provides a better picture of the degree to which dispersion of marginal tax rates in a cross 

section sample results from intertemporal variation because generalshiftsthat are not represented by the figure

would not result in cross-section variation in tax rates. 
Figure 1 

Least Squares Endogeneity Bias 

Capital Gainsnaxable Income 

I \ 

TU / 

G' 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

\ 

I 

1 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

- -

0 10 20 30 T T' 40 50 

Marginal Tax Rate on Ordinary Income (percent) 
v) 

.-c 

m 

(3 

c 

0 

X 

‘31$33 .f rii