AN ANALYSIS OF 

AGGREGATE TIME SERIES 

CAPITAL GAINS EQUATIONS 

by 

Jonathan D. Jones* 

U S. Treasury Department 

OTA Paper 65 May 1989 

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. Additional copies of this publication ma be purchased from the National Technical Information 

Service 5285 Port Royal Road, Springiield, VA 221161. Phone (703) 487-4660. 

Office of Tax Analysis U.S. Treasury Department, Room 4040 

Washington, D. C. 20220 

* The views expressed are those of the author and do not necessarily reflect those of the 

Office of Tax Analysis or the Treasury Department. The author thanks Thomas S. Neubig.

Roy Wyscarver, and 3ames R. Nunns for comments which improved presentation of the paper.

In addition, Gerald Auten, John Greenlees, Douglas Hamilton, and Lany Ozanne provided 

many comments which improved the content of the paper. 
-

TREASURYNEWS 

~opartmentof the Treasury 0 Washington, D.C. 0 Telephone riS=204i 

u 

FOR IMMEDIATE RELEASE CONTACT: Lawrence Batdorf 

May 16, 1989 202/566-2041 

New Empirical Analyses of Capital Gains Taxation 

The Treasury Department today released three new Office of Tax 

Analysis staff papers on the taxation of capital gains. The 

papers provide additional evidence supporting the Treasury

Department estimates that the President's capital gains proposal

will increase Federal tax receipts. 

These empirical papers analyze the effect of changes in capital

gains tax rates on taxpayers' capital gains realizations and 

other income sources. The papers analyze prior tax law changes

and find significant short- and long-term responsiveness of 

taxpayers' realizations to lower capital gains tax rates. 

Taxpayer responsiveness was more than sufficient to increase 

total Federal tax revenues. 

The papers use three different data sources to analyse the effect 

of capital gains tax rates on taxpayer,' realizations: (1) 

aggregate time-series data (national data for a 40 year period),

(2) pooled cross-section tax return data (four years of 

individual tax return data), and (3) panel tax return data 

(individual tax return data ,following the same taxpayers for a 

five-year period). In addition, the papers improve on the 

statistical estimation and models of prior empirical studies. 

000 

NB-274 
NEW EMPIRICAL ANALYSES OF CAPITAL GAINS TAXATION 

The Treasury Department released today three new staff papers 

on the taxation of capital gains. These empirical analyses,

prepared by the Office of Tax Analysis staff, analyze the effect 

of changes in capital gains tax rates on taxpayers' capital gains

realizations and other income sources. The papers find a significant short-term and long-term responsiveness of capital gains

realizations to lower capital gains tax rates. The papers

provide additional evidence supporting the Treasury Department

estimates that the President's capital gains proposal will 

increase Federal receipts. 

The papers use three different data sources to analyse the 

effect of capital gains tax rates on taxpayers' realizations: 

(1) aggregate time-series data (national data for a 40 year

period), (2) pooled cross-section tax return data (four years

of individual tax return data), and (3) panel tax return data 

(individual tax return data following the same taxpayers for a 

five-year period). It is important to note that significant

realization effects were found in the three different data 

sources. 

The papers make two improvements over earlier empirical

studies. First, they use more sophisticated statistical (econometric) methodologies to account for the non-linearity of the 

income tax system and the choice of taxpayers whether to realize 

gains or losses in any given year. Second, the individual tax 

return studies are the first to incorporate state marginal income 

tax rates, which also influence taxpayers' decisions on whether 

and how many gains to realize. 

The new analyses find a significant responsiveness of 

taxpayers' realizations to lower capital gains tax rates enacted 

in previous tax legislation. The increased realizations resulting from lower capital gains tax rates are more than sufficient 

to increase total Federal tax revenues after the capital gains 

rate reductions, These studies analyze prior tax law changes. 

The Papers 

The papers released today are Office of Tax Analysis (OTA)

Papers, which 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 on the papers are invited. The three 

papers are: 

OTA Paper #65: "An Analysis of Aggregate Time Series Capital

Gains Equations," by Jonathan D. Jones. 
-2-

OTA Paper #66: "New Estimates of Capital Gains Realization 

Behavior: Evidence Pooled Cross-Section Data,"

by Robert Gillingham, John S. Greenlees, and 

Kimberly D. Zieschang. 

OTA Paper #67: "Estimation and Interpretation of Capital Gains 

Realization Behavior: Evidence from Panel Data,"

by Gerald E. Auten, Leonard E. Burman, and 

William C. Randolph. 

Paper Abstracts 

An Analysis of Aggregate Time-Series Capital Gains Equations.

This paper examines the robustness of the estimates of taxpayer

responsiveness to capital gains rate changes in aggregate timeseries equations. Many prior time-series capital gain analyses

have been done without careful attention to the proper econometric specification of the equations. In particular, the paper

examines the issues of functional form, the choice of the 

dependent and explanatory variables, lag length, non-stationary

of the data, and simultaneous equation bias, After an examination of these econometric issues, the paper specifies a more 

appropriate equation for the estimation of the response of 

capital gains realizations to changes in capital gains tax rates. 

The preferred time-series equation estimates a short-run 

elasticity of -1.2 and a long-run elasticity of -0.9. These 

elasticities of capital gain responsiveness to changes in tax 

rates estimate that realizations would more than double in the 

short-run if marginal tax rates were cut in half, and realizations would nearly double in the long-run. These estimates of 

the long-run elasticity are higher than most prior time-series 

equation estimates. 

The paper finds, however, that aggregate time-series 

estimates of the taxpayer responsiveness of capital gains

realizations to changes in tax rates are not at all robust to the 

specification of the regression model. Taxpayer responsiveness 

can be large or small depending on how the estimated equation is 

specified. For instance, the use of a narrow definition of 

wealth tends to bias the estimate of taxpayer responsiveness

downward. The paper concludes that tax policy analysts should 

not rely on time-series estimation to produce definitive results 

on taxpayer responsiveness due to the sensitivity of the models 

to specification issues. 

New Estimates of Capital Gains Realizations Behavior: 

Evidence from Pooled Cross-Section Data. This paper develops and 

estimates a behavioral model of taxpayer response to capital

gains taxation'using individual tax return data from four 
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different years. The model estimates the responsiveness of 

capital gains realizations and four other capital income 

categories to changes in marginal tax rates (both federal and 

state). The paper improves the econometric specification of 

"last-dollarn marginal tax rates, the dynamic "unlocking" of 

long-term capital gains, and the decision of whether to realize 

net gains, net losses, or no gains. It also recognizes the 

importance of the entire progressive rate schedule. Perhaps most 

importantly, the data base extends over the period 1977 to 1985,

thereby including three significantly different regimes of 

capital gains taxation. 

The paper estimates the response to taxpayers to changes in 

capital gains tax rates in terms of changes in the probability of 

recipiency of gains and losses, and in terms of the dollar amount 

of the capital gains realizations conditional on recipiency. The 

paper finds significant responsiveness in both decisions. For a 

typical taxpayer, a one percentage point decrease in the marginal 

tax rate raises the probability of recipiency of gains from 7.6 

percent to 8.9 percent. Conditional on recipiency, and evaluated 

at the sample average marginal tax rate, the elasticity of the 

amount of gains with respect to the marginal tax rate is approximately -1.6. Simulation of the two effects at 1985 levels 

implies that the aggregate point elasticity of net long-term

gains, net of carryover, with respect to the effective marginal 

tax rate is approximately -3.8. Due to feedback effects, the 

alternative minimum tax and other factors, the arc elasticity of 

gains with respect to discrete changes in statutory rates would 

be substantially lower. 

The pooled cross-section data estimates imply that the 

realizations response would be sufficient to yield revenue 

increases from capital gains rate reductions. Employing a 

measure of the year-to-year change in the tax rate schedule to 

allow for temporary unlocking effects, the paper also finds a 

significant long-run tax impact. . The other primary result is 

that conversion of ordinary income to capital gain income in 

response to lower capital gains tax rates was not evident from 

this data. The existence of a large flow of unrealized gains

should provide ample theoretical plausibility to the strong

behavioral response reported in this paper. 

Estimation and Interpretation of Capital GAins Realization 

Behavior: Evidence from Panel Data. This paper partially

reconciles differences among previous individual tax return 

studies by presenting new estimates of the taxpayer response to 

changes in the capital gains tax rate. A new behavioral model 

and improved econometric techniques are applied to a panel of 

individual income tax returns in which the same taxpayers are 

followed over a five year period, 1979 to 1983. The model 
-4-

incorporates the dynamic effect of realization behavior both on 

whether to realize gains and the amount of gains realized, the 

effects on other types of capital income and losses of changes in 

their tax rates, the incorporation of wealth estimates as an 

explanatory variable, and the use of both state and federal 

marginal income tax rates. 

A simulation method was developed so that the estimated 

econometric model could be used to examine the effect of changes

in the individual income tax rates on aggregate capital gains

income and Federal tax receipts. The simulation model is 

important to capture the effect that when lower capital gains tax 

rates increase realizations, the increased realizations force 

taxpayers into higher marginal tax brackets. Ignoring the interaction of increased realizations and marginal tax rates results 

in overstated estimates of taxpayer responsiveness. The simulation at 1982 levels finds that a small change in the inclusion 

rate results in a -2.0 short-run realization elasticity and a 

-1.6 long-run realization elasticity. 

The estimation results imply that taxpayer response to lower 

capital gains rates is sufficiently large to support claims that 

lowering capital gains tax rates would increase Federal tax 

revenues. Much of the disparity between results of prior individual tax return studies is found to result from their failure 

to properly distinguish taxpayer decisions about whether or not 

to realize capital gains from their decisions about how much 

capital gains to realize. In addition, some of the disparity is 

due to lack of a proper simulation methodology that accounts for 

the simultaneous determination of capital gains realizations and 

marginal tax rates on capital gains. 

Office of Tax Analysis

Department of the Treasury

May 16, 1989 
1. INTRODUCTION 

The current debate over the direct and indirect Federal revenue effects of 

reductions in the marginal tax rate on capital gains highlights the importance of 

finding the underlying reasons for the lack of a consensus. The issue of whether 

long-term capital gains realizations should be treated as ordinary income. or be 

giwn preferential treatment. has important implications for allocative efficient!.. 

distributional equity. and the simplicity of the tax code. One possible explanation 

for the disparate views involjres clifferences in the u’aj. various time-series 

regression models are specified. since these models yield measures of the 

sensitivity of capital-asset realizations to changes in the marginal tax rate on 

capital gains. This area deserves careful attention, 

Time-series tax elasticity estimates have heen used in simulation models to 

produce revenue estimates in several recent studies dealing with the re\renue 

consequences of changes in the taxation of capital gains. For example. recent 

simulation studies by Darhy. Gillingham. and Greenlees ( 1988) and by Toder and Ozanne 

(CBO. 1988) relied on aggregate time-series estimates of the behavioral response of 

taxpayers to changes in capital gains taxation for the period 1954-1985. Obviously. 

revenue estimates are sensitive to the tax elasticity estimate that is used in the 

simulation model. While all of the aggregate time-series studies find that increases 

in the marginal tax rate discourages realizations. the magnitude of the estimated 

response varies considerably. In a recent paper. Auten. Biiman. and Randolph 1989) 

present a summary table which shows the wide range in aggregate time-series tax 

elasticity estimates. These estimates range from -0.06 to - I .5 I, With the exception 

of the study by Auerbach (1988). the importance of the specification issue to the 

capital gains debate is a matter that largely has been overlooked. 

This paper examines the robustness of capital gains tax elasticity estimates to 

alternative regression equation specifications. Only single-equation regressions

using aggregate time-series data are studied. The historical period that is examined 

spans 1948 to 1987. Specifically. functional form, choice of dependent variable. the 

explanatory variables included in the design matrix. lag length. nonstationarity of 

the data. and simultaneous equation bias are some of the issues that are addressed. 

Our intention is to use what we discover about these various aspects of equation 

specification to specify a more appropriate equation with which to estimate the 

response of realizations to changes in the marginal tax rate. 

In general. we find evidence that suggests that aggregate time-series tax 

elasticities are not at all robust with respect to specification of the regression 

model. The imqcation of our findings is that the elasticity can be macle either 

large or small depending on how the estimating equation is specified. Because of 

this troublesome sensitivity. aggregate time-series equations cannot be relied on to 

produce what could be termed a definitive elasticity estimate. This means that tax 

policy analysts must look elsewhere for more credible elasticity estimates. A better 

alternative may be the use of elasticity estimates from panel or pooled cross-section 

microdata in combination with estimates from time-series data. 

-1-
The plan of this paper is as follows. Section 2 provides a brief summary of some 

of the time-series equations used in previous studies. This discussion serves as a 

starting point for the analysis that follows. 

Section 3 discusses the importance of equation specification to valid estimation 

and statistical inference. Specification error and its implications for 

least-squares estimation and h!.pothesis testing are also discussed. A specification 

test cleveloped hjr Davidson and MacKinnon ( 198 1 ) is iised to examine the equations 

specified in recent studies bjs the Treasury Department ( 1985). Cook and O'Hare 

( 1987). Toder and Ozanne (CBO. 1988). Darby et al. ( 1988). Minarik (1988). and Kiefer 

(1988). This test should permit identification of a best. or group of hest. 

equations that can serve as a starting point for our search to find a better 

estimating equation. 

Section 4 deals with specification searching that is unclertaken to discover a 

more appropriate regression model. Modifications that involve additional explanatorj. 

\m-iahles. expectations of some of the explanatory variables. the borrowing issue. 

ancl portfolio shifting precipitated by changes in the differential between the 

marginal tax rates on ordinary and capital-asset income are examined. 

Finally. Section 5 presents the regression results for the preferred regression 

specification. The strengths and weaknesses of this equation are discussed. as well 

as several caveats about the use of aggregate time-series equations to estimate 

capital gains tax elasticities. In addition. elasticity estimates are presented for 

various combinations of al teinative capital gains measures and alternative functional 

forms for marginal tax rates in order to assess the impact on the tax elasticity 

coefficients. 

2. PREVIOUS TIME-SERIES STUDIES 

Table 1 presents the various specifications for the equations used in the six 

studies cited above, Inclucled is a description of the data and the historical period 

used. The equations are grouped according to whether the data are expressed as 

first-differences or levels. On the one hand, the equations specified by Cook and 

O'Hare. the 1985 Treasury Department study. and Minarik use first-differences of 

unlogged data. On the other hand. the equations specified by Darby et al.. Toder and 

Ozanne. and Kiefer are generally specified in terms of log-levels of the variables. 

It is shown in Section 3 that the use of differenced data to achieve stationaiity 

ancl. thereby. avoid the spurious regression phenomenon noted by Granger and Newbold 

( 1974). receives empirical support from Dickey-Fuller (1979) tests for unit roots. 

We examine four separate equations for both Darby et al. and Toder and Ozanne. 

The equations investigated for Darby et al. are the equations reported in Table 3 of 

their study. These equations use alternative functional forms from Table-A3 of the 

study by Toder and Ozanne with the 1985 Treasup study's measure of total realized 

capital gains and marginal tax rates for upper income taxpa!t.rs. In aclclition. the 

-2-
equations from the Kiefer study. although they deal with simulation and not real 

aggregate data. are examined in terms of their consistency with real data. 1 / 

The six studies differ in their choice of a dependent variable. The studies by 

Toder and Ozanne and Minarik use net long-term capital gains in excess of short-term 

losses as the measure of realizations. All the other studies employ total realized 

capital gains. This latter measure is computed as net long-term capital gains in 

excess of net short-teim losses plus net short-teim gains for those taxpajws with 

gains from the sale of capital assets.2/ 

Besides differences in the dependent variable. there also exist important 

differences in the marginal tax rates that are used. For example. Cook and 0'Hai.e 

me the maximum marginal tax rate on capital gains. Darby et al. use the 1985 

Treasury study's marginal tax rate for upper income taxpayers. while Toder and Ozanne 

employ a weighted average of marginal tax rates for all taxpayers. Valid arguments 

can be put forth to justify the use of the alternative measures of capital gains as 

well as the various definitions of the marginal tax rate variahle.3/ 

In addition. there are significant differences in the variables that are incluclecl 

in the design matrix to capture movements in economic activity and the wealth of 

taxpayers. According to economic theory. taxpayers may choose to realize capital

gains in order to rearrange their financial portfolios or to finance consumption and 

investment in consumer durables. In general. the amount of capital gains that are 

realized will be related to the stock of wealth in capital assets and economic 

acti vi ty , 

Most of the studies proxy the wealth of taxpayers with taxpayer holdings of 

corporate equity. This is done because there is no directly observable measure of 

total accrued gains since the tax basis of capital assets cannot be observed until 

the assets are sold or exchanged. Proxies for total accrued gains can be constizictecl 

with Flow of Funds data. and then can be used in estimating regressions. For 

example. Auten has constructed such an historical series for accrued capital gains 

using asset revaluation data from the Flow of Funds accounts for the post-World War 

11 period up through 1985. However. because of measurement errors, this will result 

in biased regression estimates owing to the errors-in-variables problem. 

With regard to variables that reflect change in economic activity. the studies 

use the level of GNP. changes in GNP. and some measure of the price level. such as 

the GNP deflator. or the Standard and Poor's price index. Both GNP and equity 

holdings are measured either in nominal. or real terms. depending on the study. 

Finally. Cook and O'Hare use the differential between the maximum marginal tax 

rates on ordinary and capital gains income to capture any income shifting that 

results from changes in the taxation of capital gains. 

3. SPECIFICATION ISSUES AND TESTING 

The importance of correct equation specification to \did estimation and 

inference is well known, In general. the issue of the specification of an estimating 

-3-
equation involves not only the basic structure of the regression model but also 

includes whether the standard assumptions of the classical regression model are 

satisfied. These assumptions include: (1) functional form is correct. (2) dependent

and independent variables are measured without error. (3) design matrix is coirectly 

specified in terms of the variables that are included and excluded from the 

regression. (4) regression error has a zero mean and satisfies the sphericality

conditions. (5) design matrix has full column rank. and (6) the orthogorialit), 

condition holds for all regressors. i.e.. there is a zero covariance I?etween the 

regression error and the explanatory variables. Violation of any one of these has 

important consequences for the sampling distribution of the parameter estimators nncl 

statistical inference . 

In applied econometric work. specification error is Ikvecl in a narron’er sense 

and usually falls into one of four categories: (1) incorrect functional form. (2) 

omitted relevant variables. (3) includecl irrelevant \mMles. and (4) incorrect 

specification of how the error enters the regression equation: i.e.. adcliti\~el!. or 

multiplicatively. Specification errors are important because of their adverse 

statistical consequences. These include liiasecl and inconsistent estimates. 

inefficient estimates. and incorrect inferences arising from biases in the 

\,ariance-co\iariance matrix of the estimators. For example. the omission of a 

relexint variable from the regression model can result in biased and inconsistent 

parameter estimates. In addition. incorrect hypothesis tests result because the 

constiucted confidence intervals are too wide: consequently. the null hypothesis is 

accepted too often as true. 

SDecification Tests: 

To understand better whether the time-series equations used in previous studies 

are consistent with the data. a specification test is employed to examine the 

equations detailed in Table I, The Davidson and MacKinnon ( 198 1) non-nested 

specification test is used to evaluate the equations. While there are several 

sDecification tests that can be used to examine non-nested regression models (See. 

e:g.. the special issue on specification tests in the Jour& of Econometrics. 

(1983)). the Davidson and MacKinnon test was chosen because it has correct asymptotic 

size. good asymptotic power: and. in addition. it is easy to implement. The purpose 

of the specification test is to isolate those equations that are inconsistent with 

the data. 

The equations can be divided into two groups depending on whether the data are 

expressed in levels or first-differences. Through so-called artificial nesting of 

the equations. the test is applied to all of the equations in each of the two groups. 

For those studies where there are four equations that are examined. the test is first 

used to find the equation that peiforms best in the group. This equation is then 

used to assess the relative performance of the competing specifications in the other 

studies. 

By definition. a non-nested set of quation? concists of ttqriritions that cannot he 

derived from one another through simple restrictions. such as zero restrictions. In 

-4-
other words. the union of the design matrices of the different equations is not 

identical to the design matrix for any one equation. There can be overlapping of 

explanatory variables. but there must be at least one non-overlapping explanatory 

1.ariable. See Harvey (1981) and Judge ­et al. (1984) for discussion of the difference 

lietween nested and non-nested sets of equations. 

I\/lost of the equations that are investigated are non-nested. hut there are seiwal 

exceptions. For example. the first equation for Darhy et al. is a special case of 

the third equation. Similarly. the first and third equations for Toder and Ozanne 

are nested in the second and fourth equations. respectively. Finally. the equations 

for Kiefer with one through four lags on the niarginal tax rate are all special cases 

of the equation with fixre lagged values of the tax rate. The fact that there are 

se\.eral nested equations does not ha\re an impact on the overall findings of the 

specification test. In all cases. the lower dimensional nested regression models are 

olitaineci from the general models through zero restrictions. 

Technically. a nested specification test should be used for nested equations. A 

comparison of acljustecl R-squared \.slues coirld he used as the testing procedure. For 

the nested regressions. the results of the Da\iclson and MacKinnon test were the same 

as the results from the acl-justed R-squared comparison. 

The Davidson and h4acKinnon test was applied to the equations in each group

separately. That is. the equations using differenced data were evaluated relati1,e to 

each other. and the same was done for the equations using levels of the data. 

Although the right-hand-side (RHS) variables can differ. it is necessary for the 

dependent variable to be the same, or that some transformation of the same variable 

be used. e.g.. logarithmic transfoimation, For those equations where the dependent 

variable is different. appropriate changes were macle so that the dependent variable 

was the same in conducting the test. 

The Davidson and MacKinnon test is implemented as follows. Let 

L

H, : y = Xf3, + U. u ,..N(O.uUI) 

2

H, : y = Zf3, + V. v - N(O.avI) 

represent two alternative regression models that purportedly explain movements in the 

conditional mean of y. X and Z are (T x KO) and (T x I(I) design matrices. 

respectively. f3, and f3, are (KO x 1) and CrC, x 1) location parameter vectors. 11 and v 

are both (T x 1) disturbance vectors with classical properties. and y is a (T x I) 

vector of obsemations on the dependent variable. The OLS estimators of f3, and 6, 

are clenotecl as ioand g1. 

The two regression models are tested by artifically nesting one in the other. 

This produces the following compound or mixing model 

-5-
coefficient on the fitted value from the alternative model in each 

artificially-nested equation. and is a linearized version of the Cox N-statistic. 

[See Judge ­et al. for a discussion of non-nested testing procedures that are related 

to the Cox test.] If the computed J-value is greater than the critical value for the 

test. the null regression is rejected as adequate relative to the altemative 

regression model. For a value of the J-statistic smaller than the critical value. 

the null regression model is accepted as being an adequate representation. 

Table 2 pro\.ides the coniputeci J-statistics for the equations that use le~~lxof 

the data. Several basic conclusions can he drawn. First. the four equations iisecl 17). 

Darby et al. are 1.el-y similar. and it is not possihle to identify any one equation 

that is best or worst. Intuitii,ely. this makes sense because the equations differ 

essentially only in teims of the transformation on the marginal tax rate. Because 

the capital gains tax-elasticity point estimate of -.67 produced hy Eq. (4) has heen 

the focus of some discussion. we chose this specification from the Darhy et al. strid!. 

to conduct the specification test with the equations from the other studies. 

Whether the relationship between realizations and the tax rate is 

semi-logarithmic. or double-logarithmic. does not appear to matter much in terms of 

consistency with the data. However. it does matter how the tax rate is entered if 

the elasticity is assumed to remain constant or to change as the tax rate changes. 

In general. the tax elasticity of realizations increases as the marginal tax rate 

increases. This occurs because taxpayers become more responsive to changes in the 

tax rate as the amount paid in taxes on realizations rises due to tax increases. 

Second. for the Toder and Ozanne equations. the nominal and real equations that 

include the first-difference of real GNP perform better than the equations that omit 

this variable. This suggests that inclusion of some measure of the business cycle on 

the right-hand side of the estimated equations improves the predictive power of the 

regression model, 

Third. the Kiefer specifications. using alternately lags 1 through 5 on the 

marginal tax rate. are rejected by both Darby. Gillingham. and Greenlees' Eq. (4)and 

Toder and Ozanne's Eq. (2). In estimating the Kiefer equations, the Treasury 

marginal tax rate and the value of households corporate equity holdings in the 

previous year were used as RHS variables. Finally. Darby et al.'s Eq. (4) is a 

better representation than the Toder and Ozanne equation for nominal capital gains. 

This holds whether total or net long-term realized capital gains is used as the 

dependent variable. 

Table 3 presents J-statistics for the equations using first-differences of the 

data. The Davidson and MacKinnon test was carried out on the equations used by Cook 

and O'Hare. the 1985 Treasury study. and Minarik using both total and net long-term 

gains. Real GNP and the GNP price deflator are used in the Treasury equation to 

avoid the variable problems in the original Treasury equation discussed by Darby et 

al. The results show that when total realized gains is the dependent variable. both 

the Treasury and Minarik equations are better than the Cook and O'Hare equation. In 

addition. the Treasur!' equation is found to he hetter than Minarik's equation. 

-7-
Sharply different results emerge when the dependent variable is changed to net long 

term capital gains in excess of net short-term losses. All three equations are shown 

to be inadequate representations of the process generating capital gains 

realizations. 

To summarize the results in Tables 2 and 3. the Darl?y et al. equation appears to 

outi?eiform the other equations using levels of the data in terms of its consisteiiq. 

with the data. For those equations using differenced data. the 1985 Treasiti? stttcl! 

equation is found to be better for the case in which total realized gains is the 

dependent variable. This was not the case when net long-tetm gains were used as the 

dependent variahle. 

Additional Issues in Specification: 

The topics of nonstationarity of the data. the choice of lag length. and 

simultaneous equation hias are explored in what follows. All of these represent 

potential problem areas cvhich deseive consideration in the context of equation

specification. 

Nonstationarity : 

Initial experimentation with and without a linear time trend in several of the 

equations that are examined re\vealecl a troublesome sensitivity of the regression 

results to detrending of the data. Presumably. the equations expressed in 

first-difference form were specified in such a way. in part. to acljust for 

nonstationary components in the data. There is really no way to be certain that this 

is why first-differences of the data were used. however. since there is a lack of 

discussion of the behavior of the data over time. 

The issue of nonstationarity is important because. as is well known. failure to 

account for the secular movement or low-frequency component of time series that are 

related in equations can bias re.gression results. In general. it appears that this 

issue has not been dealt with adequately in the studies under review in this paper. 

Auerbach (1988) made a similar observation about the lack of attention paid to the 

nonstationarity issue. In order to account for the nonstationarity of the data. 

Auerbach includes a linear time trend in estimating his equations using levels of the 

data. However. as we discuss below. this also results in a specification error 

because the data are found to be difference stationay time series. 

Granger and Newbold (1974) showed that the use of nonstationary data in 

regressions can result in spurious significant results. Nonstationarity causes a 

downward bias in the standard error of coefficient estimates which results in 

inflated test statistics and incorrect inferences. In order to avoid the "spurious

regression phenomenon". the use of differences is recommended. Although this is not 

a panacea. it is better than making no adjustment at all. 

-8-
In related work. Nelson and Kang (1981. 1984) have shown that inappropriately 

detrended data can lead to invalid regression results because of inflated test 

statistics. This occurs when time series that are difference stationary are 

incorrectly assumed to be time stationary. and. consequently. are detrended by being 

regressed on a time trend. or some function of time. By definition. a trend 

stationary time series is one which can be made stationary by regressing it on a 

deterministic time trend or some function of time. e.g.. a polynomial of seconcl or 

third degree. On the other hand. a difference stationary time series is macle 

stationary by differencing an appropriate numker of times. depending upon the number 

of autoregressive unit roots in the time series. 

Nelson and Plosser ( 1952) have found that most macroecononiic time series for the 

U.S. are random walks. which are a class of integrated time series processes. This 

means that these series are difference stationary. 

Tables 4 and 5 present the results of the Dickey-Fuller (1979) tests that \\ere 

carried out for all the time-series variables used in all the equations examined. 

This includes both dependent and independent variables. In all. 23 series are 

examined. Basically. what is at issue is whether a series has a unit root. and if it 

does. how many times the series must be differenced to induce stationarity. 

Table 4 presents results on whether the various series are trend stationary (TS) 

or difference stationary (DS). A TS series does not have a unit root. while a DS 

series has a unit root and must be differenced to make it stationary. The 

Dickey-Fuller test as conducted by Nelson and Plosser is used. To carry out the 

test. a first-difference of the series is regressed on a constant. a linear time 

trend. and a lagged value of the level of the series. The computed t-statistic on 

the lagged value of the series is then used to test the null hypothesis that the 

coefficient is one. To reject the null hypothesis. the t-value must be large and 

negative. The computed t'est statistics in Table 4 show that all the series are DS 

time series. Critical values for the test statistics are taken from Fuller (1976). 

Table 8.5.2. p. 373. 

Table 5 presents results of an Augmented Dickey-Fuller (ADF) test for 

stationarity. The test is conducted as done in Engle and Granger (1987). Both 

second-order and fourth-order autoregressive processes were used to conduct the test. 

Because the results were the same. the results from the second-order autoregressive 

regressions are reported. The critical values for the test statistic are taken from \ the paper by Engle and Granger. The ADF test is used to determine the degree of 

differencing necessary to induce stationarity in a DS series. Because all levels of 

the series were found to be difference stationary in Table 4. all levels of the 

series should be nonstationary. and this is found to be the case in Table 5 as well. 

Although the time series that were examined are found to have more than one unit 

root. the ADF test has low power in small samples. which results in not rejecting a 

false null hypothesis of nonstationarity. As a check on this. the autocorrelation 

functions for the first-differences of se\,eral of the series \\'ere examined. The 

autocoi-relation functions re\realecl that first clifterences \I et.? adequate to make the 

series stationan:. and that no further differencing ',\'as neeclecl. 

-9-
There are several regression strategies that can be used to avoid the spurious 

regression phenomenon. First. as recommended by Granger and Newbold. the data can be 

differenced. Second. one can make sure that all the variables that account for the 

nonstationarity of the dependent variable are included in the design matrix. This 

approach is dismissed by Nelson and Plosser as unrealistic. Third. the Engle and 

Granger approach of using co-integrating and error correction regression equations

coulcl be used. Work involiing this approach to modelling the process generating 

capital gains realizations is currently in progress. hut the results are not 

reported. The simplest strategy is the first. and this is the approach taken in (his 

study to deal with the nonstationarity issue. 

Lag Length: 

It is well known that the choice of lag length has an impact on regression 

results. This is a result of the efficiency-bias tradeoff that exists in determining 

the length of distributed lags. If significant lags of a variable are omitted. this 

will cause biased estimates. On the other hand. an excessively long lag ai.oicls the 

bias problem. but results in inefficient estimates. 

To determine if lag length has an impact on the capital gains tax-elasticities 

izielded hy time-series regressions. two different lag-length determination criteria 

ivere used to fit optimal lags to the variables used in Darby. Gillingham. nncl 

Greenlees' Eq. (4). The two criteria include Akaike's final prediction error (FPE) 

and Schwarz's Bayesian information criterion (BIC).SI Although the FPE is frequently 

used. it has a tendency to over-fit distributed lags asymtotically. This means that 

too many lags are specified. Thomton and Batten (1985) recommend its use. but Jones 

( 1989) finds evidence which does not corroborate their results. Lutltepohl ( 1985) 

recently has shown that the BIC performs well in fitting appropriate lag lengths in 

vector autoregressions. Both criteria are used to note whether the optimal lags are 

the same. 

Table 6 presents results on lag length for both the BIC and FPE. The lag lengths 

are identical for real GNP. the tax rate. and. real equity holdings. but they differ 

for the GNP deflator and lagged capital realizations. As noted above. the FPE tends 

to over-fit the lag length. which means that too many lags are included. and this 

appears to be the case here. Also included in the table is the coefficient on the 

marginal tax rate. The equation using the lag lengths specified by the BIC finds an 

elasticity of -.83. which is higher than the elasticity of -.61 for the equation 

using FPE-determined lags. 

Thus. the choice of lag length does appear to have an impact on the elasticity

estimate. The results on lag length do point out that the Darby et al. equation 

appears to be misspecified to the extent that lagged values of capital gains

realizations and the GNP price deflator are not included as RHS variables. 

- IO -
Simultaneous-Equation Bias: 

Finally. a Granger-causality test was carried out using the Darby equation for 

the sake of illustration. Both multivariate as well as bivariate tests were 

peiformed in which four lags on gains and four lags of the tax rate were used. 

E\,icIence of feedhack was found hetween capital gains realizations and tax rates. The 

strength of the feeclback is influenced b!, the equation specification. particiilarl>.. 

whether a contemporaneous term is also included as a RHS variable. There is e\klence 

of contemporaneous feedhack hetween gains and the tax rate. In addition. there is 

also e\*idenceof feedback between real GNP and equity holdings. DarbJr et al. point 

out the importance of the feeclhack hetween eqiiit), holdings and gains in their 

The results reported here support their view. 

Based on these findings for equations using leirels of the clata. it appears that 

feedhack between capital gains and a subset of the variables included in the design 

matrix is a matter that should he addressed in deciding on an appropriate estimation 

technique. That is. the question as to whether ordinary least squares (OLS). or an 

instrumental variable (IV) estimation procedure. such as two-stage least-squares. 

should be used needs to be answered. 

In previous studies. some attention has been paid to the possibility of feedhack 

between capital gains realizations and the marginal tax rate. For example. the IPS5 

Treasury study and the study by Darby et ai. report that both OLS and IV estimation 

yielded essentially the same results. Both OLS and IV estimators were used in this 

study to note the sensitivity of the results to the choice of estimation method for 

the Darby et al. specification using levels of logged clata. 

In general. the OLS and IV estimates were largely the same when the marginal tax 

rate was the only RHS variable that was instrumented. The instruments used incluclecl 

lagged real GNP. the price deflator lagged. lagged equity holdings. and the lagged 

tax rate. The OLS tax-elasticity estimate was -.59. while the IV estimate was -.54. 

which is somewhat lower. However. when the equity and GNP variables were also 

instrumented using the same set of instruments. the OLS and IV estimates varied 

considerably. More attention needs to be paid to this particular issue. and 

additional work is being done on this issue currently with equations using 

differenced data. 

The choice of instruments is a non-trivial decision. since the final regression 

estimates will be sensitive to which instruments are used. It is well known that 

instrumental variable estimators. in general. are biased in small samples. and that 

their Xrariances are difficult to establish. In addition. poor results are obtained 

if a set of instruments is chosen which are not highly correlated with the endogenous 

iariables that are instrumented in the stivctural equation. See. e.g. . Johnston 

(1984). pp. 363-366 for further discussion. 

- II -
4. SPECIFICATION SEARCHES 

In this section. we attempt to specify an improved aggregate time-series 

regression equation that can he used to estimate the capital gains tax-elasticity. 

To accomplish this task. we use the results from the previous section to guide our 

specification searching. In addition. we also examine modifications iniroli,ing 

atlditional explanatory variables. expectations on a suLiset of the explanator!. 

\.ariables. the borrowing issue. and portfolio shifting brought about 17). changes in 

the taxation of capital gains. 

Appendices A. B. and C present selected regression results of some of the 

specification searching that was done. Data for these regressions were for the most 

part taken from Flow of Funds halance sheets and the Economic Report of the 

President. 1989. Only the regressions using differenced data are reported. although 

the same basic set of equations was also estimated for levels of the data. both 

logged and unlogged. The stock repurchase series is taken from Shoven (1986). 

Appendix D presents the data that are used in the estimation. and Appendix E reports 

simple correlation coefficients for some of the \wiaLiles used in the regressions. 

In Section 3. it was shown that proper equation specification requires that the 

data be differenced in order to avoid the spurious regression phenomenon. In 

aclclition. attention needs to he paid to lag length and possible feedback between 

capital gains realizations and the marginal tax rate. as well as between realizations 

and real GNP and corporate equity. Based on the results of the Daviclson and 

MacKinnon specification test. we use Darby et al.'s Eq. (4) expressed in 

first-differences as our initial equation. The double logarithmic transformation was 

chosen to minimize potential problems with heterosceclastic regression errors. 

Whether the relationship between capital gains and the tax rate is 

double-logarithmic or semi-logarithmic is a matter that needs to be decided )y the 

researcher. since the data do not support one functional form over the other The 

choice depends on whether the elasticity is assumed to be constant or to vary as tax 

rates change. We assume in what follows that a constant elasticity holds. This 

assumption is relaxed for the results reported later in the paper in Tables 0 and 

11. 

Wealth Variables: 

Many different variables to proxy for the wealth of taxpayers were experimented 

with in the design matrix of the regression model. These variables include: Auten's 

constructed accrued gains series. stock repurchase data. the net worth of taxpayers

from the Flow of Funds accounts. and Lindsey's measure of tradable wealth. Lindse!. 

(1986) defines tradable wealth as the sum of the values of land. residential 

structures. corporate equities. and equity in non-corporate liirsinesses held Ii!, 

households. Data from the Federal Resenre Board's Flow of Funds lialance sheets vere 

used to construct this series. 

- 12 -
Although the results are not reported for all the experimentation that was clone. 

the use of a wider measure of taxpayer wealth results in an increase in the short-nrn 

tax elasticity of gains realizations. In general. the elasticity was pushed above I 

in absolute value. 

This is what would be expected to occur in the case where a rele\mt \wiablti is 

omitted from the regression model. It is well known that a relevant omitted ixriahle 

that is positi\*el>scorrelated with an included \m-iahle results in an up\vard hias in 

the coefficient estimate of the included \wiahle. See. inter alia. Knienta ( 1986) 

for discussion. For example. the inclusion of only part of tradable wealth in the 

form of corporate equit). holdings results in an elasticity estimate that is liiasecl 

tourard zero. This means that the elasticit!r estimate is snialler in ahsolute ~~alue 

than it woulcl be if the larger measure of wealth were included in the regresqion 

equation. This omitted variable problem arises because the wealth of taxpa! ers 

appears to be positively correlated with the marginal tax rate.6/ 

In the experimentation that was done with alternative wealth variables. including 

net worth. accrued gains. and tradable wealth. all had a similar impact on the 

tax-elasticity estimate. Tradable wealth was chosen as the appropriate aggregate 

measure of wealth in the preferred equation. Accrued gains was not chosen because of 

measurement errors that woulcl result in an errors-in-variables bias. Similarl>s.net 

worth was not chosen hecause it also includes taxpayers liabilities. and it could 

therefore vary without any change in potential realizable gains. The stock 

repurchase series of Shoven as well as the net equity purchase series for hoiiseholcls 

from the Flow of Funds accounts failed to have significant explanatory power. and so 

they were both dropped from further consideration. 

Expectations: 

As noted by both Auerbach ( 1988)and Toder and Ozanne (1 988). the forward-looking 

expectations of taxpayers can have a significant impact on capital gains 

realizations. In their studies. some form of instrumented single-step expectations

of the marginal tax rate were included as RHS variables. Toder and Ozanne failed to 

find a significant impact for the 1954-1985 period. Auerbach. on the other hand. 

found that the single-step expectation on the tax rate was highly significant for the 

period 1954-1986. 

The inclusion of 1986. in which there was an announcement effect which taxpaj'ers 

could take advantage of in realizing gains. explains the difference in the results 

reported by Auerbach and Toder and Ozanne. In order to avoid the much higher 

marginal tax rates of 28 and 33 percent that went into effect in 1987. taxpayers 

accelerated quite noticeably the rate at which capital gains were realized in 1986. 

For example. total realizations were $168.6 billion in 1985. while they increased to 

$335.4 billion in 1986 in anticipation by taxpayers of the higher marginal rates 

introduced by the Tax Refoim Act of 1986. 

In addition to tax-rate expectations. T,\Y consicler evpectations for other 

variables. Expectations on real GNP. the price deflator. and wealth holdings \\'ere 

- 13 -
considered. In particular. we find that the expectation of the wealth variable. 

whether it is equity. or tradable wealth. is significant when included in the 

regression model. While the coefficient on the tax rate expectation is positive. 

which means that an expected increase in the future tax rate will cause realizations 

to increase today. the coefficient on the expected wealth variable will he negative.

This means that an expected increase in the value of wealth that can be realized as 

capital gains tomorrow will cause taxpayers to decrease realizations today. 

c 

Borrowi n 4:

According to the borrowing issue. the taking of realizations hy taxpaj'er-s tocla!, 

to finance consumption and investment in consumer clurahles. results in a cliniinishecl 

stock of accrued gains and. therefore. smaller potential realizations in the future. 

Experimentation with indi\klual lagged values of realizations and transfoimations on 

lagged realizations showed that a simple three-period moving average of gains had an 

i m pact on current real izat i ons . The t h ree-pe ri ocl m oving-a\.erage i s const 1-11 ct ecl 11sin g 

three lagged values of nominal gains. A priori. we expect gains realized in the past 

to have a negative impact on gains that can he realized today. 

Portfolio Shifting: 

The portfolio shifting issue was examined by including the clifferential hetween 

the maximum marginal rates on ordinary and capital gains income. This simple measure 

was proposed by Martin Baily in unpublished work and. subsequently. was used in 

Brittain (1964) to examine income shifting for films. Cook and O'Hare also used this 

measure in their study. although their results showed that the differential had an 

insignificant impact on capital gains. Interestingly. the results that they report 

show that the differential vaiiable had a negative sign. which is not what would be 

expected a priori. One woulcl expect realizations to increase in response to an 

increase in the differential between marginal rates on ordinary and capital-asset

income. This occurs because assets yielding capital income become more attractive 

relative to assets producing ordinary income. and taxpayers rearrange their 

portfolios acccordingly . 

In addition. the significant portfolio shifting result that Cook and O'Hare 

report for the separate net interest and dividends in AGI equation that they estimate 

must be viewed with some caution. It appears that a specification error in the form 

of omitting a price index from this equation produces the finding of a significant

income shifting effect. The inclusion of the GNP deflator. which is appropriate 

since real GNP is included as a regressor. eliminates the significance of the 

differential tax term. Also. careful inspection of the data for the interest and 

dividends series that Cook and O'Hare use shows that dividends are reported in 

millions of dollars while interest is reported in billions of dollars. When the two 

series are added together without adjusting for the different units. the di\.idencl 

series dominates movement of the combined series. 

- 14 -
Portfolio shifting would occur if taxpayers rearranged the composition of their 

financial portfolios between assets yielding ordinary and capital-asset income in 

response to a change in the differential between the tax rate on ordinary and capital 

gains income. In testing for portfolio shifting. an approach similar to that used ty~ 

Cook and O‘Hare was used. but for differenced data. Some experimentation was also 

clone with expectations on the differential. It was not possilile to isolate :in!. 

significant separate effect that the differential had on realizations. To R certain 

extent. this could be the result of collinearit!. between the differential tax tt’rm 

and the marginal tax rate on capital gains. which would make the coefficient on the 

differential variable insignificant. 

5. REGRESSION RESULTS FOR THE PREFERRED EQUATION 

In this section. we report results for the regression model that incorporates the 

modifications discussed in the previous sections. Tables 7. 8. and 9 present the 

regression results . 

Most of the previous studies examined the historical period 1954-1985. with the 

exception of Auten ( 198I). who used data beginning in 195 1. This study extends the 

sample period back to 1948: and. in addition. 1986. and the preliminary estimate for 

1987. are included in the period examined. Auerbach also used 1986 in the 

regressions that he estimates. The dependent variable in all the regressions is 

total nominal realized capital gains and the marginal tax rate is that for upper 

income taxpayers used in the 1985 Treasury study. The choice of these two \ariables 

stems from their favorable perfoimance in the specification test. 

To assess the sensitivity of the tax elasticity estimates to the choice of 

capital gains and tax rate measure. Tables 10 and I1 report results for net long term 

gains and the maximum marginal rate on realizations. In addition. the 

semi-logarithmic form for the tax rate variable is also used to re-estimate the 

preferred equation specification. 

Table 7 reports results for two versions of the differenced Darby et al. 

equation. The first version includes equity holdings of taxpayers as the wealth 

variable. while the second version uses the broader tradable wealth measure. Both 

equity holdings of individuals as well as equity holdings of individuals yielded 

similar results. so only the results for the holdings of households are reported. 

Adjusted R-squared coefficients. Durbin-Watson statistics. Box-Ljung portmanteau 

Q-statistics with marginal significance levels in parentheses. and degrees of freedom 

for each equation are reported. The Q-statistics are used to test for a random 

correlogram for the regression residuals. and peimit testing for autoregressive ( AR).

moving average (MA). or some combined ARMA process for the residuals. Computed 

t-statistics for the coefficient estimates are reported in parentheses. 

The following conclusions can 17e dra:: n. First. the elmticit!. estimates arc 

sensitive to the choice of sample period. This sensitivit>r is clue to whether- 1986 is 

- 15 -
inclucled. and also is a function of whether the particular historical period that is 

examined extends back to 1948. Second. the elasticity estimates are smaller for the 

first version of the equation that includes only equity holdings. and not all of 

tradable wealth. Note that the use of tradable wealth in the specification appears 

to improve the overall fit of the regressions. The adjusted R-squared coefficients 

increase and the Q-values are generally lower. However. the Durbin-Watson statistics 

are unifornily lower with the use of the hroader measure of wealth. 

Table 8 reports results for the preferred equation specification. There are tn.o 

i.ersions of the preferi'ed specification: one with equity holdings only. and the other 

with tradable wealth, The preferred specification inclucles a three-period nioi.ing 

average on nominal realizations. the single-step expectation of the marginal tax 

rate. and the single-step expectation of the wealth variahle that is included in the 

equation. The regressions are estimated with peifect foresight expectations. This 

means that the actual values of the variables are used as the expected values. 

All jwiables are expressed as the iinweighted first-difference of natirral 

logarithms. which means that all are expressed as approximate percentage changes. 

Given this particular transfotmation on the data. our equations explain the 

percentage change in nominal capital gains realizations. Also. except for the 

moving-airerage term. all variables on the right-hand side of the regressions are 

expressed in real 1982 dollars using the GNP deflator to acl-just the nominal 

There are several interesting conclusions that can be drawn. First. the tax 

elasticity point estimate varies with the wealth variable that is used. Second. the 

expected tax rate variable is highly significant when 1986 is inclucled in the sample. 

hut becomes insignificant when 1986 is omitted. This occurs because the future tax 

rate is picking up the announcement effect in 1986. Third. the expectation on equity 

has the appropriate algebraic sign. but is never significant. 

Fourth. the expectation on tradable wealth has the right sign. and is significant 

in two of the four equations: and it is close to being significant in a third 

equation. It is interesting to note the impact that the broader measure of wealth 

has on the size of the future tax coefficient. It increases the coefficient in all , 

cases. This stems probably from the omitted variable problem discussed previously. 

Fifth. the moving average variable in general has the right sign. but it is only 

significant in one equation. 

Finally. because the inclusion of 1986 determines whether the expected tax rate 

is significant. an argument can be made that it should be viewed as an outliner year.

In order to acljust for this. an intercept dummy variable is included for those 

equations which are estimated over the period that includes 1986. The dummy takes a 

value of I in 1986 and 0 for all other years. Table 9 reports these results. but 

only for the regressions with tradable wealth. The dummy variable is found to be 

highly significant for all equations. and the significance of the expected tax rate 

drops noticeably. These results show how extremely sensitive the expected tax rate 

is to whether or not 1986 is included in the sample period. and whether a clumm?. 

variable is used to capture the structural change in the wgession nioclel [hat 

apparently takes place in 1986. 

- 16 -
Short-Run and Long-Run Elasticities: 

Discussion has focused on the quantitative difference between the short-run and 

long-rim impact of changes in the marginal capital gains tax rate. In general. the 

short-run or temporary impact is larger than the long-run or permanent effect. 

Short-rim and long-run tax elasticities are computed for ow regression model !using 

the results in both Tables 8 and 9. Only the elasticities based on the regressions 

using tradable wealth are reported. 

In Table 8. the short-run elasticity over the period 19.54-1986 is -I. 13. and for 

the 1948-1986 pei-iocl it is -1.14. The long-run elasticity. which is the sun1 of the 

coefficients on the current and expected tax rates. is -0.18 for the period 

1954-1986. and a higher -0.25 for the 1945-1986 period. These estimates. of coiii'se. 

include the effect on the future tax rate of including 1986 in the sample without any 

acl.justnient with a dummy variable. 

The short-izrn and long-run elasticities are different in Tahle 9. where there is 

a dummy i7ariable adjustment. The estimates are higher. especially for the long-run 

estimates. The shoit'-run elasticity is -I. IS over 1954-1986. and it is -I. 17 for the 

1948-I986 period. The long-run elasticities are -0.74 and -0.89 for the 19.54-I986 

and 1948-1956 periods. respectively. 

Additional Regressions. ( 1948-1987): 

This section reports regression results for the sample period extended up to 

1987. In addition. the semi-logarithmic fiirlctional form for the tax rate is used. 

and net long-term gains are substituted for total gains. This allows us to note the 

sensitivity of the tax-elasticity estimates to both changes in the preferred equation 

specification. In addition. tradable wealth. as previously defined. and a narrower 

definition of tradable wealth that includes only corporate and non-corporate equity 

are used. 

There are several aspects of the broad measure of tradable wealth that may result 

in its not being the best measure of aggregate taxpayer wealth. since it includes 

all assets that are subject potentially to the capital gains tax. For example. 

owner-occupied homes. i.e.. residential structures and land, are infrequently subject 

to the capital gains tax because of the roll-over provision and step-up in basis at 

death. In addition, non-corporate equity is measured at replacement cost. and not at 

market prices. which could have an impact on its reliability as a measure of accrued 

gains. However. an argument can be made that all capital assets in the taxpayer's 

portfolio that are potentially suLTject to the capital gains tax should be included in 

the aggregate wealth measure. 

Experimentation that was done using the double-log specification with le\,els of 

the data for the four components of tradable wealth showed that corporate equit!.. 

- 17 -
non-corporate equity. and residential structures have significant explanatory power 

for total capital gains realizations. This was true for both the 1948-1986 and 

1948-1987 periods. However. land turned out to be insignificant in the regressions 

which had the four components entered separately. On the basis of these regressions. 

it appears that the use of the broad measure of tradable wealth is justified. This 

also lends support to the earlier conclusion that the use of a narrow measure of 

wealth will result in a clownward hias of the tax-elasticity estimate. 

Tables IO and 11 report the regression results for the additional runs. The 

tahles incliicle parameter estimates with t-\,alues in parentheses. acl.iustecl R-squared. 

and also the short-run and long-run tax-elasticities. and short-run and long-run 

re\.enue-maximizing tax rates. The revenue-maximizing tax rate is @\.en 17). - 1 /I>. 

where I:, is the coefficient estimate on the tax rate in the semi-logarithmic form. 

The coefficient b gives the proportional change in realizations brought about 13). a 

1 rC point change in the tax rate. Tax-elasticities are computed hy multiplying 13 13). 

the marginal tax rate of interest. An argument can be made that it is more 

reasonable to use the semi-logarithmic functional form since the tax-elasticity of 

realizations should increase as the tax rate increases. 

Table IO presents results that compare equations using total and net long-term

gains and the semi-logarithmic form. Specifications (I) through (6) use the tax rate 

for upper income taxpayers and net long-term capital gains. Because of some cloiibt 

about the accuracy of the 1987 preliminary estimate for net long-term gains. the clata 

run up through 1986 only. Specifications (7) and (8) present results for the 

semi-logarithmic form of the preferred equation. 

Several points can be made. First. there is little difference between the 

short-nin elasticity estimate in the double-log specification ( 1 ) and that given by 

the double-log specification (8) in Table 8. The former estimate is -1.19, and the 

latter estimate is -1.14. This suggests that there is little difference between 

using total or net long-term gains with the upper income tax rate in 

double-logarithmic form. The short-run elasticity estimates also are almost 

identical when the intercept dummy is used. Second. the long-run tax elasticities 

are somewhat higher for the equations that use net long-term gains. The long-run 

elasticities run from -0.44 to -0.96 for the regressions with and without a dummy 

variable. respectively. The corresponding elasticities for the equations using total 

gains run from -0.25 to -0.89. 

Third. regarding the semi-log specification, a comparison of specifications (5) 

and (6) with (7) and (8) is informative. The former equations use net long-term 

gains. while the latter use total gains as the dependent variable. The coefficient 

estimates on the current and future tax rates are almost identical. The long-run

re\renue-maximizing tax rate is computed by summing the coefficients on the current 

and future tax rates and then computing the negative of the reciprocal of the sum. 

The only problem with these estimates is that the long-run revenue-maximizing rate 

was an implausible value in specification (7). This can he attributed to the 

omission of the intercept dummy in this specification. If we focus our attention on 

specifications (6) and (8). u.hich are the clumiii! -acl.jicsted ec1li:rtions. there ic little 

difference in the revenue-maximizing tax ratec for 130th specifications. 

- 18 -
Finally. it does not appear to matter much whether the broad or narrow measure of 

tradable wealth is used. 

Table I1 reports results for the preferred specification with the sample period 

spanning up to 1987. Both double-log and semi-log specifications are estimated. In 

addition. the sensitivity of the tax elasticity estimates to the use of the tuv 

definitions of wealth is examined. 

There are several conclusions that can he drawn. First. the ~lo~~ble-log 

specification provides a hetter fit to the data in terms of higher acl.iusted R-squared 

terms and lower Q-\dues. Second. hoth short-run and long-run tax elasticities are 

higher when 1987 is included in the sample. For- example. in specification ( 1). which 

exclucles the intercept clumm).. the short-run elasticit), is - I.79 and the long-run 

elasticity is -0.86: for specification (2). which incliides the dummy 1.ariable. the 

short-run estimate is -1.68 and the long-run elasticity is -I .49. Third. for the 

semi-log specifications. the short-run and long-run revenue-maximizing tax rates are 

lower than they are when 1957 is omitted. Depending on whether or not a dumm), 

\variable is included. these rates range from 15.4% to 17.2% for the short-run. and 

from 16.9% to 30.3% for the long-run period. 

Third. there is little difference between the elasticit). estimates when the broad 

or narrow measure of tradable wealth is used. Although the results are not reported. 

the use of just corporate equity holdings of taxpayers resulted in similar short-run 

and long-run elasticities and revenue-maximizing tax rates. Finally. it is 

interesting to note that the moving average variable on gains is significant or close 

to being significant in almost all the equations. 

Limitations of the Preferred Equation: 

There are several weaknesses of the preferred equation specification that deseive 

comment. First. only ordinary least squares estimates are reported. Because of 

collinearity problems that resulted when instrumental-variable estimation was used 

for tax rates. wealth holdings. and expected tax rates and wealth holdings. the IV 

estimates are not reported. Specifically. problems arise in coming up with a set of 

instrumental variables that avoid multicollinearity problems. Additional work is 

being done on this currently. 

Second. only perfect-foresight expectations for the marginal tax rate and the 

wealth measure were used. This means that the actual values for these variables were 

used. instead of using the one-step-ahead predicted values from an auxiliary 

equation. An improvement can he made by using the IV estimator suggested by Pagan 

( 1984) to avoid the well-known problems associated with a biased variance-covaiiance 

matrix that are encountered when generated regressors are used in equations that are 

estimated with OLS. 

Despite the weaknesses of our pi-eferrecl equation. it reprecents an inipi.orwiient 

over the other time-series equations. For example. the use of 3 I~roadermeasure of 

- 19 -
where w is the mixing or weighting coefficient. and 0 w < I, A value of zero for w 

supports the null model. H,. while a value of one syppo% the alternative. H,. To 

conduct the test. the predicted value of the alternative model is substituted in Eq. 

( 1) to produce the estimating equation 

Davidson and MacKinnon show that the t-statistic on w in Eq. (2) has an as!?mptotic 

noimal di st ri bu t i on . 

Although the Davidson and MacKinnon test is cas!' to implement. there is a 

drawhack in its use with small samples. Godfrej. and Pesaran (1983) show that the 

effecti1.e size or estimated significance level of the test in small samples can be 

much larger than the nominal size of the test. The result is that the niill 

hypothesis is rejected too often in small samples. Problems with the Da\.iclson nncl 

MacKinnon test can be expected when the following conditions hold: (i) poor fit of 

the true model. (ii) low or moderate correlations between the regressors of the two 

regression models. and (iii) the false model includes more regressors than the true 

model. 41 

With regard to the regression models examined in this study. in general. all 

models had a good fit in teims of multiple correlation coefficients. the regressors 

in the various models were highly collinear. and most of the regression models had 

approximately the same number of regressors. Most of the regression models that were 

examined used the same set or very similar sets of explanatory variables: ancl. in 

addition. most equations had a measure of economic activity. a price level measure. 

and a wealth variable as regressors. Refer to Table 1 to verify that this is the 

case. While the small sample properties can be of legitimate concern in applied 

work. it appears that those properties are of minor importance for our results. 

Tables 2 and 3 report the results of the specification tests. In carrying out 

the test. one of the equations is set up as the null hypothesis and the other 

equations represent a series of alternative hypotheses. The situation is then 

reversed. and the test is repeated, with each of the alternative hypothesis equations

serving as the null and the original null hypothesis serving as the alternative. It 

is possible for all the regression models to be rejected as adequate representations 

of the data because no one model is assumed to be the true model in conducting the 

test. Similarly. it is also possible for all the regression models to be adequate

representations of the underlying mechanism generating the data. The specification 

test is used to evaluate the consistency of each equation with the data relative to 

the equation that is specified as the null hypothesis. 

The relevant test statistic is the J-statistic. which has a stanclarcl normal 

distribution. In implementing the test. each equation is estimated. and the fitted 

or predicted values are then used as explanatory variahles in the artificiall!, nested 

equation that is estimated for the test. Thy J-statistic is the t-statistic for the 

-6-
taxpayer wealth. and the use of expectations on the wealth variable produce a tax 

elasticity estimate that is free of the omitted variable bias that would otherwise 

residt. In addition. the use of differenced data avoids the spurious regression 

phenomenon that is most likely encountered in regressions specified in terms of 

lelrels of the data. Finally. the moving average variable that models borrowing

heha\ior has the correct sign. and is significant during periods of tax rate changes 

that in\sol\.e announcement effects. such as occurred in 1986. 

6. SUMMAR? AND CONCLUSIONS 

Our preferred time-series equation estimates a short-run elasticity of - I .LO :~ncl 

a long-run elasticitj~ of -0.90 for the period 1945-1986. These elasticities of 

capital gain responsiveness to changes in tax rates show that realizations tvoulcl more 

than douhle in the short-run if marginal tax rates are cut in half. and realizations 

nmild nearly douhle in the long-run. These estimates of the long-run elasticit!, are 

higher than most prior time-series equation estimates. 

O\.erali. the results reported in this study show that the capital gains tax 

elasticity estimates produced by aggregate time-series regressions are not 

particularly robust with respect to the equation specification. Whether narrow7 

broad measures of taxpayer wealth are used. whether the data are differenced. the 

length of the sample period. and whether expectations of tax rates and other 

explanatory variables are used as adtlitional regressors are specification decisions 

that have an impact on the estimated response of capital gains realizations to 

changes in the marginal tax rate. 

One problem that has not been discussed is the possibility of aggregation bias. 

Aggregation bias results when individual relations are incorrectly aggregated into 

macrorelations for estimation and inference purposes. The use of aggregate 

time-series data to estimate tax elasticities probably suffers from such a bias. 

since it is unlikely that aggregation over individual taxpayers as well as 

aggregation over the various capital-assets that produce capital gains realizations 

is done correctly. In addition to the biases introduced by simultaneity between the 

dependent and independent variables and by omitted variables. not to mention 

errors-in-variables problems associated with the tax rates and wealth variables. the 

existence of aggregation bias also makes it a difficult undertaking to obtain an 

accurate tax elasticity estimate from aggregate time-series regressions. 

Because of the sensitivity of the elasticity estimates to the specification of 

the estimating equation and the statistical uncertainty of the estimates. it woulcl be 

advisable to avoid using only aggregate time-series estimates as a measure of 

taxpayer behavioral response to changes in capital gains taxation. A more prudent 

course would be to use estimates from panel or cross-section microdata in combination 

with estimates from time-series studies. 

- 20 -
FOOTNOTES 

1

The argument can be made that it is inappropriate to subject the nested equations 

specified by Kiefer to the specification test using real data. These equations were 

specified and estimated by Kiefer with simulated data to make the case that lagged 

mlues of the tax rate were significant. and that their omission from the estimating 

equation would overstate the tax-elasticity. This is a weak argument. however. since 

the use of lagged tax rates in the regression specification must still recti1.e 

support from actual data. 

2 Net long-teim gains in excess of short-term losses represent the realizations 

that are actually subject to the capital gains tax. On the other hand. net 

short-term gains. which are included in the total capital gains measure. are taxed as 

ordinary income. While the decisions to realize short-teim or long-term gains are 

related. it may be inappropriate to include both together in the dependent variable. 

since the relation of each to the marginal tax rate will no doubt be different. 

It should be noted that all three measures of the marginal tax rate sufffer from 3 

measurement problems that most likely result in errors-in-variables problems. There 

are measurement problems with the upper income and maximum marginal tax rates: and. 

in addition. Larry Ozanne has stated in correspondence with the author that unknown 

errors were introduced in the attempt to make the average tax rate exogenous. 

4 Under the null hypothesis. the J-statistic has an asymptotic expectation of zero. 

but a non-zero expectation in small samples. This means that the null hypothesis is 

re.jected too often in small samples. 

If zo is the test statistic under the H,. then it can be shown that 

pi2 ) +max(k, -k, ).O)) 

with 

error 

where pi (p,*< I) are the s=min (k,. k,) canonical correlations aszsociated 

the explanatory variables in the two competing regression models., uo is the 

variance for the null model. \ and k, denote the number of explanatory 

variables in the alternative and null models, respectively. and E, is the 

mathematical expectation under the null hypothesis. The correlations are given by 

the non-zero roots of the equation: 

I x'z(z'z)-lz'x - p2x.x I =o 

where X and Z denote the design matrices in the null and alternative models. 

respectively. It is clearly the case that a poor fit to the data of the null model. 

a small degree of correlation among the explanatory variables in the two models. and 

a large discrepency between k, and k, serve to make the expectation non-zero. See 

Godfrey and Pesaran ( 1983) for further discussion. 

5 In the uni\wiate case. the fcwmulas used to determine optimal lag-length are 

given by the following: 
FPE(n) = (T+n + I )/(T-n-1) SSR(n)/T 

BIC(n) = SSR(n) + (nSSR(N)/TlnT)/(T-n- I ) 

where T is the effective sample size. n is the lag-length being tested. SSR is the 

slim of squared residuals. and N denotes the maximum lag-legnth over which the search 

is carried out. Minimum FPE or BIC corresponds to the optimal lag-length. 

6 In the case where equity and tradeable wealth move in proportion. there is no 

bias. This is an unlikely case: and. moreover. there would be no admntage to adcling 

the omitted variable to the design matrix. since this would result in singularity of 

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Specification (3) (4) (7) (8)

Sainple Period 1954-1986 1954-1986 1948-1986 1948-1986 

Estiinatioii Method OLS OLS OLS OLS 

constallt -4.83 -33 -.62 4.00 

(-.51) (-. 11) (-.07) (.60) 

RGNP 4.22 3.27 3.56 2.48 

(3.64) (3.62) (3.74) (3.24) 

RGNP(-l) -.79 -1.76 -.2? -1.00 

(-. 73) (-1.96) (-.37) (-1.50) 

PGNP 1.31 .46 .96 .05 

(1.10) (.53) (.92) (.06)

Equity .65 .71 

(4.42) (5.34) 

Tradeable Wealth 2.33 2.40 

(6.16) (6.70) 

Moving Average -.33 -.05 -.50 -.32 

011 Gains (-.99) (-. 19) (-1.88) (-1.36). 

MTR -1.01 -1.15 -1.01 -1.17 

(-2.88) (-4.23) (-3.06) (-4.29) 

MTR( + 1) .02 .41 -.os .28 

(-07) (1.41) (-.26) (.99) 

Equity( + 1) -.13 -.14 

(-.91) (-1.17) 

Tradeable Wealth( + 1) -.62 -.67 

(-1.67) (-2.11) 

DM inmy 56.17 37.81 59.82 43.69 

(3.20) (2.52) (3.71) (3.03) 

Mj. R-Sq. .703 ,801 .731 .798 

Box-Ljung Q Statistic 16.41(.28) 10.55(.72) 20.06(.16) 13.09(.59) 

Degree of Freedom 19 19 25 25 
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Appendix A 

Results for Alternative Specifications 

Using Additional Variables 

(Dif f erenced Data) 

Specification: 

Sample Period: 

Estiiiiatioii Method: 

(1)

1956-1985 

OLS 

(2)

1956-1955 

OLS 

(3)

1956-1985 

OLS 

(4)

1956-1985 

OLS 

Di�Ferencecl : YES YES YES YES 

Logged: YES YES SES YES 

Dep. Var. Nom. Gains Nom. Gains Nom. Gains Noiii. Gains 

Consta 11 t -.lo .02 -.06 -.06 

(-1.40) (.32) (-1.16) (-1. I?) 

RGNP 4.89 5.15 3.32 3.32 

(5.01) (4.18) (4.22) (3.61) 

RGNP(-l) -1.25 

(-1.38) 

-2.79 

(-2.62) 

-2.09 

(-2.91) 

-2.05, 

(-2.81) 

Price Deflator 1.68 -.23 .52 .52 

(1.67) (-*21) (.69) (.67) 

Equity .73 --- --- --- (5.47) 

MTR (Upper Income) -.84 

(-2.63) 

-1.36 

(-3.50) 

-1.08 

(-7.48) 

-1.08 

(-3.95) 

Accrued Gains --- .07 --- .ooo 1 

(3.47) (.008) 

Net Worth --- --- 3.43 3.43 

(7.43) (5.28) 

Stock Repurchases --- --- --- ---

Total Non-Fin. --- --- --- --- Assets 

Total Fiiiaiicial --- --- --- --- Assets 

Adj. R-Squared 

D-W Statistic 

.651 

2.33 

.479 

2.28 

.76J 

2.21 

.754 

2.21 

Log-Likelihood 

D.F. 

27.04 

2-1 

21 .no 

2-1 

32. 

2-1 

32.Q.1 

. 23 

Notes: t-statistics are in parentheses.

The equity, accrued gains, net worth, stock repurchases, total nonfinancial and total financial asset variables are expressed in real teniis. 
--- 

--- 

Appendix A 

Results for Alternative Specifications 

Using Additional Variables 

(Differenced Data) 

Specification: 

Sample Period: (5)

1972-1985 

(6)

1972-1955 

Estimation Method: OLS ocs 

DifFere11 cwl : YES 1733 

Logged: YES YE3 

Dep. Var. Nom. Gains Nom. Gaiiis 

consta11t .20 .I7 

( 1.70) (1.63) 

RGNP 2.62 2.57 

(2.31) (3.01) 

RGNP(-1) -1.74 -1.w 

(-1.79) ( -2.2 6 1 

Price Deflator -2. os -1.92 

(-1.43) (-ln51) 

Equity --- ---

MTR (Upper Income) -1.02 -1.18 

(-3.41) (-4.31 

Accrued Gains --- .07 

(1.88) 

Net Worth --- ---

Stock Repurchases .03 -.009 

(.50) (-. 14) 

Total Non-Fin. --- --- Assets 

Total Fiiiaiicinl --- --- Assets 

Adj. R-Squared .543 .65J 

D-W Statistic 1.73 1.89 

Log-Lilceliliood 17.37 20.20 

D.F. 8 I 

/ 

Noin. Gains Nom. Gaiiis 

.12 -.l1 

(2.34) (-1.72) 

2.21 3.72 

(3.06) (3.44) 

-1.49 -1.60 

(-2.14) (-1.35) 

-1.66 1.28 

(-2-39) (1.45) 

-1.01 -.86 

(-6.52) (-2.78) 

1.88 

(4.20) 

2.61 

(5.37) 

,897 .754 

2.52 2.18 

17.2" 13.71 

6 22 
--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

Appendix A 

Results for Alternative Specifications 

Using Additional Variables 

(Differenced Data) 

Specification: (9)

Saiiiple Period: 1955-1985 

Estimation Method: OLS 

Differeiiced : YES 

Logged: YES 

Dep. Var. Nom. Gains Nom. Gains Nom. Gaiiis 

Constant 

RGNP 

RGNP(-1) 

Price Deflator 

Equity 

MTR (Upper Income) 

Accrued Gaiiis 

Net Worth 

Stock Repurchases 

Total Non-Fin. 

Assets 

Total Financial 

Assets 

Adj. R-Squared 

D-W Statistic 

Log-Likelihood 

D.F. 

-.11 .02 -.11 

(-1.84) (.28) (-1.78) 

3.81 5.22 3.62 

(4.41) (3.36) (3.81) 

-1.54 -2.7d -1.58 

(-2.05) (-2.19) (-137) 

1.33 -1.93 1.29 

(1.66) (-. 15) (1.49) 

-.84 -1.35 -.84 

(-3.02) (-3.07) (-2.98) 

.005 .07 

(.24) (3.35) 

-.11 .16 

(-.OS) (. 19) 

2.61 2.68 

(5.49) (7.48) 

* 764 .456 .764 

2.17 2.28 2.19 

33.56 21.01 33.55 

23 2.1 23 
--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

Appendix B 

Alternative Specification Results 

for Variables from Individual, 

NonFarm, and Farm Balance Sheets 

(Differences Logged Data) 

Specification: 

Saiii ple Period: 

Estiiiia tioii Met11od : 

Difkreiired: 

Logged: 

Dep. Var. 

Coilstant 

RGNP 

RGNPi-1) 

PGNP 

Corp. Equity 

MTR 

Iiivestinent Co. 

Other Equity 

Non-Residential Land 

Net Purchases 

Non-Corp. Assets 

Adj. R-Squared 

D-W Statistics 

Log-Likelihood

D.F. 

(1)

1955-1985 

OLS 

YES 

YES 

Nom. Gains 

-.IO 

(-1.42) 

4.84 

(4.68) 

-1.22 

(-1.34) 

1.68 

( 1.68) 

.74 

(5.49) 

-.83 

(-2.62) 

.652 

2.34 

11.90 

24 

(3)

1955-1985 

OLS 

YES 

YES 

Noiii. Gains 

.Oh 

(.67) 

4.37 

(2.87) 

-2.86 

(-2.20) 

-.44 

(-.32) 

-1.15 

(-2.49) 

.04 

(1.02) 

,249 

2.49 

15.52 

2-1 

(4)

1955-lQS5 

013 

1-e5 

YES 

Noiii. Gains 

-.2 I 

(-1.00) 

4.36 

(3.01) 

-1.17 

(-1.17) 

2.50 

(1.31) 

-.70 

(-2.70) 

.70 

(132) 

.ll 

(.33) 

.33 

(.46) 

.91 

1.57) 

.h64 

2.36 

2Q.00 

21 
Appendix C 

Results for One-Period-Ahead Expectations 

and Moving-Average on Lagged Gains 

(Differenced-Logged Data) 

Specification: 

Sairipie Period: 

Estiiiiatioii Method: 

(1)

19561984 

OLS 

(2)

1956-1984 

OLS 

(3)

1956-1984 

OLS 

(4)

1956-1984 

OLS 

DifFereiicecl: YES YES YES YES 

Logged: YES YES YES YES 

Dep. Var Nom. Gains Nom. Gins Nom. Gains Nom. Gains 

consta11t -.12 -.07 -.I1 -.04 

(-1.50) (-.9JI (-1.18) (-.VI 

RGNP 5.03 4.48 4.82 4.24 

(4.88) (4.52) (4.69) (4.34) 

RGNP ( - -1.29 -1.31 -1.24 -2.54 

(-1.29) (-136) (-1.21) (-2.27) 

PGNP 1*?5 1.25 1.71 -2.56 

(1.78) (1.21) (1.56) (1.12) 

MTR -.85 -1.00 -.85 -1.09 

(-2.57) (-2.99) (-2.47) (-3.31 ) 

MTR( + 1) 

Corp. Equity .73 

(5.02) 

.68 

(4.73) 

.72 

(3.90) 

.64 

(4.54) 

RGNP( + 1) ---

PGNP( + 1) 4.03 

(2.05) 

Corp. Equity --- (+1) 

AVEG 2 ---

AVEG 3 ---

AVEG 4 ---

AVEG 5 __-

Adj. R-Squared 

D-W Statistic 

.640 

2.37 

.666 

2.42 

.630 

2.33 

.689 

2.42 

Log-Likelihood 

D.F. 

26.29 

22 

27.17 

22 

25.71 

22 

28.24 

22 . 
--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

Appendix C 

Results for One-Period-Ahead Expectations 

and Moving-Average on Lagged Gains 

(Differenced Logged Data) 

Specification: 

Saiiiple Period: 

Estiiiiatioii Method: 

Differenced : 

Logged: 

Dep. Var. 

Consta 11t 

RGNP 

RGNP(- 1) 

PGNP 

MTR 

MTR( + 1) 

Corp. Equity 

RGNP( + 1) 

PGNP( + 1) 

Corp. Equity( + 1) 

AVEG 2 

AVEG 3 

AVEG 4 

AVEG 5 

Adj. R-Sqared 

D-W Statistic 

Log-Likelihood 

D.F. 

(5)

1957-1985 

OLS 

VES 

YES 

Noiii. Gains 

-.I1 

(-1.46) 

4.93 

(4.99) 

-.46 

(-.43) 

1.84 

(1.81) 

-.85 

(-2.65) 

.69 

(4.93) 

-.28 

(-1.37) 

.659 

2.25 

26.88 

22 

(7)

1959-1985 

OLS 

YES 

YES 

Noiii. Gains 

-.03 

(-35) 

4.48 

(4.14) 

-1.08 

(-1.17) 

1.70 

(1.59) 

-.93 

(-2.96) 

.75 

(5.23) 

-.75 

(-2.08) 

.670 

2.35 

26.08 

20 
Appendix C 

Results for One-Period-Ahead Expectations 

and Moving-Average on Lagged Gains 

(Differenced Logged Data) 

Specification: 

Sample Period: 

(9)

1957-1984 

(10)

1958-1984 

(11)

1959-1984 

(12)

1960-1984 

Estiiiiatioii Method: OLS OLS OLS OLS 

DiTFerencecl: YES YES YES YES 

Logged: YES YES YES YES 

Dep. Var. Nom. Gains Nom. Gains Noin. Gains Nom. Gains 

Coilsta11t -.06 -.03 -.01 

(-.,sa) (-.42) (-.ll) 

RGNP 4.47 4.11 4.27 4.45 

(4.46) (2.81) (3.82) (3.85) 

RGNP(-1) -.45 -.89 -1.21 -1.08 

(-.41) (-30) (-1.21) (-.99) 

PGNP 1.31 1.23 1.40 1.25 

(1.27) (1.11) (1.26) (1.0% 

MTR -1.05 -1.03 -1.07 -1.05 

(-3.15) (-2.95) (-3.16) (-3.08) 

MTR( + 1) --- --- ---

Corp. Equity .61 

(4.09) 

.67 

(4.03) 

.70 

(4.52) 

.65 

(4.23) 

RGNP( + 1) --- --- ---

PGNP( + 1) --- --- ---

Corp. Equity( + 1) -.23 

(-1.74) 

-.15 

(-1.02) 

-.15 

(-1.09) 

-.18 

(-1.34) 

AVEG 2 -.32 --- (-1.57) 

AVEG 3 --- -.36 --- (-1.12) 

AVEG 4 --- --- -.71 

(-1.91) 

AVEG 5 --- --- --- -.77 

(-2.01 ) 

Adj. R-Squared 

D-W Statistic 

.684 

2.28 

.633 

2.43 

,672 

2.38 

.664 

2.34 

Log-LikeIihood 

D.F. 

27.50 

20 

25.41 

19 

25.69 

18 

24.01 

17 
--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

--- 

Appendix C 

Results for One-Period-Ahead Expectations 

and Moving-Average on Lagged Gains 

(Differenced Logged Data) 

(13) (14) (15) (16) 

1957-1984 1958-1984 1959-1984 1960-1984 

Specificatioii: 

Saiiiple Period: 

Estimation Method: 

Differenced: 

Logged: 

Dep. Var. 

COllSta11t 

RGNP 

RGNP(-1) 

PGNP 

MTR 

MTR( + 1) 

Corp. Equity 

RGNP( + 1) 

PGNP( + 1-

Carp. Equity( + 1) 

AVEG 2 

AVEG 3 

AVEG 4 

AVEG 5 

Adj. R-Squared

D-W Statistic 

Log-Likelihood 

D.F. 

OLS QLS 

YES YES 

YES YES 

-*OS -.007 

(-.(it) (-.OS) 

4.38 .. 3 80

(4.41) (3.73) 

- 1.88 -2.06 

(-1048) -1.63) 

-2.24 -2.15 

(-.96) (-.9O) 

-1.10 -1.14 

(-3.31) -3.331 

.59 .59 

(4.00) (3.82) 

3.86 3.54 

(1.94) (3.82) 

-.26 

(-1.31) 

-.38 

(-1.27) 

.694 ,665 

2.29 2.40 

27.94 26.64 

20 19 

OLS QLS 

YES YES 

YES YES 

Nom. Gains Noin. Gains 

-.O0(, -.00OQ 

(-.07) (-.0OQ) 

4.12 4.30 

(3.70) (3.75) 

-2.21 -2.09 

(-1.P5) ( - 1.64 

-1.35 -1.49 

(-.54) (-.60) 

-1.12 -1.08 

(-3.31 (-3,121 

.66 .62 

(4.27) (4.03) 

2.87 2.97 

(1.36) (1.39) 

-.65 

(-1.72) 

-__ -.(7 h 

(-1 .Oh) 

,683 ,666 

2.34 2.33 

26.13 24.96 

18 17 

APPENDIX E 

This short appendix examines the correlation among various time-series variables 

used in the aggregate capital gains study. The particular issue that is addressed is 

whether the non-tax variables that are used as regressors provide infoimation that is 

sufficiently iincorrelatecl with that provided by the marginal tax rate on copital 

gains realizations. If the correlation is high. then there are problenis with 

niulticollinearity which would have an impact on hypothesis testing. The non-tax 

iwiables measure economic activity and taxpayer wealth. and include. e.�.. such 

variables as real GNP and equity holdings. Besides the non-tax explanator! 

iwiables. correlation coefficients are calculated for each of the marginal tax rates 

and total as well as net long teim capital gains realizations. 

Table I provicles simple Pearson correlation coefficients computed for a selected 

set of variables that proxi\' economic activity and taxpayer wealth. Marginal 

significance lexrels are in parentheses. In addition. Table 2 provides coi-relation 

coefficients computed for the four altemative marginal tax rates that were used in 

the regressions that were estimated. These include: Treasury's upper income tax rate 

(MTR).CBO's weighted average tax rate (CMTR).the JCT's maximum tax rate on capital 

gains (MTRJ).and the twice median income marginal tax rate (MTRM). The twice median 

income tax rate is the marginal tax rate adjusted for the appropriate exclusion rate 

applicable to a 4 person family with taxable income of twice the national median 

income. 
TabIe AI. Non-Tax Variables and Marginal Tzs Rates 

Total realized -0.406 -0.392 -0.134 0.616 

gains (.0102) (.0262) (.4797) (.0003) 

Net long term -0. I4 I -0.392 -0.128 0.6 I4 

gains (.45.52) (-0264) (.4991) (-0003) 

Real GNP 0.035 0.042 0.303 0.840 

(-8321) (.8176) (.1036) (.0001) 

Nominal GNP 0.030 -0.213 0.0 I7 0.764 

(.8749) (.2409) (.9279) (.OOOl) 

Equity holdings -0.307 -0.327 -0.056 0.568 

(.0571) (-0673) (-7665) (.0010) 

Tradeable wealth -0.274 -0.043 -0.0.54 0.723 

(.0912) (. 1754) (.7737) (.OOOl) 

NYSE 0.034 0.033 0.048 0.6 16 

(.SS61) (.8599) (.8005) (.0003) 
Table A2. Correlation Ainoiig the Marginal Tax Rates 

CMTR 0.963 

(.0001) 

MTRJ 0.957 0.910 

(.OOO I) (.OOO 1 ) 

MTRM 0.594 0.628 0.52 1 

(.0005) (.0002) (.0031 ) 

Seyeral interesting conclLisions can be drawn from examination of the correlation 

coefficients. First. in general. it appears that the non-tax variahles proikle

information that is different from that provided hy the marginal tax rates that are 

considered. The salient exception to this is the median marginal rate which is -

extremelv. highly correlated with all the non-tax variables. In addition. the simple 

correlation between the median rate and both meas~iresof realizations has the wrong 

behavioral algebraic sign, These correlation results explain why the regression 

results using the median tax rate were so poor. Because of the manner in which the 

median rate was constructed. it is inappropriate to use this rate to assess the 

impact of tax rate changes on capital gains realizations. 

It should also be noted that the correlation between equity holdings and both the 

Treasury and CBO rates is significant at almost the 5 percent level. Also. the 

Treasury rate is highly correlated with tradeable wealth. 

Second. the correlations among the four tax rates show that the Treasury. CBO. 

and maximum tax rates are very close to being perfectly collinear. while the median 

marginal rate is not. 

Finally. the significance and size of the correlation of the Treasury. CBO. and 

maximum tax rates with realizations varies considerably. For example. both the 

Treasury and CBO rates are highly correlated to about the same degree with total 

realizations. while the correlation between the maximum rate and total realizations 

is very different. For net long term realizations. the CBO tax rate remains highl!. 

correlatecl. hitt both the Treasur!, and maxinium tax rates art: not. 

-3-
REFERENCES 

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and Rhetoric - A Review. PaDer DreDared for Brookings Panel Meeting. Washington. 1 I , v v c 

D.C.. September. 1988. 

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14. Jones. J.. ”A Comparison of Lag-Length Selection Techniques in Granger-Causality 

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Economics. 1989. Vol. 2 1. forthcoming. 

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Recent OTA Papers published by

The Office of Tax Analysis

U.S. Department of the Treasury

Washington, DC 20220 

59. NonCorporate Business Taration: 

Before and Afier the Tar Reform Act of I986 

Susan C. Nelson 

May 1988 

60. llie Tar Expenditure Budget Before and A8er the 

Tar Refom Act of 1986 

Thomas S. Neubig 

David Joulfaian 

October 1988 

6 1. Recent Issues in Transfer Pricing

Barbara L. Rollinson 

Daniel J. Frisch 

November 1988 

62. Measures of Goodness of Fit for Extrapolations:

Initial Results Using tire Individual Tax Model Database 

Robert E. Gillette 

February 1989 

63. Master Limited Pannerships:

A Viewfrom Their 1986 Tat Retirnts 

Susan C. Nelson 

Joann Martens 

May 1989 

64. A History of Federal Tax Depreciatiotl Policy

David W.Brazell 

Lowell Dworin 

Michael Walsh 

May 1989 

65. An Aiialysis of Aggregate Time Series 

Capital Gains Equations

Jonathan D. Jones 

May 1989 

66. New Estimates of Capital Gains Realization Behavior: 

Evidence from Pooled Cross-Sectioii Data 

Robert Gillingham

John S. Greenlees 

Kimberly D.Zieschang

May 1989 

67, Estiniatiorr and Irtterpreration of Capital Gains Behavior: 

Evidence from Panel Data 

Gerald E. Auten 

Leonard E. Burman 

William C. Randolph

May 1989 

Copies of these OTA Papers may be purchased from the National Technical Information 

Service, 5285 Port Royal Road, Springfield, VA 22161 (phone: 703-487-4660).