Technical Paper Series 

Congressional Budget Office 

Washington, DC

ESTIMATING AND FORECASTING CAPITAL GAINS 

WITH QUARTERLY MODELS 

Jangryoul Kim 

Korea Development Institute 

Formerly: Federal Reserve Bank of Minneapolis 

Preston Miller 

Federal Reserve Bank of Minneapolis 

preston.miller@mpls.frb.org 

Larry Ozanne 

Congressional Budget Office 

larry.ozanne@cbo.gov

September 2004 

2004-14 

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

discussion and critical comment. These papers are not subject to CBO’s formal 

review and editing processes. The analysis and conclusions expressed in them are 

those of the authors and should not be interpreted as those of the Congressional 

Budget Office. References in publications should be cleared with the authors. 

Papers in this series can be obtained at www.cbo.gov/tech.cfm. 
i

Abstract 

At the end of each year, the Congressional Budget Office (CBO) estimates capital 

gains for the year ending and forecasts them for the next decade. The decade forecast is 

made using CBO’s forecast of GDP and an assumption that gains revert from their 

current size to their historical size relative to GDP. Our objective in this paper is to 

describe methods to improve CBO’s forecasts, particularly for the first year ahead. 

 We settled on two procedures. The first is similar to CBO’s method for 

forecasting gains. It uses an equation to forecast gains given forecasts of economic and 

financial variables. This procedure requires a prior step to forecast the economic and 

financial variables. The second procedure integrates the forecasting of gains and other 

variables into a single model. In this model we found it advantageous to work with 

quarterly data, so we interpolate the reported annual series on capital gains to a quarterly 

frequency. Forecasting in the prior step of the two-step method and the integrated 

quarterly method was based on Bayesian-restricted vector autoregressions. 

 Both of the procedures abstract from the effects of tax changes on forecasts of 

realizations. CBO’s baseline is required to assume that current law continues. We 

abstract from tax changes by constructing a series of capital gains realizations that 

assumes taxes remained at their 1998 level throughout the 1948-2000 period used for 

model development. This tax-adjusted series retains much of the volatility in the growth 

rate of actual capital gains. Between 1971 and 2000, the period used to test the models, 

the annual growth rate of tax-adjusted gains ranged from a high of 44 percent to a low of 

-18 percent. Its mean growth rate was 12 percent with a standard deviation of 16 percent. 

 We base our model comparisons on their root mean squared errors (RMSE) in 1-

year-ahead out-of-sample forecasts of the growth rate of tax-adjusted gains. Our 

application of CBO’s mean reversion method found a RMSE of 18.7 percentage points. 

The two-step forecasting method reduced the RMSE to 14.8 percentage points, and the 

integrated quarterly method reduced the error to 11.9 percentage points. 

Two additional findings from this investigation suggest improvements to CBO’s 

methods. First, the models we developed may help CBO improve its estimates of gains 

in the year ending. Second, the models may provide some help in forecasting a second 

year ahead, but after that, either mean reversion or a simple random walk model with 

drift appears to be as good or better.
ii

Table of Contents 

Abstract .............................................................................................. i 

1. Introduction ....................................................................................... 1 

2. Current CBO Forecasting Method ............................................................. 3 

3. Initial Steps........................................................................................ 9 

Tax-Adjusted Gains..........................................................................9 

Potential Explanatory Variables..........................................................11 

4. Two-Step Forecasting Approach..............................................................13 

Methodology ................................................................................16 

Notation and Measures of Success .............................................16 

Methodology for Estimating .....................................................18 

Methodology for Forecasting ....................................................19 

Results for Two-Step Approach ..........................................................21 

5. Integrated Quarterly Approach.................................................................24 

Three Interpolation Methods for Integrated Approach ...............................26 

Estimation of Annual Tax-Adjusted Gains .............................................29 

Forecasting of Tax-Adjusted Gains One Year Ahead .................................33 

6. Applications of the Model ......................................................................35 

Forecasting 2001 ............................................................................35 

Forecasting Multiple Years Ahead .......................................................37 

7. Conclusion ........................................................................................40 

Tables .................................................................................................43 

Figure .................................................................................................52 

Appendix A: Specifications of Forecasting Models ...........................................53 

Appendix B: Annual vs. Quarterly Univariate Models of Gains ............................56 

Appendix C: Economic Interpolation of Quarterly Gains ....................................60 

Appendix D: Actual and Interpolated Gains ....................................................66 
1

1. Introduction1

During the last quarter of each year, the Congressional Budget Office (CBO) 

forecasts revenue and spending for the fiscal year that began at the start of the quarter, 

and for the next ten fiscal years. Usually the forecasts are published in January or 

February of the next year in CBO’s Budget and Economic Outlook. The forecasts are 

made assuming current law is unchanged to provide a baseline against which the U.S. 

Congress considers proposals for change. 

A large fraction of the revenue collected in the fiscal year beginning at the time of 

the forecast comes from individual income tax liabilities for the calendar year that is 

ending. As a result, the forecast of revenue for the current fiscal year builds on an 

estimate of income tax liability for the calendar year ending. Actual information on that 

liability is incomplete, however, because the year is not quite complete and people have 

until the next year to file their tax returns, make final payments, or request refunds. As a 

result, CBO must estimate income tax liabilities for the year ending as well as forecasting 

liabilities for the coming decade. 

 CBO projects income tax liability for the population by projecting the major 

determinants of tax liability, including aggregate net capital gains realized by the sale of 

many types of assets.2

 Thus CBO must estimate gains for the calendar year ending 

(referred to here as the current year) and forecast gains for the coming 10 years. When 

the gains projections are finalized, usually in early December, CBO has preliminary 

 1

 The authors thank the Tax Analysis Division of CBO for their comments and encouragement, 

and Christopher Sims for helpful guidance. John McMurray suggested numerous improvements in the text. 2

 Net losses are also projected, but because these are subject to a limit of $3,000 on most returns, 

losses are smaller and grow relatively steadily from year to year. As a result, they are forecast separately 

from gains and are not considered in the current paper. 
2

information on gains for the previous year and most macroeconomic and financial data 

for the year ending. 

We use the term “estimation” to refer to the process of generating a figure for 

realized gains in the current year because the explanatory variables are largely known at 

the time. Currently, CBO generates that figure by using macroeconomic, stock market, 

and tax rate variables for the current year in regression equations estimated through the 

previous year. We use the term “forecasting” to refer to the process of generating figures 

for the year about to begin and the succeeding 10 years because all variables must be 

generated. Currently, the CBO generates realizations for future years by using its official 

forecast of GDP and an assumption that gains revert to their historical size relative to 

GDP. In this paper, we seek to improve forecasting of capital gains realizations by the 

CBO. Our work builds on prior work by Miller and Ozanne on estimating capital gains.3

 

 A question naturally arises: why separate estimation and forecasting? Why not 

use a single model to forecast and get estimates from it by conditioning on the observed 

values in the current year? The answer is that the variables and relationships that yield 

the best estimates can be quite different from those that yield the best forecasts. The case 

of stock prices illustrates this point. There appears to be a contemporaneous relationship 

over the year between growth in stock prices and realizations.4

 Stock prices for the 

current year are largely observed by the time that the CBO does its budget forecast, so 

they can be used to estimate current-year realizations. However, to use stock prices to 

forecast, they, too, must be forecast. Since the best forecasting model of stock prices is 

essentially a random walk with drift, they should be of little use in forecasting 

 3

 Preston Miller and Larry Ozanne, Forecasting Capital Gains Realizations, Technical Paper 

2000-5, Congressional Budget Office, August 2000. 
3

realizations.5

 The difficulty of forecasting stock prices led CBO to separate estimation 

from forecasting, a reasonable procedure. It seems that it is simpler to construct separate 

models for estimation and forecasting rather than trying to integrate separate 

informational structures into a single model. 

 In the text that follows, we first describe the current CBO forecast method. We 

then describe the evolution of our methodology with respect to current year estimation, 1-

year-ahead forecasting, and multi-year-ahead forecasting. In addition to discussing some 

things that seemed to work, we discuss some that did not. Our intent is to provide as 

much aid as we can to other researchers so that they can improve upon our method. 

2. Current CBO Forecasting Method 

CBO began predicting annual capital gains realizations at the end of 1986 for use 

in its February 1987 baseline. By the January 1989 baseline, CBO began using a single 

regression equation to estimate gains for the year ending and forecast gains for the next 

five years. That equation explained realizations in terms of the value of corporate equity 

held by households, GNP, interest rates, and a capital gains tax rate. Preliminary values 

of these explanatory variables were available to estimate gains for the year ending, and 

forecasts of them by CBO’s Macroeconomic Analysis Division were used in the same 

equation to forecast realizations for each of the next 5 years. 

 Problems with the predictions became apparent when tax return data revealed that 

realizations dropped for three successive years between 1989 and 1991. The equations 

 4

 Miller and Ozanne, Forecasting Capital Gains Realizations, pp. 11-12. 5

 We found support for both clauses in this statement. The coefficients in a monthly S&P 500 

autoregression approximate the values of a continuous-time random walk process sampled monthly. 

Although we found that stock prices were not useful in forecasting realizations when examining bivariate 

relationships, they surprisingly seemed to help marginally in some multivariate quarterly models. 
4

neither forecast the drop nor, once actual data became available, explained it. These 

errors led CBO to revise its methodology for the January 1992 baseline. New equations 

were developed to estimate realizations for the year just ending. Forecasting for future 

years was separated from the estimating equations so that forecasts of stock values would 

not be necessary and forecasts of realizations would be tied more closely to forecasts of 

general economic growth. 

The forecasting method built on the observation that realizations had grown at 

about the same average rate as GNP from the mid-1950s to the end of the 1980s, but that 

their annual growth rate frequently deviated sharply from that of GNP. As a result, the 

ratio of gains to GNP over the period showed little trend away from its average value of 

2.8 percent, but frequently jumped above or fell below that level. 

The revised CBO forecasting method incorporated this pattern by assuming that 

whatever level of gains was estimated for the year ending, gains in future years would 

trend back to their average size relative to GNP. If the level of realizations estimated for 

the year ending happened to equal the historical average relative to GNP, then in the next 

5 years, gains were forecast to grow at the same rate as GNP. If the estimated value were 

below that target, then gains would be forecast to grow faster than GNP to restore the 

average ratio by the last year of the five year forecast period. 

Over time, the method became more sophisticated by incorporating an estimated 

rate of reversion. The rate at which the ratio of gains to GNP reverted to its historical 

average was estimated by including an error-correction term in the estimating equation. 

That term’s coefficient gave an estimate of how much the gap between last year’s ratio 

and the long-run average was closed in the current year. Depending on the specification 
5

and years covered through the mid-1990s, the rate was estimated to be between 20 

percent and 30 percent. The reversion rate was used to extend the forecast to 10 years 

when the entire baseline was extended that far. In addition, the historical average ratio 

was adjusted to reflect legislated tax rates for the forecast period. When tax rates were 

below average, the target ratio was above average. Finally, in the mid-1990s when the 

National Income and Product Accounts gave greater prominence to GDP than GNP, 

domestic product became the long-run guide for the gains forecast. 

The method assumes no change in tax rates. Adjustments were made in three 

baselines when tax changes seemed imminent or had just happened. The congressional 

campaign and election in 1994 increased the likelihood that legislation reducing the 

capital gains tax rate would be passed in 1995. CBO reasoned that taxpayers might defer 

asset sales in the last two months of 1994 until 1995 in hopes of paying a lower tax rate 

on gains. Consequently, CBO shifted some of the capital gains it estimated would 

normally occur in 1994 to 1995. By early 1996, the fate a legislation reducing the capital 

gains tax rate was still unclear, so in its May 1996 baseline CBO again assumed that 

taxpayers would defer some asset sales from 1995 to 1996. Another adjustment was 

made in the January 1998 baseline to reflect the temporary surge in realizations following 

the capital gains tax reductions of 1997. The January 1997 baseline, however, did not 

anticipate the tax rate reductions that occurred that year, and therefore its forecast was too 

low. (The forecast turned out to have been too low for other reasons as well.) 

The forecasting method was called into question in the later 1990s by the failure 

of gains to revert back towards their historical size relative to GDP. Gains surpassed 

their target size relative to GDP in 1996, and continued growing faster than GDP through 
6

2000. By the end of 2000, the historical pattern showed an upward trend, and suggested 

that gains had permanently shifted to a new level relative to GDP. The error-correction 

term in the earlier estimating equations no longer was statistically different from zero, 

suggesting that no tendency to revert could be identified in the historical data. In the 

baseline of January 2001, CBO assumed that gains in the year of the baseline would 

remain at their estimated level for the prior year and then begin reverting in later years. 

It now appears that realizations in 2001 and 2002 show substantial reversion, 

which revives at least temporarily the case for incorporating mean reversion into 

forecasts of gains. It will be unclear for several years, however, whether realizations will 

continue to fluctuate around their old size relative to GDP or drift off again. 

A related question about the forecasting method has also arisen. Even if mean 

reversion ultimately reappears, could forecasts for the first year or two be improved by 

incorporating more information about the economy in the immediately preceding years? 

The current forecast method starts pulling gains back to their historical size relative to 

GDP in the first year, but the divergence recently has widened for more than a single 

year. Gains fell below their target in 1989 and then further below in 1990 and 1991. In 

addition, as noted above, gains rose further above their target each year from 1997-2000. 

Perhaps the rest of the economy can provide some guidance about the path of gains in the 

first year or two ahead. 

We address this second question directly. The primary measure of success will be 

whether new methods could have predicted gains one year ahead more accurately than 

the methods CBO used. Thus, a starting point is errors in CBO’s past forecasts for the 

year ahead. 
7

 The left half of Table 1 shows actual and forecasted growth rates from the year 

ending to the year ahead for baselines in 1987 to 2000 (Tables appear at the end of the 

text and before the appendixes). Growth rates are shown instead of dollar levels, because 

capital gains and their forecast errors tend to grow in dollar terms over time. Recall that 

the forecasts are for growth from estimated values for the year ending because hard data 

on gains are available only for the year before that. Over the 14 years, the root mean 

squared forecast error is 22.3 percentage points. Over the last 9 years, when forecasts 

were made by mean reversion, the root mean squared error is slightly lower at 22.1 

percentage points. 

The imprint of mean reversion can be seen in several years. The baselines of 

January 1992, January 1993, and January 1994 forecast that gains would grow faster than 

GNP because gains were estimated to be below their historical target in 1991, 1992, and 

1993. Baselines in January 1997 through January 2000 forecast that gains would grow 

less rapidly than GDP because gains were estimated to be above their historical target in 

each of the prior years. In baselines of 1998-2000, the estimated levels of gains for the 

year before were so far above their target values that gains were forecast to decline in the 

year of the baseline even though GDP was growing. As stated earlier, gains actually 

grew strongly in years 1997-2000, raising doubts about the mean reversion method. 

The error in 1997 is exaggerated by the tax reduction enacted and effected during 

the year. That spurred realizations beyond what could be reflected in the baseline. One 

estimate is that growth in the absence of a tax change that year would have been 22.6 

percent instead of the 39.9 percent that was observed, which would reduce the forecast 
8

error to 18 percentage points. Even this lower growth remains at variance with the 

assumption of mean reversion. 

The imprint of mean reversion is obscured in three baseline forecasts. The 

January 1995 and May 1996 baselines forecast additional gains because taxpayers were 

assumed to defer gains from the prior year to the forecast year in anticipation of a capital 

gains tax rate reduction. The decline forecasted in the baseline of January 1998 is larger 

than mean reversion would predict because taxpayers were assumed to have unlocked a 

burst of gains immediately after tax rates were reduced in 1997. 

The right half of Table 1 shows actual and predicted growth rates from two years 

before the baseline year to the baseline year. Two years are included because the 

forecasts on the left half of the table are made from estimated values for the year ending. 

At the time each forecast is made, a close approximation to actual gains is available only 

for two years before the year of the baseline. Differences between the actual and 

predicted growth rates over these two years reflect combined errors in the estimation and 

forecast methods. The root mean squared errors for these two-year predictions are about 

double those for the one-year forecasts. The errors in baselines from January 1992 on are 

slightly lower than in prior years. The error in the January 1997 baseline is overstated by 

the tax rate reductions in that year. 

 The inescapable conclusion from Table 1 is that the errors in CBO’s forecasts of 

capital gains have been large. These errors, in turn, have led to large errors in projections 

of revenue from taxing capital gains. However, to the extent that gains are driven by 

unforecastable forces, like the stock market, the potential to reduce the forecast errors 

may be limited. Nevertheless, it seemed to us to be worth trying. 
9

3. Initial Steps

Miller and Ozanne found three classes of variables to be useful in estimating 

current-year capital gains: stock market price and volume variables, macroeconomic 

variables, and capital gains tax rate variables. Our initial research strategy was to retain 

the Miller-Ozanne model for estimation and develop a new model for forecasting. Before 

we began formal modeling, we adjusted the series on capital gains realizations and 

searched for macroeconomic and financial variables that could help forecast that adjusted 

series. 

Tax-Adjusted Capital Gains 

Legislation changing capital gains tax rates can cause large changes in capital 

gains realizations, as happened, for example, in 1986 and 1997. Yet, legislative changes 

are difficult to forecast and, more importantly, CBO is legally mandated to assume that 

current policies remain in place when it does its budget projections. That assumption 

increases forecast error in years with big tax changes, yet the error isn’t the fault of the 

model. It is due to an error in assumptions about tax policy. Moreover, a model 

attempting to fit unadjusted capital gains data would mistakenly attribute the effects of 

tax changes to other variables in the model. 

 In order to address the problem of forecasting gains in years affected by tax 

changes, we removed an estimate of the tax effects from the reported series on capital 

gains. The estimate comes from the Miller-Ozanne estimating equation that contains real 

GDP, S&P 500, and tax rates. With that equation, we calculated what capital gains 
10

would have been in each year had tax rates been at their level between 1998 and 2000. 

We call this series tax-adjusted capital gains. 

 For our initial investigation conducted during the end of 2000 and the beginning 

of 2001, we had data on capital gains through 1999, although the 1999 figure was 

preliminary. Thus our initial series on tax-adjusted gains extended through 1999 and was 

estimated with data on stock prices and GDP available around the end of 2000 and the 

beginning of 2001. By the time we completed our formal forecasting models a year later, 

we had a final figure for capital gains in 1999 and a preliminary figure for 2000, and 

other data as reported early in 2002. The equation used to construct tax-adjusted gains 

through 2000 appears in Table 2. 

Tax-adjusted capital gains are compared to reported capital gains for the years 

1948-2000 in Figure 1. Adjusted gains are above reported gains in most years because 

tax rates in 1998-2000 were lower than in most years. The main exception is between 

1982 and 1986 when the top tax rate was slightly lower than in 1998. The largest single 

adjustment is in 1986. The spike in reported gains that year was caused by taxpayers 

realizing many additional gains to beat the tax rate increase legislated to begin in 1987. 

Adjusted gains iron out that spike. 

The formal models developed below will attempt to forecast the annual growth 

rates of tax-adjusted gains between 1971 and 2000. In judging their success, it will be 

helpful to keep in mind that over that period tax-adjusted gains grew at an average annual 

rate of 12.2 percent with a standard deviation of 16.4 percent. The largest decline was 

18.5 percent and the largest increase was 44 percent. 
11

Potential Explanatory Variables 

Our next step in late 2000 was to assemble a set of variables that could help to 

forecast gains. We would then incorporate these variables into a forecasting model. We 

reasoned that the greatest chance for success was at the one-year horizon. If the 

dependent variable were annual gains in year t+1 and the explanatory variables were 

quarterly values in the 4th quarter of year t and earlier, we could check out the usefulness 

of various series in forecasting gains without actually constructing a model. We could 

just deal with single equations having (growth of) annual gains on the left-hand side and 

a constant and predetermined variables on the right-hand side.6

 We initially examined 

whether adding lag distributions of alternative variables would improve the fit of an 

equation with only a constant on the right-hand-side (a random-walk specification). We 

experimented with adding more than one variable at a time and with restricting the lag 

distributions. The results from all of these specifications were discouraging: none did 

appreciably better than the random walk specification. 

We reasoned that if gains were related contemporaneously to other variables, then 

forecasts of gains would be related to forecasts of other variables. It could be that the 

failure of our first exercise reflects an inability of the univariate lag distributions we 

specified to imply adequate forecasts of the variable being examined. We sought to 

separate the ability of a variable to explain gains from the ability of that variable itself to 

be forecast. 

We attempted to gather information on the usefulness to gains forecasts of 

macroeconomic and financial variables by determining their contribution assuming that 

they, themselves, could be forecast without error. If a variable were useful in forecasting 
12

under the perfect foresight assumption, we could later direct our efforts to forecasting that 

variable as accurately as possible. However, if it were not useful, there would be no 

reason to proceed further with it. We realized that the first part of our inquiry can be 

considered estimation of tax-adjusted capital gains based on observed values of 

macroeconomic and (other than stock market) financial variables. As a result, knowledge 

gained from this first part had the potential to lead to improvements in estimation as 

carried out in Miller-Ozanne. 

The results from this exercise were surprising. Although realizations is a 

nominal, or current-dollar, series, it is best estimated with real, or constant-dollar, 

macroeconomic series. Current financial variables or prices were not useful. Whenever 

a nominal series helped to estimate gains, it was dominated by its real counterpart.7

 The 

three series among those we examined that contributed most to fit were real GDP, private 

nonfarm output-per-hour, and the ratio of real consumer durable expenditures to real 

personal disposable income. Moreover, the variables contributed the most when they 

entered jointly rather than individually. Although it is purely conjecture, we believe the 

first two variables have explanatory power for gains because they indicate something 

about the state of the economy and the profitability of firms. We believe the third 

variable has power because there is a connection, possibly going in both directions, 

between sales of financial assets and purchases of big, discretionary items. For instance, 

when people decide to buy a luxury car, they may pay for it in part by selling stock. Or, 

when people realize a big profit in the stock market, they may use some of it to purchase 

 6

 As described in Miller-Ozanne, the data seem to prefer a growth-rate specification for gains. 

7

 An exception is that the growth in nominal stock prices explains the growth of gains better than 

the growth of real stock prices. We ignored stock prices at this stage of our analysis, however, because 

they are so difficult to forecast. 
13

a luxury car. In either case, the third variable would change as the numerator is affected 

but not the denominator. 

4. Two-Step Forecasting Approach 

Since our point of departure was the Miller-Ozanne estimation model, a two-step 

approach to forecasting gains seemed logical. In estimating gains, current-year gains are 

regressed on actual values of explanatory variables. In a two-step approach to 

forecasting, next-year’s gains would be regressed on forecast values of explanatory 

variables. The explanatory variables for forecasting would be from one of all possible 

subsets of the variables found useful in our preliminary exercise: real GDP, labor 

productivity, and the ratio of consumer durables to disposable income. 

Our next task was to forecast the three explanatory variables to determine the best 

combination to forecast adjusted capital gains. Since all three potential explanatory 

variables are available quarterly, we judged that they could be best forecast using a 

quarterly model. We still had two major decisions to make: 

• what form the model should take, and 

• which variables in addition to the explanatory variables should be included to 

help forecast. 

Since our approach is not explicitly based on economic theory, the nonstructural 

forecasting methodology of vector autoregressions seemed like a natural choice for 

modeling strategy. And because we contemplated including as many as six variables 

(three explanatory variables and three auxiliary variables useful in forecasting the 

explanatory variables), improved forecasting accuracy could be expected by using 
14

Bayesian-restricted vector autoregressions (BVARs). See the procedures in Sims and 

Zha and in Robertson and Tallman.8 

The tightness of prior restrictions in a BVAR can be varied. While the data and 

the restrictions determine the estimated coefficients in a BVAR, the tightness of the prior 

determines the relative weights given to each determinant. A loose prior gives more 

weight to the data; a tight prior gives more weight to the restrictions. One set of 

restrictions essentially maintains that each variable in the BVAR can be modeled as a 

univariate random walk with drift. Another restriction takes the form of a dummy 

observation that implies the variables in the model are cointegrated. As a result, the 

BVAR retains a form of error correction, like the CBO forecasting model, but it is 

implemented in a more flexible way. 

The set of auxiliary variables we considered to aid in forecasting the three 

explanatory variables share two common properties. First, they are available on a 

quarterly basis back to 1948, which allows a significant period to estimate a model for 

annual capital gains and evaluate its out-of-sample forecasts. Second, they are related to 

variables that other researchers have found useful in forecasting business cycle data. Our 

method to select auxiliary variables was to specify four classes of variables and choose at 

most one variable from each class. The variables within each class are shown in Table 3 

along with the explanatory variables.9

 

 8

 Christopher A. Sims and Tao Zha, “Bayesian Methods for Dynamic Multivariate Models” 

International Economic Review, (1998) Vol. 39 Issue 4, pp. 949-968. John C. Robertson and Ellis W. 

Tallman, “Vector Autoregressions: Forecasting and Reality” Federal Reserve Bank of Atlanta Economic 

Review (First Quarter, 1999): Vol. 4, Issue 18. 9

 In all the models we describe in the following text, level variables enter as logarithms, while rate 

and ratio variables enter without transformation. 
15

Only the interest-rate spread is included in the first auxiliary class. It is included 

because researchers have found it useful in forecasting business cycle turning points.10

The main reason for specifying the Moody’s seasoned long-term rate in the spread is that 

it goes back to 1948. 

The second auxiliary class contains either a real wage or a labor share variable. 

Our thinking was that such a variable would close a supply-side subsector of the model 

composed of output, productivity, and wages. 

The third auxiliary class contains a mix of variables, all thought to provide 

leading indicators of demand. It is common to find either investment variables or crude 

price variables in macro BVARs.11 

We originally planned to: 

a. use the Miller-Ozanne equations to estimate gains, 

b. estimate a BVAR incorporating explanatory and auxiliary variables, and 

c. use the forecasts of explanatory variables from that BVAR in a regression 

with gains as the dependent variable. 

As work proceeded, we made two changes to this plan. The first was to estimate 

gains with our own equations rather than the Miller-Ozanne equations. The reason was 

that the dependent variable in our equations is tax-adjusted gains, while that in the latter 

is unadjusted gains. That change caused us to add the stock market as a fourth auxiliary 

 10 See, for example, Dan M. Chin, John Geweke, and Preston J. Miller, “Predicting turning points” 

Federal Reserve Bank of Minneapolis Staff Report Number 267, (June 2000). Arthuro Estrella and 

Frederic S. Mishkin, 1998. Predicting U.S. recessions: Financial variables as leading indicators. 

Review of Economics and Statistics (1998) 80, no. 1, pp. 45–61. James H. Stock, and Mark W. Watson, 

“New Indexes of Coincident and Leading Indicators” In National Bureau of Economic Research 

Macroeconomic Annual 4, edited by Olivier Blanchard and Stanley Fisher. Cambridge, Mass.: National 

Bureau of Economic Research. (1989). 11 See, for example, Robert B. Litterman, “A Bayesian Procedure for Forecasting with Vector 

Autoregressions” Massachusetts Institute of Technology Department of Economics Working Paper (1980); 
16

class of variables. Since stock market volume is not available on a quarterly basis back 

to 1948, only the S&P 500 was included in the seventh class. The second change was to 

remove any distinction among our classes of variables. We made this change because we 

found estimates and forecasts of capital gains sometimes could be improved by adding 

variables from classes other than the explanatory variables. Thus, one variable from any 

class could be included in the estimation and forecasting equations for gains. 

Next we describe our revised methodology in greater detail and then present 

evidence on the approach’s success. We distinguish in our description between 

estimation and forecasting. Experimentation confirmed our prior that it is better to use 

separate models to estimate and to forecast than to use a single model to do both. The 

models we chose are identified by the variables they contain and by the hyperparameter 

values used in the BVAR restrictions. The identifying features associated with the bestperforming models are reported in Appendix A, along with background information on 

the hyperparameters. 

Methodology 

Presentation of the methodology and discussion of the empirical results is 

facilitated by introducing some mathematical notation and our error measures. The 

notation is first used to develop our estimation methodology and then our forecasting 

methodology. 

Notation and Measures of Success. For any variable Z, we use the subscript t to denote 

year, so that Zt is the annual value of Z in year t. We use the subscript t:i to denote the ith 

 

 Robertson and Tallman, and Sims and Zha. 
17

quarter of year t, i= 1,2,3,4, so that Zt:i is the value of Z in the ith quarter of year t.

Throughout the text we denote the value of tax-adjusted capital gains in a year by CG and 

the log of CG simply by Y. We often refer to CG as “gains.” While capital gains are 

reported only annually, all the other variables used in our models are reported at least 

quarterly. We refer to the other variables as the vector X. 

Our data include annual values of CG and quarterly values of X from 1948-2000. 

These data are as reported in early 2002. At that time, we had preliminary information on 

gains in 2000. Tax-adjusted gains equaled actual gains in 2000 because tax rates on 

capital gains between 1998 and 2000 were the base rates for the series on tax-adjusted 

gains. 

Much of what we do is guided by the CBO’s process for annual revenue 

projections. The CBO finalizes its estimates and forecasts of capital gains in the late fall 

of the year. So, if s is the current year, we suppose the information at hand consists of the 

history of CG through year s-1, denoted by [CG]s-1 and the history of X through the 4th

quarter of s, denoted by [X]s:4 .12 Consequently, for any of the approaches we develop, 

we estimate (CGs)

e

 based on the information [CG]s-1 and [X]s:4. We forecast (CGs+j)

f

where j =1....10, based on the same information and conditional on (CGs )

e

. 

For the most part, the focus is on 1-year-out forecasts, and all of our estimation 

and forecasting exercises are done out-of-sample. That is, at each date t, the models’ 

coefficients are estimated on only [CG]t-1 and [X]t-1:4. The objectives and performance 

measures are stated in terms of 3 error measures: 

 12 Our information assumption deviates from the situation faced by the CBO in 2 ways. First, CBO 

had only partial data for the 4th quarter of each year, while we assume full information for the quarter. 

Second, our data set is as it existed early in 2002. When the CBO made its forecasts, it had to use data that 

existed at the time and which subsequently have been revised, perhaps, substantially. 
18

1. Estimation error, E1t = CGt 

e

/CGt-1 – CGt/CGt-1 , 

2. (Forecast Estimation) error, E2t = CGt+1

f

/CGt

e

 – CGt+1/CGt , and 

3. (Forecast Actual) error, E3t = CGt+1

f

/CGt-1 – CGt+1/CGt-1 . 

For any method, we first estimate capital gains to minimize the out-of-sample root mean 

squared error (RMSE) in terms of E1. Then, given that estimate of capital gains, we 

forecast 1-year-ahead capital gains to minimize the out-of-sample RMSE in terms of E2. 

We also report overall forecasting performance in terms of the RMSE of the 2-year error 

measure E3. 

Methodology for Estimation. In this part of the methodology, we examine the 

performance of estimation equations incorporating different sets of variables. The 

objective is to find the set of variables that minimizes the RMSE in terms of the 

estimation error E1. We begin by selecting a set of variables, with at most 1 from each of 

the 7 classes, to form the vector X. We next estimate by OLS over the period 1949-1969 

the relationship: 

(1) ∆Yt = b0 + b1(Xt – Xt-1) + εt , 
19

where the variables in X are annualized and where level variables are logged and rate or 

ratio variables are untransformed. Then using the estimated values of b0 and b1 and the 

actual X1970 – X1969, the estimate of ∆Y1970 is calculated as (∆Y1970)

e

 = b0 + b1(X1970 – 

X1969). This in turn yields the estimate of log(capital gains) in 1970 of (Y1970)

e

 = Y1969 + 

(∆Y1970)

e

, and finally (CG1970)

e

 = exp[(Y1970)

e

]. We add a year to the estimation period 

and proceed this way year-by-year until (Y2000)

e is obtained. We then compute the E1 

errors for each year and compute the RMSE in terms of these errors. We choose a 

different set of variables to be included in X and repeat this process until all possible 

subsets have been examined. We choose as best the one that has the minimum RMSE. 

Unfortunately, there is a complication due to an apparent break in the statistical 

process for gains in the 1990s. Specifications that perform best, in terms of either 

estimation or forecasting, over the whole period 1971-2000 are found to be different from 

those that perform best in the 1990s. As time passes, the question will have to be 

confronted of whether the process for gains has permanently changed to that of the 1990s 

or whether it will return to the way it was earlier. One way of dealing with this problem 

is to carry along estimates/forecasts of models best over each time period and put weights 

on them according to their relative errors, i.e., a likelihood calculation. Because we do 

not attempt to answer the question, we carry along 2 models—one best for the whole 

period 1971-2000 and one best for 1991-2000. 

Methodology for Forecasting. In this part of the methodology, we search anew for a set 

of variables that leads to the most accurate 1-year-out forecasts of gains given the current 

years’ estimates obtained above. The set of variables found to be best can differ from 
20

that selected in the estimation part. In the first step of the forecasting methodology, each 

set of variables is incorporated into a quarterly BVAR to generate forecasts over the next 

year. In the second step, these forecasts are then included in a forecasting equation for 

gains. The best set of variables for forecasting is the one that leads to the minimum 

RMSE in terms of out-of-sample E2 errors. 

As before, we begin this part by selecting a set of variables, with at most 1 

variable from each of the 7 classes, to form the vector X, and a vector of hyperparameter 

values. Given X and the hyperparameter values, we estimate a quarterly BVAR with 5 

lags from 1949:2 through 1959:4 and use that estimated model to dynamically forecast 

(X1960:1)

f

, (X1960:2)

f

, (X1960:3)

 f, and (X1960:4) f. After appropriately converting the 4 

quarterly forecasts to a forecast of (X1960)

 f, we estimate the model through 1960:4 and 

follow the same procedure to generate a forecast of (X1961)

 f. We continue this process 

iteratively until we have forecasts of (Xt)

 f conditional on [X]t-1:4 for t = 1960,....,2000. 

The forecasts of the X variables are used to forecast gains by estimating an 

equation that relates the growth of capital gains to the forecast changes of X variables: 

 (2) Yt – Yt-1 = c0 + c1[(X t)

f

 – Xt-1] + μt. 

The regression (2) is estimated by OLS initially over the eleven years 1960-1970, based 

on information available at the end of 1970. Since Y1970 is not available based on the 

information assumption, it is replaced by its estimate determined in the previous step. 

Then, based on the estimated coefficients, c0 and c1, next year’s forecasts of X, (X1971)

f

, 

and the current year’s estimate of capital gains, (Y1970)

e

, the relationship is used to 
21

forecast gains in 1971. We then add a year to the estimation period for (2) and proceed 

as before to get a forecast of gains for the 1972 (again substituting the estimated gain for 

the unobserved actual in 1971). The iteration continues until equation (2) is estimated 

through 1999 and the last forecast is for year 2000. With these thirty forecasts of gains, 

we calculate year-by-year E2 errors and the associated RMSE. Now, with the variables in 

X still fixed, the hyperparameter values are varied and the process repeated to find the 

hyperparameters that minimize the RMSE in terms of E2 errors. In practice, for each 

specification of X, we do a coarse grid search around standard hyperparameter values to 

improve forecasting results. Finally, the whole process is repeated for each set of 

variables in X to get the best model: the best set of X variables, and given those, the best 

set of hyperparameter values. 

Results for Two-Step Approach 

In Table 4, we compare the out-of-sample, one-year-ahead estimation and forecast 

RMSEs for adjusted capital gains of five alternative models. All of the models are 

conditional on [CG]t-1 and [X]t:4 . The last two models listed are the winners in the above 

process of selecting two-step models. TS-all is the combination of the model that 

estimated best and the model that forecast best for the entire 1971-2000 period. The two 

components of TS-all differ in variables included and hyperparameters, as noted above, 

but otherwise employ the same methodology. TS-90 is the corresponding combination of 

models that estimated and forecast best for the 1991-2000 period. 

The three models listed on the top of Table 4 provide standards against which the 

errors of the two-step method can be judged. The first of these, CBO, replicates the CBO 
22

method. Gains are estimated for year t using a simplified version of a CBO equation, and 

gains for t+1 are forecast using the mean reversion methodology.13 The second of these, 

RW (random walk with drift) assumes that gains grow at a constant rate and that rate is 

the average of the historical rates known at the time of the estimation and forecast. The 

third and final comparison model, AR (autoregression), regresses the log of gains on a 

constant and an unconstrained 1-year lag of gains. It uses the estimated coefficients to 

estimate and forecast gains. 

The RMSEs are shown for the E1 estimation errors, the E2 current-year estimateto-forecast, errors, and the E3 previous-year actual-to-forecast, errors. The average 

RMSEs are shown for each decade running say, from 1971-1980 and for the entire 

period, 1971-2000. 

Each two-step model estimates gains more accurately than its standards of 

comparison. That is, given actual values for exogenous variables in a year and gains in 

the prior year, the two-step models estimate gains with lower RMSEs. The simplified 

CBO equation comes in second place in the full period and in the 1990s, with RMSEs 

that are 30 percent to 35 percent higher. The better estimates of the two-step models may 

come from their having been developed to explain tax-adjusted gains while the CBO 

models were developed to explain actual capital gains. Changes in tax rates account for 

substantial changes in actual capital gains. The narrower focus of the two-step equations 

may have identified additional explanatory variables that will be helpful in explaining 

 13 The simplified CBO equation is the equation used to construct tax-adjusted capital gains, see 

Table 2, with the simplification that the two variables measuring tax rates are eliminated. The mean 

reversion forecast follows the current CBO procedure described earlier, but is applied retroactively to years 

from 1971 forward. For full details, see memo from Larry Ozanne, revised December 12, 2002. 
23

actual capital gains. The leading candidates are productivity, the interest rate spread, real 

compensation per hour, and real fixed private domestic investment. 

The two-step approach also generally forecasts better than the standards of 

comparison. Over the full period and in terms of E2 errors, TS-all forecasts have the 

lowest RMSE at 14.80 percentage points; the random-walk forecasts have the second 

lowest RMSE at 16.78 percentage points. The AR(1) forecasts, closely followed by the 

CBO mean reversion forecasts, have larger RMSEs of 18.31 percentage points and 18.57 

percentage points respectively. The TS-all forecast errors are 20 percent lower than the 

mean reversion errors. The advantage of the TS-all model occurs primarily in the 1971-

1980 and 1991-2000 decades. In the 1981-1990 decade, however, TS-all has the highest 

RMSE, although it is not much worse than the errors of the random-walk and 

autoregressive models. 

When the focus is narrowed to forecasting the decade 1991-2000, the TS-90 has 

the lowest RMSE for E2 errors at 12.55 percentage points. TS-all is second, and the 

three standards of comparison follow in the same relative ranking as for the full period. 

The poor performance of mean reversion forecasts in the 1990s is not surprising given the 

surge in gains in the later half of the decade. In fact, it was the large forecast errors CBO 

made in the 1990s, documented in Table 1, that led us to search for better forecasts.14 

The models with the lowest E2 error also have the lowest E3 error. That means 

that models that best forecast the growth rate of gains conditional on their estimate of 

 14 The RMSE from applying the CBO method to tax adjusted gains between 1992 and 2000 is only 

slightly lower than the RMSE CBO had in forecasting actual gains over the same period with its variants of 

mean reversion. Thus the primary source of error during those years was from economic and behavioral 

factors, not from unanticipated tax changes. 
24

gains in the prior year also get the closest to the actual value of gains in the year ahead. 

That outcome is not necessary, as we will see below. 

Neither two-step model forecasts better than the other over all decades. TS-90 

does considerably better than TS-all in the 90s, but it does much worse in the 70s and 

80s. 

The lower forecasting error between 1971 and 2000 of the random walk than the 

mean reversion model suggests that forecasting gains to grow at their average rate as 

known up to that year would have been superior to using the mean reversion model. 

5. Integrated Quarterly Approach 

Although the two-step approach met with some success, we reasoned that it had 

some shortcomings. One of them relates to how it deals with mixed-frequency data. 

With realizations available annually and all other macroeconomic and financial variables 

available at least quarterly, any forecasting model has to deal with a mixed-frequency 

data problem. The two most common procedures to address this problem are either to 

aggregate the high-frequency data to the lowest frequency of any variable or to 

interpolate the low-frequency data to the highest frequency of any variable. Our two-step 

approach followed the former procedure by aggregating up all quarterly data to produce 

annual growth rates or changes. A shortcoming with this procedure is that it can cause a 

loss in information. For instance, had real GDP shown no growth over the year, it could 

matter for gains whether GDP was flat over the whole year, had fallen in the first half and 

risen in the second, or risen in the first half and fallen in the second. The two-step 

approach makes no distinction among these alternatives. 
25

Another shortcoming with our two-step approach is the way it chews up degrees 

of freedom. Many observations are required in the first step to estimate the BVAR before 

it can be used to reliably generate forecasts. Then further observations are required in the 

second step to reliably estimate the regression of gains on forecast values of the 

explanatory variables. 

The second approach with interpolation of gains circumvents the shortcomings of 

our two-step approach. However, it introduces a new one. The interpolated values of 

gains must differ from the true, unreported quarterly gains. Consequently, interpolation 

introduces some measurement error. Since both the two-step approach and second 

approach using interpolation face conceptual problems, the determination of which works 

better in practice is an empirical issue. 

Our research turned to developing the second approach to estimating and 

forecasting gains. The findings are clear: interpolating gains and incorporating them into 

a unified model outperforms the two-step approach. We now describe that research. 

In order to estimate a single model that contains macroeconomic and financial 

data as well as gains, we need to construct a quarterly series of gains. We call the 

constructed series interpolated gains and require that the average of the four quarterly 

values (at annual rates) equals the reported annual value. Since there obviously are many 

reasonable ways to interpolate gains, we require a way to choose among the alternatives. 

We consider three alternatives and choose among them by which leads to the most 

accurate estimates and forecasts of capital gains in the current and next year, respectively. 

As in the previous section, the form of model used for the unified approach is taken to be 
26

a BVAR, and once again, we allow for using different models to estimate and forecast 

gains. 

With two of the alternatives for interpolating gains, we translate annual gains into 

quarterly gains once-and-for-all before we do any modeling for estimation or forecasting. 

As a result, the quarterly gains constructed from these alternatives are external to our 

models and are incorporated into them in the same way as any other quarterly series. 

With the third alternative, quarterly gains are constructed “as you go” and are internal to 

our model. Nevertheless, for all three alternative interpolation schemes, the objectives 

are: first to minimize RMSEs in terms of E1 errors conditional on the information sets 

[CG]t-1:4 and [X]t:4 , and second, to minimize the RMSEs in terms of E2 errors 

conditional on the same information plus {(CGt:1)

e

, (CGt:2)

e

, (CGt:3)

e

, (CGt:4)

e

}. 

Three Interpolation Methods for Integrated Approach 

The first interpolation alternative is linear. It is constructed by assuming that 

gains grow linearly from the fourth quarter of one year through the fourth quarter of the 

next year. The slope is set so that the average of the four quarters of the next year equals 

the actual annual amount of gains for that year.15 For example, if gains in the 4th quarter 

of one year are $100 and they total $162.50 for the next year, the path of quarterly gains 

in the next year is $125, $150, $175, and $200. This method requires an alternative way 

of interpolating gains in the first year, 1948, because no value is available for the fourth 

quarter of the prior year. The alternative we use is to assume that realizations within 

1948 grew at the average annual rate of gains over the 1948-2000 period. It is possible 

 15 Quarterly gains are at annual rates so that the average of the 4 quarters of a year equals the 

annual total. 
27

then to solve for the quarterly levels of gains consistent with both that growth rate and the 

1948 total.16

Since linear interpolation forces the changes in gains to be the same in each 

quarter of a year, these changes cannot reflect changing economic conditions within the 

year. We reasoned that there could be an advantage to letting quarterly changes in 

conditions affect quarterly patterns of gains. The next two alternatives were designed 

with that in mind. 

The second interpolation alternative that we consider is dubbed “economic 

interpolation.” Like the first alternative, annual gains are allocated at the outset to 

individual quarters in each year, and a complete quarterly series is constructed that can be 

included with all other macroeconomic and financial variables in our data set. 

To construct economically interpolated gains, we use the same Miller-Ozanne 

equation as was used to remove tax effects from the unadjusted capital gains series (see 

Table 2). The equation is used to calculate a growth rate of gains from one quarter to the 

corresponding quarter in the next calendar year. That growth depends on changes in the 

business cycle and stock prices over eight quarters. Repeated application of the equation 

provides an estimate of gains growth from each quarter to the same quarter in the 

preceding and the following calendar year. Then, by somewhat arbitrarily allocating 

actual gains for one year among four quarters, the calculated growth rates are used to 

 16 We investigated, in the context of univariate models, the difference it made to the forecasting of 

gains of using linearly interpolated quarterly gains instead of annual gains. This investigation allowed us to 

determine how much of the difference in forecasting success of the two-step and integrated approaches 

stems solely from the difference in using the annual and interpolated gains series. 

Our findings, which may seem counterintuitive, show that the interpolated series does better. The 

intuition is that since the interpolated series is just an algebraic manipulation of the original series and 

contains no additional information, the two series should do equally well. We give an explanation and 

geometrical interpretation of our results in Appendix B. 
28

estimate gains in all quarters from the first quarter of 1948 to the last quarter of 2000. In 

a final step, the amount of gains in the four quarters of each year is adjusted 

proportionately to hit the known total of gains for each year. This alternative has the 

advantage of letting the quarterly pattern of gains respond to the pattern of 

macroeconomic and financial variables. (Appendix C describes the procedure in greater 

detail and identifies key assumptions and limitations.) 

The third, and final, interpolation alternative that we consider is dubbed “inmodel” interpolation. Unlike the other two alternatives, for which the interpolated series 

is created once-and-for-all before the BVAR construction begins, this one creates the 

series on the go as the BVAR is constructed. The idea behind this alternative is to make 

the interpolation endogenous by attempting to determine the quarterly gains series, 

consistent with the model’s structure, that yields the best estimate of annual gains. The 

interpolation series using this alternative is constructed in three stages: 

a. Initial model estimation with an interpolated “seed”, 

b. Out-of-sample estimation of annual gains, and 

c. Reconciliation of estimated gains with actual annual gains. 

Since in-model interpolation is done within an estimated model, the latter must exist 

before the series construction can begin. We choose 1948-1960 as the initial estimation 

period. We refer to the initial gains series as the seed, because it is used to get the 

process started but then is extended with a new series in the period after 1960. The seeds 

are either the linearly interpolated gains series or the economic interpolated series. The 
29

estimation and reconciliation stages to construct the in-model interpolated series are 

described in the next section as part of the general discussion of estimating gains with all 

three interpolation methods. 

Appendix D compares gains interpolated by each of the three methods to limited 

data on the timing of actual transactions that generated gains. The comparison finds 

limited evidence that linear and in-model interpolation can pick up some trend in gains 

within a year, but no method can match quarter-to-quarter movement of the transactions 

data. Differences in the types of gains included in of our interpolated variable and the 

transactions data cloud our findings 

Estimation of Annual Tax-Adjusted Gains 

In this section we seek to minimize RMSEs in terms of E1 errors, when we 

estimate CGt conditional on [CG]t-1:4 and [X]t:4. For each interpolation alternative, we 

search for the best model, where a model is identified by a choice of variables and a set 

of hyperparameter values. Our method is to select a set of variables, and then given that 

set, to search for the hyperparameters that lead to the smallest out-of-sample estimation 

errors. Finally, we choose among the different models based on which estimates the most 

accurately. 

In order to estimate annual gains, we have to account for the partial information 

structure in the current year. We do this for each interpolation method by straightforward 

application of Kalman filtering. However, our method can be understood as a series of 

three steps, as described below. 
30

 The first step is to estimate the model based on complete quarterly data through 

the historical period that ends in the fourth quarter of the previous year, i.e., 

 [CG]t-1:4 and [X]t-1:4. 

The second step is to generate quarter-by-quarter estimates of gains in the current year 

based on the information set 

 [CG]t-1:4 and [X]t:q for q = 1, 2, 3, 4, 

available at the start of each quarter. We do this using a sequence of forecast and 

revision operations. Based on the data and the estimated coefficients as of the end of the 

historical period, forecasts are generated for all variables in the first quarter of the current 

year. Then, those forecasts are revised based on differences between actual and observed 

values of macroeconomic and financial variables in the quarter. Essentially, this is done 

by pre-multiplying the one-step-forecast errors for the macroeconomic and financial 

variables by their covariances with gains, where the covariance matrix is constructed 

from the estimated residuals (up to the prior on the covariance matrix of the BVAR). 

Then, the resulting revision term is added to the forecast of gains to get an initial estimate 

of gains in the first quarter. Next, we forecast all variables in the second quarter based on 

information through the first quarter, of which the data in the last quarter consist of actual 

values of macroeconomic and financial variables and the initial estimate of gains. We 

continue this process of forecasting and revising to get estimates of quarterly gains based 

on information through the fourth quarter. 

The third step involves year-end revisions to generate final quarterly estimates of 

gains. Conceptually, the initial estimates in the first three quarters of the year do not 

utilize information available later in the year when the CBO would be estimating current-
31

year gains. After observing their fourth quarter values, therefore, it is possible to 

improve further on the estimates from the second round of operations. We do this by 

Kalman smoothing, and take the average of the resulting final quarterly estimates to get 

the current-year estimate of annual gains. When we move to the next year, the process is 

repeated with estimated quarterly gains being replaced by interpolated gains, where the 

latter is based on observed annual gains for the year. 

With linear or economic interpolation, there is no ambiguity about what is meant 

by interpolated gains, but there is with in-model interpolation. With the two former 

interpolation alternatives, estimated gains for the current year are replaced by the linearly 

or economically interpolated gains contained in the data file, the model is reestimated, 

and the estimation process for the next year begins. For these two alternatives, the 

interpolated gains, which by construction add up to the observed annual gains, are in the 

data set and are treated as actual data. In contrast, with the in-model alternative, there are 

no quarterly gains in the data set for the current year. There are only estimates. By the 

fourth quarter of the next year, however, annual gains for the current year will be known, 

and the sum of the quarterly estimates will differ from it. Thus, in order to make use of 

all observations, we need to reconcile the best estimates of quarterly gains with the total 

for the year. After the estimation of gains in the current year is completed and before the 

estimation of gains in the next year begins, a reconciliation stage is required with inmodel interpolation. 

After some experimentation, we found the best way of reconciling (“best” in 

terms of lowest estimation errors) is to use the structure of the model. In concept, we 

shock the gains equation by a constant amount each quarter and let the model’s dynamics 
32

(its impulse response) determine the revised quarterly gains. We require that the constant 

shock be such that the sum of the revised gains equals the observed annual figure. Since 

the values of macroeconomic and financial variables are already observed, only gains are 

affected by this exercise. Hence, we use the lag coefficients in the gains equation to 

determine the response. Because a prior is that the log of gains follows a random walk, its 

estimated own lag coefficients will be approximately 1 on the first lag and zeros 

elsewhere. For this reason, a shock of ε applied each quarter will approximately produce 

changes in the log of gains, revised minus estimated, of ε in the first quarter, 2ε in the 

second quarter, 3ε in the third, and 4ε in the fourth. Now, based on actual values of 

macroeconomic and financial variables through the current year and reconciled gains 

figures over that period, we reestimate the model and proceed to estimate gains in the 

next year. 

In Table 5, we compare the estimation errors from each of our best models using 

the different methods. The models are the two-step annual model and the 4 quarterly 

interpolated models: linear, economic, in-model with linear seed, and in-model with 

economic seed. The three comparison models included in Table 4 are omitted because 

they generally did less well than the included two-step models. The RMSEs of the three 

comparison models shown in Table 4 can be compared to the RMSEs in Table 5. In 

panel A, we include the models that were best over the entire 1971-2000 period. In panel 

B, we include those that were best over the 1990s. 

All five models have similar estimation errors when fit over the 1971-2000 

period. The linear interpolation model is best with a RMSE of 0.1025, and the model 

with in-model interpolation from a linear seed is worst with a RMSE of 0.1190. The two-
33

step method is in the middle of the pack. All five models also have similar RMSEs when 

fit to the 1991-2000 period, but the model with in-model interpolation and a linear seed 

has moved from worst to best. Again, the two-step method is comparable to the 

interpolation models. 

Forecasting of Tax-Adjusted Gains One Year Ahead 

In this section, the objective is to minimize RMSEs in terms of E2 errors 

conditional on the information sets [CG]t-1:4 , [X]t:4 and {(CGt:1)

e

,...,(CGt:4)

e

}. Each 

forecasting model we examine uses the same gains series as the associated estimation 

model; i.e. linearly interpolated gains or in-model with economic seed gains, for example. 

However, the variables and hyperparameter values can vary in the associated models for 

estimation and forecasting. 

The models’ coefficients then are estimated using data through t-1:4. The 

forecasts are conditioned on actual values of macroeconomic and financial variables 

through t:4 and on estimated values of gains for the quarters of t. The forecasts are 

generated dynamically, quarter by quarter, for year t+1. 

All models have more difficulty forecasting a year ahead than estimating the 

current year, but some have more difficulty than others. When the models are judged by 

how well they forecast 1971-2000, the integrated model with simple linear interpolation 

does best with a RMSE of 11.92 percentage points (see Table 6). The three other models 

with interpolated gains follow with gradually higher RMSEs. The two-step model has a 

larger RMSE than all of the interpolation models. The mean reversion model that mimics 

CBO’s method has a RMSE of 18.57 percentage points. Thus, the linear interpolation 
34

model reduces the RMSE from that of mean reversion by one-third. Recall from Table 4 

that mean reversion forecasts had a lower RMSE than the two-step model for the 1981-

1990 decade. All four models with quarterly interpolation beat the mean reversion model 

in that decade as well as the other decades. 

Among models that fit the 1991-2000 years best, the superiority of integrated 

models over the two-step models is reduced (see Table 7). The model with simple linear 

interpolation again has the lowest RMSE, but the two-step model beats one of the 

interpolation models and all models are closer in their RMSEs. One interpretation of 

these results is that the advantage of the quarterly methodology over the two-step 

approach is that it conserves on degrees of freedom. That advantage would be expected 

to dissipate over time as more observations become available. 

Although the model with linear interpolation has the lowest RMSE for the E2 

error over the 1991-2000 period, it does not have the lowest RMSE for the E3 error. 

Both models with in-model interpolation have lower E3 RMSEs. The lower E3 error 

means that those models get closer to the level of gains being forecast than does the 

model with simple linear interpolation. The model with simple linear interpolation gets 

closer to the growth rate from the year of estimation to the year of forecast, but either has 

larger errors in the year of estimation or its errors from estimation and forecasting are 

more additive while the other models’ errors are more offsetting. 
35

6. Applications of the Model 

After the models were developed, we applied them to forecast tax-adjusted gains 

in 2001. We also used the integrated models to forecast more than one year ahead. In 

each case, we compared the model errors to errors of CBO’s mean reversion method. 

Forecasting 2001 

After the above models were developed, information on gains in 2001 became 

available showing that gains declined by almost 46 percent that year. The small 

reduction in tax rates on capital gains occurring in that year would have minimal effect, 

so a similar drop in tax-adjusted gains should be expected. The forecasts of the above 

models for 2001 provide a true out-of-sample test, since the data for 2001 were not 

available when the models were constructed. 

No model came close to forecasting the apparent decline, and the differences 

among the forecasts is much smaller than any of their errors. The integrated models 

forecasting best from 1971 through 2000 predict 2001 better than the ones forecasting 

best from 1991 through 2000. Three of the four models that were best over the full 

period correctly predicted the turning point in gains, although the largest predicted 

decline was only 6.9 percent (see Table 8). The four integrated models that forecast best 

over the 1991-2000 period all predicted that gains would continue to grow in 2001. In 

each group of integrated models, linear interpolation performed best. Both two-step 

models predicted gains would grow, with the one forecasting best for the full period 

predicting gains would grow a strong 13.0 percent in 2001. That was the worst forecast. 
36

CBO’s mean reversion method, as applied in this paper, predicted that gains 

would rise a slight 0.7 percent. It predicted growth because the estimated reversion rate 

available at the end of 2000 would have been just 13 percent and CBO was predicting 

strong enough growth in GDP at the time to more than offsets that slight degree of 

reversion. In its January 2001 baseline, CBO actually predicted that gains in 2001 would 

remain at the level estimated for 2000. In the three previous years CBO had continued to 

assume a 20 percent reversion rate in spite of the recent experience, and at a 20 percent 

reversion rate, gains had been predicted to decline in each of the previous 3 baselines. 

CBO dropped that methodology in the January 2001 baseline because gains had failed to 

revert in the previous years. 

While 2001 provided a true out-of-sample test, it happened to be a most difficult 

test. The 46 percent decline in 2001 was matched previously only in 1987, when gains 

subsided after the tax-induced rush to realize gains in 1986. Abstracting from taxinduced changes, the largest previous decline since 1949 was 24 percent in 1970. Not 

only is the decline in 2001 extreme, it is unexplained by standard determinants. When 

actual values of economic and financial variables for 2001 are used in equations from 

Miller and Ozanne, those equations predict declines averaging just 20 percent. That 

means some omitted variables powerfully affected realizations in 2001. The year was 

certainly unusual with the attacks on the World Trade Towers and the Pentagon and with 

the subsequent closing of stock markets. It was also unusual coming at the end of a 

dramatic bull market. While it is too early to tell what factors accounted for the sharp 

decline in realizations, it is clear that the year provided a difficult test for historically 
37

based models. In addition, a single year cannot provide a definitive test of statistical 

models. 

Forecasting Multiple Years Ahead 

The integrated quarterly models can generate dynamic forecasts over an 

arbitrarily long horizon. Thus, under the same assumptions and methodology used in the 

previous section to generate 1-year-ahead forecasts, the models can forecast gains in 

quarter t + j using previous forecasts of all variables in quarters t + j – 1, t + j – 2, ..., in 

place of actuals. With a single model, it is possible to generate forecasts of gains over the 

10-year horizon that the CBO currently is required to do. 

Although it is possible to forecast indefinitely with these models, there is good 

reason to believe that their forecast accuracy will deteriorate fairly rapidly. That is 

because the search for variables and hyperparameter values was targeted at getting the 

best 1-year forecasts of gains. If accuracy over a longer horizon were desired from the 

outset, different variables and hyperparameter values would have been found to be best. 

For instance, the weight on the cointegration dummy is almost certain to increase as the 

targeted forecast horizon is extended. 

These considerations lead to the question, for how long could these models be 

used to forecast more accurately than viable alternatives? We explore the answer to this 

question with respect to 2 alternatives. The first is an annual random walk with drift 

model. The second is the CBO mean reversion method as applied earlier in this paper, 

except that reversion is continued out additional years. 
38

We measure errors in terms of a multiyear version of E2. We define the error in 

forecast at horizon j as the difference in growth rates of forecast gains at t + j relative to 

the estimate at t: 

E2t(j) = (CGt+j)

f

/(CGt)

e

 – CGt+j/CGt, j = 1,2,....,10. 

The capital gains estimation and forecasts are all based on the standard information sets 

[CG]t-1 and [X]t:4. It is readily seen that E2(1) ≡ E2. All error calculations are done from 

out-of-sample forecasts over the years 1971-2000. 

Note that the further ahead the forecast, the fewer are the number of forecasts for 

which errors can be calculated, and more importantly, the fewer are the number of 

independent errors. The thirty-year period provides 30 tests of the one-year ahead 

forecasts, and all of them are independent. The same period provides 29 tests of the twoyear ahead forecasts, but only 15 independent ones. That is because the two-year ahead 

forecasts done for adjacent years, say 1981 and 1982, have a common year between 

them, 1982. At the extreme of 10-year ahead forecasts, only 21 forecasts can be made 

and only three of them are completely independent. Clearly, the fewer the number of 

independent forecasts, the less confidence that can be placed in the robustness of the 

RMSEs. 

The quarterly model selected for the comparison is the one with the lowest 

RMSEs over the 1971-2000 period from Table 6. It uses linear interpolation of gains. 

We did not select any of the models that were best in the 1990s because they predicted 
39

poorly in the earlier decades. And we did not select any two-step models because the 

two-step approach does not easily extend to multiyear horizons. 

 In Table 9, we compare the RMSEs in terms of E2(j) errors for CBO’s mean 

reversion method, the random walk with drift model, and the linearly interpolated model. 

The CBO forecasts go six years ahead, the other two go ten.17 The CBO forecasts are 

conditioned on estimates of gains in the base year made with a simplified CBO equation, 

as described in the above section on results from applying the two-step method. The 

random walk and BVAR models are conditioned on the same estimates from the best 

linearly interpolated estimation model. 

In terms of these RMSEs, the interpolated model outperforms the other two 

methods for only 3 years, and then falls farther behind with each successive year forecast. 

The comparison also shows that the random walk and CBO methods have similar errors. 

The random walk model’s errors are slightly lower than the mean reversion’s errors in the 

first two years forecast and become progressively higher in the next four. Although the 

difference becomes larger for five and six-year forecasts, the decreasing number of 

independent observations at these horizons makes it impossible to conclude that one 

method is systematically better than the other.18 

 17 Six years ahead are forecast because that is the number of years that CBO forecasts in its annual 

reports from 1976 through the mid-1990s. The CBO methodology described previously forecasts gains 

using the assumption that gains revert to their historically expected size relative to GDP. Thus the forecasts 

of gains rely on a forecast of GDP, or, before the mid-1990s, of GNP. The forecast errors reported here are 

based on actual CBO forecasts when they exist. In earlier years, imitations have been substituted by 

growing GNP for six years at its average rate during the prior 5 years. CBO has made 10-year forecasts in 

recent years, but too little time has lapsed to evaluate the accuracy of gains forecasts seven or more years 

ahead. 18 The limited number of independent forecasts must account for the anomaly that the random 

walk method has lower errors forecasting 10 years ahead than 9. 
40

The comparisons support our reasoning above that models selected on the basis of 

forecasting one-year ahead lose their superiority after a few years. Why they become 

worse than the alternatives probably reflects the tendency for gains to fluctuate widely 

from year to year but to, at least in the period of our data, return to its average size 

relative to the size of the economy. The BVAR models seem to capture more of the 

factors causing the annual fluctuations but at the expense of slighting the longer-range 

trends. The random walk model forecasts only the longer-run trend because its forecast 

is for gains to grow at the historical average rate known at the time the forecast is made. 

The mean reversion model captures the longer run trend through the forecast of GDP and 

some movement around trend to the extent reversion occurs in the forecast interval. 

7. Conclusion 

The primary objective of this paper has been to improve on the method of 

forecasting gains employed by the CBO. We find that both the two-step model and the 

integrated quarterly model forecast tax-adjusted gains better than a mean reversion 

method similar to that used by CBO. Both models also forecast better than two other 

basic forecasting methods: a random walk method with drift, or a slightly more general 

autoregressive model. 

 The integrated quarterly model achieves the greatest improvement in forecasting. 

Over the 1971-2000 period, the RMSE from forecasting with the best integrated quarterly 

model is 36 percent below that from forecasting with the mean reversion method. The 

RMSE from forecasting with the best two-step method is 20 percent below the error from 
41

forecasting with the mean reversion method. Similar improvements occurred for models 

best at forecasting over the 1991-2000 decade. 

 The same span of 30 years provides less opportunity to test nonoverlapping 

forecasts of two or more years ahead. Within those limits, the models developed here 

appear to forecast better than the simple models for two or three years ahead, but not for 

longer periods. An exercise similar to the one conducted here would be necessary to 

determine whether models could be found that would forecast three or more years ahead 

better than the simpler methods. 

 It can be argued that our models forecast 1971-2000 better than the simple 

methods because our models employ more variables and hyperparameters whose 

influences are determined through extensive searches for what works. We agree, but 

hope that the models succeed because they reflect actual economic behavior that will 

continue to influence capital gains in the future. In that case, our models should continue 

to forecast better than the simpler alternatives. 

 So far, only one true year of out-of-sample forecasts is available, and that year 

proved to be an historical outlier. One year is too few to draw conclusions about the 

relative success of our models and the simpler methods. The main lesson from 2001 is 

that none of the models picked up the extent of the decline in capital gains that occurred. 

 Finally, the two-step and the integrated quarterly models were able to estimate 

current year tax-adjusted gains more accurately than a simple Miller-Ozanne equation. 

This result suggests that the additional variables uncovered in our search could be used to 

improve the full Miller-Ozanne equations as used on unadjusted capital gains. The 
42

primary candidates discovered are productivity, the interest-rate spread, real 

compensation per hour, and real fixed private domestic investment. 
43

Table 1: Errors in Forecasts of Growth Rates for Baseline Year (in Percentage Points) 

 

Growth from Year Before Growth from Two Years Before

Date of 

Baseline Actual Forecast Error Actual Est-Frcst Error

Feb-87 -54.7 -47.1 -7.6 -13.7 -31.4 17.7

Feb-88 9.5 -8.3 17.9 -50.4 -59.7 9.3

Jan-89 -5.3 21.2 -26.5 3.8 33.5 -29.7

Jan-90 -19.6 13.4 -33.0 -23.9 57.1 -81.0

Jan-91 -9.8 11.2 -21.0 -27.6 24.6 -52.2

Jan-92 13.5 9.1 4.4 2.4 20.0 -17.6

Jan-93 20.2 10.6 9.6 36.4 35.2 1.3

Jan-94 0.3 10.7 -10.4 20.5 17.4 3.1

Jan-95 17.9 18.7 -0.7 18.3 22.8 -4.5

May-96 44.7 17.3 27.4 70.7 33.4 37.3

Jan-97 39.9 4.6 35.3 102.5 13.9 88.6

Jan-98 24.8 -7.1 31.9 74.6 36.0 38.6

Jan-99 21.4 -5.5 26.9 51.5 6.8 44.7

Jan-00 16.6 -3.9 20.5 41.5 9.2 32.4

 

RMSE 

1987-2000 22.3 42.0 

1992-2000 22.1 39.6 

 

NOTES: 

 Forecasts are typically completed the month before the baseline is released. 

 Forecast from year before is from estimated value in year before. 

 Est-Frcst is estimate of growth from preliminary data two years before baseline to year before, 

 and forecast from year before to year of baseline. 

 Starting in baseline for 1992, forecasts are made using a form of mean reversion. 

 Forecasts for baselines of 1995 and 1996 contain adjustments for anticipated tax changes. 

 Forecast for baseline of 1997 takes no account of tax reduction enacted in 1997. 

 Forecast for baseline of 1998 contains adjustment for responses to 1997 tax reduction. 

 

 
44

Table 2: Equation Used to Construct Tax-Adjusted Capital Gains 

Linear Regression - Estimation by Least Squares 

Dependent Variable DLRATIOFE 

Annual Data From 1949 To 2000 

Usable Observations 52 Degrees of Freedom 47 

Centered R**2 0.747748 R Bar **2 0.726280 

Uncentered R**2 0.750266 T x R**2 39.014 

Mean of Dependent Variable 0.0244196989 

Std Error of Dependent Variable 0.2455856527 

Standard Error of Estimate 0.1284861497 

Sum of Squared Residuals 0.7759084613 

Regression F(4,47) 34.8304 

Significance Level of F 0.00000000 

Durbin-Watson Statistic 2.024232 

 Variable Coefficient Std Error T-Stat Significance 

1. Constant -0.0490061 0.0208595 -2.34934 0.02306045 

2. DMTRNEXT -2.5742500 0.7479459 -3.44176 0.00122346 

3. DMTRTRANS -117.8239911 15.1468711 -7.77877 0.00000000 

4. DLGAP 3.6162592 0.7936502 4.55649 0.00003702 

5. DLSP500Q4 0.8308007 0.1262210 6.58211 0.00000004 

NOTES: 

DLRATIOFE is the change in the logarithm of the ratio of capital gains to potential Gross Domestic 

Product (GDP) 

DMTRNEXT is the change in our measure of the permanent tax rate on capital gains. 

DMTRTRANS is the change in our measure of the transitory tax rate on capital gains. 

DLGAP is the change in the logarithm of the ratio of actual to potential GDP. 

DLSP500Q4 is the change in the logarithm of the average S&P 500 closing price during the fourth quarter 

of each year. 

Equation is estimated with data available as of July 2002. 

Development of equation is described in Preston Miller and Larry Ozanne, Forecasting Capital Gains 

Realizations, Congressional Budget Office, Technical Paper 2000-5, August 2000.
45

Table 3: Explanatory and Auxiliary Variables 

Code Class Description 

X1 Explanatory Output Per Hour in Non-farm Business Sector 

X2 Explanatory Real GDP = Nominal GDP/ GDP deflator 

X3 Explanatory Real PCE on Durable Goods / Real DPI 

X4 Auxiliary 1 Spread, Moody's Seasoned Corp. Bonds (AAA) 

minus TB Rate (3M) 

X5-1 Auxiliary 2 Real Compensation per Hour in Non-farm 

Business Sector 

X5-2 Auxiliary2 Wage and Salary Disbursement (All Industries) / 

GDP 

X6-1 Auxiliary 3 Producer Price index (Crude materials) / GDP 

Deflator 

X6-2 Auxiliary 3 Producer Price index (Crude materials) / PCE 

Deflator 

X6-3 Auxiliary 3 Nominal Effective Exchange rate/PCE Deflator 

X6-4 Auxiliary 3 Real Fixed Private Domestic Investment 

X6-5 Auxiliary 3 Real Gross Private Domestic Investment 

X6-6 Auxiliary 3 Nominal Effective Exchange Rates 

X6-7 Auxiliary 3 Real Private Nonresidential Fixed investment 

X7 Auxiliary 4 S&P 500 Index 
46

 

Table 4: Root Mean Squared Errors in Estimating and Forecasting with Two-Step and 

Comparison Models (in percentage points) 

 RMSE by Decade and Full Period 

Model Type of Error 1971-1980 1981-1990 1991-2000 1971-2000 

 

CBO E1 17.46 12.30 11.94 14.13 

 E2 21.13 13.28 20.29 18.57 

 E3 26.86 22.47 27.34 25.65 

 

RW 

 E1 19.24 15.15 15.49 16.73 

 E2 19.13 15.26 15.69 16.78 

 E3 33.27 24.03 32.54 30.24 

 

AR(1)

 E1 21.51 15.20 17.06 18.12 

 E2 21.29 15.55 17.64 18.31 

 E3 38.28 25.11 36.17 33.68 

 

TS-all

 E1 9.34 11.61 11.7 10.94 

 E2 15.47 15.76 13.03 14.80 

 E3 26.92 15.05 25.14 22.98 

 

TS-90 

 E1 11.17 22.18 8.79 15.21 

 E2 28.41 21.51 12.55 21.81 

 E3 36.91 43.48 21.29 35.15 
47

Table 5: Root Mean Squared Errors in Estimating with Two-Step and 

Integrated Models (E1 Errors in Percentage Points) 

 RMSE by Decade and Full Period 

Model 1971-1980 1981-1990 1991-2000 1971-2000 

 

 Best for 1971-2000 

Annual 

 Two Step 9.34 11.61 11.70 10.94 

 

Quarterly 

 Linear 9.36 11.66 9.56 10.25 

 Economic 9.13 9.13 10.98 10.63 

 In-Model Lin. 10.73 12.22 12.66 11.90 

 In-Model Econ. 11.08 10.36 12.09 11.12 

 

Best for 1991-2000 

Annual 

 Two Step 11.17 22.18 8.79 15.21 

 

Quarterly

 Linear 17.07 15.86 8.90 14.40 

 Economic 19.24 15.23 9.78 15.25 

 In-Model Lin. 19.96 19.96 8.06 16.81 

 In-Model Econ. 11.90 16.27 10.17 13.04 
48

 

 

Table 6: Root Mean Squared Errors in Years 1971-2000 from Forecasting One Year 

Ahead (in percentage points)

 RMSE by Decade and Full Period 

Model Type of 

Error 

1971-1980 1981-1990 1991-2000 1971-2000 

Annual 

 CBO E2 21.13 13.28 20.29 18.57 

 

 Two Step E2 15.47 15.76 13.03 14.80 

 E3 26.92 15.05 25.14 22.98 

 

Quarterly

 

 Linear E2 10.57 12.47 12.62 11.92 

 E3 18.04 19.92 21.01 19.69 

 

 Economic E2 11.39 11.73 15.91 13.17 

 E3 20.14 14.92 28.36 21.85 

 

 In-Model Lin. E2 10.71 11.80 15.33 12.77 

 E3 19.45 20.31 27.79 22.82 

 

 In-Model Econ. E2 10.25 13.00 15.32 13.03 

 E3 19.50 18.60 27.86 22.38 
49

 

Table 7: Root Mean Squared Errors in Years 1991-2000 from Forecasting One Year 

Ahead (in percentage points)

 

 RMSEs by Decade and Full Period 

Model Type of 

Error 

1971-1980 1981-1990 1991-2000 1971-2000 

 

Annual 

 CBO E2 21.13 13.28 20.29 18.57 

 

 Two Step E2 28.41 21.51 12.55 21.81 

 E3 36.91 43.48 21.29 35.15 

 

Quarterly

 

 Linear E2 22.11 18.50 11.46 17.91 

 E3 38.34 32.13 20.26 31.16 

 

 Economic E2 18.85 16.27 13.41 16.33 

 E3 36.35 24.14 24.16 28.79 

 

 In-Model Lin. E2 23.85 17.26 11.72 18.30 

 E3 36.81 34.25 18.62 30.96 

 

 In-Model Econ. E2 16.17 14.97 11.55 15.54 

 E3 24.46 29.84 19.50 24.96 
50

Table 8: Gains Estimates for 2000 and Forecasts for 2001 

 (Billions of dollars)

 

 Estimate 

of 2001 

Forecast 

 of 2001 

Growth Rate 

CBO Baseline 652 652 0.0% 

CBO Reversion Methodology 563 567 0.7% 

 

Models best for 1971-2000 

 Two Step 620 701 13.0% 

 Linear 615 573 -6.9% 

 Economic 635 643 1.2% 

 In-Model Lin. 651 639 -1.9% 

 In Model Econ. 642 640 -0.2% 

 

Models best for 1991-2000 

 Two Step 657 702 6.9% 

 Linear 646 653 1.0% 

 Economic 647 707 9.2% 

 In-Model Lin. 663 699 5.4% 

 In-Model Econ. 598 611 2.2% 

 

Addendum: Actual Values 644 349 -45.8% 
51

Table 9: Root Mean Squared Errors in Multiple Year Forecasts (in percentage points)

 Years Ahead Forecast 

Method 1 2 3 4 5 6 7 8 9 10 

CBO 18.53 31.83 39.54 48.41 60.79 69.55 

RW 16.76 30.69 40.09 51.11 64.55 74.42 80.17 88.94 92.06 84.75 

BVAR 11.92 24.90 37.60 52.42 69.44 82.68 92.36 106.06 120.36 126.17 

NOTES: 

The row titled CBO forecasts with the mean reversion method described earlier. Here, however, forecasts 

of GDP are taken from CBO’s winter baselines. These baselines are usually published in January. The 

first of these baselines is for January 1976, and the last is January 1999. We substituted baseline forecasts 

for 1971-1975 in which GDP was assumed to grow at its average rate for the previous five years. The 

forecasts in the table are limited to horizons of six years because CBO forecasts covered only 6 years until 

the mid-1990s. In addition, the forecasts made before the mid-1990s were of GNP rather than GDP. 

The RMSE on one-year ahead forecasts reported for CBO is shown in the table as 18.53 while the same 

conceptual RMSE reported as E2 on Table 4 and Table 7 is slightly higher at 18.57. The difference arises 

because the mean reversion forecasts in this table are based on historical CBO forecasts of aggregate output 

whereas those in Table 4 and Table 7 are based on forecasts of aggregate output taken from a BVAR. The 

similarity of the RMSEs indicates the similarity of the independent forecasts of aggregate output. 
52

F ig u re 1: A n n u al G ain s an d T ax-A d ju sted G ain s (B illio n s o f d o llars)

0

100200300400500600700

1948

1951

1954

1957

1960

1963

1966

1969

1972

1975

1978

1981

1984

1987

1990

1993

1996

1999
53

Appendix A: Specifications of Forecasting Models 

In this appendix, we first summarize the specifications of the models compared in 

the text and then describe the role of the hyperparameters. 

Each model is identified by i) the set of variables used for current year estimation, 

ii) the set of variables used for one-year-ahead forecasting, and iii) the set of 

hyperparameters used for estimation and/or forecasting. See Table A1. As shown in the 

first column of Table 3 of the main text, variables are divided into seven categories. 

Therefore, we use the following convention for the variable IDs to describe whether a 

variable is used in the estimation or forecasting step: 

1. If the i

th element in a variable ID is 0 , no variable in the ith category is used. 

2. If the i

th element in a variable ID is j, the jth variable in the ith category is 

used. 

For example, a variable ID of [1 1 0 0 2 7 0] corresponds to a model that employs 

[X1 X2 X52 X67] in Table 3. 

The BVAR method used in the present work features a set of inexact prior 

restrictions on the coefficients and the covariance matrix, treating them as random 

quantities with given mean values. The tightness of their distributions around prior 

means is determined by a set of hyperparameters.19 Roughly, the following three prior 

restrictions are imposed: 

 19 See Robertson and Tallman for formal definitions of the hyperparameters. The six 

hyperparameters (μ1, μ2, μ3, μ4, μ5, μ6) described in this appendix correspond to (λ0, λ1, λ4, λ3, λ5, λ6) in 

that paper. 
54

i) The variables in a VAR follow a multivariate random walk with drift. In other words, 

the means of the coefficients on the first and higher-order lags are an identity and zero 

matrices, respectively. 

ii) The VAR in question is expressed entirely in terms of differenced series with an order 

lower by one. In other words, variables are difference stationary without cointegration 

among them. 

iii) The VAR system being considered has a single unit root. In other words, there is 

possible cointegration among variables. 

The three priors can coexist because none needs to be imposed completely. 

 

The first four hyperparameters control the tightness of the prior of random walk 

with drift. Among them, the first hyperparameter, μ1, controls the overall tightness of the 

random walk prior, or equivalently that of the prior on the innovation covariance matrix. 

Increasing μ1 renders less tight the random walk prior as a whole. The second 

hyperparameter, μ2, is the prior standard deviations of the diagonal elements of the 

coefficients on the first lag, reflecting how closely the random walk approximation is 

imposed. As μ2 decreases toward zero, the diagonal elements of the first lag coefficients 

get closer to one and other off-diagonal coefficients to zero. The third hyperparameter, 

μ3, determines the degree by which the intercept term shrinks toward mean of zero, 

controlling its standard deviation. The fourth hyperparameter, μ4, determines the extent 

to which coefficients on the lags beyond the first one are likely to be different from zero. 

As μ4 increases, the coefficients on higher order lags shrinks toward zero more rapidly. 
55

The fifth hyperparameter, μ5, is the weight on a VAR in terms of differenced 

series. As μ5 increases to infinity, the model tends to have as many unit roots as variables 

and no cointegration among them. 

The last hyperparameter, μ6, is the weight on the single unit root prior. As μ6

increases, more and more weight is put on a VAR in which all variables share a single 

stochastic trend and the intercept is close to zero.

Table A1: Model Specifications 

 1971-2000 1991-2000 

Model Variable ID Hyperparameters Variable ID Hyperparameters 

Two Step 

Estimation 1 1 0 1 1 4 0 NA 1 0 1 0 3 1 1 NA 

Forecasting 1 1 1 0 0 4 0 0.7 0.15 0.05 0.5 6 4 0 0 0 1 0 3 1 0.8 0.125 0.125 1.5 3 6 

 

Lin. Interpol. 

Estimation 0 0 0 1 0 4 1 0.8 0.15 0.05 0.5 7 7 0 0 1 0 0 1 1 0.6 0.05 0.075 0.5 4 7 

Forecasting 0 1 1 1 2 6 0 0.8 0.15 0.05 1.25 7 3 0 0 1 0 0 1 1 0.7 0.125 0.05 0.75 3 7 

 

Econ. Interpol. 

Estimation 1 0 0 1 0 4 0 0.8 0.15 0.125 0.5 7 7 0 0 1 0 0 4 0 0.8 0.15 0.05 1.5 7 7 

Forecasting 0 0 1 1 2 7 1 0.8 0.15 0.05 0.75 7 3 0 0 1 0 0 1 1 0.8 0.05 0.1 0.5 3 7 

 

In-Model Lin. 

Estimation 0 0 0 1 0 4 0 0.5 0.075 0.1 1.5 3 7 0 0 1 1 1 1 1 0.5 0.075 0.1 1.5 3 7 

Forecasting 0 0 1 0 0 1 1 0.7 0.15 0.075 1.25 3 7 0 0 1 0 0 1 1 0.7 0.15 0.075 1.25 3 7 

 

In-Model Econ. 

Estimation 0 0 0 1 0 4 0 0.8 0.125 0.05 0.75 7 4 0 0 0 1 1 4 1 0.4 0.075 0.1 1.5 7 7 

Forecasting 0 0 1 1 2 6 1 0.7 0.125 0.1 0.75 7 3 0 0 1 0 0 0 0 0.7 0.15 0.05 0.75 6 3 
56

Appendix B: Annual vs. Quarterly Univariate Models of Gains 

In this appendix, we offer an explanation for the forecasting superiority of a 

random walk model with drift when it is estimated with linearly interpolated quarterly 

gains instead of with actual annual gains. Our explanation is not general: it is thought to 

hold for particular series over particular times. For the gains series it holds over the 

1971-2000 period. However, it holds over neither the 1960s decade nor the period 1961-

2000. Our explanation requires that the forecast errors from the model for annual gains 

be positively serially correlated. We first show with a diagram and simple algebra why 

this condition leads to forecasting superiority of the interpolated gains model and then 

verify that this condition holds for the period 1971-2000. 

 In the diagram, we illustrate the difference in the two univariate models by 

contrasting their forecasts in the gains-time space. In this space, a random-walk-withdrift model is represented by a line with slope equal to the drift and intercept determined 

by the initial observation. 

 X(t) = Drift + X(t-1) 

To simplify the presentation, we refer to X(t) as gains in period t, however, in our actual 

estimates of the models referred to footnote 16 and in the tables, X(t) is measured by the 

logarithm of tax-adjusted gains. To further simplify, we suppose that the drift term for 

the annual model is equal to 1, and that two models have made no errors over the years 

prior to t. 

The annual model is represented in the upper graph by the two lines passing 

through midpoints (i.e., the end of the second quarters) of each year. That is, annual 
57

gains are plotted in the middle of each year, while the values in other quarters should be 

ignored. By assumption, in year t-2 the forecast from the model is accurate; gains are 

forecast to rise to 9 in year t-1 and they do. In year t-1, gains are forecast to rise to 10 in 

year t, and if no shock occurs in year t, the annual model forecasts gains will rise to 11 in 

year t+1, shown on the dashed line labeled “No shock”. 

The quarterly model is represented in the lower chart. The requirement that the 

average of the quarterly gains in a year equal the annual gains implies that the midpoint 

of a line in a year equals the annual value of gains. Since, by assumption, annual gains 

prior to t are linearly related, the quarterly model’s forecasts and actuals also coincide 

through t-1. Also, the midpoint of the “No shock” line in year t is the model’s forecast of 

gains for the year, and the midpoint in year t+1 is the model’s forecast for the following 

year assuming no forecasting error in year t. 

The difference in the two models is seen when there occurs a shock Esp(t)=1 in 

the annual value of gains for year t. Assuming the drift term is unchanged, the annual 

model now starts from the actual value of X(t)=11 at the midpoint of the year t and adds 

drift to generate a revised forecast of 12 in t+1. (The forecast is shown as Xfa(t+1) in the 

diagram.) Thus, for the annual model, the revision in its forecast for X(t+1) equals 

Eps(t), reflected by the upward shift of the line “Annual” from the “no shock” line. For 

the quarterly model, however, the error of Eps(t) must be reflected at the midpoint by the 

average of revisions for the 4 quarters of the year. Since the quarterly values of actual 

gains must lie along a line segment in year t, the revisions in forecasts from quarter 1 to 

quarter 4 must be .4Eps(t), 0.8Eps(t), 1.2Eps(t), and 1.6Eps(t), respectively. Since the 

forecast from this model starts from the 4th quarter of the year t, the line “Quarterly” 
58

shows an upward shift by 1.6Eps(t) from the “No shock” line. Its forecast for year t+1 is 

12.6, labled XFq(t+1) in the diagram. 

For both models, it is not quite right that the estimated drift terms will be 

unaffected by the error in year t. Nevertheless, we would expect the two estimated drift 

terms on an annual basis to move similarly. Moreover, the change in drift caused by one 

error should diminish as the number of observations increases, and the difference in 

forecasts caused by the difference between Eps(t) and 1.6Eps(t) should dominate. 

If the annual model is correctly specified, its errors will be uncorrelated and its 

forecasts should be more accurate than those from the quarterly model: without a further 

shock in year t+1, the quarterly model overpredicts the gain in t+1, while the annual 

forecasts accurately. However, if the annual model’s errors are sufficiently positively 

correlated, the quarterly model can be more accurate: if Eps(t) follows an AR(1) process 

with coefficient of rho, for example, the actual gain in t+1 will be higher than XFa(t+1) 

by rho times Eps(t), and the degree of underprediction by the annual model can outweigh 

that of overprediction by the quarterly model. That can occur in our example when rho is 

closer to 0.6 than to zero. 

We computed the out-of-sample forecast errors from an annual random walkwith-drift model over the years 1971-2000. The variable modeled is the logarithm of taxadjusted gains. We judge the degree of first-order serial correlation by the DurbinWatson statistic. Over this 30-year period, the statistic is 0.7822, which suggests a 

significant degree of positive serial correlation. 
59

Diagram: Forecasting with an Annual Model (Upper Graph) and an Interpolated 

Quarterly Model (lower Graph)
60

Appendix C: Economic Interpolation of Quarterly Gains 

Our initial thought on how to interpolate gains with economic data was to apply 

the Miller-Ozanne equation of Table 2 to predict the growth rate of gains from one 

quarter to the next. Applying the equation quarterly would require the assumption that the 

growth rate of stock prices and the business cycle over a quarter have the same affects on 

the growth of gains in that quarter as their growth over a year has on the growth of gains 

over the year. This assumption seemed unlikely to us. We thought that the growth rate 

of gains in one quarter was likely to depend on changes in stock prices and the business 

cycle over more than one quarter. 

 Consequently, we chose to interpolate gains to quarters by maintaining the fourquarter span of time per observation, and then applying the growth rate for that span to 

the last quarter in the span. This keeps the data used in the equation covering the same 

span of time as that used to estimate the equation, but requires us to arbitrarily assign the 

predicted growth to specific quarters. 

More specifically, we compute both explanatory variables for each span of 4 

quarters ending in the fourth quarter of 1948 through the fourth quarter of 2000. Then we 

apply the equation to predict the growth rate of gains between adjacent periods of four 

quarters. We assign the growth rate between the first and the second four-quarter span to 

the growth between the last quarter of the first span and the last quarter of the second 

span. 

For example, the growth rate of gains from the third quarter of 1962 to the third 

quarter of 1963 is computed by first computing the levels of the business cycle and stock 
61

price variables over the four quarters ending in the third quarter of 1962 and again for the 

four quarters ending in the third quarter of 1963. Next, the change in the log of each 

explanatory variable is computed and used in the equation to compute a growth rate for 

gains. (Tax rates are assumed to be constant so as to obtain a growth rate for tax-adjusted 

gains.) The growth rate computed in this manner technically applies to the growth of 

gains from the first to the second span of four quarters. However, we arbitrarily say it is 

the growth from the third quarter of 1962 to the third quarter of 1963. 

The level of the business cycle variable for the span from the fourth quarter of 

1962 through the third quarter of 1963 is computed by averaging the quarterly values of 

GDP and also of potential GDP and then taking the ratio of average GDP to average 

potential GDP. A similar computation is done for the four quarters ending in the third 

quarter of 1962. The change in the logarithm of the ratios so computed matches the time 

span of the variables in the Miller-Ozanne equation. 

The level of the stock price variable for the four quarters ending in the third 

quarter of 1963 is simply the level for the third quarter of 1963, and that for the four 

quarters ending in the third quarter of 1962 is the level for that last quarter. The reason is 

that the stock price variable in the Miller-Ozanne equation is specified as the change in 

the logarithm of the stock prices from the fourth quarter of one year to the fourth quarter 

of the next. 

The dependent variable of the Miller-Ozanne equation is the change in the 

logarithm of the ratio of gains to potential GDP. Consequently, once the equation has 

been applied to adjacent spans of four quarters, the predicted change in the logarithm of 

the ratio must be converted to the change in the logarithm gains. The conversion is done 
62

by adding to the prediction from the equation the change in the logarithm of potential 

GDP computed over the same interval. The change in the logarithm of a variable is what 

we refer to in this appendix as the growth rate of the variable. 

The process for computing the growth between the third quarters of 1962 and 

1963is repeated for all four-quarter intervals between the fourth quarter of 1948 and the 

fourth quarter of 2000. Every fourth interval corresponds to a calendar year, and so has 

the growth rate computed from the annual data. 

The next step in the interpolation converts the estimated growth rates to quarterly 

values of gains. Conversion is done by distributing the annual tax-adjusted gains for 

1960 evenly among its quarters, and then using the growth rates computed above to get 

the gains in all other quarters. We want quarterly gains to be measured at annualized 

rates, because that is the way several of the macroeconomic variables are measured that 

we will combine with quarterly gains in our integrated models. Consequently, we set the 

gain for each quarter of 1960 to the year’s total tax-adjusted gain. Applying the growth 

rates to those levels makes the predicted quarterly gains for other years measured at anual 

rates as well. 

The year 1960 is chosen because gains were plausibly realized at a constant rate 

throughout that year. Gains in 1960 were at a trough between 1959 and 1961, and growth 

rates of both the business cycle and the S&P were relatively flat during the year’s four 

quarters. Based on this evidence, we distribute annual tax adjusted gains evenly across 

the four quarters. 

Given gains in, say, the first quarter of 1960, gains in the first quarter of 1961 are 

determined by applying the computed growth rate of gains for the first quarter of 1961. 
63

And gains in the first quarter of 1959 are determined by subtracting the computed growth 

rate of gains for the first quarter of 1960 from the level of gains in that quarter. Gains in 

each other quarter of 1961 and 1959 are determined by applying the appropriate growth 

rate to gains in the corresponding quarter of 1960. Once gains are determined for each 

quarter of 1961 and 1959, a similar process determines gains for each quarter of 1962, 

and 1958. Repeating the process for each adjacent year determines gains forward 

through the fourth quarter of 2000 and backward to the first quarter of 1948. 

Finally, the four quarters of gains in each calendar year are raised or lowered 

proportionately so that their average equals the value of tax-adjusted gains for that year. 

This adjustment causes the growth rates between the fourth quarters of one year and the 

first quarter of the next to differ from the rates indicated by the equation. As a result, the 

plot of quarterly gains shows smooth changes from one quarter to the next within a year 

but frequently shows abrupt shifts between the fourth quarter of one year and the first 

quarter of the next year. This pattern creates a saw-tooth pattern in the graph of these 

interpolated quarterly gains (see Figure C1). 

While the above process arrives at a value of tax-adjusted gains for each quarter, 

the process involves strong assumptions. One assumption is that the annual growth rate 

estimated for eight quarters applies to growth to a single quarter from the same quarter in 

the preceding year. No constraint is imposed to make growth for three preceding quarters 

consistent with the annual growth rates estimated for that quarter. Some consistency 

should arise because growth rates for four adjacent quarters use overlapping quarters in 

their measures of growth in business cycles and stock prices. A second assumption is 

that unmeasured factors that cause actual tax-adjusted gains in one calendar year to differ 
64

from the average of the four quarterly gains estimated through the above process affect 

all quarters proportionately. 

In principle, economic interpolation has an unfair advantage over the two other 

interpolation methods used in this paper when it comes to testing models over the 1971-

2000 period. The Miller-Ozanne equation was estimated over the years 1948 to 2000 and 

then used to interpolate quarterly gains in all years from 1948 to 2000. Thus the 

interpolated gains in all years before 2000 reflect future information that would not have 

been available to forecasters at the time. That future information could help 

economically interpolated gains to better reflect the influence of economic activity on 

quarterly gains than could the other methods of interpolation. The better reflection could 

reduce errors in estimating and forecasting gains during the testing years of 1971-2000. 

In practice, however, models using economic interpolation never had the lowest errors for 

estimating or forecasting. Either the future information was not particularly useful or 

other limitations of economic interpolation offset this advantage. 
65

Figure C1: Economic Interpolaton of Quarterly Tax-Adjusted Gains 

(Billions of Dollars)

0

100

200

300

400

500

600

700

800

1948:11950:11952:11954:11956:11958:11960:11962:11964:11966:11968:11970:11972:11974:11976:11978:11980:11982:11984:11986:11988:11990:11992:11994:11996:11998:12000:1
66

Appendix D: Actual and Interpolated Gains 

No information on the timing of gains within a year is available for most years, 

hence the need to interpolate. In a few years, related data are available, and they provide 

limited insight to the accuracy of interpolation. A comparison of quarterly patterns in 5 

years shows a weak tendency for interpolated gains to move in the same direction as 

actual gross gains from start to finish, but no tendency to move with quarter-by-quarter 

changes in actual gross gains. These findings are limited both because of the limited 

number of years with data on timing of gains and because of conceptual difficulties in 

converting the actual annual gains variable into a quarterly version. As a result of these 

limitations, the possibility that interpolated gains provide useful quarterly variation 

cannot be rejected. 

The accuracy of interpolation is important primarily because of its contribution to 

identifying model parameters. If interpolated gains consistently reflect quarterly 

movements in actual gains, then interpolation will improve the estimation of parameters. 

If not, interpolation will add noise that estimation must overcome to arrive useful 

parameters. Even if interpolation adds only noise about movements within a year, 

quarterly models may still forecast better than annual ones if the benefits from using 

other variables on a quarterly basis is large enough. 

Net Positive Gains and Data on Individual Transactions 

Taxpayers enter information on Schedule D and related schedules about each sale 

of capital assets they have conducted during the year. The information includes the data 
67

of sale as well as the amount of gains. In addition to reporting their own sales, taxpayers 

report net gains or losses they receive from “pass-through” entities such as mutual funds, 

partnerships, and trusts. They also include unused losses carried forward from prior 

years. The form directs taxpayers to combine information from all of these sources into a 

net gain or loss for the year. If a taxpayer’s net is a gain, it is entered into adjusted gross 

income (AGI). Actually, prior to 1987, taxpayers were allowed to exclude a portion of 

their long-term gains in excess of their losses before entering the remainder into AGI. 

If the net value from the above computations is a loss, only the amount under the 

loss limit can be included in the current year’s AGI, and the remainder can be carried 

forward to future years. The loss limit was $1,000 between 1955 and 1976, $2,000 in 

1977, and has been $3,000 since 1978. 

 The annual capital gains amount that we are attempting to forecast is the sum of 

net gains across all taxpayers. The long-term gains that taxpayers excluded in years prior 

to 1987 are added back to keep the annual totals consistent over time. Net gains summed 

across all taxpayers is also the figure that we are attempting to interpolate to quarters 

within a year. (Net losses are forecast separately because the limit makes them steadier 

from year to year.) 

 In some years, the Internal Revenue Service publishes greater detail on the 

transactions taxpayers report, including the date of sale. In those years, it is possible to 

group many individual transactions by quarter of the year. The IRS or Treasury 

Department typically prepare tables of transactions data by month of sale, and that data 

has been used here for five years: 1962, 1973, 1985, 1997, and 1998. 

 The difficulties of comparing interpolated net gains to distributions of actual 
68

transactions are both conceptual and practical. A conceptual difficulty is that net gains 

include losses that are predominantly realized at the end of the year. In the five years 

analyzed here, losses in December ranged from 22 percent to 35 percent of all losses, and 

losses for the last quarter averaged around 40 percent of all losses. The concentration in 

the end of the year does not result from big drops in asset prices at the end of each year. 

Instead, it reflects a timing decision by taxpayers to realize losses so they can be counted 

for tax purposes in the current year. Since this is an annual decision, it would be 

misleading to count all losses in the quarter they occur. For this reason, we compare our 

interpolated net gains only to transactions that resulted in a gain. Fortunately these gross 

gains are much larger than gross losses. In 1998, for example, $584 billion of gross gains 

were reported compared to $152 billion of gross losses. 

 The quarterly pattern of gross gains does not show a marked pattern like that of 

gross losses. With only 5 years of data, however, we cannot be sure that a weaker but 

still regular pattern is not present, such as taxpayers selling more gains early in the year. 

If a regular pattern were present, it would tend to show up as an inconsistency between 

gross gains and our interpolated net gains. Such an inconsistency would not reflect a 

shortcoming of our interpolated gains, however, because they are intended to represent 

seasonally adjusted gains. Interpolated gains should be seasonally adjusted because they 

are used in models with other seasonally adjusted quarterly variables. In other words, if 

we find a consistent divergence between gross gains and our interpolated gains, it might 

be due to seasonal patterns rather than an inadequacy in our interpolation methods. 

 Another conceptual difficulty is that net gains include losses carried over from 

prior years. Since these losses occurred in another year, they cannot be assigned to any 
69

particular time within the year they enter net gains. Of course transactions data for the 

year to which losses are carried do not reflect those losses either. For this reason alone, 

interpolated gains cannot be expected to match transactions data on gross gains. 

Fortunately, losses carried over are usually a small component of net gains and hence the 

comparison of interpolated gains and gross gains should not be badly distorted by this 

incompatibility. 

 A practical problem in comparing interpolated net gains and actual gains is that 

taxpayers do not report dates for many transactions. In 1998, the IRS could not identify a 

date for 52 percent of gross gains. In 1985, the IRS could not date 23 percent of gross 

gains. The problem is not entirely lax reporting by taxpayers. Gains taxpayers receive 

from the pass-through entities noted above do not identify the date for taxpayers to 

report. 

 Reporting of dates is more common for stock transactions than for other assets. 

In 1998, when 52 percent of all gains lacked a date, 26 percent of stock transactions 

lacked a date. Stocks also account for a large portion of all gains; in 1998 they accounted 

for 43 percent of all gains including those without dates. As a result, stocks account for 

the preponderant share of gains with dates. For that reason, the distributions by quarter 

of stocks and of all assets look very similar for the three years in which both are 

available: 1985, 1997, and 1998. In 1962 and 1973, only gains on stocks are reported. 

For simplicity, then, only gains on stocks with valid dates are distributed by quarter and 

compared to our interpolated net gains. 
70

Comparisons 

Three types of interpolations are described in the body of the paper. Briefly, one 

is linear interpolation, where net gains are grown linearly from the last quarter of one 

year to the last quarter of the next year at a slope sufficient to make net gains for the four 

quarters of the next year average to the known annual net gain for that year. A second 

interpolation, which we call economic interpolation, is constructed with the equation in 

Table 2 of the paper (see Appendix C for a more complete explanation). The third 

interpolation is done with BVAR models as they are updated from year to year. This 

model interpolation yields many different interpolated net gains variables, depending on 

the other variables and hyperparameters used in the model. For the comparison here, the 

interpolated gains are from the model with a linear seed that had the lowest E2 RMSE 

over the period 1971-2000, as reported in Table 7. 

 The comparisons are summarized in graphs for each year. In 1962, 33 percent of 

gross stock gains with dates were realized in the first quarter. The fraction fell slowly in 

the second quarter, sharply in the third, and then barely at all in the fourth quarter, when 

18 percent of such gains were realized. Two of the interpolation schemes show similar 

total declines, but in nearly equal sized steps. One of these is linear interpolation, which 

is constrained to constant steps, and the other is model interpolation. Economic 

interpolation starts with substantially fewer gains in the first quarter (26 percent), has 

small declines in the next two quarters and then an increase in the fourth quarter to about 

the share of the first quarter. 

 In 1973, gross gains start at 25 percent, jump above in the second quarter, fall 

below in the third, and then return in the fourth. The linear and model interpolations 
71

again get the general direction (they are flat) but they miss the jump and dip. Economic 

interpolation also is essentially flat. 

 In 1985, gross gains decline from 28 percent of gains in the first quarter to 22 

percent in the third before rebounding to 25 percent in the fourth. All three interpolations 

show steady growth from around 22 percent of net gains in the first quarter to about 28 

percent in the fourth. 

 In 1997, gross gains decline slightly in the second quarter, jump to a peak in the 

third quarter, and recede slightly in the fourth quarter, still up from the first quarter. The 

linear and model interpolations again match the annual change in shares, but do so with 

constant step increases. The economic interpolation also rises steadily, but from a lower 

starting point to a higher finish. 

 The pattern for 1998 is similar to that for 1985. Gross gains decline for three 

quarters and rebound slightly in the fourth. All three interpolated gains rise linearly 

throughout the year. 

 Overall, linear and model interpolation match the change in shares between the 

first and fourth quarters 3 out of 5 times. In none of these years does model interpolation 

pick up on the quarter-to-quarter fluctuations of gross gains, and of course linear 

interpolation cannot do so. Economic interpolation is less linear than the other two, but 

its fluctuations are not synchronized with the fluctuations in gross gains. Thus, the 

comparison suggests that linear and model interpolation can catch the trend of gains 

within some years, but that no interpolation catches the quarterly fluctuations of gross 

gains. 

Model interpolation appears to be very similar to linear interpolation, for all of the 
72

machinery used to generate it. Perhaps it is simply reflecting the finding of serial 

correlation of gains among years that by coincidence makes linear interpolation 

successful. In addition, its error reconciliation process should impart a liner pattern 

within a year. 

 The limited number of years for which we have data and the conceptual and 

practical limitations of the comparisons within a year limit the degree of confidence we 

can place in our findings. 

 
73

Figure D1: Distributions of Gross Gains and 3 Interpolated Net Gains by Quarters in 1962

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1962:q1 1962:q2 1962:q3 1962:q4

Gross

Linear

Economic

Model

Figure D2: Distributions of Gross Gains and 3 Interpolated Net Gains by Quarters in 1973

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1973:q1 1973:q2 1973:q3 1973:q4

Gross

Linear

Economic

Model
74

Figure D4: Distributions of Gross Gains and 3 Interpolated Net Gains by Quarters in 1997

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1997:1 1997:2 1997:3 1997:4

Gross

Linear

Economic

Model

Figure D3: Distributions of Gross Gains and 3 Interpolated Net Gains by Quarters in 1985

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1985:1 1985:2 1985:3 1985:4

Gross

Linear

Economic

Model
75

Figure D5: Distributions of Gross Gains and 3 Interpolated Net Gains by Quarters in 1998

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1998:1 1998:2 1998:3 1998:4

Gross

Linear

Economic

Model