Technical Paper Series

Congressional Budget Office

Washington, DC

USING TIME-SERIES MODELS TO PROJECT

OUTPUT OVER THE MEDIUM TERM

Ufuk Demiroglu

Congressional Budget Office

Washington, DC

(Email: ufukd@cbo.gov)

Matthew Salomon

Joint Economic Committee

U.S. Senate

Washington, DC

(Email: Matt_Salomon@jec1.senate.gov)

September 2002

2002-1

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 by sending an email to

techpapers@cbo.gov.

1. CBO’s current procedures for making projections over the medium term are described in Congressional

Budget Office, CBO’s Method for Estimating Potential Output: An Update, CBO Paper (August 2001).

Using Time-Series Models to Project Output

Over the Medium Term

This paper examines how multivariate time-series models might be used to project

output over the medium term—that is, over a 10-year span. Fairly simple time-series

models are known to yield short-term forecasts comparable in accuracy with those

of large-scale macroeconometric models. Could the information embodied in

projections based on time-series models be informative to the Congressional Budget

Office (CBO) in constructing its own medium-term projections of output?1

Based on our examination of several time-series models, we conclude that such

models can help inform the medium-term projections in several respects. Because

time-series models generally place fewer a priori restrictions on the dynamic

properties of the projections than do CBO’s models, an independent time-series

representation might be a useful check on the efficacy of CBO’s standard, more

restrictive, modeling approach. Additionally, prospective forecast uncertainty is

readily computable for each time-series model but not for CBO’s standard

projections. Finally, implicit within the dynamic structure of a time-series model is

a transition path along which the economy moves toward its trend rate of growth.

That transition path can be compared with the transition path in CBO’s medium-term

projections, and the latter might be reassessed if the time-series predictions so

warrant. The models examined here allow ready calculation of measures of trend or

potential output, facilitating such a comparison.

This paper is only an exploratory effort to understand whether and how such
2. All the analyses in this paper make use of the data and CBO baseline available as of February 2000.

3. Congressional Budget Office, Uncertainties in Projecting Budget Surpluses: A Discussion of Data and

Methods, CBO Report (February 2002).

models could be useful for CBO's purposes. Accordingly, the focus is limited to

some basic time-series specifications, which are taken from articles that have been

published in peer-reviewed journals.

The standard errors for the alternative projections of output are estimated to

range from about 4 percent to 8 percent of CBO’s winter 2000 baseline projection for

2010.2 The uncertainty bands around the projection of real output imply a narrower

range of uncertainty for the federal surplus than was recently estimated by CBO in

its examination of errors in its surplus projections.3 However, uncertainty about real

output growth is only one source of uncertainty in the projection of the federal

surplus, which is also affected by uncertainty in the levels of revenues and spending

for given levels of real output.

CBO’s Procedures for Projecting Output Over the Medium Term

In constructing its projections of output over the medium term, CBO assumes that

actual output eventually approximates potential output. Accordingly, CBO’s

projection of actual output growth is built upon two measures: CBO’s historical

estimate and projection of potential output, as well as an assumption as to the rate at

which and the degree to which the gap between actual and potential output narrows

over time.

CBO’s Measure of Potential Output. CBO's estimates and projections of potential

2
4. Total factor productivity (TFP) is defined as that part of nonfarm business output not explained by factor

service flows. CBO assumes that the production function for nonfarm business output depends on the

flows of labor and capital services. Measures of TFP depend critically on the precise functional form

assumed for the production function as well as the many additional assumptions necessary to obtain an

estimate of the flow of capital services. Changes in any of those assumptions would alter the estimate of

TFP.

5. Whether or not there is a stable inverse relationship between the unemployment rate and changes in

inflation—a relationship inspired by the work of A. W. Phillips—has always been a matter of controversy

among economists. In recent years, the controversy has focused on the statistical precision with which the

NAIRU is measured. Some economists have argued that estimates of the NAIRU have been too imprecise

for the NAIRU to be useful to policymakers; see D. Staiger, J. H. Stock, and M. W. Watson, “The

NAIRU, Unemployment, and Monetary Policy,” Journal of Economic Perspectives, vol. 11, no. 1 (Winter

1997), pp. 33-49. Other studies have suggested that the imprecision of NAIRU estimates has been

exaggerated; see Flint Brayton, John M. Roberts, and John C. Williams, What's Happened to the Phillips

Curve? Finance and Economics Discussion Series Paper No. 1999-49 (Washington, D.C.: Federal Reserve

output are based on a model that assumes that growth in potential output equals the

weighted sum of the trend growth in labor and capital services plus the trend growth

in total factor productivity (TFP).4 To measure the trend component of potential

output (and its constituent productive factor services and TFP) in isolation from

cyclical influences, CBO relies on a variant of the well-known and widely used

relationship known as Okun’s law. According to that relationship, actual output

exceeds its potential level when the rate of unemployment is below its “trend” level.

Conversely, when the unemployment rate exceeds its trend level, output falls short

of potential. The unemployment gap—the difference between the trend and actual

rates of unemployment—is the pivotal indicator of the business cycle in models built

upon Okun’s law.

 

CBO uses as the trend level of unemployment a measure known as the

nonaccelerating inflation rate of unemployment, or NAIRU. The NAIRU is a

conceptual measure that builds upon the view that, over the short term, the

unemployment rate varies inversely with changes in inflation. In that view, periods

of high unemployment are associated with falling inflation; conversely, periods of

low unemployment are associated with rising inflation. The NAIRU is defined as

that rate of unemployment that is consistent with a constant rate of inflation.5

3
Board, September 1999). Ongoing independent analyses within CBO have yielded results that are

consistent with the latter findings.

6. The procedures are presented in greater detail in Congressional Budget Office, The Economic and Budget

Outlook: An Update (August 1994), pp. 59-63; and CBO’s Method for Estimating Potential Output: An

Update (August 2001).

1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

12

10

8

6

4

2

0

Figure 1.    Civilian Unemployment Rate and CBO's Estimate of the NAIRU

Civilian Unemployment Rate

NAIRU

Percent

CBO estimates the NAIRU by first estimating a NAIRU for married males.

CBO starts with married males because it has confirmed that demographic shifts,

such as changes in labor force participation and household formation, have distorted

SOURCES: Congressional Budget Office; Department of Labor, Bureau of Labor Statistics.

the relationship between the change in inflation and the overall civilian

unemployment rate.6 The unemployment rate for married males, however, is

relatively free of such distorting demographic shifts. Accordingly, CBO initially

estimates the NAIRU using the unemployment rate for married males. Then, CBO

constructs its estimate of the NAIRU for the entire labor force by taking into account

the demographic changes with respect to age, sex, and race. The resulting estimate

4
7. In recent years, CBO has lowered its estimate of the NAIRU below that implied by this procedure to

accommodate what has become the consensus view—namely, that the NAIRU has fallen at least

temporarily for reasons other than changes in the composition of the labor force with respect to age, sex,

and race. CBO estimates the overall NAIRU to have been about 5.2 percent in recent years.

8. In its estimate of potential output, CBO assumes that the service flow from the aggregate stock of nonfarm

business capital is always at its potential level. This assumption allows CBO’s estimate of the output

gap—though not the projected level of potential output—to be independent of the service flow of capital.

of the overall NAIRU varies slowly over time because of variations in the

composition of the labor force (see Figure 1).7

CBO uses the unemployment gap as a proxy for the business cycle in regression

equations to estimate the potential levels of the labor input and TFP. Each regression

equation breaks down the natural logarithm of the respective independent variable

(labor input or TFP) into a deterministic time trend, a cyclical component

proportional to the unemployment gap, and a random residual. Once the equations

have been estimated over the historical period, CBO identifies the estimated

deterministic trends as the so-called potential levels of labor input or TFP.

Those estimated potential levels of labor input and TFP are combined with

CBO’s estimates of the service flow of capital to yield an estimate of the level of

potential output for the nonfarm business sector.8 CBO assumes that the production

function for the nonfarm business sector has a Cobb-Douglas form, with the elasticity

of output with respect to changes in labor input equal to 0.7. Under those

assumptions, a proportionate change in the level of potential output equals the

proportionate change in potential TFP plus 70 percent of the proportionate change

in labor input flows plus 30 percent of the proportionate change in the service flow

from capital.

By construction, the proportionate gap between actual and potential output is

5
equal to the TFP gap plus 70 percent of the labor input gap. Because the TFP and

labor gaps vary directly with the unemployment gap, the output gap does as well (see

Figure 2).

Transition to the Medium Term. Given the levels of historical and projected

potential output, CBO computes a projected path for actual nonfarm business output

by "closing" the output gap at some point during the projection period. Once the gap

has been completely closed, the projected level of output grows at the same rate as

the potential level of output.

Both the degree to which the gap is closed and the pace at which it closes are

determined through assumptions made by CBO. Those assumptions have varied

throughout CBO’s history. Until recently, CBO assumed that the gap would

eventually reach and remain at its historical average of between 0.2 percentage points

and 0.4 percentage points of output, after which output was projected to grow at the

same rate as potential. Currently, however, the projection period is extended to 10

years, and CBO assumes that the gap goes to zero at least two years before the end

of the projection period and stays there. Of course, the growth rate of actual output

depends on the most recent historical estimate of the gap—if actual GDP exceeds

potential GDP, the growth rate of actual output must be less than the projected

growth rate of potential output for the gap to close.

Some Limitations of CBO’s Procedures. CBO’s approach to measuring and

projecting potential output is a version of a widely used structural model that allows

the influence of exogenous factors (such as demographic shifts) to be easily

incorporated in estimates of potential output, directly via changes in the estimate of

6
1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

12

9

6

3

0

-3

-6

-9

Figure 2.        CBO's Estimate of the Output and Unemployment Gaps

Output Gap

Unemployment Gap

Percent

labor input and indirectly via changes in the NAIRU. However, the advantages of

CBO’s approach come at the cost of some limitations. 

SOURCES: Congressional Budget Office; Department of Labor, Bureau of Labor Statistics; Department

of Commerce, Bureau of Economic Analysis.

NOTE: The output gap is measured as the difference between actual and potential GDP as a

percentage of potential GDP. The unemployment gap is measured as the difference between

the estimated NAIRU and the civilian unemployment rate.

First, the difference between the unemployment rate and the NAIRU may not be

the best method of identifying cycles in output. CBO’s standard approach to

estimating potential GDP assumes the Phillips-Okun structure. Thus, it could fail if

that structure turns out not to be a good description of reality. 

Moreover, CBO’s approach to estimating the output gap does not shed any light

on how quickly or slowly the gap tends to return to its long-run level over the

medium term. As a result, CBO has to make assumptions regarding the pace at

7
which the gap closes over the projection horizon. Although those assumptions are

informed by factors such as the most recent historical estimates of the gap and the

trend reversion patterns exhibited in the past, the determination of the gap remains

exogenous to the statistical models that are used to estimate and project potential

output.

Finally, CBO’s method depends on possibly imprecise estimates of the capital

stock. Although the use of measures of the capital stock to estimate potential output

certainly has a strong theoretical basis, such an approach comes at the cost of

possible measurement error. Capital stocks must be estimated and then linked to

CBO’s projections of business investment, requiring additional steps that introduce

the possibility of error into the process of measuring potential output for the nonfarm

business sector. For instance, conventional measures of the capital stock appear to

overstate short-run movements in the capital input.

Specification and Estimation of the Multivariate Statistical Models

In principle, multivariate time-series models offer a way around some of the

limitations of CBO’s standard procedures for projecting output over the medium

term. This section describes the three statistical models chosen for this study, the

estimation issues that arose in fitting the models to post-World War II data, and the

techniques used to isolate trend components from cyclical components in each

model’s projections.

Three Models. The three alternatives examined here are variants of models that

have appeared in leading economics journals over the past decade or so. Two of the

8
9. Olivier Blanchard and Danny Quah, “The Dynamic Effects of Aggregate Demand and Supply

Disturbances,” American Economic Review, vol. 79, no. 4 (September 1989), pp. 655-673.

10. Jordi Gali, “Technology, Employment, and the Business Cycle: Do Technology Shocks Explain Aggregate

Fluctuations?” American Economic Review, vol. 89, no. 1 (March 1999), pp. 249-271.

11. John H. Cochrane, “Permanent and Transitory Components of GNP and Stock Prices,” Quarterly Journal

of Economics, vol. 109, no. 1 (February 1994), pp. 241-265.

models are bivariate vector autoregressions (VARs), and the other is a bivariate

vector error-correction model (VECM). A VAR is a linear multivariate stochastic

model in which each variable is affected by the past and contemporaneous values of

all of the other variables in the model as well as its own past, in addition to random

influences. VECMs are an extension of VARs in which long-run relationships

among variables are explicitly built in.

• The first model is based on a seminal empirical study by Blanchard and Quah

(BQ).9 It is a dynamic version of the Okun relationship, which CBO’s method

also relies on. This model may be viewed as a VAR benchmark to CBO’s

current procedures.

• The second model is taken from a recent study by Gali.10 Unlike the

neoclassical growth-accounting framework that CBO’s current methodology

uses, this model combines TFP and the capital input into a labor productivity

variable, circumventing the need to estimate the capital stock.

• The third model, which is based on work by Cochrane, hinges on a completely

different economic theory, namely that the level of consumption is an indicator

of expected future income.11

The BQ model estimates a bivariate VAR relationship for the growth rate of real

9
12. In their original study, Blanchard and Quah used real GNP along with the unemployment rate for males

25 years and older. At the expense of a shorter sample, we’ve chosen to use the unemployment rate for

married males because it is the unemployment measure used in CBO’s estimates of potential GDP.

Additionally, we used BEA’s chained-Fisher indexes of GDP, which were not available to Blanchard and

Quah in 1989.

13. The sum of the two growth rates equals growth in real GDP for the nonfarm business sector.

GDP and the unemployment rate for married males.12 As Blanchard and Quah

originally pointed out in their 1989 study, this specification is a generalized variant

of the basic Okun relationship, which allows, potentially, for a richer range of

dynamics than does the contemporaneous Okun relationship.

Gali’s bivariate VAR uses growth in labor productivity for the nonfarm business

sector along with growth in labor hours for that sector.13 This model also allows for

the comovement of the labor input and labor productivity as indicated by the Okun

relationship.

Cochrane’s VECM imposes a cointegration relationship between the logarithms

of consumption and real GDP. The model is motivated by the permanent income

hypothesis, which postulates that households determine their consumption in relation

to their expectations of average lifetime income. One implication of that hypothesis

is that changes in income that are not accompanied by changes in consumption must

be temporary. This suggests that a given change in income may be identified as

temporary or permanent by determining the degree to which consumption also

changes. 

Specification Issues. There were two major specification issues with the models:

how the data would be transformed to make them stationary and the number of lags

to use on the right-hand side. The assumptions regarding stationarity and lag length

10
14. S. Beveridge and C. R. Nelson, “A New Approach to Decomposition of Economic Time Series Into

Permanent and Transitory Components with Particular Attention to Measurement of the ‘Business Cycle’,”

Journal of Monetary Economics, vol. 7, no. 2 (March 1981), pp. 151-174. A multivariate version of the

Beveridge-Nelson decomposition is presented in G. Evans and L. Reichlin, “Information, Forecasts, and

Measurement of the Business Cycle,” Journal of Monetary Economics, vol. 33, no. 2 (April 1994), pp.

233-254.

bear directly on the ways in which the trend, or permanent, component of output is

isolated. Although a trend-cycle decomposition is not necessary to obtain forecasts,

some defensible measure of the trend is necessary to compare the implications of the

alternative models for the estimated output gap. For the baseline specifications in

this study, we adopted the stationarity transformations and lag lengths of the original

studies.

For each of the alternative models, we estimated a measure of trend output using

a procedure first developed by Beveridge and Nelson (BN).14 Put simply, the BN

decomposition identifies the trend component of output at any point by subtracting

from output the part that the model estimates to be transient. 

In the case of our estimate of the Blanchard-Quah model, we additionally

computed another measure of trend output using the same long-run identifying

restrictions as were introduced in the original BQ study. The basic intuition

underlying those restrictions is that short-term or transitory shocks should have no

permanent effects on output growth. (Implementation of those restrictions and other

technical details on the decomposition methods are discussed in Appendix A.)

Finding a Stationary Series. The decomposition methods used here require output

to be a difference-stationary time series. Determining whether this is truly the case,

however, is a dicey matter in practice: numerous analysts over the past two decades

have examined the question for real GDP and no decisive conclusion is at hand. Part

11
15. See M. W. Watson, “Univariate Detrending with Stochastic Trends,” Journal of Monetary Economics,

vol. 18, no. 1 (July 1986), pp. 49-75.

of the reason for this uncertainty is that, using the available finite data on real GDP,

it is effectively impossible to distinguish the hypothesis of a difference-stationary

series from one that has an autoregressive root very close to unity. This

observational equivalence may be of little consequence in short-run

forecasting—indeed, it appears that these alternatives tend to produce comparable

short-run forecasts—but the implications for the very long term are radically

different.15

These difficulties are illustrated with the macroeconomic time series to be used

in the study. A first criterion for stationarity is whether the sample autocorrelation

function approaches zero sufficiently quickly for sufficiently long lags. Table 1

presents the sample autocorrelations for a number of time series, each of which is

subjected to several alternative transformations. 

Except for the unemployment rate for married males, the autocorrelations for the

untransformed series show very little dampening—all of the autocorrelations for the

logarithms or levels of the series appear to be significantly different from zero for the

first 10 quarters. When the series are differenced, however, the sample

autocorrelations for all of the series dampen very quickly—in most cases, they

approach zero after a couple of quarters.

The autocorrelations also tend to dampen when their time trends are removed

from the series. If, in addition to a break in the linear trend, a break is allowed in the

trend rate of real GDP growth, the autocorrelations dampen even more quickly. But

12
Table 1. The Autocorrelations of Selected Macroeconomic Time Series Under Alternative

Transformations

Autocorrelations by Number of Quarters Lagged

1 2 3 4 5 6 7 8 9 10

Real GDP

Level (natural logs) 0.98 0.97 0.95 0.94 0.93 0.91 0.90 0.88 0.87 0.85 

Differenced 0.33 0.18 -0.01 -0.12 -0.18 -0.11 -0.10 -0.05 0.03 0.07 

Deterministic Trend 0.96 0.90 0.84 0.77 0.71 0.66 0.61 0.57 0.53 0.49 

Deterministic Trend with Break 0.93 0.82 0.68 0.55 0.43 0.34 0.26 0.20 0.15 0.09 

Real GDP, Nonfarm Business

Level (natural logs) 0.98 0.97 0.95 0.94 0.92 0.91 0.89 0.88 0.86 0.85 

Differenced 0.24 0.16 -0.03 -0.11 -0.22 -0.11 -0.09 -0.09 0.08 0.07 

Deterministic Trend 0.94 0.85 0.75 0.65 0.57 0.50 0.45 0.41 0.37 0.33 

Deterministic Trend with Break 0.91 0.79 0.64 0.49 0.37 0.28 0.21 0.15 0.11 0.06 

Unemployment Rate, Married Males

Level (percent) 0.95 0.86 0.75 0.65 0.56 0.48 0.41 0.35 0.30 0.26

Differenced 0.57 0.19 -0.05 -0.21 -0.14 -0.05 -0.12 -0.15 -0.08 0.00

Deterministic Trend 0.95 0.85 0.74 0.63 0.53 0.45 0.38 0.31 0.26 0.22

Deterministic Trend with Break 0.95 0.85 0.74 0.63 0.53 0.45 0.38 0.32 0.27 0.22

Labor Productivity, Nonfarm Business

Level (natural logs) 0.98 0.97 0.95 0.94 0.92 0.91 0.89 0.88 0.86 0.85 

Differenced -0.14 0.14 -0.06 0.01 -0.10 -0.00 -0.07 0.04 0.15 -0.01 

Deterministic Trend 0.97 0.94 0.92 0.89 0.86 0.84 0.82 0.79 0.77 0.75 

Deterministic Trend with Break 0.85 0.74 0.61 0.50 0.39 0.32 0.26 0.22 0.19 0.13 

Labor Services, Nonfarm Business

Level (natural logs) 0.99 0.97 0.95 0.94 0.92 0.91 0.89 0.88 0.86 0.84 

Differenced 0.59 0.24 0.04 -0.15 -0.26 -0.20 -0.15 -0.18 -0.09 0.03 

Deterministic Trend 0.95 0.86 0.75 0.63 0.53 0.44 0.37 0.33 0.30 0.28 

Deterministic Trend with Break 0.93 0.79 0.62 0.44 0.28 0.15 0.06 -0.01 -0.05 -0.08 

Consumption of Real Nondurables

Level (natural logs) 0.99 0.97 0.96 0.95 0.93 0.92 0.90 0.89 0.88 0.86 

Differenced 0.13 0.11 0.14 -0.02 -0.05 0.00 0.10 -0.15 0.02 0.01 

Deterministic Trend 0.98 0.96 0.94 0.91 0.88 0.85 0.82 0.79 0.76 0.74 

Deterministic Trend with Break 0.92 0.82 0.74 0.64 0.55 0.47 0.39 0.31 0.25 0.20 

SOURCE: Authors’ calculations using data from the Department of Commerce, Bureau of Economic Analysis; and the

Department of Labor, Bureau of Labor Statistics.

the rate of dampening is still much slower than in the case of differenced output.

Running a battery of augmented Dickey-Fuller tests on the data suggests that,

with the exception of the unemployment rate, most of the series can adequately be

characterized by unit roots. However, those tests are not powerful enough to

13
16. Pierre Perron, “The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis,” Econometrica, vol.

57, no. 6 (November 1989), pp. 1361-1401.

17. N. Gregory Mankiw and Matthew D. Shapiro, “Do We Reject Too Often? Small Sample Properties of Tests

of Rational Expectations Models,” Economics Letters, vol. 20 (1986), pp. 139-145.

18. All computations of the output gaps in this report, however, reflect the basic specifications used in the

respective published works.

distinguish unit roots from mean-reversion, especially in the presence of breaks in

trend.16 Additional testing for unit roots after allowing for a break in trend in 1974

did result in mild rejections of the null hypothesis of a unit root in several cases. But

those results are not completely reliable, either, because of size distortion problems

that bias the results toward rejecting the null.17 Those problems make it difficult to

identify what the right form of stationarity transformations should be. 

For the purposes of this study, we adopted as baseline specifications the same

stationarity transformations as were used by the authors of the original studies. We

assumed that all variables, except for the unemployment rate, were stationary in first

differences of their natural logarithms (that is, growth rates). As a check on the

robustness of the basic specification, however, we also examined alternative

stationarity transformations.18

Choosing a Lag Length. The issue of lag length is critical. The efficacies of both of

the trend-cycle decompositions used in this study (BN or BQ) rely on the quality of

the empirical approximation to the long-run properties of the time-series model.

These, in turn, are sensitive to the chosen lag length. Ideally, one would like to

choose a lag length that is long enough to ensure the accuracy of the approximation

but not so long as to chew up too many degrees of freedom.

Established criteria are available to determine what the lag length should be on

14
19. See Alain DeSerres and Alain Guay, Selection of the Truncation Lag in Structural VARs (or VECMs) with

Long-Run Restrictions, Working Paper No. 95-9 (Ottawa, Canada: Bank of Canada, October 1995).

20. A lag length of eight quarters is reported for the Blanchard-Quah and Gali VARs. Cochrane does not

report a lag length in his published study; we assumed a lag length of two quarters for the basic

specification.

21. When VARs are used for forecasting, it is not actually necessary to identify the gap or make a separate

assumption about how quickly it closes. Estimating the model and simulating the reduced form regression

are sufficient to obtain forecasts. Nevertheless, the models’ estimates for the trend rate of growth and the

present deviation from the trend are worth looking into because they provide a convenient framework with

which to understand and compare forecasts of different models. They also provide a parallel to the CBO

method and therefore facilitate an understanding of how a comparison can be made with the various

components that form the CBO forecast. As discussed in Section II, CBO has historically used estimates

of the rate of growth of potential output and the output gap to guide its projections of growth in real GDP

over the medium term, by assuming that the output gap closes at a given rate over the projection period.

the basis of those considerations. It turns out that the lag lengths of the original

specifications of the three models exceeded those selected by the various information

criteria. However, a recent study has shown that the lag orders favored by those

criteria tend to be too low, resulting in inconsistent estimates of a model’s long-run

coefficients.19 That inconsistency points to the difficulty associated with determining

the appropriate lag length. On balance, we did not find a sufficiently strong case in

favor of lag orders different from the ones used in the original studies.20

Analysis of the Model Forecasts

The models are estimated using quarterly postwar data through the end of 1999 (as

published in February 2000). Those estimated models are then used to forecast real

GDP through 2010. The comparisons are made with CBO’s winter 2000 estimates

and forecasts. The range of forecasts for real GDP is substantial, rising to nearly 8

percent of the midpoint forecast by 2010. 

The analysis below breaks down the forecast differences into three components:

differences in the estimated trend rate of growth, the most recent historical value of

the output gap, and the speed of decay of the gaps through the projection period.21

15
The differences in the estimates of the output gap’s level at the end of the sample

period account for about half of the differences in output forecasts for 2010. As the

estimated gap widens, the part of current output taken to be cyclical and temporary

increases and the predictions of future growth decrease. The models vary in their

estimates of the output gap mainly because they rely on different economic principles

and choose different economic variables. The other half of the differences in 2010

forecasts is due to differences in the estimated trend rate of growth of GDP, which,

in turn, critically depend on whether or not a break in the trend in 1974 is assumed.

The speed with which the gap closes affects the short-term forecast but has

essentially no effect on 2010 forecasts.

Forecast Results. The Blanchard-Quah VAR projected real GDP to grow at an

average annual rate of 2.9 percent between 2000 and 2010. The Gali VAR projected

a 3.6 percent per year advance over the period, and the Cochrane VECM projected

growth of 3.3 percent per year, on average. The range of forecasts is substantial,

rising from just under 1 percent of the midpoint forecast in 2000 to nearly 8 percent

of the midpoint forecast by 2010. In that year, the difference between the largest and

smallest projected level of real GDP is a trillion chained-1996 dollars—those two

forecasts would yield projections of federal revenues that differed by over $450

billion in 2010, more than CBO’s current projection of the federal surplus in that

year.

Accounting for Differences in the Forecasts. More than half of the 8.3 percent

difference between the Gali and Blanchard-Quah forecasts stems from the difference

in their estimates of the gap at the end of the sample period; the remainder arises

16
22. This ignores the fact that the estimate of the output gap is not independent of the estimate of the trend rate

of growth.

23. This accumulates to a difference of approximately 3.6 percent of the level of real GDP by 2010 (see Table

1). 1.036*1.045 = 1.0826, so the 4.5 and 3.6 percentage points fully account for the 8.3 percent difference.

Note that all percentages are rounded to the nearest first decimal in the tables.

from the difference in their estimates of the trend rate of growth (see Table 2).22 The

gap at the end of the sample period in the BQ VAR is 4.5 percent higher than in the

Gali VAR. Because the gap disappears over time, that difference reduces the BQ

VAR’s projection of output growth by 4.5 percent between 1999 and 2010. The

remaining difference between the Gali VAR and the BQ VAR in the 2010 forecasts

is due to the faster (by 0.3 percent per year) trend growth rate estimated by the BQ

VAR.23

Table 2. Reasons for Differences Among the Output Forecasts for 2010 

(Percentage of 2010 GDP as estimated by the lower forecast)

Components of the Total Difference Due to

Vector Autoregressions 2010

Forecast Estimates of the Output

Gap in 1999:4

Estimates of the Trend

Growth Rate

Gali VAR vs. BQ VAR 8.3 4.5 3.6

Cochrane VECM vs. BQ VAR 5.0 2.4 2.6

Gali VAR vs. Cochrane VECM 3.1 2.0 1.0

SOURCE: Authors’ calculations using data from the Department of Commerce, Bureau of Economic Analysis; and the

Department of Labor, Bureau of Labor Statistics.

The 2010 forecast difference between the BQ VAR and the Cochrane VECM

is smaller. It is nearly equally accounted for by the difference in estimates of the

output gap at the end of the sample period and estimates of the trend rate of growth.

As for the forecast differences of the third pair (the Cochrane VECM and the Gali

17
VAR), the forecast difference is small and arises mostly from the difference in their

estimates of the output gap.

Trend Rate of Growth. The trend rates of growth range from 3.2 percent to 3.5

percent. That range is quite large considering the effects of such differences in

average growth carried out over 10 years. The range reflects the different

assumptions about a trend break in 1974. If a trend break is not assumed, as in the

Gali and Cochrane models, the estimate of the trend rate of growth is approximately

equal to the average growth rate of output in the estimation sample, which is 3.4

percent. If a trend break in 1974 is assumed, as in the Blanchard-Quah VAR, the

average real output growth for the post-1973 sample (as opposed to the full sample)

determines the estimate of the trend rate of growth. Because the growth rate of

output in the years after 1973 is substantially lower (3.2 percent) than it was prior to

1974, the trend break assumption results in a lower estimate of the trend rate of

growth. Of the three alternatives, the estimate of the trend component in the

Blanchard-Quah VAR was closest to CBO's estimate, which is not surprising given

the fact that CBO’s approach also assumes trend breaks.

Output Gap. The Beveridge-Nelson decomposition was used to estimate the output

gap for all three models. In the Blanchard-Quah VAR, the BQ decomposition was

used as well; the estimate of the output gap produced under that decomposition was

similar to the gap produced by the BN decomposition for that model. The estimated

gap shows some striking differences across models for recent years. The three

models estimate the gap at the end of the sample period (the last quarter of 1999) as

follows:

18
24. The 4.2 percent is the estimate obtained by using the BN decomposition. When long-run restrictions are

used to identify the trend, the estimate becomes 3.8 percent.

25. This can be seen from the impulse response functions for this model (Figure 3, lower panel).

BQ VAR: 4.2 percent24 (the largest boom since the 1960s)

Gali VAR: -0.2 percent (economy below potential)

Cochrane VECM: 1.8 percent (a mild boom)

By contrast, CBO’s estimate of the output gap for that quarter at the time was

approximately 3 percent, which lies midway between the BQ VAR and the Cochrane

VECM.

The differences in the gap estimates of the Blanchard-Quah VAR and the

Cochrane VECM arise from the fact that they are based on very different economic

principles. Because the BQ VAR perceives a concurrent decrease in unemployment

and increase in output as a temporary shock, it interprets the activity in the last years

of the sample period as a long and large cyclical boom.25 The cyclical increase in the

VECM gap model is not as large as that in the BQ VAR because the VECM model

estimates potential output as approximately proportional to consumption. The

increases in output in the boom of the late 1990s have at least partly been

accompanied by increases in consumption; thus, the VECM model infers that part of

the increase in output is due to an increase in potential output.

Not surprisingly, the BQ VAR’s estimate of the output gap at the end of 1999

is closest to CBO's estimates. After 1960, the output gap implicit in the BQ VAR

follows the cycle turning points of the National Bureau of Economic Research

(NBER) fairly closely; the exception is the recession of 1970 (see Figure 4). In most

19
00 05 10 15 20 25 30 35 40

1.2

0.9

0.6

0.3

0.0

-0.3

-0.6

Figure 3.        Cumulative Impulse Responses in the BQ VAR

                        Under Blanchard-Quah Restrictions

Cumulative Responses to a Permanent Shock Percent

Real GDP

Unemployment, Married Males

Quarters After Shock

00 05 10 15 20 25 30 35 40

1.2

0.9

0.6

0.3

0.0

-0.3

-0.6

Cumulative Responses to a Temporary Shock Percent

Real GDP

Unemployment, Married Males

Quarters After Shock

SOURCE: Authors’ calculations using data from the Department of Commerce, Bureau of Economic Analysis; and the

Department of Labor, Bureau of Labor Statistics.

 respects, until the early 1990s, the measure of the output gap from the BQ VAR

mirrors CBO's fairly consistently. After 1994, however, the BQ VAR estimates the

20
1960 1965 1970 1975 1980 1985 1990 1995 2000

12

9

6

3

0

-3

-6

-9

Figure 4.        Comparison of Output Gap Estimates:

                        CBO vs. the Blanchard-Quah VAR Percent

Beveridge-Nelson Procedure

Blanchard-Quah Procedure

CBO Procedure

1960 1965 1970 1975 1980 1985 1990 1995 2000

3

2

1

0

-1

-2

-3

-4

-5

Figure 5.        Comparison of Unemployment Gap Estimates:

                        CBO vs. the Blanchard-Quah VAR

Blanchard-Quah Procedure

Beveridge-Nelson Procedure

CBO Procedure

Percent

SOURCE: Authors’ calculations using data from the Department of Commerce, Bureau of Economic Analysis;

and the Department of Labor, Bureau of Labor Statistics.

SOURCE: Authors’ calculations using data from the Department of Commerce, Bureau of Economic Analysis;

and the Department of Labor, Bureau of Labor Statistics.

NOTE: The married-male unemployment gap is measured as the difference between the estimated trend and

the actual unemployment rate for married males.

21
26. If the VAR is estimated without removing a linear trend from unemployment, the gap is estimated to be

0.8 percent smaller. Removing the trend, therefore, can account for a substantial part of the 1.2 percent

difference between the gap estimates of CBO and Blanchard-Quah VAR for the end of the sample period.

(In fact, if the long-run restrictions are used to identify the trend in BQ VAR, the difference is 0.8 percent

to start with—CBO’s estimate of 3.0 percent versus the BQ VAR’s estimate of 3.8 percent—and in that

case the treatment of the unemployment trend can explain all the difference.)

1960 1965 1970 1975 1980 1985 1990 1995 2000

4.8

4.4

4.0

3.6

3.2

2.8

2.4

Figure 6.        Comparison of Estimates of the Trend Rate of Unemployment

                        for Married Males: CBO vs. the Blanchard-Quah VAR

Percent

CBO's Estimate

(NAIRU = 3.44%)

Blanchard-Quah Procedure

Beveridge-Nelson

Procedure

3.44

SOURCE: Authors’ calculations using data from the Department of Commerce, Bureau of Economic Analysis;

and the Department of Labor, Bureau of Labor Statistics.

trend growth rate somewhat below the growth rate of potential estimated by CBO.

As a result, the BQ VAR indicates a substantially larger gap than CBO does,

especially between 1995 and 1998, although the gap narrows somewhat after that.

A related observation is that the BQ VAR estimates a wider gap between the trend

and actual rates of unemployment for married males (see Figure 5). 

Part of the reason for the divergence of the output and unemployment gaps

between the BQ VAR and the CBO estimates is that the VAR has built into it a rising

trend of unemployment for married males, in contrast with CBO's assumption of a

constant married-male NAIRU (see Figure 6).26 Another reason is that CBO’s

22
27. The measure of the output gap was estimated by using the Beveridge-Nelson decomposition to calculate

the permanent component of the two variables of this VAR, which are the growth rates of labor

productivity and hours in nonfarm business, and then adding together those two rates to estimate the

growth rate of the permanent component of nonfarm business GDP. To compute the trend growth rate in

overall GDP from the estimate in the nonfarm business sector, the ratio of the two was set to grow at the

same rate as it does in CBO's projection. The underlying gaps for the two variables appear to move almost

in lockstep throughout the historical period (see Figure 8).

1960 1965 1970 1975 1980 1985 1990 1995 2000

12

9

6

3

0

-3

-6

-9

Figure 7.        Comparison of Output Gap Estimates:

                        CBO vs. the Gali VAR

CBO Procedure

Gali VAR

Percent

method, unlike the time-series methods, incorporates information on demographic

shifts on top of the information contained in past trends.

SOURCE: Authors’ calculations using data from the Department of Commerce, Bureau of Economic Analysis;

and the Department of Labor, Bureau of Labor Statistics.

NOTE: The output gap is measured as the difference between actual and potential GDP as a percentage of

potential GDP.

The Gali VAR produces a picture of the output gap that is very different from

all of the other models (see Figure 7).27 Although the estimated gap demonstrates

some propensity to follow the cycle, it is unusual in other respects. The movements

in this measure of the output gap indicate that the economy was operating below its

potential at the end of 1999—an assessment with which most public and private

23
1960 1965 1970 1975 1980 1985 1990 1995 2000

4

3

2

1

0

-1

-2

-3

Figure 8.        Measures of the Hours Gap and Labor Productivity Gap

                        Consistent with the Gali VAR Percent

Labor

Productivity

Gap

Hours Gap

forecasters would disagree. Moreover, the Gali VAR judges the 1990-1991 recession

as more severe than the back-to-back recessions of 1980 and 1982.

The measure of the output gap based on the Cochrane VECM shows a more

subdued cyclical tendency over history, though one that tracks the NBER’s turning

points fairly well (see Figure 9). This is not surprising given that the estimate of

potential output tracks consumption closely in this model and consumption is

cyclical. Consequently, the estimated potential output tends to be low when output

is low and vice versa. In other words, this model’s estimate of potential output is

cyclical—not a desirable quality for a potential output estimate. The apparent lack

SOURCE: Authors’ calculations using data from the Department of Commerce, Bureau of Economic Analysis;

and the Department of Labor, Bureau of Labor Statistics.

NOTE: The hours gap and productivity gap are measured as the difference between actual hours worked and

productivity, respectively, as a percentage of their trend levels.

24
1960 1965 1970 1975 1980 1985 1990 1995 2000

12

9

6

3

0

-3

-6

-9

Figure 9.        Comparison of Output Gap Estimates:

                        CBO vs. the Cochrane VECM Percent

CBO Procedure

Cochrane VECM

SOURCE: Authors’ calculations using data from the Department of Commerce, Bureau of Economic Analysis.

NOTE: The output gap is the difference between actual and potential GDP as a percentage of potential GDP.

of severity of the early 1980s downturns (measured by the size of the output gap) can

also be explained in the same way. In those years, consumption fell along with

output, which the model interprets as a decline in potential output, making the gap

smaller than that suggested by the other models.

Rates of Gap Closure. The speed at which the gap disappears is important for the

shorter-term forecasts, especially two- or three-year horizons, but not for the longerterm forecasts, when the gaps are closed. The “half-life” of the output gap—defined

as the number of quarters it takes for the output gap to be half closed from its most

recent historical value—varies between five and eight quarters in the three models

(see Table 3). The lag order appears to be an important determinant of the speed of

decay. (The lag length in the Cochrane VECM is two quarters as compared with

25
28. Note that the term “half-life” is typically used for processes with smooth exponential decay, but the decay

implied by a VAR is generally not smooth. Table 3 therefore reports the number of quarters it takes for the

output gap to lose 25 percent, 50 percent, 75 percent, and 90 percent of its most recent value.

29. Unlike the out-of-sample forecasts, real-time forecasts make use of the data available at the time the

forecasts were made without the benefit of later revisions. CBO’s projections of growth in real output are

presented in Congressional Budget Office, CBO’s Economic Forecasting Record (January 2002), available

at www.cbo.gov.

eight quarters in the other two models.) Although this does not have to be the case

in general, the model that is specified with a shorter lag length tends to revert to its

trend more rapidly.28

Table 3. Estimates of the Growth Rate of Potential Output and the Output Gap as

Implied by Different Models

Estimated Annual

Growth Rate of

Potential GDP

 in 1999:IV

(In percent)

Estimated Output Gap

in 1999:IV

(As a percentage of

potential GDP)

Decay Times of Output Gap

 (In quarters)

25%

closed

50%

closed

75%

closed

90%

closed

Blanchard-Quah

VAR

3.2 4.2 5 8 15 26

Gali VARa 3.5 -0.2 4 5 13 21

Cochrane VECM 3.4 1.8 4 6 9 12

SOURCE: Authors’ calculations using data from the Department of Commerce, Bureau of Economic

Analysis; and the Department of Labor, Bureau of Labor Statistics.

a. The output gap is close to zero at the end of the sample period, which may have inflated the reported

decay times.

Forecast Performance. Table 4 compares the out-of-sample forecast performances

of the three models with each other and with CBO’s annual “real-time” projections.29

Annual forecasts of real GDP beginning in 1976 were computed and compared with

CBO’s projections of the growth of real output made over the same period. For the

first quarter of each year beginning in 1976, all variants of each of the models were

estimated through the last quarter of the previous year and used to project real GDP

26
for the next five years. Projected growth rates were calculated for two and five years

ahead. Then, the process of estimation and forecasting was repeated for the next

year, and so on.

The performance of the model-based forecasts is generally similar to that of

forecasts produced by CBO over its history (see the rows showing the root mean

square errors in Table 4). The main exceptions are that the Gali VAR performs

substantially worse than CBO does in forecasts looking two years ahead, and the BQ

VAR performs better than CBO does over the five-year horizon. Large errors (such

as those occurring near turning points in the business cycle) appear in the VARs at

about the same time as they do in CBO’s real-time projections. Strikingly, the

relative accuracy of the projections of the three models improves in the longer-term

horizons. 

However, the better performance of the BQ VAR over the five-year horizon (as

compared with CBO’s projections) must be discounted because the BQ VAR

assumes a trend break in 1974. (The trend break was not yet established at the time,

so CBO’s real-time forecasts could not incorporate it.) The projections made before

1979 (the first four lines of the five-year projections in Table 4) are the ones in which

CBO’s forecasts have much larger errors than those of the BQ VAR.

27
Table 4. Comparison of Model Forecasts with CBO’s Real-Time Forecasts of Real Output

Blanchard-Quah Cochrane Gali

CBO VAR VECM VAR

Two-Year Growth

1976 - 1977 1.1 -0.4 -2.3 -3.8 

1977 - 1978 0.5 -2.1 -2.1 -5.3 

1978 - 1979 0.5 -1.0 -1.5 -4.7 

1979 - 1980 1.4 1.9 1.6 1.2 

1980 - 1981 -0.3 0.9 1.4 1.3 

1981 - 1982 2.2 3.2 2.7 4.8 

1982 - 1983 1.3 2.5 0.2 1.0 

1983 - 1984 -2.0 -0.8 -2.3 -3.8 

1984 - 1985 -0.3 -0.5 -1.1 -2.8 

1985 - 1986 0.3 0.1 -0.8 -2.3 

1986 - 1987 0.3 0.3 -0.3 -0.5 

1987 - 1988 -0.5 -0.4 -0.9 -0.8 

1988 - 1989 -1.2 -0.3 -0.6 -1.1 

1989 - 1990 0.2 0.4 0.6 0.5 

1990 - 1991 1.9 1.6 1.7 2.5 

1991 - 1992 0.8 0.6 -0.1 1.5 

1992 - 1993 0.1 0.8 -0.3 1.1 

1993 - 1994 0.0 1.1 0.2 0.9 

1994 - 1995 -0.1 0.4 -0.2 -1.3 

1995 - 1996 -0.4 -0.1 -0.3 -1.4 

1996 - 1997 -1.7 -1.4 -1.4 -2.0 

1997 - 1998 -1.8 -1.4 -1.8 -1.9 

Mean Error 0.1 0.2 -0.3 -0.8 

Mean Absolute Error 0.9 1.0 1.1 2.1 

Root Mean Square Error 1.1 1.3 1.4 2.6 

Five-Year Growth

1976 - 1980 2.0 -0.0 -1.7 -2.7 

1977 - 1981 2.3 0.1 -0.3 -1.9 

1978 - 1982 3.2 1.3 1.0 0.1 

1979 - 1983 2.5 1.2 1.6 1.7 

1980 - 1984 0.4 0.2 0.6 0.3 

1981 - 1985 0.0 0.2 -0.3 0.1 

1982 - 1986 0.1 0.2 -1.2 -1.2 

1983 - 1987 -0.3 0.2 -1.4 -2.6 

1984 - 1988 0.0 -0.2 -1.0 -1.9 

1985 - 1989 0.1 -0.1 -0.9 -1.4 

1986 - 1990 0.5 0.2 -0.2 -0.2 

1987 - 1991 0.8 0.9 0.5 0.9 

1988 - 1992 0.5 0.7 0.6 0.7 

1989 - 1993 0.6 0.8 1.0 1.1 

1990 - 1994 0.6 0.5 0.6 0.7 

1991 - 1995 0.4 0.5 0.1 0.3 

1992 - 1996 -0.2 0.3 -0.4 -0.4 

1993 - 1997 -0.3 0.2 -0.3 -0.5 

1994 - 1998 -0.7 -0.3 -0.8 -1.6 

Mean Error 0.7 0.4 -0.1 -0.4 

Mean Absolute Error 0.8 0.4 0.8 1.1 

Root Mean Square Error 1.2 0.6 0.9 1.3 

SOURCE: Authors’ calculations.

28
30. Appendix B describes how the confidence bands are computed.

Forecast Uncertainty. The standard deviation of bootstrap forecasts is about 1.3

percent to 1.9 percent of the mean bootstrap forecast for 2000, rising to 4 percent to

8 percent by 2010 (see the last three rows of Table 5, and Figures 10-12). Not

surprisingly, the forecasts’ standard errors increase as the forecast horizon lengthens,

but less than proportionally to the forecast horizon.30

Table 5. Characteristics of the Time-Series Models’ Real GDP Forecasts

Year

2000 2001 2002 2005 2010

Forecasts of Real GDP (In billions of chained 1996 dollars)

Blanchard-Quah VAR 9,170 9,340 9,540 10,380 12,080

Average Growth from 1999 (In percent) 3.4 2.7 2.5 2.7 2.9

Gali VAR 9,250 9,580 9,910 11,010 13,080

Average Growth from 1999 (In percent) 4.4 4.0 3.8 3.7 3.6

Cochrane VECM 9,200 9,430 9,710 10,720 12,690

Average Growth from 1999 (In percent) 3.8 3.2 3.1 3.2 3.3

The Range of the Forecasts (In billions of chained 1996 dollars)

Size of range 80 240 370 630 1,000

Midpoint of range 9,210 9,460 9,725 10,695 12,500

The range (As a percentage of the midpoint) 0.9 2.5 3.8 5.9 7.9

Standard Deviation of Bootstrap Forecasts (As percentage of mean bootstrap forecast)

Blanchard-Quah VAR 1.3 2.6 3.1 3.5 4.1

Gali VAR 1.9 3.4 4.1 5.7 8.0

Cochrane VECM 1.6 3.2 4.0 5.1 6.5

SOURCE: Authors’ calculations using data from the Department of Commerce, Bureau of Economic

Analysis; and the Department of Labor, Bureau of Labor Statistics.

29
1990 1995 2000 2005 2010

14.5

14.0

13.5

13.0

12.5

12.0

11.5

11.0

10.5

10.0

9.5

9.0

8.5

8.0

7.5

7.0

6.5

6.0

Figure 10.        Medium-Term Projections of Real GDP with Bootstrap

                          Error Bands:  Blanchard-Quah VAR

Trillions of Chain-Weighted 1996 Dollars

Real GDP

Mean of Bootstrap Simulations

Plus-Minus One-Sigma Error

Bands from the Bootstrap

SOURCE: Authors’ calculations using data from the Department of Commerce, Bureau of Economic Analysis;

and the Department of Labor, Bureau of Labor Statistics.

NOTE: The bootstrap is carried out by back-casting the series conditional on the last p observations of the

data. The number of replications is 2,000. See Table 5 for error percentages. 

30
1990 1995 2000 2005 2010

14.5

14.0

13.5

13.0

12.5

12.0

11.5

11.0

10.5

10.0

9.5

9.0

8.5

8.0

7.5

7.0

6.5

6.0

Figure 11.        Medium-Term Projections of Real GDP with Bootstrap

                          Error Bands:  Gali VAR

Trillions of Chain-Weighted 1996 Dollars

Real GDP

Mean of Bootstrap Simulations

Plus-Minus One-Sigma Error

Bands from the Bootstrap

SOURCE: Authors’ calculations using data from the Department of Commerce, Bureau of Economic Analysis;

and the Department of Labor, Bureau of Labor Statistics.

NOTE: The bootstrap is carried out by back-casting the series conditional on the last p observations of the

data. The number of replications is 2,000. See Table 5 for error percentages. 

31
1990 1995 2000 2005 2010

14.5

14.0

13.5

13.0

12.5

12.0

11.5

11.0

10.5

10.0

9.5

9.0

8.5

8.0

7.5

7.0

6.5

6.0

Figure 12.        Medium-Term Projections of Real GDP with Bootstrap

                          Error Bands:  Cochrane VECM

Trillions of Chain-Weighted 1996 Dollars

Real GDP

Mean of Bootstrap Simulations

Plus-Minus One-Sigma Error

Bands from the Bootstrap

SOURCE: Authors’ calculations using data from the Department of Commerce, Bureau of Economic Analysis;

and the Department of Labor, Bureau of Labor Statistics.

NOTE: The bootstrap is carried out by back-casting the series conditional on the last p observations of the

data. The number of replications is 2,000. See Table 5 for error percentages. 

The most important source of uncertainty in the forecasts for real GDP in

2010 is uncertainty in the level of potential output in that year; in contrast, the

uncertainty resulting from the cyclical position in 2010 is a minor source of error (see

columns 3 and 2, respectively, of Table 6). The standard error in the estimate of

potential output in 2010 is comparable to the error in the 2010 real GDP forecast in

all three models (see columns 1 and 3).

32
Table 6. Uncertainty of the Forecasts and Forecast Components (Standard deviations of the

models’ projections for real GDP in 2010, as a percentage of their mean forecasts)

Standard Deviation of the Forecast Components

(1) (2) (3) (4) (5) (6) (7) (8)

2010 

GDP

Forecast

2010 

GDP Gap

Estimate

2010

Potential

GDP

Forecast

Random

Walk Component of 

Potential

GDP in

2010

Trend Line

in 2010

Accumulated

Trend

Growth

Trend

Line

Intercept

Correlation of

Columns 

6 and 7

BQ VAR 4.8 2.4 4.0 2.9 2.5 2.0 1.3 (0.13)

Gali VAR 8.3 1.7 8.0 7.2 3.4 3.3 0.5 (0.19)

Cochrane 

 VECM 6.7 2.1 5.7 5.0 2.7 2.3 0.7 (0.46)

SOURCE: Authors’ calculations using data from the Department of Commerce, Bureau of Economic Analysis; and

the Department of Labor, Bureau of Labor Statistics.

NOTE: The reported values are the standard deviations as a percentage of real GDP in 2010, computed using the

natural logs of variables multiplied by 100. The GDP forecasts for 2010 (the standard deviation of which

is in column 1) equals the sum of 2010 GDP gap (whose standard deviation is in column 2) and potential

GDP in 2010 (column 3). Potential GDP has both random walk and trend components (columns 4 and 5).

The uncertainty in year 2010 position of the trend line, in turn, arises from the uncertainty in the trend growth

rate, which accumulates between the current period and year 2010 (column 6) and the uncertainty in the

intercept of the trend line (column 7). The variables in columns (4) and (5) are virtually uncorrelated. The

correlation of columns (6) and (7) is reported in column (8). The trend line intercept (column 7) is

determined by the estimate of the current output gap.

In turn, the forecast error in the projection of future potential GDP (whose

standard deviation is in column 3) can arise either from an inaccurate estimate of the

trend rate of growth (column 6) or from unexpected developments between the end

of the sample period and 2010 that will make potential GDP grow faster or slower

than the trend rate (column 4). The latter is the more important source of uncertainty

in estimating future potential GDP. This is a pessimistic result for medium-term

forecasts. It means that even if the gap and the average trend rate of growth were

known precisely, the randomness in growth of potential output, which makes the

growth of potential output in a given year differ from the average trend, would

accumulate over time and create a large amount of uncertainty. The estimate of the

intercept of the trend line (column 7), which is mainly determined by the estimate of

33
the output gap at the end of the sample period, is responsible for a small portion of

the error in the estimate of potential GDP in 2010.

The bootstrap error bands are conditional on the assumption that the models’

specifications are correct. For that reason, the bands probably understate the

uncertainty of the models’ forecasts (they might overstate the uncertainty as well).

Because the correct specifications are unknown, that uncertainty adds to the inherent

uncertainty of the models’ forecasts. Finally, as the large differences among the

forecast standard errors of the three models illustrate, different models imply

different levels of uncertainty—for example, the standard errors for the forecast of

the Gali VAR are about twice those for the BQ VAR in 2010 (see Table 5).

Similarly, none of those standard errors necessarily corresponds to the uncertainty of

CBO’s forecasts.

Sensitivity of Results to Alternative Specifications. We have examined the

sensitivity of the models’ forecasts to lag lengths and the assumptions regarding the

transformations required to make the time series stationary. As discussed in the

previous section, we adopted the lag lengths used in the original studies underlying

those alternative time-series models. It turns out, in each case, that this lag length

was at least as large as the lag length selected by a standard battery of information

criteria (see Table 7). Using the shorter lag lengths would have had a minimal effect

on the forecast for real GDP in 2010. Moreover, the out-of-sample root mean square

errors seemed to be affected relatively little by lag length for two-year forecasts, and

only small differences were manifest for the 10-year forecast horizon.

Although lag length may not have appeared to matter to a great degree, the

34
choice of the stationarity transformation does matter. In the case of the BQ VAR,

which seemed superior to the other VAR and the VECM in forecasting over the

longer horizons, the exclusion of the trend break in 1974 substantially worsened the

performance in forecasting 10 years ahead. Including the trend break did not

consistently improve the forecasts of the other models, however.

35
Table 7. Effects of Alternative Lag Lengths and Transformations on the Forecast

Selected Autoregressive Order by Information Criterion

(Lag in quarters)

Root Mean Square Errors for Out-of-Sample Projections of Real

GDP, Estimated over the Period 1970-1999

 (Percentage of historical level) 

G

 Two-Year Horizon Ten-Year Horizon 

AIC SBIC FPE HQ

Order Used in

Original Study

Smallest Lag

Selected

Order Used in

Original Study

Smallest Lag

Selected

Order Used in

Original Study

Blanchard-Quah VAR (Q denotes output of the nonfarm business sector, U denotes unemployment rate for married males)

No trend break in Qgrowth

No linear trendin U

3 2 2 3 8 3.3 3.5 6.2 6.4 

No trend break in Qgrowth

Linear trend inU

3 2 2 2 8 3.3 3.4 6.1 6.3 

Trend break in Qgrowth

No linear trendin U

6 2 2 2 8 2.9 3.0 2.9 3.2 

Trend break in Qgrowth

Linear trend inU

6 2 2 2 8 2.9 3.0 2.8 3.1 

Linear trend in log Q,with trend break

No linear trendin U

6 2 2 2 8 2.6 2.7 2.3 2.4 

Linear trend in log Q,with trend break

Linear trend inU

3 2 2 2 8 2.6 2.9 2.3 2.6 

Gali VAR ( dY/L denotes the growth rate of nonfarm business productivity, dL denotes the growth rate of hours worked in the nonfarm bu

No trend break indY/L

No trend breakin dL

3 2 2 2 8 4.7 4.6 7.9 6.6 

No trend break indY/L

Trend break indL

3 2 2 2 8 5.4 4.8 12.4 7.9 

Trend break in dY/L No trend break

in dL

3 2 2 2 8 5.2 5.3 12.2 11.7 

Trend break in dY/L Trend break in

dL

3 2 2 2 8 5.8 5.4 15.1 11.7 

Cochrane VECM (dQ denotes growth of real nonfarm business output, dC denotes growth of real consumption)

No trend break in dQ No trend break

in dC

2 1 1 1 2 3.9 3.8 11.0 10.8 

No trend break in dQ Trend break in

dC

2 1 1 1 2 3.1 3.0 5.9 6.1 

Trend break in dQ No trend break

in dC

3 1 1 1 2 3.4 3.1 6.0 5.7 

Trend break in dQ Trend break in

dC

1 1 1 1 2 3.2 3.4 8.4 9.4 

SOURCE: Authors’ calculations using data from the Department of Commerce, Bureau of Economic Analysis; and the Department of Labor, Bureau of Labor St
APPENDIX A: Alternative Approaches to Estimating the Output Gap

Because estimates of potential GDP and the output gap in the three vector

autoregression (VAR) models rely on the vector-moving-average (VMA)

representation of real GDP, we are including a brief explanation of those

calculations.

Time-series analysis relies heavily on a result known as Wold’s Decomposition

Theorem. If Zt/[Z1t, Z2t, ... , Znt] is a stationary n-vector time series, then it can be

decomposed into:

(i) a linearly deterministic component, the n-vector δt, that can be estimated

arbitrarily well given the information set through period t, and

(ii) a random component, consisting of a moving average of a white noise process.

In practice, not much is lost if it is assumed that δt is independent of time (δt = δ). 

Therefore, Z can be written in the form: 

Zt = δ + C(‹)εt (A-1)

where C is an n-by-n matrix whose elements are infinite-order polynomials in the lag

operator ‹ (that is, ‹k Zt = Zt-k for any integer k), the poles of C(‹) are outside the

unit circle, and C(0)/In so that the decomposition is unique.

Isolating the Trend Component

Two different approaches to decomposing the observed output into trend and cycle

components are used. In cases in which both methods are applied, the estimated

components turn out to be fairly similar. The first method is the Beveridge-Nelson

(BN) decomposition. It defines the cycle component as the transitory component of

output that is bound to disappear in the long run; the trend component is defined as

37
1. Some expansion of this notation would be required to accommodate the presence of cointegration

relationships among the variables in Z. For the sake of expositional simplicity, the examples in this section

assume no cointegration. The only model examined here that includes cointegrated terms is the estimate

of the Cochrane model. In an appendix to his study, Cochrane outlines his method for extracting the

permanent component of output assuming one cointegrating relationship. For a general exposition of the

themes of this section, which does not preclude the possibility of cointegrating relationships among the

variables in Z, see G. Evans and L. Reichlin, “Information, Forecasts, and Measurement of the Business

Cycle,” Journal of Monetary Economics, vol. 33, no. 2 (April 1994), pp. 233-254.

2. S. Beveridge and C. R. Nelson, “A New Approach to Decomposition of Economic Time Series Into

Permanent and Transitory Components with Particular Attention to Measurement of the ‘Business Cycle’,”

Journal of Monetary Economics, vol. 7, no. 2 (March 1981), pp. 151-174. The multivariate

decomposition is presented in ibid., G. Evans and L. Reichlin.

3. Ibid., Beveridge and Nelson, pp. 153-154.

the remaining part. The other decomposition method is long-run restrictions. In that

method, the random shocks to the system are grouped on the basis of whether they

have a permanent long-run impact. The system is simulated using only the shocks

that have no permanent effect so as to identify the part of output that the transitory

shocks are responsible for. That part is then taken as the cycle component, and the

remaining part is the trend component.

Those two basic ways of extracting the trend component of output are examined

in more detail below. The relationship (A-1) is central to each.1

The Beveridge-Nelson Decomposition

The first approach to extracting the trend component examined here comes from

Beveridge and Nelson.2

 They note that if the stationary time series Z represents the

first differences of an I(1) vector time series, say X, then the application of (A-1) to

∆X gives rise to an intuitively appealing definition of the trend, or permanent,

component of X. Beveridge and Nelson define the permanent component of Xt to be

the midpoint of the predictive distribution (at time t) for the future path of X

excluding its trend, and the transitory component of Xt to be the expected momentum

in X that is predictable at time t.3 In the present notation, the BN decomposition

38
gives:

∆X*t = δ + C(1)εt (A-2a)

∆Xt

 = ∆X*t - [(1-‹) C(j)‹k] εt (A-2b) k =

∞

∑ 0 j k >

∞

∑

in which X*t denotes the permanent component of X under the BN decomposition.

From (A-2a), the permanent component of the n-vector X follows a random walk

with drift δ. The period t innovation to the random walk, εt, has a permanent effect

on the level of X* at time t. The matrix C(1), which is the sum of the VMA

coefficients in (A-1), is the long-run impact multiplier. The product C(1)εt

cumulates the current and all future effects of εt on X and allocates them to X* at

time t.

The key to estimating the BN decomposition is estimating C(1). Following the

original studies, the assumption that C(‹) in (A-1) is an invertible matrix is imposed

so that the resulting vector autoregressive form can be estimated:

[In-A(‹)](Zt - δ) = εt (A-3)

where A(‹) /In - C-1(‹). The coefficients of the (possibly) infinite-order VAR, A(‹),

can be estimated by assuming a finite VAR order of p sufficiently large to

approximate the order of the polynomials A(‹) and by applying ordinary least

squares to the resulting specification. If those estimates of A(‹) are denoted by

AOLS(‹), then the entire sequence of VMA coefficients can be recovered as:

C(‹) – COLS(‹) / [In - AOLS(‹)]-1 (A-4)

in which the quality of the approximation depends on the lag length p used in

39
4. In the computations, a standard recursive formula was used to evaluate the VMA coefficients. See, for

example, James D. Hamilton, Time Series Analysis (New Jersey: Princeton University Press, 1994), p. 260.

Of course, the quality of the approximation depends even more critically on the invertibility assumption

that is maintained throughout this study.

5. Higher-order variants of the Blanchard-Quah decomposition are easily implemented. For an interpretation

of the BQ decomposition with n=3, see C. Dupasquier, A. Guay, and Pierre St-Amant, A Comparison

of Alternative Methodologies for Estimating Potential Output and the Output Gap, Working Paper No.

97-5 (Ottawa, Canada: Bank of Canada, February 1997).

estimating the VAR.4

Long-Run Identifying Restrictions

Although the BN decomposition’s association of the permanent component of a

series with its long-run forecast seems intuitive, the decomposition is by no means

unique. In their study, Blanchard and Quah (BQ) adopted an alternative approach to

extracting the permanent component of output by introducing identifying restrictions

on the matrix C(1). Their decomposition is easiest to describe for the case of n=2.5

In this case, the stochastic dynamics of the two-variable system are governed by two

types of random shocks, one permanent and the other transitory. The permanent and

transitory shocks are assumed to be statistically independent. If the two-dimensional

vector of structural shocks is denoted by ξ / (ξπ , ξτ)N for permanent and transitory,

respectively, the Wold form for the BQ decomposition of the stationary vector Z can

be written as

Zt = δ + B(‹)ξt (A-5)

where E(ξtξtN) is a diagonal 2-by-2 matrix, and the matrix of long-run coefficients,

B(1), is lower triangular. This lower triangularity means that ξτ (temporary shock)

has no long-run effect on the first variable in Z (taken by BQ to be growth in real

GDP).

The coefficients of the BQ structural form (A-5) are related to equation (A-1)

as follows:

40
ξt / S-1 εt (A-6a)

B(‹) / C(‹)S (A-6b)

with B(1) a lower triangular matrix. For the two-dimensional case, the nonsingular

matrix S is exactly identified by the orthogonality requirement on the structural

shocks and the requirement that the transitory shocks have no permanent effect on

the first variable in the system.

Using this structural VAR approach for identifying a two-variable model, the

permanent components of Z under the BQ decomposition are given by:

Z*1t = δ1 + B11(‹) ξπ

t (A-7a)

Z*2t = δ2 +B21(‹) ξπ

t

. (A-7b)

By design, the permanent component of Z1 is unaffected by the transitory shocks.

41

1. The procedure is more fully described in Lori A. Thombs and William R. Schucany, “Bootstrap Prediction

Intervals for Autoregression,” Journal of the American Statistical Association, vol. 85, no. 410 (June

1990), pp. 486-492.

2. The backward representation of an autoregressive system expresses the current values of the endogenous

variables in terms of their future values.

43

APPENDIX B: The Bootstrap Procedure

A bootstrap procedure is used to estimate confidence intervals for the alternative

forecasts.1 The steps are: 

• The backward representation of each model is estimated.2

• The L most recent observations in the data are taken as given (where L is the

longest lag length in the model). Holding these last L values fixed at their

historical values, artificial data are generated for the remainder of the historical

period by simulating the estimated backward representation and resampling the

residuals estimated from the backward representation. 

• The resulting artificial sample is then used, in the usual way, to generate a

projection for the variables in the model. 

The latter two steps were repeated 2,000 times. The resulting 2,000 alternative

projections were used to generate the forecast confidence bands.

The advantage of using resampling techniques in general is that they generate

nonparametric confidence intervals—confidence intervals that do not rely on

particular assumptions about the distribution of disturbance terms. Although the

residuals from the backward regression are not always serially independent, they are
3. For a discussion of this point, see Lutz Kilian and Ufuk Demiroglu, “Residual-Based Tests for Normality

in Autoregressions: Asymptotic Theory and Simulation Evidence,” Journal of Business and Economic

Statistics, vol. 18, no. 1 (January 2000), p. 41.

44

uncorrelated.3 The serial independence, which is needed to justify resampling of

residuals, is ensured, for instance, in the case of normally distributed disturbance

terms.