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\begin{document}

\title{The Income- and Expenditure-Side Estimates of U.S. Output
  Growth}\author{Jeremy J.  Nalewaik\thanks{Economist, Federal Reserve
    Board, 20th Street and Constitution Avenue, Washington, DC 20551
    (e-mail: jeremy.j.nalewaik@frb.gov).  Thanks to the economists at
    the U.S. Bureau of Economic Analysis, who answered many, many
    questions as I worked on this paper.  Thanks for helpful comments
    to Andrew Figura, Dennis Fixler, Bruce Grimm, Steve Landefeld,
    David Lebow, Deb Lindner, Jonathan Millar, Raven Molloy, Brent
    Moulton, David Romer, Dan Sichel, Stacey Tevlin, Justin Wolfers,
    and participants at the March 2010 Brookings Panel on Economic
    Activity.  Jessica Chan provided able research assistance.  The
    views expressed in this paper are solely those of the authors and
    are not necessarily those of the Federal Reserve Board.}
  \\
  \\
%  First Draft - please do not distribute.  
}
\maketitle

\doublespacing

\begin{abstract}

  The U.S. produces two conceptually identical official measures of
  its economic output, currently called Gross Domestic Product (GDP)
  and Gross Domestic Income (GDI).  These two measures have shown
  markedly different business cycle fluctuations over the past
  twenty-five years, with GDI showing a more-pronounced cycle than
  GDP.  The goal of this paper is to determine which measure better
  reflects the business cycle fluctuations in true output growth,
  and a broad range of results favor GDI.  GDI currently shows the
  2007-2009 downturn was considerably worse than is reflected in GDP.

{\it JEL classification}: C1, C82.

{\it Keywords}: GDP, statistical discrepancy, news and noise,
signal-to-noise ratios, business cycles

\end{abstract}

%\newpage
\section{Introduction}

The U.S. produces two conceptually identical official measures of its
economic output, currently called Gross Domestic Product (GDP) and
Gross Domestic Income (GDI).  These two measures have shown markedly
different business cycle fluctuations over the past twenty-five years,
with GDI showing a more-pronounced cycle than GDP.  These differences
have become particularly glaring over the latest cyclical downturn,
which appears considerably worse along several dimensions when looking
at GDI.  The aim of this paper is to determine which measure better
represents the actual business cyclical fluctuations in output growth,
and a wide variety of results suggest the answer is GDI.

%As noted in Fixler and Nalewaik (2007), national income accountants in
%general, and those at the U.S. Bureau of Economic Analysis (BEA) in
%particular, are responsible for defining an interesting and useful
%concept of aggregate economic output and then producing the best
%possible estimates of that definition of output.  While the usefulness
%of the definition employed by the BEA and other national income
%accountants is often the subject of some controversy,\footnote{%
%  ... references, including recent contributions ...} this paper does
%not address that issue.  Rather, it takes the definition of output as
%given, and examines the relative accuracy of the BEA's estimates of
%that definition of output.  In particular, the BEA reports two
%estimates of its output variable of interest.  The first is its
%featured estimate, the widely-followed estimates of Gross Domestic
%Product (GDP).  As most introductory macroeconomic texts
%discuss,\footnote{%
%  See Abel and Bernanke (2001), for example.} there are three ways to
%measure the output concept defined by BEA, and this featured estimate
%relies on the expenditure approach---that is, by measuring
%expenditures by households, businesses, governments, and the rest of
%the world.  The BEA calls its second estimate Gross Domestic Income
%(GDI), which, unsurpringly, relies on the income approach.
%\footnote{%
%  The third approach, the production or value-added approach, is used
%  by the BEA to produce industry measures of output, but BEA does not
%  produce regular top-line estimates of output based on this approach.
%  Historically, the BEA produced only annual estimates of GDP by
%  industry, but they plan on rolling out quarterly GDP-by-industry
%  estimates in the near future.  One interesting topic for future
%  research is whether the quarterly industry statistics will contain
%  information about top-line output not contained in the other two
%  estimates.} 
%What do we know about the relative reliability of these
%two sets of estimates?

In discussing the information content of these two sets of estimates,
the confusion often starts with the nomenclature.  GDP can mean either
the true output variable of interest, or an estimate of that output
variable based on the expenditure approach.  Since these are two very
different things, using ``GDP'' for both is confusing.  Furthermore,
since GDI has a different name than GDP, it may not be initially clear
that GDI measures the same concept as GDP, using the equally valid
income approach.  So, to keep things straight, this paper refers to
the true variable of interest as true output, the expenditure-side
estimate of true output as GDP(E), and the income-side estimate of
true output as GDP(I).

%After section 2 reviews some basic facts about the data, section 3
%tackles the first task.  It discusses the strengths and weaknesses of
%the source data used to compute the initial estimates, and then
%studies the revisions to the initial estimates.  The revised estimates
%incorporate better source data than the initial estimates, so the
%revised growth rates are likely closer to true output growth, and
%studying how well the initial growth rates predict the revised growth
%rates is a simple way to assess the reliability of the initial
%estimates.  Over the past 15 years or so, the initial estimates of
%GDP(I) growth contain information about the latest estimates of GDP(E)
%growth, above and beyond the information in initial GDP(E) growth.  So
%for the initial estimates of output growth, recent history suggests
%that placing at least some weight on GDP(I) is prudent, even if the
%latest estimates of GDP(E) are ultimately the only object of interest.

The paper presents results for both the initial output growth
estimates available in real time, and the later estimates that have
passed through more revisions.  After presenting some basic facts
about the estimates, section 3 discusses the initial growth rates and
shows numerous results favoring GDP(I) growth.  First, there is some
evidence that the initial GDP(I) growth predicts revisions to GDP(E)
growth, and no tendency for GDP(E) growth to predict revisions to
GDP(I) growth.  Second, initial GDP(I) growth is the better predictor
of a wide variety of business cycle indicators that should be
correlated with true output growth.  These include all measures of
output growth in subsequent periods, the change in the unemployment
rate in the current period and subsequent periods, employment growth
(measured using a household survey) in current and subsequent periods,
the manufacturing purchasing managers index in current and subsequent
periods, changes in stock prices over previous periods, the slope of
the treasury yield curve in previous periods, and forecasts of GDP(E)
growth itself from previous periods.  Each of these results suggest
GDP(E) growth is either the noisier measure of true output growth
or misses fluctuations in true output growth that appear in both
GDP(I) growth and the other business cycle indicators.  Third, initial
GDP(I) growth has identified the onset of the last few cyclical
downturns more quickly than initial GDP(E).

Section 4 discusses the latest revised growth rates.  The
section first establishes some basic facts about the discrepancies
between the fully-revised estimates.  On average, GDP(I) tends to grow
faster than GDP(E) when the economy is expanding robustly, and GDP(I)
growth falls below GDP(E) growth in recessions and in periods where
the economy is sluggish.  As such, the statistical discrepancy is
highly negatively correlated with the business cycle.  Why is this the
case?  A thorough analysis of the nature of the source data suggests
that GDP(E) misses part of the business cycle, and that GDP(I)
captures the business cycle better.  Statistical analyses reach the
same conclusion.  First, the nature of the revisions suggests they add
cyclical variation to GDP(I) that is not added to GDP(E), implying
GDP(E) misses some cyclical variation.  And second, the latest GDP(I)
growth estimates are more highly correlated with a wide range of
business cycle indicators, including changes in unemployment, the
growth rate of employment, purchasing manager surveys (both
manufacturing and non-manufacturing), changes in stock prices over
previous periods, the slope of the treasury yield curve in previous
periods, the high-yield bond spread from previous periods, and
indicator variables for NBER recessions.

Section 5 discusses the behavior of the estimates over the most recent
cyclical downturn.  Output decelerated sooner, fell at a faster rate
at the height of the downturn, and recovered less quickly when
measured by GDP(I).  Drawing on the results from the previous sections
and online Appendices, this section discusses how GDP(E) may have
missed the severity of the downturn.  Section 6 concludes with
thoughts about the implications of the results in the paper for both
data users and the BEA.

\section{Basic Facts about the Estimates}

The BEA's first GDP(E) estimate for the most recent quarter, called
the ``advance'' estimate, is released about a month after the quarter
closes.  Estimates of most components of GDP(I) for that quarter are
included in the ``advance'' release, but the BEA is not comfortable
releasing estimates of corporate profits or net income from the rest
of the world at that time.  GDP(I) first appears with the ``second''
release about two months after the quarter ends, except the estimates
for fourth quarters, when GDP(I) first appears with the ``third''
release about three months after the quarter ends.  To work with a
complete time series of the initial growth rates, we focus on these
``third'' release estimates.  However, in an online appendix, I repeat
the regression results in section 3 using the ``second'' estimates for
quarters where GDP(I) is are available, and ``advance'' GDP(I)
estimates constructed using the available income-side components and
forecasts of corporate profits and net income from the rest of the
world.

After the BEA releases its initial estimates of GDP(E) and GDP(I)
growth for any given quarter, the estimates are revised numerous
times.  Table 1 shows the variances and correlations of the initial
``third'' estimates and the latest estimates which have passed through
more revisions; here and throughout the paper $\Delta GDP(E)$ and
$\Delta GDP(I)$ are short for the annualized quarterly growth rates of
the estimates.  We focus on two samples here.  The first starts in
1978Q3, and is dictated by the start date of the time series of
``third'' growth rates employed in the paper, which is based on a
real time dataset constructed by the BEA starting in 1978.
%, which we
%extended back to 19XX using information from the Surveys of Current
%Business.  
When analyzing the latest, revised estimates, the paper
focuses on a shorter sample starting in the mid-1980s, because the
divergences between the estimates are particularly stark and highly
cyclical over this time period.\footnote{%
  The precise start date chosen here is the econometric break point
  marking the beginning of the once widely-accepted phenomenon known
  as the Great Moderation.  The precise start date is not particularly
  important, however; any start date around the mid-1980s gives
  similar results for the latest estimates.}  This second sample ends
in 2006Q4 to ensure that the latest estimates have revised to fully
incorporate all their major annual source data.  The data from the
mid-1980s to present are plotted in Figures 1 and 2.

%Latest estimates of the two growth rates shown together in Figure 3, 
% and Figure 4 shows latest year-over-year growth rates.
The top two panels of table 1 show that the correlation of the initial
estimates with the latest estimates is fairly high (0.85 and 0.82 for
one sample, and 0.68 and 0.66 for the other).  Nonetheless, the
revisions do change the estimates in important ways.  First, the
bottom panel shows the variance of the revisions is somewhat larger
for $\Delta GDP(I)$ than for $\Delta GDP(E)$.  Moreover, the revisions
tend to increase the variance of $\Delta GDP(I)$ more than the
variance of $\Delta GDP(E)$.  This suggests that the revisions add
information to latest $\Delta GDP(I)$ that is not added to latest
$\Delta GDP(E)$.  Finally, and perhaps counterintuitively, the
revisions tend to make the two measures less similar, reducing their
correlation from 0.90 to 0.60 in the shorter sample.

Given the important differences between the latest estimates after
they have passed through their revisions, this paper investigates two
questions.  The first is: what is the relative information content of
the initial growth rates of GDP(E) and GDI(I)?  Put differently, how
much weight should we place on each of these initial growth rates?
The second question is: what is the relative information content of
the GDP(E) and GDI(I) growth rates after they have passed through all
their revisions?  In other words, how much weight should we place on
each of these latest, revised growth rates?

Appendix A provides more background information about GDP(E) and
GDP(I).  Appendix B discusses the source data used to construct the
initial growth rates, while Appendix C describes the source data
incorporated at annual and benchmark revisions.

\section{The Information Content of the Initial Growth Rates}

A detailed examination of the source data used to compute the initial
``third'' growth rates of $\Delta GDP(E)$ and $\Delta GDP(I)$ shows
that both estimates suffer from similar types of measurement-error
problems; see Appendix B.  These problems include missing data for a
substantial portion of each estimate, sampling errors, and
non-sampling errors such as incomplete coverage, survey non-response,
and incomplete corrections for firm births and deaths.  A compelling
case for the superiority of either estimate cannot be made based on
such a detailed examination of source data, so this section dives
right into more-informative statistical tests.\footnote{%
  In his comments, J. Steven Landefeld suggests that, for the
  ``third'' estimates, a much greater fraction of GDP(I) than GDP(E)
  is based on ``judgmental trends'' instead of early source data.
  Almost all of the source data used to compute the ``third''
  estimates is flawed and unrepresentative in some way, and breaking
  down the data using such a binary classification scheme is a highly
  subjective exercise.  The detailed discussion of the source data in
  Appendix B suggests the evidence is less favorable to GDP(E) than
  this classification scheme suggests.  Moreover, if a much greater
  fraction of GDP(I) were based on trends, we should expect ``third''
  $\Delta GDP(I)$ to be much less variable than ``third'' $\Delta
  GDP(E)$, because trends should have less variance than the actual
  source data.  The summary statistics in table 1 show that this is not the
  case; ``third'' $\Delta GDP(I)$ is actually slightly more variable
  than ``third'' $\Delta GDP(E)$.}

Table 2 reports the main regression results examining the information
content of the initial ``third'' growth rates.  A good initial place
to start is an examination of the predictive power of the initial
estimates for the latest, revised estimates incorporating superior
source data.  Over the full sample, the initial $\Delta GDP(E)$
estimates predict well the latest estimates of $\Delta GDP(E)$, with
the initial $\Delta GDP(I)$ adding little after conditioning on
initial $\Delta GDP(E)$.\footnote{%
  Newey-West standard errors using eight lags are in parentheses below
  the estimates.}  Similarly, initial $\Delta GDP(I)$ predicts latest
$\Delta GDP(I)$, with initial $\Delta GDP(E)$ adding little
information after conditioning on initial $\Delta GDP(I)$.  However,
the final two blocs of regressions in table 2 show results for a
sample starting in 1994Q1; we stop this subsample in 2006Q4 to ensure
the latest estimates have passed through all their annual revisions,
but extending the subsample to 2008Q4 produces similar results.  The
first specification constrains the coefficients on initial $\Delta
GDP(E)$ and initial $\Delta GDP(I)$ to sum to one, while the second
does not; the results show that when initial $\Delta GDP(I)$ is one
percentage point above initial $\Delta GDP(E)$, initial $\Delta
GDP(E)$ has revised up about a third to two-fifths of a percentage
point, on average, over this time period.  Fixler and Grimm (2006)
also find some tendency for initial $\Delta GDP(E)$ to revise towards
initial $\Delta GDP(I)$, using a broader set of conditioning
variables.  The last bloc of results shows that, over this subsample,
there remains no significant tendency for initial $\Delta GDP(E)$ to
predict latest $\Delta GDP(I)$.

The initial $\Delta GDP(I)$ may have predicted revisions to the
$\Delta GDP(E)$ over this sample because they are less noisy than the
initial $\Delta GDP(E)$ estimates, or because they contain information
about true output growth missed by the initial $\Delta GDP(E)$
estimates but incorporated into latest $\Delta GDP(E)$ through
revisions.  Both of these explanations are likely part of the story.
Averaging the data into year-over-year growth rates eliminates much of
the noise in the quarterly data and shows the plausibility of the
second explanation.  These year-over-year growth rates for 4th
quarters are plotted in Figure 3 (this picture was first suggested to
me by Bill Wascher).  Broadly speaking, two periods drive the positive
relation plotted here.\footnote{%
  The line plotted is the predicted values from regressing the 13
  Q4-over-Q4 growth rates of real GDP(E) on a constant and the gap
  between the initial (third) estimates of Q4-over-Q4 GDP(I) and
  GDP(E) growth.  The coefficient on the gap is 0.98, with a standard
  error of 0.44, and an adjusted $R^2$ of 0.25.  We also experimented
  with corrections that removed the effects of major methodological
  changes from the revisions, which increased the $R^2$.} 

First, during the mid-to-late 1990s, the gap between initial $\Delta
GDP(I)$ and $\Delta GDP(E)$ was consistently positive, as the initial
estimates showed GDP(I) growing faster than GDP(E).  This phenomenon
was discussed in real time at the Council of Economic Advisors (see
the Economic Report of the President, 1997, pp. 72-74), the Federal
Reserve Board (see Greenspan, 2004), and at the BEA itself (see
Moulton, 2000), with conclusions generally favorable to GDP(I).  Those
conclusions were vindicated, since, ultimately, $\Delta GDP(E)$
revised up towards initial $\Delta GDP(I)$.  Initial $\Delta GDP(I)$
accurately captured information about the brisk pace of economic
growth that was missed by the initial $\Delta GDP(E)$ estimates and
incorporated only later through revisions (and probably only
partially---see section 4).  Second, in the period after the 2001
recession, in 2002 and 2003, the initial estimates of $\Delta GDP(I)$
showed a more sluggish recovery than the initial estimates of $\Delta
GDP(E)$, so the gap between the initial estimates was
negative.\footnote{%
  As Appendix C outlines, since 2002, the BEA has incorporated
  information from the Quarterly Census of Employment and Wages (QCEW)
  into their wage and salary estimates a couple of months after their
  ``third'' estimate.  These QCEW revisions provided much of the
  information to $\Delta GDP(I)$ on the relative sluggishnes of the
  recovery; see the discussion in Nalewaik (2007a).  The
  year-over-year growth rates for 4th quarters available in real time
  reflect these QCEW revisions.}  Ultimately, $\Delta GDP(E)$ revised
towards initial $\Delta GDP(I)$ again; the recovery was indeed quite
sluggish, and this information was reflected in $\Delta GDP(I)$ before
it appeared in $\Delta GDP(E)$.  Figure 4 shows no tendency for
$\Delta GDP(I)$ to revise towards $\Delta GDP(E)$; if anything,
$\Delta GDP(I)$ tends to revise in the opposite direction of the
initial gap between $\Delta GDP(E)$ and $\Delta GDP(I)$ over this time
period.
%This relation broke down 
%somewhat recently, especially in 2006 and 2007, but snapped back to
%form with the latest revision to 2008.\footnote{%
%  Note that subsequent annual revision may change these results for
%  2007 and 2008, and the next benchmark revision may change the
%  post-2002 results.}

This particular set of revision results occurs over a short sample,
and should be taken with a grain of salt.  However, as a robustness
check, we used past issues of the Survey of Current Business to extend
back our sample, and results reported in online Appendix D show a
marginally statistically significant tendency for initial $\Delta
GDP(E)$ to revise towards initial $\Delta GDP(I)$ over this long
sample extending from 1966Q4 to 2009Q3.  And, after the data have
passed through their first annual revision, a statistically
significant tendency for initial $\Delta GDP(E)$ to revise towards
initial $\Delta GDP(I)$ in subsequent revisions appears yet again,
using the 1978Q3 to 2009Q3 sample.  So it would probably be unwise to
ignore these revision results entirely.

A decision to ignore the revision results implies that the weight that
data users should place on the initial estimates is entirely
determined by the weight placed on the latest, fully-revised
estimates.  An analyst who believes that latest $\Delta GDP(E)$ is
more accurate than latest $\Delta GDP(I)$ should believe that initial
$\Delta GDP(E)$ is more accurate than initial $\Delta GDP(I)$, and
vice versa.  So the results outlined in the next section, addressing
the paper's second question, are also critical to answering the
first.

However, we can make considerable further progress on the paper's
first question directly, by examining the preditive power of the
initial estimates for other important cyclical indicators.  Broadly
speaking, these regressions help establish which estimate is more
informative about the busines cycle, but the regressions also help
answer the narrower question of which is the better estimate of
true output growth.  The inferior estimate of true output
growth, containing relatively more noise or classical measurement
error, should have a lower signal-to-noise ratio and be the inferior
predictor of cyclical indictors correlated with true output
growth, all else equal.  This assumes the noise in the output growth
estimates is uncorrelated with the measurement error in the cyclical
indicators, and I have chosen cyclical indicators carefully to avoid
this problem.  An estimate may be inferior not only because it is
noisier, but also because it misses more fluctuations in true
output growth---contains less news or signal about true output
growth---than the other estimate.  But, again, such an inferior
estimate should have a lower signal-to-noise ratio and be the inferior
predictor of cyclical indicators that reflect those missing
fluctuations in true output growth.

Returning to the top of table 2, we see that as a cyclical indicator
of where output growth is headed, initial $\Delta GDP(I)$ is superior
to initial $\Delta GDP(E)$.  The initial estimates of $\Delta GDP(I)$
are positively related to output growth in the next quarter, whether
output growth next quarter is measured by $\Delta GDP(E)$ or $\Delta
GDP(I)$, initial or latest.  Conditional on initial $\Delta GDP(I)$,
initial $\Delta GDP(E)$ contains no information about output growth
next quarter, and may actually be negatively related to output growth
next quarter.  This result holds two quarters ahead as well, when
output growth is measured by either initial estimate.  Following the
logic outlined above, these results may obtain because initial $\Delta
GDP(E)$ is noisier than initial $\Delta GDP(I)$, obscuring its signal
about true output growth in subsequent periods, or because initial
$\Delta GDP(E)$ misses some of the shocks that produce
serially-correlated fluctuations in true output growth, shocks
that appear in initial $\Delta GDP(I)$.

We examine next the relation of the initial estimates to other
cyclical variables that should be correlated with true output
growth.  These other variables should not be used in the construction
of either GDP(E) or GDP(I), to avoid correlated measurement errors and
spurious correlation.  As outlined in appendices, the GDP(E) and
GDP(I) estimates make little use of the Current Population Survey
(CPS), a monthly household survey used to produce the unemployment
rate.\footnote{%
  At first blush, some analysts might suspect that GDP(I) must be more
  correlated with the unemployment rate than GDP(E), because
  ``income'' is in the name GDP(I) and the unemployment rate is a
  labor market concept.  However, this reasoning is incorrect.  Of the
  various components of the two output measures, we may expect based
  on {\it a priori} considerations that compensation will have
  higher-than-averge correlation with unemployment, but the other
  components of GDP(I) should then have lower-than-average
  correlation, since all the components of GDP(I) add up to the same
  conceptual measure of output as GDP(E).  For example, stories in the
  press recently have suggested that some of the recent rebound in
  corporate profits was facilitated by weakness in the labor market,
  allowing firms to cut compensation costs.}  As one of the most
important indicators of the business cycle, the unemployment rate is a
good variable to use as a starting point for this analysis.

Table 2 shows that contemporaneously, initial $\Delta GDP(I)$ has a
strong negative relation with the change in the unemployment rate, and
negatively predicts changes to the unemployment rate one and two
quarters ahead, while the coefficients on initial $\Delta GDP(E)$ are
insignificant and have the wrong sign when conditioning on initial
$\Delta GDP(I)$.  Again, this may be because initial $\Delta GDP(E)$
is noisier than initial $\Delta GDP(I)$, or because initial $\Delta
GDP(E)$ misses fluctuations in true output that both appear in $\Delta
GDP(I)$ and are reflected in the differenced unemployment rate.

The next bloc of regressions show results using quarterly annualized
employment growth computed from the household survey data, adjusted
for breaks introduced by Census updates to the population.  Initial
$\Delta GDP(I)$ is positively related to employment growth this
quarter, as well as one- and two-quarters ahead, while initial $\Delta
GDP(E)$ contains little additional information about employment growth
beyond that contained in initial $\Delta GDP(I)$.

Broadening out the results beyond labor market variables, the next
bloc of regressions uses the Purchasing Managers' Index (PMI) from the
Institute for Supply Management (ISM) manufacturing survey.  The ISM
measure is computed quite differently from GDP(E) and GDP(I); it is an
aggregation of several diffusion indexes, so even though the companies
participating in the ISM also participate in the surveys used to
estimate GDP(E) and GDP(I), the measurement errors likely behave quite
differently.  Initial $\Delta GDP(I)$ explains the contemporaneous,
one quarter ahead, and two quarters ahead movements in the ISM measure
better than initial $\Delta GDP(E)$, with initial $\Delta GDP(E)$
providing no statistically significant information conditional on
initial $\Delta GDP(I)$.

Business cycle analysts use a host of other variables to predict
$\Delta GDP(E)$, most notably different asset prices, and since these
asset prices are not used in the construction of the output growth
estimates, they are prime candidate variables for testing the
information content of the initial estimates.  However, asset prices
typically predict output growth in subsequent quarters, rather than
being predicted by output growth, so to get the timing correct, we
regress lagged values of these asset prices on the two initial output
growth measures.  This is a somewhat odd specification, but still
quite instructive.  The results essentially tell us which initial
estimate is more consistent with market expectations of the business
cycle from earlier periods.  

The first asset-price specification regresses the log change in the
S$\&$P 500 stock price index from the end of quarter $t-4$ to the end
of quarter $t$, on the two initial output growth measures in quarter
$t$.  Initial $\Delta GDP(I)$ is strongly positively related to this
current and lagged stock price change, while the coefficient on
initial $\Delta GDP(E)$ is insignificant and negative.  The next
specification examines the slope of the yield curve, measured as the
difference in yields between ten- and two-year treasury notes.  This
variable is most closely related to the output growth measures about
two years hence; a regression of this measure from quarter $t-8$ on
the two initial output growth measures in quarter $t$ yields a
coefficient on initial $\Delta GDP(I)$ that is significant with the
correct (positive) sign, and a coefficient on initial $\Delta GDP(E)$
that is significant but the wrong sign.

The final set of testing variables employed here are median forecasts
of output growth from the Survey of Professional Forecasters (SPF).
These forecasters are trying to predict initial $\Delta GDP(E)$,
presumably inclusive of any measurement errors in $\Delta GDP(E)$.
However, if the forecasters do not yet have access to the source data
used to compute the quarter of $\Delta GDP(E)$ they are trying to
predict, their forecasts will likely reflect general information
about the state of the economy which may be better related to initial
$\Delta GDP(I)$ than to initial $\Delta GDP(E)$.  This may be the case
even for the current-quarter forecasts, because the survey occurs
relatively early in the quarter before the analysts have much GDP(E)
source data.  And the results show that those current quarter forecast
are well explained by initial $\Delta GDP(I)$, with initial $\Delta
GDP(E)$ providing no incremental explanatory power.  The SPF forecasts
for quarter $t$, made in the first half of quarter $t-1$ are also
better explained by initial $\Delta GDP(I)$ in period $t$ than initial
$\Delta GDP(E)$ in period $t$.  Forecasters' expectations for how the
economy will move in the current quarter and next quarter appear to
play out more fully in the initial $\Delta GDP(I)$ estimates than in
initial $\Delta GDP(E)$.

Given the tighter relation of initial $\Delta GDP(I)$ to all these
business cycle indicators, a business cycle analyst placing full
weight on the initial $\Delta GDP(E)$ estimates and no weight on the
initial $\Delta GDP(I)$ estimates must: (1) care about true output
growth in the current quarter only (i.e. not care about the business
cycle more broadly or even where true output growth is headed next
quarter), (2) believe the latest $\Delta GDP(E)$ estimates reflect all
available information about true output growth, so neither latest
$\Delta GDP(I)$ nor any other variable provides any additional
marginal information about true output growth, (3) believe the
superior explanatory power of initial $\Delta GDP(I)$ for all these
other cyclical indicators tells us nothing about the relative accuracy
of initial $\Delta GDP(I)$ and initial $\Delta GDP(E)$ as estimates of
true output growth, and (4) discount entirely the revisions
evidence.  This third point is clearly a stretch, and could only be
the case if initial $\Delta GDP(I)$ contained variation uncorrelated
to true output growth but correlated with all the other dependent
variables employed in table 2, including actual forecasts of output
growth.  A much more plausible explanation is that initial $\Delta
GDP(I)$ is more highly correlated with true output growth than
initial $\Delta GDP(E)$, and that true output growth is correlated
with all these other cyclical indicators.  The second point above is a
stretch as well, and would only be the case if all other variables,
including latest $\Delta GDP(I)$, provided no information about
true output growth beyond that contained in latest $\Delta
GDP(E)$.  That is quite an extreme position in favor of the accuracy
of latest $\Delta GDP(E)$, and the results in the next section suggest
latest $\Delta GDP(I)$ does contain a considerable amount information
about true output growth missed by latest $\Delta GDP(E)$.
Regarding the first point, this may be a reasonable position for the
BEA to take.  For analysts, true output growth may the only
variable of interest for some purposes, but for other purposes this
will not be the case.

The regression results in table 2 are broadly consistent with those in
Nalewaik (2007a), who uses Markov switching models to show that
$\Delta GDP(I)$ identifies cyclical turning points more quickly than
$\Delta GDP(E)$ in real time.  Specifically, at the NBER-defined start
of the 1980, 1981-2, 1990-1 and 2001 recessions, real-time estimates
of a Markov switching model using $\Delta GDP(E)$ alone put the odds
that the economy was in a low-growth state at 52$\%$, 40$\%$, 45$\%$,
and 23$\%$, respectively.  Adding $\Delta GDP(I)$ to the model
produced much-more accurate probabilities: 78$\%$, 44$\%$, 72$\%$, and
70$\%$.
%This result holds whether one
%defines turning points using the NBER's recession dating, or as
%transitions to low-growth states endogenously determined by the
%smoothed probabilities from the Markov switching models.  
Most of the research in Nalewaik (2007a) was carried out in 2005, and
the subsequent cyclical downturn was the first out-of-sample test of
the main hypotheses of the paper.  The model using $\Delta GDP(I)$
again performed much better around the start of the downturn in real
time, and also performed better than some popular models using monthly
indicators; see section 5.

While this section has focused on the ``third'' growth rates, the
information content of the preceding ``advance'' and ``second'' growth
rates are of critical importance for analysis in real time, and
Appendix D reports results for these vintages, as well as results for
the estimates once they have passed through their first annual
revision.  Briefly, when an official ``second'' $\Delta GDP(I)$
estimate is available, the results using ``second'' growth rates are
very similar to those reported in this section using the ``third''
growth rates.  And, as discussed above, the results using the first annual
revision growth rates are even more favorable to $\Delta GDP(I)$ than
the results using ``third'' growth rates, showing a statistically
significant tendency for $\Delta GDP(E)$ to revise towards $\Delta
GDP(I)$ over the full sample.

For the ``advance'' estimates, when an official $\Delta GDP(I)$ is not
available, the situation is quite different.  It should be noted that
the constructed ``advance'' $\Delta GDP(I)$ used in the Appendix
employ only lags and other ``advance'' NIPA components to forecast
profits; some companies have reported their quarterly profits numbers
at the time of the ``advance'' release, and incorporating this
information may produce a much-improved ``advance'' $\Delta GDP(I)$
estimate.  That said, these rather limited ``advance'' $\Delta GDP(I)$
estimates perform poorly compared to the official ``advance'' $\Delta
GDP(E)$ estimates, which better predict most of the business cycle
variables used in this section.  In addition, when predicting latest
$\Delta GDP(I)$, about two-thirds weight should be placed on
``advance'' $\Delta GDP(E)$, and only about one-third weight should be
placed on the constructed ``advance'' $\Delta GDP(I)$ estimates.  This
suggests that the initial estimates of corporate profits produced by
the BEA are highly informative, and cannot be easily predicted based
on lags or other available NIPA variables.  For fourth quarter
``second'' estimates, when official profits numbers remain
unavailable, this is presumably the case as well.

\section{Information Content of the Latest Growth Rates}

\subsection{The Cyclicality of the Latest Estimates}

Table 1 showed that the correlation between $\Delta GDP(E)$ and
$\Delta GDP(I)$ dropped sometime around the mid-1980s, and the
divegences between the estimates also became highly cyclical around
that time.  Figure 5 shows this using year-over-year growth rates:
GDP(I) rose faster than GDP(E) through most of the 1990s boom and the
comparatively-short boom period from 2004 to 2006, while GDP(I) growth
fell below GDP(E) growth in the 2001 recession and the latest cyclical
downturn.\footnote{%
  We should keep in mind that these data are subject to further annual
  and benchmark revisions.}  Figure 6 plots the statistical
discrepancy (GDP(E) minus GDP(I)) as a percent of GDP(E) versus the
unemployment rate; work by Charles Fleischman first examined this
relation, to my knowledge.  Fleischman and John Roberts (2010) have
studied the relation between GDP(E), GDP(I), the unemployment rate and
other variables in the context of a state space model of the business
cycle; their work points to the unemployment rate as an excellent
measure of the state of the business cycle, as well as suggesting
GDP(E) is measured with more error than GDP(I).  Figure 6 shows the
measurement errors in either GDP(I) or GDP(E) are clearly
systematically related to the business cycle, and the statistical
discrepancy is not noise, as is commonly assumed.

To understand the relation in Figure 6 better, consider a very simple
model.  It should be noted that Nalewaik (2008) shows why the type of
model outlined below is an incomplete characterization of the growth
rates of GDP(E) or GDP(I), and outlines models that fit the evidence
better.  However, the model outlined below is useful for the limited
purpose of framing the subsequent discussion.  Let true output be
$Y_t^\star$, and assume we can decompose this into trend $\tau_t$ and
cycle $\psi_t$, so $Y_t^\star = \tau_t + \psi_t$.  The unemployment
rate $U_t$ is governed by an Okun's law relation:
\begin{eqnarray*}
U_t - U_t^n = \gamma \left(Y_t^\star - \tau_t \right) = \gamma \psi_t, & \gamma < 0.
\end{eqnarray*}
Now, assume GDP(I) and GDP(E) are systematically either too cyclical
or not cyclical enough, so:
\begin{eqnarray*}
GDP(E)_t & = & \tau_t + \alpha_E \psi_t \qquad \qquad \mbox{and:} \\
GDP(I)_t & = & \tau_t + \alpha_I \psi_t.
\end{eqnarray*}
Then the statistical discrepancy $SD_t = GDP(E)_t - GDP(I)_t =
\left(\alpha_E - \alpha_I\right) \psi_t$, and assuming the systematic
mismeasurement is not identical for the two estimates, we should
observe a relation between the discrepancy and the unemployment rate:
\begin{eqnarray*}
U_t - U_t^n = \frac{\gamma}{\alpha_E - \alpha_I} \left( SD_t \right) & \gamma < 0.
\end{eqnarray*}
The strong positive relation shown in Figure 6 then implies $\alpha_E
< \alpha_I$ - i.e. that the magnitude of the cycle is smaller in
GDP(E) than in GDP(I).  Table 3 shows regressions based on this
relation.  The first panel shows that the unemployment rate captures
more than 60 percent of the variability of the discrepancy from 1984Q3
through 2006Q4, and though the statistical discrepancy is highly
autocorrelated, the unemployment rate remains significant when an AR1
term is added.
%Assuming $\gamma$
%lies between minus 0.3 and minus 0.5 (minus 0.4 is a commonly-assumed
%value), the 0.88 coefficient from the specification without the AR1
%term implies $\alpha_E - \alpha_I$ lies between 0.34 to 0.57.  The
%differences between the estimates, then, represent a third of the true
%business cycle in the best case, and more than half the cycle in the
%worst case.  
The second panel of table 3 shows specifications in first differences,
to isolate the higher-frequency variation in the data.  The first
difference exhibits some negative autocorrelation, but the coefficient
on the differenced unemployment rate remains positive, and the
relation is highly significant when the differenced unemployment rate
is lagged one quarter.  These regression results confirm that the
statistical discrepancy is not noise, even in differences.

Having established that $\alpha_E < \alpha_I$ within the context of
this very stylized model, we can consider three possibilities:
\begin{enumerate}
\item Both GDP(I) and GDP(E) are more cyclical than true output,
  so $\alpha_I > \alpha_E > 1$.  In this case, GDP(E) represents the
  cycle in true output better than GDP(I).
\item Both GDP(I) and GDP(E) are less cyclical than true output,
  so $\alpha_E < \alpha_I < 1$.  In this case,
  GDP(I) represents the cycle in true output better than GDP(E).
\item GDP(E) is less cyclical and GDP(I) is more cyclical than
  true output, so $\alpha_E < 1 < \alpha_I$.  In this case, GDP(I)
  represents the cycle in true output better than GDP(E) if
  $\alpha_I - 1 < 1 - \alpha_E$.
\end{enumerate}

These possibilities frame the detailed discussion of the source data
incorporated into the latest estimates in Appendix C.  Plenty of
evidence suggests GDP(E) misses part of the business cycle, implying
possibility 1 is unlikely.  Some of the construction components of
GDP(E) are smoothed; in particular, the additions and alterations
(adds and alts) component of residential structures is smoothed using
a three-year moving average.  This is problematic, because smoothed
estimates inherently understate the magnitude of business cycle
accelerations and decelerations.  While adds and alts is a small
component of GDP(E), it may have taken on outsized importance in the
late 2000's downturn, and may have contributed to some of the
fluctuations in the discrepancy around the 1990-1 recession.  Probably
more important, over most of this sample, the type of annual surveys
used to compute the goods-producing sector of GDP(E) simply did not
exist for most the (enormous) service-producing sector.  As such, the
BEA was forced to cobble together estimates based on trade-source,
administrative, and regulatory data that may have missed part of the
business cycle.  For example, these data sources may miss fluctuations
in the output of sole proprietors and some small businesses, highly
cyclical parts of the economy.  And the activities of many types of
financial services companies or entities may have been missed by the
regulatory data used by the BEA to compute personal consumption
expenditures (PCE) for financial services.  The magnitude of the booms
and busts in financial services, then, may not be fully reflected in
the PCE component of GDP(E) or exports of services.  However, many of
these firms and entities likely did file tax forms, so their
activities would have been represented in the tax data used to compute
GDP(I).  This could explain part of the increase in the statistical
discrepancy in 1989, 2001, and in the latest episode.

Appendix C also discusses potential reasons why GDP(I) might be too
cyclical.  It is possible that some capital gains, which should be
excluded from the BEA's definition of output, may have been
misreported to the IRS as ordinary income and thus included in the tax
data used to compute GDP(I).  Capital gains are likely highly
procyclical, so failure to exclude these gains could have made GDP(I)
more cyclical than true output.  Although the evidence on this is
thin, possibility 3 might be slightly more likely than possibility 2.
However, the evidence in favor of GDP(E) understating the cycle is
stronger than the evidence in favor of GDP(I) overstating the cycle,
so if possibility 3 holds, it is probably the case that $\alpha_I - 1
< 1 - \alpha_E$.

The last panel of table 3 shows that the statistical discrepancy is
much less cyclical prior to the mid 1980s.  Why might that be the
case?  While PCE for services has always held a relatively large share
of GDP(E), averaging 30 percent from 1947 to 1984, its share shot up
to an average of 43 percent from 1985 to 2009, and the share reached
48 percent in 2009.  As the share of services PCE has increased, the
measurement problems in GDP(E) may have become more severe, and more
plainly visible.  In addition, booms and busts in financial services
may have accounted for a much larger share of the {\it variability} of
the business cycle since the mid-1980s, with the junk bond boom and
bust (as well as the savings and loan boom and bust) from the mid to
late 1980s, the day-trading boom in the mid to late 1990s and
subsequent stock market crash from 2000 to 2002, and the mortgage
securitization boom and bust from 2002 to 2008.  GDP(E) may have
missed much of this variation.  But whatever the reason, since the
really interesting divergences between the latest estimates occur in
the post-1984Q3 period, the remainder of this section focuses on this
sample.

\subsection{Information in the Revisions about the Latest Estimates}

%Since true output growth is unobserved, some assumptions must be made to test which 
%of the two latest estimates is closer to this unobserved variable.
Consider the following hypothetical example.  We have two time series
estimating the same unobserved variable of interest.  The two time
series happen to be identical, but we know the series are subject to
considerable measurement error and may deviate a lot from the true
variable of interest.  Suppose new information becomes available that
leads us to make large revisions to one of the estimates, bringing it
closer to the truth, while the other estimate remains unrevised.
Which estimate is now better?  Obviously, the estimate that revised is
better: we now know that it was way off initially and the revisions
corrected some or all of that measurement error, and the estimate that
did not revise remains way off.  More generally, if the estimates
start out identical, or pretty close, and the revisions improve the
estimates, then the estimate that revises more, on average, will tend
to be better than the estimate that revises less.  This is the
underlying logic of Fixler and Nalewaik (2007).  Table 1 shows that
the initial estimates of $\Delta GDP(I)$ and $\Delta GDP(E)$ do start
out with a very high correlation, but $\Delta GDP(I)$ revises more.
While the evidence in section 3 suggests that $\Delta GDP(I)$ starts
out as the better estimate, if we make the relatively uncontroversial
assumption that the revisions improve the estimates, then the larger
revisions imply $\Delta GDP(I)$ expands on its lead.  Fixler and
Nalewaik (2007) use this revisions evidence to place bounds on the
optimal weights to be placed on $\Delta GDP(I)$ and $\Delta GDP(E)$,
and the bounds are favorable to $\Delta GDP(I)$.

The revisions increase the variance of $\Delta GDP(I)$ more than the
variance of $\Delta GDP(E)$, implying that they add some news, or
actual variation in true output growth, to $\Delta GDP(I)$ that is not
added to $\Delta GDP(E)$---see Mankiw, Runkle and Shapiro (1984),
Mankiw and Shapiro (1986), and Fixler and Nalewaik (2007).  This
variation in true output growth missed by latest $\Delta GDP(E)$
growth and captured by latest $\Delta GDP(I)$ is closely related to
the business cycle.  In particular, Nalewaik (2007b) shows that the
revisions tend to reduce $\Delta GDP(I)$ more than $\Delta GDP(E)$ in
low-growth states, so the extent of the weakness of true output growth
in low-growth states appears to be part of the information missing
from $\Delta GDP(E)$ but appearing in $\Delta GDP(I)$ through its
more-informative revisions.  Since this weakness in low-growth states
appears in neither initial estimate and remains missing in latest
$\Delta GDP(E)$, if latest $\Delta GDP(E)$ is correct, the revisions
showing this relative weakness in latest $\Delta GDP(I)$ must damage
the estimates.  More broadly, any suggestion that latest $\Delta
GDP(E)$ is better than latest $\Delta GDP(I)$ would seem to imply that
the variability added to $\Delta GDP(I)$ through the revisions moves
it further away from the true output growth.  This seems hard to
believe, and carried to its logical conclusion, the BEA should stop
revising $\Delta GDP(I)$, and allocate its resources elsewhere.  I do
not think anyone at the BEA would seriously advocate taking that step.
In contrast, the standard interpretation of the revisions is less
problematic for the BEA: the revisions improve both $\Delta GDP(E)$
and $\Delta GDP(I)$, but the source data incorporated into $\Delta
GDP(E)$ are just not as informative as the source data incorporated in
$\Delta GDP(I)$.  But in that case, latest $\Delta GDP(I)$ is likely
the better estimate.

%Fixler and Nalewaik (2007) make further use of the revisions evidence
%to place bounds on the weights that should be placed on $\Delta
%GDP(E)$ and $\Delta GDP(I)$ to produce an optimal estimate of true
%output growth.  The estimates contain a lot of common information
%about true output growth, but the models in Fixler and Nalewaik
%(2007) allow for the possibility that $\Delta GDP(E)$ and $\Delta
%GDP(I)$ each contain fluctuations in true output growth missed by
%the other.  The weights are not identified when we allow for this
%possibility, but using the latest estimates that have passed through
%the 2009 benchmark revision, the bounds placed on the weights by the
%revisions are fairly tight.  The maximum relative weight on $\Delta
%GDP(E)$ gives it about the same weight as $\Delta GDP(I)$, while the
%maximum relative weight on $\Delta GDP(I)$ gives it almost twice as
%much weight as $\Delta GDP(E)$.

\subsection{Relation to Other Business Cycle Variables}

The logic behind these tests is similar to the logic behind the
regression results in table 2, but table 4 switches the regression
order and reports results from pairs of regressions, one of latest
$\Delta GDP(I)$ and one of latest $\Delta GDP(E)$ on each cyclical
indictor.  The cyclical indicator is reported in the first column of
the table, while the $R^2$s in the second and third columns show that
latest $\Delta GDP(I)$ is more highly correlated with every single one
of these cyclical indicators.  Appendix D repeats these results using
annual instead of quarterly data, and the results are quite
similar.\footnote{%
  Much of the source data incorporated at annual revisions is annual
  frequency, with no information on quarterly patterns, so the
  quarterly numbers are likely less reliable than the annuals.  For
  example, the BEA is confident that employee gains from exercising
  nonqualified stock options net out of the annual GDP(I) estimates
  (since profits fall by the same amount as the increase in
  compensation), but they are concerned the quarterly pattern within
  years may be distorted.}  It is more highly correlated with lagged
stock price changes, the lagged slope of the yield curve, and the
lagged spread between high-yield corporate bonds and treasury bonds
(using a somewhat shortened sample).\footnote{%
  In his comments, J. Steven Landefeld suggests that the stock market
  may be more highly correlated with $\Delta GDP(I)$ because capital
  gains may be ``leaking'' into $\Delta GDP(I)$.  If this were the
  case, the correlation between changes in the stock market and
  $\Delta GDP(I)$ should be contemporaneous, especially at the annual
  frequency, since a rising stock market translates immediately into a
  capital gain.  Appendix D shows that the evidence does not support
  this: using the annual output growth measures, $\Delta GDP(E)$ is
  slightly more correlated with the contemporaneous change in the
  stock market, while $\Delta GDP(I)$ is more highly correlated with
  the stock market change from one year earlier.  The evidence is more
  suggestive of either the stock market anticipating changes in true
  output, or changes in the stock market affecting true output with a
  lag, with true output better represented by $\Delta GDP(I)$.  See
  also Nalewaik (2008).}  It is more highly correlated with short and
long differences of the unemployment rate, both contemporaneously and
at leads and lags; the same holds true for the household survey
measure of employment growth.  Recall that there is no reason to
suspect these measures to be spuriously correlated with $\Delta
GDP(I)$; see footnote 6.  It is more highly correlated with the
manufacturing ISM, and using a shorter sample, the non-manufacturing
ISM.  It is also more highly correlated with dummies for NBER
recessions; see also Nalewaik (2007a).\footnote{%
  Note that these higher correlations, often substantially higher, are
  evidence against the crude model outlined in section 4.1.  In that
  model, the $\Delta GDP(E)$ and $\Delta GDP(I)$ contain rescaled
  versions of the same business cycle fluctuations, in which case the
  $R^2$ must be equal across the two regressions.  That is clearly not
  the case; $\Delta GDP(I)$ contains different business cycle
  fluctuations, fluctuations that also show up in these other business
  cycle variables.  Nalewaik (2008) uses essentially this same
  argument to reject a crude rescaling model in favor of the LoSE
  model; see below.  Nevertheless, both the LoSE model and the
  rescaling model say the same thing, broadly speaking: GDP(E) growth
  misses some of the business cycle fluctuations in true output
  growth, fluctuations that show up in GDP(I) growth as well as other
  variables.  }

As in table 2, latest $\Delta GDP(I)$ may be more highly correlated
with all these variables because latest $\Delta GDP(E)$ is
contaminated with more noise.  But the interpretation of the revisions
provided in the previous subsection suggests an alternative
measurement error story, namely, that latest $\Delta GDP(E)$ misses
variation in true output growth that appears in all these cyclical
indicators and is also picked up by latest $\Delta GDP(I)$.  In either
case, latest $\Delta GDP(I)$ is the better estimate of true output
growth.  Is there any interpretation of these results where that is
not the case?  Latest $\Delta GDP(I)$ would have to contain
measurement errors uncorrelated with true output growth, but
correlated with all these other variables.  That seems highly
unlikely.

For the more econometrically-oriented reader, these regressions
provide provide formal tests of what I consider the most likely
hypothesis explaining these correlations (partially based on the
revisions evidence in the previous subsection): that latest $\Delta
GDP(E)$ is missing some of the variability of true output growth
that is reflected in latest $\Delta GDP(I)$ and these other time
series.
%Nalewaik (2008) hypothesizes that
%current and lagged stock price changes and bond spreads (spreads
%between long- and short-maturity treasury bonds, and between corporate
%bonds and treasury bonds) might contain such missing variation.  As a
%warmup, the paper starts by showing that these stock and bond prices
%predict the latest, fully-revised estimates of $\Delta GDP(I)$ and
%$\Delta GDP(E)$ better than the initial estimates, suggesting the
%asset prices contain some information about true output growth
%missed by the initial estimates but included in the latest estimates.
%Next, the paper shows that the asset prices are also better predictors
%of latest, fully-revised $\Delta GDP(I)$ than of latest, fully-revised
%$\Delta GDP(E)$, suggesting the analogous result: the asset prices
%contain information about true output growth missed by latest
%$\Delta GDP(E)$, but included in latest $\Delta GDP(I)$ .
Nalewaik (2008) derives such tests for Lack of Signal Error (LoSE), as
the paper puts it.  The maintained assumption is that the time series
used for testing captures some of the variation missing from one
estimate but included in the other.  Regressions are run of each
estimate on the testing variable.  Nalewaik (2008) shows that the LoSE
biases the regression coefficient on the testing variable towards
zero, so the regression using the estimate that contains more LoSE
yields a coefficient closer to zero.  Note that it is measurement
error of the LoSE form in the {\it dependent} variable that causes
this attenuation bias, precisely opposite the conventional wisdom
about classical measurement error (i.e. that it is measurement error
in the {\it explanatory} variable that causes attenuation bias).
Testing the equality of the coefficients on the testing variable
across the two regressions, Nalewaik (2008) rejects using the asset
price variables that are also employed in the first five
specifications of table 4, using a slightly different sample.  If the
model assumptions hold, then $\Delta GDP(I)$ contains more signal
about true output growth than $\Delta GDP(E)$, signal that is
reflected in stock and bond prices.  Table 4 shows that this missing
signal also appears in the differenced unemployment rate, household
survey employment growth, and the ISM measures.  The coefficients in
table 4 are all larger, in absolute value, when $\Delta GDP(I)$ is the
dependent variable; a relatively large amount of noise in $\Delta
GDP(E)$ cannot explain these results, but a relatively large amount of
LoSE can.

%We can run a similar set of regressions using changes to the
%unemployment rate $\Delta U_t$.  If $\Delta GDP(E)$ lacks signal about
%true output growth that is reflected in $\Delta GDP(I)$, and the
%changes in the unemployment rate pick up some of that variation, a
%regression of $\Delta GDP(E)$ on $\Delta U$ will have a smaller
%coefficient than a regression of $\Delta GDP(I)$ on $\Delta U$.  Table
%8 reports the regression results using a variety of lead-lag
%relationships and averages for the unemployment rates.  In each
%specification, the coefficient is larger in absolute value (i.e. more
%negative) when we use $\Delta GDP(I)$.  When using quarterly $\Delta
%U$, the specifications are typically too noisy to allow rejections of
%the equality of the coefficients, but when averaging the unemployment
%rate into 4-quarter changes, as in the last four specifications, we do
%reject the hypothesis that the coefficients are equal across the two
%regressions.  Changes in the unemployment rate have larger effects on
%$\Delta GDP(I)$ than on $\Delta GDP(E)$, contemporaneously, and at
%both leads and lags.  

Less formal comparisons of GDP(E) and GDP(I) with other sources of
information about the business cycle are also informative.  In
particular, we can compare the peaks and troughs in GDP(E) and GDP(I)
with the NBER peak and trough dates.  Grimm (2005) does this; Figures
7a, 7b, and 7c show the results graphically for the three recessions
prior to the most recent one.  The one case where GDP(I) seems to
differ from the NBER is the 1990-1 recession: GDP(I) starts declining
during the NBER peak quarter while GDP(E) is flat, but since the
monthly peak was July 1990, the 1990Q3 GDP(I) decline seems consistent
with the NBER dating.  In the 1981-2 recession, the NBER called the
trough in 1982Q4, the same quarter as GDP(I), while GDP(E) calls the
trough three quarters earlier.  In the 2001 recession, it is difficult
to discern any real cyclical downturn in GDP(E), while the NBER peak
and trough dates line up perfectly with GDP(I).  These peak and trough
dates summarize the information in several other reliable indicators,
and the fact that they line up better with GDP(I) is again suggestive
that GDP(I) is the better estimate.

\section{The Estimates over the 2007-2009 Cyclical Downturn}

The recent downturn looks considerably worse when output is measured
using GDP(I) instead of GDP(E).  First, the effect on output appears
sooner, with GDP(I) showing a sharp deceleration even before the NBER
peak in late 2007.  This deceleration was somewhat evident in the real
time estimates of GDP(I), but more importantly, the actual recession
itself was much more evident in the real time estimates of GDP(I) than
in the real time estimates of GDP(E).  Second, the steepness of the
plunge in output in late 2008 and early 2009 appears worse.  And
third, with the BEA's February 2010 data release, the decline in
output now appears more prolonged, extending into the summer of 2009.

Figure 8 shows levels of GDP(E) and GDP(I) as they were measured at
different dates, indexing to 2006Q1 because the levels of the two
series are different and have changed with revisions.  The light and
dark dotted lines show the GDP(E) and GDP(I) estimates towards the
start of the recession, in early 2008.  The estimates are trending up
at a similar pace through 2006 and the first half of 2007, but the
estimates then diverge considerably.  GDP(I) shows an economy in a
much more vulnerable state in late 2007, with output essentially flat
over the second half of 2007.  Meanwhile, GDP(E) showed little of this
vulnerability in the second half; although growth was weak in 2007Q4,
that weakness came on the heels of estimated annualized growth of
almost 5 percent in 2007Q3.\footnote{%
  At the end of March 2008, the bivariate Markov switching model using
  $\Delta GDP(I)$ in Nalewaik (2007a) estimated a probability of
  around 90$\%$ that the economy had downshifted to a low growth state
  by 2007Q4, and this probability remained well above 50$\%$
  throughout 2008.  At the same time, a Markov-switching model using
  $\Delta GDP(E)$ alone estimated a probability of less than 20$\%$
  that the economy had downshifted to a low-growth state, a
  probability that remained low through most of 2008 (for example
  27$\%$ at the end of September), only cracking 50$\%$ after the
  BEA's ``advance'' 2008Q3 estimates released at the end of October.
  Models based on monthly indicators did not do better: an
  implementation of the Diebold-Rudebusch (1996) monthly indicators
  model, based on Kim and Nelson (2000), did not jump above 50 percent
  until early November 2008, with the BEA's release of its initial
  personal income numbers for September 2008.  The behavior of these
  models shows that real time assessments of the state of the business
  cycle can be meaningfully improved by looking at GDP(I).}

The light and dark dashed lines of Figure 8 show the GDP(E) and GDP(I)
estimates at the end of December 2008, after the NBER had called
December 2007 a business cycle peak.  All four of the monthly
indicators the NBER uses to date business cycles had peaked in late
2007 and early 2008, and GDP(I) was trending down slightly through the
first three quarters of 2008 as well.  GDP(E) was the only anomaly,
showing continued growth at an annual rate of almost 2 percent in the
first half of 2008.

The light and dark solid lines of Figure 8 show the latest GDP(E) and
GDP(I) estimates, and we see that the initial $\Delta GDP(E)$
estimates for 2008 have since revised down considerably towards the
initial $\Delta GDP(I)$ estimates, a notable continuation of the
recent pattern in revisions discussed in section 3.  Revisions have
also reduced GDP(I), but the revisions to growth came mainly in the
first half of 2007.
%Figure 6 shows the enormous swing in the statistical discrepancy, from
%around minus two percent of GDP(E) in late 2006 to almost plus two
%percent of GDP(E) in 2009Q3.  Figure 5 shows the marked deceleration
%in GDP(I) in 2007, while GDP(E) showed little evidence of such
%deceleration until 2008.  
The latest estimates show that GDP(I) was essentially flat over the
four quarters of 2007, declining in 2007Q1 and 2007Q3.  These latest
estimates suggest the recent cyclical downturn caused a measurable
deceleration in aggregate output much earlier than is commonly
believed.  Meanwhile, GDP(E) currently shows no such early
deceleration, growing 2.5 percent over the four quarters of 2007,
about the same as in 2006.  These differences over 2007 produce the
bulk of the enormous swing in the statistical discrepancy we saw in
Figure 6, from around minus 1.9 percent of GDP(E) in late 2006 to plus
1.8 percent of GDP(E) in 2009Q3.

The current estimates show $\Delta GDP(E)$ actually slightly weaker
than $\Delta GDP(I)$ in the first three quarters of 2008, but the
current estimates of $\Delta GDP(I)$ then show a steeper downturn over
the worst part of the recession.  The current annualized $\Delta
GDP(I)$ estimates for 2008Q4 and 2009Q1 are -7.3 and -7.7 percent,
worse than the $\Delta GDP(E)$ estimates of -5.4 and -6.4 percent.

Finally, the latest $\Delta GDP(I)$ estimates for 2009Q3, released in
late February 2010 and incorporating numbers from the Quarterly Census
of Employment and Wages (see Appendix C), have called into question
the timing of the trough of the recession.  Before these numbers were
released, a conventional wisdom was emerging that the recession likely
ended late in the second quarter of 2009, perhaps in June, with the
economy resuming growth in 2009Q3.  Figure 8 shows a modest rebound in
GDP(E) in 2009Q3, but no evidence of a rebound in GDP(I).  Personal
income less transfer payments and employment---two of the four
indicators most emphasized by the NBER business cycle dating
committee---have continued to decline in 2009Q3.
%; anyone still sure of a business cycle
%trough at the end of 2009Q2 must be ignoring these income-side estimates.

What are we to make of these important differences between GDP(E) and
GDP(I) over this cycle?  It should be noted that all these estimates
remain subject to considerable future revision, but the source data is
most concrete for 2007, which happens to be the period of the greatest
widening of the statisical discrepancy.  Currently, the corporate
profits and proprietors' income components of GDP(I) incorporate IRS
tax returns data through 2007, and declines in these two income
categories account for the bulk of the deceleration in GDP(I) that
year.  Proprietors' income increased about $\$$63 billion (nominal) in
2006 and fell $\$$37 billion in 2007,\footnote{%
  The 2007 decline in the raw IRS tax numbers was larger, about $\$$66
  billion (see BEA table 7.14), but the BEA cut this down with various
  adjustments, including the inventory valuation and capital
  consumption adjustments.} a deceleration of about $\$$100 billion.
The biggest declines in 2007 were in real estate, construction,
finance and insurance, as well as (less explicably) mining; see BEA
table 6.12D.  As noted in Appendix C, it is possible that some of the
decline in proprietors' income may have represented a decline in
capital gains from house flipping, which should not be included in the
relevant concept of output.  Real estate proprietors' income fell
$\$$24 billion in 2007, but it also fell $\$$14 billion in 2006,
suggesting this type of mismeasurement cannot explain much of the
widening of the statistical discrepancy in 2007.  Construction
proprietors' income decelerated from a $\$$6 billion increase in 2006
to a $\$$14 billion decline in 2007, with the current estimates
showing a $\$$46 billion decline in 2008.  Part of this decline in
proprietors' income should probably have shown up in lower spending on
residential improvements, but as discussed earlier and in Appendix C,
the BEA's averaging of their raw source data will tend to miss such a
large deceleration.  Currently, the raw estimates of improvements
spending from Census show 4 and 14 percent declines in 2007 and 2008,
respectively, steeper than the current BEA estimates of 1 and 4
percent declines.  If the Census numbers are correct, GDP(E) should be
$\$$5 billion lower in 2007 and about $\$$22 billion lower in 2008, so
this also explains only a small portion of the widening of the
statistical discrepancy.\footnote{%
  Some other data sources suggest much larger declines in spending on
  residential improvements.  For example, Greenspan and Kennedy (2005,
  2007) use Flow of Funds data and Mian and Sufi (2009) use data from
  credit rating agencies to show that households extracted a very
  large amount of home equity in the mid-2000s, before banks cut
  credit lines in 2007 and 2008 and equity extraction dropped
  dramatically.  Using survey evidence that households spend about a
  third of extracted home equity on home improvements---see Brady,
  Canner, and Maki (2000), Canner, Dynan and Passmore (2002), and
  Greenspan and Kennedy (2007) and the references therein---updated
  Greenspan-Kennedy estimates give declines in spending on home
  improvements of $\$$65 billion in 2007 and $\$$80 billion in 2008.
  Of course, this does not necessarily imply causality from equity
  extraction to spending, because households may have found other
  financing options absent the availability of home equity lines of
  credit.}
%The deceleration in proprietors' income in 2007 was fairly
%broad-based, with the mining, finance and insurance, professional and
%business services, and manufacturing industries each contributing
%substantially.

Corporate profits' increased about $\$$152 billion in 2006, and fell
$\$$67 billion in 2007 (a deceleration of about $\$$220 billion), and
the current estimates for 2008 show a decline of $\$$181 billion.  The
biggest decline in profits in 2007 was in the finance and insurance
industry; the $\$$54 billion decline in 2007 followed an increase of
$\$$4 billion in 2006 and a massive increase of about $\$$180 billion
from 2000 to 2005.  Looking more broadly, the sum of corporate
profits, proprietors' income, and wage and salaries for the finance
and insurance industry fell close to 4 percent in 2007, while PCE for
financial services increased more than 12 percent.  While these
categories are not strictly comparable, this is difficult to reconcile
without severe measurement error in either the income measures or
PCE.\footnote{%
  The output of financial services could also have shown up in
  exports, or as an intermediate input into the production of other
  industries.}  The BEA started worrying back in late 2007 about their
ability to strip out capital losses (bad debt expenses and asset
write-downs) from their initial estimates of financial companies'
profits, but the availability of the tax data for 2007 likely made
these subtractions much easier.  With the tax data ameliorating this
issue, the problems appear more concentrated in the measurement of
financial services PCE and services more generally on the expenditure
side, as discussed in the previous section and in Appendix C.  Given
the advent of the financial crisis and the disappearance of many
securitization markets in the second half of 2007, a 12 percent growth
rate for financial services PCE seems implausibly high.  To get a
sense of the magnitudes involved, a decline in financial services PCE
of 4 percent would have lowered GDP(E) in 2007 by $\$$76 billion from
its current level, and by more if PCE missed the boom in financial
services output over prior years.  More recently, profits in the
finance and insurance industry fell an additional $\$$91 billion in
2008 (with proprietors' income and wage and salary income also
falling), while financial services PCE increased once again.  Since
the tax data have not yet been incorporated for 2008, some risk
remains that the income declines were too steep, but again it seems
implausible that financial services PCE continued its uninterrupted
growth.

Overall, this evidence suggests that although there may be problems on
both sides of the accounts, the problems are likely more severe on the
expenditure side.  Given that, the latest downturn was likely
substantially worse than the current GDP(E) estimates show.  Output
likely decelerated sooner, fell at a faster pace at the height of the
downturn, and recovered less quickly than is reflected in GDP(E), and
in conventional wisdom.

\section{Concluding Thoughts}

%and that taking a weighted average of $\Delta
%GDP(E)$ and $\Delta GDP(I)$ over these past couple of years would
%probably balance the risks and produce the best possible estimate of
%true output growth over this period. 

Considerable evidence suggests that the growth rates of GDP(I) better
represent the business cycle fluctuations in true output growth
than do the growth rates of GDP(E).  For the initial growth rates, the
revisions evidence over the past 15 years, the correlations with other
business cycle indicators, and the recent behavior of the estimates
around cyclical turning points all point to this conclusion.  For the
latest estimates that have passed through their cycle of revisions,
careful consideration of the nature of the source data, statistical
analysis of the information added by the revisions, and statistical
tests as well as informal comparisons with other businesss cycle
indicators, again all suggest GDP(I) growth is better than GDP(E)
growth.  

These results strongly suggest that economists and statisticians
interested in business cycle fluctuations in U.S. output should pay
attention to the income-side estimates, and consider using some sort
of weighted average of the income- and expenditure-side estimates in
their analyses.  The evidence in this paper clearly suggests that the
weights should be skewed towards GDP(I), and Fixler and Nalewaik
(2007) are able to place fairly tight bounds on the optimal weights
for the latest estimates, bounds favorable to GDP(I).  But a 50-50
average would be a marked improvement over an average placing all its
weight on GDP(E).  Such a 50-50 average would be following in the
footsteps of the Council of Economic Advisers, who, after concluding
GDP(I) might be better than GDP(E) in their 1997 Economic Report of
the President, have subsequently given some weight to the income-side
estimates in their productivity analyses---see the 2008 Economic
Report of the President, p. 39, and the 2009 Economic Report of the
President, pp. 47-8.

The results here also have implications for the BEA.  When a quarterly
estimate of GDP(I) growth is available, the evidence here shows it is
likely a better estimate of output growth than the corresponding
GDP(E) estimate.  However, the first GDP(E) estimate for any given
quarter, the ``advance'' estimate, is typically released about a month
before the first GDP(I) estimate, and GDP(I) is delayed an additional
month when the BEA is producing estimates for fourth quarters.  These
delays occur because the BEA has incomplete information on corporate
profits, and is not comfortable releasing earlier estimates of
profits.  In general, the profits information released by the BEA
appears tremendously useful, and the BEA does have some information on
profits at these earlier release dates.  An ``advance'' estimate of
GDP(I) based on the available profits information might be quite
helpful for real time assessment of the speed of economic growth.
Earlier release of the fourth-quarter estimates, so a GDP(I) estimate
is released at least as early as the BEA's ``second'' release, might
be similarly helpful; the BEA has still not released an estimate of
GDP(I) growth for the fourth quarter of 2009 at the time of this
paper, March 19th, 2010.  These decisions will depend on how much
information on profits is really available at these earlier dates, and
a thorough assessment of this issue seems to be in order.

The BEA, the Census Bureau, and the BLS doubtless will continue making
improvements in their estimates where feasible, and the good news is
that there have been substantial improvements over the past few years.
The data on services have taken leaps and bounds forward, with the
advent of Census's Quarterly Service Survey in 2003, the recent
expansions in the coverage of this survey, and the expansions in the
coverage of the Service Annual Survey.  Further improvements are in
train: in December 2010, the estimates from the Service Annual Survey
will roughly double in coverage, expanding to mimic the sector
coverage of the Economic Census.\footnote{%
  http://www.census.gov/services/index.html} These data should improve
the estimates of PCE and GDP(E).

However, despite these improvements, problems with the output growth
estimates will inevitably remain, and lack of coverage of services is
only one of several important limitations of GDP(E).  All the results
in this paper suggest that the current reporting practice of the BEA,
which puts nearly exclusive emphasis on GDP(E) over GDP(I), is
suboptimal statistically.  The BEA creates tremendous value by
producing an income-based estimate of output growth, but the current
BEA reporting practice downplays it so much that many analysts may not
even be aware of its existence.  The BEA's typical press release
hardly ever discusses GDP(I), and it is reported only towards the back
of the release as a nominal level, so analysts would have to deflate
and compute annualized quarterly growth rates themselves to arrive at
the number comparable to headline real GDP(E) growth.

If the BEA found the results here persuasive, they may want to
consider taking several incremental steps towards increasing the
prominence of GDP(I).  Most obviously, the BEA could report real
annualized growth rates of GDP(I) in its press releases, preferably in
table 1 of the release so they can be compared easily to the
annualized growth rates of real GDP(E).  Second, it could give those
annualized growth rates more prominence in the text of the press
releases, discussing them at a level of detail similar to its current
discussion of GDP(E).  The BEA's discussion of the corporate profits
estimates could be rolled into a more general discussin of GDP(I).
Third, the BEA could bring more balance to their statements about the
reliability of GDP(E) and GDP(I).  Landefeld, Seskin and Fraumeni
(2008) take a small step in this direction by stating: `` ... these
studies remind users that it is useful to look at growth in both GDP
and gross domestic income in assessing the current state of the
economy.''\footnote{%
  Landefeld, Seskin and Fraumeni (2008), p. 211.}

Featuring two measures of output growth in the same press release
would raise communication challenges, and the BEA might fear that
having two featured measures could be too confusing for casual
analysts.\footnote{%
  The BLS does reports two estimates of the monthly change in
  employment in its employment report, one from both the establishment
  survey and one from the household survey, but there are clear
  statistical reasons for favoring the establishment survey number at
  the monthly frequency.  For the case of GDP(E) and GDP(I), making
  the case in favor of one measure over the other is more
  complicated.}  If the BEA is concerned about the communication
issue, they may look to the example of other countries such as Great
Britain and Australia, who report an average estimate from the
different sides of the accounts as their featured output growth
measure.  The BEA has considered taking this step in the past---see,
for example, Moulton (2000)---and the BEA could report such an average
of GDP(E) and GDP(I) as GDP(A).\footnote{%
  Of course, the components of GDP(E) will not sum to the topline
  GDP(A), nor will the components of GDP(I) sum to the topline, which
  may be confusing for some analysts.  But if the evidence in this
  paper is convincing, it is the current state of affairs that the
  components of GDP(E) do not sum to true output or even the best
  estimate of true output; in fact, the sum of the components of
  GDP(E) miss important, systematic variation in true output.  The
  new reporting will simply make these facts explicit.  Over the
  long-run, allocating parts of the discrepancy to different
  components of GDP(E) and GDP(I) may be the right thing to do, but it
  is an extremely complicated task, and much research will need to be
  done before implementing anything.  But if BEA attempts to go down
  this path at some point, they should do this in a transparent and
  easily replicable fashion. The BEA is to be commended for their
  transparency in reporting the statistical discrepancy, and should do
  nothing to compromise this transparency.}  The BEA could employ
optimal weights guided by statistical analysis, as in Fixler and
Nalewaik (2007), but the results here suggest that featuring even a
straight 50-50 average would be a marked improvement over the current
practice of featuring GDP(E) alone.
%Given all the uncertainties and difficulties associated with
%measurement, featuring such an average seems the safest and soundest
%approach going forward.

\vskip 0.6cm
\begin{thebibliography}{99}
%\centerline{\bf References}
\vskip 0.6cm

\singlespacing

\bibitem {ab} Abel, A. and Bernanke, B. (2001) {\it Macroeconomics},
Fourth Edition, Boston: Addison Wesley.

\bibitem {gk2} Beaulieu, J. and Bartelsman, E. (2004) ``Integrating
  Expenditure and Income Data: What to do with the Statistical
  Discrepancy?'' Board of Governors of the Federal Reserve System,
  FEDS working paper 2004-39.

\bibitem {bcm} Brady, P., Canner, G., and Maki, D. (2000), ``The
  Effects of Recent Mortgage Refinancing.''  {\it Federal Reserve
    Bulletin} July 2000: 441-450.

\bibitem {bea1} Bureau of Economic Analysis, U.S. Department of
  Commerce (2002) ``Corporate Profits,'' Methodology Paper, September 2002.

\bibitem {bea1} Bureau of Economic Analysis, U.S. Department of
  Commerce (2008) ``Updated Summary of NIPA Methodologies,'' {\it
    Survey of Current Business}, November 2008, pp. 8-25.

\bibitem {bea2} Bureau of Economic Analysis, U.S. Department of
  Commerce (2008) ``Concepts and Methods of the U.S. National Income
  and Product Accounts (Introductory Chapters 1-4),'' July 2008.

\bibitem {cea} Council of Economic Advisers (1997), {\it Economic
    Report of the President,} Washington, D.C.: United States
  Government Printing Office.

\bibitem {cea} Council of Economic Advisers (2009), {\it Economic
    Report of the President,} Washington, D.C.: United States
  Government Printing Office.

\bibitem {cdp} Canner, G., Dynan, K., and Passmore, W. (2002),
  ``Mortgage Refinancing in 2001 and Early 2002.''  {\it Federal
    Reserve Bulletin} December 2002: 469-481.

\bibitem {dr} Diebold, Francis X. and Rudebusch, Glenn D.  ``Measuring
  Business Cycles: A Modern Perspective.'' {\it Review of Economics
    and Statistics}, 1996 (101), pp. 67-77.

\bibitem {de} Dynan, K. and Elmendorf, D. (2001), ``Do Provisional
  Estimates of Output Miss Economic Turning Points?''  Board of
  Governors of the Federal Reserve System, FEDS working paper 2001-52.

\bibitem {fg} Fixler, D. and Grimm, B. (2006) ``GDP Estimates:
  Rationality tests and turning point performance,'' {\it Journal of
    Productivity Analysis}, 25, pp. 213-229.

\bibitem {fn} Fixler, D. and Nalewaik, J. (2007) ``News, Noise, and
  Estimates of the True Unobserved State of the Economy,'' Board of
  Governors of the Federal Reserve System, FEDS working paper 2007-34.

\bibitem {fr} Fleischman, C. and Roberts, J. (2010) ``A Multivariate
  Estimate of Trends and Cycles,'' Federal Reserve manuscript.

\bibitem {fg} Greenspan, A. (2004) ``Risk and Uncertainty in Monetary
  Policy'' {\it American Economic Review}, 94, pp. 33-40.

\bibitem {gk1} Greenspan, A. and Kennedy, J. (2005) ``Estimates of
  Home Mortgage Originations, Repayments, and Debt on
  One-to-Four-Family Residences,'' Board of Governors of the Federal
  Reserve System, FEDS working paper 2005-41.

\bibitem {gk2} Greenspan, A. and Kennedy, J. (2007) ``Sources and Uses
  of Equity Extracted from Homes'' Board of Governors of the Federal
  Reserve System, FEDS working paper 2007-20.

\bibitem {grim} Grimm, B. (2005) ``Alternative Measures of U.S.
  Economic Activity in Business Cycles and Dating,'' BEA working
    paper 2005-05.

\bibitem {gw} Grimm, B. and Weadock, T. (2006), ``Gross Domestic
  Product, Revisions and Source Data.''  {\it Survey of Current
    Business} February 2006: 11-15.

\bibitem {hg} Holdren, A. and Grimm, B. (2008), ``Gross Domestic
  Income, Revisions and Source Data.''  {\it Survey of Current
    Business} December 2008: 14-20.

\bibitem {ham} Hamilton, James D.  1990.  Analysis of Time Series Subject
  to Changes in Regime.  {\it Journal of Econometrics} 45: 39-70.

\bibitem {kdp}  Koenig, Evan; Dolmas, Sheila; and Piger, Jeremy.
  ``The Use and Abuse of Real-Time Data in Economic Forecasting.''
  {\it Review of Economics and Statistics} 85 (August 2003), pp. 618-628.

\bibitem {k} Krakower, H. (2007) ``Improved Measures of the
  Misreporting of Income,'' presentation at National Economic Accounts
  Data Users' Conference, April 2007.

\bibitem {kdp}  Landefeld, E., Seskin, E., and Fraumeni, B.
  ``Taking the Pulse of the Economy: Measuring GDP.''
  {\it Journal of Economic Perspectives} 22 (Spring 2008), pp. 193-216.

\bibitem {mrs} Mankiw, N., Runkle, D., and Shapiro, M.  ``Are
  Preliminary Announcements of the Money Stock Rational Forecasts?''
  {\it Journal of Monetary Economics}, 1984 (14), pp. 15-27.

\bibitem {ms} Mankiw, N. and Shapiro, M.  ``News or
  Noise: An Analysis of GNP Revisions'' {\it Survey of Current
    Business}, May 1986, pp. 20-25.

\bibitem {sm} McCully, C., and Payson, S. (2009), ``Preview of the 2009
  Comprehensive Revision of the NIPAs.''  {\it Survey of Current Business} May 2009: 6-16.

\bibitem {fr} Mian, A. and Sufi, A. (2009) ``House Prices, Home
  Equity-Based Borrowing, and the U.S. Household Leverage Crisis,''
  manuscript, University of Chicago Booth School of Business, July
  2009.

\bibitem {fg} Moulton, B. (2000) ``Getting the 21st-Century GDP Right:
  What's Underway?'' {\it American Economic Review}, 90, pp. 253-258.

\bibitem {moy} Moylan, C. (2008) ``Employee Stock Options and the
  National Economic Accounts,'' {\it Survey of Current Business},
  February 2008, pp. 7-13.

\bibitem {nal1} Nalewaik, J., (2007a), ``Estimating Probabilities of
  Recession in Real Time Using GDP and GDI,'' Board of Governors of
  the Federal Reserve System, FEDS working paper 2007-07.

\bibitem {nal2} Nalewaik, J., (2007b), ``Incorporating Vintage
  Differences and Forecasts into Markov Switching Models.''  Board of
  Governors of the Federal Reserve System, FEDS working paper 2007-23.

\bibitem {nal3} Nalewaik, J., (2008), ``Lack of Signal Error (LoSE) and
  Implications for OLS Regression: Measurement Error for Macro Data.''
  Board of Governors of the Federal Reserve System, FEDS working paper
  2008-15.

\bibitem {ss} Perry, G. (2005), ``Gauging Employent: Is the
  Professional Wisdom Wrong?''  {\it Brookings Papers on Economic
    Activity} 2005(2): 285-321.

\bibitem {ss} Seskin, E., and Smith, S. (2009), ``Improved Estimates
  of the National Income and Product Accounts, Results of the 2009
  Comprehensive Revision.''  {\it Survey of Current Business}
  September 2009: 15-35.

\end{thebibliography}

\newpage

\singlespacing

\centerline {\bf Table 1: Variances and Correlations,}
\centerline {\bf Initial (3rd) and Latest Estimates of $\Delta$ GDP(E) and $\Delta$ GDP(I)} 
\vskip 0.5cm

\centerline {\bf Correlation Matrix, 1978Q3 to 2009Q3:}
\begin{longtable}{c c c c c}
& \multicolumn{2}{c}{Initial (3rd) Estimates} & \multicolumn{2}{c}{Latest Estimates} \\
& $\Delta$ GDP(E) & $\Delta$ GDP(I) & $\Delta$ GDP(E) & $\Delta$ GDP(I) \\
\hline \hline
\input{table_corr2.txt}
\hline \hline
\end{longtable}

\vskip 0.5cm
\centerline {\bf Correlation Matrix, 1984Q3 to 2006Q4:}
\begin{longtable}{c c c c c}
& \multicolumn{2}{c}{Initial (3rd) Estimates} & \multicolumn{2}{c}{Latest Estimates} \\
& $\Delta$ GDP(E) & $\Delta$ GDP(I) & $\Delta$ GDP(E) & $\Delta$ GDP(I) \\
\hline \hline
\input{table_corr.txt}
\hline \hline
\end{longtable}

\vskip 0.5cm
\centerline {\bf Variances:}
\begin{longtable}{c c c c}
\multicolumn{2}{c}{1978Q3 to 2009Q3} & \multicolumn{2}{c}{1984Q3 to 2006Q4} \\
$\Delta$ GDP(E) & $\Delta$ GDP(I) & $\Delta$ GDP(E) & $\Delta$ GDP(I) \\
\multicolumn{4}{c}{Initial (3rd) Estimates}\\
\hline \hline
8.53 & 8.90 & 3.88 & 3.89 \\
\hline \hline
\multicolumn{4}{c}{Latest Estimates} \\
\hline \hline
 9.44 & 10.29 & 4.23 & 4.96 \\
\hline \hline
\multicolumn{4}{c}{Revision: Latest-Initial (3rd)} \\
\hline \hline
  2.78 & 3.60 & 2.57 & 3.05 \\
\hline \hline
\end{longtable}

\newpage

\centerline {\bf Table 2: Predictive Content of Initial Growth Rates, 1978Q3-2009Q3} 
\vskip 0.1cm
{\scriptsize
\begin{longtable}{c c c c c c c c c}
\hline \hline
 & \multicolumn{7}{c}{Explanatory Variables} & \\
 & \multicolumn{3}{c}{$\Delta GDP(E)$, initial (3rd)} & \multicolumn{3}{c}{$\Delta GDP(I)$, initial (3rd)} & Constant & Adj. $R^2$ \\
Dependent Variable & $t$ & $t-1$ & $t-2$ & $t$ & $t-1$ & $t-2$ & & \\
\hline \hline
\input{table_GDP_forecasts_2.txt}
\input{table_GDI_forecasts_2.txt}
\input{table_aux_forecasts.txt} \hline
\input{table_GDP_revisions_f.txt} \hline
\input{table_GDI_revisions_f.txt}
\hline \hline
\end{longtable}}

\newpage

\centerline {\bf Table 3: Regressions Explaining Statistical Discrepancy as Percent of GDP(E)} 
\vskip 0.4cm
\centerline {\bf Levels Specifications: $SD_t = \alpha + \beta SD_{t-1} + \theta U_{t} + \varepsilon_t$} 
%\vskip 0.1cm
\begin{longtable}{ c c c c}
\multicolumn{4}{c}{1984Q3-2006Q4} \\
\hline \hline
$\beta$ & $\theta$ & $\alpha$ & Adj. $R^2$ \\
\hline \hline
\input{table_okuns_84.txt}
\hline \hline
\end{longtable}

\centerline {\bf Differences Specifications: {\large $\Delta SD_t = \alpha + \beta \Delta SD_{t-1} + \theta_0 \Delta U_{t} + \theta_1 \Delta U_{t-1}+\varepsilon_t$}} 
%\vskip 0.1cm
\begin{longtable}{ c c c c c}
\multicolumn{5}{c}{1984Q3-2006Q4} \\
\hline \hline
$\beta$ & $\theta_0$ & $\theta_1$ & $\alpha$ & Adj. $R^2$ \\
\hline \hline
\input{table_okuns_d84.txt}
\hline \hline
\end{longtable}

\centerline {\bf Levels Specifications: $SD_t = \alpha + \beta SD_{t-1} + \theta U_{t} + \varepsilon_t$} 
%\vskip 0.1cm
\begin{longtable}{ c c c c}
\multicolumn{4}{c}{1948Q1-1984Q2} \\
\hline \hline
$\beta$ & $\theta$ & $\alpha$ & Adj. $R^2$ \\
\hline \hline
\input{table_okuns_48to84.txt}
\hline \hline
\end{longtable}
\vskip 0.2cm {\bf Note:} Standard errors are Newey-West with 8 lags.

\newpage

\centerline {\bf Table 4: Regressions of $\Delta GDP(I)$ and $\Delta GDP(E)$ on}
\centerline {\bf Various Alternative Business Cycle Indicators, 1984Q3-2006Q4} 
\begin{longtable}{ c |  c c | c c c }
\hline
Explanatory Variable & \multicolumn{2}{c}{Adjusted $R^2$} & \multicolumn{2}{c}{$\beta$} & \\
 & $GDP(I)_t$ & $GDP(E)_t$ & $GDP(I)_t$ & $GDP(E)_t$ & p-val., equal $\beta$s \\
\hline
\input{table_LoSE_SP500.txt}
\input{table_LoSE_DEF.txt}
\input{table_LoSE_TERM.txt}
\input{table_LoSE_UR.txt}
\input{table_LoSE_lfce.txt}
\input{table_LoSE_ISM.txt}
\input{table_LoSE_ISMNM.txt}
\input{table_LoSE_NBER.txt}
\hline
\end{longtable}
\vskip 0.2cm {\bf Note:} Specifications using the high yield bond
spread ($r^{HY corporate} - r^{Treas.(10yr)}$) use a 1988Q3 to 2006Q4
sample.  Specifications using the non-manufacturing ISM
($ISM^{non-manuf.}$) use a 1997Q3 to 2006Q4 sample.  Standard errors
are Newey-West with 8 lags.

\end{document}
