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\begin{document}

\title{Appendices to 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).}
  \\
  \\
%  First Draft - please do not distribute.  
}
\maketitle

\doublespacing

\newpage

\doublespacing

\centerline{\large {\bf Appendix A: More Basic Data Facts}}
\vskip 0.6cm

Appendices A, B and C dig into the composition of the estimates and
their source data.  Although the main sections of the paper focus on
real growth rates, these appendices focus on the source data
underlying the nominal estimates, because any divergence between the
estimates must arise because of measurement errors in the nominal
data.  Price index measurement cannot be part of the explanation,
because GDP(E) and GDP(I) use exactly the same price deflator to
convert nominal values into real units.

Tables A.1.1 and A.1.2 examine the composition of GDP(I) and GDP(E), to
get a sense of the importance of their different subcomponents.  The
sample employed here is the same one as in section 4 of the paper,
and coincides with the period of maximum divergence between the
estimates.  The basic identity for the expenditure side will be
familiar to most readers---GDP(E) = C + I + G + NX.  GDP(I) adds up
income flows from production, which is essentially labor income plus
capital income plus taxes on production paid to the government.
Depreciation is subtracted from the measures of capital income in
GDP(I), so it must be added back to arrive at the gross domestic
product concept.  For more, see Abel and Bernanke (2001).

Table A.1.1 shows that the key components for GDP(I) are, in decreasing
order of importance, employee compensation, corporate profits, and
proprietors's income.  Compensation accounted for most of the nominal
value of GDP(I) in from the mid-1980s through 2006, and almost half of
its quarterly variance from 1984q3 to 2006q4.\footnote{%
  The variance share of a component is computed as the covariance of
  its contribution to GDP(I) growth with GDP(I) growth.  This
  covariance captures the covariation between components as well as
  the variance of the component appropriately weighted by its nominal
  value, and ensures the variance contributions sum to one.  Too see
  this, note that GDP(I) growth can be written as the sum of the
  contributions to growth, so: $\Delta GDP(I) = \sum_j C_{\Delta
    GDP(I)}^j$, where $ C_{\Delta GDP(I)}^j$ is the $j$th component's
  contribution to growth.  Then:
\begin{eqnarray*}
  \var\left(\Delta GDP(I)\right) & = & \cov\left(\Delta GDP(I),\Delta
    GDP(I)\right) \\
  & = & \cov\left(\Delta GDP(I),\sum_j C_{\Delta GDP(I)}^j\right) \\
  & = & \sum_j \cov\left(\Delta GDP(I),C_{\Delta GDP(I)}^j\right) \\
  & = & \sum_j \left(\var \left( C_{\Delta GDP(I)}^j \right) + \sum_{k \neq j} \cov\left(C_{\Delta GDP(I)}^j,C_{\Delta GDP(I)}^k\right) \right).
\end{eqnarray*}}
Corporate profits represented only 9 percent of the nominal value
of GDP(I), but about a quarter of its quarterly variability.
Proprietors' income accounted for about 7 percent of
nominal value and 8 percent of quarterly variability.  The remaining
components are less important.  Consumption of fixed capital (CFC) 
accounts for a large part of the nominal value of GDP(I) but only 3 
percent of its quarterly variability.  Taxes on production and imports 
less subsidies\footnote{%
  I've added business transfer payments to this category as
  well.} accounts for 8 percent of nominal value, but this paper has 
little to say about measurement of this category; the same is true of
net factor income.  Net interest is not a category that can explain 
the discrepancies between the estimates---see below.

For the composition of GDP(E), table A.1.2 shows nominal shares and the
quarterly variance shares computed using contributions to both nominal
and real growth.  Personal consumption expenditures (PCE) on services
comprises the largest share of the nominal value of GDP(E) by far, and
this share has been steadily growing over time (see section 4.1).
However, PCE services has accounted for a much smaller share of the
variance of GDP(E).  The fixed investment categories (residential
structures, nonresidential structures, and equipment and software)
accounted for only 16 percent of the nominal value of GDP(E) but close
to 30 percent of its real variability.  Changes in private inventories
have zero mean, on average, but account about 30 percent of the
quarterly variability of GDP(E) over this sample.\footnote{%
  This variance share does drop sharply if we switch to year-over-year
  or annual growth rates, suggesting much of this variation is
  high-frequency noise that vanishes in aggregation.}  Exports are
strongly procyclical, adding to the variability of GDP(E), but imports
are also strongly procyclical, and since they are a subtraction from
GDP(E), they reduce its variance.  Government comprises a fifth of the
nominal value of GDP(E), it contributes less to its variability.

Aside from its lack of variability, government expenditures can be
downplayed for our purposes because the majority of those expenditures
are employee compensation and consumption of fixed capital.  These
components match exactly the government portion of the CFC and
compensation on the income side, so they obviously cannot explain any
discrepancies between GDP(E) and GDP(I).  Other components of the
accounts that match exactly across the income and expenditure sides
include the housing services, financial services furnished without
payment, and insurance services components of PCE.  The PCE estimates
for housing services are spread across consumption of fixed capital,
net interest, rental income of persons, and taxes on production and
imports, among other components of GDP(I) - see BEA table 7.4.5.  Much
of financial and insurance services appears in net interest, which is
a component which mostly nets out exactly with GDP(E).

Summing the various components that are identically equal on the
income and expenditure sides of the accounts shows they account for
about 26 percent of the nominal value of GDP(E) or GDP(I) in 2008.
Appendices B and C hunt for the discrepancies between the two
estimates in the remaining 74 percent of the nominal value of the
accounts.

\newpage
\centerline{\large {\bf Appendix B: Source Data Used to Construct the Initial Estimates}}
\vskip 0.6cm

While a thorough review of the source data used to compute the initial
``third'' estimates is beyond the scope of this paper,\footnote{%
  Much of this information is contained in ``Updated Summary of NIPA
  Methodologies'', {\it Survey of Current Business}, November 2008, as
  well as the NIPA Handbook, 2008-2009.} a brief summary highlighting
some key issues may be useful.  Starting first with GDP(I), the BEA
estimates the growth rate of wage and salary accruals, which account
for more than 80 percent of employee compensation, using data from the
Current Employment Statistics (CES) establishment survey produced by
the Bureau of Labor Statistics (BLS).  The CES is a survey of
businesses drawn from the universe of firms participating in the
unemployment insurance (UI) system, which covers 98 percent of
employees in the U.S.  Compared to other business surveys, this sample
of firms is relatively large.  The survey is voluntary, although BLS
requests that all firms of a certain size (over 250 employees)
participate in the survey each month; smaller firms are selected into
the sample randomly, with the size strata reweighted to reflect the
assumed distribution of firm sizes in the population.  Firm response
rates are usually around 90 percent, but they did dip down to around
80 percent from 1999-2003.\footnote{%
  http://www.bls.gov/web/cesregrec.htm} To the extent that the
unobserved characteristics of non-responders differ from the those of
responders, non-response can impart bias to the estimates,
particularly problematic if these unobserved characteristics vary over
the business cycle.  Also, as with any random sample, sampling errors
add some noise to the CES estimates.

Until a few months ago, the CES reported data on total employment, and
the average weekly hours and hourly earnings of a subset of the
workforce---production or nonsupervisory workers.  So the initial
estimates missed variation arising from payments to managers or from
irregular payments such as bonuses, a potentially serious issue given
rising income inequality.  The BLS recently addresssed some of these
problems in rolling out new CES estimates of the compensation of all
employees, which the BEA began using immediately; unfortunately these
estimates still do not include irregular payments.  Another perennial
and much-discussed issue with the CES is the procedure for updating
the sample of establishments participating in the survey.  Firm births
and deaths are incorporated into the sample with a lag, and while the
BLS has an explicit statistical birth-death model that tries to
correct for this, the adjustments are imperfect, and there are
occasionally notable revisions to the employment counts when they are
benchmarked to the comprehensive census counts from UI records, called
the Quarterly Census of Employment and Wages (QCEW).

The BEA's initial estimates of corporate profits are based on firms'
earnings releases, reports from regulatory agencies, and data from the
Census Bureau's Quarterly Financial Reports.\footnote{%
  See the BEA's Methodology Paper ``Corporate Profits'', September
  2002.  }  The BEA must make numerous adjustments to these initial
estimates.  For example, the concept of output the BEA attempts to
measure is the value of current production, and does not include
capital gains or losses on assets acquired in previous periods.
Firms' earning releases include such gains and losses in profits, and
the BEA does not always have all the information they need to strip
out these gains and losses.  Furthermore, firms have considerable
latitude in their financial reporting on their public earnings
releases, which can introduce measurement error.  For whatever reason,
when the BEA profits data are revised to match data from official tax
records from the IRS, the revisions can be quite sizable.  

The BEA estimates of proprietors' income are ultimately based on IRS
tax data as well, but BEA extrapolates the growth rates of the initial
estimates using various indicator variables.  The indicators vary by
industry; but many appear to be based on the CES employment and
earnings data.  Of the remaining components of GDP(I), a fair amount
are extrapolated using what the BEA calls ``judgmental trends'',
typically averages of recent growth rates.  These components are of
less interest to us here.  As discussed above, most of net interest
and the rental income of persons falls within the 26 percent of GDP(I)
that nets out exactly with GDP(E), as does a sizable chunk of
consumption of fixed capital, which also accounts for little of the
variability of GDP(I).

The initial growth rates of several important components of GDP(E) are
constructed using monthly surveys from the Census Bureau, often
surveys of businesses with paid employees.  Estimates of PCE for
durable and nondurable goods are based on Census's Monthly Surveys of
Retail Trade (MRTS),\footnote{%
  The major exceptions are for auto sales, where data on unit counts
  are derived from trade source data that are census counts, and for
  sales of gasoline and oil, which are estimated from the Energy
  Information Administration.} as are estimates of food service sales
and retail inventories.  Wholesale inventories and manufacturers'
inventories are based on the Monthly Wholesale Trade Survey (MWTS) and
the monthly M3 survey of manufacturers' orders, shipments, and
inventories, which also provides shipments estimates used to compute
most of equipment and software.\footnote{%
  Software is the major exception; the third estimates of software
  currently are based, in part, on data from Census' Quarterly
  Services Survey (QSS).}  Samples are drawn from Census Bureau's
universe count of businesses with paid employees, updated with the
Economic Censuses every five years and in between with tax records.
Firms are organized by their Employer Identification Number (EIN), and
sole proprietors are not included in the surveys, a potential source
of bias.  Like the CES survey, these surveys ask firms above a certain
size to participate each month, and smaller firms are sampled randomly
with voluntary participation.  The response rate in the MRTS covers
about 77 percent of sales and 71 percent of inventories;\footnote{%
  http://www.census.gov/retail/mrts/how$\_$surveys$\_$are$\_$collected.html.}
the November 2009 MWTS reported a response rate of 75 percent.  The M3
does not describe itself as a probability sample.  The November 2009
M3 report describes the survey as a panel of several thousand
manufacturing firms with sales of $\$$500 million or more, and notes
that smaller companies are underrepresented, again a potential source
of bias.

Note that while firm births and deaths receive much more attention in
the CES employment data, they are just as much an issue for these
Census surveys employed in the initial estimates of GDP(E).\footnote{%
  Firm births and deaths may receive more attention in the CES survey
  because the BLS has an explicit birth-death model, facilitated by
  the availability of quarterly benchmarks from the QCEW.  The surveys
  employed by GDP(E) do not have quarterly or even annual universe
  counts to which they benchmark, hampering explicit birth-death
  modelling.  This suggests that the CES may do a better job than the
  Census surveys in making birth-death corrections, but this is a
  speculative point.}  New firms are added to the surveys of retail
trade with a lag of about nine months, too late to be included in the
initial quarterly estimates.  Firm deaths are an issue as well.  When
a firm currently in a survey does not report for a given month, Census
may not be able to distinguish between firms that are still in
business but not reporting, where imputing an average growth rate
would be appropriate, and firms that are no longer in business, where
imputing a growth rate of minus 100 percent would be appropriate.
Some misclassifications are inevitable, and both firm births and
deaths are likely to be systematically related to the business cycle.

The initial estimates for residential and nonresidential construction
are contructed somewhat differently; the sampling units are not firms.
The new residential contruction estimates are based on a random
sampling from a universe of 20,000 permit-issuing places in the U.S.,
which are geographic designations covering about 98 percent of new
residential construction.\footnote{%
  In contast to the surveys of businesses described above, which
  include all the sales, shipments, or inventories of the large firms
  included in the sample with certainty, these estimates will include
  only a fraction of the sales of any large firm.  However, many
  permit-issuing places are included in the sample with certainty, to
  ensure broad coverage.}  The variance of the sampling errors for
housing units started and completed are very large, but Census spreads
the spending over 12 months by ``applying fixed patterns of monthly
construction progress'', which tends to smooth out the dollar
estimates and reduce sampling variability.\footnote{%
  http://www.census.gov/const/C30/methodology.pdf} Note that the
timing here does not necessarily match that on the income side: if a
builder starts a new home and books the income at that point, it shows
up in profits or some other component of GDP(I) immediately, instead
of being spread out over several quarters as in GDP(E).  The value of
improvements to existing residential structures are estimated using a
regression model based on single-family construction expenditures,
among other things; BEA calls this a ``judgemental trend''.  The
nonresidential construction estimates are based on surveys of
construction projections above a certain value from the list of
projects compiled by F.W. Dodge of the McGraw-Hill Information Systems
company.  This list is not comprehensive, with Census Bureau
comparison studies putting its coverage rate at about 75
percent.\footnote{%
  The list tends to miss things like natural gas pipelines and wind
  farms, but not office buildings or retail projects.}  The
non-response rates to these construction surveys can be considerable;
for example, in 2008, only about 67 percent of the value of
construction put in place was based on directly-measured
data.\footnote{%
  http://www.census.gov/const/C30/reliability.pdf } As noted earlier,
such non-response and undercoverage can produce biases if these
unrepresented projects differ in systematic ways from the projects
represented in the sample.

Net exports of goods and services are compiled from U.S. Customs and
Border Protection data, while net exports of services are based on
surveys conducted by the BEA.  This paper will have little to say
about the quality of these numbers, but transfer pricing is an
important issue, and systematic biases in the trade numbers are
certainly a possibility.  As noted above, the GDP(E) estimates for
government are of comparatively little concern in this paper.

Regarding services PCE, the BEA lacks reliable source data for much of
this category at the quarterly frequency, and as such, they must
resort to a variety of methods in computing their initial current
quarterly estimates.  We can break down the third current quarterly
estimates of services PCE as follows.  About 7 percent is food
services, which is based on the MRTS surveys like most of goods PCE.
About 31 percent is the categories where the BEA ensures consistency
between the income and product side.  About 17 percent is now based on
the Quarterly Services Survey (QSS), but growth rates based on the QSS
did not exist until the last few years, so the initial quarterly
estimates from prior years (i.e. most of the sample examined here)
must have been based on some combination of the next three types of
estimates.  Currently, about 22 percent of services PCE is based on
``judgmental trends'', which surely miss a substantial portion of the
variation in true services PCE.  About 14 percent is extrapolated
using the CES estimates of total employment and the earnings of
nonsupervisory workers.  This may be a reasonable approximation for
some parts of PCE services, but for others such as the ``portfolio
management'' component of financial services, sales are likely to be
much more variable than the CES estimates.  This is because the
bonuses and payments to management missed by the CES are likely more
variable than employment or hourly wages, and because the part of
sales that shows up as corporate profits is likely much more variable
than compensation, as shown in the aggregate in table 2.  Finally,
about 9 percent is based on a variety of trade-source and
administrative data, which is generally not gathered through standard
statistical sampling techniques, so these data may have biases.  Given
that much of this source data may miss true variation in services PCE,
it is not surprising that Dynan and Elmendorf (2001) find that, among
the major categories of GDP, the initial estimates of services are
particularly prone to missing accelerations and decelerations.

Thus, all the components of initial estimates of GDP(I) and GDP(E) growth
face measurement challenges to one degree or another, and sorting
through the merits of each and aggregating is a difficult challenge.
Sampling errors are an issue, it is not clear whether they add more
noise to the initial growth rates of GDP(I) or GDP(E).
%Compared to the CES, the main survey
%employed in computing initial GDP(I), non-response rates appear higher
%in the surveys used to compute GDP(E), suggesting GDP(E) may have more
%imputation problems.  In addition, the CES has an explicit model to
%deal with firm births and deaths, while the surveys used to compute
%GDP(E) deal with firm births and deaths on a more ad-hoc basis.  This
%is another point in favor of initial GDP(I).  
One factor favoring GDP(E) is that more of initial GDP(I) growth
appears based on trends.\footnote{%
  See Grimm and Weadock (2006) and Holdren and Grimm (2008).}  But
much of this trending in GDP(I) occurs within CFC, the least variable
major component of GDP(I), so a judgmental trend may not miss much
variation.  There is little reason to suspect the smoothness in the
latest estimates of private CFC is spurious, because it is ultimately
based on reasonably-reliable tax data, and because CFC is a function
of an inherently slow-moving stock of capital.  Much of the trending
in GDP(E) occurs within services PCE, which is also quite smooth.  But
in contrast to CFC, and as discussed below, there is good reason to
suspect that some of the smoothness in services PCE is spurious,
because the BEA has not had access to particularly good data on
services PCE even after their annual revisions.  Based on {\it a
  priori} considerations, I would expect some categories of services
PCE, such as medical care, to face demand that is less elastic than
nondurable goods, but other categories, perhaps parts of travel or
recreation services, might face more elastic demand; overall, it is
not obvious that services PCE should be much less variable than
nondurable goods PCE.  The lack of data coverage for much of services
PCE is one factor suggesting the initial GDP(I) growth estimates may
have been more accurate over most of recent history.

\newpage
\centerline{\large {\bf Appendix C: Source Data Incorporated at Revisions,}} 
\centerline{\large {\bf to Construct the Latest Estimates}}
\vskip 0.3cm

\centerline{\bf Annual revisions}
\vskip 0.3cm

The initial estimates for any given year $t$ are revised in three
annual revisions, typically in the summers of years $t+1$, $t+2$ and
$t+3$.  But since 2002, the BEA incorporates information from the
Quarterly Census of Employment and Wages (QCEW) into wages and
salaries for each quarter before those annual revisions, typically two
months after the release of that quarter's third estimate.  The BEA
does make some adjustments for misreporting and for payments to
private employees not covered by the unemployment insurance (UI)
system.  But these adjustments total only a few percent of the overall
wage and salary estimate, between 1.6 and 1.9 percent of GDP(I) from
1993 to 2008 (see NIPA table 7.18), so the QCEW estimates are very
close to a full universe count.\footnote{%
  For more on the misreporting adjustments, see the proprietors'
  income paragraph below.} That a quarterly universe count can be
incorporated so quickly is remarkable and important, not least because
wages and salaries account for such a large share of both the nominal
value and variability of GDP(I).  Table A.1.1 shows that private wages
and salaries represented 38 percent of nominal GDP(I) from 1984 to
2006, roughly half of the 74 percent of GDP(I) that does not
cancel out with the expenditure side, as well as 40 percent of the
variability of quarterly GDP(I) growth.

According to these metrics, the availability of the QCEW data is a
huge advantage for the income side compared to the expenditure-side,
where the data suffer from sampling errors and various non-sampling
errors because universe counts are unavailable.  The one main
exception on the expenditure side is the universe counts of auto
sales, but this is a substantially less-important part of output by
any metric.  However, the early incorporation of the QCEW data has
caused a few glitches in the accounts.  The QCEW data include gains
from exercising nonqualified stock options, and when these gains
appear as wage and salary income, proper accounting dictates they
should be expensed by firms and subtracted from corporate profits.
The quarterly financial reports used by the BEA to compute the initial
estimates of profits do not necessarily have this information,
potentially introducing some distortions into the estimates.  This
issue is discussed forthrightly in Moylan (2008).\footnote{%
  Moylan (2008) notes that, in a perfect world, options should be
  expensed when they are granted, not when they are exercised.}
However, firms do expense the option gains in the IRS tax data the BEA
incorporates into profits at annual revisions. The IRS tax data are
not the same data as the QCEW data used to compute wages and salaries,
but the BEA's position is that the definition of wages and salaries is
very similar across the two data sources, so this problem largely
disappears at the annual frequency once the tax data are incorporated
into the estimates.  Since the tax data are not available quarterly,
some distortions may remain in the distribution of the annual data
across quarters.

The IRS tax data used to compute profits are incorporated in the
second and third annual revisions.  These are annual data, so they are
not helpful for interpolating quarterly growth patterns, but the
annual frequency is generally high enough to pick up broad business
cycle patterns.  The IRS obviously has access to the entire universe
of tax filings, but the data employed by the BEA are from the IRS's
Statistics of Income (SOI), which are samples drawn from the universe
of tax filings.  However, the coefficients of variation around the
top-line estimates of corporate net income are tiny, numbers like 0.36
of one percent for 2006 and 0.13 of one percent for 2005.\footnote{%
  http://www.irs.gov/taxstats/article/0,,id=170543,00.html } Given the
variability of corporate profits, these sampling errors are
negligible, and obviously are not part of the explanation for the
cyclicality of the statistical discrepancy.  

One example of measurement error potentially related to the business
cycle was highlighted by the BEA at the end of 2007: the difficulty of
removing bad debt expenses and asset write-downs, which are capital
losses not to be included in the BEA's concept of output, from their
initial estimates of profits at financial services companies.
However, the IRS tax forms include specific lines for bad debt
expenses and capital gains and losses, which should make these
subtractions easy, and it is not clear whether corporations could, or
why they would, misreport capital gains and losses as other types of
income.

Widespread underreporting of income on the IRS tax forms is another
problem, but the BEA has detailed data on tax settlements recommended
by IRS auditors that it uses to adjust for misreporting.  Those
adjustments are large, ranging between 1.0 and 2.2 percent of GDP from
1984 to 2006 (see BEA table 7.16), but the BEA does base these
corrections on some hard data available at the annual frequency.

The BEA also uses the IRS tax data to estimate proprietors'
income.\footnote{%
  The BEA also uses tax data to estimate most of private CFC, making
  adjustments for conceptual differences.  Note that the depreciation
  expenses used as the starting point for computing CFC are subtracted
  from corporate profits and proprietors' income on tax returns.
  Since it is the sum of CFC, corporate profits, and proprietors'
  income that matters for computing GDP(I), mismeasurement of CFC
  should not affect GDP(I), because it nets out of this sum.} The
adjustments for misreporting are again large, ranging between 2.6 and
4.0 percent of GDP(I) from 1984 to 2007 (see BEA table
7.14).\footnote{%
  The sum of the misreporting adjustments to wages and salaries,
  corporate profits, and proprietors' income range from 4.5 to 6.5
  percent of GDP(I) from 1984 to 2006.}  Unfortunately, the BEA has
fairly limited source data upon which it bases these corrections,
certainly not enough to accurately identify the cyclicality of the
unreported income.\footnote{%
  The BEA actually makes two adjustments, one for underreporting of
  income based on audits of individual tax returns, and one for
  nonreporting of income based on exact-match studies (see Krakower,
  2007 and Seskin and Smith, 2009).  For a set of individuals who
  responded to the CPS household survey, the exact-match studies
  compare their individual tax returns with the income they reported
  to the CPS.  Exact-match studies were conducted in 1977, 1990, 1996,
  1999, and 2003-2007.  The last 5 surveys actually provide a short
  panel which might shed some light on cyclicality.  However, the
  estimates suggest that nonreported income is much smaller than
  underreported income, perhaps around 10 percent.  For underreported
  income, the two most recent IRS studies are 1988 and 2001. } The
growth rates of the current corrections are uncorrelated with $\Delta
GDP(I)$ from 1985 to 2007, so the BEA is essentially assuming that the
unreported income has no cyclical component.  If this unreported
income is actually counter-cyclical, then GDP(I) would be too cyclical
- i.e. more cyclical than true output.  Note that this unreported
income is where much of the income earned in the underground economy
should appear.\footnote{%
  We should distinguish between underground activity that should
  appear in the BEA's definition of output and income that should
  not.  Legal market transactions that are not reported to the IRS
  should appear in the BEA's definition of output.  Illegal
  transactions should not, and non-market transactions such as the
  value of home production should not.}  This underground income might
be counter-cyclical if individuals move to less formal means of
generating income when they lose jobs in the formal sector during bad
economic times.  On the other hand, economic activity in the
underground economy may benefit from good economic times in the rest
of the economy.  Over the relevant sample period, proprietors' income
as a whole is more pro-cyclical than GDP(I); a regression using annual
growth rates from 1985 to 2007 of proprietors' income on GDP(I) yields
a coefficient of about 1.65 (with a standard error of about 0.47, and
an adjusted $R^2$ of 0.34).\footnote{%
  We stop in 2007 because that is the last year where tax data is
  available.  Using a longer 1930 to 2007 sample yields a somewhat
  smaller coefficient of 1.36, but a tighter standard error of 0.11,
  allowing us to reject the hypothesis that proprietors' income is not
  more procyclical than GDP(I).  The adjusted $R^2$ is 0.68.  A
  similar result obtains when we use GDP(E) as the explanatory
  variable.}  So if unreported income is similar to other types of
proprietors' income, that would suggest procyclicality, in which case
GDP(I) would not be cyclical enough.

Misreporting of capital gains as ordinary income is another potential
explanation for the pro-cyclicality of the statistical discrepancy.
For example, many people made lots of money flipping houses at the
peak of the housing bubble in the mid-2000s and then lost a lot of
money later in the decade.  Most of this income should properly be
classified as capital gains, and would be stripped out of the tax data
used in proprietors' income and GDP(I) if reported as such.  But gains
misreported by individuals as ordinary income would likely show up in
proprietors' income.  Long-term capital gains on houses held more than
a year are generally subject to a lower tax rate than ordinary income,
so individuals would have no tax incentive to misreport these types of
gains, but I know of no study of this issue, so I cannot rule out the
hypothesis that capital gains from house flipping are currently
biasing up proprietors' income in the mid-2000s.\footnote{%
  One could make the same argument about day-trading in the 1990s, but
  that argument holds less water.  These capital gains are subject to
  a quick computer audit, since the traders use brokers who must file
  tax forms which can be checked against the gains reported by the
  traders.}  However, this is a speculative point, and overall, the
evidence that GDP(I) is likely to be too cyclical seems a bit thin.

Moving to the expenditure side of the accounts, most of the monthly
surveys of businesses discussed in the previous section revise to
incorporate more-comprehensive surveys at the annual frequency.  For
example, the Monthly Surveys of Retail Trade (MRTS) revise to
incorporate information from the Annual Surveys of Retail Trade
(ARTS), which the BEA then uses to construct revised PCE and inventory
estimates at annual revisions.  The ARTS employs a noncertainty sample
about twice as large as the MRTS.  Businesses are legally required to
respond to the ARTS, and although not all do so, the response rate is
about 94 percent, much higher than the 77 percent response rate to the
MRTS.  In addition, the ARTS incorporates information from tax records
on firm births and deaths into the sampling frame and estimation.
Overall, these improvements would seem to substantially reduce the
scope for biases in the estimates.  Data from the ARTS remain subject
to sampling variability, but the variance from sampling errors is
quite small, and as discussed earlier, sampling errors obviously do
not explain the cyclical measurement errors that are our primary
interest.  The universe from which the sample is drawn remains
restricted to establishments with paid employees, and to correct this
problem, Census receives data on sales of nonemployer businesses from
IRS tax records, and adds these sales to its estimates.  These data
are essentially the same data the BEA uses to estimate proprietors'
income in GDP(I), so the prior discussion of underreporting is
applicable to these estimates.  The tax data do not provide
information on inventories, so Census imputes inventories to
nonemployers using the inventory-sales ratio for employers from its
sample.

Of the other surveys of businesses, the MWTS revises to incorporate
information from the Annual Survey of Merchant Wholesale Trade (AWTS),
the M3 incorporates information from the Annual Survey of
Manufacturers (ASM), and the QSS (and PCE for services more generally)
incorporates information from the Services Annual Survey (SAS).  The
main characteristics of these annual surveys are similar to the ARTS:
the samples are larger than the monthly surveys, response rates are
higher, and the samples are updated to incorporate firm births and
deaths.  In each case, however, certain firms remain out of the scope
of the sampling universe.  The May 2008 M3 revision reports:
``Nonemployers are defined as out of scope of the ASM,'' while the
2007 SAS reports: ``To be eligible for selection, a birth ... must
meet certain criteria regarding its number of paid employees or
quarterly payroll.''  Sole proprietors' and some small businesses are
out of the scope of these annual surveys, but the SAS makes
corrections similar to the ARTS, adding to their estimates the
revenues of nonemployer businesses which they obtain from IRS tax
records.  Some small businesses with employees may be missed by the
SAS, however.
%Census does have annual nonemployer statistics surveys - does BEA use
%this information for the GDP(E)?

The annual revisions to the residential and nonresidential structures
components of GDP(E) do not incorporate information from any new,
improved surveys.  The construction put in place surveys are based on
the same relatively small and noisy samples as the initial estimates;
the only information incorporated at annual revisions is from any
additional responses to the surveys that have continued to trickle in.
The response rate increases from 67 percent to 79 percent, but the
non-response rate remains quite high compared to other surveys like
the ARTS, leaving greater scope for biases.  This is particularly true
for nonresidential structures, where the 75 percent project coverage
rate of the universe combined with a response rate of about 79 percent
yields coverage of below 60 percent.  Unfortunately, the extent of any
biases and their effect on the cyclicality of GDP(E) is unclear.  The
mismatched timing across the income and expenditure sides remains an
issue in the fully revised data, with construction spending spread out
over several quarters in GDP(E), and income from construction probably
not as spread out in GDP(I).

Perhaps the most poorly estimated component of structures is
improvements to residential structures.  The Census Bureau estimates
are ultimately based on data from the Consumer Expenditure Survey, a
relatively small survey of a few thousand households.\footnote{%
  The BEA uses this Census Bureau data for improvements to
  owner-occupied housing.  For rental units, the BEA used to employ a
  survey of landlords that is now discontinued.}  These data are
likely to be quite noisy, given the small sample size and the fact
that large home improvement projects tend to be infrequent and lumpy.
Perhaps for this reason, the BEA takes a ``weighted 3-year moving
average'' of the Census Bureau data.  While such averaging is sensible
from the standpoint of minimizing the variance from sampling errors,
it has the negative side effect of cutting down the true cyclical
variation in the underlying data, especially around business cycle
turning points such as the most recent one - see section 5.  Such
smoothing will tend to make GDP(E) less cyclical than it should be.

Finally, as noted earlier, the annual revisions incorporate the
Service Annual Survey (SAS) data into the BEA's estimates of services
PCE.  However, the coverage of the SAS is incomplete, even at the
annual frequency.  About 36 percent of the nominal value of 2008
services PCE was based on the SAS after the data was incorporated,
more than half of the 62 percent of the nominal value that neither
nets out with the income side nor is estimated with ARTS data
(i.e. food services).  This coverage rate improved considerably with
the 2009 benchmark revision, which expanded the use of the SAS to
hospitals and telecommunication services, and the SAS for those
categories extends back to 2003 only.  Excluding these categories, the
SAS covers only 24 percent of the nominal value of services PCE.  As
of the third annual revision, about 4 percent is currently based on
judgmental trends or employee compensation (down from about 12 percent
as of the first annual revision), and about 7 percent is based on
estimates from the annual surveys of government finances (which are
only fully available as of the third annual revision, and require
adjustments from fiscal years to calendar years).

The remaining roughly 26 percent of services PCE has been largely
based on administrative data and trade source data, which can suffer
from numerous drawbacks.  Indeed, the recent shift of the estimates of
hospital services to SAS data from trade source data from the American
Hospital Association (AHA), was motivated by the following: ``... the
AHA data are fiscal year data which must be converted to a calendar
year basis.  In addition, the receipts of taxable hospitals used for
the PCE estimates must be derived by converting AHA expenses data
using fixed ratios, while the SAS directly provides receipts data for
taxable hospitals.''\footnote{%
  McCully and Payson, 2009, p.10} So the trade source data likely
missed some variation in outlays to hospitals that resulted in
profits, which may have been more variable than expenses.  For
telecommunications providers, BEA used FCC data prior to the SAS, but
``... the SAS data capture broadband telephone services, which the FCC
data on regulated carriers do not.''\footnote{%
  McCully and Payson, 2009, p.10} More generally, since the trade
source and administrative data are not necessarily gathered for the
purpose of producing consistent estimates of the quantities of
interest, for the most part, they likely do not do things like
reweighting to correct for lack of coverage of sole proprietors or
small firms.  For example, the SAS corrections using IRS tax data to
capture the revenues of some nonemployer businesses are the type of
thing that is surely not done in trade source data and some
administrative data.  As pointed out earlier, proprietors' income was
more cyclical than output from 1985 to 2008, so not including the
sales of these businesses in the annual growth rates of GDP(E) should
tend to make GDP(E) less cyclical than ``true'' output.  These and
other biases could result in excessive smoothness for services PCE and
GDP(E).

One industry where use of the trade source and administrative data
likely has been particularly problematic is financial
services.\footnote{%
  Beaulieau and Bartelsman (2004) attempt to allocate the statistical
  discrepancy across industries, and their methodology places much of
  the discrepancy in financial services.}  For securities commissions,
the BEA must make assumptions about the share of trading done by
individuals, which may have varied considerably over the past couple
of decades with the rise of day trading in the 1990s and subsequent
stock market collapse.  Moreover, the BEA estimates rely heavily on
administrative data from the Securities and Exchange Commission (SEC),
which likely has serious coverage problems.  Many banks were using
off-balance sheet entities (i.e. SIVs) to deal in structured credit
products, which may not have been covered by the SEC data.  Many
financial companies, particularly hedge funds and private equity
firms, appear to have avoided registering with the SEC, at least until
recently, which may have biased down the estimates of
commissions.\footnote{%
  In particular, it has been widely reported in the media that hedge
  funds and private equity firms have used exemptions 3(c)(1) and
  3(c)(7) to the Investment Company Act of 1940 to avoid SEC
  registrations.  In early 2009, senators Grassley and Levin
  introduced a bill that would require firms that use these exemptions
  to register with the SEC.  }

Most of the fees and charges of these types of firms would likely fall
under the BEA's ``portfolio management and investment advice
services'' line, instead of securities commissions.  The BEA uses SAS
data here, whose sample of firms may include some hedge funds and
private equity firms, but it seems doubtful that these estimates
captured the boom in financial services over the past couple of
decades.  The SAS data extend back to 1998 only, and so would have
missed the boom in financial services through most of the 1990s.
Moreover, the magnitudes reported by the BEA seem too small.  Trade
sources estimate that, globally, hedge fund assets were reported to
have reached around $\$2$ trillion before the financial collapse, and
the typical ``2 and 20'' fee structure would imply $\$40$ billion in
base fees and an additional $\$40$ billion in return fees assuming a
10 percent return.  The total PCE for ``portfolio management and
investment advice services'' reported by the BEA in 2006 was $\$90$
billion; adding in mutual funds (who manage more than five times the
assets of hedge funds) and the investment advisory services of
commercial banks and other financial services companies, and it seems
the number should be much higher.  The majority of these fees were
probably charged to wealthy individual investors and non-profit
institutions serving households (NPISHs) such as colleges and
universities, so they should count as services PCE.  The BEA has
limited data on other components of financial service charges and fees
as well; for credit card fees, the BEA constructs annual estimates
using bank card dollar-volume data from a trade source, essentially
assuming the ratio of credit card fees to dollar-volume usage is
constant.  Any variation in credit card fees charged by banks is
missed.

Overall, there seems ample reason to believe that GDP(E) missed much
of the boom and bust cycles in financial services over the past couple
of decades, while GDP(I), which relies on IRS tax data, probably
picked up much more of this activity.  More systemic measurement
problems also suggest GDP(E) has missed some business cycle variation,
and is likely not cyclical enough.

\vskip 0.3cm
\centerline{\bf Benchmark revisions}
\vskip 0.3cm

Every five years GDP(E) is benchmarked to data from the quinquenial
Economic Censuses that are close to comprehensive, and the BEA's
internally consistent input-output tables.  Not all the categories
employ data from the Censuses; residential investment does not, and a
very small portion (less than 10 percent) of PCE services does not.
Of course, no data source is fully comprehensive, and the Censuses
probably face underreporting issues similar to those faced by the
comprehensive data used in GDP(I).  Nevertheless, the GDP(E) estimates
are undoubtedly of much higher quality in the years where they are
benchmarked to the Economic Censuses.  The availability of the
Censuses is part of the reason the BEA has taken an official position
that ``... the source data for the expenditure components are
considered more reliable than those for the income components ...''
(Concepts and Methods of the U.S. National Income and Product
Accounts, July 2008, p 2-10).

Unfortunately, those benchmarks do not provide much aid for
business cycle analysis, the focus of this report.
%The BEA's estimates of true output are useful for many different
%purposes, and in assessing the reliability of GDP(E) and GDP(I), we
%must be clear: are we interested in levels or growth rates, and over
%what time period?  The BEA has taken an official position that
%``... the source data for the expenditure components are considered
%more reliable than those for the income components ... '' (Concepts
%and Methods of the U.S. National Income and Product Accounts, July
%2008, p 2-10).  If one queries the BEA about this official stance,
%their rationale, as explained to me in private correspondence, leans
%heavily on the fact that 
%Every five years GDP(E) is benchmarked to
%data from economic censuses and the BEA's internally consistent
%input-output tables.  BEA says they prefer GDP(E) because the economic
%censuses provide close to complete enumerations of most of the
%component expenditures.  As we shall see, much of the data used to
%contruct GDP(I) is based on essentially complete enumerations as well,
%so the BEA's rationale is not immediately compelling.  But more
%importantly, this line of reasoning for the superiority of GDP(E)
%applies only to the level of output in benchmark years and the average
%growth rate of output between benchmark years.
%This report is focused on evaluating the usefulness of GDP(E) and
%GDP(I) for business cycle analysis, and the benchmarks provide very
%little information about that.  
They provide no information about the past 7 to 12 years, depending on
the timing of the last benchmark; for example, the last year to which
GDP(E) is currently benchmarked is 2002, and that benchmark was
implemented in the summer of 2009.  Nor are the benchmarks
particularly informative about earlier business cycles.  The level of
true output may be more accurately measured by GDP(E) in 2002, as
may the average growth rate between 1997 (the last prior benchmark)
and 2002, but the benchmarks provide no information about how to
distribute the growth between those years.  How fast was output
growing in the late 1990s?  How deep was the 2001 recession, and when
did it start and end?  Answers to these questions require
higher-frequency growth rates, at least annual and preferably
quarterly, and it is hard to see how the BEA's arguments for the
superiority of GDP(E) apply to these growth rates.  Instead, we must
look at the quality of the annual and quarterly data the BEA uses to
interpolate growth rates between benchmark years, and compare that to
the quality of the annual and quarterly data used to compute GDP(I).

%Benchmark revisions also make methodological changes to the estimates
%that can extend back many years.  Ideally, these changes should be
%consistent across the income and expenditure sides, but there is
%reason to doubt that this has been the case.  As shown in table 7,
%recent benchmarks have tended to make the statistical discrepancy
%positive, on average, prior to about 1987.  The 1996 switch in
%%treatment of government spending on structures and equipment to
%investment, and the 1999 switch in the treatment of software to
%%investment, may not have been carried out consistently across the two
%sets of accounts.  The table shows that there has been little tendency
%for recent benchmarks to move the discrepancy towards zero, and the
%benchmarks have actually increased the variability of the discrepancy.
%Figure 7 shows the effect of the latest (2009) benchmark revision.
%While there is still considerable volatility in the discrepancy prior
%to 1987 (a standard deviation of 0.65 percent), the volatility is
%considerably higher after 1987 (a standard deviation of 1.1 percent).

\newpage 
\centerline{\large {\bf Appendix D: Further Regression Results}} \vskip 0.6cm

Table A.2.1 shows the same set of regression results as table 2, using
``advance'' vintage estimates instead of ``third'' vintage, while
table A.3 reports some additional results on revisions.  As discussed
in section 3, unofficial ``advance'' $\Delta GDP(I)$ estimates are
constructed using the components of GDP(I) that are available with the
``advance'' release, and forecasts of the missing corporate profits
and net factor income components.  See Nalewaik (2007b) for details on
how these are constructed; no information from the companies that have
reported their quarterly profits numbers at the time of the
``advance'' release is employed here; using such information may
produce a much-improved ``advance'' $\Delta GDP(I)$ estimate.  Table
A.2.1 shows that ``advance'' $\Delta GDP(E)$ better predicts most of
the business cycle variables examined in the paper, although the
constructed ``advance'' $\Delta GDP(I)$ estimates retain some
predictive power, above and beyond that contained in ``advance''
$\Delta GDP(E)$, for output growth next period and the contemporanous
changes in the labor market variables.  But ``advance'' $\Delta
GDP(E)$ largely dominates; most importantly, the best forecast of
latest $\Delta GDP(I)$ places about two-thirds weight on ``advance''
$\Delta GDP(E)$ and only about one-third weight on the constructed
``advance'' $\Delta GDP(I)$ estimate.  Table A.3 shows that this
result holds over the more recent 1994Q1 to 2006Q4 sample as well,
although over this shorter sample, the best forecast of latest $\Delta
GDP(E)$ does places about one third weight on the constructed
``advance'' $\Delta GDP(I)$ estimate.
 
Table A.2.2 shows the a similar set of regressions as table 2, using
``second'' vintage estimates instead of ``third'' vintage, with table
A.3 reporting additional results on revisions.  The samples employed
here are restricted by the availability of the ``second'' $\Delta
GDP(I)$ estimates.\footnote{%
  For the regressions using the contemporaneous values of output
  growth, only 1st, 2nd, and 3rd quarters are in the sample.  For
  regressions using output growth estimates lagged one quarter, only
  2nd, 3rd, and 4th quarters are in the sample.  For regressions using
  the output growth estimates lagged two quarters, only 3rd, 4th, and
  1st quarters are in the sample.}  Table A.3 shows that, from 1994Q1
to 2006Q4, a quite sizable weight of about 0.6 should be placed on
``second'' $\Delta GDP(I)$ when predicting latest $\Delta GDP(E)$,
although the standard errors here are large.  The results in table
A.2.2 are quite similar to those reported in table 2.


Table A.2.3 repeats the results in table 2 using the ``third''
estimates, but as a robustness check, we used past issues of the
Survey of Current Business to extend the sample back to
1966Q4.\footnote{%
  The information required to convert GNP to GDP was not always
  available farther back.}  Some of the initial estimates from the
earlier period appear a bit odd, certainly not behaving like efficient
initial estimates.  For example, the ``third'' estimates of $\Delta
GDP(E)$ and $\Delta GDP(I)$ in 1974Q4 and 1975Q1 were both around
minus 10 percent at an annual rate, and these were subsequently
revised to show declines of half that size.  So these results should
also be taken with a grain of salt, but as noted in the main paper,
the revision results are certainly of note.  In predicting latest
$\Delta GDP(E)$ in the second specification, it is optimal over this
43 year sample to place more weight on current and lagged values of
``third'' $\Delta GDP(I)$ than on current and lagged values of
``third'' $\Delta GDP(E)$.  The remainder of the results are broadly
similar to those reported in table 2.

Table A.2.4 reports results using estimates that have passed through
one annual revision.  Again, these are very similar to the results
using the ``third'' estimates, with the revisions evidence providing a
notable exception.  Over the full 1978Q3 to 2008Q4 sample, there is
statistically significant evidence that the first annual revision
estimates of $\Delta GDP(E)$ revise towards the first annual revision
estimates of $\Delta GDP(I)$, so an optimal forecast of latest $\Delta
GDP(E)$ would place some weight on $\Delta GDP(I)$ after the estimates
have passed through one annual revision.  Indeed, the second
specification shows that, in predicting latest $\Delta GDP(E)$, it is
optimal to place as much weight on current and lagged values of
$\Delta GDP(I)$ as on current and lagged values of $\Delta GDP(E)$.

Table A.4 repeats the results in table 4 using annual growth rates of
the output variables, and annual average levels and growth rates of
all the explanatory variables (here we drop the specification using
the non-manufacturing ISM, because the sample is too short, and the
specification using NBER recession dates).  As noted in the text, the
annual growth rate of the stock market is contemporaneously more
highly correlated with annual $\Delta GDP(E)$ than annual $\Delta
GDP(I)$, but the growth rate of the stock market the previous year is
much more highly correlated with annual $\Delta GDP(I)$.  The
remainder of the variables are all more correlated with annual $\Delta
GDP(I)$, and the regression coefficients are all larger in absolute
value using $\Delta GDP(I)$, with F-tests rejecting the equality of
these coefficients.


\newpage

\singlespacing

\begin{landscape}
\centerline {\bf Table A.1.1: Composition of GDP(I)}
\begin{longtable}{c c c c}
GDP(I) Components & Nominal Share & Nominal Variance Share \\
                  & 1984-2006 & quarterly $\%$ changes, 1984Q3-2006Q4 \\
\hline \hline
Employee Compensation & 57$\%$ & 42$\%$ \\
Corporate Profits & 9$\%$ & 25$\%$ \\
Proprietors' Income & 7$\%$ & 8$\%$ \\
Net Interest & 6$\%$ & 9$\%$ \\
Rental Income of Persons & 2$\%$ & 1$\%$ \\
Consumption of Fixed Capital & 12$\%$ & 3$\%$ \\
Taxes on Production and Imports, plus misc & 8$\%$ & 4$\%$ \\
Net Factor Income & -1$\%$ & 7$\%$ \\ \hline
memo: Wages and Salaries, excluding government & 38$\%$ & 40$\%$ \\
\hline \hline
\end{longtable}


\centerline {\bf Table A.1.2: Composition of GDP(E)}
\begin{longtable}{c c c c}
GDP(E) Components & Nominal Share & Nominal Variance Share & Real Variance Share \\
                  & 1984-2006 & \multicolumn{2}{c}{quarterly $\%$ changes, 1984Q3-2006Q4} \\
\hline \hline
PCE, durable goods            & 9$\%$ & 8$\%$ & 13$\%$ \\
PCE, nondurable goods         & 16$\%$ & 13$\%$ & 9$\%$ \\
PCE, services                 & 42$\%$ & 12$\%$ & 11$\%$ \\
Nonresidential stuctures      & 3$\%$ & 7$\%$ & 6$\%$ \\
Equipment and Software        & 8$\%$ & 11$\%$ & 14$\%$ \\
Residential Structures        & 5$\%$ & 7$\%$ & 10$\%$ \\
Change in Private Inventories & 0$\%$ & 30$\%$ & 29$\%$ \\
Exports                       & 10$\%$ & 18$\%$ & 14$\%$ \\
Imports                       & -12$\%$ & -13$\%$ & -15$\%$ \\
Government                    & 19$\%$ & 6$\%$ & 8$\%$ \\
\hline \hline
\end{longtable}
\end{landscape}


\newpage

\centerline {\bf Table A.2.1:  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
% & $\Delta GDP(I)_t - \Delta GDP(E)_t$ & $\Delta GDP(E)_t$ & $\Delta GDP(E)_{t-1}$ & $\Delta GDP(E)_{t-2}$ & $\Delta GDP(I)_t$ & $\Delta GDP(I)_{t-1}$ & $\Delta GDP(I)_{t-2}$ &  $GDP(E)_t - GDP(I)_t$ & Constant & Adj. $R^2$ \\
% & \multicolumn{8}{c}{Explanatory Variables, All Initial (Advance) Estimates} & & \\
 & \multicolumn{7}{c}{Explanatory Variables} & \\
 & \multicolumn{3}{c}{$\Delta GDP(E)$, initial (advance)} & \multicolumn{3}{c}{$\Delta GDP(I)$, initial (advance)} & Constant & Adj. $R^2$ \\
Dependent Variable & $t$ & $t-1$ & $t-2$ & $t$ & $t-1$ & $t-2$ & & \\
\hline \hline
\input{table_GDP_forecasts_2adv.txt}
\input{table_GDI_forecasts_2adv.txt}
\input{table_aux_forecasts_adv.txt}
\hline \hline
\end{longtable}}

\newpage

\centerline {\bf Table A.2.2:  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
% & $\Delta GDP(I)_t - \Delta GDP(E)_t$ & $\Delta GDP(E)_t$ & $\Delta GDP(E)_{t-1}$ & $\Delta GDP(E)_{t-2}$ & $\Delta GDP(I)_t$ & $\Delta GDP(I)_{t-1}$ & $\Delta GDP(I)_{t-2}$ &  $GDP(E)_t - GDP(I)_t$ & Constant & Adj. $R^2$ \\
% & \multicolumn{8}{c}{Explanatory Variables, All Initial (3rd) Estimates} & & \\
 & \multicolumn{7}{c}{Explanatory Variables} & \\
 & \multicolumn{3}{c}{$\Delta GDP(E)$, initial (2nd)} & \multicolumn{3}{c}{$\Delta GDP(I)$, initial (2nd)} & Constant & Adj. $R^2$ \\
Dependent Variable & $t$ & $t-1$ & $t-2$ & $t$ & $t-1$ & $t-2$ & & \\
\hline \hline
\input{table_GDP_forecasts_2pre.txt}
\input{table_GDI_forecasts_2pre.txt}
\input{table_aux_forecasts_pre.txt}
\hline \hline
\end{longtable}}


\newpage

\centerline {\bf Table A.2.3:  Predictive Content of Initial Growth Rates, 1966Q4-2009Q3} 
\vskip 0.1cm
{\scriptsize
\begin{longtable}{c c c c c c c c c}
\hline \hline
% & $\Delta GDP(I)_t - \Delta GDP(E)_t$ & $\Delta GDP(E)_t$ & $\Delta GDP(E)_{t-1}$ & $\Delta GDP(E)_{t-2}$ & $\Delta GDP(I)_t$ & $\Delta GDP(I)_{t-1}$ & $\Delta GDP(I)_{t-2}$ &  $GDP(E)_t - GDP(I)_t$ & Constant & Adj. $R^2$ \\
% & \multicolumn{8}{c}{Explanatory Variables, All Initial (3rd) Estimates} & & \\
 & \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_66.txt}
\input{table_GDI_forecasts_66.txt}
\input{table_aux_forecasts_66.txt} 
\hline \hline
\end{longtable}} \vskip 0.2cm {\bf Note:} Specification using the
slope of the yield curve spread ($r^{Treas(10yr)} - r^{Treas(2yr)}$)
uses a 1978Q2 to 2009Q3 sample.  Specifications using SPF forecasts use
a 1969Q3 to 2009Q3 sample.  Standard errors are Newey-West with 8 lags.

\newpage

\centerline {\bf Table A.2.4:  Predictive Content of Initial Growth Rates, 1978Q3-2008Q4} 
\vskip 0.1cm
{\scriptsize
\begin{longtable}{c c c c c c c c c}
\hline \hline
% & $\Delta GDP(I)_t - \Delta GDP(E)_t$ & $\Delta GDP(E)_t$ & $\Delta GDP(E)_{t-1}$ & $\Delta GDP(E)_{t-2}$ & $\Delta GDP(I)_t$ & $\Delta GDP(I)_{t-1}$ & $\Delta GDP(I)_{t-2}$ &  $GDP(E)_t - GDP(I)_t$ & Constant & Adj. $R^2$ \\
% & \multicolumn{8}{c}{Explanatory Variables, All Initial (Advance) Estimates} & & \\
 & \multicolumn{7}{c}{Explanatory Variables} & \\
 & \multicolumn{3}{c}{$\Delta GDP(E)$, 1st annual revision} & \multicolumn{3}{c}{$\Delta GDP(I)$, 1st annual revision} & Constant & Adj. $R^2$ \\
Dependent Variable & $t$ & $t-1$ & $t-2$ & $t$ & $t-1$ & $t-2$ & & \\
\hline \hline
\input{table_GDP_forecasts_2_1a.txt}
\input{table_GDI_forecasts_2_1a.txt}
\input{table_aux_forecasts_1a.txt}
\hline \hline
\end{longtable}}

\newpage

\centerline {\bf Table A.3: Regressions explaining Revisions, 1994Q1-2006Q4} 
\vskip 0.1cm
{\scriptsize
\begin{longtable}{c c c c c c c c c c c}
\hline \hline
 & \multicolumn{3}{c}{Explanatory Variables, Initial (2nd, 3rd) or 1st Annual} & & \\
 & $\Delta GDP(I)_t - \Delta GDP(E)_t$ & $\Delta GDP(E)_t$ & $\Delta GDP(I)_t$ & Constant & Adj. $R^2$ \\
\hline \hline
\input{table_GDP_revisions_3.txt}
\hline \hline
\input{table_GDI_revisions_3.txt}
\hline \hline
\end{longtable}}

\newpage

\centerline {\bf Table A.4: Regressions of Annual $\Delta GDP(I)$ and $\Delta GDP(E)$ on}
\centerline {\bf Various Alternative Business Cycle Indicators, 1985-2006} 
\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_annual.txt}
\hline
\end{longtable}
\vskip 0.2cm {\bf Note:} Specifications using the high yield bond
spread ($r^{HY corporate} - r^{Treas.(10yr)}$) use a 1986-7 to 2006 
sample.  Standard errors are Newey-West with 2 lags.


\end{document}
