News, Noise and Estimates of the "True" Unobserved State of the Economy

News, Noise and Estimates of the “True” Unobserved State of the Economy

Dennis J. Fixler and Jeremy J. Nalewaik

WP2004-08
January 31, 2006
















The views expressed in this paper are solely those of the author and
not necessarily those of the U.S. Bureau of Economic Analysis
or the U.S. Department of Commerce.

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News, Noise, and Estimates of the “True”
Unobserved State of the Economy
Dennis J. Fixler and Jeremy J. Nalewaik∗
Draft, Comments Welcome
January 31, 2006
Abstract
Which provides a better estimate of the “true” state of the U.S. economy,
gross domestic product (GDP) or gross domestic income (GDI)? Past work has
assumed the difference between each estimate and the “true” state of the economy
is pure noise, taking greater variability to imply lower reliability. However each
difference may be pure news instead; then greater variability implies higher infor-
mation content and greater reliability. We analyze various vintages of estimates,
developing models for combining GDP and GDI under the differing assumptions,
and use revisions to the estimates to show the news assumption is probably more
accurate.
JEL classification: C1, C82.
Keywords: GDP, statistical discepancy, news and noise, signal-to-noise ratios,
optimal combination of estimates, business cycles
∗Fixler: U.S. Bureau of Economic Analysis, 1441 L Street NW, Washington, DC 20230 (e-mail:
dennis.fixler@bea.gov); Nalewaik: U.S. Bureau of Economic Analysis, 1441 L Street NW, Washington,
DC 20230 (e-mail: jeremy.nalewaik@bea.gov). We thank Erick Sager for research assistance, and Ben
Bridgman, Ufuk Demiroglu, Bob Dennis, Bruce Grimm, Steve Landefeld, Keith Phillips, Matt Pritsker,
Marshall Reinsdorf, Mark Watson, and seminar participants at the Bureau of Economic Analysis, the
Washington Statistical Society, the Board of Governors of the Federal Reserve System, and the 2005
NBER SI Macroeconomics and Productivity workshop for comments. The views expressed in this paper
are soley those of the authors and are not necessarily those of the U.S. Bureau of Economic Analysis
or the U.S. Department of Commerce.
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For analysts of economic fluctuations, estimating the true state of the economy from
imperfectly measured official statistics is an ever-present problem. As most economists
agree that no one statistic is a perfect gauge of the state of the economy, many have
proposed using instead some type of weighted average of multiple imperfectly measured
statistics. Examples include the composite index of coincident indicators, and averages
of different measures of aggregate economic activity such as GDP and GDI. While the
precise meaning of the state of the economy can vary from case to case, in this paper we
fix ideas, taking it to mean the growth rate of the size of the economy as traditionally
defined in the U.S. National Income and Product Accounts (NIPAs).1
The main point of our paper is as follows. Prior attempts to produce such a weighted
average of imperfectly measured statistics have made a strong implicit assumption that
drives their weighting: that the difference between the true state of the economy and
each measured statistic is pure noise, or completely uncorrelated with information about
the true state of the economy.2 We examine a different assumption that produces an
essentially opposite weighting: that the difference between the true state of the economy
and each measured statistic is pure news, or pure information about the true state of
the economy. Under the noise assumption, a statistic with greater variance is given
smaller weight because it is assumed to contain more noise; under the news assumption,
a statistic with greater variance is given larger weight because it is assumed to contain
more news about the true state of the economy.
We play out these contrasting assumptions in our empirical application of choice.
The most widely used statistic produced by the U.S. Bureau of Economic Analysis
(BEA) is its expenditure-based estimate of the size of the economy, gross domestic
product (GDP); however it also produces an income-based estimate of the size of the
economy, gross domestic income (GDI), from different information. Computing the
value of GDP and GDI would be straigthforward if we could record the value of all the
underlying transactions included in the NIPA definition of the size of the economy, and
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GDP and GDI would coincide in this case. However all the underlying transactions are
not recorded, and measured GDP and measured GDI do not coincide, giving rise to
the statistical discrepancy. BEA relies on various surveys, censuses and administrative
records, each imperfect, to compute the estimates; differences between the data sources
used to produce GDP and GDI, as well as other measurement difficulties, leads to the
statistical discrepancy.
Some economists3 have estimated the growth rate of “true” unobserved GDP as
a combination of growth rates of BEA’s latest available time series on GDP and GDI,
generally concluding that GDI growth should be given more weight than measured GDP
growth. Is GDI really the more accurate measure, implying that the emphasis on GDP
is misplaced? We argue for caution, as the results in these papers are driven by the
noise assumption: the models implicitly assume that since GDP growth has higher
variance than GDI growth, it must be noisier, and so should receive a smaller weight.
However GDP may have higher variance because it contains more information about
“true” unobserved GDP (this is the essence of the news assumption); then measured
GDP should receive the higher weight.
We emphasize that, since we never observe the “true” state of the economy, assump-
tions about news vs. noise are inherently untestable. Indeed, the general version of our
model allows the difference between each measured statistic and the “true” state of the
economy to be a mixture of news and noise, and we proceed to show that virtually any
set of weights can be rationalized by making untestable assumptions about the mix-
tures. More information must be brought to bear on the problem; otherwise the choice
of weights will be totally arbitrary.
We argue that it is possible to gather additional information that may shed some light
on whether the differences between “true” GDP and our two measures are mostly news
or noise. BEA releases its first national accounts estimates for any given quarter about
a month after the quarter closes, and subsequently revises the initial estimates a number
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of times over the next three to four years (as well as periodically at approximately five
year intervals after that).4 The revisions incorporate more comprehensive and accurate
source data, and each revision presumably brings the national accounts estimates closer
to the “truth.” Following N. Gregory Mankiw and Matthew D. Shapiro (1986), we
show that these revisions are in fact mostly news, not noise, and based on this result
we make two points. First, when combining the first few releases of GDP and GDI
(the estimates that have not been revised many times, and that are of most interest to
analysts attempting to gauge the current state of the economy), we should favor the
assumptions of the news model, as we know that at least some of the difference between
each estimates and “true” GDP is news - the part that will be eliminated later through
revisions. Second, we argue that, since the differences between the early release and most
current estimates are largely news, the differences between the most current estimates
and “true” unobserved GDP are likely news as well. The argument is based on the
assumption that observed patterns would continue: if BEA did ultimately acquire exact
knowledge of “true” unobserved GDP, we hypothesize that its (hypothetical) ultimate
revision from the most current estimates to the truth would be similar to other revisions
we have observed in the past. This second point is more speculative than the first,
however.
We then proceed to estimate models and compute estimates of “true” unobserved
GDP growth under the news and noise assumptions, distinguishing between the first
few estimates and the later, more heavily revised estimates that are typically used for
historical research. For the first few estimates, the variance of measured GDP exceeds
the variance of GDI, so the news assumption dictates that measured GDP receives the
higher weight. If we take parameters to be fixed over our full sample, the same holds true
for the heavily revised estimates, as the variance of GDP exceeds the variance a GDI, the
same result found in prior research on this topic. However the variance of our measured
estimates drops dramatically after the early 1980s - see Margaret M. McConnell and
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Gabriel Perez-Quiros (2000) - and if we account for this phenomenon, we find that the
variance ordering of the two estimates changes: the variance of GDI then exceeds the
variance of GDP after the early 1980s. Whether this result will continue going forward
remains to be seen, but assuming that it does, when combining the heavily revised later
vintages of data in the post-1984 time period, the news assumptions would place more
weight on GDI.
Figure 1 plots data from the latest recession and recovery, the 1999-2002 growth
rates of the most current BEA data on GDP and GDI, and estimated “true” GDP from
the news model.5 We see that these news model estimates reflect some characteristics of
averaging, with the patterns in 2000 in “true” GDP being less erratic than the patterns
in either GDP or GDI; indeed, the combined estimates show a smooth downward trend
into recession. But we also see in the fourth quarter of 1999, the quarter with the biggest
late-cycle boom, “true” GDP growth exceeds the growth rate of either GDP or GDI, and
in the third quarter of 2001, the nadir of the recession, “true” GDP growth again takes
on a value more extreme than either GDP or GDI. The examples are not anomalies:
the variance of our estimated “true” GDP growth exceeds the variance of either GDP
growth or GDI growth. This is a natural outcome of the news hypothesis, and since
there likely exists other unobserved information about “true” GDP reflected in neither
measured GDP or GDI, this estimated variance of “true” GDP represents a lower bound
on the actual variance of “true” GDP. If the news hypothesis is true, then, it has clear
implications for real business cycle and other economic models that use as an input the
variance of GDP.
The rest of the paper is organized as follows. Section I discusses the various models
in a simple bivariate setup, drawing out their implications for constructing weighted
averages. Section II describes BEA’s national income accounts data that we employ in
our empirical work, and analyzes whether revisions from the early to later vintages of
GDP and GDI are mostly news or noise. Section III reports results from constructing
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weighted averages of GDP and GDI under both the news model and the noise model. In
addition to reporting results from the most current available data, it also reports results
from earlier vintages, as these are important and widely followed indicators of the state
of the economy. We draw our conclusions in Section IV.
I. Theory: The Competing News and Noise Models
A. Review of News, Noise, and Covariance Assumptions
Let ∆y⋆ be the true growth rate of the economy, let ∆yk be one of its measured
t
t
estimates, and let εk be the difference between the two, so:
t
∆yk = ∆y⋆ + εk.
t
t
t
The noise model makes the classical measurement error assumption that cov ∆y⋆, εk =
t
t
0; this is the precise meaning of the statement that εk is noise. One implication of a
t
noisy estimate ∆yk is that it’s variance is greater than the variance of the true growth
t
rate of the economy, or var ∆yk > var (∆y⋆).
t
t
In contrast, if an estimate ∆yk were constructed efficiently with respect to a set of
t
information about ∆y⋆ (call it Fk), then ∆yk would be the conditional expectation of
t
t
t
∆y⋆ given that information set:
t
∆yk = E ∆y⋆|Fk .
t
t
t
Writing:
∆y⋆ = ∆yk + ζk,
t
t
t
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the term ζk represents the information about ∆y⋆ that is unavailable in the construc-
t
t
tion of ∆yk. Then ∆yk and ζk represent mutually orthogonal pieces of news about
t
t
t
∆y⋆, employing the terminology in Mankiw and Shapiro (1986), and cov ∆yk, ζk = 0.
t
t
t
This leads us to an implication of the news model that we employ later, namely that
cov ∆yk, ∆y⋆ = var ∆yk . We also have var (∆y⋆) > var ∆yk , an implication oppo-
t
t
t
t
t
site to that of the noise model.
The news model can be written with the notation of the noise model if we take −ζk =
t
εk and switch this term to the other side of the equation, but the covariance assumption of
t
the noise model will be violated; in fact the error will be perfectly negatively correlated
with the missing piece of information about the true growth rate of the economy, so
cov ∆y⋆, εk = cov ∆yk + ζk, −ζk = − var εk . The variance ordering of the news
t
t
t
t
t
t
assumption, var (∆y⋆) > var ∆yk , will still hold, as:
t
t
var ∆yk
= var (∆y⋆) + var εk + 2 cov ∆y⋆, εk
t
t
t
t
t
= var (∆y⋆) − var εk .
t
t
Writing the models in this common notation, and differentiating them by assumptions
about the covariance of εk with ∆y⋆, will be useful in discussing the empirical results in
t
t
the paper.
The pure news and pure noise assumptions are extremes; many intermediate cases
could be considered where εk is part news and part noise, implying differing degrees of
t
negative covariance between ∆y⋆ and εk. We consider a general model that encompasses
t
t
these intermediate cases in the next subsection.
B. The Mixed News and Noise Model
We consider a model with two estimates of true unobserved GDP, each an efficient
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estimate plus noise:
∆y1 = E ∆y⋆|F1 + ε1,
and
t
t
t
t
∆y2 = E ∆y⋆|F2 + ε2.
t
t
t
t
The noise variables ε1 and ε2 are mutually uncorrelated and, naturally, uncorrelated with
t
t
true unobserved GDP. Taking ∆y1 to be GDP and ∆y2 to be GDI, the information in
t
t
F1 likely would consist of personal consumption expenditures, investment, net exports,
t
and the other components that sum to GDP, while the information in F2 likely would
t
consist of wage and salary income, corporate profits, proprietors’ income, and the other
components that sum to