A Diffusion Index for GDP
by P. Cross*
Measuring how widespread, or diffuse, an economic phenomenon has
become is a basic analytic tool. It is important to know whether
growth or recession is widespread or confined to certain sectors.
Measuring diffusion also allows analysts to trace how a change in
one sector or region spreads to others: for example, an increase
in auto output will boost demand upstream for suppliers and supply
downstream for distributors of motor vehicles, along with indirect
spin-offs in increased consumer and corporate incomes and spending.
A diffusion index measures the share of industries
experiencing an increase in activity (measured by output, employment,
prices, profits or virtually any other relevant variable) over a
given time span. Developed by the National Bureau of Economic Research
(NBER)1, diffusion indices look
only at the direction, not the rate of change.
The pioneering NBER research on diffusion indices was motivated
by two considerations. First, the original reference dates for peaks
and troughs in the business cycle compiled by Burns and Mitchell
were based on the concept of “clusters” of turning points
in a plurality of indicators, a variant of the diffusion indices
compiled by the NBER for a number of economic time series. As well,
the NBER hoped that diffusion would have leading indicator properties
to anticipate the severity of the business cycle (Moore, 1955, p.14).
Neither of these original motivations is very relevant today.
The development of more sophisticated statistical measures of the
macro-economy, notably GDP, has rendered obsolete the cluster approach
to dating business cycles. As well, diffusion has not differentiated
well between mild and severe swings in the business cycle. This
is not surprising, since diffusion indices measure only whether
industries are expanding or contracting, not the degree to which
they are expanding or contracting. As Broida (an analyst for the
Federal Reserve Board) argued, “It probably would be difficult
to make a general case in favor of discarding information on amplitude
as a means of improving forecasts” (p.15).
Still, business cycle analysts use diffusion indexes as one of the
three criteria (depth, diffusion, and duration) in classifying business
cycle reference dates. They are also used to isolate drops in output
due to irregular or industry-specific factors (such as strikes or
bad weather) from the incremental factors normally associated with
cyclical fluctuations (Moore 1955; Burns and Mitchell 1946; see
Cross, 1982, for an application in the Canadian context).These extraordinary
events can cause output in an industry to abruptly drop towards
zero, a change in amplitude that the diffusion index helps analysts
Business cycle research evolved from the descriptive classification
of cycles in many time series towards more fundamental research
into their causes and mechanisms, and interest in diffusion indices
began to grow. Diffusion indices are a useful tool for studying
the transmission of a cyclical change in one sector of the economy
to other sectors. Diffusion indices “throw light on the sequential
and cumulative aspects of cyclical developments” (Broida,
p.7). They are implicitly used in surveys of purchasing managers,
the business conditions survey conducted by Statistics Canada, and
the attitudinal surveys of consumers and businesses conducted by
the Conference Board in Canada. These all measure whether people
are feeling more positive or negative, but not the intensity of
Like all summary measures of the economy, diffusion indices also
hide important information. One cannot tell from their aggregate
values the industries that are driving the overall change, their
relative importance, their recent trend, or how fast they are changing.
Constructing an index
The technical aspects of the construction of a diffusion index have
been extensively debated in the literature. Since few of the questions
can be addressed strictly from a theoretical perspective, their
final resolution has been largely determined by what works best
in practice. Generally speaking, the empirical evidence points to
a number of guidelines.
Weighted and unweighted diffusion indices
produce very similar results, and most indices employ the simple
measure of diffusion where each industry has an equal weight, regardless
of its actual importance in the total economy (see Hickman, 1958;
Stekler, 1961). Hickman did find that amplitude weighting (that
is, weighting each industry’s movement by the change in that
industry) served to heighten the cyclical amplitude of diffusion
indices, but not their turning points.2
Moreover, as Broida observed (p.15), “the desirability of
ignoring amplitude, however, is at the heart of the diffusion approach”
and most statisticians have followed this path. These include the
two most widely used diffusion indices in the US: the Federal Reserve
Board’s for industrial production and the Bureau of Labor
Statistics’ for payroll employment.3
The relationship of GDP to its diffusion index is analogous to the
different variants of lighting a room: the diffusion index is like
a simple on-off switch; amplitude-weighting is the dimmer switch
that varies the intensity of light; the final step of allowing for
the variable wattage of each bulb is analogous to weighting each
component of GDP by its relative importance.
The analyst must also decide which industries are to be covered
as well as their weights. For Canada, the selection of industries
is largely determined by what is available seasonally adjusted at
a 2 or 3 digit-level of detail back to 1981 in constant dollars
(the chain-weighted measures are not used as they are available
only from 1997). The industries included in the index and the corresponding
weight of each sector in the calculation of value-added GDP are
presented in Table 1.
Table 1: Sectorial Distribution of GDP and its Diffusion Index
|Primary and Construction
|Transportation and Trade
The most striking difference between the distribution
of the diffusion index and GDP itself is in goods-producing industries
(notably manufacturing), which have a share of about two-thirds
in the diffusion index compared with about one-third in value-added
GDP. The higher proportion of goods in the diffusion index can be
justified by their greater sensitivity to fluctuations in demand.4
The counterpart to the much greater weight of manufacturing (36
percentage points more in diffusion than GDP) is a 21-point shortfall
for business services, which includes information and culture (notably
telecommunications), finance and real estate, administrative and
management services, and the amorphous professional, scientific
and technical services industries. Other sectors receiving less
weight in the diffusion index include goods-handling industries
(trade and transportation) and government. These are less problematic
for analysis, as the fortunes of the former are largely driven by
the course of goods production, while the latter is often relatively
inert on a monthly basis.
In comparing diffusion indices calculated separately for goods
and services (Figure 1), what is most striking is the similarity
between the two up until 1998–except for more widespread drops
in goods during the recessions in 1990 and, to a lesser extent,
in 1981. Since overall output in services is clearly less cyclical
than in goods, the similarity of the two diffusion indexes suggests
that the service industries included were indeed quite cyclical,
at least before 1998. This raises the separate question of why declines
in services are less common after 1998–explosive growth in
ICT services and fewer cutbacks in government may explain some of
The index is based on 83 industries, each with
an equal weight. Each series is assigned a value of 100, 50 or 0,
depending on whether it is rising, unchanged,5
or falling. These values are then summed up and divided by the number
of series to get the diffusion index.
A value of 50 does not necessarily mean that 50% of industries are
growing, since output in some industries could be unchanged. Instead,
50 means that equal numbers of industries are expanding and contracting.
Over 50 means more industries are growing than contacting; less
than 50, that a plurality of industries are shrinking. (In practice,
a value of less than 50 in the smoothed index has always been associated
with least one quarter of declining GDP). An index level of 60 means
that 20% more industries are expanding than contracting, using the
formula [60-(100-60) = 20].
Another variable is the time span over which growth is calculated,
which can be anywhere from one month to several years, which can
help smooth the index. The Federal Reserve Board calculates its
index over 1-month, 3-month, and 6-month spans. For GDP in Canada,
the standard deviation of the 6-month span is actually higher than
for 3-months (13.5 versus 11.1). We have found that smoothing is
better accomplished by a 5-month moving average, with a standard
deviation of 6.1 (Figure 2).
Even discriminating between industries that are expanding or contracting
(effectively setting the bar at 0% growth) is arbitrary. It is just
as easy to calculate a diffusion index for industries surpassing
or lagging behind the long-run monthly growth average of about 0.2%
(as is done in Figure 3). But raising the bar like this basically
just shifts down the overall level of the diffusion index, without
changing any of its other statistical properties (such as turning
points or lead times), as the two series have a correlation of 0.96.
Broadly speaking, raising the bar to 0.2% made more of a difference
to the level of the diffusion index in the 1980s and early 1990s
than in the last decade, implying that more industries were growing
by less than 0.2% in the earlier years.
The diffusion index for GDP
Figure 4 shows the unsmoothed diffusion index
since 1981 and smoothed with a 5-month moving average.6
The first noteworthy point is the relatively narrow range in which
the index moves: even at the best of times, the index rarely breaks
70 (its record high of 78 was set in the run-up to the new millenium
in November 1999), while its low of 27 was set during the worst
of the recession in 1982. This highlights that at any given moment
the economy contains large numbers of industries going in opposite
directions. As Burns and Mitchell noted, business cycle conditions
are “never strictly uniform, and [are] at times markedly different”
(p.456). Anecdotal evidence of industries shrinking during boom
times, and growing during recessions, is easy to turn up.
The narrow band of values for the diffusion index also highlights
the tightrope the economy walks between expansion and contraction.
A change in the fortunes of a small number of key industries can
tip this balance, with a swing for just a dozen of the 83 industries representing the difference between boom and bust.
The diffusion index moves closely with the monthly change in GDP,
with a correlation coefficient of 0.90 since 1981 (both smoothed
with a 5-month moving average in Figure 5). The high correlation
of GDP growth and its diffusion is instructive about business cycle
dynamics: periods of expansion and recession occur because the impulse
to grow or contract is widely dispersed through the economy, not
because a few industries are posting exceptional results. Slower
growth in GDP almost always reflects an increasing number of industries
contracting instead of expanding, not slower growth in many industries.
In statistical terms, the diffusion effect is much stronger than
the amplitude effect (which is also why amplitude-weighting adds
little to the analytical power of these indices). As Hickman (1959)
pointed out, this has important implications for the business cycle,
as industries where output is falling are more likely to cut investment
One exception to discounting amplitude is sudden movements in
diffusion. Sharp drops in the unsmoothed diffusion index are usually
a sign of temporary forces at work in the economy, not the more
gradual spreading of cyclical forces through the economy. Some of
the sharpest decreases were due to the Ice Storm in January 1998
(when the one-month index fell from 66 to 51) and the blackout in
August 2003 (when it plunged from 62 to 30, one of its lowest readings
ever). In both cases, output recovered quickly; indeed the smoothed
index was hardly affected in either case. The lesson is that sharp
monthly movements in diffusion should be regarded suspiciously from
a business cycle point of view.
The business cycle tends to work gradually through the economy.
The turning point in diffusion usually occurs at the mid-point of
an expansion or recession. The lowest levels attained in the recession
that began in 1990 was plumbed a half a year after the downturn
began. Similarly, diffusion hit its peak levels in 1985 and 1994,
about three years after GDP began to recover. But the largest changes
in diffusion did occur just after the turning point of the business
Leading indicator properties
While the diffusion index closely tracks the monthly change in GDP,
it is not a reliable guide to what is about to occur in the economy.
Neither the level of the diffusion index nor the magnitude of its
recent change bears a consistent relationship with the subsequent
course of GDP.
First, a given level of the diffusion index has accompanied a wide
range of growth outcomes. Diffusion did dip to or below 40 in the
recessions beginning in 1981 and 1990. But the index fell below
50 in 1986, 1995 and 2003, none of which were associated with outright
recession. Conversely, the index fell only to 51.2 in September
2001, which was at least as difficult a period for the economy as
the three periods of slowdown.
Second, the size of the change in the diffusion index early in
the business cycle has little bearing on the magnitude of the subsequent
change in the economy. For example, the smoothed index rose about
40% early in the recoveries in 1983 and 1991, but the ensuing GDP
upturn was much slower in 1991 (1.2%) than in 1983 (5.6%). A drop
of 10 points in diffusion in 1990 signalled the onset of recession,
but declines of 13 points in both 1986 and 2001 were not followed
by recessions (although GDP slowed on each occasion, with the deceleration
in year-over-year growth ranging from 6% in 1986 to 3% in 2001).
Table 2: The Diffusion Index for GDP at Cyclical Turning Points*
||Diffusion Index at Troughs
||8 months later
||Change of GDP
||Diffusion Index at Peaks
||6 months later
||Change of GDP
|*The turning points in
the diffusion index for 1986 and 2001 correspond to slowdowns,
In terms of lead times at turning points, the diffusion index
should also be interpreted cautiously. Comparing the turning points
of the diffusion index with those for GDP shows they were identical
emerging from the recession in 1982, while the diffusion index led
slightly in the 1990-1992 cycle (three months at the peak, one at
the trough). But offsetting this are the large number of false signals,
where the diffusion index turned but the economy did not (notably
in 1986, 1995, 2000 and 2003).
One reason for the lack of discriminating power of the diffusion
index in predicting the qualitative movement of business cycles
is that the components were selected for their sensitivity to cyclical
movements in the economy. As such, they are not representative of
the amplitude of cyclical fluctuations in other sectors of the economy.
The dearth of leading indicator properties in our diffusion index
is consistent with the findings for the US payroll data, where analysts
concluded that its “leading indicator properties currently
appear tenuous” (Getz and Ulmer, p. 15). Broida and Valvanis
express similar reservations for the US industrial production diffusion
Our new diffusion index for GDP is a useful measure of how widespread
are the economic impulses at work in the economy. They highlight
that business cycles occur because expansion or contraction is widely
transmitted among industries, not that a few industries dominate
the cycle. They also show that sudden lurches in diffusion usually
reflect non-cyclical forces, such as poor weather or supply disruptions.
Cyclical movements in the economy tend to be slower moving and widely
The diffusion index is also a useful reminder that at any given
point of the business cycle, a significant number of industries
are always showing opposing movements. Citing a selective litany
of sectors that are expanding or contracting proves little about
the overall state of the economy.
But diffusion indexes also have shortcomings. Most importantly,
a given level or change in diffusion has no consistent relationship
with whether a change in the business cycle is impending or the
severity of that change. As well, lead times in signaling turning
points are short or non-existent, while false signals are common.
The inability of diffusion indices to discriminate between periods
of slowdown and recessions (or weak versus strong recoveries) reflects
the preponderance of goods-producing industries, which make up two-thirds
of the index but just one-third of GDP. This is especially relevant
in the current situation in Canada, where the exchange rate has
the potential to drive a wedge between a growing service sector
and a shrinking manufacturing base. Diffusion indices are best used
as summary measures of the cumulative process of how changes in
the economy are transmitted between industries. They are also useful
in predicting severe fluctuations in the amplitude of a cyclical
expansion or contraction, although their ability is muted for milder
Broida, A. Diffusion Indexes. The American Statistician,
Vol. 9, No. 3, June 1955.
Burns, A. and W. Mitchell. Measuring Business Cycles. National
Bureau of Economic Research, 1946.
Chaffin, C. and W. Talley. Diffusion indexes and a statistical
test for predicting turning points in business cycles. International
Journal of Forecasting, 5 (1989).
Cross, P. The Business Cycle in Canada, 1950-1981. Current
Economic Analysis, Statistics Canada Catalogue 13-004E, March 1982.
Getz, P. and M. Ulmer. Diffusion indexes: a barometer of the
economy. Monthly Labor Review, April 1990.
Hickman, B. An Experiment With Weighted Indexes of Cyclical
Diffusion. The Journal of the American Statistical Association,
Vol. 53, No. 281, March 1958.
Hickman, B. Diffusion, Acceleration, and Business Cycles.
The American Economic Review, Vol. XLIX, No. 4, Sept. 1959.
Moore, G. Diffusion Indexes: A Comment. The American Statistician,
Vol. 9, No. 4, Oct. 1955.
Moore, G. The Diffusion of Business Cycles. Reprinted in
Business Cycle Indicators, Volume I, G. Moore ed., Princeton University
Press, Princeton, 1961.
Stekler, H. Diffusion Index and First Differences Forecasting.
The Review of Economics and Statistics, Vol. XLII, No. 2, May 1961.
Valvanis, S. Must the Diffusion Index Lead? The American
Statistician, Vol. II, No. 4, Oct. 1957.
* Current Analysis; for the data in this
series, contact S. Iliadis (613) 951-1789 or firstname.lastname@example.org.
1. See p. 453-4 in W.C. Mitchell,
Business Cycles. University of California Press, Berkeley,
2. By then weighting these
percent changes with their share in GDP, the calculation reverts
to GDP itself.
3. The US Bureau of Economic
Analysis provided diffusion indices for a dozen major series in
its Business Conditions Digest journal, ranging from new orders
to prices, profits, weekly hours and the stock market. Most of these
series were terminated when this publication was folded into the
Survey of Current Business in April 1990.
4. Statisticians are often
defensive about the greater wealth of detail available for goods,
especially manufacturing. But this also has advantages, as manufacturing
is often at the centre of developments ranging from the business
cycle to globalization. For the latter, the scope for carving up
the production process into smaller parts is much greater than for
natural resources or services, where production is comparatively
simple. This is why trading patterns have changed much more for
5. The number of industries
showing no change was never less than 6% of the total through most
of the 1980s, but has fallen to 1% in recent years.
6. Most of the figures and
tables show the smoothed version, but the text refers to unsmoothed
data, unless otherwise noted.