A Diffusion Index for GDP

by P. Cross*

Introduction

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.

Origins

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 to ignore.

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 this sentiment.

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

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 this.

Figure 1

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).

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.

Figure 3

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.

Figure 4

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 sharply.

Figure 5

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 cycle.

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*

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 index.

Conclusion

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 felt.

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 fluctuations.

Bibliography

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.

Notes

* Current Analysis; for the data in this series, contact S. Iliadis (613) 951-1789 or ceo@statcan.ca.

¹. See p. 453-4 in W.C. Mitchell, Business Cycles. University of California Press, Berkeley, 1913.

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 manufacturers.

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.