Guide for Using Diffusion Indexes and Composite Indexes

   The indexes of business conditions are summary measures for aggregate economic activity. They are designed to be a useful tool for analyzing current state and forecasting future economic conditions. They are indexes that combine the behavior of key cyclical indicators that represent widely differing activities of the economy such as production, employment, etc.
The composite indexes are used to identify the volume of overall business activities by composing percentage changes of selected indicators. On the other hand, diffusion indexes are used to determine turning points of the business cycle, among other purposes, by counting changes in directions of selected indicators.

1.Overview: How To Use the Composite Indexes

(Objective)
   The composite indexes mainly aim to measure the tempo and the magnitude ("the volume") of economic fluctuations. They compose the quantitative changes in indicators that are sensitive to business cycle movements.

(Using composite indexes)
   There are three types of composite indexes.
1) The leading CI
This tends to precede the coincident CI by a few months. This is used to anticipate changes in the direction of the economy.
2) The coincident CI
This coincides with the business cycle. This is used to identify the current state of the economy.
3) The lagging CI
This tends to lag behind the coincident CI by about six months. This is used to confirm turning points and business cycle phases.


In general, increasing coincident index reflects that the economy is in an expansion phase, and decreasing coincident index reflects that the economy is in a contraction phase. The magnitude of the changes in the coincident index reflects the tempo of the expansion or contraction phases. However, not only the composite indexes but also the diffusion indexes should be considered in order to identify whether the economy is in the expansion or contraction phase, or to date business cycle turning points, the details of which are described below. Furthermore, it should be noted that, because the composite indexes are the summary measures of quantitative movement of component indicators selected for their being sensitive to business cycles, they do not cover the full range of economic activities. Currently, the composite indexes use the same selected series of indicators as the diffusion indexes, i.e. 29 series of indicators in total: 12 leading indicators, 11 coincident indicators, and six lagging indicators. The list of selected series of indicators is reviewed each time the economy goes through one complete cycle. The current 29 series were selected in November 2004, when the reference date of the 13th cycle was determined as the best components of the diffusion indexes to achieve their objective of identifying business cycles accurately.

An increasing or decreasing of the composite indexes for an extremely short period of time should not be interpreted as a sign of economic expansion or contraction. Moreover, the composite indexes must move up or down with a certain magnitude in order to interpret that the economy has taken a upward or downward turn. However, as mentioned above, the ultimate measures to determine the cyclical phase are the diffusion indexes. (The business cycle reference date is determined based on the historical diffusion indexes.)

When analyzing month-to-month movements of the composite indexes, it is advisable to smooth idiosyncratic fluctuations using a moving average, although the outliers of the individual indicators are truncated to eliminate the effects of them to the changes in the composite indexes. The three-month backward moving average, which is useful to monitor the short-term tendency, as well as the seven-month backward moving average, which is useful to confirm that the cyclical changes, should be considered for the assessment of the coincident CI.

(Differences from the diffusion indexes)

The diffusion indexes are used to determine the turning points in the business cycles. They measure the proportion of the component indicators that are improving. When the diffusion indexes are above the 50 percent threshold, the economy can be interpreted to be in an expansion phase; when below, in a contraction phase.

On the other hand, the composite indexes provide a quantitative measurement of economic strength. The composite indexes differentiate between small and large overall movements. They are referred to as the indexes that indicate the "volume" of economic activities, e.g. the amplitude of economic peaks and troughs, as well as the speed of economic expansion or contraction. For the better comparison of changes in different economic areas, it is recommended to use both the contributions of the component series to the change in the composite indexes and the diffusion indexes.

(Calculating the composite indexes)
   The composite indexes are calculated by composing month-to-month percentage changes in multiple economic indicators. As a simplified example, assume that one of the composite indexes is constructed from two indicators, Indicator A and Indicator B ("y₁" and "y₂" in the illustration below). In a certain month, Indicators A and B are higher than the previous month by 1 percent and 0.5 percent respectively ("γ₁" and "γ₂" in the same illustration). These change rates are averaged, and the average is multiplied by the previous month's level of the composite index to obtain the current month's.

The change rates with different volatilities are subjected to the process called "normalization" before averaging so that they can be evaluated on a common basis. Assume a situation where Indicator A shows an upward trend and large monthly fluctuations, while Indicator B shows a flat trend and small monthly fluctuations. In this situation, the change rates have the different meanings between Indicator A and B.

The normalization process is performed taking into consideration two types of elements, i.e. trend and amplitude. Assume that Indicator A has a trend of 2 percent and an amplitude of 0.5 percent, the normalized percentage change rate for Indicator A is calculated as follows:

(Change rate for the current month 1 - Trend 2) / (Amplitude 0.5) = -2.

Similarly, assume that Indicator B has a trend of 0 percent and an amplitude of 0.2 percent, the normalized percentage change rate for Indicator B is calculated as follows:

(Change rate for the current month 0.5 - Trend 0) / (Amplitude 0.2) =2.5.

Then, the "composite normalized percentage change rate," Z, is calculated as follows, by averaging the normalized percentage change rates of Indicator A and B:

Z = (-2 + 2.5) / 2 = 0.25.

Because Z is an absolute number with no dimension, it should be adjusted back to the percentage change of the original economic indicators by taking the following two steps: (1) the composite normalized percentage change Z is multiplied by the composite amplitude σ, which is obtained by averaging the amplitudes of Indicator A and B; and (2) the composite trend μ is added to the result of the step (1). The result of the two steps represents the composite " month to month percentage change " V. Then, the current month's composite index is obtained by multiplying V by the previous month's level of the composite index. Individual indicators' month-to-month percent change rate is calculated using a "symmetric percentage change." The symmetric percentage change uses for the denominator an average of the previous and current month values (mean value), instead of the previous month level, as in the ordinary calculation of month-to-month ratios. When calculating the composite index based on V, the symmetric percentage change formula is used inversely.

Composite index calculation flow and examples of values (when the composite index is constructed from two indicators)

2. The calculation method

   The calculation method is as follows.

   Step 1: The previous formula is used for calculating the symmetric percent change ri(t) of individual series yi(t), as in the following. In the notation below, i subscript refers to the number assigned to each indicator.

   If the given time series is zero or a negative value, or is already in percentage form, simple arithmetic differences are calculated.

   Then, outliers are trimmed using the following formula.

   is the first quartile in the interquartile range and is the third quartile in the interquartile range.

Step 2: The trend of individual series (mean percent change ) is calculated by the trimmed 60-month backward moving average.

Next, percent change normalized by interquartile range () is calculated by applying the following formula.

Step 3: Composite percentage change (V(t))is calculated by adding up trend (composite mean percent change,) and the mean of percent change normalized by interquartile range (composite percent change normalized by interquartile range,). In this process, composite percent change normalized by interquartile range is multiplied by the mean of interquartile ranges (composite interquartile range,) so that the levels of the trend component and the cyclical component coincide.

   
where n represents the number of indicators.

Step 4: As in the previous calculation method of composite indexes, composite percent change is cumulated. Finally, the index is rebased so that the value for the reference year is 100.

    ( is the average of the reference year )

Notes for calculation

•    For calculating the leading composite index and the lagging composite index, the trend component of coincident composite index is applied as in the previous method.
•    The threshold for trimmed mean, k, is uniformly set as 1.65, by which 5 percent of the data in the coincident series from January 1980 to December 2005 are regarded as outliers.
•    With regard to dealing with missing data of indicators because of late publication in calculating composite indexes, the trend is calculated by using the available data of all the indicators at that time. For example, if an indicator has data for only 57 months, then the 57- month data are used in calculating the trend. On the other hand, when calculating composite percentage change normalized by interquartile range, indicators with missing data are excluded. As a result, the latest composite percent change normalized by interquartile range does not include all of the indicators.

Reference: Yoshizoe et al., "Expansion of Traditional Indicators," in "New Developments in Business Cycle Indicators," Economic Analysis, No. 166, December 2001, Economic and Social Research Institute, Cabinet Office.

3. Other

Revision of the consumer confidence index

(i)With regard to the consumer confidence index, which is a selected leading index, since the consumer confidence survey is now conducted nationwide on a monthly basis, this series has been revised retroactively from the preliminary release of April 2004.
(ii)Indexes applied from April 2004 onward are those for total households of the consumer confidence survey (nationwide, monthly); between November 2001 and March 2004 are indexes calculated using four items, excluding the item "how prices rose" of the mentioned survey (Tokyo, monthly); between April 1982 and October 2001 are indexes calculated using four items, excluding "how prices rose" of the mentioned survey (nationwide, quarterly); prior to March 1982 are indexes from the mentioned survey (nationwide, quarterly). Where gaps occur, prior to March 2004, prior to October 2001 and prior to August 1982, link coefficients are compiled respectively to connect the periods.

4. The Reference Dates of Business Cycle

   The reference date of a business cycle is first discussed in the Investigation Committee for Business Cycle Indicators, based on historical diffusion indexes, composed of all selected series of coincident diffusion indexes. Consecutively, the President of ESRI determines the reference date. The historical diffusion indexes determine the peak and trough for each selected time series of diffusion indexes (this is referred to as the individual turning point), and diffusion indexes are calculated by marking the period from trough to peak with a plus, and the period from peak to trough with a minus. Since the change in direction is determined by smoothing irregular month-to-month movements of individual time series, the historical diffusion index calculated from these values is relatively smooth, and reflects the basic movement of the business cycle. The last month when the historical diffusion index compiled from a selected series of coincident indexes stays below the 50-percent line corresponds to the cyclical trough; the last month when this index stays above the 50-percent line corresponds to the cyclical peak.
   In addition, the peaks and troughs of each individual time series is dated by applying the Bry-Boschan method, which was developed in the U.S. National Bureau of Economic Research (NBER). In simple terms, this method determines the cyclical peak or trough by providing a series of rules. Two examples of this rule: that five months or more are required in the period between peak and trough, and that the duration of one cycle must be 15 months or more. This procedure, which also involves multiplication of the 12-month moving average, was presented along with the computer program to actually run it.

   Reference: Bry & Boschan (1971) Cyclical Analysis of Time Series: Selected Procedures and Computer Programs, NBER, New York.