Calculating the Composite Indexes
revised 01/01
The procedure for calculating the composite indexes has five distinct
steps. In the notation below, [t] and [t1] refer to the current and prior
month respectively. Also, [x] and [m] refer to a particular component of
the index and notation such as ({sum over [x]} w_{x}) means that the "w"s
for each [x] are added together.
(1) Monthtomonth changes are computed for each component.
If the component X is in percent change form or an interest rate, simple
arithmetic differences are calculated: x _{t}=X _{t}  X _{t1}. If the component
is not in percent change form, a symmetric alternative to the conventional
percent change formula is used: x _{t} = 200 * (X _{t}  X _{t1})/(X _{t} + X _{t1}). (See
below for details on this formula.)
(2) The monthtomonth changes are adjusted to equalize the volatility
of each component. Standard deviations v_{x} of the changes in
each component are computed. These statistical measures of volatility are
inverted (w_{x} = 1/v_{x}), their sum is calculated (k={sum over [x]} w_{x}), and they are restated so the index's component standardization factors
sum to one (r_{x}=(1/k) * w_{x}). The adjusted change in each component is
the monthtomonth change multiplied by the corresponding component standardization
factor (m_{t} = r_{x} * x_{t}).
(3) The level of the index is computed using the symmetric percent
change formula The first month's value is I_{1} = (200+i_{1}/(200i_{1}). The second month's value I_{2} = I_{1} * (200+i_{2})/(200i_{2}) and this formula is used recursively to compute the index levels for each month that data
are available.
(4) The index is rebased to average 100 in 1996 The history of the index is multiplied by 100 and divided by the average for the twelve months of 1996.
Updating the indexes Steps 1 through 5 are used to compute the composite indexes for a long historical period. The indexes are updated for the latest and previous
six months of data using the predetermined factors from the sample period.
Revisions in the components that fall outside of the moving sixmonth window
are not incorporated in the index until the entire index is recomputed.
(The Conference Board plans to update the standardization factors and recompute the
entire history of the three composite indexes once a year.) Also, when data for a particular
indicator is not available, the standardization factors ( r_{x} ) for the
other components are recomputed that month so that they continue to sum
to one. No change is made to the index factor (f).
Click here for an example.
Computing the component contributions Steps 2 and 3 can be combined to compute the contributions of each component as m_{t} = f * r_{x} * x_{t}.
2001 Revisions
Prior to 2001, an additional adjustment was made to equalize the volatility of
the composite indexes. For the U.S. leading and lagging indexes, each monthly
sum (i_{t}) was multiplied by an index standardization factor (f) that
equalizes the volatility these indexes relative to the coincident index.
This factor is the ratio of the standard deviation of the percent changes for
the coincident index (vcoin) to the standard deviation of the unadjusted
percent changes for the particular composite index (flead = vcoin/vlead, flag =
vcoin/vlag). The Conference Board decided to remove this step as it was
proven not have any meaningful difference to the composite indexes' analytical
value.
The leading, coincident and lagging indicators that are not available at the
time of publication are estimated using statistical imputation. An
autoregressive model is used to estimate each component. The resulting
indexes are constructed using real and estimated data, and will be revised as
the data unavailable at the time of publication become available. Such
revisions are part of the monthly data revisions, now a regular part of the U.S.
and global business cycle indicators program. The main advantage of this
procedure is to utilize available data sooner. Empirical research by The
Conference Board suggests there are real gains in adopting this procedure to
make all the indicator series as uptodate as possible.
Additional technical details:
Symmetric percent changes The
formula, 200 * (X_{t}  X_{t1})/(X_{t} + X_{t1}), treats positive and negative changes symmetrically. When it shows a one percent increase followed by
a one percent decrease, the level of X has returned to its original value.
This is not true with the more conventional formula, 100 * (X_{t}  X_{t1})/X_{t1}, the same percent increase and decrease would leave X at slightly lower
value. The symmetric percent change formula has been used since the public
debut of the composite indexes in the late 1960s. Both formulas, as well
as a third, increasingly popular alternative based on logarithmic differences,
produce very similar cyclical patterns.
Rounding is avoided wherever possible until the final
step when the index is reported at one decimal place. (One exception is
the standardization factors, which are calculated to three decimal places.)
The final rounding, together with the symmetric percent change formula
in step 4, is the reason the rounded sum of the reported contributions
from each component does not always equal the simple percent change in
the rounded index.
Changes in procedures Prior to the December 1996 revision, the first revision made by The Conference
Board to the U.S. composite index, average absolute changes were used, instead of standard deviations, to measure
the volatility of each component The remaining procedures follow those
developed by the Department of Commerce before the composite index program
was transferred to the Board. (For an alternative description, see the
Survey of Current Business, October 1993.)
Prior steps not reinstated Two additional stepsuse of performancebased factors and reverse trend adjustmentswere part of the
U.S. composite index calculations at one time, but were dropped by the Department of Commerce before The Conference Board's involvement. In 1989, the use of additional and performancebased component weighting factors, which were derived from a cyclical scoring system, was discontinued. In 1993, trend adjustments that equalized the growth rates of the three indexes were discontinued. The Conference Board considered reinstating these steps, but found the added complications to outweigh their benefits.
