Calculating the Composite Indexes

The procedure for calculating the composite indexes has five distinct steps. In the notation below, [t] and [t-1] 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) Month-to-month 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 _t-1. 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 _t-1)/(X _t + X _t-1). (See below for details on this formula.)

(2) The month-to-month 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 month-to-month change multiplied by the corresponding component standardization factor (m_t = r_x * x_t).

(3) Add the adjusted month to month changes across the components for each month to obtain the growth rate. This step results in the sum of the adjusted contributions (i_t={sum over [x]} m_x,t) which is the monthly growth rate of the index.

(4) The sum of the adjusted contributions, i.e., growth rates, of the composite index are adjusted to equate their trends to that of the coincident index. . This is accomplished by adding an adjustment factor, a, to the growth rates of the index each month (i_t`= i_t+ a). The trend adjustment factor for the leading index is computed by subtracting its average monthly growth rate (sum over [t] i_t/ T where T is the number of observations in the sample) from the average monthly growth rate of the coincident index.

(5) The level of the index is computed using the symmetric percent change formula. The index is calculated recursively starting from an initial value of 100 for the first month of the sample period (i.e. January 1959). The first month's value is I₁=100. The second month's value I₂ = I₁ * (200+i₂`)/(200-i₂`) and this formula is used recursively to compute the index levels for each month that data are available.

(6)The index is rebased to average 100 in 2004. The history of the index is multiplied by 100 and divided by the average for the twelve months of 2004.

Updating the indexes Steps 1 through 6 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 six-month window are not incorporated in the index until the entire index is recomputed. (The Conference Board updates the standardization factors and recomputes 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.

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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 up-to-date as possible.

Additional technical details:

Symmetric percent changes The formula, 200 * (X_t - X_t-1)/(X_t + X_t-1), 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_t-1)/X_t-1, 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. (Two exceptions are the standardization factors and the trend adjustment factor, which are calculated to four 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.)

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