# Calculating the Composite Indexes

## Please refer to January 26, 2012 technical notes for details on the comprehensive revisions.

The procedure for calculating the composite indexes has six 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.) If the component X is a diffusion index (e.g. ISM New Orders Index) or an interest rate spread the monthly level is used x _{t}=X _{t} (Diffusion indexes are first normalized by subtracting their sample mean and dividing by their standard deviation).

(2) **The monthly contributions of the components 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 contribution in each component is the monthly contribution multiplied by the corresponding component standardization factor (m_{t} = r_{x} * x_{t}).

(3)** Add the adjusted monthly contribution across the components for each month to obtain the growth rate of the index.** 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 indexes 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). For example, 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 I_{1}=100. The second month's value I_{2} = I_{1} * (200+i_{2}`)/(200-i_{2}`) 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 the base year.** The history of the index is multiplied by 100 and divided by the average for the twelve months of the based year, currently 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, usually in January.) 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.

## 2012 Comprehensive Benchmark Revisions

In addition to regular annual revisions, The Conference Board implemented a comprehensive revision of Leading Economic Index (LEI) for the United States effective with the January 26, 2012 release. The last time the LEI had comprehensive revisions was in 1996 after The Conference Board received the responsibility for the LEI and the Business Cycle Indicators program from the Bureau of Economic Analysis at the U.S. Department of Commerce.

These comprehensive revisions are the result of an extensive reevaluation of existing components of Leading Economic Index for the United States. Following discussions with the Business Cycle Indicators Advisory Panel and other experts, The Conference Board has decided to replace three of the ten components and make a minor adjustment to another component. The composition changes reflected in the new LEI address structural changes that have occurred in the U.S. economy in the last several decades.

The changes in the LEI composition include:

1) incorporating the new Leading Credit Index ^{ TM } (LCI) and omitting the real money supply (M2) component starting in 1990 (real M2 remains in the index before 1990);

2) replacing the ISM Supplier Delivery Index with the ISM New Orders Index;

3) replacing the Reuters/University of Michigan Consumer Expectations Index with an equally weighted average of consumer expectations of business and economic conditions using questions from Surveys of Consumers conducted by Reuters/University of Michigan and Consumer Confidence Survey by The Conference Board (after 1978, Reuters/University of Michigan Consumer Expectations Index remains in the index before 1978 ); and

4) replacing “New Orders for (nondefense) Capital Goods” with “New Orders for (nondefense) Capital Goods excluding Aircraft.”

In addition to these major changes to the composition, The Conference Board has implemented changes in the methodology and procedures used in the calculation process.

These modifications are:

1) normalized levels of the indicator rather than its monthly changes will be used to calculate the component contributions of components based on diffusion indexes such as the ISM New Orders Index;

2) when component data are missing, autoregressions in log differences instead of levels will be used to calculate the statistical imputation of the missing months;

3) trend adjustment will be done in two periods: 1959-1983 and 1984-2010 (same as the volatility adjustment); and

4) LCI contributions to the LEI are calculated from its levels (not monthly changes) and it is inverted

As a result of these changes, the history of the revised indexes and their month-over-month changes will no longer be directly comparable to those issued prior to the comprehensive benchmark revision. Based on its performance since 1990, and especially before and during the 2008-2009 recession, the new LEI should provide more accurate predictions of business cycle peaks and troughs.

Working papers on these changes are posted on The Conference Board Economics Program Working Paper Series: EPWP 1105 and EPWP 1106.

## 2005 Revisions

The July 2005 release incorporated two major revisions to The Conference Board Leading Economic Index (LEI): 1) a trend adjustment to the LEI and 2) a new method for calculating the contribution of the yield spread in the LEI. Click here for an article which describes these revisions and compares the current version of the LEI (old LEI) with the new version released in July 2005 (new LEI). The trend adjustment facilitates interpretation and use of the LEI. The new measure of the yield spread improves the performance of the LEI by better reflecting the way the yield spread anticipates cyclical turning points. There were no changes to the composition of the coincident or lagging indexes. Further information on methodology and revisions can be found here.

## 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.)