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Bureau of Labor Statistics > Consumer Price Index > Methods > Additional Resources

Technical Appendix to Improvements to the CPI Health Insurance Index: Transitioning from an unsmoothed to a smoothed relative

One of the changes to the CPI health insurance index involves switching from using annual unsmoothed retained earnings data to using a multi-year average. Switching from an unsmoothed to a smoothed relative will create a permanent distortion in the index.

To see how this distortion arises, and how the BLS has derived the correction term consider the following example. The RE ratio starts at 0.20 in the first two periods, increases to 0.40 in period three, and returns to 0.25 the following period before increasing 25% per period for the next two periods before stabilizing. In this example, unlike in the real world data, the RE ratio starts at a stable value and finishes at different stable value. So, the smoothed and unsmoothed indexes should finish at the same level.

Table 1 and figure 1 show what would happen if we switched from using the annual RE ratio to the smoothed RE ratio (as a two period simple moving average) immediately after showing the large increase (in period four). Column two shows the unsmoothed RE ratio. Column four shows the relative calculated from the unsmoothed relative, and column six shows the index calculated from the unsmoothed relative. Column three shows the smoothed RE ratio, column five shows the relative calculated from the smoothed RE ratio, and column seven shows the associated index. Column eight shows the index that starts by using the unsmoothed relative and switches to using the smoothed relative starting in period four.

Table 1: Example showing the impact of switching from an unsmoothed relative to a smoothed relative
Period	RE ratio	RE ratio smoothed (2 yr average)	Relative	Smoothed Relative	Index	Smoothed index	Index – smooth starting in period four
1	0.200				100.000	100.000	100.000
2	0.200	0.200	1.000	1.000	100.000	100.000	100.000
3	0.400	0.300	2.000	1.500	200.000	150.000	200.000
4	0.250	0.325	0.625	1.083	125.000	162.500	216.670
5	0.313	0.281	1.250	0.865	156.250	140.630	187.500
6	0.391	0.352	1.250	1.250	195.310	175.780	234.380
7	0.391	0.391	1.000	1.111	195.310	195.310	260.420

The smoothed index shows a further increase in period four after the spike in RE in period three, which is appropriate since the smoothed index shows less of an increase initially. However, if the switch happens after the spike is already reflected in the index, then switching to the smoothed relative in the fourth period will generate a persistent positive divergence in the index.

The relative in period four for the switch index needs to be the smoothed relative with an adjustment term to return the index to the correct level. The adjustment term can be calculated as the gap between the indexes assuming the retained earnings remains constant after the switch (starting in period five). Table 2 shows how the gap can be derived by holding the RE ratio constant in period five. The first four period values are the same as table 1. In period five, the RE ratio is held constant at period four values. The unsmoothed and smoothed indexes converge to the same value. The index that starts out unsmoothed and switched to smoothed in period four converges to a different value.

Table 2: Example deriving the gap in the indexes from switching from an unsmoothed to smoothed relative
Period	RE ratio	RE ratio smoothed	Relative	Smoothed Relative	Index	Smoothed Index	Switch to smooth in period four index
1	0.200				100.000
2	0.200	0.200	1.000		100.000	100.000
3	0.400	0.300	2.000	1.500	200.000	150.000	200.000
4	0.250	0.325	0.625	1.083	125.000	162.500	216.670
5	0.250	0.250	1.000	0.769	125.000	125.000	166.670
Note: The first four periods are the same as the example. In period five, the RE ratio is assumed to remain constant at the period four value.

This gap in the indexes is the adjustment term. So, the corrected relative in period four will be the smoothed relative times the adjustment term (125/166.6667 = 0.75). It is also possible to calculate the adjustment term using the period four relative and the period five relatives assuming no change in the retained earnings ratio in period five, and adjustment term is equal to the period four unsmoothed relative divided by the period four smoothed relative times the period five smoothed relative assuming no change in the RE ratio (0.625/(1.0833*0.7692) = 0.75). Figure two shows the effect of using this corrected period four relative. After the correction is applied in period four, the switch index is identical to the smoothed index.

Last Modified Date: August 21, 2023