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Consumer Price Index

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.