Rebasing and Linking Price Index Series

Why Rebase or Link a Price Index?

A price index is a relative measure, not a price quoted in dollars. Its numerical scale depends on the chosen reference period. Rebasing changes that scale so a selected period has an index value such as 100. Linking is used when two index series overlap but have different scales: one series is adjusted to meet the other at an agreed overlap period, allowing a continuous series to be displayed.

These operations are useful for presenting a long history on a convenient reference base or joining periods covered by separate index series. They do not, by themselves, change the underlying prices, baskets, weights, or measured movements. The earlier material on price indexes and chain-weighted indexes provides the context for treating index values as relative measures; here the focus is on changing their scale and joining their records.

Definition: To rebase an index series is to multiply every value by the same positive constant so that the value in a chosen reference period becomes the desired reference level, commonly 100. To link two index series is to rescale one series so its value equals the other series’ value in a chosen common period, called the overlap period, and then join the adjusted series.

Rebasing an Index Series

Let \(I_t\) be the original index in period \(t\), and let \(b\) be the selected reference period. If the rebased series is to equal 100 in period \(b\), define the rebased index \(R_t\) by dividing each original value by the value in period \(b\), then multiplying by 100.

R_t = \frac{I_t}{I_b}\times 100

The choice of 100 is conventional, not essential. To set the reference-period value to another positive number \(K\), use \(R_t=(I_t/I_b)K\). Every period receives the same multiplier, so the index’s relative movements are retained.

Why rebasing preserves measured changes: For any two periods \(s\) and \(t\), rebasing gives \(R_t/R_s=(I_t/I_b)/(I_s/I_b)=I_t/I_s\). The reference-period denominator cancels. Therefore, the ratio between any two index values is unchanged. In particular, the proportional change from \(s\) to \(t\), \((I_t-I_s)/I_s\), is the same when calculated from \(R_s\) and \(R_t\). Rebasing changes the displayed level, not these relative movements.

Worked Example: Set a New Reference Period

A hypothetical household price index has the values 80 in 2018, 88 in 2019, 96 in 2020, and 108 in 2021. Rebase the series so that 2020 equals 100. Then check that the change from 2019 to 2021 is preserved.

Method. Divide every original value by the 2020 value, 96, and multiply by 100.

For 2018, \(R_{2018}=(80/96)\times100=83.3333\ldots\). For 2019, \(R_{2019}=(88/96)\times100=91.6666\ldots\). For 2020, \(R_{2020}=(96/96)\times100=100\). For 2021, \(R_{2021}=(108/96)\times100=112.5\).

Rounded to two decimal places, the rebased series is 83.33, 91.67, 100.00, and 112.50. The requested change from 2019 to 2021, calculated from the original series, is \((108-88)/88=20/88=0.2272727\ldots\), or 22.73%. From the unrounded rebased values it is \((112.5-91.6666\ldots)/91.6666\ldots=0.2272727\ldots\), also 22.73%.

Check. The 2020 value is exactly 100 before rounding, and the 2019-to-2021 proportional change agrees on both scales. The method avoids treating the reference value as an amount to subtract: rebasing is multiplication by a common factor, not an additive shift.

Linking Two Series at an Overlap Period

Suppose an older series \(A_t\) and a newer series \(B_t\) both report a value for period \(o\). To express the newer series on the older series’ scale, calculate the linking factor as the older series’ overlap value divided by the newer series’ overlap value. Multiply every value in the newer series by that factor.

L = \frac{A_o}{B_o}, \qquad B_t^{*}=L B_t

Here \(B_t^{*}\) is the linked value. At the overlap period, \(B_o^{*}=LB_o=(A_o/B_o)B_o=A_o\), so the two series meet exactly. Also, for any two periods \(s\) and \(t\) within the newer series, \(B_t^{*}/B_s^{*}=(LB_t)/(LB_s)=B_t/B_s\). Thus the linking operation preserves the newer series’ ratios and proportional changes.

The direction of the factor matters. The factor shown above puts the newer series on the older series’ scale. If instead the older series is to be expressed on the newer series’ scale, the factor is \(B_o/A_o\). State which scale is being retained before calculating.

Worked Example: Join Two Series at a Common Year

A hypothetical regional price index has an older series with values 135 in 2019, 144 in 2020, and 150 in 2021. A revised series reports 120 in 2021, 126 in 2022, and 132 in 2023. Link the revised series to the older series at 2021, then verify the 2021-to-2023 movement.

Method. Retain the older series’ scale. The overlap period is 2021, so the linking factor is \(L=150/120=1.25\). Multiply every revised-series value by 1.25.

For 2021, \(120\times1.25=150\). For 2022, \(126\times1.25=157.5\). For 2023, \(132\times1.25=165\). The joined series is therefore 135 in 2019, 144 in 2020, 150 in 2021, 157.5 in 2022, and 165 in 2023.

The revised series’ 2021-to-2023 proportional change is \((132-120)/120=12/120=0.10\), or 10%. In the linked series it is \((165-150)/150=15/150=0.10\), also 10%.

Check and common mistake. The linked value in 2021 equals the older value, 150, and the later movement is unchanged. A common error is to reverse the factor and calculate \(120/150=0.8\). That would give a 2021 linked value of \(120\times0.8=96\), which does not match 150. The correction is to use older-over-newer, \(150/120=1.25\), when expressing the newer series on the older scale.

Linking Several Series in Sequence

A continuous record may require more than one link. Each new series is first adjusted to match the already linked series at its overlap period. The adjustment is then applied uniformly to every value in that new series. Keep track of which scale is being carried forward: the target at each step is the value of the existing linked series in the overlap period, not an unrelated base value.

Choose the retained scale.
Identify the existing series whose levels will anchor the joined record.

Find the overlap period.
Use the period reported by both the existing linked series and the new series.

Calculate the factor.
Divide the existing linked value by the new series’ value at that overlap.

Scale and check.
Multiply all values in the new series by the factor, verify equality at the overlap, and check that its internal ratios are unchanged.

Worked Example: Link Three Series in Sequence

Three hypothetical index records cover successive periods. Series A reports 100 in 2019, 110 in 2020, and 120 in 2021. Series B reports 100 in 2021, 110 in 2022, and 120 in 2023. Series C reports 120 in 2023 and 126 in 2024. Starting with Series A’s scale, construct a continuous series through 2024.

First link, at 2021. The existing value is 120 and Series B’s overlap value is 100. The factor is \(120/100=1.2\). Multiplying B’s values gives 2021: \(100\times1.2=120\), 2022: \(110\times1.2=132\), and 2023: \(120\times1.2=144\).

Second link, at 2023. The linked value already established for 2023 is 144, while Series C reports 120. The factor is \(144/120=1.2\). Thus C’s linked 2023 value is \(120\times1.2=144\), and its 2024 value is \(126\times1.2=151.2\).

Joined result. The continuous series is 100 in 2019, 110 in 2020, 120 in 2021, 132 in 2022, 144 in 2023, and 151.2 in 2024. Each overlap agrees: both linked records give 120 in 2021 and 144 in 2023. As a movement check, Series C changes by \((126-120)/120=6/120=5\%\); its linked values change by \((151.2-144)/144=7.2/144=5\%\).

Check. Applying the second factor to Series C’s overlap value reproduces the existing 2023 value, and the 2024 movement is retained. The method avoids scaling C directly to Series A’s 2021 value: the correct target for the second link is the already linked 2023 value.

What Rebasing and Linking Do Not Fix

A numerical match at an overlap does not establish that two series measure exactly the same concept. They may use different baskets, weights, population coverage, collection procedures, or quality adjustments. Linking makes their levels meet at a selected point; it does not remove any differences in how the series were constructed. A change in measured movement near the joining point may therefore reflect a methodological difference as well as price experience.

If a series contains a missing overlap period, the direct linking calculation above cannot be made from the supplied values alone. A factor requires a common period with values in both series, or a separately justified method for establishing a comparable overlap. Choosing a factor without such support would impose an arbitrary relationship between the scales.

Keep the operations distinct. Rebasing selects a new reference period within a series. Linking aligns separate series at an overlap. Either can be followed by the other, but the formula and target must be stated each time. These transformations are useful for consistent presentation and comparison; the resulting index values remain index numbers, not prices or direct measures of a household’s personal cost of living.

Key takeaway: Rebase by dividing every value by the chosen reference-period value and multiplying by the desired reference level. Link by dividing the retained series’ overlap value by the new series’ overlap value, then multiplying the new series by that factor. Both operations use a common multiplier, so they preserve each transformed series’ internal ratios and measured proportional changes.

Check Your Understanding

For each question, identify the target scale or reference period before applying a factor.

A hypothetical index reports 72 in 2020, 81 in 2021, and 90 in 2022. Rebase it so 2021 equals 100. Give all three rebased values to two decimal places. (Answer: 2020: 88.89; 2021: 100.00; 2022: 111.11.)
An older series has values 160 in 2022 and 168 in 2023. A newer series has values 200 in 2022 and 220 in 2023. Link the newer series to the older scale at 2022. What is the linked 2023 value, and what is the 2022-to-2023 proportional change? (Answer: Linking factor \(160/200=0.8\); linked 2023 value \(220\times0.8=176\); change \((176-160)/160=10\%\).)
A series has values 90 in period A and 108 in period B. After rebasing, the corresponding values are 75 and 90. Check the period A-to-B proportional change on both scales. (Answer: Original change \((108-90)/90=20\%\); rebased change \((90-75)/75=20\%\).)
Series M reports 50 in 2021 and 55 in 2022. Series N reports 80 in 2022 and 88 in 2023. Starting with M’s scale, link N at 2022 and report the joined values for 2021–2023. (Answer: Factor \(55/80=0.6875\); joined values are 50 in 2021, 55 in 2022, and \(88\times0.6875=60.5\) in 2023.)
Series P reports 120 in 2023 and 126 in 2024. Series Q reports 150 in 2023 and 165 in 2024. If Q is linked to P’s scale at 2023, calculate the factor, linked 2024 value, and 2023-to-2024 proportional change. (Answer: Factor \(120/150=0.8\); linked 2024 value \(165\times0.8=132\); proportional change \((132-120)/120=10\%\).)

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