Rebasing and Splicing an Output Index: Laspeyres versus Paasche, and a Seasonal "Collapse"

Laspeyres fixes the 2015 basket; Paasche uses the current (2024) basket. The category whose relative price rose most - patented biologics, up about 60% - also saw its basket share fall from 30% to 12%. Buyers substituted away from the item that became relatively more expensive.

A fixed-weight index keeps a heavy weight on that item, so it rises faster. Laspeyres 138.2 > Paasche 127.5 is the expected direction, and the 10.7-point gap (about 8.4% of the Paasche level) is the substitution bias.

Interpretation of the bounds: under normal substitution behaviour, a Laspeyres index tends to overstate the change in the cost of the basket and Paasche to understate it, so the "true" index lies between them. A symmetric summary is the Fisher index, √(138.2 × 127.5) = √17,620.5 ≈ 132.7.

Which to use depends on the question:

• Wage or pension indexation over nine years with a 60% relative price shift: use a superlative or chained index, or the fixed Laspeyres will systematically overcompensate. The gap of 10.7 points over nine years is roughly 0.9 points per year - material in a payout formula.
• Deflating nominal revenue: you need a price index with current-period quantity weights, i.e. a Paasche-type or implicit deflator, because nominal value is Σpₜqₜ and must be divided by Σpₜqₜ/Σp₀qₜ. Dividing by a fixed-basket Laspeyres overstates real growth once substitution has occurred.
• Publishing a headline: state the formula and the base; either is defensible when labelled, but never present one as if the choice were neutral.

Check the arithmetic first: 109.8/121.4 = 0.9044, and 138.2 × 0.9044 = 125.0. The arithmetic is right; the inference is wrong.

Ratio splicing forces the spliced series to reproduce the old series' growth rates. Old-series growth 2023→2024 is 138.2/121.4 − 1 = 13.8%. Applying that to 109.8 gives exactly 125.0. The official new series implies 123.0/109.8 − 1 = 12.0%. The 1.6% gap between 125.0 and 123.0 is not rounding - it is the re-weighting (plus any coverage or methodology change) introduced at rebase. A new base is not merely a change of scale.

Consequences:

• The spliced series is a legitimate way to present a continuous history at the old weighting, and its 13.8% growth is the old-basket growth. It is not an estimate of the official series and must not be used to overwrite or extend it; a 1.6% discrepancy in a published index is a substantive error, not a rounding note.
• The comparison should be presented as "old-weight growth" versus "new-weight growth" and the difference attributed to substitution, not quietly reconciled.

Correct practice:

• Use the office's official linked or annually chained series, including any published back-series, for current analysis.
• If splicing is unavoidable, link on a single overlapping period (e.g. December 2023, or the 2023 annual average if monthly data are unavailable) rather than the whole annual average, apply one factor, and document the link period and factor.
• State that levels before and after the rebase are not strictly comparable at the old weights, and present growth rates rather than levels across the break.

The claim compares December with January - two different seasonal positions in a series with a strong 12-month pattern (December peak, January trough). A −19.0% month-on-month fall is a recurring calendar movement, not news.

Correct comparisons:

• Year on year: January 2025 (98.0) against January 2024 (96.5) = +1.6%. The "collapse" disappears entirely; output is slightly up.
• Seasonally adjusted series: use X-13ARIMA-SEATS or STL to estimate and remove the seasonal and trading-day components, then quote month-to-month changes on the adjusted series, naming the method and the revision policy. Adjusted month-to-month changes are the only ones in which a monthly movement can be read as a change in trend.
• If unadjusted figures must be quoted, always compare the same month across years and give both the year-on-year rate and the index level.

Checks before interpreting any single month:

• Number of working days and the timing of public holidays (a lunar-calendar shift can move a whole month's output).
• Whether the December figure is routinely revised downward and the January figure upward; monthly indices are estimates and revisions can reverse the sign of a one-month change.
• The historical December→January pattern over the previous five years; if the drop is always about 18–20%, the January value carries no information about current conditions.

Conclusion: the journalist has made a specification error in the comparison, not a discovery about the industry. A 19% one-month move in a strongly seasonal index should never be reported without either year-on-year comparison or seasonal adjustment.