1. Background
Consumers Price Index (CPI) trimmed mean and weighted percentile analytical measures are available in supplementary tables released with the quarterly CPI Hot Off The Press. The annual measures are calculated by directly weighting annual percentage changes, rather than by aggregating the four relevant quarterly percentage changes.
As part of a major review of the CPI (see the information paper Consumers Price Index Review for details), the CPI basket of goods and services was reselected and reweighted at the June 2006 quarter. This resulted in a transition period during which the percentage changes used to calculate the annual trimmed mean and weighted percentile measures spanned the June 2006 quarter price reference period.
As the June 2006 quarter reweight, based on the 2003/04 Household Economic Survey (HES), resulted in some significant relative changes in expenditure weights (such as for petrol and the purchase of new housing), Statistics New Zealand made available two sets of annual trimmed mean and weighted percentile measures during the transition across the June 2006 quarter reweight. The two sets of figures were based on:
- 'price-updated' June 2002 quarter expenditure weights
- 'price-backdated' June 2006 quarter expenditure weights.
- Two sets of figures were released for the September 2006, December 2006 and March 2007 quarters. From the June 2007 quarter onwards, only the June 2006 quarter expenditure weights will be used.
Statistics NZ considers that the transition across the June 2006 quarter reweight has resulted in a structural break in the annual trimmed mean and weighted percentile time series. For this reason, the annual time series based on the June 2002 quarter expenditure weights, and on the June 2006 quarter expenditure weights, will be presented as separate time series, with an overlap period of three quarters. The annual series based on the June 2002 quarter expenditure weights cease at the March 2007 quarter. The annual series based on the June 2006 quarter expenditure weights commence with figures for the September 2006 quarter and will be updated on an ongoing basis.
This article explains the issue and gives details of how the annual series will be presented on an ongoing basis.
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2. Transition across a reweight period
When the composition and relative importance of the CPI basket of goods and services is reviewed, the expenditure weights and price movements of the updated basket are linked to the previous basket at an overlap quarter. For the 2006 reweight, the CPI was compiled, for periods up to and including the June 2006 quarter, using the previous (that is, June 2002 quarter) basket and set of expenditure weights.
The September 2006 quarter CPI was then compiled from price movements for the updated and reweighted basket from the June 2006 quarter to the September 2006 quarter, and linked to the index for the June 2006 quarter. Linking the updated basket in this way ensures that reweighting the index does not in itself result in a shift in the index.
Quarterly and annual movements that span a reweight period
Movements in the CPI from one quarter to the next reflect the quarterly price change and price-updated relative importance of the goods and services in the basket. The relative importance of the goods and services is determined by the unique set of expenditure weights being used at the time. As new expenditure weights are linked at a single quarter when a reweight is implemented, an immediate transition is made for quarterly changes in the CPI from the old set of weights to the new set of weights. The weights applied to quarterly price changes from the March 2006 quarter to the June 2006 quarter were based on June 2002 quarter expenditure weights price updated to the March 2006 quarter. In the following quarter, the weights applied to quarterly price changes from the June 2006 quarter to the September 2006 were the new June 2006 quarter expenditure weights.
However, as new expenditure weights are linked at a single quarter, the CPI annual price changes from a period prior to the linking quarter to a period after the linking quarter are implicitly calculated by 'compounding' one or more quarterly percentage changes based on old weights and one or more quarterly percentage changes based on new weights. For example, the CPI annual percentage change from the September 2005 quarter to the September 2006 quarter implicitly contained one quarter using the new weighting pattern, and three quarters based on the previous weighting pattern. It will not be until the June 2007 quarter that annual movements fully incorporate the new weighting pattern.
Note: Successive quarterly increases of 1.0, 2.0, 3.0 and 4.0 percent result in an annual increase for the latest quarter, compared with the same quarter a year earlier, of 10.4 percent. This result is best obtained by directly comparing the index numbers for the two quarters a year apart, but can also be obtained by compounding (rather than simply adding) the quarterly percentage changes. Compounding involves expressing the percentage changes as ratios of change, multiplying them together, then expressing the result as a percentage. That is: ((1.01 x 1.02 x 1.03 x 1.04) - 1) x 100 = 10.4 percent.
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Quarterly and annual trimmed mean and weighted percentile movements
The trimmed mean measures involve calculating the weighted mean of price movements in the central core of the ranked distribution, in effect cutting off the tail at each end of the distribution of price movements. This distribution is ranked by price change at the 'item' level of the basket, from smallest (greatest negative) to largest. The items that are excluded from the trimmed mean from period to period depend on the distribution of price change and the level of trim.
Trimmed means are published for trims of 5, 10, 15, 20, 25 and 30 percent. Each item is given its price-updated weight in the distribution, and it is this weight that determines which items are trimmed from the distribution at each level of trim. For example, for the 10 percent trimmed mean measure, items with 5 percent of the expenditure weight are trimmed from the lower end of the distribution of price change, and items with 5 percent of the expenditure weight are trimmed from the upper end of the distribution of price change.
The weighted median measure is the movement of the item that is in the middle of the ranked distribution of price change (also known as the 50th percentile). That is, items with half of the price-updated weight have price changes that are below that of the median, and items with the remaining weight have price changes that are above the median. This is similarly calculated for the 10th, 25th, 75th and 90th percentiles.
The weights used each quarter to calculate the trimmed mean and weighted percentile measures are price updated from the relevant price reference period to the earlier of the two quarters in the relevant comparison. Price updating involves multiplying the price reference period expenditure weight by a ratio of the price index number for the earlier of the two quarters in the comparison to the index number for the price reference period, as discussed in the appendix.
When the CPI is reweighted, as happened at the June 2002 quarter and again at the June 2006 quarter, there is a transition period during which the percentage changes used to calculate the annual trimmed mean and weighted percentile measures span the price reference period. The annual measures are calculated directly from the annual percentage movements. This means that the annual trimmed means are calculated directly by trimming the tails of the annual price change distribution.
During the transition across the June 2002 quarter CPI reweight, the weights used to calculate the September 2002, December 2002 and March 2003 quarter annual trimmed means and weighted percentiles were the June 1999 quarter price reference period expenditure weights, price updated to the September 2001, December 2001 and March 2002 quarters, respectively. The June 2002 quarter price reference period expenditure weights were then used in the annual trimmed mean and weighted percentile calculations for the June 2003 quarter.
A similar situation arose with the June 2006 quarter CPI reweight. The same approach as was taken in 2002 to make the transition across the June 2006 quarter CPI reweight resulted in the September 2006, December 2006 and March 2007 quarter annual trimmed means and weighted percentiles being calculated by using the June 2002 quarter price reference period expenditure weights, price updated to the September 2005, December 2005 and March 2006 quarters, respectively.
An alternative approach to calculating the annual trimmed mean and weighted percentile measures during the transition across the June 2006 quarter price reference period is to 'price backdate' expenditure weights that relate to the new price reference period of the June 2006 quarter. Price backdating involves multiplying the June 2006 quarter expenditure weight by a ratio of the price index number for the earlier of the two quarters in the annual comparison (for example, the December 2005 quarter) to the index number for the June 2006 quarter price reference period.
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3. Presentation of results
As the June 2006 quarter reweight resulted in some significant relative changes in expenditure weights (such as for petrol and the purchase of new housing), Statistics NZ made two sets of annual trimmed mean and weighted percentile measures available during the transition across the June 2006 quarter reweight. The two sets of figures were based on:
- price-updated June 2002 quarter expenditure weights
- price-backdated June 2006 quarter expenditure weights.
In supplementary tables 1 and 2, under the heading 'percentage change from same quarter of previous year' two sets of figures were released for the September 2006, December 2006 and March 2007 quarters. From the June 2007 quarter onwards, only the June 2006 quarter expenditure weights will be used.
Statistics NZ considers that the transition across the June 2006 quarter reweight has resulted in a structural break in the annual trimmed mean and weighted percentile time series. For this reason, the annual time series based on the June 2002 quarter expenditure weights, and on the June 2006 quarter expenditure weights, will be presented as separate time series, with an overlap period of three quarters. The annual series based on the June 2002 quarter expenditure weights cease at the March 2007 quarter. The annual series based on the June 2006 quarter expenditure weights commence with figures for the September 2006 quarter and will be updated on an ongoing basis.
Supplementary tables show the way that the annual trimmed mean and weighted percentile time series will be presented for the June 2007 and subsequent quarters.
Figures will also be held on Statistics NZ's online time series database INFOS as separate series. The INFOS references for the full set of trimmed mean and weighted percentile series are given at the end of this article.
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4. Reconciling annual trimmed means and weighted percentiles over the reweight period
As two separate time series exist during the transition across the June 2006 quarter price reference period, users should keep in mind two considerations when interpreting the two parallel time series for the annual trimmed means and weighted percentiles. Firstly, consistency with the published all groups CPI, and secondly, relevance of the expenditure weights.
Consistency with the all groups CPI
Given the discussion above concerning the interpretation of the annual percentage change in the all groups CPI when the comparison spans a price reference period, it can be seen that the annual percentage change in the all groups index progressively shows more influence from the new set of expenditure weights during the transition across the reweight period.
As the two separate annual trimmed mean and weighted percentile time series each uses only one set of weights in their calculation, consistency with the all groups index can guide their interpretation. For example, for the September 2006 quarter all groups CPI annual percentage movement, the new set of expenditure weights was only used for the movement between the June and September 2006 quarters. The old set of expenditure weights was used for quarterly movements for the December 2005, March 2006 and June 2006 quarters.
Considering this, the September 2006 quarter annual trimmed means and weighted percentiles using price-updated June 2002 weights have three quarterly movements in common with the all groups CPI. However, for the December 2006 and March 2007 quarters, this falls to two quarters and one quarter, respectively. This means that, during the transition period, the annual trimmed means and weighted percentiles based on price-backdated June 2006 quarter weights became progressively more consistent with the all groups CPI; the price-updated June 2002 quarter trimmed mean and weighted percentile annual movements became progressively less consistent with the all groups CPI.
Annual movements for the all groups CPI and the two 10 percent trimmed mean series (based on the June 2002 quarter and June 2006 quarter expenditure weights) are shown in the following graph.
Relevance of expenditure weights
Another consideration is which of the two sets of expenditure weights for calculating annual trimmed means and weighted percentiles in 2006 and 2007 is more relevant – the June 2002 quarter weights or the June 2006 quarter weights?
Price updating weights accounts for any price change between the price reference period and the earlier period in each annual comparison. However, when a new set of expenditure weights is introduced, the weights reflect the underlying quantities of the new weight reference period. The weight reference periods (and main weighting source) corresponding to the June 2002 quarter and June 2006 quarter price reference periods are the 2000/01 HES and the 2003/04 HES, respectively.
The quantities underlying the June 2006 quarter expenditure weights are considerably more recent than those underlying the June 2002 quarter expenditure weights. Because they are more recent, the June 2006 quarter expenditure weights (based on 2003/04 quantities) could be considered more relevant for calculating annual trimmed mean and weighted percentile inflation in 2006 and 2007, than the June 2002 quarter expenditure weights (based on earlier, 2000/01 quantities).
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5. Further reading on trimmed means and weighted percentiles
Information paper published with the first release by Statistics NZ of the trimmed mean and weighted percentile measures of price change, "Trend Measures of Price Level Change".
http://www2.stats.govt.nz/domino/external/pasfull/pasfull.nsf/web/Hot+Off+The+Press+Consumers+Price+Index+September+2003+quarter+Information+Paper?open
Paper presented at the 2004 New Zealand Association of Economists conference, "Trend Measures of Price Level Change: Trimmed Means and Weighted Medians". top
6. List of trimmed mean and weighted percentile measures available on INFOS
| INFOS reference: CPIQ.SE9______ |
Series description |
| NS4000 |
Quarterly 5 percent trim |
| NS4005 |
Quarterly 10 percent trim |
| NS4010 |
Quarterly 15 percent trim |
| NS4015 |
Quarterly 20 percent trim |
| NS4020 |
Quarterly 25 percent trim |
| NS4025 |
Quarterly 30 percent trim |
| NS4050 |
Annual 5 percent trim |
| NS4055 |
Annual 10 percent trim |
| NS4060 |
Annual 15 percent trim |
| NS4065 |
Annual 20 percent trim |
| NS4070 |
Annual 25 percent trim |
| NS4075 |
Annual 30 percent trim |
| NS4100 |
Quarterly weighted 10th percentile |
| NS4105 |
Quarterly weighted 25th percentile |
| NS4110 |
Quarterly weighted median |
| NS4115 |
Quarterly weighted 75th percentile |
| NS4120 |
Quarterly weighted 90th percentile |
| NS4150 |
Annual weighted 10th percentile |
| NS4155 |
Annual weighted 25th percentile |
| NS4160 |
Annual weighted median |
| NS4165 |
Annual weighted 75th percentile |
| NS4170 |
Annual weighted 90th percentile |
| NS4250 |
Annual 5 percent trim – June 2006 qtr weights |
| NS4255 |
Annual 10 percent trim – June 2006 qtr weights |
| NS4260 |
Annual 15 percent trim – June 2006 qtr weights |
| NS4265 |
Annual 20 percent trim – June 2006 qtr weights |
| NS4270 |
Annual 25 percent trim – June 2006 qtr weights |
| NS4275 |
Annual 30 percent trim – June 2006 qtr weights |
| NS4350 |
Annual weighted 10th percentile – June 2006 qtr weights |
| NS4355 |
Annual weighted 25th percentile – June 2006 qtr weights |
| NS4360 |
Annual weighted median – June 2006 qtr weights |
| NS4365 |
Annual weighted 75th percentile – June 2006 qtr weights |
| NS4370 |
Annual weighted 90th percentile – June 2006 qtr weights |
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Appendix – Price updating weights
For all periods following a reweight, the weight assigned to each individual good or service's quarterly percentage change depends not only on its original price reference period weight, but also on its price change, relative to other goods and services in the basket, since the price reference period. As the quantity of an item purchased in the weight reference period is fixed, the relative expenditure share of that item in the CPI will implicitly rise if its price rises faster than the (weighted) average of all other goods and services in the CPI, or fall if its price change is below that of the remainder of the CPI. This implicit weight is termed the price-updated weight.
The price reference period expenditure weights of goods and services in the basket can be price updated to a specific quarter and then used to weight price movements from that quarter to a subsequent quarter. For example, the price-updated weights for the September 2006 quarter can be applied to the price movements from the September 2006 quarter to the December 2006 quarter to calculate the index for the December 2006 quarter.
An example illustrating the calculation of price-updated weights is set out below.
|
Price reference period o |
Period t |
Price
(Po)
($) |
Expenditure weight
(Eo)
($) |
Relative
expenditure weight
(percent) |
Price
(Pt)
($) |
Price-updated
expenditure weight
(Et)
($) |
Relative price-
updated expenditure
weight
(percent) |
| Milk |
2.50 |
25,000 |
28.06 |
2.60 |
26,000 |
27.07 |
Eggs |
3.52 |
1,760 |
1.98 |
3.57 |
1,785 |
1.86 |
Bread |
3.17 |
47,550 |
53.38 |
3.50 |
52,500 |
54.67 |
Butter |
1.97 |
14,775 |
16.59 |
2.10 |
15,750 |
16.40 |
Total |
... |
89,085 |
100.00 |
... |
96,035 |
100.00 |
Symbol:
... not applicable
As the price of bread (up 10.4 percent) has risen faster than the (weighted) average of all items (up 7.8 percent), its relative weight at period t has increased.
This can be illustrated by comparing the weight of petrol for the June 2002 quarter with the June 2002 weight after price updating to the June 2006 quarter. Petrol prices rose faster than the CPI from the June 2002 quarter to June 2006 quarter, so quarterly petrol price movements had a larger influence on the CPI before the 2006 reweight than after the 2002 reweight. At the June 2006 quarter, the price-updated weight of petrol was 4.41 percent, compared with the June 2002 weight of 3.12 percent.
Tables
Supplementary tables (Excel, 35kb)
The downloadable attached file is in Microsoft Excel format. If you do not have access to Excel use the Excel file viewer to view, print and export the contents of these files.