We are implementing annually chain-linked business prices in line with international best practice and to improve consistency with other price indices such as the Consumer Prices Index (CPI). This is a significant improvement to the weighting and linking of business inflation statistics, which we previously announced as part of a consultation in 2017. The implementation of chain-linking is recommended by Eurostat over the current method of rebasing for price statistics, as the weighting structures are updated more frequently.
This article focuses on the methodology and practical implementation of chain-linking for business prices, including the technical process of price-updating sales data to forecast more representative weights. This article is part of a collection of articles we are publishing. Other articles published are:
- producer price inflation methods changes: this outlines the move from net to gross basis to measure the headline producer price index, removal of duty and the sources used to compile the weights required for chain-linking
- services producer price inflation methods changes: this outlines the sources used to compile the weights required for chain-linking and a change to the classification framework
- producer price weight change impacts: this discusses the impact of introducing chain-linking and the other new methods on weights used in the Producer Price Indices (PPIs)
- services producer price weight change impacts: this discusses the impact of introducing chain-linking and the other new methods on weights used in the Services Producer Price Index (SPPI)
To complete the collection of articles, we will publish a further two articles to provide the impact of implementing the new methods on the PPI and SPPI. We are planning to publish the PPI and SPPI using the new methods towards the end of 2020.Back to table of contents
Business prices are a collection of inflation statistics that measure the inflation across the manufacturing and service sectors and include Producer Price Index (PPI), Export Price Index (EPI), Import Price Index (IPI) and Services Producer Price Index (SPPI). To meet international regulations, the weighting structure has been updated historically every five years to reflect changes in the economy. Annual chain-linking is the method of updating weights on an annual basis and statistically linking them to produce a continuous time series. The method changes described in this article apply to the following statistics:
Our aim is to construct indices that track producer price movements at several different levels of detail in the manufacturing and service sectors.
We collect price data for many manufacturing and service products in the form of a basket of goods. These are weighted together to form indices that measure the price behaviour of broad groupings, up to the headline Producer Price Index (PPI) and Services Producer Price Index (SPPI). These measure the price movements in the manufacturing sector and part of the service sector, respectively.
We construct weights using sales data from Office for National Statistics (ONS) surveys and administrative data for specific periods. The sales data represent the turnover generated by UK companies selling each manufactured product (in the PPI) or service (in the SPPI) to the UK market. For the Import Price Index (IPI) and Export Price Index (EPI), sales of manufactured products are sourced from HM Revenue and Customs’ (HMRC’s) records. More information about the sales data is available in Services producer prices weights changes: 2020 and Producer prices weights changes: 2020.
The weight for any product group into the higher grouping is equal to the proportion of the products’ sales within the total sales for that group. The higher the sales value for a product, the higher its weight into the aggregated price index. See Appendix 1 and Appendix 2 for further details.Back to table of contents
In the current method, business price indices are calculated using the Laspeyres index formula, which weights prices in proportion to their quantities sold in the base period:
pt,i is the price of item i in the current period t.
p0,i is the price of item i in the base period 0.
q0,i is the quantity sold of item i in the base period 0.
This can be expressed as the sum of price relatives weighted by expenditure share from the base period:
w0,i is the expenditure share or weight of item i in the base period 0.
Rebasing is a generic process that updates one or more of the following:
- the weights in an index
- the price reference period of an index number series
- the index reference period of an index number series
As explained, in the current method for producing business prices, weights are updated and indices are re-referenced every five years. Price relatives are also updated so that in the base year, the price relative is 100, and therefore price changes in other periods are comparable to this base year. To update a price relative, the original price relative is divided by the price relative of that item at the new reference period, then multiplied by 100.
To avoid any step change when an index is rebased, a link factor is applied to the historical series to link the new series on, which ensures movements in an index only reflect price movements. The link factor, L, is calculated as follows:
INew,l is the index under the new base conditions in the link period.
IOld,l is the index under the old base conditions in the link period.
This link factor is applied to construct a continuous series over time (Figures 1 and 2).
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While the underlying methodology for both annual chain-linking and rebasing is the same, there is a difference in the frequency of updating weights. To adopt annual chain-linking, one of the main challenges to overcome is the requirement for sales data from the previous year to produce the weights for the current year (for example, 2019 sales data are needed for weights used in the compilation of 2020 indices). Sales data are normally provided with a lag, so is not normally available within the timeframe required.
A secondary issue is that sales data to be used as weights need to represent transactions during the link period, which is Quarter 4 (Oct to Dec) for the Services Producer Price Index (SPPI) and December for the Producer Price Index (PPI). It is not always possible to obtain sales data representing a single quarter or month for each service product’s or manufacturing product’s transactions, as many relevant surveys gather sales data from transactions that occur over the course of a year. For quinquennial rebasing of both the SPPI and PPI, this was not an issue because a whole year of transactions was used as weights.
To align the sales data to the period required, we need to update them. To overcome these issues, a method for estimating the sales during the link period is employed, referred to as price updating. This approach uses the annual total sales, which represent an entire year of transactions, and updates them to represent the sales during the link period (that is, Quarter 4 for the SPPI or December for the PPI).
This is done by calculating and applying a measure of inflation between the year that the given sales cover and the link period. The measure of inflation is determined using the relevant indices from the SPPI or PPI. The price updating formula for a product or service is:
Sl are the estimated sales during the link period, l. This is referred to as “price updated sales”.
Sy are annual sales for year y.
I ̅y is the average price index value during year y.
Il is the price index value during the link period, l.
The index values Il and I ̅y must share the same reference period.
Price updating normally is applied at the Classification of Products by Activity (CPA) six-digit index level, which is the lowest level of aggregation in the PPI and SPPI. However, there is an issue when the index value coverage for a CPA six-digit is not sufficient to enable price updating of its corresponding sales figure. This is most often because of the CPA six-digit product group not having an available index value (for example, because of small sample sizes) for the link period or for the entire year. Where either is the case, alternative approaches are taken that generally require use of a price index from a higher level of aggregation as a proxy.
For example, to produce weights for the 2019 PPI gross sector output indices, UK manufacturers’ sales by product (ProdCom) sourced sales are price updated from the annual sales totals for 2016 to estimate sales during December 2018. The price updated sales for December 2018 are then used as weights from January 2019 to December 2019. For further details on these approaches, please see Appendix 3.
A consequence of price updating is a delay in weight movement, as changes in the volume estimates are not considered as part of the method. Once actual sales are received, they may be different to price-updated sales as they include both price and volume changes. However, price updating is an internationally agreed method to be used as part of annual chain-linking to address limitations in the timeliness of the weights data. Price updating is used also in the Consumer Prices Index (CPI) measure (see Consumer price inflation, updating weights: 2020).Back to table of contents
Weighted indices are calculated using sales data as weights and price indices. A weighted index in mathematical form is calculated as:
This is the price index I for some grouping of products h at time t, denoted It,h.
The sum is over all the components (subgroups of products i that belong in the higher-level grouping h).
It,i is the index value of subgrouping i within h at time t.
wt,i is the weight of subgrouping i within h at time t.
We need an appropriate and consistent way of calculating wt,i, which determines the importance of each price component within an index. To understand the problem, observe that the aggregated indices are effectively measurements of the price of a basket of goods. This basket needs to be representative of activity in the economy for the index to be a good indicator of price behaviour.Back to table of contents
Let the sales of a product grouping i be xi. Then, the weight of that grouping in the broader grouping h is:
h* are all the subgroupings within h for which we have price data.
xr(t),i are the sales of subgrouping i at time r(t). r is a function of the time t for which we are calculating index values. It defines the time r we are taking sales data from, given the index value we want to calculate is at time t.
Therefore, the weight of any product grouping into a higher grouping is equal to the proportion of its sales within the total sales of that group.Back to table of contents
The following methods are used when there is not appropriate coverage to carry out price updating at the Classification of Products by Activity (CPA) six-digit level.
Services Producer Price Index and Producer Price Index
The index value used for price updating will be an unweighted average across all the CPA six-digits that share the same CPA four-digit parent of the given CPA six-digit whose sales are being price updated. This applies to both the link period index value and the average index values from the sales year (average of a set of averages), if either are zero or dead for a given CPA six-digit. CPA six-digit index values are based on a Laspeyres methodology.
If the link period index value and/or the average index value from the sales year is still zero after calculating unweighted averages across all CPA six-digit with the same CPA four-digit parent, then we consider all the six digits nested within the next parent index above the CPA four-digit. This process continues ascending from the lowest tier parent until a non-zero index value for both the link period and yearly average is found. These two non-zero values are then used to carry out the price updating process.
Import Price Index and Export Price Index
The coverage in the Import Price Index (IPI) and Export Price Index (EPI) is currently more limited compared to the PPI; therefore, price updating uses a different method. By default, price updating will be performed by the unweighted index value average across all CPA six-digits that share the same CPA two-digit parent as the given CPA six-digit’s allocated sales being price updated. This applies to both the index used for the link period index and the sales reference year. If the CPA two-digit parent averages provide a zero value, the next higher-level index will be considered in the same manner as the PPI for price updating.
The reason for using unweighted averages for the SPPI, PPI, IPI and EPI (as opposed to weighted averages) is there being no guarantee that the dead CPA six-digit’s siblings that are alive are proportionally representative of the weighted group. It also resolves the circularity issue of using price-updated sales to create weighted averages to generate the next set of price-updated sales. Use of weighted averages could result in cumulative error over time, as price-updated sales would become a function of previous price-updated sales.
In summary, when price updating using groups of CPA six-digits to calculate index averages, we apply the following formula:
Sl denotes the price updated sales, estimating the sales during link period, l, for a given CPA six-digit.
h is the set defined by all CPA six-digits sharing the same parent as the given CPA six-digit being price updated that are alive.
n denotes the number of elements in the set h.
i is the index identifier, which refers to an element belonging to the set h.
Sy denotes the annual sales for a given year, y, for the given CPA six-digit being price updated.
Ii,l denotes the CPA six-digit, i, index value during the link period l.
I ̅i,y denotes the CPA six-digit, i, average index value during the year y.Back to table of contents
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