Statistical Office of the European Communities / Directorate General Statistics
16 April 2007
TF-SAQNA-18
Task Force on Seasonal Adjustment
of Quarterly National Accounts
Second Meeting
Agenda Item 6
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Chain-Linking and Seasonal Adjustment
Properties of Unadjusted and Adjusted Time Series
Draft contribution to a paper on chain-linking by Robert Kirchner
Addendum by Marcus Scheiblecker
Deutsche Bundesbank28 March 2007
Robert Kirchner
Draft
Chain-linking of Quarterly National Accounts and its implications for seasonal adjustment
- Conceptual considerations
In order to be able to assess the implications of the various chain index concepts which are currently used in Europe for seasonal adjustment, the new index methods for the unadjusted values need to be described first. In the following section, they are briefly developed and presented on the basis of the critique of the previous fixed-price-base method.
The subsequent presentation of the new index concepts makes no claim to being exhaustive. Rather, only those properties are addressed which are also significant in the analysis of seasonally adjusted results and also play a role in the analysis of the current economic dynamics:
- The comparison with previous periods, which is used in the description of current developments
- The decomposition of the movement of an aggregate (such as gross domestic product) into the development of its components (such as private consumption) to explain the overall dynamics
- The consistency between quarterly and annual figures (time consistency)
- Properties of unadjusted data
- Introduction
The new chain-linked concepts for capturing volume movements are most easily described in contradistinction to the previous method.
In the past volumes were calculated by valuing the current quantities () at the average prices of the base year (). The formula for the volume for the first quarter in year t was thus
(I),
where the sum-total is calculated across the different goods.
This method of representation sparked the criticism, however, that the information content of the current volume movements declines with the growing distance of the current reporting period from the base period. This is because the representativeness of the price structures from the base period decreases with the growing distance from the reporting period. For example, technical goods (such as computers) have become cheaper whereas energy prices have tended to increase. Hence, calculating current volumes on the basis of a price structure dating back many years can no longer be considered meaningful.
In order to overcome this problem, therefore, the distance between the price base year and the current period should not be too large. In the national accounts the past year is thus used for calculating the price base.
(II)
If the movement within a year is being analysed, the price base does not change. Therefore, the quarter-on-quarter pure volume movement can be shown as
(III),
During transition from the fourth quarter of one year to the first quarter of the following year, however, not only the quantity component but also the price component changes. Hence the pure volume movement is not shown; instead a statistically induced break occurs caused by the change in the price base.
(IV),
Such breaks should not appear in a pure volume series. They should be eliminated. However, that is easier said than done as this raises the question of which break should be eliminated.
- Is the effect of the change in the price base on the rates of change of the annual figures compared with the previous period to be eliminated?
- Or should the rates of change of the quarterly values vis-à-vis the previous period always be shown without any breaks?
- Or should the rates of change of the quarterly values vis-à-vis the past year be adjusted?
Actually, the change in the price base should not have any influence on any of the volume changes mentioned. This requirement was fulfilled by the previous fixed-price-base concept. However, owing to the aforementioned problem of outdated price bases this method is now no longer an option.
The methods described below aim to resolve the problem of the statistical break using different chain-linking approaches. No method exists which satisfies all three demands simultaneously.Rather, each method can only tackle one of the questions above, leaving the other demands either only approximately satisfied or not satisfied at all.
- Annual overlap
In the case of the annual overlap approach, the current quantities are valued at the average prices of the past year and compared with the average nominal values of the past year. But to construct an index value which reflects the distance to the reference period, it is necessary to repeat this step several times for the annual figures. Finally, the index value is calculated by multiplying all these growth factors together. The index value for the first quarter of 2003 in a series beginning in 2000 with 2000 as the reference year can be expressed as
(V)
- Time consistency
As the annual average of the thus calculated quarterly figures is the same as the autonomously calculated annual index, there is no need to adjust the quarterly results to the annual results using benchmarking techniques as is the case with other approaches.
Furthermore, the percentage changes in the annual figures show no statistical breaks.
(VI)
As the same price base is used in the current and past year, the annual overlap gives the pure volume changes when comparing the annual figures with those from the previous period.
- Quarter-on-quarter comparison
Pure volume changes are also calculated for the quarter-on-quarter movements within a year, as the following example shows.
(VII)
By contrast, breaks occur in the transition from the fourth quarter of one year to the first quarter of the following year.
(VIII)
This rate of change is thus composed of two components (in the formula after the right-hand equals sign). First, the change factor of the quantities for the first quarter of the current year valued at past-year prices is calculated vis-à-vis the quantities from the fourth quarter of the past year valued at the prices of the preceding year (t-2). As a result of the change in the price base, however, this is not a pure volume comparison. In order to at least approximate this, the price base change effect must be factored out. This is done using the Paasche price index at the extreme right of the last formula. Yet as the deflator relates to whole years but the volume development is to be measured for quarters, the deflator is not fully congruent with the variable that is to be deflated. As a result, a certain statistical contamination remains during the transition from the fourth quarter of one year to the first quarter of the following year. This results in a break in the series.
The relationship between the index change and the pure volume change can be used to express the statistical break as
(IX)
The size of the statistical break can thus be derived from the relationship between two Paasche price indices, one based on the quantities from the fourth quarter of the past year and one based on the quantities over the entire past year.
Hence the more the quantity structure in the fourth quarter differs from that of the entire year, the larger the break in the volume series. Or, in other words, if the quantity of each single good over the whole year was always λ-times the quantity in the fourth quarter, there would be no break. This is because assuming the following holds:
(X)
There is likewise no statistical break if there is no change in the relative prices. If this is the case, holds for all goods prices and hence
(XI)
The change-of-year break caused by the change in the price base is, therefore, smaller, the less the relative prices fluctuate from year to year or the less the quantity structure in the fourth quarter of the past year differs from the quantity structure of the entire past year.
- Aggregation and decomposition
The aggregation of chain-linked indices and the decomposition of aggregates into sub-indices is not as easy as in the case of the earlier fixed-price-base concept. Chain-linked indices are not additive, ie their aggregates do not simply equal the sum of the components, nor do they result from the addition of their components multiplied by constant weights. Nevertheless, the aggregates are functionally dependent on their components. This can be used in the case of the annual overlap approach to calculate aggregates using components or to decompose aggregates into sub-indices.
For example, to calculate an aggregate from components, it is first necessary to de-chain the time series for the components, ie annual index links must be calculated. Using the structure of the nominal figures from the respective previous year, these can be summarised into annual links for the aggregate. Finally, for the aggregate, the annual components must be chain-linked.
For this purpose the Deutsche Bundesbank has produced an Excel macro which is available to users on request. The individual steps of the algorithm are described in the annex.
Interested parties can use this method to themselves calculate the variables in which they are interested from the official statistics. The only thing which is needed for the calculation are the published nominal and real figures.
The following relationship applies between values, volumes and deflators
Index value = Annual overlap Laspeyres volume chain-linked index x
annual overlap Paasche price chain-linked index / 100
- Quarterly overlap
In the case of the quarterly overlap approach, the current chain link is derived by valuing the quantities of the current quarter at average prices of the past year and then comparing them with the quantities of the fourth quarter of the past year valued at average prices of the past year. In order to show the distance from the reference year, this step must be repeated, related each time to the quantities for the fourth quarter. The change factors which result from this process are then multiplied by one another. The index value for the volume in the first quarter of 2003 in a time series beginning in 2000 with 2000 as the reference year is thus calculated as follows
(XII)
- Time consistency
The year-on-year rates of change for the annual figures of an index calculated in this way for 2003 in comparison with 2002 are derived as follows
(XIII)
As a rule, they deviate from autonomously calculated annual rates (Equation VI). In practice this raises the requirement that the quarterly figures calculated using the quarterly overlap approach need to be forced to equal the autonomously calculated annual figures (benchmarking). Overall, therefore, the quarterly overlap approach does not unambiguously denote a method which is used in practice. Instead, the specification of the benchmarking procedure is additionally necessary for an unambiguous description of the method.
If all quarterly figures within a calendar year are adjusted to the annual figures using the same factor (pro rata technique), the annual overlap is calculated. It is therefore a special case of quarterly overlap with benchmarking.
- Quarter-on-quarter comparison
Without benchmarking the quarter-on-quarter rates of change calculated using the quarterly overlap method always show the pure volume change without a break not only within a year but also between the fourth quarter and the first quarter of the following year.
(XIV)
This property has, however, been lost at least partially as a result of the introduction of benchmarking techniques. Depending on the extent of the differences between the average of the quarterly results and the autonomously calculated annual figures and also depending on the choice of the benchmarking method used, deviations from the pure volume comparison occur. In a pro rata approach, breaks can only occur during the change from the fourth quarter to the first quarter of the following year. In other methods breaks can also occur during the quarterly movement within a particular year (albeit to a lesser extent than the pro rata break at the turn of the year).
- Aggregation and decomposition
Figures calculated using the quarterly overlap approach without benchmarking can basically be aggregated using the same procedure as for the annual overlap method. However, it is not the structure of the past-year nominal figures that is used as a weight for the annual links but rather the structure of the quantities of the fourth quarter of the past year valued at average past-year prices. These figures are not generally released as part of the normal publishing programme of volume indices, deflators and figures at current prices. If users wish to calculate their own aggregates or disaggregates, then the quantities in the fourth quarter of the past year valued at past-year prices must be published separately and saved separately by the user. This complicates publication policy and increases the costs of providing and managing data.
Aggregation and decomposition are more difficult using benchmarking techniques (unless a pro rata approach is taken). This is because in these circumstances the user needs both the figures calculated without benchmarking, the aforementioned weights as well as the benchmarking algorithm. The official statistics office would have to provide this information in addition to the benchmarked quarterly overlap results. The problems resulting from the complicated data storage and subsequent calculation steps could easily lead to a situation where third parties are not able to calculate correct aggregates or sub-components of time series.
The following relationship applies between values, volumes and deflators in the case of quarterly overlap without benchmarking.
Index value = Quarterly overlap Laspeyres volume chain-linked index x
quarterly overlap Paasche price chain-linked index / 100
With benchmarking this relationship, too, is not necessarily satisfied. In fact, the validity of this equation depends on the approach adopted.
- Over-the-year technique
In the case of the over-the-year technique, the current chain link is expressed as a relationship between the current quantities valued at average past-year prices and the quantities of the same quarter of the past year, valued at the same prices. Corresponding growth factors are also calculated for earlier years. Finally, the current chain-linked index value is the product of all these factors. Therefore, the index value for the first quarter of 2003 in a series beginning in 2000 with 2000 as the reference year can be expressed as:
(XV)
- Time consistency
The formal comparison between the annual indices calculated using the over-the-year method and the autonomously calculated annual results is made more difficult by the fact that the corresponding formulae for the over-the-year approach are very long and not easily interpreted. Applied studies do, however, show that although results calculated using this method are not fully identical with autonomously calculated annual figures, they do not differ greatly. Benchmarking is, therefore, necessary although its effects on the results are likely to be limited.
- Quarter-on-quarter comparison
Even without benchmarking the formula for quarter-on-quarter comparison, also within a particular year, is very complicated:
(XVI)
Consequently, the currently reported quarter-on-quarter rate depends on the entire history of these quarters. For example, if just one value from the first quarter of 2001 changes, this has an effect on the changes of the unadjusted values in the first quarter of 2003 compared with the previous quarter.
The formula, which cannot be simplified further, also proves that the same price base is not used in the comparison of the volume indices of two consecutive quarters. Instead, the historical sequence of all price bases is included in the calculation. Therefore, the results calculated using the over-the-year technique are far from the ideal of a pure volume comparison. Hence, statistically induced breaks constantly occur.
The introduction of benchmarking techniques causes further complications (use also of the history of the volumes of the other quarters). In effect, the theoretical penetration of the expanded concept for the comparison with previous periods is becoming almost impossible.
- Aggregation and decomposition
Aggregates and disaggregates can also be calculated using the over-the-year method. For this, however, the chain links for each individual quarter must first be calculated (that is, the relationship between the quantities in the quarter in question valued at average past-year prices and the quantities from the same quarter of the past year valued at the same prices). The structure of the variables in the denominator of this relationship can be used as weights.