- SNA, 2008, para. 15.44.
- The terms Laspeyres-type and Fisher-type indexes are used to describe quarterly indexes with annual weights.
- SNA, 2008, paras. 15.53-15.54.
Annex A Deriving chain volume indexes
Latest release
Australian System of National Accounts: Concepts, Sources and Methods
Reference period
2020-21 financial year
Released
9/07/2021
6A.1 The following provides a detailed description of the various chain volume measures and the issues associated with using them
Different index formulae
6A.2 The general formula for a Laspeyres volume index from year \(y-1\) to year \(y\) is given by:
\(\large {L_Q} = \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^y} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }},\) - - - - - - - (1)
where \(P_i^y\) and \(Q_i^y\) are prices and quantities of the \(i^{th}\) product in year \(y\) and there are \(n\) products. The denominator is the current price value of the aggregate in year \(y-1\) and the numerator is the value of the aggregate in year \(y\) at year \(y-1\) average prices.
6A.3 A Paasche volume index from year \(y-1\) to year \(y\) is defined as:
\(\large {P_Q} = \frac{{\sum\limits_{i = 1}^n {P_i^yQ_i^y} }}{{\sum\limits_{i = 1}^n {P_i^yQ_i^{y - 1}} }},\) - - - - - - - (2)
6A.4 A Fisher index is derived as the geometric mean of a Laspeyres and Paasche index:
\(\large {F_Q} = {\left( {{L_Q}{P_Q}} \right)^{1/2}}\) - - - - - - - (3)
6A.5 A Paasche price index from year \(y-1\) to year \(y\) is defined as:
\(\large {P_P} = \frac{{\sum\limits_{i = 1}^n {P_i^yQ_i^y} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^y} }},\) - - - - - - - (4)
6A.6 When this Paasche price index is divided into the current price index from year \(y-1\) to year \(y\) a Laspeyres volume index is produced:
\(\large \frac{{\sum\limits_{i = 1}^n {P_i^yQ_i^y} }}{{\frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }}{{{P_P}}}}} = \frac{{\frac{{\sum\limits_{i = 1}^n {P_i^yQ_i^y} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }}}}{{\frac{{\sum\limits_{i = 1}^n {P_i^yQ_i^y} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^y} }}}} = \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^y} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y-1}} }} = {L_Q}\) - - - - - - - (5)
6A.7 Evidently, Laspeyres volume indexes and Paasche price indexes complement each other, and vice versa
| Sales of beef and chicken | |||||
|---|---|---|---|---|---|
| Quantity (kilos) | Year 1 | Year 2 | Year 3 | Year 4 | |
| Beef | 20 | 18 | 16 | 17 | |
| Chicken | 10 | 12 | 14 | 17 | |
| Price per kilo ($) | |||||
| Beef | 1.00 | 1.10 | 1.20 | 1.30 | |
| Chicken | 2.00 | 2.00 | 2.10 | 2.15 | |
| Value ($) | |||||
| Beef | 20.00 | 19.80 | 19.20 | 22.10 | |
| Chicken | 20.00 | 24.00 | 29.40 | 36.55 | |
| Total | 40.00 | 43.80 | 48.60 | 58.65 | |
| Laspeyres volume index: year 1 to year 2 using year 1 prices | |||||
| Values at year 1 prices ($) | |||||
| Year 1 | Year 2 | Volume index | Growth rate | ||
| Beef | 20.00 | 18.00 | 0.900 | -10.0% | |
| Chicken | 20.00 | 24.00 | 1.200 | 20.0% | |
| Total | 40.00 | 42.00 | 1.050 | 5.0% | |
| Laspeyres volume index: year 2 to year 3 using year 2 prices | |||||
| Values at year 2 prices ($) | |||||
| Year 2 | Year 3 | Volume index | Growth rate | ||
| Beef | 19.80 | 17.60 | 0.889 | -11.1% | |
| Chicken | 24.00 | 28.00 | 1.167 | 16.7% | |
| Total | 43.80 | 45.60 | 1.041 | 4.1% | |
| Laspeyres volume index: year 3 to year 4 using year 3 prices | |||||
| Values at year 3 prices ($) | |||||
| Year 3 | Year 4 | Volume index | Growth rate | ||
| Beef | 19.20 | 20.40 | 1.063 | 6.3% | |
| Chicken | 29.40 | 35.70 | 1.214 | 21.4% | |
| Total | 48.60 | 56.10 | 1.154 | 15.4% | |
| Paasche volume index: year 1 to year 2 using year 2 prices | |||||
| Values at year 2 prices ($) | |||||
| Year 1 | Year 2 | Volume index | Growth rate | ||
| Beef | 22.00 | 19.80 | 0.090 | -10.0% | |
| Chicken | 20.00 | 24.00 | 1.200 | 20.0% | |
| Total | 42.00 | 43.80 | 1.043 | 4.3% | |
| Paasche volume index: year 2 to year 3 using year 3 prices | |||||
| Values at year 3 prices ($) | |||||
| Year 2 | Year 3 | Volume index | Growth rate | ||
| Beef | 21.60 | 19.20 | 0.089 | -11.1% | |
| Chicken | 25.20 | 29.40 | 1.167 | 16.7% | |
| Total | 46.80 | 48.60 | 1.038 | 3.8% | |
| Paasche volume index: year 3 to year 4 using year 4 prices | |||||
| Values at year 4 prices ($) | |||||
| Year 3 | Year 4 | Volume index | Growth rate | ||
| Beef | 20.80 | 22.10 | 1.063 | 6.3% | |
| Chicken | 30.10 | 36.55 | 1.214 | 21.4% | |
| Total | 50.90 | 58.65 | 1.152 | 15.2% | |
| Comparisons of the volume indexes | |||||
| Year 1 to 2 | Year 2 to 3 | Year 3 to 4 | |||
| Laspeyres | 1.050 | 1.041 | 1.154 | ||
| Paasche | 1.043 | 1.038 | 1.152 | ||
| Fisher | 1.046 | 1.040 | 1.153 | ||
6A.8 The following table provides an example of deriving Laspeyres volume indexes by deflation.
| Sales of beef and chicken | |||||
|---|---|---|---|---|---|
| Paasche price index: year 1 to year 2 using year 2 quantities | |||||
| Values at year 2 quantiles ($) | |||||
| Year 1 | Year 2 | Price index | Growth rate | ||
| Beef | 18.00 | 19.80 | 1.100 | 10.0% | |
| Chicken | 24.00 | 24.00 | 1.000 | 0.0% | |
| Total | 42.00 | 43.80 | 1.043 | 43.0% | |
| Paasche price index: year 2 to year 3 using year 3 quantities | |||||
| Values at year 3 quantiles ($) | |||||
| Year 2 | Year 3 | Price index | Growth rate | ||
| Beef | 17.60 | 19.20 | 1.091 | 9.1% | |
| Chicken | 28.00 | 29.40 | 1.050 | 5.0% | |
| Total | 45.60 | 48.60 | 1.066 | 6.6% | |
| Paasche price index: year 3 to year 4 using year 4 quantities | |||||
| Values at year 4 quantiles ($) | |||||
| Year 2 | Year 3 | Price index | Growth rate | ||
| Beef | 20.40 | 22.10 | 1.083 | 8.3% | |
| Chicken | 35.70 | 36.55 | 1.024 | 2.4% | |
| Total | 56.10 | 58.65 | 1.045 | 4.5% | |
| Laspeyres volume indexes derived by deflation | |||||
| Year 1 to 2 | Year 2 to 3 | Year 3 to 4 | |||
| Value index | 1.095 | 1.110 | 1.207 | ||
| Paasche price index | 1.043 | 1.066 | 1.045 | ||
| Laspeyres volume index | 1.050 | 1.041 | 1.154 | ||
Chain volume indexes
6A.9 Annual chain Laspeyres and Paasche volume indexes can be formed by multiplying consecutive year-to-year indexes:
\(\large L_Q^y = \frac{{\sum\limits_{i = 1}^n {P_i^0Q_i^1} }}{{\sum\limits_{i = 1}^n {P_i^0Q_i^0} }} \times \frac{{\sum\limits_{i = 1}^n {P_i^1Q_i^2} }}{{\sum\limits_{i = 1}^n {P_i^1Q_i^1} }} \times \frac{{\sum\limits_{i = 1}^n {P_i^2Q_i^3} }}{{\sum\limits_{i = 1}^n {P_i^2Q_i^2} }} \times ..... \times \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^y} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }}\) - - - - - - - (6)
\(\large P_Q^y = \frac{{\sum\limits_{i = 1}^n {P_i^1Q_i^1} }}{{\sum\limits_{i = 1}^n {P_i^1Q_i^0} }} \times \frac{{\sum\limits_{i = 1}^n {P_i^2Q_i^2} }}{{\sum\limits_{i = 1}^n {P_i^2Q_i^1} }} \times \frac{{\sum\limits_{i = 1}^n {P_i^3Q_i^3} }}{{\sum\limits_{i = 1}^n {P_i^3Q_i^2} }} \times ..... \times \frac{{\sum\limits_{i = 1}^n {P_i^yQ_i^y} }}{{\sum\limits_{i = 1}^n {P_i^yQ_i^{y - 1}} }},\) - - - - - - - (7)
6A.10 Chain Fisher indexes can be derived by taking their geometric mean:
\(\large F_Q^y = {\left( {L_Q^yP_Q^y} \right)^{1/2}}\) - - - - - - - (8)
6A.11 All of these indexes can be re-referenced by dividing them by the index value in the chosen reference year and multiplying by 100 to produce an indexed series, or by multiplying by the current price value in the reference year to obtain a series in monetary values.
The case for using chain indexes
6A.12 Frequent linking is beneficial when price and volume relativities progressively change. For example, volume estimates of gross fixed capital formation are much better derived as chain indexes than as fixed-weighted indexes (i.e. constant price estimates) mainly because of the steady decline in the relative prices of computer equipment and the corresponding increase in their relative volumes. While chain Fisher indexes perform best in such circumstances and are a much better indicator than fixed-weighted indexes, chain Laspeyres indexes capture much of the improvement from frequent linking.
6A.13 Conversely, frequent chaining is least beneficial when price and volume relativities are volatile. All chained series are subject to drift (see box below) when there is price and volume instability, but chain Fisher indexes usually drift less than either chain Laspeyres or chain Paasche indexes.
Drift and long-term accuracy
Suppose the prices and quantities are \(p_i^t\) and \(q_i^t\) at time t and \(p_i^{t+n}\) \(n\) periods later at time \(t+n\).
Further suppose that the price in year \(t+n\) \((𝑝^{𝑡+𝑛})\) returns to the same level that it was in year \(t (𝑝𝑡) \)after having diverged from \(𝑝𝑡\) during the intervening years (\(𝑡^2\) to \(𝑡^{𝑛−1}\)). Similarly, the quantity in year \(t+ n\) (\(𝑞^{𝑡=𝑛}\)) also returns to its original level (\(𝑞^𝑡\)) after having diverged between those years. Direct Laspeyres, Paasche and Fisher volume indexes from year \(t\) to year \(t+ n\) would equal 1.
However, it is unlikely that the values of a chain volume index would be identical in these years because of the cumulative effects of changes in the prices and volumes during the intervening years. The extent of the difference (usually expressed as the quotient of the two values) is a measure of the “drift” in the chain volume index between the two time periods.
In reality it is very uncommon for prices and volumes to return to the values observed in an earlier period. Therefore, in practice, the drift and long-term accuracy of a chain or fixed-weighted index can be assessed over a period of time by comparing it with a direct Fisher index; that is, a Fisher index calculated directly from the first to the last observation in a period.
6A.14 Table A.3 below compares the chain Laspeyres, chain Paasche and chain Fisher indexes of meat sales. It shows that in this example:
- the chain Fisher index and the Fisher index calculated directly from the first year to the fourth year show almost the same growth rate over the four year period; that is, the chain Fisher index shows very little drift; and
- both the chain Laspeyres and chain Paasche indexes come much closer to the two Fisher indexes than their fixed-weighted counterparts.
6A.15 It is important to note that this is just an example. In the real world, the differences between the different indexes are usually much less.
6A.16 For aggregates such as gross value added of mining and agriculture, and maybe exports and imports, where volatility in price and volume relativities are common, the advantages of frequent linking may be doubtful, particularly using the Laspeyres (or Paasche) formula. For reasons of practicality and consistency, the same approach to volume aggregation has to be followed throughout the accounts. So when choosing which formula to use, it is necessary to make an overall assessment of drift, accuracy and practical matters.
6A.17 In considering the benefits of chain volume indexes against fixed-weighted indexes, the 2008 SNA concludes that:
. . . it is generally recommended that annual indexes be chained. The price and volume components of monthly and quarterly data are usually subject to much greater variation than their annual counterparts due to seasonality and short-term irregularities. Therefore, the advantages of chaining at these higher frequencies are less and chaining should definitely not be applied to seasonal data that are not adjusted for seasonal fluctuations.³⁶
| Laspeyres | Passche | Fisher | ||||||
|---|---|---|---|---|---|---|---|---|
| Chain volume indexes | ||||||||
| \(\large L_{CV}^1\) | = 100.0 | = 100.0 | \(\large P_{CV}^1\) | = 100.0 | = 100.0 | \(\large F_{CV}^1\) | = 100.0 | = 100.0 |
| \(\large L_{CV}^2\) | = 100.0 X 1.050 | = 105.0 | \(\large P_{CV}^2\) | = 100.0 X 1.043 | = 104.3 | \(\large F_{CV}^2\) | \({\left( {105.0 \times 104.3} \right)^{0.5}}\) | = 104.6 |
| \(\large L_{CV}^3\) | = 105.0 X 1.041 | = 109.3 | \(\large P_{CV}^3\) | = 104.3 X 1.038 | = 108.3 | \(\large F_{CV}^3\) | \({\left( {109.3 \times 108.3} \right)^{0.5}}\) | = 108.8 |
| \(\large L_{CV}^4\) | = 109.3 X 1.154 | = 126.2 | \(\large P_{CV}^4\) | = 108.3 X 1.152 | = 124.8 | \(\large F_{CV}^4\) | \({\left( {126.2 \times 124.8} \right)^{0.5}}\) | = 125.5 |
| Direct volume indexes | ||||||||
| \(\large L_{DV}^4\) | \(\large \frac{{17 \times 1.00 + 17 \times 2.00}}{{40.00}}\) | = 127.5 | \(\large P_{DV}^4\) | \(\large \frac{{58.65}}{{20 \times 1.30 + 10 \times 2.15}}\) | = 123.5 | \(\large F_{DV}^4\) | \({\left( {127.5 \times 123.5} \right)^{0.5}}\) | = 125.5 |
Deriving annual chain volume indexes in the national accounts
6A.18 It is recommended in the 2008 SNA that the annual national accounts should be balanced in both current prices and in volume terms using S-U tables. In most cases, the volume estimates are best derived in the average prices of the previous year rather than some distant base year. This is for two key reasons:
- assumptions of fixed relationships in volume terms are usually more likely to hold in the previous year’s average prices than in the prices of some distant base year: and;
- so that the growth rates of volumes and prices are less affected by compositional change.
6A.19 The compilation of annual S-U tables in current prices and in the average prices of the previous year lends itself to the compilation of annual Laspeyres indexes and to the formation of annual chain Laspeyres indexes.
6A.20 In order to compute annual Fisher indexes from data balanced in a S-U table, it is conceptually desirable to derive both Laspeyres and Paasche indexes from that data. The former requires balancing the S-U tables of the current year \((y)\) in current prices \((y)\) and in the average prices of the previous year \((y-1)\) and the latter requires balancing S-U tables in the previous year \((y-1)\) in the average prices of that year \((y-1)\) and in the average prices of the current year \((y)\). Thus, the compilation of annual chain Fisher indexes, at least in concept, is somewhat more demanding than compiling annual chain Laspeyres indexes.
Deriving quarterly chain indexes in the national accounts
6A.21 Computationally, the derivation of quarterly chain indexes from quarterly data with quarterly base periods is no different to compiling annual chain indexes from annual data with annual base periods. As recommended by the 2008 SNA, if quarterly volume indexes are to have quarterly base periods and be linked each quarter, then it should only be done using seasonally adjusted data. Furthermore, if the quarterly seasonally adjusted data are subject to substantial volatility in relative prices and relative volumes, then chain indexes should not be formed from indexes with quarterly base periods at all. Even if the quarterly volatility is not so severe, quarterly base periods and quarterly linking are not recommended using the Laspeyres formula because of its greater susceptibility to drift than the Fisher formula.
6A.22 A way round this problem is to derive quarterly volume indexes from a year to quarters. In other words, use annual base years (i.e. annual weights) to derive quarterly volume indexes. Consider the Laspeyres annual volume index in formula 1. It can be expressed as a weighted average of elemental volume indexes:
\(\large {L_Q} = \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^y} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }} = \sum\limits_{i = 1}^n {\left( {\frac{{Q_i^y}}{{Q_i^{y - 1}}}} \right)} s_i^{y - 1},\;where\;s_i^{y - 1} = \frac{{P_i^{y - 1}Q_i^{y - 1}}}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }}\) - - - - - - - (9)
\(s_i^{y - 1}\) is the share, or weight, of the \(i^{th}\) item in year \(y-1\).
6A.23 Paasche volume indexes can also be expressed in terms of a weighted average of the elemental volume indexes, but as the harmonic, rather than arithmetic, mean.
6A.24 A Laspeyres-type³⁷ volume index from year \(y-1\) to quarter \(c\) in year \(y\) takes the form:
\(\large L_Q^{(y - 1) \to (c,y)} = \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}4q_i^{c,y}} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }} = \sum\limits_{i = 1}^n {\frac{{4q_i^{c,y}}}{{Q_i^{y - 1}}}} s_i^{y - 1},\) - - - - - - - (10)
where \({q_i^{c,y}}\) is the volume of product \(i\) in the \(c^{th}\) quarter of year \(y\). In this case the annual current price data in year \(y-1\) are used to weight together elemental volume indexes from year \(y-1\) to each of the quarters in year \(y\). The “4” in formula 10 is to put the quarterly data onto a comparable basis with the annual data. Note that constant price (or fixed-weighted) volume indexes are traditionally formed in this way, but the weights are kept constant for many years.
6A.25 2008 SNA describes how chain Fisher-type indexes of quarterly data with annual base periods can be derived:
"Just as it is possible to derive annually chained Laspeyres-type quarterly indices, so it is possible to derive annually chained Fisher-type quarterly indices. For each pair of consecutive years, Laspeyres-type and Paasche-type quarterly indices are constructed for the last two quarters of the first year, year \(y-1\) and the first two quarters of the second year, year \(y\). The Paasche-type quarterly indices are constructed as backward-looking Laspeyres-type quarterly indices and then inverted. This is done to ensure that the Fisher-type quarterly indices are derived symmetrically. In the forward-looking Laspeyres-type indices the annual value shares relate to the first of the two years, whereas in the backward-looking Laspeyres-type indices the annual value shares relate to the second of the two years.
For each of the four quarters a Fisher-type index is derived as the geometric mean of the corresponding Laspeyres-type and Paasche-type indices. Consecutive spans of four quarters can then be linked using the one-quarter overlap technique. The resulting annually chained Fisher-type quarterly indices need to be benchmarked to annual chain Fisher indices to achieve consistency with the annual estimates."³⁸
Choosing between chain Laspeyres and chain Fisher indexes
6A.26 There are several advantages in using the Laspeyres formula:
- its adoption is consistent with compiling additive S-U tables in both current prices and in the prices of the previous year;
- quarterly chain volume estimates of both seasonally adjusted and unadjusted data can be derived;
- it is unnecessary to seasonally adjust volume data at the most detailed level, if desired; and
- it is simpler and lower risk to construct chain Laspeyres indexes than Fisher indexes.
6A.27 The advantages of using the Fisher formula are:
- it is more accurate than the Laspeyres formula; and
- it is more robust and less susceptible to drift when price and volume relativities are volatile.
6A.28 In practice, it is generally found that there is little difference between chain Laspeyres and Fisher indexes for most aggregates. The major threat to the efficacy of the use of the Laspeyres formula in the National Accounts has been computer equipment. The prices of computer equipment relative to improvements in quality have been falling rapidly and the volumes of production and expenditure have been rising rapidly for many years. Consequently, the chain Laspeyres and chain Fisher indexes for aggregates for which computer equipment is a significant component are likely to show differences. Until now, these differences have been insufficient to cause concern and have not been considered to outweigh the advantages of using the Laspeyres formula. This is largely due to the fact that a country such as Australia does not produce a large volume of computers domestically, and as such GDP is unaffected.
6A.29 There is one other reason why the ABS has chosen to derive chain volume estimates using the Laspeyres formula. A requirement of using quarterly base periods is the availability of quarterly current price data (see formula 9). While there are quarterly current price estimates of final expenditures in the ASNA, there are no quarterly current price estimates of gross value added by industry at the moment. Hence, it is currently not possible to derive chain volume estimates with quarterly base periods for the production measure of GDP.
Deriving annually-linked quarterly Laspeyres-type volume indexes
6A.30 While there are different ways of linking annual Laspeyres volume indexes, they all produce the same result. But this is not true when it comes to linking annual-to-quarter Laspeyres-type volume indexes for consecutive years. Paragraphs 15.46 -15.50 of the 2008 SNA discuss three methods for linking these Laspeyres-type volume indexes; they are:
- Annual overlap;
- One-quarter overlap: and
- Over the year.
6A.31 When a Laspeyres-type quarterly volume index from year \(y-1\) to quarter \(c\) in year \(y\) is multiplied by the current price value for year \(y-1\) divided by four, then a value for quarter \(c\) is obtained in the average prices of year \(y-1\).
\(\large \sum\limits_{i = 1}^n {\frac{{4q_i^{c,y}}}{{Q_i^{y - 1}}}} s_i^{y - 1}\frac{1}{4}\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} = \sum\limits_{i = 1}^n {\frac{{4q_i^{c,y}}}{{Q_i^{y - 1}}}} \frac{{P_i^{y - 1}Q_i^{y - 1}}}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }}\frac{1}{4}\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} = \sum\limits_{i = 1}^n {q_i^{c,y}P_i^{y - 1}} \) - - - - - - - (11)
6A.32 Hence, the task of linking quarterly Laspeyres-type volume indexes for two consecutive years, year \(y-1\) and year \(y\), amounts to linking the quarterly values of year \(y-1\) in year \(y-2\) average prices with the values of year \(y\) in year \(y-1\) average prices.
Annual overlap method
6A.33 One way of putting the eight quarters described in the previous paragraph onto a comparable valuation basis is to calculate and apply a link factor from an annual overlap. Values for year \(y-1\) are derived in both \(y-1\) prices and \(y-2\) prices and then the former is divided by the latter; thus, giving an annual link factor for year \(y-1\) to year \(y\) is equal to:
\(\large \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }}{{\sum\limits_{i = 1}^n {P_i^{y - 2}Q_i^{y - 1}} }}\) - - - - - - - (12)
6A.34 Multiplying the quarterly values for year \(y-1\) at year \(y-2\) average prices with this link factor puts them on to a comparable valuation basis with the quarterly estimates for year \(y\) at year \(y-1\) prices. Note that this link factor is identical to the one that can be used to link the annual value for year \(y-1\) at \(y-2\) average prices with the annual value for year \(y\) at year \(y-1\) average prices. Therefore, if the quarterly values for every year \(m\) at year \(m1\) average prices sum to the corresponding annual value, then the chain-linked quarterly series will be temporally consistent with the corresponding chain-linked annual series.
One-quarter overlap method
6A.35 The one-quarter overlap method, as its name suggests, involves calculating a link factor using overlap values for a single quarter. To link the four quarters of year \(y-1\) at year \(y-2\) average prices with the four quarters of year \(y\) at year \(y-1\) average prices, a one-quarter overlap can be created for either the fourth quarter of year \(y-1\) or the first quarter of year \(y\). The link factor derived from an overlap for the fourth quarter of year \(y-1\) is equal to:
\(\large \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}q_i^{4,(y - 1)}} }}{{\sum\limits_{i = 1}^n {P_i^{y - 2}q_i^{4,(y - 1)}} }}\) - - - - - - - (13)
6A.36 Multiplying the quarterly values for year \(y-1\) at year \(y-2\) average prices with this link factor puts them on to a comparable valuation basis with the quarterly estimates for year \(y\) at year \(y-1\) prices.
6A.37 A key property of the one-quarter overlap method is that it preserves the quarter-to-quarter growth rate between the fourth quarter of year \(y-1\) and the first quarter of year \(y\) - unlike the annual overlap method. The “damage” done to that growth rate by the annual overlap method is determined by the difference between the annual and quarter link factors. Conversely, this difference also means that the sum of the linked quarterly values in year \(y-1\) differ from the annual-linked data by the ratio of the two link factors. Temporal consistency can be achieved by benchmarking the quarterly chain volume estimates to their annual counterparts.
6A.38 The following table illustrates the methods used to deriving link factors
| Sales of beef and chicken | |
|---|---|
| Annual overlap method | |
| Year 2 to Year 3 | Year 3 to Year 4 |
| \(\Large \frac{{\sum\limits_{i = 1}^2 {P_i^2Q_i^2} }}{{\sum\limits_{i = 1}^2 {P_i^1Q_i^2} }}\) | \(\Large \frac{{\sum\limits_{i = 1}^2 {P_i^3Q_i^3} }}{{\sum\limits_{i = 1}^2 {P_i^2Q_i^3} }}\) |
| \(\Large \frac{{(1.1x18) + (2x12)}}{{(1x18) + (2x12)}} = 1.043\) | \(\frac{{(1.2x16) + (2.1x14)}}{{(1.1x16) + (2x14)}} = 1.066\) |
| One-quarter overlap method | |
| Quarter 4 in Year 2 | Quarter 4 in Year 3 |
| \(\Large \frac{{\sum\limits_{i = 1}^2 {P_i^2q_i^{4,2}} }}{{\sum\limits_{i = 1}^2 {P_i^1q_i^{4,2}} }}\) | \(\Large \frac{{\sum\limits_{i = 1}^2 {P_i^3q_i^{4,3}} }}{{\sum\limits_{i = 1}^2 {P_i^2q_i^{4,3}} }}\) |
| \(\Large \frac{{(1.1x6) + (2.0x3)}}{{(1.0x6) + (2.0x3)}} = 1.05\) | \(\Large \frac{{(1.2x4) + (2.1x3)}}{{(1.1x4) + (2.0x3)}} = 1.0673\) |
Over the year method
6A.39 The over-the-year method requires compiling a separate link factor for each type of quarter. Each of the quarterly values in year \(y-1\) at year \(y-2\) average prices is multiplied by its own link factor. The over-the-year quarterly link factor for year \(y-1\) at average year \(y-2\) prices to year y at average year \(y-1\) prices for quarter c is equal to:
\(\large \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}q_i^{c,(y - 1)}} }}{{\sum\limits_{i = 1}^n {P_i^{y - 2}q_i^{c,(y - 1)}} }}\) - - - - - - - (14)
6A.40 The over-the-year method does not distort quarter-on-same quarter of previous year growth rates, since the chain-links refer to the volumes of the same quarter in the respective previous year valued at average prices of that year. However, it does distort quarter-to-quarter growth rates. In addition, the linked quarterly data are temporally inconsistent with the annual-linked data and so benchmarking is needed. Given these shortcomings, the over-the-year method is best avoided.
6A.41 The following tables provide examples of using the annual and one-quarter overlap methods.
| Sales of beef and chicken | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Year | 2 | 3 | 4 | |||||||||
| Quarter | 1 | 2 | 3 | 5 | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 |
| Beef (kilos) | 5 | 4 | 3 | 6 | 4 | 5 | 3 | 4 | 4 | 4 | 5 | 4 |
| Chicken (kilos) | 2 | 3 | 4 | 3 | 2 | 4 | 5 | 3 | 3 | 4 | 6 | 4 |
| Price of beef in previous year ($) | 1.00 | 1.00 | 1.00 | 1.00 | 1.10 | 1.10 | 1.10 | 1.10 | 1.20 | 1.20 | 1.20 | 1.20 |
| Price of chicken in previous year ($) | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.10 | 2.10 | 2.10 | 2.10 |
| Value of beef at previous year's prices ($) | 5.00 | 4.00 | 3.00 | 6.00 | 4.40 | 5.50 | 3.30 | 4.40 | 4.80 | 4.80 | 6.00 | 4.80 |
| Value of chicken at previous year's prices ($) | 4.00 | 6.00 | 8.00 | 6.00 | 4.00 | 8.00 | 10.00 | 6.00 | 6.30 | 8.40 | 12.60 | 8.40 |
| Total sales of meat in previous year's prices ($) | 9.00 | 10.00 | 11.00 | 12.00 | 8.40 | 13.50 | 13.30 | 10.40 | 11.10 | 13.20 | 18.60 | 13.20 |
| Link factor year 2 to 3 | 1.0429 | 1.0429 | 1.0429 | 1.0429 |
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| Linking year 2 to year 3 ($) | 9.39 | 10.43 | 11.47 | 12.51 | 8.40 | 13.50 | 13.30 | 10.40 |
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| Link factor year 3 to 4 | 1.0658 | 1.0658 | 1.0658 | 1.0658 | 1.0658 | 1.0658 | 1.0658 | 1.0658 |
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| Linking year 2 and 3 to year 4 ($) | 10.00 | 11.12 | 12.23 | 13.34 | 8.95 | 14.39 | 14.18 | 11.08 | 11.10 | 13.20 | 18.60 | 13.20 |
| Factor to reference to year 2 | 0.9383 | 0.9383 | 0.9383 | 0.9383 | 0.9383 | 0.9383 | 0.9383 | 0.9383 | 0.9383 | 0.9383 | 0.9383 | 0.9383 |
| Referenced to year 2 ($) | 9.39 | 10.43 | 11.47 | 12.51 | 8.40 | 13.50 | 13.30 | 10.40 | 10.41 | 12.39 | 17.45 | 12.39 |
| Annualised ($) | 43.80 |
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| 45.60 |
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| 52.64 |
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| Quarterly growth rate (%) |
| 11.11 | 10.00 | 9.09 | -32.88 | 60.71 | -1.48 | -21.80 | 0.14 | 18.92 | 40.91 | -29.03 |
| Sales of beef and chicken | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Year | 2 | 3 | 4 | |||||||||
| Quarter | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 |
Beef (kilos) | 5 | 4 | 3 | 6 | 4 | 5 | 3 | 4 | 4 | 4 | 5 | 4 |
| Chicken (kilos) | 2 | 3 | 4 | 3 | 2 | 3 | 5 | 3 | 3 | 4 | 6 | 4 |
| Price of beef in previous year ($) | 1.00 | 1.00 | 1.00 | 1.00 | 1.10 | 1.10 | 1.10 | 1.10 | 1.20 | 1.20 | 1.20 | 1.20 |
| Price of chicken in previous year ($) | 2.0 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.10 | 2.10 | 2.10 | 2.10 |
| Value of beef at previous year's prices ($) | 5.00 | 4.00 | 3.00 | 6.00 | 4.40 | 5.50 | 3.30 | 4.40 | 4.80 | 4.80 | 6.00 | 4.80 |
| Value of chicken at previous year's prices ($) | 4.00 | 6.00 | 8.00 | 6.00 | 4.00 | 8.00 | 10.00 | 6.00 | 6.30 | 8.40 | 12.60 | 8.40 |
| Total sales of meat in previous year's prices ($) | 9.00 | 10.00 | 11.00 | 12.00 | 8.40 | 13.50 | 13.30 | 10.40 | 11.10 | 13.20 | 18.60 | 13.20 |
| Link factor year 2 to 3 | 1.05 | 1.05 | 1.05 | 1.05 | ||||||||
| Linking year 2 to year 3 ($) | 9.45 | 10.50 | 11.55 | 12.60 | 8.40 | 13.50 | 13.30 | 10.40 | ||||
| Link factor year 3 to 4 | 1.0673 | 1.0673 | 1.0673 | 1.0673 | 1.0673 | 1.0673 | 1.0673 | 1.0673 | ||||
| Linking year 2 and 3 to year 4 ($) | 10.09 | 11.21 | 12.33 | 13.45 | 8.97 | 14.41 | 14.20 | 11.10 | 11.10 | 13.20 | 18.60 | 13.20 |
| Factor to reference to year | 0.9306 | 0.9306 | 0.9306 | 0.9306 | 0.9306 | 0.9306 | 0.9306 | 0.9306 | 0.9306 | 0.9306 | 0.9306 | 0.9306 |
| Referenced to year 2 ($) | 9.39 | 10.43 | 11.47 | 12.51 | 8.34 | 13.41 | 13.21 | 10.33 | 10.33 | 12.28 | 17.31 | 12.28 |
| Annualised ($) | 43.80 | 45.29 | 52.20 | |||||||||
| Quarterly growth rate (%) | 11.11 | 10.00 | 9.09 | -33.33 | 60.71 | -1.48 | -21.80 | 0.00 | 18.92 | 40.91 | -29.03 | |
Deriving chain volume estimates of time series that are not strictly positive
6A.42 Some quarterly national accounts series can take positive, negative or zero values, and so it is not possible to derive chain volume estimates for them. The best-known example is changes in inventories, but any variable which is a net measure is susceptible. While it is not possible to derive true chain volume estimates for variables that can change sign or take zero values, it is possible to derive proxy chain volume estimates. The most commonly used approach is to:
- identify two strictly positive series that when differenced yield the target series;
- derive chain volume estimates of these two series expressed in currency units; and
- difference the two chain volume series.
6A.43 The same approach can be used to derive seasonally adjusted proxy chain volume estimates except that after step 2 the two series are seasonally adjusted before proceeding to step 3.
6A.44 In the case of changes in inventories, the obvious candidates for the two strictly positive series are the opening and closing inventory levels. The chain volume index of opening inventories is referenced to the opening value in the reference year expressed at the average prices of the reference year. Likewise, the chain volume index of closing inventories is referenced to the closing value of inventories expressed at the average prices of the reference year. This ensures that the value of the proxy chain volume measure of changes in inventories is equal to the current price value in the reference year.
6A.45 Seasonally adjusted current price estimates of changes in inventories are obtained by inflating the proxy chain volume estimates by a suitable price index centred on the middle of each quarter and with the same reference year as the volume estimates.