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Chain Fisher volume index - Methodology

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On May 31, 2001, the quarterly Income and Expenditure Accounts adopted the Fisher index formula, chained quarterly, as the official measure of real gross domestic product in terms of expenditures. This formula was also adopted for the Provincial Economic Accounts on October 31, 2002. 

There were two reasons for adopting this formula: to provide users with a more accurate measure of real GDP growth between two consecutive periods and to make the Canadian measure comparable with the Income and Product Accounts of the United States, which has used the chain Fisher index formula since 1996 to measure real GDP. 

This document sets out the theory and the application of the formula to the Canadian economic accounts.

Table of contents:

  • Choice of index
  • Application in the Income and Expenditure Accounts
  • Annual and Quarterly National Economic and Financial Accounts
  • The problem of inventory
  • Application at the provincial level
  • The problem of non-additivity
  • Questions and answers
  • Introduction

    Growth in the gross domestic product (GDP) or any other nominal value aggregate can be decomposed into two elements: a "price effect", or the part of the growth linked to inflation, and a "volume effect", which covers the change in quantities, quality and composition of the aggregate. The volume effect is presented in the National Accounts by what is referred to as the "real" series (such as the real GDP).

    In the Canadian National Accounts, the volume effect is determined using the deflation method, which eliminates the price effect from each component of the aggregate and then aggregates the components thus deflated to obtain the "total" volume effect.

    There are several ways to aggregate the components of a total in order to calculate the volume effect. Index number theory offers a wide range of tools to this end. Since spring 2001, Statistics Canada has preferred the chain Fisher index. This measure is theoretically superior to the former fixed base Laspeyres measure and also makes the Canadian data comparable with the United States' official measures of economic activity. Furthermore, it offers compliance with the recommendations of the System of National Accounts 19931 (SNA '93).

    The following paragraphs provide a simplified explanation of the methods that will henceforth be used by the National Accounts to measure the country's real economic activity.

    Building an index and chaining

    A given nominal aggregate (GDP or other) represents a summation of quantities evaluated in the same monetary unit, at the prices of the current period. To use GDP as an example, this summation can be expressed as GDP = Σpq , which is the sum of all quantities of goods and services transacted in the economy, multiplied by their respective prices. The change or variation in nominal GDP, between a period o and a period t, can therefore be expressed in index2 form by:

    Equation 1 - GDP value index
    where: ΔGDPt/o is the GDP value index.
    pt is the price at time t
    po is the price at time o
    qt is the quantity at time t
    qo is the quantity at time o

    Example 1: A wine and cheese economy.

    The change obtained by this formula may theoretically be divided into a change in prices and a change in volume. If there were an "average" GDP price then it would be quite simple to divide the change in GDP (given by Equation (1)) by this average price to obtain the average change in quantities. Most of the time in the National Accounts, there is no such average price. Thus, the total change in quantities can only be calculated by adding the changes in quantities in the economy.

    However, creating such a summation is problematic in that it is not possible to add quantities with physically different units, such as cars and telephones, even two different models of cars. This means that the quantities have to be re evaluated using a common unit. In a currency-based economy, the simplest solution is to express quantities in monetary terms: once evaluated, that is, multiplied by their prices, quantities can be easily aggregated.

    An intuitive way to measure changes in quantity over time is to take the prices available for a given period and to multiply the quantities from the subsequent periods by these same prices. It amounts to re evaluating current quantities at prices fixed in time, which essentially "removes" the price effect. In mathematical terms, this can be expressed by the formula for the fixed-base Laspeyres index:

    Equation 2 - Laspeyres quantity index, fixed-base
    where: LQt/o is the Laspeyres quantity index
    po is the price at time o
    qt is the quantity at time t
    qo is the quantity at time o

    The only difference from Equation (1) is found in the numerator, where the quantities at time t are here multiplied by the prices at time o.

    Example 2: Wine and cheese production without the price effect.

    It is quite clear with such a formula that the results are highly dependent on the structure of prices at time o. Should this structure change with time, for example as a result of a drop in the price of one component compared with the others, then the index from Equation (2) will eventually be biased by the fact that it is dependent on an outdated price structure.

    One way to overcome this type of problem is to periodically update the weighting base to bring it in line with the current period. This technique was used in the past by the Canadian System of National Accounts (CSNA) when the real series was rebased every five or six years to reflect changes in the price structure.

    It is possible, however, for the price structure to change more quickly than usual. The weighting base then becomes outdated quickly, perhaps making it necessary to increase the frequency of the rebasing. Ultimately, the weighting base can be systematically moved from period to period so that it is defined as being the period preceding the current period:

    Equation 3 - Laspeyres quantity index, moving base

    Where we find, in place of po from Equation (2), pt-1. For the current period t, this moving base index gives the growth in volume weighted according to prices t-1. To some extent, it incorporates the frequency of the rebasing, thereby eliminating the arbitrariness of a rebasing done only on an "as required" basis.

    This type of index has a short-term base but it can be adapted to cover several periods. Variations between successive periods (as calculated by Equation (3)) can be cumulated by multiplying as we do in a compound interest calculation. This is how indexes are chained. The prices used for weighting in the resulting chain are very recent prices and never become obsolete. Using our example, a chain index would have the following form:

    Equation 4 - Laspeyres quantity index, chained
    where: n is the number of periods over which the chain index extends.

    Example 3: Another way to remove the price effect.

    The System of National Accounts 1993 recommends using chain indexes. Statistics Canada has been following this recommendation since spring 2001 for the quarterly National Economic and Financial Accounts and since fall 2002 for the Provincial Economic Accounts and for the monthly GDP by Industry. Systematic chaining allows for constant renewal of the weighting base, thus avoiding the problem of outdated data associated with a fixed-base index.

    Choice of index

    The previous examples refer to a Laspeyres type index. However, index number theory provides numerous other indices that differ in the way the components are weighted. For example, although the quantities in the Laspeyres index are weighted with the prices of a previous period, in the Paasche index they are weighted with the prices of the current period:

    Equation 5 - Paasche quantity index
    where: PQt/o is the Paasche quantity index

    This index is in fact the reciprocal of the Laspeyres index. Used in its fixed-base form, it presents the same problem as that described earlier, but the inverse: it does not adequately reflect changes in the structure of the economy for previous periods. However, the Paasche index can be chained in the same way as the Laspeyres index (as in Equation (4)).

    It can be shown that, in general, a Laspeyres quantity index will generate a larger increase over time than a Paasche quantity index. This occurs when prices and quantities are negatively correlated, that is, when goods or services that had become relatively more expensive are replaced by goods and services that have become relatively less expensive. This is a common substitution effect. In such cases economic theory says that the Laspeyres and Paasche indexes set upper and lower limits for a theoretically ideal, less biased, index.

    This theoretical index can be approached by a Fisher type index, representing the geometric mean of a Laspeyres and Paasche index:

    Equation 6 - Fisher quantity index
    where: FQt/o is the Fisher quantity index

    This index is not only superior theoretically, but it also includes a number of desirable properties from the standpoint of the National Accounts. For example, it is "reversible over time", that is, the index showing the change between period o and period t is the reciprocal of the index showing the change between period t and period o. Another interesting feature is the "reversibility of factors" by which the product of the price and quantity indexes is equal to the index of the change in nominal values:

    Reversibility of factors

    This brings us back to our index of nominal change in Equation (1) and the decomposition of the "price effect" and "volume effect" discussed at the start of this paper. From there, it is quite easy to find the implicit Fisher price of GDP by dividing GDP in current dollars by real GDP using the Fisher formula. The Laspeyres and Paasche indexes do not have either of these two properties.

    Statistics Canada uses the chain Fisher index as a measure of real GDP. Following the same sequence that we used with Equation (4), chaining Equation (6) gives us :

    (7) Equation 7 - Fisher quantity index, chained

    This is the formula used as the basis of the calculations of real GDP at both the national and provincial levels.

    High technology and the price structure of the economy in the 90s.

    Application in the Income and Expenditure Accounts

    In practice, the formulas provided above cannot be used as is by the Income and Expenditure Accounts, given the absence of data on quantities and price levels. Only the current value (C) series and price indexes (relative prices) are available. Formulas have to be transformed using the fact that the price multiplied by the quantity (ptqt) equals the series in current dollars (Ct). We then get formulae expressed in terms of nominal series (Ct) and relative prices (pt/pt-1 or the reverse). This then gives us, for Laspeyres (using Equation (3)) :

    Equation 8 - Laspeyres volume index

    . for Paasche (using Equation (5)):

    Equation 9 - Paasche volume index

    . and lastly, for Fisher (geometric mean of Equations (8) and (9)):

    Equation 10 - Fisher volume index

    It is this formula, chained, that is used in practice. The detail of the transformations can be examined here.

    Since the series are no longer expressed here in terms of quantities, we now refer to the volume index. The concept of volume is broader than that of quantity, because it includes variations in quality and ultimately, changes in the composition of the economy.

    Example 4: In the real world: the deflation method.

    Annual and Quarterly National Economic and Financial Accounts

    Real aggregates published by Income and Expenditure Division (IEAD) are calculated using Equation (10) shown above. For each real aggregate, an index is calculated from component series, then chained quarter by quarter as shown in examples 3 and 4. The chained index series thus obtained is then benchmarked to a reference year in order to express it in dollars. Benchmarking consists in putting the level of the chained index series to a level such that, for a given reference year, it is equal to the corresponding aggregate in current dollars, while maintaining the quarterly growth rates intact.

    The level of detail - that is, the number of components used in each of the aggregates - is determined by the availability of data and by certain determinants of overall quality (such as the stability of seasonality). At the national level, 435 series in current dollars and the same number of corresponding price series are used to calculate real GDP using the chain Fisher index. The table level of detail at the national level shows how these series are distributed between the various aggregates presented in Table 3 of the publication National Income and Expenditure Accounts, Quarterly Estimates (13-001).

    None of the Fisher index calculations are done on an annual basis. The real annual aggregates are simple averages of the year's four quarters. These are the official measures of real annual national GDP.

    The problem of inventory

    For most of the items in Table 3 and other tables in real terms of the publication National Income and Expenditure Accounts, Quarterly Estimates (13-001), the Fisher calculation does not present any real technical problems. This is not the case for the investment in inventory series, which are first difference series. Since these series fluctuate around zero, the Laspeyres and Paasche indexes take opposite signs; since Fisher is the geometric mean of these two indexes, it becomes indeterminate.

    As published by IEAD, real investment in inventories is not the result of a direct chained Fisher calculations as shown above, but rather an approximation. The approach used by IEAD is based on the fact that an investment in inventory represents the variation of a total stock, which is always positive. In principle, a Fisher index can be calculated on a total stock series. Once this index is benchmarked to the dollar value of a reference year, one can suppose that the first differences of this series in dollars represents an estimate of the real series of the investment in inventories.

    If such a method is easily applicable to the calculation of the real inventory series, it is however unusable in the context of the calculation of real GDP. Indeed, the calculation of real GDP should be done with series of investment in inventories, and not with series of total stock. To by-pass this problem, IEAD uses two series of total stock rather than one for each series of investment in inventories: a first series of the stock in the current period (with a positive sign); and a second series, of the stock in the previous period (with a negative sign). This last series is in fact a series of total stock with a one-period lag. At any time t, the difference between these two series corresponds to the investment in inventories during the same period. If in the GDP we replace every series of investment in inventories by these two series of stocks, one positive and the other negative and lagged, we can calculate the real GDP with the chained Fisher index formula. The prices used for total stocks are those of investment in inventories.

    The calculation of the real aggregates of investment in inventories involves the same series of total stock. For each aggregate of investment in inventories, a chain Fisher index is obtained from the series of total stock in the current period and another one from the series of the lagged total stock. Once benchmarked to the reference period, these chained Fisher series can be subtracted from each other to simulate a real series of investment in inventories. This is the way that the real aggregates of the investment in inventories are calculated by the Income and Expenditure Accounts Division.

    This methodology for calculating the investment in inventory explains the fact that, at the national level, there are 110 inventory series used in the Fisher calculation, of which 76 are non agricultural and 34 agricultural (when, in fact, there are 55 inventory series published in current dollars, of which 38 are non agricultural and 17 agricultural).

    Application at the provincial level

    At the provincial level, real values are calculated in the same way as they are at the national level, but on a annual basis. Investment in inventory is calculated according to the methodology described above, on an annual basis, using average prices for the year.

    The level of detail of the provincial accounts differs from that of the quarterly national accounts. For each province, 502 series are used in calculating real GDP. The table level of detail at the provincial level  shows the distribution of these series through the items in Table 3 of the publication Provincial Economic Accounts (13-213). This distribution is slightly different than the national structure because of the different availability and quality of provincial data.

    Sources of bias between the national and the provincial systems

    The problem of non-additivity

    The chain Fisher series published by the Income and Expenditure Accounts are not additive, and this problem increases with distance from the reference period. Non-additivity of real series comes both from chaining and from the Fisher formula itself. Chaining destroys the additive consistency of accounting equations and the Fisher formula (as opposed to the Laspeyres formula) doesn't have the additivity property.

    The fact that the real series are not additive makes them more difficult to manipulate than in the past, when the calculations were based on fixed-base Laspeyres index. For example, it becomes difficult to measure the contribution of an individual aggregate to a bigger whole knowing that the sum of the aggregates does not add up to the total. It is also imprudent to create aggregates from other aggregates.

    There are a variety of ways to overcome this additivity problem. For some summary analysis, current dollar data may be enough and even desirable, because they reflect the economic structure at current prices. This is especially true if the aggregates being studied do not exhibit large price variations or if these variations are relatively uniform.

    For those who want to use real data and create aggregations, one solution is to calculate Fisher indexes using existing Fisher data. In 1979 Erwin Diewert demonstrated that a Fisher index was approximately consistent, and that therefore it was possible to calculate Fisher indexes from aggregates already in Fisher, what he called a "Fisher of Fishers". This solution provides a valid approximation provided that the aggregates used in the calculation are relatively consistent in terms of prices (this solution should not be used, for example, if the calculation involves inventory series).

    A more "structural" solution is to play with the benchmarking frequency. Since additivity decreases with distance from the reference year, rebenchmarking the series to bring the reference year closer may alleviate part of the problem without, however, making the whole strictly additive. It is important to note that, in the case of real data based on chain Fisher index calculations, changing the reference period does not have any impact on the growth rates of real series.

    Since it is not possible to make the levels additive, Statistics Canada, following the lead of the Bureau of Economic Analysis in the United States, suggests a strictly additive decomposition of the variations of the aggregates for tables published from real data. The formula used reweights the contributions to the series in such a way that they become strictly additive at the total variation of the aggregate:

    Equation 11 - Contribution to growth formula

    or, in a form that applies to nominal series and to prices,

    Contribution to growth formula, with nominal series

    This formula is the basis for the percentage contribution to variation series published by Income and Expenditures Accounts Division. A detailed mathematical transformation is available here.

    Contribution to a GDP growth series

    Questions and answers

    What is the difference between a reference period and a base period?

    The prices used to compile the volume indexes are prices from the base period, while the period in which the value of a series in constant dollars is equal to the value of said series in current dollars is the reference period. In the former real GDP measure using the fixed-base Laspeyres method, the reference period and the base period were the same. In a chain volume measure, however, the two periods are not necessarily the same. For example, the chain Fisher series in our publication are referenced to 1997 (current dollars equal constant dollars for 1997) but the base corresponds to a combination of the current period and the period immediately before the current period, because it is a chain Fisher index. The reference period serves only to benchmark the series and a change in the reference period does not change any aspect of the growth rates of the series or the aggregates. The only change occurs with the levels, which are benchmarked on a different value.

    For this reason, the chain Fisher series currently published cannot be said to be "at 1997 prices", because the prices of the reference period are not used in any way in the calculation of the quarters preceding or following the reference year. However, it can be said that these are series expressed in real terms, thereby easing the price effects, at a level at which they are equal to the nominal aggregate level for 1997. In other words, a real series in which the reference year is 1997 is the equivalent of a nominal series in which the price effect has been removed since 1997.

    How can we make period over period comparisons when the periods are far apart in time?

    The formula for percentage contribution to variation presented earlier applies only to a single period. To use the same formula over a longer period of time, a Fisher non chained value is required where the weighting bases correspond to the periods to be analysed. For example, to analyse the growth in durable consumer goods between the fourth quarter of 1996 and the fourth quarter of 2000, it is possible to calculate a Fisher index in which the weighting is explicitly a function of the prices in the fourth quarter of 1996 and of the fourth quarter of 2000. To some degree, it amounts to a fixed-base Fisher index. Once this index has been calculated, the percentage contribution to variation formula can be used directly.

    Users can do such calculations themselves, if they have all of the series included in the aggregate. Otherwise, they can be done by Statistics Canada on request.

    Moreover, it is always possible to use the nominal series. The ratios in these series reflect the position of a sector to the prices at that time, and not at the prices of an arbitrarily selected reference year.

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