2.3 Other aspects of the Income and Expenditure Accounts data

Measures of volume and prices in the Income and Expenditure Accounts

2.122 Growth in the current dollar gross domestic product (GDP) or any other nominal value aggregate can be decomposed into two elements: a price element, or the part of the growth linked to inflation, and a volume element, which covers the change in quantities and quality of the aggregate. The volume element is presented in the System of National Accounts by what is referred to as the real economic activity series (such as the real GDP). Real GDP is the key economic indicator, on the trend-cycle growth pattern of the economy. Calculating real GDP and its expenditure components is, therefore, an important undertaking.

2.123 The volume element is measured to a very large extent by deflating the nominal value of a given series by an appropriate price index.¹ This is referred to as the deflation method.

2.124 Within the Canadian System of National Accounts the deflation of a series (such as exports of machinery and equipment) occurs at the lowest possible level for which a representative price index can be found. The deflated series can then be aggregated together to derive various totals. For example, 435 series are deflated and then aggregated together in order to calculate real quarterly GDP.

2.125 There are several ways to aggregate deflated series in order to calculate the volume aggregate. Index number theory offers a wide range of tools to this end. Since the spring of 2001, the Canadian System of National Accounts (CSNA) has adopted 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 U.S. official measure of real economic activity. Furthermore, it offers compliance with the recommendations of the System of National Accounts 1993 (SNA 1993).

2.126 The following paragraphs provide a simplified explanation of the methods used by the CSNA to measure the country's real economic activity.

2.127 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. Table 2.11 shows how these series are distributed between the various aggregates presented in Table 3 of the publication National Income and Expenditure Accounts.

 

2.128 Real gross domestic product (GDP) is estimated from 435 series. The values in those series are derived from surveys and administrative sources (see the chapters on sources and methods). The price indexes associated with each of the 435 series are taken from various sources, primarily consumer price indexes, industry product price indexes, raw materials price indexes, farm price indexes, export price indexes, import price indexes, foreign price indexes, and wage rates.

2.129 Personal consumption expenditure on goods and services includes 130 categories of goods and services. In most cases, consumer price indexes are associated with those goods and services.

2.130 There are 24 series for government current expenditure on consumer goods and services (see Chapter 8 for information on the deflation of government data). Volume estimates for the 10 labour income series are derived from hours worked, and the price indexes for the series are implicit. The 10 series for government capital consumption allowances are deflated with indexes based on information from the Investment and Capital Stock Division, while a number of sources are used to construct the other 4 price indexes.

2.131 The 32 series that make up gross fixed capital formation are mainly based on industry product price indexes, raw materials price indexes, machinery and equipment price indexes and import price indexes.

2.132 Business investment in inventories includes 55 components,² whereas government investment in inventories is based on just one series. The price indexes are mainly industry price indexes and farm price indexes.

2.133 International imports consists of 68 import price indexes, while 69 different price indexes are used to deflate international exports.

2.134 The price index associated with the value of the statistical discrepancy is the aggregate implicit expenditure-side GDP price index (constructed by excluding the statistical discrepancy).

Chain Fisher volume indexes

2.135 The quarterly Income and Expenditure Accounts adopted the Fisher index formula, chained quarterly, as the official measure of expenditure based real gross domestic product in May 2001. This formula was also adopted for the Provincial Economic Accounts, chained annually, in October 2002.

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

Building an index and chaining

2.137 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. Note: When we refer to quantities we are implicitly including any quality change as well. 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 0 and a period t, can therefore be expressed in index³ form by:

Equation 2.1

where:

• ΔGDP_t/o is the GDP variation index
• p_t is the price at time t
• p₀ is the price at time 0
• q_t is the quantity at time t
• q₀ is the quantity at time 0

2.138 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 2.1) by this average price to obtain the change in quantities. In the Canadian System of 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.

2.139 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. 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.

2.140 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 2.2

where:

• LQ_t/0 is the Laspeyres quantity index
• p₀ is the price at time 0
• q_t is the quantity at time t
• q₀ is the quantity at 0

2.141 The only difference from Equation 2.1 is in the numerator, where the quantities at time t are multiplied this time by the prices at time 0.

2.142 It is clear with such a formula that the results are highly dependent on the structure of prices at time 0. 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 2.2 will eventually be biased by the fact that it is dependent on an outdated price structure.

2.143 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 ten years to reflect changes in the price structure.

2.144 It is possible, however, for the price structure to change more quickly. The weighting base then becomes outdated quickly, making it preferable 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 2.3

where we find p_t-1, in place of p₀ from Equation 2.2. For the current period t, this mobile-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 rebasing done only on an as required basis.

2.145 In the short term, this type of index can be adapted to cover several periods. Equation 2.3 can be chained by successive multiplications, that is, in each period, it can be multiplied by the results obtained from the preceding period. This is called chaining. 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 2.4

where:

• n is the number of periods over which the chain index extends.

Equation 2.4 can also be written as:

Equation 2.5

where:

• LQ_c/0 = Q₀ x p _q1
• Q₀ is an arbitrary constant, usually 1 or 100, used as a reference point.
• LQ_c/0 is the chain Laspeyres quantity index for period 0 to period n.

2.146 The System of National Accounts 1993 recommends using chain indexes. 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 – Towards the chain fisher index

2.147 The previous section refers to a Laspeyres-type index. However, index number theory provides numerous other indices that differ in the way the components are weighted. For example, while 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 2.6

where:

• PQ_t/0 is the Paasche quantity index.

2.148 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 2.4).

2.149 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 common substitution effect implies that the Laspeyres and Paasche indexes set upper and lower limits for a theoretically ideal, less biased, index.

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

Equation 2.7

where:

• FQ_t/0 is the Fisher quantity index.

2.151 This index is not only superior theoretically, but it also includes a number of desirable properties from the standpoint of the Canadian System of National Accounts. For example, it is reversible over time, that is, the index showing the change between period 0 and period t is the reciprocal of the index showing the change between period t and period 0. 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 current values:

Equation 2.8

2.152 This brings us back to our index of nominal change in Equation 2.1 and the decomposition of the price element and volume element discussed earlier. 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.

2.153 Statistics Canada uses the chain Fisher index as a measure of real GDP. Following the same sequence that we used with Equation 2.4, chaining Equation 2.7 gives us:

Equation 2.9

where:

• n is the number of periods covered by the chain index.
• Q₀ is an arbitrary constant, usually 1 or 100.
• FQ_c/0 is the chain Fisher quantity index for period 0 to period n.

2.154 This is the formula used as the basis of the calculations of real GDP for the national and provincial accounts.

Application to the Income and Expenditure Accounts

2.155 In practice, the formulas provided above cannot be used as is, given the absence of data on quantities and price levels. The accounts have only current value (C) series and price indexes (thus, relative prices). Formulas have to be transformed using the fact that the price multiplied by the quantity (p_tq_t) equals the series in current dollars (C_t). We then get formulas expressed in terms of nominal series (C_t) and relative prices (p_t/p_t-1 or the reverse). This then gives us, for Laspeyres (using Equation 2.3):

Equation 2.10

… for Paasche (using Equation 2.6):

Equation 2.11

… and lastly, for Fisher (geometric mean of equations 2.10 and 2.11):

Equation 2.12

2.156 It is this formula, chained, that is used in practice. Since the series are no longer expressed in terms of quantities, we refer to this as a 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.

2.157 Table 2.12 provides an example of the calculation that the Income and Expenditure Accounts Division (IEAD) performs to produce the GDP volume estimates. The calculation is based on current-dollar data and price indexes for four commodities. In the second part of the table, the Paasche, Laspeyres and Fisher volume indexes, both chained and non-chained, are computed for the AB and CD aggregates and the total. The third part of the table deals with converting volume indexes into chained dollars.

Inventories

2.158 For most of the items, the Fisher calculation does not present any real technical problems, however, this is not the case for the investment in inventory series, which are first-difference series. Since these series fluctuate around zero, it happens that the Laspeyres and Paasche indexes take opposite signs—since Fisher is the geometric mean of these two indexes, it becomes indeterminate.

2.159 As published by the Income and Expenditure Accounts Division, real investment in inventories is not the result of a direct chained Fisher calculation as shown above, but rather an approximation. The approach used by Income and Expenditure Accounts Division is based on the fact that an investment in inventories represents the variation of a total stock, which is always positive. In principle, a Fisher index can be calculated on a total stock series. One for the inventory level at the end of period t (e.g., March 31, 2005) and the other for the inventory level at the beginning of period t (e.g., January 1, 2005). The difference between the two real stock estimates (end-of-period inventories minus beginning-of-period inventories) represents the real investment in inventories. Chapter 10 (Investment in inventories) contains a section on how to calculate the Fisher index for investment in inventories.

Provincial and Territorial Economic Accounts

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

2.161 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. Table 2.13 shows the distribution of these series through the items in Table 3 of the publication Provincial Economic Accounts. This distribution is slightly different than the national structure because of the different availability and quality of provincial data.

Estimates in “millions of chained (2002) dollars”

2.162 The base period of a volume index is the period for which the relative prices are chosen for the purpose of evaluating the index. For example, when quantity changes are valued at 2002 prices, the index is a Laspeyres volume index with base year 2002. With chained indexes, the base period is changed periodically. The period in which the value of a series in real terms is equal to the value of the same series in current dollars is the reference period.

2.163 In the old measure of real GDP by the fixed-base Laspeyres method, the reference period and the base period were the same. In a chain volume measure, on the other hand, the two periods do not necessarily coincide. For example, the chain Fisher series in our publication currently has 2002 as their reference year (estimates in current dollars are equal to estimates in real terms for 2002), but the base period is a combination of the current period and the period immediately preceding the current period, since they are Fisher indexes, which are chained quarterly. Consequently, the reference period is used only to re-calibrate the indexes, and a change in the reference period has absolutely no effect on growth rates. The only change is in the levels, which are re-calibrated to a different scale.

2.164 It cannot be said that the currently published chain Fisher series are at 2002 prices, since the prices for the reference period are not included in the calculations for the quarters preceding or following the reference year. It is, however, safe to say that the Fisher series are expressed in real terms, that is, free of price effects, at a level such that the series are equal to the level of the nominal aggregate for 2002, which is why the title of Table 3 in the publication refers to millions of chained (2002) dollars.

Non-additivity

2.165 The chain Fisher series published by the Income and Expenditure Accounts Division are not additive, and this non-additivity increases with the 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. Table 2.14 shows explicitly the non-additivity between the sum of components and the total in the following example:

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

2.167 There are a variety of ways to overcome this additivity issue. For some summary analysis, current dollar data may be sufficient or 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.

2.168 For those who want to use real data and create aggregations, one solution is to calculate Fisher indexes using existing Fisher data. In 1978 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”.⁴ 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).

2.169 A more structural solution is to accelerate the benchmarking frequency. Since additivity decreases with distance from the reference year, re-benchmarking 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.

2.170 Since it is not possible to make the levels additive, the Income and Expenditure Accounts Division, 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 re-weights the contributions to the series in such a way that they become strictly additive at the total variation of the aggregate:

Equation 2.13

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

Equation 2.14

2.171 This formula is the basis of the contribution to change series published by the CSNA.

2.172 The contribution of an aggregate to the percentage change in GDP in real terms is presented in Table 4 of the quarterly publication.⁵ Contribution-to-change tables are also calculated for various large aggregates (see Tables 18, 21, 24 and 27 of the same publication).

2.173 Each of these tables follows the layout of the corresponding real data table. Instead of real data, they show the contribution to the percentage change of the reference aggregate mentioned in the table's title. For example, Table 4 of our quarterly publication follows the layout of Table 3 and shows the contribution of the aggregates in Table 3 to the percentage change in real GDP. These contributions are not presented as proportions, but directly as percentage points. A contribution of the aggregate of personal consumption expenditure of 0.453 to real GDP growth of 1.473% for example means that 0.453 percentage points of the 1.473 are due to personal consumption expenditure.

2.174 The formula for contribution to percentage change (Equation 2.13) 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.

The Provincial Economic Accounts

2.175 Statistics Canada produces an annual provincial and territorial counterpart to the National Income and Expenditure Accounts. The estimates appear in the publication Provincial Economic Accounts, which provides measures of income-based and expenditure-based GDP for the ten provinces and three territories. The Provincial Economic Accounts also provide a detailed revenue and expenditure account for the government sector. A personal sector table is also provided.

2.176 While the conceptual framework of the Provincial Economic Accounts is similar to that of the national accounts, there are conceptual and statistical issues of measurement and allocation that are specific to the provincial accounts, notably with respect to the federal government, corporation profits and inter-provincial trade.

2.177 Estimates are produced for the ten provinces and three territories. Provincial estimates go back to 1981, whereas territorial estimates go back to 1981 for Yukon and the combined Northwest Territories and Nunavut and to 1999 for the new Northwest Territories and Nunavut.

2.178 It is important to note that the sum of the output of the provinces and territories is somewhat less than the output for Canada. This is because the GDP of Canada includes, on the income side, the income of Canadians temporarily posted abroad (diplomatic and military personnel and employees in the private sector) and on the expenditure side, current public expenditure abroad. There is thus a portion of Canada's domestic output that is not produced within the borders of a province or territory. This portion is simply omitted from the provincial accounts rather than being allocated arbitrarily to one region or another. More specifically, the foreign portion of income and expenditures is omitted from the GDP tables, and in the case of wages, salaries and supplementary labour income, it is also omitted from the table on personal income. However, governments' receipts and expenditures abroad are shown separately under the region Outside Canada in the supplementary tables on government. There they serve as a balancing item, for the aggregates involved, between Canada and the sum of provinces and territories.

The satellite accounts

2.179 The satellite accounts exist because of the need to flexibly extend the analytical capabilities of the national accounts core components in particular areas. These accounts are related to the other accounts in the Canadian System of National Accounts and through them, to integrated economic statistics in general. Furthermore, they relate to a particular field (tourism) or theme (non-profit organizations) and are also linked to the information system specific to it. Since they are closely linked to the Canadian System of National Accounts, they facilitate the analysis of some subjects in the context of macroeconomic accounts and analyses. In short, the satellite accounts play a double role, as instruments of analysis and means of statistical co-ordination.

The tourism satellite account and tourism economic indicators

2.180 The Income and Expenditure Accounts Division produces a tourism satellite account both for Canada and for the provinces and territories. The satellite account explicitly defines the tourism industry within the statistical system of the national accounts and measures its contribution to the economy. Because it is based on the System of Canadian National Accounts framework, the satellite account makes it possible to compare tourism with other industries. It provides benchmark data used to estimate quarterly indicators of tourism activity in Canada. This quarterly product is called the National Tourism Indicators.

The satellite account of non-profit institutions and volunteering

2.181 This satellite account is primarily intended to account explicitly for the economic activities of the non-profit sector and shed more light on its interaction with other areas of the economy within a macroeconomic framework.

2.182 The satellite account includes a standard set of economic accounts for the non-profit sector, along with a non-market component used to determine the value of volunteer work. These accounts provide information on a number of economic characteristics of the non-profit sector, including income and outlays. They serve to establish the economic profile of Canada's non-profit sector and answer major questions such as: what percentage of GDP does the non-profit sector represent in Canada? In what fields do non-profit organizations contribute to the well-being of Canadian society? How financially stable or vulnerable is the non-profit sector? How much does this sector depend on paid work rather than the contribution of volunteers?

Pension satellite account

2.183 The pension satellite account is currently under development. It will allow for enhanced analysis of the economic impact(s) of an aging population.

Continue

• Appendix 2A The presentation of product and output in the Canadian System of National Accounts Input-Output Tables
• Appendix 2B Interest and miscellaneous investment income in the income and outlay accounts of the institutional sectors
• 3 Wages, salaries and supplementary labour income

Back to

• 2.2 Institutional sectors dimension of the Canadian System of National Accounts – The Income and Expenditure Accounts
• 2 Concepts and definitions

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Notes

1. Exceptions to the deflation approach in the estimation of volume measures will be noted where relevant in the chapters dealing with expenditure side GDP components (refer also to 2.140).

2. Each component is used twice in computing the aggregates, once as beginning-of-period inventory and a second time as end-of-period inventory.

3. That is, by establishing a ratio between the value of the current period and the value of a preceding period.

4. Diewert, W. E., "Superlative Index Numbers and Consistency in Aggregation," Econometrica 46(4), 1978, pp. 883-900.

5. National Income and Expenditure Accounts, catalogue no. 13-001.