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Statistical methods:  Chain Fisher Volume Index


Chain Fisher Volume Index


In May 2001, the quarterly Income and Expenditure Accounts adopted the Fisher index formula, chained quarterly, as the official measure of real expenditure-based Gross Domestic Product. There are two reasons for the adoption of this particular formula: it produces the most accurate measure of quarter to quarter growth in GDP and its components; and, the change brings the Canadian measure in line with the US quarterly Income and Product Accounts which also use the chain Fisher formula to measure real GDP.



Top of listWhy change?

Previously, the real expenditure-based GDP estimate was calculated using a Laspeyres formula chained periodically. The Laspeyres formula is fixed-weighted using weights based on some fixed point in the past. This formula basically adds up the volume changes in GDP by using the price levels of the base year as the weights. It is chained by changing the base year periodically. The base years were 1961, 1971, 1981, 1986 and 1992. Each time span is added up in its own base year prices and then the spans are linked together by measuring each base year on two price levels (old base and new base) and using the ratio of the two to link them together. Under the old method, the Canadian accounts were due to be rebased to 1997 price levels in May 2001.

With the rapid expansion of the Information and Communication Technology (ICT) industries in Canada, the Laspeyres volume measure produced significantly biased results. Growth was overestimated because prices of equipment and services related to this fast growing sector of the economy declined dramatically since the base year 1992, as a result of rapid technological change. The Laspeyres index measures changes in GDP by adding up the quantities produced of these commodities using 1992 price levels as weights. This is the equivalent of giving these commodities roughly four times the weight they would have at price levels of 2000. This over-weighting of the Laspeyres index, called "substitution bias", is why the Laspeyres formula produces a growth rate of GDP that is at the upper limit of possible measures. It does not compensate for substitution to lower priced commodities.

The degree of the bias depends on the amount of price dispersion. If, for example, all prices are rising at about the same rate as was the case for the period from the mid seventies to the early eighties, the bias is minimal. But, if technological change results in prices of one sector of the economy falling relative to other prices, as is the case from the mid eighties until 2000, the bias becomes significant.

A summation of the differences in the growth between the two formulas from 1981 until 2000 gives an estimate of bias of 2.5% in growth of GDP, with 1% of the differential occurring in the six quarters leading up to the third quarter of 2000  (graph 1). This may not seem significant, but because the high-technology revolution has had a large impact on investment (which is largely imported), the estimate of bias is as much as 40% in investment  (graph 2) and 12% in exports  (graph 3), offset to a large degree by a bias in imports (graph 4). When the ICT industries are growing rapidly in Canada, the bias in imports no longer offsets that of investment and exports and the bottom line of expenditure based GDP is also affected. 

Changing the index formula to a Paasche index where current prices would always be used as a weight base, was not a solution. This results in a bias opposite to that of the Laspeyres, with a tendency to understate growth in GDP as ICT prices fall monotonically. Growth in the base year would be measured using current prices, which is as faulty as the reverse. So with a Laspeyres index producing an upper bound to measurement of economic growth and a Paasche index producing the lower bound, the Fisher, which is the geometric average of the two, follows a more stable middle path.

Therefore, the real GDP is now calculated using the Fisher formula, which is the middle ground between the Laspeyres and the Paasche, rebased each quarter to minimise the bias introduced by dispersion.

For the time period 1981 to 2000, in all but a handful of exceptional quarters, the differences between the Paasche, Fisher and Laspeyres indexes in any individual quarter are fairly small. As well, in a chain index, the differences between using the prior weights, current weights or a combination of the two are greatly reduced. This is because the chain index approximates an "ideal index" for which weights change continuously.

Top of listHow is the Fisher formula calculated?

The Fisher formula calculates each link in the chained volume index as the geometric mean of the Laspeyres (fixed-weighted) index:

VtLasp = sum(Pt-1 Qt ) / sum(Pt-1 Qt-1 )

and the corresponding Paasche (current-weighted) index:

VtPaas = sum(Pt Qt ) / sum(Pt Qt-1 )

Where Pt and Qt are the price and quantity series and t refers to the reference quarter. They are summed across the entire commodity detail of expenditure based GDP.

The formulas above cannot be used as written because, in practice, price levels (as opposed to price indexes) and quantity series are not available.

To derive usable formulas, first note that Pt Qt (price times quantity) is the current dollar (Ct) series. Secondly, modify the formulas to replace the price series by price relatives, that is, the ratios of prices from one period to another. These are readily available using price index series.

This gives the equivalent formulas:

VtLasp = sum(Ct (Pt-1 / Pt)) / sum(Ct-1 )

VtPaas = sum(Ct) / sum(Ct-1 (Pt / Pt-1))

These are the formulas that are used in practice.

More details

Top of listAt what level of detail is the calculation done?

The summation is applied to the components of GDP at the most detailed level possible, consistent with published data. At the national level, 380 components of GDP are used: 

Personal expenditure on consumer goods and services 130
  Durable goods 22
  Semi-durable goods 15
  Non-durable goods 14
  Services 79
Government current expenditure on goods and services 24
Government gross fixed capital formation 14
Government investment in inventories 1
Business gross fixed capital formation 18
  Residential structures 4
  Non-residential structures and equipment 14
    Non-residential structures 4
    Machinery and equipment 10
Business investment in inventories 55
  Non-farm 38
  Farm 17
Exports of goods and services 69
Imports of goods and services 68
Statistical discrepancy 1
Gross Domestic Product at market prices 380
Final domestic demand 186

The 380 components represent the level of detail for which stable seasonal patterns can be identified, to provide publishable quality seasonally adjusted data.

Even at the level of detail used for the summations, price data must be derived implicitly as the ratio of current to constant price values. For most series this works well. For the inventory series however, the implicitly derived price series are not valid. This is because the inventory current and constant price values fluctuate about zero and it is not uncommon for the constant price values to be close to zero or negative while the current dollar value is not. In such cases, the ratios of the two series take on extremely large values or negative values, and these are not acceptable in a price series. Therefore, for inventories, the method is modified.

Previously, the published real (constant price) values for inventories were the quarterly changes in values of the corresponding fixed based Laspeyres real values for stocks of inventories. More accurately, fixed based real values were calculated on a series by series basis and these values were added, implicitly applying the Laspeyres formula since such addition can be done.

The Fisher real values (or chained real values) for inventories can not be obtained simply by adding individual series. Therefore, one must go back one step and apply the Fisher formula to the stock values of inventories. This can be done because the stock series are always large positive values and therefore valid implicit price indexes can be calculated and used in the Fisher formula. The published chain Fisher real values for inventories are then the quarterly changes in values of the corresponding chained Fisher real values for stocks of inventories.

The statistical error also takes negative value, but since it is a single series (as opposed to an aggregate series), it remains unchanged, in other words the current valued series is deflated by a price series.

While net trade also fluctuates about zero, in this case we have two distinct components (imports and exports) for which we can calculate indexes. The import components are entered in the summations as negative values, offsetting the export series.

Top of listThe calculation at the provincial level

Essentially the same methodology is used to calculate national or provincial real Fisher series. The differences are in the level of detail (447 components at the provincial level; see the table below) and that the provincial calculations are done on an annual basis. As such, the provincial Fisher indexes are chained annually rather than quarterly.

Personal expenditure on consumer goods and services 130
  Durable goods 22
  Semi-durable goods 15
  Non-durable goods 14
  Services 79
Government current expenditure on goods and services 24
Government gross fixed capital formation 3
Government investment in inventories 1
Business gross fixed capital formation 5
  Residential structures 3
  Non-residential structures and equipment 2
    Non-residential structures 1
    Machinery and equipment 1
Business investment in inventories 55
  Non-farm 38
  Farm 17
Exports of goods and services 114
Imports of goods and services 114
Statistical discrepancy 1
Gross Domestic Product at market prices 447
Final domestic demand 162

Top of listAre there any drawbacks?

An important consequence of using chain indexes is that the associated volume measures are not additive. That is, the sum of the chained values for each component of an aggregate does not equal the chained value of the aggregate. The values associated with the Laspeyres volume index are additive from the base year forward but not for the years prior to the base year which were calculated on other weight bases and chained together. Chaining produces non-additive components. Since the new Fisher measure is chained every quarter, it is also non-additive every quarter.

While this seemingly complicates analysis, conclusions drawn using published data on the Laspeyres formula, even in the years when the data is additive, could be very misleading. For example, at a quick glance, from the 1992 prices data it appears that computers and related equipment accounted for about 25% of total exports of goods in the year 2000. The question this really answers is: What share of exports would computers and related equipment account for in the year 2000 if they were sold at 1992 prices? In fact, at 2000 prices this commodity group accounts for about 10% of exports. This is a more relevant ratio in that it relates to the resources used and the incomes gained from the production of these commodities given 2000 price levels.

What can be more informative from the Fisher formula is the contribution to growth of any commodity group. Since the data is non-additive, this is not a straightforward calculation and this is why "contribution to change" calculations will be provided with all published chained dollar data.

Our primary reference has been the United Nations "System of National Accounts 1993", chapter 16 which recommends the use of chain Fisher indexes for measuring volume changes in GDP. There are many references available in this issue. 

If you have further questions, do not hesitate to contact the Income and Expenditure Accounts information service at (613) 951-3640 or by email.

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Date Modified: 2008-11-16 Important Notices