Chain Fisher volume index - Methodology

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Sources of bias between the national and provincial systems

The national and provincial systems are additively consistent when expressed in nominal value. However, in real terms, the properties of the chain Fisher index are such that this consistency can no longer be guaranteed.

First, real series based on a chain Fisher calculation are not additive. This means that for each province and for Canada as a whole, the sum of the aggregates will not equal the main aggregate. Another consequence of this non-additive problem is that for each aggregate, the sum of the provinces will not equal the national level (for example, the sum of expenditure on consumer goods and services in all of the provinces will not equal the national expenditure on consumer goods and services).

Second, real series are calculated differently at the national and provincial levels. While the national annual series represent an average of the quarters, the provincial annual series represent a chain Fisher index calculated on the year. These two different methodologies produce different results.

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Third, theoretically there are two ways to calculate real series at the national level. Fisher indexes can be calculated with the national series (these being the sum of the provincial series), or calculated directly with the provincial series. Since Fisher is an index sensitive to the number of series involved in the calculation, the two calculations do not produce exactly the same result.

Lastly, the level of detail is different between the national and provincial calculations (respectively 435 and 502 series). Since Fisher is an index sensitive to the number of series involved in the calculation, the difference in level of detail produces an inevitable bias between the national and provincial calculation.

For these reasons, it is unlikely that the provincial real GDP series will be additively consistent with the national series.