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Current Analysis (613-951-4886). The idea for this paper came from discussions with Philip Smith, former Assistant Chief Statistician, Statistics Canada.

Several analysts have discussed growth rates and touched on the potential relationship between growth rates of different frequencies: Stephen Gordon, Université de Laval economics professor, posted a discussion on growth rates on the Worthwhile Canadian Initiative blog, May 3, 2009: http://worthwhile.typepad.com/worthwhile_canadian_initi/2009/05/a-preliminary-estimate-for-canadian-2009q1-gdp-growth.html. A short discussion on fourth-quarter-over-fourth-quarter growth and annual average growth is in Gene Epstein, "The Upward Slog Continues," Barron's, March 12, 2011: http://online.barrons.com/article/SB50001424052970203594204576194791734188776.html#articleTabs%3Darticle. This relationship is also alluded to in Bank of Canada, "Monetary Policy: Measuring Economic Growth" (http://www.bankofcanada.ca/monetary-policy-introduction/measuring-economic-growth/), and in Bruce Little "To understand growth figures, look to the past," The Globe and Mail, October 8, 2001 (https://secure.globeadvisor.com/servlet/WireFeedRedirect?cf=sglobeadvisor/config&date=20011008&slug=RAMAZ&archive=gam).

Annual average growth rates can also be closely approximated as the average of the year-over-year growth rates for all four quarters (or 12 months) of the year.

For example, annual GDP is the sum of four raw quarterly levels, and the annual CPI is the average of the twelve monthly levels.

The annual average growth rate and the fourth-quarter-over-fourth-quarter growth may produce the same number in a given year, but, in light of the different dynamics discussed in this article, it becomes evident that this is simply coincidence.

The annual average growth rate of the leading indicator, in contrast, is not important as the purpose of this indicator is to examine short-term trends.

Quarterly growth rates must be entered into the equation in the form of 1.10 for a 10% increase and 0.90 for a 10% decrease in order for the equation to yield the annual average growth rate.

The seven relevant growth rates of 2007 and 2008 (as shown in Figure 3.3) are entered into the equation as follows: Annual average growth rate

_{2008 =}(((1.008 x 1.006 x 1.005 x 0.999) + (1.008 x 1.006 x 1.005 x 0.999 x 1.001) + (1.008 x 1.006 x 1.005 x 0.999 x 1.001 x 1.002) + (1.008 x 1.006 x 1.005 x 0.999 x 1.001 x 1.002 x 0.991)) / (1+ 1.008 + (1.008 x 1.006) + (1.008 x 1.006 x 1.005)) – 1) =1.007 = 0.7%A second (and slightly more exact method as it reflects the effect of compounding) is to input the quarterly growth rates one at a time into the equation and calculate the difference in the annual average growth rate resulting from each subsequent growth rate addition. The difference between the resulting growth rates offers a sense of how much the additional impact each quarterly growth rate divided by the quarters' growth rate has on the annual average growth rate. As the impact may vary slightly according to the growth rates in the two adjacent years, it is useful to repeat this exercise for several years to estimate the impact (as measured by the pass-through rate) of each quarter on annual growth.

When the quarterly growth rates were inputted into the equation sequentially and the difference noted, the rates of impact for this hypothetical example were 25%, 50%, 75%, 101%, 77%, 52%, and 26%.

These are approximations for large percent changes, due to the asymmetric impact of large percentage changes on levels. A 50% decrease in sales of $1000 in one month will not recover its previous level with a 50% increase in sales the following month. The initial 50% drop will leave sales at $500 and the 50% increase will raise sales to $750. Large percentage increases and decreases or alternating increases and decreases over the seven relevant quarters will show a very different impact pyramid than the 1% growth in each quarter example provided. A sequence of 25% declines, for example, leaves the last periods' level so low that it has almost no impact on the annual average.

The share of annual average growth accounted for by the different quarters becomes less clear in cases in which quarterly growth rates switch between positive and negative growth.

The magnitude of revisions for a series has an impact on the equation's calculations of annual average growth rate. Revisions can alter the monthly or quarterly growth rates and therefore have a large impact on the annual average growth rate, according to the months or quarters in which the revisions are concentrated. The equation can be useful, however, in determining how the quarterly revisions affected the annual average growth rate.

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