Growth Rates

A growth rate shows the percentage change from one period to another/the next, or the average change over a number of periods.

The one period growth rate can be used on all kind of data (--except for the case where the initial value is zero), to measure growth in economic indicators like GDP or CPI, population growth etc. from one period to the next.

Since:



It follows that:


where g is average percentage growth/change from period 0 to t

The compound growth rate is used when averaging growth, interest and rate of return over discrete periods. Most economic phenomena are measured only at intervals (month, quarter or year) for which the compound growth model is appropriate.

Compound growth rates do not take into account intermediate values of the series, thus, is sometimes named end-point growth rates.

Annualized, or compounded sub-annual growth rates

When measuring growth in sub-annual time series, e.g., quarterly GDP or monthly CPI, there is always a choice of how to measure and present the growth. Growth can be presented (i) as the change from the same period the of previous year ( 4.98 to 4.99 ), (ii) as the change from the previous period ( 3.99 to 4.99 ), or (iii) as the change from the previous period ( 3.99 to 4.99 ) at annualized, or compounded rate of change.



The purpose of annualizing the rates of change is to present period-to-period rates of change for different period lengths on the same scale, and thus to make it easier for the layman to interpret the data, e.g., to realize that a 0.8 percentage growth from one month to the next is equivalent to a 2.4 percentage growth from one quarter to the next, or an annual growth of 10.0 percent.

However, annualizing growth rates also means that the irregular effects are being compounded. Since short-term statistics are prone to larger erratic movements than annual data, annualizing may result in an exaggeration of the short term movements in the series, at the expense of the overall trend.

If the frequency of compounding is considered to be continuous, the growth is called exponential. That is, the growth is approximately exponential when c in the formula for the compounding growth rate is so high that r/c becomes ‘very small’.

While the compound growth rate is used to measure average growth for discrete data/periods, the exponential growth rate is used for continuous data (like population, labor force, negative growth in analfabethism, continuous compounding interest, and radioactive disintegration).

Exponential growth rates are used in Bank publications to measure average growth over time in labor-force and population.

Least-squares growth rates can be used whenever there is a sufficiently long time series to permit a reliable calculation. The least-square growth rate is estimated by fitting a linear regression trend line to the logarithmic (annual) values of the variable in the relevant period.

The regression equation takes the form:

which is equivalent to the logarithmic transformation of the compound growth equation:

where x is the variable, t is time, and a = ln x₀ and b = ln ( 1 + r ) are the parameters to be estimated.

If b is the estimate of b, then the average annual growth rate, r, is obtained as:


An average growth rate estimated as a least-squares growth rate represents a trend, and it is not necessarily equal to any of the actual growth rates between any two periods.

Least squares growth rates are used in Bank publications when measuring trend-wise growth in economic variables such as GDP, and GNP per capita.

It should be noted that most national statistical offices use the compounding growth rate as an average growth rate, not the least squares growth rate.