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# 3  Calculating Growth Rates

For a given time series of annual real GDP per capita data, how should the average growth rate for the entire data period, or a particular sub-period of interest, be calculated?[7] There are a number of potential ways of constructing an average growth rate for a particular period. This paper focuses on four alternatives: (a) least squares growth rates; (b) a differenced logarithmic model; (c) the average annual growth rate; and (d) the geometric average growth rate. This section explains the procedures involved in computing growth rates using these alternative techniques. As will be shown, deriving these alternative growth rate estimators algebraically highlights that the different estimates obtained from these methods are all some variant of the average of the annual growth rates. The alternative approaches differ in terms of the averaging technique used on these annual growth rates.

The annual growth rate for a series of T annual observations, say Y1, Y2, Y3, … ,YT, is defined as:

(1)

where Yt is the observation for year t.[8]

## 3.1  Least Squares Growth Rates

One common approach to measuring growth rates is the Least Squares or Ordinary Least Squares (OLS) approach. In fact Kakwani (1997) notes that this is the most commonly used procedure for estimating growth rates.

The OLS approach is based on the compound growth formula:

(2)

The compound growth formula states that the value of real GDP per capita at time t is equivalent to the value of real GDP per capita at time 1 grown at a constant annual rate r (with compounding occurring annually over t-1 years).

Taking natural logs of (2) gives:

(3)

Adding a disturbance term , and letting and yields equation (4):

(4)

By regressing on t (time) using OLS we obtain an estimate of the slope coefficient () that provides an estimate of the instantaneous growth rate (). The compound rate of growth can be obtained as follows[9]:

(5)

It can be shown (see Appendix A.1) that the OLS estimator of can be expressed as:

(6)

where

That is, is a weighted average of the’s with the ’s serving as weights. As the OLS estimator for approximates a weighted average of the proportional changes in the series of interest (eg, in the case of annual data, a weighted average of the annual growth rates).

However, it is worth focusing on the weights. Note that the formula for the weights () is a quadratic in s. This weighting scheme means that the weights on the annual growth rates first increase with s, until reaching a maximum when , and then decrease symmetrically until .

To illustrate the differing weights applied to the (approximations of) the annual growth rates, Figure 2 plots the weights that would apply if one was working with sample of size T=20. When T=20, the weight given to the annual growth rate in the middle of the sample is 5.26 times the weight given to the growth rates at the end points of the sample. In general the ratio of the highest weight used to the lowest weight used is . The ratio of the highest weight to the lowest weight for values of T between 2 and 100 are shown in Figure 3.

#### Notes

• [7]The discussion that follows is equally relevant for estimating the growth rate in any series. The data does not necessarily need to be annual; what is important is that it is available for regular intervals over time.
• [8]It is possible to construct T-1 annual growth rates from series that has T annual observations.
• [9]Solving ln(1 + r) = β for r gives r = eβ - 1 hence equation (5).
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