3.2 Log Difference Model Growth Rates
As just discussed, the commonly used least square regression approach results in a quadratic weighting scheme. The use of a different model enables us to obtain an estimator based on a simpler weighting scheme. Consider the model:
(7) 
By noting that this is equivalent to 
, recursive substitution enables (7) to be rewritten as:
(8) 
Note that the model shown in equation (8) is identical to the model shown in (4) except for the error term[10]. Therefore, the argument (based on manipulating the compound growth formula) that 
can be interpreted as a growth rate also holds for this model. In the model shown in (8) the error term is described by a moving average process.
The OLS estimator of in the model described by equation (7) can be expressed as (see Appendix A.2):
(9) 
So 
is just an average of the T-1 
terms, with each term being given an equal weighting of 
.
As was the case in section 2.1, the estimate of the slope coefficient () provides an estimate of the instantaneous growth rate. The compound rate of growth can be obtained as follows:
(10) 
3.3 The Average Annual Growth Rate
Noting that approximates the annual growth rate implies that in equation (9) is approximately equal to the average of the annual growth rates. In fact, using the actual annual growth rates rather than their counterparts gives us another simple way of calculating the annualised rate of growth over a period. This approach is referred to as the “average annual growth rate” (AAGR) approach. The average annual growth rate can therefore be specified as:
(11) 
3.4 Geometric Average Growth Rates
Another way of calculating the average growth rate for a period when an annual time series of data (Y1 to YT) is available is to directly utilise the compound growth formula by using the data points Y1 and YT as follows:
(12) 
solving the expression in (12) for r gives:
(13) 
Here r is the rate of growth required to grow Y1 so that it equals YT in T-1 years when compounding occurs annually. This approach is referred to as the geometric average approach. The fact that this approach only uses the values of the two endpoints of the series of interest is often considered a weakness. The reason for referring to this approach as the geometric average approach is that it is possible to express 1+r as follows (see appendix A.3 for details):
(14) 
The expression shown in (14) states that 
is the geometric average of one plus the annual growth rates obtainable from the data.
Notes
- [10]Equation (4) is ln Yt = α + βt + εt where α = ln Y1 - ln(1 + r) and β = ln(1 + r) thus equation (4) can be written as
which is the same as equation (8) except for the error term.
