Appendix A
A.1 Weighting Scheme of the OLS Growth Rate Estimator
As explained in Section 3.1 one way of estimating a growth rate is to estimate the model:
The OLS estimator for 
can be written as:
(A.1.1)
where 
and 
Now 
which implies:
(A.1.2)
We can therefore rewrite (A.1.1) as:
(A.1.3)
by noting that 
due to (A.1.2) and 
The next step is to note that:
note that the last term in a bracket cancels with the first term in the following bracket so:
substituting this into (A.1.3) gives:
(A.1.4)
where 
, ie a set of weights.
Using the definition of 
we get:
(A.1.5)
By making use of 
and 
it can be shown that:
(A.1.6)
where 
Therefore:
(A.1.7)
A.2 Obtaining the growth rate estimator in a log-difference model
The OLS estimator for the model 
can be found as follows:
(A.2.1)
Therefore:
(A.2.2)
Differentiating with respect to gives:
(A.2.3)
Minimisation requires setting (A.2.3) to zero. Therefore:
(A.2.4)
Thus:
(A.2.5)
and so:
(A.2.6)
A.3 The geometric Average Growth Rate
Solving the compound growth formula for the growth rate r is one possible way of calculating the average annual growth rate over a period. Outlined below is why this solution for r is known as a geometric average growth rate. We begin with the compound growth rate formula:
(A.3.1)
which can be rewritten as:
(A.3.2)
This is equivalent to:
(A.3.3)
as 
Note: the numerator in each fraction cancels with the denominator in the following fraction, except at the endpoints.
Therefore by adding and subtracting 
to the numerator:
(A.3.4)
so that:
(A.3.5)
(A.3.6)
A.4 Proof that Log Difference Regression and Geometric Growth Rates are Equal
The log difference regression estimate of the growth rate can be found using the following expression:
(A.4.1)
now 
as 
and 
(by using the rules 
and 
). Consequently (A.4.1) can be rewritten as:
(A.4.2)
which can be simplified further to:
(A.4.3)
Noting that 
and observing that the numerator in each fraction cancels with the denominator in the following fraction, except at the endpoints, we get:
(A.4.4)
