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)