3.5  The four approaches summarised

Sections 3.1 to 3.4 have identified four approaches that can be used to measure the average growth rate over a period or sub-period of interest. Table 2 summarises how these approaches measure the growth rate as a function of the time series observations of the series for which average growth rates are being constructed.

Table 2 – Four techniques to construct a growth rate

	Technique	Construction
1	OLS ()	where
2	Log Difference Regression ()	where
3	Average Annual Growth Rate ()
4	Geometric Average ()

For the time series Y={Y₁, Y₂, Y₃, … , Y_T}

It is possible to show that techniques 2 and 4 are equivalent so that (see Appendix A.4 for details). That is, the average growth rate for a period calculated by the log difference regression technique would be the same as the average growth rate calculated by the geometric average approach. Another point worth noting is that the log difference regression rate is approximately equal to the used in its construction. It was noted that estimated in the log difference regression approximately equals the average annual growth rate and therefore the average annual growth rate is approximately equal to the log difference growth rate (and subsequently the geometric average growth rate).[11] That is:

(15)    

Note: The GDP per Capita series is GDP per head at the price levels and exchange rates of 1995 (US dollars) as published in (OECD 2002). Data was obtained electronically to 3 decimal places and data to this level of accuracy was used in the growth rate calculations.

AAGR, GEO, OLS and LD refer to the growth rate obtained using the Average Annual Growth Rate, Geometric Average, Ordinary Least Squares and Log Difference techniques respectively.

Table 3 illustrates the results obtained by using the four growth rate techniques outlined earlier. Estimates of the average growth rate in New Zealand real GDP per capita over a number of different 10-year windows are computed as well as the average growth rate of the entire period (1970-2000). This means that for each window a sub-series of 11 data points is used. Not surprisingly, the growth rates calculated using the geometric average and log difference techniques are identical and only differ to the average annual growth rates by up to 5 one hundredths of a percent. There is substantial variation between the growth rate computed using the OLS technique and the other three techniques. In some cases the difference between the growth rates obtained using the OLS technique and the average annual growth rate technique is greater than the average annual growth rate value. The quadratic weighting scheme used in the OLS technique results in the OLS growth rates being materially different from the growth rates obtained using the other techniques, even when the rates are calculated over the entire sample.

Figure 4 – A comparison of different growth rate construction techniques (NZ growth rates measured over different 10 year windows)

Source: Author’s growth rate calculations based on OECD data as shown in Table 4.

Figure 4 provides an alternative representation of the data in Table 3 and plots the average growth rate in New Zealand’s real GDP per capita for moving 10-year periods as measured using the different growth rate approaches. Thus the first growth rate plotted for each series is for the period 1970 to 1980, the second for the period 1971 to 1981 and so on. Figure 4 again highlights that the growth rate estimates for a particular period can vary significantly depending on the technique used, with the OLS growth rate at times differing substantially from the rates obtained using other approaches.

3.6  Choice of Method

Section 3.5 highlighted that growth rates obtained from the OLS approach sometimes differed substantially to those obtained from the other 3 methods. Given this, what is the most appropriate way of calculating a growth rate? This depends on the data generating process underlying the data being used. In the case that the log of GDP per capita is stationary around a deterministic trend and hence does not contain a unit root, then it is appropriate to use the OLS approach. On the other hand when the log of GDP per capita is integrated of order one (I(1)) the log difference approach is more appropriate.[12] As the log difference approach provides the same results as the geometric average approach, and is approximately equal to the average annual growth approach, there is little in it when choosing between these three approaches.

The average annual growth rate approach involves a weighting structure (standard arithmetic weights) that makes it intuitively simple. The geometric average approach (and consequently log difference approach) is also quite intuitive and has the advantage that if one takes the value of real GDP per capita at the start of the sample period of interest and grow it at the geometric average growth rate for the appropriate number of years, the value obtained will be that of the final value of real GDP per capita in the sample period of interest. In general this will not be the case when the other growth rate approaches are used.

As shown in Table 4, the natural log of all the New Zealand real GDP per capita series used in this paper are integrated of order 1 (I(1)). What this means is, that with New Zealand data at least, the use of the OLS approach to calculate an average growth rate should be avoided and one of the other 3 approaches used. Due to its simplicity, when growth rates are computed in the rest of this paper the average annual growth rate has been used.

Both a constant and trend were included in the Augmented Dickey-Fuller (ADF) tests when conducted on levels data. Numbers in brackets in the second and third columns indicate number of lags used in these tests. The lag lengths were determined using the Schwarz criterion.

* signifies a unit root null is rejected at the 5% significance level

** signifies a unit root null is rejected at the 1% significance level