## R. C. O. Matthews, C. H. Feinstein, and J. Odling-Smee

Print publication date: 1982

Print ISBN-13: 9780198284536

Published to British Academy Scholarship Online: November 2003

DOI: 10.1093/0198284535.001.0001

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# (p.556) Appendix B The Measurement of Trend Rates of Growth

Source:
British Economic Growth 1856-1973
Publisher:
Oxford University Press

There is no ultimate truth in the choice between alternative measures of trend rates of growth within periods, or in the demarcation of the periods. If the behavior of the system over time is irregular, there are bound to be different ways of summing it up. Moreover, the same measure is not necessarily appropriate for the two purposes for which economists have commonly used the concept of a trend rate of growth, namely as a summary of the actual behavior of output and as an indicator of the growth of productive potential.

Throughout this study, except where stated otherwise, the trend rate of growth of any series during a particular period is measured as the annual compound rate of growth between the first and last years of the period. This method ignores all the intervening years and therefore depends very heavily on the exact period chosen.

Wherever possible we have chosen periods such that the end‐years are roughly comparable from a cyclical point of view. The chosen periods usually begin and end with years of high levels of activity (low unemployment), because the peak years provide the best approximation to productive potential, and are also easy to identify. Some peaks, of course, were stronger than others, and a comparison between peaks of different strength may not provide a very good measure of secular change. Whenever such a distortion might alter the conclusions to be drawn from the data, we have said so in the text, but in general the point can be forgotten.

There are strong practical and analytical reasons for using growth rates between end‐years in preference to the alternative method of using growth rates derived by fitting an exponential trend to all the years in the period. In the first place, growth rates between end‐years have a clear meaning, whereas fitted trends are a statistical artefact. In the second place, unlike fitted trends, growth rates between end‐years allow consistency between the growth rates of disaggregated elements and the growth rates of their total, both for additive disaggregation (the sum of the output of different industries equals GDP) and for multiplicative disaggregation (employment times productivity equals production). And finally, they are, of course, less laborious to calculate. (p.557)

Table B.1 Alternative Measures of the Trend Growth of Gross Domestic Product at Factor Cost, 1856–1973

(Annual percentage growth rates)

Period

Between end‐years of period

Exponential trend fitted by least squares

Least squares R2

Peacetime phases:

1856–1913

2.0%

1.9%

0.99

1856–1873

2.2

2.1

0.97

1873–1913

1.8

1.8

0.98

1924–1937

2.2

1.8

0.80

1920–1938

1.9

2.0

0.88

1951–1973

2.8

2.8

0.99

1948–1973

2.8

2.8

0.99

Wartime phases:

1913–1924

−0.1

−1.3

0.24

1937–1951

1.8

1.1

0.30

However, growth rates between end‐years are not ideal measures, for the reasons stated above, and if substantial differences exist between them and fitted trend rates, their use requires to be justified. In the rest of this appendix we therefore compare the two types of measure.

Growth rates calculated in the two ways are shown in Table B.1, together with the corrected coefficient of determination of the least‐squares trend. The GDP series is the mean of the output‐, income‐, and expenditure‐side estimates, as in Table 2.1. The most obvious point that emerges is the great difference between peacetime and wartime phases. In the peacetime phases the growth rates calculated in the two ways are similar, and the fit of the least squares is good. In the wartime phases the fitted trend growth rates are much lower than the peak‐to‐peak rates, and the fit is poor.

The explanation for the wartime results is that in both cases the war occurred nearer the beginning of the period than the end. Since GDP as measured includes military expenditure, the level of GDP during the war years was high,* and this pulls up the fitted trend toward the beginning of each period. The result is that the first years of both phases, being prewar, are below the fitted trend, and the last years are above, thus causing the growth rates between end‐years to be higher than the fitted trend rates.

In this study we are not really concerned with what happened during the wars, except insofar as postwar economic growth was affected. Growth over the transwar periods is interesting, not because of what was happening during those periods, but because it indicates the position of the economy at the beginning of (p.558) the ensuing peacetime periods compared with its position at the end of the previous peacetime periods. For this purpose, the growth rates between endyears are more suitable since they are not affected by abnormal wartime events, which are anyway only approximately reflected in the rough statistical estimates covering the war years.

Let us turn, then, to the peacetime phases. It can be seen from Table B.1 that comparisons between the phases lead to very similar conclusions whichever set of growth rates is used. The two measures are closest for the pre–World War I and post–World War II periods, for which the fits of the least‐squares trends are very good. For the 1856–73 period the peak‐to‐peak rate of growth was slightly higher than the fitted‐trend rate. Since 1873 is known to have been a stronger peak than 1856, 2.1 percent may be a better measure of the secular rate of growth in 1856–73 than 2.2 percent. For the 1873–1913 period there is, for all practical purposes, no difference between the two measures; in 1856–1913 as a whole the 0.1 percentage point difference associated with the weak peak of 1856 appears again.

Similarly, there is no difference to speak of for the postwar phase. Two sets of growth rates are shown: in addition to the one for the usual peak‐to‐peak period, 1951 to 1973, we show the rate for 1948 to 1973, since 1948 is the first year reasonably free of wartime distortions for which data are available. The end‐to‐end growth rates are the same for the two periods, even though 1948 was not a peak year. They are also the same as the fitted‐trend growth rate. Furthermore, the $R ¯$ 2's are very high, indicating that the fluctuations were not severe, and the cycles are known to have been fairly regular. The 1951–73 peak‐to‐peak rate can therefore be held to be a good measure of postwar growth.

Two possible periods are used to represent “the interwar phase”: 1924 to 1937, which begins and ends in years with similar levels of economic activity, and 1920 to 1938, which are the first and last interwar years for which reliable data exist. For 1924–37, the fitted‐trend rate turns out lower, probably because the cyclical downswing of the 1930's tends to pull the trend line down. However, whereas the end‐to‐end rate of growth is lower for 1920–38 than for 1924–37, because 1920 was a boom year and 1938 was not, the use of fitted trend growth rates makes 1920–38 appear to have been a period of more rapid growth than 1924–37. The explanation for this is probably that the inclusion of the period of fairly rapid growth from the recession of 1921 until 1924 more than compensates for the inclusion of the two years of 1920 and 1938 and tends to increase the slope of the trend line.

There are thus three reasonable alternative measures of the trend rate of growth in the interwar period: the fitted rate 1924–37, the fitted rate 1920–38, and the end‐to‐end rate 1924–37. (The end‐year rate of growth between 1920 and 1938 cannot be considered reasonable, because the two years are not cyclically comparable.) Which is the best? The interwar period was marked by greater fluctuations than the other peacetime periods shown in Table B.1; this is reflected in the lower $R ¯$ 2's. Furthermore, there was only one major cycle, with a peak in 1929 and a trough in 1932, and this gives a rather lopsided appearance (p.559) to the period and pulls down the fitted trend. The 1924–37 peak‐to‐peak growth rate is not affected by the cycles, and so probably represents more satisfactorily the secular rate of growth.

To sum up, differences between end‐to‐end growth rates and fitted growth rates are substantial only for the war periods. For our purposes, the end‐to‐end growth rates are the more appropriate ones for those periods. As far as concerns the small differences found for peacetime periods, the evidence of the fitted trends suggests that the end‐to‐end growth rate for 1856–73 is too high by 0.1 of a percentage point. No amendment is indicated for 1873–1913, 1924–37, or 1951–73.

It obviously follows from this that as far as total output is concerned we can use the end‐to‐end measures with a clear conscience. We have not repeated the exercise for all the other series we shall be considering, and no doubt there are cases when the end‐to‐end growth rates give a faulty indication of underlying trends. In the course of the study we try to avoid misleading ourselves and our readers in such cases by supplementing growth‐rate data with charts of annual data and other summary measures.

## Notes:

(*) Real GDP was higher throughout World War I than in 1924; and in World War II it was higher in one year (1943) than in 1951.