Fractional calculus in economic growth modelling: the Spanish and Portuguese cases

Abstract

This work presents a fractional order approach to model the growth of national economies, namely, their gross domestic products (GDPs). Land area, arable land, population, school attendance, gross capital formation, exports of goods and services, general government final consumption expenditure and money and quasi money are taken as variables to describe GDP. The particular cases of the national economies of Spain and Portugal are studied along the last five decades. Results show that fractional models have a better performance than the other alternatives considered in the literature.

Appendix

The economic data used in this work can be found in Tables 5 and 6. Sources for the economic data in Table 5 are as follows:

Table 5 Spanish economic data for years 1960–2012

Table 6 Portuguese economic data for years 1960–2012

• $x_1$ is taken from [37]. The data concerns what is currently the territory of Spain only, and not what are now Equatorial Guinea and Western Sahara, which were always separate national economies. Slight variations in area, found in the database, which are spurious, since the territory of Spain did not change in the period considered, were discarded. This input is thus constant.

• $x_2$ and $x_3$ are taken from [37].

• $x_4$ is taken from [38]. As the data has a 5-year sampling time (starting in 1960), a third-order spline interpolation was used for intercalary years.

• $x_5$, $x_6$ and $x_7$ are taken from [37], in current euros. The price index mentioned below was used to convert values to 2012 euros.

• $x_8$ is taken from [39] in current euros in the 1999–2012 period. In the 1962–1968 period, it is taken from [37] also in current euros. These two series are clearly coherent. [40] has data for 1941–1970 in current pesetas; values for 1962–1970 are consistently 60 % of those in [37]: and so for 1960–1961 we used the values of [40] converted to euros and divided by 0.6. The price index mentioned below was used to convert values to 2012 euros.

• The price index mentioned several times above is the one implicit in [37], that for several variables provides values in current euros and in constant euros.

Sources for the economic data in Table 6 are as follows:

• $x_1$ is taken from [37]. The data concerns what is currently the territory of Portugal only, and not the former colonies, then overseas provinces, granted independence in the 1974–1976 period, and which were always separate national economies. A slight variation in 2004, in all probability spurious, found in the database, was kept; otherwise this input is constant.

• $x_2$ is taken from [37]. As the series begins in 1961, the value for that year was also assumed to be that of 1960.

• $x_3$ is taken from [37].

• $x_4$ is taken from [41] in the 1960–1990 period, when the series ends. In the 1998–2012 period, the value is a weighted average of the percentages of labour force with primary, secondary and tertiary education (to which the weights of 4, 12 and 18 years were assigned, according to the criteria of [41]), taken from [37]. Data in [37] for the 1992–1997 were neglected, as they are clearly inconsistent with figures for the following years (there are abrupt changes in values from 1997 to 1998 that can only result from different criteria used by the source, claimed to be the Eurostat.) The values for 1991–1997 were quadratically interpolated from those in the rest of the series (the resulting fit has a very convincing R$^2=0.9964$).

• $x_5$ is taken from [41] in the 1960–1993 period, in current PTE (Portuguese escudos). In the 1994–2012 period, it is taken from [37], in current euros. Data was converted to euros and the price index mentioned below used to convert values to 2012 euros. [37] has data from 1970 on, and its series coincides notably with that in [41] in the 1970–1993 period, without being precisely equal.

• $x_6$ and $x_7$ are taken from [37], in current euros. The price index mentioned below was used to convert values to 2012 euros.

• $x_8$ is taken from [42] in the 1960–1998 period, when the series ends, in current PTE. Since then Portugal belongs to the Eurozone, making it difficult to build a coherent series. Consequently data for deposits in the 2005–2010 period from [43] was used. These two series were cubically interpolated and extrapolated for 1999–2004 and 2011–2012. All values were converted to euros and the price index mentioned below was used to convert values to 2012 euros.

• The price index mentioned several times above is taken from [42] for the 1960–2008 period, and extended in the 2009–2012 period using the price index published by the Instituto Nacional de Estatística.