A Wavelet Investigation of Periodic Long Swings in the Economy: The Original Data of Kondratieff and Some Important Series of GDP per Capita
Abstract
:1. Introduction
2. Data and Methodology
- -
- England—Index number of commodity prices 1780–1922 (to simplify correct identification, we marked them with a capital letter, and begin with A);
- -
- France—Index number of commodity prices 1858–1922 (B);
- -
- USA—Index number of commodity prices 1791–1922 (C);
- -
- England—Quotations of interest-bearing securities 1816–1922 (D);
- -
- France—Quotations of interest-bearing securities 1814–1922 (E);
- -
- England—Index of Weekly wages in agriculture 1789–1913 (F) and Cotton Textiles 1807–1913 (G);
- -
- France—Foreign trade 1827–1913 in per capita francs (H);
- -
- England—Coal production 1855–1917 in t/1000 inhabitant (I);
- -
- France—Coal consumption 1827–1913 in t/1000 inhabitant (J);
- -
- England—Pig iron production 1840–1914 in t/1000 inhabitant (K);
- -
- England—Lead production 1855–1920 in t/1000 inhabitant (L).
- -
- Brazil (1850–2018; Barro and Ursua 2008);
- -
- France (1280–2018; Ridolfi 2016);
- -
- Germany (1850–2018);
- -
- India (1884–2018);
- -
- Italy (1310–2018; Baffigi 2011; Malanima 2010);
- -
- Japan (1885–2018; Fukao et al. 2015);
- -
- Turkey (1913–2018);
- -
- UK (1252–2018; Broadberry et al. 2015);
- -
- USA (1800–2018; Sutch 2006);
- -
- Former USSR (1885–2018; Gregory 1982 and Markevich and Harrison 2011).
- -
- T = the last term of the discrete series;
- -
- e = Euler’s number (also known as Nepier’s constant equal to 2.71…);
- -
- i = is the conventional for imaginary part;
- -
- = is the radians representation of the frequency (fnt).
- -
- the functions ϕ and φ satisfy conditions (4) and (5).
- -
- j = 1, 2, …, J indexes the maximum scale sustainable with the data to process (each scale represents a fixed interval of frequencies);
- -
- k indexes the translation parameter;
- -
- are the trend smooth coefficients in the wavelet transform capturing the underlying behavior of the data at the coarsest scale;
- -
- are the detail wavelet coefficients representing deviations from the smooth behavior.
3. Results
- -
- e = Euler’s number (also known as Nepier’s constant equal to 2.71…);
- -
- i is the conventional for imaginary part;
- -
- ω is the angular frequency in radians per time unit (equivalent to 2).
- -
- * represent the complex conjugate;
- -
- τ is the parameter to localize the position of the particular daughter wavelet in the time domain by an equal increment of dt (in our case dt = 1);
- -
- s represents the scale value used in the FFT algorithms to evaluate (13) in an efficient way (Torrence and Compo 1998). The choice of the set of scales as fractional powers of 2 defines the wavelet coverage of the series in the frequency domain (Rösch and Schmidbauer 2018).
4. Conclusions
Funding
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Scale Level J | Scale Crystals (Detail Level dj) | Annual Frequency Resolution |
---|---|---|
1 | d1 | 2–4 |
2 | d2 | 4–8 |
3 | d3 | 8–16 |
4 | d4 | 16–32 |
5 | d5 | 32–64 |
6 | d6 | 64–128 |
7 | d7 | 128–256 |
Series | A | B | C | D | E | F |
N | 143 | 65 | 132 | 107 | 109 | 125 |
Linearity test | ||||||
Keenan test | 14.37 | 30.68 | 10.56 | 2.23 | 3.81 | 4.80 |
p-value | 0.00 * | 0.00 * | 0.00 * | >0.05 | 0.05 * | 0.03 * |
BDS test p-value | 0.00 * | 0.00 * | 0.00 * | 0.00 * | 0.00 * | 0.00 * |
Series | G | H | I | J | K | L |
N | 107 | 87 | 63 | 87 | 75 | 66 |
Linearity test | ||||||
Keenan test | 0.25 | 0.29 | 5.15 | 0.22 | 9.85 | 0.91 |
p-value | >0.05 | >0.05 | 0.03 * | >0.05 | 0.00 * | >0.05 |
BDS test p-value | 0.00 * | 0.00 * | 0.00 * | 0.00 * | 0.00 * | 0.00 * |
Series | Brazil | France | Germany | India | Italy | Japan |
N | 169 | 709 | 169 | 135 | 709 | 134 |
Linearity test | ||||||
Keenan test | 9.06 | 21.38 | 1.75 | 64.97 | 13.84 | 37.17 |
p-value | 0.00 * | 0.00 * | >0.05 | 0.00 * | 0.00 * | 0.00 * |
BDS test p-value | 0.00 * | 0.00 * | 0.00 * | 0.00 * | 0.00 * | 0.00 * |
Series | Turkey | UK | USA | USSR | ||
N | 98 | 767 | 219 | 129 | ||
Linearity test | ||||||
Keenan test | 4.00 | 0.60 | 2.37 | 0.03 | ||
p-value | 0.05 * | >0.05 | >0.05 | >0.05 | ||
BDS test p-value | 0.00 * | 0.00 * | 0.00 * | 0.00 * |
Series | Scale Crystals (Detail Level dj) | Annual Frequency Resolution |
---|---|---|
A | d5 | 32–64 |
B | d5 | 32–64 |
C | d5 | 32–64 |
D | d5 | 32–64 |
E | d5 | 32–64 |
F | >d7 | >128 |
G | >d7 | >128 |
H | >d7 | >128 |
I | >d7 | >128 |
J | >d7 | >128 |
K | >d7 | >128 |
L | >d7 | >128 |
Serie | Scale Crystals (Detail Level dj) | Annual Frequency Resolution |
---|---|---|
Brazil | >d7 | >128 |
France | >d7 | >128 |
Germany | d6 | 64–128 |
India | >d7 | >128 |
Italy | >d7 | >128 |
Japan | >d7 | >128 |
Turkey | >d7 | >128 |
UK | >d7 | >128 |
USA | d6 | 64–128 |
Former USSR | >d7 | >128 |
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Focacci, A. A Wavelet Investigation of Periodic Long Swings in the Economy: The Original Data of Kondratieff and Some Important Series of GDP per Capita. Economies 2023, 11, 231. https://doi.org/10.3390/economies11090231
Focacci A. A Wavelet Investigation of Periodic Long Swings in the Economy: The Original Data of Kondratieff and Some Important Series of GDP per Capita. Economies. 2023; 11(9):231. https://doi.org/10.3390/economies11090231
Chicago/Turabian StyleFocacci, Antonio. 2023. "A Wavelet Investigation of Periodic Long Swings in the Economy: The Original Data of Kondratieff and Some Important Series of GDP per Capita" Economies 11, no. 9: 231. https://doi.org/10.3390/economies11090231