# 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);
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- 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).

_{n}= the complex number resulting from the DFT formed by a real (a) and an imaginary part identified by a lower-case i (ib);

- -
- 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 $\sqrt{-1}$ for imaginary part;
- -
- $\frac{2\pi tn}{T}$ = is the radians representation of the frequency (f
_{n}t).

_{k}) can be mathematically derived as

_{k}and b

_{k}are the coefficients of the numbers Z

_{n}for k = 1, 2, …, K (K the last time period until the Nyquist–Shannon frequency, i.e., the minimum sampling period needed in order to identify a possible periodicity, usually represented as: 0 ≤ f ≤ 0.5 f).

_{j}

_{,k}(t) and $\phi $

_{j}

_{,k}(t) are assumed to be orthogonal and represented as

- -
- 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;
- -
- ${s}_{j,k}$ are the trend smooth coefficients in the wavelet transform capturing the underlying behavior of the data at the coarsest scale;
- -
- ${d}_{j,k}$ are the detail wavelet coefficients representing deviations from the smooth behavior.

_{t}is given by

^{J}). However, considering that the highest scale (lowest-frequency crystal) can only just be resolved, it is usually recommended to decrease the number of crystals to be considered by one additional unit (Crowley 2007).

## 3. Results

_{0}= linearity), while the BDS test always rejects it. These outcomes corroborate the adoption of nonlinear data analysis methods.

- -
- e = Euler’s number (also known as Nepier’s constant equal to 2.71…);
- -
- i is the conventional $\sqrt{-1}$ for imaginary part;
- -
- ω is the angular frequency in radians per time unit (equivalent to 2$\pi f$).

- -
- * 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).

_{0}= NO periodicity) is set equal to 1000 (method = white.noise, p-value = 0.05 for the red line and p-value = 0.10 for the blue line). We include a period span between 16 and 128 years. Even if we have longer series which could be suitable for detecting higher crystals (as in the case of ITA and UK), for our research purpose, the range 16–128 (crystal d

_{6}) is appropriate. The AWP (also called percent energy by crystal d for scale j E

_{j}on total energy E) is given by (Crowley 2007).

_{5}). The other diagrams do not confirm this outcome. In the I, K, and L Series diagrams, the highest peak is in the d

_{7}crystal (periodicity over the 128 years), while the second highest is at d

_{6}crystal (period between 64 and 128). The Series G and H have the highest common peak in the d

_{7}crystal (over 128), and, finally, the Series F and J share the d

_{7}and the d

_{5}crystal as the highest and second most relevant peak (over 128 and 32–64 period). The results are summarized in Table 3. For Kondratieff’s original time series, there is a majority of cases that rejects the theoretical K-wave hypothesis. As pointed out in the previous section remarking the presence of price data within this dataset, we can highlight the different types of results based on the different natures of the original data. In fact, both the series of data on prices (A, B and C) and the series of quotations (prices) of interest-bearing securities (D,E) confirm the original results achieved by Kondratieff. On the other hand, there is no confirmation from the “physical” series of consumption and production (I–L). The same rejection of the K-wave hypothesis applies to the wage series (F and G). Although we observe a coherent result for the price series, overall, we can say that our results do not confirm those of the original author. Differently, all of Kondratieff’s results were homogeneous in supporting his thesis.

_{5}crystal is never the prevailing periodicity. For all cases (excluding the Germany and the USA), the d

_{7}crystal represents the highest AWP peak (with a period over 128 years), while both for Germany and the USA, it is possible to note the d

_{6}crystal (64–128 years) as the most significant one. For some countries (France, Italy and the UK) there is no second relevant periodicity in AWPs. For all those countries showing a second higher AWP peak (Brazil, Germany, India, Japan, Turkey and USSR), the most frequent periodicity is represented by the d

_{6}crystal. All results are summarized in Table 4. Overall, from the visual inspection of the graphs, we can deduce that the long-wave hypothesis is not supported by the GDP per capita investigation. Indeed, we never detect a periodicity consistent with the hypothesis. In addition, we can point out that it is not possible to observe a common pattern across countries. Two countries (Germany and the USA) have shorter fluctuation cycles. Curiously, they are the nations with the highest total GDP in the Western World. Probably the effects of the changes that occur in their economies spread and are trasmitted with inertia to the economic systems of others countries. They play what is called “the leadership role” both internationally and within their specific areas of economic influence.

_{6}instead of d

_{5}).

## 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 d_{j}) | Annual Frequency Resolution |
---|---|---|

1 | d_{1} | 2–4 |

2 | d_{2} | 4–8 |

3 | d_{3} | 8–16 |

4 | d_{4} | 16–32 |

5 | d_{5} | 32–64 |

6 | d_{6} | 64–128 |

7 | d_{7} | 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 d_{j}) | Annual Frequency Resolution |
---|---|---|

A | d_{5} | 32–64 |

B | d_{5} | 32–64 |

C | d_{5} | 32–64 |

D | d_{5} | 32–64 |

E | d_{5} | 32–64 |

F | >d_{7} | >128 |

G | >d_{7} | >128 |

H | >d_{7} | >128 |

I | >d_{7} | >128 |

J | >d_{7} | >128 |

K | >d_{7} | >128 |

L | >d_{7} | >128 |

Serie | Scale Crystals (Detail Level d_{j}) | Annual Frequency Resolution |
---|---|---|

Brazil | >d_{7} | >128 |

France | >d_{7} | >128 |

Germany | d_{6} | 64–128 |

India | >d_{7} | >128 |

Italy | >d_{7} | >128 |

Japan | >d_{7} | >128 |

Turkey | >d_{7} | >128 |

UK | >d_{7} | >128 |

USA | d_{6} | 64–128 |

Former USSR | >d_{7} | >128 |

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## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Focacci, 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