Abstract
<p>A maximum likelihood method for estimating the power-law exponent verifies that the positive and negative tails of the Colombian stock market index (IGBC) and the Colombian peso exchange rate (TRM) approximate a scale-free distribution, whereas none of the heavy tails of a local sovereign securities index (IDXTES) are a plausible case for such distribution. Results also (i) support critiques regarding the flaws of ordinary least squares estimation methods for scale-free distributions; (ii) question the validity of Zipf’s law; (iii) suggest that IGBC and TRM display the scale-free nature documented as a stylized fact of financial returns, and that they may be following a gradually truncated Lévy flight; and (iv) suggest that local financial markets are self-organizing systems.</p>
Highlights
The magnitude of earthquakes, the population of cities, the intensity of wars, the level of rivers, the size of avalanches, the number of connections in most social and biological networks, and the usage of words in written language share a common feature
The left panel shows that the distribution of inhabitants across the Colombian territory is inhomogeneous, with the population of municipalities extremely varying across observations
The inadequacy of the first two moments of a Gaussian distribution to fit the population of Colombian municipalities or the assets of local financial institutions is even more prominent when using a standard (i.e. Gaussian) Monte Carlo method for simulating municipalities or financial institutions
Summary
The magnitude of earthquakes, the population of cities, the intensity of wars, the level of rivers, the size of avalanches, the number of connections in most social and biological networks, and the usage of words in written language share a common feature. The most wellwealth: in the wake of the twentieth century, based on tax record data from Basel (Switzerland) and Augsburg (Germany), rental income from Paris, personal income from Britain, Prussia, Saxony, Ireland, Italy and Peru, Vilfredo Pareto found that the straight line that fitted plotting income against the number of people had a particular slope, γ 3/2, which was consistent with much wealth concentrated in very few hands (Mandelbrot and Hudson, 2004) Another well-known phenomenon that has been documented as approximating a power-law with γ 3/2 pertains to geology, and is related to the magnitude of earthquakes and their frequency. Based on his findings Mandelbrot would contend the Brownian motion assumption with its generalized version (i.e. fractional Brownian motion), and would suggest changing from the Gaussian hypothesis for price changes to the stable Paretian or stable Lévy hypothesis, in which the exponent that determines the height of the tails (γ) is the most important for comparing the goodness-of-fit against the traditional Gaussian hypothesis (Fama, 1963).
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