# A case study of speculative financial bubbles in the South African stock market 2003-2006

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## Abstract

We tested 45 indices and common stocks traded in the South African stock market for the possible existence of a bubble over the period from Jan. 2003 to May 2006. A bubble is defined by a faster-than-exponential acceleration with significant log-periodic oscillations. The faster-than-exponential acceleration characteristics are tested with several different metrics, including nonlinearity on the logarithm of the price and power law fits. The log-periodic properties are investigated in detail using the first-order log-periodic power-law (LPPL) formula, the parametric detrending method, the $(H,q)$-analysis, and the second-order Weierstrass-type model, resulting in a consistent and robust estimation of the fundamental angular log-frequency $\omega_1 =7\pm 2$, in reasonable agreement with previous estimations on many other bubbles in developed and developing markets. Sensitivity tests of the estimated critical times and of the angular log-frequency are performed by varying the first date and the last date of the stock price time series. These tests show that the estimated parameters are robust. With the insight of 6 additional month of data since the analysis was performed, we observe that many of the stocks on the South Africa market experienced an abrupt drop mid-June 2006, which is compatible with the predicted $t_c$ for several of the stocks, but not all. This suggests that the mini-crash that occurred around mid-June of 2006 was only a partial correction, which has resumed into a renewed bubbly acceleration bound to end some times in 2007, similarly to what happened on the S&P500 US market from Oct. 1997 to Aug. 1998.

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