Chaos Theory Case Study

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Abstract
While most traditional science deals with supposedly predictable phenomena like gravity, electricity, or chemical reactions, Chaos Theory deals with nonlinear things that are effectively impossible to predict or control, like turbulence, weather, the stock market, our brain states, and so on. It focuses on non-randomness, nonlinearity and chaotic characteristics. In recent times such nonlinear dynamics and chaotic dynamics have augmented in the field of financial analysis. This paper studies the extent to which the daily return data from the Indian Stock Exchange Indices (Nifty & Sensex) exhibit these non-linear, non-random characteristics. The Hurst exponent in Rescaled range analysis rejects the hypothesis that the index return series
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Arguments on whether deterministic structure appears in the economic and financial series are being continued in recent years. Up to now, most empirical findings are contradictory. Scheinkman and LeBaron (1989) report the evidence of determinism in weekly stock returns. Willey Non-random and Chaotic Behavior of Chinese Equities Markets (1992) finds the deterministic characteristic appears in both daily S&P 100 Index returns and NASDAQ Index returns. On the other hand, Howe et al. (1997) find no evidence of deterministic patterns in Australia and Hong Kong equity returns.
3. Research Methodologies
3.1. Rescaled range (R/S) analysis
Whether the stock price movement follows a random walk or not can be detected by the rescaled range analysis or R/S analysis. The R/S analysis is an ideal statistical tool for analyzing the occurrence of rare events and is robust to possible nonlinear process that normality assumption may not be needed. The result of the R/S analysis is the Hurst exponent, which is a measure of the bias or trend in a time series.
The dataset is portioned into a sequential non-overlapping blocks, as , where, A is the number of partition, N is the amount of data in the series and n is the amount of data in each partition. The data in each block is . The mean of for the block of data is defined as
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This hypothesis testing uses the test statistics in which mechanism is based on the correlation integrals. The BDS test is a powerful tool for detecting serial dependence in time series. It tests the null hypothesis of independent and identically distributed (I.I.D.) against an unspecified alternative. The null and alternative hypothesis is as follows:
H0:The data are independently and identically distributed (I.I.D.).
H1: The data are not I.I.D.; this implies that there may be some serial dependence. If the linear dependence has been removed in the time series, the serial dependence is thus nonlinear.
However, BDS test is unable to distinguish between nonlinear deterministic chaos and nonlinear stochastic systems. BDS test cannot test chaos directly but only nonlinearity, provided that any linear dependence has been removed from the data (e.g. using traditional ARIMA-type models or taking a first difference of natural logarithms). Nevertheless, since a nonlinear process is one of the indications of chaos, we may use the BDS test to detect such indication. The BDS Test Statistics is defined by (13)
This test will be repeated at different values of ‘ε’ and ‘m’. Lin (1997) suggested that the appropriate values of range from 0.5 to 2, Brock et al. (1987) pointed out the appropriate values of m are between 2 and

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