An Investigation of the Chaos Complexity Theory in the Cost of Bahar Azadi Gold Coin in Iran

The savings becomes to invest in the capital market and then import into the production cycle and helps to the development and growth of countries. However, inefficient capital markets, cause savings to flow into real assets. Gold is a real asset, liquidity with high strength, and a suitable replacement for money. This wealth is a booming market in Iran. Fluctuations in the price of gold in addition to the influence of other markets can also affect other markets. Therefore, it is important for the state and the people to understand the trend in the price of gold and gold coins. The gold price forecast will help policymakers to make the right decisions. On the other hand, it is difficult and complicated to accurately predict the real variables. We need to recognize the structural nature is predictable pattern. In this article, chaos theory was used to identify the structural nature of the time series of Bahar Azadi gold coin.
Theoretical Framework: Chaos theory analysis of the systems that have non-linear relationships and irregular time series. Economic time series variables follow a stochastic process and thus are not predictable. However, the series are not random, and are expected in the short term. There are tests for chaos in time series, such as correlation dimension, BDS, and Lyapunov exponent maximum test. Results of the study by Kim et al. (2003) showed that the BDS test is more efficient than other tests.
For the purpose of this study, the non-linearity of the BDS test, and the Lyapunov exponent maximum test of the chaotic time series were used. BDS test was conducted in three stages: the original data, the residual of ARIMA, and the residual of GARCH. To determine the structure of time series of Bahar Azadi gold coin, 1670 observation was divided into 8 groups of the two hundred. Null hypothesis test is the IID and independent data. The Lyapunov exponent maximum test check on all data. Positive values of the statistics indicated the existence of chaos in the system. R and MATLAB software were used for data analysis.
First, the stationary data were checked. Dickey-Fuller test the null hypothesis is accepted, which implies the existence of a unit root. The first stage of BDS test was performed on the original data in the dimensions inscribed. The results showed that the null hypothesis was rejected, except the first group. As a result, the original data were not IID, and linear or non-linear dependence exists between them. Before the second phase of the test, the appropriate ARIMA model was selected. The unit root test was performed on the residual of ARIMA, and the null hypothesis was rejected. As a result, BDS test was conducted on the residual ARIMA. In the third stage, first the variance heterogeneity was checked, white test the null hypothesis is rejected, thus confirming the heterogeneity of variance. Then, the existence of ARCH effect was checked. ARCH effect in the first five groups, GARCH effect in the next three tests by Ljung-Box and LM-ARCH was confirmed. According to the BDS test conducted on the residual of GARCH, the null hypothesis was rejected, which residual IID, and linear and nonlinear dependence does not exist, thus confirming the process of chaotic time series data structure of Bahar Azadi gold coin. Wolfe algorithm was used in this study to calculate the Lyapunov exponent maximum test. The results showed that the Lyapunov exponent was small and positive for all aspects and intervals.
As a result, time series of Bahar Azadi gold coin is possessed of a chaotic process. So we can predict future prices with the non-linear model in this series.