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Volatility, fitting with first-order Landau expansion, stationarity, and causality of the Taiwan stock market (TAIEX) are investigated based on daily records. Instead of consensuses that consider stock market index change as a random time series we propose the market change as a dual time series consists of the index and the corresponding volume. Therefore, causalities between these two time series are investigated. Our results suggest the volume time series is of second-order importance than the index time series. The index time series receives slightly stronger influence from the previous 67th trading day, while the volume time series is slightly stronger influenced by the previous 62nd trading day.

Physicists are interested in studying stock markets as complex systems. Almost all questions asked can be summarized as searching for price formation theories. Previous researches shown the distribution of price change has pronounced tail distribution in contrast to Gaussian distribution expected. Furthermore, the auto-correlation of price change decays exponentially with a characteristic time scale around View the MathML source. Stock crash or rally have also been identified by physicists as a kind of herd behaviour [1]. Johansen and Sornette considered fitting most stock markets for the bubbles using Landau expansions of the index [2]. They showed evidence that market crashes as well as large corrections are preceded by speculative bubbles with two main characteristics: a power-law acceleration of the market price decorated with log-periodic oscillations. For most markets the log frequency ω/2π is close to unity. However, most data analysis were done for the index of more mature market like S&P 500. Not all market indexes are defined in the same way. Could different weighting methods reach the same conclusion? Will government intervention play a role in the conclusion? Johansen and Sornette further found emergent markets have larger fluctuations. They extended the expansion up to third order and successfully predicted Nikkei raise in the year 2000 [3]. However, why do these fluctuations exist? The log-periodic oscillation appeared in a wide class of out of equilibrium dynamic systems, like ruptures in heterogeneous media, historic analysis of earthquakes data, and world population. Canessa tried to establish universality for the exponents from a renormalization group theory [4], and used a stochastic theory to show that the log periodicities are a consequence of transient clusters introduced by an entropy-like term. As entropy in thermodynamics corresponds to the information in an information theory. The possibility to arrive at the log-periodic oscillation therefore suggests the log-periodic oscillation is a consequence of information exchange between different species of a large system [5]. The effect of volume is less analyzed in the literature. Volume is a measure of market liquidity while index means the price. Gopikrishnan et al. [6] analyzed the statistical properties of number of shares traded of a particular stock at a given time interval from an empirical rule saying that it takes volume to push the index. Bonanno et al. [7] also analyzed the number of shares traded of selected stocks and find a power spectrum of approximately 1/f. In the present work we considered the volume effects of Taiwan stock market (TAIEX). In particular, the cause-effect relation between the volume time series and the index time series is analyzed. Taiwan stock market is one of the largest emerging markets. Johansen and Sornette did not studied it because of availability of trading information. In the present work we tried to make up this missing piece. When people talk about the volume involved, sometimes they mean the number of shares traded, sometimes the amount of money involved. We consider money flow to be more important than share number flow if the whole market is considered, since the person or institute involved should have fixed amount of money.

We tried to find the correlation between the volume of transactions involved and stock market fluctuations beyond simply considering temporal price series, using Taiwan stock market as an example in the present work. We find TAIEX volatility spread relatively uniformly in the past 10 years, which can also be found in the stationarity test. The distribution of price change shows fat-heel behavior. The Landau expansion gives us confidence for the future. Our results on causality suggest all studies using only index time history can be viewed as a first-order approximation. If accurate modeling is necessary concerning the time series of the stock market dual-time series model could be considered. A slightly higher influence coefficient is found in the index difference fitting with volume. The time lag is 62 trading days. Johansen and Sornette's result suggests us to try regression with log-periodic functional forms in the future. However, it does not seem meaningful to go beyond 90 days for the time lag, since companies will have their seasonal reports by that time.