Chaos and Nonlinear Dynamics: Application to Financial Markets

After the stock market crash of October 19, 1987, interest in nonlinear dynamics, especially deterministic chaotic dynamics, has increased in both the financial press and the academic literature. This has come about because the frequency of large moves in stock markets is greater than would be expected under a normal distribution. There are a number of possible explanations. A popular one is that the stock market is governed by chaotic dynamics. What exactly is chaos and how is it related to nonlinear dynamics? How does one detect chaos? Is there chaos in financial markets? Are there other explanations of the movements of financial prices other than chaos? The purpose of this paper is to explore these issues.     

Nonlinearity and Low Deterministic Chaotic Behavior in Insurance Portfolio Stock Returns

This article presents new empirical evidence indicating a deterministic component in the portfolio return dynamics of life‐health and property‐liability insurance company stocks. Our research is motivated by the fact that nonlinearities are a fact of economic life for many financial applications the source of which is logically apparent, yet empirical evidence of their existence is at best weak. The primary reason attributed to the weak findings of nonlinearities reported in previous research is the use of aggregate data that can hide nonlinearities at the micro level. Insurance sector stock returns are analyzed because unique institutional characteristics indicate the possibility of identifying nonlinear dynamics. Tests based on the correlation dimension partially confirm the presence of nonlinearity. However, the more powerful Brock, Dechert, and Scheinkman (BDS) statistic strongly suggests the presence of nonlinearities in the insurance stock portfolio data. The BDS statistic applied to the standardized residuals of exponential generalized auto regressive conditional heteroskedasticity (EGARCH) models strongly rejects the null of independent and identically distributed, indicating that conditional heteroskedasticity is not responsible for the presence of the nonlinear structures in the data. In addition, tests for chaos based on locally weighted regressions indicate that insurance stock portfolio returns indicate low‐complexity chaotic behavior. This is an important result since most previous research has failed to report evidence of chaotic behavior in the time series of stock returns. Important contributions of this article are the application of tests of nonlinearities and chaos to more desegregated data sets and the findings of statistically significant evidence indicating nonlinearities and low‐deterministic chaotic behavior in insurance stock portfolio returns.     

Contagion effect in financial markets after the South-East Asia Tsunami

Powerful earthquakes may cause heavy damage to the financial markets of individual countries (regions), and may even spillover to other countries (regions). Using 26 international stock indexes and exchange rates, this study examines whether any contagion effect occurred across financial markets after the strong earthquake in South-East Asia on December 26, 2004. Using heteroscedasticity biases based on correlation coefficients to examine the existence of the contagion effect, this study shows that no individual country stock market suffered from the contagion effect, but that the foreign exchange markets of some countries (namely India, Philippines and Hong Kong) did suffer from the contagion effect.     

NONLINEAR DYNAMICS AND THE DISTRIBUTION OF DAILY STOCK INDEX RETURNS

Three alternative models of daily stock index returns are considered: (1) a diffusion-jump process; (2) an extended generalized autoregressive conditional heteroskedasticity (GARCH) process; and (3) a combination of the GARCH and jump processes. Non-nested tests between the diffusion-jump process and a GARCH(1.1) process with t-distributed errors reject the diffusion-jump process, but do not always reject the GARCH process. Kolmogorov-Smirnov tests of fit, however, reject the GARCH(1,1)-t process for all cases. Nonlinear dependence is not removed for the value-weighted index and the S&P 500 stock index; therefore, deterministic chaos cannot be dismissed.     

Multiscale analysis of economic time series by scale-dependent Lyapunov exponent

Economic time series usually exhibit complex behavior such as nonlinearity, fractal long-memory, and non-stationarity. Recently, considerable efforts have been made to detect chaos and fractal long-memory in finance. While evidence supporting fractal scaling in finance has been accumulating, it is now generally thought that financial time series may not be modeled by chaos or noisy chaos, since the estimated Lyapunov exponent (LE) is negative. A negative LE amounts to a negative Kolmogorov entropy, and thus implies simple regular dynamics of the economy. This is at odds with the general observation that the economy is highly complicated due to nonlinear and stochastic interactions among component systems and hierarchical regulations in the world economy. To resolve this dilemma, and to provide an effective means of characterizing fractal long-memory properties in non-stationary economic time series, we employ a multiscale complexity measure, the scale-dependent Lyapunov exponent (SDLE), to characterize economic time series. SDLE cannot only unambiguously distinguish low-dimensional chaos from noise, but also detect high-dimensional and intermittent chaos, as well as effectively deal with non-stationarity. With SDLE, we are able to show that the reported negative LE may correspond to large-scale convergence, but not imply the absence of small-scale divergence or noisy chaos in the world economy. Using US foreign exchange rate data as examples, we further show how SDLE can readily characterize fractal, persistent or anti-persistent long-range correlations in economic time series.     

Stock market performance and manufacturing capability of the fifth long-cycle industries

Nearly 80 years ago Russian economist, Kondratieff, introduced the theory of economic long-cycles. Since from the start, this theory has faced controversial acceptance; for example, in the future studies researchers have used it to develop further specific applications, but in economics some leading scientists reject the entire idea still. Although, this theory is well developed, there does not exist research from the examination of relation between stock market performance, and leading innovation cycle industries manufacturing capacity addition and utilization. Based on the system dynamics model, called world dynamics, capacity addition and utilization have earlier been identified as the leading indicators of long-cycles.Our research results in this paper indicate that capacity utilization of computer manufacturing in US, and in some cases of US semiconductors, has influence on the stock market indexes of Nasdaq, S&P500 and Dow. However, it should be noted that capacity investment changes of these three examined industries (semiconductors, computers and telecommunications) are involved in the proposed regression models too. Further analysis reveals, that we are able to build regression models for all three stock indexes, containing only two variables. Notably, these two variables are capacity addition change in semiconductors and computers. This observation further increases discussion, whether we should be interested only about capacity addition changes of innovation wave industries, and possibly give secondary importance for the utilization.     

On the long-term or short-term dependence in stock prices: Evidence from international stock markets

This study examines the short- and long-term dependence in the United States and 21 international equity market indexes. Two heteroscedastic-robust testing methods, the modified rescaled range analysis and the rescaled variance ratio test, are employed to test for the existence of dependence. The evidence consistently reveals the absence of long-term dependence in these 22 stock returns indexes. The random walk hypothesis for most, but not all, stock returns indexes is not rejected. When the random walk hypothesis is rejected, the evidence supporting the rejection is weak and the stochastic dependence occurs mainly in short-horizon, rather then long-horizon holding period returns.     

The search for evidence of chaos in FTSE-100 daily returns

This study provides an elementary discussion of deterministic chaos as it applies to security returns. the study demonstrates a simple technique, well known in the physical sciences, for discriminating between random and chaotic time-series. Applying the technique to a time-series of daily returns on the FTSE-100, an index comprised of the stocks of the 100 largest British firms, results in evidence that the time-series is random, not chaotic.     

Stock market crashes,firm characteristics,and stock returns

A number of studies have investigated the causes and effects of stock market crashes. These studies mainly focus on the factors leading to a crash and on the volatility and co-movements of stock market indexes during and after the crash. However, how a stock market crash affects individual stocks and if stocks with different financial characteristics are affected differently in a stock market crash is an issue that has not received sufficient attention. In this paper, we study this issue by using data for eight major stock market crashes that have taken place during the December 31, 1962–December 31, 2007 period with a large sample of US firms. We use the event-study methodology and multivariate regression analysis to study the determinants of stock returns in stock market crashes.     

How does Monetary Policy Influence Capital Markets? Using a Threshold Regression Model

This study indicates that the effects of interest rate changes on stock prices could be twofold and that the net effect is determined by which effect is dominant. The study employs a threshold regression model to see if, before and after the central banks cut the interest rates, there is a nonlinear relation between interest rates and the stock index. Based on traditional economic theory, stock prices should be inversely related to interest rates. However, the present study finds that as interest rates start to increase or decrease, the stock index prices are significantly and positively related to the interest rates. The changes in interest rates affect stock indexes inversely only after interest rates have crossed a certain threshold. The inverse U-shaped relationship between interest rates and stock indexes differs from the traditional wisdom. It could make interest rates more valuable in forecasting stock indexes, and it holds implications for monetary policies of central banks. To avoid the spurious regression problem, this study uses a cointegration test and an error correction model to confirm the results from the threshold regression model and finds that there is a significant cointegration relationship before and after central banks cut interest rates.