رفتار پر هرج و مرج در شاخص های بازار سهام ملی: شواهد جدید از آزمون بازده نزدیک

Attempts have been made to detect chaotic behaviour in financial markets data using techniques which require large, clean data sets. Although such data are common in the physical sciences where these tests were developed, financial returns data typically do not conform. The close returns test is a recent innovation in the literature and is better suited to testing for chaos in financial markets. This paper tests for the presence of chaos in a wide range of major national stock market indices using the close returns test. The results indicate that the data are not chaotic, although considerable nonlinearities are present. The commonly used BDS test is also applied to the data and, in comparison, the close returns test provides substantially more evidence of nonlinearity compared to the BDS test.

The discovery, that deterministic nonlinear equations could generate data which appear random, provided a major breakthrough in the way scientists viewed a wide range of physical processes and natural phenomena. While by no means a complete list, chaos has been identified in hydrodynamic turbulence, lasers, electrical circuits, chemical reactions, disease epidemics, biological reactions, and climatic change.1 Buoyed by the research of their physical science counterparts, financial market researchers have attempted to establish whether the apparently random nature of asset prices and economic time series could also be explained by the presence of chaotic behaviour. One of the most commonly applied tests for nonlinearity is the BDS test of Brock, Dechert, and Scheinkman (1987), details of which may be found in Dechert (1996). Subsequent to its introduction, the BDS test has been generalised by Savit and Green (1991) and Wu, Savit, and Brock (1993) and more recently, DeLima (1998) introduced an iterative version of the BDS test. The BDS test is a statistical test of the null hypothesis of IID and is based on the Grassberger and Procacia (1983) correlation integral. As such, the BDS procedure may be considered as a test for linear and nonlinear departures from IID rather than a specific test for chaos. It is in this latter context however, that the test has most commonly been applied usually in conjunction with the estimation of entropy, Lyapunov exponents, or correlation dimensions. The BDS test has been used to test for nonlinear behaviour in a wide range of financial data including national stock market indices (see Abhyankar et al., 1995, Abhyankar et al., 1997, Ahmed et al., 1996, Barkoulas & Travlos, 1998, Hsieh, 1991, Mayfield & Mizrach, 1992, Olmeda & Perez, 1995, Philippatos et al., 1993, Scheinkman & LeBaron, 1989, Sewell et al., 1996 and Willey, 1992), exchange rates (see Cecen & Erkal, 1996, Chiarella et al., 1994, Hsieh, 1989, Serletis & Gogas, 1997 and Vassilicos et al., 1992), futures data (see Chwee, 1998, Eldridge & Coleman, 1993, Kodres & Papell, 1991 and Vaidyanathan & Krehbiel, 1992), and commodity prices (see Frank & Stengos, 1988 and Kohzadi & Boyd, 1995). In general, the BDS test results furnished by this literature provide substantial empirical evidence of nonlinear structure in a wide range of financial asset prices. The BDS test belongs to the metric invariant class of tests for chaotic behaviour, which were developed for application in the physical sciences where long, clean data sets are the norm. In finance, however, small noisy data sets are more common and the application of the BDS test to such data presents a number of problems (a discussion of which is presented in Section 2). A recent development in the literature has been the introduction of the close returns test, which is a topological invariant testing procedure.2 Compared to the existing metric class of testing procedures including the BDS test, the close returns test is better suited to testing for chaos in financial and economic time series. Despite these advantages, in comparison to the BDS test, the close returns test has largely been overlooked in the finance literature. Gilmore, 1993a and Gilmore, 1993b applied the close returns test to weekly CRSP data sampled over the period 1962–1989 and the results suggest the presence of nonlinear structure in the data, which was not chaotic. US macroeconomic, treasury bill, and exchange rate data were also considered with similar results. Gilmore (1996) extended the earlier analysis of the CRSP data set and found that linear filters and GARCH modelling did not fully capture all of the structure in the data.3Gilmore (2000) used the close returns test to test for the presence of chaos in daily foreign exchange returns and while no evidence of chaos was found, the data did exhibit nonlinearity. The purpose of this article is to apply the close returns test for low-dimensional chaos to daily data sampled from 12 national stock market indices. To allow comparisons to be made, the BDS test will also be applied to these data. Thus, this study will augment the evidence on nonlinearity in stock markets provided by Gilmore, 1993a, Gilmore, 1993b and Gilmore, 1996 for CRSP data to encompass a wider range of markets. The application of the BDS test to the same data will allow comparisons to be drawn between the results derived using a topological and a metric method. Further, as some of the markets included in this study have been tested previously in the literature using the BDS methodology, comparisons may also be drawn to these prior results. For example, Willey (1992) examines daily S&P 100 and NASDAC index returns sampled over the period 1982–1988 using the BDS statistics and concludes “… no underlying dependence, chaotic or otherwise, is present in the data” (p. 72). In contrast, Sewell et al. (1996) reject the null of IID for stock market indices for Japan, Hong Kong, Singapore, and the S&P 500 using weekly data sampled over the period 1980–1994. Abhyankar et al. (1997) strongly reject IID behaviour in real-time returns data for the S&P 500, NIKKEI, DAX, and FTSE sampled over the period September–November 1991. The rest of this paper proceeds as follows. 2 and 3 introduce and discuss the limitations and prospects of the BDS and close returns test, respectively. Section 4 formally introduces the close returns test procedure for low-dimensional chaos. Section 5 discusses the data to be tested and presents the estimation results. The BDS test is initially applied to the data followed by the close returns test and comparisons drawn. Finally, Section 6 presents some concluding comments and discusses areas for further research.

Until recently, financial market researchers were ill equipped to detect the presence of chaos. The most commonly used nonlinear testing procedure was the BDS test, which is poorly suited for application to the small, noisy data sets common in finance. The introduction of the close returns test procedure for chaotic behaviour, however, has provided researchers with an exciting new tool for detecting chaos in finance data. This paper applied the BDS test procedure to 12 national stock market indices and the results suggest the presence of nonlinearities for all indices except the Australian market index. The close returns testing procedure was also applied to these data and the results furnished strong evidence of nonlinearity although it was not found to exhibit sensitive dependence on initial conditions, i.e., was not chaotic. In comparison to the BDS test results, the close returns test procedure provided much stronger evidence of nonlinearity in the data. The results provided by the BDS and close returns testing procedures indicate the presence of nonchaotic nonlinear behaviour in stock markets. Based on this evidence, future research may best be spent establishing on how best to explain and model such nonlinearities.