# Chaos and Order in the Capital Markets: A New View of Cycles, Prices, and Market Volatility.

NONLINEAR DYNAMICS is one of the significant developments applied to
economic theory during the past decade. What is needed is a translation
of the theoretical work into applications that can be utilized by the
economist concerned with empirical issues. Edgar Peters' book
outlines the theoretical foundations of nonlinear dynamics, provides
empirical evidence from the capital markets supporting nonlinear
dynamics, and discusses the implications of nonlinear dynamics to the
traditional capital market analysis tools such as the Capital Asset
Pricing Model (CAPM).

Chaos theory is a subset of nonlinear dynamics. Nonlinear dynamics refers to the concept that a certain system is governed by nonlinear parameters. Nonlinear systems can be stochastic. Chaos theory arose when researchers discovered that a completely deterministic system can produce a time path nearly indistinguishable from a random time series. Even though the process is exactly governed by a set of deterministic equations, the time path may not be predictable. Chaos by James Gleick is an introduction to chaos theory presented in a readable format.

Formally, deterministic chaos is defined with two characteristics:

1. The time path of the system never returns to the same exact value.

2. Slight changes in the initial conditions lead to widely divergent time paths.

Weather and economic systems may display these two characteristics. Long-range forecasting of chaotic systems is impossible because in no two periods are the conditions ever quite equivalent, and slight variations in the starting conditions will result in completely different growth paths after a period of time. The difficulty of generating accurate forecasts has plagued both economists and climatologists.

Economists are interested in forecasting future conditions, and, more importantly, economists are interested in using the proper tools to gain an understanding of the current conditions and the limits of future forecasts. Edgar Peters provides a description of the quantitative techniques used in nonlinear analysis. He then applies these tools to the analysis of stock markets, interest rates, currency markets, and business cycles. Peters' quantitative analysis indicates that movements in these economic variables follow nonperiodic cycles. The average S&P 500 cycle as measured by Peters is around forty-eight months. The cycles are not exact; the cycles are statistical in nature. A single S&P 500 cycle may be shorter or longer than forty-eight months. Because the cycle is nonperiodic, linear ARIMA models and spectral analysis will not discern the cycle parameters. Technical analysis may not be productive. Conditional heteroscedasticity models, such as the ARCH and GARCH specifications, are better suited to the analysis of nonperiodic cycles.

Nonperiodic cycles in economic variables leads to the conclusion that past values of the economic series have some value for predicting future values; the autocorrelation function is not zero. The existence of nonperiodic cycles contradicts the efficient market hypothesis. The efficient market hypothesis is the foundation for many of the standard asset pricing models including the CAPM, Arbitrage Pricing Theory (APT), and the Black-Scholes option pricing model.

Edgar Peters explores the implications of nonlinear analysis to the beta values of the CAPM in Chapter 8. Peters suggests that the Hurst statistic used in nonlinear analysis may be applied to generate risk measures of individual investments. Portfolio diversification does reduce risk according to both the Hurst statistic and the beta of the CAPM; however Peters suggests that the Hurst statistic may be more valuable than the beta in assessing the risk of individual stocks. Peters does not argue with the development of the quantitative models mentioned earlier; he submits that the assumptions of a normal distribution and no serial correlation may be too restrictive. Peters is not intent on destroying the past forty years of work in economics and finance. In fact, Peters' work builds on the statistical analysis of asset and commodity prices showing that the distribution of price changes is leptokurtotic; the distribution has "fat tails." Some of this same analysis has indirectly led to the development of alternative asset pricing models, such as APT. Peters contends that the simplifying assumptions of the standard asset pricing models ignore much of the information embedded in time series data. Peters indicates that finance theory has greatly benefited from the efficient market hypothesis, and it is time to extend pricing analysis beyond the efficient market hypothesis to an understanding of the processes underlying market dynamics.

Section three of the book explains some of the nonlinear empirical techniques that are currently used to analyze economic time series. One of the most revealing is the scrambling test or "shuffle diagnostics," where an economic time series is scrambled to remove any time-dependent characteristics. The same statistics are then applied to the time-ordered data set and to the shuffled data set. Any time-dependent characteristics of the data will be highlighted by differences in the statistics. Shuffle diagnostics will work with linear methods and need not be restricted to nonlinear analysis.

The final chapters present some alternative models that may explain the nonperiodic cycles in economic time series. Serial correlation is an important aspect of the proposed models. Alternative market models are reviewed in the spring 1990 issue of Economic Perspectives.

The book includes a glossary and complete bibliography for those wishing to consult the original articles. Some of the BASIC computer code used in the nonlinear analysis is included in the appendixes to the text. The computer programs included in the appendix are interesting, but the appendix could be more complete. The text relies heavily on the calculation of the Hurst statistic, and the associated computer code is not included in the appendix. The code included in the appendix is either very simple (bifurcation diagrams from the logistic equation and generation of a biased random walk), or deals with complex routines (estimation of correlation integrals and the Wolf algorithm to estimate the Lyanpunov exponent). My suggestion to readers is to use another program when calculating correlation integrals, and to consult the original article by Wolf to estimate the Lyanpunov exponent. The Wolf article contains the program code in FORTRAN (Wolf, A. et al, "Determining Lyapunov Exponents from a Time Series," Physica 16D, July 1985). My attempts to obtain additional information of the program code used in the Peters' text or to talk with the author were ignored by the publisher, John Wiley.

One of the best DOS system programs available to calculate the correlation integral and BDS statistics can be obtained from W. Davis Dechert (Department of Economics, University of Houston, Houston, Texas, 77030, 713-743-3800). The MacIntosh or IBM version can be obtained from MIT-press for $15 (617-253-3172), written by Blake LeBaron at the University of Wisconsin. The Dechert program is well written and very well optimized. Although I could duplicate the results of the Dechert program, I could not duplicate the speed with my own optimized program code.

Chaos and Order in the Capital Markets will be useful to economists and finance professionals who are interested in the practical economic implications of nonlinear dynamics. Peters' book is designed to provide an overview of the field and to stimulate thought. The book is written in an informal style without rigorous mathematical proofs, and, although economic material does not allow for a bestseller writing style, I did find the book hard to put down once I began reading the text.

Thomas A. Noll Idaho Power Corporation

Chaos theory is a subset of nonlinear dynamics. Nonlinear dynamics refers to the concept that a certain system is governed by nonlinear parameters. Nonlinear systems can be stochastic. Chaos theory arose when researchers discovered that a completely deterministic system can produce a time path nearly indistinguishable from a random time series. Even though the process is exactly governed by a set of deterministic equations, the time path may not be predictable. Chaos by James Gleick is an introduction to chaos theory presented in a readable format.

Formally, deterministic chaos is defined with two characteristics:

1. The time path of the system never returns to the same exact value.

2. Slight changes in the initial conditions lead to widely divergent time paths.

Weather and economic systems may display these two characteristics. Long-range forecasting of chaotic systems is impossible because in no two periods are the conditions ever quite equivalent, and slight variations in the starting conditions will result in completely different growth paths after a period of time. The difficulty of generating accurate forecasts has plagued both economists and climatologists.

Economists are interested in forecasting future conditions, and, more importantly, economists are interested in using the proper tools to gain an understanding of the current conditions and the limits of future forecasts. Edgar Peters provides a description of the quantitative techniques used in nonlinear analysis. He then applies these tools to the analysis of stock markets, interest rates, currency markets, and business cycles. Peters' quantitative analysis indicates that movements in these economic variables follow nonperiodic cycles. The average S&P 500 cycle as measured by Peters is around forty-eight months. The cycles are not exact; the cycles are statistical in nature. A single S&P 500 cycle may be shorter or longer than forty-eight months. Because the cycle is nonperiodic, linear ARIMA models and spectral analysis will not discern the cycle parameters. Technical analysis may not be productive. Conditional heteroscedasticity models, such as the ARCH and GARCH specifications, are better suited to the analysis of nonperiodic cycles.

Nonperiodic cycles in economic variables leads to the conclusion that past values of the economic series have some value for predicting future values; the autocorrelation function is not zero. The existence of nonperiodic cycles contradicts the efficient market hypothesis. The efficient market hypothesis is the foundation for many of the standard asset pricing models including the CAPM, Arbitrage Pricing Theory (APT), and the Black-Scholes option pricing model.

Edgar Peters explores the implications of nonlinear analysis to the beta values of the CAPM in Chapter 8. Peters suggests that the Hurst statistic used in nonlinear analysis may be applied to generate risk measures of individual investments. Portfolio diversification does reduce risk according to both the Hurst statistic and the beta of the CAPM; however Peters suggests that the Hurst statistic may be more valuable than the beta in assessing the risk of individual stocks. Peters does not argue with the development of the quantitative models mentioned earlier; he submits that the assumptions of a normal distribution and no serial correlation may be too restrictive. Peters is not intent on destroying the past forty years of work in economics and finance. In fact, Peters' work builds on the statistical analysis of asset and commodity prices showing that the distribution of price changes is leptokurtotic; the distribution has "fat tails." Some of this same analysis has indirectly led to the development of alternative asset pricing models, such as APT. Peters contends that the simplifying assumptions of the standard asset pricing models ignore much of the information embedded in time series data. Peters indicates that finance theory has greatly benefited from the efficient market hypothesis, and it is time to extend pricing analysis beyond the efficient market hypothesis to an understanding of the processes underlying market dynamics.

Section three of the book explains some of the nonlinear empirical techniques that are currently used to analyze economic time series. One of the most revealing is the scrambling test or "shuffle diagnostics," where an economic time series is scrambled to remove any time-dependent characteristics. The same statistics are then applied to the time-ordered data set and to the shuffled data set. Any time-dependent characteristics of the data will be highlighted by differences in the statistics. Shuffle diagnostics will work with linear methods and need not be restricted to nonlinear analysis.

The final chapters present some alternative models that may explain the nonperiodic cycles in economic time series. Serial correlation is an important aspect of the proposed models. Alternative market models are reviewed in the spring 1990 issue of Economic Perspectives.

The book includes a glossary and complete bibliography for those wishing to consult the original articles. Some of the BASIC computer code used in the nonlinear analysis is included in the appendixes to the text. The computer programs included in the appendix are interesting, but the appendix could be more complete. The text relies heavily on the calculation of the Hurst statistic, and the associated computer code is not included in the appendix. The code included in the appendix is either very simple (bifurcation diagrams from the logistic equation and generation of a biased random walk), or deals with complex routines (estimation of correlation integrals and the Wolf algorithm to estimate the Lyanpunov exponent). My suggestion to readers is to use another program when calculating correlation integrals, and to consult the original article by Wolf to estimate the Lyanpunov exponent. The Wolf article contains the program code in FORTRAN (Wolf, A. et al, "Determining Lyapunov Exponents from a Time Series," Physica 16D, July 1985). My attempts to obtain additional information of the program code used in the Peters' text or to talk with the author were ignored by the publisher, John Wiley.

One of the best DOS system programs available to calculate the correlation integral and BDS statistics can be obtained from W. Davis Dechert (Department of Economics, University of Houston, Houston, Texas, 77030, 713-743-3800). The MacIntosh or IBM version can be obtained from MIT-press for $15 (617-253-3172), written by Blake LeBaron at the University of Wisconsin. The Dechert program is well written and very well optimized. Although I could duplicate the results of the Dechert program, I could not duplicate the speed with my own optimized program code.

Chaos and Order in the Capital Markets will be useful to economists and finance professionals who are interested in the practical economic implications of nonlinear dynamics. Peters' book is designed to provide an overview of the field and to stimulate thought. The book is written in an informal style without rigorous mathematical proofs, and, although economic material does not allow for a bestseller writing style, I did find the book hard to put down once I began reading the text.

Thomas A. Noll Idaho Power Corporation

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Author: | Noll, Thomas A. |
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Publication: | Business Economics |

Article Type: | Book Review |

Date: | Apr 1, 1993 |

Words: | 1206 |

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