Empirical and theoretical evidence of economic chaos

Empirical and theoretical investigations of chaotic phenomena in macroeconomic systems are presented. Basic issues and techniques in testing economic aggregate movements are discussed. Evidence of low dimensional strange attractors is found in several empirical monetary aggregates. A continuous time deterministic model with delayed feedback is proposed to describe the monetary growth. Phase transition from periodic to chaotic motion occurs in the model. The model offers an explanation of the multiperiodicity and irregularity in business cycles and of the low-dimensionality of chaotic monetary attractors. Implications in monetary control policy and a new approach to forecasting business cycles are suggested. System Dynamics Review Evidence of Economic Chaos 2 In recent years, there has been rapid progress in the studies of deterministic chaos, random behavior generated by deterministic systems with low dimensionality. This progress has been made not only in theoretical modelling, but also in experimental testing [Abraham , Gollub, and Swinney 1984]. Chaotic models have been applied to a variety of dynamic phenomena in the areas of fluid dynamics, optics, chemistry, climate and neurobiology. Applications to economic theory have also been developed, especially in business cycle theory [see review article: Grandmont and Malgrange 1986]. Over the last century, the nature of business cycles has been one of the most important issues in economic theory [Zarnowitz 1985]. Business cycles have several puzzling features. They have elements of a continuing wave-like movement; they are partially erratic and at the same time serially correlated. More than one periodicity has been identified in business cycles in addition to long growth trends. Most simplified models in macroeconomics address one of these features [Rau 1974] , while system dynamics models describe economic movements in terms of a large number of variables [Forrester 1977]. Two basic questions arise in studies of business cycles. Are endogenous mechanism or exogenous stochastics the main cause of economic fluctuations? And can complex phenomena be characterized by mathematical models as simple as, say, those for planetary motion and electricity? The early deterministic approach to business cycles with well-defined periodicity mainly discussed the endogenous mechanism of economic movements. A linear deterministic model was first proposed by Samuelson [1939], which generated damped or explosive cycles. Nonlinearities were introduced in terms of limit cycles to explain the self-sustained wavelike movement in economics [Goodwin 1951]. A stochastic approach seems to be convenient for describing the fluctuating behavior in economic systems [Osborne 1959; Lucas 1981]. The problem with the stochastic models, however, lies in the fact that random noise with finite delay terms (usually less than ten lags, in practice) only explains the short term fluctuating behavior. Most aggregate economic data are serially correlated not only in the short term but also over System Dynamics Review Evidence of Economic Chaos 3 long periods. Two methods dealing with long correlations are often used: longer lags in regression studies and multiple differencing time series in ARIMA models. Longer lags require estimating more "free parameters", while ARIMA models are essentially whitening processes that wipe out useful information about deterministic mechanism. Actually, fluctuations may be caused by both intrinsic mechanism and external shocks. An alternative to the stochastic approach with a large number of variables and parameters, is deterministic chaos, with few variables or low-dimensional strange attractors [Schuster 1984]. This is the approach adopted in the present article. Newly developed numerical techniques of nonlinear dynamics also shed light on a reasonable choice of the number of variables needed in characterizing a complex system. An increasing number of works examine economic chaos. Most theoretical models are based on discrete time [Benhabib 1980; Stutzer 1980; Day 1982; Grandmont 1985; Deneckere and Pelikan 1986; Samuelson 1986], only one long wave model is based on continuous time [Rasmussen et al 1985]. On-going empirical studies are conducted by a few economists [Sayers 1986; Brock 1986; Scheinkman and Le Baron 1987; Ramsey and Yuan 1987; Frank and Stengos 1987]. Some clues of nonlinearities have been reported, but no solid evidence of chaos has yet been found by these authors. Two efforts were made to fit nonlinear discrete models with empirical data [Dana and Malgrange 1984; Candela and Gardini 1986], but the parameters were found outside the chaotic regions. We started search for empirical evidence of chaos in economic time series in 1984. The main features of deterministic chaos, such as complex patterns of phase portraits and positive Lyapunov exponents, have been found in many economic aggregate data such as GNP and IPP, but most of our studies have failed to identify the dimensionality of attractors because of limited data. Then we tested monetary aggregates at the suggestion of W. A. Barnett. Low-dimensional strange attractors from weekly data were found in 1985, and a theoretical model of low-dimensional monetary attractors was developed in 1986 [Chen 1987]. A brief description of comprehensive studies of economic chaos is presented here for general readers. System Dynamics Review Evidence of Economic Chaos 4 In this article, a short comparison between stochastic and deterministic models is introduced. Positive evidence of low-dimensional strange attractors found in monetary aggregates is shown by a variety of techniques. A continuous-time model is suggested to describe the delayed feedback system in monetary growth. The period-doubling route to chaos occurs in the model [Feigenbaum 1978]. The model offers an explanation for the low dimensionality of chaotic monetary time series and for the nature of business cycles and long waves. Finally, the implications of deterministic chaos in economics and econometrics are discussed. Simple pictures of deterministic and stochastic processes To what extent economic fluctuations around trends should be attributed to endogenous mechanism (described by deterministic chaos) or exogenous shocks (described by stochastic noise) is a question that can be addressed by empirical tests. There are at least four possible candidates in describing fluctuating time series: linear stochastic process, discrete deterministic chaos, continuous deterministic chaos, and nonlinear deterministic chaos plus noise. The test of the last one is only in its infancy, because a high level of noise will easily destroy the subtle signal of deterministic chaos. We mainly discuss the first three candidates here and give numerical examples of white noise and deterministic chaos as the background for further discussions. The linear autoregressive AR(2) model adopted in explaining the fluctuations of log linear detrended GNP time series [Brock 1986] is demonstrated as an example of a linear stochastic process. For deterministic chaos, two models are chosen: the discrete logistic model [May 1976], which is widely used in population studies and economics, and the continuous spiral chaos model [Rossler 1976]. The time sequences of these models are shown in Figure 1. They seem to be equally capable to describe economic fluctuations when appropriate scales are used to match real time series. But closer examination reveals the differences among them. System Dynamics Review Evidence of Economic Chaos 5 Phase space and phase portrait From a given time series X(t), an m-dimensional vector V(m,T) in phase space can be constructed by the m-history with time delay T: V(m,T) = {X(t), X(t+T), . . , X[t+(m1)T] }, where m is the embedding dimension of phase space [Takens 1981]. This is a powerful tool in developing numerical algorithms of nonlinear dynamics, since it is much easier to observe only one variable to analyze a complex system. The phase portrait in two-dimensional phase space X(t+T) versus X(t) gives clear picture of the underlying dynamics of a time series. With the fixed point solution (the socalled zero-dimensional attractor), the dynamical system is represented by only one point in the phase portrait. For periodic solution (the one-dimensional attractor), its portrait is a System Dynamics Review Evidence of Economic Chaos 6 closed loop. Figure 2 displays the phase portrait of the three models. The nearly uniform cloud of points in Figure 2a closely resembles the phase portrait of random noise (with infinite degree of freedom). The curved image in Figure 2b is characteristic of the onedimensional unimodal discrete chaos. The spiral pattern in Figure 2c is typical of a strange attractor whose dimensionality is not an integer. Its wandering orbit differs from periodic cycles. Long-term autocorrelations The autocorrelation function is another useful concept in analyzing time series. The autocorrelation function AC(I) is defined by System Dynamics Review Evidence of Economic Chaos 7 AC(I) = AC(t'-t) = cov [ X(t'), X(t) ] / E [ (X(t) M)2] (1) Where M is the mean of X(t) and cov[ X(t'), X(t) ] is the covariance between X(t') and X(t). They are given by