Detecting Macroeconomic Chaos Juan D. Montoro & Jose V. Paz Department of Applied Economics, Umversidad de Valencia,

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1 Detecting Macroeconomic Chaos Juan D. Montoro & Jose V. Paz Department of Applied Economics, Umversidad de Valencia, Abstract As an alternative to the metric approach, two graphical tests (close returns and histogram) are implemented in Mathematica for detecting and visualising chaotic behaviour in macroeconomic time series. We show its ability to distinguish chaos over different simulated dynamic systems including a multisector kaldorian economic model. Finally, using them against some monetary aggregates we found no evidence of chaos in the Divisia M2 series, in contrast with the results obtained in several previous works. 1 Introduction The aggregated economic activity (observed through macroeconomic time series) shows aperiodic fluctuations over time known as business cycles. Deterministic nonlinear dynamic systems allow us to generate endogenous business cycles, with some of them being able to exhibit a kind of behaviour that resembles a random process: chaotic motion or just chaos. The properties of chaotic systems provide a quite plausible alternative to the dominant 5Wz&?/-Fn,sc/% paradigm (based on linear systems) that stresses the role of exogenous random shocks to explain the fluctuations of economic variables. While the stochastic approach emphasizes the equilibrium characteristics of the economic system (with the economy moving to the steady state in the absence of external perturbations), nonlinear and chaotic models stress their instabilities. We follow with an example in which business cycles are generated by an unstable investment demand.

2 354 Innovation In Mathematics 1.1 A multisector chaotic model Following Lorenz [1], we consider a kaldorian economy with n different sectors: Yi = ai(lf(yi,yj,ki)-si(yi)) i^j (1) Ki = Ii(Yi,KJ-6iKi (2) with YI being the total product, Ki the capital stock, If is the investment demand (equal to the internal demand /% plus the rest of the sectors demand), Si total savings, 0,1 the adjustment parameter, ^ the depreciation rate, and i l,2...n. Consider a sigmoidal investment function of the type: and a linear savings function: 3 %! (3) Si(Yi) = aft (4) Lorenz showed that for certain values of the adjustment parameter a, each sector may exhibits limit cycle behaviour; when coupling between sectors is introduced (b^ ^ 0), the system may exhibit chaotic motion. Figure 1 shows the numerically integrated system in Mathematica, using the following parametric set: n=3; <% = 5.0; & = 0.05; 62 = ; 63 = ; ^ = 10-5; ^ = 0.29; ^ = 5.0; q = 25.0; ^ = 0.015; e^ = 0.05; ^ = 320.0; ^ = 3.0; 612 = 0.011; 613 = 0.015; 633 = 0.012; 621 = 631 = 632 = 0. 2 Testing for chaos with Mathematical the close returns test If business cycle may be generated by chaotic processes in the variables involved, the complex dynamics of the system will appear under the procedures developed to detect chaos. During last decade economists have tried to verify the presence of chaos in economic time series (specifically in financial and macroeconomic series). Nevertheless the results of the search for chaos has found several methodological difficulties as the tests applied, directly imported from the natural sciences, have proved to be of limited validity when using economic data. Focusing our attention to macroeconomic time series, we find three objections to the indiscriminate application of these tests to macroeconomic data:

3 Innovation In Mathematics 355 Y2 Yl Figure 1: System (l)-(3): strange attractor in the onto the Y\-Ki space (right). space (left); projection 1. The limited length of the available series: in the best case we manage several hundreds of observations in contrast to the amount of experimental data that can be generated in the laboratory. 2. The presence of noise as a constitutive part of the processes analysed; this may be aggravated in the aggregation process of the macroeconomic series (some authors consider that this not only increases the level of noise in the resulting series but it also may destroy the evidence of chaotic signal in any of the individual series). 3. Finally the fact that many of the economic time series are nonstationary may yield to invalid results when applying some test to the unaltered data. By the other hand the classical solution to this problem, by applying linear filters, may increase the amount of noise with the problems that this procedure may imply as mentioned above. The metric approach for detecting chaos, although highly contested in its application to economic data, has been the main method employed by the researchers working on this topic. As an alternative, the topologic approach analyses the organization of the strange attractor: in this paper we follow the initial work of Gilmore [2] in this field, applying the close returns test to economic series. This is the first step in the topological analysis and searches for unstable aperiodic orbits in the attractor, allowing the researcher to detect if a time series exhibits chaotic behaviour. The main advantages of this method are: 1. It is robust even in the presence of noise in the series. In the case of a high noise to signal ratio, it is posible to filter away the noise and obtain the characteristic graph defining the process.

4 356 Innovation In Mathematics 2. Moreover, it is possible to apply the method with a reduced set of data; we find that the method gives reliable enough results in series consisting of 300 observations. For shorter series the coded graph does not show enough definition. 3. The method detects the presence of unit roots; this, in contrast to the metric approach, makes it suitable for analysing macroeconomic time series. The mentioned properties make this a good method, as afirstapproach, for testing against chaos in macroeconomic series, as it overcomes the problems found in the metric tests when applied to economic data. 2.1 Implementing the test The close returns test is a graphic test that tries to identify the presence of recurrences in the data by means of the search of patterns in the close returns graph. Consider a chaotic time series Xi\ the method searches for unstable aperiodic cycles. In doing so we compare and code the distance between observation pairs: 0(e-.%;-Z(+r ) with Z = 1,2...7V and T = l,2... (5) with 0 the heaviside function. The results of (5) are plotted in a graph where the horizontal axis are the ordered observations of Xt, and the vertical axis shows the period r. Close returns are indicated by the areas of horizontal line segments in the graph. A chaotic series will show a number of these segments as unstable periodic orbits are present on it. By contrast, a stochastic one will generate a blurred pattern in which no structure may be inferred. In addition to the close returns graph the close returns histogram is defined as: -Xt+r\) (6) and it summarizes the ocurrences of close returns for different values of r. We implemented both graphics in Mathematica taking advantage of its graphics programming features. The package Returns. m defines the functions CloseReturns [e,x] and Histogram [e,x] ; these implement the graphic tests decribed above to the time series x comparing each pair of observations at a distance e = e*(max[x]-min[x] ). In terms of efficiency, these functions were fast enough for limited data sets (as those found in macroeconomic series of not more than 500/600 observations at best): in a PowerMacintosh 6100/66 with 16 Mb of RAM and running Mathematica v2.2, it took 74.6 seconds to calculate the matrix and render the graph for a 400 observations time series; this timing was increased up to seconds when a 600 data set was used. Obviously this implies an exponential increase in time when increasing the size of the sample. Certainly, even in

5 Innovation In Mathematics 357 the case of economic data, we find longer series (in financial data, as daily exchange rates or continuous stock market variables, that may imply thousands of observations); for these cases a C external function implemented via Mathlink has been developed. While the speed gains are important in the histogram (taking less than a second for a 900 observations series in contrast to the seconds that involves the same calculation using the Mathematica package), these are limited in the case of the close returns graph (for example, with the 600 time series, it took seconds to render the graph). We suspect that the exponential growth in the matrix that returns the function (n x n) in addition to the limited availability of memory in our system, could be in the origin of these results. 3 Some examples 3.1 Application to simulated data We follow with some example applications of the close returns test. In order to show its ability to detect complex behaviour we have simulated with Mathematica different dynamic systems: first we have simulated a stochastic variable that follows a 7V(0,1) distribution; second we obtained the data corresponding to Y\ in the system defined by equations (l)-(3), with the specification used in figure 1. Applying the test to both series we obtained the results as per figure 2. The graph and the histogram allow to infer the dynamics involved in the generation of the processes analysed. In the case of the random variable no pattern is recognize in the graph; on the other hand, the chaotic variable displays a clear structure in its close returns with some horizontal segments being clearly defined. The histogram supplies the researcher with the same information so it can be used instead of the graph: in the case of the stochastic data no isolated peaks seem to be significant, while this is evident for the chaotic series. With these results we will apply the test to empirical data. 3.2 Application to monetary aggregates Barnett and Chen [3], Chen [4] and DeCoster and Mitchell [5], have obtained some interesting conclusions regarding the possibility of chaos in some monetary aggregates for the U.S. economy. By using the metric approach these works conclude that the dynamics of of them were chaotic. We will use the test described in the previous sections in order to validate or pose our objections to their results. We will apply it to the Divisia monetary aggregate M2 (U.S. monthly data from 1960 to 1992) Elaborated by Thornton and

6 358 Innovation In Mathematics t Observation t! r\\ v \V \ v & Observation H(t) t H(t) t Figure 2: Close returns graph (above) and histogram (below) for a stochastic (left) and chaotic series (right). Yue (data are available in the International Divisia Database, maintained at the University of Mississippi). In order to filter the original index two usual techniques are employed: taking first differences of the log series and detrending the series according to the following expression: where St is the original series and xt is the detrended one. As the above authors used one (or both) of these methods, we will replicate them in our test. In this way, after filtering the Divisia M2, we get two series. The results of the test are shown in figure 3. From both graphics inspection no evidence of chaos is found. Furthermore, the structure that seems to appear in the detrended series is that of a unit root process, and, as already mentioned, applying metric methods

7 Innovation In Mathematics x:%.::##&# 150# <#;^m# : ;--^j,r>.;-. $ -g- ^. ^, _.-> ;-, '? JM tioo t Observation Observation H(t) t t Figure 3: Close returns graph (above) and histogram (below) for demand Divisia M2 first (log) differences (left) and detrended series (right). to nonstationary processes may yield inaccurate results. Based on this evidence, this (quiet inexpensive) graphical test may be used before proceeding with other tests that may be more time/computing-power consuming. 4 Conclusions A Mathematica package, Returns.m, implementing the clos.e returns and histogram tests for detecting the presence of chaos in economic time series is presented. These graphical tests take full advantage of Mathematical power and graphics capabilities to identify recurrences in the data. For long series the computing time may be large, and for this reason an installable (via Mathlink) C program (returns. c and returns, tm) has been developed. We illustrate the use of the package analyzing simulated (stochastic and

8 360 Innovation In Mathematics chaotic series) and empirical data (monetary aggregate Divisia M2). In the latter no signs of chaos was found in the close return graphs, in contrast to previous works that use the metric approach. Furthermore, this test revealed the existence of a unit root process that may invalidate the former results. Based on this evidence, we could say that this quite inexpensive graphical test: robust even in the presence of noise, that does need very much observations, and prevents us against nonstationarity is higly suitable as a first approach for macroeconomic series. References 1. Lorenz, H.W. Strange attractors in a multisector business cycle model, o/ Economic Be/muiowr orw On;omzof%07z, 1987, 8, Gilmore, C.G. A new test for chaos, Journal of Economic Behaviour Orgomzoh'on, 1993, 22, Barnett, W.A. & Chen, P. The aggregation- theoretic monetary aggregates are chaotic and have strange attractors: an econometric application of mathematical chaos in Dynamic Econometric Modelling, Proceedmf/s o/ Z/te JW /nzerna^otw 5?/mpo,smm on.eyxmormc 27%eory and Econometrics (ed. W.A. Barnett, E. Berndt & H. White), pp Cambridge, 1988, Cambridge University Press. 4. Chen, P. Searching for economic chaos: a challenge to econometric practice and nonlinear tests, in Nonlinear Dynamics and Evolutionary Econoymcg (ed. Day R.H. Day & P. Chen), pp , Oxford, 1991, Oxford University Press. 5. DeCoster, G.P. & Mitchell, D.W. Nonlinear monetary dynamics, Journal of Business and Economic Statistics, 1991, 9,