35

Alison Butler

Al/eon Butler is an economist at the Federal Reserve Bank ofSt. Louis. Lora Ho/man provided research assistance. Theauthor would like to thank William Barnett and Jennifer E/lisfor helpful comments and suggestions.

A Methodological Approach toChaos: Are Economists Missingthe Point?

“A very slight cause which escapes our notice determines a considerableeffect which we cannot fail to see, and then we say that this effect isdue to chance.”

—Poincar~

HERE IS INCREASING interest among econ-omists in a new field of study that may offeran alternative explanation for the seeminglyrandom behavior of many economic variables.This research, which originated in the physicaland biological sciences, concerns a phenomenoncalled deterministic chaos.’

Contrary to the common usage of the word,chaos in this context describes the behavior of avariable over time which appears to follow noapparent pattern but in fact is completely deter-ministic, that is, each value of the variable overtime can be predicted exactly. In fact, one“chaologist” describes chaos as “... lawlessbehavior governed entirely by law.”

To demonstrate the difficulty in determiningwhether a variable is random or chaotic, figuresla and lb show two time series of a variable;one series is a random variable, whose actualvalue cannot be known with certainty, and the

other is a chaotic variable, whose value can bepredicted with certainty. Even the most prac-ticed observer, however, would have difficultydetermining which of these series, if any, is notrandom. As a result, most economists wouldmodel or estimate both time series as randomprocesses. The chaotic series is described by avery simple deterministic equation and identifiedlater in this paper.

Often, behavior that cannot be explained bystandard theories and modeling techniques is at-tributed to random forces, even when there isno theoretical reason to do so. This paper arguesthat economists are perhaps not using the ap-propriate types of models and empirical techni-ques to explain the behavior of some economicvariables and that the choice of methodologyneeds to be more closely examined.

The study of chaos is a recent phenomenon inthe biological and physical sciences and is just

1The terms “deterministic chaos” and “chaos” are used in-terchangeably here, although deterministic chaos is themore precise description.

2Stewart (1989), page 17.

35

now beginning to be applied to economics. Un-fortunately, many of the empirical tests forchaos are imprecise and, because of mathemati-cal constraints, the theoretical models used togenerate chaos are generally limited to systemswith only one or two explanatory variables.Both of these factors restrict the usefulness ofapplying chaos to economic systems. Neverthe-less, the theory of deterministic chaos has at-tracted a great deal of attention, both in thepopular press and in academic circles. The dis-cussion that follows attempts to clarify some ofthe issues and suggests some ways to incorporatechaos into economics.

This article first reviews how economic vari-ables typically are modeled by describing andevaluating several techniques of economic model-ing using a simple model of output and popula-tion growth.3 Next, chaos is defined and its pro-perties demonstrated. The advantages and pit-falls of applying the theories of chaos to eco-nomics are then discussed and illustrated,

ECONOMIC MODELING

There are many different ways to buildeconomic models. Four such possibilities are ex-amined here for the case in which all variablesare completely deterministic.~The types of mod-els examined here are static linear, static non-linear, dynamic linear and dynamic nonlinear. Afurther distinction, which proves to be signifi-cant, is also drawn between discrete and contin-uous time dynamic models. A simple model ofoutput, where labor is the only input, is used toillustrate each approach to modeling as well asthe restrictiveness of many common modelingtechniques. In addition, focusing on economicmodeling allows us to show that chaotic dynam-ics can only arise in certain types of modelsthat have often been excluded, a priori, byeconomists.

Static MAdeIsThe simplest type of economic model is a

static linear model, in which variables do not

change over time and are related in a propor-tionate manner. Consider, for example, thefollowing simple production function, which hasonly labor as an input:

(l)Y=AN A>0,

where Y is output, which is completely consumedby workers (there is no saving or investment), Nis labor employed and A is the productivityparameter. This equation states that output ispositively related to the amount of labor em-ployed. Given the value of A and the labor sup-ply, the exact value of output can be determined.

This type of model is highly restrictive; anychange in labor changes output by a constantpercentage. Hence, the production function ex-hibits constant returns to scale.

Allowing the model to be nonlinear (that is,not necessarily proportionate) provides a moregeneral model in which equation 1 is a specialcase. An example of a nonlinear productionfunction is given by:

(2) Y = AN° A > 0, a> 0.

If a = 1, this model is identical to the oneshown in equation I - By not restricting a toequal one, however, this model can be used toexamine the case in which output can varydisproportionately with respect to changes inlabor. This is illustrated in figure 2, whichshows the relationship between output and laborfor different values of a (for simplicity, A = 1).Notice that if a is between zero and one, theproduction function exhibits decreasing returnsto scale (that is, output increases less than pro-portionately with respect to a change in labor);if a is greater than one, production is character-ized by increasing returns to scale (output in-creases more than proportionately with respectto an increase in labor). Empirical tests of actualproduction relationships can be performed todetermine if a is actually different from orequal to one.

‘There is a growing body of theoretical literature incor-porating chaos into many different types of economicmodels. These models include Benhabib and Day (1981),Deneckere and Pelikan (1986), Grandmont (1985), DeGrauwe and Vansanten (1990), Kelsey (1988), Day andShafer (1985) and Stutzer (1980). For surveys of thetheoretical literature, see Kelsey (1988) and Baumol andBenhabib (1989).

ness used in econometric models into the theoretical litera-ture. Recent papers also look at the properties of chaos inthe presence of a random component see, for example,Kelsey (1988)]. For simplicity, this paper focuses only onpurely deterministic systems.

‘There is also a burgeoning field in stochastic (random)modeling, which incorporates the assumption of random-

39

Figure 2Linear vs. Nonlinear ProductionFunctionsy=ANaV3

2

0

tinear Dv.nanne MAdeis

One disadvantage of these static models is thatthey can be used to describe the relationshipbetween output and employment only if thelabor force or population remains constant overtime.’ Suppose instead that we want to examinethe behavior of output over time as it is relatedto a continuously changing labor force. A stan-dard equation borrowed from Haavelmo (1954),used to describe the growth of the labor forcewhere that growth is dependent on the level ofoutput, is given by:

(3) N(t)IN(t) = C —DN(t)IY(t) C > 0, D > 0,

where C and U are constants, and a dot over avariable means the change in the variable withrespect to a very smali change in time. Thistype of equation is called a differential equation.

Equation 3 states that the percentage rate ofchange of the labor force IN(t)/N(t)I, where timeis continuously changing, equals the differencebetween the rate of birth, C, and the rate ofdeath, given by DN(t)IY(t), where N(t)IY(t) is thenumber of individuals who have to subsist oneach good at time t.

Using the linear production function given inequation I and substituting it into equation 3provides a linear specification of the percentagechange in the population:

(4) N(t)IN(t) = C—D/A.

Notice that when the production function islinear the rate of death, DfA, is constant.

Solving equation 4 yields the following solu-tion for the population:

(5) N(t) = Ke(C-DIAh,

N where K is an arbitrary constant.6

This solution has the property that, unless therate of birth (C) is ezcactly equal to the rate ofdeath (D/A)—in which case the population willequal K—the population will either rise exponen-tially or fali to zero. This result is highly restric-tive, however, because the likelihood of eitherthe two rates being identical or the populationincreasing infinitely is, in reality, very small. Inother words if C U/A, the system is unstable.’Unfortunately, in models of other types of eco-nomic variables, results that greatly restrict thepossible values of the parameters of the modelsare not uncommon. In addition, because of thecomplexity of many economic models, the im-plications of restricting the value of the parame-ters to determine the solution or to ensure a

‘For expositional ease, the terms “population” and “laborforce” are used interchangeably.‘Equation 4 is solved by the variable separable method ofsolving differential equations found in most calculus books.K is the constant of integration, which can be determinedby choosing an initial condition.

‘In fact, stability is an important issue which is frequentlyignored or abstracted from in economics. Stability is impor-tant because, for example, an unstable equilibrium is not asustainable equilibrium. Stability is also important in thechoice between linear and nonlinear models. Linear models

have three possible cases: stable converging dynamics(such as when C = D/A in the model above), unstabledynamics (when C * D/A) and cyclical dynamics, which isthe least common of the three. In nonlinear models, how-ever, cyclical dynamics are far more common, and ex-ploding dynamics may not occur. Thus, it is also importantto consider the desirable and realistic stability propertieswhen choosing a model. Obviously the nonlinear case ismore general and the most realistic for variables that ex-hibit cyclical variation. For the purpose of this paper,however, the issue of stability is ignored.

0 I 2 3

stable solution are not always as obvious as inthe population growth model. Because linear dif-ferential equations are far simpler to solve thannonlinear differential equations, and becausetheir solutions are more often stable and easierto interpret, however, they are used in econom-ic models more often than may be appropriate.

\imIfr,p~r~ IJvnam.ie Models

Combining the nonlinear production functiongiven by equation 2 with the description ofpopulation growth given in equation 3 providesa less restrictive model of population growth:

(6) N(t)IN(t) = C — DN(t)’~fA.

Unlike equation 4, equation 6 allows the laborforce to vary more or less as the current laborforce changes. Unfortunately, the price of thegenerality provided by such nonlinear differen-tial equations is that most either cannot be solvedor have solutions so complex the results cannotbe interpreted. Not surprisingly, economistsoften avoid these types of models.

The model used here, however, was chosen forits tractability and can be solved for the value oflabor at any time t.’ All that is necessary for astable solution is that the production function ex-hibits decreasing returns to scale (0 < a -C 1).Regardless of the value of the other parameters,if a is between zero and one, the populationwill reach a stable equilibrium level. Hence, incontrast to the dynamic linear model discussedpreviously, the results of this model are morerealistic and provide a more general descriptionof population and output growth.

.th.serete .Modets

One problem with using continuous timemodels in economics is that data are availableonly in distinct intervals (daily, weekly, monthly,etc.). One approach typically taken by econom-ists, therefore, is to convert these continuoustime models into discrete time models. Discrete

dynamic models are called difference equations;they measure time in distinct intervals ratherthan the d4’ferential equations used above, whichmeasure time continuously. Equation 6 can betransformed into a difference equation by let-ting the rate of change of N (previously givenby N) equal the difference between the value ofN at time t and t + I. Thus, equation 6 becomes

(7) (N~÷,—N~)IN1

= C — D(N,IY,),

where Y, = AN:.

Combining these equations and simplifying theresult yields:

(8) N,~,= N, [(1+C) — DN,-~/A]

which, following Stutzer (t980), can be rewrittenusing a change of variables as

(9) X~.÷,= k x,(1—x~’°),

where k = 1 + C.9

The models shown in equations 8 and 9 de-scribe the most general specification of popula-tion and output growth given the assumptionsmade above. Behavior is not restricted to beinglinear, nor is population or output restricted toremaining constant over time. On the otherhand, as noted earlier, these more generalmodels often cannot be solved or have solutionswithout any economic interpretation. Neverthe-less, unless there are theoretical reasons forassuming relationships are static or linear,dynamic nonlinear models, which provide themost general specification of behavior, should atleast be considered in economic analysis. Al-though generality for its own sake is not a de-sirable goal, using a more general model wouldbe appropriate when simpler models have solu-tions that are highly unrealistic or when param-eters have to be restricted beyond reason (as inthe model presented here). In addition, if eco-nomists want to test their models and resultsempirically, then these variables should bemodeled in the form in which they are esti-mated—discrete form.b0 As it turns out, thesetypes of nonlinear dynamic models can exhibitchaos.

aFor the solution and discussion of this model, seeHaavelmo (1954), pp. 24-29.

9Allowing N, [A(1 +C)ID]”-” X, only changes the scaleof the population and has no effect on the general charac-teristics of the solution. For further discussion of this pro-cedure and the solution, see Stutzer (1980).

since in discrete time this model can generate chaoticdynamics for certain parameter values. Differential equa-tions can also exhibit chaos, although only in more com-plicated models. This is discussed in greater detail later.

‘°Althoughthe model given by equation 6 is stable in con-tinuous time, it is not necessarily stable in discrete time

IN~TRO]flLJf71Tit) IT) THEJfl;))fl)( OF (I~.H.r4.OS

The possibility that chaos exists in economicvariables has strong implications for the way inwhich economics is modeled. For example, somevariables that appear to be random processes,like one of the variables shown in figure 1,

might in fact not be random at all; instead itmight be completely explained using the ap-propriate deterministic model. This sectiondemonstrates the properties of chaos, using asimple model.

In the most general sense, the term chaos isused to describe the behavior of a variable overtime that appears random but, in fact, is deter-ministic; more precisely, given the initial valueof the variable, all future values of the variablecan be calculated with exact precision.” In con-trast, the value of a random variable can neverbe predicted with certainty.

More formally, a function is chaotic if, forcertain parameter values, the following two con-ditions hold: First, the function never reachesthe same point twice under any defined intervalof time. In this case, the function is said to ex-hibit aperiodic behavior. Second, the time pathis sensitive to changes in the initial condition, sothat a small change in the value of the initialcondition will greatly alter the time path of thefunction.”

Chaos only arises in certain types of nonlineardynamic systems, although not all nonlinear dy-namic equations are chaotic. Moreover, equa-tions that can be characterized as chaotic neednot exhibit chaos for all parameter values. Rath-er, functions that can exhibit chaos will do soonly for certain parameter values. This is ex-plained by example below.

And j3iow for Soniethint;E..7Ainphi.etv tEll rent

The properties of chaos can be demonstratedusing a simple mathematical equation, called thelogistic growth equation. While this model hasno particular economic interpretation, it is the

Figure 3The Logistic Growth Curve ForVarious Values of k

= kX~(1 — Xt)

simplest model that exhibits chaos and providesreasonably interpretable graphical results. Thisequation is given by:

(10) X,~,= k X~(I—X,),0 < X, < 1,0 -C k < 4.

Equation 10 describes the time path of a vari-able, X (which for expositional purposes is calleda population), that is a function of its previousvalue and a parameter k. To demonstrate chaoticbehavior in this simple framework, the value ofX can only take on values between zero and one.The value of k, the only parameter in the equa-tion, is called the ‘tuning” parameter; it deter-mines the steepness of the function. Figure 3shows the function given in equation 10 forvarious values of k. Increases in the popula-tion below X increase future values of X morethan proportionately. Past this point, the popula-tion begins to decrease.” For larger values of k,

“For simplicity. only single-variable equations are discussed.Although chaos exists in multivariate economic systems,tests for chaos in these systems are just beginning to bedeveloped, and the mathematics of such systems are ex-tremely complex.

12Thei-e are many different characterizations of deterministicchaos, but they all include the one used here. For morerigorous definitions and discussion of the different defini-

tions, see Li and Yorke (1975), Brock and Dechert (1988)and Melese and Transue (1986). For a good mathematicaldescription of chaos and the mathematical tools used inthe theory of chaos, see Devaney (1989).

“This behavior is similar to that of a total product curvewhere, once the marginal product becomes negative, fur-ther increases in an input decreases output.

0.0 0.5 1.0I xt

42

Figure 4A Stable Time Path for aLogistic Growth CurveX~÷1= 3Xt(1 — Xt)X0 = .20 t=1 to 500

S

0:5

0.0

the absolute value of the rate of change of X islarger.

For certain values of the tuning parameter(k 3), the system is stable; this means thepopulation will reach some sustainable steady-state value which differs from X.

Figure 4 illustrates how the time path for X,÷,is solved graphically. The parabola representsequation 10 when k is equal to three; all valuesof X, and X,~must lie on this curve. The 45-degree line depicts the points where X,~1= X,,

which is required for a steady-state equilibrium.In this example the initial value (when t = 0) is.20. To determine the value of X1, draw a linebetween the initial value (X0) and the parabola(line segment X0B). To find the value of X,, set

= X by drawing a line from point B to the45-degree line (point C) - Then draw a straightline from point C to the parabola. This is thevalue of X, (point D). This process, called itera-tion, can be used to determine as many subse-quent values of X as is desired, once the initialvalue is determined.14 As we can see in figure 4,

14Notice that X,~,,which must always lie on the parabola,can be either above the 45-degree line (as in point D) orbelow it (as in point F). For precision, the equation issolved numerically and then graphed.

xt+l

1.0

0.0 Xo X0.5 X~X~ 1.0xt

43

the population appears to be converging to asteady-state equilibrium value at 2/3 (X).

If the value of k increases past three, how-ever, the equilibrium point becomes unstableand the time path exhibits a two-period cycle,where the variable alternates between twovalues. Further increases in k produce a four-period cycle (that is, the time path repeats thesame sequence of numbers every fifth iteration),then an eight-period cycle, and so on, with theperiodicity increasing by z~(n = 1,2,3, . .). If kincreases past a certain point called the “pointof accumulation” (for this function, it occurs atk = 3.5700), the time path enters into a regionin which the function can exhibit chaos,” In thechaotic region (3.57 < k < 4 for this function),there can be both an infinite number of periodiccycles and an infinite number of initial condi-tions that produce an aperiodic time path.16 Us-ing this simple example, we can demonstratesome of the properties of chaos in graphicalform.

Properties of Chaos

An example of aperiodic behavior is seen infigure 5. The first 500 iterations are shown inthis figure (that is, t = 1, 2, - . - 500), and nosingle point is ever reached twice.17 In fact, nomatter how many times this equation is iterated,

never has the same value twice.18 If the dataare plotted as a time series, it would look similarto figure la, the chaotic series in figure 1, andone might conclude that the data are generatedby a random process, such as figure lb. becausethey follow no obvious pattern. This is not thecase here; the data in figure la and figure 5were generated from models without a randomcomponent and therefore are completelydeterministic.

The other characteristic of a chaotic functionis that its time path is sensitive to the choice ofinitial values. An example of how changing the

Figure 5A Logistic Growth CurveExhibiting ChaosX~÷1= &82840 Xt( 1 — Xt)X0 = .0101 t=1 to 500

1.oj / all

ICinitial condition can affect the time path isshown in figure 6. In this figure, the values ofX are plotted against time, as in figure 1. Thisdiagram demonstrates how changing the initialvalue, X,,, at the fourth decimal place (from.0101 to .0100) causes the time paths generatedby equation 10 to deviate substantially fromeach other.” Although not all sections of thetime path differ as dramatically as the oneshown here, figure 5 graphically demonstratesthat the choice of an initial condition or, forforecasting purposes, the choice of a time inter-val (that is, determining where to start the sam-ple), can greatly alter the results. In fact, despite

I 1St

“This process of increasingly complex periodicity is calledbifurcation and is discussed in most papers on chaos. Fora nontechnical discussion of bifurcation, see Gleick (1987)and Stewart (1989). For a more analytical treatment ofbifurcation, see May (1976) and Baumol and Benhabib(1989). A more rigorous discussion of the relationship be-tween bifurcation and chaos is given in Li and Yorke(1975).

“Notice that not every initial condition gives rise to anaperiodic time path.

icity is required for chaos, tests for chaos in actual datatake a different approach, thus avoiding the problem.

“The time path can avoid having repeat values because thenumber of possible points between zero and one is infinite.

“Although it sometimes looks like the function is periodic,this appearance is a result of the lack of precision of theprinter and the scale of the graph. In fact, there are noperiodic points in this function.

r ‘“““St

05

o.o~~2t—

0.5 I .0xt

I7This property is unlikely to be found in actual data, how-ever, because of rounding. Although theoretically, aperiod-

44

Figure 6A Segment of the Time Trend Showing Sensitivityto Initial ConditionsX~+i=3.82840Xt(1— Xt)

X0=.01o1 X0=AtiOO

xt1.0

0.8

0.6

0.4

0.2

0.0600

seemingly trivial differences in the initial condi-tions, the time path produced by one initial valuewill not necessarily be similar to the time pathgenerated by a marginally different initial value.In general, the two time paths that arise from thedifferent initial values wifi have periods duringwhich they are arbitrarily close together andperiods during which they deviate substantially.

Chaotic functions also exhibit sensitivity tovery small changes in the parameter values. Athird. or fourth-order change in the value of aparameter can alter the time path from stableto chaotic or vice versa.

Sensitivity to changes in the parameter valuesis illustrated in figure 7. Here, a fifth-orderchange in the value of the tuning parameter(from 3.82840 to 3.82844) produces not only asubstantially different time path from the one infigure 5, but also one that exhibits periodicrather than chaotic behavior.20

Are Attractors Strange?

Another feature often found in chaos,although neither necessary or sufficient forchaos, is a strange attractor.2’ The properties ofattractors and strange attractors are best illus-

20flecall that when a function is in a chaotic region (that is,when the parameters are such that the function can ex-hibit chaos), there can be both periodic and aperiodic timepaths.

21The only examples of chaos without the presence of astrange attractor are found in certain types of dissipativesystems.

610 620 630 640I:

650

43

Figure 7A Logistic Growth Curve WithPeriodic PointsXt.j=3.82844Xt(1 — Xt)

= .0101 t=1 to 500

xt+1

//

/ N/

/ .

II //

trated by example. In a stable system, the timepath converges to an equilibrium point (for ex-ample, X’ in figure 4). The equilibrium point isalso called the attractor, because the time pathis “attracted” to the equilibrium point. Anotherpossibility is that the time path has two attract-ors, and the system oscillates between them,never remaining at one equilibrium point. Thisis found in predator/prey population models,where the population grows until it is so largeit begins to die off and then shrinks to a levelso small it begins to grow again.

A “strange attractor” is the name given to thecase where there is a region, rather than afinite set of points, that attracts the time pathof the variable. That is, after some number ofiterations, which varies depending on the func-tion, the time path of the variable is completelycontained in this region (the strange attractor).Thus, even though the path is aperiodic and

Figure 8The Strange Attractor for aChaotic Function

= 3.82840X~(1— Xt)X0 = .0101 t=1 to 1500

xt.I

1.0

0-s

0.0

0.0

therefore never reaches an equilibrium in thestandard sense, it also never leaves the strangeattractor and therefore is not unstable (for ex-ample, never goes to positive or negative infini-ty). An example of this is shown in figure 8,which takes the same numerical example as infigure 5, but iterates it 1500 rather than 500times. In this picture, the values of X are stillcontained in the same area as in figure 5, butthe distribution of points is becoming denser.The bounded region (shown by the dotted linein figure 8) is the strange attractor for thisfunction, If the function is iterated further, thearea within the bounded region would appearto be a solid block, although the function wouldnever have the same value twice. In fact, theexistence of a strange attractor is an importantway to distinguish between a random andchaotic time path.22

22Another definition of a strange attractor is an attractor withfractal dimension. In fact, if a strange attractor exists, thevariable has fractal dimension. Random variables have in-finite dimension, however. As a result, tests for dimensionare one of the main ways data are tested to determine if

they are chaotic. For the purpose of this paper, however,the issue of fractals and fractal dimension will be ignored.For a discussion of these topics, see Mandelbrot (1983)and Gleick (1987).

1.0

0.5

/

/

0.00.0 0.5

xt1 .0

0.5 1.0

XI

LESSONS •EROM P11505

Although economists are beginning to incor-porate chaos into their economic and econo-metric models, there has been little discussionof the ways in which chaotic dynamics are use-ful or realistic for economic models. Clearly,chaos holds considerable appeal for economistswho are looking for a deterministic explanationof the apparent randomness in economic vari-ables. Economists frequently assume randomnesswhen they are unable to explain the behavior ofan economic variable empirically. The presenceof an alternative explanation, chaos, will requirethem to consider more carefully the rationalebehind their assumptions.

One problem with incorporating chaos intoeconomics is that, while economists can eitherpostulate an equation and test it for the presenceof chaos or, alternatively, see if the data them-selves are chaotic, it is especially difficult toidentify the correct functional form that charac-terizes the data. The choice of a functional formis always a problem in economics, but, as pre-viously discussed, it is particularly difficult tomodel nonlinear dynamics. This problem is ex-acerbated because, as a result of the mathemat-ics required, the study of nonlinear dynamicshas, until recently, been relatively limited ingeneral and largely ignored in economics.23

Even when it is possible to estimate nonlineardynamic equations, the models themselves oftencannot be solved analytically. Without explicitsolutions to these models, their usefulness is ex-tremely limited. Obviously, the difficulty of de-termining the “true” underlying model from adata series is a problem whether or not chaosexists. The “discovery” of chaos, however, hasfocused much more attention on this problem,especially if the data are nonlinear.

Econonn.c Modeling an.d Chaos

The study of chaos emphasizes the impor-tance of rigorously modeling the dynamics of asystem rather than merely taking a static model(like equations I and 2) and adding time sub-

scripts and an error term. Although these sim-pler models may be more likely to have solutionswith explicit results that can be tested empirical-ly, the dynamics that arise may not capture thebehavior of the variable of interest. The richnessof a model may be found in explaining the be-havior of a variable over time as much as in thedirect, time-independent (or time-constant) rela-tionship between the variables.

In addition, the study of deterministic chaos il-lustrates some of the pitfalls of first differen-cing a dynamic model to convert it to discretetime, as was done in the model of populationgrowth presented above. This practice is com-mon in economics because data are onlyavailable in discrete intervals.

As is shown in Stutzer (1980) and demon-strated here, there are first-order differentialequation models (such as equation 6) which con-verge to a steady-state equilibrium that are cha-otic when expressed in discrete time (equation9). Thus, the dynamic properties of the discreteanalog of a differential equation cannot be as-sumed to be the same. In fact, it has beenshown that, although chaos can arise in first-order difference equations, it can only arise inthird-order or higher differential equations.’~Asa result, an economist must be careful abouteither converting a continuous time dynamicmodel into a discrete model (such as convertingequation 6 into equation 7), or taking a staticmodel and simply adding a time subscript,rather than postulating a model that is dynamic(in either discrete or continuous time) and esti-mating or solving it in that form. The choice ofthe appropriate type of model should depend onthe economic variables being described ratherthan analytical convenience. This issue is partic-ularly important if a continuous-time dynamicmodel is estimated in discrete time using thesteady-state equilibrium properties of thecontinuous-time solution. The discrete-time equa-tion that is being estimated may not reach asteady state at all, or the solution could differqualitatively from that found in the continuous-time version of the model.

“For recent work in nonlinear dynamics, see Grandmont(1987).

24The “order” of an equation refers, for a differential equa-tion, to the highest power attained by the derivative and,for a difference equation, the highest degree of differen-cing. For a more complete discussion, see Chiang (1984).

Empirical Applications of Chaos in Economics1 here are generally t~to approaches used in For- these tests to be accurate, the data need

the empirical literature to test for the pres- to be especially exact. This degree ot preci-ence of deternunistic chaos in economic and sion presents a particular problem for cc-a-financial data ‘l’lie first approach tests for nomics, whei-e controlled experiments arethe presence of nonlinearities in the data essentially impossible, especially on the macroSince chaos only arises in nonlinear systems, level. Data collection is far from perfect, andfinding nonlinearities in the (laid suggests the quality of the data declines as the degreethat lest ing cli ccci ly for’ the presence of chaos of aggregation increases introducing measure-is appropriate. In addition. the presence of ment error in the data In addition, becausenonlinear ities in the data pi-ovicles inforina- of rounding, the data are not as precise asHon to theorists niodeling these types of they should be. For this reason, tests for chaoseconomic systems. Because testing for non- are not simply tests for aperiodicity.linearities in the dat a is in u cli simpler (aridless controversial) than testing for chaos, ‘lhe (]uarititv of high-quality data is also cx-these tests are often performed firsl - tremnely important. Even if the results sho½

aperiodic its’ for a sample of 100 ohsc-,rvations,Many macroeconomic time series have been the system need not be aperiodic The existing

found to beha’~e in a nonlinear manner. Brock empirical tests br the presence of chaos re-and Sayers (1988) find such n idence in data quire an ext rernel~large number of highlyfor quartet ly eniplovment (1930-83), quarterly accurate data. Barely are both of these avail-unemployment (1949-82). monthly post-war able to econometricians. As a result, any cvi-industrial production arid pig-i on pr-oduc’tion dence from tests for chaos should he % iewed(1877-] 937). Nonlinearities have also been with caution

found in the Dit isia Ml monetary aggregatesAOther studit’s ha~c’ found nonlI rieacit ic’s ~ Ci ten these cat eats, sortie statistical tests,financial data as well. I or example, Hinich originating in the physical sciences, do lookand Patterson (1985a. 1985W find strong cvi- for the presence of chaos in economic dataclencc’ of nonlinearity in daily stock returns. ‘I hese tests are run on variables that have

- long time series available, art not aggregateThe second approach is to test directly for variables and are thus more likek to provide

he pr eseru e of chaos.’ There are ma iw pro- a ccu rat c’ restilt s. ‘lest s I tave km rid] evidenc-eblenis with testing directly for chaos using consistent with chaos in exchange rates (Ellis,economic data. however. ‘1 he most obvious, it 990), daily gold and silver prices on the 1 on-arid perhaps t lie most iniport ant, is the sen- clan market (Frank and S terigos, 1 988) atid insitivity of chaotic systems to sniall changes in the Di~isia monetary aggregates (Bai nelt andthe par amnetcm- ~‘altm~s and iiii I ial c-onditions. (hen, 1988) .~

- For a discussion of the tests used, see Brock and Sayers drawbacks see Barnett and Hinich (forthcoming), Brock(1986). as well as the other papers cited above (1986) and Ramsey (1989)

‘For a definition and discussion of the Divrsia monetary 4This is by no means a comprehensive survey of theaggregates. see Barnett and Spindt (1982). empirical literature applying chaos to economic and

‘A descr’ptton of the actual ernprricat techniques used to financial data. For a more comprehensivc discussion oftest for chaos is beyond the scope of this paper. For a the empirical work on chaos, see Barnett and Hinrchdescription of the tests available for chaos and their (forthcoming) and Ramsey (1989).

iom’ecasting i rio-c’ to impossible -,incc a smallI he stuc]~-ot deterministic chaos also olters error iii the talLie ol the initial condition can

set i’m ,d les on~for c’ronomctr’ iaris. It torerast— leach to hiMhlv utac curate precut tion. (see, lot’jog i.. a goal cit cc orioiriit’ niocleling. inappi opt iatc’ e~aiiipli’ figurc’ 6) simil,ri’lv, an error ri ant

modeling techniques in Itt1’ presence oF chaos par’iirmietem abc’ can also pm ocluce inc or rc’ct lore—lnq-oiiu’ tnorc’ costlt - IF Itu’ data arc’ chaotic’ c,i. ts (see tigtim’c’s3aticl It ‘thus. it is itnpor tant

to realize the limitations of economic forecastsin the presence of chaotic variables.

Chaos does not have to be present in the datato find the sort of fluctuating behavior (althoughwithout any clearly defined periodicity) that isoften found in economic data. Nonlinear non-chaotic models often can generate time pathsthat appear random, and testing for nonlineari-ties is the likely next step for future research inthis area, In fact, empirical economists are be-ginning to test for both nonlinearities and chaosin economic data (see insert on page 47). As aresult, more work needs to be done in under-standing nonlinear estimation so that economicmodels can describe a greater variety of be-havior and be more accurate as well. Tn addi-tion, the existence of chaos suggests that econo-mists might want to try nonlinear specificationsof a variable before resorting to modeling it as arandom variable. This in turn will help to im-prove the quality of economic forecasts in thepresence of nonlinear variables,

In addition, the use of a random componentin estimation does not necessarily imply that thevariable itself is random, but rather that otherrelevant variables might be excluded from theregression. Although each of these other vari-ables could have a small influence on the systemby itself, the total effect of these excluded vari-ables could be substantial. Given both the diffi-culty in detecting what these missing variablesmight be and data limitations, such a complexsystem might best be approximated by a randomvariable, even if there is no true randomness inthe variable being estimated. In fact, some argue(see, for example, Kelsey, 1988) that, since eco-nomic models do not include such (chaotic) phe-nomena as weather and other biological factorswhich can influence economic variables, it“seems inevitable that we will have randomterms in our equations.’25

(X)NCLUSI.ON

The study of deterministic chaos and its subse-quent application to economics has opened anew realm of possibilities for economists tryingto explain cyclical or erratic behavior in eco-nomic variables. As discussed above, chaos hasimplications for both theoretical modeling andempirical applications in economics. By illustrat-

ing explicitly how restrictive the assumption oflinearity can be, the study of chaos emphasizesthe importance of allowing for the possibility ofnonlinear behavior. The use of chaos in econom-ics also has offered new explanations for behav-ior that, until recently, has been able to be ex-plained only by random forces.

The techniques that have arisen from thestudy of chaos in the physical and biologicalsciences are in their infancy. As these techni-ques become more refined, and economists be-come better trained in working with these typesof models, their ability to explain the behaviorof variables such as exchange rates, businesscycles and stock prices is likely to improve.That possibility alone is sufficient reason foreconomists to take a closer look at deterministicchaos in particular and nonlinear dynamics ingeneral.

D..E.FE:R.]EJ~ItiESBarnett, William A., and Ping Chen. “The Aggregation-

Theoretic Monetary Aggregates are Chaotic and haveStrange Attractors: An Econometric Application ofMathematical Chaos?’ in William A. Barnett, Ernst R.Berndt, and Halbert White, eds., Dynamic EconometricModeling, Proceedings of the Third International Sym-posium in Economic Theory and Econometrics (CambridgeUniversity Press, 1988), pp. 199-245~

Barnett, William A., and Melvin J. Hinich. “Has Chaos BeenDiscovered with Economic Data?” in Ping Chen andRichard Day, eds., Evolutionary Dynamics and NonlinearEconomics (Oxford University Press, forthcoming).

Barnett, William A., and Paul A. Spindt. “Divisia MonetaryAggregates: Compilation, Data, and Historical Behavior?’Board of Governors of the Federal Reserve System, StaffStudies 116 (May 1982).

Baumol, William J., and Jess Benhabib. “Chaos: Significance,Mechanism, and Economic Applications?’ Journal ofEconomic Perspectives (Winter 1989), pp.77-lOS.

Benhabib, Jess, and Richard H. Day. “Rational Choice andErratic Behaviour?’ Review ofEconomic Studies, (July 1981),pp. 459-71.

Brock, W. A. “Distinguishing Random and DeterministicSystems: Abridged Version?’ Journal of Economic Theory,(October 1986), pp. 186-95.

Brock, W. A., and W. D. Dechert. “Theorems on Distinguish-ing Deterministic from Random Systems?’ in William A.Barnett, Ernst A. Berndt, and Halbert White, eds., DynamicEconometric Modeling, Proceedings of the Third Interna-tional Symposium in Economic Theory and Econometrics(Cambridge University Press, 1988),pp. 247-65.

Brock, W. A., and Chera L. Sayers. “Is the Business CycleCharacterized by Deterministic Chaos?” Journal ofMonetary Economics, (July 1988), pp. 71-90.

‘tmKelsey (1988), p. 12.