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Predictive accuracy for chaotic economic models
*
Silvano Bordignon, Francesco Lisi
University of Padova, Department of Statistics, Via S. Francesco 33, 35122 Padova, Italy
Received 5 August 1999; accepted 15 July 2000
Abstract
In this work we present a technique to obtain prediction intervals for chaotic data. Using nearest neighbors method we give estimates of local variance and percentiles of the prediction error distribution. This allows to define an interval containing a future value with a given probability. Its effectiveness is shown with data
generated by a chaotic economic model. 2001 Elsevier Science B.V. All rights reserved.
Keywords: Prediction; Nonlinear economic models; Chaotic dynamics; Interval prediction
JEL classification: C53; C22
1. Introduction
The possibility of chaos in economic systems brought an enormous amount of interest in the recent literature. The concepts of limited forecastability and complex dynamical properties have a very strong intuitive appeal for economics. In fact, chaotic dynamics supplies an alternative explanation for at least some part of economic fluctuations and provides some reasons for the difficulty in forecasting. The range of potential applications of chaos theory is very broad: from forecasting movements in foreign exchange rates and stock markets, to understanding international business cycles. Furthermore, it has been shown that very-standard equilibrium models could easily generate cycles and chaos and some examples of theoretical economic models which behave in a chaotic way have been provided in the literature.
A first example is that developed by De Grauwe et al. (1993) for exchange rates. Other examples of economic models generating chaos can be found in the class of the so-called overlapping generations models with or without production (see, e.g., the contributions of Benhabib and Laroque, 1988; Jullien, 1988; Medio and Negroni, 1996). Finally, chaos and complex dynamics can occur in
*Corresponding author: Tel.:139-49-827-4173; fax: 139-49-875-3930. E-mail address: [email protected] (F. Lisi).
nonlinear cobweb type models with adaptive production adjustment, as shown, for example, by Onozaki et al. (2000).
One reason for the attention that chaotic dynamics received in the economic literature concerns prediction. In effect this can provide an explanation for economists’ difficulties with forecasting. However, while it is well known that unpredictability due mainly to sensitive dependence on initial conditions becomes significant only after some time periods, predictability in the short run is not excluded in chaotic systems. Furthermore, suitably exploiting the chaotic features of the economic system provides also a way to quantify predictive accuracy. This latter point has been rather neglected in the economic literature dealing with chaotic time series, where most attention has been paid just to point prediction (see, e.g., Lisi and Medio, 1997; Soofi and Cao, 1999). Point prediction alone is not enough to get good forecasts, since it is known that predictive accuracy is not constant when dealing with a chaotic time series and could vary considerably with the state of the system, and thus with the time.
To take into account this important aspect of chaotic prediction we suggest to support and complete any point prediction with a suitable interval which is expected to contain the future value of the series with high probability. In particular, the main idea of this work is that of exploiting the local variability pattern in the phase-space to obtain an estimate of the local temporal variability, that we call local predictive variance. This estimate, in turn, can be used to build a prediction interval, that is an interval containing a future observation with a given probability.
It is worth mentioning that, besides interval forecasting, the local predictive variance can be useful in other situations. For example, it could be employed as a measure of volatility for financial time series, and an application in this context is provided by Le Baron (1992), whose results show that some forecasting improvements can be obtained using jointly nonlinear techniques and a simple volatility index based on local variability. Further examples can be found in all those situations where volatility forecasts are useful for portfolio and risk management (see, e.g., Sengupta and Zheng, 1994, and, more recently, Christoffersen and Diebold, 2000).
The paper is organized as follows: in the next section local predictive variance and prediction intervals for chaotic time series are introduced; in Section 3 a non-linear economic model (the CARAL model) is briefly presented; in Section 4 results of the validation procedure on the previously introduced model are given, while some conclusions are contained in Section 5.
2. Local variability and prediction intervals
T
In the following we assume that the time serieshXt tj51 is generated by a chaotic dynamical system. First we introduce a measure of local variability, then we give its predictive version and finally we use it for building prediction intervals.
2
then st represents a local variance and quantifies how much variability there is about X .t
It is possible to exploit this characteristic for forecasting. Expression (1) can not yet be used for predictive purposes, because the future state is not known in this context; thus a modification of (1) must be considered.
Given the embedding dimension d, the point prediction of XT11 can be obtained by using the nearest neighbors method, widely used in the chaotic literature (Farmer and Sidorowich, 1987). It works by building subdomains defined by the sets 65hX ,X , (i51, . . . ,k)j and by
approx-T(i ) T(i )11
imating the dynamics, within each of them, by a (linear) polynomial. Thus, given the nearest 2
ˆ
neighbors prediction XT11, we can consider the predictive version of st, that we call the local ˆ
predictive variance, by putting XT11 instead of X in (1) and considering the iterates of the neighborst
XT(i )
The basic idea is that the local predictive variance should be small where orbits do not diverge (i.e., this happens in correspondence of negative local Lyapunov exponents) and, consequently, the predictability is good. On the opposite, the local predictive variance should be large where the system is not well predictable, that is where local Lyapunov exponents are positive and near states tend to locally diverge.
P
Let us now turn to the construction of prediction intervals by means of sT. For analogy with the traditional statistical approach, we suggest to consider as a prediction interval for XT11 at a level
P P
ˆ ˆ ˆ
(12a) the interval [XT111za/ 2sT, XT111z12a/ 2sT], where XT11 is the nearest neighbors point prediction of XT11, za is the estimated a2th percentile of the standardized forecasting error
P
distribution and sT is the local predictive variance. To estimate the percentiles, the observed time series of length T has to be divided into 2 parts of length n and n , respectively, with Tf t 5nf1n . Thet first part is used to fit the model and the second one to obtain n one-step-ahead predictions. Thist allows us to compute n prediction errors that can be used to estimate the distribution and hence thet percentiles.
3. The CARAL model
In this section we introduce the nonlinear economic model which will be subsequently employed for the validation of our previously proposed procedure. The model considered here is the so-called CARAL (Constant Absolute Risk Aversion with production function of Leontief type) model and belongs to the broader class of overlapping generations models with or without production. The main characteristics of this model are given in the following. Let us consider an economy composed of two overlapping generations. The members of each generation live for two periods, youth and old age, and work only when young, but consume in both periods of life. If c and ct t11 are consumption levels for the two periods and l is labor at period t, the general problem faced by the young (representative)t
agent at the beginning of period t is to choose c , ct t11 and l that maximize an overall utility functiont subject to some budget constraints.
the first and second period, and with v(l ) the disutility of labor, let us consider the following
The first expression represents an utility function of the constant absolute risk aversion type, while the remaining two are of the constant relative risk aversion type. Solving the constrained optimization problem stated above (see Medio and Negroni, 1996, for the technical details), the dynamics of consumption can be described by
g 2ct1 /a
ct115
s
lt 2rc etd
. (3)Eq. (3) represents the optimal evolution of consumption derived from consumer’s intertemporal choice of consumption and leisure only.
Finally, introducing a linear Leontief technology into the model, the labor dynamics is given by
lt115b lst2c .td (4)
Eqs. (3) and (4) together represent the evolution of the system that is compatible with intertemporal optimization and equilibrium conditions in a Leontief economy. The parameter g .1 denotes the elasticity of the (dis)utility of labor; 0,a,1 is the elasticity of utility of consumption ‘tomorrow’, b is a productivity coefficient and r represents the steepness of the derived utility function.
It can be shown (see Medio and Negroni, 1996) that the CARAL system has a unique positive equilibrium and that it can lose its stability through a flip or a Neimark bifurcation. In particular, suitably changing the parameters values, the dynamic behavior of the model shows a very reach scenario, where periodic, aperiodic and chaotic behaviors can occur in many possible combinations. Furthermore, chaotic attractors can be generated by different types of transitions: period doubling, quasiperiodicity, intermittency routes to chaos. For example, keeping the parametersg, r anda fixed and increasing the productivity coefficient b we can obtain, passing through a Neimark bifurcation, the chaotic attractor depicted in Fig. 1.
4. Validation of the procedure
For the validation of our proposed procedure, we construct prediction intervals for time series generated by the CARAL model (3)–(4), which, for convenience, has been rewritten in the following form
g 2xt 1 /a
xt115( yt 2rx et ) (5)
yt115b( yt2x ).t
Fig. 1.. The attractor of the CARAL system.
After having generated scalar time series of length T, starting from a random initial condition belonging to the attraction basin, the data have been divided in three parts: the first n data points weref used to fit the model; the successive n were used to make internal validation and to estimate thet percentiles; the remaining n points were used for out of sample interval prediction; np f1nt1np5T. In particular, we consider nf51000, nt5500, np5500, T52000.
Both clean and noisy data are considered; in the latter case a gaussian noise w , was added to thet
2 2 1 / 2 2
clean data. The proportion of added noise p is such that p5
s
sw/sxd
, where sx is the variance of 2the data and sw the variance of the noise.
The embedding dimension and the number of neighbors have been estimated by minimizing the prediction error on the second subset. The estimated parameters are given in the second column of Table 1.
Table 1
Results of prediction intervals with a nominal coverage of 0.95 for data generated by the CARAL model
ˆ ˆ
Noise (d,k ) Real MIW Percentiles
coverage
x-Component
Noise-free 2,7 0.992 0.156 22.04 2.08
p50.02 2,8 0.964 0.182 21.95 2.05
p50.05 2,8 0.950 0.225 21.92 1.96
p50.10 2,10 0.958 0.397 22.10 2.00 y-Component
Noise-free 2,7 0.996 0.168 22.02 2.03
p50.02 2,8 0.964 0.186 21.97 1.94
p50.05 2,8 0.958 0.265 22.02 1.95
P
Fig. 2.. Behaviour of sTas a function of the x-component. The solid line is a nonparametric curve estimate, depicting the mean behaviour P
ofsT.
Table 1 is relative to prediction intervals for the x and the y components of the model, and shows that the method works quite well in term of points that are within the intervals; in the noise-free case the percentage of future values contained between the upper and lower limits is near to 100%, while for noisy data the coverage percentage is, correctly, about 95%. From this point of view, a first conclusion is that this method is quite robust with respect to the presence of noise and performs satisfactorily also in presence of a not ignorable amount of noise.
However, the number of future values within the prediction intervals is not enough to say that the method is really useful. In fact, for a given coverage, a prediction interval is useful only if it is not too large. Since these intervals are time varying, it is not easy to quantify this aspect. To this purpose we
P
employ a summary measure given by the normalized mean interval width, MIW5mean
s
sT(z12a/ 22za/ 2) / 2s
d
x.The third column of Table 1 gives the MIW for different amounts of noise. While the coverage is quite good also for not too small noises, the width, as expected, increases fast with the noise and for a 10% noise is about 0.4 times the standard deviation of the data.
P
Fig. 2 shows the behavior of sT in function of the state for the x-component of the system, while Fig. 3 shows an example of the observed values and the relative one-step-ahead prediction intervals. It
P
is manifest thatsT varies considerably with the state of the system. This, in turn, produces prediction intervals of varying width and, consequently, clusters of points with different predictive accuracy.
5. Conclusions
Fig. 3.. An example of one-step-ahead prediction intervals for the x-component. Points are observed values.
Sidorowich (1987) on point prediction and is based on the simple idea of using the nearest neighbors to give an estimate of the local variability. This estimate, together with the empirical distribution of the prediction error, allows to build prediction intervals. Since the procedure is essentially computational, it does not depend on a specific model, thereby avoiding any distributional assumption on the prediction error. In the paper detailed results about the nominal vs. real coverage of the prediction intervals and about the mean interval widths are given for a chaotic economic model, both for clean and noisy data. These results are quite encouraging because the method, in spite of its simplicity, seems to work well with noise-free data and is rather robust also for noisy data.
Acknowledgements
The authors would like to acknowledge the helpful comments of an anonymous referee.
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