Determinism in Financial Time Series

The attractive possibility that financial indices may be chaotic has been the subject of much study. In this paper we address two specific questions: “Masked by stochasticity, do financial data exhibit deterministic nonlinearity?”, and “If so, so what?”. We examine daily returns from three financial indicators: the Dow Jones Industrial Average, the London gold fixings, and the USD-JPY exchange rate. For each data set we apply surrogate data methods and nonlinearity tests to quantify determinism over a wide range of time scales (from 100 to 20,000 days). We find that all three time series are distinct from linear noise or conditional heteroskedastic models and that there therefore exists detectable deterministic nonlinearity that can potentially be exploited for prediction. The hypothesis that deterministic chaos may underlie apparently random variation in financial indices has been received with enormous interest (Peters 1991, LeBaron 1994, Benhabib 1996, Mandelbrot 1999). However, initial studies have been somewhat inconclusive (Barnett and Serletis 2000) and there have been no reports of exploitable deterministic dynamics in financial time series (Malliaris and Stein 1999). Contradictory interpretations of the evidence are clearly evident (Barnett and Serletis 2000, Malliaris and Stein 1999). In his excellent review of the early work in this field, LeBaron (1994) argues that any chaotic dynamics that exist in financial time series are probably insignificant compared to the stochastic component and are difficult to detect. In some respects the issue of whether there is chaos in financial time series or not is irrelevant. Some portion of the dynamics observed in financial systems is almost certainly due to stochasticity1. Therefore, we restrict our attention to whether in addition there is significant nonlinear determinism exhibited by these data, and how such determinism may be exploited. Crack and Ledoit (1996) showed that certain nonlinear structures are actually ubiquitous in stock market data, but represent nothing more than discretization due to exchange imposed tick sizes. Conversely, in his careful analysis of nonlinearity in financial indices, Hsieh (1991) found evidence that stock market log-returns are not independent and identically distributed (i.i.d.), and deterministic fluctuation in volatility could be predicted. Hence, if we are to exploit nonlinear determinism in financial time series we should first be able to demonstrate that it exists. If we find that these data do exhibit nonlinear determinism then we must assess how best to exploit this. Before reviewing significant advances within the literature we will define precisely what we mean by “determinism” and “predictable”. Let {xt}t denote a scalar time series with E(xt) = 0. • A time series xt is predictable if E(xt |xt−1,xt−2,xt−3, . . .) 6= 0. • A time series xt is deterministic if there exists a deterministic (i.e. closed-form, non-stochastic) function f such that xt = f (xt−1,xt−2,xt−3, . . .). • A time series xt contains a deterministic component if there exist a deterministic f such that xt = f (xt−1,xt−2,xt−3, . . . ,et−1,et−2,et−3, . . .)+ et where et are mean zero random variates (not necessarily i.i.d.). • Determinism is said to be non-linear if f is a non-linear function of its arguments. For example, we note that according to these definitions a GARCH model contains determinism but it is not deterministic. We do not (as Hsieh (1991) did) classify a system that is both chaotic and stochastic as stochastic. Such a system we call stochastic with a non-linear deterministic component. Our choice of nomenclature emphasizes that we are interested in whether there is significant structure in this data that cannot be adequately explained using linear stochastic methods. In an effort to quantify determinism, predictability and nonlinearity, a variety of nonlinear measures have been applied to a vast range of economic and financial time series. Typical examples include 1By stochastic we mean either true randomness or high dimensional deterministic events that, given the available data, are in no way predictable. 1 Small and Tse: Determinism in Financial Time Series Produced by The Berkeley Electronic Press, 2003 the estimation of Lyapunov exponents (Schittenkopf, Dorffner, and Dockner 2001); the correlation dimension (Harrison, Yu, Lu, and George 1999, Hsieh 1991); the closely related BDS statistic (Brock, Hsieh, and LeBaron 1991); mutual information and other complexity measures (Darbellay and Wuertz 2000); and nonlinear predictability (Agnon, Golan, and Shearer 1999). The rationale of each of these reports was to apply a measure of nonlinearity or “chaoticity” to the supposed i.i.d. returns (or logreturns). Deviation from the expected statistic values for an i.i.d. process is then taken as a sign of nonlinearity. However, it is not immediately clear what constitutes sufficient deviation. In each case, some statistical fluctuation of the quantity being estimated is to be expected. For example, the correlation dimension of an i.i.d. process should be equal to the embedding dimension2 de. Although not a signature of chaos, for a deterministic chaotic time series, the correlation dimension is often a non-integer. But for practical purposes it is impossible to differentiate between the integer 1 and the non-integer 1.01. Furthermore, for de 1 and time series of length N < ∞, one observes that the correlation dimension is systematically less than de, even for i.i.d. noise. These issues are complicated further by the fact that it is almost certain that financial time series contain a combination of deterministic and stochastic behavior (Schittenkopf, Dorffner, and Dockner 2001). The problem is that for any nonlinear measure applied to an arbitrary experimental time series one does not usually have access to the expected distribution of statistic values for a noise process. In the case of the correlation dimension, the BDS statistic goes some way towards addressing this limitation (Brock, Hsieh, and LeBaron 1991). However, the BDS statistic is an asymptotic distribution and only provides an estimate for i.i.d. noise. Of course, this can be generalized to more sophisticated hypotheses by first modeling the data and then testing whether the model prediction errors are i.i.d. For example, one could expect that linearly filtering the data and testing the residuals against the hypothesis of i.i.d. noise can be used to test the data for linear noise. However, testing model residuals has two problems. Firstly, one is now testing the hypothesis that a specific model generated the data. The statistical test is now (explicitly) a test of a the particular model that one fitted to the data rather than a test of all such models. Secondly, linear filtering procedures have been shown to either remove deterministic nonlinearity or introduce spurious determinism in a time series (Mees and Judd 1993, Theiler and Eubank 1993). A linear filter is often applied to time series to pre-whiten3 the data. For linear time series so-called “bleaching” of the time series data is known not to be detrimental (Theiler and Eubank 1993). However, Theiler and Eubank (1993) show that linear filtering to “bleach” the time series can mask chaotic dynamics. For this reason they argue that the BDS statistic should never be applied to test model residuals. Conversely, Mees and Judd (1993) have observed that nonlinear noise reduction schemes can actually introduce spurious signatures of chaos into otherwise linear noise time series. In the context of nonlinear dynamical systems theory, the method of surrogate data has been introduced to correct these deficiencies (Theiler, Eubank, Longtin, Galdrikian, and Farmer 1992). The method of surrogate data is clearly based on the earlier bootstrap techniques common in statistical finance. However, surrogate methods go beyond both standard bootstrapping and the BDS statistic, and provide a non-parametric method to directly test against classes of systems other than i.i.d. noise. 2The embedding dimension, de, is a parameter of the correlation dimension estimation algorithm and will be introduced in section 1.1. 3That is, make the data appear like Gaussian (“white”) noise. 2 Studies in Nonlinear Dynamics & Econometrics Vol. 7 [2003], No. 3, Article 5 http://www.bepress.com/snde/vol7/iss3/art5 Surrogate data analysis provides a numerical technique to estimate the expected probability distribution of the test statistic observed from a given time series for: (i) i.i.d. noise, (ii) linearly filtered noise, and (iii) a static monotonic nonlinear transformation of linearly filtered noise. Some recent analyses of financial time series have exploited this method to provide more certain results (for example (Harrison, Yu, Lu, and George 1999) and (Kugiumtzis 2001)). However, it has been observed that for non-Gaussian time series the application of these methods can be problematic (Small and Tse 2002a) or lead to incorrect results (Kugiumtzis 2000). But it is widely accepted that financial time series are leptokurtotic and it is therefore this problematic situation that is of most interest. Despite the commonly observed “fat” tails in the probability distribution of financial data the possibility that these data may be generated as a static nonlinear transformation of linearly filtered noise has not been adequately addressed. In this paper we report the application of tests for nonlinearity to three daily financial time series: the Dow-Jones Industrial Average (DJIA), the US$-Yen exchange rate (USDJPY), and the London gold fixings (GOLD). We utilize estimates of correlation dimension and nonlinear prediction error. To overcome some technical problems4 (Harrison, Yu, Lu, and George 1999) with the standard Grassberger-Proccacia correlation dimension estimation algorithm (Grassberger and Procaccia 1983a), we utilize a more recent adaptation (Judd 1992) that is robust to noise and short time series. To estimate the expected distribution of statistic values for noise processes, we apply the method of surrogate data (Theiler, Eubank, Longtin, Galdrikian, and Farmer 1992). By examining the rank distribution of the data and rescaling, we avoid problems with these algorithms (Kugiumtzis 2000, Small and Tse 2002a). For each of the three financial indicators examined we find evidence that the time series are distinct from the three classes of linear stochastic system described above. Hence, we reject the random walk model for financial time series and also two further generalizations of this. Financial time series are not consistent with i.i.d. noise, linearly filtered noise, or a static monotonic nonlinear transformation of linearly filtered noise. Therefore, i.i.d. (including leptokurtotic and Gaussian distributions), ARMA process, or a nonlinear transformation of these do not adequately model this data. Although this result is consistent with the hypothesis of deterministic nonlinear dynamics, it is not direct evidence of chaos in the financial markets. As a further test of nonlinearity in this data we apply a new surrogate data test (Small, Yu, and Harrison 2001). Unlike previous methods, this new technique has been shown to work well for non-stationary time series (Small and Tse 2002a). This new test generates artificial data that are both consistent with the original data and a noisy local linear model (Small, Yu, and Harrison 2001, Small and Tse 2002a). This is therefore a test of whether the original time series contains any complex nonlinear determinism. In each of the three time series we find sufficient evidence that this is the case. We show that this new test is able to mimic ARCH, GARCH and EGARCH models of financial time series but is unable to successfully mimic the data. Therefore, financial time series contain deterministic structure that may not be modeled using these conditional heteroskedastic processes. Therefore, state dependent linear or weakly nonlinear stochastic processes are not adequate to completely describe the determinism exhibited by financial time series. Significant detail is missing from these models. 4These technical issues and their remedies will be discussed in more depth in section 1.1. 3 Small and Tse: Determinism in Financial Time Series Produced by The Berkeley Electronic Press, 2003 To verify this result and in an attempt to model the deterministic dynamics we apply nonlinear radial basis modeling techniques (Judd and Mees 1995, Small and Judd 1998a, Small, Judd, and Mees 2002) to this data both to quantify the determinism and to attempt to predict the near future evolution of the data. We find that in each of the three cases, for time series of almost all lengths, the optimal nonlinear model contains nontrivial deterministic nonlinearity. In particular we find that the structure in the GOLD time series far exceeds that in the other two data sets. Our conclusion is that these data are best modeled as a noise driven deterministic dynamical system either far from equilibrium or undergoing deterministic bifurcation. In section 1 we will discuss the mathematical techniques utilized in this paper. Section 2 presents the results of our analysis and discusses the implications of these. Section 3 is our conclusion. 1. Mathematical Tests for Chaos In this section we provide a review of the mathematical techniques employed in this paper. Whilst these tests may be employed to find evidence consistent with chaos, it is more correct to say that these are mathematical tests for nonlinearity. Nonlinearity is merely a necessary condition for chaos. In section 1.1 we discuss correlation dimension and section 1.2 describes the estimation of nonlinear prediction error. Section 1.3 describes the method of surrogate data and section 1.4 introduces the new pseudoperiodic surrogates proposed by Small, Yu, and Harrison (2001). 1.1. Correlation dimension Correlation dimension (CD) measures the structural complexity of a time series. Many excellent reviews (for example (Peters 1991)) cover the basic details of estimating correlation dimension. However, the method we employ here is somewhat different. Let {xt} t=1 be a scalar time series, consisting of N observations, xt being the t-th such observation. In the context of this work xt = log pt+1 pt is the difference between the logarithm of the price on successive days. Nonlinear dynamical systems theory requires that the time series xt is the output of the evolution on a deterministic attractor in de-dimensional space (Takens 1981). Taken’s embedding theorem (Takens 1981) (later extended by many other authors) gives sufficient conditions under which the underlying dynamics may be reconstructed from {xt} t=1. The basic principle of Taken’s embedding theorem is that (under certain conditions) one may construct a time delay embedding xt 7−→ (xt ,xt−1,xt−2, . . . ,xt−de+1) (1.1) = zt such that the embedded points zt are homeomorphic to the underlying attractor5, and their evolution is diffeomorphic to the evolution of that underlying dynamical system6. The necessary conditions of this theorem require that xt is a sufficiently accurate measurement of the original system. This is an 5The attractor is the abstract mathematical object on which the deterministic dynamic evolution is constrained. 6Those adverse to mathematics may substitute “the same as” for both “diffeomorphic to” and “homeomorphic to”. 4 Studies in Nonlinear Dynamics & Econometrics Vol. 7 [2003], No. 3, Article 5 http://www.bepress.com/snde/vol7/iss3/art5 assumption we are forced to make. Application of the theorem and the time delay embedding then only requires knowledge of de. In general there is no way of knowing de prior to the application of equation (1.1). One may either apply (1.1) for increasing values of de until consistent results are obtained, or employ the method of false nearest neighbors or one of its many extensions (see for example (Kennel, Brown, and Abarbanel 1992)). For the current application we find that both methods produce equivalent results and that a value of de ≥ 6 is sufficient. Ding, Grebogi, Ott, Sauer, and Yorke (1993) demonstrate that the requirements on de for estimating correlation dimension are not as restrictive as one might otherwise expect. Embedding aside, correlation dimension is a measure of the distribution of the points zt in the embedding space Rde . Define the correlation function CN(ε) by CN(ε) = ( N 2 )−1 ∑ 1≤i< j≤N Φ(‖zi− z j‖< ε) (1.2) where Φ(X) = 1 if and only if X is true. The correlation function measures the fraction of pairs of points zi and z j that are closer than ε; it is an estimate of the probability that two randomly chosen points are closer than ε. The correlation dimension is then defined by: dc = lim ε→0 lim N→∞ logCN(ε) logε . (1.3) The standard method to estimate correlation dimension from (1.3) is described by Grassberger and Procaccia (1983a). The Grassberger-Proccacia algorithm (Grassberger and Procaccia 1983a, Grassberger and Procaccia 1983b) estimates correlation dimension as the asymptotic slope of the log-log plot of correlation integral against ε. Unfortunately this algorithm is not particularly robust for finite time series or in the case of noise contamination (see for example the discussion in (Small, Judd, Lowe, and Stick 1999)). Careless application of this algorithm can lead to erroneous results. Several modifications to this basic technique have been suggested in the literature (Judd 1992, Diks 1996). The method we employ in this paper is known as Judd’s algorithm (Judd 1992, Judd 1994) which is more robust for short (Judd 1994) or noisy (Galka, Maaß, and Pfister 1998) time series. Judd’s algorithm assumes that the embedded data {zt}t can be modeled as the (cross) product of a fractal structure (the deterministic part of the system) and Euclidean space (the noise). It follows (Judd 1992) that for some ε0 and for all ε < ε0 CN(ε) ≈ εc p(ε) (1.4) where p(ε) is a polynomial correction term to account for noise in the system8. Note that, correlation dimension is now no longer a single number dc but is a function of ε0: dc(ε0). A discussion of the 7Oftentimes, a second parameter, the embedding lag τ is also required. The embedding lag is associated with characteristic time scales in the data (such as decorrelation time) and is employed in the embedding (1.1) to compensate for the fact that adjacent points may be over-correlation. However, for inter-day recording of financial indicators, a natural choice of embedding lag is τ = 1. 8The polynomial p(ε) has order equal to the topological dimension of the attractor. In numerical applications linear or quadratic p(ε) is sufficient. 5 Small and Tse: Determinism in Financial Time Series Produced by The Berkeley Electronic Press, 2003 physical interpretation of this is provided by Small, Judd, Lowe, and Stick (1999). The parameter ε0 may be thought of as a viewing scale: as one examines the features of the attractor more closely or more coarsely, the structural complexity and therefore apparent correlation dimension changes. The concept is somewhat akin to the multi-fractal analysis introduced by Mandelbrot (1999)9. 1.2. Nonlinear prediction error Correlation dimension measures the structural complexity of a time series. Nonlinear prediction error (NLPE) measures certainty with which one may predict future values from the past. Nonlinear prediction error provides an indicator of the average error for prediction of future evolution of the dynamics from past behavior. The value of this indicator will depend crucially on the choice of predictive model. The scheme employed here is that originally suggested by Sugihara and May (1990). Informally, NLPE measures how well one is able to predict the future values of the time series. This is a measure of critical importance for financial time series, but, even in the situation where predictability is low, this measure is still of interest. The local linear prediction scheme described by Sugihara and May (1990) proceeds in two steps. For some embedding dimension de the scalar time series is embedded according to equation (1.1). To predict the successor of xt one looks for near neighbors of zt and predicts the evolution of zt based on the evolution of those neighbors. The most straightforward way to do this is to predict zt+1 based on a weighted average of the neighbors of zt : < zt+1 > = ∑ τ denotes the prediction of zt+1 and α(·) is some (possibly nonlinear) weighting function10. The scalar prediction < xt+1 > is simply the first component of < zt+1 >. Extensions of this scheme include employing a local linear or polynomial model of the near neighbors zτ. Another common application of schemes such as this is for estimation of Lyapunov exponents (Wolf, Swift, Swinney, and Vastano 1985). To estimate Lyapunov exponents one must first be able to estimate the system dynamics. Estimating Lyapunov exponents requires additional computation and the estimates obtained are known to be extremely sensitive to noise. For these reasons we chose to only estimate nonlinear prediction error in this paper. Applying equation (1.5) one has an estimate of the deterministic dynamics of the system. NLPE is computed as the deviation of these predictions from reality ( 1 N N ∑ t=1 (< xt+1 >−xt+1) ) 1 2