Miyano, Takaya

Tatsumi, Kenichi

Kyoto University Research Information Repository

TitleNonlinear Time Series Analysis of Stock Return Variation (The8th Workshop on Stochastic Numerics)Author(s)Miyano, Takaya; Tatsumi, KenichiCitation数理解析研究所講究録 (2009), 1620: 27-38Issue Date2009-01URL http://hdl.handle.net/2433/140225RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityNonlinear Time Series Analysis of StockReturn VariationTakaya Miyano* and Kenichi Tatsumi\dagger*Department of Micro System Technology, Ritsumeikan University1-1-1 Noji-Higashi, Kusatsu, Shiga 525-8577, Japan\dagger Graduate School of Economics, Gakushuin University1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, JapanAbstractWe tested for the days-of-the-week effect on stock retum variation using NIKKEI 225stock index data ranging from the 4th of January in 1989 to the 29th of August in 2003.Nonlinear time series analysis with the surrogate method suggests that there may be aparticular weekly pattem in stock return variation such that stock retum is significantlylower on Monday.1 IntroductionThere has been a long controversial conjecture that stock return might display dependenceon the days of the week [1]. We may call this as the days-of-the-week effect, which is one ofanomalies that cannot be explained within the framework of the standard theories of economics.In the United States, there has been an uncertain observation such that retums on investmentsin stocks might be lower on Monday, which is called the Monday effect. Possible existenceof the days-of-the-week effect has been investigated using regression analysis with inconclusiveresults in previous literature.In this study, we test for the days-of-the-week effect on daily stock return in the Japanesestock market using nonlinear time series analysis. Our approach is practical but has not beentaken in previous literature. We use a time series of NIKKEI 225 stock index, i.e., mean priceover 225 stocks at the Tokyo Stock Exchange. The index data are deficient on Saturdays,Sundays and the national holidays when the stock market is closed. To recover the missingdata, linear interpolation is applied to the time series. As the methods for time series analysis,1620 2009 27-38 27we apply the diagnostic algorithm developed by Wayland et al. [2] and the surrogate methodintroduced by Theiler et al. [3, 4]. These methods have been devised to test for chaos fromcomplex time series. We may not expect that stock retum variation represents chaotic behavior.Nevertheless, the days-of-week effect is supposed to be captured in terms of the degree ofcomplexity in stock return variation. These mathematical methods are suitable for this purpose.This paper is organized as follows. In section 2, we describe the method for interpolatingdata, i.e., linear interpolation. Linear interpolation smoothly recovers the data deficiencies,in particular, those on between Friday and Monday. This property is very important for ourpurpose, as discussed in this paper. The mathematics for the time series analysis are presentedin section 3. Results of numerical analysis are given in section 4. We discuss the results andmake conclusions in section 5.2 Interpolating dataWe used a time series of daily closing stock index ranging from the 4th (Wednesday) of Januaryin 1989 to the 29th (Friday) of August in 2003, issued by the Tokyo Stock Exchange. The stockdata are deficient on Saturdays, Sundays and the national holidays. Linear interpolation wasused to recover these missing data points: We synthesized $n$ successive missing data points,denoted as $x(t+i)(i=1, \cdots, n)$ , between stock data $x(t)$ and $x(t+n+1)$ by$x(t+i)= \frac{n-i+1}{n+1}x(t)+\frac{i}{n+1}x(t+n+1)$ (1)We thus obtained an interpolated time series consisting of $N=5348$ data points, as shown inFig. 1. Daily stock return $r(t)(t=2, \cdots, N)$ was estimated from the interpolated time seriesby$r(t)= \frac{x(t)-x(t-1)}{x(t)}$ . (2)Figure 2 shows the time series $\{r(t)\}_{t=2}^{N}$ .3 Methods for time series analysisGiven a time series $\{x(t)\}_{t=1}^{N}$ as a set of observational data for a single observable quantityin a system of $Q$ degrees of freedom, we can represent the dynamical behavior of the sys-tem using embedology [5, 6]: We reconstruct trajectories as lagged sequences of data points$x(t)=(x(t),$ $x(t+\Delta t),$ $\cdots,$ $x(t+(D-1)\Delta t)$ where $D$ is the embedding dimension and $\Delta t$ is28Figure 1: NIKKEI stock index with interpolations on Saturdays, Sundays, and national holi-days.an appropriately chosen time lag. The sufficient condition for a map of the trajectories from Q-dimensional space to D-dimensional reconstructed space to be a diffeomorphism is $D\geq 2Q+1$[5]. In practical applications, $Q$ is usually unknown and $D$ is determined by trial-and-error.The optimal time lag $\Delta t$ can be inferred from autocorrelation function or mutual information.In this work, we determined $\Delta t$ on the basis of mutual information [7, 8].3.1 Mutual informationMutual information as a function of the time distance between data points of a time series,introduced by Fraser and Swinney [7, 8], provides useful information for determining the optimaltime lag for embedding as well as for measuring the rate of spontaneous decay of informationfrom the time series. Let $U$ and $V$ be sets of realizations of the random variables $u$ and$v$ , respectively. We approximate the probability density functions $p_{U}(u)$ and $p_{V}(v)$ and thejoint probability density function $p(u, v)$ from l-dimensional and 2-dimensional histogramsgenerated from sets of sample data $\{u_{i}\}_{i=1}^{N}$ and $\{v_{i}\}_{i=1}^{N}$ . Here, $N$ is the number of examples.The information entropies $H(U)$ and $H(V)$ and the joint information entropy $H(U, V)$ are29Figure 2: NIKKEI stock retum.calculated by$H(U)$ $=$ $- \sum_{1=1}^{N}p_{U}(u_{i})\log_{2}p_{U}(u_{i})$ , (3)$H(V)$ $=$ $- \sum_{1=1}^{N}p_{V}(v_{i})\log_{2}p_{V}(v_{i})$ , (4)$H(U, V)$ $=$ $- \sum_{i,j=1}^{N}p(u_{i}, v_{j})\log_{2}p(u_{i}, v_{j})$ . (5)The average mutual information $I(V;U)$ is given by$I(V;U)=H(U)+H(V)-H(U, V)$ . (6)$I(V;U)$ represents the amount of information that one can know about $V$ given $U$ . In thiswork, the time series $\{x(t)\}_{t=1}^{N-\Delta t}$ is substituted into $U$ and its time-delayed counterpart $\{x(t+$$\Delta t)\}_{t=1}^{N-\Delta t}$ is substituted into $V$ . Then, the mutual information is a function of the time lag$\Delta t$ , denoted by $I(\Delta t)$ , and represents the spontaneous loss of information caused by the timeevolution of the system in terms of $\Delta t$ . The optimal value for $\Delta t$ in embedding is estimated asthe first minimum of $I(\Delta t)$ or the $\Delta t$ corresponding to $I(\Delta t)/e$ .303.2 Wayland testThe algorithm developed by Wayland et al. captures visible determinism underlying a timeseries in terms of the parallelness of neighboring trajectories reconstructed in the phase space[2]. The central idea of the algorithm is that visible determinism achieves smooth trajectoriesof time evolution in the phase space according to the definition of determinism that similarcauses give rise to similar effects. Hence, for deterministic processes, neighboring trajectorieswill point in similar directions. In contrast, stochastic processes will give rise to large diversityin the directions of the trajectories. Thus, the degree of diversity depends on how visibledeterminism is in a time series. Practical procedures of this algorithm are as follows.We randomly choose a vector $x(t_{0})$ and find its $K$ nearest neighbors $x(t_{k})$ , for instance, inthe sense of Euclidean distance. Next, we map each vector to $x(t_{k}+T)(k=0, \cdots, K)$ withan appropriately chosen time interval $T$ . The diversity of directions of neighboring trajectoriescan be measured in terms of the translation error:$E_{trane}$ $=$ $\frac{1}{K+1}\sum_{k=0}^{K}\frac{\Vert v(t_{k})-\overline{v}\Vert}{\Vert\overline{v}\Vert}$ , (7)$v=$ $\frac{1}{K+1}\sum_{k=0}^{K}v(t_{k})$ , (8)$v(t_{k})$ $=$ $x(t_{k}+T)-x(t_{k})$ . (9)The difference vectors $v(t_{k})$ approximate tangential vectors of the trajectories. The more visibledeterminism is, the smaller $E_{trans}$ will be. To reduoe stochastic error in estimating $E_{trans}$ , weuse the following procedures. We seek the medians of $E_{trans}$ for $S$ sets of $M$ randomly chosen$x(t_{0})$ and estimate the mean over the $S$ medians. The mean is taken as a representative of$E_{tran\epsilon}$ .The previous numerical work [9] suggested the following tentative criteria to infer the dy-namical nature of a time series from an estimate of $E_{trans}$ as a function of embedding dimension.$E_{tr\alpha n\epsilon}$ below 0.1 indicates the visible determinism of the underlying dynamics regardless of howcomplex a time series appears. For $E_{lrans}$ exceeding 0.5, the time series represents stochas-tic processes. In particular, white noise (temporally uncorrelated random noise) generates$E_{tran\epsilon}\approx 1$ independently of the embedding dimension. In case $E_{tran\epsilon}$ lies between 0.1 and 0.5,the time series may represent either temporally correlated stochastic processes or deterministicprocesses contaminated with observational random noise. In this work, we utilize these criteriato interpret estimates of $E_{trans}$ .313.3 Surrogate methodThe surrogate method is an ingenious way of evaluating the statistical significance of an estimateof a statistic made from a single time series [3, 4]. Given a time series as a single set ofrealizations of certain dynamics, we make a null hypothesis about the dynamics. We thensynthesize surrogate time series sharing the statistical properties relevant to the null hypothesiswith the original time series. This procedure may be viewed as if we reobserved the dynamicalsystem under the restriction of the null hypothesis.In this work, we made the null hypothesis that daily stock retum is subject to an independentidentically distributed $(i. i. d.)$ random process and hence displays no particular weekly pattern.Under the null hypothesis, forty surrogate time series were synthesized by random-shuffling theoriginal stock prices within each week (from Monday to Friday, totally 764 weeks) using $i$ . $i$ . $d$.random numbers followed by the interpolation of the data and the estimation of stock returnsor by random-shuMing the original stock returns within each week. Each surrogate time serieshas the same statistical moments as those of the original time series.We next ranked the stock retums within each week and gave scores 1 to 5 to the stockreturn of each day of the week for both the original and the surrogate data. To test the nullhypothesis, a statistical t-test was applied to the mean rank score on each day of the week. Letus denote the mean rank scores for the original time series as $s_{0}$ , those for the surrogate timeseries as $s$ , and their mean and variance as $\overline{s}$ and $\sigma_{s}^{2}$ , respectively. Then the two-sided t-teststatistic is given as$t= \frac{|s_{0}-\overline{s}|}{\sigma_{s}}$ (10)In the case of forty surrogate time series, the critical value of $t$ is 2.02 at a two-sided level of5% reliability. If $t>2.02$ , the null hypothesis is rejected, which means that the original timeseries is unlikely to be realized by the dynamics of the null hypothesis.4 ResultsWe estimated the statistical moments of the first (mean), second (variance), third and fourthorder for the stock returns $(N=5347)$ . Similar estimates were made for the stock returns oneach day of the week $(N=764)$ . Results are summarized in Table 1. The stock retum dataare expressed as annual rate in percentage, i.e., $r(t)x360\cross 100$ .Mean stock return takes a negative large value with the smallest variance on Monday, and32it is the lowest with a large variance on Friday. It turns out that this statistical tendency isnot due to interpolation, from time series analysis based on the Wayland test.Table 1 Statistical moments of annual stock returns (in %)We next estimated the mutual information as a function of time lag for the stock returns. Theprobability density function and the joint probability density function were approximately es-timated from a one-dimensional histogram with 32 partitions and a two-dimensional histogramwith 32 $x32$ partitions, respectively. Estimates are shown in Fig. 3. The mutual informationtakes the first minimum at $\Delta t=4$ , which corresponds to the optimal time lag in embedding.Small peaks appearing every 7 days of time lag are artifacts due to the interpolation of thedata. No such artifacts appear in autocorrelation function, as shown in Fig. 4.The Wayland test was applied to the stock retums with $\Delta t=4,7,$ $K=4,$ $T=1,5,6,7$,8, $M=301$ and $S=20$ . $\Delta t=4$ is the optimal time lag in terms of the mutual informationand $\Delta t=7$ corresponds to one week shift. Results are shown in Figs. 5 and 6. In the case of$\Delta t=4$ , determinism in stock return is as invisible as that of Gaussian white noise at $T\leq 7$ .At $T=8$ , determinism in stock return becomes as visible as those in temporally correlatedstochastic processes, unlike those at $T=1,5,6,7$ , and takes the minimum at $D=6$ . Such atendency is also observed for $\Delta t=7,$ $T=7$ and $D=4$ , as shown in Fig. 6.We next applied the Wayland test to the stock returns on each day of the week. Each timeseries of stock retum consists of 764 data points. Results under $K=4,$ $T=5,$ $M=51$ and$S=10$ are presented in Fig. 7. Standard deviation in $E_{tran\epsilon}$ was estimated to be 0.04–0.14at $D\geq 8$ . We chosen $T=5$ because of the smallest variance in $E_{trans}$ with respect to thetime translation $T$ . The results in Fig. 7 should be compared with those at $T=5$ in Fig. 6.These results tell us that determinism is the most invisible on Monday. As discussed in thenext section, this tendency is not due to the interpolation of data.33Figure 3: Mutual information of NIKKEI retum as a function of time lag.Estimates of the two-sided t-test statistic for the rank scores of stock retum on each dayof the week are summarized in Table 2. The critical t-value at 5 % reliability is 2.02. Theestimated t-values exceed 2.02 on Monday and Friday. Hence, the null hypothesis can berejected. The weekly pattem in stock return variation is unlikely to be ascribed to an $i$ . $i$ . $d$ .stochastic process.kble 2 Two-sided t-test statistics on the rank scores for the original and forty surrogate data.Upper row: Estimates for the stock returns calculated from randomly shuffled stock prices withineach week. Lower row: estimates for randomly shuffled stock returns within each week. The criticalt-value for rejecting the null hypothesis is 2.02.34Figure 4: Autocorrelation function of NIKKEI return.Figure 5: Wayland test for NIKKEI return. $T=1,5,6,7,8$ and $\Delta t=4$ .35Figure 6: Wayland test for NIKKEI return. $T=1,5,6,7,8$ and $\Delta t=7$ .Figure 7: Wayland test for NIKKEI retums on each day of the week $(T=5)$ .365 DiscussionThe Wayland test indicates that NIKKEI stock return displays the most complex dynamicalbehavior on Monday. This is not due to linear interpolation applied to the data, because linearinterpolation generates smooth sequences of data points and should lower $E_{trans}$ . These resultssuggest that the significantly low stock return with a small variance on Monday is likely torepresent a particular weekly pattern in stock return variation.This interpretation is supported by the surrogate analysis. In fact, the null hypothesis thatthe stock retum variation represents an $i$ . $i$ . $d$. stochastic process has been rejected by statisticalt-test on the rank order of the stock retums on Monday and Riday. The surrogate analysistells us that there may be a Riday effect on stock return, as well.In conclusion, the present analysis suggests that a particular weekly pattern in stock returnvariation, i.e., the Monday effect as well as the Riday effect, is likely to exist in the Japanesestock market. The economic origin of such an anomaly is not unclear. However, we conjectureone possible mechanism such that the dynamics governing stock retum variation might bealtered or be different on Saturday and Sunday because of missing stock index as the benchmarkfor trading. This would be investigated in our future work.References[1] W. Tong, “International Evidence on Weekend Anomalies”, Joumal of Financial Research,23, 495522, 2000.[2] R. Wayland, D. Bromley, D. Pickett and A. Passamante, “Recognizing Determinism in aTime Series”, Phys. Rev. Lett., 70, 580-582, 1993.[3] J. Theiler, S. Eubank, A. Longtin, B. Galdrikian and J. D. Farmer, ”Testing for nonlin-earity in time series: The method of surrogate data”, Physica D, 58,77-94, 1992.[4] J. Theiler and D. Prichard, “Constrained-realization Monte-Carlo method for hypothesistesting”, Physica D, 94,221-235, 1996.[5] F. Takens, “Detecting strange attractors in turbulence”, Lecture Notes in Mathematics(Springer-Verlag), 898, 366-381, 1981.[6] T. Sauer, J. A. Yorke and M. Casdagli, Embedology’$\rangle$ , J. Stat. Phys., 65, 579-616, 1991.37[7] A. M. Fraser and H. L. Swinney, “Independent Coordinates for Strange Attractors fromMutual Information”, Phys. Rev. A, 33,1134-1140, 1986.[8] A. M. Fraser, “Information and Entropy in Strange Attractors”, IEEE Trans. InformationTheory, 35, 245-262, 1989.[9] T. Miyano, “Time Series Analysis of Complex Dynamical Behavior Contaminated withObservational Noise”, Int. J. Bifurcation and Chaos, 6, 2031-2045, 1996.38

Nonlinear Time Series Analysis of Stock Return Variation (The 8th Workshop on Stochastic Numerics)

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Nonlinear Time Series Analysis of Stock Return Variation (The 8th Workshop on Stochastic Numerics)

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