University of Pretoria Department of Economics Working Paper Series

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1 University of Pretoria Department of Economics Working Paper Series A Historical Analysis of the US Stock Price Index using Empirical Mode Decomposition over Aviral K. Tiwari IFHE University Arif B. Dar University in Ghaziabad Niyati Bhanja University of Petroleum & Energy Studies Rangan Gupta University of Pretoria Working Paper: 1-88 November 1 Department of Economics University of Pretoria, Pretoria South Africa Tel:

2 A Historical Analysis of the US Stock Price Index using Empirical Mode Decomposition over Aviral K. Tiwari *, Arif B. Dar **, Niyati Bhanja ***, and Rangan Gupta **** Abstract In this paper, the dynamics of Standard and Poor's (S&P ) stock price index is analysed within the time-frequency framework over a monthly period of 1791:8-1:. Using the Empirical Mode Decomposition technique, the S&P stock price index is divided into different frequencies known as intrinsic mode functions (IMFs) and one residual. The IMFs and the residual are then reconstructed into high frequency, low frequency and trend components using the hierarchical clustering method. Using different measures, it is shown that the low frequency and trend components of the stock prices are relatively important drivers of the S&P index. These results are also robust across various subsamples identified based on structural break tests.the US stock prices are, therefore, mostly driven by fundamental laws rooted in economic growth and long-term returns on investment. JEL Classification: C, G1 Keywords: Empirical Mode Decomposition, Stock Prices, S&P Index; United States * Faculty of Management, IBS Hyderabad, IFHE University, Donthanapally Shankarapalli Road, Hyderabad, Andhra Pradesh 13, India. aviral.eco@gmail.com. ** Institute of Management Technology, Rajnagar, Ghaziabad, Delhi, 11, India. billaharif@gmail.com. *** Department of Economics and IB UPES, Dehradun India niyati.eco@gmail.com. **** Corresponding author. Department of Economics, University of Pretoria, Pretoria,, South Africa. rangan.gupta@up.ac.za. 1

3 1. Introduction In recent years, the analysis of stock prices within the time-frequency framework has attracted lot of attention from the academicians and market practitioners. The intrinsic complexities of the stock markets have made them least analysis worthy using the conventional time-domain tools. The obvious reason being that the stock prices are determined by the traders who deal at different frequencies. While institutional investors and central banks constitute as the low-frequency traders, speculators and market makers fit in the category of high-frequency traders in the stock markets. The price formation in the stock markets can be attributed to the trading of heterogeneous traders with different frequencies. Therefore, some appealing events may remain hidden under different frequencies if stock prices are analysed within the time-domain framework. In the literature of financial economics, number of frequency based approaches have been used to unravel the hidden characteristics of financial time series. Zhang et al. (8) have used the Empirical Mode Decomposition (EMD) to unravel the price characteristics of crude oil at different frequencies. Zhu et al. (1) have analysed the price formation in the carbon markets by using the EMD. Using wavelet based approaches, studies like Tiwari et al. (1, 13), and Chang et al. (1) and references cited therein have used the wavelet decompositions to study the behaviour of financial variables like, oil prices, exchange rates, inflation and stock prices at different frequencies. Nevertheless, EMDs till date have not been used to study the behaviour of stock prices. In this paper, we therefore try to study, for the first time, conduct a time-frequency analysis using EMDs for the Standard & Poor's (S&P ) index covering a long monthly sample of 1791:8-1:. This will allow us to determine, the various frequency components that drive stock prices in the US over a prolonged historical sample. EMDs have an advantage over the wavelets because their decompositions are based on local characteristic time scales and have characteristics of self-adapting. We try to identify the frequencies which have a substantial impact on the stock prices. The Ensemble EMD (EEMD) introduced by Huang et al. (1998) is used to decompose the stock price data into different intrinsic modes. The IMFs and the residual extracted are then reconstructed into high-frequency, low-frequency and trend components using the hierarchical clustering method. Then different measures are used to assess the importance of each frequency for the overall stock prices series.

4 The rest of the scheme in this paper organized as: Section provides the information about the methodology followed in the paper. Section 3 provides the discussion of the data and the results, and Section 4 concludes with the main findings.. Methodology 1 An EMD algorithm for extracting Intrinsic Mode Functions (IMFs) is followed as: In the first step, the minima and maxima of a time series x(t) are identified. Then with the cubic spline interpolation upper e min (t) and lower e max (t) envelopes are generated. In the third step, the point-by- point mean (m(t)) is calculated from lower and upper envelopes as: m (t) = (e min (t) + e max (t)) /. The mean form the time series is calculated in step 4 and d (t) as the difference of x (t) and m (t) is calculated as: d (t) = x (t) m (t). The properties of d (t) are checked in step. If, for example, it is an IMF, the ith IMF is denoted by d (t). The x (t) is replaced by the residual given as: r (t) == x (t) d (t). Often the ith IMF is denoted by c i (t) where I is interpreted as index. If d (t) is not an IMF, it is replaced by d (t). These five steps are repeated till the residuals satisfy some conditions known as stopping criteria. Contrary to the EMD, Ensemble EMD proposed by Wu and Huang (9) avoids the limitation of mode mixing associated with EMD. The procedure involves an additional step of adding white noise series too targeted data followed by the decomposition to generate the IMFs. The procedure is repeated by adding different white noise series each time to generate the Ensemble IMFs from the decompositions as an end product. 3. Results and Discussion Our analysis is based on a historical data set of US stock prices. The monthly data on the S&P, covering the period of 1791:8 to 1:, has obtained from the Global Financial Database (GFD). The natural logarithmic values of the data have been plotted in Figure A1 in the Appendix. Through EEMD, four data samples of the US stock prices are decomposed into (IMFs) and residuals. The data sets include the full sample ranging from 1791:8 to 1: and threesub samples of 1791:8 to 186:1, 1863:1 to 194:4 and 194: to 1:. The subsamples are identified by applying Bai and Perron (3) test of structural breaks in both mean and trend of the natural logarithms of the S&P stock index. The division of data in 1 For more on methodology please refer to Zhu et al. (1). For the stopping criteria please refer to Zhang et al. (8). 3

5 to three sub-samples thus gives a better idea of how the dynamics of the US stock market has evolved over time as well as adds to the robustness of the results. The IMFs along with the residual are shown in Figures A, A3, A4 and A in the Appendix. The IMFs are generated in the order from highest to lowest frequency. The IMFs are then analysed by three measures. First, the mean period of each IMF, defined as the value extracted by dividing total number of points by the number of peaks in the dataset is calculated. Second, the pairwise correlation between the original data series and the IMFs is estimated by using Pearson and Kendall rank correlation. Third, the variance and variance percentage of each IMF is calculated. These results are shown in Tables 1,, 3 and 4 For example, it can be seen that both the Pearson and Kendall coefficients between the original and high frequency IMFs is low. The correlation however is higher between the low frequency IMFs and the original series. It can also be seen that the variances of lower (higher) frequencies contribute Table 1.Measures of IMFs and residuals with full sample (1791:8-1:) Mean Pearson Kendall Variance Variance as a % of Observed Variance as % of (ΣIMFs +residual) Original Series IMF IMF IMF3-8.76E IMF4 -..3*.4*** IMF.8.91***.7*** IMF ***.3*** IMF ***.13*** IMF ***.*** IMF ***.4*** IMF *** -.*** Residual ***.84*** SUM Table. Measures of IMFs and residuals for the sub-sample 1791:8-186:1 Mean Pearson Kendall Variance Variance as a %ge of Observed Original Series E- IMF1 6.9E-.73**.4* IMF.3.137***.7*** IMF ***.198*** IMF4.1.87***.1938*** IMF.14.3***.313***.97E IMF6-6.3E-.7***.449** Variance as % of (ΣIMFs +residual) 4

6 IMF7.3.44***.367** IMF ***.34*** 6.47E Residual.96.7***.337*** SUM Table 3. Measures of IMFs and residuals for the sub-sample 1863:1-194:4 Original Series Mean Pearson Kendall Variance Variance as a %ge of Observed IMF1-3.4E-6.4* IMF **.34** IMF3 1.76E-.1496***.9*** IMF ***.18*** IMF ***.18*** IMF ***.169*** IMF ***.1376*** IMF Residual SUM Variance as % of (ΣIMFs +residual) Table 4. Measures of IMFs and residuals for the sub-sample 194:-1: Original Series Mean Pearson Kendall Variance Variance as a %ge of Observed IMF IMF IMF IMF4.96.9***.8*** IMF ***.16*** IMF *** -.1*** IMF ***.*** IMF *** -.469*** Residual ***.916*** SUM Variance as % of (ΣIMFs +residual) substantially (lesser) to the total variability. Within these decompositions, however, the residues are the dominant modes. Their contribution to the total variability is highest and the correlation with the original data series is also highest. The residue referred as the deterministic long term trend by Huang et al. (1998) indicate very high correlation and also account for very high variability in the original series. A noteworthy observation here is that the correlation of long term trend with the data and the variability contribution increases for

7 the more recent samples. Since, the continuing increasing trend of US stock market is consistent with development of the US economy over the decades, it can be said that the long term price behaviour of US stocks is determined by the long term growth of the US. We then use the hierarchical clustering analysis and subsequently the Euclidean distance to group the IMFs and residuals into the high-frequency, low-frequency and trend components. 3 The extracted components for all the time series are shown in Figure 1.. Full sample: 1791:8-1: 1791:8-186: High Frequency component Low Frequency component SP Trend High Frequency component Low Frequency component SP Trend 1863:1-194:4 194: 1: High Frequency component Low Frequency component SP Trend High Frequency component Low Frequency component SP Trend Figure 1.Three components of the S&P Each component in these diagrams shows the distinct features. For example, the residuals show the slow variation around the long term trend. Hence, it is considered as a long term trend of a time series. The effect of medium to high frequencies is captured by other two frequencies, with high frequency components reflecting the effect of short term market fluctuations. For the moment of observed stock price series, the most important components are the low frequency component and the trend. The Pearson and Kendall correlation between 3 We have followed Zhu et al. (13) to extract the different time series components. For the sake of brevity we are not showing the results here, nevertheless, can be produced on demand. 6

8 the different frequency components and the original series shown in Table varies between samples. For example, it is comparatively higher for lower frequency and trend components of a time series especially during the recent periods. This holds for the variance contribution also. The variance contribution is relatively more from the low frequency and trend components of the time series. This is especially true for the more recent periods. The results obtained are robust to the sub-samples. In nutshell, we do not find any evidence of the US stock prices being driven by short term irrational behaviour. Our results support the view that the US stock market is driven mostly by fundamentals, which, in turn, are most likely rooted in economic growth and long term returns on investment (Rapach and Zhou, 13). Table.Correlation and variance of components for the S&P index Full Sample: 1791:8-1: Pearson Kendall Variance as % of Variance as % of Variance Mean Correlation Correlation observed (ΣIMFs +residual) ORIGINAL_SERIES HFRQ.9.18***.19*** LFRQ ***.8*** RESIDUAL ***.84*** :8-186:1 Pearson correlation Kendall correlation Mean Variance Variance as % of observed Variance as % of (ΣIMFs +residual) ORIGINAL_SERIES HFRQ.9419***.77*** LFRQ.73***.3373*** RESIDUAL.7***.3373*** :1-194:4 Pearson correlation Kendall correlation Mean Variance ORIGINAL_SERIES Variance as % of observed HFRQ.334***.19*** LFRQ.974***.83*** RESIDUAL.877***.686*** Variance as % of (ΣIMFs +residual) 7

9 194: 1: Pearson correlation Kendall correlation Mean Variance Variance as % of observed ORIGINAL_SERIES..4 HFRQ LFRQ.9997***.918*** RESIDUAL.983***.9196*** Variance as % of (ΣIMFs +residual) 4. Conclusion In this note, the data of S&P index was decomposed into number of IMFs and residuals using the EEMD. The monthly data sets include the full sample ranging from 1791:8 to 1: and three sub samples of 1791:8 to 186:1, 1863:1 to 194:4 and 194: to 1: for the US stock prices. The division of data in to three sub-samples gives a better idea of how the dynamics of the US stock market has evolved over time, as well as the robustness of the results. The IMFs were generated in the order from highest- to lowestfrequency. The IMFs were analysed by three measures of mean, correlation with the original series and the contribution to the variability of original series. It was shown that the residuals and low frequency IMFs indicate a very high correlation and also account for very high variability in the original series. Also, it was found that the correlation of long term trend with the data and the variability contribution increases for the more recent samples. The IMFs and residuals were reconstructed into high-frequency, low-frequency and trend components for the same full and sub samples. Again it was found that thepearson and Kendall correlation is comparatively higher for the lower-frequency and trend components of a time series especially during the recent periods. The variance contribution was also found to be relatively more from the low-frequency and trend components of the time series. The sub-sample results were found to be corroborating the full-sample results. It was concluded that, in general, the US stock prices are not driven by short term irrational behaviour of investors, but seems to be driven mostly by fundamentals, though, it is true that there have been episodes of bubbles as indicated by Phillips et al. (1). References Bai, J., and Perron, P., (3). Computation and analysis of multiple structural change models. Journal of Applied Econometrics, 18, 1. 8

10 Chang, T., Li, X-L., Miller, S. M., Balcilar, M., and Gupta, R. (1). The Co-Movement and Causality between the U.S. Real Estate and Stock Markets in the Time and Frequency Domains. International Review of Economics and Finance, 38 (1), -33. Huang, N. E., Shen, Z., and Long, S. R. (1998).The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis.proceedings A of the Royal Society of London, 44(1971), Phillips, P.C.B., Shi, S-P., and Yu, J., (1). Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S&P. International Economic Review, 6 (4), Rapach, D., Zhou, G., 13. Forecasting stock returns. In: Elliott, G., Timmermann, A., Eds. Handbook of Economic Forecasting, , Amsterdam: Elsevier. Tiwari, A. K., Dar, A. B., and Bhanja, N. (13a). Oil price and exchange rates: A wavelet based analysis for India. Economic Modelling, 31(1), Tiwari, A. K., Dar, A. B., and Bhanja, N. (13). Stock market integration in Asian countries: Evidence from wavelet multiple correlations. Journal of Economic Integration, 8(3), Wu, Z., and Huang, N.E., (9). Ensemble Empirical Mode Decomposition: a Noiseassisted Data Analysis Method. Advances in Adaptive Data Analysis, 1 (1), Zhang, X., Lai, K. K., and Wang, S. Y. (8). A new approach for crude oil price analysis based on empirical mode decomposition. Energy Economics, 3, Zhu, B., Wang, P., Chevallier, J., and Wei, Y. (1). Carbon Price Analysis using Empirical Mode Decomposition. Computational Economics, 4,

11 Appendix M8 18M4 181M1 183M8 1834M4 1844M1 18M8 1866M4 1876M1 1887M8 1898M4 198M1 1919M8 193M4 194M1 191M8 196M4 197M1 1983M8 1994M4 4M1 S&P Figure A1: Natural Logarithms of S&P Index (1791:8-1:) signal IM F 1 IM F IM F3 IM F 4 IM F IM F 6 IM F 7 IM F8 IM F 9 IM F 1 re s. EEMD 1 1 Figure A. IMFs for the Full-Sample (1791:8-1:) 1

12 IMF8 IMF7 IMF6 IMF IMF4 IMF3 IMF IMF1 signal EEMD res Figure A3. IMFs for the period 1791:-186:1 Amplitude IMF IMF4 IMF3 IMF IMF1 signal EEMD IMF6. -. IMF7. -. IMF res Figure A4. IMFs for the 1863:1-194:4 11

13 IMF8 IMF7 IMF6 IMF IMF4 IMF3 IMF IMF1 res. signal EEMD Figure A. IMFs for the period 194:-1: 1