Wavele Variance, Covariance and Correlaion Analysis of BSE and NSE Indexes Financial Time Series Anu Kumar 1*, Sangeea Pan 1, Lokesh Kumar Joshi 1 Deparmen of Mahemaics, Universiy of Peroleum & Energy Sudies, Dehradun-48007, India Deparmen of Applied Mahemaics, Faculy of Engineering and Technology Gurukul Kangri Vishwavidyalaya, Haridwar-49404, India * Corresponding auhor: anu4march@gmail.com (Received February 6, 016; Acceped March 8, 016) Absrac The mosly used measure o analyze he sock marke behavior is wavele correlaion analysis. Cross-counry correlaions have been largely used o obain a saic esimae of he comovemens of acual reurns across counry. In his paper wavele based variance, covariance and correlaion analysis of BSE and NSE indexes financial ime series have been done using index daa from April 1990 o March 006. Keywords-sock markes, MODWT, financial ime series, BSE & NSE indexes, wavele variance and covariance. 1. Inroducion A ime series is a sequence of daa poins, measured ypically a successive imes, spaced a uniform ime inervals. Time series analysis comprises mehods ha aemp o undersand such ime series, ofen eiher o undersand he underlying heory of daa poins or o make forecass (predicions). Examples of ime series are he gross naional produc, seel producion, income per capia, ec. Waveles are a relaively new way of analyzing ime series. Wavele analysis is in some cases complemenary o exising analysis echniques (e.g. correlaion and specral analysis) and in cases capable of solving problems for which lile progress has been made prior o he inroducion of waveles (Percival and Walden, 000). Tradiional ime series analysis echniques can be represened as auoregressive inegraed moving average models (Bowerman and Connell, 1987; Box and Jenkins, 1976). The radiional models can provide good resuls when he dynamic sysem under invesigaion is linear or nonlinear. However, for cases in which he sysem dynamics are highly nonlinear, he performance of radiional models migh be very poor (Cichocki and Unbehauen, 1993; Weigend and Gershenfeld, 1994). The analysis of ime series has ofen been difficul when daa do no conform o well sudy heoreical conceps. One of he mos common saisical properies violaed by ime series daa is saionariy. A ime series is considered (weakly or second-order) saionary when i has a mean and auo covariance sequence ha do no vary wih ime. I is no uncommon o encouners deparures from saionariy in financial ime series. Is effecs are no limied o he mean of a financial ime series, bu may also ener ino he variance. Oher ime series exhibi a persisence of correlaion much longer han can be explained by shor memory (ARIMA) models; hey are known as long memory process. The exisence of daa, such as hese, ha defy curren saisical mehods moivaes researchers o develop beer heories and beer ools wih which o analyze hem. Anoher concep which arises in he ime series analysis is he noion of muliscale feaures i. e., an observed financial ime series may conain several phenomena, each occurring in differen ime scales (hese correspond o ranges of frequencies in he Fourier 6
domain). Wavele echniques possess a naural abiliy o decompose financial ime series ino several sub-series which may be associaed wih paricular ime scales. Hence, inerpreaion of feaures in complex financial ime series may be alleviaed by firs applying a wavele ransform and subsequenly inerpreing each individual sub-series (Kumar e al., 010, 011).. Mehodology Variabiliy and associaion srucure of cerain sochasic processes can be represened wih he help of wavele mehods on a scale-by-scale basis. For a given saionary process {X} wih variance, he wavele variance a scale have he relaionship (Saii e al., 014): 1 X (1) X Thus, as represens he conribuion of he changes a scale o he overall variabiliy of he process. Wih he help of he above relaionship he variance of a ime series can be decomposed ino componens ha are associaed o differen ime scales. Specral densiy decomposes he variance of he original series wih respec o frequency f in he similar manner he wavele variance decomposes he variance of a saionary process wih respec o he scale a h level i.e. 1 1/ 1/ f X S X df () where S(.) denoes he specral densiy funcion. By definiion he ime independen wavele variance a scale, is given by he variance of - ~ var. level wavele coefficiens W, A ime-independen wavele variance may be defined no only for saionary processes bu also for non-saionary processes wih saionary d h order differences wih local saionariy (Gallegai, 008). As he wavele filer {h l} represens he difference beween wo generalized averages and is relaed o a difference operaor, wavele variance is ime-independen in case of non-saionary processes wih saionary d h order differences, provided ha he lengh L of he wavele filer is ~ large enough. L d is a sufficien condiion o make he wavele coefficiens W, of a sochasic process saionary whose d h order backward difference is saionary. As MODWT employs circular convoluion, he coefficiens generaed by boh beginning and 1 L 1 ending daa could be spurious. Thus, if he lengh of he filer is L, here are 1 coefficiens affeced for -scale wavele and scaling coefficiens (Schleer-van and Gellecom, 014). If N L 0, hen an unbiased esimaor of he wavele variance based on he MODWT may be obained by removing all coefficiens affeced by he periodic boundary condiions using ~ ~ 1 N ~ ~ W,, where N N L 1 is he number of maximal overlap coefficiens a N L 7
scale and L 1L 11 is he lengh of he wavele filer for level. Thus, he h scale wavele variance is simply he variance of he non-boundary or inerior wavele coefficiens a ha level. Scaling of BSE index financial ime series has been done and shown ha i is monofracal and can be represened by a fracional Brownian moion (Razadan, 00). The MODWT-based esimaor has been shown o be superior o he DWT-based esimaor alhough boh can decompose he sample variance of a ime series on a scale-by-scale basis via is squared wavele coefficiens. To deermine he magniude of he associaion beween wo financial ime series of observaions X and Y on a scale-by-scale basis he noion of wavele covariance is used. The wavele covariance a wavele scale can be defined as he covariance beween scale wavele coefficiens of X and Y, i.e. ~ ~ X ~ Y cov W (3) XY, W,, An unbiased esimaor of he wavele covariance using MODWT can be obain by removing all wavele coefficiens affeced by boundary condiions and given by 1 ~ ~, ~ 1 N ~ N X Y XY W, W, (4) N L 1 The MODWT esimaor of he wavele cross-correlaion coefficiens for scale and lag may be obained by making use of he wavele cross-covariance ~, XY,, and he square roo of he wavele variances ~ and ~ Y, by ~ ~, XY,, XY, ~ ~ (5) Y, The wavele cross-correlaion coefficiens ~, XY,, us as he usual uncondiional crosscorrelaion coefficiens are beween 0 and 1 and provide he lead/lag relaionships beween he wo processes on a scale-by-scale basis. Saring from specrum S of scale wavele coefficiens, i is possible o deermine he asympoic variance V of he MODWT-based esimaor of he wavele variance and consruc a random inerval which forms a 100(1-p) % confidence inerval. 3. Resuls and Conclusion Here, he auhors have presened he variance of a process on a scale basis wih he help of wavele analysis. Plo of ~ agains scale indicaes which scales conribue more o he process variance. Fig. 1 shows he MODWT-based variance of he BSE Index and NSE Index ploed on a log-log scale. In his figure he sraigh line U and L represen he upper and lower bounds for he 95% approximae confidence inerval and he sraigh line shows he valued wavele variance. Reflecion boundary condiion has been applied for he calculaion of wavele 8
variance. Due o his, we have sufficien number of non-boundary coefficiens o approximae wavele variance up o scale 6. As wavele analysis have he abiliy o decompose a financial ime series ino is ime scale componens. I is also advanageous in analyzing condiions in which he degree of associaion beween wo financial ime series is likely o change wih he ime-horizon. The lead/lag relaionship beween wo financial ime series of BSE and NSE has been analyzed on a scale-by-scale source by using wavele cross-correlaion analysis. Fig. 1. Wavele variance for he BSE index and NSE index a log-log scale Scale Fig.. Wavele specrum of NSE (curve a) and BSE (curve b) indices 9
BSE Sensex Value 1000 11000 10000 9000 8000 7000 6000 5000 4000 3000 BSE Index Time Series NSE Sensex Value 6000 5500 5000 4500 4000 3500 3000 500 000 1500 NSE Index Time Series 000 1000 1000 500 0 0 1-Nov-91 1-Jul-93 1-Mar-95 1-Nov-96 1-Jul-98 Time 1-Mar-00 1-Nov-01 1-Jul-03 1-Mar-05 -- 1-Nov-91 1-Jul-93 1-Mar-95 1-Nov-96 1-Jul-98 Time 1-Mar-00 1-Nov-01 1-Jul-03 1-Mar-05 -- Fig. 3. Financial ime series of BSE index Fig. 4. Financial ime series of NSE index Fig. shows he Wavele specrum of NSE and BSE indices and Fig. 3 & Fig. 4 shows he financial ime series of BSE and NSE indexes. Fig. 5 exhibis he MODWT-based wavele correlaions and cross-correlaion coefficiens wih he corresponding imprecise confidence inervals. For example, scale 1 is associaed o 4 monh periods, scale o 4 8 monh periods, scale 3 o 8 16 monh periods, and so on. The magniude of he associaion beween he wo variables a he shores scales; i.e. scales 1 o ; is generally close o zero a all leads and lags, whereas a scales 4 and 5, such connecion become sronger. There is a low magniude of associaion beween BSE and NSE indexes a scales 3 and 4 as he value of wavele correlaion coefficien a lag zero indicaes. On he oher hand, he cross-correlaion wavele coefficiens 0.5 and 0.7 a scale 4 and 5 reveal high posiive foremos relaionship beween BSE and NSE indexes wih he leading period increasing as he ime scale increases. I is clear ha he larges cross-correlaion coefficiens going on a leads 6 for wavele scale 4, ha is 16 3 monh periods, and 10 for wavele scale 5, ha is 3 64 monh periods. Table 1 and Table represens he monhly average of BSE and NSE Sensex closing index The averages are based on daily BSE Sensex closing index. (Base: 1978-79 = 100) The averages are based on daily closing index. (Base: 1983-84 = 100) So, i is clear from he above discussion ha correlaion beween BSE and NSE indices is scale dependen. 30
Fig. 5. Wavele cross-correlaions beween BSE and NSE indexes 31
Year/Monh Apr. May Jun. Jul. Aug. Sep. Oc. Nov. Dec. Jan. Feb. Mar. 1 3 4 5 6 7 8 9 10 11 1 13 1990-91 780.18 785.57 80.45 938 1116.19 1307.87 1354.0 1306.09 1161.87 996.45 1100.78 1180.7 1991-9 155.5 191.74 195.15 1440.7 173.8 1833.34 1789.5 1890.09 187.31 073.6 464.74 3487.19 199-93 4131.01 3366.55 3088.59 797.7 89.96 343.19 3075.8 618. 535.64 53.86 708.7 398.7 1993-94 05.37 48.01 81.95 190.34 556.16 708.39 688.51 850.35 3301.85 3813.74 4039.4 3811.5 1994-95 384.75 3756.1 4135.67 4106.95 4407.4 4511.34 4351.16 4139.06 3949.78 3651.59 3474.9 3408.9 1995-96 3359.9 306.86 3336.46 3334.86 340.81 3396.37 358.1 317.0 3060.05 979.3 3405.56 337.33 1996-97 3599.66 373. 3906.7 3668.1 3449.17 3390.11 3159.79 3044.8 918.68 3410.3 3453.4 376.5 1997-98 3681.5 3740.95 4001.47 456.11 476.31 3944.79 3991.75 3611.83 3515.54 347.87 3413.49 3816.87 1998-99 4114.66 3911.95 3317.49 371.73 988.4 3089.88 866.55 91.39 945.99 375.05 389.4 3689.4 1999-00 3455.05 3880.37 4066.84 456.5 466.84 474.96 4835.47 4588.53 480.0 5407.14 5650.66 3689.4 000-01 4905.3 453.11 4675.4 4647.34 4330.31 4416.61 3819.69 398.1 4081.4 415.39 4310.13 561.77 001-0 3480.94 3613.84 3439.01 3346.88 3304.99 918.8 933.55 3164.5 3314.88 3353.31 358.58 3807.64 00-03 3435.13 330.91 357.03 314.87 3053.16 3085.53 949.76 3058.19 3315.84 337.66 378.85 3580.73 003-04 3036.66 3033.47 3386.89 3665.46 3977.86 4314.74 474.3 4951.1 544.67 5954.15 586.74 3155.7 004-05 5809.01 504.65 483.87 497.88 5144.17 543.7 5701.61 5960.75 6393.83 6300.76 6595.05 6679.18 005-06 6379.9 648.7 695.86 7336.7 776.03 87.3 80.45 855.09 916.07 9539.67 10090.08 10857.03 Table 1. Monhly average of BSE sensex Year/Monh Apr. May Jun. Jul. Aug. Sep. Oc. Nov. Dec. Jan. Feb. Mar. 1 3 4 5 6 7 8 9 10 11 1 13 1990-91 417.99 43.79 48.68 486 573.49 669.06 67.79 644.79 573.77 506.56 553.4 593.69 1991-9 67.7 641.3 639.17 703.0 85.33 87.98 853.77 898.09 880.5 960.14 1138.17 1579.04 199-93 1850.94 1481.01 1351.06 167.88 160.8 1444.95 1376.05 1194.63 116.9 1160.0 11.6 1081.17 1993-94 993.63 109.84 1057.45 1030.1 1199.31 183.5 19.7 1368.88 1589.5 187.17 1945.4 184.8 1994-95 1855.81 18.5 1967.76 1947.56 080.67 133.49 054.54 1968.58 1876.13 1755.38 1683.04 1658.97 1995-96 1631.55 1539.44 1570.48 1550.57 1568.33 1555.07 1603.84 144.44 1406.95 1369.94 1556.09 1539.14 1996-97 1649.6 1701.15 1771.88 1676.6 1575.49 15.67 1409.83 1356.4 190.1 150.66 1504.97 169.43 1997-98 1586.13 1610.98 1716.56 1844.63 1863.6 1717.5 17.58 1563.46 155.78 151.7 1467.54 1654.9 1998-99 1804.55 178.93 1459.7 1437.4 1333.8 1371.49 181.38 198.19 1307.34 145.71 1450.6 160.74 1999-00 1506.84 168.65 1755.07 1960.83 075.59 156.8 7.13 161.39 49.71 8.05 3394.88 3109.03 000-01 663.53 10.93 334.7 344.9 180.79 49.43 1931.61 017.59 113.84 140.09 03.99 189.3 001-0 1641.89 1753.46 1661.6 157.67 1559.95 1373.77 1357.64 1486.33 1587.9 1601.9 1711.43 1746.78 00-03 1715.11 1661.1 1658.78 163.07 1536.74 151.96 1466.79 1510 163.19 164.07 16.58 1559.54 003-04 1504.6 1538.65 179.15 1843.86 055.64 4.97 43.87 543.09 813.58 314.3 3003.89 956.07 004-05 3101.76 77.81 563.78 653.04 748.3 908.81 3049.8 3188.9 3455.8 3403.5 3558.11 3595. 005-06 3431.03 3483.19 3697. 390.1 4139.35 4407.48 4353.07 4508.71 485.99 5048.54 5303.59 5686.04 References Table. Monhly average of NSE index Bowerman, B. L. & O Connell, R. T. (1987). Time series forecasing. New York PWS. Box, G. E. P. & Jenkins, G.M. (1976). Time series analysis, forecasing, and conrol. San Francisco, CA: Holden-Day. Cichocki, A. & Unbehauen, R. (1993). Neural neworks for opimizaion and signal processing. New York: Wiley. Gallegai, M. (008). Wavele analysis of sock reurns and aggregae economic aciviy. Compuaional Saisics & Daa Analysis, 5(6), 3061-3074. Kumar, A., Joshi, L. K., Pal, A. K. & Shukla, A. K. (010). A new approach for generaing parameric orhogonal wavele. Journal of Wavele Theory and Applicaions, 4(1), 1-8. Kumar, A., Joshi, L. K., Pal, A. K. & Shukla, A. K. (011). MODWT based ime scale decomposiion analysis of BSE and NSE indexes financial ime series. Inernaional Journal of Mahemaical Analysis, 5(7), 1343-135. Percival, D. B. & Walden, A. T. (000). Wavele mehods for ime series analysis. Cambridge Universiy Press. Razadan, A. (00). Scaling in he Bombay sock exchange index. Pramana, 58(3), 537-544. 3
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