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Wavelet analysis of financial datasets

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The document discusses using wavelet analysis techniques to identify hidden patterns in financial time series data. It presents the wavelet transform approach for filtering time series data and decomposing it into different time scales. Several examples are given showing how wavelet analysis can be used to study structural patterns in datasets like NASDAQ and for momentum-based trading strategies.

Economy & Finance
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Recognition of hidden patterns in financial datasets The wavelet approach Boryana Bogdanova, PhD Department of Statistics and Econometrics, Faculty of Economics and Business Administration, Sofia University bpelova@gmail.com https://www.researchgate.net/profile/Boryana _Bogdanova
Wavelet analysis of financial datasets
Popular filters for financial time series
Popular filters for financial time series
Financial data through the lens of the wavelet transform 6 months ………………… 2 years 1 day ………………… Time series
Wavelet analysis of financial datasets
Wavelet filtering • The function 𝜓 𝑡 is a wavelet if: • 𝜓 𝑡 is decaying sufficiently fast; • Ψ 0 = −∞ ∞ 𝜓 𝑡 𝑑𝑡 = 0
Some typical wavelets
Main reference and toolbox Aguiar-Conraria, L. & Soares, M., 2014. The continuous wavelet transform: moving beyond uni-and bivariate analysis. Journal of Economic Surveys, 28(2), pp. 344-75.
5 year cyclic component 2-year cyclic component
5 year cyclic component 2-year cyclic component
5 year cyclic component 2-year with a break to 3-year cyclic component
5 year cyclic component 2-year with a break to 3-year cyclic component
The Continuous Wavelet Transform 𝜓 𝑡 𝜓𝜏,𝑠 = 1 𝑠 𝜓 𝑡 − 𝜏 𝑠 , 𝑠, 𝜏 ∈ ℝ, 𝑠 ≠ 0 𝑊𝑥;𝜓 𝜏, 𝑠 = −∞ ∞ 𝑥 𝑡 1 𝑠 𝜓∗ 𝑡 − 𝜏 𝑠 𝑑𝑡
𝑊𝑃𝑆 𝑥 = 𝑊𝑥 2
Case I: NASDAQ structural patterns
Case II: Momentum analysis J K Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners 0.0323 7.1447 0.0000 0.0247 7.0244 0.0000 0.0229 7.5064 0.0000 0.0233 8.4151 0.0000 0.0226 8.8303 0.0000 0.0190 7.9918 0.0000 Losers 0.0177 4.7705 0.0000 0.0211 6.6860 0.0000 0.0223 7.8883 0.0000 0.0200 8.2088 0.0000 0.0199 8.4945 0.0000 0.0175 7.8737 0.0000 WML 0.0146 2.6522 0.0086 0.0036 0.9848 0.3259 0.0006 0.2507 0.8023 0.0033 2.0835 0.0385 0.0027 2.3467 0.0200 0.0015 1.7920 0.0751 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners 0.0269 6.3134 0.0000 0.0236 6.7914 0.0000 0.0223 7.2336 0.0000 0.0232 8.3037 0.0000 0.0230 8.8314 0.0000 0.0192 8.0397 0.0000 Losers 0.0220 6.3427 0.0000 0.0231 7.1983 0.0000 0.0217 7.6831 0.0000 0.0194 8.2568 0.0000 0.0193 8.4815 0.0000 0.0169 7.9192 0.0000 WML 0.0049 0.9608 0.3378 0.0005 0.1331 0.8942 0.0006 0.2159 0.8293 0.0039 1.8690 0.0632 0.0037 2.3830 0.0182 0.0022 2.1851 0.0304 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners 0.0204 5.3413 0.0000 0.0187 5.6522 0.0000 0.0212 6.5992 0.0000 0.0224 7.9281 0.0000 0.0227 8.2819 0.0000 0.0190 7.6339 0.0000 Losers 0.0228 6.5782 0.0000 0.0221 6.7299 0.0000 0.0185 6.7247 0.0000 0.0164 7.0421 0.0000 0.0173 8.0145 0.0000 0.0154 7.7591 0.0000 WML -0.0024 -0.5154 0.6068 -0.0034 -0.8658 0.3876 0.0027 0.8396 0.4021 0.0060 2.6236 0.0094 0.0054 2.7814 0.0060 0.0036 2.6212 0.0096 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners 0.0241 6.0682 0.0000 0.0225 6.5348 0.0000 0.0225 6.8938 0.0000 0.0239 7.7602 0.0000 0.0236 8.0562 0.0000 0.0186 7.3310 0.0000 Losers 0.0181 5.6930 0.0000 0.0180 5.9843 0.0000 0.0143 5.7088 0.0000 0.0144 6.4174 0.0000 0.0153 7.0734 0.0000 0.0133 6.6971 0.0000 WML 0.0060 1.3296 0.1852 0.0045 1.1541 0.2498 0.0082 2.4471 0.0153 0.0094 3.1704 0.0018 0.0083 3.2157 0.0016 0.0054 2.8855 0.0045 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners 0.0271 7.0184 0.0000 0.0257 7.2742 0.0000 0.0248 7.1517 0.0000 0.0239 7.2276 0.0000 0.0238 7.4947 0.0000 0.0179 6.5884 0.0000 Losers 0.0140 5.4076 0.0000 0.0145 5.7164 0.0000 0.0140 5.9406 0.0000 0.0135 6.0022 0.0000 0.0142 6.6320 0.0000 0.0116 5.7974 0.0000 WML 0.0131 3.1493 0.0019 0.0112 2.9654 0.0034 0.0108 2.9730 0.0033 0.0104 3.2030 0.0016 0.0095 3.3357 0.0010 0.0063 2.8391 0.0052 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners 0.0237 5.9885 0.0000 0.0227 6.0682 0.0000 0.0234 6.4116 0.0000 0.0231 6.7986 0.0000 0.0217 6.7706 0.0000 0.0142 5.3461 0.0000 Losers 0.0114 4.7190 0.0000 0.0117 4.9251 0.0000 0.0109 4.7895 0.0000 0.0096 4.4271 0.0000 0.0095 4.5935 0.0000 0.0073 4.0119 0.0001 WML 0.0123 3.0589 0.0026 0.0110 3.0281 0.0028 0.0125 3.6440 0.0004 0.0134 4.2987 0.0000 0.0121 4.2779 0.0000 0.0069 2.8628 0.0049 1 2 4 8 1 2 4 8 13 26 13 26
Case II: Momentum analysis J K Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners 0.0039 1.4443 0.1499 -0.0022 -0.9564 0.3398 -0.0037 -1.8617 0.0638 -0.0040 -2.2884 0.0230 -0.0030 -1.8100 0.0716 -0.0024 -1.4614 0.1454 Losers -0.0017 -0.7157 0.4748 0.0029 1.2795 0.2019 0.0021 0.9777 0.3292 0.0003 0.1269 0.8991 -0.0005 -0.2589 0.7959 -0.0014 -0.7384 0.4611 WML 0.0056 1.9243 0.0554 -0.0051 -2.4068 0.0168 -0.0058 -4.9694 0.0000 -0.0042 -4.2570 0.0000 -0.0025 -2.6254 0.0092 -0.0010 -1.5703 0.1179 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners -0.0051 -2.1201 0.0350 -0.0075 -3.8342 0.0002 -0.0063 -3.4597 0.0006 -0.0051 -2.9606 0.0034 -0.0037 -2.2322 0.0265 -0.0025 -1.5228 0.1293 Losers 0.0043 1.7003 0.0903 0.0057 2.3520 0.0194 0.0029 1.2652 0.2070 0.0012 0.5501 0.5827 0.0003 0.1352 0.8925 -0.0011 -0.5713 0.5684 WML -0.0094 -3.2411 0.0013 -0.0133 -6.3158 0.0000 -0.0092 -5.8404 0.0000 -0.0063 -4.7560 0.0000 -0.0040 -3.4845 0.0006 -0.0014 -1.9590 0.0514 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners -0.0080 -3.1651 0.0017 -0.0089 -4.2518 0.0000 -0.0074 -3.9632 0.0001 -0.0047 -2.7821 0.0058 -0.0033 -2.0078 0.0458 -0.0021 -1.2909 0.1982 Losers 0.0068 2.6630 0.0082 0.0069 2.7406 0.0066 0.0033 1.3894 0.1660 0.0020 0.9015 0.3682 0.0002 0.1177 0.9064 -0.0011 -0.5600 0.5761 WML -0.0148 -5.1928 0.0000 -0.0159 -7.2208 0.0000 -0.0107 -5.9253 0.0000 -0.0067 -4.6062 0.0000 -0.0036 -2.9956 0.0030 -0.0010 -1.1963 0.2330 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners -0.0090 -4.3041 0.0000 -0.0098 -4.9768 0.0000 -0.0070 -3.8889 0.0001 -0.0047 -2.8312 0.0050 -0.0032 -2.0262 0.0439 -0.0021 -1.3317 0.1845 Losers 0.0074 2.7496 0.0064 0.0065 2.4111 0.0166 0.0037 1.4941 0.1364 0.0012 0.5025 0.6158 -0.0008 -0.3786 0.7053 -0.0010 -0.5246 0.6004 WML -0.0165 -6.1991 0.0000 -0.0163 -6.7317 0.0000 -0.0107 -4.9861 0.0000 -0.0058 -3.0903 0.0022 -0.0024 -1.6427 0.1018 -0.0010 -0.9037 0.3672 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners -0.0071 -3.7641 0.0002 -0.0074 -4.2318 0.0000 -0.0051 -3.0222 0.0028 -0.0033 -2.0159 0.0450 -0.0021 -1.3588 0.1756 -0.0013 -0.8516 0.3955 Losers 0.0051 1.9639 0.0507 0.0043 1.7052 0.0894 0.0024 1.0177 0.3098 -0.0003 -0.1345 0.8931 -0.0012 -0.5341 0.5938 -0.0006 -0.3217 0.7480 WML -0.0123 -4.7470 0.0000 -0.0118 -4.9468 0.0000 -0.0075 -3.4206 0.0007 -0.0030 -1.4182 0.1575 -0.0010 -0.5275 0.5984 -0.0007 -0.4486 0.6542 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners -0.0030 -1.5814 0.1151 -0.0041 -2.2877 0.0231 -0.0031 -1.7892 0.0749 -0.0015 -0.9007 0.3687 -0.0004 -0.2703 0.7872 -0.0002 -0.1689 0.8661 Losers 0.0026 0.9858 0.3253 0.0025 0.9875 0.3244 0.0011 0.4463 0.6558 0.0004 0.1673 0.8673 0.0001 0.0387 0.9692 0.0011 0.5721 0.5679 WML -0.0056 -2.0451 0.0420 -0.0067 -2.5423 0.0117 -0.0042 -1.7141 0.0879 -0.0019 -0.8374 0.4033 -0.0005 -0.2455 0.8063 -0.0013 -0.7370 0.4620 1 2 4 8 1 2 4 8 13 26 13 26
Case II: Momentum analysis
Case II: Momentum analysis
Thank you for your attention! Ivanov, I., Kabaivanov, S., Bogdanova, B., 2016. Stock market recovery from the 2008 financial crisis: The differences across Europe. Research in International Business and Finance, 37(May 2016), pp. 360-74. http://www.sciencedirect.com/science/article/pii/S027553191630006X Impact metrics: 2016 SJR (Scopus): 0.694; 2016 Source normalized impact per paper: 1.819 Bogdanova, B. & Ivanov, I., 2015. A wavelet-based appraoch to the analysis and modelling of financial time series exhibiting strong long-range dependence: the case of Southeast Europe. Journal of Applied Statistics, 43(4), pp.655-73. http://www.tandfonline.com/doi/full/10.1080/02664763.2015.1077370 Impact metrics: 2016 Impact factor (Thomson Reuters): 0.664 Bogdanova, B., 2015. A wavelet-based discussion on the Greek stock market integration during the last decade. Journal of Engineering Science and Technology Review, 8(1), Special Issue on Econophysics, pp. 8-11. http://www.jestr.org/downloads/Volume8Issue1/fulltext28115.pdf Journal of Engineering Science and Technology Review is indexed SCOPUS, EBSCO, DOAJ, CERN, ACS. Bogdanova, B. & Ivanov, I., 2014. Adaptive and relative efficiency of stock markets from Southeastern Europe: a wavelet approach. Applied Financial Economics, 24(10), pp. 705-722, http://www.tandfonline.com/doi/full/10.1080/09603107.2014.899669#.U2spB01ZrDc Impact metrics: 2014 SJR (Scopus): 0.265 Bogdanova, B., 2014. The Crimean crisis through the lens of international stock market co- movements. Vanguard Scientific Instruments in Management, 9(2), pp. 16-27. http://vsim-journal.info/static_cont/vsim_e-journal_vol_09-2014-2_screen_ver2.pdf Vanguard Scientific Instruments in Management is indexed in EBSCO and Google Scholar.
The Discrete Wavelet transform • The following sampling rule is applied: • 𝑠 = 2 𝑗 𝑎𝑛𝑑 𝜏 = 𝑘2 𝑗 • The father and the mother wavelet: 𝜙𝐽,𝑘 = 2− 𝐽 2 𝜙 𝑡−2 𝐽 𝑘 2 𝐽 , where −∞ ∞ 𝜙 𝑡 𝑑𝑡 = 1 , 𝐽 ∈ ℕ 𝜓𝑗,𝑘= 2− 𝑗 2 𝜓 𝑡−2 𝑗 𝑘 2 𝑗 , 𝑗 = 1,2, … , 𝐽 , where −∞ ∞ 𝜓 𝑡 𝑑𝑡 = 0 , 𝑠 = 2 𝑗 , 𝑗 = 1, … , 𝐽
The father and the mother wavelet
The DWT

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Wavelet analysis of financial datasets

  • 1. Recognition of hidden patterns in financial datasets The wavelet approach Boryana Bogdanova, PhD Department of Statistics and Econometrics, Faculty of Economics and Business Administration, Sofia University bpelova@gmail.com https://www.researchgate.net/profile/Boryana _Bogdanova
  • 3. Popular filters for financial time series
  • 4. Popular filters for financial time series
  • 5. Financial data through the lens of the wavelet transform 6 months ………………… 2 years 1 day ………………… Time series
  • 7. Wavelet filtering • The function 𝜓 𝑡 is a wavelet if: • 𝜓 𝑡 is decaying sufficiently fast; • Ψ 0 = −∞ ∞ 𝜓 𝑡 𝑑𝑡 = 0
  • 9. Main reference and toolbox Aguiar-Conraria, L. & Soares, M., 2014. The continuous wavelet transform: moving beyond uni-and bivariate analysis. Journal of Economic Surveys, 28(2), pp. 344-75.
  • 10. 5 year cyclic component 2-year cyclic component
  • 11. 5 year cyclic component 2-year cyclic component
  • 12. 5 year cyclic component 2-year with a break to 3-year cyclic component
  • 13. 5 year cyclic component 2-year with a break to 3-year cyclic component
  • 14. The Continuous Wavelet Transform 𝜓 𝑡 𝜓𝜏,𝑠 = 1 𝑠 𝜓 𝑡 − 𝜏 𝑠 , 𝑠, 𝜏 ∈ ℝ, 𝑠 ≠ 0 𝑊𝑥;𝜓 𝜏, 𝑠 = −∞ ∞ 𝑥 𝑡 1 𝑠 𝜓∗ 𝑡 − 𝜏 𝑠 𝑑𝑡
  • 15. 𝑊𝑃𝑆 𝑥 = 𝑊𝑥 2
  • 16. Case I: NASDAQ structural patterns
  • 17. Case II: Momentum analysis J K Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners 0.0323 7.1447 0.0000 0.0247 7.0244 0.0000 0.0229 7.5064 0.0000 0.0233 8.4151 0.0000 0.0226 8.8303 0.0000 0.0190 7.9918 0.0000 Losers 0.0177 4.7705 0.0000 0.0211 6.6860 0.0000 0.0223 7.8883 0.0000 0.0200 8.2088 0.0000 0.0199 8.4945 0.0000 0.0175 7.8737 0.0000 WML 0.0146 2.6522 0.0086 0.0036 0.9848 0.3259 0.0006 0.2507 0.8023 0.0033 2.0835 0.0385 0.0027 2.3467 0.0200 0.0015 1.7920 0.0751 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners 0.0269 6.3134 0.0000 0.0236 6.7914 0.0000 0.0223 7.2336 0.0000 0.0232 8.3037 0.0000 0.0230 8.8314 0.0000 0.0192 8.0397 0.0000 Losers 0.0220 6.3427 0.0000 0.0231 7.1983 0.0000 0.0217 7.6831 0.0000 0.0194 8.2568 0.0000 0.0193 8.4815 0.0000 0.0169 7.9192 0.0000 WML 0.0049 0.9608 0.3378 0.0005 0.1331 0.8942 0.0006 0.2159 0.8293 0.0039 1.8690 0.0632 0.0037 2.3830 0.0182 0.0022 2.1851 0.0304 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners 0.0204 5.3413 0.0000 0.0187 5.6522 0.0000 0.0212 6.5992 0.0000 0.0224 7.9281 0.0000 0.0227 8.2819 0.0000 0.0190 7.6339 0.0000 Losers 0.0228 6.5782 0.0000 0.0221 6.7299 0.0000 0.0185 6.7247 0.0000 0.0164 7.0421 0.0000 0.0173 8.0145 0.0000 0.0154 7.7591 0.0000 WML -0.0024 -0.5154 0.6068 -0.0034 -0.8658 0.3876 0.0027 0.8396 0.4021 0.0060 2.6236 0.0094 0.0054 2.7814 0.0060 0.0036 2.6212 0.0096 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners 0.0241 6.0682 0.0000 0.0225 6.5348 0.0000 0.0225 6.8938 0.0000 0.0239 7.7602 0.0000 0.0236 8.0562 0.0000 0.0186 7.3310 0.0000 Losers 0.0181 5.6930 0.0000 0.0180 5.9843 0.0000 0.0143 5.7088 0.0000 0.0144 6.4174 0.0000 0.0153 7.0734 0.0000 0.0133 6.6971 0.0000 WML 0.0060 1.3296 0.1852 0.0045 1.1541 0.2498 0.0082 2.4471 0.0153 0.0094 3.1704 0.0018 0.0083 3.2157 0.0016 0.0054 2.8855 0.0045 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners 0.0271 7.0184 0.0000 0.0257 7.2742 0.0000 0.0248 7.1517 0.0000 0.0239 7.2276 0.0000 0.0238 7.4947 0.0000 0.0179 6.5884 0.0000 Losers 0.0140 5.4076 0.0000 0.0145 5.7164 0.0000 0.0140 5.9406 0.0000 0.0135 6.0022 0.0000 0.0142 6.6320 0.0000 0.0116 5.7974 0.0000 WML 0.0131 3.1493 0.0019 0.0112 2.9654 0.0034 0.0108 2.9730 0.0033 0.0104 3.2030 0.0016 0.0095 3.3357 0.0010 0.0063 2.8391 0.0052 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners 0.0237 5.9885 0.0000 0.0227 6.0682 0.0000 0.0234 6.4116 0.0000 0.0231 6.7986 0.0000 0.0217 6.7706 0.0000 0.0142 5.3461 0.0000 Losers 0.0114 4.7190 0.0000 0.0117 4.9251 0.0000 0.0109 4.7895 0.0000 0.0096 4.4271 0.0000 0.0095 4.5935 0.0000 0.0073 4.0119 0.0001 WML 0.0123 3.0589 0.0026 0.0110 3.0281 0.0028 0.0125 3.6440 0.0004 0.0134 4.2987 0.0000 0.0121 4.2779 0.0000 0.0069 2.8628 0.0049 1 2 4 8 1 2 4 8 13 26 13 26
  • 18. Case II: Momentum analysis J K Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners 0.0039 1.4443 0.1499 -0.0022 -0.9564 0.3398 -0.0037 -1.8617 0.0638 -0.0040 -2.2884 0.0230 -0.0030 -1.8100 0.0716 -0.0024 -1.4614 0.1454 Losers -0.0017 -0.7157 0.4748 0.0029 1.2795 0.2019 0.0021 0.9777 0.3292 0.0003 0.1269 0.8991 -0.0005 -0.2589 0.7959 -0.0014 -0.7384 0.4611 WML 0.0056 1.9243 0.0554 -0.0051 -2.4068 0.0168 -0.0058 -4.9694 0.0000 -0.0042 -4.2570 0.0000 -0.0025 -2.6254 0.0092 -0.0010 -1.5703 0.1179 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners -0.0051 -2.1201 0.0350 -0.0075 -3.8342 0.0002 -0.0063 -3.4597 0.0006 -0.0051 -2.9606 0.0034 -0.0037 -2.2322 0.0265 -0.0025 -1.5228 0.1293 Losers 0.0043 1.7003 0.0903 0.0057 2.3520 0.0194 0.0029 1.2652 0.2070 0.0012 0.5501 0.5827 0.0003 0.1352 0.8925 -0.0011 -0.5713 0.5684 WML -0.0094 -3.2411 0.0013 -0.0133 -6.3158 0.0000 -0.0092 -5.8404 0.0000 -0.0063 -4.7560 0.0000 -0.0040 -3.4845 0.0006 -0.0014 -1.9590 0.0514 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners -0.0080 -3.1651 0.0017 -0.0089 -4.2518 0.0000 -0.0074 -3.9632 0.0001 -0.0047 -2.7821 0.0058 -0.0033 -2.0078 0.0458 -0.0021 -1.2909 0.1982 Losers 0.0068 2.6630 0.0082 0.0069 2.7406 0.0066 0.0033 1.3894 0.1660 0.0020 0.9015 0.3682 0.0002 0.1177 0.9064 -0.0011 -0.5600 0.5761 WML -0.0148 -5.1928 0.0000 -0.0159 -7.2208 0.0000 -0.0107 -5.9253 0.0000 -0.0067 -4.6062 0.0000 -0.0036 -2.9956 0.0030 -0.0010 -1.1963 0.2330 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners -0.0090 -4.3041 0.0000 -0.0098 -4.9768 0.0000 -0.0070 -3.8889 0.0001 -0.0047 -2.8312 0.0050 -0.0032 -2.0262 0.0439 -0.0021 -1.3317 0.1845 Losers 0.0074 2.7496 0.0064 0.0065 2.4111 0.0166 0.0037 1.4941 0.1364 0.0012 0.5025 0.6158 -0.0008 -0.3786 0.7053 -0.0010 -0.5246 0.6004 WML -0.0165 -6.1991 0.0000 -0.0163 -6.7317 0.0000 -0.0107 -4.9861 0.0000 -0.0058 -3.0903 0.0022 -0.0024 -1.6427 0.1018 -0.0010 -0.9037 0.3672 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners -0.0071 -3.7641 0.0002 -0.0074 -4.2318 0.0000 -0.0051 -3.0222 0.0028 -0.0033 -2.0159 0.0450 -0.0021 -1.3588 0.1756 -0.0013 -0.8516 0.3955 Losers 0.0051 1.9639 0.0507 0.0043 1.7052 0.0894 0.0024 1.0177 0.3098 -0.0003 -0.1345 0.8931 -0.0012 -0.5341 0.5938 -0.0006 -0.3217 0.7480 WML -0.0123 -4.7470 0.0000 -0.0118 -4.9468 0.0000 -0.0075 -3.4206 0.0007 -0.0030 -1.4182 0.1575 -0.0010 -0.5275 0.5984 -0.0007 -0.4486 0.6542 Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Return t-stat p-val Winners -0.0030 -1.5814 0.1151 -0.0041 -2.2877 0.0231 -0.0031 -1.7892 0.0749 -0.0015 -0.9007 0.3687 -0.0004 -0.2703 0.7872 -0.0002 -0.1689 0.8661 Losers 0.0026 0.9858 0.3253 0.0025 0.9875 0.3244 0.0011 0.4463 0.6558 0.0004 0.1673 0.8673 0.0001 0.0387 0.9692 0.0011 0.5721 0.5679 WML -0.0056 -2.0451 0.0420 -0.0067 -2.5423 0.0117 -0.0042 -1.7141 0.0879 -0.0019 -0.8374 0.4033 -0.0005 -0.2455 0.8063 -0.0013 -0.7370 0.4620 1 2 4 8 1 2 4 8 13 26 13 26
  • 19. Case II: Momentum analysis
  • 20. Case II: Momentum analysis
  • 21. Thank you for your attention! Ivanov, I., Kabaivanov, S., Bogdanova, B., 2016. Stock market recovery from the 2008 financial crisis: The differences across Europe. Research in International Business and Finance, 37(May 2016), pp. 360-74. http://www.sciencedirect.com/science/article/pii/S027553191630006X Impact metrics: 2016 SJR (Scopus): 0.694; 2016 Source normalized impact per paper: 1.819 Bogdanova, B. & Ivanov, I., 2015. A wavelet-based appraoch to the analysis and modelling of financial time series exhibiting strong long-range dependence: the case of Southeast Europe. Journal of Applied Statistics, 43(4), pp.655-73. http://www.tandfonline.com/doi/full/10.1080/02664763.2015.1077370 Impact metrics: 2016 Impact factor (Thomson Reuters): 0.664 Bogdanova, B., 2015. A wavelet-based discussion on the Greek stock market integration during the last decade. Journal of Engineering Science and Technology Review, 8(1), Special Issue on Econophysics, pp. 8-11. http://www.jestr.org/downloads/Volume8Issue1/fulltext28115.pdf Journal of Engineering Science and Technology Review is indexed SCOPUS, EBSCO, DOAJ, CERN, ACS. Bogdanova, B. & Ivanov, I., 2014. Adaptive and relative efficiency of stock markets from Southeastern Europe: a wavelet approach. Applied Financial Economics, 24(10), pp. 705-722, http://www.tandfonline.com/doi/full/10.1080/09603107.2014.899669#.U2spB01ZrDc Impact metrics: 2014 SJR (Scopus): 0.265 Bogdanova, B., 2014. The Crimean crisis through the lens of international stock market co- movements. Vanguard Scientific Instruments in Management, 9(2), pp. 16-27. http://vsim-journal.info/static_cont/vsim_e-journal_vol_09-2014-2_screen_ver2.pdf Vanguard Scientific Instruments in Management is indexed in EBSCO and Google Scholar.
  • 22. The Discrete Wavelet transform • The following sampling rule is applied: • 𝑠 = 2 𝑗 𝑎𝑛𝑑 𝜏 = 𝑘2 𝑗 • The father and the mother wavelet: 𝜙𝐽,𝑘 = 2− 𝐽 2 𝜙 𝑡−2 𝐽 𝑘 2 𝐽 , where −∞ ∞ 𝜙 𝑡 𝑑𝑡 = 1 , 𝐽 ∈ ℕ 𝜓𝑗,𝑘= 2− 𝑗 2 𝜓 𝑡−2 𝑗 𝑘 2 𝑗 , 𝑗 = 1,2, … , 𝐽 , where −∞ ∞ 𝜓 𝑡 𝑑𝑡 = 0 , 𝑠 = 2 𝑗 , 𝑗 = 1, … , 𝐽
  • 23. The father and the mother wavelet