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Financial Time Series Analysis with Wavelets

Financial Time Series Analysis with Wavelets. Rishi Kumar Baris Temelkuran. Agenda. Wavelet Denoising Threshold Selection Threshold Application Applications Asset Pricing Technical Analysis. Denoising Techniques. 4 choices to make Wavelet Haar, Daub4 Threshold Selection

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Financial Time Series Analysis with Wavelets

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  1. Financial Time Series Analysis with Wavelets Rishi Kumar Baris Temelkuran

  2. Agenda • Wavelet Denoising • Threshold Selection • Threshold Application • Applications • Asset Pricing • Technical Analysis

  3. Denoising Techniques • 4 choices to make • Wavelet • Haar, Daub4 • Threshold Selection • Application of Thresholding • Depth of Wavelet Decomposition • 1, 2

  4. Threshold Selection • Universal Threshold • Minimax • Stein's Unbiased Risk • Hybrid of Stein’s and Universal

  5. Threshold Selection • Universal Threshold • Let z1,…,zN be IID N(0,σε2) random variables

  6. Threshold Selection • Minimax • Does not have a closed formula. • Tries to find an estimator that attains the minimax risk • Does not over-smooth by picking abrupt changes

  7. Threshold Selection • Stein's Unbiased Risk • Threshold minimizes the estimated risk

  8. Threshold Application • Hard Thresholding • Soft Thresholding

  9. Asset Pricing • Fama French Framework • Cross sectional variation of equity returns • Sensitivity to various sources of risk • Market Risk (1 factor) • Systematic Factor Risk (2 factors) • Factors should be proxies for real, macroeconomic, aggregate, nondiversifiable risk

  10. Asset Pricing • Fama French Framework • Pricing Relation • Regression

  11. Wavelet Denoising • High Frequency Data: daily • Use Denoising to Clean • Predictor Variables • Response Variables • Goals • Improve Regression Fit • Decrease Out-of-Sample Error of Expected Excess Return

  12. Data • Daily returns: 19630701 to 20021231 • Factors: • market return - risk free return • (small - big) market cap returns • (high - low) book to market returns • Assets • IBM, GE, 6 Fama-French portfolios

  13. Model Fit Tests • R-square • Regress using sliding window (e.g. 2 year) • Compute Rsquare • Mean Square Error in forecasting • Regress using sliding window • Forecast using regression Betas for 14 days • Compare MSE of with actuals • Pricing Relation Test • Compute mean of excess return for out-of-sample data (e.g. 1 year forward) • Compare with estimated expected excess return

  14. Results • Expected • Soft thresholding will work better • Daub4 will work better than Haar • Empirical • General: no statistically significant improvement • Few odd cases: improved R-square • FF portfolio using Daub4, soft, universal and heuristic

  15. Technical Analysis • Charting, pattern watching • Common practice among traders • Not well studied in academia • Our work modeled after seminal paper by Lo et al

  16. Goal • Determine if Technical Patterns have information content • Distribution of conditional returns (post-pattern) is different from distribution of unconditional returns • Replace Lo’s Kernel regression based smoothing algorithm (for pattern recognition) with wavelet denoising

  17. Common Technical Patterns

  18. Pattern Recognition • Parameterize patterns • Characterize patterns by geometry of local extrema • Need denoised price path for securities

  19. Defining Patterns • Defined in terms of sequences of local extrema • e.g. head and shoulders • e1 is a max • e3 > e1, e3 > e5 • e1 and e5 within 4% of their average • e2 and e4 within 4% of their average

  20. Wavelet Smoothing • Smooth out noise for pattern recognition • Mimics human cognition in extracting regularity from noisy data

  21. Information Content • Measure 1 day conditional return after completion of pattern • continuously compounded • lagged by 3 days to allow for reaction time to pattern • Measure 1 day unconditional return • Random sample, periodic sample • Check if both return series are from the same distribution

  22. Data and Testing • Data • Stocks from Nasdaq 100 index • 19950101 to 19991231 • Daily price • Goodness-of-fit • Normalize returns from each stock • Combine all conditional returns to increase strength of test • Kolmogorov-Smirnov goodness-of-fit test

  23. Example Detected Pattern

  24. Results • About 300 Head&Shoulders pattern detected in 5 year data per denoising technique • Distribution of conditional returns found significantly different from the distribution of unconditional returns • Patterns have information content!

  25. Conclusion • Wavelet analysis seems to add little value in asset pricing paradigm • Wavelet smoothing might prove useful in cognitive/behavioral finance studies in its ability to mimic human cognition

  26. The End Questions?

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