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  Application of Thresholding  Depth of Wavelet Decomposition 1, 2

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

Threshold Selection Universal Threshold  Let z 1,…,z N be IID N(0,σ ε 2 ) random variables

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

Threshold Selection Stein's Unbiased Risk Threshold minimizes the estimated risk

Threshold Application Hard Thresholding Soft Thresholding

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

Asset Pricing Fama French Framework  Pricing Relation  Regression

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

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

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

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

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

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

Common Technical Patterns

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

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

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

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

Data and Testing Data  Stocks from Nasdaq 100 index  to  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

Example Detected Pattern

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!

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

The End Questions?