Financial Time Series Analysis with Wavelets

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 Steins and Universal

5
Threshold Selection

• Universal Threshold
• Let z1,,zN be IID N(0,se2) 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 Los 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 gt e1, e3 gt 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 HeadShoulders 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?