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 z1,…,zN be IID N(0,σε2) random

variables

NU log2ˆ

N

NzPt

t

as

1log2)max(

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

),ˆ( sup inf)(~

ˆxxRF

xx

Threshold Selection

Stein's Unbiased Risk

Threshold minimizes the estimated risk

),(min}:{#2),(1

2

k

iii zzikzSURE

),(minarg0

zSURES

Threshold Application

Hard Thresholding

Soft Thresholding

otherwise 0

if

ttH oo

, ) )(( ttS oosign

0 if0

0 if)(

x

xxx

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

HMLi

SMBi

Mi

ft

it hsRRE )(

it

HMLti

SMBti

ft

Mtii

ft

it RhRsRRRR )(

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: 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

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 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

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?