return forecasting by quantile regression qwafafew december 20101 larry pohlman and lingjie ma
TRANSCRIPT
Return Forecasting by Quantile Regression
QWAFAFEWDecember 20101
Larry Pohlman and Lingjie Ma
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Outline
• The Math• Examples• Multivariate Model• Results
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The Math and Code
• Model
• OLS Estimation
• QR Estimation
• R, S+, Stat, SAS
uXby
2)(minˆ bxy TiibOLS
Tii
Tiixy bxy
Tii
TiibQR bxybxy
))(1()(minˆ
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What does QR do?
T
uT
y
uTT
y
TT
xFxxQ
FxxxF
uδxβxy
))((
)(
)(
1
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Sample Quantile:
Conditional Quantile:
1,0,:
,yYProb
y
y
FyfQ
F
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Example: Book to Price
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Why QR?
A natural question is under what conditions will QR be “better” than OLS?
1. Full picture view: heterogeneity
If there is heterogeneity, then QR will provide a more complete view of the relationship between variables through the effects of independent variables across quantiles of the response distribution.
2. Robustness: fat-tail distribution
If the conditional return distribution is not Gaussian but fat-tailed, the QR estimates will be more robust and efficient than the conditional mean estimates
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Example Price Momentum
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Example Return on Equity
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Multivariate Model
• Book to Price• Earnings to Price• Cashflow to Enterprise Value• Balance sheet Accruals• Return on Equity• Price Momentum (9 months)• Earnings Momentum (9 months)
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Model Variable Plots
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Results: Equal Weight Quintiles
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Results: Cap Weighted Qunitiles
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Optimized Portfolios TE=3%
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Optimized Portfolios TE=6%
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Conclusion
■ Conditional mean method is still attractive
■QR provides a full-picture distributional view
■ Link between distribution estimates and point portfolio.