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.