Return Forecasting by
Quantile Regression
QWAFAFEW
December 20101
Larry Pohlman and Lingjie M
Outline
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•
•
•
The Math
Examples
Multivariate Model
Results
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The Math and Code
y  Xb  u
• Model
• OLS Estimation
• QR Estimation
ˆ
OLS


T
2
 minb  ( y i  xi b) 




T
T
ˆ
QR    minb    ( yi  xi b)   (  1)( yi  xi b)
 yi  xiT 

yi  xiT b
• R, S+, Stat, SAS
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What does QR do?
Sample
Quantile:
Fy  P robY  y ,
Q  f y : Fy  ,   0,1
Conditional
Quantile:
y  xT β  ( xT δ )u
Fy1  x   xT   xT Fu1 ()
Qy  x   xT (   Fu1 ())  xT  
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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
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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.
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