1. Optimal Adjustment of Attributes in Cross-Sectional Prediction Models
Global Asset Allocation and Stock Selection
Optimal Adjustment of Attributes in Cross-Sectional Prediction Models
Campbell R. Harvey

2. Cross-Sectional Prediction
Look at one point in time
rt = d0+ d1P/Bt-1 + et
Example: Regress 1000 different equity returns in January 2004 on their P/B in December Estimate two coefficients.

3. Cross-Sectional Prediction
Given the estimated coefficients, we can predict February 2004
rFeb2004 = d0+ d1P/BJan2004
Sort by predictions. Implement Long-Short
Buy 100 highest predicted returns
Sell 100 lowest predicted returns

4. Cross-Sectional Prediction
Problem is that ‘d1’ changes through time
Indeed, ‘d1’ could flip sign!
What to do?
Ad hoc solution of averaging ‘d1’ over time
Use our factor model

5. Cross-Sectional Prediction
Dynamic linear factor model:
rit = ai0+ bitFt + vit
Assume beta is a function of price to book
bit = coi + ci1 (P/B)i,t-1
Substitute this for the usual beta

6. Cross-Sectional Prediction
Dynamic linear factor model:
rit = ai0+ [coi + ci1 +(P/B)i,t-1] Ft + vit
Rewrite
rit = ai0+ coi Ft + ci1 (P/B)i,t-1Ft + vit

7. Cross-Sectional Prediction
Dynamic linear factor model:
rit = ai0+ coi Ft + ci1 Ft (P/B)i,t-1+ vit
Compare this to the cross-sectional regression
ci1 Ft = d1
This explains why d1 unstable through time!
The usual model is misspecified.

8. Cross-Sectional Prediction
What to do?
Run dynamic linear factor model, firm by firm
rit = ai0+ coi Ft + ci1 Ft (P/B)i,t-1+ vit
Collect ci1 for each of the 1000 firms

9. Cross-Sectional Prediction
What to do?
Scale each firm’s P/B by the estimated coefficient
P/B*it-1 = ci1 P/Bi,t-1

10. Cross-Sectional Prediction
Re-estimate cross-sectional prediction model
rt = d0+ d1P/B*t-1 + et