Optimal Adjustment of Attributes in Cross-Sectional Prediction Models

1. 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 2003. 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