How can you interpet the downside risk factor that we used

Question

Tasks:

1. Run the Fama-MacBeth procedure to estimate the risk premia of a 3- factor model. In the first step, for each portfolio estimate the time series regression

r_i,t = α_i + β_i,Mr_M,t + β_i,SMBr_SMB,t + β_i,HMLr_HML,t + ∈_i,t,

on rolling windows of length 84 months (from period t - 83 to t), where r_i,t denotes the excess return of portfolio i in period t, r_M,t the excess return of the market, and r_SMB,t, r_HML,t are the size and value factors, respectively.

In the second step use the estimated βˆ_i,F,t in cross-sectional regressions

r_i,t = βˆ_i,M,tγ_M,t + βˆ_{i,SM B,t}γ_SMB,t + βˆ_{i,HM L,t}γ_HML,t + a_i,t

to obtain the estimates for realized factor risk premia γ_F,t for each time t. For each factor, compute the average risk premium, its standard error, and provide an approximate 95% confidence interval. Analyze the pricing errors, which are by a_i,t.

2. Re-run the entire Fama-MacBeth procedure as before, but now include in addition to the three factors from above a downside-risk factor. The first stage now is:

r_i,t = α_i + β_i,M r_M,t + β_i,SMB r_SMB,t + β_i,HMLr_HML,t + β_i,DRr_DR,t + ∈_i,t,

Downside risk is given by r_DR,t which is equal to r_M,t if r_M,t is more than one standard deviation below its mean and 0 otherwise. Calculate the downside risk measure on each window separately.

3. How can you interpet the downside risk factor that we used here? Is there any problem? Could we do better? Hint: In their JF paper, Jurek and Stafford (2015) explain hedge fund returns using downside risk. Would a variant of their proxy for downside risk also make sense in our analysis?

4. Explain in simple language what you have learned from your calculations about the cross-section of expected returns.