For estimation, we consider the parametric probit model alolng with the semiparametric single-index model estimator of Ichimura (1993).
Single-index models, on the other hand, can accommodate certain forms of heteroskedasticity (general but known form and unknown form if the distribution of the error term depends on x only through the index, i.
Optimization-based estimation methods have been developed for single-index models without making distributional assumptions and thus avoiding misspecification.
Note that we need a location-scale normalization for identification purposes in single-index models.
In single-index models, the asymptotic distribution of the normalized and centered estimator does not depend on the smoothing parameter, so asymptotically, any sequence of smoothing parameters is going to give the same estimate as long as it satisfies certain conditions.
Ichimura, 1993, Optimal Smoothing in Single-Index Models, The Annals of Statistics, 21: 157-178.
Moreover, comparing these results with those of Table 2, it appears that unconditional alphas from the single-index model are less useful in predicting future performance as measured by excess returns than by their own future values.
By comparing the results from unconditional and conditional alphas, we conclude that in the case of the single-index model it seems that conditional alphas lead to stronger evidence of performance persistence.