Cross-validation for selecting the penalty factor in least squares model averaging

Asymptotic properties of least squares model averaging have been discussed in the literature under two different scenarios: (i) all candidate models are under-fitted; and (ii) the candidate models include the true model and may also include over-fitted ones. The penalty factor ϕn in the weight selection criterion plays a critical role. Roughly speaking, ϕn=2 is usually preferred in the first scenario but it does not achieve asymptotic optimality in the second scenario as ϕn=log(n) does. It is difficult in the practice to select an appropriate penalty factor since the true scenario is unknown. We propose a non-trivial cross-validation procedure to select the penalty factor that leads to an asymptotically optimal estimator in an adaptive fashion for both scenarios.