Model Confidence set

Question

Is anyone here familiar with Hansen & Lunde's Model Confidence set? If yes, can you briefly explain how it is linked with loss functions? I basically have to calculate Quasi Likelihood loss functions for different volatility forecasting models (QLK basically compares each forecast with a volatility proxy) however, I can't seem to draw the link between the loss function and MCS. Any input would be appreciated, thanks!

Answer 1

From their paper:

where $$ \mu_{ij} = \mathbb{E}[L_{i,t} - L_{j,t}] \quad \forall i,j \in \mathcal{M}^0$$ with $\mathcal{M}^0$ the universe of models and $L_{i,t}$ the loss for model $i$ in period $t$. So the models in $\mathcal{M}^\ast$ have equal expected loss, which gives you the link to the loss function. Since expected loss is not observed, $\mathcal{M}^\ast$ needs to be inferred at a certain level of confidence (hence Model Confidence Set), which is what the paper is about.