Are combination forecasts of S&P 500 volatility statistically superior?

where ft are individual forecasts obtained from individual models (and the
VIX) along with combination forecasts based on equal weights or regression
weights. The total number of candidate forecasts will be denoted as mo,
therefore the competing forecasts, individual and combination are given by
ft, i = 1, 2,...,rno. While there are many alternative loss functions, Patton
(2006) shows that MSE and QLIKE belong to a family of loss functions that are
robust to noise in the volatility proxy, RV_t+22 in this case. Each loss function
has somewhat different properties, MSE is symmetric whereas QLIKE penalises
under-prediction more heavily than over-prediction.

While these loss functions allow forecasts to be ranked, they give no indica-
tion of whether the performance of the forecasts are significantly different. The
model confidence set (MCS) approach allows for such conclusions to be drawn.
The interpretation attached to an MCS is that it contains the best forecast
with a given level of confidence. The MCS may contain a number of models
which indicates they are of equal predictive ability (EPA). The construction of
an MCS is an iterative procedure in that it requires a sequence of tests for EPA.
The set of candidate models is trimmed by deleting models that are found to
be inferior. The final surviving set of models in the MCS contain the optimal
model with a given level of confidence and are not significantly different in terms
of their forecast performance.

The procedure starts with a full set of candidate models Mo = {1, ∙∙∙, m_o}.
The MCS is determined by sequentially trimming models from Mo therefore
reducing the number of models to m < mo. Prior to starting the sequential
elimination procedure, all loss differentials between models i and j are com-
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