  Volatilities, in high-dimensional panels of economic time series with a
dynamic factor structure on the levels or returns, typically also admit a
dynamic factor decomposition. We consider a two-stage dynamic factor model
method recovering the common and idiosyncratic components of both levels and
log-volatilities. Specifically, in a first estimation step, we extract the
common and idiosyncratic shocks for the levels, from which a log-volatility
proxy is computed. In a second step, we estimate a dynamic factor model, which
is equivalent to a multiplicative factor structure for volatilities, for the
log-volatility panel. By exploiting this two-stage factor approach, we build
one-step-ahead conditional prediction intervals for large $n \times T$ panels
of returns. Those intervals are based on empirical quantiles, not on
conditional variances; they can be either equal- or unequal- tailed. We provide
uniform consistency and consistency rates results for the proposed estimators
as both $n$ and $T$ tend to infinity. We study the finite-sample properties of
our estimators by means of Monte Carlo simulations. Finally, we apply our
methodology to a panel of asset returns belonging to the S&P100 index in order
to compute one-step-ahead conditional prediction intervals for the period
2006-2013. A comparison with the componentwise GARCH benchmark (which does not
take advantage of cross-sectional information) demonstrates the superiority of
our approach, which is genuinely multivariate (and high-dimensional),
nonparametric, and model-free.
