  The time-varying kernel density estimation relies on two free parameters: the
bandwidth and the discount factor. We propose to select these parameters so as
to minimize a criterion consistent with the traditional requirements of the
validation of a probability density forecast. These requirements are both the
uniformity and the independence of the so-called probability integral
transforms, which are the forecast time-varying cumulated distributions applied
to the observations. We thus build a new numerical criterion incorporating both
the uniformity and independence properties by the mean of an adapted
Kolmogorov-Smirnov statistic. We apply this method to financial markets during
the COVID-19 crisis. We determine the time-varying density of daily price
returns of several stock indices and, using various divergence statistics, we
are able to describe the chronology of the crisis as well as regional
disparities. For instance, we observe a more limited impact of COVID-19 on
financial markets in China, a strong impact in the US, and a slow recovery in
Europe.
