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The time-varying kernel density estimation relies on two free parameters: thebandwidth and the discount factor. We propose to select these parameters so asto minimize a criterion consistent with the traditional requirements of thevalidation of a probability density forecast. These requirements are both theuniformity and the independence of the so-called probability integraltransforms, which are the forecast time-varying cumulated distributions appliedto the observations. We thus build a new numerical criterion incorporating boththe uniformity and independence properties by the mean of an adaptedKolmogorov-Smirnov statistic. We apply this method to financial markets duringthe COVID-19 crisis. We determine the time-varying density of daily pricereturns of several stock indices and, using various divergence statistics, weare able to describe the chronology of the crisis as well as regionaldisparities. For instance, we observe a more limited impact of COVID-19 onfinancial markets in China, a strong impact in the US, and a slow recovery inEurope.

Estimation of time-varying kernel densities and chronology of the impact of COVID-19 on financial markets