Once reverted the Merton/Vasicek formula I could compute the $PD^{PIT}$ for IFRS9 as

$PD^{PIT}_i(z) = \Phi \left( \phi^{-1}(PD^{TTC}_i) \sqrt{1-\rho_i} + \sqrt{\rho_i}z\right)$

The main issue is to extract a proper set of scenario $z$ as it is not the GDP or any other observable quantity but rather a factor intrinsic of the model.

This paper of BoE suggests to use the Kalman filter that seems really appropriate in this context.

I am struggling figuring how to apply the Kalman filter as I am not using it since the Signals exam at university.

How should procedure work?

Should I consider the hts of an observable variable (eg. GDP) and extrapolate this hidden state from that?



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