# IFRS 9 PiT-PD model when the lagged dependency exceeds one year

It is a common approach to model the point-in-time PD (PiT PD; meaning that the PD depends on the current or lagged economy) by regressing default rates on current or past macro variables (such as GDP or unemployment rate). The specific functional form (e.g., regression of logits) is not relevant for my question. Let us assume that we found a model of the following form: $$DR_{t+1} = f(DR_t, \Delta GDP_{t-5})$$ where $$t$$ measures quarters. This means that the default rate of the following year depends on the current default rate (a sensible assumption) and the change of GDP five quarters ago.

In order to incorporate forward looking information into our PiT-PD estimate of the following year we consider $$\Delta GDP$$ forecasts for $$t+1, \ldots, t+4$$. Looking at the above equation and due to the lag in the reaction, these forcasts have, according to our model, impact on $$DR_{t+7}, DR_{t+8}, DR_{t+9} \text{ and } DR_{t+10}.$$ Finally, this means that the whole PiT-PD for the coming year is indpendent of the forecast and is rather a deterministic calculation using observed GDP changes from $$t-5$$ until $$t-1$$.

While this looks simple, we can not use this for the following use cases for the coming year:

• incorporating various forward looking forecasts (scenarios) to get different PDs in these scenarios.
• Stresstesting by assuming that a future GDP decrease impacts next years default rate in our portolio.

My question, thus, is:

• Can we use PiT-models with lagged relations of more than one year at all for PiT-modelling. It looks as this then only helps for the lifetime view.
• Should we restrict feasible models to such where a timely reaction is assumed?