Hypotheses An effective and you may B relate with the initial phase

• d P ( R ninety + i he said, t = step 1 | A beneficial we , t , Letter we , t , A beneficial ? i , t , Letter ? i , t ) d An effective i , t > 0 and you can P ( R ninety + we , t = 1 | A good i , t , A beneficial ? we , t , Letter i , t , N ? we , t ) ? 0
• d P ( Roentgen 90 + we , t = step 1 | A we , t , Letter i , t , An excellent ? i , t , Letter ? i , t ) d Good i , t ? 0
• d P ( F i , t = step 1 | A beneficial i , t , N we , t , A ? i , t , N ? we , t , R ninety + i , t ? 1 = step one ) d An effective we , t > 0 and P ( F we , t = step one | A great we , t , An excellent ? i , t , N i , t Letter ? we , t , Roentgen ninety + i , t ? step one = step 1 ) ? 0
• d P ( F i , t = step 1 | A beneficial we , t , N i , t , A great ? i , t , N ? i , t , Roentgen 90 + we , t ? step one = step 1 ) d A beneficial we , t ? step one = 0

Hypothesis A states that the probability of a loan entering 90+ day arrears is increasing in the size of the ability-to-pay shock and is close to 0 where the size of the shock does not exceed the borrowers’ ability-to-pay threshold. Hypothesis B states that the marginal probability of a loan entering 90+ day arrears is at best weakly related to negative equity. Under the double-trigger hypothesis, negative equity itself does not cause borrowers to enter arrears. However, previous research has suggested that borrowers may be less willing to cut back on their consumption to remain current on their repayments when they have negative equity (Gerardi et al 2018). If this is the case, then threshold A ? i , t may be a function of N_we,t and the derivative in Hypothesis B may be positive.

Hypotheses C and you can D interact with the second stage. Theory C claims your probability of property foreclosure are increasing inside brand new the total amount off bad equity, due to the fact the mortgage has been around arrears, it is near to 0 in which the extent of negative collateral try below the expense of property foreclosure. Hypothesis D states that once financing keeps arrears from 90+ weeks, the size of the ability-to-spend shock doesn’t have affect the possibilities of foreclosures (except if the newest surprise is actually subsequently stopped).

5.2 Cox Proportional Danger Designs

We take to the brand new hypotheses detail by detail above playing with a-two-phase Cox proportional possibilities model build having competing dangers. Following the framework lay out above, the original phase examines records so you can ninety+ date arrears, due to the fact 2nd stage estimates changes to help you foreclosures, relieving and you can complete cost.

Cox proportional hazard designs try most often found in the new biomedical books, but i have already been regularly estimate the end result out of covariates on probability of loans entering arrears (e

grams. Deng et al 1996; Gerardi ainsi que al 2008). It imagine the result out-of a change in an effective vector from variables to your quick chances (or possibility) you to definitely an event of interest is observed, as enjoy hasn’t already been observed (Cox 1972).

The fresh Cox proportional hazard design is right when the likelihood of a meeting transform more a while dimensions (including time since mortgage origination), financing are located in the some other items with each other this time around dimension, and the ones financing with not even experienced the event you are going to however do so afterwards (labeled as correct censoring). An important advantage of one’s Cox model would be the fact this time around dimensions is part of brand new inherent construction of the design, in the place of binary or multinomial possibilities activities that are included with the fresh go out measurement due to the fact an extra parts having a specific useful function. With this specific big date-situated construction, the brand new Cox design isnt biased by the devoid of factual statements about the near future; all that will become necessary is actually experience in whether the experience got taken place by the point where the loan was observed.