# Logistic regression for observations on a monthly grid - impact of overlapping information of a given customer

My question is based on the usual setting in credit scoring. Assume we have historical monthly observations of customers, their risk factors $$R_{i,t}$$ and the flag whether they are currently in default $$ND_{i,t}$$ and whether they will default within the forthcoming 12 months $$D_{i,t}$$.

Then we can estimate a logistic regression based on the risk factors of non-defaulted customers and the flag. The following questions arise concerning the inclusion of observations in the set from which we estimate the logisitc regression:

1. Can we use repeated observations of a specific customer in a given year? If the customer did not default in, say, January we have them in the set with the target observation 0/no default. Can we include the observation from February again in or estimation set? Obviously, the event of no default within a year starting in January is not independent from the event of no default within a year starting in February. However, the conditional distribution of the default event given the risk factors could be assumed to be independent.

2. How do we deal with customers that have first observations with no default and then a default which is an absorbing state for many months. E.g. the observation in January with the no-default flag and February and many following with the default flag. What I would do is, to use the observations without default flag (depending on the answer to question 1 above) and the first default event, and all those observations would be contained in the estimation set.

My question: what is the statistically appropriate approach here? Unconditional independence is not given but conditional dependence can be assumed. Is the situation different if there is a default event included (2. above). Are there good references in the literature to this standard problem?

Thank you!

EDIT: To clarify as suggested by a commenting user:

• The risk factors $$R_{i,t}$$ contain information up to time $$t$$. This can be some recent account behavior (e.g., arrears of the last 6 months) or some balance sheet information available at time $$t$$.
• The information whether a customer already is in default, $$ND_{i,t}$$, is only used to exclude them from the estimation.
• What we want to predict, is the default event within the year after $$t$$ (i.e., at any monthly observation date $$t+1, t+2, \ldots, t+12$$) given information up to $$t$$.

EDIT: I further discussed this issue with colleagues. It turned out that in the statistical literature, the constellation is refered to as "marginal modelling of correlated, clustered responses". It will take a while for me to understand this theory. I assume that it offers strategies to handle bias in coefficients, and/or underestimates of standard errors of coefficients.

What I am still wondering is, what a typical solutions in a market practice setting with statistical rigor looks. This problem must be faced by many model developers in banks ...

• I've been facing the problems as you did before. Have you found the solution to this problem? Many thanks. Commented Apr 2 at 6:39
• @Aurora Unfortunately, I was not able to perform a complete mathematical research. At any rate, I think that the approach is valid. You use all 1 year observations when the customer is initially performing including those where he/she defaults. Whenever a customer is already in default at the beginning of the period, you exclude this observation. Commented Apr 3 at 10:56

It turns out your question isn't well defined.

Are you modeling the risk of default (Cox model for time-to-event outcome) or the probability of being in default (Logistic regression for probability with random effects, or lagged effects for prior default)? Furthermore, how are the covariates input to the model? You have many possible expressions for a covariate history $$R_{i,t}$$. It can be as simple as modeling last month's income as a predictor of this month's risk of default - a time varying covariate in the Cox model. Hopefully no subject is in default at study timepoint 1, since you have a missing data problem in that case. Two lagged effects give more power and flexibility at the cost of higher dimensional analysis, and missing data for those who default at timepoint 2.

• Hi, thank you for your comment. I clarified as you suggested. Do you know scoring models? My setting is the standard setting in scorecard development. I hope, it clear now. The main issue is that we have several observations of the same customer and whether we can use all of them in the estimation. I know various practices here, but I was wondering what theory exactly says. Commented Oct 7, 2022 at 6:21

Usually the following steps are taken in credit risk modelling (PD models in your case).

1. Collect snapshot data - the same month in a given year (usually December) for all client IDs
2. Identify the in-default IDs at the beginning of this one-year observation period - your ND binary feature. You only need the performing IDs (non-defaulted)
3. Identify whether these non-defaulted credit complexes have at least one default event during the one-year observation period: In order to gain this information, we create the variable “todef_current” to indicate that the credit complex will default in the coming 12 months. The steps to create this variable “todef_current” are as follows: Using the default information (start and end date of the default which are available), if a default start date is in between the snapshot date and snapshot date + 1 year, the variable “todef_current” will be set to 1, otherwise 0. For example, if a default event is in May 2017, the default is merged to the December 2016 snapshot. Provided the contract ID and the reporting date match, it is stated that there is a default event in the 1-year performance period after the snapshot date.

At the end, the todef_current varialbe would be your target.