Semi-structured multi-state delinquency model for mortgage default

Victor Medina-Olivares; Wangzhen Xia; Stefan Lessmann; Nadja Klein

stat.APMar 27, 2026

Semi-structured multi-state delinquency model for mortgage default

Authors: Victor Medina-Olivares, Wangzhen Xia, Stefan Lessmann, Nadja Klein

Organizations: Business School, The University of Edinburgh, UK. · School of Management, Technische Universität München, Germany. · School of Business and Economics, Humboldt-Universität zu Berlin, Germany. · Scientific Computing Center, Karlsruhe Institute of Technology, Germany.

Abstract

We propose a semi-structured discrete-time multi-state model to analyse mortgage delinquency transitions. This model combines an easy-to-understand structured additive predictor, which includes linear effects and smooth functions of time and covariates, with a flexible neural network component that captures complex nonlinearities and higher-order interactions. To ensure identifiability when covariates are present in both components, we orthogonalise the unstructured part relative to the structured design. For discrete-time competing transitions, we derive exact transformations that map binary logistic models to valid competing transition probabilities, avoiding the need for continuous-time approximations. In simulations, our framework effectively recovers structured baseline and covariate effects while using the neural component to detect interaction patterns. We demonstrate the method using the Freddie Mac Single-Family Loan-Level Dataset, employing an out-of-time test design. Compared with a structured generalised additive benchmark, the semi-structured model provides modest but consistent gains in discrimination across the earliest prediction spans, while maintaining similar Brier scores. Adding macroeconomic indicators provides limited incremental benefit in this out-of-time evaluation and does not materially change the estimated borrower-, loan-, or duration-driven effects. Overall, semi-structured multi-state modelling offers a practical compromise between transparent effect estimates and flexible pattern learning, with potential applications beyond credit-transition forecasting.

Figures & tables

Figure 1: Delinquency states and permissible transitions over a one-month period.

Figure 2: Realisation of a process path for two randomly selected subjects in the training set.

Figure 3: Empirical transition probabilities (Aalen–Johansen estimator) for the U.S. mortgage loan dataset. The y-axis is on a log scale for readability. Solid lines show the raw estimates, and the overlaid smooth curves (LOESS) highlight the underlying time pattern.

Training Set
		To state
		0	1	2	3	Total
From state	0	95,866	12,476	-	-	108,342
	0	(88.5%)	(11.5%)	-	-	(100%)
	1	8,245	2,674	5,860	-	16,779
	1	(49.1%)	(15.9%)	(34.9%)	-	(100%)

Table 2: Frequency of loans for different transition types.

Figure 4: Proportion of loans in training set in each state at month-end since origination.

Variable	Min.	Mean	Median	Max.	SD.
Credit Score	386.00	747.81	756.00	832.00	46.72
Original Interest Rate (%)	2.25	3.96	3.99	6.12	0.50
Original Loan-to-Value Ratio (%)	6.00	73.85	79.00	339.00	17.34
Mortgage Insurance (%)	0.00	6.60	0.00	40.00	11.50
Original Loan Term (months)	96.00	318.42	360.00	360.00	74.08
Original UPB ($)	14,000.00	225,947.27	204,000.00	985,000.00	117,011.18

Table 3: Summary statistics for numerical variables in the training set. Further variable descriptions are available in the General User Guide for the Freddie Mac Single-Family Loan-Level Dataset. Numerical variables are normalised prior to modelling.

Variable(s)	Cardinality
Number of Borrowers , First Time Homebuyer , Super Conforming	2
Channel , Loan Purpose , Mortgage Insurance Cancellation , Occupancy Status	3
Number of Units	4
Original Debt-to-Income Ratio , Property Type	5
Seller Name , Servicer Name	31
US state or territory	54

Table 4: Cardinality of categorical loan characteristics in the training set. Further variable descriptions are available in the General User Guide for the Freddie Mac Single-Family Loan-Level Dataset.

Figure 5: Baseline recovery. Facets are arranged with transitions in rows and sample sizes in columns. Red curves show the true baseline functions; black curves are estimates from 100 simulation runs.

Figure 6: Linear-effect recovery (same facet layout as Figure 5 ). Red points indicate the true coefficients; boxplots summarise estimates from 100 simulation runs.

Figure 7: Nonlinear-effect recovery (same facet layout as Figure 5 ). Red curves show the true functions; black curves are estimates from 100 simulation runs.

	MSE (continuous-time)			MSE (discrete-time)
Transition	10K	50K	100K	10K	50K	100K
0 $\to$ 0	0.0016 (0.00024)	0.0015 (0.00026)	0.0015 (0.00022)	0.0007 (0.0002)	0.0006 (0.00017)	0.0006 (0.00015)
0 $\to$ 1	0.0013 (0.00017)	0.0013 (0.00014)	0.0012 (0.00014)	0.0006 (0.00013)	0.0005 (0.00013)	0.0005 (0.00012)
0 $\to$ 2	0.0006 (0.00006)	0.0006 (0.00005)	0.0006 (0.00005)	0.0002 (0.00003)	0.0001 (0.00003)	0.0001 (0.00003)
0 $\to$ 3	0.006 (0.00075)	0.0057 (0.00059)	0.0055 (0.0006)	0.0018 (0.00045)	0.0015 (0.00043)	0.0015 (0.00038)
	MAE (continuous-time)			MAE (discrete-time)

Table 5: MSE and MAE comparison for continuous and discrete-time transformation methods.

#	Macro. Vars.	Transition	Initial LR	Decay Steps	Decay Rate	Optimizer	First-moment	Epochs	Batch Size
1	No	0 $\to$ 1	0.00046416	10,000	0.99	Adam	0.80	37	32
2	No	1 $\to$ 0	0.01	10,000	0.80	Adam	0.99	40	32
3	No	1 $\to$ 2	0.01	15,000	0.99	Adam	0.80	27	32
4	No	2 $\to$ 0	0.01	15,000	0.90	Adam	0.99	47	32
5	No	2 $\to$ 1	0.01	10,000	0.80	Adam	0.99	99	32
6	No	2 $\to$ 3	0.01	15,000	0.99	Adam	0.90	39	32

Table 6: Final hyperparameter configurations for each transition model.

Name	0 $\to$ 1	1 $\to$ 0	1 $\to$ 2	2 $\to$ 0	2 $\to$ 1	2 $\to$ 3
channelC	-0.12 (-0.17, -0.07)	0.06 (-0.04, 0.16)	0.08 (-0.05, 0.21)	-0.25 (-0.53, 0.02)	0.22 (-0.14, 0.57)	-0.30 (-0.52, -0.08)
channelR	-0.27 (-0.32, -0.22)	0.09 (-0.00, 0.19)	0.24 (0.12, 0.36)	-0.13 (-0.38, 0.12)	0.08 (-0.26, 0.42)	-0.21 (-0.41, -0.01)
cnt_borrone	0.48 (0.45, 0.51)	-0.19 (-0.25, -0.13)	0.05 (-0.03, 0.13)	-0.02 (-0.18, 0.14)	0.04 (-0.17, 0.25)	-0.03 (-0.15, 0.10)
cnt_units	-0.04 (-0.10, 0.02)	0.17 (0.02, 0.32)	0.17 (-0.01, 0.34)	0.01 (-0.36, 0.37)	-0.13 (-0.63, 0.37)	0.03 (-0.24, 0.29)
cov_dummy	1.04 (1.00, 1.08)	-0.15 (-0.23, -0.07)	1.51 (1.42, 1.61)	0.14 (-0.05, 0.32)	-0.44 (-0.67, -0.22)	1.15 (1.00, 1.30)
dti1	0.15 (0.07, 0.23)	-0.16 (-0.33, 0.01)	-0.05 (-0.28, 0.17)	-0.31 (-0.80, 0.18)	-0.26 (-0.86, 0.35)	-0.16 (-0.57, 0.25)

Table 7: Linear covariate effects (mean and 95% interval). GAM without macroeconomic variables.

Name	0 $\to$ 1	1 $\to$ 0	1 $\to$ 2	2 $\to$ 0	2 $\to$ 1	2 $\to$ 3
channelC	-0.14 (-0.20, -0.11)	-0.01 (-0.13, 0.10)	0.08 (-0.07, 0.20)	-0.24 (-0.48, -0.03)	0.20 (-0.05, 0.50)	-0.16 (-0.37, 0.04)
channelR	-0.27 (-0.32, -0.22)	0.08 (-0.03, 0.19)	0.23 (0.10, 0.36)	-0.17 (-0.40, 0.08)	0.11 (-0.20, 0.42)	-0.21 (-0.41, -0.04)
cnt_borrone	0.47 (0.43, 0.51)	-0.20 (-0.26, -0.13)	0.03 (-0.04, 0.13)	-0.02 (-0.17, 0.13)	0.06 (-0.17, 0.25)	0.00 (-0.10, 0.12)
cnt_units	-0.04 (-0.10, 0.00)	0.11 (-0.01, 0.26)	0.09 (-0.11, 0.28)	-0.16 (-0.49, 0.13)	-0.17 (-0.75, 0.25)	-0.10 (-0.28, 0.13)
cov_dummy	1.02 (0.98, 1.06)	-0.16 (-0.24, -0.06)	1.49 (1.39, 1.57)	0.16 (-0.04, 0.34)	-0.43 (-0.66, -0.22)	1.20 (1.05, 1.36)
dti1	-0.04 (-0.10, 0.03)	-0.25 (-0.41, -0.07)	-0.08 (-0.35, 0.18)	-0.39 (-0.77, -0.01)	-0.23 (-0.78, 0.35)	-0.37 (-0.77, -0.03)

Table 8: Linear covariate effects (mean and 95% interval). Semi-structured without macroeconomic variables.

Figure 8: Estimated time-dependent baseline risk terms $f_{kl}(t)$ (log-odds scale) by transition for the GAM and the semi-structured model, estimated without macroeconomic covariates. Positive values indicate higher baseline propensity for the corresponding transition at time $t$ .

Figure 9: Estimated time-dependent baseline risk terms $f_{kl}(t)$ (log-odds scale) by transition for the GAM and the semi-structured model, estimated with macroeconomic covariates. Patterns are similar to the no-macro specification, indicating that macro covariates do not materially alter the duration dependence in the transition dynamics.

	GAM					Semi-structured
Span	MultiAUC	AUC1vsA	Brier	ECE	ACC	MultiAUC	AUC1vsA	Brier	ECE	ACC
6-12	0.741	0.827	0.002	0.002	0.995	0.753	0.838	0.002	0.003	0.995
12-18	0.791	0.871	0.004	0.002	0.992	0.800	0.882	0.004	0.003	0.992
18-24	0.755	0.841	0.005	0.005	0.989	0.755	0.840	0.005	0.008	0.988

Table 9: Performance by span at

h=6

months (no macro): GAM vs. semi-structured.

	GAM					Semi-structured
Span	MultiAUC	AUC1vsA	Brier	ECE	ACC	MultiAUC	AUC1vsA	Brier	ECE	ACC
6-18	0.708	0.778	0.002	0.003	0.995	0.715	0.783	0.002	0.005	0.995
12-24	0.720	0.780	0.004	0.003	0.993	0.721	0.778	0.004	0.005	0.993
18-30	0.727	0.797	0.005	0.007	0.988	0.727	0.796	0.005	0.010	0.988

Table 10: Performance by span at

h=12

months (no macro): GAM vs. semi-structured.

	GAM					Semi-structured
Span	MultiAUC	AUC1vsA	Brier	ECE	ACC	MultiAUC	AUC1vsA	Brier	ECE	ACC
6-12	0.736	0.822	0.002	0.002	0.995	0.740	0.825	0.002	0.002	0.995
12-18	0.785	0.864	0.004	0.002	0.992	0.785	0.861	0.004	0.002	0.992
18-24	0.755	0.843	0.005	0.006	0.989	0.745	0.829	0.005	0.006	0.989

Table 11: Performance by span at

h=6

months (with macro): GAM vs. semi-structured.

	GAM					Semi-structured
Span	MultiAUC	AUC1vsA	Brier	ECE	ACC	MultiAUC	AUC1vsA	Brier	ECE	ACC
6-18	0.705	0.776	0.002	0.003	0.995	0.706	0.779	0.002	0.003	0.995
12-24	0.718	0.781	0.004	0.003	0.993	0.713	0.777	0.004	0.003	0.993
18-30	0.730	0.802	0.005	0.008	0.988	0.723	0.793	0.005	0.007	0.988

Table 12: Performance by span at

h=12

months (with macro): GAM vs. semi-structured.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Variable	Min.	Mean	Median	Max.	SD.
Credit Score	479.00	744.82	752.00	832.00	46.81
Original Interest Rate (%)	2.88	4.70	4.75	6.88	0.49
Original Loan-to-Value Ratio (%)	6.00	75.48	80.00	135.00	17.67
Mortgage Insurance (%)	0.00	8.12	0.00	35.00	12.16
Original Loan Term (months)	96.00	329.79	360.00	360.00	65.69
Original UPB ($)	16,000.00	213,876.98	188,000.00	1,124,000.00	117,053.28

Appendix

Table 13: Summary statistics for numerical variables in the test set. More detail on these variables can be found in the General User Guide for the Freddie Mac Single-Family Loan-Level Dataset. Numerical variables are normalised prior to modelling.

Variable(s)	Cardinality
Number of Borrowers , First Time Homebuyer , Super Conforming	2
Channel , Loan Purpose , Mortgage Insurance Cancellation , Occupancy Status	3
Number of Units	4
Original Debt-to-Income Ratio , Property Type	5
Seller Name	32
Servicer Name	26

Appendix

Table 14: Cardinality of categorical variables in the test set. More detail on these variables can be found in the General User Guide for the Freddie Mac Single-Family Loan-Level Dataset.

Name	0 $\to$ 1	1 $\to$ 0	1 $\to$ 2	2 $\to$ 0	2 $\to$ 1	2 $\to$ 3
cci_lag	-1.19 (-1.57, -0.81)	-1.27 (-1.98, -0.56)	-1.88 (-2.87, -0.89)	-0.65 (-2.76, 1.47)	0.14 (-2.34, 2.63)	-5.05 (-6.89, -3.21)
channelC	-0.12 (-0.17, -0.07)	0.05 (-0.05, 0.15)	0.08 (-0.05, 0.21)	-0.26 (-0.54, 0.01)	0.21 (-0.14, 0.57)	-0.30 (-0.52, -0.08)
channelR	-0.27 (-0.32, -0.22)	0.09 (-0.01, 0.18)	0.24 (0.12, 0.36)	-0.14 (-0.40, 0.11)	0.09 (-0.25, 0.43)	-0.21 (-0.41, -0.01)
cnt_borrone	0.48 (0.45, 0.51)	-0.19 (-0.25, -0.12)	0.05 (-0.02, 0.13)	-0.03 (-0.20, 0.13)	0.03 (-0.17, 0.24)	-0.02 (-0.15, 0.10)
cnt_units	-0.04 (-0.10, 0.02)	0.17 (0.02, 0.32)	0.15 (-0.03, 0.33)	0.03 (-0.33, 0.40)	-0.12 (-0.63, 0.38)	0.05 (-0.22, 0.32)
cov_dummy	1.10 (1.05, 1.15)	0.01 (-0.08, 0.10)	1.43 (1.32, 1.54)	0.35 (0.11, 0.59)	-0.36 (-0.65, -0.08)	1.34 (1.14, 1.54)

Appendix

Table 16: Linear covariate effects (mean and 95% interval). GAM with macroeconomic variables.

Name	0 $\to$ 1	1 $\to$ 0	1 $\to$ 2	2 $\to$ 0	2 $\to$ 1	2 $\to$ 3
cci_lag	-0.74 (-1.54, 0.16)	-1.01 (-1.76, -0.27)	-1.31 (-2.52, -0.04)	-0.94 (-2.16, 0.49)	-0.53 (-1.71, 0.83)	-2.06 (-3.12, -0.65)
channelC	-0.16 (-0.20, -0.11)	0.01 (-0.13, 0.14)	0.04 (-0.08, 0.17)	-0.18 (-0.44, 0.04)	0.23 (-0.06, 0.55)	-0.24 (-0.45, -0.04)
channelR	-0.29 (-0.33, -0.25)	0.07 (-0.06, 0.17)	0.20 (0.08, 0.33)	-0.11 (-0.33, 0.14)	0.10 (-0.19, 0.41)	-0.15 (-0.35, 0.03)
cnt_borrone	0.47 (0.45, 0.49)	-0.24 (-0.31, -0.16)	0.03 (-0.04, 0.13)	-0.03 (-0.22, 0.14)	0.04 (-0.21, 0.21)	-0.00 (-0.13, 0.11)
cnt_units	-0.07 (-0.13, -0.02)	0.12 (-0.01, 0.28)	0.13 (-0.04, 0.30)	-0.04 (-0.50, 0.27)	-0.13 (-0.68, 0.30)	-0.02 (-0.19, 0.22)
cov_dummy	1.02 (0.95, 1.10)	-0.05 (-0.17, 0.07)	1.47 (1.36, 1.59)	0.21 (-0.00, 0.40)	-0.41 (-0.65, -0.18)	1.17 (0.96, 1.34)

Appendix

Table 17: Linear covariate effects (mean and 95% interval). Semi-structured with macroeconomic variables.

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University of Edinburgh, Edinburgh, UK · Humboldt-Universität zu Berlin, Berlin, Germany

Interpretable ML Credit Scoring Algorithmic Fairness Logistic Regression Semiparametric Inference

Open paper AskPDF arXiv

Credit risk models increasingly need to combine predictive accuracy with transparent explanations and auditable fairness constraints. Logistic regression remains attractive because its coefficients are easy to interpret, but it can miss nonlinear structure. Flexible models can improve prediction, but their explanations are often post-hoc and may not describe the decision rule itself. We introduce

\texttt{findr}

, short for flexible, interpretable deep regression, a semi-structured framework for binary credit risk modelling that decomposes the logit into an interpretable structured component and an orthogonal neural residual. The orthogonalisation separates coefficient-based effects from residual nonlinear variation, while an in-processing Wasserstein penalty mitigates group disparities by comparing score distributions during training. The framework also includes diagnostics that measure the structured component's contribution to logit variation, decision agreement, and local directional consistency. We evaluate

\texttt{findr}

in a simulation study and on eight public credit datasets using score-level accuracy-fairness frontiers. The results show that

\texttt{findr}

behaves close to logistic regression when the signal is approximately linear, while recovering much of the predictive gain of neural models when nonlinear structure is relevant. The diagnostics identify when coefficient-based explanations remain close to the full fitted model and when residual variation must also be examined. These findings support semi-structured modelling as a practical way to make performance, fairness, and interpretability trade-offs explicit in credit risk decisions.

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