Spatio-Temporal Modelling of Credit Default Data

Transcription

1 1/20 Spatio-Temporal Modelling of Credit Default Data Sathyanarayan Anand Advisor: Prof. Robert Stine The Wharton School, University of Pennsylvania April 29, 2011

2 2/20 Outline 1 Background 2 Conditional Autoregressive (CAR) Model 3 Results on Credit Default Data 4 Simulation Based Analysis 5 Work In Progress

3 3/20 Problem Statement & Data Description To predict credit default rates aggregated over some geographical region, and understand the underlying spatial and temporal structures

4 3/20 Problem Statement & Data Description To predict credit default rates aggregated over some geographical region, and understand the underlying spatial and temporal structures Quarterly Data from the Fed County, state & national levels 71 quarters from 1992 to 2009 County Level Default rates of many different kinds & time limits. Ex: mortgage, revolving, etc. Economic indicators, such as poverty, income, unemployment, etc. Time series 3000 x 71 per default rate; plus covariates State & National Levels All the variables at county level; plus many more rates & economic indicators State level 50 x 71, National level 1 x 71 per default rate; plus covariates

5 4/20 Dependencies in Default Rates REPB60M Medians (Smoothed) 1.0 REPB60M Mississippi Georgia North Carolina National California Minnesota Correlation CreditType Installment 60-Day Mortgage 60-Day Revolving 60-Day time Time

6 5/20 Issues to Think About Revolving Default Rate (2006.1) REPB60M Spatial Structure? Markovian? How many layers to use? Geographic distance? Or in covariate space? Outliers? Overfitting and predictive power Bob s motto: regression work s best

7 6/20 A Naive Model We focus on 60-day Revolving Debt Default Rate for Pennsylvania from here on Let p st be the default rate for location s at time t

8 6/20 A Naive Model We focus on 60-day Revolving Debt Default Rate for Pennsylvania from here on Let p st be the default rate for location s at time t A simple OLS model can be written as logit(p st) = ψ + φ s + α t + J β jt x jst + ɛ st j=1 iid ( ) ɛ st N 0, σ 2 e Fixed effects, φ s for space and α t for time For each covariate, one slope coefficient β jt across space at a given time t No spatial structure captured Why logit? keeps predicted default rates to between 0 and 1 Is it actually logit?

9 7/20 Is It Logit? Inverse response plot from the logit model looks linear (Simon Sheather) Fitted Observed

10 8/20 CAR Specification The CAR model is based on the logit model previously specified. logit(p st) N ( µ st, σs 2 ) ; s = 1, 2,..., S J µ st = ψ + φ s + α t + β jt x jst j=1 T s.t. α t = 0 and S φ s = 0 t=1 s=1

11 8/20 CAR Specification The CAR model is based on the logit model previously specified. logit(p st) N ( µ st, σ 2 s ) ; s = 1, 2,..., S µ st = ψ + φ s + α t + s.t. T α t = 0 t=1 Spatial and temporal prior distributions: J β jt x jst j=1 and S φ s = 0 s=1 φ i φ i N ρ c w ij σc 2 φ j, w j i+ w i+ α i α i N ρ τ v ij στ 2 α j, v j i+ v i+

12 9/20 Interpreting the CAR Model φ i φ i N ( ρ c j w ij w i+ φ j, σ 2 c w i+ ) w ij = 1 if nodes i and j are neighbors, and 0 otherwise Node mean is a scaled average of its neighbors values Node variance is inversly proportional to the number of neighbors

13 9/20 Interpreting the CAR Model φ i φ i N ( ρ c j w ij w i+ φ j, σ 2 c w i+ ) w ij = 1 if nodes i and j are neighbors, and 0 otherwise Node mean is a scaled average of its neighbors values Node variance is inversly proportional to the number of neighbors Let W be the adjacency matrix of the graph. Then, φ N S ( 0, σ 2 c (E w ρ c W ) 1) where E w = diagonal matrix of row sums of W Note: the covariance matrix is invertible if ρ c (λ 1 n, λ 1 1 ), where λn (negative) and λ 1 (positive) are the minimum and maximum eigenvalues of W.

14 10/20 Estimating the CAR Model We derive the CAR prior specification under the zero-sum constraint to get w ij φ j σc φ i ψ, φ i N ρ 2 c + ψ(ρ c 1), w i+ j i These terms are now contrasts from the global intercept ψ, with φ 1 = 0. w i+

15 10/20 Estimating the CAR Model We derive the CAR prior specification under the zero-sum constraint to get w ij φ j σc φ i ψ, φ i N ρ 2 c + ψ(ρ c 1), w i+ j i These terms are now contrasts from the global intercept ψ, with φ 1 = 0. Estimation notes: Estimate models separately for each time point t and predict for t + 1 Each spatial point s gets its own variance term, σ 2 s Each covariate j gets its own variance, σ 2 jb CAR scaling parameter ρ c is fixed to its highest permissible value i.e. λ 1 1 This is the commonly used method in other papers Adaptive rejection metropolis sampling proposed but computationally intensive (Olivera 2008) Solution: good old friend, MLE (in a few slides, time permitting) w i+ All posteriors have closed form expressions Gibbs sampling Number of parameters to be estimated O(S + J T ), for S spatial locations, J covariates, and T time points

16 11/20 Prediction of Default Rates: RMSEs CAR Model Blue: Fitted RMSE, Pink: Predicted RMSE Logit Model Red: Fitted RMSE, Green: Predicted RMSE Model RMSE 0.3 Logit Fitted Logit Predicted CAR Fitted CAR Predicted TimePoint

17 12/20 Simulation Experiments With real data, we can compare different models but no baseline truths Understand how the CAR model behaves under lower signal to noise ratios Full scale simulated datasets of Pennsylvania 67 counties with the same neighborhood structure Covariates (income, unemployment, etc.) kept the same Generate slopes for covariates, one for each time point Generate the spatial and temporal CAR-based intercepts Add in noise Noise levels 2-5 times higher than the actual dataset Both models estimated in the same fashion

18 Quarter 13/20 Estimation for a Random Time Point Estimated & True Covariate Slopes Beta

19 Quarter Quarter Estimation for a Random Time Point Estimated & True Covariate Slopes Beta Estimated & True α t Alpha 13/20

20 Quarter Quarter Estimation for a Random Time Point Estimated & True Covariate Slopes Estimated vs. True φ s Beta True Phi Estimated & True α t Alpha Estimated Phi 13/20

21 Quarter Quarter Actual Estimation for a Random Time Point Estimated & True Covariate Slopes Estimated vs. True φ s Beta True Phi Estimated & True α t Alpha Estimated Phi Predicted vs. Actual Response Predicted 13/20

22 14/20 Prediction of Default Rates: RMSEs RMSE Knowing True Model: Pink CAR Model Dark Green: Fitted RMSE, Blue: Predicted RMSE Logit Model Red: Fitted RMSE, Olive: Predicted RMSE Model Logit Fitted RMSE 2.5 Logit Predicted CAR Fitted CAR Predicted True TimePoint Predictive performace is much worse than the RMSE of fitted values

23 Type Predicted True /20 Predicting the Temporal CAR Parameter The temporal CAR term is specified as ( α t α t N ρ τ v tu α u, v u t+ When predicting for time T + 1, the only neighbor is time T, and hence ˆα T +1 = ρ c α T Compare predicted ˆα T +1 from model built for time T, and fitted α from model built using the entire dataset σ 2 τ v t+ ) Alpha Absolute Difference Period Period

24 16/20 Understanding the Temporal CAR Parameter 3.5 Blue Bars: Alpha Difference RMSE Model a CAR Fitted a CAR Predicted a True TimePoint Difference in α explains about 64% of the difference between fitted and predicted RMSEs Need a predictive model for the temporal intercepts on top of the CAR structure

25 17/20 Work In Progress: MLE for CAR Covariance Let X i N S (0, Σ) ; i = 1, 2,..., T The log-likelihood is given by log(l(σ Y )) = C T 2 log(det(σ)) 1 2 T i=1 X i Σ 1 X i Under the CAR structure, Σ = σ 2 (E w ρw ) 1, and log-likehood reduces to log(l(σ 2, ρ Y )) = C+ T 2 log(det(ew ρw )) ST 2 log(σ2 ) 1 tr((ew ρw )V ) 2σ2 where V = T equations. i=1 X i X i. Solving for the MLE, we get the following set of tr ( ) (ˆσ 2 T (E w ˆρW ) 1 V ) W = 0 ˆσ 2 = 1 tr ((Ew ˆρW ) V ) ST

26 18/20 Work In Progress: Utilizing the Spatial CAR Structure Spatial structure is useful if the future can be partially observed Knowing future values for k counties, update predictions of remaining counties Plots: X-axis = percentage of known values, Y-axis = updated RMSE TimePoint 1.6 RMSE 2.0 TP21 TP35 TP43 TP61 RMSE TimePoint TP33 TP59 TP Peek Peek Seems to work for most time points, but no improvement for some Could be the basis for a test of changing spatial structure

27 19/20 Work In Progress MLE for CAR covariance structure Likelihood ratio test for CAR structure with power calculations under two alternative spatial structures, robustness to the multivariate t distribution, and scalability with graph size Predictive layers on the CAR model Compare to Bob and Dean s α-investing procedure Spatial clustering of counties using Hierarchical DP Summary statistic for amount of spatial correlation : extending Geary s C to spatio-temporal data A second application: Understanding patient access to care in the face of reduction or closure of hospital service lines Extension of previous work, The Effect of California s Safe-Staffing Law (AB394) on Hospital Behavior, to be submitted to the Journal of Health Economics, May With Emi Terasawa and Scott Harrington, Healthcare Management Department

28 20/20 Thank You