Credit scoring models based on accepted applications may be biased and their consequences can have a statistical and economic impact. Reject inference is the process of attempting to infer the creditworthiness status of the rejected applications. In this research, we use deep generative models to develop two new semi-supervised Bayesian models for reject inference in credit scoring, in which we model the data generating process to be dependent on a Gaussian mixture. The goal is to improve the classification accuracy in credit scoring models by adding reject applications. Our proposed models infer the unknown creditworthiness of the rejected applications by exact enumeration of the two possible outcomes of the loan (default or non-default). The efficient stochastic gradient optimization technique used in deep generative models makes our models suitable for large data sets. Finally, the experiments in this research show that our proposed models perform better than classical and alternative machine learning models for reject inference in credit scoring.
Heckman's bivariate two-stage model @cite_22 @cite_10 has been used in different reject inference studies The Heckman's model, named after Nobel Laureate James Joseph Heckman, has been extended or modified in different directions. See @cite_15 for a chronological overview of the model evolution and its early applications. It was in @cite_23 where the Heckman's approach was first applied to credit scoring where the outcome is discrete. . This approach simultaneously models the accept reject and default non-default mechanisms. Assuming that the error terms in these processes are bivariate normally distributed with unit variance and correlation coefficient @math , the selection bias arises when @math and it is corrected using the inverse of the Mills ratio.
@cite_15 @cite_10 @cite_22 @cite_23
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Credit scoring models are commonly developed using only accepted Known Good Bad (G B) applications, called KGB model, because we only know the performance of those accepted in the past. Obviously, the KGB model is not indicative of the entire through-the-door population, and reject inference precisely attempts to address the bias by assigning an inferred G B status to rejected applications. In this paper, we discuss the pros and cons of various reject inference techniques, and pitfalls to avoid when using them. We consider a real dataset of a major French consumer finance bank to assess the effectiveness of the practice of using reject inference. To do that, we rely on the logistic regression framework to model probabilities to become good bad, and then validate the model performance with and without sample selection bias correction. Our main results can be summarized as follows. First, we show that the best reject inference technique is not necessarily the most complicated one: reweighting and parceling provide more accurate and relevant results than fuzzy augmentation and Heckmans two-stage correction. Second, disregarding rejected applications significantly impacts the forecast accuracy of the scorecard. Third, as the sum of standard errors dramatically reduces when the sample size increases, reject inference turns out to produce an improved representation of the population. Finally, reject inference appears to be an effective way to reduce overfitting in model selection. Many researchers see the need for reject inference in credit scoring models to come from a sample selection problem whereby a missing variable results in omitted variable bias. Alternatively, practitioners often see the problem as one of missing data where the relationship in the new model is biased because the behaviour of the omitted cases differs from that of those who make up the sample for a new model. To attempt to correct for this, differential weights are applied to the new cases. The aim of this paper is to see if the use of both a Heckman style sample selection model and the use of sampling weights, together, will improve predictive performance compared with either technique used alone. This paper will use a sample of applicants in which virtually every applicant was accepted. This allows us to compare the actual performance of each model with the performance of models which are based only on accepted cases. One of the aims of credit scoring models is to predict the probability of repayment of any applicant and yet such models are usually parameterised using a sample of accepted applicants only. This may lead to biased estimates of the parameters. In this paper we examine two issues. First, we compare the classification accuracy of a model based only on accepted applicants, relative to one based on a sample of all applicants. We find only a minimal difference, given the cutoff scores for the old model used by the data supplier. Using a simulated model we examine the predictive performance of models estimated from bands of applicants, ranked by predicted creditworthiness. We find that the lower the risk band of the training sample, the less accurate the predictions for all applicants. We also find that the lower the risk band of the training sample, the greater the overestimate of the true performance of the model, when tested on a sample of applicants within the same risk band — as a financial institution would do. The overestimation may be very large. Second, we examine the predictive accuracy of a bivariate probit model with selection (BVP). This parameterises the accept–reject model allowing for (unknown) omitted variables to be correlated with those of the original good–bad model. The BVP model may improve accuracy if the loan officer has overridden a scoring rule. We find that a small improvement when using the BVP model is sometimes possible. Reject inference is one of the key processes required to build relevant credit scorecard models. Reject inference is used to infer the good or bad loan status to credit applicants that were rejected by the financial institution. If rejected applicants data is not used in the updating of the credit scoring model, the model is biased because it will not be representative of the entire applicant population. Many reject inference techniques perform an extrapolation method to infer the good or bad loan status of the rejected applicants. The issues with extrapolation are discussed, and this study provides a novel reject inference technique in which the rejected applicants are included in the model estimation process. The extrapolation problem is avoided using the methodology in this paper. The newly proposed reject inference technique is shown to outperform the standard extrapolation technique using a simulation study.