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.
Despite the popularity of Heckman's model, it is unclear whether this model can correct the selection bias or improve model performance. Some studies claim either higher model performance or different model parameters after using Heckman's model @cite_28 @cite_0 @cite_43 @cite_6 @cite_33 . These results, as explained by @cite_15 , depend upon whether the selection and default equations are correlated. On the other hand, @cite_47 @cite_11 @cite_45 state that the model parameters are inefficient, and the main criticism is that the Heckman's model fails to correct the selection bias when it is strong. This happens either when the correlation between the error terms in the selection and outcome equations is high or the data has high degree of censoring @cite_47 .
@cite_47 @cite_33 @cite_28 @cite_6 @cite_0 @cite_43 @cite_45 @cite_15 @cite_11
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This paper gives a short overview of Monte Carlo studies on the usefulness of Heckman's (1976, 1979) two-step estimator for estimating selection models. Such models occur frequently in empirical work, especially in microeconometrics when estimating wage equations or consumer expenditures. It is shown that exploratory work to check for collinearity problems is strongly recommended before deciding on which estimator to apply. In the absence of collinearity problems, the full-information maximum likelihood estimator is preferable to the limited-information two-step method of Heckman, although the latter also gives reasonable results. If, however, collinearity problems prevail, subsample OLS (or the Two-Part Model) is the most robust amongst the simple-to-calculate estimators. Copyright 2000 by Blackwell Publishers Ltd This paper investigates the effect of including the customer loan approval process to the estimation of loan performance and explores the influence of sample selection bias in predicting the probability of default. The bootstrap variable reduction technique is applied to reduce the variable dimension for a large data-set drawn from a major UK retail bank. The results show a statistically significant correlation between the loan approval and performance processes. We further demonstrate an economically significant improvement in forecasting performance when taking into account sample selection bias. We conclude that financial institutions can obtain benefits by correcting for sample selection bias in their credit scoring models. We examine three models for sample selection that are relevant for modeling credit scoring by commercial banks. A binary choice model is used to examine the decision of whether or not to extend credit. The selectivity aspect enters because such models are based on samples of individuals to whom credit has already been given. A regression model with sample selection is suggested for predicting expenditures, or the amount of credit. The same considerations as in the binary choice case apply here. Finally, a model for counts of occurrences is described which could, in some settings also be treated as a model of sample selection. # 1998 Elsevier Science B.V. AbstractTechnology evaluation has become a critical part of technology investment, and accurate evaluation can lead more funds to the companies that have innovative technology. However, existing processes have a weakness in that it considers only accepted applicants at the application stage. We analyse the effectiveness of technology evaluation model that encompasses both accepted and rejected applicants and compare its performance with the original accept-only model. Also, we include the analysis of reject inference technique, bivariate probit model, in order to see if the reject inference technique is of use against the accept-only model. The results show that sample selection bias of the accept-only model exists and the reject inference technique improves the accept-only model. However, the reject inference technique does not completely resolve the problem of sample selection bias. 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. 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. Reject inference is a method for inferring how a rejected credit applicant would have behaved had credit been granted. Credit-quality data on rejected applicants are usually missing not at random (MNAR). In order to infer credit-quality data MNAR, we propose a flexible method to generate the probability of missingness within a model-based bound and collapse Bayesian technique. We tested the method's performance relative to traditional reject-inference methods using real data. Results show that our method improves the classification power of credit scoring models under MNAR conditions. Abstract In many situations one needs to know which action one should take with a customer to yield the greatest response. Typically, estimates of the response functions of different actions will be based on the responses of customers previously assigned to each action. Often, however, the previous assignments will not have been random, so that estimates of the response functions will be biased. We examine the case of two possible actions. We look at the error arising from using the simple OLS estimate ignoring the selection bias, and also explore the possibility of using the Heckman model to allow for the sample selectivity. The performance of Heckman’s model is then compared with the simple OLS through simulation.