Introduction

Academics, practitioners and regulators have routinely used models to predict the bankruptcy of companies. For example, the discriminant analysis model (DAM) has been a popular technique for studying the financial health of a corporate; see Altman (1968). Other frequently referred models include the models by Ohlson (1980) and Zmijewski (1984). The former bankruptcy prediction method is based on a linear logit model (LLM). The latter, on the other hand, is based on a probit model. Grice and Dugan (2001) recently cautioned the routine application of these two probabilistic mod-els of bankruptcy. Their study showed that using the prediction modmod-els to time periods and industries other than those used to develop the models may result in significant decline in prediction accuracies.

Bankruptcy prediction methods using other models or concepts include, for exam-ple, the recursive partition model (Frydman, Altman, and Kao 1985), expert systems (Messier and Hansen 1988), chaos theory (Lindsay and Campbell 1996), neural networks (Koh and Tan 1999), survival analysis (Lane, Looney, and Wansley 1986; Shumway 2001; Chava and Jarrow 2004), rough set theory (McKee 2003), KMV-Merton model (KMV; Merton 1974; Bharath and Shumway 2004; Vassalou and Xing 2004), and sup-port vector machines (Härdle et al., 2006). Basically, these methods are more compli-cated in computation and interpretation than the above probabilistic models.

The bankruptcy prediction model in Ohlson (1980) postulates that the logit function of bankruptcy probability is a linear function of the predictors. Nine predictors were selected for developing his model because they appeared to be the ones most frequently mentioned in the literature. The main reason of using the LLM is due to its simplicity in computation and interpretation. There are many software packages having logistic regression capabilities, for example, BMDP, EGRET, GAUSS, GLIM, and SAS, etc.

Thus LLM can be easily updated or revised as long as there are new observations of the same predictors or new predictive variables available for analysis. For a detailed introduction of the LLM, see the monograph by Hosmer and Lemeshow (1989).

When appropriate, the LLM has definite advantages. For example, the correspond-ing inferential methods usually have nice efficiency properties. Also, the parameters generally have some physical meaning which makes them interpretable and of interest in their own right. If the assumed linear logit function is grossly in error, then the advantages of the LLM will not be realized. Thus, there are few benefits from using a poorly specified LLM. See the discussion and Figure 2 of Härdle, Moro, and Schäfer (2006). Their results show that the relation between the bankruptcy probability and predictors, such as net income change and company size, may not be monotonic. The LLM is most appropriate when theory, past experience, or other sources are available that provide detailed knowledge about the data under study. Sometimes, based on previous experience, there are reasons for modelling the logit function of bankruptcy probability as a particular parametric function of predictors, which may not be linear.

However, a general drawback of such parametric modelling is that if one chooses a parametric family that is not of appropriate form, at least approximately, then there is still a danger of reaching erroneous inference.

The first focus of this dissertation is to consider a robust method, against misspec-ification of the parametric logit model relation, by introducing a semiparametric logit model (SLM; Zhao, Kristal, and White 1996) for predicting bankruptcy. This model is basically very similar to the LLM, except that some unspecified function replaces the linear function to model the relation between the predictors and the logit function of bankruptcy probability. Thus, clearly, the SLM is much more general and flexible in predicting the bankruptcy of a firm. Since the SLM is developed without assuming a parametric form for the logit function, there is some loss in the interpretability and efficiency of estimators obtained in this fashion. In contrast to physics or engineer-ing, it may not be often appropriate to give a specific functional relationship between

the probability of bankruptcy and the predictors in finance fields. Härdle, Moro, and Schäfer (2006) also propose a flexible but fully nonparametric approach for predicting bankruptcy. They use support vector machines to generate nonlinear score function of predictors, and then employ nonparametric technique to map scores into bankruptcy probabilities. Their work presents a new trend in bankruptcy analysis.

On the other hand, there is another potential pitfall of the LLM. It is static in nature, since it uses only one set of predictor values collected at a specific time point for each firm. The static model is generally not appropriate for predicting bankruptcy because it ignores both facts that the characteristics of firms change through time as well as bankruptcy does not often occur. For more discussions of the drawback to the static model, see Shumway (2001).

To avoid the drawback to the static model, Shumway (2001) applies the idea of time survival analysis (Cox and Oakes 1984) to develop the so-called discrete-time hazard model. The model has the advantage of using all available historical information to determine each firm’s bankruptcy risk at each point in time, hence it is a dynamic forecasting model. It is also called the discrete-time survival model (DSM) in Allison (1982). The values of parameters in Shumway’s dynamic prediction model are estimated by using the same approach as those in the multiperiod logit model (Pagano, Panetta, and Zingales 1998). However, theoretically, the multiperiod logit model assumes the predictor values collected for each firm at all time points are independent. Clearly, such predictor values are dependent, and the assumption does not hold in practice. Thus, asymptotic properties of the resulting estimates of parameters in Shumway’s dynamic prediction model can not be obtained from the multiperiod logit model.

The second focus of this dissertation is to employ directly the DSM to predict bank-ruptcy, and ignore the estimation procedure of the multiperiod logit model. The values of parameters in the DSM are estimated by the maximum likelihood method. The advantages of direct employment of the DSM include, for example, using all available

historical information to determine each firm’s bankruptcy risk at each point in time, assuming the predictor values of each firm are dependent. Hence, it is more general and flexible for the DSM to predict bankruptcy. The DSM has been successfully ap-plied in many fields including, for example, social science (Allison 1982), econometrics (Lancaster 1990), education (Singer and Willett 1993), and biostatistics (Klein and Moeschberger 1997).

The rest of this chapter is organized as follows. In Section 1.2, three important sampling schemes including the prospective (simple random), the case-control (choice-based), and the discrete-time survival data for bankruptcy prediction study are de-scribed. The data of the first two types are for static forecasting models, the LLM, the SLM, the KMV, and the DAM, and the data of the third type are for the dynamic forecasting model, the DSM. In Sections 1.3-1.7, five bankruptcy prediction models, the LLM, the SLM, the KMV, the DAM, and the DSM, are introduced respectively. In Sec-tion 1.8, bankruptcy predicSec-tion devices based on the five bankruptcy predicSec-tion models are presented. Finally, Section 1.9 contains a brief summary of the results obtained.