Empirical comparison of hazard models in predicting SMEs failure

Abstract

This study aims to shed light on the debate concerning the choice between discrete-time and continuous-time hazard models in making bankruptcy or any binary prediction using interval censored data. Building on the theoretical suggestions from various disciplines, we empirically compare widely used discrete-time hazard models (with logit and clog-log links) and the continuous-time Cox Proportional Hazards (CPH) model in predicting bankruptcy and financial distress of the United States Small and Medium-sized Enterprises (SMEs). Consistent with the theoretical arguments, we report that discrete-time hazard models are superior to the continuous-time CPH model in making binary predictions using interval censored data. Moreover, hazard models developed using a failure definition based jointly on bankruptcy laws and firms’ financial health exhibit superior goodness of fit and classification measures, in comparison to models that employ a failure definition based either on bankruptcy laws or firms’ financial health alone.

Keywords:

• Bankruptcy
• SMEs
• Discrete hazard models
• Cox proportional hazard
• Financial distress

JEL Classification Codes:

• G33
• C25
• C41
• C53

Acknowledgements

We render our sincere gratitude to the Editors of Quantitative Finance, two anonymous referees, Anup Srivastava from the Tuck School of Business (discussant), Taufiq Choudhry from the University of Southampton (discussant), Sushanta Mallick from the Queen Mary University of London, and seminar participants at the annual meetings of the European Financial Management Association (Amsterdam 2015) and the 28th Asian Finance Association Annual Conference (Bangkok 2016) for their useful comments and suggestions that improved this paper significantly.

Notes

¹ See Kleinbaum and Klein (2012) for detailed understanding about various tests of proportional hazards assumption for time-independent covariates. A Cox model with time-dependent covariates does not need to satisfy the proportional hazards assumption and is called an Extended Cox model. However, if the model employs both time-dependent and time-independent covariates, then PH assumption for time-independent covariates must be satisfied.

² Although the law provides other provisions, we consider only Chapter 11 and Chapter 7 as the vast majority of the financially distressed firms file for either of these two.

³ We are aware of the fact that the US Small Business Administration (SBA) defines SMEs differently. Broadly it considers a firm as an SME if it has less than 500 employees and annual turnover of less than $7.5 million. However, their precise definition varies across industrial sectors to reflect industry differences. For instance, a mining firm with less than 1000 employees, a general building and heavy construction firm with annual turnover of less than $36.5 million and a manufacturing firm with less than 1500 employees are all classified as small businesses as per SBA (https://www.sba.gov/contracting/getting-started-contractor/make-sure-you-meet-sba-size-standards/summary-size-standards-industry-sector; accessed on May 18, 2016). This may not be a convenient workable definition from the lender’s point of view. Many of these firms are too big to be called SMEs in the real sense, despite being classified as small firms as per SBA. They do this primarily to determine the eligibility of a firm for SBA financial assistance, or for its other programs. Thus we follow a more appropriate and popular definition of SMEs provided by the European Union for this study. The most popular study on predicting bankruptcy of US SMEs by Altman and Sabato (2007) also follows the European Union’s definition of SMEs. They consider firms as SMEs if they report sales revenue of less than $65 million (approximately €50 million, as suggested by the European Union).

⁴ It could be increasing, decreasing, and then increasing or any shape we may imagine. But it assumes that whatever the general shape of the hazard function, it is same for all subjects.

⁵ In our analysis the risk set keeps decreasing with successive failures. Efron's (1977) method reduces the weight of contributions to the risk set from the subjects which exhibit tied event times in successive risk sets.

⁶ The event is experienced in continuous-time but we only record the time interval within which the event takes place.

⁷ While calculating the ratio EBITDAIE, zero interest expense (IE) for all firm-year observations is replaced with $1 to avoid missing values.

⁸ See Cleves et al. (2010) for a detailed description of Kaplan-Meier curves.

⁹ See for example http://www.ats.ucla.edu/STAT/stata/seminars/stata_survival/default.htm (accessed May 18, 2016). Also see Cleves et al. (2010) for a more thorough understanding.

¹⁰ In non-linear regression analysis, Marginal Effects are a useful way to examine the effect of changes in a given covariate on changes in the outcome variable, holding other covariates constant. These can be computed as marginal change (it is the partial derivative for continuous predictors) when a covariate changes by an infinitely small quantity and discrete change (for factor variables) when a covariate changes by a fixed quantity. Average Marginal Effects (AME) of a given covariate is the average of its marginal effects computed for each observation at its observed values. Alternatively, AME can be interpreted as the change in the outcome (financial distress = 1, in our case) probabilities due to unit change in the value of a given covariate, provided other covariates are held constant. See Long and Freese (2014) for detailed discussion on this topic.

¹¹ Akaike Information Criterion (AIC) is defined as: AIC = −2 × L + 2 × (p + 1) where L is the log-likelihood of the fitted model and p is the number of regression coefficients estimated for non-constant covariates. In general, models with lower values of AIC are preferred to larger ones.

¹² For most covariates and their respective time lags in table 8, absolute values of coefficients are highest for logit estimates, followed by cloglog estimates, and least for Cox estimates (i.e. |logit| > |cloglog| > |Cox|). However, based on this it shall be inappropriate to conclude that, for a unit change in the value of a covariate logit estimates lead to highest change in the outcome probability than its alternative counterparts. This generalization may only be valid if their Average Marginal Effects (AME) or other similar estimate also show this pattern.

¹³ Table 1 shows that the earliest age that a firm can experience a distress event under all three default definitions is one year. However, the hazard curves start from somewhere around five years. This difference is due to the fact that the ‘sts graph’ command in Stata performs an adjustment of the smoothed hazard near the boundaries. In case of the default kernel function of -sts graph- (Epanechnikov kernel), the plotting range of the smoothed hazard function is restricted to within one bandwidth of each endpoint. The same is true for other kernels, except the epan2, biweight, and rectangular kernels, in which case the adjustment is performed using boundary kernels. If we wish to plot an estimate of the hazard for the entire range, we could use a kernel without a boundary correction. Alternatively, we can use the -noboundary- option, but this will produce an estimate that is biased near the edges. See ‘help sts graph’ in Stata and Silverman (1986) for further details. This will not affect the empirical analysis if one uses a fully non-parametric method of baseline hazard specification. However, one needs to be careful while using piecewise-constant specification.

¹⁴ These results are not reported in this paper; however, it may be made available from the authors.

¹⁵ Coefficients with a negative sign become positive and vice versa.

¹⁶ This might result in misleading estimates of AUROC. Thus one needs to be careful when drawing inferences regarding out-of-sample predictive ability of the forecasting model.