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1 Introduction, Economists and accountants have been forecasting bankruptcy for decades Most re 1. searchers have estimated single period classi cation models which I refer to as static mod. els with multiple period bankruptcy data By ignoring the fact that rms change through. time static models produce bankruptcy probabilities that are biased and inconsistent es. timates of the probabilities that they approximate Test statistics that are based on static. models give incorrect inferences I propose a hazard model that is simple to estimate. consistent and accurate, Static models are inappropriate for forecasting bankruptcy because of the nature of. bankruptcy data Since bankruptcy occurs infrequently forecasters use samples that span. several years to estimate their models The characteristics of most rms change from. year to year However static models can only consider one set of explanatory variables. for each rm Researchers that apply static models to bankruptcy have to select when to. observe each rm s characteristics Most forecasters choose to observe each bankrupt rm s. data in the year before bankruptcy They ignore data on healthy rms that eventually go. bankrupt By choosing when to observe each rm s characteristics arbitrarily forecasters. that use static models introduce an unnecessary selection bias into their estimates. I develop a simple hazard model that uses all available information to determine each. rm s bankruptcy risk at each point in time While static models produce biased and. inconsistent bankruptcy probability estimates the hazard model proposed here is consistent. in general and unbiased in some cases Estimating hazard models with the accounting. variables used previously by Altman 1968 and Zmijewski 1984 reveals that half of these. variables are statistically unrelated to bankruptcy probability I develop a new bankruptcy. model that uses three market driven variables to identify failing rms My new model. outperforms alternative models in out of sample forecasts. 1 See Altman 1993 for a survey of forecasting models. 2 For example Altman s 1968 original bankruptcy sample spans twenty years The sample used in this. paper includes bankruptcies observed over 31 years. 3 Hazard models are described in Kiefer 1988 and Lancaster 1990. 1 1 Advantages of Hazard Models, Hazard models resolve the problems of static models by explicitly accounting for time The. dependent variable in a hazard model is the time spent by a rm in the healthy group. When rms leave the healthy group for some reason other than bankruptcy e g merger. they are considered censored or no longer observed Static models simply consider such. rms healthy In a hazard model a rm s risk for bankruptcy changes through time and. its health is a function of its latest nancial data and its age The bankruptcy probability. that a static model assigns to a rm does not vary with time. In econometric terms there are three reasons to prefer hazard models for forecasting. bankruptcy The rst reason is that static models fail to control for each rm s period at. risk When sampling periods are long it is important to control for the fact that some. rms le for bankruptcy after many years of being at risk while other rms fail in their. rst year Static models do not adjust for period at risk but hazard models adjust for it. automatically The selection bias inherent in static bankruptcy models is a result of their. failure to correct for period at risk, The second reason to prefer hazard models is that they incorporate time varying co. variates or explanatory variables that change with time If a rm deteriorates before. bankruptcy then allowing its nancial data to reveal its changing health is important. Hazard models exploit each rm s time series data by including annual observations as. time varying covariates Unlike static models they can incorporate macroeconomic vari. ables that are the same for all rms at a given point of time Hazard models can also. account for potential duration dependence or the possibility that rm age might be an. important explanatory variable, The third reason that hazard models are preferable is that they may produce more.
e cient out of sample forecasts by utilizing much more data The hazard model can be. thought of as a binary logit model that includes each rm year as a separate observation. Since rms in the sample have an average of ten years of nancial data approximately. ten times more data is available to estimate the hazard model than is available to estimate. corresponding static models This results in more precise parameter estimates and superior. 1 2 Empirical Issues, Hazard models are preferable to static models both theoretically and empirically Compar. ing the out of sample forecasting ability of hazard models to that of Altman 1968 and. Zmijewski 1984 I nd that hazard models perform as well as or better than alternatives. Furthermore hazard models often produce dramatically di erent statistical inferences than. static models For example estimating hazard models reveals that about half of the ac. counting ratios that have been used to forecast bankruptcy are not statistically related. to failure Since previous models use independent variables with little or no explanatory. power I search for a new set of independent variables to develop a more accurate model. The most accurate out of sample forecasts that I can generate are calculated with a. hazard model that uses both market driven and accounting variables to identify bankrupt. rms The market variables include market size past stock returns and the idiosyncratic. standard deviation of stock returns I combine these market variables with the ratio of. net income to total assets and the ratio of total liabilities to total assets to estimate a. model that classi es 75 percent of failing rms in the top decile of rms ranked annually. by bankruptcy probability,1 3 Related Research, Precise bankruptcy forecasts are of great interest to academics practitioners and regula. tors Regulators use forecasting models to monitor the nancial health of banks pension. funds and other institutions Practitioners use default forecasts in conjunction with models. like that of Du e and Singleton 1997 to price corporate debt Academics use bankruptcy. forecasts to test various conjectures like the hypothesis that bankruptcy risk is priced in. stock returns e g Dichev 1997 Given the broad interest in accurate forecasts a superior. forecasting technology is valuable, Most previous bankruptcy forecasting models are subject to the criticism of this paper. The models of Altman 1968 Altman Haldeman and Narayanan 1977 Ohlson 1980. Zmijewski 1984 Lau 1987 and those of several other authors are misspeci ed Some. authors have addressed the de ciencies of existing bankruptcy models Queen and Roll. 1987 and Theodossiou 1993 develop dynamic forecasting models This paper builds. on the work of these researchers by explicitly addressing the bias in static models and. developing a consistent model, Bankruptcy forecasters are not the only researchers that can bene t from the results of. this paper Forecasters of corporate mergers have also applied static models to multiple. period data sets In particular the merger model of Palepu 1986 is biased and inconsistent. in the same way as the bankruptcy studies listed above Other authors such as Pagano. Panetta and Zingales 1998 and Denis Denis and Sarin 1997 estimate multiple period. logit models that can be interpreted as hazard models This paper concentrates on the. bankruptcy forecasting literature because it includes some of the most obvious misapplica. tions of single period models but the results reported here are relevant for other areas of. empirical nance as well,2 Hazard versus Static Models.
It is important to specify exactly what sort of bankruptcy data is available before discussing. alternative models For simplicity I assume that bankruptcy can only occur at discrete. points in time t 1 2 3 Most bankruptcy samples contain data on n rms that all. existed for some time between t 1 and t T Each rm either fails during the sample. period survives the sample period or it leaves the sample for some other reason such as a. merger or a liquidation De ne a failure time ti for each rm indexed by i as the time. when the rm leaves the sample for any reason Let a dummy variable yi equal one if rm. i failed at ti and let it equal zero otherwise and let the probability mass function of failure. be given by f t x where represents the vector of parameters of f and x represents a. vector of explanatory variables used to forecast failure. 2 1 Similarities Between Hazard and Static Models, To facilitate comparison between static and hazard models only maximum likelihood mod. els are discussed in this section The static models considered here have likelihood functions. of the form,L F ti xi y 1 F ti xi y, where F is the cumulative density function CDF that corresponds to f t x While. there are a number of models with likelihood functions of this form I refer to all models. that pertain to this family as logit models for simplicity. Describing hazard models requires a few more de nitions Following hazard model. conventions the survivor function S t x and the hazard function t x are de ned. S t x 1 f j x t x 2, The survivor function gives the probability of surviving up to time t and the hazard function. gives the probability of failure at t conditional on surviving to t The hazard model s. likelihood function is,L ti xi y S ti xi, A parametric form for the hazard function ti xi is often assumed The model can. incorporate time varying covariates by making x depend on time. Hazard and static models are closely related To make the relation between the models. clear I de ne a multiperiod logit model as a logit model that is estimated with data on. each rm in each year of its existence as if each rm year were an independent observation. The dependent variable in a multiperiod logit model is set equal to one only in the year in. which a bankruptcy ling occurred The following proposition illustrates the link between. hazard and multiperiod logit models, Proposition 1 A multiperiod logit model is equivalent to a discrete time hazard model.
with hazard function F t x, Since a multiperiod logit model is estimated with the data from each rm year as if it. were a separate observation its likelihood function is. L F ti xi yi 1 F j xi A 4, As a CDF F is strictly positive and bounded by one Since F depends on t and it is. positive and bounded it can be interpreted as a hazard function Replacing F with the. hazard function 0 1,L ti xi y 1 j xi A, Finally Cox and Oakes 1984 show that the survivor function for a discrete time hazard. model satis es,S t x 1 j x 6, Substituting equation 6 into 5 demonstrates that the likelihood function of a multiperid. logit model is equalivant to the likelihood function of a discrete time hazard model 3. with hazard rate t x F t x 2, 2 2 Econometric Properties of Hazard and Static Models.
Given the relationship between hazard and static models explained above it is possible to. see both the source and the e ect of the selection bias in previous bankruptcy forecasting. models This section illustrates the bias with a simple example It also presents a fairly. general argument for the inconsistency of static models and the consistency of hazard. models Finally it discusses problems of statistical inference and e ciency inherent in. static models,2 2 1 Consistency A Simple Example, Suppose that there are two periods in which bankruptcy is possible A dummy variable. yit is set to one if rm i goes bankrupt in period t In each period each rm has a. nonstochastic covariate xit which only takes on values of zero or one The covariate is. related to the rm s bankruptcy probability by,Prob yit 1 xit 7. There are N rms for which both yit and xit are observable in period one In period two. only rms that did not go bankrupt in period one are observable Each rm s observation is. assumed to be independently and identically distributed i i d The problem is to estimate. given the available data, Consider rst the hazard model estimator for The model of bankruptcy assumed. above stipulates that a rm s risk is independent of its age The discrete time hazard. model described by Proposition 1 has a hazard rate equal to the CDF of y Thus the. hazard function for this problem is equal to the probability of bankruptcy F xit. and the log likelihood function for the model is,LH ln H xi1 yi1 1 H xi1 H xi2 yi2 1 H xi2 1 yi2. The terms involving values in period two are raised to the power 1 yi because they are1. only observed when the rm does not go bankrupt in period one. The rst order condition for the maximization of this likelihood function is. xi yi 1 yi xi 0,H H 1 H xi H 1 H xi, Using the fact that both xit and yit can only take values of zero or one this expression can.
be simpli ed to,N 1 y x 1 y 1 y x,i 1 H i 1 1 H,which leads to the maximum likelihood estimator. PN y 1 y y,PN 1 1 1 2,i 1 yi1 1 yi1 yi2 1 yi1 xi1 1 yi1 1 yi2 xi2. Since rms with xit 0 have no probability of failure and rms with yi1 1 are not. observed in period two this can be simpli ed to,H PNi yi yi. Notice that this is a natural estimate of bankruptcy probability The numerator is equal. to the total number of failures observed while the denominator is the total number of rms. bankruptcy mo dels Queen and Roll 1987 and Theo dossiou 1993 dev elop dynamic forecasting mo dels This pap er builds on the w ork of these researc hers b y explicitly addressing bias in static mo dels and dev eloping a consisten t mo del Bankruptcy forecasters are not the only researc hers that can b ene t from the results of this pap er F orecasters of corp orate mergers ha v e also

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