Academic journal article
*Akron Business and Economic Review*

# Causality, Regression, Discriminant Analysis, and Research on Failure

## Article excerpt

Causality, Regression, Discriminant Analysis, and Research on Failure(*)

It is natural that lenders, investors, and regulators would like to have some empirical insight into the default risk of the firms and financial institutions with which they are involved. This interest has led to substantial empirical research on the failure of such firms and institutions.(1) While other variables have been employed in this research, the most widely-used variable set has been the financial ratios of failed and nonfailed firms and institutions.(2) Statistical methodologies used to relate these ratios to failure have included various types of regression and maximum-likelihood estimation, but by far the most common analysis technique has been discriminant analysis.(3) This has led to a series of papers citing serious methodological, empirical, and interpretational problems in published failure studies where discriminant analysis was used[4, 13, 20, 21]. Among the problems cited are misinterpretation of the classification outputs, violation of the statistical assumptions underlying the discriminant model, misinterpretation of the implications of discriminant statistics regarding the importance of individual variables, unnecessary reduction in dimensionality and variable elimination, and problems in application over time.

In investigating the association between ratio and other measures and failure, the researcher must choose an analysis methodology: regression (or another related maximum-likelihood method), discriminant analysis, or another technique. Several considerations are relevant to this choice of methodology. One consideration is the different statistical requirements of discriminant analysis and regression[10, 23, 26]. However, these different analysis techniques also reflect different assumptions regarding the underlying causal structures. Thus, causality should also be a consideration in the choice and application of analysis methods. Few default studies have explicitly considered the causal issue. This article discusses causality relative to the empirical investigation of failure and the choice of analysis technique.

In the next section of this paper, the implied causality in regression and in discriminant analysis is discussed. Potential assumptions regarding causality in the failure process are then reviewed, with particular reference to financial ratios and their relation to failure. These assumptions are related to the implied causality in regression and in discriminant analysis and thus to methodology choice. The problems of interpretation of individual variables and of dimensionality reduction in discriminant analysis are discussed relative to these assumptions about failure. Examples of prior research are presented where the causality implied in the researchers' hypotheses was not matched to the statistical techniques used and where some of these problems consequently occurred. Two example analyses are performed on a small data set to illustrate the proper use of the two methodologies in failure research. The final section presents conclusions and implications for future research on failure.

CAUSALITY IN REGRESSION AND IN DISCRIMINANT ANALYSIS

In the regression model, the implied causality runs from the independent variables (the Xs), which are exogenously determined, to the dependent variable (Y), which is hypothesized to be determined by the independent variables plus a random disturbance term[27]. In such a model, it is appropriate to form hypotheses regarding the signs and magnitudes of the estimated coefficients of the independent variables, since the postulated causal situation leads to expectations regarding these signs and magnitudes that the model is (in part) testing. The postulated causal situation will also suggest a limited set of potential independent variables, and it is appropriate in the regression model to include only those variables that the researcher believes determine the dependent variable; extraneous variables increase estimation time requirements. The regression procedure estimates a unique set of coefficients for the independent variables. Thus the estimated coefficient interpretation and unnecessary dimensionality reduction cited as problems with discriminant studies of failure are not applicable in regression.(4)

In discriminant analysis, the causal flow is postulated to run from the Y to the Xs. That is, the Xs are the symptoms of underlying Y (group membership)[4]. Here, it is inappropriate to make statements about the signs and magnitudes of the estimated coefficients for two reasons. First, the estimated coefficients in multiple discriminant analysis are not unique. Second, in order to make such statements, the researcher must have a theory relating the symptoms to their cause (where the cause is group membership; this is akin to relating symptoms to the underlying disease in medicine). Colinearity is irrelevant (since it is inappropriate to interpret the signs or to perform significance tests on individual X variables, and the major difficulty inherent in colinearity concerns such signs and tests), and the appropriate evaluation statistic is classification accuracy (since it is membership in the groups that is being researched). Dimensionality reduction (deleting Xs for one reason or another, particularly before statistical analysis on colinearity grounds) is also inappropriate since potential symptoms are not known a priori.

Causal considerations are, of course, not part of the discriminant analysis or regression algorithms. Instead, causality is reflected in what the researcher does: the methodology selected, the data used, and how the output of the algorithm is interpreted. The standard statistical packages provide output for each algorithm that is intended to aid this interpretation process in line with the implied causality. For example, in regression analysis, t statistics are computed for each estimated regression coefficient. These statistics are then interpreted as significance tests of the variables. In multiple discriminant analysis, an estimated standardized discriminant function coefficient (beta weight) is calculated. This is not intended for interpretation as a significance test statistic, however, but as an indicator of the relative importance of individual variables[25]

While regression and discriminant analysis model different causal situations and their results are structured to allow for different interpretational methods, it is possible to use an inappropriate model and generate an estimated function that may have reasonable explanatory power. Success depends largely on the characteristics of the data set. However, if prediction is a concern, the method that is most likely to lead to the best results outside the estimation sample would seem to be the one most in keeping with causality in failure. If hypothesis testing is the concern, a statistical methodology that is designed to directly test the researcher's hypotheses is most likely to produce easily interpretable and appropriate results.

CAUSALITY, FAILURE, AND FINANCIAL RATIOS

With the exception of Wilcox's[32, 33] gambler's ruin studies, few empirical studies of failure have explicitly considered the causal issue.(5) No study has addressed the causality question via the available econometric techniques for assessing causality.(6) Some have merely postulated an association between financial ratios and default (with the functional relationship unspecified) and then have proceeded to apply some type of statistical analysis.(7) To evaluate prior studies, a discussion of the relationship between the descriptor variables used in these studies (financial ratios) and the default event is in order.(8)

Businesses default because they are unable to pay on schedule one or more short- or long-term creditors. Financial theory generally suggests that ratios are related to this inability to pay since they measure the amount of the firm's debts, the liquidity of its assets, and the level of its cash flows; such financial dimensions of the firm are critical determinants of the ability to pay debts as they come due. This is, however, not the only possible view of causality in default. Let us illustrate different perspectives by postulating two (admittedly extreme and unrealistic) potential views of the relationship between ratios and the default event.(9)

Firms Set Risk Levels; Exogenous Events then Determine Failure

Here, firms explicitly decide on their risk and return position along such dimensions as capital structure, liquidity structure, debt maturity structure, fixed cost structure, etc. A series of random exogenous events then occur.(10) Based on these events, the firm's cash flow is determined, and the firm will or will not be able to pay maturing debt from the earnings stream (see Wilcox[32, 33] and Gordon [16]; this is a states-of-the-world approach). The firm can then restructure its debt via out-of-court negotiating with creditors, selling assets, selling new equity[19], or defaulting and seeking court protection. In this scenario, ratio analysis performed between the time of management's risk-level decision and the time of the random events' occurrence would focus on ratios that are indices of the various risks involved: liquidity ratios, for example, might be used to measure the risk of default on short-term debt. If exogenous events are serially correlated, trend analysis or analysis involving prior years' cash flow results could also be used. Here, since ratios measure the firm's risk levels and thus its susceptibility to default, they are indices of the causes of default. Because of this causal link, regression (or a similar maximum-likelihood method) is the suggested analysis method. Also, it is appropriate to formulate hypotheses regarding the significance and level of estimated coefficients and to avoid the inclusion in the function of "wrongly-signed" variables since these would be suspect in their predictive ability in the general population (if predictivity, as well as hypotheses testing, is a goal of the research).

Management's Ability Determines Failure

In this assumption set, firms fail primarily for reasons related to the firm's marketing and management capabilities. Firms are either in the "extinction mode" or they are not[2]. Financial risks as such are a good deal less important in this view, since any amount of debt can be serviced with a sufficient level of sales (given a positive margin), and sales levels are seen here not as a partly random event but as the result of management's ability to run the business successfully. Financial ratios, if they are useful as descriptors, must relate primarily to this management ability. The ratios are symptoms, and it is difficult to state hypotheses relating them to the underlying management-related causes of success or failure. For example, following the logic of the first assumption set, it might be expected (all other things being equal) that the total debt to total assets ratio would be positively related to default, since relatively more debts must be serviced. But under the second assumption set, this ceteris parabus assumption is not appropriate. For example, a firm whose management has developed strategies that lead them to expect substantial growth might opt for higher leverage to better translate these sales gains into increases in earnings per share. Firms in the same industry with fewer growth prospects might choose to use less leverage to reduce financial risk. In fact, this strategy with regard to growth and leverage is recommended in finance textbooks (see, for example, Weston and Brigham[31]). However, the fewer growth opportunities may be a symptom of poor management, and these firms may default even on the lower amount of debt. It would seem that in this second view, the appropriate statistical procedures would involve searching for symptoms of whether the firm's management is capable or not. Given the current state of management theory, it would seem difficult to say which financial ratios might be related to management ability, and a search of potential variables based on a classification accuracy criterion would be appropriate. These are the methods of discriminant analysis. Table 1 summarizes the proper analysis techniques for these two scenarios.

CRITICAL REVIEW OF SELECTED PRIOR STUDIES

By intent, these two assumption sets represent extreme views of causality in failure. Actual defaults may contain elements of each. However, given the parallel between the first assumption set and much thinking in finance and economics, one might expect that maximum-likelihood estimation would be the prevalent failure analysis technique displayed in the literature. In fact, discriminant analysis has been much more common. Much failure research appears to have involved the assumption and interpretation of a causal situation as that in the first assumption set but the choice of analysis methodology as if the causality were that of the second assumption set. That is, many studies of failure have used discriminant analysis but interpreted the results as if a maximum-likelihood procedure had been used. This confusion has been manifested in at least three ways: (1) researchers have tried to interpret the signs of discriminant function coefficients, (2) dimensionality reduction based on colinearity has occurred, and (3) inclusion/exclusion criteria other than classification ability have been used. As illustrations, we examine three widely-cited studies that exhibit problems of this sort: the pathbreaking research of Altman[3], Edminster's[12] work on small business default, and Dambolena and Khoury's[11] research on ratio stability and failure.

Altman's stated purpose was to assess the empirical validity of ratio analysis [3]. However, it seems from his discussion of the ratios selected for his discriminant function that his implied hypothesis involves causality running from the ratios to the default/nondefault event[3]. His discussion of the relationship between the equity/debt ratio and insolvency is evidence of this view; he argues that lower debt/equity ratios combined with asset value variability increase default probability[3]. Multiple discriminant analysis is, however, the statistical analysis method chosen. In interpreting the estimated discriminant function, he states that" ... the discriminant coefficients of equation (1) display positive signs, which is what one would expect"[3]. Also, intercorrelations were, in part, used to reduce dimensionality and determine which variables were to be included in the estimation of this equation[3]. These are inappropriate procedures in the use of discriminant analysis.

However, Altman's study properly stresses classification ability as a primary criterion for variable selection. Several studies use a variable selection criterion that is not totally in keeping with this goal, such as minimization of Wilk's lambda. The use of the Wilk's lambda criterion in these studies may stem from the interpretation of one minus Wilk's lambda as a goodness-of-fit measure similar to [R.sup.2] in regression analysis. In a failure research context, however, one aim is typically the development of a function for examining the characteristics associated with membership in the defaulted or nondefaulted groups. Consequently, the decision criteria for variable inclusion/exclusion should focus on classification.(11)

As with Altman, Edminster's[12] discussion of ratios seems to embrace the causality described in the first assumption set. "The first hypothesis is that a ratio's level is a predictor of small business failure. For example, a firm is believed less likely to fail if its current ratio is 3/1 rather than 1/1"[12]. However, discriminant analysis is apparently chosen as the statistical methodology. This study contains extensive concern for multicolinearity, which is irrelevant in a discriminant analysis context. This concern results in the selection of an unusual stepwise variable selection criterion based on correlation with variables already in the equation[12] rather than on classification.

Edminster also attempts to interpret the signs of the estimated discriminant analysis coefficients[12]; in discriminant analysis, this is not an appropriate procedure. Thus, his "surprising result" of a negative sign for the trend in the quick ratio[12] is, in discriminant analysis, neither surprising nor a result.

Research by Dambolena and Khoury interprets ratios as "...explanatory variables in the derivation of a discriminant function"[11]. Also, it is clear from the authors' other discussion[11] that they believe that causality runs from the ratios to the default/nondefault event. For discriminant analysis to be the appropriate technique, however, explanation and causality must instead run from group membership to the ratios. Estimated discriminant coefficients are not interpreted in this reference or dimensionality reduced on colinearity grounds. However, variables are selected from the candidate set in a stepwise fashion based on the minimization of Wilk's lambda rather than a criterion oriented more toward classification.

In retrospect, it is clear that errors were made in the use and interpretation of statistical algorithms in some prior studies of failure. Our knowledge of failure stems to a great extent from these studies; if their results are flawed, so is this knowledge. But did these errors make a difference? What if Altman had not used inter-correlations to reduce dimensionality? What if Edminster had ignored multicolinearity? What if Dambolena and Khoury had used a more classification-based inclusion criterion for variables? What if all these researchers had directly set forth hypotheses relating various ratios to failure and tested them via maximum-likelihood estimation? If different procedures had been used, variables other than those found to be critical in failure may have been highlighted, and other (better or worse) statistical results in explaining failure may have occurred. A complete response to these questions requires both a replication of prior research using properly-applied discriminant analysis techniques and a second replication that would test theories of failure via maximum-likelihood estimation. Such an empirical effort poses a substantial challenge for future research but is beyond the scope of this paper. Instead, in the following section, we illustrate what we believe to be the proper procedures relating causality, the selection of statistical methodology, and the interpretation of statistical output for a small set of actual data on failed and nonfailed firms.(12)

EXAMPLE ANALYSES

In this section, example analyses will be performed under each of the causality approaches previously outlined. The same data set will be used in each analysis. This data set consists of the financial ratios and asset sizes of eleven failed and eleven nonfailed custom plastic injection molders; it is presented in Table 2.(13) The financial ratios presented are a combination of those found to be useful in a predictive sense in default studies and those that may be theoretically associated with default. The data are from the early 1970s; the defaulted sample consists of all the defaults to a firm selling to this industry during this period. The nondefaulted sample is a random sample of nondefaulted customers from the same period. In executing the two example analyses, we will sacrifice some sophistications in technique for the sake of simplicity. These sophistications, which would have to be considered in a substantive study of failure, will be discussed in footnotes.

Let us first deal with the case in which the researcher believes that the financial ratios measure causes of default (scenario 1). Maximum-likelihood analysis is suggested by this causality. Candidate explanatory variables would be selected based on their relationship to the theoretical causes of default. For the example analysis, five potential independent variables were selected from the data set based on such theoretical considerations:

1. Ratio of total debt to total assets (TD/TA) - based on the idea that the more

debt a firm has, the more likely is default on that debt.

2. Ratio of current assets to total assets (CA/TA) - because the greater

proportion of near-cash assets a firm has, the easier it is to liquidate assets and

avoid default.

3. Ratio of last year's operating cash flow to toal debt (NCF/TD) - where

operating cash flow is earnings plus depreciation. If there are trends over time,

last year's cash flow is a predictor of cash flow in the current period. The

larger the cash flow and the smaller the debt, the less the chance of default.

4. Current ratio (CA/CL) - because one potential cause of default is the

inability to pay short-term creditors, and this liquidity ratio measures the amount

of short-term debt relative to near-cash assets.

5. Total assets in thousands of dollars (TA)-because larger firms are thought

for several reasons to be less prone to default [8, pp. 1628-29].

For simplicity, we estimated a linear probability function using ordinary least squares regression.(14) Since the research hypothesis is that each of the explanatory variables measure a different type of risk (or measure a different type of hedge against risk) and we want to test their joint effect, all the variables were entered directly into the regression; stepwise procedures were not used. The dependent variable here is default, with defaulted firms assigned a value of 1 on this variable and nondefaulted firms assigned a value of zero. With this coding of the dependent variable, theory would lead us to expect a positive estimated coefficient of the ratio of total debt to total assets and negative coefficients for the other variables. The results of the regression estimation are presented in Table 3. (15)

We see from these results that the current ratio and level of total assets are not significant in this sample; the implication is that they are not significant affectors of default in this particular industry, allowing for the usual caveats about the ability of the sample to represent the industry as a whole as well as any colinearity effects. The ratio of last year's net cash flow to total debt is significant but wrongly signed. There are several possible explanations for this wrong sign; for example, the logic previously presented regarding this ratio may be incorrect (for example, trends in net cash flow for this industry may not persist). The theoretic relationship here would, however, suggest that the incorrect sign is a data artifact. If predictivity (in addition to hypothesis testing) is a goal of the research, one possible procedure at this point would be to try another ratio relating cash flow and debt. Another would be to eliminate the cash flow ratio altogether on the grounds that, while the variable is statistically significant within the sample, its estimated coefficient would lead us to believe that it would probably not be of aid in forecasting default in the general population. These interpretational and statistical techniques, while applicable in regression, are not appropriate in discriminant analysis.

Let us now proceed with a parallel analysis using the discriminant analysis approach (scenario 2). Here, the financial ratios are not causally related to default but are signs of the firm's membership in the extinction or nonextinction modes; mode membership is primarily a function of the firm's management. We may use stepwise discriminant analysis as a pure search technique over the entire variable set to reduce the size of this set, given the goal of prediction and viewing the ratios as symptoms [4, p. 145]. After this search process is completed, the accuracy of the estimated function is validated on a holdout sample or via other validation techniques. Since prediction is the goal, a variable-entry conditioning scheme based on classification error rates should be used. Urbakh [30] shows that the probability of misclassification for two-group discriminant analysis is directly proportional to Mahalanobis distance. We thus used an entry scheme based on the variable's contribution to this statistic. We also allowed for included variables to be deleted later in the stepwise procedure.(16) The results are presented in Table 4. It should be noted that given the group centroids, the theory of scenario 1 would suggest that the ratios of net cash flow to total debt (NCF/TD), net cash flow to sales, and the level of total assets "have the wrong sign." However, this interpretation is incorrect here since it is inappropriate to interpret the estimated discriminant coefficients in this way and since the theory of scenario 2 does not imply any signs for the coefficients. Given the research hypothesis, the appropriate criterion for assessing the function is its ability to classify, and the validation statistics for the estimated equation indicate that it performs quite well by this criterion.

CONCLUSIONS AND RESEARCH IMPLICATIONS

In this paper, the relationship between causality in studies of the failure of firms and banks and the choice of estimation methodology in these studies have been discussed. The literature contains studies using both discriminant and regression-type approaches. We have argued that the two methods are oriented toward the examination of hypotheses involving different causal structures. Thus, the causality in the researcher's hypotheses should be a consideration in the choice and application of a methodology. The confusion of implied causality between the two techniques may have led to the inappropriate use and interpretation of discriminant analysis in published research on failure.

The reasoning presented in this paper, along with the current state of knowledge, suggests two future avenues for research on failure. The first avenue would entail rigorous testing of hypotheses regarding the financial causes of failure. This avenue involves a reexamination of prior failure studies in a rigorous framework so as to allow for unclouded conclusions relating financial position and failure. In this avenue, the default event being studied (bankruptcy/nonbankruptcy, reorganization/liquidation, loan default/nondefault, etc.) would first be precisely defined.(17) A precise specification of the event is necessary to form clear hypotheses regarding the processes that may lead to the event and thus the variables on which the occurrence or nonoccurrence of the event depends. Specific hypotheses regarding causes of the default event would then be stated and tested in a maximum-likelihood framework. Ratios may be useful as explanatory variables in testing these hypotheses since they are reasonable proxies in the measurement of many aspects of the firm's financial position, including its risk levels and hedges. This avenue would improve on prior studies by properly matching causality and analysis technique and would lead to a clearer understanding of which aspects of the firm's financial position contribute to default (and therefore require careful management) and which do not.

The second avenue for future research would attempt to search for links between management's ability and default. Indices of management characteristics would first be developed. Included here might be the length of time that the firm's managers have worked in the industry or for the firm, the educational characteristics of these managers, their backgrounds (Is the CEO from finance? Sales? etc.), and similar contextual variables. Discriminant analysis would then be applied to assess which characteristics are associated with failure/nonfailure. Ratios might be included in such inquiries, buy they would be regarded as indicators of management ability rather than indices of various levels of financial risk. This is a virgin area of research; prior empirical studies of failure have concentrated almost exclusively on ratio data, though other studies of failure usually cite managerial variables as being critical [6].

These new challenges in failure research are substantial areas for future work. By pursuing these research opportunities, while keeping in mind their hypotheses regarding causality and default, researchers will add substantially to our knowledge regarding failure and its causes. [Tabular Data 1 to 4 Omitted]

(*)The author gratefully wishes to acknowledge the assistance of Professor Robert A. Eisenbeis (University of North Carolina) and two anonymous reviewers, whose comments were of great help in the development of this paper.

(1)Classic references include Altman[3], Altman, Haldeman, and Narayanan[5], Beaver[7], Edminster[12], and Myers and Pifer[24]. More recent research includes Aharony, Jones, and Swary[1], Dambolena and Khoury[11], and Collins and Green[10]. A review of the literature through 1980 is provided in Altman, et al.[4, chapters VII and VIII]. (2)Interestingly, while numerous studies have attempted to differentiate between the characteristics of "failed" and "nonfailed" firms, there has been relatively little consistency in the definition of "failure." For example, Beaver[7] defines failure as bankruptcy, bond default, an overdrawn bank account, or nonpayment of preferred stock dividends; Dambolena and Khoury[11] define it as bankruptcy; and Altman[3] defines it as involuntary bankruptcy. There is quite a difference between an overdrawn bank account (which can be easily remedied via borrowing) and involuntary bankruptcy in terms of the likelihood of continuation of the firm. In most of this paper, we use the terms "failure" and "default" in the same broad sense that these terms have been used in prior studies; we will discuss the issue of defining failure in the final section of this paper. (3)A linear probability function was estimated in Chesser[9] and in Myers and Pifer[24]. Logit functions were used in Wilcox[32, 33] and probit in Hanweck[18]. Almost all other empirical studies use some form of multiple discriminant analysis. (4)One caveat is useful here. Where the independent variables are correlated or where the model is not completely specified (that is, where there are left-out variables), there will be certain biases in the estimated coefficients and their associated significance tests. These problems often make interpretation of regression results difficult. However, these problems are irrelevant in a discriminant analysis situation where hypotheses regarding the coefficients of the Xs are not appropriate. (5)The reason that prior studies have ignored causality in the selection of analysis methods is unknown to the author. However, one possible explanation is that Altman[3] used the discriminant analysis methodology in his pathbreaking study, and many researchers after him continued with this procedure without considering alternatives. (6)The pioneering and widely-cited work in the empirical assessment of causality is Granger[17]. The investigation of causality in default via methods of this sort would appear to be a fruitful area of future empirical inquiry. (7)Stepwise procedures are often used in this analysis. Such a methodology usually produces good statistical results because of the relatively large number of candidate predictor variables often used. This methodology has been criticized by Foster[15] and others. Also see Scherr[29] for discussion of stepwise procedures and failure research. We will argue later in this paper that stepwise procedures are appropriate in failure research when the research calls for discriminant analysis but not when the research calls for maximum-likelihood estimation. (8)In this discussion, we concentrate on financial ratios and their relation to failure because this has been the major thrust of prior studies. While these studies have shown that financial ratios are associated with default, better statistical results might be achieved in future studies via the incorporation of other data, such as industry, business cycle, and other affectors of default. See Argenti[6] for a discussion of the financial and nonfinancial causes of failure and Scherr[29] for some analysis of the effects of industry membership and business cycles on failure. (9)For ease of exposition, we frame our discussion in terms of the default of industrial firms, but parallel arguments can be made regarding the failure of financial institutions. (10)These events may include the level of economic activity, interest rate levels, war, default of debtors in the firm's accounts receivable portfolio, technological innovation, etc. (11)Statistics such as Wilk's lambda focus primarily on the separation of group means. For a discussion of the relationship between classification accuracy and these statistics, see Eisenbeis[13] and Eisenbeis and Avery[14]. (12)By choosing to illustrate the proper techniques via a small problem, we are able to present all the data within this article. The data can then be used inexpensively by other researchers to replicate our results, to experiment, and to illustrate methodologies in research on failure to students. We consider this a major advantage over illustrating discriminant and maximum-likelihood techniques on a data set that may not be easily available to all readers. Of course, a data set that is small enough to reprint in an article is far too small to yield substantive research results; it is utilized for illustrative purposes only. (13)In the usual failure research situation, the researcher would choose one method to be used based on the causality believed to be present, statistical considerations, and other factors. We perform both types of analysis here only to contrast the two methodologies. (14)In actual research applications, the linear model is probably not the best one to choose, and the more complex logit or probit methods should be used; see [4, pp. 3-32]. (15)While we are using t tests here to test for the significance of individual variables in default, this is not precisely the hypotheses tested when a dummy variable is used as the dependent variable. See [4, p. 163]. (16)This conditioning method is discussed in Nie et al.[25, pp. 447-54]. For a discussion of conditioning methods, including the advantages of using a metric in stepwise that allows for the later deletion of previously included variables, see [4, pp. 135-44]. (17)Bankruptcy/nonbankruptcy is a convenient event from a data collection standpoint but is a relatively poor event to study from a theoretical standpoint since bankruptcy is the outcome of a complex series of subprocesses (default on debts, negotiation with creditors, court proceedings, etc.); see Scherr[28] for a model of bankruptcy and the subprocesses within it.

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FREDERICK C. SCHERR is an Associate Professor of Finance at West Virginia University.

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