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ESTIMATING THE PROBABILITY OF FINANCIAL DISTRESS:

INTERNATIONAL EVIDENCE

Julio Pindadoa , Luis Rodriguesb,*, Chabela de la Torrea

a Dpt. Administración y Economía de la Empresa, Universidad de Salamanca, Spain b Dpt. Management, Escola Superior de Tecnologia de Viseu, Portugal

Abstract

This paper focuses on the development of a model for the probability of financial

distress that is intended to be stable across periods and countries. Our approach consists in firstly testing the specification of the proposed model by using panel data methodology to eliminate the unobservable heterogeneity. The model is then estimated cross-sectionally to obtain an indicator of the probability of financial distress that incorporates the specificity of each company. Our results confirm the stability of our model, in terms of significance and sign of the coefficients, as well as its classification power for the different periods and countries analyzed. JEL classification: G33 Keywords: Financial insolvency, probability of financial distress, logit analysis

* Corresponding author. Escola Superior de Tecnología de Viseu, Dpt. Gestão, Campus Politécnico de Repeses. 3504-510 Viseu, Portugal. Telephone number: +351 232 480500 ext. 1280. Fax number: +351 232 242651. E-mail address: luisfr@dgest.estv.ipv.pt. Pindado and de la Torre are grateful to the research agency of the Spanish Government, DGI (Project SEJ2004-06627) and the Junta de Castilla y Leon (Project SA079A05) and Rodrigues to the European Union (PRODEP Program) for financial support.

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ESTIMATING THE PROBABILITY OF FINANCIAL DISTRESS:

INTERNATIONAL EVIDENCE

1. Introduction

The static theory of capital structure postulates that the optimum debt level arises

from the trade-off between tax advantages of borrowed money and financial distress

costs. This theory has led many scholars (see, for instance, Mackie-Mason, 1990;

Graham, 1996; and, more recently, Leary and Roberts, 2005) to incorporate a measure

of the probability of financial distress (hereafter PFD) based on Altman’s (1968)

Z-score model when studying capital structure. Miguel and Pindado (2001) go a step

forward when arguing that financial distress costs should include two components; not

only a measure of the probability of a firm filing for bankruptcy, but also the

consequences for the firm if bankruptcy occurs. The latter would be closely linked to

asset specificity in that some assets, such as intangible assets for instance, would lose

their value whenever the firm files for bankruptcy. However, the measure of the PFD

provided by Miguel and Pindado (2001) could be improved, since its values do not

range from 0 to 1. A similar criticism applies to the studies that rely on a PFD based on

Altman’s (1968) paper.

The potential contribution to financial research of a measure of financial distress

probability goes beyond capital structure literature. For instance, Nash et al. (2003) use

Altman’s Z-score when evaluating the costs and benefits of restrictive covenants in

bonds in that the probability of financial distress could drive to the use of some specific

covenants. Another example is Denis and Mihov (2003), who study the choice among

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bank debt, non-bank private debt and public debt. They argue that a firm facing a high

PFD (computed by means of Altman’s Z-score) is likely to borrow non-bank private

debt owing to the role of private lenders in renegotiations. Finally, a more recent

example is provided by Bhagat et al. (2005). They analyze the relationship between the

investment and internal funds of distressed firms, which are identified by using

Altman’s Z-score as well as Ohlson’s (1980) bankruptcy probabilities. The study by

Bhagat et al. (2005), as well as the ones by Dichev (1998) and Grice and Dugan (2001),

suggest the use of other methods and models than the one proposed by Altman (1968)

when computing bankruptcy probabilities.

Let us then consider these different models and methodologies. Initially, Altman

(1968) applies Linear Discriminant Analysis by using a paired sample of manufacturing

companies. In 1980, Ohlson substitutes Logistic Analysis for Linear Discriminant

Analysis as the estimation method. Following this trend, Zmijewski (1984) opted for a

Probit Analysis1.

These authors have raised a number of interesting issues that have been widely

discussed in the literature. First, Zmijewsky’s (1984) model is quite parsimonious in the

sense that it only includes three explanatory variables while it maintains identical

accuracy compared to Altman’s and Ohlson’s models2. Second, Ohlson (1980) poses “a

basic and possibly embarrassing question: why forecast bankruptcy?” in that, he argues,

bankruptcy is only one among a richer set of possible outcomes. Moreover, while firms

which experience financial distress are more likely to declare bankruptcy than other

firms, most financially distressed firms are not likely to declare bankruptcy.

Accordingly, a wider definition of financial distress, such as the one proposed by

1 A comparison of the mentioned models concerning methodologies, number of companies, period of estimation, and type of variables can be found in Appendix 1. 2 We must take into account that Ohlson (1980) explicitly recognizes that the nine explanatory variables of his model are derived from only four basic factors affecting the PFD; namely, the size of the company, its performance, liquidity and financial structure.

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Beaver (1966) as “the point where the firm will be unable to pay its obligations as they

mature”, would be more appropriate.

More recent studies (see for instance, Begley et al., 1996; Grice and Dugan, 2001)

have focused on re-estimating the above models in order to learn whether these models

remain useful for predicting bankruptcy in more recent and longer periods and, more

importantly, for predicting other financial distress conditions besides bankruptcy.

Empirical evidence provided by the these papers suggests that these models are still

useful for predicting financial distress, but indicates that the models’ accuracy is

significantly lower in recent periods. These results are improved when the models are

re-estimated, but the magnitude and significance of the re-estimated coefficients differ

from those reported by their original application. In short, these studies reveal that there

is not a stable pattern in the coefficients of the seminal models when applied to more

recent and longer periods and when used to predict financial distress conditions besides

bankruptcy.

These extensions of the seminal models for predicting financial distress have

encouraged renewed research on the topic. However, there are three important issues

that, as far as we know, have not been addressed yet. First, empirical evidence is largely

US-based, so it would be very interesting to learn whether the models developed for US

firms also apply to foreign firms and what the differences are. In fact, nowadays there is

a growing interest in conducting research using data from several countries at the same

time. A pioneering example is the study by Rajan and Zingales (1995), who investigate

the determinants of capital structure in the G-7 countries. Within this context, the

following question arises: are the probabilities of financial distress based on seminal

models valid for other-than-US countries? Second, the evidence in Begley et al. (1996)

and Grice and Dugan (2001) points to the need for a more stable model for the PFD that

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offers a consistent pattern in terms of significance and sign of the coefficients and a

consistent classification power across different periods and countries, as well. Finally,

the most important issue refers to the methodology. There have been several

methodological advances since the publication of the seminal models summarized in

Appendix 1. Specifically, Arellano and Honoré (2001) highlight some recent

developments that could be applied to improving the logit methodology, which is

currently the most widely applied methodology. They specifically focus on the

estimation of panel data models for discrete dependent variables in order to control for

unobservable heterogeneity. This advance is especially important in that seminal models

are developed using data from heterogeneous firms. Consequently, there are always

characteristics influencing financial distress which are difficult to measure or hard to

obtain and as a result are not entered into the model. Therefore, seminal models could

be improved by using the panel data methodology, which allows us to check the correct

specification of the model once the bias arising from the unobservable heterogeneity has

been eliminated.

Thus, the aim of this paper is to extend the seminal models by developing a new

approach for estimating the PFD that can be applied to more recent and longer periods

as well as to different countries. To achieve this aim, we begin by specifying a PFD

model according to financial theory which is intended to be stable when applied to

different periods and countries. We then test for the specification of the model by using

the panel data methodology. This methodology allows us to eliminate the unobservable

heterogeneity, and to solve the problem of choosing the estimation year before the crisis

by using the maximum annual data for each firm and thus improving the accuracy of the

model. Finally, and once the correct specification of our model has been tested, it is

used to perform a cross-sectional analysis to predict the PFD which incorporates the

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specificity of each company.

Our US-based empirical evidence validates the econometric specification of the

proposed model. The estimated coefficients in panel data models yield the expected sign

using both fixed and random effects panel data methodologies. Specifically, we find

that the PFD is significantly explained by a number of theoretically underpinned factors

which is smaller than has generally been assumed, namely the company’s returns on

assets, and the trade-off between this way of generating funds and the company’s need

to comply with its financial expenses during the financial year. The results of the cross-

sectional analysis strongly confirm the accuracy of our model, and show high

percentages of correct classification for all the years. In short, these US-based results

show that our model for the PFD is useful for predicting financial distress in more

recent and longer periods compared with the results provided by Altman (1968), Ohlson

(1980) and Zmijewsky (1984).

Moreover, the stability of our model, in terms of significance and sign of the

coefficients, and its classification power is corroborated by two robustness checks that

are intended to validate our approach. The first robustness check consists in replicating

our approach for estimating the PFD by using data on other-than-US G7 countries. The

results obtained reveal that our model for the PFD also applies to countries other than

US. The second robustness check consists in re-estimating the most representative

benchmark models to check whether they report consistent results for periods, sectors

and countries other than those in their original application, and whether they remain

useful to predict financial distress conditions other than bankruptcy. The results show

that these models remain useful for predicting financial distress in more recent and

longer periods. However, the stability of the models’ coefficients across countries and

periods is significantly lower than ours. Particularly, the coefficient on leverage does

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not show a consistent pattern of significance and sign supporting our idea of replacing

leverage by the burden of interest.

The remainder of the paper is organized as follows. Section 2 describes the data

set used in our study. In Section 3, a model for estimating the PFD is specified, for

which we then propose an innovative estimation strategy that incorporates panel data

methodologies as well as cross-sectional analyses in Section 4. Throughout Section 5

we present and discuss the US-based estimation results of our PFD model, as well as the

results of two robustness checks that validate our approach. Section 6 presents

conclusions which can be drawn from this study.

2. Data

According to our approach, data from several countries are needed in order to

make sure that our model for the PFD works regardless of the data used to estimate it.

We have thus used an international database, the Compustat Global Vantage, as our

source of information.

For each country we constructed a panel of firms with information for at least six

consecutive years from 1990 to 2002. There are only a few countries for which samples

with this structure can be selected, which is the case of the G-7 countries. Note that we

have a 13-year sample period and that the selected countries represent a great variety of

institutional environments, which allows us to check the stability of our model over

recent and longer periods, and also across different institutional and legal contexts. The

distribution by number of companies and number of annual observations per country is

provided in Table 1.

We have thus constructed an unbalanced panel with between six and thirteen

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years of data for each company, combining the available Compustat Global Industrial

Active files containing information on active companies, with Compustat Global

Industrial Research files which provide data on companies suspended from quotation

for some reason after a certain period in the capital market3. This data structure allows

the number of observations to vary across companies and thus represents added

information for our model. This way we can use the largest number of observations

reducing the possible survival bias arising when the observations in the initial cross-

section are independently distributed and subsequent entries and exits in the panel occur

randomly. The financial companies in Compustat Global were excluded because they

have their own specificity. As shown in Table 1, the selected samples contain 1,583

companies (15,702 observations) for the US, and 2,250 companies (18,160

observations) for the rest of the G-7 countries.

We follow Ohlson (1980) and Zmijewsky (1984) who, unlike the original

Altman’s Z-score, do not use a paired sample for estimating the PFD. In fact, Ohlson

(1980) indicates that sample selection procedures such as the matched-pairs design give

rise to biases. Particularly, Zmijewsky (1984) provides evidence of biased parameters

when the probability of a firm entering the sample depends on dependent variable

attributes, and that this bias decreases as the probability of bankruptcy in the sample

approaches that of the population. Consequently, we use samples in which the

percentage of distressed firms is representative of the population. This percentage is

subsequently used as the cut-off point to classify the firms, according to the obtained

PFD. Table 2 provides the classification of the annual observations4 as normal and

3 Firms that filed for bankruptcy are an example. However, companies in such a situation only represent a small percentage of the available data and, even in these cases, the available information is of poor quality as a natural consequence of the degradation of information flow characterizing severe crises. 4 Since our approach requires a long time period, a company may be financially distressed in some of the thirteen years which comprise our sample but not in the remainder. For this reason we report the number of observations instead of the number of companies.

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financially distressed. The US sample comprises 7.6% of financially distressed

observations, whereas this percentage is smaller, 4.1%, in the sample for other-than-US

G-7 countries. Note that the percentage of US distressed companies is similar to the

ones reported in the pioneering studies by Ohlson (1980) and Zmijewsky (1984) (see

Appendix 1).

3. A model for the probability of financial distress

In this section we develop a model that, in accordance with our aim, provides an

indicator of the PFD. Our approach is characterized by three features that clearly

differentiate it from others outlined in this strand of literature. First, the binary

dependent variable of our model is based on a financial definition of distress. Second,

the selection of the explanatory variables relies on financial theory. Third, like in most

of the recent papers, our model is a logistic regression. However, our strategy and

method of estimation take advantage of the panel data methodology to control for

unobservable heterogeneity, as well as of the cross-sectional estimation to obtain a PFD

that incorporates the specificity of each company. In what follows we offer a more

detailed discussion of the features characterizing our model, and Section 4 describes our

strategy and method of estimation.

3.1. A financial-based definition of financial distress

According to the aim of our study, we focus on financial distress regardless of

the legal consequences of this situation. This choice is based on the fact that our

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primary objective is to obtain a measure of the PFD, and not to predict the event of

bankruptcy. Note the relevance of this point in our approach because, as Barnes (1987,

1990) indicates, the failure to meet financial obligations does not necessarily lead to

bankruptcy. Moreover, Ward and Foster (1997) point out that studying only bankruptcy

leads to an important bias, because companies usually get into a financial distress cycle,

lack financial flexibility and incur serious financial distress costs several years before

going bankrupt. Additionally, Altman (1984) highlights the importance of using a

definition of financial crisis regardless of its final outcome. We have thus used a

financial criterion when defining a crisis, particularly because definitions of financial

distress based on the company’s failure to face its financial obligations are coherent

with our ex-ante approach. Therefore, we are considering that financial distress costs are

not limited to bankruptcy, as pointed out by Clark and Weinstein (1983).

Specifically, following Wruck (1990), Asquith et al. (1994), Andrade and Kaplan

(1998) and Whitaker (1999), we adopt a definition of financial distress that evaluates

the company’s capacity to satisfy its financial obligations. A company is thereby

classified as financially distressed not only when it files for bankruptcy, but also

whenever both of the following conditions are met: i) its EBITDA is lower than its

financial expenses for two consecutive years5, leading the firm into a situation in which

it cannot generate enough funds from its operational activities to comply with its

financial obligations; and ii) a fall in its market value takes place between these two

periods6. This criterion allows us to divide the samples into two groups and to construct

5 The relevance of the firm’s financial behavior in the previous two periods was already taken into account in Ohlson’s model. This model includes two variables that explicitly account for a firm’s performance during two years: the first one verifies if the operational income was negative during the two periods, and the second one is intended to measure the change in firm value between these two periods. 6 We decided to consider a firm as financially distressed also in the year that immediately follows the occurrence of these facts. In fact, a company that suffers from the previously mentioned operational fund deficit is expected to be negatively assessed by the market and its stakeholders, hence it will suffer the negative consequences of the financial distress situation until the improved economic condition is recognized once again.

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a binary dependent variable which takes value one for financially distressed companies,

and zero otherwise. According to Wood and Piece (1987), who questioned the accuracy

of ex-post models when predicting financial distress ex-ante, we follow an ex-ante

approach and, consequently, our definition of the dependent variable mainly based on

the condition of the EBITDA being lower than financial expenses is the most

appropriate.

3.2. A theoretically-based selection of explanatory variables

A noteworthy feature of our study refers to the selection of the explanatory

variables in the model. According to Scott (1981), the selection of explanatory variables

should not be based on sequential processes of elimination of variables according to a

maximum prediction capacity criterion, since this method often leads to over-adjusted

models with counter-intuitive coefficient signs and results. Consequently, we have

selected our explanatory variables relying on a theoretical justification.

Additionally, this selection of variables allows us to specify a logistic model that

is intended to be stable and parsimonious and that reduces the problems concerning the

choice of countries and periods to consider. Actually, a parsimonious selection of the

explanatory variables is expected to provide a more stable model in terms of magnitude,

sign and significance of the variables. In fact, reviewing previous discriminant models

(see, for instance, Zmijewsky, 1984; Pindado and Rodrigues, 2004) we can conclude

that this kind of model does not require a huge set of variables in order to reach its

maximum level of efficiency. Consequently, our PFD model includes a small set of

variables, for which the financial theory suggests a strong relationship with financial

distress. The selected variables are: Earning Before Interest and Taxes (EBIT),

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Financial Expenses (FE) and Retained Earnings (RE) 7. Note that these variables showed

the highest discriminatory power in the models of Altman (1968), Altman et al. (1977),

and Ohlson (1980), as well as in the subsequent studies by Begley et al. (1996), Dichev

(1998) and Cleary (1999).

Our first explanatory variable, Earnings Before Interest and Taxes

(EBITit /RTAit-1), captures the capacity of the firm to efficiently manage its assets in

order to generate enough funds to face its financial obligations. Profitability ratios have

typically been used as measures of firm performance in the research on financial

distress (see, for instance, Altman, 1968; Ohlson, 1980; Zmijewski, 1984). Our ratio, in

particular, is a measure of the productivity of the firm’s assets, independent of any tax

or leverage factors, and it is also the main driver of liquidity. In fact, creditors habitually

rely on measures of profitability when extending credit or renegotiating repayments in

order to estimate the return generated by the firm on borrowed capital (Claessens et al.,

2003). Note that our profitability ratio appears to be particularly appropriate for studies

dealing with financial insolvency because its value has persistently been shown to be

fundamentally determined by the earning power of a firm’s assets, thus outperforming

other profitability measures, including cash flow (see Altman, 2000). Taking all this

into account, we expect this variable to negatively influence the PFD.

Second, Financial Expenses (FEit/RTAit-1) has been chosen instead of debt stock

ratios because the latter seem to lose explanatory power as compared to the chosen flow

variable. Indeed, the research on PFD reveals the advantages of using a variable that

considers the flow of financial expenses instead of the stock of debt. In fact, since the

revision of the Z-score carried out by Altman et al. (1977), many other subsequent

7 A detailed description of all the variables used in this analysis can be found in Appendix 2. All these variables are scaled by the replacement value of total assets at the beginning of the period (RTAit-1). Note that this scaling factor is less biased than the book value of total assets, which is particularly dependent on the accounting principles. Table 3 reports summary statistics for these variables.

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studies, such as Andrade and Kaplan (1998), point out that debt variables are less an

explanatory variable of financial distress than variables of financial expense. Asquith et

al. (1994) also show how the leverage effect can be absorbed by the financial expense

effect, which constitutes one of the most frequent causes of financial distress in addition

to the individual and sectoral components of economic crisis. In fact, Mackie-Mason

(1990) points to the advantage of using a measure of financial distress obtained by

means of a model that does not include debt as an explanatory variable in order to avoid

problems of endogeneity in capital structure models. Additionally, Begley et al. (1996)

point out that since the 1980s, firms have been continuously increasing their debt levels

without necessarily increasing their probability of distress. Actually, Altman et al.

(1977) replace the leverage variable by a debt service variable in their model, which

allows them to take into account the potential benefits of leverage (see Jensen, 1986,

1989). In fact, Begley et al. (1996) justify a better performance of the re-estimated Z-

score, relative to the classification made by using the coefficients of the original model,

owing to the correction of this leverage bias, which is in turn translated into a reduced

contribution of the leverage variable to the total discriminating power of the model8.

This trend would explain the declining performance of the models proposed by Altman

(1968) and Ohlson (1980). In short, recent literature shows that the flow of financial

expense imposes stricter limits on the company’s policies than the stock of debt.

Therefore, we include the variable of financial expense scaled by the replacement value

of total assets (FEit/RTAit-1) in our model, expecting a positive relation with the PFD.

Third, retained earnings (REit/RTAit-1) are the total reinvested earnings and/or

losses of a firm over its entire life. This is a measure of cumulative profitability over

time that remains one of the most crucial predictors of financial crisis. Particularly,

8 In fact, these authors examine the extent to which the original models misclassify firms when using data from recent periods, and they conclude that Altman’s original model tends to misclassify highly leveraged non-bankrupt firms.

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Routledge and Gadenne (2000) highlight the usefulness of past profitability in

predicting future results and capacity for self-financing. Moreover, Mayer (1990)

concludes that retained earnings constitute a privileged source of funds for companies in

eight countries (namely, the G7 countries and Finland). We have thus introduced this

ratio of cumulative profitability to replacement value of total assets into our model. The

pecking order theory, proposed by Myers and Majluf (1984), highlights the company’s

preference for internal funds suggesting a negative relation between cumulative

profitability and the PFD.

3.3. Econometric specification

This study proposes a model to obtain a PFD that includes the variables

described above. We use explanatory stock variables evaluated at the beginning of the

period and flow variables of the period, as suggested by Cleary (1999).

Given an objectively obtained binary dependent variable, the logistic regression

technique determines the extent to which a set of variables containing useful

information are able to classify every firm in our sample in one category or the other

(financially distressed or non-financially distressed). Let us consider that y* is a linear

function of xi (i=1,...,n) explanatory variables9, and µi is a random term capturing

individual characteristics other than the explanatory variables. That is,

y* = β0+β1x1+…+βnxn + µi

Then there is a critical level of y* such that if this level is exceeded the firm is

financially distressed. Consequently, our binary dependent variable is

yi = 1 if y*>0 9 We must take into consideration that whereas the predicted probabilities of bankruptcy can be evaluated empirically, the event of insolvency is not observable.

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yi = 0 else

Subsequently,

P(yi = 1| β’xi ) = P(µi > - β’xi ) = 1-F(- β’xi ), where F( ) is the cumulative

distribution function for µ, that is assumed to be logistically distributed. Finally, the

logistic regression is expressed in terms of the odds ratio, which concerns the

probability of being financially distressed according to the criterion described in Section

3.1. The explanatory variables in our logistic regression, whose effect on the probability

of the firm being financially distressed was theoretically justified in Section 3.2, are:

Earnings Before Interest and Taxes (EBIT), Financial Expenses (FE) and Retained

Earnings (RE).

Accordingly, the logistic model we used to estimate the PFD is as follows:

nt)Prob(noeve

)Prob(eventlog =β0+β1EBITit/RTAit-1+β2FEit/RTAit-1+β3REit-1/RTAit-1+dt+ηi+uit (2)

where all variables are indexed by an i for the individual cross-sectional unit (i= 1,…,N)

and a t for the time period (t= 1,…, T). Additionally, dt is the time effect, ηi denotes the

individual effect, and the random disturbance is uit~IID(0,σ2).

The coefficients β1, β2 and β310 can be interpreted as follows. The first one (β1) is

associated with the capability of assets to generate returns, and is thus expected to be

negative. The second one (β2) is predicted to be positive, since we expect the PFD to

increase as the company’s risk of not being able to comply with its financial obligations

rises. Finally, the third coefficient (β3) is expected to be negative in that the economic

agents’ expectations are based on past profitability and self-financing.

10 It is worth noting that, although the marginal effect that any explanatory variable in the regression has on the probability of the firm being financially distressed does not come directly from the beta coefficients, the interpretation of the signs of these coefficients is similar to that of the ordinary least squares regression.

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4. Strategy and methods of estimation

In this section, we describe the strategy implemented in order to obtain an

indicator of the PFD. Figure 1 portrays the several stages followed in our new approach.

The estimation methods of our models are also discussed in this section.

4.1. Strategy of estimation

Our strategy consists firstly in developing the econometric specification of the

model according to financial theory, as has already been described in the previous

section.

In a second stage, our study presents the innovation of estimating this model by

using panel data methodology. There are several advantages in using this methodology.

First, although the traditional maximum likelihood estimator of the βs is consistent, it

will be inefficient, generating biased standard errors of βs and, consequently, leading to

an incorrect specification of the model. Second, the definition of financial distress

adopted in this study can only be appropriately addressed by the use of panel data

methodology, and not by merely pooling data across years, thus diminishing the

problem of financial distress as a rare event.

Specifically, we estimate panel data models with a discrete dependent variable,

since this methodology allows us to verify the significance of the model coefficients

through estimating fixed and random effects panel data models that are robust to

unobservable heterogeneity. Note that the implementation of this second stage requires

the selection of a sample that allows us to work with data panels in which companies

are chosen according to their financial distress situation in each period. Additionally,

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our concept of financial distress is compatible with large data panels, allowing us to use

panel data methodology in order to consistently estimate the models of financial distress

probability.

However, our measure of the PFD does not stem from these panel data models,

since they eliminate unobservable heterogeneity. In other words, the panel data

estimation removes the individual effects from the error term and, consequently, it does

not account for the firm-specific contribution to the prediction of the PFD. The third

stage in our approach addresses this issue. Specifically, once the robustness and the

correct specification of the model have been tested for, we estimate a cross-sectional

regression for each year, thus obtaining an appropriate indicator of the PFD for each

company and year.

4.2. Estimation Methods

Logit analysis is an appropriate explanatory technique for our study since our

dependent variable is a binary variable. The research carried out in the 1980s

consolidated logit analysis as a better estimation methodology than discriminant

analysis, since the hypotheses on which the latter relies do not generally hold11.

Consequently, we prefer to use logit analysis instead of discriminant analysis for several

reasons12. First, as discussed by Karels and Prakash (1987), discriminant analysis

requires strict multivariate normality and homoskedasticity across groups, whereas the

logit analysis does not strictly require these assumptions. Second, logit analysis is often

preferred even when these assumptions hold, mainly because of its ability to incorporate

11 These hypotheses refer to the homoskedasticity of variances and covariances matrices and to the multinormality of the variables. See Einsenbeis (1977) and Joy and Tollefson (1975) for an extensive discussion of the problems related to the use of discriminant analysis. 12 Note that some authors that began using the discriminant analysis recognize the advantages of logistic regression in their more recent work (see, for instance, Altman and Sabato, 2005).

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non-linear effects, and because of other technical features (see Hair et al., 1995).

Finally, discriminant analysis is not suitable for dealing with unobservable

heterogeneity and other characteristics common to panel data samples.

Panel data models for discrete dependent variables allow us to correct the

specification of the model by eliminating the bias of omitted variables that arises when

the unobserved individual-specific effects (ηi) are correlated with the explanatory

variables. In this respect, it is necessary to distinguish between fixed effects models –

those in which a relationship between the individual effects and the remaining right-

hand side variables is not assumed – and random effects models – those in which this

relation is functionally specified.

In fixed effects models, the conditional likelihood estimator proposed by

Chamberlain (1980), when feasible, allows us to obtain consistent estimates of the

parameters that are no longer dependent on the individual effects. On the other hand,

Chamberlain (1984) proposes a random effects estimator that specifies the conditional

distribution of ηi on explanatory variables. Specifically, this procedure is based on a

parameterization of the correlation between the individual effects and the regressors, in

such a way that the latter are considered time-variant explanatory variables of the

former and of a random time-invariant term. This functional assumption made on the

individual effects makes the model less general than the fixed effects model. However,

although a fixed effects framework seems appropriate, the random effects estimator may

still be preferred. The reason is that ηi and xit may be correlated, in which case the fixed

effects approach, ignoring this correlation, would lead to inconsistent estimators due to

omitted variables bias.

To sum up, the choice between fixed and random effects models depends on the

characteristics of the explanatory variables. On the one hand, when all the explanatory

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variables are expected to be strictly exogenous, the fixed effects model would yield

good results if the estimation sample (observations for which there is a change in the

regime between sample periods) is large enough and there is temporal variation in the

explanatory variables in order to identify the individual effects. On the other hand,

Arellano and Honoré (2001) highlight that the random effects model works better if

explanatory variables are not strictly exogenous, samples show insufficient changes, or

the contribution to the maximum likelihood function of the variation in explanatory

variables is not enough.

Therefore, the preference for one of these models basically depends on the

assumptions about the dependence of the error distribution on the explanatory variables.

Given the difficulty in establishing this relation, we follow Arellano and Honoré (2001)

in suggesting the convenience of estimating both models.

However, as Arellano and Honoré (2001) point out, the knowledge of the

parameters of interest does not allow us to infer all the “quantities of interest” and,

unfortunately, this consistent estimator of the βs does not allow us to obtain the

probability distribution of the dependent variable either. Actually, although these

models provide robust estimates of the parameters, they do not allow us directly to

obtain a PFD because they do not take into account the individual effects. Overcoming

this limitation can only be indirectly obtained by cross-sectionally estimating the PFD

for each year.

5. Results

In this section, we first present our US-based results. We discuss the estimation

results of the random and fixed effects logistic regression models, as well as the results

20

of the cross-sectional estimation of our model for the PFD for all the years13. We next

tabulate the main statistics of the estimated probabilities and the percentages of correct

classification produced by the cross-section models. We then test the accuracy of our

approach by means of two robustness checks. The first one verifies whether our model

for the PFD is stable, in terms of sign and significance of the coefficients, when applied

to other-than-US G7 countries. The second test consists in re-estimating other

benchmark models and finding whether these models also provide stable results when

applied to recent and longer periods, as well as to different countries.

5.1. US-based results

Table 4 presents the results for the fixed and random effects models, respectively.

The goodness of fit tests point to the high explanatory power of all the variables in both

the fixed effects models (see likelihood ratios, LR) and in the random effects models

(see Wald tests). Additionally, Wald tests of the joint significance of the time dummies

are presented, which validate the use of such variables in both models and for all

countries, thus confirming, as was expected, that there have been fluctuations in the

financial distress processes over time. These results show that the consideration of these

dummy variables is important, since they allow us to accommodate the impact of

changes in the macroeconomic environment.

The estimation of random effects models includes additional tests that verify the

existence of unobservable heterogeneity. As shown in Table 4, the additional panel-

level variance component is parameterized as ln(σ2η). The standard deviation, ση, is also

reported in Table 4, and it is used to obtain a third indicator, ρ = σ2η / (σ2

η+1), which is

13 Note that the same set of variables we proposed in Section 3 is always used when estimating such models.

21

the proportion of the total variance contributed by the panel-level variance component.

When ρ is approximately zero, the panel-level variance component is unimportant and

the panel estimator is not different from the pooled estimator. In our study, the null

hypothesis of equality of both estimators is rejected, and the existence of unobservable

heterogeneity is thus confirmed. According to these results, we conclude that the

proposed model needs to be validated by using a panel data methodology in order to

control for unobservable heterogeneity.

We now turn our attention to the estimated coefficients in our panel data models.

Since they are statistically significant and of the theoretically-expected sign using both

methodologies, we shall describe the results jointly.

First, the variable that captures profitability (EBIT it/RTAit-1) negatively affects the

PFD. This evidence is consistent with all the studies referred to in Appendix 1. Second,

the positive effect of financial expenses (FEit/RTAit-1) confirms our expectations about

the capacity of this variable to capture the firm’s financial vulnerability, particularly in

periods of low inflation and low interest rates, such as the one we study here, where the

leverage constraints are lower. Note that this result is consistent with Begley et al.

(1996), and that the variable of financial expense was already significant in prior studies

(see, for instance, Altman et al., 1977). Finally, the coefficient on cumulative

profitability (REit/RTAit-1) is negative, which confirms the consequences of past

profitability in determining the firm’s financial structure.

Once we have checked that our panel data model is correctly specified and that the

variables used to explain the PFD are validated and supported by financial theory, the

next step is to cross-sectionally estimate this correctly specified model for each year.

The results are presented in Table 5. The stability of the model, in terms of sign and

significance of the coefficients, is supported by the cross-sectional estimation. We find a

22

negative and significant effect of profitability and retained earnings on the PFD for all

the years, and the effect of financial expenses remains positive and significant except

for the two last years, where this effect becomes non-significant.

Overall, these results reveal that the PFD is explained in essence by the

company’s efficiency in extracting returns from its assets, and by the trade-off between

this way of generating funds and the need to comply with its financial expenses during

the financial year. We also find that higher historical profitability tends to reduce the

company’s PFD, which can serve as a cushion to provide wider financial solutions to

the crisis.

As we have already explained, the cross-sectional estimation of our model of the

PFD is particularly useful in that it allows us to obtain an indicator of such a probability

that takes the individual effects into account. In other words, the relevant output from

our cross-section models is the PFD obtained for each company and year. Table 6

provides the basic statistics (Mean and Standard Deviation) of the estimated

probabilities of financial distress. As can be seen in the table, the mean of this

probability is quite low (around .075), which is reasonable since we are computing the

ex-ante probability of financial distress. Another indicator of the accuracy of our

approach is the standard deviation shown by the PFD, which is very small. Moreover,

these basic statistics are found to be quite stable along the sample period. In short, these

results strongly support the accuracy of our approach for estimating the PFD.

As shown in Table 6, the high percentage of correct classification also supports

our approach. Additionally, this percentage is quite stable across years with a mean

value of 87%. Note that the percentages of Type I error go beyond a naive

classification, thus showing that our model is quite accurate in terms of classification,

especially if we take into account that the same cut-off point is used for all the years. In

23

fact, we consider the percentage of financially distressed firms in our sample as

representative of the population which in turn define the cut-off point for the

classification of firms into normal and financially distressed according to the estimated

PFD. That is, we do not try to find an optimal cut-point for the different years, mainly

because our discussion is not focused on the percentage of correct classification, and

because our main concern is not to predict financial distress but to offer a model of its

probability (Palepu, 1986)14.

Overall, our US-based results strongly support the accuracy of our ex-ante

approach for estimating the PFD. Note that these ex-ante estimations of the PFD are

crucial in financial models that need to incorporate a measure of ex-ante financial

distress costs. Therefore, works by Opler and Titman (1994), Andrade and Kaplan

(1998) or Dichev (1998) can be extended by making use of a concept of ex-ante

financial distress costs as the product of the PFD and the ex-post financial distress cost

perceived by investors. In this way, our approach is a step forward, since it allows

researchers to obtain a good measure of the PFD with a parsimonious stable model.

Next we provide two tests of the accuracy of our approach. First, we check

whether our model for the PFD also applies to countries other than US. Second, we

verify whether the most representative benchmark model (Altman’s 1968 model) also

provides consistent estimations of the PFD when applied to recent and longer periods,

and to different countries.

14 In fact, this percentage depends on the cutoff point, and the most common criticism relies on the fact that this point is usually determined ex-post, by a process of trial and error, without taking into account the fact that the probability of failure for the sample is not the same as that of the population. This process of classification can be particularly misleading when the loss functions of the errors are quite asymmetrical (see Hsieh, 1993) and, consequently, maximizing the percentage of correct classification can be quite different from minimizing the total error costs.

24

5.2. First robustness check: Different Institutional contexts

In this section we replicate our approach for estimating the PFD by using data on

other-than-US G7 countries in order to find whether our model for the PFD remains

stable when applied to different institutional and legal contexts15. The estimation results

of the fixed and random effects models are provided in Table 7. It is worth noting that

the estimated coefficients remain significant and have the expected sign in both models;

that is, the PFD is negatively affected by a firm’s profitability and retained earnings,

and positively affected by its financial expenses. Moreover, the stability of these

relations along the sample period is supported by the results of the cross-sectional

estimation of the model, provided in Table 8. The consistency of our evidence strongly

validates our idea of selecting the explanatory variables relying on a theoretical

justification in order to specify a stable model that reduces the problems concerning the

choice of periods and countries to consider.

The basic statistics of the estimated probabilities of financial distress are reported

in Table 9. Consistent with the US-based evidence, the mean of the PFD for the other

G7 countries is quite low (around .0397), as well as the standard deviation (around

.0949). The percentage of correct classification, also shown in Table 9, has a mean

value of 83%, and it is quite stable across years in spite of using the same cut-off point

for all the years. Additionally, the percentages of Type I error confirm the high

classification power of our model.

We can thus conclude that the evidence provided by this robustness check

validates our approach, and confirms that it is possible to build a more parsimonious

15 We provide the results of estimating our model for the PFD by using the whole non-US sample. However, we have also replicated our approach by using the individual samples for each of the other-than-US G7 countries, except for France and Italy, since their samples are very small. Individual results are consistent with the ones we report here, and are available on request from the authors.

25

model leading to a general and stable indicator of the PFD that can be used in different

institutional and legal contexts.

5.3. Second robustness check: Re-estimating Altman’s model

There are already several studies (see, for example, Begley et al., 1996; Grice and

Dugan, 2001) that re-estimate the benchmark models described in Appendix 1 (mainly

Altman’s model) in order to assess whether these models remain useful for predicting

bankruptcy in recent periods (Begley et al., 1996), as well as their capacity to predict

financial distress situations other than bankruptcy (Grice and Dugan, 2001; Grice and

Ingram, 2001). The empirical evidence provided in these studies suggests that the

models’ accuracy is significantly lower in recent periods. Moreover, the coefficients of

the re-estimated models differ from those reported in the original studies. This

instability, in terms of magnitude, significance and sign of the coefficients raises

important questions regarding the appropriateness of applying this kind of model for

predicting bankruptcy over different periods.

In this section, we extend the analysis performed by the above studies, and we re-

estimate Altman’s Z-score model16 to check whether it reports consistent results for

other periods, sectors and countries than in their original application, and whether it

remains useful to predict financial distress conditions other than bankruptcy.

Table 10 presents the estimation results of the panel data models (fixed and

random effects logistic regressions), and the results of the cross-sectional estimation of

the model for the PFD are provided in Tables 11 and 12 for the US and the rest of the

16 Our re-estimation of Altman’s Z-score is performed by means of a logistic regression, instead of a discriminant analysis. This choice is based on reasons related to consistency with our approach. It is also justified by Altman and Sabato (2005), who explicitly recognize the potential advantage of re-estimating the original model within a logistic regression structure.

26

G7 countries, respectively. Taken as a whole, the estimated coefficients reveal that only

profitability and retained earnings maintain their significance in the re-estimated

Z-score model across countries and years. In other words, these variables are the only

ones which present a consistent pattern of significance and sign of their effect on the

PFD across the different countries and years.

The basic statistics of the estimated probabilities and the percentages of correct

classification, presented in Tables 13 and 14, are similar to those obtained for our model

for the PFD. These results indicate that the re-estimated Z-score model remains useful

for predicting financial distress conditions other than bankruptcy when applied to

periods, sectors and institutional and legal contexts other than their original application.

However, similarly to the findings of Begley et al. (1996) and Grice and Dugan (2001),

we find that the leverage coefficient does not show a consistent pattern of significance

and sign (see Tables 11 and 12). This lack of consistency supports our idea of replacing

leverage by the burden of interest in that the latter is shown to be more crucial in

predicting financial distress processes than the former.

Overall, a global comparison between the results for re-estimated Z-score and our

approach reveals that our model for the PFD is more stable, in terms of sign and

significance of the coefficients, and provides more consistent coefficients on the

variables determining the PFD, regardless of the period, sector, and institutional context

considered17.

17 We have also re-estimated Ohlson’s and Zmijewski’s models, obtaining evidence consistent with that of the re-estimated Altman’s Z-score. Additionally, our evidence is consistent with Grice and Dugan (2001) suggesting that these models are sensitive to financial distress situations other than those used to develop the models. Although they are not provided in this paper, the results of the re-estimation of these alternative models are available on request from the authors.

27

6. Conclusions

This paper offers a new approach for estimating the probability of financial

distress which can be applied to different periods and countries. To achieve this aim, we

have first developed a theoretically supported model relying on a financial criterion of

financial distress that is independent of legal institutions. This model is intended to be

more stable than the re-estimated seminal models, in terms of significance and sign of

the variables, when applied to more recent and longer periods as well as to countries

other than US. We have then tested the specification of the resulting logistic model by

using panel data methodology in order to eliminate the unobservable heterogeneity. The

results obtained confirm the proposed specification of the model, and reveal that all the

coefficients are statistically significant and of the expected sign for both the US and

other G7 countries, and using both fixed and random effects methodologies.

Specifically, we find that the probability of financial distress is accurately explained by

the company’s returns on assets, and the consequent trade-off between this way of

generating funds and the company’s need to comply with its financial expenses during

the financial year.

The need to incorporate the specificity of each company into the final estimates of

the probability of financial distress has motivated a cross-sectional estimation of this

correctly specified model. The results definitely confirm the accuracy of our model, and

show a very high percentage of correct classification for all years and for both the US

and other G-7 countries.

In short, these results corroborate the stability of our model, in terms of

significance and sign of the coefficients, and its classification power across different

periods and countries. Additionally, a comparison between our model and the re-

28

estimated benchmark models reveal that, although these models remain useful for

predicting financial distress in more recent and longer periods, the stability of the

coefficients across countries and periods is significantly lower than ours. Particularly,

the coefficient on leverage does not show a consistent pattern of significance and sign

supporting our idea of replacing leverage by the burden of interest.

Finally, it is worth remarking that our approach goes a step forward in the

literature, since it provides an appropriate measure of the probability of financial

distress that facilitates the calculation of ex-ante financial distress costs as the product

of such a probability and the ex-post financial distress costs. In this way, we contribute

to the research that requires a measure of the probability of financial distress, such as

the one summarized in the introduction and which motivated this paper.

Appendix 1: Benchmark models

This appendix provides a comparison of the seminal models that are usually

considered benchmark models in bankruptcy prediction.

In the first model, Altman (1968) uses the discriminant analysis to obtain a firm’s

Z-score, which is higher the lower its probability of bankruptcy and which includes five

categories of ratios: liquidity, profitability, leverage, solvency, and activity.

Consequently,

Z= 1.2X1+ 1.4X2+ 3.3X3+ 0.6X4+ 0.999X5

where X1 is the working capital over total assets, X2 stands for the retained earnings

over total assets, X3 is earnings before interest and taxes over total assets, X4 denotes

market value of equity over book value of total debt, and X5 is sales over total assets.

Ohlson (1980) uses the logistic analysis to derive his bankruptcy prediction model

29

by using nine measures of a firm’s size, leverage, liquidity, and performance. That is,

Y=-1.3 - 0.4X1 + 6.0X2 - 1.4X3 + 0.1X4 - 2.4X5 - 1.8X6 + 0.3X7 -1.7X8 - 0.5X9

where X1 is calculated as the log(total assets/GNP price-level index), X2 stands for total

liabilities over total assets, X3 is the working capital over total assets, X4 denotes current

liabilities over current assets, X5 takes value one if total liabilities exceed total assets,

and zero otherwise, X6 is the net income over total assets, X7 stands for funds provided

by operations over total liabilities, X8 takes value one if net income was negative for the

last two years, and zero otherwise, and X9 is a measure of the change in net income.

Finally, Zmijewski (1984) proposes a probit model that incorporates three

financial ratios measuring a firm’s performance, leverage, and liquidity.

X = -4.3 - 4.5X1 + 5.7X2 - 004X3

where X1 is the net income over total assets, X2 stands for total debt over total assets,

and X3 denotes current assets over current liabilities.

We next summarize the main features of these three studies.

Author Altman, E. Ohlson, J. Zmijewski, M.

Year of the study 1968 1980 1984

Estimation LDA LOGIT PROBIT

Samples 33 FD, 33 N 105 FD, 2058 N 129 FD, 2241 N

Sample period 1946-1965 1970-1976 1972-1978

Overall prediction accuracy (estimation sample)

T-1 : 95 T-2 : 72 T-3 : 48 T-4 : 29 T-5 : 36

T-1 : 96,12 T-2 : 95,55 T-3 : 92, 84 and other results for other cut-off points

Estimations for different percentages of insolvent firms for T-1

where LDA denotes Linear Discriminant Analysis, and N and FD stand for normal and

financially distressed, respectively.

30

Appendix 2: Variable definitions

- Replacement value of total assets: RATit=RFit+(TAit -BFit), where RFit is the

replacement value of tangible fixed assets, TAit is the book value of total assets, and BFit is

the book value of tangible fixed assets. The last two terms were obtained from the firm’s

balance sheet, and the first one was calculated according to Perfect and Wiles (1994):

,t

it i t -1 itit

1+= +RF RF I

1+

φδ

for t>t0 and RFit0=BFit0, where t0 is the first year of the chosen period, in our case 1990.

On the other hand, δit=D it/BFit and φt=(GCGPt-GCGPt-1)/GCGPt-1, where GCGPt is the

growth of capital goods prices reported in the Main Economic Indicators published by the

Organization for Economic Cooperation and Development (OECD).

- Earnings before interest, taxes and amortization: EBITDAit=RVTit-XOPRit , where RVTit

is the total revenue, and XOPRit denotes operating expenses.

- Return ratio: EBITit /RTAit-1, where EBITit is earnings before interest and taxes.

- Financial Expenses ratio: FEit /RTAit-1, where FEit stands for financial expenses.

- Cumulative Profitability ratio: REit-1 /RTAit-1, where REit denotes retained earnings.

31

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Table 1 Structure of the samples

country US CAN GER FRA UK ITA JPN Other-than-US G7 countries

G7

No. of annual

observations per company obs. firms obs. firms obs. firms obs. firms obs. firms obs. firms obs. firms obs. firms obs. firms

6 1,074 179 288 48 234 39 408 68 720 120 132 22 1,254 209 3,036 506 4,110 685 7 1,113 159 175 25 105 15 938 134 308 44 84 12 5,635 805 7,245 1,035 8,358 1,194 8 2,984 373 216 27 184 23 0 0 248 31 192 24 0 0 840 105 3,824 478 9 981 109 288 32 306 34 0 0 711 79 0 0 0 0 1,305 145 2,286 254

10 690 69 180 18 70 7 0 0 150 15 0 0 0 0 400 40 1,090 109 11 561 51 99 9 66 6 0 0 231 21 0 0 0 0 396 36 957 87 12 720 60 60 5 216 18 0 0 216 18 0 0 0 0 492 41 1,212 101 13 7,579 583 1,144 88 897 69 0 0 2,405 185 0 0 0 0 4,446 342 12,025 925

Total 15,702 1,583 2,450 252 2,078 211 1,346 202 4,989 513 408 58 6,889 1,014 18,160 2,250 33,862 3,833 Data of companies for which the information is available for at least six consecutive years between 1990 and 1999 were extracted. The resultant unbalanced panels comprise 1,583 companies (15,702 observations) for the US, 252 companies (2,450 observations) for Canada, 211 companies (2,078 observations) for Germany, 202 companies (1,346 observations) for France, 513 companies (4,989 observations) for the UK, 58 companies (408 observations) for Italy and 1,014 companies (6,889 observations) for Japan.

35

Table 2 Classification of the annual observations into normal and financially distressed

US sample Other-than-US G7 countries sample G-7 sample Year N FD Total % FD N FD Total % FD N FD Total % FD 1990 414 18 432 4.2 705 61 766 8.0 1119 79 1198 6.6 1991 441 20 461 4.3 741 65 806 8.1 1182 85 1267 6.7 1992 462 24 486 4.9 761 70 831 8.4 1223 94 1317 7.1 1993 498 25 523 4.8 815 61 876 7.0 1313 86 1399 6.1 1994 669 25 694 3.6 910 75 985 7.6 1579 100 1679 6.0 1995 760 31 791 3.9 1,279 107 1,386 7.7 2039 138 2177 6.3 1996 1,764 57 1,821 3.1 1,364 94 1,458 6.4 3128 151 3279 4.6 1997 2,148 79 2,227 3.5 1,431 108 1,539 7.0 3579 187 3766 5.0 1998 2,119 94 2,213 4.2 1,408 108 1,516 7.1 3527 202 3729 5.4 1999 2,119 81 2,200 3.7 1,392 94 1,486 6.3 3511 175 3686 4.7 2000 2,084 85 2,169 3.9 1,328 105 1,433 7.3 3412 190 3602 5.3 2001 2,037 94 2,131 4.4 1,241 115 1,356 8.5 3278 209 3487 6.0 2002 1,924 88 2,012 4.4 1,139 125 1,264 9.9 3063 213 3276 6.5 Total 17,439 721 18,160 4.1 14,514 1,188 1,5702 7.6 31,953 1,909 33,862 5.9

N and FD stand for normal and financially distressed, respectively. Total is the number of observations available for each sample year. % FD is the percentage of financially distressed observations for each year.

36

Table 3 Summary statistics

This table provides the summary statistics of the explanatory variables of our model for the PFD and of Altman’s Z-score model. Summary statistics of the explanatory variables of Ohlson’s (1980) and Zmijewski’s (1984) models are available on request from the authors. EBITit is earnings before interest and taxes, FEit denotes financial expenses, REit stands for retained earnings, WCit is the working capital, MVit stands for market value of equity, SALESit denotes sales, and TAit and RTAit stand for the book value and replacement value of total assets, respectively. For each variable we report the number of observations, and the values of the following statistics: Mean, Standard Deviation, Minimum and Maximum.

EBIT it / RTA it-1 FE it / RTA it-1 RE it-1 / RTAit-1 WC it/ TA it-1 REit/ TA it-1 EBIT it/ TA it-1 MV it/ TL it-1 SALES it-1 / TA it-1

USA Other USA Other USA Other USA Other USA Other USA Other USA Other USA Other

Observations 15702 18160 15702 18160 15702 18160 15702 18160 15702 18160 15702 18160 15702 18160 15702 18160

Mean .073 .058 .024 .015 .083 .115 .213 .146 .106 .186 .082 .065 2.957 1.599 1.210 1.171

Std. deviation .131 .088 .020 .012 .530 .196 .222 .197 .530 .472 .139 .096 5.037 2.254 .757 .652

Minimum -.901 -.350 .000 .000 -4.555 -1.295 -.590 -.501 -3.818 -1.109 -.662 -.415 .015 .034 .057 .050

Maximum .593 .427 .144 .095 .838 .646 .755 .691 .914 4.726 .470 .444 45.953 24.143 4.440 4.726

37

Table 4 US-based estimation results of the panel data models

Fixed effects model Random effects model

EBIT it/ RTA it-1 -.877 (.124)* -1.646 (.096)*

FEit/ RTA it-1 3.910 (2.829) 1.354 (2.396)*

RE it-1 / RTAit-1 -4.262 (.366)* -7.123 (.352)*

Time χ2 (12) 93.46* 71.92*

LR χ2 (15) 387.17*

lnσσσσ2ηηηη 1.451 (-.061)

σσσσηηηη 2.066 (.064)

ρρρρ .565 (.015)

ρρρρ = 0 χ2 (1) 1299.96 *

Wald χ2 (15) 842.49*

The regressions are performed by using the US panel described in Table 1. The dependent variable is PFD, a dummy variable that takes value one for financially distressed firms, and zero otherwise. The explanatory variables are described in Table 3. The rest of the information needed to read this table is: i) Heteroskedasticity consistent asymptotic standard error in parentheses; ii) *, **, *** indicate significance at 1, 5 and 10%, respectively; iii) Time is a Wald test of the joint significance of the time dummy variables, asymptotically distributed as χ2 under the null of no relationship; degrees of freedom in parentheses; iv) LR is the Maximum likelihood ratio test of goodness of fit, asymptotically distributed as χ2 under the null of no joint significance of the coefficients; degrees of freedom in parentheses; v) ρ = 0 is a test of the joint significance of individual effects, asymptotically distributed as χ2 under the null of no joint significance; degrees of freedom in parentheses; vi) Wald is a test of goodness of fit, asymptotically distributed as χ2 under the null of no joint significance of the coefficients; degrees of freedom in parentheses.

38

Table 5 US-based estimation results of the cross-sectional model

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

-1.001 -.919 -1.137 -1.047 -1.063 -1.129 -.937 -1.17 -1.199 -.921 -1.297 -1.549 EBIT it/ RTA it-1

-1.3479

(.3980)* (.335)* (.298)* (.267)* (.259)* (.193)* (.185)* (.198)* (.177)* (.180)* (.151)* (.169)* (.225)*

22.7979 2.037 31.107 24.34 14.779 1.261 17.458 23.094 1.159 8.208 11.944 6.656 -2.672 FEit/ RTA it-1

(5.7842)* (6.026)* (7.187)* (7.197)* (7.307)** (5.343)*** (3.919)* (4.689)* (4.317)** (4.889)*** (4.812)** (6.200) (7.559)

-1.718 -6.026 -11.517 -5.999 -9.253 -8.446 -4.634 -7.183 -5.025 -4.070 -5.090 -1.122 -12.260 RE it-1 / RTA it-1

(1.6016)* (1.196)* (1.638)* (1.237)* (1.170)* (.922)* (.689)* (.839)* (.700)* (.876)* (.738)* (.979)* (1.238)*

Pseudo R-squared .273 .1758 .3028 .2349 .317 .317 .2994 .3788 .3113 .296 .2789 .401 .4028

LR 116.20 79.44 145.46 104.02 168.14 266.56 208.76 296.29 242.40 207.46 209.48 315.75 328.58

Observations 766 806 831 876 985 985 1458 1539 1516 1486 1433 1356 1264

The regressions are performed by using the US panel described in Table 1. The dependent variable is PFD, a dummy variable that takes value one for financially distressed firms, and zero otherwise. The explanatory variables are described in Table 3. The rest of the information needed to read this table is: i) Heteroskedasticity consistent asymptotic standard error in parentheses; ii) *, **, *** indicate significance at 1, 5 and 10%, respectively; iii) pseudo-R2 is a measure of the goodness of fit of the model

that is equivalent to the R2. null

fullnull

LL

LLLLRpseudo

2

)2(22

−

−−−=− , where -2LL is the likelihood value and where the null model is the one including only the constant; iv) LR

is the Likelihood Ratio statistic that tests the joint significance of the independent variables in the model, which is asymptotically distributed as a χ2 with degrees of

freedom in parentheses under the null of the lack of joint significance; v) Observations stands for the number of observation included each year to run the cross-sectional logit model.

39

Table 6 Results for the estimation of the probabilities of financial distress Year Financial status Type I

error Type II error

Correct classification

Mean Standard deviation

1990 N (705); FD (61) 13 (21) 125 (18) 628 (82) .0796 .1364 1991 N (741); FD (65) 12 (18) 169 (23) 625 (78) .0806 .1115 1992 N (761); FD (70) 13 (19) 154 (20) 664 (80) .0842 .1512 1993 N (815); FD (61) 18 (30) 128 (16) 730 (83) .0696 .1210 1994 N (910); FD (75) 17 (23) 114 (13) 854 (87) .0761 .1503 1995 N (1,279); FD (107) 25 (23) 149 (12) 1,212 (87) .0772 .1602 1996 N (1,364); FD (94) 24 (26) 131 (10) 1,303 (89) .0645 .1334 1997 N (1,431); FD (108) 14 (13) 138 (10) 1,387 (90) .0702 .1583 1998 N (1,408); FD (108) 21 (19) 143 (10) 1,352 (89) .0712 .1462 1999 N (1,392); FD (94) 22 (23) 102 (7) 1,362 (92) .0633 .1356 2000 N (1,328); FD (105) 19 (18) 145 (11) 1,269 (89) .0733 .1403 2001 N (1,241); FD (115) 11 (10) 150 (12) 1,195 (88) .0848 .1793 2002 N (1,139); FD (125) 10 (8) 165 (14) 1,089 (86) .0989 .1924 Total N (14,514); FD (1,188) 219 (18) 1,813 (12) 13,670 (87) .0757 .1509

The information needed to read this table is: i) N and FD denote normal and financially distressed companies, respectively (number of observations in parenthesis); ii) Type I error stands for the number and percentage (in parenthesis) of normal observations classified as financially distressed; iii) Type II error stands for the number and percentage (in parenthesis) of financially distressed observations classified as normal; iv) Correct classification is the number and percentage (in parenthesis) of observations correctly classified into normal and financially distressed; v) Mean is the mean value of the estimated probabilities of financial distress; vi) Standard deviation is the standard deviation of the estimated probabilities of financial distress.

40

Table 7 Estimation results of the panel data models: Evidence from other-than-US G7 countries Fixed effects model Random effects model

EBIT it/ TA’ it-1 -3.706 (.404)* -4.400 (.277)*

FEit/ TA’ it-1 26.766 (6.641)* 26.539 (5.328)*

RE it-1 / TA’ it-1 -7.714 (.708)* -1.967 (.595)*

Time χ2 (12) 11.62 5.48

LR χ2 (15) 285.53*

lnσσσσ2ηηηη 1.376 (.057)

σσσσηηηη 1.980 (.057)

ρρρρ .546 (-.014)

ρρρρ = 0 χ2 (1) 94.91*

Wald χ2 (20) 647.91*

The regressions are performed by using the panel for other-than-US G7 countries described in Table 1. The dependent variable is PFD, a dummy variable that takes value one for financially distressed firms, and zero otherwise. The explanatory variables are described in Table 3. The rest of the information needed to read this table is i) Heteroskedasticity consistent asymptotic standard error in parentheses; ii) *, **, *** indicate significance at 1, 5 and 10%, respectively; iii) Time is a Wald test of the joint significance of the time dummy variables, asymptotically distributed as χ2 under the null of no relationship; degrees of freedom in parentheses; iv) LR is Maximum likelihood ratio test of goodness of fit, asymptotically distributed as χ2 under the null of no joint significance of the coefficients; degrees of freedom in parentheses; v) ρ = 0 is a test of the joint significance of individual effects, asymptotically distributed as χ2 under the null of no joint significance; degrees of freedom in parentheses; vi) Wald is a test of goodness of fit, asymptotically distributed as χ2 under the null of no joint significance of the coefficients; degrees of freedom in parentheses.

41

Table 8 Estimation results of the cross-sectional model: Evidence from other-than-US G7 countries

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

-.2521 -2.601 -.709 -2.404 -2.514 -1.528 -2.646 -4.615 -5.472 -2.935 -2.308 -3.644 -3.509 EBIT it/ RTA it-1

(1.297) (1.027)** (.869) (.678)* (.704)* (.746)** (.954)* (.845)* (.678)* (.530)* (.451)* (.491)* (.459)*

24.2466 2.644 28.444 1.4 28.401 46.306 38.989 36.844 8.398 16.205 13.882 8.569 17.642 FEit/ RTA it-1

(15.209) (19.526) (16.069)*** (13.454) (14.948)*** (13.524)* (13.073)* (12.005)* (11.265) (12.711) (11.372) (11.720) (12.021)

-17.698 -11.977 -12.147 -6.476 -9.508 -8.505 -11.092 -13.119 -1.251 -11.302 -1.541 -11.076 -9.846 RE it-1 / RTA it-1

(3.9256)* (3.021)* (2.585)* (2.005)* (2.364)* (1.790)* (1.751)* (1.488)* (1.258)* (1.357)* (1.290)* (1.276)* (1.237)*

Pseudo R-squared .3577 .316 .317 .225 .293 .249 .19 .3 .298 .28 .237 .285 .277

LR .3577 .316 .317 .225 .293 .249 .19 .3 .298 .28 .237 .285 .277

Observations 432 461 486 523 694 791 1821 2227 2213 2200 2169 2131 2012

The regressions are performed by using the panel for other G7 countries described in Table 1. The dependent variable is PFD, a dummy variable that takes value one for financially distressed firms, and zero otherwise. The explanatory variables are described in Table 3. The rest of the information needed to read this table is: i) Heteroskedasticity consistent asymptotic standard error in parentheses; ii) *, **, *** indicate significance at 1, 5 and 10%, respectively; iii) pseudo-R2 is a measure of the

goodness of fit of the model that is equivalent to the R2. null

fullnull

LL

LLLLRpseudo

2

)2(22

−

−−−=− , where -2LL is the likelihood value and where the null model is the one

including only the constant; iv) LR is the Likelihood Ratio statistic that tests the joint significance of the independent variables in the model, which is asymptotically distributed as a χ2 with degrees of freedom in parentheses under the null of the lack of joint significance; v) Observations stands for the number of observation included each year to run the cross-sectional logit model.

42

Table 9 Results for the estimation of the probabilities of financial distress: Evidence from other-than-US G7 countries Year Financial status Type I

error Type II error

Correct classification

Mean Standard deviation

1990 N (414); FD (18) 3 (17) 64 (15) 365 (84) .0417 .1127 1991 N (441); FD (20) 7 (335) 74 (17) 380 (82) .0434 .1132 1992 N (462); FD (24) 2 (8) 90 (19) 394 (81) .0494 .1131 1993 N (498); FD (25) 3 (12) 107 (21) 413 (79) .0478 .0938 1994 N (669); FD (25) 6 (24) 95 (14) 593 (85) .0360 .0932 1995 N (760); FD (31) 9 (29) 138 (18) 644 (81) .0392 .0861 1996 N (1,764); FD (57) 17 (30) 294 (17) 1,510 (83) .0313 .0620 1997 N (2,148); FD (79) 13 (16) 340 (16) 1,874 (84) .0355 .0907 1998 N (2,119); FD (94) 12 (13) 436 (21) 1,765 (80) .0425 .1031 1999 N (2,119); FD (81) 21 (26) 313 (15) 1,866 (85) .0368 .0947 2000 N (2,084); FD (85) 18 (21) 344 (17) 1,807 (83) .0392 .0888 2001 N (2,037); FD (94) 13 (14) 327 (16) 1,791 (84) .0441 .1044 2002 N (1,924); FD (88) 14 (16) 318 (17) 1,680 (83) .0437 .1004 Total N (17,439); FD (721) 138 (19) 2,940 (17) 15,082 (83) .0397 .0949

The information needed to read this table is: i) N and FD denote normal and financially distressed companies, respectively (number of observations in parenthesis); ii) Type I error stands for the number and percentage (in parenthesis) of normal observations classified as financially distressed; iii) Type II error stands for the number and percentage (in parenthesis) of financially distressed observations classified as normal; iv) Correct classification is the number and percentage (in parenthesis) of observations correctly classified into normal and financially distressed; v) Mean is the mean value of the estimated probabilities of financial distress; vi) Standard deviation is the standard deviation of the estimated probabilities of financial distress.

43

Table 10 Estimation results of the panel data models: Evidence from the re-estimated Z-score model

US-based results Results for other

G7 countries

Fixed effects model

Random effects model

Fixed effects

model

Random effects

model

WC it/ TA it-1 -.229 (.110)** -.158 (.070)** -2.466 (.384)* -1.963 (.290)*

REit/ TA it-1 -.634 (.092)* -1.397 (.086)* .034 (.035) .013 (.030)

EBIT it/ TA it-1 -3.385(.306)* -5.962 (.304)* -5.888 (.565)* -1.538 (.525)*

MV it/ TL it-1 -.002 (.004) .002 (.005) -.027 (.014)** -.013 (.012)

SALES it-1 / TA it-1 -.498 (.137)* -.114 (.070) -.069 (.152) .057 (.087)

Time χ2 (12) 81.15* 64.53* 10.87 225.5*

LR χ2 (17) 383.37* 229.50*

lnσσσσ2ηηηη 1.458 (.062) 1.397 (.057)

σσσσηηηη 2.073 (.065) 2.011 (.058)

ρρρρ .566 (.015) .551 (.014)

ρρρρ = 0 χ2 (1) 1316.8 * 461.66*

Wald χ2 (17) 746.08*

Wald χ2 (22) 1019.7*

The regressions are performed by using the panels described in Table 1. The dependent variable is PFD, a dummy variable that takes value one for financially distressed firms, and zero otherwise. The explanatory variables are described in Table 3. The rest of the information needed to read this table is i) Heteroskedasticity consistent asymptotic standard error in parentheses; ii) *, **, *** indicate significance at 1, 5 and 10%, respectively; iii) Time is a Wald test of the joint significance of the time dummy variables, asymptotically distributed as χ2 under the null of no relationship; degrees of freedom in parentheses; iv) LR is Maximum likelihood ratio test of goodness of fit, asymptotically distributed as χ2 under the null of no joint significance of the coefficients; degrees of freedom in parentheses; v) ρ = 0 is a test of the joint significance of individual effects, asymptotically distributed as χ2 under the null of no joint significance; degrees of freedom in parentheses; vi) Wald is a test of goodness of fit, asymptotically distributed as χ2 under the null of no joint significance of the coefficients; degrees of freedom in parentheses.

44

Table 11 Estimation results of the cross-sectional model: Evidence from US for the re-estimated Z-score model

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

.7404 .466 -.032 .108 3.131 1.789 1.227 -.233 -.03 .064 -.221 -1.036 .129 WC it/ TA it-1

(.764) (.560) (.423) (.658) (.659)* (.558)* (.595)** (.496) (.097) (.202) (.298) (.376)* (.506)

-1.9473 -1.107 -1.401 -1.14 -1.326 -1.168 -1.495 -1.178 -1.006 -1.201 -.933 -1.062 -1.354 REit/ TA it-1

(.4002)* (.318)* (.274)* (.236)* (.262)* (.198)* (.194)* (.178)* (.139)* (.154)* (.145)* (.167)* (.200)*

-9.1309 -6.385 -9.431 -5.672 -7.955 -7.059 -4.788 -5.94 -4.074 -2.678 -3.787 -7.947 -9.781 EBIT it/ TA it-1

(1.4618)* (1.198)* (1.414)* (.995)* (1.089)* (.817)* (.675)* (.744)* (.596)* (.726)* (.607)* (.846)* (1.060)*

-.0746 -.568 -.014 -.023 -.048 -.004 -.102 -.008 .014 -.003 .01 .017 -.033 MV it/ TL it-1

(.061) (.194)* (.019) (.025) (.021)** (.009) (.027)* (.013) (.012) (.020) (.012) (.010)*** (.025)

.2928 .184 .101 .06 -.143 -.061 .317 .016 .115 -.536 -.178 -.157 -.323 SALES it-1 /TA it-1

(.180) (.157) (.175) (.168) (.200) (.163) (.117)* (.139) (.090) (.207)* (.162) (.152) (.185)***

Pseudo R-squared .2509 .202 .265 .206 .349 .349 .318 .367 .301 .321 .277 .397 .411

LR 101.52 88.03 123.03 88.82 182.93 271.00 213.23 280.52 229.06 221.83 206.19 308.03 330.995

Observations 727 767 793 838 948 948 1412 1493 1470 1445 1397 1321 1232

The regressions are performed by using the panels described in Table 1.The dependent variable is PFD, a dummy variable that takes value one for financially distressed firms, and zero

otherwise. The explanatory variables are described in Table 3. The rest of the information needed to read this table is: i) Heteroskedasticity consistent asymptotic standard error in parentheses; ii)

*, **, *** indicate significance at 1, 5 and 10%, respectively; iii) pseudo-R2 is a measure of the goodness of fit of the model that is equivalent to the R2. null

fullnull

LL

LLLLRpseudo

2

)2(22

−

−−−=− ,

where -2LL is the likelihood value and where the null model is the one including only the constant; iv) LR is the Likelihood Ratio statistic that tests the joint significance of the independent

variables in the model, which is asymptotically distributed as a χ2 with degrees of freedom in parentheses under the null of the lack of joint significance; v) Observations stands for the number

of observation included each year to run the cross-sectional logit model.

45

Table 12 Estimation results of the cross-sectional model: Evidence from other-than-US G7 countries for the re-estimated Z-score model

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

.7404 .466 -.032 .108 3.131 1.789 1.227 -.233 -.03 .064 -.221 -1.036 .129 WC it/ TA it-1

(.764) (.560) (.423) (.658) (.659)* (.558)* (.595)** (.496) (.097) (.202) (.298) (.376)* (.506)

-1.9473 -1.107 -1.401 -1.14 -1.326 -1.168 -1.495 -1.178 -1.006 -1.201 -.933 -1.062 -1.354 REit/ TA it-1

(.4002)* (.318)* (.274)* (.236)* (.262)* (.198)* (.194)* (.178)* (.139)* (.154)* (.145)* (.167)* (.200)*

-9.1309 -6.385 -9.431 -5.672 -7.955 -7.059 -4.788 -5.94 -4.074 -2.678 -3.787 -7.947 -9.781 EBIT it/ TA it-1

(1.4618)* (1.198)* (1.414)* (.995)* (1.089)* (.817)* (.675)* (.744)* (.596)* (.726)* (.607)* (.846)* (1.060)*

-.0746 -.568 -.014 -.023 -.048 -.004 -.102 -.008 .014 -.003 .01 .017 -.033 MV it/ TL it-1

(.061) (.194)* (.019) (.025) (.021)** (.009) (.027)* (.013) (.012) (.020) (.012) (.010)*** (.025)

.2928 .184 .101 .06 -.143 -.061 .317 .016 .115 -.536 -.178 -.157 -.323 SALES it-1 /TA it-1

(.180) (.157) (.175) (.168) (.200) (.163) (.117)* (.139) (.090) (.207)* (.162) (.152) (.185)***

Pseudo R-squared .2509 .202 .265 .206 .349 .365 .318 .367 .301 .321 .277 .397 .411

LR 54.64 53.17 60.78 40.45 49.14 58.50 85.18 174.43 167.24 171.12 154.71 152.45 150.71

Observations 727 767 793 838 948 1342 1412 1493 1470 1445 1397 1321 1232

The regressions are performed by using the panels described in Table 1.The dependent variable is PFD, a dummy variable that takes value one for financially distressed firms, and zero otherwise. The explanatory variables are described in Table 3. The rest of the information needed to read this table is: i) Heteroskedasticity consistent asymptotic standard error in parentheses; ii) *, **, *** indicate significance at 1, 5 and 10%, respectively; iii) pseudo-R2 is a measure of the goodness of fit of the model that is equivalent to the R2.

null

fullnull

LL

LLLLRpseudo

2

)2(22

−

−−−=− , where -2LL is the likelihood value and where the null model is the one including only the constant; iv) LR is the Likelihood Ratio statistic that tests

the joint significance of the independent variables in the model, which is asymptotically distributed as a χ2 with degrees of freedom in parentheses under the null of the lack of joint significance; v) Observations stands for the number of observation included each year to run the cross-sectional logit model.

46

Table 13 Results for the estimation of the probabilities of financial distress: US-based evidence from the re-estimated Z-score model

The information needed to read this table is: i) N and FD denote normal and financially distressed companies, respectively (number of observations in parenthesis); ii) Type I error stands for the number and percentage (in parenthesis) of normal observations classified as financially distressed; iii) Type II error stands for the number and percentage (in parenthesis) of financially distressed observations classified as normal; iv) Correct classification is the number and percentage (in parenthesis) of observations correctly classified into normal and financially distressed; v) Mean is the mean value of the estimated probabilities of financial distress; vi) Standard deviation is the standard deviation of the estimated probabilities of financial distress.

Year Financial status Type I error

Type II error

Correct classification

Mean Standard deviation

1990 N (705); FD (61) 12 (20) 225 (32) 529 (69) .0798 .1319 1991 N (741); FD (65) 12 (18) 196 (26) 598 (74) .0821 .1144 1992 N (761); FD (70) 17 (24) 142 (19) 672 (81) .0858 .1430 1993 N (815); FD (61) 19 (31) 115 (14) 742 (85) .0716 .1148 1994 N (910); FD (75) 17 (23) 122 (13) 846 (86) .0791 .1600 1995 N (1,279); FD (107) 19 (18) 142 (11) 1,225 (88) .0790 .1661 1996 N (1,364); FD (94) 28 (30) 128 (9) 1,302 (89) .0637 .1359 1997 N (1,431); FD (108) 21 (19) 118 (8) 1,400 (91) .0710 .1579 1998 N (1,408); FD (108) 26 (24) 131 (9) 1,359 (90) .0721 .1457 1999 N (1,392); FD (94) 24 (26) 102 (7) 1,360 (92) .0644 .1423 2000 N (1,328); FD (105) 19 (18) 128 (10) 1,286 (90) .0752 .1428 2001 N (1,241); FD (115) 17 (15) 141 (11) 1,198 (88) .0863 .1797 2002 N (1,139); FD (125) 10 (8) 173 (15) 1,081 (86) .1006 .1960 Total N (14,514); FD (1,188) 241 (20) 1,863 (13) 13,598 (87) .0744 .1509

47

Table 14 Results for the estimation of the probabilities of financial distress: Evidence from the re-estimated Z-score model for other-than-US G7 countries Year Financial status Type I

error Type II error

Correct classification

Mean Standard deviation

1990 N (414); FD (18) 2 (11) 61 (15) 369 (85) .0423 .1193 1991 N (441); FD (20) 6 (30) 91 (21) 364 (79) .0441 .1162 1992 N (462); FD (24) 3 (13) 88 (19) 395 (81) .0483 .1115 1993 N (498); FD (25) 5 (20) 115 (23) 403 (77) .0470 .0858 1994 N (669); FD (25) 8(32) 135 (20) 551 (79) .0368 .0800 1995 N (760); FD (31) 8 (26) 160 (21) 623 (79) .0399 .0822 1996 N (1,764); FD (57) 17 (30) 269 (15) 1,535 (84) .0315 .0581 1997 N (2,148); FD (79) 18 (23) 335 (16) 1,874 (84) .0357 .0833 1998 N (2,119); FD (94) 18 (19) 533 (25) 1,662 (75) .0427 .0865 1999 N (2,119); FD (81) 28 (35) 331 (16) 1,841 (84) .0370 .0856 2000 N (2,084); FD (85) 24 (28) 361 (17) 1,784 (82) .0394 .0812 2001 N (2,037); FD (94) 16 (17) 408 (20) 1,707 (80) .0438 .0844 2002 N (1,924); FD (88) 23 (26) 338 (18) 1,651 (82) .0434 .0856 Total N (17,439); FD (721) 176 (24) 3,225 (18) 14,759 (81) .0394 .0845

The information needed to read this table is: i) N and FD denote normal and financially distressed companies, respectively (number of observations in parenthesis); ii) Type I error stands for the number and percentage (in parenthesis) of normal observations classified as financially distressed; iii) Type II error stands for the number and percentage (in parenthesis) of financially distressed observations classified as normal; iv) Correct classification is the number and percentage (in parenthesis) of observations correctly classified into normal and financially distressed; v) Mean is the mean value of the estimated probabilities of financial distress; vi) Standard deviation is the standard deviation of the estimated probabilities of financial distress.

48

Figure 1

Stages of the new approach for estimating the probability of financial distress

PROBABILITY OF FINANCIAL DISTRESS

CROSS-SECTION ESTIMATION OF THE MODEL

ESTIMATION BY USING PANEL DATA

METHODOLOGY

TESTING FOR THE SPECIFICATION OF THE

MODEL

ECONOMETRIC SPECIFICATION OF THE MODEL