Jones-Predicting financial distress logit mode

1. 1
Predicting Firm Financial Distress: A Mixed Logit Model
Stewart Jones
David A. Hensher
The University of Sydney
ABSTRACT: Over the past three decades the literature on financial distress prediction
has largely been confined to simple multiple discriminant analysis, binary logistic or
probit analysis, or rudimentary multinomial logit models (MNL). There has been a
conspicuous absence of modeling innovation in this literature as well as a failure to keep
abreast of important methodological developments emerging in other fields of the social
sciences. In particular, there has been no recognition of major advances in discrete
choice modeling over the last 15 years which has increasingly relaxed behaviorally
questionable assumptions associated with the IID condition and allowed for observed
and unobserved heterogeneity. In contrast to standard logit, the mixed logit model fulfils
this purpose and provides a superior framework for explanation and prediction. We
explain the theoretical and econometric underpinnings of mixed logit and demonstrate
its empirical usefulness in the context of a specific but topical area of accounting
research, financial distress prediction. Comparisons of model-fits and out-of-sample
forecasts indicate that mixed logit outperforms standard logit by significant margins.
While mixed logit has valuable applications in financial distress research, its potential
usefulness in other areas of accounting research should not be overlooked.
Keywords: Financial distress prediction; Mixed logit; Multinomial logit.
Data Availability: All data available from public and commercial sources identified in
the paper.
We appreciate the helpful comments of two anonymous referees, participants at the University of Sydney
research seminar series, Bill Greene, Ken Train, Rob Stambaugh, Mike Bradbury, David Johnstone, Paul
Bahnson, Raj Srivastava and Marcus O’Connor.

2. 2
I. INTRODUCTION
The prediction of firm financial distress has been of considerable interest to
accountants and financial economists over the last three decades (Altman 2001). The
continual development of conceptually richer and more accurate forecasting models is
of importance to regulators, practitioners and academics (Shumway, 2001). Distress
forecasts are now widely used for a range of purposes, including monitoring of the
solvency of financial and other institutions by regulators, assessment of loan security,
going concern evaluations by auditors, the measurement of portfolio risk, and the
pricing of defaultable bonds, credit derivatives and other securities exposed to credit
risk (Scott, 1981; Shumway, 2001; Altman, 2001; Duffie and Singleton, 2003). Given
the broad interest in distress prediction, an improved modeling approach can enhance
the state of art in this field as well as providing a new analysis technique having
potentially broad application to other areas of accounting research.
While an extensive literature on financial distress prediction has emerged over
the past three decades, innovative modeling techniques have been slow to develop. In
fact, many commonly used techniques would rate as primitive and dated in other
fields of the social sciences. Much of the literature has relied on relatively simplistic
multiple discriminant analysis (MDA), binary logistic or probit analysis or
rudimentary multinomial logit models (MNL) (see e.g., Altman 1968; Altman,
Haldeman, and Narayanan, 1977; Ohlson, 1980; Zmijewski, 1984; Lau, 1987). The
relative merits of binary logit, MDA and to a lesser extent probit and tobit models
have been discussed in an extensive literature (see e.g., Efron, 1975; McFadden, 1976;
Ohlson, 1980; Jones, 1987; Maddala, 1991). Leclere (1999) advanced the literature
by providing an interpretative analysis of accounting research which has employed
some form of n-chotomous qualitative response variable (such as multinomial logit
and ordered probit) (see also Elliot and Kennedy, 1988; and Kennedy, 1992).
Methodological discussions are yet to progress beyond this point. The major
limitation of the financial distress literature (and other related accounting literatures)
is that there has been no recognition of the major developments in discrete choice
modeling over the last 15 years which has increasingly relaxed behaviorally
questionable assumptions associated with the IID condition (independently and
identically distributed errors) and allowed for observed and unobserved
heterogeneity.1
The latter in particular has been shown in other fields of the social
sciences to have an important role to play in the understanding and prediction of
choice outcomes. This literature is reviewed in Louviere, Hensher and Swait (2000),
Hensher, Louviere and Swait (1999), Train (2003) and Hensher, Rose and Greene
(2005).
This study uses an advanced discrete choice model known as mixed or random
parameter ordered logit. The mixed logit design is one of the latest and most
advanced techniques now being used in many fields of the social sciences to model
discrete choices or outcomes. We seek to explain the conceptual underpinnings of the
mixed logit model and how it improves on standard logit.2
We also attempt to
demonstrate the empirical usefulness of the model in the specific but topical area of
financial distress prediction, a field of inquiry highly amenable to this form of
analysis. Using a broad cross section of financial variables employed by researchers
over the past thirty years, we compare the predictive performance of mixed logit with
standard logit in terms of model-fit and out-of-sample forecasting accuracy.3
The remainder of this paper is organized as follows. Section II discusses the
theoretical significance of mixed logit as well as the econometric properties of this
model. Section III outlines the research methodology. Section IV provides the results

3. 3
and Section V addresses further empirical considerations relating to the estimation
and observed performance of the mixed logit model. Concluding remarks and
directions for future research are discussed in Section VI.
II. MIXED LOGIT: THEORETICAL AND ECONOMETRIC
CONSIDERATIONS
Conceptual Foundations of the Mixed Logit Model
Mixed logit is the latest among a new breed of econometric models being
developed out of discrete choice theory (Train, 2003). Discrete choice theory is
concerned with understanding the discrete behavioral responses of individuals to the
actions of business, markets and government when faced with two or more possible
outcomes (or choices) (Louviere, Hensher and Swait, 2000). Its theoretical
underpinnings are derived from microeconomic theory of consumer behavior, such as
the formal definition of agent preferences as inputs into a choice or outcome setting as
determined by the utility maximization of agents. Given that the analyst has
incomplete knowledge on the information inputs of the agents being studied, the
analyst can only explain a choice outcome up to a probability of it occurring. This is
the basis for the theory of random utility (see Louviere, Hensher and Swait, 2000 for a
review of literature).4
While random utility theory has developed from economic
theories of consumer behavior it can be applied to any unit of analysis (e.g., firm
failures) where the dependent variable is discrete.5
The concept of behavioral heterogeneity (individual variations in tastes and
preferences), and how this impinges on the validity of various theoretical and
empirical models has been the subject of much recent attention in this literature.6
However, econometric techniques to model heterogeneity have taken time to develop,
despite a long standing recognition that failure to do so can result in inferior model
specification, spurious test results and invalid conclusions (Louviere, Hensher and Swait,
2000; Train, 2003). Starting with the simple binary logit model, research progressed
during the 1960s and 1970s to the multinomial logit (MNL) and nested logit models,
the latter becoming the most popular of the generalized logit models. Although more
advanced choice models such as mixed logit existed in conceptual and analytical form
in the early 1970s, parameter estimation was seen as a practical barrier to their
empirical usefulness. The breakthrough came with the development of simulation
methods (such as simulated maximum likelihood estimation) that enabled the open-
form models such as mixed logit to be estimated with relative ease (e.g., Stern, 1997).
Mixed logit and its variants have now supplanted simpler models in many
areas of economics, marketing, management, transportation, health, housing, energy
research and environmental science (Train, 2003). This can largely be explained in
terms of the substantial improvements delivered by mixed logit over binary logistic and
MNL models. Considering the case of firm failures, the main improvement is that
mixed logit models include a number of additional parameters which capture
observed and unobserved heterogeneity both within and between firms.7
For a mixed
logit model, the probability of failure of a specific firm in a sample is determined by
the mean influence of each explanatory variable with a fixed parameter estimate
within the sampled population, plus, for any random parameters, a parameter weight
drawn from the distribution of individual firm parameters estimated across the
sample. This weight is randomly allocated to each sampled firm unless there are
specific rules for mapping individual firms to specific locations within the distribution
of firm specific parameters.8
In contrast, the probability of failure for an individual
firm using a binary logistic or MNL model is simply a weighted function of its fixed

4. 4
parameters (i.e., assumption of homogeneous preferences) with all other behavioral
information assigned (incorrectly) to the error term.9
As noted by Hensher and Greene
(2003), parameter estimation in the mixed logit model maximizes use of the behavioral
information embedded in any dataset appropriate to the analysis. Ultimately, these
conceptual advantages afford the analyst with a substantially improved foundation for
explanation and prediction.10
The important theoretical advantages of the mixed logit
model are further considered in the formal specification and analysis of the model
which now follows.
An Ordered Mixed (Random Parameters) Logit Model
Like any random utility model of the discrete choice family of models, we
assume that a sampled firm (q=1,…,Q) faces a ‘choice’ amongst I alternatives in each
of T occasions. Within the context of financial distress, since firms do not choose to
fail per se, we prefer to use the phrase outcome domain (or simply outcome) as the
descriptor of the observed choice outcome. A firm q is assumed to recognize the full
set of alternative outcomes in occasion t and to focus on business strategies designed
to result in the delivery of the outcome associated with the highest utility (i.e.,
nonfailure). The (relative) utility associated with each outcome i as evaluated by each
firm q in occasion t is represented in a discrete outcome model by a utility expression
of the general form:
Uitq = ßqXitq + eitq (1)
Xitq is a vector of explanatory variables that are observed by the analyst (from any
source) and include attributes of the alternative outcomes (where observed),
characteristics of the firm and descriptors of the decision context in occasion t, ßq and
eitq are not observed by the analyst and are treated as stochastic influences.
To provide an intuitive explanation of how equation (1) operates in an
outcome setting, think of the task as being one of representing sources of variance that
contribute to explaining a specific outcome. For a specific firm, equation (1) has
variance potential associated with the coefficient attached to each observed
characteristic (i.e., ß), to each observed characteristic itself (i.e., X) and the
unobserved effects term (e). We could expand this equation out to reflect these
sources of variance for three characteristics, defining ‘0’ as observed and ‘U’ as
unobserved, as (dropping the q and t subscripts):11
Ui = (ß01X01 + ßu1Xu1) + (ß02X02 + ßu2Xu2) + (ß03X03 + ßu3Xu3) + ei (1a)
Each characteristic is now represented by a set of observed and unobserved
influences. In addition each parameter and characteristic can itself be expressed as
some function of other influences, giving more depth in the explanation of sources of
variance. As we expand the function out we reveal deeper parameters to identify. In
the most restrictive (or simplistic) versions of the utility expression, we would gather
all the unobserved sources together and replace (1a) with (1b):
Ui = ß01X01 +ß02X02 + ß03X03 + (ßu1Xu1+ ßu2Xu2 + ßu3Xu3 + ei) (1b)
and would collapse the unobserved influences into a single unknown by assuming that
all unobserved effects cannot be related in any systematic way with the observed
effects:

5. 5
Ui = ß01X01 +ß02X02 + ß03X03 + ei (1c)
Furthermore by defining a utility expression of the form in (1c) for each
alternative outcome i and imposing a further assumption that the unobserved
influences have the same distribution and are independent across alternatives we can
remove the subscript i attached to e. What we have is the functional form for the
utility expressions of a multinomial logit (MNL) model. This intuitive discussion has
highlighted the way in which an MNL model restricts through assumption the
opportunity to reveal the fuller range of potential sources of influence on utility as
resident throughout the full dimensionality of equation (1a). Explaining these fuller
sources is equivalent to explaining the broader set of sources of observed and
unobserved heterogeneity on an outcome domain.
A condition of the MNL model is that eitq is independent and identically
distributed (IID) extreme value type 1.12
IID is clearly restrictive in that its does not
allow the error components of different alternative outcomes to be correlated. We
would want to be able to take this into account in some way. One way to do this is to
partition the stochastic component into two additive (i.e., uncorrelated) parts. One part
is correlated over alternative outcomes and heteroskedastic, and another part is IID
over alternative outcomes and firms as shown in equation (2) (ignoring the t subscript
for the present).
Uiq = β′xiq + (ηiq +εiq) (2)
where ηiq is a random term with zero mean whose distribution over firms and
alternative outcomes depends in general on underlying parameters and observed data
relating to alternative outcome i and firm q; and εiq is a random term with zero mean
that is IID over alternative outcomes and does not depend on underlying parameters or
data.
The Mixed Logit class of models assumes a general distribution for η and an
IID extreme value type 1 distribution for ε.13
That is, η can take on a number of
distributional forms such as normal, lognormal, and triangular. Denote the density of
η by f(η|Ω) where Ω are the fixed parameters of the distribution. For a given value of
η, the conditional probability for outcome i is logit, since the remaining error term is
IID extreme value:
Li(η) = exp(β′xi + ηi) / ∑jexp(β′xj + ηj). (3)
Since η is not given, the (unconditional) outcome probability is this logit formula
integrated over all values of η weighted by the density of η is as shown in equation
(4).
Pi=∫Li(η) f(η|Ω)dη (4)
Models of this form are called mixed logit because the outcome probability
Li(η) is a mixture of logits with f as the mixing distribution. The probabilities do not
exhibit the well known independence from irrelevant alternatives property (IIA), and
different substitution patterns are obtained by appropriate specification of f. This is
handled in two ways. The first way, known as random parameter specification,

6. 6
involves specifying each ßq associated with an attribute of an alternative outcome as
having both a mean and a standard deviation (i.e., it is treated as a random parameter
instead of a fixed parameter). The second way, known as the error components
approach, treats the unobserved information as a separate error component in the
random component. We have chosen to use the former interpretation and hence the
reference to a mixed or random parameter model.
The standard deviation of a ß parameter accommodates the presence of
preference heterogeneity in the sampled population of firms. While one might handle
this heterogeneity in the context of a fixed ß parameter through data segmentation
(e.g., a different model for each industry segment) and/or attribute segmentation (e.g.,
separate ßs for different industry segments), in contrast to treating it all as random, the
challenge of these (deterministic) segmentation strategies is in picking the right
segmentation criteria and range cut-offs that account for statistically significant
sources of preference heterogeneity. A random parameter representation of preference
heterogeneity is more general; however such a specification carries a challenge in that
these parameters have a distribution that is unknown. Alternative analytical
distributions can be evaluated such as the normal, lognormal and triangular.
The identification of the parameter estimates in a mixed logit model is
complex. Unlike the simple multinomial logit model which has a closed form solution
and guarantees a unique globally optimal set of parameter estimates, the mixed logit
model can produce a range of solutions, only one of which is globally optimal.14
In
particular we now have an additional set of parameters to identify associated with the
unobserved heterogeneity across the sampled firms. This information is captured in
the distribution of parameters that are assigned random in contrast to fixed. To take
this into account we use Bayes Rule to define the conditional distribution as equation
(5) (formally derived in Train (2003)).
Hq(ß|θ) = Lq(ß)g(ß|θ)/Pq(θ) (5)
Lq(ß) is the likelihood of a firm’s outcome domain if they had this specific ß; g(ß|θ) is
the distribution in the population of ßs (or the probability of a ß being in the
population), and Pq(θ) is the outcome probability function defined in open-form as:
Pq(θ) = ∫ Lq(ß)g(ß|θ) dß (6)
The outcome probability in (4) or (6) cannot be calculated exactly because the
integral does not have a closed form in general. The integral is approximated through
simulation. For a given value of the parameters, a value of η is drawn from its
distribution. Using this draw, the logit formula (3) for Li(η) is calculated. This process
is repeated for many draws, and the mean of the resulting Li(η)’s is taken as the
approximate outcome probability giving equation (7).
SPi = (1/R) 1
R
r=∑ Li(ηir) (7)
R is the number of replications (i.e., draws of η), ηir is the rth
draw, and SPi is the
simulated probability that a firm chooses alternative outcome i.15
There are typically
constraints on the η components across alternative outcomes; in particular the values
of η are the same for the calculation of the probabilities of each alternative outcome i
for a specific outcome occasion.16

7. 7
An Application to Financial Distress Prediction
The mixed logit framework presented above is sufficiently general to apply to
all forms of discrete outcome models with an extreme value distribution (type 1) for
the IID component of the unobserved elements representing the random component of
each utility expression. In the current application to financial distress prediction we
propose an ordered set of alternatives and hence need to introduce the idea of an
ordered logit specification.17
The ordered outcome scale adopted in this study has three levels (described in
detail in the next section). Treating this (ranking) scale as if it were both continuous
and an interval scale and applying ordinary least squares regression to analyze the
relationship between the failure levels and the explanatory variables is not a
statistically valid approach. A number of theoretical studies have questioned the
validity of the linearity assumption when using a rank ordered response (e.g., Henry
1982; Winship and Mare 1984; Crask and Fox 1987). An appropriate approach which
recognizes the nonlinearity of a ranking scale is an ordered response model which
defines points on the observed rating scale as thresholds.
Empirical ranking scales are best viewed as discrete realizations of
unmeasured continuous variables. The ordered logit model allows one to include
ordinal dependent variables into the outcome model in a way that explicitly
recognizes their ordinality and avoids arbitrary assumptions about their scale. The
essence of the approach is an assumed probability distribution of the continuous
variable that underlies the observed ordinal dependent variable. Ordered logit takes
into account the ceiling and floor restrictions on models which include ordinal
variables, whereas a linear regression does not.
In specifying an appropriate firm failure model, we assume that the three-point
ranking scale is a monotonic transformation of an unobserved interval variable. Thus
one or more values of an interval-level variable are mapped into the same value of a
transformed ordinal variable. An underlying continuous variable is mapped into
categories that are ordered but are separated by unknown distances. We cannot, for
example, say that the difference between ranks 0 and 1 is identical to the difference
between 1 and 2.
Formally, let Y* denote an unobserved (or latent) continuous variable (-∞ <
Y* < +∞), and µ0 µ,..., µJ-1, µJ denote the cut-off or threshold points in the
distribution of Y*, where µ0= -∞ and µJ = +∞. Define Y to be an ordinal (observed)
variable such that Y = j if, and only if, µJ-1 ≤ Y* ≤ µJ; j = 1,2,...,3. Since Y* is not
observed but Y is observed, its mean and variance are unknown. Statistical
assumptions must be introduced such that Y* has a mean of zero and a variance of
one. To operationalize the model, we need to define a relationship between Y* and
Y.
The ordered outcome model is based on the following specification: There is
a latent regression equation (8).
Yi* = β′xi + εi, εi ~ F(εi |θ), E(εi) = 0, Var(εi) = 1 (8)
The observation mechanism results from a complete censoring of the latent dependent
variable as follows:

8. 8
Yi = 0 if Yi ≤ μ0,
= 1 if μ0 < Yi ≤ μ1,
= 2 if μ1 < Yi ≤ μ2, (9)
...
= J if Yi > μJ-1.
The probabilities which enter the log likelihood function are given by equations (10)
and (11).
Prob(Yi = j) = Prob(Yi* is in the jth range) (10)
Prob(Yi = j) = F((μj - β′xi)) - F((μj-1 - β′xi)), j = 0,1,...,J (11)
Let F(.) equal the CDF of the logistic distribution. The log likelihood function is then
given by equation (12).
log L = Σi log Li = Σi log Prob(Yi = yi) (12)
where Yi = the theoretical random variable
and yi = the observed value of Yi for firm i
The first derivatives are:
f((μj-1 - β′xi)) - f((μj - β′xi))
∂log Li/∂β = ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ xi
F((μj - β′xi)) - F((μj-1 - β′xi)) (13)
where f(.) denotes the appropriate density Λ(1-Λ) for logistic. For convenience,
denote fj = f((μj- β′xi))
and Fj = F((μj- β′xi)),
and likewise for j-1. By convention, f-1 = F-1 = fJ = 0, and FJ = 1.
Then, ∂log Li/∂μj = (fj / (Fj - Fj-1))
and ∂log Li/∂μj-1 = -(fj-1 / (Fj - Fj-1)) .
These imply that,
∂log Li/∂μm = 0 if m = -1, 0, or J.
Explanatory variables can be readily incorporated into the mixed or random
parameter ordered logit model in the usual manner that they are incorporated into a
regression equation. Estimates are obtained for both the parameters associated with
each of the firm-specific variables, and the threshold parameters. The threshold
parameters indicate the extent to which the categories of the ranking scale are equally
spaced in the logit scale.18
All parameters are obtained by iterative maximum
likelihood estimation, using the Davidson-Fletcher Powell optimization method with

9. 9
the first derivatives used to define the variance matrix. Starting values are obtained by
ordinary least squares regression on a binary dependent variable. Sample cell
frequencies on the observed rankings are used to initially divide up the real line in
order to define the starting threshold values on a normalized scale. The lowest
threshold is normalized to zero. The preceding expressions are summed over all
outcomes. We use the BHHH estimator19
for the covariance matrix, so the second
derivatives are not computed.
III. METHOD
A Three-State Classification Model
This study models financial distress in three states:
State 0: non-failed firms;
State 1: insolvent firms. For the purposes of this study, insolvent firms are
defined as: (i) failure to pay Australian Stock Exchange (ASX) annual
listing fees as required by ASX Listing Rules20
; (ii) a capital raising
specifically to generate sufficient working capital to finance continuing
operations; (iii) loan default, (iv) a debt/total equity restructure due to a
diminished capacity to make loan repayments.21, 22
State 2: firms who filed for bankruptcy followed by the appointment of
liquidators, insolvency administrators or receivers.23
Sample Selection
We develop two samples for the purposes of model estimation and validation.
The estimation sample is based on firm financial distress data collected between 1996
and 2000. Over this period we collect a sample of failed firms (state 2) and a sample
of firms with solvency problems (state 1). Firms were observed to fail or have
solvency difficulties at different times over this period. We attempted to collect up to
five annual reporting periods of data24
on all firms in categories 0,1 and 2, unless
certain conditions described below were not met. The sample of nonfailed firms is
drawn over the same time period range as the firms in states 1 and 2, and the
proportion of failed to nonfailed firms sampled is approximately equal across each of
the years the data are collected. To avoid the backcasting problem noted by Ohlson
(1980), data are collected only from the financial statements already in the public
domain on the date the failure is first made known to the market.25
The same
procedure is followed for firms in state 1. To avoid over sampling problems and error
rate biases associated with matched pair designs (see e.g., Casey and Bartczak, 1985;
Gentry, Newbold and Whitford 1985; Jones 1987) we use a sample of failed and
nonfailed firms which better approximates actual fail rates in practice26
(Zmijewski,
1984). This produces a final useable sample of 2,838 firm years in the nonfailed state
0; 78 firms years in state 1; and 116 firm years in state 2.27
Following the approach of Joy and Tollefson (1975), a validation sample is
collected for the period 2001-2003 using the same definitions and procedures applied
to the estimation sample.28
This produces a final useable sample of 4,980 firm years
in the nonfailed state 0; and 119 and 110 firms years in states 1 and 2, respectively.
Checks are made to ensure that failed/insolvent firms identified in states 1 and 2 for
the estimation sample are not also included in the validation sample. Following a
standard practice in the literature, failed/insolvent firms in the validation sample are
also excluded from the estimation sample. The larger sample of nonfailed firms in the
validation sample reflects a significant increase in the number of new listings on the

10. 10
ASX over this period and the fact that the more recent financial distress data was
found to be relatively more complete than for the estimation sample.29
Only publicly listed firms on the ASX are included in the estimation and
validation samples. Furthermore, only firms who reported cash flow information
under requirements of the Approved Australian Accounting Standard AASB 1026
“Statement of Cash Flows” are included in both samples.30
Compliance with AASB
standards has been mandatory in Australia under the requirements of the existing
Corporations Act (1991) of the day, hence we could find very few instances of
noncompliance with AASB 1026. In a very small number of cases, firms are deleted
from both samples because no financial statement records could be found. Following
the approach of Ohlson (1980) no firm is deleted simply because it is newly or
recently listed, and some firms in both our samples only had one or two years of
financial reports. Consistent with Ohlson (1980), if a firm produced its annual report
after the announcement of failure, then its published financial report of the previous
reporting period is used. In the estimation sample, the average lead time between the
date of the previous annual report and the announcement of failure was approximately
11.2 months (and 10.4 months for validation sample) which is broadly consistent with
the lead time reported in other studies (Altman, 1968; Ohlson, 1980). However, the
variance was quite high, with one firm having a lead time of nearly 40 months. The
median lead time was 9.1 months for the estimation sample and 8.7 months for the
holdout sample.31
Table 1 shows the ‘lead time’ frequency distribution for the sample
of failed and insolvent firms used in the estimation sample.
Insert Table 1
With respect to the sample of insolvent firms, we employ the same data
collection procedures used for failed firms. The financial report prior to the
indication of the firm’s solvency problem is used for estimation purposes. Whether a
firm experienced a solvency problem (as defined in this study) is ascertained from the
analysis of the ASX’s Signal G releases.
For the estimation sample, financial statement data is collected on firms in
each of the three states from four major sources: (i) Aspect Financial Pty Ltd’s
Financial Analysis Database (2003) and DatAnalysis Database (2003) – two leading
Australian financial database sources which contains up to 15 years of historical data
on all listed companies in Australia (ii) Huntley’s Delisted Company Database (1993-
1999), which contains all delisted firms in Australia up until 1999; (iii) ASX Market
Comparative Analysis, 2003; and (iv) company financial statements collected from
the Australian Securities and Investment Commission (ASIC). For the validation
sample, all the financial data and failure statistics are generated from a customised
data feed provided commercially to the authors by AspectHuntley Pty Ltd.32
Dependent Variable
For the purposes of this study, states 0-2 are mutually exclusive categories,
this being a condition of the ordered mixed logit model. Furthermore, these states
represent an increasing severity of financial distress. State 2 is differentiated from
states 0-1 insofar as these firms are no longer going concerns. The financial health of
these firms deteriorated to the point where filing for bankruptcy was the only option
available to the firm. State 1 represents firms which are going concerns, but have
experienced financial difficulties serious enough to warrant being described as
‘financially distressed’. State 0 describes firms which are going concerns but have
shown no evidence of being financially distressed as defined in this study.
Differences in the financial condition of firms in each state is reflected in the

11. 11
descriptive analysis of results (see Table 2). Using a number of variables which
featured most prominently in the mixed logit and MNL analysis (Tables 3 & 4), Table
2 indicates a noticeable deterioration in the performance of firms across states 0-2 for
most variables.
Insert Table 2
Explanatory Variables
We test a number of financial variables used in research over the last three
decades (see e.g., Beaver, 1966; Altman, 1968; Altman, Haldeman and Narayan 1977;
Ohlson, 1980; Zemjewski, 1984; Casey and Bartczak 1985; Gentry, Newbold and
Whitford, 1985; Jones 1987). These financial measures include ratios based on cash
position; operating cash flow (CFO); working capital; profitability and earnings
performance; turnover; financial structure; and debt servicing capacity. Furthermore,
we test reported CFO rather than estimates of CFO. Hribar and Collins (2002)
document evidence that earnings management, capital market and other studies using
estimates of CFO (as opposed to reported CFO) are prone to significant measurement
errors which can potentially contaminate the empirical results of some of these
studies. Previous financial distress research testing the predictive performance of CFO
has invariably used estimates rather than reported CFO. This is not surprising
considering that many of these studies predate the introduction of SFAS 95 (e.g.,
Casey and Bartczak, 1985). However, in light of Hribar and Collins (2002), this
could also potentially explain why so few of these studies have found any supporting
evidence for the usefulness of CFO in distress prediction33
(see e.g., Neill, Schaefer,
Bahnson and Bradbury, 1991). Ratio measures based on reported CFO are the net
operating cash flow number extracted from company cash flow statements prepared
under AASB 1026.34
A correlation matrix of key variables found to be significant in the mixed logit
and MNL analysis is provided in the Appendix Table 1. An examination of the partial
correlations indicates generally weak correlations across most of these covariates,
suggesting that predictor variables all provide distinct and unique information.
We now briefly comment on the contextual variables. In contrast to previous
research, which has tended to be restricted to industrial or manufacturing firms (Jones,
1987), this study tests the predictive value of financial variables on four major sectors:
the old economy sector; the new economy sector; the resources sector and the
financial services sector. This classification approach has been adopted for a variety
of reasons: (1) it recognizes that industry sectors are structurally different and have
different financing, operating and investing characteristics that can undermine inter-
sector comparability and generalizability (Ohlson, 1980). We attempt to capture
sector-specific affects in our modeling in order to make determinations about the
generalizability of our results as well as assess the predictive value of our model to
specific sectors; (2) the classification approach gives explicit recognition to the
economic characteristics of Australian industry, particularly the emerging importance
of the New Economy sector in Australia over the past decade, which now amounts to
more than 60% of the market capitalization of the ASX (ASX Market Comparative
Analysis, 2003). Firms in the New Economy sector are classified according to the
ASX industry classification guidelines, outlined in the ASX Market Comparative
Analysis (2003).35
These are: (i) health and biotechnology; (ii) high technology; (iii)
internet firms; and (iv) telecommunications. Furthermore, in Australia, the resources
sector constitutes the largest and single most important export industry – nearly 30%
of all listed firms in Australia are resource companies (see ASX Market Comparative

12. 12
Analysis, 2003). The resources sector is classified by the ASX as: (i) gold
companies; (ii) other metals and (iii) diversified resources. Financial services are
defined by the ASX as banks and finance houses, insurance companies and
investment and financial services companies. Old economy firms are defined for the
purpose of this study as all firms not being in the new economy, resources and
financial services sectors;36
and (3) our classification scheme is sufficiently broad to
preserve a statistically sufficient sample size for each major sector, while maintaining
meaningful contextual information about the impact of industry sector in our analysis.
IV. EMPIRICAL RESULTS
Several mixed ordered logit models are estimated to identify the statistically
significant influences on the probability of a firm failing for various reasons or not
failing in specific years. Table 3 summaries the overall model system for both the
mixed logit model and MNL. Both models were developed from the full set of
variables in the data set, with some common variables occurring in both models as
well as some different effects (e.g., sales revenue to total assets for MNL and total
debt to total equity for mixed logit). The mixed logit has delivered a very good overall
goodness of fit. The log-likelihood has decreased from -2057 (assuming no
information other than random shares) to -776. However, for the MNL, the model fit
is significantly less impressive. The MNL log-likelihood has only decreased from -
2057 to -1972. Using a likelihood ratio test we can calculate the likelihood ratio as -
2*(1972-776) = -2392 at 8 degrees of freedom. This is chi-square distributed and at
any level of significance the mixed logit is a substantial improvement over MNL.
The mixed logit results in Table 3 indicate that some variables have a single fixed
parameter whereas other variables (four of them - cash resources to total assets, net
operating cash flow to total assets, total debt to total equity and cash flow cover) have
up to three parameters representing their role. What is impressive is a recognition that
unobserved heterogeneity as represented by the standard deviation parameters is
statistically significant for all four financial variables. If the analyst was to estimate a
simple multinomial or binary logit model, the opportunity to establish the role of the
mean and variance influence of a particular variable (through the structure of its
parameter space) would be denied. This is an important finding and recognition of the
amount of information loss that is caused by rigid model specifications. The ability to
capture important relationships through a random parameter specification has meant
that such information has not been assigned (incorrectly) to the IID random
component as exists for a standard logit model.
Insert Table 3
In addition we find that for total debt to total equity and cash flow cover, the
interaction with the new economy dummy variable has produced a contextual effect
suggesting that membership of the new economy has a differential influence on the
role of these variables to the failure outcome.
Although the interpretation of the significance of various variables is
important and possible in Table 3, the magnitude and direction of impact of each
explanatory variable on the probability outcome for each state of financial distress and
nonfailure requires us to derive the marginal effects to which we now turn.
Results: Marginal Effects
A direct interpretation of the parameter estimates reported in Table 3 is not
possible given the logit transformation of the outcome dependent variable required for
model estimation. We therefore provide the marginal effects, defined as the

13. 13
derivatives of the probabilities which have substantive behavioral meaning (Louviere,
Hensher and Swait, 2000). A marginal effect is the influence a one unit change in an
explanatory variable has on the probability of selecting a particular outcome, ceteris
paribus. This holds for continuous variables only. For dummy (1,0) variables, the
marginal effects are the derivatives of the probabilities given a change in the level of
the dummy variable and thus represent the influence of a change in level of the
variable upon the probability of choosing a given outcome, ceteris paribus.
The marginal effects need not have the same sign as the utility parameters.
Hence the statistical significance of a utility parameter does not imply the same
significance for the marginal effect.
∂Prob(yq = j)/∂x = Pj(βj - β ), β = ΣjPjβj. (defined below as δj ) (14)
Neither the sign nor the magnitude of δj need bear any relationship to those of βj. The
asymptotic covariance matrix for an estimator of δj would be computed using
Asy.Var.( δ
∧
j ) = Gj Asy.Var( β
∧
)Gj (15)
where β
∧
is the full parameter vector, Gj is a matrix of first derivatives used
to establish standard errors of betas,
and Asy.Var.(δ
∧
j ) = Σl Σm Vjl Asy.Cov.( β
∧
l,β
∧
m')Vjm', j = 0,...,J, (16)
where Vjl = (1(j=l) - Pl){PjI - δjx′} - Pjδlx′
and 1(j=l) = 1 if j=l, and 0 otherwise.
βj, equal to ∂log(Pj/P0)/∂x, has been suggested as an interpretation of the utility
parameters.37
Marginal effects in the mixed logit model are quite involved. Since there
is no meaningful conditional mean function to manipulate, we consider, instead, the
effects of changes in the covariates on the cell probabilities. These are:
∂Prob(cell j)/∂xi = (f(μj-1 - β′xi) - f(μj - β′xi)) × β (17)
where f(μj-1 - β′xi) is the logistic density, Λ(μj-1 - β′xi)(1-Λ(μj-1 - β′xi).
The marginal effects of each explanatory variable for each of the three
outcomes is shown in Table 4 for both mixed logit and MNL. These results suggest
that cash resources to total assets, net operating cash flow to total assets, working
capital to total assets and cash flow cover have a strong statistical impact on the
probability of each of the performance outcomes. Total debt to gross operating cash
flow and total debt to total equity are both statistically significant but their t values are
smaller than the other variables. It is noteworthy that all random parameters in the
mixed logit analysis have a consistent and logical sign across all levels of the response
variable. For example, net operating cash flow to total assets has a positive
coefficient for Y = 0 (increases in this ratio increase the probability of nonfailure) but
negative coefficients across the two financial distress categories. Likewise, total debt
to gross operating cash flow has a negative coefficient for Y = 0 (increases in total

14. 14
debt to gross operating cash flow reduce the probability of nonfailure) but positive
coefficients across the two financial distress categories, indicating that increases in
this value increase the probability of financial distress across each distress state. The
marginal effects for the MNL analysis does not produce logical or consistent signs for
all parameter estimates, even though nearly all the financial and contextual variables
are statistically significant as we found in the mixed logit analysis.
It is noteworthy that measures based on reported CFO produced statistically
significant parameters which were relatively stable across each of the three response
categories. One potential explanation for why our results differ from previous distress
research is suggested by Hribar and Collins (2002), who conclude that estimated CFO
is generally a poor proxy for reported CFO. The results could also be attributed, at
least in part, to methodological improvements introduced by the mixed logit model.38
In addition the membership of the new economy sector is particularly notable,
with the probability of failure increasing when a firm belongs to this sector (primarily
IT and technology firms). This result is likely to reflect the pronounced demise of the
technology sector in Australia in 2000 (and worldwide), and the subsequent financial
collapse of many of these companies. The influence is not uniform across all distress
classes with the marginal effects for most financial and contextual variables declining
sharply as we move to failure state 2 (outright failure). These results are intuitive,
reflecting differences in the financial positions that we would expect of firms in each
of the distress states. When a firm reaches more advanced stages of financial distress,
such as outright failure, the financial position of these firms would have deteriorated
to a more significant extent than firms classified ‘0’ or ‘1’. Likewise, we would
expect on average firms in ‘2’ to have deteriorated to a more significant extent than
‘1’ (see Table 2). Hence, significantly larger changes in the magnitude of financial
variables, such as operating cash flow, would be required to have any impact on
changes to the outcome probability, particularly for firms in the highest distress state
(‘2’). Another interpretation of the declining marginal effects across response
categories is that once firms enter into structurally ‘irreversible’ decisions, such as
liquidation, financial statements, including cash flows, assume less overall marginal
significance.
Insert Table 4
V. FURTHER CONSIDERATIONS
The analysis in previous sections incorporated the entire data set of 5 annual
reporting periods per firm in the estimation sample. If there was an observed failure it
occurred after the last reporting period of data collected in this study. Each observation
point on each firm was treated as an independent observation. This suggests two further
foci of investigation: (i) a model in which the distress level is the dependent variable but
in which the explanatory variables in all 5 reporting periods are linked through a
structure such as a geometric distributed lag and (ii) a cross section model estimated on
only the last reporting period of financial data before failure.
We estimate a model for each alternative specification and report the findings
below. The first modification is to recognize the possibility that performance leading
up to failure/insolvency has an accumulating impact on the response outcome. To
investigate this we propose a geometric distributed lag of the moving average form:39
β(1-λ)(xt + λXt-1+λ2
Xt-2+….)

15. 15
Given a known value for λ, the function is linear in the parameters. We
undertook a grid search on λ (over the range 0 to 2), selecting the value which
maximizes the log-likelihood of the model. The best fit λ was 0.23. The model in
which we allow all financial variables to be the moving average form of geometric
distributed lag are summarized in Table 5 below.
Insert Table 5
Comparing Table 5 with Table 3 (the initial model in which we pool all
reporting periods and treated each as an independent cross section) it can be seen that
none of the fixed or random parameter variables are significant (other than the
constant term). Marginal effects are not reported on the variables reported in Table 5
as none were found to be significant, suggesting that these variables have no affect on
the probability of the response outcome.
We also estimated a mixed logit model for the last reporting period in the data
set. Again, this specification produced a model statistically and behaviorally inferior
to the one reported in Table 3. All but one random parameter variable had a
statistically significant standard deviation, and none of the fixed parameters were
significant. Once again, the marginal effects are not reported here, as no variables
were found to be statistically significant.
These results suggest that the pooled data model reported in Table 3 is
preferred to a cross sectional model. Further, the independence of years specification
for our model reported in Table of 3 outperforms a geometric lagged model
specification which captures time dependencies in the explanatory variables.
Model Validation: Probability Outcomes
The overall predictive performance of the mixed logit model and MNL can be
investigated by deriving the predicted probabilities for each firm for each outcome on
our validation sample. In deriving the probability outcomes for the mixed logit model we
have to recognize that some explanatory variables are a composite function of a mean
parameter, a distribution around the mean and decomposition of the mean by some
contextual effect (in our case it is the new economy effect). Each individual firm is
‘located’ in parameter space on the normal distribution for four financial variables (cash
resources to total assets, net operating cash flow to total assets, total debt to total
equity and cash flow cover). The precise formulation used to derive the contribution to
relative utility of each outcome is:
Preference Distribution for cash resources to total assets
= -.06082089+.08277721*normal density
Preference Distribution for net operating cash flow to total assets
= -.01711009+.01229528* normal density
Preference Distribution for total debt to total equity
= .00090835-.00766415*new_econ+.00509659* normal density
Preference Distribution for cash flow cover ratio
= -.00519331+.008199972*new_econ+.00519331*normal density
where normal densities have mean zero and unit standard deviation. For each
individual we randomly draw a location on the distribution given the mean and
standard deviation and derive their overall contribution to ‘relative utility’. This is

16. 16
derived a repeated number of times and averaged per firm. We calculate the set of
three outcome probabilities using the formula set:
P0 = Phi(-Xb)
P1 = Phi(Mu1-Xb) - P0
P2 = 1 - P1-P0
Importantly we focus on a sample enumeration methodology which recognizes
that the estimated model is based on a sample drawn from a population and the
application of the model must preserve the full distribution of information obtained
from the model system (Horowitz and Louviere 1993). This includes the outcome
probabilities. That is, given that the focus is on a sample drawn from a population of
firms, the parameterization used to establish the probabilistic outcomes is a
representation of the preference profile of a sample. Thus it is necessary to aggregate
the probabilities associated with each outcome across the entire sample to obtain the
predicted outcome values (i.e. the absolute number of firms predicted to be in each
outcome category or outcome category shares). However, while the practice of
identifying the outcome probability associated with each outcome for each firm is
common and supported practice in accounting research, the use of threshold
probabilities to assign firms to specific predicted outcomes is not supported by a
correct application of outcome models. The literature on discrete choice analysis (e.g.,
Hensher, Rose and Greene, 2005) shows that imposing a cut-off allocation rule such
as assigning an outcome probability of 1.0 to the alternative whose model derived
outcome probability is greater than a specific value (e.g., 0.5 for a binary outcome) is
arbitrary and not consistent with the intent of the model (whether binary, MNL or
mixed logit).40
Implementing a sample enumeration strategy on our hold out sample, we can
compare the predictive performance of mixed logit and MNL. Table 6 compares the
forecasting accuracy of both models on pooled data, and on data 1, 3 and 5 reporting
periods prior to failure. The overall results indicate that mixed logit has substantially
better predictive accuracy than MNL across the pooled results, and in all reporting
periods prior to failure. It is noted that the MNL is particularly poor in classifying
financial distressed firms. For instance, in predicting state ‘2’ or outright failure, the
MNL model’s best performance is only 5% accuracy based on the pooled
observations and its best result is 6.4% accuracy three reporting periods prior to
failure. The MNL performs better in predicting state 1 (insolvent firms), though the
accuracy rate is only 24% on the pooled data with a peak accuracy rate of 29% five
reporting periods from the public announcement of insolvency problems. In contrast,
mixed logit predicts state ‘2’ with 95% accuracy based on the pooled observations,
and is 95% accurate up to three reporting periods prior to failure, with accuracy rates
falling to 78% five reporting periods from failure. The model performs very well on
predicting nonfailures, and the overall accuracy for predicting state 1 is also
impressive, with over 90% accuracy three and five reporting periods prior to the
public announcement of insolvency problems. Mixed logit has an overall forecasting
accuracy (in terms of predicting accurately across all distress states) of 99.16% on the
pooled data, 98.73% from the last reporting period, 99.6% accuracy from the third
reporting period, and 98.9% accuracy from the fifth reporting period.
Insert Table 6
VI. CONCLUSIONS

17. 17
In this paper we seek to advance the state of art in financial distress modeling
by explicating the theoretical and empirical value of the mixed logit model. With
few exceptions, the financial distress literature to date has employed relatively
primitive binary logistic models, and in a few cases a rudimentary MNL approach.
The literature has not kept abreast of considerable theoretical and empirical
developments made in discrete choice modeling in recent years. The focus of recent
discussions in the econometric literature has been on improving the behavioral realism
of discrete choice models by relaxing the rigid assumptions associated with IID error
terms in a manner that is conceptually enriching, computationally tractable and
practical. The mixed logit model is one approach that allows the analyst to relax the
very rigid assumptions associated with IID, and allows a meaningful interpretation of
the role of the mean and variance influence of a particular variable through the
structure of its parameter space. Such refinements hold much promise in this field of
research. The results of this study confirm the superiority of mixed logit over
standard approaches such as MNL. After adjusting for the number of parameters,
mixed logit produced a substantially improved model-fit compared with MNL.
Furthermore, the out-of-sample forecasting accuracy of the mixed logit design was
much superior to multinomial logit.
There are several possibilities for future research arising from this study.
First, the benefits of mixed logit are by no means confined to the financial distress
literature, but can be usefully applied to many other areas of accounting research.
Mixed logit is a highly flexible model and can be applied to a variety of data sources
and structures where use of a discrete dependent variable is contemplated, including
cross-section, time-series, panel and experimental data. Elliot and Kennedy (1988)
and Kennedy, (1992) note several applications in accounting research for use of a
discrete dependent variable with multiple alternatives. Further, Leclere (1999) cites at
least 21 studies in the leading accounting journals which use an n-chotomous
dependent variable such as multinomial logit or ordered probit. Studies cover such
diverse fields as predicting management’s choice of accounting method, pension
reporting, loan classifications, bond ratings, lobbying positions, disclosure, capital
structure analysis and auditing switching (see also Kennedy, 1992).41
As pointed out
by Hensher and Greene (2003), estimation of an MNL model is the first step in
establishing a more robust mixed logit specification. Such research domains can be
readily extended and enriched using a mixed logit framework. Another field where
mixed logit could have considerable application is in the broad field of human
judgment and decision making, such as cognitive processes and judgment tasks
(Libby and Lipe, 1992). As with other areas of accounting research, behavioral
accounting research has tended to be narrowly focused on general linear models, such
as ANOVA and multiple regression. However, there are now comprehensive
illustrations available of the empirical benefits of nonlinear models such as mixed
logit in the context of experimental research and behavioral decision making
(Louviere, Hensher and Swait, 2000, Hensher, Rose and Greene, 2005).
A second direction for future research could be to evaluate the mixed logit
model against other innovative techniques used in credit risk modeling. One other
modeling approach becoming more widely used in this literature is structural equation
modeling (Duffie and Singleton, 2003). SEM focuses on relationships among latent
constructs rather than on the manifest variables used to measure these constructs. It
was shown in Section II that mixed logit performs well in capturing latent effects by
incorporating observed and unobserved heterogeneity in parameter estimates. Given
some conceptual similarities between mixed logit and SEM, a formal comparison of the

18. 18
forecasting performance of these models is needed if either approach is to be relied upon
in future research.42
Extending the current model to include other distress categories such as
distressed mergers (Clark and Ofek, 1994) would provide a further opportunity to
examine the empirical usefulness of mixed logit in forecasting across multiple states of
financial distress.
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1
A related problem is that most studies to date have modeled failure as a simplistic binary
classification of failure or nonfailure (see Jones, 1987). This has been called into question, one reason
being that the strict legal concept of bankruptcy may not always reflect the underlying economic reality
of corporate financial distress (Lau, 1987; Delaney, 1991). The two state model can also conflict with
underlying theoretical models of financial failure and limits the generalizability of empirical results
(Scott, 1981; Bahnson and Bartley, 1992; Hill et al., 1996). Others have argued that the practical risk
assessment decisions by lenders and other parties normally cannot be reduced to a simple pay off space
of failed and nonfailed (Ward, 1994; Ohlson, 1980). Lau (1987) improved on the methodology of
dichotomous prediction models by using a five-state model, but the study has a number of limitations.
For instance, the MNL approach selected is not robust to violations of the IID assumption and some of
the distress states are hampered by unacceptably small sample sizes. Using a holdout sample, Lau’s
model predicted the nonfailure state with reasonable accuracy, but was less impressive in predicting
the different states of financial distress.
2
The estimated standard logit model herein is referred to in the literature as multinomial logit (MNL)
in contrast to conditional logit. Although there are some interpretative differences between conditional
logit and MNL, they have the same functional form. An MNL model can be described by a situation
where the main influences on the choice outcome are the characteristics of the observations (i.e., the
firms). This is in contrast to conditional logit where the primary influences are the attributes of
alternatives (i.e., financial ratios). Our data are described only by the characteristics of firms whose
attributes do not vary across outcomes (although such variation exists across the sample). For example,
an individual firm’s total debt to total equity ratio (attribute) does not vary across failure outcomes for
that individual firm, but it does for the sample as a whole. We have chosen an ordered specification of
the outcomes within which it is possible to allow a single characteristic to impact on all outcomes,
since the notion of an alternative is not applicable. All model specifications are members of the general
family of random utility models (RUM).
3
In advocating the mixed logit model, we acknowledge certain limitations of “static” or single period
distress models which draw on multiple-period distress data (Shumway, 2001; Leclere, 2000). Much of the
literature is of this genre (see Altman, 2001 for a survey). Shumway (2001) concluded that static models
can result in inconsistent and biased estimates. Firm characteristics can change over time, but such
problems are not expected to be pronounced in this study given the relatively short time frame in which our
estimation sample is drawn. While a hazard analysis is not deemed appropriate to the modeling focus of
this study, we did meet some of Shumway’s (2001) concerns by undertaking an assessment of a
geometric lag specification to test time dependencies in our estimation sample (see Section V).
4
In the theory of discrete choices, an essential departure from traditional microeconomic theory
originated in the now classical works of Lancaster (1966, 1971) and Rosen (1974). This is the postulate
that utility is derived from the properties or characteristics of things, rather than the goods per se.
Discrete choice theory incorporates the work of the standard Lancaster-Rosen model, but modifies this
approach further by assuming that individuals maximize their utility on the basis of their perceptions of
characteristics, rather than the characteristics per se (see Louviere, Hensher and Swait, 2000 for an
overview).
5
Random utility theory (RUT) is a very general theory of how the analyst represents the preferences of
agents where elements of information (known to the agents) are not observed by the analyst. While
RUT has gained particular recognition within discrete choice theory in recent years, RUT is not
restricted to choice theory and can be implemented in a wide range of possible decision contexts.
6
The modeling of behavioral heterogeneity has been important in many fields of inquiry, including
recent economics literature (see Grandmont, 1987, 1992; Kneip, 1999).
7
In addition to fixed parameters, mixed logit models include estimates for the standard deviation of
random parameters, the mean of random parameters and the heterogeneity in the means (discussed further
below and in Section IV).
8
The moments of an individual firm’s coefficient cannot be observed from a single data point, but
rather estimated by assuming a distribution for the coefficients of any particular attribute across all
firms in the sample (see Train, 2003, 262-263).
9
A fixed parameter essentially treats the standard deviation as zero such that all the behavioral
information is captured by the mean. Standard logit models assume the population of firms is
homogeneous across attributes with respect to domain outcomes (i.e., levels of financial distress). For

22. 22
instance, the parameter for a financial ratio such as total debt to total equity is calculated from the sample of
all firms (thus it is an average firm effect), and does not represent the parameter of an individual firm.
10
A variety of studies have now demonstrated the superior forecasting accuracy of mixed logit
compared to standard logit (see, as examples, Brownstone, Bunch and Train; 2000; Revelt and Train,
2000; Sándor and Wedel, 2002).
11
Based on Hensher, Rose and Greene (2005).
12
Extreme value type 1 (EV1) is a commonly used distribution in discrete choice analysis. The phrase
‘extreme value’ arises relative to the normal distribution. The essential difference between the EV1 and
normal distributions is in the tails of the distribution where the extreme values reside. With a small
choice set such as two alternatives this may make little difference because the resulting differences in
the choice probabilities between the normal and EV1 is usually negligible. When there are more than
two alternatives, however, there can be a number of very small choice probabilities. As a result,
differences between the distributions can be quite noticeable (see Hensher, Rose and Greene, 2005).
13
The proof in McFadden and Train (2000) that mixed logit can approximate any choice model
including any multinomial probit model is an important message. The reverse cannot be said: a
multinomial probit model cannot approximate any mixed logit model, since multinomial probit relies
critically on normal distributions. If a random term in utility is not normal, then mixed logit can handle
it and multinomial probit cannot.
14
The mixed logit model has a likelihood surface that is capable of producing local optima in contrast
to a single unique global optimum from MNL. Using the MNL parameter estimates as starting values
produces a global solution since it begins the gradient search at a location of the nonlinear surface that
tends to be the best starting location for determining the global optimum.
15
By construction, SPi is an unbiased estimate of Pi for any R; its variance decreases as R increases. It
is strictly positive for any R, so that ln (SPi) is always defined in a log-likelihood function. It is smooth
(i.e., twice differentiable) in parameters and variables, which helps in the numerical search for the
maximum of the likelihood function. The simulated probabilities sum to one over alternative outcomes.
Train (2003) provides further commentary on this.
16
The simulation method was initially introduced by Geweke (and improved by Keane, McFadden,
Börsch-Supan and Hajivassiliou - see Geweke et al 1994, McFadden and Ruud 1994) to compute random
variates from a multivariate truncated normal distribution. Although it fails to deliver unbiased multivariate
truncated normal variates (as initially suggested by Ruud and detailed by Börsch-Supan and Hajivassiliou
(1990), it does produce unbiased estimates of the choice probabilities. The cumulative distribution function
in their research is assumed to be multivariate normal and characterized by the covariance matrix M. The
approach is quick and generated draws and simulated probabilities depend continuously on the parameters
β and M. This latter dependence enables one to use conventional numerical methods such as quadratic hill
climbing to solve the first order conditions for maximizing the simulated likelihood function (equation 5)
across a sample of q=1,…,Q firms; hence the term maximum simulated likelihood (MSL) (Stern 1997).
17
The most common application of mixed logit involves unordered alternative outcomes.
18
The threshold parameters are essentially constants that redefine the utility scale across the set of
outcomes to recognize that the utility scale is nonlinear between outcomes.
19
This is the Berndt, Hall, Hall, Hausman algorithm which uses the sums of squares and cross products
of the first derivatives to obtain the asymptotic covariance matrix of the maximum likelihood estimator
which contains the parameters of interest.
20
Occasionally we found a financially healthy firm which failed to pays its listing fees. These firms
were removed from the sample.
21
Three firms fell into category (iv). These companies were carefully examined to ensure that their
financial condition was well differentiated from state 2 (outright failure). For instance, it was found that
none of these firms involved creditors granting concessions (such as forgiven principal or reduced
interest rates). In two cases creditors converted debt to total equity to alleviate the liquidity problems of
the firm. In one case, the firm sought to refinance current debt commitments because of short term
liquidity problems.
22
It is noted that other studies (e.g., Lau, 1987; DeAngelo & DeAngelo, 1990) have used reduction in
dividends as a measure of financial distress. Lau (1987) for example, examined firms with a greater
than 40% dividend per share drop, after a period of three years of stable dividend record. Following
Lau’s approach, only a small number of firms (eight in total) satisfied this requirement. More than half
of these firms were not associated with any degree of financial distress, and the other half were already
absorbed into the other distress categories. As acknowledged by Lau (1987) dividend reductions can
be associated with reasons other than financial distress, such as capital investment activities.

23. 23
23
Included in our sample are three principal forms of bankruptcy proceeding available under the
legislative provisions of the Australian Corporations Act (2001): (i) voluntary administration (first
introduced in Australia in June 1993 under the Corporate Law Reform Act, 1992); (ii) liquidation and
(iii) receivership. Voluntary administration has similarities with Chapter 11 provisions in the US,
where the company is effectively given a period of time or “breathing space” to reorganize and/or
reconstruct. Under Australian voluntary administration laws, once appointed the insolvency
administrator has a limited period (28 days) to assess the company and recommend to the creditors
whether the company should be wound up or enter into a deed of arrangement (this is a contract that
binds the company and creditors and includes such issues as the order by which creditor claims are to
be settled from sale of assets). If deed of arrangement stage is not reached, the legislation provides for
an automatic transition to liquidation. With respect to (ii) liquidation, there are essentially two types of
winding up procedure available: a creditors’ voluntary winding up (decided by special resolution of the
company) and a court winding up. In the case of (iii) receiverships, the Corporations Act (2001)
provides that a secured creditor, in the event of a firm’s insolvency can appoint a receiver (or a receiver
and manager) to recover outstanding claims against the company. We note that the great majority of
our failed firm sample fell into categories (i) and (ii).
24
The term ‘reporting periods’ is more accurate than ‘years’. For instance, the last reporting period
before failure is not strictly speaking one year before failure. For some firms, the last reporting period
may be only 3 or 4 months before failure, and for others it might be longer. Likewise, 5 reporting
periods before failure may not necessarily equate to exactly 5 years before failure.
25
The backcasting problem is where a company files for bankruptcy at some point in time after the
fiscal year date, but prior to publicly releasing the financial statements. All company financial
statements in Australia, including the date of their release to the market, can be checked on the ASX’s
Signal G. Signal G disclosures are regulated by the ASX Listing Rule 3.1 which identifies the types of
information which Australian companies must disclose to the market on a timely basis. Examples include:
the appointment of a receiver or liquidator; information relating to mergers and takeovers; capital
reorganizations; loan defaults; share issues; failure to pay listing fees and any other information which
could have a material affect on the value of a company’s securities (see also Sections 674 and 675 of the
Corporations Act, 2001). Information for release to the market must be lodged by companies with the
ASX’s company announcement office (CAO), which is then immediately released to the market under
Signal G (see Explanatory Note, ASX Listing Rules, Chapter 3, 97,575; and Chapter 15 of the listing
rules). Because all Signal G market releases are electronically dispatched (in ‘real time’), an accurate
determination could be made whether a firm’s financial statements were released before or after the
announcement of failure.
26
Our own database indicates an average historical failure rate in Australia of between 2-3% over the
entire sample period 1996-2003.
27
We screened the nonfailed sample to assess whether there were nonevent (i.e nonfailed) companies
that possessed possible characteristics of insolvency or failure that were not captured by the specific
indicators defined in states 1 and 2. Distressed firms subject to a merger or takeover is one such
example (see Clark and Ofek, 1994). Clark and Ofek (1994) define a distressed merger/takeover
according to any or all of the following factors: poor stock price performance; financial distress, such
as severe deficiencies in working capital to fund continuing operations; corporate reorganizing in order
to rationalize non-performing key assets or the business as a whole; high turnover in directors and
senior management. We identified 93 firm years from the estimation sample, and 76 firm years from
the validation sample which satisfied the Clark and Ofek definition of a distressed merger. These firm
years were removed from the estimation and validation samples accordingly.
28
Joy and Tollefson (1975) note that holdout samples drawn from the same time period as the original
sample will not provide an intertemporal validation, and hence cannot provide an adequate test of a
model’s predictive ability.
29
The validation sample was generated from a single customized data feed provided commercially to
the authors by AspectHuntley Pty Ltd. The data feed had quite complete records on firm failures and
delistings, including nonfailed company records between 2001 and 2003. The estimation sample had to
be drawn from several databases, resulting in a number of incomplete records which had to be removed
from the final sample.
30
AASB 1026 was issued by the AASB in 1991, with a mandatory operative date of June 1992. In
most respects, AASB 1026 is very similar to the requirements of SFAS 95. The major difference is that
AASB 1026 requires the direct method of reporting operating cash flow (see paragraph 6.2 of AASB
1026).

24. 24
31
A typical example is the telecommunications company One.Tel, which announced failure to the
market on May 30, 2001, but its last annual report was released to the market on the September 12,
2000, a lead time of over 8.5 months.
32
AspectHuntley was created in 2003 from a merger of two leading data providers in Australia –
Aspect Financial Pty Ltd and Huntley’s Financial Services.
33
The value relevance of reported CFO relative to estimated CFO has been confirmed in other studies –
see e.g., Chang, Liu and Schaefer (1997).
34
Except for the total debt to gross operating cash flow ratio (gross operating cash flow is defined
under AASB 1026 as gross receipts from customers less payments to suppliers).
35
Further industry statistics below are all based on the ASX Market Comparative Analysis, 2003.
36
Industries include: Alcohol & Tobacco; Building Materials; Chemicals; Developers & Contractors;
Diversified Industrials; Energy; Engineering; Food & Households; Infrastructure & Utilities; Media;
Miscellaneous Industrials; Paper & Packaging; Property Trusts; Retail; Tourism & Leisure; and
Transport.
37
See W.H. Greene, Econometric Software (2003) Limdep 8 Econometric Modeling Guide Volume 2,
Chapter 19, Econometric Software Inc., New York and Sydney.
38
A direction for future research is to formally test various CFO estimates and reported CFO in the
context of financial distress prediction.
39
Except for finance, resource and new economy.
40
Applying a model estimated on a sample tends to produce less reliable firm-specific predictions than
estimating a model based on data for a single firm. In reality, there is usually insufficient data to
estimate a unique model for each individual firm. Hence, we draw on samples which provide us the
richness to predict at the sample level. Even with the mixed logit model, which provide estimates for
firm specific parameters, coefficients for the standard deviation of random parameters are calculated
from a distribution of all coefficients in the sample. Hence, inferences can only be validly made at the
sample level. By implication, one can still obtain firm-specific predictions, but they are relatively
limiting.
41
Some UK studies have used multinomial logit designs in the context of predicting the likelihood of
mergers and takeovers (Powell, 1997) and in assessing the determinants of audit reports (Ireland,
2003).
42
Future research can also evaluate the performance of the mixed logit against other widely used advanced
discrete outcome models, such as multinomial nested logit. The nested logit retains the advantages of a
close ended form but is more flexible than MNL in dealing with the restrictive IID condition because
through partitioning (or nesting), potential differences in sources of unobserved heterogeneity can be
investigated. However, nested logit only relaxes the IID condition to some extent. We would only totally
relax this condition if we allowed all variances and covariances to be different, up to identification, as
in the case of the mixed logit model.

25. 25
TABLE 1
Lead Times Between Date of Last Published Annual Report and Date of Failure Announcement
Months 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 12-13 13-14 14-15 15-16 16-17 17-18 18-19 19-20 >20
# of
Reports
- 2 3 2 2 2 1 4 1 1 3 - 2 - 1 2 - 2 - 6
Mean: 11.2 months; Median 9.1 months, Mode 8 months.
Lead Times Between Date of Last Published Annual Report and Date Solvency Problem First Identified
Months 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 12-13 13-14 14-15 15-16 16-17 17-18 18-19 19-20 >20
# of
Reports
1 3 3 2 2 4 2 2 - 2 2 1 2 - - - - - - -
Mean: 6.24 months; Median 6.4 months, Mode 6.5 months.

26. 26
TABLE 2
Descriptive Statistics for Significant Covariates in Distress States 0-2
Variables
Distress
State N Mean Std. Dev. Min. Max.
Net operating cash flow to
total assets Nonfailed 2838 -1.26 26.33 -404.76 256.64
Insolvent 78 -5.12 52.17 -168.17 263.77
Failed 116 -17.14 37.53 -193.68 21.87
Cash resources to total
assets Nonfailed 2838 19.08 25.10 0.00 177.00
Insolvent 78 7.75 32.40 0.00 99.84
Failed 116 1.49 32.07 0.00 98.65
Cash flow cover Nonfailed 2838 8.24 79.65 -1014.99 1134.00
(net operating cash flow to
annual interest payments) Insolvent 78 -1.61 79.39 -475.00 180.23
Failed 116 -25.05 125.48 -988.00 101.35
Sales revenue to total
assets Nonfailed 2838 65.96 104.59 0.00 1734.6
Insolvent 78 55.04 78.47 0.00 332.23
Failed 116 54.58 70.49 0.00 327.76
Total debt to total equity Nonfailed 2838 42.69 72.89 0.00 698.87
Insolvent 78 53.64 97.84 0.00 511.90
Failed 116 97.30 134.38 0.00 616.65
Total debt to gross
operating cash flow Nonfailed 2838 4.21 13.91 0.00 229.52
Insolvent 78 6.16 16.99 0.00 137.15
Failed 116 13.82 31.08 0.00 185.69
Working capital to total
assets (wc = current assets
– current liabilities) Nonfailed 2838 6.12 28.13 -850.67 323.00
Insolvent 78 0.77 42.01 -197.40 167.00
Failed 116 -22.21 43.43 -215.26 32.06

27. 27
TABLE 3
Fixed and Random Parameter Estimates and Model-Fit Summary for Final Mixed Logit and Multinomial
Logit Models (100 Halton draws)
Variablesa
Mixed Logit Parameter
Estimates (t-values)
MNL Parameter
Estimatesb
(t-values)
Fixed parameters:
Constant -2.85 (-39.2) -2.6703 (-33.75)
Total debt to gross operating cash flow 0.00895 (4.08) -.001654 (-6.7)
Working capital to total assets -0.0119 (-10.765) 0.00186 (4.62)
Resources sector (1,0) -0.5063 (-3.032) nsc
New economy sector (1,0) 0.725 (4.101) 0.40586 (3.12)
Finance sector (1,0) ns 0.33873 (2.38)
Sales revenue to total assets ns -.00095 (-3.06)
Cash resources to total assets - 0.002186 (4.98)
Net operating cash flow to total assets - -.00091 9-2.87)
Total debt to total equity - ns
Cash flow cover - 0.00042 (3.8)
Random parameter means:
Cash resources to total assets -0.0608 (-7.65)
Net operating cash flow to total assets -0.0171 (-9.21)
Total debt to total equity 0.0009 (2.68)
Cash flow cover -0.0051 (-7.93)
Standard deviations of random
parameters:
Cash resources to total assets 0.0827(13.173)
Net operating cash flow to total assets 0.0122 (6.104)
Total debt to total equity 0.00509 (12.66)
Cash flow cover 0.0048 (6.836)
Heterogeneity in means:
Total debt to total equity*New_Econd
-0.0076 (-5.75)
Cash flow cover*New_Econ 0.00819 (5.287)
Threshold Parameters:
μ (0 to 1) Mu(0) 0 0
μ (1 to 2) Mu(1) 1.2611 (10.1) 1.0398 (16.6)
Log-likelihood at zero -2057.46 -2057.46
Log-likelihood at convergence -776.17 -1971.95
Sample size 2838 2838

28. 28
TABLE 3 (Continued)
Descriptive Statistics
Variable Mean Std.Dev. Minimum Maximum
New economy sector 0.165 0.371 0.000 1.000
Finance sector 0.138 0.345 0.000 1.000
Resources sector 0.204 0.403 0.000 1.000
Net operating cash flow to
total assets -2.159 26.910 -404.760 256.640
Cash flow cover 7.218 81.220 -1014.986 1134.000
Cash resources to total
assets 19.050 25.124 0.000 177.000
Total debt to gross
operating cash flow 4.375 14.367 0.000 229.520
Total debt to total equity 56.836 152.620 0.000 698.870
Working capital to total
assets 3.751 77.675 -850.000 323.000
Sales revenue to total
assets 66.050 104.070 0.000 1734.600
a
The full list of financial and contextual variables examined in the analysis included: net operating cash flow to total assets;
net operating cash flow to sales revenue; cash flow cover (net operating cash flow to annual interest payments); total debt to
gross operating cash flow (where gross operating cash flow equals total receipts from customers minus payments to
suppliers); two annual periods of negative CFO, coded 1 = yes; 0 = no; three annual periods of negative CFO; coded 1 equal
yes; 0 = no; cash resources (cash, deposits and marketable securities) to total assets; cash resources (cash, deposits and
marketable securities) to current liabilities; current assets to current liabilities; working capital (current assets – current
liabilities) to total assets; total debt to total equity; total liabilities to total equity; total debt to total assets; total liabilities to
total assets; market value of equity to book value of debt; interest cover (reported EBIT to annual interest payments);
reported EBIT to total assets; return on equity (net profit after tax to total equity); return on assets (net profit after tax to total
assets); annual growth in sales revenue; sales revenue to total assets; retained earnings to total assets; natural log of total
assets (a control variable for size); new economy sector (if a new economy sector firm coded 1, 0 otherwise); resources
sector (if a resources sector firm coded 1, 0 otherwise); old economy sector (if an old economy sector firm coded 1, 0
otherwise); finance sector (if a finance sector firm coded 1, 0 otherwise).
b
Note that all MNL parameters are fixed.
c
ns = not significant
d
Interaction of total debt to total equity and the new economy sector variable (coded 1 if a new economy firm, 0 otherwise).

29. 29
TABLE 4
Marginal Effects for Significant Covariates in Final Mixed Logit and Multinomial Logit Models
Panel A: Mixed Logit
Prob (Y=0) Prob (Y=1) Prob (Y=2)
Variables (t-values) (t-values) (t-values)
Total debt to gross operating cash flow -.00029 (-4.053) .0002 (4.056) .000087(3.800)
Working capital to total assets .00039 (9.24) -.00027 (-9.3) -.00011 (-9.54)
Resource sector (1,0) .01409 (4.05) -.0099 (-6.74) -.0041 (-5.4)
New Economy sector (1,0) -.0315 (-214.15) .021 (9.88) .009 (14.3)
Cash resources to total assets .00199 (9.05) -.0014 (-9.5) -.00059 (-11.17)
Net operating cash flow to total assets .00056 (8.73) -.00039 (-8.77) -.00016 (-8.31)
Total debt to total equity -.0000298 (-2.59) .000021 (2.60) .0000088(2.43)
Cash flow cover .00017 (7.66) -.00011 (-7.69) -.00005 (-7.2)
Panel B: MNL
Prob (Y=0) Prob (Y=1) Prob (Y=2)
Variable (t-values) (t-values) (t-values)
Total debt to gross operating cash flow .0001006 (6.32) -.0000618 (-6.37) -.0000387 (-6.16)
Working capital to total assets -.000113 (-4.48) .0000695 (4.50) .0000436 (6.16)
Sales revenue to total assets .0000575 (3.02) -.000035 (-3.0) -.000022 (-3.31)
Finance sector (1,0) -.0234 (-8.24) .01426 (4.40) .00914 (4.74)
New Economy sector (1,0) -.02847 (-11.5) .01733 (5.19) .011146(15.41)
Cash resources to total assets -.0001331 (-4.77) .0000818 (4.80) .0000512 (4.95)
Net operating cash flow to total assets .00005511 (2.82) -.0000338(-2.83) -.0000212 (-2.82)
Cash flow cover -.0000256 (-3.3) .0000157 (3.3) .0000098 (3.8)

30. 30
TABLE 5
Fixed and Random Parameter Estimates and Model-Fit Summary for
(Distributed Lag) Mixed Logit Model (100 Halton draws)
Variables
Mixed Logit Parameter
Estimates (t-values)
Fixed parameters:
Constant -5.85 (-4.613)
Resource sector (1,0) -26.32 (.000)
New Economy sector (1,0) -16.732 (-.091)
Finance sector (1,0) -39.77 (-.004)
Random parameter means:
Cash resources to total assets -0.94808 (-.878)
Net operating cash flow to total assets -0.0679 (-1.155)
Total debt to total equity -.0036 (-.831)
Cash flow cover -.099 (-3.305)
Working capital to total assets -.0646(1.762)
Total debt to gross operating cash flow .0067(.09)
Standard deviations of random parameters:
Cash resources to total assets 0.056 (.07)
Net operating cash flow to total assets 0.004 (.08)
Total debt to total equity 0.0004 (.316)
Cash flow cover 0.0001 (.005)
Working capital to total assets 0.001 (.069)
Total debt to gross operating cash flow .0009(.037)
Log-likelihood at zero -166.46
Log-likelihood at convergence -14.5
Sample size 1515

31. 31
TABLE 6
Forecasting Performance of Final Mixed Logit and Multinomial Logit Models across Distress States 0-2
Pooled Data (Reporting Periods 1-5)
Nonfailure (0) Insolvent (1) Outright Failure (2)
Model Actual Predicted Actual Predicted Actual Predicted
Mixed 95.60% 95.90% 2.28% 1.84% 2.10% 2.20%
MNL 95.50% 99.30% 2.34% 0.58% 2.15% 0.11%
Last reporting period prior to failure
Nonfailure (0) Insolvent (1) Outright Failure (2)
Model Actual Predicted Actual Predicted Actual Predicted
Mixed 95.70% 96.20% 2.37% 1.71% 1.91% 2.02%
MNL 95.70% 99.37% 2.37% 0.59% 1.91% 0.02%
Third reporting period prior to failure
Nonfailure (0) Insolvent (1) Outright Failure (2)
Model Actual Predicted Actual Predicted Actual Predicted
Mixed 95.70% 95.90% 2.02% 1.97% 2.17% 2.04%
MNL 95.60% 99.27% 2.32% 0.59% 2.03% 0.13%
Fifth reporting period prior to failure
Nonfailure (0) Insolvent (1) Outright Failure (2)
Model Actual Predicted Actual Predicted Actual Predicted
Mixed 96.20% 95.80% 1.93% 1.77% 1.84% 2.37%
MNL 96.20% 99.38% 1.93% 0.57% 1.84% 0.04%

32. version 2.0 32
APPENDIX TABLE 1
Pearson Product-Moment Correlations Between Significant Covariates
Net
operating
cash flow to
total assets
Cash
flow
cover
Total debt
to gross
operating
cash flow
Cash
resources
to total
assets
Working
capital to
total
assets
Total debt
to total
equity
Sales
revenue to
total
assets
New
economy
sector
Resources
sector
Old
economy
sector
Finance
sector
Net operating cash
flow to total assets
1 0.2428 0.0267 -0.1592 0.1523 0.0097 0.1725 -0.1575 -0.0671 0.1383 0.0402
Cash flow cover 1 -0.0174 -0.0199 0.0379 -0.0041 0.1095 -0.0336 -0.0488 0.041 0.021
Total debt to gross
operating cash flow
1 -0.0949 -0.0026 0.2328 -0.0052 -0.0324 -0.058 0.0641 0.0174
Cash resources to total
assets
1 -0.1047 -0.1529 -0.1595 0.1299 0.0827 -0.2624 0.1523
Working capital to
total assets
1 0.0036 0.0637 -0.0007 -0.0407 0.0353 -0.006
Total debt to total
equity
1 0.0646 -0.0394 -0.0718 0.0912 -0.001
Sales revenue to total
assets
1 -0.0285 -0.2283 0.2525 -0.0794
New economy sector 1 -0.2252 -0.4225 -0.1772
Resources sector 1 -0.4851 -0.2035
Old economy sector 1 -0.3817
Finance sector 1