# Machine Learning for Zombie Hunting:
Predicting Distress from Firms’ Accounts and Missing Values 00footnotetext: We
are grateful to participants to the Bank of Italy/CEPR/EIEF conference on
‘Firm Dynamics and Economic Growth’, to the Bank of England/King’s College
conference on ‘Modelling with Big Data and Machine Learning’, to the Annual
Conference of the JRC Community of Practice in Financial Research organized by
the European Commission, and to the workshops on ‘Data Science for Impact
Evaluation’ jointly organized by KU Leuven and IMT School for Advanced
Studies. We want to thank Tommaso Aquilante, Nicola Benatti, Kristina
Bluwstein, Elena Cefis, Giulio Bottazzi, Dimitrios Exadaktylos, Nicoló
Fraccaroli, Mahdi Ghodsi, Andreas Joseph, Francesca Lotti, Francesco Manaresi,
Juri Marcucci, Andrea Mina, Chiara Osbat, Gianmarco Ottaviano, Giacomo Rodano,
Andrea Roventini, Gabriele Rovigatti, Abhishek Samantray, Federico Tamagni,
Francisco Queiro, Gias Uddin, Nicolas Woloszko and Nicoló Vallarano for their
valuable comments.

Falco J. Bargagli-Stoffi Mail to<EMAIL_ADDRESS>Harvard
University, 677 Huntington Ave, Boston, MA 02115, United States. Fabio Incerti
Mail to<EMAIL_ADDRESS>Laboratory for the Analysis of Complex
Economic Systems, IMT School for Advanced Studies, piazza San Francesco 19 -
55100 Lucca, Italy. Massimo Riccaboni Corresponding author, mail to:
<EMAIL_ADDRESS>Laboratory for the Analysis of Complex Economic
Systems, IMT School for Advanced Studies, piazza San Francesco 19 - 55100
Lucca, Italy. Armando Rungi Mail to<EMAIL_ADDRESS>Laboratory for
the Analysis of Complex Economic Systems, IMT School for Advanced Studies,
piazza San Francesco 19 - 55100 Lucca, Italy.

In this contribution, we propose machine learning techniques to predict zombie
firms. First, we derive the risk of failure by training and testing our
algorithms on disclosed financial information and non-random missing values of
304,906 firms active in Italy from 2008 to 2017. Then, we spot the highest
financial distress conditional on predictions that lies  above a threshold for
which a combination of false positive rate (false prediction of firm failure)
and false negative rate (false prediction of active firms) is minimized.
Therefore, we identify zombies as firms that persist in a state of  financial
distress, i.e., their forecasts fall into the risk category above the
threshold for at least three consecutive years. For our purpose, we implement
a gradient boosting algorithm (XGBoost) that exploits information about
missing values. The inclusion of missing values in our predictive model is
crucial because patterns of undisclosed accounts are correlated with firm
failure. Finally, we show that  our preferred machine learning algorithm
outperforms (i) proxy models such as Z-scores and the Distance-to-Default,
(ii) traditional econometric methods, and (iii) other widely used machine
learning techniques. We provide evidence that zombies are on average less
productive and smaller, and that they tend to increase in times of crisis.
Finally, we argue that our application can help financial institutions and
public authorities design evidence-based policies— e.g., optimal bankruptcy
laws and information disclosure policies.
Keywords: zombie firms; machine learning; financial constraints; bankruptcy;
missing data
JEL Codes: C53; C55; G32; G33; L21; L25

##  1 Introduction

In this paper, we propose machine learning techniques as suitable tools to
make predictions about business failure when information about firm viability
is partially undisclosed. Therefore, we define zombies as firms that persist
in a high-risk status because their predicted probability of failure  is above
a threshold for which a combination of the false positive rate (i.e.,
predicting a firm that did not fail as failed) and the false negative rate
(i.e., predicting a firm that failed as not failed) is minimized. We prove
that the latter is also the distributional segment in which the probability of
transitioning to a lower risk of failure is minimal. According to our machine
learning approach, the Italian proportion of zombie firms in the analysis
period is between  1.5% and 2.5%. Zombie firms are on average  19% less
productive and  17% smaller than viable firms. Interestingly, we find that
zombies are countercyclical, as their share increases in times of crisis and
decreases in times of economic recovery.

The problem of identifying nonviable firms is important to scholars and
practitioners, whether to assess the credit risk of an individual firm or to
identify the part of an entire economy that is in trouble.  Originally, the
notion of zombie firms was associated with the phenomenon of “zombie lending”,
in which banks extend credit to otherwise insolvent borrowers. In some cases,
zombie lending is a deliberate strategy to avoid a bank’s budget restructuring
while only apparently complying with capital standards set by financial
regulators (Bonfim et al., 2020), e.g., the case of Japanese banks in the
1990s (Peek and Rosengren, 2005; Caballero et al., 2008). More recently,
Schivardi et al. (2021) studied Italian companies during the 2008 financial
crisis and found that the misallocation of credit under zombie lending
increased the default rate of otherwise healthy firms while decreasing the
default rate of nonviable firms. This was because undercapitalized banks could
restrict lending to more viable projects to avoid disclosing nonperforming
loans in their portfolios. Paradoxically, severely distressed companies can
appear resilient in times of financial crisis thanks to continued access to
financial resources.

From a more general perspective, recent studies introduce zombies as firms
that have persistent problems meeting their interest payments (Andrews et al.,
2017; McGowan et al., 2018; Banerjee and Hofmann, 2018; Andrews and
Petroulakis, 2019), thus expanding the original category to include firms that
are in some form of financial distress. Based on proxy indicators from
financial accounting, they show that zombies account for a non-negligible
share of modern economies— up to 10% of incumbent firms— while absorbing up to
15%, 19%, and 28% of the capital stock in countries such as Spain, Italy, and
Greece, respectively. When market exit or restructuring delays occur, they
drag down aggregate productivity by hindering the reallocation of resources in
favor of healthier firms and preventing the entry of potentially more
innovative and younger firms. It is therefore argued that identifying non-
viable, financially distressed firms can be particularly useful in avoiding
the misallocation of productive and financial resources.

Against this background, we argue that the empirical problem of assessing
whether a firm is a zombie is closely related to the more general problem of
determining its credit risk. Ultimately, a zombie is a non-viable firm that
may escape bankruptcy despite its extreme financial distress— i.e., despite
scoring the highest credit risk. From another perspective, healthier firms are
the furthest from bankruptcy and zombie status. Traditionally, credit risk has
been studied from the perspective of a financial firm, which must assess the
health of a company using information, albeit limited, from financial books
and public records.111The seminal reference is to a departure from the
Modigliani and Miller (1958) theorem, according to which capital structure
should not be relevant to a company’s value if there are no market frictions,
including bankruptcy costs. Thus, a firm’s ability to raise external financing
should depend solely on the profitability of its investment projects. However,
since financial market frictions cannot be eliminated, a firm’s capital
structure actually provides information about the profitability of the firm
and its assets. See also the discussion of Rajan and Zingales (1995) for an
international perspective Thus, for decades, academics and practitioners have
attempted to determine a firm’s profitability after benchmarking exercises on
firm-level indicators of financial constraints— e.g., in estimating Z-scores
(Altman, 1968; Altman et al., 2000), Distance-to-Default (Merton, 1974), or
investment- to- cash- flow sensitivity (Fazzari et al., 1988). However,
information on corporate viability may be incomplete due to strategic
disclosure of relevant information and simplified financial records for
certain categories of unlisted companies.

Thus, we propose a machine learning approach to predict credit risk and zombie
status from incomplete financial accounts. To show the potential of our
approach, we work with a sample of 304,906 Italian firms over the period
2008-2017. Italy is a compelling example of a country that hosts a relevant
share of inefficient firms that hinder the growth potential of the economy
(Calligaris et al., 2018).

The underlying intuition is simple: based on the experience of firms that
failed in previous periods, we derive predictions about the risk of failure of
active firms. Each time we compare the observed outcomes with the predicted
outcomes, the algorithm updates and reduces the prediction errors in the next
periods after processing new “in-sample” information about the financial
accounts. In the end, we obtain a probabilistic measure of the likelihood that
a business will fail. Our machine learning framework improves upon existing
benchmark models by leveraging a rich set of firm-level economic and financial
indicators that potentially contain diverse information about both the firm’s
core economic activity and its ability to meet financial obligations.

Importantly, we find that emerging patterns of missing financial accounts are
correlated with firm failure, possibly due to the fact that managers are more
likely to conceal accounts when they are in financial distress. We provide
evidence that most missing variables are often those that have been used as
proxies for zombies or financial constraints in the previous literature.
Therefore, we implement our missing-aware methodologies incorporating patterns
of undisclosed accounts and test whether a substantial improvement in
prediction occurs when missing data are properly handled. Using the missing
values information as another predictor of outcome, we show that the best
predictive algorithm in our setting, eXtreme Gradient Boosting (XGBoost) (Chen
and Guestrin, 2016), explains up to $0.97$ of the Area-Under-the-Curve (AUC),
and its Precision-Recall (PR) performance reaches $0.76$. XGBoost outperforms
credit scoring models (i.e., Z-score and Distance-to-Default models), standard
econometric methods (i.e., logistic regression), and also other machine
learning techniques (i.e., Classification and Regression Tree and Random
Forests).

We argue that asymmetric and undisclosed (missing) information is ubiquitous
in corporate financial accounts. Therefore, simply omitting records with
missing values would severely impair the search for predictors of zombies and
lead to a loss of precision and bias (Little and Rubin, 2019). If the missing
information is correlated with firm failures, excluding these firms would
significantly reduce the number of failures in the sample and thus hinder the
algorithm’s learning for these instances. This problem could potentially be
avoided by a missing data imputation approach that would allow the inclusion
of these firms in the training sample. On the other hand, the missing
information per se constitutes relevant information to learn from failures.
This may be the case if nonviable firms have the option of not disclosing
information in their financial accounts. As a result, the mechanism in the
data is Missing-Not-At-Random (MNAR), and the complete records are not a
random sample of the population of interest. In this second case, if some
companies consistently avoid disclosing part of their information, imputation
of missing data would not be sufficient to solve the problem. Indeed, we know
from previous literature that techniques for imputing data, such as mean or
median imputation, can fail in the presence of MNAR because they distort the
empirical distributions.222Further limitations of traditional approaches to
imputing missing data are discussed in (He et al., 2010; White et al., 2018;
Little and Rubin, 2019).

Based on the previous considerations, we choose an approach that can
simultaneously incorporate missing data into our analysis while being robust
to MNAR. We implement two methods specifically designed to deal with non-
random patterns of missing data: XGBoost and Bayesian Additive Regression
Trees with Missing Incorporated in Attributes (BART-MIA) (Kapelner and Bleich,
2015). Although these methods differ in implementation, they have very similar
routines for dealing with MNAR patterns. XGBoost uses default directions, also
known as block propagation (Josse et al., 2019), to group all incomplete
observations and send them to one side of the tree. The MIA method (Twala et
al., 2008), extended by Kapelner and Bleich (2015, 2016), strengthens block
propagation so that missingness can be used as an explicit feature to compute
the best splits. According to Josse et al. (2019), MIA can handle both
informative and non-informative missing values. Our results suggest that both
XGBoost and BART-MIA effectively capture the direct influence of missing
values as either implicit (block propagation) or explicit (MIA) predictors of
the response variable. Finally, in the following analyzes, we show that
XGboost has a significantly lower computational cost and relatively higher
predictive power than BART-MIA.

To shed more light on the information that contributes most to prediction, we
use an interpretable framework that has its roots in game-theoretic Shapley
values, introduced by Strumbelj and Kononenko (2010). Shapley values have
recently been proposed to identify the economically meaningful nonlinearities
learned by machine learning models (Buckmann et al., 2021). We find that no
single financial indicator predicts failure better than the ensemble of
predictors used in our machine learning approach.

Economically informative groups of variables have heterogeneous predictive
power. In particular, indicators of firms’ financial constraint or previously
used indicators of zombies are important. Nevertheless, they are of secondary
importance when compared to information on firms’  financial accounts,
indicators of corporate governance, and the presence of non-random missing
values. Our results confirm that machine learning techniques perform better
than single indicators when incorporating as much valuable in-sample
information as possible while updating each time there is new out-of-sample
information because prediction errors dynamically decrease after independent
tests that minimize the discrepancies between realized and predicted outcomes
(Athey, 2018).

Our framework is of particular interest to policymakers in designing optimal
bankruptcy laws.333See also the suggestions by the European Directive
2012/30/EU, and the recent Italian Law on business failures on October 19,
2017, n. 155, which provides the legal basis for early notification of
corporate crises to improve targeted interventions. Tracking a company’s
bankruptcy risk allows all stakeholders, not just creditors, to understand
whether there is an opportunity for restructuring and, if not, to prevent
incumbent, albeit non-viable, firms from wasting additional economic
resources. Evidence-based, interpretable methods are even more important after
the recent pandemic crisis, as we believe that financial support must be
targeted to companies that have a real chance of recovering and staying on
their feet in normal times to avoid misallocation of resources.444For a first
analysis of the impact of the pandemic crisis on Italian companies, see
Schivardi and Romano (2020)

##  2 Data and preliminary evidence

We obtain the financial accounts from the ORBIS database555ORBIS firm-level
data (Orbis, 2020) have become a common source of global financial accounts.
For previous use of this database, see Gopinath et al. (2017) and Cravino and
Levchenko (2016), among others. Coverage of smaller firms and some financial
accounts may change across countries as national business registries impose
different filing requirements, as observed in the validation exercises of
Kalemli-Ozcan et al. (2015) and Gal (2013). In the case of Italy, the original
information provider for Italian financial accounts is CERVED, a credit rating
agency. Bureau Van Dijk standardizes and translates the original financial
accounts to make them comparable between countries. Note that, unlike other
platforms of the same Bureau Van Dijk (e.g., AIDA or AMADEUS), ORBIS does not
drop exiting firms in our analysis period. It supplements the financial
accounts with other information from various sources on ownership, management
and intellectual property rights, which we also use for predictions., compiled
by the Bureau Van Dijk, for manufacturing firms active in Italy for at least
one year from 2008-2017. Italy is a compelling case to study business failure
and zombie firms: it is a country where relatively inefficient firms hinder
the economy’s growth potential (Calligaris et al., 2016; Bugamelli et al.,
2018), perpetuating geographic divergence (Rungi and Biancalani, 2019), and
are studied extensively by international organizations (McGowan et al., 2018;
Andrews and Petroulakis, 2019).

For our purpose, we use two main variables that help us identify business
failure: the status of a firm and the date on which it becomes inactive. Table
1 shows our sample coverage by firm status over in the period of analysis. We
assume that a firm has failed in the first year if it is reported as
“bankrupt”, “dissolved”, or “in liquidation”, as in the original data.
Overall, the share of exiting firms accounts for about 5.7% of the total
sample, which is close to the average official 6.3% obtained by ISTAT, the
national statistics office, for the same period.

Figure 1 shows the proportions of firm failures by NUTS 2-digit region. As
expected, we find a higher concentration of failures in the north and center
of the country, where economic density is higher. It is noteworthy that we
fully represent the whole Italian territory since we detect business failures
in every region during our period of analysis.

Table 1: Firms by status

Status | Active | Bankrupted | Dissolved | In Liquidation | Total
---|---|---|---|---|---
Sample | 287,586 | 1,533 | 8,540 | 7,221 | 304,906
Percentage | 94.33% | 0.50% | 2.80% | 2.37% | 100%

Figure 1:  Geographic coverage Proportion of business failures to the total
number of firms for NUTS 2-digit regions. We assume that a firm fails when it
is reported as bankrupt, dissolved, or in liquidation.

We use a set of economic and financial indicators to train our predictive
models. The battery of predictors includes (i) original financial accounts at
the firm level; (ii) widely used indicators to proxy firm-level financial
constraints; (iii) indicators previously used to detect zombie firms; (iv)
indicators included as warnings of corporate crises in the recent Italian
bankruptcy law.666A recent reform of the bankruptcy law (L. 155/2017 and DL.
14/2019) proposes an early warning system based on indicators identified by
practitioners, the purpose of which is to identify companies in distress in
time to intervene to preserve entrepreneurial capabilities and find a way out
of the crisis. It delegated practitioners (in particular the National
Association of Chartered Accountants: Consiglio Nazionale dei Dottori
Commercialisti e degli Esperti Contabili ) to draw up a list of indicators
that could help assess the state of crisis of a company Each predictor we
consider is described in detail in Table A.1 in the Appendix A.

Note that many indicators we select as predictors have been used in various
frameworks to assess the extent to which a firm is in trouble. From our
machine learning perspective, they cannot be interpreted as drivers of
failure. It is sufficient that they contribute, albeit in a small way, to the
assessment of a company’s health. In our predictive framework, they might even
border on multicollinearity, e.g., in the case of different measures of
efficiency, liquidity, and solvency ratios. Since we are not interested in
identifying a causal contribution to firm failure, high collinearity does not
pose a problem for our predictions (Makridakis et al., 2008; Shmueli et al.,
2010). On the contrary, we will discuss in Section 3.4 how, by construction,
one cannot separate the empirical contribution of an indicator from the entire
set in the context of a pure prediction problem. For this reason, the
predictors we use should be considered as an inseparable ensemble, and a
discussion of the statistical significance of the individual predictors is not
relevant in our framework.

In addition to mandatory and basic information (volume of activity, profits,
location, industry affiliation, ownership, and intellectual property rights),
many other financial accounts have different patterns of missing values over
time.  In the Appendix A, Figure A.1, we visualize a map of missing values in
our sample. The frequency of missing values affects all firms in the data,
both active and failed, and it is more concentrated in the financial accounts
of recent years. In addition, liquidations exhibit less pronounced patterns
over time than the other failure categories. After running a series of chi-
squared tests (see Table A.3), we find a positive statistical relationship
between the patterns we observe in the sample and the event of a firm’s
failure. In short, a firm is more likely to fail if a pattern of missing
financial accounts is observed. The exercise performed in Table 2 clearly
shows such correlations. We run a simple logistic regression using the
observed failure of a firm as the dependent variable. We then add a binary
regressor equal to one if the predictor is missing at least once in the three
years prior to failure. Fixed effects by NUTS 3-digit region and NACE 2-digit
industry are included. We report results for each predictor per row in Table
2.

Table 2: Missing predictors and firms’ failures

Missing predictor | Odds ratio | Std. Error | N. obs. | Pseudo R2
---|---|---|---|---
Interest Coverage Ratio | 1.91*** | (0.30) | 304,906 | 0.026
Interest Benchmarking | 1.29*** | (0.20) | 304,906 | 0.019
value added | 2.37*** | (0.37) | 304,906 | 0.033
Z-score | 2.55*** | (0.38) | 304,906 | 0.037
Total Factor Productivity | 2.37*** | (0.39) | 304,906 | 0.034
Profitability | 1.91*** | (0.30) | 304,906 | 0.027

Odds ratios according to a logit specification in which the dependent variable
is a firm failure and the binary regressor equals one if at least one missing
value was found in the last three years. Fixed effects at the region and
industry level. Errors are clustered by industry.

The above correlations are particularly relevant to the scope of our analyses.
The main problem is sample selection when observations are selectively missing
for some categories of firms. In this case, there are two potential sources of
sample selection bias: (i) distressed firms vis à vis firms that are not in
distress, as the former may have the incentive to disclose less information
than the latter; (ii) smaller companies firms vis à vis bigger firms because
the first are often exempted from a complete financial report, in accordance
with Italian Regulation.777Under Italian civil law, companies that do not list
financial activities on the stock exchange have the option to provide more
aggregate financial reports if their size does not simultaneously exceed two
of the following thresholds in one or two consecutive periods: i) 4,400,000
euros in total assets; ii) 8,800,000 euros in operating revenues; iii) 50
employees. A simplified financial statement always includes the most
significant items in the first or second position digit of the aggregation. Of
course, the two sets of firms may overlap, as smaller companies may also be
the ones that are proportionally more affected by financial difficulties.
Using a missing-aware procedure, as described in Section 3, allows us to
consider both sources of sample selection when patterns of missing financial
accounts emerge since the algorithm considers such patterns as another
predictor of firm failure.

Interestingly, we note that most of the missing variables are also those used
in previous work as proxies for zombies or financial constraints. Take, for
example, the case of the Interest Coverage Ratio (ICR), which is derived as
the ratio between a firm’s earnings before interest and taxes (EBIT) and its
interest expense. If the ICR is less than one, Bank of England (2013) assumes
that a company is a zombie because it is having trouble meeting its financial
obligations. In our sample, we find that about 19% of firms have an ICR
smaller than one, but at the same time, there are 62.50% firms whose ICR
information is not available at all. Moreover, according to Bank of Korea
(2013), negative value added is the most appropriate indicator for evaluating
a zombie status, as it indicates that intermediate inputs have a higher market
value than the firm’s output. In the case of Italy, about 64.27% of
enterprises do not report their value added in at least one period, while
about 3% of them report a negative value. A negative value added is, of
course, a more severe condition than a negative profit, since a company can
make no profits without destroying economic value. Indeed, firms’
profitability is at the heart of two similar proxies for zombies used by
Schivardi et al. (2021) when comparing firm-level profits to an external
benchmark. Repeating the same exercises, we find that about 3% of Italian
firms are distressed, while a large part of the sample (62.50%) does not
report any information on the predictor. Finally, both Caballero et al. (2008)
and McGowan et al. (2018) perform another benchmarking exercise, comparing the
interest a firm pays to raise external finance with the cost opportunity to
invest in alternative, safer investments. In our case, when we try to
reproduce the same exercise with the yields of Italian government bonds with a
ten-year maturity, we find that there is a high proportion (60.29%) of
companies for which we have no information in at least one period in which
they were active.

We also include an estimate of Total Factor Productivity (TFP) as a predictor,
following the methodology proposed by Ackerberg et al. (2015), to account for
the simultaneity bias arising from ex-post adjustments in combining factors of
production. In this respect, firm-level TFP allows us to make predictions
based on the ability to transform inputs and sell output in the market.
Indeed, the relationship between financial constraints and productivity is one
of the most debated issues (see, among others, Aghion et al., 2019; Ferrando
and Ruggieri, 2018). The simple assumption for our forecasting models is that
less productive firms are the ones that have more difficulty surviving in the
market. At the same time, zombie firms have also often been defined in terms
of (lack of) productivity (McGowan et al., 2018; Andrews et al., 2017; Andrews
and Petroulakis, 2019; Schivardi et al., 2020).888Please note how White et al.
(2018) recently highlighted the limitations of using missing imputation
techniques for the variables used to calculate TFP in the US.

##  3 Empirical strategy

It is difficult to identify non-viable companies for obvious reasons. If
financial accounts are in bad shape, one could argue that it is only a matter
of time before they become more competitive if conditions are right. If the
balance sheets are good, one could argue that the worst is yet to come because
bad management decisions will show up later. Trivially, only the already
bankrupt companies were certainly not viable at some point. Still, an outside
observer will never know when that happened because the manager of a company
in trouble has an incentive not to disclose private information.

In principle, an analyst would like to observe the entire event horizon to
discount all possible scenarios and understand the value of a company and its
investment projects. However, this is not possible because it is the typical
double problem of a financial institution facing uncertainty in the presence
of information asymmetries. On the one hand, the company has a clear
information advantage in its investment plans. On the other hand, both the
financial institution and the company have a limited ability to predict future
economic shocks, which can have either positive or negative effects.

In their seminal works Altman (1968) and Ohlson (1980) apply standard
econometric techniques—i.e., multiple discriminant analysis (MDA) and logistic
regression—to assess the probability of firm bankruptcy. Following these
contributions and the Basel Accord II in 2004, default forecasts are based on
standard reduced-form linear regression approaches. However, these approaches
may fail because their limited complexity precludes nonlinear interactions
among predictors, while their ability to handle large sets of predictors is
limited due to potential multicollinearity problems.

Machine learning algorithms compensate for these shortcomings by providing
flexible models that allow nonlinear interactions in the space of predictors
and the inclusion of a large number of predictors without the need to invert
the covariance matrix of the predictors, thereby circumventing
multicollinearity (Linn and Weagley, 2019). In addition, machine learning
models are directly optimized to perform the prediction task, resulting in
better prediction performance in many complex situations.

The data science literature has already developed exercises to predict
corporate failures using financial accounts, but without a clear economic and
financial framework. In this context, Bargagli-Stoffi et al. (2021) provides a
comprehensive review of the recent literature on the use of machine learning
for the analysis firm dynamics. The authors highlight that the majority of
work dealing with the prediction of bankruptcy or financial distress uses
either decision tree-based techniques (see, e.g., Behr and Weinblat, 2017;
Linn and Weagley, 2019; Moscatelli et al., 2019; De Martiis et al., 2020;
Davies et al., 2023; Incerti et al., 2022) or neural network-based methods
(see, e.g., Alaka et al., 2018; Brédart, 2014; Hosaka, 2019; Sun and Li, 2011;
Tsai and Wu, 2008; Tsai et al., 2014; Wang et al., 2014; Lee et al., 1996;
Udo, 1993).

With this in mind, we propose a machine learning procedure that uses past
information about previously failed firms to estimate the probability that
another firm in similar condition will go bankrupt. The broader the variety of
past experiences on which we can draw, the more accurate the prediction about
the health - or lack thereof - of a company (Kleinberg et al., 2015).
Ultimately, our perspective is on a firm’s (lack of) resilience, using
potentially any observable data that might hold information about the firm’s
viability. In the end, we obtain a probabilistic measure at the firm level,
ranging from 0 to 1, which tells us the probability that a firm will exit the
market in the next period, given that other firms in a similar situation have
done so. As shown in Figure 2, we can assess the distance of each company from
the highest financial distress.

Let us consider a generic predictive model in the form:

${f}(\mathbf{X}_{i,t-1})={Pr}(Y_{i,t}=1\>|\>\mathbf{X}_{i,t-1}=x)$ (1)

where $Y_{i,t}$ is the binary realization of the outcome at time $t$ that
takes the value 1 if the $i$th firm exits the market and takes the value zero
otherwise, while $\mathbf{X}_{i,t-1}$ is the $P$-dimensional vector of firm-
level predictors in the previous time period, where $P$ is the number of
predictors included in the model. The functional form linking the predictors
to the outcomes is determined by the generic supervised machine learning
procedure used to predict out-of-sample information. In short, the generic
algorithm chooses the best in-sample loss-minimizing function in the form:

$\operatorname*{arg\,min}\sum_{i=1}^{N}L(f(x_{i,t-1}),y_{i,t})\>\>\>\>over\>\>\>\>f(\cdot)\in
F\>\>\>\>\>\>\>\>s.\>t.\>\>\>\>\>\>\>\>R\big{(}f(\cdot)\big{)}\leq c$ (2)

where $F$ is a function class from where to pick $f(\cdot)$, and
$R\big{(}f(\cdot)\big{)}$ is the generic regularizer that summarizes the
complexity of $f(\cdot)$ (see also Mullainathan and Spiess, 2017). In our
case, the function $f(\cdot)$ is an element from the family of classification
trees or a combination of them. The set of regularizers, $R$’s, will change
according to the standards adopted by each method. Ultimately, each algorithm
takes a loss function $L({f}(x_{i}),y_{i})$ as input and searches for the
function that minimizes the prediction losses.

To determine the region of highest financial distress, we show in the
following analysis that a cutoff of 0.9 minimizes the combination of false-
positive (false prediction of firm failures) and false-negative (false
prediction of active firms) rates. This empirical result supports the choice
of the highest decile of the predicted risk distribution as the optimal
threshold for identifying cases of critical financial distress. We will also
prove that using the highest decile to identify zombie firms is successful, as
this is the segment where prediction accuracy is highest, and after which the
probability of transitioning back to lower levels of financial distress is
minimal.

Figure 2: Fictional distribution of failure’s probability

We would like to emphasize that we are not interested in identifying the
causes of firm failure, since we are dealing with a pure prediction problem,—
i.e., the probability of the failure of firm $i$ at time $t$. Nevertheless, we
perform variable selection to evaluate the contribution of the predictors to
the estimated risk. We show how the predictors of failure can change over
time, under the circumstances in which we make the predictions, each time
there is an update with new out-of-sample information.

###  3.1 Decision tree learning for failure prediction

Consistent with the literature on bankruptcy and firm exit predictions
presented above, we derive—given new out-of-sample information— a prediction
for the failure of each firm based on its current financial accounts, both for
established firms that have operated in previous periods and for firms
entering the market for the first time. From another perspective, we interpret
this probability range as the degree of risk of an investor who has no
information other than that contained in the current financial accounts.

Our study introduces an innovative approach to failure prediction by
demonstrating the remarkable effectiveness of tree-based machine learning
models, especially in the presence of missing data. While neural networks are
widely considered the best option for prediction tasks with homogeneous data
such as images and text, they often fail for “problems with heterogeneous
features, noisy data, and complex dependencies” (Prokhorenkova et al., 2018).
Decision tree-based algorithms are considered suitable tools in these cases
due to their flexibility and high performance. The Classification and
Regression Tree (CART) algorithm, first introduced by Breiman et al. (1984),
is a widely used decision tree algorithm that constructs binary trees where
each node is divided into only two branches. Figure 3 shows how binary
partitioning works in practice, using a simple example with only two
predictors. Ensemble methods have been developed to improve the stability of
decision tree estimates, which combine multiple weak learners into one strong
learner. Bagging and boosting are two popular methods for this purpose
(Breiman, 1996; Freund et al., 1996). In line with recent developments in the
literature, we perform a comparative evaluation of firm failure prediction
using two state-of-the-art approaches: XGBoost (Chen and Guestrin, 2016) and
BART-MIA (Chipman et al., 2010). Both statistical models are specifically
designed to deal with missing values, which makes them ideal for dealing with
severe cases of missing financial data in corporate accounts. As illustrated
in Section 2, the absence of financial data does not follow random patterns,
or at least does not follow the failure of a company completely randomly. When
a company is financially distressed and/or smaller, it is more likely that
data is missing from some accounts. Simply discarding the missing observations
would introduce selection bias and exclude certain categories of firms with a
higher probability of failure. Crucially, both XGBoost and BART-MIA include
patterns of undisclosed accounts as a feature of the model.

Figure 3: An example of binary tree

$x_{1}<0.6$$l_{1}$$x_{2}>0.2$$l_{2}$$l_{3}$ (a) Binary tree

(b) Feature space

In Figure (3a), the internal nodes are labeled by their splitting rules and
the terminal nodes by the corresponding parameters $l_{i}$. Figure (3b) shows
the corresponding partition of the feature space.

XGBoost is an ensemble method based on decision trees with a gradient boosting
framework. It has proven to be highly competitive and reliable on various
prediction tasks, mainly due to its efficient use of computation time and
memory resources (Gumus and Kiran, 2017; Abbasi et al., 2019; Li et al.,
2020). It is equipped with various optimization techniques and tools, such as
a percentile-based split-finding algorithm, parallelized tree building, a
depth-first approach to tree pruning, and efficient handling of missing values
through default directions. In addition, it uses regularization to prevent
overfitting (Bentéjac et al., 2021).

BART-MIA is a robust Bayesian ensemble of trees methodology that combines the
traditional BART (Chipman et al., 2010) with a variant called MIA (Twala et
al., 2008). This variant was specifically designed to deal with MNAR (Missing
Not At Random) patterns in the data. BART is a well-established method that
has shown excellent performance on various prediction tasks (Murray, 2017;
Linero and Yang, 2018; Linero, 2018; Hernández et al., 2018). This can be
attributed to two key factors: first, BART contains a noise component that
mitigates the overfitting problem common to random forest methods, and second,
it has shown consistently strong performance on standard model specifications,
avoiding time-consuming hyperparameter tuning procedures. This property is
particularly important because it reduces the researcher’s dependence on
parameter choice and minimizes the computational time and cost associated with
cross-validation. See the appendix B for the technical details of the two
models.

###  3.2 Validation against other machine learning techniques

We compare the predictive performance of our methods that account for missing
values, XGBoost and BART-MIA, with the Conditional Inference Tree (Hothorn et
al., 2006), the Random Forest (Breiman, 2001), and the Super Learner (Van der
Laan et al., 2007). The Conditional Inference Tree we use is a simple variant
of the Classification and Regression Tree (CART) algorithm (Breiman et al.,
1984), based on a significance testing procedure that avoids bias towards
variables with many possible splits (see Odén et al., 1975; Loh, 2002; Hothorn
et al., 2006). The Random Forest is an ensemble method that combines different
trees to obtain stronger predictive power. Each tree is created by randomly
selecting different variables from all possible predictors and randomly
selecting a subset of the total number of observations (see also Breiman,
2001). The Super Learner (Van der Laan et al., 2007) is based on a weighted
combination of other algorithms.  We build it as a convex combination of the
following models: Logistic regression, CART, Random Forest, BART, and XGBoost.

To evaluate the performance of the models, we use a standard five-fold cross-
validation, where the dataset is divided into five groups and the model is
trained on four of them, while the fifth group is used as a validation set,
iteratively repeating this process. In this way, we can comprehensively
evaluate the predictive performance of the model over the entire time series.

To ensure comparability with the other models that do not consider missing
values, we perform a Complete-Case analysis. The latter is a widely used
approach for dealing with missing values in models that are not designed to do
so. In this approach, all observations with at least one missing value in the
predictors are discarded. Although XGBoost and the BART-MIA models could use
the entire dataset without excluding missing observations, we perform a
Missing-Aware analysis to ensure fair performance evaluation of all models: we
train and test XGBoost and BART-MIA using five-fold cross-validation with
identical dimensions as the original folds but allow missing observations to
be included in the sample. For clarity, we use the terms Missing-Aware XGBoost
(MA-XGBoost) and BART-MIA for the models trained on observations with missing
values. In this case, MA-XGBoost uses default directions to deal with missing
values, while BART-MIA uses the MIA procedure. On the other hand, we will
simply refer to the models trained on Complete-Case data as XGBoost and BART.

Table 3:  Models’ horse race: performance measures

(a) Complete-Case analysis Method | AUC | PR | F1-Score | BACC | $R^{2}$ | Time
---|---|---|---|---|---|---
Logit | 0.8966 | 0.4542 | 0.1833 | 0.7504 | 0.2658 | 9.13
Ctree | 0.8957 | 0.4444 | 0.1987 | 0.7668 | 0.2640 | 572.46
Random Forest | 0.9117 | 0.5233 | 0.1907 | 0.7595 | 0.3135 | 261.62
XGBoost | 0.9140 | 0.5170 | 0.1833 | 0.7504 | 0.3126 | 43.66
BART | 0.9185 | 0.5221 | 0.1843 | 0.7533 | 0.3179 | 1249.05
Super Learner | 0.9231 | 0.5464 | 0.1844 | 0.7535 | 0.3373 | 4147.87

(b) Missing-Aware analysis Method | AUC | PR | F1-Score | BACC | $R^{2}$ | Time
---|---|---|---|---|---|---
MA-XGBoost | 0.9685 | 0.7591 | 0.2070 | 0.7646 | 0.5243 | 24.70
BART-MIA | 0.9681 | 0.7516 | 0.2092 | 0.7676 | 0.5178 | 1126.88

All algorithms are trained with five-fold cross-validation. The training and
test sets include 95,970 and 19,194 observations in each iteration,
respectively. All metrics correspond to the five-fold average. Time indicates
the average seconds required to train the model in each fold. Figure 4:
Goodness-of-Fit Scores

The ROC and PR curves for the Complete-Case Logit and the MA-XGBoost models.
Each plot shows the five-fold cross-validated mean curves, with the mean taken
with respect to the ROC and PR curves of each validation set along the cross-
validation routine.

Table 3 shows the results of the horse race of the models. It is clear that
the missing-aware models consistently outperform the state-of-the-art methods
involved in the Complete-Case analysis. To compare the predictive power of the
different methods, we show five different performance measures commonly used
for classification problems: the Area Under the receiver operating
characteristic Curve (AUC); the area under the Precision-Recall (PR) curve;
F1-Score; Balanced ACCuracy (BACC); adjusted $R^{2}$. Both AUC and PR vary
between 0 and 1, with 0 indicating complete misclassification and 1 indicating
perfect prediction. The AUC (Hanley and McNeil, 1982) is a general measure of
predictive power that tells us the extent to which we are able to classify
failures vis à vis non-failures, hence with an accent on the false discovery
rate (FDR). PR is particularly useful for our scope because it takes into
account both the total proportion of true failures that we are able to predict
in the data (i.e., the sensitivity/recall of the predictions) and for the
proportion of predicted failures that turn out to be true failures (i.e., the
precision of the predictions). Indeed, assessing sensitivity/recall alone
could be misleading in a zero-inflation environment such as ours, where the
number of non-failures systematically exceeds the number of failures.999For
more details, see Saito and Rehmsmeier (2015); Fawcett (2006). Figure 4
reports the average ROC and PR curves over the five-folds for the Complete-
Case Logit and  MA-XGBoost. The F1 score (Van Rijsbergen, 1979) and the BACC
(Brodersen et al., 2010) are used for cases of unbalanced data. The former is
based on a harmonic mean of precision and recall, while the latter is a simple
average between the rate of true positives and the rate of true negatives from
our predictions.

In the Complete-Case analysis, the best performing model is the Super Learner,
which, however, stops at $0.9231$ and $0.5463$ in terms of AUC and PR,
respectively.101010It is noteworthy that the Super Learner performs better
than the Random Forest in terms of accuracy, but not in terms of F1 score. The
reason is that the Super Learner algorithm (Van der Laan et al., 2007) is
optimized to find a convex combination of algorithms that minimizes the
accuracy of the ensemble method. The latter strategy is not optimal in our
case because the dataset is unbalanced. The results of the Missing-Aware
analysis differ significantly from those of the Complete-Case ones. MA-XGBoost
shows a PR of $0.7591$ and an AUC of $0.9685$, followed by BART-MIA, whose
performances are almost identical. Interestingly, the most notable improvement
comes from higher precision. This is reflected in the larger departure of the
Missing-Aware models with respect to the performance measures that use
precision (namely, PR and F1 score). This is critical to our objective as it
means that Missing-Aware models have better predictive power in identifying
companies that will close within a year based on the predictions. However,
training the BART-MIA model is computationally intensive and takes 45 times
longer than MA-XGBoost. Due to its significantly lower computational cost and
higher predictive power, we choose MA-XGboost as the optimal prediction
algorithm for the following analysis.

To ensure the robustness of our results, we perform a number of robustness
checks in Appendix F. We investigate whether (i) imputing missing observations
could improve the performance of predictive models that do not include
information on missing attributes; (ii) whether excluding liquidations from
the set of firm failures would affect the performance of the model. For (i),
we use out-of-range and median imputation methods (Josse et al., 2019). We
find that the performance of traditional models increases significantly when
missing values are considered, especially for tree-based methods. This result
is consistent with our observation that missing entries signaled by a constant
value either over the entire dataset (out-of-range imputation) or over a
specific feature (median imputation) are included in tree splitting. As a
result, the model can now account for missingness in a similar way to the
standard directions and MIA techniques, leading to comparable results. With
respect to (ii), we test whether predictive performance depends on the type of
failure (e.g., bankruptcy, dissolution, M&A, and liquidation). We find that
excluding the largest group of business failures – i.e., liquidations – has
little effect on the performance of the models, suggesting that the dependence
on the type of failure is negligible.

###  3.3 Validation against proxy models of credit scoring

So far, we have compared the predictions of failures from different
econometric and machine learning techniques and concluded that  MA-XGBoost is
the best choice in terms of predictive power and computational burden. Here,
we compare our baseline predictions with widely known proxies for firm-level
credit scores: the Z-scores (Altman, 1968) and the Distance-to-Default
(Merton, 1974).

Z-scores involve putting a selection of financial ratios (profitability,
leverage, liquidity, solvency, and volumes of activity) into an equation with
some weights to proxy their relative importance. The weights are taken from
the literature and from previous scholarly estimates of the relative
importance of these indicators in assessing a firm’s distress. In this way, a
threshold is obtained, the crossing of which indicates a high probability of
future bankruptcy. Unlike Z-scores, the Distance-to-Default (DtD) by Merton
(1974) focuses specifically on a firm’s ability to meet its financial
obligations. The original intuition is that a firm’s equity can be modeled as
a call option on its assets. Thus, to build such a model, one must combine
firm-level accounts (firm assets, debt, market value) and information from
financial markets (risk-free interest rate, standard deviation of stock
returns). Finally, one puts the variables into an equation that gives the
value of a theoretically fair call option.111111Following the insights of the
distance-to-default model, Black and Scholes (1973) developed their widely
known model based on the observation that one can eliminate a systemic risk
component by hedging an option.

We evaluate the predictive power of both Z-scores and Distance-to-Default
(DtD). The Precision and False Discovery Rate (FDR) are given in Table 4.
Similar to our previous analysis, we performed five-fold cross-validation. We
start with the non-missing observations in Z-scores and DtD and randomly split
them into five folds. Based on the Z-scores and DtD measures, distressed firms
have lower scores. Therefore, we used the first ten percentiles of the in-
sample scores as cutoffs for classifying the out-of-sample observations. We
found that the DtD predictions have higher precision (0.3314 vs. 0.2239) and
lower False Discovery Rate (0.6686 vs. 0.7761) than the Z-scores. We then
train a MA-XGBoost model with the same cross-validation routine and use the
in-sample percentiles as cutoffs to classify the out-of-sample observations.
The MA-XGBoost model considers the highest risk firms to be in the right tail,
unlike Z-scores and DtD. Therefore, the first column of Table 4 represents the
bottom ten percentiles (1-10) for DtD and Z-scores, while for MA-XGBoost the
top ten percentiles (99-90) symmetrically. It is clear that MA-XGBoost
outperforms both DtD and Z-scores in all percentiles.

Table 4:  Goodness-of-fit: Distance-to-Default (DtD), Z-scores and MA-XGBoost

| DtD | Z-Scores | MA-XGBoost
---|---|---|---
Percentile | Precision | FDR | Precision | FDR | Precision | FDR
1 | 0.3314 | 0.6686 | 0.2239 | 0.7761 | 0.9850 | 0.0150
2 | 0.3314 | 0.6686 | 0.2102 | 0.7898 | 0.9054 | 0.0946
3 | 0.3314 | 0.6686 | 0.2070 | 0.7930 | 0.8316 | 0.1684
4 | 0.3314 | 0.6686 | 0.1986 | 0.8014 | 0.7517 | 0.2483
5 | 0.3020 | 0.6980 | 0.1937 | 0.8063 | 0.6715 | 0.3285
6 | 0.2723 | 0.7277 | 0.1882 | 0.8118 | 0.6037 | 0.3963
7 | 0.2497 | 0.7503 | 0.1875 | 0.8125 | 0.5447 | 0.4553
8 | 0.2334 | 0.7666 | 0.1831 | 0.8169 | 0.5018 | 0.4982
9 | 0.2226 | 0.7774 | 0.1769 | 0.8231 | 0.4600 | 0.5400
10 | 0.2139 | 0.7861 | 0.1745 | 0.8255 | 0.4312 | 0.5688

For DtD and Z-Scores, a firm is predicted to fail if its score falls below a
certain threshold. The thresholds for classification are determined by the
first ten percentiles of their in-sample score distribution. In contrast, MA-
XGBoost predicts failure when the score is above a certain threshold, and
classification cutoffs are determined by the top ten percentiles of the own
in-sample score distribution.

###  3.4 Unboxing the black box: Shapley values for predictors

In the previous sections, we validated our proposed  MA-XGBoost model against
state-of-the-art financial indicators, traditional econometric models, and
widely used machine learning techniques. Despite its excellent predictive
performance, the nonlinear relationships learned by  MA-XGBoost are not
directly observable, leading to the recurring criticism that machine learning
models are black boxes despite their improved accuracy. However, there are a
number of tools to shed light on complex nonlinear relationships in machine
learning predictions to make models interpretable (Lundberg et al.,
2020).121212Interpretability is a non-mathematical concept, but is often
defined as the degree to which a human can understand the cause of a decision
or consistently predict the outcomes of the model (see, e.g., Kim et al.,
2016; Miller, 2019; Stoffi et al., 2021; Lee et al., 2020; Bargagli-Stoffi et
al., 2022; Bargagli-Stoffi, 2020; Bargagli Stoffi and Gnecco, 2020). Popular
methods for assessing predictor importance include permutation importance or
Gini importance for tree-based models (Breiman, 2001), LIME (Ribeiro et al.,
2016), DeepLIFT (Shrikumar et al., 2017), and Shapley values (Strumbelj and
Kononenko, 2010).

We chose to use Shapley values because they are not only able to reveal the
complex patterns linking the predictors and the outcome, but also guarantee a
number of desirable properties: efficiency, null payoffs, symmetry,
monotonicity and linearity (Rozemberczki et al., 2022). In the case of a
general ML model, these properties state that: the importance of individual
variables must add up to the goodness-of-fit of the model trained with the
entire set of variables (efficiency)131313This property is crucial because it
allows quantifying the contribution of each variable to the overall
performance of the model.; a variable that does not improve the goodness-of-
fit of the model is assigned a contribution of zero (null payoff); two
variables that make the same marginal contribution to the global goodness-of-
fit have the same importance (symmetry); if variable A consistently
contributes more to the global goodness-of-fit of the model than variable B,
the Shapley value of A must be higher than that of B (monotonicity); for two
subsets of the same data set, the global Shapley value of a variable is equal
to the sum of the Shapley values calculated separately for the same variable
in the two subsets (linearity).

An intuitive way to understand how Shapley values for variable importance work
is to include each variable in the predictive model in random order. All
variables in the model contribute to the final predictions (and thus to the
goodness-of-fit of the model). The Shapley value of a variable is the average
change in prediction (and goodness-of-fit) experienced by the set of variables
already in the model when that variable is included (Molnar, 2020). From a
mathematical perspective, assume that $S$ is a $q$-dimensional subset of
variables, $m$ is a generic variable in $S$ ($m\subset S$), and $v(T)$ is a
generic value function that takes in the subset $S$ and returns real-valued
payoff of the model (e.g., the goodness-of-fit) created using $S$ or subsets
thereof. Then the Shapley value $\phi_{m}(v)$ for a generic variable $m$ is:

$\phi_{m}(v)=\frac{1}{q}\sum_{S\subseteq\\{1,...,q\\}\setminus\\{m\\}}[v(S\cup\\{m\\})-v(S)]\frac{|S|!(q-|S|-1)!}{q!}.$
(3)

Using (3), we see how the Shapley value is computed by calculating a weighted
average gain in payoff (read: gain in goodness-of-fit) that the variable $m$
yields when included in all subsets of variables that exclude $m$.

Figure 5:  Shapley values for the variables in the predictive model Figure 6:
Shapley values for the groups of variables in the predictive model

The results of our Shapley value analyses are shown in Figures 5 and 6. Figure
5 reports the results of the average Shapley Values over the time frame of our
analysis for each variable used as input. Figure 6 shows the same value, but
this time all variables are aggregated into some economically relevant
groups.141414A full description of how variables are aggregated into groups is
described in Appendix Table A1. Governance variables are indicated by (G),
financial constraints by (FC), financial accounts by (FA), zombie indicators
by (ZI), area by (A), innovation by (I), productivity by (PDC), sector by
(SE), profitability by (PFT), size by (SI).

First, we note that no single financial indicator has better predictive power
than the aggregate of indicators. On their own, most indicators have
relatively low predictive power. The financial indicators with higher Shapley
Values – i.e., that contribute most to the model’s predictions – are the
corporate control indicator (Corporate control), the missingness in the
financial accounts (Missingness), some profit and loss characteristics (Net
income, Taxes, Revenues), the size and age indicators (Employees, Size-age),
the resources invested in the company (Shareholders funds) and the long-term
physical assets (Fixed assets).151515See Table A.1 for a full description of
the predictors and their construction. When analyzing the groups of
indicators, it appears that original financial accounts with no further
elaboration carry the greatest explanatory power. This result is reasonable
considering that most predictors fall into this category. In addition, the
governance, size, and financial constraint indicators also have relatively
high predictive power. Although they compete with groups consisting of
multiple predictors, missingness remains an important factor. On the other
hand, firms’ productivity, sector, and geographic location of firms also have
high predictive power, but are of secondary importance compared to the
aforementioned indicators.

Finally, to check for robustness, we conduct a logit-LASSO analysis to
evaluate, using a shrinkage method, the most frequently selected variables
that predict a firm’s distress. We find substantial overlap, with corporate
control, profit and loss characteristics, and firm size-age indicators among
the most important predictors. The full analysis can be found in the Appendix
C.

We conclude this Section with a discussion of the possibility that firms
manipulate financial accounts and the predictive power of missing values in
this case. We argue that there are two reasons for the occurrence of missing
values in firms’ financial accounts:

  1. 1.

Smaller firms are often exempt from reporting complete information to national
registries if they fall below certain size thresholds because it may be too
costly for them to implement complex accounting procedures.161616In footnote 7
of the manuscript, we indicate the combination of size thresholds that the
Italian law provides.

  2. 2.

Some financial accounts are optional each year or may be incomplete (e.g.,
number of employees, management of inventories, cash prospects, etc.).

In any case, there is room for manipulation by firms that know that banks and
policymakers do not use random missing values to predict their financial
distress. They have two choices: they can provide truthful information and
thereby disclose their true financial distress, or they can provide false
information and thereby potentially commit accounting fraud. In the first
case, the missing values are no longer meaningful, but our algorithm can still
rely on the disclosed financial accounts for prediction. In the second case,
we would have false negatives, i.e., companies that are predicted to be
financially viable but are not. More generally, we conclude that a limitation
of our approach is that we assume that companies do not report false
information; otherwise, we would have statistical noise that reduces
prediction accuracy.

##  4 A case for zombie firms

In our analysis, we showed how we can use machine learning to predict firms’
failures. But how do we spot a zombie firm? There is no consensus on the exact
meaning of the zombie status, other than its suggestive power. To date,
scholars have merely adopted various thresholds based on a proxy assessment of
one or more available financial indicators in the absence of more precise
theoretical guidance. Ideally, a company’s competitiveness and financial
constraints should be viewed from a dynamic perspective that considers the
entire horizon of future events. Considering all future threats and
opportunities and their impact on profit and net cash flow at the firm level
could be the theoretical equivalent of “zombie firms”. Without the latter,
empirical identification is left to the creativity of academics and
practitioners. Caballero et al. (2008) define zombies as firms that receive
subsidies in the form of bank loans after observing how interest payments
compare to an estimated benchmark of debt structure and market interest rates.
McGowan et al. (2018) assume that zombies are old firms that have persistent
problems meeting their interest payments, although they focus a policy
discussion on the macroeconomic impact of low-productivity firms. Bank of
Korea (2013) explicitly examines when the interest coverage ratio (ICR) is
below one over three years. Bank of England (2013) disregards financial
management and considers firms that have both negative profits and negative
value added, thus focusing on a firm’s core activity.

In our view, the direction of the work so far is clear: scholars and
practitioners want to infer companies’ future viability from their current
financial accounts. If they do not appear to be in good shape, it is likely
that the company will be in trouble in the near future. From this perspective,
the empirical classification of zombie firms is a perfect case study for
applying machine learning techniques to firm-level data. It is a call to use
in-sample information to predict an out-of-sample event.

Strengthened by the previous intuition, we propose an identification of
zombies, starting from the predictions of failure made in Section 3. We
propose to classify as zombies those firms that are at the right end of the
risk distribution and for which the chances of recovering from financial
distress are minimal for at least three years.

In notation, we first consider the deciles along the predictions
${f}(\mathbf{X}_{i,t-1})$, where each $q_{j,t}$ is the threshold for the $j$th
decile of the probability of default at time $t$:

$Q_{j,t}=\begin{cases}1&\text{if ${f}(\mathbf{X}_{i,t-1})\geq
q_{j,t}\>\cap\>Y_{i,t}\neq 1$},\\\ 0&\text{otherwise}.\end{cases}$

where $Y_{i,t}\neq 1$ indicates that the $i$th firm did not fail yet at time
$t$.

Table 5:  Transitions across deciles of risk

$t\>/\>t+1$ | 9th decile $t+1$ | 8th decile $t+1$ | 7th decile $t+1$ | 6th decile $t+1$ | Below 6th decile $t+1$ | Total $t+1$
---|---|---|---|---|---|---
9th decile $t$ | 0.46 | 0.22 | 0.12 | 0.08 | 0.12 | 1.00
8th decile $t$ | 0.22 | 0.22 | 0.17 | 0.13 | 0.26 | 1.00
7th decile $t$ | 0.11 | 0.16 | 0.19 | 0.15 | 0.39 | 1.00
6th decile $t$ | 0.07 | 0.11 | 0.15 | 0.18 | 0.49 | 1.00
Below 6th decile $t$ | 0.03 | 0.04 | 0.06 | 0.09 | 0.78 | 1.00

In Table 5, we report a transition matrix for the firms that did not fail,
based on elaborations over the entire period of analyses in 2008-2017. We
observe that a significant share of firms that our predictions locate beyond
the 9th decile in a representative year $t$ do not have a high chance to
improve in $t+1$. In fact, most of them (46%) remain stuck in the same
highest-risk category, and only 12% are able to recover and reach a more
reasonable level of financial distress, i.e. below the 6th decile.
Interestingly, the 9th decile is quite difficult to reach from the bottom of
the distribution, as only 11%, 7%, and 3% of firms from lower deciles,
respectively, are observed to transit to a situation of highest distress.
Moreover, about 78% of companies stay below the 6th decile in the entire
analysis period.

In general, we can say that, according to the information we have observed, a
viable firm does not easily shift into financial distress, but if it does, it
is difficult to recover from it. With this in mind, it makes sense to set an
appropriate threshold that realistically reflects the most difficult
situations in business life. We obtain this threshold by determining the
cutoff that minimizes the combination of false positive and negative rates. We
accomplish this task by maximizing the BACC, since it corresponds to a convex
combination of the true positive and negative rates, which in turn are
complementary to the false positive and negative rates. Moreover, the concept
of zombie firms is closely related to the concept of false positives, since
these firms are expected to fail but remain active. Therefore, focusing on the
false positive and negative rates within this framework is natural. According
to our results in Figure 7, the BACC peaks at about 0.9.

The latter result provides a first justification for using the 9th decile as a
cutoff. Nevertheless, we want to give firms a trial period to discount the
break-even strategies of some firms in the short run, e.g., in the case of
newly formed firms and start-ups. For all the above reasons, we define zombie
firms those firms that persist at least three years beyond the 9th decile of
the risk distribution:

$\mathbbm{1}(\sum_{t=1}^{3}Q_{j,t}=3)$ (4)

where the risk distributions are estimated with the MA-XGBoost, as proposed in
Section 3.

Figure 7:  BACC at different cutoffs along the distribution of predictions

Finally, Figure 8 shows a transition analysis of zombie firms based on our
definition from Equation 4. We calculate the fraction of zombies that either
fail, remain in zombie status, transition to a lower (but still significant)
risk category, or transition to a non-distressed state in the following year.
These results suggest that the most common outcome is remaining in zombie
status, which applies to nearly 50% of them over multiple years. In addition,
a substantial proportion of zombies transition to a lower-risk category.
However, only a small fraction of them completely deviate from their current
state, resulting in either failure or absence of distress.

Figure 8:  Transitions after predictions of a zombie status

The bars of the diagram show the transition of zombie firms in the years
following the prediction: i) to failure (light red); ii) to remaining in a
zombie status (dark red); iii) to relatively lower distress, i.e. between the
6th and 9th deciles (dark gray); iv) to a range of no distress, i.e., below
the 6th decile (light grey). Note that we cannot report 2017 because we cannot
compare it to actual observations in subsequent years.

###  4.1 Zombies in Italy

In this section we give some coordinates for the phenomenon of zombies in
Italy. First, we provide an overview of their evolution along the
macroeconomic cycle. Then, we show how they differ from the rest of the viable
firms that survive the market. Finally, we provide a comparison across
geography and industry taking into account their financial performance and
their ability to generate added value.

In Figure 9, we plot the share of zombies in our analysis period against the
GDP growth rates observed over the same period. Interestingly, we find that a
range between  $1.48\%$ and $2.55\%$ of manufacturing firms are on the verge
of bankruptcy. The share of zombies is higher immediately after the financial
crisis in 2011 and then decreased from 2013. In fact, the presence of zombies
seems to be related to the business cycle. The latter is an interesting
finding for further analysis but beyond the scope of this paper. We presume it
makes sense for zombie firms to be countercyclical: many firms can be pushed
to the brink of bankruptcy in times of crisis, while a few financially
distressed firms can find a way to recover when the recovery begins.

To fully capture their economic importance, we briefly show in Table 6 what
distinguishes zombies from the other firms. We find that they have lower
productivity on average, i.e., 19.7% and 45% when measured as TFP and labor
productivity, respectively. They are also consistently smaller on average,
selling about 200% less and employing 17% less than the otherwise
representative healthy firm. The latter results are consistent with the idea
that these firms drag down aggregate productivity while hindering the
redistribution of resources (Andrews et al., 2017; McGowan et al., 2018).

Figure 9:  Zombie firms and the economic cycle

The share of zombies on the left axis is compared with nominal GDP growth
rates on the right axis, obtained from the World Bank for the period
2011-2017. Zombie firms are firms that are at the right end (9th decile) of
the predicted risk distribution for at least three consecutive years. Table 6:
Productivity and size premia for zombies vs. healthy firms

Indicator (in logs) | Coeff. | Std. Error | N. obs. | Adj. R squared
---|---|---|---|---
Total Factor Productivity | -0.197*** | (.037) | 600,771 | .967
Labor Productivity | -0.450*** | (.014) | 559,315 | .110
Sales | -2.066*** | (.053) | 1,234,750 | .116
Employees | -0.170*** | (.039) | 1,119,486 | .157

The table shows the coefficients of the linear models for panel data with
fixed effects at the region and industry levels. We use pooled OLS estimation
with cluster-robust standard errors to account for possible correlations
within regions and industries. The dependent variable is a measure of firm
productivity or size. The main covariate is an indicator that takes the value
of one if the firm is classified as zombie in a given year of our sample, and
zero otherwise. Zombie firms are defined as firms that are at the right end of
the predicted risk distribution (above the 9th decile) for three consecutive
years.

Eventually, we lay out how the segment of zombies that we find according to
our working definition compares to the segments of firms that:

  1. 1.

have problems in making their interest payments because their Interest
Coverage Ratio (ICR);

  2. 2.

have problems in their core economic activity because they are destroying
value, i.e., their value added is negative.

Interestingly, both interest coverage ratio and negative value added have
occasionally been proposed as indicators of zombieness in previous literature
(see Appendix A for more details on these indicators and their previous use).

In Figure 10, the rays of the radar show for NUTS 2-digit regions the
proportions of zombie firms that we detect with MA-XGBoost (circles) compared
to firms that have an ICR of less than one (triangles in panel a) and firms
that experience negative value added (triangles in panel b). We find that
there is indeed common support when firms are financially distressed because
their ICR is too low, and at the same time they are also classified as zombies
according to our algorithm. This is evident from a common core region in the
radar graph bounded by square nodes. However, the two segments do not
coincide. If we consider only the ICR, we may have zombies that are obviously
not in distress but are classified as such by our algorithm. It is noticeable
that this occurs mostly in southern and central Italy. Abruzzo (ITF1),
Basilicata (ITF5), Calabria (ITF6), Campania (ITF3), Lazio (ITI4), Molise
(ITF2), and Puglia (ITF4) host a relatively large proportion of zombies
according to MA-XGBoost.171717Not surprisingly, patterns of missing values are
also more pronounced in southern and central Italy. Between 2008 and 2016, ICR
values are missing for 81.79% of firms in central Italy and for 76.24% of
firms in southern Italy on average. In these cases, the cause of distress may
lie in other aspects of business activity, as our methodology allows us to use
information from a wide range of financial predictors covering different
aspects of firms’ economic life.

At the same time, panel (a) of Figure 10 also identifies firms that have
liquidity problems because their ICR is less than one, but our algorithm does
not classify them as zombies, possibly because they are still solvent thanks
to profitable economic activity that allows them to eventually meet their
financial obligations and thus overcome temporary liquidity constraints
shortages. This is particularly evident in North and Insular Italy, including
Emilia-Romagna (ITH5), Friuli-Venezia Giulia (ITH4), Lombardy (ITC4), Piemonte
(ITC1), Sardinia (ITG2), Tuscany (ITI1), Valle d’Aosta (ITC2) and Veneto
(ITH3).

Figure 10:  Zombie firms and geography

(a) Zombies and interest payments

(b) Zombies and value destruction

Note: The rays of the radar show, at the regional level (NUTS 2-digit), the
proportion of zombie firms versus firms that have an Interest Coverage Ratio
(ICR) of less than one (panel a) and versus firms that have negative value
added (panel b). The square nodes indicate the common areas where the segments
overlap. The circles represent the fraction of zombies that we detect using
MA-XGBoost. The triangles indicate the fraction of firms identified with
$ICR<1$ or negative value added. Along each ray, the values of the squares,
circles, and triangles sum to one. Panel (a) and (b) include a total of 30,380
and 24,351 observations, respectively.

On the other hand, in panel (b) of Figure 10, we show that a relatively small
segment of firms destroy economic value because what they sell has less value
than what they buy as inputs. They are problematic; in fact, most of them are
classified as zombies. However, there is a significant portion of zombies that
show positive value added, but are still struggling financially and are
therefore detected as zombies by our algorithm. This scenario extends over the
entire country, but peaks in some central and southern regions: Abruzzo
(ITF1), Basilicata (ITF5), Calabria (ITF6), Campania (ITF3), Lazio (ITI4),
Molise (ITF2) and Puglia (ITF4). The descriptive statistics in Figure 10
confirm our intuition that only a comprehensive battery of predictors,
including missing values, can provide a clear picture of the relevance of the
phenomenon of zombie firms. Finally, in the Appendix A, we report in FigureA.2
and TableA.5 a snapshot of the proportion of zombies in different industries
(2-digit digits NACE Rev. 2). As largely expected, firms that have problems
with interest payments ($ICR1$) are always a larger segment than firms that we
identify as zombies. On the other hand, the share of firms with negative value
added is generally smaller than the share of zombie firms.

##  5 Conclusions

In this contribution we show how statistical learning can infer non-trivial
information on a set of financial indicators, and we successfully classify
firms into risk categories after training on past failures. Our preferred
algorithm is  XGBoost, which outperforms other well-known econometric and
machine learning methods, especially in the presence of missing patterns of
financial accounts.

Thanks to our machine learning approach, we can also reduce prediction errors
compared to traditional credit scoring tools such as the Z-scores and the
Distance-to-Default. Therefore, we propose to classify as zombie firms those
firms that remain in high-risk status because they are at the right end of the
distribution of our predictions for at least three years, beyond the 9th
decile of risk, where we find that the chances of recovery to a smaller risk
of distress are minimal.

The preceding evidence suggests that identifying zombies may be crucial for
financial institutions to avoid wasting credit resources on insolvent firms
when the latter survive the market only thanks to some adverse selection
mechanisms that arise from imperfect financial markets. However, from a more
general perspective, we believe that our exercise can be useful to identify a
part of an economy that is in trouble. The issue is all the more critical
because recent studies discuss how zombies can hinder the growth potential of
many countries by impeding the redistribution of productive resources in
modern economies (Andrews et al., 2017; Banerjee and Hofmann, 2018; Andrews
and Petroulakis, 2019).

Indeed, we find that Italian zombies have lower levels of total factor
productivity and are mainly found in smaller firms. Interestingly, we note a
possible relationship with the business cycle that would be worth
investigating further in the future: the share of zombies was higher after the
two financial crises in 2008 and 2011, but then declined in recent years as
the recovery took hold.

It is beyond the scope of this article to make predictions for the post-
pandemic scenario. However, we anticipate that the problem of separating
companies that can stand on their own two feet from those that hide their
inability to pay will become all the more urgent now that policymakers have
finally withdrawn financial support programs. The challenge is to avoid
further misallocation of resources, which would slow down the much-needed
economic recovery.

## References

  * Abbasi et al. (2019) Abbasi, R. A., Javaid, N., Ghuman, M. N. J., Khan, Z. A., Ur Rehman, S., 2019. Short term load forecasting using xgboost. In: Web, Artificial Intelligence and Network Applications: Proceedings of the Workshops of the 33rd International Conference on Advanced Information Networking and Applications (WAINA-2019) 33. Springer, pp. 1120–1131.
  * Ackerberg et al. (2015) Ackerberg, D. A., Caves, K., Frazer, G., 2015. Identification properties of recent production function estimators. Econometrica 83 (6), 2411–2451.
  * Aghion et al. (2019) Aghion, P., Bergeaud, A., Cette, G., Lecat, R., Maghin, H., 2019. Coase lecture \- the inverted-u relationship between credit access and productivity growth. Economica 86 (341), 1–31.
  * Ahrens et al. (2019) Ahrens, A., Hansen, C. B., Schaffer, M. E., 2019. lassopack: Model selection and prediction with regularized regression in stata. arXiv preprint arXiv:1901.05397.
  * Ahrens et al. (2020) Ahrens, A., Hansen, C. B., Schaffer, M. E., 2020. lassopack: Model selection and prediction with regularized regression in stata. The Stata Journal 20 (1), 176–235.
  * Alaka et al. (2018) Alaka, H. A., Oyedele, L. O., Owolabi, H. A., Kumar, V., Ajayi, S. O., Akinade, O. O., Bilal, M., 2018. Systematic review of bankruptcy prediction models: Towards a framework for tool selection. Expert Systems with Applications 94, 164–184.
  * Altman (1968) Altman, E. I., 1968. Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. The Journal of Finance 23 (4), 589–609.
  * Altman et al. (2000) Altman, E. I., et al., 2000. Predicting financial distress of companies: revisiting the z-score and zeta models. Stern School of Business, New York University, 9–12.
  * Andrews et al. (2017) Andrews, D., McGowan, M. A., Millot, V., et al., 2017. Confronting the zombies: Policies for productivity revival. Tech. rep., OECD Publishing.
  * Andrews and Petroulakis (2019) Andrews, D., Petroulakis, F., Feb. 2019. Breaking the shackles: Zombie firms, weak banks and depressed restructuring in Europe. Working Paper Series 2240, European Central Bank.
  * Athey (2018) Athey, S., 2018. The impact of machine learning on economics. In: The Economics of Artificial Intelligence: An Agenda. University of Chicago Press.
  * Banerjee and Hofmann (2018) Banerjee, R., Hofmann, B., 2018. The rise of zombie firms: causes and consequences. BIS Quarterly Review Spetember.
  * Bank of England (2013) Bank of England, 2013. Inflation report. Tech. rep., Bank of England, London.
  * Bank of Korea (2013) Bank of Korea, 2013. Financial stability report. Tech. rep., Bank of Korea, Seoul.
  * Bargagli-Stoffi (2020) Bargagli-Stoffi, F. J., 2020. Essays on applied machine learning.
  * Bargagli-Stoffi et al. (2020) Bargagli-Stoffi, F. J., Cevolani, G., Gnecco, G., 2020. Should simplicity be always preferred to complexity in supervised machine learning? In: 6th International Conference on machine Learning, Optimization Data science (LOD2020). Lecture Notes in Computer Science. Springer.
  * Bargagli Stoffi et al. (2022) Bargagli Stoffi, F. J., Cevolani, G., Gnecco, G., 2022. Simple models in complex worlds: Occam’s razor and statistical learning theory. Minds and Machines 32 (1), 13–42.
  * Bargagli-Stoffi et al. (2022) Bargagli-Stoffi, F. J., De Witte, K., Gnecco, G., 2022. Heterogeneous causal effects with imperfect compliance: a bayesian machine learning approach. The Annals of Applied Statistics 16 (3), 1986–2009.
  * Bargagli Stoffi and Gnecco (2020) Bargagli Stoffi, F. J., Gnecco, G., 2020. Causal tree with instrumental variable: an extension of the causal tree framework to irregular assignment mechanisms. International Journal of Data Science and Analytics 9 (3), 315–337.
  * Bargagli-Stoffi et al. (2021) Bargagli-Stoffi, F. J., Niederreiter, J., Riccaboni, M., 2021. Supervised learning for the prediction of firm dynamics. In: Data science for economics and finance. Springer, Cham, pp. 19–41.
  * Behr and Weinblat (2017) Behr, A., Weinblat, J., 2017. Default patterns in seven eu countries: A random forest approach. International Journal of the Economics of Business 24 (2), 181–222.
  * Bellman (1961) Bellman, R. E., 1961. Adaptive Control Processes. Princeton University Press.
  * Belloni et al. (2016a) Belloni, A., Chernozhukov, V., Hansen, C., Kozbur, D., 2016a. Inference in high-dimensional panel models with an application to gun control. Journal of Business & Economic Statistics 34 (4), 590–605.
  * Belloni et al. (2016b) Belloni, A., Chernozhukov, V., Wei, Y., 2016b. Post-selection inference for generalized linear models with many controls. Journal of Business & Economic Statistics 34 (4), 606–619.
  * Bentéjac et al. (2021) Bentéjac, C., Csörgő, A., Martínez-Muñoz, G., 2021. A comparative analysis of gradient boosting algorithms. Artificial Intelligence Review 54, 1937–1967.
  * Black and Scholes (1973) Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of political economy 81 (3), 637–654.
  * Bonfim et al. (2020) Bonfim, D., Cerqueiro, G., Degryse, H., Ongena, S., 2020. On-site inspecting zombie lending. Available at SSRN 3530574.
  * Brédart (2014) Brédart, X., 2014. Bankruptcy prediction model using neural networks. Accounting and Finance Research 3 (2), 124–128.
  * Breiman (1996) Breiman, L., 1996. Bagging predictors. Machine learning 24, 123–140.
  * Breiman (2001) Breiman, L., 2001. Random forests. Machine learning 45 (1), 5–32.
  * Breiman et al. (1984) Breiman, L., Friedman, J. H., Olshen, R. A., Stone, C. J., 1984. Classification and regression trees. Belmont, CA: Wadsworth & Brooks.
  * Brodersen et al. (2010) Brodersen, K. H., Ong, C. S., Stephan, K. E., Buhmann, J. M., 2010. The balanced accuracy and its posterior distribution. In: 2010 20th International Conference on Pattern Recognition. IEEE, pp. 3121–3124.
  * Buckmann et al. (2021) Buckmann, M., Joseph, A., Robertson, H., 2021. An interpretable machine learning workflow with an application to economic forecasting.
  * Bugamelli et al. (2018) Bugamelli, M., Lotti, F., Amici, M., Ciapanna, E., Colonna, F., D’Amuri, F., Giacomelli, S., Linarello, A., Manaresi, F., Palumbo, G., Sc, F., Jan. 2018. Productivity growth in Italy: a tale of a slow-motion change. Questioni di Economia e Finanza (Occasional Papers) 422, Bank of Italy, Economic Research and International Relations Area.
  * Caballero et al. (2008) Caballero, R. J., Hoshi, T., Kashyap, A. K., 2008. Zombie lending and depressed restructuring in japan. American Economic Review 98 (5), 1943–77.
  * Calligaris et al. (2016) Calligaris, S., Del Gatto, M., Hassan, F., Ottaviano, G. I., Schivardi, F., et al., 2016. Italy’s productivity conundrum. a study on resource misallocation in italy. Tech. rep., Directorate General Economic and Financial Affairs (DG ECFIN), European Commission.
  * Calligaris et al. (2018) Calligaris, S., Del Gatto, M., Hassan, F., Ottaviano, G. I. P., Schivardi, F., 09 2018. The productivity puzzle and misallocation: an Italian perspective. Economic Policy 33 (96), 635–684.
  * Chen and Chen (2008) Chen, J., Chen, Z., 2008. Extended bayesian information criteria for model selection with large model spaces. Biometrika 95 (3), 759–771.
  * Chen and Guestrin (2016) Chen, T., Guestrin, C., 2016. Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd acm sigkdd international conference on knowledge discovery and data mining. pp. 785–794.
  * Chipman et al. (2010) Chipman, H. A., George, E. I., McCulloch, R. E., et al., 2010. Bart: Bayesian additive regression trees. The Annals of Applied Statistics 4 (1), 266–298.
  * Cravino and Levchenko (2016) Cravino, J., Levchenko, A. A., 11 2016. Multinational Firms and International Business Cycle Transmission. The Quarterly Journal of Economics 132 (2), 921–962.
  * Davies et al. (2023) Davies, L., Kattenberg, M., Vogt, B., 2023. Predicting firm exits with machine learning: Implications for selection into covid-19 support and productivity growth. Tech. rep., CPB Netherlands Bureau for Economic Policy Analysis.
  * De Martiis et al. (2020) De Martiis, A., Heil, T. L., Peter, F. J., 2020. Are you a zombie? a supervised learning method to classify unviable firms and identify the determinants.
  * Fawcett (2006) Fawcett, T., 2006. An introduction to roc analysis. Pattern Recognition Letters 27 (8), 861–874.
  * Fazzari et al. (1988) Fazzari, S. M., Hubbard, R. G., Petersen, B. C., 1988. Financing Constraints and Corporate Investment. Brookings Papers on Economic Activity 19 (1), 141–206.
  * Ferrando and Ruggieri (2018) Ferrando, A., Ruggieri, A., 2018. Financial constraints and productivity: Evidence from euro area companies. International Journal of Finance & Economics 23 (3), 257–282.
  * Freund et al. (1996) Freund, Y., Schapire, R. E., et al., 1996. Experiments with a new boosting algorithm. In: icml. Vol. 96. Citeseer, pp. 148–156.
  * Gal (2013) Gal, P. N., 2013. Measuring total factor productivity at the firm level using oecd-orbis.
  * Gopinath et al. (2017) Gopinath, G., Kalemli-Özcan, Ş., Karabarbounis, L., Villegas-Sanchez, C., 2017. Capital allocation and productivity in south europe. The Quarterly Journal of Economics 132 (4), 1915–1967.
  * Gumus and Kiran (2017) Gumus, M., Kiran, M. S., 2017. Crude oil price forecasting using xgboost. In: 2017 International conference on computer science and engineering (UBMK). IEEE, pp. 1100–1103.
  * Hadlock and Pierce (2010) Hadlock, C. J., Pierce, J. R., 2010. New evidence on measuring financial constraints: Moving beyond the kz index. The Review of Financial Studies 23 (5), 1909–1940.
  * Hanley and McNeil (1982) Hanley, J. A., McNeil, B. J., 1982. The meaning and use of the area under a receiver operating characteristic (roc) curve. Radiology 143 (1), 29–36.
  * He et al. (2010) He, Y., Zaslavsky, A. M., Landrum, M., Harrington, D., Catalano, P., 2010. Multiple imputation in a large-scale complex survey: a practical guide. Statistical methods in medical research 19 (6), 653–670.
  * Hernández et al. (2018) Hernández, B., Raftery, A. E., Pennington, S. R., Parnell, A. C., 2018. Bayesian additive regression trees using bayesian model averaging. Statistics and Computing 28 (4), 869–890.
  * Hosaka (2019) Hosaka, T., 2019. Bankruptcy prediction using imaged financial ratios and convolutional neural networks. Expert systems with Applications 117, 287–299.
  * Hothorn et al. (2006) Hothorn, T., Hornik, K., Zeileis, A., 2006. Unbiased recursive partitioning: A conditional inference framework. Journal of Computational and Graphical statistics 15 (3), 651–674.
  * Incerti et al. (2022) Incerti, F., Bargagli-Stoffi, F. J., Riccaboni, M., 2022. A two-country study of default risk prediction using bayesian machine-learning. In: International Conference on Machine Learning, Optimization, and Data Science. Springer, pp. 188–192.
  * Josse et al. (2019) Josse, J., Prost, N., Scornet, E., Varoquaux, G., 2019. On the consistency of supervised learning with missing values. arXiv preprint arXiv:1902.06931.
  * Kalemli-Ozcan et al. (2015) Kalemli-Ozcan, S., Sorensen, B., Villegas-Sanchez, C., Volosovych, V., Yesiltas, S., 2015. How to construct nationally representative firm level data from the orbis global database. Tech. rep., National Bureau of Economic Research.
  * Kapelner and Bleich (2015) Kapelner, A., Bleich, J., 2015. Prediction with missing data via bayesian additive regression trees. Canadian Journal of Statistics 43 (2), 224–239.
  * Kapelner and Bleich (2016) Kapelner, A., Bleich, J., 2016. bartmachine: Machine learning with bayesian additive regression trees. Journal of Statistical Software, Articles 70 (4), 1–40.
URL https://www.jstatsoft.org/v070/i04

  * Kim et al. (2016) Kim, B., Khanna, R., Koyejo, O. O., 2016. Examples are not enough, learn to criticize! criticism for interpretability. In: Advances in Neural Information Processing Systems. pp. 2280–2288.
  * Kleinberg et al. (2015) Kleinberg, J., Ludwig, J., Mullainathan, S., Obermeyer, Z., 2015. Prediction policy problems. American Economic Review 105 (5), 491–95.
  * Lee et al. (2020) Lee, K., Bargagli-Stoffi, F. J., Dominici, F., 2020. Causal rule ensemble: Interpretable inference of heterogeneous treatment effects. arXiv preprint arXiv:2009.09036.
  * Lee et al. (1996) Lee, K. C., Han, I., Kwon, Y., 1996. Hybrid neural network models for bankruptcy predictions. Decision Support Systems 18 (1), 63–72.
  * Li et al. (2020) Li, M., Fu, X., Li, D., 2020. Diabetes prediction based on xgboost algorithm. In: IOP conference series: materials science and engineering. Vol. 768. IOP Publishing.
  * Linero (2018) Linero, A. R., 2018. Bayesian regression trees for high-dimensional prediction and variable selection. Journal of the American Statistical Association 113 (522), 626–636.
  * Linero and Yang (2018) Linero, A. R., Yang, Y., 2018. Bayesian regression tree ensembles that adapt to smoothness and sparsity. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 80 (5), 1087–1110.
  * Linn and Weagley (2019) Linn, M., Weagley, D., 2019. Estimating financial constraints with machine learning. Available at SSRN 3375048.
  * Little and Rubin (2019) Little, R. J., Rubin, D. B., 2019. Statistical analysis with missing data. Vol. 793\. John Wiley & Sons.
  * Loh (2002) Loh, W.-Y., 2002. Regression tress with unbiased variable selection and interaction detection. Statistica sinica, 361–386.
  * Lundberg et al. (2020) Lundberg, S. M., Erion, G., Chen, H., DeGrave, A., Prutkin, J. M., Nair, B., Katz, R., Himmelfarb, J., Bansal, N., Lee, S.-I., 2020. From local explanations to global understanding with explainable ai for trees. Nature machine intelligence 2 (1), 56–67.
  * Makridakis et al. (2008) Makridakis, S., Wheelwright, S. C., Hyndman, R. J., 2008. Forecasting methods and applications. John wiley & sons.
  * McGowan et al. (2018) McGowan, M. A., Andrews, D., Millot, V., 2018. The walking dead? zombie firms and productivity performance in oecd countries. Economic Policy 33 (96), 685–736.
  * Merton (1974) Merton, R. C., 1974. On the pricing of corporate debt: The risk structure of interest rates. The Journal of finance 29 (2), 449–470.
  * Miller (2019) Miller, T., 2019. Explanation in artificial intelligence: Insights from the social sciences. Artificial Intelligence 267, 1–38.
  * Modigliani and Miller (1958) Modigliani, F., Miller, M. H., 1958. The cost of capital, corporation finance and the theory of investment. The American Economic Review 48 (3), 261–297.
  * Molnar (2020) Molnar, C., 2020. Interpretable machine learning. Lulu. com.
  * Moscatelli et al. (2019) Moscatelli, M., Narizzano, S., Parlapiano, F., Viggiano, G., et al., 2019. Corporate default forecasting with machine learning. Tech. rep., Bank of Italy, Economic Research and International Relations Area.
  * Mullainathan and Spiess (2017) Mullainathan, S., Spiess, J., 2017. Machine learning: an applied econometric approach. Journal of Economic Perspectives 31 (2), 87–106.
  * Murray (2017) Murray, J. S., 2017. Log-linear bayesian additive regression trees for categorical and count responses. arXiv preprint, arXiv:1701.01503.
  * Nickell and Nicolitsas (1999) Nickell, S., Nicolitsas, D., 1999. How does financial pressure affect firms? European Economic Review 43 (8), 1435–1456.
  * Odén et al. (1975) Odén, A., Wedel, H., et al., 1975. Arguments for fisher’s permutation test. The Annals of Statistics 3 (2), 518–520.
  * Ohlson (1980) Ohlson, J. A., 1980. Financial ratios and the probabilistic prediction of bankruptcy. Journal of accounting research, 109–131.
  * Orbis (2020) Orbis, 2020. Orbis company information across the globe. Bureau van diijk, A Moody’s Analytics Company.
URL https://orbis.bvdinfo.com/

  * Peek and Rosengren (2005) Peek, J., Rosengren, E. S., 2005. Unnatural selection: Perverse incentives and the misallocation of credit in japan. American Economic Review 95 (4), 1144–1166.
  * Prokhorenkova et al. (2018) Prokhorenkova, L., Gusev, G., Vorobev, A., Dorogush, A. V., Gulin, A., 2018. Catboost: unbiased boosting with categorical features. Advances in neural information processing systems 31.
  * Rajan and Zingales (1995) Rajan, R. G., Zingales, L., December 1995. What Do We Know about Capital Structure? Some Evidence from International Data. Journal of Finance 50 (5), 1421–1460.
  * Ribeiro et al. (2016) Ribeiro, M. T., Singh, S., Guestrin, C., 2016. Model-agnostic interpretability of machine learning. arXiv preprint arXiv:1606.05386.
  * Riccaboni et al. (2021) Riccaboni, M., Wang, X., Zhu, Z., 2021. Firm performance in networks: The interplay between firm centrality and corporate group size. Journal of Business Research 129, 641–653.
  * Rozemberczki et al. (2022) Rozemberczki, B., Watson, L., Bayer, P., Yang, H.-T., Kiss, O., Nilsson, S., Sarkar, R., 2022. The shapley value in machine learning. arXiv preprint arXiv:2202.05594.
  * Rungi and Biancalani (2019) Rungi, A., Biancalani, F., 2019. Heterogeneous firms and the north–south divide in italy. Italian Economic Journal, 1–23.
  * Saito and Rehmsmeier (2015) Saito, T., Rehmsmeier, M., 2015. The precision-recall plot is more informative than the roc plot when evaluating binary classifiers on imbalanced datasets. PloS one 10 (3).
  * Schivardi and Romano (2020) Schivardi, F., Romano, G., 2020. A simple method to estimate firms liquidity needs during the COVID-19 crisis with an application to Italy. CEPR Covid Economics: Vetted and Real-Time Paper.
  * Schivardi et al. (2020) Schivardi, F., Sette, E., Tabellini, G., 2020. Identifying the real effects of zombie lending. The Review of Corporate Finance Studies 9 (3), 569–592.
  * Schivardi et al. (2021) Schivardi, F., Sette, E., Tabellini, G., 2021. Credit misallocation during the european financial crisis credit misallocation during the crisis. The Economic Journal.
  * Shmueli et al. (2010) Shmueli, G., et al., 2010. To explain or to predict? Statistical science 25 (3), 289–310.
  * Shrikumar et al. (2017) Shrikumar, A., Greenside, P., Kundaje, A., 2017. Learning important features through propagating activation differences. In: International Conference on Machine Learning. PMLR, pp. 3145–3153.
  * Stoffi et al. (2021) Stoffi, F. J. B., De Beckker, K., Maldonado, J. E., De Witte, K., 2021. Assessing sensitivity of machine learning predictions. a novel toolbox with an application to financial literacy. arXiv preprint arXiv:2102.04382.
  * Strumbelj and Kononenko (2010) Strumbelj, E., Kononenko, I., 2010. An efficient explanation of individual classifications using game theory. The Journal of Machine Learning Research 11, 1–18.
  * Sun and Li (2011) Sun, J., Li, H., 2011. Dynamic financial distress prediction using instance selection for the disposal of concept drift. Expert Systems with Applications 38 (3), 2566–2576.
  * Tsai et al. (2014) Tsai, C.-F., Hsu, Y.-F., Yen, D. C., 2014. A comparative study of classifier ensembles for bankruptcy prediction. Applied Soft Computing 24, 977–984.
  * Tsai and Wu (2008) Tsai, C.-F., Wu, J.-W., 2008. Using neural network ensembles for bankruptcy prediction and credit scoring. Expert systems with applications 34 (4), 2639–2649.
  * Twala et al. (2008) Twala, B., Jones, M., Hand, D. J., 2008. Good methods for coping with missing data in decision trees. Pattern Recognition Letters 29 (7), 950–956.
  * Udo (1993) Udo, G., 1993. Neural network performance on the bankruptcy classification problem. Computers & industrial engineering 25 (1-4), 377–380.
  * Van der Laan et al. (2007) Van der Laan, M. J., Polley, E. C., Hubbard, A. E., 2007. Super learner. Statistical Applications in Genetics and Molecular Biology 6 (1).
  * Van Rijsbergen (1979) Van Rijsbergen, C. J., 1979. Information retrieval. London: Butterworths.
  * Wang et al. (2014) Wang, G., Ma, J., Yang, S., 2014. An improved boosting based on feature selection for corporate bankruptcy prediction. Expert Systems with Applications 41 (5), 2353–2361.
  * White et al. (2018) White, T. K., Reiter, J. P., Petrin, A., 2018. Imputation in us manufacturing data and its implications for productivity dispersion. Review of Economics and Statistics 100 (3), 502–509.

Appendix

##  Appendix A Tables and graphs

Table A.1: Panel (A): List of predictors for firms’ failures. Variables |  Description
---|---
Material Costs (FA), Costs of Employees (FA), Added Value (FA), Taxation, Tax and Pensions’ Payables (FA), Revenues (FA), Financial Expenses (FA), Interest Payments (FA), Cash Flow (FA), Fixed Assets (FA), Current Assets (FA), Shareholders’ Funds (FA), Retained Earnings (FA), Long-Term Debt (FA), Loans (FA), Current Liabilities (FA), EBITDA (Earnings before interest, Taxation, Depreciation and Amortization) (PFT), Intangible Fixed Assets (FC), Total Assets (SI), Net Income (SI), Number of Employees (SI) |  Original accounts (expressed in euro)
|
Capital Intensity (FA) |  Fixed Assets/Number of employees.
|
Corporate Control (G) |  A binary variable equal to one if a firm belongs to a corporate group.
|
Consolidated Accounts (G) |  A binary variable equal to one if the firm consolidates accounts of its subsidiaries
|
Number of Patents (I) |  The portfolio of patents granted to a firm by patent offices (Dummy Patents equal to 0 if the firm issued no patents, and 1 otherwise).
|
Number of Trademarks (I) |  The total number of trademarks issued to the firm by national or international trademark offices (Dummy Trademarks equal to 0 if the firm issued no trademarks, and 1 otherwise).
|
NACE rev. 2 (SE) |  A 4-digit industry affiliation following European classification NACE rev. 2.
|
NUTS 2 regions (A) |  The region in which the company is located.
|
TFP (PDT) |  It is the Total Factor Productivity of a firm computed as in Ackerberg et al. (2015).
Table A.2: Panel (B): List of predictors for firms’ failures. Variables |  Description
---|---
Interest Benchmarking (ZI) |  It is a zombie proxy proposed by Caballero et al. (2008) and calculated as $R^{*}=rs_{t-1}BS_{i,t-1}+(\frac{1}{5}\sum_{j=1}^{5}rl_{t-j})BL_{i,t-1}+rcb_{5y,t}\cdot Bonds_{i,t-1}$, where $BS_{i,t-1}$ are short-term bank loans, $BL_{i,t-1}$ are long-term bank loans, $rs_{t-1}$ are the average short-term prime rate in year $t$, $rl_{t-j}$ is the average long-term prime rate in year $t$, $Bonds$ are the total outstanding bonds, $rcb_{5y,t}$ is the minimum observed rate on any convertible corporate bond issued over the previous five years.
Interest Coverage Ratio (ZI) |  It is calculated as EBIT/Interest Expenses. When it is less than one for three consecutive years and the firm is at least ten years old, then Bank of Korea (2013), McGowan et al. (2018) and Banerjee and Hofmann (2018) assume a firm is a zombie.
Financial Misallocation (ZI) |  It is a binary indicator adopted by Schivardi et al. (2021) for catching zombie lending, based on both $ROA\frac{\frac{1}{3}\sum_{t=1}^{3}EBITDA_{t}}{Total\>Assets}<prime$ and $Leverage=\frac{Financial\>Debt}{Total\>Assets}>\tilde{L}$, where $prime$ is the measure of the cost of capital for firms with a Z-score equal to 1 or 2, and where $\tilde{L}$ is the median value of leverage in the current year for firms that exited in two following years.
Negative value added (ZI) |  It is a binary variable to spotzombie firms (McGowan et al., 2018), equal to one when the value added is negative, i.e. when the value of sold output is less than purchases of intermediate inputs.
Profitability (ZI) |  Calculated as $EBITDA/Total\>Assets$, and adopted by Schivardi et al. (2021) as a control for zombie lending
Financial Constraints (FC) |  It is a proxy of financial constraints as in Nickell and Nicolitsas (1999), calculated as a ratio between interest payments and cash flow
Size-Age (FC) |  It is a synthetic indicator proposed by Hadlock and Pierce (2010), equal to $-0.737*\log(totalassets)+0.043*\log(totalassets))^{2}-0.040*age$.
Financial Sustainability (FC) |  It is a ratio calculated as Financial Expenses over Operating Revenues.
Capital Adequacy Ratio (FC) |  It is a ratio of Shareholders’ Funds over Short and Long Term Debts.
Liquidity Ratio (FA) |  (Current/Assets - Stocks)/Current/Liabilities
Solvency Ratio (FA) |  (Shareholders funds / (Non current liabilities + Current liabilities)) * 100
Liquidity Returns (FA) |  It is the ratio of cash flow over total asset
Tax and Pension Payables (FA) |  It is the ratio of the sum of tax and pension payables over total assets.
Table A.3: Missing predictors and firms’ failures - Chi-square tests

| Firm’s failure |
---|---|---
| 0 | 1 | Test Statistic
| $N=287587$ | $N=17319$ |
Interest Benchmarking : 0 | 38%  (110524) | 61%  (10530) |  $\chi^{2}_{1}$=3414.25, P<0.001
Interest Benchmarking : 1 | 62%  (177063) | 39%  (6789) |
Interest Coverage Ratio : 0 | 37%  (105907) | 49%  (8422) |  $\chi^{2}_{1}$=970.93, P<0.001
Interest Coverage Ratio : 1 | 63%  (181680) | 51%  (8897) |
Negative value added : 0 | 34%  ( 98014) | 63%  (10915) |  $\chi^{2}_{1}$=5958.81, P<0.001
Negative value added : 1 | 66%  (189573) | 37%  (6404) |
Financial Constraint : 0 | 37%  (105904) | 49%  (8419) |  $\chi^{2}_{1}$=968.27, P<0.001
Financial Constraint : 1 | 63%  (181683) | 51%  (8900) |
Financial Misallocation : 0 | 39%  (112560) | 54%  (9276) |  $\chi^{2}_{1}$=1415.82, P<0.001
Financial Misallocation : 1 | 61%  (175027) | 46%  (8043) |
Total Factor Productivity : 0 | 36%  (104345) | 38%  (6600) |  $\chi^{2}_{1}$=23.52, P<0.001
Total Factor Productivity : 1 | 64%  (183242) | 62%  (10719) |
Solvency Ratio : 0 | 41%  (118851) | 63%  (10897) |  $\chi^{2}_{1}$=3115.5, P<0.001
Solvency Ratio : 1 | 59%  (168736) | 37%  (6422) |
Liquidity Ratio : 0 | 42%  (119357) | 72%  (12543) |  $\chi^{2}_{1}$=6362.72, P<0.001
Liquidity Ratio : 1 | 58%  (168230) | 28%  (4776) |
Size-Age : 0 | 42%  (120260) | 75%  (12989) |  $\chi^{2}_{1}$=7310.19, P<0.001
Size-Age : 1 | 58%  (167327) | 25%  (4330) |
Liquidity Returns : 0 | 39%  (112561) | 54%  (9277) |  $\chi^{2}_{1}$=1416.88, P<0.001
Liquidity Returns : 1 | 61%  (175026) | 46%  (8042) |
Labour Productivity : 0 | 34%  ( 97253) | 36%  (6221) |  $\chi^{2}_{1}$=32.23, P<0.001
Labour Productivity : 1 | 66%  (190334) | 64%  (11098) |
Profitability : 0 | 37%  (105907) | 49%  (8422) |  $\chi^{2}_{1}$=970.93, P<0.001
Profitability : 1 | 63%  (181680) | 51%  (8897) |
Financial Sustainability : 0 | 41%  (117294) | 64%  (11119) |  $\chi^{2}_{1}$=3673.94, P<0.001
Financial Sustainability : 1 | 59%  (170293) | 36%  (6200) |
Capital Intensity : 0 | 36%  (104122) | 42%  (7325) |  $\chi^{2}_{1}$=261.17, P<0.001
Capital Intensity : 1 | 64%  (183465) | 58%  (9994) |

  * •

Note: Chi-square tests for the null hypothesis that missing predictors do not
correlate with the event of failure. Number of observations in parentheses.

Figure A.1:  Patterns of missingness over time across firms

Panel (A): Share of missing values out of 287,787 non-failing firms.

Panel (B): Share of missing values out of 7,221 firms in liquidation.

Panel (C): Share of missing values out of 4,718 bankruptcies and other
dissolved firms.

Table A.4:  Zombies, geography and selected financial indicators

|  | Interest payments | Value generation
---|---|---|---
Region | NUTS 2 | Common support | Zombies | ICR$<1$ | Common support | Zombies | Value added$<0$
Abruzzo | ITF1 | 0.16 | 0.48 | 0.36 | 0.15 | 0.57 | 0.27
Basilicata | ITF5 | 0.19 | 0.53 | 0.29 | 0.12 | 0.66 | 0.22
Calabria | ITF6 | 0.20 | 0.50 | 0.30 | 0.13 | 0.64 | 0.23
Campania | ITF3 | 0.16 | 0.57 | 0.27 | 0.12 | 0.62 | 0.25
Emilia-Romagna | ITH5 | 0.14 | 0.29 | 0.57 | 0.14 | 0.46 | 0.39
Friuli-Venezia Giulia | ITH4 | 0.15 | 0.26 | 0.59 | 0.14 | 0.49 | 0.37
Lazio | ITI4 | 0.15 | 0.55 | 0.31 | 0.13 | 0.62 | 0.25
Liguria | ITC3 | 0.20 | 0.37 | 0.43 | 0.14 | 0.54 | 0.32
Lombardia | ITC4 | 0.15 | 0.28 | 0.57 | 0.14 | 0.48 | 0.38
Marche | ITI3 | 0.17 | 0.41 | 0.42 | 0.16 | 0.53 | 0.31
Molise | ITF2 | 0.13 | 0.46 | 0.41 | 0.15 | 0.57 | 0.29
Piemonte | ITC1 | 0.15 | 0.29 | 0.56 | 0.15 | 0.46 | 0.39
Puglia | ITF4 | 0.19 | 0.49 | 0.33 | 0.15 | 0.58 | 0.26
Sardegna | ITG2 | 0.17 | 0.30 | 0.52 | 0.15 | 0.54 | 0.31
Sicilia | ITG1 | 0.18 | 0.41 | 0.41 | 0.14 | 0.53 | 0.33
Trentino-Alto Adige | ITH1/2 | 0.18 | 0.46 | 0.36 | 0.17 | 0.53 | 0.30
Toscana | ITI1 | 0.16 | 0.25 | 0.59 | 0.12 | 0.42 | 0.46
Umbria | ITI2 | 0.21 | 0.36 | 0.43 | 0.17 | 0.46 | 0.37
Valle d’Aosta | ITC2 | 0.06 | 0.28 | 0.66 | 0.14 | 0.45 | 0.41
Veneto | ITH3 | 0.15 | 0.30 | 0.54 | 0.15 | 0.45 | 0.40
Num. obs. |  | 30,380 | 24,351

We report the proportions of zombies against firms with ICR lower than one and
against firms with negative value-added for each NUTS 2-digit region in Italy.
Common support indicates the overlap between zombie firms and the firms that
have problems with interest payments ($ICR<1$) and have a negative value-
added, respectively.

Figure A.2:  Zombie firms and industries

(a) Zombies and interest payments

(b) Zombies and value destruction

The rays of the radar show, at the industry level (NACE Rev. 2, 2-digit
manufacturing codes), zombie firms against firms that report Interest Coverage
Ratio (ICR) of less than one (panel a) and the proportion of firms reporting
negative value added (panel b). The square nodes indicate the common areas
where the segments overlap. The circles represent the fraction of zombies that
we detect with MA-XGBoost. The triangles indicate the fraction of firms
identified with $ICR1$ or negative value added, alternatively. Along each ray,
the values of the squares, circles, and triangles sum to one. Panel (a) and
(b) include a total of 30,380 and 24,351 observations, respectively.

Table A.5:  Zombies, industries and viability indicators

|  | Interest payments | Value generation
---|---|---|---
Manufacturing of | NACE | Common support | Zombies | ICR$<1$ | Common support | Zombies | Value added$<0$
Food products | 10 | 0.18 | 0.31 | 0.50 | 0.15 | 0.45 | 0.40
Beverages | 11 | 0.33 | 0.33 | 0.33 | 0.12 | 0.50 | 0.38
Tobacco products | 12 | 0.19 | 0.35 | 0.46 | 0.16 | 0.55 | 0.29
Textiles | 13 | 0.19 | 0.49 | 0.31 | 0.17 | 0.51 | 0.32
Wearing apparel | 14 | 0.15 | 0.53 | 0.32 | 0.13 | 0.55 | 0.32
Leather | 15 | 0.17 | 0.34 | 0.49 | 0.17 | 0.49 | 0.34
Wood and cork products | 16 | 0.12 | 0.31 | 0.57 | 0.13 | 0.56 | 0.31
Paper products | 17 | 0.15 | 0.32 | 0.53 | 0.16 | 0.54 | 0.30
Printing and reproduction of recorded media | 18 | 0.14 | 0.34 | 0.52 | 0.11 | 0.44 | 0.45
Coke and refined petroleum products | 19 | 0.13 | 0.34 | 0.53 | 0.14 | 0.46 | 0.41
Chemicals and chemical products | 20 | 0.19 | 0.36 | 0.45 | 0.10 | 0.49 | 0.41
Pharmaceutical products and preparations | 21 | 0.12 | 0.30 | 0.57 | 0.13 | 0.53 | 0.34
Rubber and plastic products | 22 | 0.18 | 0.31 | 0.52 | 0.17 | 0.51 | 0.32
Other non metallic mineral products | 23 | 0.12 | 0.26 | 0.61 | 0.12 | 0.49 | 0.39
Basic metals | 24 | 0.14 | 0.39 | 0.46 | 0.14 | 0.59 | 0.27
Fabricated metal products | 25 | 0.16 | 0.41 | 0.44 | 0.11 | 0.53 | 0.35
Computer, electronic and optical products | 26 | 0.16 | 0.39 | 0.44 | 0.14 | 0.54 | 0.32
Electrical equipment | 27 | 0.15 | 0.37 | 0.48 | 0.14 | 0.52 | 0.34
Machinery and equipment n.e.c. | 28 | 0.17 | 0.33 | 0.50 | 0.16 | 0.50 | 0.35
Motor vehicle, trailers and semi-trailers | 29 | 0.18 | 0.45 | 0.37 | 0.18 | 0.49 | 0.32
Other transport equipment | 30 | 0.19 | 0.35 | 0.46 | 0.19 | 0.52 | 0.30
Forniture | 31 | 0.19 | 0.40 | 0.41 | 0.16 | 0.51 | 0.33
Other manufacturing | 32 | 0.12 | 0.51 | 0.37 | 0.10 | 0.55 | 0.35
Num. obs. |  | 30,380 | 24,351

We report for 2-digit industries (NACE Rev. 2) the shares of zombies compared
to firms with an ICR of less than one and compared to firms with negative
value added. The common column contains the proportions of firms classified as
zombies by our algorithm that have either an ICR of less than one or negative
value added.

##  Appendix B Methodologies

###  B.1 XGBoost

XGBoost is a gradient- boosting algorithm introduced by Chen and Guestrin
(2016). The algorithm uses a standard boosting method where $J$ decision trees
are sequentially created to approximate the outcome. Each tree uses the
information learned from the previous trees, and the final model can be
expressed as follows:

$Y_{i,t}=\sum_{j=1}^{J}\mathcal{T}_{j}\left(\mathbf{X}_{i,t-1};\mathcal{D}_{j},\mathcal{W}_{j}\right)+\epsilon_{i,t-1}$

where
$\mathcal{T}_{j}\left(\mathbf{X}_{i,t-1};\mathcal{D}_{j},\mathcal{W}_{j}\right)$
corresponds to an independent tree with structure $\mathcal{D}_{j}$ and leaf
weights $\mathcal{W}_{j}$. Note that $\epsilon_{i,t-1}$ is typically assumed
to be zero-mean, but no probabilistic assumptions are made about it. The model
approximation is built additively, minimizing the loss function iteratively.
The loss function includes a regularization term to penalize the complexity of
the model and avoid overfitting, and has the following form:

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}L\left(\hat{Y}_{i,t},Y_{i,t}\right)+\sum_{j=1}^{J}\Omega\left(\mathcal{T}_{j}\right)$
$\displaystyle\Omega(\mathcal{T}_{j})$ $\displaystyle=$ $\displaystyle\gamma
T_{j}+\frac{1}{2}\lambda\|\mathcal{W}_{j}\|^{2}$

where $T_{j}$ and $\mathcal{W}_{j}$ represent the number and weights of the
leaves of the $j$-th tree, respectively, while $\gamma$ and $\lambda$ are
regularization parameters used to reduce complexity and avoid overfitting.

XGBoost is characterized by the fact that it includes several tools that
increase the training speed and deal with sparsity in the data (Bentéjac et
al., 2021). In traditional algorithms for splitting data, finding the best
splits is quite time consuming. To be efficient, each variable is sorted and
all potential splits are explored to determine the optimal one. XGBoost
overcomes this problem by using a compressed column-based structure with
parallel processing to avoid repeated sorting of the data. The algorithm for
finding the best split is now based on the percentiles of the features, so
only a subset of the possible splits is examined, which significantly
increases the training speed. In addition, XGBoost uses default directions to
handle sparsity patterns. For missing values, if a feature required for a
split is missing, XGBoost assigns the observation to a default direction (left
or right) learned from the data. This approach is useful for handling and
incorporating information about missing values into the model.

###  B.2 BART-MIA

BART is a Bayesian sum-of-trees ensemble algorithm and its approach is based
on a fully Bayesian probability model (Kapelner and Bleich, 2015, 2016). The
revised version of the BART model we use for our predictions can be expressed
as follows:

$\centering
Y_{i,t}=\sum_{j=1}^{J}\mathcal{T}_{j}(\mathbf{X}_{i,t-1};\mathcal{D}_{j},\mathcal{M}_{j})+\epsilon_{i,t-1},\>\>\>\>\>\>\>\>\>\epsilon_{i,t-1}\sim\mathcal{N}(0,\sigma^{2}),\@add@centering$
(5)

where each
$\mathcal{T}_{j}(\mathbf{X}_{i,t-1};\mathcal{D}_{j},\mathcal{M}_{j})$ denotes
a unique binary tree. Each $\mathcal{T}_{j}$ is a function that sorts each
unit into one of the sets of $m_{j}$ terminal nodes associated with mean
parameters $\mathcal{M}_{j}\\{\mu_{1},...,\mu_{m_{j}}\\}$ based on a set of
decision rules, $\mathcal{D}_{j}$. The error terms $\epsilon_{i,t-1}$ are
usually assumed to be independent and identically normally distributed if the
outcome is continuous (Chipman et al., 2010). The Bayesian component of the
algorithm is incorporated in a set of three different priors on: (i) the
structure of the trees, $\mathcal{D}_{j}$ (this prior aims to limit the
complexity of each tree $\mathcal{T}$ and serves as a regularization tool);
(ii) the distribution of the outcome in the nodes, $\mathcal{M}_{j}$ (this
prior aims to reduce the node predictions to the center of the distribution of
the response variable $Y$); (iii) the error variance $\sigma^{2}$ (which
bounds away $\sigma^{2}$ from very small values that would cause the algorithm
to overfit the training data)181818The choice of priors and the derivation of
posterior distributions are discussed in detail in Chipman et al. (2010) and
Kapelner and Bleich (2016). Namely: (i) the prior on the probability that a
node splits at depth $k$ is $\beta(1k)^{-\eta}$, where
$\beta\in(0,1),\eta\in[0,\infty)$ (these hyperparameters are generally chosen
to be $\eta 2$ and $\beta 0.95$); (ii) the prior on the probability
distribution in the nodes is a normal distribution with zero mean:
$\mathcal{N}(0,\sigma^{2}_{q})$ where $\sigma_{q}\sigma_{0}/\sqrt{q}$ and
$\sigma_{0}$ can be used to calibrate the plausible range of the regression
function; (iii) the prior on the error variance is $\sigma^{2}\sim
InvGamma(v/2,v\lambda/2)$ where $\lambda$ is determined from the data such
that the BART improves the RMSE of an OLS model in 90% of cases.. The goal of
these priors is to regularize the algorithm and prevent individual trees from
dominating the overall fit of the model. This property is considered important
to balance the tendency of tree-based methods to overfit the training data
(Kapelner and Bleich, 2016).

BART-MIA extends the original BART algorithm by including additional
information from patterns of missing values (Kapelner and Bleich, 2015). This
is done by introducing the possibility to share on a feature for missing
values in each binary tree component of the BART algorithm, $\mathcal{T}$.
Figure B.1 illustrates the intuition behind the MIA procedure in a simple case
with only one variable (male). In the presence of missing values in this
variable, the algorithm creates a new feature (missing) – namely, an indicator
variable that takes the value 1 when the $i$th observation for the variable is
missing – and uses it to perform the data split. As shown in B.1, the
algorithm in this case has three possible splits. As in the usual CART
procedure, the algorithm chooses the one that minimizes the prediction error
in the generated leaves $l_{1}$ and $l_{2}$.

As shown by Twala et al. (2008), this splitting rule allows trees to better
capture the direct influence of missing values as another predictor of the
response variable. Furthermore, after a theoretical and empirical comparison
of different missing value strategies in trees, Josse et al. (2019) show that
MIA can handle both non-informative and informative missing values.

Figure B.1:  MIA splitting rules

Total sample$=1$$\mu_{l_{1}}$$=1$$\mu_{l_{2}}$

Total sample$=0$$\mu_{l_{1}}$$=1$$\mu_{l_{2}}$

Total sample$=0$$\mu_{l_{1}}$$=1$$\mu_{l_{2}}$

The three potential trees from the MIA procedure in a simple case with only
one binary variable ( Control $\in\\{0,1\\}$).

##  Appendix C Selection of predictors with LASSO

In our analysis, we asserted that we need as much in-sample information as
possible to reduce out-of-sample prediction errors. However, especially in
small sample analyses, overfitting problems can arise as the dimensionality of
the inputs increases. The so-called curse of dimensionality is an obstacle
when working with finite data samples and many variables. The basic reference
is the work of Bellman (1961), who introduced the notion of dimensionality
reduction. Bargagli-Stoffi et al. (2020) and Bargagli Stoffi et al. (2022)
discuss the role of regularization and dimensionality reduction in the context
of machine learning.In this Appendix, we show what happens when we sift
through firm-level data and attempt to extract a set of predictors with the
highest ability to detect financial distress. The natural candidate for
reducing the dimensionality of a matrix of predictors is the LOGIT-LASSO
(Ahrens et al., 2019), whose functional form in a panel setting is the
following:

$\displaystyle\operatorname*{arg\,min}_{{\beta}\in\mathbb{R}^{p}}\>\>\frac{1}{2N}\sum_{i=1}^{N}\left(y_{i,t}(x_{i,t-1}^{T}\beta)-log(1+e^{(x_{i,t-1}^{T}\beta)})\right)^{2}\>\>\>{\rm
subject\,\,to\,\,}\|{\beta}\|_{1}\leq k.$ (6)

where $y_{i,t}$ is a binary variable equal to one if a firm $i$ failed at time
$t$ and zero otherwise. Each $x_{i,t-1}$ is a lagged predictor chosen in
$\mathbb{R}^{p}$ at time $t-1$, while
$\|{\beta}\|_{1}\sum_{j1}^{p}|\beta_{j}|$ and $k0$. The constraint
$\|{\beta}\|_{1}\leq k$ limits the complexity of the model to avoid
overfitting, and $k$ is chosen following Ahrens et al. (2019) as the value
that maximizes the Extended Bayesian Information Criteria (Chen and Chen,
2008). We use the rigorous penalization introduced by Belloni et al. (2016a)
to account for the possible presence of heteroskedastic, non-Gaussian, and
cluster-dependent errors. Table C.1 shows the ten highest ranked predictors.

Although the number of predictors varies over time, up to a maximum of twenty-
one characteristics in 2015, we find a core set of frequently selected
predictors. This stable set includes indicators of financial distress
(Liquidity Returns, Interest Coverage Ratio, Interest Benchmark, Financial
Constraint) and indicators of firms’ core economic activities (Negative value
added, Total Factor Productivity, Size-Age). Apparently, firms controlled by
parent companies (Corporate Control) are less likely to fail in each period,
i.e. the predictor enters the algorithm with a negative coefficient191919For
an in-depth analysis of the impact of corporate control on the performance of
Italian firms, see Riccaboni et al. (2021).. The same is true for the Dummy
Trademarks and the Dummy Patents since it makes sense that intangible assets
reduce the probability of exit.

Some of the top predictors we showed were used to measure either financial
distress or zombie lending. However, the rankings change over time, and we
cannot discern a meaningful pattern in these changes. In this context, we
cannot rely on a single indicator (or set of them) to derive predictions about
defaults. If we did, we would have a higher rate of false positives (when our
forecasts suggest that a firm is at risk of failure, but it is not) and a
higher rate of false negatives (when forecasts incorrectly suggest that a firm
in trouble deserves a loan).

Indeed, at this stage, we cannot rule out the possibility that a different
ranking over time is due to better use of the newly acquired information from
the in-sample information after new failures have been observed. It could also
be that a change in rankings reflects a change in the business environment in
which firms operate. We presume applications to different aggregates
(countries, regions, industries) may result in different rankings. We conclude
that it is better to keep the full battery of predictors, provided there is no
dimensionality problem in the predictors when we can train with a large number
of observed firm-level outcomes.

Table C.1: Top 10 predictors for firms’ failures - Results from a rigorous
LOGIT-LASSO

Rank | 2017 | 2016 | 2015 | 2014 | 2013 | 2012 | 2011 | 2010 | 2009
---|---|---|---|---|---|---|---|---|---
1 | Liquidity Returns | Negative value added | Negative value added | Negative value added | Liquidity Returns | Negative value-added | Negative value added | Negative value added | Negative value added
2 | Negative value added | Liquidity Returns | Corporate Control | Liquidity Returns | Negative value added | Profitability | Liquidity Returns | Liquidity Returns | Liquidity Returns
3 | Corporate Control | Corporate Control | Financial Constraint | Solvency Ratio | Solvency Ratio | Financial Constraint | Financial Constraint | Profitability | Financial Constraint
4 | Interest Coverage Ratio | Financial Constraint | Interest Coverage Ratio | Profitability | Profitability | Corporate Control | Corporate Control | Financial Constraint | Profitability
5 | Financial Constraint | Interest Coverage Ratio | Profitability | Financial Constraint | Corporate Control | Solvency Ratio | Solvency Ratio | Corporate Control | Corporate Control
6 | Solvency Ratio | Size-Age | Solvency Ratio | Corporate Control | Financial Constraint | Interest Coverage Ratio | Size-age | Solvency Ratio | Solvency Ratio
7 | Size-age | Solvency Ratio | Size-age | Size-age | Size-Age | Liquidity Returns | Interest Coverage Ratio | Region (NUTS 2) | Interest Coverage Ratio
8 | Profitability | Profitability | Interest Benchmark | Interest Coverage Ratio | Interest Coverage Ratio | Size-age | Financial Misallocation | Dummy Trademarks | Dummy Trademarks
9 | Interest Benchmark | Interest Benchmark | Liquidity Ratio | TFP | TFP | Dummy Patents | Dummy Trademarks | Interest Coverage Ratio | Size-age
10 | Liquidity Ratio | Liquidity Ratio | Capital Intensity | Liquidity Ratio | Dummy Patents | Dummy Trademarks | Dummy Patents | Dummy Patents | Capital Intensity

  * •

Rankings are determined after conducting a rigorous LOGIT-LASSO (Ahrens et
al., 2020; Belloni et al., 2016b) each year on the entire battery of
predictors described in Figure A.1. Only the first ten selections are
reported. The procedure selects a different number of predictors each year, up
to a maximum of 21.

##  Appendix D Imputation of missing predictors

In this Appendix, we show the performance of machine learning algorithms when
we impute the missing values of predictors using two possibilities known in
the literature: Out-of-Range and Median imputation. In this way, we can use
all the observations available in the dataset. We first conduct the analysis
keeping all categories of firm failures and then exclude the cases of
liquidations, since the latter can sometimes be due to reasons other than
financial distress. The training and test datasets include 238,148 and 59,537
observations in each iteration, respectively, when liquidations are excluded.
On the other hand, when liquidations are included, the training and test data
sets comprise 243,924 and 60,982 observations, respectively.

###  D.1 Imputation of out-of-range values

We impute all missing values in the dataset with an out-of-range value, i.e.
$10^{20}$.

(a) Liquidations exluded Method | ROC | PR | F1-Score | BACC | $R^{2}$ | Time
---|---|---|---|---|---|---
Logit | 0.9657 | 0.6689 | 0.1270 | 0.7588 | 0.4558 | 56.49
Ctree | 0.9712 | 0.6999 | 0.1323 | 0.7686 | 0.4840 | 1628.08
Random Forest | 0.9343 | 0.7000 | 0.2011 | 0.8371 | 0.4999 | 608.33
XGBoost | 0.9785 | 0.7592 | 0.1270 | 0.7586 | 0.5427 | 40.28
BART | 0.9770 | 0.7482 | 0.1296 | 0.7635 | 0.5327 | 2736.74
Super Learner | 0.9788 | 0.7574 | 0.1289 | 0.7626 | 0.5454 | 11765.18

(b) Liquidations included Method | AUC | PR | F1-Score | BACC | $R^{2}$ | Time
---|---|---|---|---|---|---
Logit | 0.9496 | 0.6482 | 0.2039 | 0.7647 | 0.4173 | 76.89
Ctree | 0.9652 | 0.7230 | 0.2264 | 0.7919 | 0.48907 | 2835.230
Random Forest | 0.9410 | 0.7313 | 0.2680 | 0.8214 | 0.5130 | 695.48
XGBoost | 0.9733 | 0.7815 | 0.2039 | 0.7648 | 0.5525 | 45.29
BART | 0.9719 | 0.7718 | 0.2048 | 0.7659 | 0.5412 | 4056.36
Super Learner | 0.9748 | 0.7896 | 0.2039 | 0.7649 | 0.5620 | 13992.57

Table D.1:  Performance of the model with out-of-range values imputed data.
All algorithms were trained with five-fold cross-validation. All metrics
correspond to the five-fold average. Time indicates the average seconds taken
to train the model in each fold.

###  D.2 Median imputation

We impute the missing values of a variable $q$ with its median $q_{.5}$.

(a) Liquidations exluded Method | ROC | PR | F1-Score | BACC | $R^{2}$ | Time
---|---|---|---|---|---|---
Logit | 0.8825 | 0.2690 | 0.1465 | 0.7540 | 0.1495 | 50.77
Ctree | 0.9270 | 0.4017 | 0.1492 | 0.7962 | 0.2364 | 1975.38
Random Forest | 0.7767 | 0.4337 | 0.1487 | 0.7060 | 0.2667 | 855.37
XGBoost | 0.9584 | 0.5246 | 0.1271 | 0.7579 | 0.3281 | 39.51
BART | 0.9485 | 0.4936 | 0.1328 | 0.7688 | 0.3014 | 4345.19
Super Learner | 0.9595 | 0.5393 | 0.1269 | 0.7576 | 0.3389 | 13927.68

(b) Liquidations included Method | ROC | PR | F1-Score | BACC | $R^{2}$ | Time
---|---|---|---|---|---|---
Logit | 0.8907 | 0.3966 | 0.2282 | 0.7667 | 0.2302 | 60.44
Ctree | 0.9237 | 0.5059 | 0.2187 | 0.7808 | 0.3082 | 2429.78
Random Forest | 0.8281 | 0.5453 | 0.2227 | 0.7304 | 0.3420 | 1167.71
XGBoost | 0.9536 | 0.6175 | 0.2037 | 0.7636 | 0.3942 | 40.93
BART | 0.9440 | 0.5890 | 0.2063 | 0.7663 | 0.3696 | 4702.54
Super Learner | 0.9551 | 0.6295 | 0.2043 | 0.7643 | 0.4051 | 15218.50

Table D.2:  Performance of models with column-wise median imputed data. All
algorithms are trained with five-fold cross-validation. All metrics correspond
to the five-fold average. The time indicates the average seconds taken to
train the model in each fold.