A Markov Model of Bank Failure Estimated Using an Information-Theoretic Approach 

Dennis Glennon *

Risk Analysis Division, Office of Comptroller of the Currency

250 E Street, SW, Washington, DC 20219

Tel.: (202) 874-4725

Fax: (202) 874-5394

Email: dennis.glennon@occ.treas.gov, 

Amos Golan 

Economics Department, The American University

4000 Massachusetts Ave., Washington, DC 20016

Tel: (202) 885-3783

Fax: (202) 885-3790

Email: agolan@american.edu

March 2003 

*

 The authors are Senior Financial Economist, Risk Analysis Division, Office of Comptroller of the Currency, and 

Research Professor of Economics, The American University. Golan thanks the OCC for partially supporting this 

research. The opinions expressed in this paper are those of the authors and do not necessarily reflect those of the 

Office of the Comptroller of the Currency. All errors are the responsibilities of the authors. 
A Markov Model of Bank Failure Estimated Using an Information-Theoretic Approach 

Abstract: In this paper, we develop an early-warning bank failure model (EWM) designed specifically 

to capture the dynamic process underlying the transition from financially sound to closure. We model the 

transition process as a stationary Markov model and estimate the transition probabilities using a 

Generalized Maximum Entropy (GME) estimation technique. The GME estimation method is a member 

of the class of information-theoretic methods, is semi-parametric, and is better suited for estimating 

models in which the data are limited (e.g., few events, and data availability problems), highly collinear, 

and measured with error – conditions that often exist with micro-level banking data. In addition, this 

method allows us to incorporate prior information and impose fewer distributional assumptions relative to 

conventional maximum likelihood (or full information maximum likelihood) methods. We report 

estimates of the transition probabilities for nine transition states for the population of nationally chartered 

banks incorporating the effect of bank-specific and macroeconomic variables from 1984 through 1999. 

JEL code: C13, C14, C25, C51, G21

Keywords: Bank Failure, Early Warning Model, Markov Process, Generalized Maximum Entropy 

Estimators 
I. Introduction 

In this paper, we develop an early-warning bank failure model (EWM) designed specifically to 

capture the dynamic process underlying the transition from financially sound to failure/closure. We 

model the transition process as a first-order stationary Markov process in which failure is but one of 

several possible financial states (e.g., financially sound, distressed, insolvent, and closure). The multistate design has several advantages over the more conventional binary-state approach generally found in 

the bank-failure literature (Altman, et al., 1981; Fissel, et al., 1996; Looney, et al., 1989; and Kola ri, et al., 

2001). First, the model is not dependent solely on an outcome state derived at the discretion of the 

regulators (i.e., failure/closure). Second, this approach captures the problem-bank/failure transition 

process over several states of financial distress; a model design that better serves the bank supervisors’ 

early intervention objectives. Moreover, it differentiates between banks that remain healthy, those that 

exhibit distress but recover, and those that eventually fail. 

We estimate the Markov transition model using a Generalized Maximum Entropy (GME) 

estimation procedure. The GME approach is an information-theoretic modeling technique that is better 

suited for estimating models, in which the data are limited (e.g., few events and data availability 

problems), highly collinear, and measured with error (Golan, et al., 1996a). The GME is a semiparametric, robust estimator that is based on fewer distributional assumptions than conventional 

maximum likelihood (or full information maximum likelihood) methods. Furthermore, this approach 

allows us to incorporate prior information about the transition process directly into the estimation process. 

More importantly for our purpose, the GME procedure allows us to incorporate bank-specific 

time-series, cross-section data as well as time-varying macroeconomics and financial market variables 

directly into estimation of the transition probabilities; a data design significantly broader than the crosssection data design generally found in the bank-failure literature. Our objective, then, is to model the 

transition process conditional on bank-specific characteristics, exogenous macroeconomic effects, and 

changing financial market conditions. The latter two categories are especially important because they 

1
capture the effects of changing market conditions on the failure probabilities generally ignored in the 

empirical bank-failure literature. 

The remainder of the paper is organized as follows. In Section II we provide a brief overview of 

the limitations of the conventional EWM design. In Section III, we outline our proposed first-order 

Markov model and discuss in more detail the GME approach (see also Appendix 1). In Section IV we 

summarize the data and report our main results. 

II. Limitations of Conventional Early-Warning Bank Failure Models 

Although general agreement exists on the fundamental objective of an early warning model (i.e., 

a timely and accurate list of financially fragile banks), there is far less agreement on the development and 

design of these models (Stengel, et al., 2000). The early work in this area used conventional statistical 

techniques (e.g., discriminant analysis, logit models, and factor analysis) to develop models primarily for 

failure-classification purposes (see Altman, et al., 1981). Those models were designed specifically to 

partition the population of banks at a point in time into distinct groups that reflect the relative financial 

strength of the institutions. More recently, multinomial logit, survival analysis, neural networks, and 

other pattern recognition methods (Fissel, et al., 1996, Kolari, et al., 1996) have been used to improve on 

the classification power and overall reliability of these models. 

The underlying data design supporting the estimation of those models, however, has not changed 

since the early work of Altman (1968), Beaver (1968), Sinkey (1975), and Eisenbeis (1977) on corporate 

bankruptcies. A cross-section sample of banks, at time t, is used to predict the performance (i.e., closure) 

over the interval [t, t+n] (n=1,2 years forward) conditioned on initial balance sheet and income ratios. 

Under that design, the financial ratios are used to differentiate between those banks that correctly, from 

those that incorrectly, position themselves to withstand credit, liquidity, or market shocks. 

Unfortunately, credit, liquidity, and market shocks are environment or state-specific events that 

are only observed over time. Models developed on cross-section data, by design, omit time-varying 

2
factors and, as a result, fail to capture the underlying dynamics of the failure/survival process.1 Although 

these models tend to perform well at classifying banks within sample, they generally perform poorly outof-sample, especially as banks reposition their portfolios and le nding strategies to correspond to 

contemporaneous economic and industry conditions. For that reason, bank-failure classification models 

have not fared well as supervisory tools. 

A time-series/cross-section sample design, however, introduces its own set of problems that 

makes it difficult to model performance using statistical procedures found in the bank-failure literature. 

Not all methods or techniques that are used with cross-section data work well with time-series/crosssection data. For example, survival analysis procedures could be extended to capture the impact of timevarying covariates on the probability of failure. However, those procedures require a relatively large 

number of events in each time period. Although the failure rates over the 1987-92 observation period 

were relatively large, the number of total failures since 1992 has declined dramatically making it difficult 

to satisfy this requirement.2 In fact, the relatively small number of bank failures in general underlies the 

overwhelming choice of a pooled (often over several years), cross-section sample design in the bankfailure literature. That condition, by itself, suggests that a simple fail/nonfail model design is inadequate 

for our purpose. 

For those reasons we propose an alternative model design in which we estimate the transition 

probabilities instead of failure-classification probabilities identified under a conventional EWM design. 

The multi-state, transition-based approach allows us to better capture the dynamic process leading to 

financial distress that cannot be achieved under a classification-based model design. Unfortunately, the 

multistate design does not address the data limitation problem. For that reason, we use a GME procedure 

1

 For example, banks adjust their assets and liabilities in anticipation of specific movements in economic (i.e., 

business and interest-rate cycles) conditions. The success or failure of their asset-liability strategy (ex post) is 

conditional on the actual outcome of the macro economy over time (e.g., an asset-liability strategy in which a bank 

that maintains a high concentration of variable-rate C&I loans funded with seasoned long-term, fixed-rate liabilities 

will perform much differently in an expanding economy with interest rates rising than if the macro economy is 

slowing down and interest rates are falling). Models developed on cross-section data fail to capture this potentially 

important aspect of the failure process that is identified through behavior over time. 

2 For example, no national banks failed in more than 60 percent of the time periods from first quarter 1993 to fourth 

quarter 1999. 

3
to recover the transition probabilities. The GME approach is better suited for estimating models in which 

data are limited (e.g., few events). Moreover, the GME procedure allows us to incorporate firm-specific 

strategic objectives (e.g., portfolio composition, funding, liquidity measures) and general economic 

conditions (e.g., economic growth, interest rate effects) at different points in the business cycle directly 

into the estimation process through the use of both time-series and cross-section data. Furthermore, the 

GME procedure uses minimal distributional assumptions, is easy to apply, and is computationally 

efficient. 

III. An Alternative EWM Model: A Markov Transition Approach 

We define the various states (e.g., financially sound, distressed, insolvent/failure, and closure) in 

terms of the equity-asset ratio, using a book-value equity measure. The transition probabilities measure 

the probability that a bank with equity capital yt,j (state j, in time t) will have equity capital yt+1,k (state k , 

in time t+1) in the next period. These transitions probabilities capture the likelihood that a bank will 

exhaust or increase its equity capital in period t+n conditional on its initial state, and that of the macro 

economy, in time t. 

As a logical starting point for our analysis, we define the transition states using the book equity 

(i.e., leverage) capital zones defined under the prompt correction action provision of the Federal Deposit 

Insurance Corporation Improvement Act (FDICIA). In addition, we include an insolvency state (negative 

book-valued equity) and three terminal or absorption states – merged with an affiliated (i.e., within 

holding company) bank, merged with an unaffiliated bank, and failure for a total of nine financial states 

(see Table 1). The latter absorption state (i.e., failure) is of greatest interest to supervisors and the 

primary focus of the following analysis. 

4
III.1 A Markov Transition Model: The Basic Model Design 

We define the binary random variable yitj = 1 if the ith bank (i=1, 2, ..., n) is in state j (j=1, 2, ..., 

K) at time t (t=1, 2, ..., T), and yitj = 0 for all other K-1 states. Under this assumption, our model is a 

generalization of a qualitative response variable model discussed in Amemiya (1985). However, to 

simplify the analysis, we formulate our model on the mean behavior of all banks within each state in time 

t, rather than the specific behavior of each bank. This allows us to capture the same information, but with 

a much lower dimensionality of data. 

We define yt as a K-dimensional vector of proportions in the kth Markov state in period t. The 

elements of yt are the proportions of banks in each state, in period t, calculated directly from the sample 

data. Similarly, y t +1 is a K-dimensional vector of proportions representing the fraction of banks in the 

kth Markov state in period t+1. We start with periods two and three and condition the estimate of the 

transition probabilities on the first-period proportions. Our objective is to estimate the K · K Markov 

transition probabilities P, using information from the full sample 1984 through 1999. 

Following convention, we represent the general linear relationship between y t +1 , yt and the 

matrix of transition probabilities P as 

K 

yt +1, j = � pkj ytk , (1a) 

k =1 

where the pkj are the stationary Markov probabilities over the relevant periods, and 

K 

� pkj = 1 for j, k = 1, 2, ..., K, (1b) 

j =1 

imposes the condition that the pkj are proper probabilities. 

A model such as that proposed in equation (1), in which the only available information/data are 

given by the outcome states yt , is the most elementary design of a Markov probability model. This base5
� 

� 

case model serves our purpose only insofar as it provides the foundation for the generalized model, which 

incorporates both bank-specific and macroeconomic factors. 

Under this simple design, the number of unknowns may exceed the number of data points if the 

number of periods T is small. As a result, the base-case model is underdetermined in the sense that there 

are infinitely many stationary Markov solutions that satisfy the basic relationship in equation (1). An 

underdetermined model always exists regardless of the number of periods available when generalizing the 

approach to incorporate noise in the data. This type of problem can be addressed by (i) incorporating 

additional restrictions such as distributional assumptions, (ii) imposing a decision rule to select one of the 

infinitely many solutions, or (iii) both. The former approach is valid only when the assumptions or 

restrictions are consistent with the data generating process. Therefore, to avoid imposing arbitrary 

assumptions or restrictions, we employ the latter approach using an entropy decision criterion. 

Using the entropy decision criterion to estimate the transition probability matrix P of equation 

(1), yields the classical maximum entropy (ME) method: 

�pˆ = arg max -� pkj log pkj � k , j 

ME = � s.t. . (2) 

�yt +1,j = � pkj ytk ; � pkj = 1 

� k j 

The solution to the ME problem in equation (2) is 

exp�

�

-� ytk l

ˆ 

tj �

� 

pˆ kj = Ł t ł (3) 

�exp�

�

-� ytk lˆ 

tj �

� 

j Ł t ł 

where ˆl is the vector of K Lagrange multipliers associated with equation (1a). See Appendix 1 for 

further background on the ME framework. 

6
III.2 The General Model and Estimation Procedure 

The simple ME model outlined in equation (2) forms the basis of a more general model, in which 

we incorporate bank-specific and macroeconomic variables. In addition to the information on the 

proportion of banks in the kth state in time t, we incorporate balance sheet and income statement 

information as well as macroeconomic variables by imposing additional constraints in the maximization 

problem. In this section, we outline the development of the full GME model used to estimate our bankfailure model. However, for a more detailed discussion, and additional variations of the model, see 

Glennon and Golan (2001). 

Let zti be a G-dimensional vector of bank-specific covariates (e.g., asset, lia bility, and off-balance 

sheet composition, asset quality/non-performing indicators, etc.) with individual elements zgti. In 

addition, for each period t, let st be an L-dimensional vector of macro-level variables.3 It is important to 

note that the bank-specific covariates vary by both state and time; however, the macro-level variables 

vary only by time. We can represent the bank-specific and macro variables more compactly as 

X = [Z, S] with elements xnti (n = 1, 2, ..., N; N = G+L) and the L macro variable held constant across 

states.4 The exact functional relationship describing the effect of these two types of variables (i.e., Z and 

S) on the transition probabilities is unknown. For that reason, we follow the instrumental variable (IV) 

literature and capture the information related to these variables via the cross moments: 

T T -1 K 

�� ytj xntj = �� pkj ytk xntk (4) t= 2 j t =1 k =1 

The bank-specific data are measured at their historic or book, instead of market, values. In 

addition, the sample is composed of a large number of banks tracked over a relatively long period of time. 

As a result, the data are inherently noisy. Because we are interested in stationary estimates of the 

3 The macro-level variables may include macroeconomic, industry/market, and policy variables that either directly 

or indirectly affect the banking industry. For example, changes in national or regional income, oil shocks, or large 

swings in the federal funds rate (to name a few) are likely to affect the financial health of banks; the actual impact 

depends on the degree to which banks protect themselves from such shocks. 

4

 For a slightly more general approach for introducing the macro effects, see Glennon and Golan (2001). The 

approach we use here was tested against the more general one that was statistically rejected for these data. 

7
transition probabilities, we must accommodate the noise as part of the modeling process. Unfortunately, 

the ME model in equation (2) does not capture the effect of random noise on the estimates. However, 

following Golan, Judge and Miller (1996a), we can incorporate the effect of noise in the data into the ME 

framework by reformulating the model as a generalized maximum entropy (GME) problem. More 

specifically, we can rewrite equation (1a) as 

K 

yt+1, j = � pkj ytk + etj 

k =1 (1a') K M 

= � pkj ytk +� wtjmvm 

k =1 m=1 

M 

whereetj is a random error term for each state j and period t, etj ” � wtjmvm , v is a symmetric-aroundm=1 

M 

zero support space for each random error etj , � wtjm = 1, and M ‡ 2 . Furthermore, because ytj ̨[0,1], 

m=1 

it follows that etj  ̨[-1,1] for all t, j, and that the errors’ support has natural bounds vm  ̨ [-1,1]. Using 

this more general formulation of the basic Markov transition model, we reformulate the data in equation 

(4) to be consistent with the specification outlined in equation (1a'): 

T T -1 K T -1 

�� ytj xntj = �� pkj ytk xntk +��etj xntk 

t= 2 j t =1 k =1 t =1 i (4a)

T -1 K T -1 

= �� pkj ytk xntk + ��� wtkmxntkvm 

t =1 k =1 t =1 k m 

The data specified in equation (4a) incorporate both the bank-specific and macroeconomic data, and 

account for possible noise in the data. We refer to this version of the model as a generalized maximum 

entropy – instrumental variable (GME-IV) estimator. 

Following the method proposed by Golan, Judge, and Miller (1996a), we select values for the 

support space vm and use the modified or generalized ME method to estimate the values for the unknown 

pkj and wtkm. The resulting GME-IV estimation rule for our Markov problem is 

8
� � � 

� 

� 

�Max �-� pkj log pkj - � wtjm log wtjm � , � , , { p w} � k , j tjm � �s t.. 

GME - IV = � T T -1 K T -1 (5) 

��� ytjxntj = �� p ytk xntk +��� wxv tjm ntk m � t =2 i t=1 k =1 

kj 

t=1 k m 

�� pkj = 1; �wtjm = 1 

� j m 

The solution to the GME-IV problem is 

� T -1

ˆ � � T -1 

exp �-�� ytkxntkl jn � exp �-�� ytk xntkl

ˆ 

jn �

� 

Ł t =1 n ł ” Ł t =1 n ł pˆ = (6) kj 

� exp �

�

-

T 

�� 

-1 

ytk xntklˆ 

jn �

� Wk 

j Ł t =1 n ł 

and 

exp �

�

-� xntkvm lˆ 

jn �

� exp ��

-� xntkvm lˆ 

jn �

� 

wtjm = Ł n ł = Ł n ł ˆ (7)

exp �

�

-� xntk vm l

ˆ 

jn 

�

� Yitj � 

m Ł n ł 

with eˆ tj = � wˆ tjmvm .

5 

m 

The concentrated-dual (unconstrained) GME-IV method can be derived from the Lagrangian of 

equation (5), and is 

5 Although we use discrete values for w, continuous values may be used as well (e.g., Golan and Gzyl, 2001). 

9
T K Ø T -1 �ø l(l ) = ��� ytjxntjlnj + �log Œ�exp �

�-�� ytk xntklnj �œ t = 2 j =1 n k o j Ł t =1 n łß 

+� log Œ

Ø

�exp �

�-� xntjvmlnj �

�

œ

ø (8) 

t , j o m Ł n łß 

T K 

= ��� ytjxntjlnj +�log Wk (l) +�log Ytj (l) t = 2 j=1 n k t , j 

where Wk and Ytj are defined in equation (6) and (7). 

Minimizing equation (8) and solving for l, yields the estimated l

ˆ , which in turn yield the 

optimal probabilities pˆ kj and wˆ tjm via equations (6) and (7). It is important to note that this model is 

computationally as efficient as the maximum likelihood (ML) approach. Moreover, because the estimates 

are unique functions of the Lagrange multipliers, l, this method (which is a generalization of the ML) has 

the same level of complexity as the ML. 

Interestingly, the solution to the maximum entropy (ME) problem in equation (2) subject to the 

moment constraints in equation (4) is equivalent to the maximum likelihood-logit (ML-logit) model (e.g., 

Amemiya, 1985, chapter 11). Moreover, the solution to the GME-IV problem in equation (5) converges 

to the ME/ML-logit solution as the noise approaches zero.6 As a result, the GME-IV method can be 

viewed as a generalized ML-logit model, in which the conventional ML solution is a special case when all 

errors in equation (1a') are zero. That condition exists, however, if, and only if, the Markov process is 

stationary, and there is no noise in the data; an assumption that is generally inconsistent with the data. 

6

 The log-likelihood for a conventional logit model is: 

n xB n 

ln L = �[ yi ln(1 + 

e 

exB ) + (1 - yi )ln(1 + 

1 

exB )] = �[ yi {ln(exB ) -ln(1 + exB )} + (1 - yi ){ln(1) -ln(1 + exB )}] i i 

n 

= �[ yi xB - ln(1 + exB )] i 

If we set j=2, etj=0, or v=0, for all t,j, redefine Bj=-lj, and impose the standard identification condition that B1=0, the 

dual GME-IV in equation (8) collapses to the log-likelihood of the conventional maximum likelihood-logit model 

(see Golan, et al., 1996a). 

10
III.3 Introducing Prior Information – A General Cross-Entropy (GCE) Approach 

Lastly, we consider the case in which prior knowledge or information on the underly ing values of 

P are available. In general, such information may come from a previous sample, a previous study, or 

derived from the underlying theory. The prior information (P0

) is easily incorporated into the GME-IV 

approach via the cross-entropy or Kulback-Liebler information-divergence measure (see Appendix 1), 

and, as a result, represents a natural generalization of the entropy measure. Rewriting the objective 

function of our generalized entropy estimator in equation (5) as 

0 0 I (P,W ; P0

,W 0 )= � pkj log (pkj / pkj )+ � wtjm log (wtjm / wtjm ) (9) 

k , j t , j ,m 

and minimizing equation (9) with respect to (i) the moment constraints in equation (4a) and (ii) the 

requirements that pkj and wtjm are proper probabilities (�pkj=1 and � wtjm=1), yields the optimal solutions 

for the generalized cross entropy – instrumental variable (GCE-IV) problem 

0 � T -1 ~ � 0 � T -1 ~ � pkj exp ��� ytk xntkl jn � pkj exp ��� ytk xntkl jn � 

~ = Ł t =1 n ł ” Ł t =1 n ł p (10) kj 

pkj exp � 0 � T 

�� 

-1 

ytk xntkl ~ 

jn 

�

� Wk

� 

j Ł t =1 n ł 

and 

0 � ~ � 0 � ~ � wtjm exp �� xntkvm l jn � wtjm exp �� xntkvm l jn � 

~ Ł n ł ” Ł n ł w (11) tjm = 

� 0 � � ~ � Ytj wtjm exp � xntkvml jn � m Ł n ł 

~ ~ 0 where, as before, etj ” � wtjmvm , and the priors for the noise terms, wtjm , are assumed to be uniformly 

m 

distributed. A uniform distribution for the priors is used when prior information does not exist. In that 

case, the GCE-IV and GME-IV methods are equivalent.7 Although additional information from 

7 Although prior information for the transition probabilities is available from a previous sample, priors for the error 

terms do not exist. For that reason, we use uniform priors for the error terms in equations (9) and (11). In that case, 

11 
economic theory, or institutional arrangements, can be incorporated easily by imposing additional 

restrictions (i.e., equalities, inequalities, etc.) when solving equation (9), we impose only information 

from the prior probabilities at this time. 

Using the GME estimation rule in equation (5) with equation (9) substituted for the objective 

function, we derive the concentrated-dual GCE-IV 

T K 

l(l ) = ��� ytj xntjlnj -�log W k -� log Ytj (l) (12) 

t = 2 j =1 n k t , j 

where both normalization factors Wk (l ) and Ytj (l ) are defined in equations (10) and (11). Equation 

(12) is used to estimate the ?

% . Substituting the estimated ?% into equation (10) and using the mean values 

for the covariates, we can solve for the optimal transition probabilities for a set of representative or 

benchmark banks across all initial states. 

IV. Data and Empirical Results 

Bank-specific data were collected directly from the quarterly Call Reports for all national banks 

that existed from 1984.1 through 1999.4 (excluding a relatively small number of national banks that 

switched charters during the sample period). The full sample covers a total of 64 quarters: t = 1, 2, ..., 

64. Failed and merged banks were identified using the FDIC’s Mergers and Failures data set. Closure 

due to merger was labeled as a within-holding company merger if the regulatory top holding company 

identifier of the closed bank matched that of the acquiring bank; all other closures because of merger 

were classified as mergers with nonaffiliated banks.8 

the results are the same with or without imposing the uniform prior. We include the priors for the error terms for 

completeness. 

8 At this stage in our development of a Markov transition-based EWM approach, we do not thoroughly investigate 

merger behavior. The current design recognizes that mergers within a holding company are more likely to be 

affected by changes in state banking laws and the potential lower costs associated with merging affiliates under a 

single corporate identity. Mergers among nonaffiliates however may represent a nonfailure exit for a bank that may 

have failed had it not merged. For those reasons, we differentiate between the two types of mergers, but postpone 

our analysis of this aspect of the EWM for future research. 

12
IV.1 Bank-Specific Covariates and Macroeconomic Variables 

Following the literature, we identified several bank-specific ratios based on publicly available 

data that reflect credit, interest rate, and liquidity risks. In addition, we evaluate the effect of several 

institutional factors on the dynamics of a bank’s transition to alternative states, all else equal. We also 

selected a small set of macroeconomic variables that reflect general economic conditions, major industry 

effects, and interest rate expectations. Our initial selection of bank-specific and macroeconomic variables 

is consistent with those generally found in the bank-failure literature or commonly identified as indicators 

of a fragile/strong banking system. 

We assume banks continuously reposition their portfolios and redevelop their lending/funding 

strategies in response to changes in both current and anticipated market conditions. These may involve 

modifying their credit risk exposure through adjustments in their concentration of assets in, say, C&I 

loans during the expansionary period of the business cycle, or controlling their interest-rate exposure by 

adjusting their reliance on core deposits and the average duration of their loan portfolio in anticipation of 

a flattening of the yield curve. In Table 2 we report the average value (on a pooled basis) for each bankspecific covariate included in the model for both the full sample and three sub-periods.9 The trend in the 

average values across sub-periods suggests that, as a group, banks adjusted their behavior quite 

extensively to changes in market condition over the sample period; a result reflected in the rather 

pronounced change in the average values of the covariates between the 1987-92 and 1993-99 sub-periods. 

9 The post 1992 sub-period begins soon after the passage of FDICIA, FIRREA, and implementation of the new riskbased capital rules. It covers a period in which relatively few national banks failed (i.e., on average, roughly four 

national banks failed per year during this period). The 1987-92 sub-period, however, was a period in which by 

historical standards a relatively large number of national banks failed (i.e., there were on average 70 national bank 

failures per year during this period with more than 100 in 1989 alone). The large number of failures during this 

period reflects the general state of the financial/banking market affected by the savings and loans crisis. A relatively 

moderate number of failures occurred during the pre 1987 sub-period (i.e., on average, roughly 35 per year). We 

use this sub-period to generate the prior transition probabilities in the following model. 

13
In Table 3, we report the average values of the bank-specific covariates by failure status.10 There 

are significant differences in the average (pooled) values by failure status over the full sample; a result 

that suggests that not all bankers correctly position themselves to survive credit, interest rate, and liquidity 

shocks. For example, failed banks over the full sample period were relatively more concentrated in C&I 

loans; held lower quality loans (i.e., higher non-performing to total loans); maintained lower levels of 

liquid assets to liabilities, net loans to deposits, market-sensitive assets (i.e., trading account assets and 

securities), and long-term assets; were less likely to be affiliated with a strong holding company; and 

were more likely to be a new bank relative to the average nonfailed bank. 

We also report in Table 3 the average values of the covariates by failure status for each of the 

three sub-periods in which there were a moderate (i.e., 1984-86), large (i.e., 1987-92), and small (i.e., 

1993-99) number of failures. These results show that the relationship between nonfailed and failed banks 

continues to hold over time, even as banks, as a group, adjust to changes in the market. The average 

values by failure status are significantly different for each sub-period, except for asset growth and longterm asset ratio in the later sub-period. 

In Table 4 we list the macroeconomic variables used in the model. In addition, we report the 

average values over the full sample and sub-periods. These results show that the macroeconomic 

conditions have changed significantly over time – with lower unemployment, agricultural, oil, and interest 

rates, and tighter interest-rate spreads in the later period of our sample. The sub-periods were defined 

with respect to the volume of bank failures and, as such, do not correspond directly with the business 

cycle. However, the relatively large difference in the average values of the macroeconomic variables 

across the sub-periods suggests that macroeconomic conditions are contributing factors in the overall 

condition of the banking system; a result that implies traditional bank-failure or EWM models that omit 

macroeconomic variables are under-identified. 

10 We identify the timing of failure as the period in which the failed bank files its last Call Report. Banks that fail 

are considered “failed” for the full sample period. Under this classification rule, a bank that fails at any time during 

the sample period is classified as a failed bank. As a result, these banks contribute to the average for each quarter 

over the full sample period in which they file a Call Report, not merely the period in which they fail. 

14
IV.2 Estimation Results 

The GCE-IV procedure requires information on the prior transition probabilities (P0), the row 

shares (yt), and bank-specific and macroeconomic data. The prior probabilities are derived from the 

quarterly transition frequencies over the first three years of our sample. More specifically, the prior 

transition probabilities (P0

) are calculated as the mean of the percentage of banks in state i in time t that 

fall in state j in time t+1, i,j = 1, 2, ..., 9, and t = 1, 2, ..., 12. The prior probability matrix, based on the 

transition frequencies over the sample period 1984.1 through 1986.4, is presented in Table 5. 

The proportions of banks in the kth Markov state in each time t, t=13, 14, ..., 64 (i.e., 1987.1 

through 1999.4) are used to define the row shares yt. We estimated the transition probabilities using a 

GCE procedure with only the row shares and prior probabilities (i.e., excluding the instrumental 

variables). The estimated transition probabilities based on this information only are illustrative of the 

potential effect of a GCE approach. For example, we see in Table 6 that the likelihood of remaining in 

the same state increases (i.e., each pii, i=1, 2, ..., 6 – the probabilities along the main diagonal – increases) 

and the failure probability declines over all six states (i.e., each pi,9 for all i=1,2, ..., 6, decreases) after 

incorporating the proportion of banks in each state over time – the row shares – via the constraints 

outlined in equation (1a'). These results reflect the relative stability of the banking industry since 1993. 

They do not, however, reflect the effect of bank-specific or macroeconomic conditions on the process that 

underlies the multistate Markov transition process. 

The GCE-IV transition probabilities are estimated using equation (12). We introduce onequarter lagged, bank-specific covariates and macroeconomic instruments through the moment constraints 

derived in equation (4a). We evaluated the effect of additional lags (up to two periods) for the macro 

level variables. However, we found that the additional information was not statistically significant and 

therefore was not included in the finalmodel. Table 7 shows that the transition-probability mass becomes 

more concentrated around the main diagonal under a GCE-IV approach. For example, in Table 6, without 

taking into account bank-specific and macroeconomic conditions (i.e., the GCE approach), an 

undercapitalized bank (state 4) in time t has a 42 percent probability of remaining undercapitalized in time 

15
t+1 (i.e, p44|GCE = 0.42996), a 11 percent probability of moving to a well capitalized state (i.e., p46|GCE = 

0.1076), and a 10 percent probability of becoming critically undercapitalized (i.e., p42|GCE = 0.1037) within 

one quarter. After incorporating the bank-specific covariates and macroeconomic variables (Table 7), a 

representative bank in state 4 (i.e., undercapitalized) is significantly more likely to remain in state 4 (i.e., 

p44|GCE -IV = 0.674) and far less likely to become well capitalized (i.e., p46|GCE -IV = 0.0004) or critically 

undercapitalized (i.e., p42|GCE -IV = 0.0535) within one quarter. These results suggest that a bank’s strategic 

structure (e.g., asset concentration and management efficiency ) contributes significantly to the likelihood 

a bank will undergo an extreme jump in states over a short period of time (i.e., a single quarter). 

The effect of a change in a bank-specific or macroeconomic variable on the estimates of the 

transition probabilities is difficult to sign a priori. In most cases, it will depend on the initial state. For 

example, the effect of an increase in asset growth on a well-capitalized (state 6), well-managed bank is 

likely to be different from, and, possibly, of different sign, than the same increase in the asset growth rate 

of a significantly undercapitalized (state 3), critically undercapitalized (state 2), or insolvent (state 1) 

bank. 

The incremental or marginal effect of a change in bank-specific (zntk) or macroeconomic variable 

(stl) on the transition probabilities (pkj) can be derived from equation (10). The marginal effect of each 

bank-specific covariate and macroeconomic variable on the estimated transition probabilities is derived 

by differentiating equation (10) with respect to xtin: 

¶pkj Ø ø 

¶xntk 

= pkj ytk Œ

o 

l jn -� pkj l jn œ

ß 

(13) 

j 

and evaluating at the means, or at any other value of interest, to capture the “dynamic” effects of the 

market. 

In Table 8 we report the marginal effects of including several of the bank-specific covariates and 

macroeconomic variables as factors in the estimates of the transition probabilities. The marginal effects 

show the direct effect of an increase in bank-specific covariates or macroeconomic variables on each of 

the transition probabilities. For example, the probability the representative bank in state 1 (i.e., book 

16 
insolvent) in time t will remain in state 1 in t+1 increases 0.000392 for a unit increase in a bank's 

concentration in C&I loan; and, the probability a critically undercapitalized bank (state 2) transition to a 

failure state (state 9) increases by 0.005149 for an unit increase in the quarter-over-quarter asset growth 

rate. 

These results are especially interesting insofar as they show that the effect of a change in bankspecific and macroeconomic conditions is state specific. For example, in Table 8 we find that an increase 

in asset growth tends to increase the likelihood of failure for all banks except those that are initially well 

capitalized (i.e., state 6).11 In addition, the results in Table 8 show that the probability that a wellcapitalized bank in time t remains well-capitalized in time t+1 (i.e., p6,6) rises as asset growth increases. 

These results suggest that growth for a well-capitalized bank may improve the overall strength of the bank 

(at least in the short run). However, for banks that are initially significantly undercapitalized or worse 

(i.e., states 1-3), growth increases the probability of failure primarily by reducing the probability of 

remaining in the current state. Interestingly, asset growth increases the probability that a bank that is 

significantly undercapitalized or worse transitions to the higher state of undercapitalized (i.e., state 4) in a 

relatively short period of time. This suggests some problem banks may in fact improve their capital 

position through growth; however, relative to the initial transition probabilities, the impact is small 

especially when evaluated in terms of the effect that growth has on the likelihood of transitioning to a 

failure state. 

Similar results hold for a decrease in the holding company-level equity-asset ratio. The resources 

of the holding company are expected to act as a source of strength for problem banks. As such, an 

increase in the equity-asset ratio of the holding company decreases the likelihood of transitioning to 

failure. Moreover, the results in Table 8 show that an increase in the holding company-level equity-asset 

ratio increase the transition probabilities associated with remaining in the same, or moving to, a higher 

capital state (i.e., dpij > 0 for i £ j; i,j = 1,2, ..., 6) and decrease those associated with lower states (i.e., dpij 

< 0 for i > j; i = 2, ..., 6; j = 1, 2, ..., 5). 

11 See Table 8, Asset Growth series: column 9, rows 1 through 5 are positive, row 6 is zero 

17
In general, an increase in the national unemployment rate lowers (raises) the likelihood of moving 

to a higher (lower) state. The results in Table 8 show that the likelihood of moving to a higher state from 

an initial state of significantly undercapita lized or better (i.e., states 3-6) declines, in general, and except 

for moving to the book-insolvent state, the likelihood of moving to a lower state increases. The decrease 

in the transition probability to a book-insolvent state is partially a reflection of the increased failure 

probabilities. Interestingly, the probabilities that book-insolvent or critically undercapitalized banks 

move to moderately higher states (i.e., states 2 and 3) rise as the national unemployment rate increases. 

These results appear to be counterintuitive; we expect the likelihood of moving to a higher state to fall as 

the unemployment rate increases. These results, however, are consistent with the hypothesis that a slow 

down in the macro economy is more likely to affect the ability of severely distressed banks (i.e., initial 

states 1 and 2) to move to the higher states 4 through 6. As a result, a proportion of the banks that would 

have transitioned to the higher-level states 4 through 6, in the absence of an increase in the national 

unemployment rate, are more likely to transition to less desirable higher states 2 and 3. As such, the 

decline in the likelihood of severely distressed banks reaching the higher states 4 through 6 has the effect 

of increasing the marginal effect on the intermediate-state transition probabilities (i.e., p12, p13, and p23). 

IV.3 Information, Diagnostics and Model Reliability 

The amount of information captured by the GCE or GCE-IV model can be measured by using a 

normalized entropy (information) measure: 

~ ~ - �� pkj ln pkj

~ S(P) ” 

k j 

~0 ~0 (14) - �� pkj ln pkj 

k j 

with the ~pkj representing estimated transition probabilities under either a GCE or GCE-IV estimation 

~ ~0 rule, and the pkj are the prior probabilities. S(P) is bounded between 0 and 1, with 1 reflecting 

uniformity (complete ignorance) of the estimates and 0 reflecting perfect knowledge (Golan, Judge and 

18
~ Perloff, 1996b). In that way, S(P) captures the amount of information in the data relative to the initial 

knowledge reflected in the priors (see also Soofi, 1996). 

In addition, a likelihood ratio test of model fit can be constructed that is analogous to that 

developed under a maximum likelihood (ML) procedure. That is, let IW be the value of the optimal GCE 

objective function where the data (or parameters of interest) are not constrained (e.g., equation 9). Let Iw 

be the constrained GCE model where, say, b = l = 0 , which is equivalent to minimizing (9) subject to 

no constraints. Then, the Entropy Ratio (ER) statistic is defined as 2| IW - Iw |. Under the null hypothesis, 

2 ER converges in distribution to c( K-1) . The ER ratio can also be used to derive a pseudo- R2

measure 

(McFadden, 1974) based on the normalized entropy S: 

~ pseudo-R2 ” 1 - S(P) ; 

a common measure of within-sample prediction (i.e., goodness-of-fit). Incorporating bank-specific and 

macroeconomic data into the model under a GCE-IV approach greatly increased the amount of 

~ information above that captured by the priors and row shares alone. We estimate S( P) |

GCE -IV = 0.657; a 

normalized entropy measure that converts to a pseudo-R2 = 0.343 and a high value for the ER statistic that 

2 exceeds the critical value of cdf =8,a = 0.05=15.51. In contrast, the normalized entropy for the GCE model 

~ using prior probabilities and row shares only (i.e., Table 6 results) is S( P) |

GCE = .9697 with a 

corresponding pseudo-R2 = .0303. 

In addition, we evaluate the reliability of the estimated Markov transition matrix in terms of the 

predicted one-quarter, one-year, two-year, and three-year ahead failure rates. The cumulative probability 

of failure of banks initially in states 1-6 are plotted in Figure 1. The plots are consistent with our 

expectation that problem banks are more likely to fail, and that the likelihood of failure increases over 

time (all else equal). For example, although only one-in-five book-insolvent banks (initial state 1) fails 

within one quarter, nearly one-in-two fail within one year, and nearly three out of four fail within three 

19
years. These results suggest that, although in the short-run it appears that the resiliency of book-insolvent 

banks is relatively high, the long-run likelihood of recovery is low. In contrast, an adequately capitalized 

bank (initial state 5) has only a 30 bps likelihood of failing within one quarter, and a two percent 

likelihood of failure within one year. The likelihood that the average or representative, adequatelycapitalized bank fails within three years, however, increases to nearly 13 percent of the initial number of 

banks within that state (all else equal). 

In Figure 2, we show the cumulative probability of moving to a higher or nonfailure lower state, 

or remaining in an unchanged state conditioned on the bank’s initial state (note that states 7, 8 and 9 are 

terminal states and, therefore, they cannot represent initial states). For example, the probability that the 

representative bank initially in state 3 will move to a higher state is 0.187 – the sum of the probabilities a 

bank initially in state 3 moves to either states 4, 5 or 6; the probability that a bank initially in state 3 

moves to a lower state is 0.215 – the sum of the probabilities of moving to either states 1 or 2. These 

results show that the probability of moving to a lower state, short of failure, is greater than that of moving 

to a higher state for all banks irrespective of their initial condition, except for state 1 – the lowest nonfailure state. 

In Figure 3 we show the cumulative probability of remaining in the same or moving to a higher 

state within one quarter, one year, two years, and three years ahead, conditional on the bank's initial state. 

The one-quarter ahead results are equivalent to the sum of the unchanged and higher curves in Figure 2 

and reflect the condition that in the short-run, a large percentage of the severely distressed banks tend to 

remain unchanged or move to a higher state. Similar results hold for undercapitalized banks (i.e., states 2 

through 4). Over longer periods, however, the probability of remaining or moving to a higher state falls 

off for all, but the well capitalized banks (i.e., state 6).12 As expected, the probability of remaining or 

moving to a higher state declines more quickly for distressed banks (i.e., banks initially in the lower 

financial states). By the end of year 1, one out of two of the undercapitalized or insolvent banks (i.e., 

12 State 6 (i.e., well capitalized) is the highest financial state. For that reason, the probability of remaining in the 

same or moving to a higher state for each of the time horizons in Figure 3 is nearly equivalent to the probability of 

remaining in the same state. 

20
states 1 through 4) move to a lower state or close. However, after three years, four out of five of the 

insolvent banks, but only two out of three of the undercapitalized and significantly undercapitalized 

banks, move to a lower state or close. Adequately capitalized banks (i.e., state 5) fare much better. Only 

one out of four after one year, and after three years (all else held constant) one out of two, move to a 

lower state or close. 

The results to this point have been based on the mean values of the bank-specific variables within 

each financial state. However, within each of these states, there are banks better positioned, whether 

because of the composition of their portfolio, their asset-liability structure, or growth potentia l, to recover 

(or fail) over time. More specifically, we expect that banks in the upper tail of the distribution of values 

for some or all of the bank-specific variables to have lower probabilities of failure than that of the 

representative bank (or those with values for the bank-specific variables in the lower tail of the 

distribution). We test this hypothesis by constructing a set of hypothetical banks based on the observed 

90th- and 10th-percentile values for a selected subset of bank-specific variables. 

It is unlikely a specific bank will have values in the upper tail of the distributions for each of the 

bank-specific variables. For that reason, we substitute the upper (lower) values for only five of the eleven 

bank-specific variables used in our model: C&I loans to total loans, asset growth, income earned but not 

received to total loans, nonperforming loans to total loans, and the equity capital ratio of the holding 

company. We use the 10th-percentile values to identify the upper tail for each of the covariates, except for 

the equity capital ratio of the holding company in which we use the 90th-percentile value. We reverse the 

percentiles values to define the lower tail of the distributions. 

To illustrate the impact, we graph the 1-quarter, 1-year, 2-year, and 3-year ahead failure 

probabilities using the upper, mean, and lower tail values for the bank-specific variables (all other values 

held constant at their mean values) for banks initial in states 2 and 3 (see Figures 4 and 5). The mean 

value in each graph are from Figure 1. The failure probabilities for the upper and lower tail banks are 

consistent with our expectations. The failure probabilities for the representative bank generally fall 

between those for the upper and lower tail banks. The effect is greatest for those in the upper tail; as 

21
reflected in the initial lower failure probability and a much flatter growth in the failure probabilities over 

time. The difference in the failure probabilities between the upper and lower tailed banks is nearly 20 

percentage points for the hypothetical state 2 banks; and nearly more than 15 percentage points for the 

state 3 banks. 

V. Conclusion 

In this paper, we develop and estimate an alternative early-warning bank failure model. The 

model is designed as a stationary, first-order Markov transition process estimated, using a Generalized 

Maximum Entropy approach. We show that by incorporating bank-specific and macroeconomic variables 

through the constraints on the estimates of the transition probabilities, we can identify the marginal or 

incremental effects of a bank’s financial structure and the economic environment on the transition 

probabilities. 

We find that the effect of changes in the concentration of assets, management efficiency, and 

liquidity, and general economic conditions are conditional on the initial state of the banks – an intuitive 

result, but one that has not been empirically verified in the banking literature. 

22
Table 1: Transition States1 

States Label/Zone Criteria 

1 Insolvent (Book-Value) 

2 Critically Undercapitalized 

3 Significantly Undercapitalized 

4 Undercapitalized 

5 Adequately Capitalized 

6 Well Capitalized 

7 Merged with Affiliate 

8 Merged with Non-Affiliate 

9 Failure 

Equity/Assets < 0 

0 < Equity/Assets < 2% 

2% < Equity/Assets < 3% 

3% < Equity/Assets < 4% 

4% < Equity/Assets < 5% 

5% < Equity/Assets 

Merged within the Holding Company 

Merged with Unaffiliated Bank 

Closed by Primary Supervisor 

1. States 7, 8, and 9 are absorbing states ; banks never leave these states once they enter. For example, banks that 

fail in time t remain in that state in time t+1. Banks that begin the period in states 7 or 8 are censored and 

remain in that state in t+t, t=1, 2, ..., ¥. 

23
Table 2: Bank-Specific Summary Statistics 

Category1 Description of the Measure2 Average Values3

Credit Risk Commercial and Industry Loans (C&I) / Loans 

Asset Growth (percentage change) 

Non-performing Loans / Loans 

Income Earned but Not Received / Loans 

Net Interest Income / Loans 

Liquidity On-Hand Liquid Assets / Liabilities 

Net Loans / Deposits 

Trading Account and Securities / Assets 

Interest Rate Long-Term Assets / Assets 

Institutional Equity / Assets (within holding company) 

New Bank (less than five years) 

84.1-99.4 84.1-86.4 87.1-92.4 93.1-99.4 

0.2129 0.2642 0.2149 0.1746 

0.0089 0.0224 0.0080 0.0016 

0.0218 0.0271 0.0262 0.0129 

0.0127 0.0164 0.0126 0.0103 

0.0106 0.0005 0.0002 0.0302 

0.8480 0.6469 0.9014 0.9231 

3.385 1.5722 3.0560 4.5337 

0.2815 0.2544 0.2808 0.3009 

0.1262 0.1050 0.1229 0.1447 

0.0936 0.0878 0.0859 0.1067 

0.1101 0.1822 0.1102 0.0604 

1. Source: Quarterly Call Report data (Report of Condition and Income) filed by national banks 1983.1 – 1999.4. 

2. Asset growth: quarter-to-quarter percentage change; on-hand liquidity: interest-bearing bank balances, federal 

funds sold and securities less federal funds purchased and pledged securities; net loans: total loans less allowance for 

loan and leases; long term assets: fixed- rate loans and securities with remaining maturity over five years and 

floating rate loans and securities with re -pricing frequency less than every five years; new banks: banks that were 

chartered within the past five years; equity and assets of the holding company: the sum over all national banks held 

under the same the holding company; and deposits: demand deposits , MMDA, NOW, and other savings deposits. 

3. Based on the results of a pair-wise test of the hypothesis of no difference between sub-period means, we reject 

the null at the 5 percent level for all combinations except three cases: (i) the difference between means asset growth 

between 87.1-92.4 and 93.1-99.4; (ii) the difference between the mean values of net interest income ratio 84.1-86.4 

and 87.1-92.4; and (iii) the difference in the mean liquid asset to liabilities ratio 87.1-92.4 and 93.1-99.4. 

24
Table 3: Bank-Specific Summary Statistics by Failure States 

Variables1 1984 Q1 – 1999 Q4 1984 Q1 – 1986 Q4 1987 Q1 – 1992 Q4 1993 Q1 – 1999 Q4 

Non-Failed Failed Non-Failed Failed Non-Failed Failed Non-Failed Failed 

C&I / Loans2 0.207 0.331 * 0.254 0.358 * 0.210 0.304 * 0.174 0.254 * 

Asset Growth 0.009 0.010 0.021 0.031** 0.009 -0.011 * 0.002 -0.004 

Non-performing Loans / Loans 0.020 0.056 * 0.025 0.042 * 0.024 0.073 * 0.013 0.048 * 

Income Earned but Not Received / Loans 0.013 0.016 * 0.016 0.019 * 0.013 0.014 * 0.010 0.012 * 

Net Interest Income / Loans 0.012 -0.017 * 0.003 -0.019 * 0.001 -0.016 * 0.030 -0.007 * 

On-Hand Liquid Assets / Liabilities 0.879 0.204 * 0.698 0.190 * 0.939 0.221 * 0.925 0.181 * 

Net Loans / Deposits 3.519 0.683 * 1.669 0.710 * 3.560 0.651 * 4.644 0.728 * 

Trading Account and Securities / Assets 0.289 0.132 * 0.268 0.132 * 0.289 0.129 * 0.301 0.201 * 

Long-Term Assets / Assets 0.129 0.073 * 0.109 0.066 * 0.125 0.077 * 0.145 0.143 

Equity / Assets (within holding company) 0.095 0.065 * 0.088 0.085 * 0.088 0.042 * 0.107 0.073 * 

New Bank 0.098 0.359 * 0.155 0.426 * 0.100 0.298 * 0.060 0.045 * 

1. See footnotes to Table 2. 

2. H0: mnf = mf assuming unequal variance. * (**) denote significant at the 5 percent (10 percent) level. 
Table 4: Macroeconomic Variables 

Variables Source Description Summary Statistics 

General Business Conditions 

Unemployment Rate 

Agricultural Prices 

Oil Prices 

Interest Rate/Yield Curve 

Bank Prime Loan Rate 

10-year Spread 

30-year Spread 

BLS 

USDA 

BEA 

DOE 

BEA 

FRB 

FRB 

FHLB 

FRB 

Civilian Unemployment Rate (SA) 

Agricultural prices received by farmers: all farms 

(1990-92 = 100) deflated by GDP price index

Refiners’ acquisition cost of crude oil: composite 

($/bbl) deflated by GDP price index

Prime rate – average of daily figures

Spread of 10-year Treasury rate over the

3-month Treasury rate

Spread of 30-year mortgage rate over the 

3-month Treasury rate

84.1 - 99.4 84.1 - 86.4 87.1 - 92.4 93.1 - 99.4 

6.018 7.233 6.147 5.386 

1.126 1.269 1.1671 1.030 

21.575 31.947 21.615 17.052 

8.653 10.103 8.852 7.862 

1.761 2.321 1.852 1.444 

3.348 4.229 3.433 2.900 

26
Table 5: Prior Transition Probabilities (P0

) - Average percentage of banks in each transition state 1984.1 through 1986.4 

Book 

Insolvent 

Critically 

Undercapital 

Significantly 

Undercapital Undercapital

Adequately 

Capitalized 

Well 

Capitalized 

Merged with 

Affiliate 

Merged with 

Non-Affiliate Failure 

States time [t+1] 

1 2 3 4 5 6 7 8 9 

1 0.4925059 0.044193 0.0708333 0.0611111 0.0167625 0.0916362 0.0086207 0 0.2143373 

2 0.1581899 0.4485589 0.0572696 0.074592 0.0339627 0.0856976 0.0066667 0.012037 0.1230256 

3 0.0252646 0.2138889 0.410961 0.1496769 0.0523097 0.0944088 0 0.0027778 0.0507123 

4 0.0164488 0.1052244 0.0954008 0.4109943 0.212435 0.1288807 0 0.0036811 0.026935 

time [t] 5 0.0044846 0.0193092 0.026702 0.0855492 0.522016 0.3195045 0.0058906 0.0040465 0.0124972 

6 0.0003299 0.0007536 0.0007154 0.0023621 0.0120795 0.9757902 0.002628 0.0049354 0.0004058 

7 0 0 0 0 0 0 1 0 0 

8 0 0 0 0 0 0 0 1 0 

9 0 0 0 0 0 0 0 0 1 

1. The percentage of banks in state i during time [t] that fall in state j during time [t+1]; [t+1] represents one quarter ahead. States 7, 8, and 9 are terminal 

states; e.g., a bank in state 9 in time [t] will not transition to any other state in [t+1]. This implies that the transition probabilities along the main diagonal 

for these initial states are one and zero for the off diagonal elements. 
Table 6: GCE Estimates – Based only on share data 1987.1-1999.4 and the prior states from 1984.1-1986.4 

Book 

Insolvent 

Critically 

Undercapital 

Significantly 

Undercapital Undercapital

Adequately 

Capitalized 

Well 

Capitalized 

Merged with 

Affiliate 

Merged with 

Non-Affiliate Failure 

States time [t+1] 

1 2 3 4 5 6 7 8 9 

1 0.504043 0.045738 0.073228 0.062982 0.017972 0.085537 0.008504 0.0001 0.201895 

2 0.1596 0.46297 0.059293 0.078339 0.037775 0.075585 0.006385 0.011701 0.108351 

3 0.025035 0.216375 0.416643 0.153989 0.056369 0.083473 9.68E-05 0.002697 0.045322 

4 0.015764 0.103792 0.095017 0.419958 0.232337 0.107614 9.17E-05 0.003417 0.02201 

time [t] 5 0.004088 0.018761 0.02639 0.089048 0.617065 0.228484 0.004837 0.003412 0.007915 

6 .000044 0.000199 0.000302 0.001054 0.003496 0.993914 0.000833 0.000158 0 

7 0 0 0 0 0 0 1 0 0 

8 0 0 0 0 0 0 0 1 0 

9 0 0 0 0 0 0 0 0 1 

1. See footnote to Table 5. 

28
Table 7: Generalized Cross-Entropy-Instrumental Variable Model – Incorporating P0

, bank-specific, and macroeconomic variables1 

Book 

Insolvent 

Critically 

Undercapital 

Significantly 

Undercapital Undercapital

Adequately 

Capitalized 

Well 

Capitalized 

Merged with 

Affiliate 

Merged with 

Non-Affiliate Failure 

States time [t+1] 

1 2 3 4 5 6 7 8 9 

1 0.593600 0.112387 0.061766 0.024834 0.001318 0.000059 0.004713 0.000083 0.201241 

2 0.165668 0.605370 0.064082 0.051062 0.004946 0.000022 0.003897 0.010388 0.094566 

3 0.025340 0.189740 0.547806 0.165623 0.021031 0.000710 0.000082 0.002863 0.046805 

4 0.014130 0.053542 0.150941 0.673824 0.081888 0.000366 0.000072 0.003922 0.021316 

time [t] 5 0.000320 0.000067 0.001285 0.059956 0.926716 0.000060 0.004980 0.004144 0.002473 

6 0 0 0.000008 0 0.000164 0.999329 0.000292 0.000208 0 

7 0 0 0 0 0 0 1 0 0 

8 0 0 0 0 0 0 0 1 0 

9 0 0 0 0 0 0 0 0 1 

1. See Tables 3 and 4 for a list of bank-specific and macroeconomic variables. 

29
1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

Table 8: Marginal Effects – The incremental effect of a change in bank-specific and macroeconomic variables 

Selected Series 

States 

time[ t] 1 2 

C&I Loans 0.000392 -0.001359 

0.002231 -0.006228 

0.000006 -0.003606 

-0.000066 -0.002125 

0.000011 -0.000003 

Asset Growth 

-0.004731 -0.000531 

-0.001946 -0.003448 

-0.000178 -0.000399 

-0.000535 -0.001549 

0.000002 0.000002 

Equity/Assets Hld Co. 

0.009308 -0.023771 

0.035217 -0.127552 

-0.003148 -0.088691 

-0.009754 -0.070382 

-0.001073 -0.000319 

Unemployment Rate 

-0.000299 0.000106 

-0.000337 0.000402 

-0.000046 0.000070 

-0.000013 0.000163 

-0.000001 0.000000 

10-year Treasury Spread 

-0.000350 0.000043 

-0.000270 0.000109 

-0.000032 0.000040 

-0.000051 -0.000051 

0.000001 0.000001 

3 

0.000168 

0.001109 

0.001844 

0.000152 

0.000059 

-0.002445 

-0.000592 

-0.000945 

-0.005173 

-0.006381 

-0.000004 

-0.000106 

0.008310 

0.027814 

0.030303 

-0.054887 

-0.003353 

-0.002562 

0.000068 

0.000060 

0.000323 

0.000521 

0.000008 

0.000006 

-0.000006 

-0.000046 

-0.000282 

-0.000342 

0.000009 

-0.000011 

time [t+1] 

4 5 

0.000098 -0.000007 

0.001000 0.000011 

0.000864 -0.000186 

0.002948 -0.001735 

0.003229 -0.004092 

-0.050131 

0.000292 -0.000014 

0.001275 -0.000083 

0.003769 -0.000233 

0.010983 -0.003702 

0.007825 -0.008988 

-0.002213 

0.007757 0.000756 

0.039078 0.006190 

0.053648 0.015106 

0.084192 0.068972 

-0.089694 0.129018 

-0.026320 

-0.000015 0.000000 

-0.000112 -0.000005 

6 7 8 9 

0.000050 0.000031 0.000000 0.000627 

0.000035 0.000096 0.000041 0.001706 

0.000912 0.000001 -0.000022 0.000185 

0.000853 0.000001 -0.000074 0.000045 

0.000319 0.000353 0.000005 0.000120 

0.204352 -0.088225 -0.063551 

0.000002 0.000028 0.000000 0.005547 

0.000001 0.000055 -0.000059 0.005149 

0.000033 0.000001 -0.000006 0.002184 

0.000021 0.000000 -0.000113 0.001274 

0.000014 0.000466 0.000115 0.000568 

0.006909 -0.002225 -0.002365 

0.000060 0.000643 0.000004 -0.003067 

0.000046 0.001705 0.002847 0.014655 

0.000993 0.000005 -0.000213 -0.008002 

0.000761 -0.000026 -0.002351 -0.016525 

0.000176 -0.012938 -0.013042 -0.008775 

0.194003 -0.092695 -0.072425 

0.000000 -0.000008 0.000000 0.000147 

0.000000 -0.000016 -0.000018 0.000025 

-0.000322 -0.000020 -0.000002 0.000000 -0.000005 0.000002 

-0.000778 0.000057 -0.000001 0.000000 -0.000002 0.000053 

-0.000235 0.000278 0.000000 -0.000051 -0.000010 0.000011 

0.000052 -0.000009 -0.000082 0.000033 

0.000030 -0.000002 0.000000 -0.000021 0.000000 0.000306 

0.000089 -0.000015 0.000000 -0.000035 -0.000050 0.000217 

0.000243 -0.000052 0.000004 -0.000001 -0.000011 0.000090 

0.000904 -0.000476 0.000003 -0.000001 -0.000033 0.000046 

0.000923 -0.000850 0.000002 -0.000102 -0.000027 0.000043 

-0.000298 0.001613 -0.000859 -0.000445 
0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

1 2 3 4 5 6 

initial states 

Figure 1: 

1-quarter 1-year 2-year 3-year 

Probability of Failure by Initial State 
Figure 2: 

State (one-quarter ahead) 

0 

0.25 

0.5 

0.75 

1 

1 2 3 4 5 6 State 

Probabiity 

unchanged higher lower 

Probability of Moving to Higher, Lower, or Unchanged 

32
0.00 

0.20 

0.40 

0.60 

0.80 

1.00 

Probability 

1 2 3 4 5 6 

State 

Figure 3: Probability of Remaining in or Moving to a Higher State 

1-quarter 1-year 2-years 3-years 

33
Figure 4: 

(State 2) 

0 

0.2 

0.4 

0.6 

0.8 

1-quarter 1-year 2-years 3-years 

prob of failure 

lower mean upper 

Upper and Lower Tails of the Distribution 

34
Figure 5: 

(State 3) 

0 

0.2 

0.4 

0.6 

1-quarter 1-year 2-years 3-years 

prob of failure 

lower mean upper 

Upper and Lower Tails of the Distribution 

35
References 

Altman, E.I., R.B. Avery, R.A. Eisenbeis, and J.F. Sinkey, Jr. 1981. Application of classification 

techniques in business, banking, and finance. Greenwich, CT : JAI Press. 

Amemiya, T. 1985. Advanced Econometrics, Harvard University Press, Cambridge, MA. 

Beaver, W.H. 1968. “Market Prices, Financial Ratios, and Prediction of Failure,” Journal of Accounting 

Research, 6, 179-192. 

Eisenbeis, R.A. 1977. “Pitfalls in the Application of Discriminant Analysis in Business, Finance, and 

Economics,” Journal of Finance, 32, 875-900. 

Fissel, G., G.A. Hanweck, and J. O’Keefe. 1996. “A Comparative Analysis of Modeling Methodologies 

of Financially Distressed Banking Firms,” Working Paper. 

Glennon, D., and A. Golan. 2001. “Identifying Problem Banks Using Information-Theoretic Methods.” 

OCC draft manuscript. 

Golan, A., G.G. Judge, and D. Miller. 1996a. Maximum Entropy Econometrics: Robust Estimation with 

Limited Data, John Wiley & Sons, New York. 

Golan, A., G. G. Judge and J. Perloff. 1996b. “A Generalized Maximum Entropy Approach to Recovering 

Information From Multinomial Response Data,” Journal of the American Statistical Association, 

Vol. 91, pp. 841-853. 

Golan, A. and H. Gzyl, 2001. “A Generalized Maxentropic Inversion Procedure for Noisy Data,” Applied 

Mathematics and Computation (Forthcoming). 

Jaynes, E.T. 1957a. Information Theory and Statistical Mechanics, Physics Review, 106, 620–630. 

Jaynes, E.T. 1957b. Information Theory and Statistical Mechanics II, Physics Review, 108, 171–190. 

Kolari, J., M. Caputo, and D. Wagner. 1996. Trait recognition: An alternative approach to early warning 

systems in commercial banking. Journal of Business, Finance and Accounting 23: 1415-1434. 

Kolari, J., D. Glennon, H. Shin, and M. Caputo. 2001. “Predicting Large U.S. Commercial Bank 

Failures,” Journal of Economics and Business, 54, 361-387. 

Kullback, J. 1959. Information Theory and Statistics, John Wiley & Sons, New York. 

Looney, S.W., J.W. Wansley, and W.R. Lane. 1989. “An Examination of Misclassifications with Bank 

Failure Prediction Models,” Journal of Economics and Business, 41, 327-336. 

Shannon, C.E. 1948. A Mathematical Theory of Communication, Bell System Technical Journal, 27, 

379–423. 

Sinkey, J. E., Jr. 1975. A multivariate statistical analysis of the characteristics of problem banks. 

Journal of Finance, 20: 21-36. 

36
Soofi, E. 1996. Information Theory and Bayesian Statistics, , in D. A. Berry, K. M. Chaloner and J. K. 

Geweke (eds.), Bayesian Analysis in Statistics and Econometrics: Essays in Honor of Arnold 

Zellner, John Wiley, New-York, 179-189. 

Stengel, M., G. Whalen, P. Swamy, and T. Lutton. 2000. “Early Warning Models,” OCC draft 

manuscript. 

37
Appendix 1. The Classic Maximum Entropy Model: A Review 

To provide a basis for understanding the philosophy of the ME approach, we consider the 

following example. Let Q = {q1,q 2, ...,q M }be a finite set and p be a probability mass function on Q . 

M

The Shannon’s (1948) information criterion, called entropy, is H ( p) = -�i=1 pi log pi with 

0 log 0 ” 0 . This information criterion measures the uncertainty, or informational content, in Q which is 

implied by p. The entropy-uncertainty measure H(p) reaches a maximum when 

p1 = p2 = ... = pM = 1/ M and a minimum with a point mass function. Given the entropy measure and 

structural constraints in the form of moments of the data (distribution), Jaynes (1957a, 1957b) proposed 

the maximum entropy (ME) method, which is to maximize H(p) subject to these structural constraints. If 

no constraints (data) are imposed, H(p) reaches its maximum value and the distribution of the p’s is a 

uniform one. Thus, if we have partial information in the form of some moment conditions, Yt (t=1, 2, ..., 

T) , where T<M, the maximum entropy principle prescribes choosing the p(q i ) that maximizes H(p) 

subject to the given constraints (moments) of the problem. The solution to this underdetermined problem 

is 

� � 

pˆ(q i ) � exp �-�ltYt (q i )� (A1.1) 

� t � 

where l are the T Lagrange multipliers. 

If prior information, qi , concerning the unknown pi exists, then one alternative to the ME 

approach is to minimize the Kullback-Leibler (K-L) entropy-distance between the post-data weights and 

the priors (Gokhale and Kullback, 1978). Under this criterion, known as cross entropy (CE), the problem 

of recovering p, may be formulated as minimizing the CE subject to the relevant structural constraints 

(moments). The resulting solution is 

~ � �

p(q i ) � qi exp ��ltYt (q i )� . (A1.2) 

� t � 

38 
When the prior information qi has uniform mass, the optimal solutions of the ME and CE 

problems are identical. 

To relate the ME formalism to the more familiar linear model, consider a special case of this 

model where there assumed to be no noise in the observed moments: 

y=Xp (A1.3) 

and p is a K-dimensional proper probability distribution. The ME formulation is 

� � � �pˆ = arg max �-� pk log pk � � � k � ME = � . (A1.4) 

�s.t. y = Xp and � pk = 1 �

� k

Similarly, the CE formulation is just 

�~ � � �p = arg min �� pk log( pk qk ) � � � k � CE = � (A1.5)

�s.t. y = Xp and � pk =1 �

� k

where I (p, q) = � pk log( pk qk ) is the Kullback-Leibler information, or CE, measure. 

k 

The exact CE solution is 

� T ~ � T ~ 

qk exp ��li xik � qk exp�

�

�li xik �

� 

~ 

pk = Ł i =1 ł ” Ł i=1 ł (A1.6) � T ~ � W �qk exp ��li xik � 

k Ł i =1 ł 

The dual CE counterpart is 

Inf I (p,q) = Sup{ l¢y - log W( X ¢l)} (A1.7) 

p ̨P l  ̨D 

where P = {p: Xp = y} is a set of proper (normalized) distributions satisfying the linear constraints 

~ (A1.3), and D is the set { l  ̨  ́T : W(X '? ) << ¥}. Having solved for l , one gets p% via Eq. (A1.6). 

39