# Do financial regulators act in the public’s interest? A Bayesian latent
class estimation framework for assessing regulatory responses to banking
crises

Padma Sharma111The views expressed here are the opinions of the authors and
should not be attributed to the Federal Reserve Bank of Kansas City or the
Federal Reserve System. and Trambak Banerjee1
*Federal Reserve Bank of Kansas City and 1University of Kansas

###### Abstract

When banks fail amidst financial crises, the public criticizes regulators for
bailing out or liquidating specific banks, especially the ones that gain
attention due to their size or dominance. A comprehensive assessment of
regulators, however, requires examining all their decisions, and not just
specific ones, against the regulator’s dual objective of preserving financial
stability while discouraging moral hazard. In this article, we develop a
Bayesian latent class estimation framework to assess regulators on these
competing objectives and evaluate their decisions against resolution rules
recommended by theoretical studies of bank behavior designed to contain moral
hazard incentives. The proposed estimation framework addresses the unobserved
heterogeneity underlying regulator’s decisions in resolving failed banks and
provides a disciplined statistical approach for inferring if they acted in the
public interest. Our results reveal that during the crises of 1980’s, the U.S.
banking regulator’s resolution decisions were consistent with recommended
decision rules, while the U.S. savings and loans (S&L) regulator, which
ultimately faced insolvency in 1989 at a cost of $132 billion to the taxpayer,
had deviated from such recommendations. Timely interventions based on this
evaluation could have redressed the S&L regulator’s decision structure and
prevented losses to taxpayers.

Keywords: Bank failures, Federal Deposit Insurance Corporation (FDIC), Federal
Savings and Loans Insurance Corporation (FSLIC), Bayesian inference, collapsed
Gibbs sampler, Latent class models.

##  1 Introduction

During financial crises when a large number of banks fail, actions of
financial regulators receive substantial public scrutiny. The global financial
crisis of 2008 is one such recent example that led to widespread bank
failures, reviving debate over how financial regulators might preserve
immediate financial stability while also safeguarding against future moral
hazard. During a financial crisis, regulators determine and administer the
bailout, sale or liquidation of failed banks. Regulators bail out banks when
they place greater emphasis on preserving financial stability and liquidate
institutions when they are more attentive to the curtailment of moral hazard
incentives. Critical assessments of these actions are essential to ensuring
regulators balance the competing concerns in a manner that serves the public
interest. However, the public typically criticizes specific regulatory
decisions instead of evaluating how closely their overall decision framework
serves the public interest. Individuals disfavor bailouts because they
represent transfers from taxpayers to shareholders222“The firms we rescued
were usually not gracious about the terms of their rescues, while the
overwhelming sentiment among the public was that they shouldn’t have been
rescued at all.”(Bernanke et al., 2019). The public also criticizes bank
liquidations because they are costly for depositors and loan customers (Isaac,
2010). How can the public and their elected representatives comprehensively
assess the actions of regulators against their competing objectives of
preserving financial stability and restraining moral hazard?

Theoretical studies of regulator and bank behavior develop decision rules that
resolve the trade-off between the two objectives in a manner that maximizes
the value of output generated by the banking sector, and thereby provide a
benchmark for evaluating such agencies. Broadly, these studies recommend
state-dependent decision rules that vary in response to economic and industry-
wide conditions that accompanied bank failures (see for example Cordella and
Yeyati (2003); Acharya and Yorulmazer (2008); DeYoung et al. (2013) and the
references therein). For instance, theoretical models recommend that
regulators adopt distinct decision rules for handling banks that failed in the
midst of economic distress, and those that failed in normal economic
conditions. However, data on bank resolutions do not contain details on the
extent to which bank-specific and broader economic conditions were considered
in each decision. Furthermore, even though distress in the economy and in the
banking industry are observable through measures such as unemployment rate,
and growth in output, it is not immediately clear (1) what threshold
regulators may have used to distinguish between periods of distress and
normalcy, and (2) what aspects of a failed bank’s overall health ultimately
inform the regulator’s resolution decision.

In this paper, we develop a Bayesian latent class estimation framework to
compare regulators’ decisions against theoretical decision rules that foster
financial stability while restraining moral hazard. Theoretical benchmarks
recommend applying distinct decision rules that vary by economic and industry
conditions, but the thresholds used by regulators to categorize banks into
distinct decision rules are unobservable. For instance, when one bank failed
in Kansas, and another in Kentucky in 1988 where unemployment rates were 5%
and 8.5% respectively, it is not apparent whether regulators considered only
one, both or neither banks to have failed amid high distress. The latent class
model developed in this paper incorporates such uncertainty by assigning banks
into distinct decision rules, or classes with probability, rather than
certainty. The proposed model consists of a hierarchical structure (see figure
2) in which the first layer is a probabilistic class-membership model that
assigns failed banks to classes that correspond to the two distinct states of
nature, such as high or low underlying economic distress. Conditional on class
membership, the second layer specifies the relationships between the
resolution type, namely assistance, sale or liquidation of failed banks, and
bank-specific covariates, such as size and asset quality. These relationships
are homogeneous within and heterogeneous across the latent classes when the
classes are statistically different from each other. Our approach serves as a
classification algorithm that determines whether regulators assigned distinct
decision rules to classes of banks that failed in disparate economic and
industry conditions, and to systematically infer if they acted in the public
interest.

###  1.1 Regulators of the U.S. banking sector, their resolution methods and
the crises of 1980’s

In this article we assess regulators from two sub-sectors of the U.S. banking
industry, commercial banks and Savings and Loans (S&L) institutions333“ A
Savings and Loans institution is a financial institution that ordinarily
possesses the same depository, credit, financial intermediary, and account
transactional functions as a bank, but that is chiefly organized and primarily
operates to promote savings and home mortgage lending rather than commercial
lending. Also known as a savings bank, a savings association, a savings and
loan association, or an S&L.”FDIC (1998), during their simultaneous crises of
the 1980’s. The Federal Deposit Insurance Corporation (FDIC) serves as the
regulatory authority for commercial banks while the Federal Savings and Loans
Insurance Corporation (FSLIC) was the counterpart for S&L’s until its failure
in 1989. The two regulators are comparable on account of the fundamental
similarities across banks and S&L’s in that both institutions offer loans and
deposits, undertake maturity transformation, monitor information and offer
liquidity and payments services (Freixas and Rochet, 2008).

Figure 1: Number of failed banks resolved by the FDIC and number of failed
S&L’s resolved by the FSLIC from 1934 to 2020.

During the crises of 1980’s, these two related sectors of the U.S. banking
industry, namely commercial banks and S&L’s, witnessed the highest number of
failures since the Great Depression as depicted in Figure 1. Notably, the FDIC
and FSLIC underwent contrasting trajectories following the crises. While the
FDIC survived the crisis, albeit with depleted insurance funds, the FSLIC
faced insolvency by the end of the crisis and was closed in 1989 at a cost of
$132 billion to the taxpayer (FDIC, 1998).

When banks and S&Ls failed, the FDIC and FSLIC applied one of the following
three resolution methods (Walter, 2004):

  1. 1.

Type I: Open Bank Assistance (OBA) - Under this resolution method, the
regulator provides financial assistance to acquirers toward the purchase of a
failing bank or grants direct assistance to the failing bank.

  2. 2.

Type II: Purchase and Assumption (P&A) - Resolutions under this category
consist of acquiring a part of the assets and liabilities of a failed bank by
a participating institution.

  3. 3.

Type III: Deposit Payout (PO) - Under this resolution category, the regulator
liquidates the failed institution and pays out its insured depositors from the
insurance fund.

Each of the three resolution methods described above involve a progressively
more severe breakdown of relationships between the bank and its customers
(Ashcraft, 2005). For instance while a Type I resolution method ensures
continuity of banking relationships, a Type III resolution terminates all such
relationships. We compare the decision rules employed by the FDIC and FSLIC in
assigning of the three resolution methods to failed institutions during the
crises of 1980’s in addition to assessing their decisions against recommended
decision rules from theoretical studies. Our results expose specific
weaknesses in the FSLIC’s decision structure and the FDIC’s relative strengths
that are likely to have contributed to the former’s failure in 1989 and the
latter’s continued survival.

The crises of the 1980’s are particularly suitable for comparing the
resolution decisions of FDIC and FSLIC against theoretically recommended
state-dependent decision rules for several reasons. First, the simultaneous
crises in banking and S&L industries in the 1980’s provided a basis to compare
the two regulators, and to identify the stronger of the two approaches to
resolving failed institutions. Second, bank and S&L failures in this period
occurred against the backdrop of shocks in specific sectors, namely,
agriculture, real-estate and energy that resulted in regional crises (FDIC,
1998). For instance, the major sectoral crises that occurred during this
period were the recessions following the collapse of energy prices in Texas,
Louisiana and Oklahoma, the agricultural recession in Kansas, Iowa and
Nebraska and the real-estate-led downturns in California, the Southwest and
the Northeast (FDIC, 1997). Third, banks were subject to varying levels of
branching restrictions and operated either within state borders or across
states that had entered into reciprocal arrangements (Kroszner and Strahan,
1999). Specifically, this period predates the elimination of the interstate
branching restrictions mandated by the 1994 Riegle-Neal Interstate Banking and
Branching Efficiency Act (Medley, 2013). Thus, the combination of sectoral
crises that were regionally contained and branching restrictions that limited
the geographic scope of banking markets entailed that certain bank failures
occurred amid economic and financial distress, and others, in relatively
normal economic conditions. This provided an ideal setting for the two
regulators towards implementing the theoretically recommended state-dependent
decision rules for resolving failed banks during this crisis.

###  1.2 Recommended decision rules from theoretical studies and testable
hypotheses

In this section we first discuss the decision rules that are recommended by
the branches of theoretical literature for the resolution of failed banks
under two state-dependent scenarios - economic distress and banking industry
distress. Thereafter, we identify the specific testable hypotheses for each of
these scenarios as wells as the hypothesis to infer the impact of political
influence on regulators’ resolution decisions.

Economic distress and recommended decision rule for bank resolutions -
Cordella and Yeyati (2003) determined a resolution strategy, or a decision
rule, in which the regulator provides bailouts to banks if their failure
occurred under macroeconomic distress, when bank failures are less likely to
have arisen due to their unsound portfolio decisions and more likely to have
arisen due to exogenous factors. Correspondingly, in the event of bank
failures under normal economic conditions, their theoretical model recommended
liquidating such banks.

Hypothesis $H_{1}$: The testable hypothesis is that the FDIC and FSLIC applied
different decision rules for banks that failed in normal economic conditions
and those that failed amid macroeconomic distress. Conditional on the presence
of two distinct rules, the subsequent statistical inference centers on testing
the hypothesis that the probability of receiving a Type I resolution was
higher for banks that failed amid high economic distress relative to those
that failed amid low distress.

Banking industry distress and recommended decision rule for bank resolutions -
Acharya and Yorulmazer (2007, 2008) propose resolution strategies for the
“too-many-to-fail” problem or, equivalently, the simultaneous failure of many
banks. Their recommended decision rule consisted of facilitating acquisitions
of failed banks when such failures were small in number but providing bailouts
and financial assistance when there were a large number of failures.444Acharya
and Yorulmazer (2007) note that it is optimal for regulators to commit to not
assisting failed banks on an ex-ante basis. However, when a large number of
banks fail, it becomes optimal to forgo that commitment and assist banks
rather than liquidate them. This is the time-inconsistency between regulators’
ex-ante and ex-post decisions.

Hypothesis $H_{2}$: The first testable hypothesis is that regulators applied
distinct rules in the presence and absence of banking industry distress.
Second, the decision rule employed in the presence of banking industry
distress designated a higher proportion of resolutions as Type I compared to
the rule applied in its absence.

Political influence on regulators’ decision rule for bank resolutions -
Several empirical studies (Igan et al., 2012; Duchin and Sosyura, 2012) have
revealed the evidence of political and lobbying influence on regulators’
decisions for bank resolutions. For instance, DeYoung et al. (2013) find that
when a regulator experiences political pressure to place greater emphasis on
maintaining current liquidity, they will provide more bailouts than when the
regulator prioritizes the prevention of future moral hazard.

Hypothesis $H_{3}$: The primary hypothesis is that the presence of political
influence induces a separate decision rule that is distinct from the rule
applied in its absence. Subsequently, inferences center on whether the
decision rule utilized under political influence resulted in a higher
probability of receiving a Type I resolution relative to the rule applied in
the absence of political influence.

###  1.3 Our contributions and connections to existing works

We consider the three hypotheses from Section 1.2 and test for the presence of
state-dependent resolution strategies, either recommended by theory or arising
from political interference, in the decision rules employed by the FDIC and
FSLIC to resolve failed banks and S&Ls during the crises of 1980’s. Our
findings reveal that regular assessments of financial regulators by lawmakers
and the public can uncover gaps between observed and recommended resolution
rules and provide guidelines for corrective actions. For instance, timely
assessments of the FSLIC could have revealed that the agency had provided
excessive assistance to institutions that failed amid relatively normal
economic conditions and to institutions that received political support
(Section 5). Interventions based on these assessments could have potentially
prevented both, the failure of FSLIC in 1989 and the ensuing costs to the
taxpayer. Conversely, the decision structure of the FDIC identified in this
paper (Section 4) provides a road-map for newer resolution agencies that face
widespread failures from systemic shocks. The ensuing discussion summarizes
our main contributions.

  1. 1.

This paper contributes to the empirical literature examining bank resolution
decisions in two ways. First, we statistically evaluate how regulators’
decisions align with recommended decision rules from alternative theoretical
models, as well as the extent to which political economy factors interfere
with those recommended decisions. Second, and to the best of our knowledge,
this is the first article to compare between the decision rules of the FDIC
and FSLIC during the simultaneous crises in the banking and S&L industries
during the 1980’s.

  2. 2.

To evaluate the three hypotheses in Section 1.2, we develop a new methodology
for assessing regulators in the form of a Bayesian latent class estimation
framework for ordered outcomes. The proposed framework detects unobserved
heterogeneity in regulators’ resolution decisions based on underlying economic
and political conditions. For estimating this model and conducting inference,
we design a novel collapsed Gibbs sampler algorithm that provides a technique
for efficient sampling from the posterior distribution relative to standard
approaches by reducing the autocorrelations across successive draws. Our
method provides a statistical framework to compare parameters across the
latent classes, and additionally allows for inferences on all estimated
quantities of interest, including marginal effects and the probability of
class membership.

  3. 3.

We consider hypothesis $H_{1}$ and evaluate whether the FDIC and FSLIC
provided bailouts to banks or S&L’s that failed amid macroeconomic distress
and withheld such assistance for failures in normal economic conditions. Our
results reveal that the FSLIC deviated from and the FDIC adhered to this
theoretically recommended rule. Specifically, banks that failed amid high
economic distress received financial assistance, or bailouts from the FDIC
with an average probability of 25% compared to 3% for banks that failed amid
low distress. The FSLIC, on the contrary, assigned Type I assistance with
probabilities 68% and 70% to the two groups of S&L institutions that did not
statistically differ across measures of macroeconomic distress.

  4. 4.

We examine hypothesis $H_{2}$ on whether the two regulatory agencies
experienced a too-many-to-fail problem and responded to it in the form of a
greater reliance on bailouts and financial assistance to acquiring
institutions. Our results show that the FDIC’s decision rules aligned with the
theoretical rules as the agency provided Type I assistance with a probability
of 27% for failures amid economic and banking industry distress and a
statistically lower probability of 4% for failures amid low levels of such
distress. The FSLIC assigned Type I assistance with probabilities of 76% and
70% among groups of institutions that did not statistically differ by industry
and economic distress.

  5. 5.

We evaluate hypothesis $H_{3}$ and assess the extent to which political
pressures influenced the resolution decisions of the two agencies. Whereas the
previous two hypotheses examined the extent to which regulators followed
recommended theoretical rules, this assessment examines potential
institutional weaknesses. We find that political support for the banking
industry played a limited role in the FDIC’s decisions, but a more salient
role in the FSLIC’s decisions. Notably, the FSLIC assigned assistance to S&L’s
that likely received a higher degree of political support and failed amid
lower economic distress at a statistically higher probability of 92% relative
to 59% for S&L’s that failed amid low political support and in a climate of
higher economic distress.

A rich literature examined the financial costs incurred by the FDIC (Bennett
and Unal, 2014; Balla et al., 2015) and the weaknesses of the FSLIC (Kane,
1989; Akerlof et al., 1993; Romer and Weingast, 1991; White, 1991). The
empirical results in this paper align with the sources of weaknesses in the
FSLIC’s decision structure discussed in previous literature. However, prior
literature has not formally assessed both agencies against resolution rules
recommended by theoretical models. Relatedly, the decision rules of the two
agencies have not been compared with each other. Our paper addresses both
these gaps in the literature.

Greene and Hensher (2010) developed a classical method to estimate latent
class models with ordered outcomes by way of an Expectation-Maximization
algorithm. Heckman and Singer (1984) proposed latent class models as a
nonparametric alternative to random coefficients models in addressing
unobserved heterogeneity without the problems of “over-parameterization” and
excessive sensitivity to distributional assumptions associated with the latter
method. Latent class models have since been developed for a range of outcomes
including multinomial (Greene and Hensher, 2003; Burda et al., 2008), count
(Wang et al., 1998; Wedel et al., 1993; Deb and Trivedi, 1997; Nagin and Land,
1993) and ordered (Greene et al., 2014) responses. These works apply latent
class models to study heterogeneity in fields ranging across healthcare,
marketing and transportation. Here we provide a new interpretation of latent
class models as a tool to assess banking regulators and develop a framework
for evaluating regulators in the event of future crises.

###  1.4 Organization

The rest of the article is organized as follows. In Section 2 we discuss the
data used in this article. Section 3 presents the proposed Bayesian latent
class estimation framework for evaluating the regulatory decision rules for
resolving bank failures against recommended decision rules from theoretical
studies. In Section 4 we discuss the results of our analysis pertaining to the
FDIC’s resolution decisions while Section 5 is dedicated to analyzing the
FSLIC’s resolution decisions. The article concludes with a discussion in
Section 6. Additional technical details, numerical results and background
information are relegated to the supplementary materials.

##  2 Data

We examine bank resolutions by the FDIC between 1984 and 1992 and S&L
resolutions by the FSLIC from 1984 until the agency’s closure in 1989. The
period between 1984 and 1992 is particularly suited to evaluate the decisions
of the FDIC since the agency was subject to restrictions in applying Type I
resolutions before 1982 and after $1993$ 555Prior to 1982, the FDIC was
restricted to providing assistance to a bank only when the institution’s
continued existence was deemed to be essential to the community in which it
operated. The Garn-St.Germain Depository Institutions Act of 1982 dropped this
essentiality test. After 1993, new legislation prohibited the FDIC from using
its funds to provide assistance to failing institutions, particularly if such
assistance resulted in benefits to the troubled institution’s shareholders
(Walter, 2004). Consequently, the FDIC was authorized to autonomously provide
assistance under Type I resolutions during 1984-1992, the time period
considered in this study.. The FSLIC, on the contrary, retained this authority
from the start of the sample period until its closure in 1989.

The data used in this paper includes variables that can be broadly divided
into seven categories that we describe below.

  1. 1.

Data on resolution methods - the data on resolution methods applied to failed
banks and S&L’s are obtained from the Historical Statistics on Banking (HSoB)
maintained by the FDIC. The sample consists of 1385 banks, of which there are
118, 1175 and 92 institutions resolved under resolution types I, II and III
respectively. There are 389 S&L institutions in the sample of which 270, 104
and 15 institutions underwent resolution methods I, II and III respectively.

  2. 2.

Bank and S&L-level characteristics - failed banks from the HSoB are matched
with call report data from the Federal Reserve Bank of Chicago to obtain
information from the financial statements of each institution. We aggregate
the call reports by certificate number, which the FDIC uniquely assigns to
each head office of depository institutions, and use this identifier to merge
the two datasets. To allow for the duration of 90 to 100 days (FDIC, 1998)
between the FDIC receiving notification of an institution that is in danger of
failing and determining the resolution method, call reports from two quarters
prior to the date of failure are used in the study.

The failed S&L institutions in the sample are matched with Thrift Financial
Reports as of six months prior to failure from the Research Information System
(RIS) of the FDIC. The data on S&L institutions is less extensive than the
corresponding bank-level data due to differences in the reporting requirements
for banks and S&L’s. Specifically, data on Agricultural loans, Nonperforming
loans and Core Deposits are not available for S&L institutions for the period
under study.

  3. 3.

Insurer characteristic - we obtain the year-end data on outstanding balances
on the deposit insurance fund of the FDIC from annual reports from the
agency’s website666https://www.fdic.gov/about/financial-
reports/reports/index.html. The balances on the FSLIC’s insurance fund are
obtained from FDIC (1997)777The balances are reported in Table 4.1 of Chapter
4. These balances remain constant across banks and S&L’s, and vary by year. We
use this information to calculate the amount of deposit insurance fund as a
percentage of total deposit for both FDIC and FSLIC. This characteristic
measures the extent of insurance funds available to the two agencies relative
to the maximum value of their potential insurance payouts.

  4. 4.

State characteristics - our data hold several dimensions of information
pertaining to the states in which the banks and S&L’s operated. Specifically,
the data on quarterly housing starts at the state level have been obtained
from IHS Global Insight, the data on annual unemployment at the state level
were obtained from the Iowa Community Indicators Program of Iowa State
University and the information on branching deregulation laws was collated
using the table in Strahan et al. (2003).

  5. 5.

County characteristics - the data pertaining to county economic
characteristics in which the banks and S&L’s operated have been collated from
the Bureau of Economic Analysis. These characteristics consist of per capita
growth in gross domestic product (GDP) and the share of employment in each
sector, which measure the economic output of each county and the importance of
each sector to the county’s economy respectively.

  6. 6.

County-level characteristics of bank distress -county-level statistics on the
banking industry are obtained by aggregating bank-level data from the Research
Information System (RIS) of the FDIC, which is available starting from 1984.

  7. 7.

State-level political economy characteristics - Congressional voting data were
obtained from the website of GovTrack (https://www.govtrack.us) and converted
into state-level percentages of representatives who voted in favor of each
bill evaluated in this study. The description of these bills is available in
Section A of the supplement.

Table 3 in Section A of the supplement provides a description of the variables
available under each of these seven categories. Tables 4 and 5 in Section A
provide summary statistics of the data across these seven categories for FDIC
and FSLIC resolution decisions, respectively. In these tables, the Texas Ratio
for a bank is defined as,

$\mbox{Texas Ratio}=\frac{\mbox{Non-performing Assets}}{\mbox{Tangible Equity
+ Loan Loss Reserves}}.$ (1)

The Texas-Ratio is a measure of distress as it identifies institutions whose
capital would be insufficient to absorb losses that could emanate from
nonperforming assets.

##  3 Bayesian latent class estimation framework

In this section, we propose a Bayesian latent class estimation framework for
ordinal outcomes to represent the decision rules of FDIC and FSLIC in
resolving failed banks and to evaluate these decision rules against
recommended rules from theoretical studies.

###  3.1 Ordering of resolution decisions

We model the primary outcome of interest, the resolution decision of the FDIC
and FSLIC, as an ordered variable by specifying the three resolution methods
of Section 1.1 as ordered categories. Previous studies have shown that these
resolution methods resulted in progressively more severe effects on economic
outcomes (Ashcraft, 2005) and on the level of liquidity (DeYoung et al.,
2013). In particular, Ashcraft (2005) points out that each of the three
resolution categories entail an increasingly severe breakdown of relationships
between the bank and its customers. The provision of Type I assistance allows
a bank to continue functioning in its present form. A Type II acquisition or
purchase results in certain loan and deposit relationships continuing within
the acquiring bank’s books. A Type III liquidation and deposit payout results
in the termination of all banking relationships. The specification of
resolution methods as an ordered outcome variable also allows for a decision
structure in which the regulators order banks by their franchise value888A
bank’s franchise value is the present discounted value of its future stream of
profits and incorporates the value of its customer relationships and resulting
informational advantages. and assign Type I resolutions to the most valuable
and Type III resolutions to the least valuable banks. Such an ordering of
banks and S&Ls by franchise value is consistent with a cost-minimization
objective, which was relevant to both FDIC and FSLIC since they were required
to preserve their insurance funds by controlling their costs of resolution
(FDIC, 2007).

In the following discussion, bank $i$ refers to a representative bank or S&L
without loss of generality. Let $y_{i}$ be the resolution method applied on
bank $i$ where $y_{i}$ takes values $1,2$ and $3$ to denote resolution types
I, II and III respectively.

###  3.2 Latent class model for ordinal outcomes

We use a latent class model to represent state-dependent rules in the two
regulators’ decision structure. Figure 2 depicts the structure of the latent
class model. The class indicator $s_{i}$ is introduced into the model to
denote assignment of bank $i$ into one of the two classes. Within each latent
class, the regulator applies a class-specific decision rule on the failed bank
$i$ to assign it one of the three available resolution methods.

Figure 2: Structure and components of the latent class model for bank
resolutions. Here index $i$ refers to a representative bank or S&L $i$.

The distinguishing feature of the latent class model is that classes are
determined with probability and not deterministically. This feature is
relevant to the current problem since the true assignment of banks into
distinct classes by regulators is not observable as the data record the final
decision made by the regulators but not the rationale that motivated each
decision. Specifically, $y_{i}$ is observed but $s_{i}$ is not. As a result,
the probabilistic assignment of banks into classes addresses the researcher’s
uncertainty on class assignments by the regulatory agencies.

Relatedly, the recommended decision rules from theoretical models described in
Section 1.2 consisted of state-dependent rules, which entailed heterogeneity
in relationships between the outcome and covariates across sub-groups of banks
that failed in different states of nature. Moreover, each distinct decision
rule recommended by theoretical studies represents a distinct latent class in
our hierarchical model. Therefore, the latent class structure permits a direct
comparison between theoretical decision rules and the observed decisions of
the FDIC and FSLIC.

###  3.3 Random utility framework

The random utility representation of this model is based on the framework
developed by Marschak (1974). We model the regulator’s problem of assigning
bank $i$ to one of the two latent classes as a binary discrete choice problem
with a latent outcome $s_{i}$. To apply the random utility representation to
this discrete choice problem, we introduce a continuous latent variable
$l_{i}$, which represents the difference in utilities or value to the
regulator from assigning bank $i$ to latent class 2 relative to latent class
1. We express $l_{i}$ as,

$l_{i}=\bm{w}_{i}^{\prime}\bm{\alpha}+\nu_{i},$ (2)

where $\bm{w}_{i}$ is a $p-$dimensional vector of covariates, $\bm{\alpha}$
are parameters and $\nu_{i}$ is the error term. Finally, the relationship
between the discrete variable $s_{i}$ and the continuous variable $l_{i}$ is
expressed via the following threshold crossing framework,

$s_{i}=\begin{cases}1&\text{if}\ l_{i}\leq 0\\\
2&\text{otherwise}\end{cases}.$

The covariates $\bm{w}_{i}$ in Equation (2) are determined by the three
hypotheses of interest described in Section 1.2. For instance, in testing
hypothesis $H_{1}$, the covariates consist of economic indicators pertaining
to the state and the county in which bank $i$ operates, for hypothesis
$H_{2}$, they contain measures of the banking industry’s health in the county
in which bank $i$ operates and for hypothesis $H_{3}$, the covariates include
the percentage of Congressional representatives in banks $i$’s state who voted
for various bills that were favorable to the banking industry.

Within latent class $s_{i}$, the regulator’s utility function $z_{i,s_{i}}$
determines the final resolution method applied on bank $i$ where,

$z_{i,s_{i}}=\begin{cases}\bm{x}_{i}^{\prime}\bm{\beta}_{1}+\epsilon_{i,1},&\text{if}~{}s_{i}=1\\\
\bm{x}_{i}^{\prime}\bm{\beta}_{2}+\epsilon_{i,2},&\text{if}~{}s_{i}=2\end{cases}.$
(3)

Here $z_{i,s_{i}}$ is the utility that the regulator derives from preserving
bank $i$’s franchise value as discussed in Section 3.1. The $q-$dimensional
covariate vector $\bm{x}_{i}$ consists of bank $i$’s characteristics that are
representative of its financial health and importance, salient among which are
its size, the quality of its assets and composition of risky asset classes.
Note that in Equation (3), $\bm{x}_{i}^{\prime}\bm{\beta}_{s}$ and
$\epsilon_{i,s}$ represent the observable and unobservable components of
utility respectively (Train, 2009) for $s\in\\{1,2\\}$. The relationship
between the observed outcome $y_{i}$ and the latent utility $z_{i,s_{i}}$ is
represented using the following threshold-crossing framework,

$y_{i}=\begin{cases}3:\hskip 5.0pt\text{Type III},&\text{if}\
-\infty<z_{i,s_{i}}\leq\gamma_{1,s_{i}}\\\ 2:\hskip 5.0pt\text{Type
II},&\text{if}\ \gamma_{1,s_{i}}<z_{i,s_{i}}\leq\gamma_{2,s_{i}}\\\ 1:\hskip
5.0pt\text{Type I},&\text{if}\ \gamma_{2,s_{i}}<z_{i,s_{i}}\leq\infty\\\
\end{cases}.$ (4)

The regulator selects a resolution method that preserves more of the bank’s
franchise value as $z_{i,s_{i}}$ crosses a progressively larger threshold.
When $z_{i,s_{i}}$ is below the lowest threshold, $\gamma_{1,s_{i}}$, bank $i$
loses all its franchise value as the regulator’s utility level corresponds to
liquidation under a Type III resolution.

###  3.4 Likelihood function

The likelihood contribution $P_{ij}$ of bank $i$ receiving resolution type
$j=1,2,3$ is the sum of the likelihood contribution based on each latent class
weighted by the marginal probability of belonging to each of the two latent
classes,

$P_{ij}=\sum_{s=1}^{2}P_{ij|s}Q_{is},$ (5)

where $P_{ij|s}$ is the probability of $y_{i}$ taking a particular value $j$
conditional on belonging to class $s_{i}=s\in\\{1,2\\}$ and $Q_{is}$ is the
corresponding probability of bank $i$ belonging to class $s$. With $\nu_{i}$
in Equation (2) distributed independently as $\mathcal{N}(0,1)$, we obtain the
following binary probit representation of the class membership model,

$Q_{is}={\Phi(\bm{w}_{i}^{\prime}\bm{\alpha})}^{s^{\prime}}\Bigl{\\{}1-\Phi(\bm{w}_{i}^{\prime}\bm{\alpha})\Bigr{\\}}^{1-s^{\prime}},s^{\prime}=s-1,\hskip
10.0pts\in\\{1,2\\}.$ (6)

On specifying a $\mathcal{N}(0,\sigma^{2}_{s})$ distribution for the
unobserved component $\epsilon_{i,s}$ in Equation (3), the probability of
$y_{i}$ taking a particular value $j$ conditional on class
$s_{i}=s\in\\{1,2\\}$ from Equation (4) is,

$P_{ij|s}=\begin{cases}\Phi\Big{(}\dfrac{\gamma_{1,s}-\bm{x}_{i}^{\prime}\bm{\beta}_{s}}{\sigma_{s}}\Big{)},~{}&\text{if}~{}j=3\\\
\Phi\Big{(}\dfrac{\gamma_{2,s}-\bm{x}_{i}^{\prime}\bm{\beta}_{s}}{\sigma_{s}}\Big{)}-\Phi\Big{(}\dfrac{\gamma_{1,s}-\bm{x}_{i}^{\prime}\bm{\beta}_{s}}{\sigma_{s}}\Big{)},~{}&\text{if}~{}j=2\\\
1-\Phi\Big{(}\dfrac{\gamma_{2,s}-\bm{x}_{i}^{\prime}\bm{\beta}_{s}}{\sigma_{s}}\Big{)},~{}&\text{if}~{}j=1\end{cases}.$
(7)

In estimating the ordinal outcome model $P_{ij|s}$ conditional on class
membership $s$, we use the identification scheme in which the cut-points
$\gamma_{1,1}$ and $\gamma_{1,2}$ are restricted to 0 and the cut-points
$\gamma_{2,1}$ and $\gamma_{2,2}$ are restricted to 1 (Jeliazkov and Rahman,
2012).This identification restriction eliminates the need for estimating cut-
points and allows the scale parameter $\sigma_{s}$ to be estimated as a free
parameter in each latent class. As discussed in Jeliazkov and Rahman (2012)
and Greene and Hensher (2010), identification restrictions are required in
estimating ordinal models as neither the scale nor the location of the latent
variable $z_{i}$ is identified in this category of models.

Denote
$\bm{\Theta}=\\{\bm{\beta}_{1},\bm{\beta}_{2},\sigma_{1}^{2},\sigma_{2}^{2},\bm{\alpha}\\}$.
The likelihood function is obtained as,

$\mathcal{L}(\bm{\Theta})=\prod_{i=1}^{n}\Bigl{\\{}\prod_{j=1}^{3}\Big{(}P_{ij}\Big{)}^{\mathbb{I}\\{y_{i}=j\\}}\Bigr{\\}},$

where $P_{ij}$ and its components are as defined in equations (5), (6), (7),
and $\mathbb{I}\\{y_{i}=j\\}=1$ if $y_{i}=j$ and $0$ otherwise.

###  3.5 Augmented posterior

The augmented posterior for the parameters and latent variables in this model
is obtained by augmenting the likelihood with the latent variables
$\bm{z}=(z_{1,s_{1}},\ldots,z_{n,s_{n}})$ and $\bm{u}=(s_{1},\ldots,s_{n})$
using the method of Albert and Chib (1993). Denote

$\mathcal{B}_{i}=\begin{cases}(-\infty,0],~{}&\text{if}~{}y_{i}=3\\\
(0,1],~{}&\text{if}~{}y_{i}=2\\\ (1,\infty),~{}&\text{if}~{}y_{i}=1\\\
\end{cases}.$

Using equations (2)-(7), the resulting expression for the augmented posterior
is,

$f(\bm{\Theta},\bm{z},\bm{u}|\bm{y})\propto\Big{[}\prod_{i=1}^{n}\Bigl{\\{}\mathbb{I}\\{z_{i,1}\in\mathcal{B}_{i}\\}f_{\mathcal{N}}(z_{i,1}|\bm{x}_{i}^{\prime}\bm{\beta}_{1},\sigma_{1})Q_{i1}+\mathbb{I}\\{z_{i,2}\in\mathcal{B}_{i}\\}f_{\mathcal{N}}(z_{i,2}|\bm{x}_{i}^{\prime}\bm{\beta}_{2},\sigma_{2})Q_{i2}\Bigr{\\}}\Big{]}h(\bm{\Theta}),$
(8)

where $\bm{y}=(y_{1},\ldots,y_{n})$,
$\mathbb{I}\\{z_{i,s}\in\mathcal{B}_{i}\\}$ takes the value $1$ if
$z_{i,s}\in\mathcal{B}_{i}$ and $0$ otherwise,
$f_{\mathcal{N}}(z_{i,s}|\bm{x}_{i}^{\prime}\bm{\beta}_{s},\sigma_{s})$ is the
density of a normal distribution with mean $\bm{x}_{i}^{\prime}\bm{\beta}_{s}$
and standard deviation $\sigma_{s}$ for $s\in\\{1,2\\}$ and $h(\bm{\Theta})$
is the joint probability density function of the prior distribution of the
parameters in $\bm{\Theta}$. We assign a $q-$dimensional multivariate normal
prior to $\bm{\beta}_{s}$ that has mean $\bm{\beta}_{0,s}$ and covariance
$B_{0,s}$, and an Inverse Gamma prior to $\sigma_{s}^{2}$ with shape parameter
$v/2$ and scale parameter $d/2$. Finally, we assign a $p-$dimensional
multivariate normal prior to $\bm{\alpha}$ with mean $\bm{\alpha}_{0}$ and
covariance $A_{0}$. Since the priors are independent, their joint density
$h(\bm{\Theta})$ in Equation (8) can be represented as,

$h(\bm{\Theta})=\Bigl{\\{}\prod_{s=1}^{2}f_{\mathcal{N}}(\bm{\beta}_{s}|\bm{\beta}_{0,s},B_{0,s})f_{\mathcal{IG}}\Big{(}\sigma_{s}^{2}\Big{|}\dfrac{v}{2},\dfrac{d}{2}\Big{)}\Bigr{\\}}f_{\mathcal{N}}(\bm{\alpha}|\bm{\alpha}_{0},A_{0}),$

where $f_{\mathcal{IG}}(\sigma_{s}^{2}|v/2,d/2)$ is the density of an Inverse
Gamma distribution with shape parameter $v/2$ and scale parameter $d/2$.

###  3.6 MCMC algorithm

A standard approach to developing an MCMC algorithm results in a Gibbs sampler
that draws from the full conditionals of all parameters as well as the two
latent variables $\bm{u}$ and $\bm{z}$. In this section we present a Collapsed
Gibbs sampler (Liu, 1994) which provides an efficient technique for sampling
from the posterior distribution relative to standard sampling approaches by
reducing autocorrelations across successive draws. Specifically, in our
algorithm, we do not have to draw from the conditionals of $\bm{u}$ as we
sample for the parameters of the class membership model, $\bm{\alpha}$,
independently of this latent variable. We first present the algorithm for our
collapsed Gibbs sampler and then discuss the details underlying each step in
the sampler.

Algorithm: Collapsed Gibbs Sampler

  1. 1.

Sample $\bm{\beta}_{s}$ from the distribution
$\bm{\beta}_{s}|\bm{z},\bm{u},\sigma^{2}_{s}$ for $s\in\\{1,2\\}$.

  2. 2.

Sample $\sigma^{2}_{s}$ from $\sigma^{2}_{s}|\bm{\beta}_{s},\bm{z},\bm{u}$ for
$s\in\\{1,2\\}$.

  3. 3.

Sample $\bm{\alpha}$ from $\bm{\alpha}|\bm{\beta},\bm{\sigma}^{2},\bm{y}$
where $\bm{\sigma}^{2}=(\sigma^{2}_{1},\sigma^{2}_{2})$ and
$\bm{\beta}=(\bm{\beta}_{1},\bm{\beta}_{2})$.

  4. 4.

Sample $s^{\prime}_{i}$ from
$s^{\prime}_{i}|\bm{\alpha},\bm{\beta},\bm{\sigma}^{2},\bm{y}$, where
$s_{i}^{\prime}=s_{i}-1$ for $i=1,\ldots,n$.

  5. 5.

Sample $z_{i,s_{i}}$ from
$z_{i,s_{i}}|\bm{\beta},\bm{\sigma}^{2},\bm{y},\bm{u}$ for $i=1,\ldots,n$.

For the following discussion, denote $X$ to be the $n\times q$ matrix with the
vector of $q$-dimensional covariates $\bm{x}_{i}^{\prime}$ from Equation (3)
in its rows.

Sampling coefficients $\bm{\beta}_{s}$ of the ordinal model -  The
coefficients $\bm{\beta}_{s}$ of the ordinal model are sampled for the two
latent classes, i.e., for $s\in\\{1,2\\}$ from their respective conditional
posterior distributions. We have
$\bm{\beta}_{s}|\bm{z},\bm{u},\sigma^{2}_{s}\sim\mathcal{N}\bigl{(}\hat{\bm{\beta}}_{s},\hat{B}_{s}\bigr{)}$,
where
$\hat{B}_{s}=\bigl{(}B_{0,s}^{-1}+X_{s}^{\prime}X_{s}/\sigma_{s}^{2}\bigr{)}^{-1}$
and
$\hat{\bm{\beta}}_{s}=\hat{B}_{s}\bigl{(}B_{0,s}^{-1}\bm{\beta}_{0,s}+X_{s}^{\prime}\bm{z}_{s}/\sigma_{s}^{2}\bigr{)}$.
Here $X_{s}$ is the $n_{s}\times q$ submatrix of $X$ that includes those rows
of $X$ for which $s_{i}=s$ and $n_{s}$ is the number of observations in class
$s$ which is updated in every MCMC iteration. Similarly, $\bm{z}_{s}$ denotes
the length $n_{s}$ subvector of $\bm{z}$ that includes those elements of
$\bm{z}$ for which $s_{i}=s$. In this sampling step, the computations
involving $X_{s}$ are efficient as they only require working with matrices of
reduced dimension $n_{s}\times q$, without having to preserve the full
$n\times q$ matrix $X$.

Sampling the variance $\sigma^{2}_{s}$ of the ordinal model -  The variances
are sampled using the conditionals
$\sigma^{2}_{s}|\bm{z},\bm{u},\bm{\beta}_{s}\sim\mathcal{IG}({\sf
shape=}~{}\hat{\nu}_{s},{\sf scale=}~{}\hat{d}_{s})$ for $s\in\\{1,2\\}$,
where $\hat{\nu}_{s}=(\nu+n_{s})/2$ and
$\hat{d}_{s}=\bigl{\\{}d+(\bm{z}_{s}-X_{s}\bm{\beta}_{s})^{\prime}(\bm{z}_{s}-X_{s}\bm{\beta}_{s})\bigr{\\}}/2$.
Here $X_{s}$ and $\bm{z}_{s}$ are retained from the previous step.

Sampling coefficients $\bm{\alpha}$ of the class membership model -  The
coefficients $\bm{\alpha}$ of the class membership model are sampled from
$\bm{\alpha}|\bm{\beta},\bm{\sigma}^{2},\bm{y}$, marginally of $\bm{u}$, by
using a Metropolis Hastings (MH) step with a tailored proposal distribution.
We use a $p-$dimensional $t$ distribution with location parameter
$\hat{\bm{\alpha}}$, covariance matrix $\mathcal{V}$ and degrees of freedom
$v$ as the tailored proposal distribution where
$\hat{\bm{\alpha}}=\operatorname*{arg\,max}_{\bm{\alpha}}f(\bm{y}|\bm{\alpha},\bm{\beta},\bm{\sigma}^{2})f_{\mathcal{N}}(\bm{\alpha}|\bm{\alpha}_{0},A_{0})$,
$\mathcal{V}$ is the inverse of the negative Hessian of
$\log\\{f(\bm{y}|\bm{\alpha},\bm{\beta},\bm{\sigma}^{2})f_{\mathcal{N}}(\bm{\alpha}|\bm{\alpha}_{0},A_{0})\\}$
evaluated at $\hat{\bm{\alpha}}$ and

$f(\bm{y}|\bm{\alpha},\bm{\beta},\bm{\sigma}^{2})=\prod_{i=1}^{n}\Big{[}\Bigl{\\{}1-\Phi(\bm{w}_{i}^{\prime}\bm{\alpha})\Bigr{\\}}P_{y_{i}|1}+\Phi(\bm{w}_{i}^{\prime}\bm{\alpha})P_{y_{i}|2}\Big{]},$

with

$P_{y_{i}|s}=\begin{cases}\Phi\Big{(}\dfrac{\gamma_{1,s}-\bm{x}_{i}^{\prime}\bm{\beta}_{s}}{\sigma_{s}}\Big{)},~{}&\text{if}~{}y_{i}=3\\\
\Phi\Big{(}\dfrac{\gamma_{2,s}-\bm{x}_{i}^{\prime}\bm{\beta}_{s}}{\sigma_{s}}\Big{)}-\Phi\Big{(}\dfrac{\gamma_{1,s}-\bm{x}_{i}^{\prime}\bm{\beta}_{s}}{\sigma_{s}}\Big{)},~{}&\text{if}~{}y_{i}=2\\\
1-\Phi\Big{(}\dfrac{\gamma_{2,s}-\bm{x}_{i}^{\prime}\bm{\beta}_{s}}{\sigma_{s}}\Big{)},~{}&\text{if}~{}y_{i}=1\end{cases},$
(9)

for $s\in\\{1,2\\}$. This MH step enhances the efficiency of the overall
algorithm by circumventing the need for additional data augmentation through
the latent variable $l_{i}$ from Equation (2).

The proposed draw $\bm{\alpha}^{\dagger}$ from this proposal is accepted with
probability,

$\Upsilon_{MH}(\bm{\alpha},\bm{\alpha}^{\dagger})=\min\Bigg{\\{}1,\dfrac{f(\bm{\alpha}^{\dagger}|\bm{\beta},\bm{\sigma}^{2},\bm{y})q(\bm{\alpha}|\bm{\beta},\bm{\sigma}^{2},\bm{y})}{f(\bm{\alpha}|\bm{\beta},\bm{\sigma}^{2},\bm{y})q(\bm{\alpha}^{\dagger}|\bm{\beta},\bm{\sigma}^{2},\bm{y})}\Bigg{\\}},$

where $q(\bm{\alpha}|\bm{\beta},\bm{\sigma}^{2},\bm{y})$ is the density of the
tailored proposal distribution and the expression
$f(\bm{\alpha}|\bm{\beta},\bm{\sigma}^{2},\bm{y})$ in the display above is
proportional to the product of
$f(\bm{y}|\bm{\alpha},\bm{\beta},\bm{\sigma}^{2})$ and the prior probability
density of $\bm{\alpha}$.

Sampling the class membership indicator $\bm{u}$ \-  The vector $\bm{u}$ of
class membership indicators $s_{i}$ identifies the latent class
$s\in\\{1,2\\}$ to which each observation $i$ belongs. These indicators are
sampled from a Bernoulli distribution by introducing the binary variable
$s^{\prime}_{i}=s_{i}-1$, where
$s_{i}^{\prime}|\bm{\alpha},\bm{\beta},\bm{\sigma}^{2},\bm{y}\sim Bern(K_{i})$
for $i=1,\ldots,n$ and,

$K_{i}=\dfrac{\Phi(\bm{w}_{i}^{\prime}\bm{\alpha})P_{y_{i}|2}}{\Phi({\bm{w}_{i}^{\prime}\bm{\alpha}})P_{y_{i}|2}+(1-\Phi({\bm{w}_{i}^{\prime}\bm{\alpha}}))P_{y_{i}|1}}.$

The values $P_{y_{i}|1}$ and $P_{y_{i}|2}$ are retained from the previous step
and are computed using Equation (9).

Sampling the latent variable $\bm{z}$ \-  The sampling of continuous latent
variables $z_{i,s_{i}}$ is based on the data augmentation step from Albert and
Chib (1993), resulting in
$z_{i,s_{i}}|\bm{\beta},\bm{\alpha},\bm{\sigma}^{2},\bm{y}$ having a truncated
normal distribution with mean $x_{i}^{\prime}\bm{\beta}_{s_{i}}$, variance
$\sigma_{s_{i}}^{2}$ and truncated between
$(\gamma_{y_{i}-1,s_{i}},\gamma_{y_{i},s_{i}})$ for $i=1,\ldots,n$. The second
subscript $s_{i}$ in $(\bm{\beta}_{s_{i}},\sigma_{s_{i}}^{2})$ is added to
establish that the sampling scheme augments just the continuous outcomes
associated with the class $s_{i}$ to which each observation belongs and does
not require the augmentation based on the counterfactual latent class. This
approach minimizes storage requirements and permits the sampling of the entire
vector $\bm{z}$ in one step.

In the Collapsed Gibbs sampler described above, the discrete latent variable
$\bm{u}$ is marginalized out of the conditional distribution for
$\bm{\alpha}$. This novel approach to marginalization results in a sharper
decline in autocorrelations across successive lags of sample draws.
Consequently, the draws from this algorithm are close to independently and
identically distributed early in the chain. In Section B of the supplement we
consider two simulation settings and demonstrate that the reduction in
autocorrelations gained from our collapsed Gibbs sampler is substantial when
compared to those from a full Gibbs sampler.

###  3.7 Estimation of model with $J>3$ values of the ordered outcome

The Bayesian latent class estimation framework developed in Sections 3.1 – 3.6
relies on the ordered outcome variable $y_{i}$, the resolution method, taking
three values. However, in several practical applications $y_{i}$ can take
$J>3$ values. For instance, consumer ratings of products and credit ratings
assigned to firms are ordered outcomes that typically span over five or more
categories. In this section, we provide an extension of our latent class model
that allows the ordered outcome variable $y_{i}$ to take $J>3$ values and
develop a collapsed Gibbs sampler for posterior inference in that model.

The sampling algorithm is based on the identification scheme used in Section
3.4 where we set $\gamma_{1,s}=0$ and $\gamma_{J-1,s}=1$ for $s\in\\{1,2\\}$.
In order to ensure that the ordering of the $J-1$ cut-points, namely
$\gamma_{1,s}<\cdots<\gamma_{J-1,s}$, is preserved without having to resort to
the introduction of computationally intensive constraints into the estimation
procedure, the following transformation proposed in Chen and Dey (2000) is
used,

$\delta_{j,s}=\log{\dfrac{(\gamma_{j,s}-\gamma_{j-1,s})}{(1-\gamma_{j-1,s})}},~{}2\leq
j\leq J-2,~{}s\in\\{1,2\\}.$

We propose a collapsed Gibbs sampler algorithm that uses a MH step to sample
$\bm{\beta}_{s}$ and $\bm{\delta}_{s}=(\delta_{2,s},\ldots,\delta_{J-2,s})$ in
one block along the lines of the examples provided in Chib and Jeliazkov
(2001). A multivariate normal prior is assigned to $\bm{\delta}_{s}$ that has
mean $\bm{\delta}_{0,s}$ and covariance matrix $D_{0,s}$.

Algorithm: Collapsed Gibbs Sampler for model with cut-points

  1. 1.

Sample $\bm{\beta}_{s}$ and $\bm{\delta}_{s}$ jointly from
$(\bm{\beta}_{s},\bm{\delta}_{s})|\bm{y},s,\sigma^{2}_{s}$ for
$s\in\\{1,2\\}$.

  2. 2.

Sample $\sigma^{2}_{s}$ from $\sigma^{2}_{s}|\bm{\beta}_{s},\bm{z},\bm{u}$ for
$s\in\\{1,2\\}$.

  3. 3.

Sample $\bm{\alpha}$ from $\bm{\alpha}|\bm{\beta},\bm{\sigma}^{2},\bm{y}$ for
where $\bm{\sigma}^{2}=(\sigma^{2}_{1},\sigma^{2}_{2})$ and
$\bm{\beta}=(\bm{\beta}_{1},\bm{\beta}_{2})$.

  4. 4.

Sample $s^{\prime}_{i}$ from
$s^{\prime}_{i}|\bm{\alpha},\bm{\beta},\bm{\sigma}^{2},\bm{y}$ for
$i=1,\ldots,n$ and $s_{i}^{\prime}=s_{i}-1$.

  5. 5.

Sample $z_{i,s_{i}}$ from
$z_{i,s_{i}}|\bm{\beta},\bm{\sigma}^{2},\bm{y},\bm{u}$ for $i=1,\ldots,n$.

Steps (b)–(e) are identical to the algorithm described in Section 3.6. Step
(a) of this algorithm is described below.

Sampling coefficients $\bm{\beta}_{s}$ and cut-points $\bm{\delta}_{s}$ of the
ordinal model -  Sample
$(\bm{\beta}_{s},\bm{\delta}_{s})|\bm{y},s,\sigma^{2}_{s}$ by drawing
$(\bm{\beta}_{s}^{\dagger},\bm{\delta}_{s}^{\dagger})$ from a tailored
proposal distribution. We use a $q+J-3$ dimensional $t$ distribution with
location
$(\hat{\bm{\beta}}_{s},\hat{\bm{\delta}}_{s})\coloneqq\operatorname*{arg\,max}_{(\bm{\beta}_{s},\bm{\delta}_{s})}f(\bm{y}|\bm{\beta}_{s},\bm{\delta}_{s},\sigma^{2}_{s},s)f_{\mathcal{N}}(\bm{\beta}_{s}|\bm{\beta}_{0,s},B_{0,s})f_{\mathcal{N}}(\bm{\delta}_{s}|\bm{\delta}_{0,s},D_{0,s})$,
covariance matrix $\mathcal{V}$ being the inverse of the negative hessian of
the logarithm of the maximand evaluated at
$(\hat{\bm{\beta}}_{s},\hat{\bm{\delta}}_{s})$ and degrees of freedom $v$.
Here,

$f(\bm{y}|\bm{\beta}_{s},\bm{\delta}_{s},\sigma^{2}_{s},s)=\prod_{i=1}^{n}\Big{(}P_{y_{i}|s_{i}}\Big{)}^{\mathbb{I}\\{s_{i}=s\\}}$

for $s\in\\{1,2\\}$ and

$P_{y_{i}|s}=\begin{cases}\Phi\Big{(}\dfrac{-\bm{x}_{i}^{\prime}\bm{\beta}_{s}}{\sigma_{s}}\Big{)},~{}&\text{if}~{}y_{i}=1\\\
1-\Phi\Big{(}\dfrac{1-\bm{x}_{i}^{\prime}\bm{\beta}_{s}}{\sigma_{s}}\Big{)},~{}&\text{if}~{}y_{i}=J\\\
\Phi\Big{(}\dfrac{\gamma_{y_{i},s}-\bm{x}_{i}^{\prime}\bm{\beta}_{s}}{\sigma_{s}}\Big{)}-\Phi\Big{(}\dfrac{\gamma_{y_{i}-1,s}-\bm{x}_{i}^{\prime}\bm{\beta}_{s}}{\sigma_{s}}\Big{)},~{}&\text{if}~{}y_{i}\in\\{2,\ldots,J-1\\}\end{cases}.$

The proposed draw $(\bm{\beta}_{s}^{\dagger},\bm{\delta}_{s}^{\dagger})$ is
accepted with probability
$\Upsilon_{MH}\\{(\bm{\beta}_{s},\bm{\delta}_{s}),(\bm{\beta}_{s}^{\dagger},\bm{\delta}_{s}^{\dagger})\\}$

$\displaystyle=\min\Bigl{\\{}1,\dfrac{f(\bm{y}|\bm{\beta}_{s}^{\dagger},\bm{\delta}_{s}^{\dagger},\sigma_{s}^{2},s)f_{\mathcal{N}}(\bm{\beta}_{s}^{\dagger}|\bm{\beta}_{0,s},B_{0,s})f_{\mathcal{N}}(\bm{\delta}_{s}^{\dagger}|\bm{\delta}_{0,s},D_{0,s})q(\bm{\beta}_{s},\bm{\delta}_{s}|\bm{y},s,\sigma_{s}^{2})}{f(\bm{y}|\bm{\beta}_{s},\bm{\delta}_{s},\sigma_{s}^{2},s)f_{\mathcal{N}}(\bm{\beta}_{s}|\bm{\beta}_{0,s},B_{0,s})f_{\mathcal{N}}(\bm{\delta}_{s}|\bm{\delta}_{0,s},D_{0,s})q(\bm{\beta}_{s}^{\dagger},\bm{\delta}_{s}^{\dagger}|\bm{y},s,\sigma_{s}^{2})}\Bigr{\\}},$

where $q(\bm{\beta}_{s},\bm{\delta}_{s}|\bm{y},s,\sigma_{s}^{2})$ is the
density of the tailored proposal distribution.

##  4 Bank resolutions by the FDIC

This section provides an assessment of the FDIC’s resolution decisions over
the period 1984-1992 by evaluating the agency’s decision rules against
recommended rules from theoretical studies. We perform this assessment by
interpreting the results from the latent class ordinal model developed in
Section 3 to test the three hypotheses derived from theoretical studies as
summarized in Section 1.2. In Section 4.1 we discuss the results for
hypotheses $H_{1}$ while the results pertaining to hypotheses $H_{2}$ and
$H_{3}$ are presented in sections G and H of the supplement. The prior
distributions that we consider in this analysis are as follows:
$\bm{\alpha}\sim\mathcal{N}_{p}(\bm{0},3I_{p}),\bm{\beta}_{s}\sim\mathcal{N}_{q}(\bm{0},I_{q})$
and $\sigma_{s}^{2}\sim\mathcal{IG}({\sf shape=}~{}4.3,{\sf scale=}~{}1.3)$
for $s\in\\{1,2\\}$. The hyperparameters for $\sigma_{s}^{2}$ are chosen to
result in an uninformative prior with a mean of approximately 0.4 and prior
standard deviation of 0.26. The collapsed Gibbs sampler algorithm of Section
3.6 is run for $11,000$ iterations and we use $G=10,000$ post-burn in samples
for posterior inference.

###  4.1 Regional economic distress and FDIC’s decisions for bank resolutions

The period of this study, 1984-1992, presents a unique set of economic and
banking conditions that facilitate the test for presence of the recommended
resolution strategy from Cordella and Yeyati (2003) in the FDIC’s decisions.
By virtue of the combination of regionally-contained sectoral crises and
branching restrictions during this period (see discussion in Section 1.1), the
FDIC simultaneously administered both, bank failures that occurred amid high
and low economic distress. We find that the FDIC’s responses supported
Hypothesis $H_{1}$ as the agency provided assistance to banks that failed amid
economic distress with a higher probability than to banks that failed in low
economic distress. Furthermore, our results reveal that within the class of
high regional distress, the FDIC targeted assistance to banks with relatively
healthy balance sheets that were more likely to recover and operate as going
concerns and arranged for the sale or liquidation of the remaining banks.

####  4.1.1 Class-membership Model

The class membership model is represented in the first level of the decision
structure in Figure 2. We perform a Bayesian model comparison, described in
Section E of the supplement, to select the specification of the class-
membership model that is most decisively supported by the data. The covariates
in the resolution type model, the second level of the hierarchy in Figure 2,
are constant across all the specifications considered and are discussed in
Section 4.1.3.

Table 1 summarizes the covariate effects from the class membership model for
four specifications that include indicators of state and county-level economic
performance along with controls for institutional features underlying the
resolution decision. The expression for the covariate effects are derived in
Section D of the supplement. The values of log marginal likelihood reported in
the last row of Table 1 point to specification (3) as the selected model as it
has the highest posterior odds among the four candidate specifications. This
selected specification highlights a statistically important role for state-
level unemployment in assigning banks into two different classes. The other
covariates that inform the assignment of banks to latent classes are county-
level indicators of economic performance along with a control variable for the
amount of insurance fund available per dollar of insured deposit in the
banking system.

Among alternative specifications considered in Table 1, specifications (1) and
(2) of the model entirely consist of state and county-level indicators of
economic performance and controls for county-level shares of employment by
sector. Note that specification (2) is a more parsimonious setting that is
nested within specification (1). Specification (4) augments specification (2)
with indicators for the charter status of failed banks since chartering
agencies, namely, the OCC for federally chartered banks and state banking
departments for state-chartered banks, retain the final authority to enforce
closure. The reference group in this class membership model consists of
nationally chartered banks that are supervised by the OCC.

In the following discussion, latent class 1 is labeled as the class of
failures under “High Regional Distress (HRD)” and latent class 2, as “Low
Regional Distress (LRD)”. In the model for class membership in Equation (6),
the event of success in the binary probit model (where the latent binary
indicator $s_{i}$ equals 1) is represented by a bank belonging to latent class
2. Therefore, the negative signs associated with unemployment in specification
(3) and the positive signs for covariate effects of GDP growth rate and
housing starts in Table 1 show that latent class 2 contains banks that failed
during periods of low unemployment or periods of relatively low regional
economic stress whereas banks that failed amid high regional distress belong
to latent class 1. These findings support the first element of hypothesis
$H_{1}$ by confirming that the FDIC distinguished across banks based on
economic distress in applying its resolution decisions.

Table 1:  Covariate effects from class-membership models for specifications of latent classes based on regional economic distress. The reported values are posterior means of the covariate effects. Posterior standard deviations are in parentheses. | (1) | (2) | (3) | (4)
---|---|---|---|---
State-level characteristics |  |  |  |
Unemployment | -0.12 (0.06) | -0.11 (0.05) | -0.1 (0.04) | -0.15 (0.08)
Housing starts | 0.11 (0.05) | 0.09 (0.05) | 0.05 (0.05) | 0.1 (0.05)
County-level characteristics |  |  |  |
Per capita GDP growth | 0.04 (0.05) | 0.05 (0.05) | 0.04 (0.05) | 0.04 (0.05)
Farm, agri, mining | 0.11 (0.09) | 0.07 (0.05) | 0.06 (0.04) | 0.07 (0.05)
Manufacturing | 0.03 (0.05) | - | - | -
Construction | 0.02 (0.04) | - | - | -
Fin Serv Transport | 0 (0.07) | - | - | -
Government | 0.04 (0.05) | - | - | -
Insurer characteristics |  |  |  |
Dep. Ins. Fund/ Total Deposits | - | - | -0.05 (0.03) | -
Bank-level characteristics |  |  |  |
State charter Fed member | - | - | - | 0.03 (0.07)
State charter non-Fed member | - | - | - | -0.06 (0.05)
log Marginal Likelihood | -703.35 | -701.10 | -699.79 | -700.19

####  4.1.2 Heterogeneity in Decision Rules

The results from the second level of the decision structure in Figure 2 show
that the average probability of the FDIC assigning a Type I resolution was
statistically higher among HRD banks than among LRD banks. These findings
confirm that the FDIC’s decisions fully aligned with hypothesis $H_{1}$ in
that the agency was more likely to provide assistance to banks when their
failure was accompanied by regional economic distress.

The average probability of each resolution type for the HRD and LRD failures
is computed as follows.

$\mbox{Avg. Prob}(Y=j|s)=\dfrac{1}{n}\sum_{i=1}^{n}P_{ij|s}^{(g)},~{}j=1,2,3,$
(10)

where $s=1$ and $s=2$ correspond to the results for the class of HRD and LRD
failures respectively and $g$ is the index for the $G$ post burn-in MCMC
draws. The values $P_{ij|s}^{(g)}$ are computed for each MCMC iteration using
Equation (7).

Figure 3: Posterior distribution of the average probability of the FDIC
assigning each resolution method within classes based on regional distress.
The horizontal axis represents the probability of assigning a resolution
method and the vertical axis represents the posterior density associated with
that probability based on a kernel density estimate. The solid vertical lines
represent the means of these posterior distributions across the $G$ MCMC
draws.

Figure 3 provides the density of the full posterior distribution of the
average probability of receiving each resolution method across the two latent
classes. The average probability of receiving a Type I resolution among HRD
banks was 24.6% compared to 3.3% for LRD banks. The fully disjoint posterior
densities of the average probability of receiving a Type I resolution for the
HRD and LRD classes shows that the difference between their averages is
statistically important. This observation continues to hold for Type II
resolutions with statistically important differences in average resolution
probabilities across HRD and LRD classes at 72.1% and 87.9% respectively. The
theoretical recommendation from Cordella and Yeyati (2003) does not explicitly
address the decision to facilitate partial or whole acquisitions of failed
banks and the findings from this estimation exercise provide new insights into
the differences in the probabilities of implementing the Type II resolution
method under varying levels of economic distress. Finally, in a further
confirmation of the predictions of the theoretical model, the average
probability of being liquidated under a Type III resolution was 8.7% for LRD
banks compared to 3.2% for HRD banks. This difference is also statistically
important, as evidenced by the minimal overlap in posterior densities of the
two classes.

####  4.1.3 Resolution Type

The next stage of the empirical analysis centers on the results from the
ordinal probit models represented in Equation (7) and depicted in the second
level of the decision structure in Figure 2. These models estimate separate
relationships between the resolution method $y_{i}$ and bank-level financial
indicators $\bm{x}_{i}$ in the LRD and HRD classes. If the FDIC responded
differently to LRD and HRD failures for the same change in bank financial
characteristics, this would manifest in different magnitudes of covariates
across the two classes and provide conclusive evidence of the presence of two
different decision rules implemented by the FDIC. In this section, we report
the six largest covariate effects from the selected ordered response model
(specification 3 in Table 1) in Figure 4. The covariate effects of the
remaining variables are provided in Section F of the supplement.

Figure 4: Covariate effects from the models for resolution type for banks in
the class of High Regional Distress (HRD) and Low Regional Distress (LRD).

From Figure 4, the financial variables, Real Estate Loan Ratio and
Nonperforming Loans Ratio, both exhibit qualitatively similar covariate
effects. A unit standard deviation increase in Real Estate Ratio and
Nonperforming Loans Ratio is associated with a reduced probability of
obtaining assistance under a Type I resolution among HRD failures and an
increased probability of such banks undergoing Type II and III resolutions.
Since nonperforming loans provide a succinct measure of the quality of the
failed bank’s assets and real estate loans represent a risky asset category,
these results reveal that the FDIC provided assistance under Type I to banks
that had relatively healthier balance sheets even among those banks that
failed amid economic distress, which is consistent with the theoretical
recommendations of Cordella and Yeyati (2003). The effects of these covariates
on banks within the class of LRD failures, on the other hand, are not
statistically important.

The Interest Receivable Ratio is seen to be important in the FDIC’s evaluation
of bank health, with elevated levels of this ratio eliciting more stringent
resolution methods from the FDIC. Balla et al. (2015) originally identified
the Interest Receivable Ratio to be highly predictive of both bank failure and
loss subsequent to failure in their study. Accordingly, an increase in
Interest Receivable Ratio among HRD failures resulted in a reduction in the
probability of Type I resolution and a corresponding increase in the
probability of a Type II resolution, entailing partial or whole acquisitions
of the failed institutions. Banks that belonged to the LRD class of failures
experienced a more severe response in the form of an increased probability of
a Type III resolution and hence, complete liquidation, along with a decreased
probability of the other two resolution methods.

Larger banks were less likely to be liquidated under a Type III resolution
across both latent classes. Among HRD failures, a standard deviation increase
in log of assets was also associated with a decreased probability of a Type II
resolution and a compensatory increase in the probability of assistance under
Type I resolution. LRD failures experienced an increase in the probability of
both, Type I and II resolutions concomitantly with a decrease in the
probability of a Type III resolution. The increased probability of Type I
resolutions associated with a larger bank reveals that the “too-big-to-fail”
doctrine was present in the decisions of FDIC during the crisis of the 1980’s.

The estimation results provide new insights into the role of Loan Loss Reserve
Ratio, an accounting variable that records the amount of funds set aside to
meet expected losses. An increase in Loan Loss Reserves ratio was associated
with a higher probability of receiving Type I resolution in the class of HRD
failures. Contrarily, among LRD failures, an equivalent increase in this
measure was associated with an increase in their probability of receiving a
Type III resolution and being liquidated. A possible explanation for this
disparity is that in the HRD class of banks, where failures are more likely to
have occurred due to systemic factors, changes in loss reserve ratios can be
attributed to the deterioration in asset quality resulting from market-wide
fluctuations. However, among LRD failures, the FDIC is likely to have viewed
the increase in this ratio as a signal of deterioration in asset quality
arising from issues idiosyncratic to the failed bank.

Interstate is an indicator variable that identifies whether interstate banking
was legal in the state in which a bank is located in the year of failure and
is derived from summary tables in Strahan et al. (2003). Intuitively,
interstate banking laws are likely to affect resolution outcomes as they
determine the breadth of demand for assets of failed banks. For instance, the
model in Acharya and Yorulmazer (2008) predicts that interstate banking, by
expanding the set of available acquiring banks in the event of a failure,
should be associated with an increase in the probability of a Type II
resolution and an equivalent decline in the probability of a Type I
resolution. In the bottom right panel of Figure 4, an increased probability of
Type II resolutions is observed among both, HRD and LRD classes, even though
the LRD class of failures also underwent a modest increase in the probability
of Type I resolutions.

Overall, Figure 4 reveals that the magnitudes of the covariate effects from
the selected model, specification (3), described in the preceding subsection,
are larger for banks that belong to the latent class “High Regional Distress
(HRD)” relative to banks in the class labeled “Low Regional Distress (LRD)”.
This pronounced difference in the effects of each covariate on the FDIC’s
decisions across the two classes confirms the presence of two distinct
decision rules in the agency’s resolution procedure. The larger covariate
effects for banks in the class of HRD failures indicate that within the group
of banks that failed amid regional economic distress, the FDIC ordered banks
based on their financial characteristics and provided Type I resolutions to
relatively healthier banks and Type II and III resolutions to relatively
weaker banks. The smaller covariate effects in the LRD class suggest that the
FDIC evaluated banks that failed in low economic distress on a case-by-case
basis rather than appraising their relative financial strength. This approach
potentially included evaluating unobservable individual circumstances, which
are captured by the error term in Equation (3). These results reveal a smaller
role for observed financial statement information in determining resolution
methods within the LRD class relative to the HRD class of banks and further
support the hypothesis that banks in the former group are likely to have
failed due to largely idiosyncratic factors.

##  5 Savings and loans resolution by FSLIC

The Federal Savings and Loans Insurance Corporation (FSLIC) insured S&L’s and
served as a receiver for failed institutions, both of which were functions the
FDIC performed within the banking industry. The FSLIC, however, was ultimately
dissolved following unsustainable resolution losses that resulted in its
insolvency in 1989. This section compares the decision rules of the FSLIC
against theoretically recommended rules summarized in Section 1.2. To compare
the FDIC with the FSLIC, we analyze the latter’s decisions through the same
empirical lens and estimate the specifications described in Section 4 for FDIC
decisions.

In Section 5.1 we discuss the results for hypotheses $H_{1}$ while the results
pertaining to hypotheses $H_{2}$ and $H_{3}$ are presented in sections K and L
of the supplement. The period under study covers the S&L failures that
occurred over 1984-1989. The industry had also undergone an earlier wave of
failures in the early 1980’s. Section I in the supplement provides an overview
of the history of the crisis in the S&L industry and distinguishes between the
two waves of S&L failures. In this analysis, we continue to use the same
specifications as those described in Section 4 for the prior distributions,
their hyperparameters and the number of post-burn in MCMC samples.

###  5.1 Regional economic distress and FSLIC’s decisions for S&L resolutions

We find that FSLIC’s designation of S&L institutions into classes based on
regional economic distress is ambiguous and does not support Hypothesis
$H_{1}$. The FSLIC’s decisions deviated from the decision structure implied by
Hypothesis $H_{1}$ in a notable way. The agency did not develop distinct
decision rules to resolve failures that occurred amid high and low regional
distress and is observed to have adopted a common decision rule for both
groups of S&L’s. The implication of using a common decision rule is that
institutions that failed due to weaknesses in their own risk management as
well as due to economic factors are likely to have received assistance with
similar probabilities. The theory underlying this hypothesis suggests that by
not fully disentangling economic and idiosyncratic factors while providing
assistance, the FSLIC potentially fostered moral hazard among S&L
institutions.

####  5.1.1 Class-membership model

Table 2 summarizes the covariate effects from estimating the specifications
reported in Section 4.1.

Table 2:  Covariate effects from class-membership models of S&L’s for specifications of latent classes based on regional distress. The reported values are posterior means of the covariate effects. Posterior standard deviations are in parantheses. | (1†) | (2†) | (3†) | (4†)
---|---|---|---|---
State-level characteristics |  |  |  |
Unemployment | -0.08 (0.06) | -0.71 (0.14) | -0.17 (0.48) | -0.72 (0.11)
Housing starts | -0.21 (0.1) | -0.01 (0.04) | -0.03 (0.09) | -0.02 (0.04)
County-level characteristics |  |  |  |
Per capita GDP growth | -0.13 (0.09) | 0.02 (0.06) | 0.00 (0.09) | 0.01 (0.05)
Farm, agri, mining | -0.09 (0.08) | 0.04 (0.04) | -0.01 (0.09) | 0.04 (0.03)
Manufacturing | 0 (0.07) | - | - | -
Construction | 0.09 (0.05) | - | - | -
Fin Serv Transport | -0.11 (0.12) | - | - | -
Government | -0.16 (0.1) | - | - | -
Insurer characteristics |  |  |  |
Dep. Ins. Fund/Total Dep. | - | - | 0.01 (0.08) | -
S&L-level characteristics |  |  |  |
State charter | - | - | - | -0.07 (0.15)
log Marginal Likelihood | -302.32 | -305.40 | -302.02 | -305.44

Specification (3†) is determined to be the model selected by the data by
virtue of its marginal likelihood being the largest among candidate models.
The covariate effects for unemployment, per capita income and housing starts
in this specification are, however, not statistically important. Accordingly,
the latent classes generated by this model cannot be distinguished as
representing “high” or “low” regional distress and are labeled as “Regional
Distress Class 1” and “Regional Distress Class 2”.

####  5.1.2 Heterogeneity in Decision Rules

Figure 5 shows the posterior densities of the average probability of the FSLIC
assigning each resolution method to S&L’s in the two latent classes. These
distributions offer two main insights into the decisions of the FSLIC. First,
on comparing with Figure 3, it is clear that the FSLIC relied more heavily on
Type I resolutions relative to the FDIC. The average probabilities of the
FSLIC assigning a Type I resolution were 67.5 % and 69.6% in class 1 and 2
compared with probabilities of 24.6 % and 3.3% of the FDIC assisting banks
that failed in high and low regional distress respectively. Second, the FSLIC
recognizably deviated from the recommended resolution strategy developed in
Cordella and Yeyati (2003) since the posterior densities for the two classes
overlap across all three resolution methods. Accordingly, the average
probabilities of receiving a Type I resolution are not statistically different
across the two classes. This finding signifies that the FSLIC did not
distinguish between S&L institutions that failed amid high and low economic
distress in assigning Type I assistance.

Figure 5: Posterior distribution of the average probability of the FSLIC
assigning each resolution method within classes based on regional distress.
The horizontal axis represents the probability of assigning a resolution
method and the vertical axis represents the posterior density associated with
that probability based on a kernel density estimate. The solid vertical lines
represent the mean of these posterior distributions across the $G$ MCMC draws.

####  5.1.3 Resolution Type

Figure 6: Covariate effects from the models for resolution type for S&L’s in
the classes based on regional distress.

The box plots for covariate effects from specification (3†) in Figure 6 show
that the FSLIC adopted a common decision rule in resolving institutions in the
two classes. The covariate effects of S&L financial characteristics are
homogeneous across the two latent classes. This contrasts with the covariate
effects on the FDIC’s responses in Figure 4, which are statistically different
across the classes of banks that failed amid high and low regional distress.
In both latent classes in the FSLIC’s decision structure, a standard deviation
increase in Real Estate Loan Ratio is seen to result in an average decline of
around 7% in the probability of receiving a Type I resolution and a
corresponding increase in the probability of receiving the other two
resolution methods. Since real estate prices collapsed across several regions
during this period, institutions with excess concentrations in real estate
lending would have been exposed to heightened risks of defaults and losses.
Thereby, the FSLIC likely considered such institutions to be incapable of
being revived through assistance. Between the two latent classes, the FSLIC
responded to Larger Loan Loss Reserve Ratios among institutions in “Regional
Distress Class 2” by withholding assistance and instead facilitating their
acquisition or liquidating them. This response is similar but more muted in
“Regional Distress Class 1”. This suggests that the FSLIC viewed larger loss
reserves as a signal of greater deterioration in the asset quality of the
failed institution.

The FSLIC did not respond to changes in the Interest Rate Receivable Ratio,
which has been shown to be correlated with the losses resulting from the
failed institution to the resolution agency (Balla et al., 2015). The average
change in the probability of assigning each resolution method is close to zero
in response to a standard deviation increase in this measure. This finding
suggests that the FSLIC did not distinguish across institutions that held
larger or smaller shares of loans on which payments were past due, on which
interest accrues and is thereby included in our measure of receivable
interest. The FSLIC’s decisions with respect to bank size were similar to
those of the FDIC. Larger institutions were more likely to be assisted by the
agency and less likely to be sold or liquidated across both latent classes,
thereby reflecting the “too-big-to-fail” doctrine in the agency’s decisions.
The FSLIC assigned Type I assistance to a greater extent in states where
interstate banking was permitted, but this effect was solely present in class
1. Average covariate effects were close to zero for all three resolution types
in class 2. The variable “Interstate banking” pertains only to the
deregulation within the banking industry as the S&L industry was not subject
to such restrictions (Roster, 1985). Since S&L’s would have faced more intense
competition from the banking industry in states where interstate banking was
permitted, the FSLIC likely provided assistance with greater probability in
such states as such competition would have adversely affected even those S&L’s
that were healthy. The covariate effects of the remaining variables are
provided in Section J of the supplement.

Overall, the FSLIC’s decision rules did not differ by the extent of regional
distress that accompanied the failure of S&L’s. Moreover, the covariate
effects in the ordinal models for the FSLIC’s decisions on resolution types
were lower in magnitude relative to the equivalent values for the FDIC’s
decisions. These findings suggest that the FSLIC did not adhere to the
recommendation from theoretical studies of assigning S&L’s to distinct
decision rules depending on whether the failure was likely related to broader
economic factors or the institution’s idiosyncratic weaknesses. In addition,
the FSLIC relied on S&L’s financial characteristics to a lesser extent than
the FDIC in undertaking resolution decisions, thereby suggesting that the
agency assigned resolution methods on a case-by-case basis rather than by
adopting a consistent data-driven rule.

##  6 Discussion and contemporary relevance

During banking crises, financial regulators intervene to bail out certain
failed institutions and liquidate others. Regulators are expected to meet the
dual objectives of preserving financial stability and discouraging moral
hazard in the process of reaching such decisions. However, they may deviate
from socially optimal resolution decisions. Furthermore, as bailouts typically
entail transfers from taxpayers to bank depositors and equity holders, these
actions evoke public disapproval even when they are carried out in the public
interest. The risk of regulatory transgressions creates a need for the public
to regularly evaluate regulators’ actions. Additionally, in order to mitigate
biases in assessments from unduly strong subjective beliefs against public
assistance or liquidation, an objective framework of assessment is essential.

An important line of inquiry in assessing the appropriateness of resolution
decisions consists of evaluating whether regulatory agencies applied two
different decision rules for bank failures amid economic distress and under
normal economic conditions. Since the true classification of banks into
distinct decision protocols is unobservable, we have developed a Bayesian
latent class estimation framework to detect unobserved heterogeneity in the
resolution decisions of banks based on underlying economic conditions. This
flexible estimation approach permits inferences on whether the decision rules
across the latent classes are statistically different. Bayesian model
comparison exercises predicated on posterior odds inform the selection of
models that best explain the decision rules of regulatory agencies.

We utilize this modeling framework to assess the responses of the FDIC and the
FSLIC to bank and S&L failures respectively during the crises of the 1980’s in
the two industries. Our results show that the decision rules of the FSLIC,
which subsequently faced insolvency at a significant cost to taxpayers, were
inconsistent with the decision rules recommended by theoretical studies that
address the trade-offs between financial stability and moral hazard. The FDIC,
which survived the earlier crisis as well as the financial crisis of 2008, was
found to have undertaken decisions that were consistent with those recommended
rules. These findings consequently validate the applicability of this approach
to assess regulators.

Were there structural differences between the FDIC and FSLIC that contributed
to these distinct resolution strategies? First, diluted standards for closing
S&L institutions implied that the financial position of S&L’s was more
critical than that of banks at the time of failure. Second, deregulation and
forbearance resulted in a deeper crisis in the S&L industry than in the
banking industry (9% S&L failures relative to 2% bank failures). Following
deregulation in the mid-1980’s, S&L’s that previously operated purely within
the retail loan and deposit market were permitted entry into the commercial
loan market. Despite their entry into a line of business that was historically
serviced by banks, capital requirements were lower for S&L’s than they were
for banks. This forbearance resulted in the rapid expansion of the S&L
industry accompanied by augmented risk-taking, which ultimately led to
widespread failures.

A regular assessment of resolution authorities by lawmakers and the public can
uncover gaps between observed and recommended resolution rules and also point
out the source of such gaps. Assessments of the FSLIC could have revealed the
issues arising from deregulation of the S&L industry and prompted a review of
the structure of the agency, preventing both, the failure of the agency and
the ensuing costs to the taxpayer. The insights from this study can
potentially guide the development of resolution strategies among newer
resolution agencies such as the Single Resolution Board under the European
Central Bank.

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Supplementary materials – Do financial regulators act in the public’s
interest? A Bayesian latent class estimation framework for assessing
regulatory responses to banking crises

This supplement is organized as follows. Section A includes additional details
about the data used in this paper, Section B provides a simulation study to
assess the performance of the collapsed Gibbs sampler discussed in Section
3.6, Section C provides the full Gibbs sampler for posterior inference in the
our latent class ordinal model of Section 3, Section D describes the
calculation of covariate effects and Section E provides additional details
about model comparison. Sections F, G and H include, respectively, additional
covariate effects, and results for hypotheses $H_{2},~{}H_{3}$ pertaining to
FDIC’s resolution decisions. The corresponding discussion related to FSLIC’s
resolution decisions are provided in Sections I (history of S&L failures), J
(additional covariate effects), K (results for hypothesis $H_{2}$) and L
(results for hypothesis $H_{3}$).

##  Appendix A Summary statistics and description of bills

Table 3 provides the data dictionary and Tables 4, 5 present descriptive
statistics of the data described in Section 2.

In this paper, we consider the following five bills that were favorable to the
banking or S&L industry.

  1. 1.

Bill 1 - reform, recapitalize and consolidate the federal deposit insurance
system as well as to enhance the powers of federal regulatory agencies.

  2. 2.

Bill 2 - introduce additional checks on the banking industry by proposing the
restoration of civil penalties for criminal offenses involving financial
institutions

  3. 3.

Bill 3 - restructure the S&L industry and recommended that the FDIC insure
deposits held at S&L institutions in addition to commercial banks following
the failure of the FSLIC. This bill also authorized the establishment of the
Resolution Trust Corporation (RTC) to resolve failed S&L institutions that had
been within the purview of the FSLIC and any additional failures that arose
within the next three years.

  4. 4.

Bill 4 - The Competitive Equality Bank Act (CEBA) of 1987 provided the FDIC
with the option to establish a temporary national bank or a bridge bank for a
maximum period of three years. This option served as an alternative to
liquidation when acquirers were not forthcoming for purchasing a bank in the
period immediately following its failure (Huber, 1988)

  5. 5.

Bill 5 - disclosure of ratings assigned to banks and thrifts under the
Community Reinvestment Act (CRA).

With the exception of Bill 4, all other bills described above are components
of the Financial Institutions Reform, Recovery and Enforcement Act(FIRREA) of
1989. In tables 4 and 5, the category ‘% vote for Bill $k$’ represents the
percentage of Congressional representatives from the state in which the bank
(S&L) operated who voted for bill $k$ that was favorable to the banking or S&L
industry, where $k=1,\ldots,5$.

Table 3:  The data dictionary. All state level and county level
characteristics pertain to those U.S. states and counties where the banks
(S&L’s) failed during the crises of 1980’s.

Covariates Description Bank and S&L level characteristics C&I Loan Ratio ratio
of commercial and industrial loans to total loans CLD Loan Ratio ratio of
construction and land development loans to total loans Real Estate Loan Ratio
ratio of real estate loans to total loans Loan Loss Reserves Ratio ratio of
reserves set aside against expected losses to total loans Nonperforming loans
Ratio (for banks) ratio of loans past due and loans in non-accruing status to
total assets Interest Receivable Ratio ratio of interest income earned but not
collected to total assets Securities Ratio ratio of securities (e.g.,
Treasuries, mortgage-backed securities) to total assets Core Deposits Ratio
(for banks) ratio of the sum of transaction deposits of individuals,
governments, corporations, money market deposit accounts, and time deposits
with balances less than $100,000 relative to total liabilities Earnings (for
banks) ratio of net income to total assets Size(Assets mlns.) total assets in
millions of dollars State charter Fed member (for banks) indicator depicting
state-chartered banks that are members of the Federal Reserve system State
charter non-Fed member (for banks) indicator depicting state-chartered banks
that are not members of the Federal Reserve system State charter (for S\&L’s)
indicator depicting state-chartered S&L’s Insurer characteristic Dep. Ins.
Fund/ Total Deposits ratio of insurance funds available to total deposits
State Characteristics Interstate banking whether interstate banking is legal
in the state in which the bank (S&L) failed Unemployment unemployment rate in
the state in which the bank (S&L) failed Housing starts number of new house
constructions in the state in which the bank (S&L) failed County
characteristics Per capita GDP growth per capita year-on-year growth of county
GDP Farm, Agri and Mining quarterly share of employment in farming,
agriculture and mining Manufacturing quarterly share of employment in
manufacturing Construction quarterly share of employment in construction Fin
Serv and Transport quarterly share of employment in financial services and
transportation Government quarterly share of employment in government County-
level characteristics of bank distress % Assets in Banks with Texas Ratio $>$
100% percent of assets in banks with Texas Ratio $>$ 100% relative to total
banking assets in the county % Deposits in Banks with Texas Ratio $>$ 100%
percent of deposits in banks with Texas Ratio $>$ 100% relative to total bank
deposits in the county % banks with Texas Ratio $>$ 100% percent of banks in
the county with Texas Ratio $>$ 100% Previous Closures number of bank (S&L)
closures in the county in the previous year State-level political economy
characteristics % Republicans in 1987 % of Republicans in Congress in 1987 %
Republicans in 1989 % of Republicans in Congress in 1989 % vote for Bill 1:
enhancing reg. agencies powers % of Congressional votes in favor of Bill 1 %
vote for Bill 2: restoring civil penalties for fin. Inst. % of Congressional
votes in favor of Bill 2 % vote for Bill 3: recommitting S&L restructuring
bill % of Congressional votes in favor of Bill 3 % vote for Bill 4: CEBA % of
Congressional votes in favor of Bill 4 % vote for Bill 5: disclosure of CRA
ratings % of Congressional votes in favor of Bill 5

Table 4:  Descriptive statistics of bank, county and state-level
characteristics in the data sample for FDIC resolution decisions. See Section
A of the supplement for a description of the bills evaluated in this analysis
under the state-level political economy characteristics.

Type I (OBA) Type II (P&A) Type III (PO) Mean Std. dev Mean Std. dev Mean Std.
dev Bank-level characteristics C&I Loan Ratio 27% 13% 27% 16% 28% 16% CLD Loan
Ratio 6% 8% 5% 7% 4% 8% Real Estate Loan Ratio 39% 17% 42% 21% 31% 20% Loan
Loss Reserves Ratio 6% 6% 4% 3% 5% 7% Nonperforming loans Ratio 6% 6% 8% 5% 8%
6% Interest Receivable Ratio 1% 1% 1% 1% 2% 1% Securities Ratio 12% 12% 13%
10% 14% 11% Core Deposits Ratio 63% 18% 73% 13% 74% 15% Earnings -3% 6% -3% 4%
-4% 4% Size(Assets mlns.) 212 654 171 893 47 96 State charter Fed member 4%
20% 7% 26% 14% 35% State charter non-Fed member 50% 50% 53% 50% 48% 50%
Insurer characteristic Dep. Ins. Fund/ Total Deposits 14% 11% 16% 23% 10% 20%
State Characteristics Interstate branching 0.86 0.34 0.77 0.42 0.51 0.50
Unemployment 8% 1% 7% 2% 6% 1% Housing starts 13% 7% 11% 13% 16% 17% County
characteristics Per capita GDP growth 3% 3% 5% 5% 6% 7% Farm, Agri and Mining
8% 8% 11% 11% 16% 14% Manufacturing 11% 5% 11% 7% 8% 5% Construction 6% 1% 5%
2% 5% 2% Fin Serv and Transport 39% 7% 36% 9% 36% 11% Government 15% 6% 16% 7%
16% 6% County-level characteristics of bank distress % Assets in Banks with
Texas Ratio $>$ 100% 13% 16% 5% 11% 7% 15% % Deposits in Banks with Texas
Ratio $>$ 100% 12% 13% 5% 11% 7% 15% % banks with Texas Ratio $>$ 100% 8% 8%
6% 9% 7% 11% Previous Closures 4.00 6.27 2.62 6.15 1.54 4.72 Count 118 - 1175
- 92 - State-level political economy characteristics % Republicans in 1987 41%
12% 43% 17% 52% 22% % Republicans in 1989 35% 14% 40% 18% 47% 19% % vote for
Bill 1: enhancing reg. agencies powers 83% 14% 80% 21% 83% 18% % vote for Bill
2: restoring civil penalties for fin. Inst. 93% 20% 88% 23% 82% 28% % vote for
Bill 3: recommitting S&L restructuring bill 97% 2% 98% 3% 98% 3% % vote for
Bill 4: CEBA 99% 3% 98% 5% 98% 4% % vote for Bill 5: disclosure of CRA ratings
33% 17% 37% 23% 33% 26% Count 118 - 1170 - 92 -

Table 5:  Descriptive statistics of S&L, county and state-level
characteristics in the data sample for FSLIC resolution decisions. See Section
A of the supplement for a description of the bills evaluated in this analysis
under the state-level political economy characteristics.

Type I (OBA) Type II (P&A) Type III (PO) Mean Std. dev Mean Std. dev Mean Std.
dev S&L-level characteristics C&I Loan Ratio 3% 5% 4% 6% 5% 9% CLD Loan Ratio
14% 20% 27% 25% 20% 25% Real Estate Loan Ratio 93% 11% 96% 17% 103% 20% Loan
Loss Reserves Ratio 5% 8% 9% 11% 4% 5% Interest Receivable Ratio 1% 1% 2% 3%
2% 1% Securities Ratio 20% 15% 13% 9% 12% 10% Size(Assets mlns.) 471 1961 316
484 201 243 State charter 11% 31% 8% 27% 7% 25% Insurer characteristic Dep.
Ins. Fund/ Total Deposits -1% 18% 0% 22% 7% 22% State Characteristics
Interstate branching 0.69 0.46 0.54 0.50 0.33 0.47 Unemployment 8% 2% 8% 2% 8%
2% Housing starts 16% 17% 19% 22% 23% 25% County characteristics Per capita
GDP growth 4% 4% 4% 4% 6% 3% Farm, Agri and Mining 7% 8% 8% 8% 9% 7%
Manufacturing 14% 8% 13% 7% 14% 7% Construction 5% 3% 5% 2% 5% 1% Fin Serv and
Transport 37% 8% 37% 9% 34% 10% Government 15% 6% 16% 7% 20% 15% Count 270 -
104 - 15 - County-level characteristics of S&L distress % Assets in Banks with
Texas Ratio $>$ 100% 3% 8% 2% 5% 3% 6% % Deposits in Banks with Texas Ratio
$>$ 100% 3% 8% 2% 5% 3% 6% % banks with Texas Ratio $>$ 100% 4% 8% 3% 7% 3% 5%
Previous Closures 0.24 0.29 0.27 0.56 0.76 0.77 Count 270 - 102 - 15 - State-
level political economy characteristics % Republicans in 1987 41% 11% 41% 15%
43% 10% % Republicans in 1989 35% 14% 40% 18% 47% 19% % vote for Bill 1:
enhancing reg. agencies powers 83% 14% 80% 21% 83% 18% % vote for Bill2:
restoring civil penalties for fin. Inst. 93% 20% 88% 23% 82% 28% % vote for
Bill 3: recommitting S&L restructuring bill 97% 2% 98% 3% 98% 3% % vote for
Bill 4: CEBA 97% 5% 97% 6% 94% 6% % vote for Bill 5: disclosure of CRA ratings
33% 17% 37% 23% 33% 26% Count 267 - 103 - 15 -

##  Appendix B Simulation Study

In this simulation study we assess the performance of the Collapsed Gibbs
sampler of Section 3.6 for posterior inference in our latent class ordinal
model. We consider two simulation settings, the first of which contains latent
classes with disparate means and the other, in which the means overlap. To
evaluate the algorithm’s accuracy in recovering parameters, we generate
differences in means by considering alternative values of coefficients
$\bm{\beta}_{s},s\in\\{1,2\\}$ across the two settings, and leave the
covariates unchanged. The simulation exercise has been performed on a sample
of size $n=1200$ observations under both the settings with $p=2$ and $q=4$. In
each setting, the first component of the covariates
$\bm{w}_{i},~{}i=1,\ldots,n$ is an intercept and the second component is drawn
independently from a standard normal distribution. Similarly, in the ordinal
model, the first component of the covariates $\bm{x}_{i}$ is an intercept, and
the second, third and fourth components are drawn independently from
$\mathcal{N}(0.5,1)$, $\mathcal{N}(0.5,1)$ and $\mathcal{N}(0,0.8)$
respectively. The true values of parameters in the two settings are provided
in the second column of Table 6. Under these specifications, the mean under
the two latent classes for the two settings are obtained as follows:

  * •

Setting 1:
$\sum_{i=1}^{n}\bm{x}_{i}^{\prime}\bm{\beta}_{1}\mathbb{I}\\{s_{i}=1\\}/\sum_{i=1}^{n}\mathbb{I}\\{s_{i}=1\\}=-0.04$,
$\sum_{i=1}^{n}\bm{x}_{i}^{\prime}\bm{\beta}_{2}\mathbb{I}\\{s_{i}=2\\}/\sum_{i=1}^{n}\mathbb{I}\\{s_{i}=2\\}=-0.52$.

  * •

Setting 2:
$\sum_{i=1}^{n}\bm{x}_{i}^{\prime}\bm{\beta}_{1}\mathbb{I}\\{s_{i}=1\\}/\sum_{i=1}^{n}\mathbb{I}\\{s_{i}=1\\}=0.01$,
$\sum_{i=1}^{n}\bm{x}_{i}^{\prime}\bm{\beta}_{2}\mathbb{I}\\{s_{i}=2\\}/\sum_{i=1}^{n}\mathbb{I}\\{s_{i}=2\\}=-0.02$.

We continue to use the same prior distributions and hyperparameters as those
specified in Section 4.

Table 6:  Credibility intervals from simulation studies based on two parameter specifications | Setting 1: Disparate Means | Setting 2: Common Means
---|---|---
| True Values | Cred. Int. | True Values | Cred. Int.
| Class Membership
$\alpha_{1}$ | -0.3 | [-0.42,-0.26] | -0.3 | [-0.50,-0.22]
$\alpha_{2}$ | 1.5 | [1.29,1.53] | 1.5 | [1.14,1.48]
| Latent class 1
$\beta_{11}$ | 0.6 | [0.57,0.64] | 0.6 | [0.59,0.67]
$\beta_{21}$ | -0.7 | [-0.69,-0.61] | -0.6 | [-0.67,-0.57]
$\beta_{31}$ | -0.6 | [-0.62,-0.53] | -0.6 | [-0.69,-0.60]
$\beta_{41}$ | 0.5 | [0.48,0.58] | 0.5 | [0.48,0.57]
$\sigma_{1}^{2}$ | 0.25 | [0.20,0.27] | 0.25 | [0.22,0.29]
| Latent class 2
$\beta_{12}$ | 0.1 | [0.06,0.17] | 0.1 | [-0.03,0.08]
$\beta_{22}$ | 0.6 | [0.56,0.66] | -0.1 | [-0.12,-0.04]
$\beta_{32}$ | 0.2 | [0.18,0.26] | -0.1 | [-0.11,-0.02]
$\beta_{42}$ | 0.8 | [0.79,0.94] | 0.8 | [0.79,0.92]
$\sigma_{2}^{2}$ | 0.25 | [0.21,0.29] | 0.25 | [0.17,0.24]

Table 6 summarizes the one-standard deviation credibility intervals from the
estimation of all parameters under the two specifications. The two-standard
deviation credibility intervals, not shown here, contain the true values for
all parameters under both settings. The more conservative one-standard
deviation credibility intervals under Setting 1 contain the true values of
parameters with the exception of $\beta_{21}$, for which the credibility
interval lies marginally above the true value. Under Setting 2, the true
values of parameters lie marginally beyond credibility intervals for
$\alpha_{2}$, $\beta_{12}$ and $\sigma_{2}^{2}$. The credibility intervals are
also narrower under Setting 1 relative to Setting 2, demonstrating the
enhanced precision of estimates under more separated classes.

Figure 7: Autocorrelation in the posterior sample of
$\bm{\alpha}=(\alpha_{1},\alpha_{2})$ from a full Gibbs sampler under
simulation Setting 1. The full Gibbs sampler algorithm is available in Section
C of the supplement. Here $\alpha_{1}$ is the intercept and $\alpha_{2}$ is
the coefficient of the second component of $\bm{w}_{i}$ within the class
membership model considered in this simulation exercise. Figure 8:
Autocorrelation in the posterior sample of
$\bm{\alpha}=(\alpha_{1},\alpha_{2})$ from the collapsed Gibbs sampler
(Section 3.6) under simulation Setting 1. Here $\alpha_{1}$ is the intercept
and $\alpha_{2}$ is the coefficient of the second component of $\bm{w}_{i}$
within the class membership model considered in this simulation exercise.

Figures 7 and 8 display the autocorrelations in the posterior samples of
$\bm{\alpha}$ under a full Gibbs sampler (Section C) and the proposed
Collapsed Gibbs sampler (Section 3.6), respectively, for Setting 1. The
figures show that the reduction in autocorrelations gained from the latter
method is striking. The autocorrelations from the full Gibbs sampler are close
to 1 at lower lags and decay slowly whereas the autocorrelations from the
Collapsed Gibbs sampler are negligible even at lower lags and taper to zero
within 5 lags. This demonstrates that the proposed Collapsed Gibbs sampler
algorithm introduced in Section 3.6 provides an efficient technique for
sampling from the posterior distribution relative to standard sampling
approaches by reducing autocorrelations across successive draws.

The results from this simulation exercise have been corrected for label-
switching using Papastamoulis (2016). This issue affects MCMC algorithms
constructed for the estimation of finite mixture models and results in
switching of class labels in the course of the chain. The class labels have
been reassigned post-estimation using an ordering constraint on the intercept
in both classes.

##  Appendix C Full Gibbs sampler

Here we present the full Gibbs sampler algorithm for posterior inference in
the latent class ordinal model discussed in Section 3.

Algorithm: Full Gibbs Sampler

  1. 1.

Sample $\bm{\beta}_{s}$ from the distribution
$\bm{\beta}_{s}|\bm{z},\bm{u},\sigma^{2}_{s}$ for $s=1,2$.

  2. 2.

Sample $\sigma^{2}_{s}$ from $\sigma^{2}_{s}|\bm{\beta}_{s},\bm{z},\bm{u}$ for
$s=1,2$.

  3. 3.

    1. (a)

Sample $\bm{\alpha}$ from
$\bm{\alpha}|\bm{u},\bm{\beta},\bm{\sigma}^{2},\bm{y}$, where
$\bm{\beta}=(\bm{\beta}_{1},\bm{\beta}_{2})$ and
$\bm{\sigma}^{2}=(\sigma_{1}^{2},\sigma_{2}^{2})$.

    2. (b)

Sample $l_{i}|\bm{\alpha},\bm{u}$, where for $i=1,\ldots,n$.

  4. 4.

Sample $s^{\prime}_{i}$ from
$s^{\prime}_{i}|\bm{\alpha},\bm{\beta},\bm{\sigma}^{2},\bm{y}$ for
$i=1,\ldots,n$.

  5. 5.

Sample $z_{i,s_{i}}$ from
$z_{i,s_{i}}|\bm{\beta},\bm{\sigma}^{2},\bm{y},\bm{u}$ for $i=1,\ldots,n$.

Steps (a), (b), (d) and (e) are identical to the algorithm described in
Section 3.6. Step (c) of this algorithm is described below.

Sampling coefficients $\bm{\alpha}$ of the Class Membership Model -  (i) The
coefficients $\bm{\alpha}$ of the class membership model are sampled from the
full conditional $\mathcal{N}(\hat{\bm{\alpha}}_{0},\hat{A})$ where
$\hat{A}=(A_{0}+W^{\prime}W)^{-1}$,
$\hat{\bm{\alpha}}_{0}=\hat{A}(A_{0}^{-1}\bm{\alpha}_{0}+W^{\prime}l)$ and $W$
is the $n\times p$ matrix that has the vector of $p$-dimensional covariates
$\bm{w}_{i}^{\prime}$ from Equation (2) in its rows.

(ii) The latent variable $l_{i}$ is sampled using the data augmentation
approach in Albert and Chib (1993) and drawing from the full conditional
distribution, which is a truncated normal distribution with mean
$\bm{w}_{i}^{\prime}\bm{\alpha}$, variance $1$ and region of truncation
$\mathcal{C}_{i}$, where

$\mathcal{C}_{i}=\begin{cases}(0,\infty),&\text{if}\ s_{i}=2\\\
\left(-\infty,0\right]&\text{if}\ s_{i}=1\\\ \end{cases}.$

##  Appendix D Covariate Effects

An intuitive interpretation of the relationship between covariates and
outcomes is provided by covariate effects in models with discrete dependent
variables. Here, we discuss the calculation of covariate effects for the
resolution-type model, the second level model in Figure 2. Consider any
covariate, $x_{k}\in\mathcal{X}=(x_{1},\ldots,x_{q})$, whose effects on the
outcome $y$ marginally of the other variables in $\mathcal{X}$ is of interest.
Denote $\mathcal{X}_{\setminus k}$ to be the $q-1$ dimensional covariate
vector obtained by excluding the $k^{th}$ covariate for $k=1,\ldots,q$ and let
$\bm{\theta}_{s}=(\bm{\alpha},\bm{\beta}_{s},\sigma_{s}^{2})$. In our
resolution-type model, the covariate effect measures the change in the
marginal probability of observing $y=j$ for a given change in $x_{k}$
conditional on latent class $s$.

The framework for computing covariate effects while addressing both data
variability and parameter uncertainty described in Jeliazkov and Vossmeyer
(2018) has been adapted to the latent class model for ordinal outcomes. Each
of these issues is overcome by averaging over the sample and the posterior
distribution of parameters respectively. Equation (11) details how the output
from the MCMC algorithm developed in Section 3.6, which consists of $G$ draws
from the posterior distribution of parameters, is used in evaluating covariate
effects for each of the two latent classes. We have, for $s\in\\{1,2\\}$,

$\displaystyle\begin{split}Pr(y=j|{x_{k}^{\dagger}},s)-Pr(y=j|{x_{k}^{\ddagger}},s)=&\int\Bigl{\\{}Pr(y=j|{x_{k}^{\dagger}},\mathcal{X}_{\setminus
k},\bm{\theta}_{s})\\\ -&Pr(y=j|{x_{k}^{\ddagger}},\mathcal{X}_{\setminus
k},\bm{\theta}_{s})\Bigr{\\}}f(\mathcal{X}_{\setminus
k})f(\bm{\theta}_{s}|y)\mbox{d}\mathcal{X}_{\setminus
k}\mbox{d}\bm{\theta}_{s}\\\
\approx&\frac{1}{nG}\sum_{i=1}^{n}\sum_{g=1}^{G}\Bigl{\\{}Pr\left(y_{i}=j|{x_{ik}^{\dagger}},\bm{x}_{i,\setminus
k},\bm{\theta}_{s}^{(g)}\right)\\\
-&Pr\left(y_{i}=j|{x_{ik}^{\ddagger}},\bm{x}_{i,{\setminus
k}},\bm{\theta}_{s}^{(g)}\right)\Bigr{\\}}\end{split}$ (11)

where $\bm{x}_{i,\setminus k}$ denotes the $i^{th}$ row of $X$ excluding the
$k^{th}$ element. The probabilities in the parentheses in the final step of
Equation (11) are obtained by using the expression in Equation (7).

##  Appendix E Model Comparison

Subsequent to estimating the model, the empirical objective is to then
identify the specification of the model that is corroborated by the data most
decisively. Accordingly, this section presents the procedure for the
comparison of posterior probabilities of estimated models. This is a method of
model comparison that conforms to the Bayesian principle of representing
uncertainty in the form of probability statements. Specifically, in comparing
models $\mathcal{M}_{i}$ and $\mathcal{M}_{j}$, the posterior odds ratio,
$\mathcal{P}_{ij}$, is evaluated to select between the pair of models, where,

$\mathcal{P}_{ij}=\frac{P(\mathcal{M}_{i}|\bm{y})}{P(\mathcal{M}_{j}|\bm{y})}=\frac{m(\bm{y}|\mathcal{M}_{i})}{m(\bm{y}|\mathcal{M}_{j})}\frac{P(\mathcal{M}_{i})}{P(\mathcal{M}_{j})}.$

The first term on the right hand side of the second equality is the Bayes
factor and the second term is the prior odds. The Bayes factor is the ratio of
marginal likelihoods of models $i$ and $j$ and following standard convention
in which the $a$ $priori$ probability of each model occurring is equal, this
quantity singularly determines the evidence in favor of one model against the
other. Therefore, Bayesian model selection among $L$ models,
$\\{\mathcal{M}_{1},\mathcal{M}_{2},...,\mathcal{M}_{L}\\}$, proceeds by
comparing the marginal likelihood across these models.

The basic marginal likelihood identity recognized by Chib (1995) allows for
the exact evaluation of the marginal likelihood by MCMC methods. This identity
expresses the marginal likelihood of model $l$ as

$m(\bm{y}|\mathcal{M}_{l})=\frac{f(\bm{y}|\mathcal{M}_{l},\bm{\theta}_{l})\pi(\bm{\theta}_{l}|\mathcal{M}_{l})}{\pi(\bm{\theta}_{l}|\bm{y},\mathcal{M}_{l})},$

where $\bm{\theta}_{l}$ is a parameter vector specific to model $l$. The
computation of the marginal likelihood simply requires the evaluation of this
ratio for a given $\bm{\theta}_{l}^{*}$, typically the posterior mean or mode.
The likelihood $f(\bm{y}|\mathcal{M}_{l},\bm{\theta}_{l})$ and prior ordinate
$\pi(\bm{\theta}_{l}|\mathcal{M}_{l})$ at $\bm{\theta}_{l}^{*}$ can be
evaluated analytically for the latent class model with ordered outcomes. The
posterior ordinate $\pi(\bm{\theta}_{l}|\bm{y},\mathcal{M}_{l})$ at
$\bm{\theta}_{l}^{*}$ is estimated to obtain
$\hat{\pi}(\bm{\theta}_{l}^{*}|\bm{y},\mathcal{M}_{l})$ using methods outlined
in Chib and Jeliazkov (2001) and Chib (1995).

###  E.1 Evaluating the Marginal Likelihood

Here we present the algorithm to evaluate the estimated posterior ordinate
$\hat{\pi}(\bm{\theta}^{*}_{l}|\bm{y},\mathcal{M}_{l})$. Henceforth, we will
drop the subscript $l$ for notational ease.

In the latent class model $\mathcal{M}$ with ordered outcomes, the parameter
vector $\bm{\theta}$, excluding any latent variables, consists of the
coefficients $\bm{\alpha},\bm{\beta}=(\bm{\beta}_{1},\bm{\beta}_{2})$ and the
error variances $\bm{\sigma}^{2}=(\sigma_{1}^{2},\sigma_{2}^{2})$. The
objective of this exercise is to evaluate the posterior ordinate at the
posterior mean,
$\bm{\theta}^{*}=(\bm{\alpha}^{*},\bm{\beta}^{*},\bm{\sigma}^{2*})$. The law
of total probability is used to decompose the posterior ordinate at
$\bm{\theta}^{*}$ as,

$\pi(\bm{\alpha}^{*},\bm{\beta}^{*},\bm{\sigma}^{2*}|\bm{y},\mathcal{M})=\pi(\bm{\alpha}^{*}|\bm{y},\mathcal{M})\pi(\bm{\beta}^{*}|\bm{\alpha}^{*},\bm{y},\mathcal{M})\pi(\bm{\sigma}^{2*}|\bm{\alpha}^{*},\bm{\beta}^{*},\bm{y},\mathcal{M})$

The order of this decomposition has been chosen to minimize computational time
and effort. The first component, $\pi(\bm{\alpha}^{*}|\bm{y},\mathcal{M})$ is
estimated using the method introduced in Chib and Jeliazkov (2001). By
conditioning the other two densities on $\bm{\alpha}^{*}$, these ordinates can
be estimated using reduced Gibbs samplers described in Chib (1995). The
following reduced Gibbs sampler provides the estimated ordinate
$\hat{\pi}(\bm{\beta}^{*}|\bm{\alpha}^{*},\bm{y},\mathcal{M})$.

  1. 1.

Sample $\bm{\beta}_{s}$ from the distribution
$\bm{\beta}_{s}|\bm{z},\bm{u},\bm{\sigma}^{2}_{s}$ for $s=1,2$.

  2. 2.

Sample $\bm{\sigma}^{2}_{s}$ from
$\bm{\sigma}^{2}_{s}|\bm{\beta}_{s},\bm{z},\bm{u}$ for $s=1,2$.

  3. 3.

Sample $s^{\prime}_{i}$ from
$s^{\prime}_{i}|\bm{\alpha}^{*},\bm{\beta},\bm{\sigma}^{2},\bm{y}$, where
$s_{i}^{\prime}=s_{i}-1$ for $i=1,\ldots,n$.

  4. 4.

Sample $z_{i,s_{i}}$ from
$z_{i,s_{i}}|\bm{\beta},\bm{\sigma}^{2},\bm{y},\bm{u}$ for $i=1,\ldots,n$.

The ordinate
$\hat{\pi}(\bm{\sigma}^{2*}|\bm{\alpha}^{*},\bm{\beta}^{*},\bm{y},\mathcal{M})$
is obtained by iterating over the following reduced Gibbs sampler.

  1. 1.

Sample $\bm{\sigma}^{2}_{s}$ from
$\bm{\sigma}^{2}_{s}|\bm{\beta}_{s}^{*},\bm{z},\bm{u}$ for $s=1,2$.

  2. 2.

Sample $s^{\prime}_{i}$ from
$s^{\prime}_{i}|\bm{\alpha}^{*},\bm{\beta}^{*},\bm{\sigma}^{2},\bm{y}$ for
$i=1,\ldots,n$.

  3. 3.

Sample $z_{i,s_{i}}$ from
$z_{i,s_{i}}|\bm{\beta}^{*},\bm{\sigma}^{2},\bm{y},\bm{u}$ for $i=1,\ldots,n$.

##  Appendix F Regional economic distress and FDIC’s decisions for bank
resolutions

In this section, we report the covariate effects of the remaining variables
from the selected ordered response model in specification 3 of Table 1.

Figure 9: Additional covariate effects from the models for resolution type for
banks in the class of High Regional Distress (HRD) and Low Regional Distress
(LRD).

In Figure 9, a standard deviation increase in Commercial and Industrial (C&I)
loan ratio is associated with an increased probability of Type II and Type III
resolutions and a corresponding decline in the probability of a Type I
resolution among HRD failures. These findings are consistent with the higher
levels of risk attributed to increases in this ratio by the FDIC (FDIC, 1998).
A standard deviation increase in Construction and Land Development Loans (CLD)
is associated with an increased probability of Type I resolutions among HRD
failures and of Type III resolutions among LRD failures. In the period under
study, C&I loans constitute a higher concentration of bank balance sheets
(27.5% of assets) than CLD loans (4.6% of assets). Accordingly, an increase in
C&I loans represents a more acute concentration in that loan category and
elicits a more stringent response than an equivalent increase in CLD loans.

The remaining covariates, securities ratio, core deposits ratio and earnings
ratio resulted in covariate effects that were close to zero upon controlling
for other balance sheet items pertaining to size, asset quality and interstate
branching restrictions. An increase in each of these characteristics is
associated with an increase in the bank’s franchise value and is accordingly
expected to result in a lower probability of liquidation under Type III
resolutions.

##  Appendix G Banking industry distress and FDIC’s decisions for bank
resolution

We examine whether the FDIC’s responses supported Hypothesis $H_{2}$, i.e.,
whether the agency provided assistance with higher probability to banks that
failed amid distress within the banking industry. Our results show that the
agency’s decision rules qualitatively aligned with this hypothesis as it
provided Type I resolutions with a marginally higher probability when the
local banking industry experienced failures. However, measures of regional
economic distress remained the most important determinants of membership into
classes that received statistically different levels of Type I resolution.

Table 7:  Covariate effects from class-membership models for specifications of
latent classes based on regional and banking industry distress. The reported
values are posterior means of the covariate effects. Posterior standard
deviations are in parantheses.

| (5) | (6) | (7) | (8) | (9) | (10)
---|---|---|---|---|---|---
State-level characteristics |  |  |  |  |  |
Unemployment | - | - | - | -0.07 (0.04) | -0.08 (0.04) | -0.1 (0.05)
County-level characteristics |  |  |  |  |  |
Housing starts | - | - | - | 0.03 (0.05) | 0.04 (0.05) | 0.06 (0.06)
Per capita GDP growth | - | - | - | 0.04 (0.05) | 0.04 (0.05) | 0.07 (0.06)
Farm, agri, mining | - | - | - | 0.03 (0.04) | 0.04 (0.04) | 0.06 (0.05)
Banking industry characteristics |  |  |  |  |
Previous closures | -0.08 (0.08) | -0.07 (0.06) | -0.08 (0.07) | -0.02 (0.01) | -0.02 (0.01) | -0.01 (0.02)
% Assets in distressed banks | -0.03 (0.04) | - | - | -0.03 (0.02) | - | -
% Dep. in distressed banks | - | -0.03 (0.02) | - | - | -0.03 (0.02) | -
% distressed banks | - | - | -0.01 (0.01) | - | - | -0.02 (0.03)
Insurer characteristics |  |  |  |  |  |
Dep. Ins. Fund/ Total Dep. | - | - | - | -0.05 (0.03) | -0.05 (0.03) | -0.05 (0.04)
log Marginal Likelihood | -719.29 | -705.30 | -719.14 | -697.22 | -697.78 | -701.21

Table 7 summarizes the covariate effects and log marginal likelihood from
model specifications (5) through (7) that are based purely on banking industry
distress and specifications (8) through (10) that incorporate a combination of
banking industry and regional economic distress. The data favor the latter
three specifications over the former three as evidenced by their higher
marginal likelihood. Specifically, the Bayesian model selection procedure
based on posterior odds selects specification (8), which defines distress in
the banking industry using previous closures and the percent of assets in
distressed banks within a county. Here distressed banks are those institutions
whose Texas ratio, defined in Equation (1), exceeded 100% based on previous
literature that utilize this measure (Cooke et al., 2015; Siems et al., 2012).
On account of the negative covariate effect of unemployment, previous closures
and percent of assets in distressed banks and the positive signs for housing
starts and per capita income growth, latent class 2 contains banks that failed
amid relatively low regional or banking distress and banks that failed amid
high regional and banking distress belong to latent class 1. As a result, in
the following discussion, latent class 1 will be labeled as the class of
failures under “High Regional and Banking Distress (HRBD)” and latent class 2,
as the class of failures under “Low Regional and / or Banking Distress
(LRBD)”.

Figure 10: Posterior distribution of the average probability of the FDIC
assigning each resolution method within classes based on regional and banking
distress. The horizontal axis represents the probability of assigning a
resolution method and the vertical axis represents the posterior density
associated with that probability based on a kernel density estimate. The solid
vertical lines represent the mean of these posterior distributions across the
$G$ MCMC draws.

Figure 10 provides the density of the posterior distribution of the
probability of the FDIC assigning each resolution category under the selected
model, specification (8). These results qualitatively align with the
recommendations from Shleifer and Vishny (1992) and Acharya and Yorulmazer
(2008) represented by Hypothesis $H_{2}$, which posited a greater reliance on
public financial assistance in the form of Type I resolutions when the local
banking industry experienced distress. The figure shows that the average
probability of a Type I resolution was 26.8% under banking and regional
distress and 3.5% under low regional and banking distress. These probabilities
are marginally higher than the probability of Type I resolution of 24.6% and
3.3% under high and low regional distress respectively, depicted in Figure 3.
Correspondingly, the average probability of Type II resolutions under banking
and regional distress was 69.3%, which was lower than the equivalent
probability under regional distress at 72.1%. Liquidations under Type III
resolutions remained largely unchanged with probabilities of 3.9% and 3.2%
respectively under classes based on high regional and banking distress and
solely regional distress. The inclusion of measures of banking distress to
indicators of regional distress from Section 4.1.1 primarily resulted in
substitutions between Type I and II resolutions. Overall, the densities based
on the two models show that measures of regional distress continue to be the
most important determinants of heterogeneity in the FDIC’s decision rules and
that measures of bank distress marginally augment the separation across the
two classes. These findings show that banking industry distress in addition to
regional distress likely contributed to a “too-many-to-fail” response from the
FDIC in line with predictions from the theoretical literature (Acharya and
Yorulmazer, 2007).

##  Appendix H Political economy factors and FDIC’s decisions for bank
resolution

In this section we address constraints to the FDIC’s decision-making in the
form of political pressures to provide Type I resolutions represented by
Hypothesis $H_{3}$. We find that political factors played a limited role in
the FDIC’s decisions as the average probability of Type I assistance to banks
increased marginally from 24.6% amid economic distress to 26.5% in the
presence of political support and economic distress.

The role of political economy considerations in the FDIC’s resolution
decisions during the 1980’s has not been examined in detail in previous
studies. The immediate reason for the sparse attention to this question in
previous periods is that data on lobbying did not become available until after
the Lobbying Disclosure Act of 1995. We address the paucity of data on direct
measures of lobbying in two ways. First, we measure political support for
financial institutions by way of the percentage of Congressional
representatives from each state who voted for a bill that is favorable to the
banking or S&L industry. This approach recognizes votes in favor of
legislation that benefits the banking industry as indicative of lobbying
efforts by the industry, which is consistent with the theoretical model of
Becker (1983) in which pressure groups compete for political favors. This
approach also follows from Kroszner and Strahan (1999) and Economides et al.
(1996), who provide evidence of private interest groups influencing the voting
behavior of elected representatives on legislation pertaining to the banking
industry. Second, we include the measures of congressional voting in the class
membership model that classifies banks into groups that are subject to
different decision rules rather than in the model for resolution types.
Consequently, instead of associating political support with the resolution
method applied on specific banks, this specification suggests that such