86

Dirk SchoenmakerContagion Risk in Banking
Dirk Schoenmaker*
Ministry of Finance, the Netherlands

Abstract
A controversial issue in the literature on banking regulation is whether there iscontagion risk, or not. This paper derives a framework to test for contagion risk
and applies it to a data set of monthly bank failures under the US National Banking System from 1880 till 1936. To capture the count nature of bank failure data,
an autoregressive Poisson model is used. The empirical results indicate that there
is contagion risk in banking. An initial failure could generate further failures
without intervention by the authorities. This finding suggests that there may be arole for the central bank as lender of last resort to assist ailing banks, whose
failure is expected to have a systemic impact.
In wild periods of alarm, one failure makes many, and the best way toprevent the derivative failures is to arrest the primary failure whichcauses them.
Bagehot, Lombard Street (1873, p.51-2)

1

Introduction

The debate on the need for banking regulation and supervision depends criticallyon the question whether there is contagion risk in banking, or not. Contagionrisk—which is also referred to as systemic risk—is here defined as the risk that
financial difficulties at one or more bank(s) spill over to a large number of otherbanks or the financial system as a whole. There has been little attempt so far tomeasure contagion risk. Reviewing the limited empirical evidence of bank contagion, Kaufman (1994) comes to the conclusion that concerns about systemicstability have been greatly exaggerated.
The paper highlights some shortcomings of previous studies on contagion risk.
*
The author is grateful to Charles Goodhart for helpful suggestions and many discussions on
this topic. He would also like to thank Kevin Dowd, Philipp Hartmann, Richard Payne, as well as
seminar participants at the Bank for International Settlements and the London School of Economics
for valuable comments. This paper was written while the author was at the Financial Markets Groupof the London School of Economics. The views expressed, as well as any errors, are those of theauthor.

Contagion risk in banking

87

First, some of these studies (e.g. Grossman, 1993) use ordinary least squares
regressions to analyse bank failures. But a least squares regression may lead tomis-specifications, since bank failure time series consist of count data—that are
non-negative integers—recording the number of events occurring in a given
interval. To deal explicitly with the count nature of bank failure data, we use an
autoregressive Poisson regression model developed by Shephard (1995).
More importantly, some of these mainly US studies (e.g. Aharony and Swary,
1983; Swary, 1986) examine possible contagion effects during periods in which
the central bank played an active role as lender of last resort to prevent contagiousbank failures. Consequently, the lack of evidence of contagion from such periods
does not disprove the possibility of bank contagion. To avoid this pitfall, we
examine data on bank failures under the US National Banking System, founded in
1864, some fifty years before the establishment of the Federal Reserve System.
Interestingly, the results from our empirical study are consistent with theexistence of contagion risk and are thus opposite to those of Kaufman (1994). An
initial failure could generate further failures without intervention by the authorities. This seems to support the view that lender of last resort assistance from thecentral bank to individual banks may be justified in some cases to prevent thepotential spillover effects of bank failures (e.g. Solow, 1982; Goodhart, 1987;
Summers, 1991). If there is an important role for public intervention in the formof emergency assistance in times of crisis, then it is arguable that the banking
sector, as recipient of this liquidity support, should be regulated and supervised(i.e. monitored) to contain the adverse selection and moral hazard effects of suchintervention (e.g. Folkerts-Landau and Garber, 1992; Goodhart and Schoenmaker,
1995).
The paper proceeds as follows. In Section 2, we review the theoretical framework for bank contagion. Although the literature on bank runs (e.g. Diamond andDybvig, 1983; Chari and Jagannathan, 1988) provides explanations for runs on
individual banks, it does not explain why depositors at many banks withdraw
their deposits simultaneously, generating a banking panic. The possible channels
for the spread of contagion are explored. Next, we discuss some of the earlier
empirical studies on contagion risk. In Section 3, we develop a methodology totest for contagion risk. The idea is that we can speak of contagion risk when bankfailures are found to be dependent (i.e. correlated) after controlling for common
factors, such as macro-economic variables. The model is tested with bank failure
data from the pre-Federal Reserve period. As indicated above, the results are
consistent with the existence of bank contagion. The conclusions follow in Section 4.

2

Contagion risk in banking

The phenomenon of bank runs has been rigorously modelled in the literature.1 A
1

See Bhattacharya and Thakor (1993) for a recent and comprehensive survey of banking theory.

first set of models (e.g. Diamond and Dybvig, 1983; Postlewaite and Vives, 1987)
assumes consumption risk—reflected in a stochastic withdrawal of deposits—andriskless, but illiquid, investments. The illiquidity of these investments provides
the rationale for the existence of banks and for their vulnerability to runs. Excessive withdrawals of deposits would force a bank into costly liquidation. Hence, if
a depositor expects that others will withdraw, he will also withdraw to avoid
losses from such a liquidation. The Diamond-Dybvig model gives rise to multiple
equilibria, including a bank run equilibrium. A bank run is caused by a shift in
expectations, which can depend on some commonly observed factor such as asunspot (it need not be anything fundamental about the bank’s condition).
In a more realistic setting, a second set of bank run models (e.g. Chari and
Jagannathan, 1988; Gorton, 1985) introduces investment risk in addition to consumption risk. Asymmetric information between the bank and its depositors on
the true value of loans is a key element of these models. In the Chari-Jagannathanmodel, only a fraction of depositors receives information on the prospects ofloans. Uninformed depositors, however, do not know whether large deposit
withdrawals are caused by an increase in the fraction of early consumers and/or
by information on a low outcome of loans. Given that a long withdrawal queue
could be caused by bad information about a bank’s solvency, the rational responseof the uninformed depositor is to join the queue as well and withdraw early. An
information-induced bank run can thus occur, even if no one receives a bad signal.
The Chari-Jagannathan and Diamond-Dybvig models differ in that bank runs in
the former start with fears of insolvency of particular banks, while bank runs in
the latter are based on self-fulfilling beliefs.
Although the bank run literature provides explanations for runs on individual
banks, it does not explain why depositors at many banks withdraw their deposits
simultaneously, generating a banking panic (Calomiris and Gorton, 1991). The
contagion effects of bank runs need to be treated explicitly in a model of banking
panic. The risk of contagion in banking—also referred to as systemic risk—ishere defined as the risk that financial difficulties at one or more bank(s) spill over
to a large number of other banks or the financial system as a whole. Contagion
can spread either through the information channel or the credit channel.
Starting with the information channel, Aharony and Swary (1983) make a
distinction between pure (industry specific) contagion and noisy (firm specific)
contagion. Pure contagion occurs when negative information—such as fraud orlosses on specific risky investments—about one bank adversely affects all otherbanks, including those that have nothing in common with the first bank. Noisy or
firm specific contagion arises when the failure of one bank reveals a bad (butnoisy) signal regarding other banks with common characteristics. If one bank fails,
then other banks with a similar asset and liability structure—and therefore vulner-

Contagion risk in banking

89

able to the same economic shocks—may also face a run.2 In a world with imperfect information, runs on other banks can be triggered by perceived—and thus notnecessarily actual—similarities with the failing bank.
Turning to the credit channel, there is a complex web of linkages between
banks in the interbank funding market, the over-the-counter (OTC) derivatives
market, and the payment system (e.g. Guttentag and Herring, 1987; Schoenmaker,
1995). The size of interbank credit lines is exempt from large exposure rules andis usually related to the size of the borrowing bank and not of the lending bank.
Surviving banks can thus have substantial claims on the failing bank and may
subsequently fail.3 Moreover, there is a lack of timely data on interbank exposures.
Market participants know that interbank positions may be very large, but the sizeof particular bilateral positions is not known. In the event of a bank failure,
market participants do not know which banks have unsatisfied claims against thefailing bank. This in turn may generate a general loss of confidence in the interbank market.

2.1

Earlier studies on contagion risk

Although the theoretical framework to show the possibility of contagion in banking has been well-developed, little effort has been expended to quantify the
likelihood and intensity of contagious bank runs. In a recent paper, Kaufman(1994) reviews the empirical evidence of bank contagion. The first question
examined is how broad contagion can spread within the banking sector. A numberof studies have measured the breadth of spillover from a bank failure by the loss
to shareholders of surviving banks as evidenced in share returns (e.g. Aharony
and Swary, 1983; Swary, 1986; Peavy and Hempel, 1988). Using stock market
data, these studies examine the post announcement share performance. Negative
abnormal returns are an indicator for contagion effects. Following the earliermentioned distinction between industry specific and firm specific contagion,
Kaufman finds only some support for the latter in these empirical studies. An
initial failure does not cause further failures directly, but information about thefirst, or first few, bank(s) in difficulties reveals information about (some) otherbanks. The problems causing defaults stem thus from a common factor.
In a similar review of the empirical literature, Saunders (1987) finds little
evidence of contagion. However, a major drawback of these (mainly US) studies
is that most of them examine possible contagion effects during periods in which
the Federal Reserve played an active role as lender of last resort to prevent conta-

2
A recent example of contagion via the information channel is provided by the liquidation ofBarings in February 1995. After the Barings collapse, most other UK merchant banks faced funding
problems in the wholesale market.
3See, for example, the failure of Continental Illinois in 1984. Continental acted ascorrespondent bank for nearly 1,000 banks at the time. Sixty-six banks had uninsured deposits
exceeding 100% of their capital, and another 113 banks had deposits between 50 and 100% of theircapital. We will discuss this case in more detail below.

gious bank failures.4 Consequently, this evidence does not disprove the possibilityof bank contagion. In the next section, we discuss some recent studies (e.g.
Grossman, 1993; Hasan and Dwyer, 1994) investigating bank contagion in the
19th century, before the Federal Reserve was founded. The empirical findings inthese studies indicate the existence of contagion risk, though the contagion effects
are not found to be very large.
Another mechanism of contagion examined by Kaufman (1994) is the credit
channel. Interconnections between banks—via the fed funds market and correspondent balances—speed up the transmission of losses from affected banks to
many other banks. A prime example is the earlier-mentioned failure of Continental Illinois in 1984. Continental had a large correspondent banking network andnearly 1,000 banks had deposits at Continental at the time it failed. Sixty-six ofthese banks had uninsured deposits exceeding 100 percent of their capital, andanother 113 had deposits between 50 and 100 percent of their capital (US Department of the Treasury, 1991).
But uninsured depositor losses are unlikely to be very large with a timely
closure, since the value of the bank’s total assets is unlikely to drop suddenly
anywhere close to zero, as Kaufman correctly argues. The full amount was therefore not to be lost, but uncertainty about the precise size of the losses and about
the time needed to recover (part of) Continental’s assets could have generated
runs on these banks.5 If the uninsured deposits in Continental had not been protected by the Federal Reserve and the Federal Deposit Insurance Corporation(FDIC), its failure might have caused a chain of bank failures.
Finally, it is not sufficient to show that banks face low probabilities of failure
in order to downplay possible contagion effects. Benston et al. (1986, p.58), forexample, provide evidence that the average annual rate of bank failures was not
greatly different from the rate of business failures in the 1875-1929 period (i.e.
before the multiple bank failures during the Great Depression). Repeating theircalculations in Table 1, we find an even stronger result: the average annual bankfailure rate (0.43 percent) was less than half of the average annual business failure
rate (1.00 percent) during the 1875-1929 period. This difference is due to the factthat our sample of bank failures covers only national banks, while the sample in
Benston et al. (1986) covers both national and state-chartered banks.

4
Saunders (1987) also notes that these studies must be interpreted in light of the fact that theFederal Reserve has clearly sought to prevent contagious bank runs. Thus the lack of clear evidenceof contagion may simply indicate that the Fed has largely succeeded.
5
Kaufman (1994, p.131) asserts that ‘actual losses at the Continental were less than five centson the dollar, which was also a reasonable estimate at the time of the failure’ (italics added), but
fails to provide evidence to support this claim.

Contagion risk in banking

91

Table 1: Bank and business failures in the USA, 1875-1935
PeriodMeanStandard
deviation

Coefficient ofvariation

Bank failure rate*
1875-1935
1875-1929
1930-1935

0.84
0.43
4.61

1.58
0.43
3.03

2.97
0.43
1.99

Business failure rate*
1875-1935
1875-1929
1930-1935

1.01
1.00
1.05

0.24
0.23
0.38

0.06
0.05
0.14

Source: Historical Statistics of the United States Colonial Times to 1970, US Department ofCommerce, Washington DC (1975).
*
The annual bank (business) failure rate—reported in percent—is calculated as the number of bank(business) failures divided by the total number of banks (concerns in business).

Although the mean of bank failures may be lower than that of business failures, the variation is significantly higher. Following Benston et al. (1986), we
have calculated the coefficient of variation in order to adjust for the differences inmeans between the two series. This statistic is 0.43 for the rate of bank failuresand only 0.05 for the rate of business failures from 1875 through 1929. Thishigher variation might, but does not need to, indicate larger contagious effects inbanking than in other business sectors.
Reviewing our discussion of the literature, we conclude that a rigorous test ofcontagion risk can only be done with data from a period without a central bankacting as lender of last resort. In other words, contagion is allowed to spreadfreely through the banking system. In the next section, we make an attempt toquantify contagion risk with this constraint in mind.

3

Empirical test of contagion risk

3.1

Methodology

In this section, we develop a measure to test for contagion risk, which is definedas the risk that an initial (bank) failure may spill over to the rest of the (banking)industry and cause further (bank) failures. Bank failure time series consist of
count data—that are non-negative integers—recording the number of eventsoccurring in a given interval. Some previous studies on contagion (e.g. Grossman,1993) use ordinary least squares regressions to analyse bank failures. But given
the predominance of zeroes and small values and the discrete nature of bankfailure data, a least squares regression may lead to mis-specifications. To deal
explicitly with the non-negative and discrete nature of count data, such as bankfailures, we use the Poisson regression model.6
The standard Poisson process assumes that the number of bank failures lremains constant and that bank failures are independent. In reality, the probabilityof bank failure may vary over time, violating the first assumption. A generalisation of the Poisson process is to allow the parameter l to depend on time
(Lancaster, 1990). The probability function of the time-dependent Poisson distribution is as follows:
Prob ( yt ; l t ) =e -lt lytt
yt !

(1)

where yt = 0,1,2,… is the number of bank failures at time t, and lt > 0 is theaverage number of bank failures per unit of time at time t.
In an international survey of bank failures, Heffernan (1995) finds the following macro-economic indicators to be important explanatory variables for theobserved variation in the number of bank failures: GDP growth, inflation, interest
rates, and exchange rates. Accordingly, we can model the number of failures todepend on a set of macro-economic variables. Since lt must be positive, thestandard log-linear link function can be applied:
ln lt = bx't

(2)

where x't denotes a vector of macro-economic variables at time t, and b is thecorresponding vector of parameters.
Given a set of independent observations, the likelihood and log-likelihood
functions are as follows:
e -l t lytt
yt !
t =1
T

L=Õ

(3)

T

T

T

t =1

t =1

t =1

ln L = -å l t + å yt ln l t - å In( yt !)(4)

The log-likelihood function specified in (4) can be maximised by differentiatingwith respect to the parameter vector b:¶ ln L T= å xt ( yt - l t ) = 0
¶b
t =1

(5)

6
Another model appropriate for the analysis of discrete dependent variables is the probit model.
In a recent study, Hasan and Dwyer (1994) estimate probit equations for bank closings during theFree Banking Period in the USA from 1837 through 1863.

Contagion risk in banking

93

The maximum likelihood estimator b̂ can be found using the Newton-Raphson
algorithm (Hamilton, 1994).
The second assumption of the standard Poisson model is that bank failures are
independent. This characterises the absence of contagion risk. Individual bankruns are not causally related, though they may have a common factor (such as theabove-mentioned macro-economic indicators). By contrast, the notion of bankcontagion means that an initial failure may cause further failures by banks which
are exposed to the originally failing bank and thus violate the condition of independence. While the initial failure may follow a stochastic process, the secondary
failures are then partly determined by the initial failure. An extreme form ofdependency is the domino effect: a failure of any single bank entails a chain
reaction, such that eventually all the other banks in the system will also fail(Paroush, 1988). A less stringent form of dependency is the situation in which theprobability of bank failure conditional on the number of previous failures increases with the number of previous bank failures.
While the standard Poisson model assumes that the dependent variable yt is
independently distributed, Shephard (1995) provides an extension to autoregressive Poisson models.7 The procedure proposed by Shephard allows the lagged
observations to enter additively on a log-linear scale. But the log specification
cannot be used as it leads to singularities at yt = 0. To solve this problem,
Shephard (1995) approximates ln yt with a first-order Taylor series around lt:z ( yt ) = ln l t +yt - l t
lt(6)

The link function for the autoregressive Poisson model then becomes:pln l t = b xt¢ + å g j z ( yt - j )

(7)

j =1

where g is the coefficient for the lagged number of bank failures, and p denotes
the number of lags. The second term on the right-hand side of (6) can be rewritten
as follows: ct = (yt – lt)/lt, where ct is a martingale difference sequence. Substituting (7) into (6), we get a non-linear autoregression for bank failures yt withmartingale difference errors:pz ( yt ) = b xt¢ + å g j z ( yt - j ) + ct

(8)

j =1

According to (8), the number of bank failures is dependent on various macroeconomic indicators and the number of failures in the previous period(s). Recent
examples of banking crises with multiple failures are the savings & loans debacle
7See, for example, Ljungqvist (1995) for an application of the autoregressive Poisson model todata on initial public offerings (IPO) in Germany from 1970 through 1994.

in the US (1980s) and the Scandinavian banking crisis (early 1990s). It is anempirical question whether the multiple bank failures are caused by a common
macro-economic variable, e.g. rising interest rates in the US and a sharp fall in
GDP in Scandinavia, perhaps in conjunction with a structural variable such as
deregulation, or by interdependency between banks. But to test whether there isrisk of contagion in banking (i.e. bank failures are dependent), we have to examine periods in which there was no rescue of banks whose failure was expected tohave a systemic impact. In other words, neither the government nor the central
bank (as lender of lender resort) assumed an active role in the management ofbanking crises during these times.

3.2

Data

In order to test for risk of contagion, we examine the US National Banking System, which was established in 1864, fifty years before the foundation of theFederal Reserve System.8 The Annual Reports of the Comptroller of the Currency,
the regulator of the national banks, are a rich source of information and report theexact date of each bank failure.9 The contagion effect, if any, can probably beobserved within weeks or months, but not years, after the initial bank failure(s).
While our bank failure data are on a daily basis, macro-economic data are only
available on a monthly basis. We therefore use monthly series from January 1865
till December 1940. The dependent variable in our Poisson regression model isthe number of bank failures per month (BF), which is a non-negative integer.
The macro-economic variables are the level of real output (GDP), the index ofstock prices (SP), the price level (P), and the short-term interest rate (R). The
level of real output is proxied by pig iron production, the stock price index is thevalue weighted index of the New York Stock Exchange, the price level is proxiedby Snyder’s Index, and the interest rate is the commercial paper rate. The Appendix provides a more detailed description of the data and their sources.
Using an augmented Dickey-Fuller test we cannot reject a unit root in the
output (GDP), stock price (SP), and price level (P) series. But a unit root can berejected in the changes of these variables. Accordingly, the regressions are runwith the monthly changes in the logarithm of these variables (the differential istaken on a twelve-month basis to eliminate seasonal influences). We also takemonthly changes in interest rates (R), as we expect changes rather than the levelof interest rates to determine the possible number of bank failures.

3.3

Empirical results

Based on (8), the following model is used for the regressions:8
In its early years, the Federal Reserve did not play an active role as lender of last resort to stembanking crises. This period (1914-33) is known as the real-bills era (see Chapter 17 of Timberlake,1993).
9
The Comptroller’s Annual Reports from 1865 till 1940 are available in the Bank of Englandlibrary. We would like to thank the Bank for their assistance.

Contagion risk in bankingz(BFt) = a + b1DGDPt + b2DSPt + b3DPt + b4DRt + g1z(BFt–1) + ct

95

(9)

The null hypothesis is that bank failures are independent, which implies g1 = 0.
Only when gamma is significantly different from zero while controlling for
macro-economic influences, we can reject our null hypothesis.
In the estimation of the regression model, it appears that some of the macroeconomic variables are too highly intercorrelated to allow precise analysis of theirindividual effects. The correlation coefficient between the monthly change in our
GDP proxy and the monthly change in stock prices is 0.63. The two regressors
show the symptoms of multicollinearity: coefficient estimates of the two variables
have high standard errors, which can produce wide swings in response to smallchanges in the sample. One remedy suggested by Greene (1993) is to drop one ofthe variables suspected of causing the problem from the regression. Further
specification analysis reveals that the GDP regressor induces higher multicollinearity among all explanatory variables than the stock price index. We therefore
drop the GDP proxy from the regression.
Figure 1 shows the annual bank failure rate—that is the number of bankfailures as a percentage of the total number of banks—from 1865 till 1940. The
bank failure rate is rather stable from 1865 till 1920, apart from occasional banking panics. Examples of the various banking panics in this period are the crises of
1873, 1884 and 1893 (see, for example, Sprague, 1910). The bank failure rate
starts to increase in the 1920s, cumulating in multiple bank failures in 1930-33.
Testing for structural change, we divide the overall sample into two sub-sample
periods: 1880-1919 and 1920-1936 (most of our macro-economic data series startin January 1877 and stop in January 1937). We conduct the likelihood ratio test
for structural change on the model before and after 1920. The likelihood ratio test
is based on the maximum likelihood method (Hamilton, 1994).

Figure 1: Annual bank failure rate, 1865-1940

To perform the test we define a dummy variable, which is zero for each month
until December 1919 and one for each month from January 1920 onwards. In the
restricted version of the model, each pre-1920 coefficient is imposed to be equal
to the corresponding post-1920 coefficient. The unrestricted version allows fordifferent behaviour in the post-1920 period. Column 1 of Table 2 reports the
restricted version and column 2 the unrestricted version of the model. It can beseen that for most variables, there is a significant change of slope indicating a
structural break.10 The likelihood ratio test statistic is computed as two times thedifference between the log-likelihood for the unrestricted version and the loglikelihood for the restricted version: 2[lnLUR – lnLR] ~c2(m). The relevant value inour case is: 285.36. As c2(7,.99) = 18.48, we can clearly reject the null hypothesisof no structural change.

10
The post-1920 slope coefficients for each variable are found by adding the overall coefficientand the dummy coefficient in the unrestricted model. For stock prices, for example, the coefficientin the 1920-1936 period is -0.307 (= -0.652 + 0.345).

Contagion risk in banking

97

Table 2: Test for structural change in number of bank failures, 1880-1936
Explanatory variables

Restricted

Unrestricted

CONSTANT

0.431***(26.92)-0.092(1.19)-1.509***(4.30)
0.022**(2.01)
0.291***(17.23)0.289***(17.05)0.162***(10.11)0.109**(2.48)-0.652*(1.94)-4.431***(3.30)
0.049(1.54)
0.145***(6.34)
0.175***(6.55)
0.077***(3.09)
0.266***(3.58)
0.345(0.99)
4.518***(3.22)-0.082**(2.20)
0.260***(7.78)
0.134***(3.62)
0.033(0.93)-1575.08-1432.40

DSPt
DPt
DRt
BFt–1
BFt–2
BFt–3
DUMMY * CONSTANT
DUMMY * DSPt
DUMMY * DPt
DUMMY * DRt
DUMMY * BFt–1
DUMMY * BFt–2
DUMMY * BFt–3
Log-likelihood

DUMMY = 0 for 1880-1919 and DUMMY = 1 for 1920-1936
Maximum likelihood estimates of coefficientst-statistics in parentheses (absolute values)***
significant at 1% level**significant at 5% level*significant at 10% level

The empirical results are presented in Table 3. Starting with the first period from1880 to 1919, all coefficients have the expected sign. The coefficient for stock
prices (DSP) is negative—a fall in stock prices generates more bank failures—andTable 3: Regressions of number of bank failures, 1880-1936
Explanatory variables

1880-1919

1920-1936

CONSTANT

0.109***(2.66)-0.645*(1.90)-4.467***(3.21)
0.049(1.58)
0.146***(7.50)
0.174***(7.14)
0.077***(3.09)

0.374***(6.54)-0.307***(3.24)
0.086(0.20)-0.033*(1.77)
0.404***(15.64)0.307***(11.40)0.113***(4.50)-658.110.17
16.09-774.50
0.59
13.63

DSPt
DPt
DRt
BFt–1
BFt–2
BFt–3
Log-likelihood
Pseudo-R2
Ljung-Box (13 lags)

Maximum likelihood estimates of coefficientst-statistics in parentheses (absolute values)***
significant at 1% level**significant at 5% level*significant at 10% level
The pseudo-R2 is computed as: 1 – lnL (W) / lnL (w), where lnL (W) is the value of the
log-likelihood function evaluated at the maximum likelihood estimates and lnL (w) is the
maximum value of the likelihood function under the hypothesis that the coefficients of
all regressors are zero.
The Ljung-Box statistic for serial correlation is calculated for 13 lags to allow forseasonal influences. The statistic is distributed as: c2(13) . The reported statistic in both
periods is below the critical value of 19.88 at the 90% level, so there is no evidence ofserial correlation.

is significant at the 10% level. The rate of inflation (DP), i.e. the first difference in prices, is negatively correlated (at the 1% level of significance) withthe bank failure rate. During periods of (unexpected) increasing inflation theburden of debt diminishes for a bank’s borrowers, while it rises in times of de-Contagion risk in banking

99

creasing inflation or deflation.11 The impact of interest rates (DR) is close to the
10% significance level (p = 0.11): an increase in short-term rates raises funding
costs for banks and causes a higher bank failure rate. We have run the regressions
with different lags for the macro-economic variables, but the contemporaneous
impact of the variables appears to be the strongest.
After controlling for macro-economic influences, we still find a strong formof dependency in the bank failure data. The coefficients of the lagged number ofbank failures (for 1 to 3 lags) are all significant at the 1% level. The persistence ofbank failures is thus up to three months. In our application of the autoregressive
Poisson model, g1 can be interpreted as the elasticity of the current number ofbank failures, BFt, with respect to the number of failures in the previous month,
BFt-1. As displayed in Table 3, the elasticities with respect to BFt-1, BFt-2, and BFt-3
are 0.146, 0.174, and 0.077, respectively. A 1% increase in the number of bankfailures in the previous month will lead to a 0.15% rise in the number of failuresthis month, ceteris paribus. The combined impact of a 1% increase in the numberof failures in each of the three preceding months is a 0.40% rise in the current
number of failures.
The results for the second period, 1920-1936, are also reported in Table 3.
Again, as expected, the number of bank failures is negatively correlated withstock prices at the 1% level of significance. The correlation coefficient of inflation has a positive sign, but is not significant (t = 0.20). The relationship betweenbank failures and interest rates is negative, which is surprising. Lower funding
costs (in the form of lower short term interest rates) generate more bank failures,and vice versa. However, the coefficient is only significant at the 10% level. The
overall assessment is that the explanatory power of macro-economic factors in thesecond period (1920-1936) is less strong than in the first period (1880-1919).
On the other hand, bank failures showed more dependency during the Great
Depression than in the first period. This indicates that the multiple bank failuresduring the 1930s are more driven by the spread of contagion than by macroeconomic factors. The elasticities of the current number of bank failures withrespect to the number of failures in each of the three preceding months is 0.404,
0.307, and 0.113, respectively (with all three coefficients significant at the 1%
level). The combined impact of a 1% increase in the number of failures in each of
the three preceding months is a 0.83% rise in the current number of failures. The
contagion effect in the second period (0.83%) is thus about twice as much as in
the first period (0.40%).
To shed light on the ‘too big to fail’ problem, we also examine in both periods
whether the lagged size of the failed banks (measured by the amount of outstanding deposits at the date of failure) in addition to the lagged number of failed
banks has any impact on the failure rate. Consistent with Davutyan (1989), we do

11This result is consistent with Goodhart (1995) and Heffernan (1995). Goodhart reports thatbank failures are high when asset (especially housing and property) prices have peaked, and havestarted to decline, and when the rate of inflation of current flow prices is starting to decline.

not find a significant result for the size variable.12 It should be noted, however,that there are missing values in the data on outstanding deposits which are estimated by data on paid-in capital of failing banks (see the Appendix). The usedproxy for size may therefore not be very reliable.
Reviewing our results, we conclude that macro-economic variables have
explanatory power in our model of bank failure. Furthermore, the results indicatethat bank failures are dependent, providing support for the lender of last resortrole performed by the central bank (e.g. Goodhart, 1987). The social cost of abank failure (i.e. the potential contagious effects) can be higher than the private
cost, justifying lender of last resort intervention. Put differently, it may be cheaperin certain cases to stop the initial failure upfront rather than to let the contagion
spread unfettered.
Linking our results to earlier studies, Grossman (1993) reports a smaller
contagion effect than we do. Grossman uses OLS regressions to assess the contagion effect under the National Banking System. Using quarterly data on bankfailures from 1875 to 1914, he finds a coefficient on lagged bank failures in the
failure equation of 0.260. The current number of failures will rise by 0.26% in
response to a 1% increase in the number of failures in the previous quarter. To
compare our results with those of Grossman, we have repeated our MLE estimations for the 1878-1914 period. The respective parameter estimates of g1, g2, andg3 are 0.142, 0.178, and 0.058. The current number of failures will rise by 0.38%
in response to a 1% increase in the number of failures in each of the three
preceding months. The higher frequency of our data (monthly) enables us to
identify more clearly the contagion effect.13 In addition, the use of OLS regressions to analyse discrete data may lead to mis-specifications (see above).
Another recent study (Hasan and Dwyer, 1994) examines bank contagion by
estimating probit equations for bank closings during the Free Banking Era from1837 till 1863. Analysing periods with large numbers of banks closing, Hasan and
Dwyer find evidence of bank contagion in the state of New York during the crisesof 1841-42 (significant at the 10% level) and 1854 (at the 5% level). But they do
not observe contagion effects during the 1854 crisis in Indiana and the 1861 crisisin Wisconsin. The Free Banking Period ended with the introduction of the National Currency Act in 1863 and thus precedes our period of analysis, for the
National Banking System was established in 1864.
Finally, a further test could be designed to investigate whether banking is‘special’ with regard to contagion risk. Contrasting data on business failures andbank failures, it can be analysed whether business failures show less dependency,
i.e. gamma in equation (9) is lower than for banking and, perhaps, close to zero.
This procedure would allow us to test whether the risk of contagion is more
12
Analysing a dataset of US bank failures from 1947 till 1986, Davutyan (1989) reports that thecoefficient for size (also measured by deposits in failed banks) is always insignificant.
13
It should be noted that we use several macro-economic variables to measure common factors
explaining bank failures, while Grossman uses only one variable (GDP). If anything, our estimatedcontagion effect should be smaller than that of Grossman, ceteris paribus.

Contagion risk in banking

101

prominent in banking than in other industries. Unfortunately, monthly data on
business failures are difficult to obtain for our reference period (1880-1936). Thisis an area for future research.

4

Concluding remarks

In this paper, we develop a model to measure possible contagion risk, which isdefined as the risk that an initial bank failure may spill over to the rest of thebanking industry and cause further bank failures. An autoregressive Poissonmodel is used to deal explicitly with the count nature of bank failure data. Although most earlier empirical studies (e.g. Aharony and Swary, 1983; Swary,1986) find little evidence of contagion, this evidence is not conclusive as some ofthese US studies examine possible contagious effects during periods in which theFederal Reserve acted as lender of last resort. To avoid this pitfall, we use datafrom the US National Banking System, which was founded some fifty years
before the establishment of the Federal Reserve.
The empirical results indicate that bank failures are dependent after controlling for macro-economic influences. These results are consistent with the existence of contagion risk in banking. An initial failure could generate further failures without intervention by the authorities. The results are compatible withrecent studies on contagion during the pre-Federal Reserve years (e.g. Grossman,
1993; Hasan and Dwyer, 1994). In spite of including more macro-economic
variables to control for common factors than Grossman, we find stronger contagion effects. First, our methodology of Poisson regressions is better suited to deal
with discrete data, such as bank failures, than Grossman’s OLS regressions.
Second, the use of monthly, instead of quarterly, data allows us to capture more
precisely possible effects of contagion, which is a fast-moving phenomenon.
Our empirical findings underpin the view that lender of last resort assistanceto individual banks may be justified in certain circumstances to prevent thepotential contagious effects of bank failures (e.g. Solow, 1982; Goodhart, 1987).
The central bank as lender of last resort should, however, not preserve all banksfrom failure, but should only prevent those failures that are expected to have asystemic impact. Furthermore, the banking sector that receives liquidity support
may need to be regulated and supervised to contain the moral hazard effects ofsuch lender of last resort intervention.

Data appendix
Bank failures: In our empirical study of contagion risk in Section 3, we examine
the period from 1865 till 1940. Data on bank failures are taken from various
issues of the Annual Report of the Comptroller of the Currency. The 1931 Annual
Report (Table 46, p.311) contains a survey of all failures from the inception of the
National Banking System up to 1930. The date of each failure and the amount ofoutstanding deposits at the date of failure are compiled. There were, however,about 10% missing values in the data on outstanding deposits. The missing values
are estimated by examining the paid-in capital of a failing bank. It should be
noted that paid-in capital is a poor proxy for the size of a bank, as banks that
expand do not always increase their capital base accordingly. The total number ofbanks and total amount of deposits are taken from Banking and Monetary Statistics, 1914-1941 (Board of Governors of the Federal Reserve System, 1943, Table4, p.20-1).
Macro-economic variables: The level of real output is proxied by pig iron production in the US. Monthly data on pig iron production (in thousands of grosstons) from Jan. 1877 till Jan. 1937 are published in The Movements of Interest
Rates, Bond Yields and Stock Prices in the United States since 1856 (Macaulay,
1938, Table 27, column 4, p.A252-70). Stock prices are measured by a value
weighted index of stock prices on the New York Stock Exchange with cash dividends reinvested. The data before 1871 are not meaningful and comprise mainly
railroad stocks. The monthly series on stock prices covers 1871-1937 and ispublished in Common Stock Indices, 1871-1937 (Cowles, 1938, Series C-1,
p.167-9). Prices are measured by Carl Snyder’s Index of General Price Level,from Jan. 1875 till Jan. 1937 (Macaulay, 1938, Table 27, column 5, p.A252-70).
The short-term interest rate is the commercial paper rate in New York City. From1865 till 1923, the CP rate is the ‘choice 60-90 day two name paper,’ and from
1924 till Jan 1937 the CP rate is the ‘4 to 6 month prime double and single namepaper’ (Macaulay, 1938, Table 10, column 3, p.A141-61).

References
Aharony, Joseph, and Itzhak Swary, 1983, “Contagion Effects of Bank Failures:
Evidence from Capital Markets,” Journal of Business 56, 305-322.
Bagehot, Walter, 1873, Lombard Street. Kegan, Paul & Co, London.
Benston, George, Robert Eisenbeis, Paul Horvitz, Edward Kane, and George
Kaufman, 1986, Perspectives on Safe & Sound Banking: Past, Present, andFuture. MIT Press, Cambridge, MA.
Bhattacharya, Sudipto, and Anjan Thakor, 1993, “Contemporary Banking
Theory,” Journal of Financial Intermediation 3, 2-50.
Board of Governors of the Federal Reserve System, 1943, Banking and Monetary
Statistics, 1914-1941, Washington, DC.
Calomiris, Charles, and Gary Gorton, 1991, “The Origins of Banking Panics:
Models, Facts, and Bank Regulation,” in Glenn Hubbard, ed., Financial
Markets and Financial Crises. University of Chicago Press, Chicago.
Chari, V., and Ravi Jagannathan, 1988, “Banking Panics, Information, and Rational

Contagion risk in banking

103

Expectations Equilibrium,” Journal of Finance 43, 749-761.
Comptroller of the Currency, Annual Report, various issues, Washington, DC.
Cowles, Alfred, 1938, Common Stock Indexes, 1871-1937, Principia Press,
Bloomington, Indiana.
Davutyan, Nurhan, 1989, “Bank Failures as Poisson Variates,” Economics Letters29, 333-338.
Diamond, Douglas, and Philip Dybvig, 1983, “Bank Runs, Deposit Insurance, andLiquidity,” Journal of Political Economy 91, 401-419.
Folkerts-Landau, David, and Peter Garber, 1992, “The ECB: A Bank or A
Monetary Policy Rule?,” in Matthew Canzoneri, Vittorio Grilli and Paul
Masson, eds., Establishing a Central Bank: Issues in Europe and Lessons
from the US. Cambridge University Press, Cambridge.
Goodhart, Charles, 1987, “Why Do Banks Need a Central Bank?,” Oxford
Economic Papers 39, 75-89.
Goodhart, Charles, 1995, “Price Stability and Financial Fragility,” in Kuniho
Sawamoto, Zenta Nakajima and Hiroo Taguchi, eds., Financial Stability in a
Changing Environment. Macmillan Press, London.
Goodhart, Charles, and Dirk Schoenmaker, 1995, “Should the Functions ofMonetary Policy and Banking Supervision be Separated?,” Oxford Economic
Papers 47, 539-560.
Gorton, Gary, 1985, “Bank Suspension of Convertibility,” Journal of Monetary
Economics 15, 177-193.
Greene, William, 1993, Econometric Analysis. Macmillan Publishing Company,
New York, Second Edition.
Grossman, Richard, 1993, “The Macroeconomic Consequences of Bank Failures
under the National Banking System,” Explorations in Economic History 30,
294-320.
Guttentag, Jack, and Richard Herring, 1987, “Emergency Liquidity Assistance forInternational Banks,” in Richard Portes and Alexander Swoboda, eds.,
Threats to International Financial Stability. Cambridge University Press,
Cambridge.
Hamilton, James, 1994, Time Series Analysis. Princeton University Press,
Princeton.
Hasan, Iftekhar, and Gerald Dwyer, 1994, “Bank Runs in the Free Banking
Period,” Journal of Money, Credit, and Banking 26, 271-288.
Heffernan, Shelagh, 1995, “An Econometric Model of Bank Failure,” Economic
and Financial Modelling 2, 49-83.

Kaufman, George, 1994, “Bank Contagion: A Review of the Theory andEvidence,” Journal of Financial Services Research 8, 123-150.
Lancaster, Tony, 1990, The Econometric Analysis of Transition Data. Cambridge
University Press, Cambridge.
Ljungqvist, Alexander, 1995, “When Do Firms Go Public? Poisson Evidence
from Germany,” mimeo, Merton College, Oxford.
Macaulay, Frederic, 1938, The Movements of Interest Rates, Bond Yields andStock Prices in the United States since 1856. NBER, New York.
Paroush, Jacob, 1988, “The Domino Effect and the Supervision of the Banking
System,” Journal of Finance 43, 1207-1218.
Peavy, John, and George Hempel, “The Penn Square Bank Failure,” Journal of
Banking and Finance 12, 141-150.
Postlewaite, Andrew, and Xavier Vives, 1987, “Banks Runs as an Equilibrium
Phenomenon,” Journal of Political Economy 95, 485-491.
Saunders, Anthony, 1987, “The Interbank Market, Contagion Effects andInternational Financial Crises,” in Richard Portes and Alexander Swoboda,
eds., Threats to International Financial Stability. Cambridge University Press,
Cambridge.
Schoenmaker, Dirk, 1995, “A Comparison of Alternative Interbank Settlement
Systems,” LSE Financial Markets Group Discussion Paper No 204, London.
Shephard, Neil, 1995, “Generalized Linear Autoregressions,” mimeo, Nuffield
College, Oxford.
Solow, Robert, 1982, “On the Lender of Last Resort,” in Charles Kindleberger
and Jean-Pierre Laffargue, eds., Financial Crises: Theory, History and Policy.
Cambridge University Press, Cambridge.
Sprague, O M W, 1910, History of Crises under National Banking System,
National Monetary Commission, Government Printing Office, Washington,
DC.
Summers, Lawrence, 1991, “Planning for the Next Financial Crisis,” in Martin
Feldstein, ed., The Risk of Economic Crisis. Chicago University Press,
Chicago.
Swary, Itzhak, 1986, “Stock Market Reaction to Regulatory Action in theContinental Illinois Crisis,” Journal of Business 59, 451-473.
Timberlake, Richard, 1993, Monetary Policy in the United States: An Intellectualand Institutional History. University of Chicago Press, Chicago.
US Department of the Treasury, 1991, Modernizing the Financial System:
Recommendations for Safer, More Competitive Banks. Washington, DC.