Understanding Liquidity and Credit Risks in the Financial Crisis

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scottish institute for research in economics

SIRE DISCUSSION PAPER

SIRE-DP-2011-26

Understanding Liquidity and Credit Risks in the Financial

Crisis

Deborah Gefang

University of Lancaster

Gary Koop

University of Strathclyde

Simon M. Potter

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Understanding Liquidity and Credit Risks in

the Financial Crisis

Deborah Gefang

Department of Economics University of Lancaster email: d.gefang@lancaster.ac.uk

Gary Koop

Department of Economics University of Strathclyde email: Gary.Koop@strath.ac.uk

Simon M. Potter

Research and Statistics Group Federal Reserve Bank of New York

email: simon.potter@ny.frb.org

October 2010

Abstract

This paper develops a structured dynamic factor model for the spreads between London Interbank O¤ered Rate (LIBOR) and overnight index swap (OIS) rates for a panel of banks. Our model involves latent factors which relect liquidity and credit risk. Our empirical results show that surges in the short term LIBOR-OIS spreads dur-ing the 2007-2009 …nancial crisis were largely driven by liquidity risk. However, credit risk played a more signi…cant role in the longer term (twelve-month) LIBOR-OIS spread. The liquidity risk factors are more volatile than the credit risk factor. Most of the familiar events in the …nancial crisis are linked more to movements in liquidity risk than credit risk.

The views expressed in this paper are those of the authors and do not necessarily re‡ect the views of the Federal Reserve Bank of New York or the Federal Reserve System. Gary Koop is a Fellow of the Rimini Center for Economic Analysis.

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1 Introduction

One of the most obvious signs of the …nancial crisis was a jump in the rates on uncollateralized loans across banks in Europe and the US. A standard measure of the rates on these uncollateralized loans is the daily submission of borrowing rates by major banks to the British Bankers Association used to form the standard set of London Interbank O¤er Rates (LIBOR). LIBOR rates are referenced in a large number of …nancial contracts in the global economy. There are three main reasons why LIBOR rates can change: i) central banks can change expectations of their policy rate, thereby repricing most short-term loans between banks; ii) banks can require a higher com-pensation for default risk on loans and iii) liquidity in the inter-bank loan market can change in ways unrelated to the open market operations of cen-tral banks. The last two of these reasons are called credit risk and liquidity risk. In order to focus on the roles of credit and liquidity risks, it is common to take out market expectations of future central bank policy rates by sub-tracting the overnight index swap (OIS)1 rate from the LIBOR rate, leading to the OIS spread. A number of studies emphasize that the LIBOR-OIS spread contains credit risk and liquidity risk premia (e.g., McAndrews et al, 2008, Michaud and Upper, 2008, Sengupta and Tam, 2008 and Hui et al, 2010).2 _{Detailed chronicles of the sharp increases in LIBOR-OIS spread}

along with the unfolding of …nancial turmoil can be found in Sengupta and Tam (2008), Brunnermeier (2009), Thornton (2009) and many other papers. The LIBOR-OIS spread has attracted a lot of attention in the literature on the …nancial crisis. For instance, McAndrews et al (2008), Taylor and Williams (2008a, 2008b) and Wu (2008) investigate whether the Federal Re-serve’s Term Auction Facility (TAF) has helped to reduce the liquidity risk component embodied in the LIBOR-OIS spread. However, the credit risk and liquidity risk components are rarely explicitly disentangled in the liter-ature. For example, researchers testing the TAF e¤ect on liquidity premium

1_{The OIS rate is a commonly-used measure of investor expectations of the e¤ective} federal funds rate and should not re‡ect credit or liquidity risk (see, e.g., Sengupta and Tam, 2008).

2_{According to the Basel Committee on Banking Supervision (2000, 2001), credit risk} mostly re‡ects the uncertainty that a bank borrower or counterparty will fail to repay the loan according to the agreed terms. However, credit risk also arises from various banking activities other than loans, including in the banking book and in the trading book, and both on and o¤ the balance sheet. Liquidity risk arises from banks’potential inability to fund increases in assets and meet obligations when they come due.

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typically look into the impact of TAF on the LIBOR-OIS spread after in-cluding an observed measure, such as the credit default swap (CDS) rate, to control for the credit premium. As noted in McAndrews et al (2008), this practice risks underestimating the TAF e¤ect if liquidity and credit risk pre-miums is positively correlated. Furthermore, most of the literature has used a single aggregate LIBOR-OIS time series, not exploiting the cross-bank, cross-term and cross-currency variation in LIBOR rates. In the present pa-per, we develop an econometric model which exploits these extra dimensions of variation in LIBOR and CDS rates and explicitly includes separate latent variables to model credit and liquidity risks. The main goal is to obtain a better understanding of the evolution of credit and liquidity risks during the …nancial crisis and investigate the importance of “Good” and “Bad” states for these risks.

Our data includes LIBOR-OIS spreads in the USD (US Dollar), EUR (Euro) and GBP (UK Pound Sterling) markets at di¤erent terms for a large panel of banks. Note that the British Bankers Association (BBA) published LIBOR rate (which is used in much of the literature), is calculated based on the trimmed average of the submission rates of all banks listed in the LIBOR panel. In contrast to this, we focus on individual banks’ quoted rates.3 Submission rates of a bank on a certain LIBOR currency panel are the lowest perceived rates for that bank to get unsecured interbank loans in that currency. Our assumption is that an individual bank’s submitted LIBOR rates re‡ect the bank’s own exposure to the latent credit and liquidity risk factors which operate in the market as a whole.

Our empirical results suggest that the widened short term (one-month and three-month) LIBOR-OIS spreads in the …nancial crisis were largely due to sharp increases in liquidity risks, but the surges in the longer term (twelve-month) spread were associated with both credit and liquidity risk. Furthermore, the latent liquidity risk factors are found to be much more variable than the credit risk factors and their variation is associated with familiar events in the …nancial crisis.

The rest of the paper is organized as following. Section 2 presents our modelling framework which section 3 discusses the econometrics. Our Bayesian econometric methods for estimating our models combine standard algorithms, such as Gibbs sampling methods for the dynamic factor and Markov

switch-3_{Information on banks’ LIBOR rates submission and how published BBA LIBOR is} calculated can be found in the BBA LIBOR Ltd’s website http://www.bbalibor.com

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ing models. Accordingly, we do not provide all econometric details in this paper. Instead a Technical Appendix, available on the website

http://personal.strath.ac.uk/gary.koop/, provides complete details of the posterior simulation algorithm along with details of the prior. This appendix also includes additional empirical results including a prior sensitivity analysis (see Section 4.4 for details). Section 4 describes the empirical results and section 5 concludes.

2 The Modelling Framework

LetSijkt be the spread between an individual bank’s (j = 1; :::; J will denote

banks) quoted LIBOR rate at term i (for i = 1; :::; I) and the OIS rate of the same term in currency k (for k = 1; :::; K) at date t (for t = 1; :::; T). This spread is assumed to depend on latent variables measuring liquidity and credit risks which re‡ect the state of the market as a whole. Liquidity risk is allowed to vary across currencies (since liquidity risk in di¤erent curren-cies will be in‡uenced by country-speci…c e¤ects and actions of local central banks), but credit risk does not. Thus, let Lkt denote liquidity risk in

cur-rencyk andCt denote the counterparty credit risk amongst banks. Lkt and

Ct are unobserved factors that we are seeking to estimate. To try and

disen-tangle liquidity and credit risk, we model the LIBOR-OIS spread jointly with a second dependent variable which is the CDS rate of an individual bank,

Djt. The main part of our model can be written as:

Sijkt = SijkLkt+ SijCt+ 0 ikXt+"Sijkt (1) Djt = CjCt+ 0 Zt+"Djt (2) where "S

ijkt IIDN(0; 2ijkS), "Djt IIDN(0; 2jD) and these errors are

in-dependent of each other. Note that (2) also allows for observed explanatory variables, Xt and Zt, which can in‡uenceSijkt and Djt, respectively.

Note that this model embodies many assumptions which allow us to iden-tify the liquidity and credit risks. Most importantly, the CDS rate of a bank is assumed not to depend directly on the liquidity risk in the market as a whole. Next, note the subscripts on the factor loadings. These vary across terms, to capture the e¤ect that, in the …nancial crisis, banks were especially reluctant to lend to each other at longer maturities. They also vary across

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banks. However, the credit risk factor has an impact of the LIBOR-OIS spread which is the same across currencies. Given our previous interpreta-tion of the credit risk factor as it impacts upon global banks, we argue the latter assumption is sensible.

Dynamic factor models such as ours require identi…cation restrictions. In this paper, we impose the following restrictions on factor loadings in order to uniquely identify and interpret the factors:

S ijk >0; S 11k = 1 C j >0; C 1 = 1 S ij >0; S 11 = 1

Note that the inequality restrictions above ensure that increases in Lkt and

Ct increase the LIBOR-OIS spread and the CDS rate (i.e. high values of

these latent variables indicate a bad liquidity/credit risk situations). Setting

S

11k = C

1 =

S

11 = 1 is a standard was of ensuring identi…cation in dynamic

factor models such as ours.4 _{We also require an identi…cation restriction on}

ik which is done by setting I1 = 0.

To complete the dynamic factor model, we have to specify equations describing the evolution of Lkt and Ct. We assume they evolve over time

according to Markov-switching AR(1) processes. The AR choice allows for us to model persistence in the states, but the Markov-switching allows for the abrupt switches which may have occurred in the …nancial crisis. Formally, we allow the latent factors, Ct and Lkt, to depend on Markov-switching states,

sC

t 2 fGC; BCg and sLt 2 fGL; BLg, where our G and B notation indicates

“good” and “bad” states for credit and liquidity risks:

Lkt = Lk0(s L t) + L k1(s L t)Lk;t 1+ kL(sLt)v L kt; (3) Ct= C0(sCt ) + C 1(sCt )Ct 1+ C(sCt )vCt ; (4) where vL

kt and vtC are independent standard Normal (i.e. independent over

time and of each other). To ensure stationarity of the dynamics of the linear factors, we impose the restrictions:

0 C₁(sC_t )<1

4_{Our identi…cation scheme normalizes the factor loading for the term} _i _{= 1}_{to unity.} We experimented with normalizing oni= 2or 3and results were virtually unaltered.

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0 L₁(sL_t)<1:

With regards to the Markov switching part of the model, note that there are four possible combinations of the good and bad credit and liquidity states. Thus, we can de…ne st as

st2 f(GL; GC);(BL; GC);(GL; BC);(BL; BC)g:

For instance,(BL; BC)is when the bad state rules in both credit and liquidity

markets which can be interpreted as the …nancial crisis state. The transition probabilities for the four-state Markov-switching process can be expressed through the 4 4matrix M:

8 > > < > > : m11 m12 m13 m14 m21 m22 m23 m24 m31 m32 m33 m34 m41 m42 m43 m44 9 > > = > > ; where 04M = 0 4, with 4 = [1;1;1;1] 0

ensures that probabilities sum to one. To ensure that G and B represent good and bad states we restrict the unconditional means ofCtand Lktto be lower in the good states than in the

bad: C 0(GC) 1 C₁(GC) < C 0(BC) 1 C₁(BC) L 0(GL) 1 L₁(GL) < L 0(BL) 1 L₁(BL) E[ C(GC)]< E[ C(BC)] E[ kL(GL)]< E[ kL(BL)]

That is, spreads and credit risk are higher and more volatile on average in the bad states.

Note that the liquidity factors across currencies are linked through this Markov state, since we assume the same st holds for all currencies. Thus,

it captures the inter-linked nature of liquidity. However, by having di¤erent coe¢ cients for di¤erent currencies (e.g. L_k₁(sL

t) and other parameters di¤er

across currencies), we allow for the impacts of di¤erences in monetary policy implementation and provision of central bank liquidity.

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3 Bayesian Estimation Methods

Bayesian estimation of our model can be done using a Markov Chain Monte Carlo (MCMC) algorithm which combines familiar algorithms for dynamic factor and Markov switching models. Accordingly, we do not provide details here, but provide only a sketch of how the algorithm proceeds. A Technical Appendix containing full details of our MCMC algorithm is available on http://personal.strath.ac.uk/gary.koop/. Our model depends on the latent credit and liquidity risk factors (LktandCt), and the Markov switching states

(sL

t and sCt ). Conditional on all these latent variables, the model de…ned by

(1), (2), (3) and (4) is a multivariate Normal linear regression model and our MCMC algorithm can draw on standard methods to draw the factor loadings and other parameters of the model. Conditional on draws of these parameters and draws of sL_t and sC_t , standard methods for drawing factors can be used to drawLkt andCt. We use the algorithm described in Kim and

Nelson (1999, chapter 8).

Conditional on draws ofLktandCtand the parameters in (1), (2), (3) and

(4), we use methods for drawing from Markov switching models described in Kim and Nelson (1999, chapter 9).

Our prior imposes the inequality and equality restrictions on the parame-ters described in the previous section, but is otherwise relatively noninforma-tive. Precise details are provided in the Technical Appendix to this paper. We have carried out an extensive prior sensitivity analysis which shows our results are robust, even to reasonably large changes in the prior. Results of this prior sensitivity analysis are also available in the Technical Appendix.

We use the posterior expectation of the Bayesian information criterion (BIC) and Akaike’s information criterion (AIC) to compare models.

4 Empirical Results

4.1 Data

Our daily data runs from 2 January 2007 to 15 December 2009 for the fol-lowing 12 banks (acronyms used in tables in parentheses) that listed in the LIBOR panels for USD, GBP and EUR: Barclays (barclays), Bank of Tokyo-Mitsubishi UFJ (btmufj), Citibank (citibank), Deutsche Bank (deutschebank), JPMorgan Chase (jpmc), HBOS (hbos), HSBC (hsbc), Lloyds TSB Bank

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(lloyds), Rabobank (rabobank), Royal Bank of Scotland (rboscotland), UBS (ubs), and WestLB (westlb). For each bank, we consider three term rates: one-month, three-month and twelve-month.

With regards to the explanatory variables in (1) and (2), for Zt we use

the implied volatility of overall stock prices (VIX). Djt is a measure of

bank-speci…c credit risk and inclusion of the VIX controls for volatility in the market as a whole (e.g. Hakkio and Keeton, 2009). In the equations with the LIBOR-OIS spreads as dependent variables, we set Xt to be the

commonly-used Term Auction Facility (TAF) dummy variable (which equals one for auction dates) when the Fed was injecting liquidity into the markets. We include Xt only in the equations involving the USD.

Both LIBOR-OIS spread and CDS rate are measured in basis points, while VIX is a percentage. To have all the data of approximately the same magnitude, we divide both CDS and VIX data by 100 and demean all the variables. Excluding common holidays and weekends,5 _{we have 727}

observa-tions for each variable.

4.2 Model Comparison

In addition to the very ‡exible unrestricted model described in Section 2, we estimate several restricted models which reduce the number of latent liquidity factors and/or remove the Markov switching part of the model. Altogether, we estimated eight models:

Model A (the Full Model): One latent credit risk factor, three latent liquidity risk factors in USD, GBP and EUR, respectively.

Model B: One latent credit risk factor, one common latent liquidity risk factor in USD and GBP, one unique latent liquidity risk factor in EUR.

Model C: One latent credit risk factor, one unique latent liquidity risk factor in USD, one common latent liquidity risk factor in GBP and EUR.

Model D: One latent credit risk factor, one common latent liquidity risk factor in USD, GBP and EUR.

5_{Four of the banks have a small number of missing observations. We simply omit these} observations for these banks.

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Model E: Same as Model A but without Markov switching. Model F: Same as Model B but without Markov switching. Model G: Same as Model C but without Markov switching. Model H: Same as Model D but without Markov switching.

Before presenting empirical results relating to …nancially-relevant features of interest, it is important to present statistical information on which models are supported by the data. Table 1 reports AIC and BIC for these eight models. The …rst point that stands out is that neither of the information criteria vary by a large amount across models. If we were to do a Bayesian model averaging exercise using the information criteria to construct model weights, then all of the models would receive appreciable probability. Sec-ondly, Models A (the full unrestricted model) and E (which is the same as Model A except removes the Markov switching component of the model) are the two models which perform best according to both information criteria. AIC chooses the Model A as being best whereas the more parsimonious BIC chooses Model E.

Both Model A and Model E include three liquidity risk factors, one for each of the currency markets (USD, EUR, GBP). Note that Table 1 indicates little support for the idea that it is statistically acceptable to reduce the number of liquidity risk factors. Models with a single liquidity risk factor perform worst in Table 1. Models with two liquidity risks factors, one for the USD and the second for the EUR and GBP, do not perform too poorly, but are clearly out-performed by models with three liquidity risk factors. Thus, statistically-speaking, we are …nding evidence to support our practice of using information in the three currency markets, as opposed to working solely with the USD market (as it typically done in the literature).

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Table 1: Model Comparison BIC AIC Model A -3.118 -4.217 Model B -2.739 -3.812 Model C -2.901 -3.974 Model D -2.474 -3.527 Model E -3.168 -4.191 Model F -2.815 -3.831 Model G -2.942 -3.959 Model H -2.508 -3.518

4.3 Financial Features of Interest

4.3.1 The Factors

In light of the …ndings of the preceding sub-section, in our discussion of …nancial features of interest, we will restrict our attention to the Full Model (Model A) and the Best Model (Model E) selected by BIC.

There are many …nancially-interesting features of interest in our modelling framework. In addition to the parameters in the model, we can present the latent factors, Lkt and Ct, which shed light on liquidity and credit risk. For

the Full Model, we can also present the probabilities of our four states (i.e. f(GL; GC);(BL; GC);(GL; BC);(BL; BC)g) relating to the Markov switching

aspect of the model

Furthermore, to investigate the relative roles of credit and liquidity risks in the …nancial crisis, we also present a variance decomposition based on the LIBOR-OIS equations. As explained in McAndrews et al (2008), credit and liquidity risks might be correlated. Additionally, the latent risk factors might not be orthogonal to Xk. The existence of possibly correlated components

renders variance decomposition a less straightforward exercise. From (1), we have the following:

var(Sijkt) = Sijk

2 var(Lkt) + Sij 2 var(Ct) + 0 ik 2

var(Xt) +var "Sijkt(5)

+2 S_ijk S_ijcov(Lkt; Ct) + 2 Sijk ijcov(Lkt; Xt) + 2 Sij

0

ikcov(Ct; Xt):

Following, e.g., Campbell and Ammer (1993), we report all the components on the right hand side of equation (5) as a proportion of var(Sijkt). This

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decomposition will allow us to see the the relevant importance of the di¤erent terms on the right hand side of (1).

The posterior mean of the latent credit and liquidity risks are presented in Figure 1 (for the Full Model) and Figure 2 (for the Best Model).6 _Figure

1 and Figure 2 exhibit almost identical patterns, indicating that the addition of Markov switching states adds little to our understanding of credit and liquidity risk. The three liquidity risk factors tend to move together (although there are some important di¤erences between them which we will note below) in a manner that is quite di¤erent from the credit risk factor.

First let us consider credit risk. This tended to grow gradually from August 2007 to a peak in December 2008, before gradually decreasing. By the end of 2009, credit risk was still at a level higher than in August 2007, the time when the …nancial turmoil …rst emerged. However, it is interesting to note the two clear dips in credit risk before December 2008. The …rst happened in December 2007. This was around the time when many banks were forced to take additional large write downs. Furthermore, the Fed’s term auction facility (TAF) …rst opened on December 17, 2007 (although the dip in Figures 1 and 2 began well before December 17). It seems that immediate response in the market to these events was that the perceived credit risk dropped, though this drop only lasted for the month of December 2007 and by January 2008 credit risk returned to where it had been before the dip began. The second smaller drop (or a plateau in the gradual increase) in credit risk happened in the summer of 2008. However, by late summer of 2008, the gradual increase in credit risk resumed (even before Lehman Brothers …nally went bankrupt in September 2008 and AIG disclosed that it faced serious liquidity shortage). It worth mentioning that credit risk remained at a high level throughout 2009, only slightly lower than its December 2008 peak. But it is worth stressing that the movements in the latent credit risk tend to be gradual and there is no huge spike in credit risk in autumn 2008.

In contrast to the credit risk, liquidity risks rose and dropped more abruptly. Figures 1 and 2 show that there were three major upsurges in liquidity risk, with by far the largest being at the time when the …nancial crisis was at its worst (i.e. in autumn 2008). The other two peaks in liquidity risk occurs in August 2007 and December 2007. August 2007 is often cited as the beginning of the …nancial crisis when Northern Rock’s problems became clear. Also in August, central banks implemented a wide range of

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dented liquidity easing approaches. For instance, European Central Bank (ECB) injected 95 billion Euros in overnight credit and the Fed injected $24 billion. Our methodology is …nding all liquidity risks to have dropped after an August 2007 peak, although this drop was much more rapid in the USD than the EUR market. The second peak, in early to mid-December 2007 is just before the time that the Fed’s term auction facility, TAF, began. It …rst opened on December 17, 2007 and our methodology is …nding liquidity risk to have fallen shortly after this.

During all of these upsurges, liquidity risks in the three di¤erent currencies behaved broadly similarly to one another. However, there are many minor di¤erences in their behavior (which presumably accounts for why models with fewer than three liquidity factors performed poorly in Table 1). One notable di¤erence is that the EUR liquidity risk exhibited a fourth upsurge in May and June 2008 which does not appear in the USD and GBP liquidity risks. Furthermore, as noted above, after the August 2007 increase in liquidity risk (common to all currencies), liquidity risk remained high in the EUR market well after it declined in the other currencies.

Another interesting di¤erence between currencies can be noted at the time that the credit crunch was at its worst. In October 2008, when all the three liquidity risks skyrocketed, liquidity risk in USD led those of the other currencies (and especially the GBP). Furthermore, the USD liquidity risk reached a higher level than the EUR liquidity risk. Interestingly, after reaching the peak, liquidity risks in USD and EUR dropped much more rapidly than those in the GBP currency market.

As noted above, the comparison between the evolution of credit risk and the liquidity risks is quite informative. Credit risk tends to move quite slowly, whereas we have seen liquidity risks swayed up and down more rapidly and in a manner consistent with the familiar events of the …nancial crisis. This suggests the importance of liquidity risk relative to credit risk in the evolution of the …nancial crisis. We will investigate this point in greater depth below.

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Figure 2

4.3.2 The Markov Switching States

Table 1 and the similarity of results in Figures 1 and 2 suggest that Markov switching is not an empirically vital aspect of our model. It seems that al-lowing for the gradual evolution of factors, as state equations such as (3) and (4) would allow for (even if we removed the Markov switching state variable), is adequate to capture the main empirical features of the data. Nevertheless, our Full Model which includes the Markov switching does receive apprecia-ble support in Taapprecia-ble 1, so it is worthwhile to brie‡y present empirical results relating to this aspect of the model.

Table 2 presents information about the posterior of the matrix of tran-sition probabilities. We …nd that 90% of the time the model is either in the (GL; GC) or (BL; BC) state. This accounts for why the probabilities for

remaining in the other states are quite low and imprecisely estimated (as revealed by large posterior standard deviations). The point estimates imply that the good credit and good liquidity state is quite persistent (i.e. given it is currently in this state, the probability of remaining in it is 95%). The bad liquidity and bad credit state is somewhat less persistent (i.e. the point

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estimate tells us the probability of remaining there is 74%). Table 2: Posterior Mean of Transition Probabilities

(Posterior standard deviations in parentheses)

(GL; GC) (BL; GC) (GL; BC) (BL; BC) (GL; GC) 0:945 (0:013) 0:275 (0:152) 0:178 (0:111) 0:102 (0:052) (BL; GC) 0:014 (0:009) 0:268 (0:151) 0:079 (0:079) 0:095 (0:070) (GL; BC) 0:016 (0:010) 0:088 (0:086) 0:635 (0:166) 0:066 (0:043) (BL; BC) 0:025 (0:011) 0:369 (0:182) 0:108 (0:094) 0:736 (0:081)

Figures 3 through 6 plot posterior means of the states themselves for the four states. Figures 3 and 6 are of most interest since, as just discussed, the states plotted in Figures 4 and 5 (i.e. (BL; GC) and (GL; BC)) rarely occur

in our data set. Given this fact, Figures 3 and 6 can be seen (approximately) to be mirror images of each other. Our main …ndings are that, at the heart of the credit crunch (from mid-September through mid-November 2008), the bad liquidity and bad credit state holds (with a few exceptions). The other periods when the bad liquidity and credit state often has high probability are similar to periods discussed previously and associated with the liquidity risk upsurges noted in Figures 1 and 2. These are late August through mid-September, 2007 and December 2007.

Thus, Table 2 and Figures 3 through 6 indicate that the major periods of the credit crisis were associated with increases in both credit and liquidity risk.

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Figure 6

4.3.3 Variance Decompositions

Our results thus far suggest that both liquidity and credit risk played a role in the …nancial crisis, although perhaps the former played a more important role than the latter. The variance decomposition of (5) can be used to formalize these …ndings. Noting that this decomposition relates tovar(Sijkt), it follows

that we will have a variance decomposition for each bank, term and currency (i.e. each i; j and k). These are presented in Tables 3, 4 and 5 for the Full Model. Results using the Best Model are similar and will not reported here for the sake of brevity.

Note …rst that results in these tables indicate our model is …tting well, with the error rarely accounting for more than 10% of the variation in spreads and typically accounting for much less (especially at the 3 month term). Fur-thermore, there are substantive di¤erences across banks, terms and currencies which provides us with strong evidence of the need for studies such as the present one which exploits these panel dimensions in the econometric model. The pattern in the tables is for liquidity risk to play the predominant role at short terms (1 month or 3 months). Particularly for one month

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LIBOR-OIS spreads, we have strong evidence that liquidity risk has been the major driver. For the GBP, the proportion of the variability in one month spreads attributable to the liquidity risk factor is typically very near to one (e.g. numbers like 0.99 or 0.98 are common) and never less than 0.80. For the USD and EUR markets the role of liquidity risk, although still high, is somewhat lower (e.g. numbers like 0.83 and 0.85 are common).

At the three month term, a similar pattern holds but the role of liquidity risk is somewhat less than at the one month term. But there are important di¤erences across currencies. That is, in the USD markets, liquidity risk typically accounts for 75 or 80% of the variability in spreads. But in the GBP market these numbers tend to be around 65% and in the EUR markets even lower than this.

At the 12 month term, however, a di¤erent picture emerges. At this longer term, credit risk assumes a much more important role. Although there is some variation across currencies and over banks, we are tending to …nd credit risk to account for about 50% of the variability in spreads in the USD markets (results for the GBP are slightly lower and EUR slightly higher than this). Liquidity risk is still important, tending to account for 20 or 30% of the variability in spreads in the USD markets (with results for the GBP being slightly higher and EUR slightly lower than this). However, credit risk is clearly the more important factor at this longer term in the vast majority of cases.

The role of the covariance between the liquidity and credit risk factors tends to be fairly small, accounting for typically around 10% of the variability in spreads. The other covariance terms in the variance decomposition play a negligible role. The dummy explanatory variable containing the TAF auction dates also explains little of the variation in spreads.

Note that the main dependent variable in our study isSijkt, as opposed to

most of the related literature which works withSt. The preceding discussion

of the variance decompositions (as well as our earlier empirical results) have shown that there are substantive di¤erences across currencies and terms, em-phasizing the importance of having ik subscripts on our dependent variable. But what about thej subscript? Are there bene…ts from working with a panel of banks, as we do, rather than just working with an aggregate LIBOR-OIS spread which averages over banks? With one exception, an examination of Tables 3 to 5 indicates that, although there is some variation across banks, it tends to be fairly minor. The one exception is HBOS, results for which are substantively di¤erent from all the other banks. The role of the credit

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risk factor for HBOS, at all terms and in all currencies, consistently is very low. Especially at longer horizons they are much lower than the other banks in the sample.

4.4 Robustness to Modelling Assumptions

In this paper, we have presented results using a very ‡exible model and compared the full model to several restricted versions of it. We have used a proper, but relatively noninformative prior and a sensible identi…cation scheme. We have investigated other modelling assumptions and have found our results to be very robust. Since these investigations do not substan-tively alter the empirical conclusions of this paper (and for the sake of brevity), we do not present these additional empirical results here. In-stead they are included in the Technical Appendix to this paper available at http://personal.strath.ac.uk/gary.koop/.

In particular, this appendix contains results from a prior sensitivity analy-sis and di¤erent identi…cation assumptions. Furthermore, we also estimated the model using data of the eight banks for which we do not have any miss-ing observations. These are barclays, citibank, deutschebank, jpmc, lloyds, rabobank, rboscotland, ubs. The latent risk factors derived from the eight bank panel are found to be very similar to what we get from the twelve bank panel.

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Table 3: Proportion of Variance of Spreads Attributable to Each Component (USD) var(L) var ( C) var(X) cov(L,C) cov(L,X) cov(C,X) var(e)

barclays1m 0.8918 0.0000 0.0008 0.0000 0.0020 0.0000 0.1054 barclays3m 0.8208 0.0720 0.0000 0.0966 -0.0002 0.0000 0.0108 barclays12m 0.2720 0.4217 0.0000 0.1345 0.0000 0.0000 0.1718 btmuf j1m 0.8755 0.0042 0.0011 0.0239 0.0023 0.0000 0.0930 btmuf j3m 0.7801 0.1027 0.0000 0.1125 -0.0003 0.0000 0.0050 btmuf j12m 0.2129 0.5345 0.0000 0.1340 0.0000 0.0000 0.1186 citibank1m 0.8963 0.0015 0.0013 0.0146 0.0025 0.0000 0.0836 citibank3m 0.7715 0.1101 0.0000 0.1158 -0.0003 0.0000 0.0029 citibank12m 0.1991 0.5585 0.0000 0.1325 0.0000 0.0000 0.1099 deutschebank1m 0.9094 0.0000 0.0014 0.0002 0.0026 0.0000 0.0864 deutschebank3m 0.7876 0.0982 0.0000 0.1106 -0.0003 0.0000 0.0039 deutschebank12m 0.2908 0.4672 0.0000 0.1465 0.0000 0.0000 0.0956 hbos1m 0.8682 0.0000 0.0009 0.0001 0.0020 0.0000 0.1287 hbos3m 0.8352 0.0038 0.0000 0.0221 -0.0002 0.0000 0.1391 hbos12m 0.6070 0.0496 0.0000 0.0688 0.0000 0.0000 0.2747 hsbc1m 0.9085 0.0001 0.0013 0.0028 0.0025 0.0000 0.0848 hsbc3m 0.7876 0.0992 0.0000 0.1111 -0.0003 0.0000 0.0024 hsbc12m 0.2660 0.4843 0.0000 0.1426 0.0000 0.0000 0.1070 jpmc1m 0.9009 0.0000 0.0015 0.0003 0.0027 0.0000 0.0946 jpmc3m 0.7692 0.1076 0.0000 0.1144 -0.0003 0.0000 0.0091 jpmc12m 0.1849 0.5484 0.0000 0.1265 0.0000 0.0000 0.1403 lloyds1m 0.9186 0.0000 0.0013 0.0005 0.0026 0.0000 0.0770 lloyds3m 0.7716 0.1103 0.0000 0.1160 -0.0003 0.0000 0.0024 lloyds12m 0.1903 0.5624 0.0000 0.1300 0.0000 0.0000 0.1173 rabobank1m 0.8999 0.0006 0.0015 0.0089 0.0027 0.0000 0.0864 rabobank3m 0.7460 0.1237 0.0000 0.1208 -0.0003 0.0000 0.0098 rabobank12m 0.2110 0.5224 0.0000 0.1319 0.0000 0.0000 0.1346 rboscotland1m 0.9163 0.0001 0.0012 0.0036 0.0024 0.0000 0.0764 rboscotland3m 0.7686 0.1099 0.0000 0.1155 -0.0003 0.0000 0.0062 rboscotland12m 0.1867 0.5159 0.0000 0.1233 0.0000 0.0000 0.1741 ubs1m 0.9148 0.0001 0.0012 0.0045 0.0024 0.0000 0.0769 ubs3m 0.7560 0.1217 0.0000 0.1206 -0.0003 0.0000 0.0021 ubs12m 0.1929 0.5353 0.0000 0.1276 0.0000 0.0000 0.1441 westlb1m 0.8708 0.0015 0.0013 0.0145 0.0024 0.0000 0.1095 westlb3m 0.7369 0.1307 0.0000 0.1234 -0.0003 0.0000 0.0093 westlb12m 0.1818 0.5561 0.0000 0.1263 0.0000 0.0000 0.1358

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Table 4: Proportion of Variance of Spreads Attributable to Each Component (GBP) var(L) var ( C) var(X) cov(L,C) cov(L,X) cov(C,X) var(e)

barclays1m 0.9964 0.0000 0.0000 0.0000 0.0000 0.0000 0.0036 barclays3m 0.6936 0.1117 0.0000 0.1493 0.0000 0.0000 0.0454 barclays12m 0.3393 0.4224 0.0000 0.2031 0.0000 0.0000 0.0352 btmuf j1m 0.9363 0.0081 0.0000 0.0466 0.0000 0.0000 0.0091 btmuf j3m 0.6632 0.1297 0.0000 0.1574 0.0000 0.0000 0.0497 btmuf j12m 0.4042 0.3832 0.0000 0.2112 0.0000 0.0000 0.0013 citibank1m 0.9627 0.0025 0.0000 0.0263 0.0000 0.0000 0.0084 citibank3m 0.6614 0.1204 0.0000 0.1515 0.0000 0.0000 0.0668 citibank12m 0.3860 0.4004 0.0000 0.2110 0.0000 0.0000 0.0027 deutschebank1m 0.9893 0.0000 0.0000 0.0004 0.0000 0.0000 0.0103 deutschebank3m 0.6811 0.1194 0.0000 0.1530 0.0000 0.0000 0.0464 deutschebank12m 0.4663 0.3144 0.0000 0.2055 0.0000 0.0000 0.0138 hbos1m 0.8170 0.0000 0.0000 0.0003 0.0000 0.0000 0.1828 hbos3m 0.6717 0.0046 0.0000 0.0295 0.0000 0.0000 0.2943 hbos12m 0.5569 0.0366 0.0000 0.0764 0.0000 0.0000 0.3301 hsbc1m 0.9705 0.0001 0.0000 0.0057 0.0000 0.0000 0.0236 hsbc3m 0.6560 0.1306 0.0000 0.1571 0.0000 0.0000 0.0563 hsbc12m 0.4222 0.3599 0.0000 0.2092 0.0000 0.0000 0.0087 jpmc1m 0.9895 0.0000 0.0000 0.0005 0.0000 0.0000 0.0099 jpmc3m 0.6919 0.1078 0.0000 0.1465 0.0000 0.0000 0.0538 jpmc12m 0.4497 0.3364 0.0000 0.2087 0.0000 0.0000 0.0052 lloyds1m 0.9978 0.0000 0.0000 0.0009 0.0000 0.0000 0.0012 lloyds3m 0.6791 0.1208 0.0000 0.1537 0.0000 0.0000 0.0465 lloyds12m 0.3867 0.4010 0.0000 0.2113 0.0000 0.0000 0.0010 rabobank1m 0.9702 0.0008 0.0000 0.0149 0.0000 0.0000 0.0141 rabobank3m 0.6683 0.1207 0.0000 0.1524 0.0000 0.0000 0.0587 rabobank12m 0.4470 0.3355 0.0000 0.2078 0.0000 0.0000 0.0097 rboscotland1m 0.9907 0.0002 0.0000 0.0067 0.0000 0.0000 0.0024 rboscotland3m 0.6555 0.1367 0.0000 0.1606 0.0000 0.0000 0.0471 rboscotland12m 0.3503 0.4340 0.0000 0.2093 0.0000 0.0000 0.0064 ubs1m 0.9861 0.0003 0.0000 0.0084 0.0000 0.0000 0.0052 ubs3m 0.6411 0.1411 0.0000 0.1614 0.0000 0.0000 0.0564 ubs12m 0.4013 0.3863 0.0000 0.2113 0.0000 0.0000 0.0011 westlb1m 0.9665 0.0025 0.0000 0.0265 0.0000 0.0000 0.0045 westlb3m 0.6500 0.1459 0.0000 0.1653 0.0000 0.0000 0.0388

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Table 5: Proportion of Variance of Spreads Attributable to Each Component (EUR) var(L) var ( C) var(X) cov(L,C) cov(L,X) cov(C,X) var(e)

barclays1m 0.8953 0.0000 0.0000 0.0000 0.0000 0.0000 0.1047 barclays3m 0.6521 0.1986 0.0000 0.1391 0.0000 0.0000 0.0101 barclays12m 0.2183 0.5940 0.0000 0.1392 0.0000 0.0000 0.0485 btmuf j1m 0.8509 0.0154 0.0000 0.0441 0.0000 0.0000 0.0896 btmuf j3m 0.6115 0.2391 0.0000 0.1478 0.0000 0.0000 0.0015 btmuf j12m 0.2308 0.5804 0.0000 0.1415 0.0000 0.0000 0.0474 citibank1m 0.8573 0.0045 0.0000 0.0240 0.0000 0.0000 0.1142 citibank3m 0.6327 0.2208 0.0000 0.1445 0.0000 0.0000 0.0020 citibank12m 0.2288 0.5784 0.0000 0.1406 0.0000 0.0000 0.0522 deutschebank1m 0.8600 0.0000 0.0000 0.0004 0.0000 0.0000 0.1396 deutschebank3m 0.5914 0.2377 0.0000 0.1449 0.0000 0.0000 0.0260 deutschebank12m 0.2786 0.5370 0.0000 0.1495 0.0000 0.0000 0.0348 hbos1m 0.8276 0.0000 0.0000 0.0003 0.0000 0.0000 0.1721 hbos3m 0.7510 0.0077 0.0000 0.0290 0.0000 0.0000 0.2123 hbos12m 0.5575 0.0610 0.0000 0.0711 0.0000 0.0000 0.3104 hsbc1m 0.8616 0.0003 0.0000 0.0051 0.0000 0.0000 0.1330 hsbc3m 0.6458 0.2079 0.0000 0.1416 0.0000 0.0000 0.0047 hsbc12m 0.2908 0.5170 0.0000 0.1499 0.0000 0.0000 0.0423 jpmc1m 0.9018 0.0000 0.0000 0.0005 0.0000 0.0000 0.0977 jpmc3m 0.6562 0.2013 0.0000 0.1405 0.0000 0.0000 0.0020 jpmc12m 0.2787 0.5280 0.0000 0.1483 0.0000 0.0000 0.0450 lloyds1m 0.8918 0.0000 0.0000 0.0009 0.0000 0.0000 0.1073 lloyds3m 0.6309 0.2220 0.0000 0.1447 0.0000 0.0000 0.0024 lloyds12m 0.2037 0.6015 0.0000 0.1353 0.0000 0.0000 0.0595 rabobank1m 0.8816 0.0015 0.0000 0.0137 0.0000 0.0000 0.1032 rabobank3m 0.6043 0.2301 0.0000 0.1442 0.0000 0.0000 0.0214 rabobank12m 0.2736 0.5323 0.0000 0.1475 0.0000 0.0000 0.0467 rboscotland1m 0.8773 0.0003 0.0000 0.0060 0.0000 0.0000 0.1164 rboscotland3m 0.6107 0.2350 0.0000 0.1465 0.0000 0.0000 0.0078 rboscotland12m 0.1847 0.6213 0.0000 0.1309 0.0000 0.0000 0.0630 ubs1m 0.8353 0.0005 0.0000 0.0078 0.0000 0.0000 0.1564 ubs3m 0.5832 0.2578 0.0000 0.1499 0.0000 0.0000 0.0091 ubs12m 0.2277 0.5860 0.0000 0.1412 0.0000 0.0000 0.0451 westlb1m 0.8539 0.0043 0.0000 0.0234 0.0000 0.0000 0.1184 westlb3m 0.5929 0.2518 0.0000 0.1494 0.0000 0.0000 0.0059 westlb12m 0.2337 0.5835 0.0000 0.1427 0.0000 0.0000 0.0400

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5 Conclusion

In this paper, we have motivated and developed a statistical model which uses a panel of LIBOR-OIS spreads and bank CDS rates to disentangle liq-uidity and credit risk. The panel dimensions of the spreads include variation across banks, currencies and terms. The existing literature almost always ignores these panel dimensions and simply works with one average LIBOR-OIS spread. From a statistical point of view, our empirical results show that there are bene…ts from exploiting these panel dimensions in terms of increasing our understanding of liquidity and credit risk.

Our empirical results indicate that liquidity and credit risks both played an important role in the …nancial crisis. However, especially at the 1 month and 3 month terms, the role of liquidity risks is much more important. At the 12 month term, credit and liquidity risks both play an important role. The latent liquidity risk factors are much more variable than the credit risk factor and their variation is associated with familiar events in the …nancial crisis.

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References

Basel Committee on Banking Supervision, 2000, Sound practices for man-aging liquidity in banking organisations, consultative document available at http://www.bis.org/publ/bcbs69.pdf.

Basel Committee on Banking Supervision, 2001, The new Basel capital accord, consultative document available at http://www.bis.org/publ/bcbsca03.pdf.

Brunnermeier, M., 2009. Deciphering the liquidity and credit crunch 2007–2008. Journal of Economic Perspectives. 23, 77-100.

Campbell, J., Ammer, J., 1993. What moves the stock and bond markets? A variance decomposition for long-term asset returns. Journal of Finance. 48, 3-37.

Hakkio, C., Keeton, W., 2009. Financial stress: What is it, how can it be measured, and why does it matter? Federal Reserve Bank of Kansas City Economic Review. Third Quarter, 5-50.

Hui, C., Chang, T., Lo, C., 2010. Using interest rate derivative prices to estimate LIBOR-OIS spread dynamics and systemic funding liquidity shock probabilities. Hong Kong Monetary Authority, working paper 04/2010.

Kim, C. and Nelson, C., 1999. State Space Models with Regime Switch-ing. MIT Press, Cambridge, MA.

McAndrews, J., Sarkar, A., Wang, Z., 2008. The e¤ect of the term auction facility on the London inter-bank o¤ered rate. Federal Reserve Bank of New York Sta¤ Report No. 335.

Michaud, F., Upper, C., 2008. What drives interest rates? Evidence from the Libor panel. BIS Quarterly Review, March, 47-58.

Sengupta, R., Tam Y., 2008. The LIBOR-OIS spread as a summary indicator. Monetary Trends, Federal Reserve Bank of St. Louis, Nov. 25.

Taylor, J., and Williams, J., 2008a. A black swan in the money market. Working Paper, Stanford University and the Federal Reserve Bank of San Francisco 2008-4.

Taylor, J., and Williams, J., 2008b. Further results on a black swan in the money market, Stanford Institute for Economic Policy Research Working Paper 07-046.

Thornton, D., 2009. What the Libor-OIS spread says, Economic Syn-opses, Federal Reserve Bank of St. Louis.

Wu, T., 2008. On the e¤ectiveness of the Federal Reserve’s new liquidity facilities, Federal Reserve Bank of Dallas, Working Paper 0808.

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Technical Appendix to: Understanding Liquidity

and Credit Risks in the Financial Crisis

Deborah Gefang Department of Economics University of Lancaster email: d.gefang@lancaster.ac.uk Gary Koop Department of Economics University of Strathclyde email: Gary.Koop@strath.ac.uk Simon M. Potter

Research and Statistics Group Federal Reserve Bank of New York

email: simon.potter@ny.frb.org

October 2010

1 The Gibbs Sampler

This section describes the detailed Gibbs sampler for Bayesian estimation. For all notation, please see the original paper.

1.1 Generating Lkt and Ct

To generateLktand Ct, we …rst transform the dynamic factor model into a

state space form. Let S_ijkt = Sijkt

0

ikXt; and Djt = Djt 0Zt. The measurement

The views expressed in this paper are those of the authors and do not necessarily re‡ect the views of the Federal Reserve Bank of New York or the Federal Reserve System. Gary Koop is a Fellow of the Rimini Center for Economic Analysis.

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equation of the model can be written as: 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ D₁_t D₂_t ::: D_{J t} S₁₁₁_t S₂₁₁_t ::: S_I₁₁_t S₁₂₁_t S₂₂₁_t ::: S_I₂₁_t ::: S_IJ₁_t S₁₁₂_t S₂₁₂_t ::: S_IJ₂_t ::: S₁₁_Kt S₂₁_Kt ::: S_{IJ Kt} 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A = 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ C 1 0 0 ::: 0 C 2 0 0 ::: 0 ::: ::: :::: C J 0 0 ::: 0 S 11 S111 0 ::: 0 S 21 S111 0 ::: 0 ::: ::: :::: S I1 SI11 0 ::: 0 S 12 S121 0 ::: 0 S 22 S221 0 ::: 0 ::: ::: :::: S I2 SI21 0 ::: 0 ::: ::: :::: S IJ SIJ1 0 ::: 0 S 11 0 S112 ::: 0 S 21 0 S212 ::: 0 ::: ::: :::: S IJ 0 SIJ2 ::: 0 ::: ::: :::: S 11 0 0 ::: S11K S 21 0 0 ::: S21K ::: ::: :::: S IJ 0 0 ::: SIJ K 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A 0 B B @ Ct L1t :: LKt 1 C C A+ 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ "D₁_t "D₂_t ::: "D J t "S 111t "S 211t ::: "S_I₁₁_t "S₁₂₁_t "S₂₂₁_t ::: "S_I₂₁_t ::: "S_IJ₁_t "S₁₁₂_t "S₂₁₂_t ::: "S_IJ₂_t ::: "S₁₁_Kt "S₂₁_Kt ::: "S_{IJ Kt} 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A (1 ) (yt=H t+et) (2)

where et i:i:d:N(0; R), the size of yt is J +IJ K by 1, the size of H is

J+IJ K by 1 +K, and the size of _t is 1 +K by 1. The transition equation can be written as

0 B B @ Ct L1t ::: LKt 1 C C A= 0 B B @ C 0(sCt) L 10(sLt) ::: L K0(sLt) 1 C C A+ 0 B B @ C 1(sCt) 0 ::: 0 0 L₁₁(sL_t) ::: 0 ::: 0 0 ::: L_K₁(sL_t) 1 C C A 0 B B @ Ct 1 L1;t 1 ::: LK;t 1 1 C C A+ 0 B B @ C(sCt) Ct 1L(sLt) L1t ::: KL(sLt) LKt 1 C C A (3 ) ( _t= _t+F_{t t} ₁+ t) (4)

where t i:i:d:N(0; Qt), the size of t is1 +K by1, the size of t is1 +K

by1, and the size ofFt is1 +K by1 +K.

Let _eyt = (y1; y2; :::; yt)0. Following Kim and Nelson (1999, Ch. 8), we

can draw the latent risk factors in the following two steps.

First run Kalman …lter to calculate _t_j_t=E( _t_jyet)and Ptjt=Cov( tjyet)

fort= 1;2; :::; T : tjt 1 = t+Ft t 1 P_t_j_t ₁=FtPtjt 1F 0 t +Qt tjt= tjt 1+Ptjt 1H 0 (HPtjt 1H 0 +R) 1(yt H tjt 1) Ptjt=Ptjt 1 Ptjt 1H 0 (HPtjt 1H 0 +R) 1HPtjt 1

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Next, we draw _T based on the last iteration of the Kalman …lter:

TjeyT N( TjT; PTjT)

Let _t₊₁= (Ct+1; L1;t+1; :::; LK;t+1)0. We have

t+1 = t+1+Ft+1 t+ t+1

De…ningQt+1 the variance of t+1, fort=T 1; T 2; :::;1, we can derive

tjeyT backward from tjyet; t+1 N( tjt; _t₊₁; Ptjt; t+1) where tjt; t+1 = tjt+PtjtF 0 t+1fFt+1PtjtF 0 t+1+Qt+1g 1( t+1 t+1 Ft+1 tjt) Ptjt; t+1 =Ptjt PtjtF 0 t+1fFt+1PtjtF 0 t+1+Qt+1g 1Ft+1Ptjt

There are many observations missing, especially in the CDS data. To treat the missing elements inyt, following Durbin and Koopman (2008, Ch.

4), we replaceyt,H andR byyt,Ht and Rt, where yt =Wtyt,Ht =WtH

andR_t =WtRW

0

t, withWtbeing the known matrix whose rows are a subset

of rows of the identity matrixIJ+IJ K that correspond to the observed values

in yt. Thus, we adjust the dimensions of H and R to account for the fact

that there is no new information added to the Kalman …lter when data are missing.1

1.2 Draw Parameters for the Measurement Equations

Generating 2

ijkS, Conditional on Sijkt, Sijk, Sij, ik, Lkt and

Ct.

Prior

hijkS G(s_ijkS2 ; ijkS)

where hijkS = ijkS2 . G denotes a Gamma distribution. Details of the

distribution can be found in Koop (2003, p.326).

1_{Since generated missing observations contain no new data information aside from the}

parameters, we don’t take them as observed in the Kalman …lter. In practice, we …nd that otherwise the Gibbs sampler hardly converges. Using balanced panel data where rows with missing observations simply deleted, we …nd very similar results for latent risk factors.

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Posterior

hijkSjSijkt; Sijk; Sij; ik; Lkt; Ct G(s_ijkS2 ; ijkS)

where

ijkS=T TSmijk+ ijkS

and

s2_ijkS =

PT

t=1(eSijkt)2+ ijkSs2ijkS ijkS

whereeS_ijkt=Sijkt SijkLkt SijCt

0

ikXt ifSijkt is observed,TSmijk is

the number of missing observations inSijk.

Generating S_ijk, Conditional on Sijkt, Sij, ik, hijkS, Lkt and

Ct.

LetS_ijkt[ =Sijkt

0

ikXt SijCtifSijktis observed. We transform each

measurement equation into the following form:

S_ijkt[ = S_ijkLkt+"Sijkt (5)

Prior S ijk N(b S ijk; V S ijk) with S ijk>0 Posterior S

ijkjSijkt; Sij; ik; hijkS; Lkt; Ct N(b S ijk; V S ijk)with S ijk >0 where V S ijk =fV 1 S ijk +hijkSL 0 kLkg 1 and b S ijk =V S ijkfV 1 S ijk b S ijk +hijkSL 0 kSijk[ g

whereS_ijk[ = [S_ijk[ ₁; S_ijk[ ₂; :::; S_ijkT[ ]0 and Lk= [Lk;1; Lk;2; :::; LkT]

0

.

Generating S_ij, Conditional on Sijkt, Sijk, ik, hijkS, Lkt and

Ct.

LetS_ijkt\ =Sijkt SijkLkt

0

ikXtifSijktis observed. The corresponding

measurement equation can be rewritten as

S_ijkt\ = S_ijCt+"Sijkt (6) Prior S ij N(b S ij; V Sij) with S ij >0

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Posterior

S

ijjSijkt; Sijk; ik; hijkS; Lkt; Ct N(b S

ij; V Sij) with S ij >0 where V S ij =fV 1 S ij + k_X=K k=1 hijkC 0 C_g 1 and b S ij =V SijfV 1 S ij b S ij+ k_X=K k=1 hijkC 0 S_ijk\ _g

whereS_ijk\ = [S_ijk\ ₁; S_ijk\ ₂; :::; S_ijkT\ ]0 and C= [C1; C2; :::; CT]

0

.

Generating _ik, Conditional on Sijkt, Sijk, Sij, hijkS, Lkt and

Ct.

LetS_ijkt] =Sijkt SijkLkt SijCtifSijktis observed. The corresponding

measurement equation can be transformed into:

S_ijkt] = 0_ikXt+"Sijkt (7)

Prior

ik N(b ik; V ik)

Posterior

ikjSijkt; Sijk; Sij; hijkS; Lkt; Ct N(b ik; V ik)

where V _ik =_fV 1 ik+ j=J X j=1 hijkX 0 X_g 1 and b _ik =V _ik_fV 1 ikb ik + j=J X j=1 hijkX 0 S_ijk] _g

whereS_ijk] = [S_ijk] ₁; S_ijk] ₂; :::; S_ijkT] ]0 and X= [X1; X2; :::; Xt]0.

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Prior hjD G(sjD2; jD) wherehjD = jD2 Posterior hjDjDjt; Cj ; ; Ct G(sjD2; jD) where jD =T TDmjt + jD and s_jD2 = PT t=1(eDjt)2+ jDs2jD jD where T_Dm

jt is the total number of missing observations inDj, e

D jt =Djt C j Ct 0 Zt ifDjt is observed.

Generating C_j , Conditional on Djt, , hjD and Ct.

Let D[_jt = Djt 0Zt if Djt is observed. We rewrite the measurement

equation for the credit default variable as:

D[_jt = C_j Ct+"Djt (8) Prior C j N(b C j; V Cj) with C j >0 Posterior C j jDjt; hjD; ; Ct N(b C j; V Cj) with C j >0 where V C j =fV 1 C j +hjDC 0 C_g 1 and b C j =V CjfV 1 C j b C j +hjDC 0 D[_j_g whereD[_j = [D[_j₁; D[_j₂; :::; D_jT[ ]0 and C= [C1; C2; :::; CT]0.

Generating , Conditional on Djt, Cj, hjD and Ct.

LetD\_jt =Djt Cj Ct ifDjt is observed. We rewrite the corresponding

measurement equation as:

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Prior N(b ; V ) Posterior jDjt; hjD; Cj ; Ct N(b ; V ) where V =_fV 1+ j=J X j=1 hjDZ 0 Z_g 1 and b =V _fV 1b + j=J X j=1 hjDZ 0 D\_j_g whereD\_j = [D\_j₁; D\_j₂:::; D_jT\ ]0 andZ = [Z1; Z2; :::; ZT] 0 .

1.3 Draw Parameters for the Latent Risk Equations

Without loss of generality, we setsL_t = 0 ifsL_t is in ‘good’state for liquidity risk, and sL_t = 1 if sL_t is in ‘bad’ state for liquidity risk. Likewise, we set

sC_t = 0 if sC_t is in ‘good’ state for credit risk, and sC_t = 1 if sC_t is in ‘bad’ state for credit risk.

For notation convenience, we rewrite the state equations as follows:

Lkt= Lk00+ Lk01sLt + ( Lk10+ kL11sLt)Lk;t 1+ kL(sLt)vktL (10) Ct= C00+ C01sCt + ( C10+ C11sCt )Ct 1+ C(sCt )vtC (11) where L k00= Lk0(GL), Lk00+ Lk01= Lk0(BL) L k10= Lk1(GL), Lk10+ Lk11= Lk1(BL) and C 00= C0(GC), 00C + C01= C0(BC) C 10= C1(GC), 10C + C11= C1(BC)

Considering that latent risks are more volatile on average in the bad states, we de…ne 2 kL(BL) = 2kL(GL)(1 +hkLB) 2 C(BC) = 2C(GC)(1 +hCB) wherehkLB >0and hCB >0.

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Generating L_k₀₀, L_k₀₁, _kL₁₀, L_k₁₁, Conditional on Lkt and kL(sLt).

Dividing both sides of the state equation by kL(sLt), we have

Ly_kt= L_k₀₀xy₁_t+ L_k₀₁xyy₁_t+ L_k₁₀xy₂_t+ L_k₁₁xyy₂_t+vL_kt (12) where Ly_kt= Lkt kL(sLt) xy₁_t= 1 kL(sLt) xyy₁_t= s L t kL(sLt) xy₂_t= Lk;t 1 kL(sLt) xyy₂_t= Lk;t 1s L t kL(sLt)

In matrix notation, equation (12) can be written as

Ly_k=X_kLy b _kL+VkL VkL N(0; IT) (13)

where

X_kLty = [xy₁_t; x₁yy_t; xy₂_t; xyy₁_t]

and

b _kL = [ L_k₀₀; L_k₀₁; L_k₁₀; L_k₁₁]0

We assume Normal prior for b _kL:

Prior

b _kL N(b

kL; V kL)

with the elements ofb _kL subject to the following restrictions:

L k102(0;1) Lk10+ Lk112(0;1) L k00 1 L_k₁₀ < L k00+ Lk01 1 L_k₁₀ L_k₁₁

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Posterior b _kL N(b _kL; V _kL)R(b _kL) where V _kL =_fV 1 kL+ (X y kL) 0 X_kLy _g 1 and b _kL =V _kL_fV 1 kLb 1 kL+ (X y kL) 0 Ly_k_g Generating 2_kL(GL), Conditional on Lkt, Lk00, Lk01, Lk10, Lk11 and hkLB.

To generate 2_kL(GL), we divide both sides of equation (10) by

p 1 +hkLBsLt Lz_kt= L_k₀₀xz₁_t+ L_k₀₁xzz₁_t+ L_k₁₀xz₂_t+ L_k₁₁xzz₂_t+ kL(GL)vktL (14) where Lz_kt= p Lkt 1 +hkLBsLt xz₁_t= p 1 1 +hkLBsLt xzz₁_t= s L t p 1 +hkLBsLt xz₂_t= _p Lk;t 1 1 +hkLBsLt xzz₂_t= Lk;t 1s L t p 1 +hkLBsLt Prior hkL G(s_kL2; kL) Posterior hkL G(s_kL2; kL) where kL=T+ kL and s2_kL= kLs 2 kL+ Pt=T t=1(Lzkt L k00xz1t L k01xzz1t L k10xz2t L k11xzz2t)2 kL

GivenhkL, we can derive kL(GL) by kL(GL) = p_h1

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GeneratinghkLB, Conditional onLkt, Lk00, Lk01, Lk10, Lk11 and

kL(GL).

To draw hkLB, we divide both sides of equation (10) by kL(GL):

Lx_kt= L_k₀₀x₁x_t+ L_k₀₁xxx₁_t+ L_k₁₀xx₂_t+ _kL₁₁xxx₂_t+vL_kt q 1 +hkLBsLt (15) where Lx_kt= Lkt kL(GL) xx₁_t= 1 kL(GL) xxx₁_t = s L t kL(GL) xx₂_t= Lk;t 1 kL(GL) xxx₂_t = Lk;t 1s L t kL(GL) Lethx_kLB = ₁₊_h1

kLB, we set Gamma prior for h

x

kLB:

Prior

hx_kLB G(s_kL2_x; _kLx)

subject to the restriction that hx_kLB <1.

Posterior

hx_kLB G(s_kL2_x; _kLx)

subject to the restriction that hx_kLB <1. where kLx = _kLx+Tx s2_kL_x = kLxs 2 kLx+ PTx (Lx_kt L_k₀₀xx₁_t L_k₀₁xxx₁_t L_k₁₀xx₂_t L_k₁₁xxx₂_t)2 kLx

Note that Tx =_ft: sL_t = 1_g, also hx_kLB depends only on the values of Lx_kt

for which sL_t = 1.

Once hx_kLB is drawn from the above posterior, we can calculatehkLB by

hkLB = _hx1

kLB 1.

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Dividing both sides of equation (11) by C(sCt ), we have C_ty= C₀₀z₁y_t+ C₀₁z₁yy_t+ C₁₀z₂y_t+ C₁₁z₂yy_t+vC_t (16) where C_ty= Ct C(sCt ) z₁y_t= 1 C(sCt ) zyy₁_t = s C t C(sCt) z₂y_t= Ct 1 C(sCt ) zyy₂_t = Ct 1s C t C(sCt)

In matrix notation, equation (12) can be written as

Cy=Z_Cyb _C+VC VC N(0; IT) (17)

where

Z_Cty = [z₁y_t; z₁yy_t; z₂y_t; zyy₁_t]

and

b _C = [ C₀₀; C₀₁; C₁₀; C₁₁]0

We assume Normal prior for b _C:

Prior

b _C N(b

C; V C)

with the elements ofb _C subject to the following restrictions:

C 102(0;1) C10+ C112(0;1) C 00 1 C₁₀ < C 00+ C01 1 C₁₀ C₁₁

To save space, we useR(b _C) to denote these additional restrictions.

Posterior b _C N(b _C; V _C)R(b _C) where V _C =_fV 1 C + (Z y C) 0 Z_Cy_g 1 and b _C =V _C_fV 1 Cb 1 C+ (Z y C) 0 Cy_g

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Generating 2_C(GC), Conditional on Ct, C00, C01, C10, C11 and

hCB.

To generate 2_C(GC), we divide both sides of equation (11) by

p 1 +hCBsCt C_tz= C₀₀z₁z_t+ C₀₁z₁zz_t+ C₁₀z₂z_t+ C₁₁zzz₂_t+ C(GC)vtC (18) where C_tz= p Ct 1 +hCBsCt z₁z_t= p 1 1 +hCBsCt z₁zz_t = s C t p 1 +hCBsCt z₂z_t= p Ct 1 1 +hCBsCt z₂zz_t = Ct 1s C t p 1 +hCBsCt Prior hC G(sC2; C)where hC = C2(GC) Posterior hC G(s_C2; C) where C =T+ C and s2_C = Cs 2 C+ Pt=T t=1(Ctz C00z1zt C01z1zzt C10zz2t C11z2zzt)2 C

GivenhC, we can derive C(GC) by C(GC) = p1_h

C.

Generating hCB, Conditional on Ct, C00, C01, C10, C11 and

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To draw hCB, we divide both sides of equation (11) by C(GC): C_tx= C₀₀z₁x_t+ C₀₁z₁xx_t+ C₁₀z₂x_t+ ₁₁Cz₂xx_t+v_tC q 1 +hCBsCt (19) where C_tx= Ct C(GC) zx₁_t= 1 C(GC) z₁xx_t = s C t C(GC) zx₂_t= Ct 1 C(GC) z₂xx_t = Ct 1s C t C(GC) Lethx_CB = ₁₊_h1

CB, we set Gamma prior forh

x

CB:

Prior

hx_CB G(s_C2_x; _Cx)

subject to the restriction that hx_CB <1.

Posterior

hx_CB G(s_C2_x; _Cx)

subject to the restriction that hx_CB <1. where Cx = Cx+T_Cx s2_Cx = Cxs2_Cx+ PT_Cx₍_Cx t C00z1xt C 01z1xxt C 10z2xt C 11z2xxt)2 Cx Note thatT_Cx =_ft:sC

t = 1g, andhxCB depends only on the values of Ctx for

which sC_t = 1.

Once hx_CB is drawn from the above posterior,hCB can be calculated by

hCB = _hx1

CB 1.

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1.4 Generating sL

t, sCt and M

Generate sL_t and sC_t , Conditional on M, Lkt, Ct and the other

model parameters

We use the multimove Gibbs-sampling method described in Kim and Nelson (1999, Ch 9) to draw the latent Markov-Switching states.

To summarize, we …rst run Hamilton’s (1989) …lter to get the probability

f(stjLekt;Cet)fort= 1;2; :::; T and save them, whereLekt= [Lk1; Lk2; :::; Lkt]

0

and Cet = [C1; C2; :::; Ct]0. Then we generate sT from the last iteration of

the …lter. After that, we recursively generate st conditional on Lekt, Cet and

st+1 fort=T 1; T 2; :::1.

Generate M, Conditional on sL

t, sCt , Lkt, Ct and the other

model parameters

As explained in Chib (1996), given _esT, the posterior distribution of the

transition matrix M can be derived without regard to the sampling model. Let vector mb be the bth column of M. We elicit a Dirichlet prior for

mb:

mb D( 1;b; 2;b; :::; a;b)

Thus, the posterior formb is:

mbjesT D( 1;b+q1;b; 2;b+q2;b; :::; a;b+qa;b)

whereqa;b is the number of transitions fromst=btost+1=ain the vector

e

sT.

2 Priors and Prior Sensitivity Analysis

2.1 Elicited Priors

The results presented in the paper involve the following priors:

s_ijkS2 = 10; _ijkS = 0:1 b S ijk = 1; V Sijk = 10 b S ij = 1; V Sij = 10 b = 0; V = 10 b ik = 0; V ik = 10 s_jD2 = 10; _jD = 0:1

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b C j = 1; V Cj = 10 b kL= 04 1; V kL= 10 I4 s_kL2= 10; _kL= 0:1 s_kL2_x = 0:1; _kLx = 10 b C = 04 1; V C = 10 I4 s_C2= 10; _C = 0:1 s_C2_x = 0:1; _Cx = 10 a;b= 1 fora; b2 f1;2;3;4g. 2.2 Prior Sensitivity Analysis

The following priors are used for prior sensitivity analysis.

s_ijkS2 = 1; _ijkS = 1 b S ijk = 0; V S ijk = 100 b S ij = 0; V Sij = 100 b = 0; V = 100 b ik = 0; V ik = 100 s_jD2 = 1; _jD = 1 b C j = 1; V Cj = 100 b kL= 04 1; V kL= 4 I4 s_kL2= 1; _kL= 1 s_kL2_x = 0:1; _kLx = 10 b C = 04 1; V C = 4 I4 s_C2= 1; _C = 1 s_C2_x = 0:1; _Cx = 10 a;b= 2 fora; b2 f1;2;3;4g.

Empirical results using these new priors are as follows. Table 1: Model Comparison

BIC AIC Model A -2.802 -3.901 Model B -2.492 -3.565 Model C -2.664 -3.737 Model D -2.321 -3.369 Model E -2.909 -3.932 Model F -2.601 -3.617 Model G -2.744 -3.760 Model H -2.386 -3.396

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Table 2: Posterior Mean of Transition Probabilities (Posterior standard deviations in parentheses)

(GL; GC) (BL; GC) (GL; BC) (BL; BC) (GL; GC) 0:972 (0:007) 0:231 (0:127) 0:246 (0:132) 0:221 (0:096) (BL; GC) 0:009 (0:005) 0:258 (0:140) 0:235 (0:132) 0:198 (0:111) (GL; BC) 0:009 (0:005) 0:187 (0:113) 0:251 (0:128) 0:149 (0:087) (BL; BC) 0:010 (0:005) 0:324 (0:150) 0:267 (0:142) 0:432 (0:145)

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Table 3: Proportion of Variance of Spreads Attributable to Each Component (USD) var(L) var ( C) var(X) cov(L,C) cov(L,X) cov(C,X) var(e)

barclays1m 0.8918 0.0000 0.0008 0.0000 0.0020 0.0000 0.1054 barclays3m 0.8208 0.0720 0.0000 0.0966 -0.0002 0.0000 0.0108 barclays12m 0.2720 0.4217 0.0000 0.1345 0.0000 0.0000 0.1718 btmuf j1m 0.8755 0.0042 0.0011 0.0239 0.0023 0.0000 0.0930 btmuf j3m 0.7801 0.1027 0.0000 0.1125 -0.0003 0.0000 0.0050 btmuf j12m 0.2129 0.5345 0.0000 0.1340 0.0000 0.0000 0.1186 citibank1m 0.8963 0.0015 0.0013 0.0146 0.0025 0.0000 0.0836 citibank3m 0.7715 0.1101 0.0000 0.1158 -0.0003 0.0000 0.0029 citibank12m 0.1991 0.5585 0.0000 0.1325 0.0000 0.0000 0.1099 deutschebank1m 0.9094 0.0000 0.0014 0.0002 0.0026 0.0000 0.0864 deutschebank3m 0.7876 0.0982 0.0000 0.1106 -0.0003 0.0000 0.0039 deutschebank12m 0.2908 0.4672 0.0000 0.1465 0.0000 0.0000 0.0956 hbos1m 0.8682 0.0000 0.0009 0.0001 0.0020 0.0000 0.1287 hbos3m 0.8352 0.0038 0.0000 0.0221 -0.0002 0.0000 0.1391 hbos12m 0.6070 0.0496 0.0000 0.0688 0.0000 0.0000 0.2747 hsbc1m 0.9085 0.0001 0.0013 0.0028 0.0025 0.0000 0.0848 hsbc3m 0.7876 0.0992 0.0000 0.1111 -0.0003 0.0000 0.0024 hsbc12m 0.2660 0.4843 0.0000 0.1426 0.0000 0.0000 0.1070 jpmc1m 0.9009 0.0000 0.0015 0.0003 0.0027 0.0000 0.0946 jpmc3m 0.7692 0.1076 0.0000 0.1144 -0.0003 0.0000 0.0091 jpmc12m 0.1849 0.5484 0.0000 0.1265 0.0000 0.0000 0.1403 lloyds1m 0.9186 0.0000 0.0013 0.0005 0.0026 0.0000 0.0770 lloyds3m 0.7716 0.1103 0.0000 0.1160 -0.0003 0.0000 0.0024 lloyds12m 0.1903 0.5624 0.0000 0.1300 0.0000 0.0000 0.1173 rabobank1m 0.8999 0.0006 0.0015 0.0089 0.0027 0.0000 0.0864 rabobank3m 0.7460 0.1237 0.0000 0.1208 -0.0003 0.0000 0.0098 rabobank12m 0.2110 0.5224 0.0000 0.1319 0.0000 0.0000 0.1346 rboscotland1m 0.9163 0.0001 0.0012 0.0036 0.0024 0.0000 0.0764 rboscotland3m 0.7686 0.1099 0.0000 0.1155 -0.0003 0.0000 0.0062 rboscotland12m 0.1867 0.5159 0.0000 0.1233 0.0000 0.0000 0.1741 ubs1m 0.9148 0.0001 0.0012 0.0045 0.0024 0.0000 0.0769 ubs3m 0.7560 0.1217 0.0000 0.1206 -0.0003 0.0000 0.0021 ubs12m 0.1929 0.5353 0.0000 0.1276 0.0000 0.0000 0.1441 westlb1m 0.8708 0.0015 0.0013 0.0145 0.0024 0.0000 0.1095 westlb3m 0.7369 0.1307 0.0000 0.1234 -0.0003 0.0000 0.0093 westlb12m 0.1818 0.5561 0.0000 0.1263 0.0000 0.0000 0.1358

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Table 4: Proportion of Variance of Spreads Attributable to Each Component (GBP) var(L) var ( C) var(X) cov(L,C) cov(L,X) cov(C,X) var(e)

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Table 5: Proportion of Variance of Spreads Attributable to Each Component (EUR) var(L) var ( C) var(X) cov(L,C) cov(L,X) cov(C,X) var(e)

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Figure 6

3 Empirical Results Involving Only Eight Banks

Empirical results reported in this section using data of the eight banks for which we do not have any missing observations.

Table 6: Model Comparison

BIC AIC Model A -3.227 -4.325 Model B -2.853 -3.926 Model C -3.013 -4.086 Model D -2.593 -3.641 Model E -3.273 -4.296 Model F -2.886 -3.902 Model G -3.093 -4.109 Model H -2.620 -3.630