Credit Spread, Financial Market and Real
Activities under Financial Instability: Empirical
Evidence with MS-SBVAR

Satoshi Tezuka
Yoichi Matsubayashi

March 2018
Discussion Paper No.1812

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANCredit Spread, Financial Market and Real
Activities under Financial Instability: Empirical Evidence with MS-SBVAR
Satoshi Tezuka
Yoichi Matsubayashi

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANCredit Spread, Financial Market and Real Activities
under Financial Instability: Empirical Evidence with MS-SBVAR
Satoshi Tezuka and Yoichi Matsubayashi

Abstract

The purpose of the paper is to show how widening credit spreads in ”unstable periods”
influence the primary markets, the lending markets, and production activities, in comparison with stable periods. The MS-SBVAR identifies the 2008 global financial crisisand the 2011 great East Japan earthquake as unstable periods. During unstable periods, negative shocks influence industrial activities and bond issuance, while outstandingloans are aﬀected by positive shocks, which results from the banks in Japan remaining
their financial health. In addition, marginal research is conducted, using a ”modified
credit spread,” which eases the excess impact of the great East Japan earthquake on
credit spreads. It is confirmed that the results are constant, although the regime of the
disturbance terms corresponds to other events.

1

Introduction

Financial systems play a primary role in the economy. In particular, corporate bonds, along with the direct
finance market, have been growing in Japan. The corporate bond market consists of the primary market,
where firms issue bonds, and the secondary market, where institutional investors trade bonds.
However, the financial system may not function well during ”unstable periods.” For instance, a financial
crisis might alter the economic fundamentals. The studies about the global financial crisis suggest that the
responses of the financial systems and the production sectors might diﬀer. Regarding bank lending, Bassett,et al. (2014) observed that a worsening credit supply reduced real GDP and U.S. lending. In the production
sector, Caldara, Dario, et al. (2016), suggested that in the case of a global financial crisis, financial andeconomic uncertainty contributes to a drop in industrial production and in stock prices. Additionally, Naifar(2011) reported industrial production and credit default swaps as an indicator of default risk from 2006 to2009 has highly negative-correlation only during the 2008 financial crisis in Japan.
An ”unstable period” in the paper is defined as when a sudden and drastic widening of credit spreads
in the secondary market changes the behavior of financial systems and industrial production. Credit spreads
mean the diﬀerence in yield between Japanese Governmental Bonds (JGB) and straight corporate bonds(SB) with the same maturity.
CSi,t = SBi,t − JGBi,t(1)

For each variable, i denotes maturity at time t. The few studies that discuss credit spreads in the Japanese
secondary market include the following. Nakashima and Saito (2009) opined credit spreads reflect the debtto-equity ratios, the volatility of firms’ value, and the term to maturity at the firm level, and market liquidity.
Shirasu (2014) documented that Japanese bond spreads are aﬀected by credit risk, macroeconomics, market
liquidity, the primary bondholders, and the issuer’s liquidity. Therefore, credit spreads includes not only firms’
credit situations but the whole financial condition, and widening credit spreads imply financial condition isunpreferable.
This paper aims to research the response of the primary bond, banking, and production sectors to thewidening credit spreads between financially unstable periods and stable periods. To capture the diﬀerenceof changes in financial sector and production sector between two periods, a Markov Switching Structural
Bayesian VAR (hereafter, MS-SBVAR) is implemented to allow time variation in the multivariate timeseries model, and investigates relationships between corporate bond, average loan balances, and industrial
production. The model was developed by Hamilton (1989), Chib (1996), and Kim and Nelson (1999). Sims,1

et al. (2008) relaxed the model for the multivariate MS-SBVAR.
Our results verify that the MS-SBVAR identifies the global financial crisis and the great East Japan
earthquake as unstable periods, and others as stable periods. Furthermore, diﬀerent responses of variables
is found to widening credit spreads between the two periods. A novelty of this paper is that observationof the responses to the shocks from Japan’s primary market to the secondary market, and the MS-SBVAR
identifies two static yet clearly diﬀerent financial conditions in responding to credit spreads in Japan.
This paper proceeds as follows. Section 2 introduces the methodology and selects the best-fit models.
Section 3 provides the estimated results and analyzes and interprets the estimated impulse response functions.
Section 4 researches the secondary market responses to three variables, using a ”modified credit spread” toremove the excessive impact of the 2011 earthquake on credit spreads. Section 5 provides a conclusion.

2

Methodology

2.1

Data

To discuss the state-dependent relationships between financial systems and industrial production, the following variables are used: credit spreads (CS) as a secondary market and financial stress indicator, outstandingof straight bonds (SB) as a primary market indicator, average loan balances (LV ) as a banking sector indicator, and the index of industrial production (IIP ) as a production sector indicator. The data sample runs
from January 2000 to June 2017. The data frequency is monthly, and details of data follows:• Industrial production (IIP ): the logarithm of the seasonally adjusted industrial production (May, 2010=100) (Source: Ministry of Economy and Industry)• Average loan balance (LV ): the logarithm of banks’ and credit unions’ average loan balances (Source:Bank of Japan)• Outstanding straight bonds (SB): the logarithm of outstanding straight bonds (Source: Japan Securities Dealers Association)• Credit spreads (CS): 1-year A-rated credit spreads, rated by Rating & Investment Information (Source:
Japan Securities Dealers Association)

2.2

Model

A class of MS-SBVAR models, developed by Sims, et al. (2008), is the following structural VAR models
that allows structural shocks and coeﬃcients to change independently in accordance with unobserved state2

variables:yt′ A0 (sct ) =

ρ
∑

′
yt−i
Ai (sct ) + C (sct ) + ϵ′t Ξ−1 (svt )

(2)

i=1

• y denotes n × 1 vector of endogenous variables.
In this paper, MS-SBVAR orders y ≡ [IIP, LV, SB, CS]′ , discussed further in the later identification
section. ρ stands for lag length.
• A0 (k): n × n matrix of parameters, describing contemporaneous relationships between the elements ofyt .
• Ai (k): n × n regular matrix of the endogenous variables.
• ϵt : n × n unobservable disturbance terms• Ξ (k): n × n diagonal matrix• st = (sct , svt ): composite Markov process with sct and svt as independent regime variable. Both are latentvariables. svt corresponds to variances in the disturbance term, and sct corresponds to the constant term
and coeﬃcients.
• qi,j : probability of st =i and st+1 =j
st evolves according to a first-order Markov process with the following state probabilities:
qi,j = P r (st = i|st−1 = j) (qi,j ≥ 0∑qi,j = 1)i∈H• Q: Markov transition matrix
The general form is expressed as follows:
q1,1(1 − q2,2 ) /2
1 − q1,1
q2,2
Q = (qi,j )i∈H =  0(1 − q2,2 ) /2
 .
..
 .
.
.
0
0

···
..
.
..
.
..
.
···

0
..
.

0
..
.

(1 − qk−1,k−1 ) /20

1 − qk−1,k−1(1 − qk−1,k−1 ) /2

1 − qk,k
qk,k










(3)

In our estimation1 , VAR imposes two assumptions, identification and normalization. Identification is the
restriction on the contemporaneous coeﬃcient matrix A0 to understand the relationships among endogenous
variables. The Choleski decomposition is used on a variance-covariance matrix in this paper, and A0 is set as
an upper triangular matrix. Recursive identification schemes, including the Choleski decomposition, assume
that variables are ordered, along with exogeneity of variables. For instance, the variable ordered first is
1 Results are estimated by ”ms-sbvar,” based on Sims, et al. (2008), using Dynare open software. For the estimation, thecodes of Lhuissier (2017) is referred, available on his homepage (http://www.stephanelhuissier.eu).

3

assumed to be contemporaneously uncorrelated to all other variables in an upper triangular matrix A0 . This
paper orders IIP, LV, SB, CS for the following reasons:• Credit Spreads(CS): Credit spread, which responds simultaneously to all other variables, is ordered
last, because it is forward-looking and includes information about the future economy.
• Outstanding Straight Bonds (SB) and Average Loan Balance (LV ): In general, firms raise funds by
issuing bonds or by borrowing from banks. Hosono, et al (2013) verifies a hold-up hypothesis between
lending and bond issuance in Japan. Because of imperfect contracts, banks exclusively hold information
about borrowers and exploit firms’ profits when the hypothesis is verified. Therefore, firms prefer issuingbonds under this condition, and LV orders before SB.
• Industrial Production (IIP ): Following Leeper, et al (1996), the production sector does not respondcontemporaneously with other variables and, therefore, orders first in our model.
Secondary, normalization is required. Waggoner and Zha(2003) point out that incorrect normalization on
VAR leads to a misinterpretation of the impulse response function analysis. To prevent this problem, thispaper imposes diag (A (st )) > 0, where diag (X) describes the diagonal of matrices in X.

2.3

Model Selection

In this paper, the best-fit model is compared among several models, using the marginal data density (hereafter,
MDD) as the criteria. The MDD, a likelihood function, integrates whole parameters, as in the following
equation:∫p (YT ) =p (YT |θ) p (θ) dθ

(4)p (YT |θ) is a likelihood function and p (θ) is prior. The larger value of the likelihood function is preferable.
Chib’s (1995) method is employed for constant parameter modeling and Sims, et al (2008) for time-variant
models (see the Appendix for details). For the first step, the lag order of MS-SBVAR is chosen, using
time-invariant model Mconstant . Table 1 reports MDD of Mconstant with each lag. ρ = 3 and ρ = 4 shows
the largest MDD. ρ = 3 is selected to reduces the cost of calculation. Next, various models with ρ = 3
are estimated. Table 2 reports each MDD of various time-variant models. Compared to the time-invariant
model, time-variant models improve MDD by more than 10. M2c2v has the largest MDD among the models
and it is selected as the best fit.

4

Table 1: MDD of Mconstant with Each Lag
ρlog MDD

32729.742729.752708.6

62687.872679.1

Table 2: MDD of Time-Variant Models with ρ=3
Model
M1c1v
M1c2v
M1c3v
M2c2v
M2c3v

Description
Time-invariant model
2-regimes in shock variances
3-regimes in shock variances
2-regimes in all equation coeﬃcients and 2-regimes in shock variances
2-regimes in all equation coeﬃcients and 3-regimes in shock variances

3

Estimated Results and Analysis

3.1

Posterior Distribution

Log MDD
2729.72862.72920.52924.72898.9Figure 6, 7 shows the estimated probabilities from the best fit M2c2v . The generated probabilities aresmoothed by Kim’s (1994) method. In this section, we arbitrarily segregate each regime for analysis. Probabilities in Figure 6, 7 are clearly equal to either 1 or 0. Statistically speaking, the regimes are clearly
identified. sct = 1 corresponds to the coeﬃcients, and svt = 2 corresponds to the variance of disturbanceterm, showing high probabilities in two periods from January 2008 to January 2009, and from March 2011
to August 2011. The former is the time of the global financial crisis, and the latter is the time of the great
East Japan earthquake.
[][]
202.11 −6.50 14.98
11.39
54.99 21.88−83.62−12.52cc
0387.1548.43−30.510526.21−204.78−259.44A0 (st = 1) =A0 (st = 2) =00366.94 −35.72
00441.76193.22
0
0
0
24.42
0
0
0
44.50
c
The contemporaneous coeﬃcient matrix A0 (st ) is reported above for both stable and unstable periods.
Coeﬃcients, beginning in the left column, list IIP, LV, SB, and CS equations. The absolute value of thecoeﬃcients in the equation of CS in unstable periods are larger than in a stable period, which indicatesthat the secondary market substantially responses to other variables in the unstable periods. Table 3, 4
report relative shock variance in each regime. All values in svt = 2 except IIP in A0 (sct = 2) are larger
than in svt = 1. From these fact, sct = 1 is defined as the high-coeﬃcient-stress period, and svt = 2 as thehigh-volatility period. Furthermore, the duration, in which sct = 1 and svt = 2 overlap, are defined as unstable
period, and, sct = 2 and svt = 1 are defined as stable period.
Q has latent variables sct and svt in our model. The model is expressed as
[ c
] [ v
]
c
v
q
1 − q2,2
q1,1
1 − q2,2
Q = Qc ⊗ Qv = 1 −1,1
⊗
c
c
v
v
q
q
1−q
q
1,1

2,2

1,1

2,2

(5)

Qc is the transition matrix which governs the coeﬃcients and Qv is the transition matrix which governs the5Table 3: The Relative Shock Variance A0 (sct = 1) Diagonal Components are Normalized to 1

svt = 1
svt = 2IIP
1.845.E-02
1.903.E-02

LV
1.900.E-03
2.706.E-03

SB
2.264.E-03
4.132.E-03

CS
2.221.E-01
3.003.E+00

Table 4: The Relative Shock Variance A0 (sct = 2) Diagonal Components are Normalized to 1

svt = 1
svt = 2IIP
4.948.E-03
3.227.E-03

LV
2.583.E-03
3.678.E-03[
Qc = 0.976
0.023

0.083
0.916

SB
2.725.E-03
4.975.E-03

CS
4.093.E-02
5.535.E-01

variance of disturbance term.
][]
0.980 0.053
Qv = 0.009
0.946

(6)

Equation 6 reports the value of the transition matrix. The probability of high stress coeﬃcient periodscc(q22= 0.976) is 6% lower than the probability of low stress coeﬃcient periods (q11
= 0.916). The former

lasts for 11.90 months, and the latter lasts for 41.66 months2 . Regarding volatility, the probability of highvcvolatility periods (q22
= 0.946) is 3.4% lower than the probability of low volatility periods (q11
= 0.980).

The former lasts for 18.51 months, and the latter lasts for 50 months. This is consistent with the fact that
unstable periods are not long-lasting.

3.2

Impulse Response

Using the impulse response, credit-spreads shocks versus other variables are analyzed in respective regimes.
The impulse response functions show how one variable’s shocks in one disturbance term of one variable aﬀect
other variables.
Before viewing the results, we note that our assumptions about the relationship of variables, in both
stable and unstable financial situations, could be made by referring to similar studies using MS-SBVAR in
Europe and in the U.S., (Hubrich and Tetlow (2015), Hartmann, et al (2015), and Lhuissier (2017)). The
variables are not aﬀected by the widening credit spreads during stable periods. In unstable periods, thevariables are aﬀected by negative shocks. Banks holding 40% of outstanding bonds in Japan might suﬀer
a loss from a widening credit spread. Simultaneously, a widening credit spreads informs borrowers about
economic conditions. In this situation, banks might take funds back from borrowers, or stop new financings.
Firms are forced to cancel new issues for constraint on the primary market. Under these conditions, themacroeconomic outlook turns negative and industrial production decreases.
1
2 Average duration of state i is calculated by E (D ) =
i
1−q

ii

6

Figure 12 reports the responses to widening credit spreads during both stable and unstable periods.
Each period shows a remarkably diﬀerent response. In a stable period, outstanding corporate bonds keeps
increasing over the entire period. The average loan balances decline for six months after the shock, but
industrial production increases during the same six-month period. During an unstable period, SB, in the
short run, experiences a subtle negative shock, but this turns positive in the long run. Surprisingly, LV is
positive for the first six months and remains the credible interval on the zero boundary. Industrial production
plunges instantly and remains below zero for eight months. The response of variables counters our assumption
from the previous studies.
Our prior assumption was that credit shocks would never aﬀect other variables during a stable period
and would decline during an unstable one. However, the impulse responses in this research display that SB
and IIP are positive and LV is negative in the former case, while SB and IIP are negative and LV is
positive in the latter case. In the following sections, we interpret this response in detail and investigate thecause.
3.2.1

Interpretation

The research above proves that the global financial crisis and the great East Japan earthquake are identified
as unstable period. In stable period, the impulse responses in this research display that SB and IIP are
positive and LV is negative. Credit spreads shocks have a positive eﬀect on average loan balances and anegative eﬀect on the outstanding straight bonds and industrial production during an unstable period. We
now interpret these responses in detail.
Although the previous studies in Europe and in the U.S. have noted that credit spread shocks during
stable periods have minimal eﬀects to other variables, a widening credit spreads significantly aﬀect the othersin our research. This is interpreted as the result from the monetary policy that lowers JGB yields during
stable periods. The changes in the JGB yields, conducted by BoJ, are greater than those in corporate bonds,
then credit spreads relatively widen. Regarding the monetary policy from an issuer’s viewpoint, Graham andHarvey, (2001), Bancel and Mittoo (2004), and Barry (2005) prove that the dollar amount of issues increases
when yields are lower than in the past. These facts support our interpretation of the impulse response in
primary market during stable periods. When fundraising is conducted by issuing bonds smoothly, firms are
disincentivized to lend from banks. Additionally, industrial production is activated in stable periods whenthe economic outlook is favorable.
In unstable periods, our research indicates that positive shock has aﬀected lending and negative shock7

has aﬀected corporate bond issuance for the first six months. This is because banks in Japan stay healthy
and continue to lend during unstable periods. In the primary market, institutional investors are unwilling tomarginally invest, and firms cut down on issuing bonds. As seen in Figure 12, the impulse response of SB
is positive eight months after a credit spread shock. This is interpreted as a reverse eﬀect, due to a strong
demand from firms that had postponed issuances. In the production sector financing constraints and future
declines in demand suppress production activities.
The 2008 global financial crisis and the 2011 great East Japan earthquake are captured as financial
instability by the MS-SBVAR. The former was a financial crisis that originated in the U.S., and the latter
was a natural disaster. Both negatively aﬀected the production sectors and the financial markets. Although
we interpreted the impulse response above, it is unsure that it is consistent with the facts. Next, we will
confirm the interpretation by checking the facts.
The 2008 global financial crisis reduced firms’ industrial production, which acts as a proxy variable for the
production sector, fell drastically in comparison to Europe and the U.S. Imports declined by approximately
25%, mostly from Europe and from the U.S., in the first quarter of 2009. As Figure 10 shows, corporate
bonds rated below ”A” could not be issued. Regarding bank lending, lending increased in Japan, although
Ivashina and Scharfstein (2010) documented that new financing in the fourth quarter in 2008 decreased by
47% in the U.S. compared with the previous quarter. Uchino (2011) found that banks remained financiallyhealthy during the 2008 crisis, and that firms shifted from issuing bonds to bank borrowings.
The 2011 earthquake in Japan caused a tsunami and disabled a nuclear generator in Fukushima. Production activities ceased not only where the earthquake hit but all over Japan. Turmoil is also created in thecorporate bond market. Electric power bonds accounted for 21% of the total outstanding Japanese corporate
bonds in March 2011. After the earthquake, the heightened credit risk in the electric power industry widened
the credit spreads for the entire market. Furthermore, the uncertain outlook for nuclear generation restrained
further marginal investment in electronic power companies. As a result, the issuance of new electric powerbonds was suspended for 11 months except for a few issues. Figure 11 presents the changes in long-term debt,
versus electric power firms’ public bond issuance. The figure indicates that the electric power sector borrowed
more from financial institutions after 2011. Those changes are interpreted as the shift of fundraising methodfrom public bond issuance to bank lending in the electric power sector.
In summary, these separate 2008 and 2011 disasters are consistent with the interpretation of estimatedimpulse response. The reactions from the financial and production sectors during these unstable periods

8

diﬀer from their respective reactions in stable periods. Firms, locked out of the primary market, borrowed
from financial institutions, along with a decline in the production sector. However, the impact of the wider
credit spreads may be overstated, since the physical damage to electric power firms primarily widened the
A-rated credit spreads. Therefore, our credit spreads are modified to confirm the conclusions in the next
section.

4

Modified Credit Spreads

As reported in the previous section, a nuclear disaster occurred at Tokyo Electric Power Co.’s (TEPCO)
Fukushima 1st Nuclear Power Plant on March 11, 2011. Before the earthquake, TEPCO was one of thelargest bond issuers in Japan. TEPCO bonds totaled 4.974 trillion yen, accounting for 0.5% of Japan’s total
outstanding corporate bonds in 2010). The disaster raised TEPCO’s yield sharply, and the company’s credit
rating was downgraded from AA- to A on April 7th, 2011, and to B on October 7th. Since the data is endof-month one, TEPCO’s bond existed in A-rate and might influence on whole A-rated credit for 6 months.
TEPCO’s impact was reflected in a credit spread. increase, from 0.408% in March, 2011 to 0.83% in April,
2011, which shrank approximately 1% when TEPCO’s rating was lowered to BBB. TEPCO’s situation clearlyskewed all A-rated fixed-income securities. A modified credit spreads was implemented to ease TEPCO’s
impact. Following Okimoto and Takaoka (2017), the Thomson Reuters Bond Credit Curve3 is applied tomodify credit spreads after January 2011. CStX denotes credit spreads rated by X at time t and n is sample
size.

1 CStR&I − CStReuter
CStmodif ied = CStReuter + CStR&I Σn
CStR&I(7)The MS-SBVAR with [IIP, LV, SB, M CS]′ is estimated, with others constant. Figure 8, 9 displays
the probability of each Markov Switching process’ estimated regime from model M2c2v with modified credit
spread. The regime of coeﬃcient sct = 1 responds to the duration from May 2008 to June 2009 and from March2011 to August 2011. By contrast, the variance of disturbance period svt = 2 corresponds to the collapseof the IT bubble (October 2001 to January 2012); the global financial crisis (June 2008 to January 2010);
the European debt crisis (April 2012 to August 2012); and implementation of quantitative and qualitative
monetary easing with negative interest rates (January 2016 to September 2016). From these facts, sct = 1 isregarded as a high-coeﬃcient-stress period and svt = 2 as a high-volatility period. The impulse responses inhigh-coeﬃcient-stress period and low-coeﬃcient stress period diﬀer minimally from the original (Figure 13).
3 The index is provided from August 2010 and bond yields are smoothed by B-spline curve

95Conclusion

The purpose of the paper is to assess how widening credit spreads during stable and unstable periods influence
primary bond markets, lending markets, and production activities. The MS-SBVAR identifies the 2008 global
financial crisis and the 2011 great East Japan earthquake as unstable periods. The shocks by widening credit
spread in each period are diﬀerent. In a stable period, the monetary policy might widen credit spreads.
Corporate bonds and industrial production see positive eﬀects, while average loan balances decline. In
unstable periods, higher credit spreads indicate unfavorable economic conditions, which negatively aﬀectindustrial activities and bond issuance. Interestingly, it is found that a widening credit spreads increased
lending. In addition, the marginal research is concluded with modified credit spreads, which eases the impactof the great East Japan earthquake on financial markets. The results are constant, although the regime ofthe disturbance period responds to other events.
However, further research is needed. First, a financial shift from bonds to banks, which we observed during
unstable periods, might not happen during times of serious financial instability. For instance, commercial
banks suﬀering a deterioration of shareholders’ equity, limited lending during the 1991- 93 collapse of Japan’s
real estate bubble. Production could also be aﬀected, as it occurred. Secondary, the relationship between
demand and supply in the primary market both in stable and unstable periods is unclear. In other words, ourstudy uses limited empirical research and does not provide a theoretical interpretation. It is still open question”which is the determinants of issue amounts, whether the risk capacity of investors on the demand side, or
the investment interest of issuers on the supply side.” Hence, additional theoretical research is required.

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13

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14

Appendix
A

MS-SBVAR

A.1

Assumption for Estimation

Conditional posterior ϵt is assumed asp (ϵt |Yt−1 , St , θ, q) = normal (ϵt |On , In )(A-1)

where On : n × 1 vector of 0, In : n × n identity matrix, and θ denotes all coeﬃcients in the model without q:
θ = {A, F, Ξ}
A = {A (1) , ..., A (h)}, F = {F (1) , ..., F (h)}, Ξ = {Ξ (1) , ..., Ξ (h)}As disturbance terms defined above, this assumption is equivalent top (yt |Yt−1 , St , θ, q) = normal (yt |µt (k) , Σt (k))()′µt (k) = F (k) A−1 (k) xt

(A-2)

(
)−1
Σ (k) = A (k) Ξ2 (k) A′ (k)In short, yt is mean values explained by coeﬃcients and explanatory variables at t with variance of disturbanceterms.
A conditional likelihood function, which follows a normal distribution (Equation A-1 ) with mean µt (k),variance Σ (k), represents
p (yt |Yt−1 , St , θ, q) =

(
)
1
′ −1
exp
−
(y
−
µ
(s
))
Σ
(s
)
(y
−
µ
(s
))
t
t
t
t
t
1
2
(2π) |Σ (k) | 2
1n2≈ 1 and the equation in exp is expanded:
(
)
1 ′
′
2
′
′
p (yt |Yt−1 , St , θ, q) = |A (st ) Ξ (st ) | exp − (yt A (st ) − xt F (st )) Ξ (st ) (A (st ) yt − F (st ) xt )
2
)
(
n
∏
ξj2 (st ) ′(yt aj (st ) − x′t fj (st ))p (yt |Yt−1 , St , θ, q) = |A (st ) ||ξj (st ) | exp −2
j=1
As n is suﬃciently large,

(A-3)

1
n
(2π) 2

(A-4)
(A-5)

Hence, the overall likelihood function of YT is given byp (YT |θ, q) =T ∑∏[p (yt |Yt−1 , θ, q, st ) p (st |Yt−1 , θ, q)](A-6)

t=1 st ∈H

A.1.1

Time-Variant Restriction

As estimated sample numbers and latent variables increase, computational processes exponentially swell (thecurse of dimensionality). To deal with this issue, time-variant restriction is imposed as follows:
F (sct ) = G (sct ) + S̄A (sct )

(A-7)

[
]
S̄ = In , 0(m−n)×n
gj (Stc ) is the jth column of G (Stc ), which consists of a time varying factor gδj (h) and a regime-independent15

factor gΨj , are expressed by
(
)
(
(
)
′
′)
diag gj (1) , .., gj (h) = diag [gδ′ j (1) , ..., gδ′ j (h) ]′ diag gΨj
A.1.2

(A-8)

Identification Restriction

MS-SBVAR representing a simultaneous equation itself would not be identified without any restriction.
Following Waggoner and Zha(2003), Rj [ajfj ]′ = 0 is applied as linear restriction. where Rj is any(n + ρn + m) × (n + ρn + m) matrix and is not full of rank. To deal with over-parameterization, aj (k)and fj (k) are given by4
aj (k) = Uj bj (k)(A-9)

fj (k) = Vj gj − Wj Uj bj (k)(A-10)

where Uj : n× orthonormal matrix of qj , V j: (ρn + m) × orthonormal matrix of rj , Wj : (ρn + m) × n freeparameter matrix.
Substitute Equation A-9 , A-10 into Equation A-6 and transform it to the following form:
((
))
n
∏
)2
ξj (k) ( ′|ξj (k)| exp−
p (yt |Yt−1 , St , θ, w) = |A (k)|yt Uj bj (k) − x′j (Vj gj − Wj Uj bj (k))
2
j=1

p (yt |Yt−1 , St , θ, w) = |A (k)|

n
∏
j=1

A.2

((
|ξj (k)| exp)
ξj (k) ( ′yt + x′j Wj Uj bj (k) − x′j Uj gj−
2(A-11))2 )(A-12)

Settings of Priors• The Prior of aj ,gj follows normal distribution below:()
p (aj (k)) = normal aj (k) |0, Σ̃aj()
( )p gψj = normal gψj |0, Σ̃gψj(A-13)(A-14)

Sims (1992) points out that large share of the sample period fluctuation accounts for deterministic
components in multi-variate time series model without dummy observations in the prior. To confront
this, Sims and Zha (1998) suggest n + 1 dummy observations from variables, introduced in part of theprior. Let VAR model with mth equations, i any value up to m, s any value described with j = 1, ..., m,lag l = 1, ., p and, constant periods, dummy observation on overall equation are given by()
Yd A0 = Xd GΨ + ŜA (k) + Êd Yd = {y (i.j)} , Xd = {x (i.s)}(A-15)

where Yd : (n + 1) × n dummy observation matrix, Xd : (n + 1) × m dummy observation matrix, GΨ :(ρn + m) × n matrix comprised of gΨ , Ēd : (n + 1) × n matrix.
4 Refer to Appendix D in Sims et al.(2008) for transformation of equations and proof in detail.

16Given Equation A-13 , Equation A-14 , and Equation A-15 , the prior is transformed as follows:()
p (aj (k)) = normal aj (k) |0, Σ̃aj
(A-16)()( )p gψj = normal gψj |0, Σg˜ψj(A-17)()−1Σ̄gψj = Ih1 ⊗ Σ̄g , Σ̄g = Xd′ Xd + Σ̃−1g
• The Prior of gψj , bj , ψj : linear restriction aj (k) = Uj bj (k) , gψj = Ψj ψj and Equation A-16 , Equation
A-17 leads to()−1
Σ̄bj = Uj′ Σ−1
U
j
aj
(
)−1
Σ̄ψj = Ψ′j Σ−1
Ψ
ψj j

(
)
p (bj (k)) = normal bj (k) |O, Σ̄bj

(A-18)()
p (ψj ) = normal ψj |O, Σ̄ψj

(A-19)• The Prior of δj follows a normal distribution:()
p (δj (k)) = normal δj (k) |0, Σ̃δj (k)Σ̄δj (k) = σδ2 Irδ,j(A-20)1 ¯ α¯j α¯j −1 −βj x
βj xe
Γ (α¯j )(A-21)• The Prior of ξj follows a Gamma distribution:()()
p ξj2 (k) = gamma ξj2 (k) |α¯j , β¯j =• The Prior of qj follows a Dirichlet distribution:p (qj ) = dirichlet (qj |α1,j , ...., αk,j ) =

(

Γ (Σ
α )
∏ i∈H i,j
Γ
(α
i,j )
i∈H

)∏

α

(qi,j ) i,j−1(A-22)i∈H

Following Litterman (1986), the hyperparameter of Σ̄bj is set where MDD of the constant VAR model ismaximized. In an optimization of MDD, we refer to the value from previous studies.5

A.3

Conditional Posterior

1. p (θ|YT , q, ST )To approximate the joint posterior density p (θ, w|YT ), alternatively sampling from the following conditional posterior:p (bj |yt , St , bi (k))(A-23)p (gj (k) |yt , St )(A-24)()
p ξj2 (k) |yt , St

(A-25)

5 In estimations, hyperparameters are set to µ =0.70: overall tightness of the random walk prior on A , F , µ =0.30: relative
1
0
2
tightness of the random walk prior on F , µ3 =0.1: relative tightness of random walk prior on the constant periods, µ4 =1.0:
erratic sampling eﬀects on lag coeﬃcients, µ5 =2.0: belief about unit roots, µ6 =2.0: belief in cointegration. The prior on δ(i, j, st )for each i, j and st is set to a normal distribution with mean 0 and a standard deviation of 50. The prior on each element of
the diagonal of Ξ2 (st ) is a gamma distribution, represented by Gamma(ᾱ, β̄) with ᾱ = 1 and β̄ = 1.

17

(a) p (bj |yt , St , bi (k))
Reproduce Equation A-23 by employing the MetroPolis-Hastings method:()
1
p (bj |yt , St , bi (k)) = exp − b′j (k) Σ̄−1
b
(k)
bj j
2
[
((
))]
∏
)2
ξj (k) ( ′×|A0 (k)| exp−
yt aj (k) − x′j (k)2(A-26)

t∈t:st =k(b) p (gj (k) |yt , St ) Equation A-24
Generate Equation A-24 from a multi-variate normal distribution:()
p (gj (k) |yt , St ) = normal gj (k) |µ̃gj (k) , Σ̃gj (k)(A-27)()(c) p ξj2 (k) |yt , St
Generate Equation A-25 from a Gamma distribution:()()
p ξj2 (k) |yt , St = gamma ξj2 (k) |α̃j (k) , β̄j (k)α̃j (k) = ᾱj +β̃j (k) = β̄j +1
2∑(A-28)

T2,k2(yt′ aj (st ) − x′t fj (st ))2t∈t:s2t =k

T2,k denotes {t : s2t = k}
2. p (ST |YT , q, θ)
Due to computation costs, it is getting diﬃcult to sample latent variable st at the same time with otherestimation as sample size T is large. A multi-move sampler is employed in order to sample eﬃciently.
The free parameter sT is sampled fromp (st |yt , St ) =∑p (St |YT , θ, q, st+1 ) p (St+1 |YT , θ, q)(A-29)

St+1 ∈H

3. p (q|YT , ST , q, θ)The posterior of qj follows
p (qj |Yt , St ) =

h
∏
i=1

18

n

(qi,j ) i,j+βi,j −1

(A-30)

Figure 1: A-rated One-Year Credit Spreads (Source: Datastream)

Figure 2: Outstanding of Straight Bonds (Source: Japan Securities Dealers Association)

Figure 3: Average Loans Outstanding (Source: Bank of Japan)

19

Figure 4: Industrial Production (Source: Ministry of Economy, Trade and Industry)

Figure 5: Modified Credit Spreads (Source: DataStream)

20

Figure 6: The Posterior svt with Credit Spread

Figure 7: The Posterior sct with Credit Spread

Chain 1

1

Chain 2

1
Regime 1
Regime 2

0.9

Regime 1
Regime 2

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0
Jan-00Mar-04

May-08

Jul-12

Sep-16

Jan-00Mar-04

May-08

Jul-12

Sep-16

Figure 8: The Posterior svt with Modified Credit Spread Figure 9: The Posterior sct with Modified Credit Spread
Chain 1

1

Chain 2

1
Regime 1
Regime 2

0.9

Regime 1
Regime 2

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0
Jan-00Mar-04

May-08

Jul-12

Sep-16

Jan-0021

Mar-04

May-08

Jul-12

Sep-16

Figure 10: Amount of Issues for A-rated and BBB-rated Bonds (Source: Japan Securities Dealers Association)

Figure 11: Changes in Issues of Electric Power Companies and Long-Term Debt from the Previous Year
(Source: Japan Securities Dealers Association)

22

Figure 12: Impulse Response to a of Credit Spread Shock

Impulse response from an identified MS-SBVAR on an unstable regime (first column) and a stable regime(second column). The median is the dotted lines. A 70% credible interval in the solid lines is in each column.

23

Figure 13: Impulse Response to a Modified Credit Spread Shock

Impulse response from an identified MS-SBVAR on an unstable regime (first column) and a stable regime(second column). The median is the dotted lines. A 70% credible interval in the solid lines is in each column.

24