PHASE DATING AND CONTAGION IN
THE GFC: A SMOOTH TRANSITION
STRUCTURAL GARCH APPROACH
1
George Milunovich – Macquarie University
Susan Thorp – University of Technology Sydney
Minxian Yang – University of New South Wales
MOTIVATION


Real estate shocks preceded the 2007-2009
financial crisis but other asset classes including
debt and equities received, transmitted and
possibly amplified the shocks.
We dissect the crisis at the level of structural
shocks, tracking changes in simultaneous links
between equities, T-bonds and real estate.
Stocks (SP 500)
 Real Estate (FTSE NAREITs)
 T-Bonds (BOA Merrill Lynch US Treasury Index)

2
DATA AND COMPLICATIONS

Data sample:
Time Period:
 Sampling Frequency:
 No. of Observations:



June 2001 – September 2010
Daily
2296
Investigate possible breaks in the structural
relationships due to the GFC
Modeling Challenges:
Endogenous data
 Possibility of several regime shifts during the period
of the GFC

3
SP500
15
Included observations: 2296 after adjustments
SP500
TBOND
10
5
0
REIT
Mean
Maximum
Minimum
0.002861
10.24540
-9.459519
0.022405
2.117925
-1.957185
0.038848
16.35494
-20.59137
Std. Dev.
Skewness
Kurtosis
1.360837
-0.340865
10.44911
0.340871
-0.172506
5.003489
2.269040
-0.096049
16.29085
Jarque-Bera
Probability
5352.939
0.000000
395.3904
0.000000
16902.72
0.000000
Observations
2296
2296
2296
-5
-10
1000
1500
2000
2500
TBOND
3
2
1
0
-1
-2
1000
1500
2000
2500
Correlation
Probability
SP500
SP500
1.000000
-----
TBOND
TBOND
-0.349065
0.0000
1.000000
-----
REIT
0.743164
0.0000
-0.220409
0.0000
REIT
REIT
20
10
0
-10
1.000000
-----
4
-20
-30
1000
1500
2000
2500
MODEL

Basic Structure for filtered returns   L  yt  rt
rSP 500,t  12 rTBond ,t  13 rREIT ,t  u1,t
rTBond ,t   21rSP 500,t   23 rREIT ,t  u2,t
rREITs ,t  31rSP 500,t  32 rTBond ,t  u3,t

Or in vector notation:
Brt  ut
ut I t 1 ~ N  0, Gt 
 Gt is diagonal
 diagonal elements follow GARCH(1, 1): gi ,t  i   i ui ,t  i gi ,t 1
5
ENDOGENOUS DATING AND ESTIMATING
THE IMPACT OF THE GFC

In order to account for possible regime shifts in
the relationships across the three markets we
extend the model as follows
Brt  ut
Bt rt  ut
where


Bt  1  S3  1  S2  1  S1  B0  S1B1   S2 B2  S3B3

S j  1 e

 j xt c j


1
for xt 
t
and j  1, 2,3
T
6
SMOOTH TRANSITION FUNCTIONS SJ

Shape of the transitions function S j  1  e

  j xt  c j


1
depends on:
1.0
1.0
1.
the speed of transition through γ >
0. As γ →∞ transition becomes
abrupt and the model jumps
between the states.
2.
the location of transition through
c > 0. We allow up to three changes
in regime, i.e. four phases
0<c1<c2<c3<1. For a large value of γ
if c1≤ xt <c2 then Bt=B1 etc.
yyy0.9
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00
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xx5
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yy 0.9
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
For information on smooth
transition models see Granger
(1993), van Dijk, Terasvirta,
Frances (2002), Silvennoinen and
Terasvirta (2009), amongst others
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.1
-5
-4
-3
-2
-1
0
5
x
7
IDENTIFICATION STRATEGY

When the error vector ut=Byt is homoskedastic, the structural matrix
B cannot be recovered from the reduced VAR without identifying
restrictions.


Examples of such restrictions include a) exclusion restrictions, see for
example Sims(1980) and Bernanke (1986), b) sign constraints on
parameters in B (Blanchard and Diamond (1989)) or c) assumptions about
long-run multipliers (Blanchard and Quah (1989))
Recently a number of papers used identification via heteroskesticity
to avoid imposing such constraints






Sentana and Fiorentini (2001) provide sufficient conditions for
identification of factor models in which the factors are heteroskedastic
Rigobon (2003) uses discrete regime shifts in volatility to identify SVAR
models
Rigobon and Sack (2003) suggest that ARCH in structural errors could be
used to identify structural VAR models but do not provide exact conditions
for identification.
Lanne et al (2010) obtain identification of a structural model where
heteroskedasticity follows a Markov switching process
Klein and Vella (2010) and Lewbel (2010) exploit relationships between
heteroskedasticity and exogenous explanatory variables to prove
identification
Milunovich and Yang (2010) prove joint identification of all structural
parameters of SVAR models with ARCH variances
8
IDENTIFICATION STRATEGY


In this paper we use Milunovich and Yang (2010)
arguments and extend them to take into account the
possibility of regimes shifts as described in this paper.
All structural parameters are locally identified at any
regular point in the parameter space
1.
γ is sufficiently large
2.
B0, B1, B2, B3 are all invertible and different
3.
at least n-1 structural shocks have ARCH effects
9
rSP500  12 rTBond  13rREIT
13 Sep – bailout of Northern Rock
18 Sep – lowering of Fed Funds rate
1 Oct – UBS announces a large write-down of
its portfolios
5 Oct – Merrill Lynch reports large losses
10 Oct – establishment of the HOPE NOW
alliance to stave off mortgage foreclosures
ESTIMATED CRISIS
REGIME DATES
rTBond   21rSP500  23rREIT
9 Aug – large European banks report falls in
earnings of between 28%-63% one
year after the start of the crisis
7 Sep – rFannie
Mae and Freddie Mac passed
REITs  31rSP 500  32 rTBond
into conservatorship, $100bn
provided to each company , both
CEOs replaced
10 Sep – Lehman announces $3.9bn loss in
3rd quarter
15 Sep – Lehman files for bankruptcy, BOA
buys Merrill Lynch , AIG debt
downgraded by all three major
rating agencies
10
11
12
VARIANCE DECOMPOSITIONS


Since the structural parameters are identified and we
obtain the estimates of the B matrices in the next
step is to try to identify the structural shocks rt  Bt1u t
We use the following strategy developed in Dungey et
al (2010)
A shock is named after the market to which it contributes
the largest fraction of its variance
 Two variable example

Vart  j|t  ri    tVart  j|t  u1   tVart  j|t  u2 
VDtu1 j|t 
VDtu2 j|t 

If
VDtu1 j|t
 VDtu2 j|t
 tVart  j|t  u1 
 tVart  j|t  u1   tVart  j|t  u2 
tVart  j|t  u2 
 tVart  j|t  u1   tVart  j|t  u2 
then u1 is called the ri variable shock.
13
14
VARIANCE DECOMPOSITIONS
15
MODEL FIT – RESIDUAL DIAGNOSTICS
16
CONCLUSIONS

We develop an identified Structural GARCH model with
smooth transition functions


Significant changes are found in the linkages between gov’t
debt, real estate and equity which persist into the postGFC period





We are able to endogenously date 3 structural breaks and 4
regimes
Direct linkages to and from T-bonds and the other two
markets become insignificant over the crisis
Impact of equities on real estate increases dramatically during
the first phase of the GFC and remains high
Impact of real estate on stocks doesn’t change over the crisis
but almost halves over the post-GFC period
Impact from T-bonds on REITs corrects sign from in the postGFC period
Variance decompositions illustrate the propagation of risk
across the three assets, with real estate shocks starting to
grow in importance in 2003-2004 period.
17