Time Series Convergence within I(ch20:eqn20.2) Models: the Case of Weekly Long Term Bond Yields in the Four Largest Euro Area Countries
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218
G. Passamani
13
GEYLD10
ITYLD10
12
FRYLD10
SPYLD10
11
10
9
8
7
6
5
4
3
1995
2000
2005
Fig. 1 Ten-year zero-coupon bond yields for Germany, France, Italy and Spain
Source: Ehrmann et al. (2007)
driving force had led them to a point from which they have moved together, still
sharing the same driving force. The point corresponds to the time of the introduction
of euro. Therefore, the idea of convergence we have in mind is the one according
to which series, following different dynamic behaviours and time paths, get to
converge through narrowing their differences.
It’s to note that from the figure it’s possible to see some evidence, at least
over the first sub-period, of a certain similarity with the behaviour of zero-coupon
bonds yields of different maturities, where the German and French series variability
resembles the behaviour of yields of long-term maturities, whereas the Italian and
Spanish series resemble the behaviour of short-term yields, although the long-term
yields should typically be higher in level than the short-term yields. If we applied the
expectations theory of the term structure to investigate the relations among the series
represented in the figure, we would expect to find evidence of some broken trend
stationary relations representing the co-movements between pairs of series, where
the German series can be considered as the benchmark series to which referring the
others.
The data chosen for the empirical analysis are weekly long-term zero-coupon
bond yields for France, Germany, Italy and Spain. They cover the period from
1993 to 2008, that is through the years leading to monetary union in 1999, and
to monetary unification in 2002, and before the financial crisis of 2008.1
1
The persistence properties of each observed variable have been analyzed in terms of the
characteristic roots of its autoregressive polynomial. Allowing for a certain number of significant
Time Series Convergence within I(2) Models
219
2 The Analysis Within the I(1) Model
In order to analyze the process of convergence, we assumed the observed vector
time series xt to be generated by a cointegrated vector auto-regression (CVAR)
model with broken linear trends, both in the data and in the cointegrating relations
(Juselius 2006, p.297):
xt D
k 1
X
k xt
k
0
C ’“Q xQ t
1
C ˆDt C 0 C ©t ; ©t
Np .0;
/;
(1)
kD1
where “Q0 D “0 ; “11 ; “12 , xQ 0t D x0t ; t; tDt and t D 1993 W 01 W 08; : : : ; 2008 W
07 W 25. The observation times correspond to the last day of the working weeks.
When applying2 this cointegration model to our data set, made up of p D 4
variables, according to our expectations we should find one common stochastic
trend driving the system of variables towards convergence, and possibly three
stationary cointegrating relations. Considering t as the first week of 1999, which
corresponds to the beginning date of the third stage of EMU, and analyzing the
data3 within model (1), the simulated critical values of the rank test statistic
(Johansen 1997) gave some evidence, at 10%, of the existence of three cointegrating
relations. When trying to identify them as spreads between bond yields, the relative
hypotheses were rejected. But, when considering them as simple stationary relations
between pairs of bond yields and identifying them through the restriction that the
deterministic trends have the same coefficients with opposite sign over the second
sub-period, that is just a broken linear trend in the first sub-period and no trend
component since the beginning of 1999, the relative hypotheses were accepted with
a p-value D 0:191. The identified relations are represented in Fig. 2, where they
lags (lags 1 and 15 for both Italy and Spain; lags 1, 2, 3, 4, 6, 7, 8, 10 and 12 for France; lags 1,
2, 13 and 14 for Germany), the modulus of the largest root, 1 , satisfies 1 D 1:0, and the next
root, 2 , is less than 1.0, but not too far from it, i.e.: 0.89 for Italy, 0.90 for Spain, 0.85 for both
France and Germany. As Juselius (2010, p.9) observes: “. . . whether a characteristic root can be
interpreted as evidence of persistent behaviour or not depends both on the sample period and the
observational frequency.” Therefore our univariate series can be considered as generated either by
an I(1) process, or by a near I(2) process.
2
The empirical analysis was performed using the subroutine CATS, which needs the software
RATS to be run (Dennis 2006).
3
The number K D 2 of lags chosen is the one suggested by the information criteria and the LR
lag reduction tests, when starting from 5 lags. Dt is a vector of three impulse dummies, which
take the value one the weeks ending on 1995:03:10, 1996:03:29 and 2003:03:21. A shift dummy
is introduced by the program when specifying the broken trend in the cointegrating relations. The
misspecification tests for the unrestricted VAR(2) model with dummies, take the following values:
the LM(1) test for first order autocorrelation is equal to 20.175 with a p-value of 0.212, while the
LM(2) test for second order autocorrelation is equal to 21.778 with a p-value of 0.151. The tests
for normality and ARCH effects show some problems, but adding more dummies is not a solution.
As VAR results are reasonably robust anyway, we continue with this specification.
220
G. Passamani
0.0
B1*X(t)
-0.5
1995
2000
2005
3
B2*X(t)
2
1
2
1995
2000
2005
B3*X(t)
1
0
1995
2000
2005
Fig. 2 The three identified cointegrating relations within the I(1) model
show some evidence of non-stationary behaviour, at least over the first part of the
observation period. Such evidence of non stationarity can be attributed to the fact
that the choice of a cointegration rank r D 3, leaves in the model a root very close
to unit4 , which means that one very persistent component in the data has been
considered as stationary.
Therefore, in a further analysis, we chose a cointegration rank r D 2, that is two
cointegrating relations and .p r/ D 2 common stochastic trends, corresponding to
the two near unit roots. Trying to identify the long-run structure underlying the two
relations, a reasonable supposition we made is that some linear combinations of the
spreads could emerge as stationary, instead of the spreads themselves, as we made
for rank r D 3. The final structure is given by the following normalized relations,
accepted with a p-value D 0:141:
“OQ 01 xQ t D 0:837.FRYLD10t
C 0:002t08W01W1999
“OQ 02 xQ t D 0:823.FRYLD10t
C 0:002t08W01W1999
GEYLD10t /
0:163.ITYLD10t
RYLD10t /
0:177.ITYLD10t
FRYLD10t /
0:002t
GEYLD10t /
0:002t
These relations show that, when corrected for the slope coefficients of the broken
deterministic trends, weighted differences between pairs of spreads seem to be
4
The largest characteristic roots of the unrestricted VAR are: 0.985, 0.966, 0.938, 0.840, 0.292.
Time Series Convergence within I(2) Models
221
-0.4
B1*X(t)
- 0.6
- 0.8
-1.0
1995
2000
2005
B2*X(t)
- 0.4
- 0.6
- 0.8
1995
2000
2005
Fig. 3 The two identified cointegrating relations within the I(1) model
stationary, and it’s quite interesting to note that the spread between France and
Germany and either the one between Italy and France, or the one between Spain
and France, result to cointegrate, as if French yields were the linking factor.
Similar results have been found by Giese (2008), where she analyzes monthly US
treasury zero-coupon bond yields of different maturities. Giese found that weighted
differences between pairs of spreads of short, medium and long-term maturities
become stationary with the medium-term yields as the linking factor.
The two final relations are represented in Fig. 3. As we can see, they provide
evidence of some stationarity in the first part, though some long swings are still
present. Analyzing the estimated adjustment dynamics of the system, we found
that both Italian and Spanish yields show significant adjusting behaviours to the
two identified relations, as we would expect, whereas German yields satisfy the
restriction of being weakly exogenous with respect to the system of variables, with
a p-value D 0:312, that is the German series results to be non-equilibrium correcting
within the system and its cumulated residuals form a common trend.
The results obtained within the I(1) model are, anyway, unsatisfactory from
different points of view: the number of unit or near unit roots larger than expected
and, therefore, a smaller number of cointegrating relations; the non-stationarity
of the yields spreads by themselves, whereas linear combinations of the spreads
are much more stationary; the results of recursive tests, suggested by Hansen and
Johansen (1999) showing that coefficients in the restricted model are not really
stable; finally, the clear indication, coming from the last relations, that we should
consider the possibility that spreads, which are non stationary over the observation
period, could become stationary when combined with other spreads or with non
stationary variables, which could be the first differences of the same yields making
up the spreads, or any other combination of the same yields.
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G. Passamani
Table 1 The trace test for the determination of the I(2) rank indices
.p r/
r
s2 D 4
s2 D 3
s2 D 2
2
2
111:510
.0:000/
1
3
s2 D 1
49:118
.0:001/
17.076
.0:129/
s2 D 0
27:878
.0:026/
5.995
.0:471/
3 The Analysis Within the I(2) Model
The methodological approach of analysis just described, that is trend-adjusting
for the change in regime and treating the series as I(1), has proven unsatisfactory
in terms of both identification and stationarity of the cointegrating relations. In
particular, their graphs, together with the graphs of the data in first and second
differences, and the values of the model characteristic roots, are clear signals that
we should consider also the possibility of a double unit roots in the time series,
that is analysing them as I(2) variables5 . As Juselius (2006, p. 293) explains: “. . .
the typical smooth behaviour of a stochastic I(2) trend can often be approximated
with an I(1) stochastic trend around a broken linear deterministic trend . . . ”
Moreover, Juselius (2010, p.7) argues: “. . . in a p-dimensional VAR model of
x0 t D Œx1;t ; : : : ; xp;t , the number of large roots in the characteristic polynomial
depends on the number of common stochastic trends pushing the system, (p r/,
and whether they are of first order, s1 , or second order, s2 . To determine s1 and s2 ,
we can use the I(2) test procedure . . . ” Table 1 reports the I(2) trace test results for
both choices, r D 2 and r D 3.
As we can see from the table, the sequential testing of the joint hypothesis
(r, s1 , s2 ) for all values of r, s1 and s2 , has given as the first non rejection6 r D 3,
s1 D 0 and s2 D 1 with a p value D 0:129. Therefore, the two unit roots of the
model should be captured by an I(2) common stochastic trend.
These results suggest to model the regime change stochastically within a
cointegrated I(2) model, as follows (Juselius (2006, p.319):
0
2 xt D ’.“Q xQ t
1
0
C •Q Qxt
1/
Qt
C —£Q 0 X
1
C ˆDt C ©t ; ©t
Np .0;
/;
(2)
where the number r of stationary polynomially cointegrating relations - the relations
within round brackets in (2) -, the number s1 of I(1) common stochastic trends and
the number s2 of the I(2) ones among the (p r) non stationary components,
have been determined by the trace test. We started with the estimation of the
unrestricted model and, quite surprisingly, the iterative procedure converged to the
5
6
A formal treatment of I(2) models and relative tests can be found in Johansen (1997).
The second non rejection is just the case r D 3, (p r/ D 1.
Time Series Convergence within I(2) Models
223
final estimates in few iterations, while for any other choice for r, s1 and s2 the
number of iterations taken was really very high.
Before proceeding with the identification of the polynomially cointegrating
relations, we followed the approach adopted by Johansen et al. (2008), of testing
a set of non-identifying hypotheses.
First we tested the restriction whether a linear trend is needed in the sub-period
going from the beginning of 1993 to the end of 1998, but not in the following subperiod, that is the deterministic trend characterizing the relations is just a broken
trend ending in 1998. This is a test of the hypothesis that the variables t and tDt
have got the same coefficient with opposite sign since the beginning of 1999. The
results show that for each polynomially cointegrating relation the hypothesis is
largely accepted with an overall p-value D 0:296. This can be interpreted as a
clear signal that the convergence in the long-term bond yields has been achieved
by January 1999. After that date, the data show no significant linear trends and
the eventual deterministic components imply only that the equilibrium means are
different from zero. Then we tested separately four hypotheses, each of which, if
not rejected, implies that the variable in question is at most I(1). The hypothesis
was borderline rejected for German and for French yields, but strongly rejected for
Italian and for Spanish yields, implying that the last two variables can be considered
I(2), while the other two are only borderline I(2).
Another interesting hypothesis is the one of no long-run levels feed-back from
the variable considered, that is the variable is rather pushing the system than
adjusting. The testing results were such that the null hypothesis was accepted with a
p-value D 0:312 for the German long term bond yields and with a p-value D 0:134
for the French ones. As regards purely adjusting variables, Spanish yields seem such
a variable with a p-value D 0:696, while Italian yields are such only borderline, with
a p-value D 0:073.
Moving to the identification of the polynomially cointegrating relations, we were
interested to see whether relations between pairs of yields, corrected for short-run
dynamics and deterministic trends, could be considered stationary within the I(2)
model. Therefore, we identified the long-run structure by imposing the restrictions
that each vector making up the “Q matrix represents a relation between pairs of yields,
with a broken linear trend whose effect ends at the beginning of 1999. The LR
test statistic on the over-identifying restrictions gave the value ¦23 D 3:698, with a
p-value D 0:296, making the restrictions largely accepted. The estimated identified
dynamic long-run equilibrium relations are the following:
“OQ 01 xQ t C •OQ 01 Qxt D FRYLD10t
1:119GEYLD10t
0:0005t02W01W1999 C 0:0005t
C 3:111GEYLD10 C 3:481FRYLD10 C 5:307ITYLD10
C 5:147SPYLD10
“OQ 02 xO t C •OQ 02 Qxt D ITYLD10t
0:646t04W01W1999 C 0:793t
1:711GEYLD10t
0:012t02W01W1999 C 0:012t
C 11:656GEYLD10 C 13:044FRYLD10 C 19:927ITYLD10
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G. Passamani
MCI1*X(t)
50
0
-50
1995
2000
2005
MCI2*X(t)
100
0
-100
1995
2000
2005
MCI3*X(t)
100
0
-100
1995
2000
2005
Fig. 4 The three polynomially cointegrating relations within I(2) model
C 19:319SPYLD10
“OQ 03 xO t C •OQ 03 Qxt D SPYLD10t
2:184t04W01W1999
1:659GEYLD10t
0:820t
0:009t02W01W1999 C 0:009t
C 10:329GEYLD10 C 11:558FRYLD10 C 17:647ITYLD10
C 17:111SPYLD10
1:970t04W01W1999
0:179t
These equations show the existence of significant stationary relations between
Germany yields and any other country yields if in the same relations we allow for
the presence of differenced variables, that is the resulting polynomially cointegrated
relations need the differenced variables to become stationary. The estimated relations are plotted in Fig. 4.
If we compare this figure with Fig. 2, we can see the importance of the
differenced variables in making the cointegrating relations stationary over the
sample period.
We then identified the long-run structure by imposing the restrictions that each
vector making up the “Q matrix represents a spread between pairs of yields, in
particular, a spread between German yields and any other country’s yields. The LR
test statistic on the over-identifying restrictions gave the value ¦26 D 10:571, with
a p-value D 0:103, making the restrictions still accepted. As the results in terms
of estimated identified dynamic long-run equilibrium relations, were very similar to
the relations already plotted in Fig. 4, we don‘t report them.
Time Series Convergence within I(2) Models
225
Table 2 The estimated matrix of adjustment coefficients ’O and the estimated vector ’
O ?2 . Bold
numbers denote coefficients significant at 5%
DDGEYLD
DDFRYLD
DDITYLD
DDSPYLD
’
O1
0:030
0:076
0.076
0.020
’O 2
0.018
0.018
0:070
0.069
’O 3
0:020
0:009
0.040
0.100
’O ?2
1.000
0:494
0:046
0:175
When trying to analyze the adjustment dynamics of the system, we have to
keep in mind that the adjustment structure embodied by the I(2) model is much
more complex than the one embodied by the I(1) model. Within the polynomially
cointegrating relations, if •ij “ij > 0, then the differences xi;t are equilibrium
correcting to the levels xi;t 1. It is, therefore, interesting to note that first differences
in German, French, Italian and Spanish bond yields, are equilibrium correcting
to the levels of the normalized variable in each relation. Within the model, the
2
estimated ’O matrix contains information on the adjustment dynamics between xi;t
2
and the variables in levels and in differences, in particular if ’ij •ij < 0, then xi;t
is equilibrium correcting in xi;t . As we can see from the values in Table 2, French
yields are significantly equilibrium correcting in the first relation, Italian yields
are significantly equilibrium correcting in the second and Spanish are equilibrium
correcting in the third, as expected.
From the table we can have interesting information also on the I(2) stochastic
trend component which can be considered as the driving force for the system. The
estimated vector ’O ?2 shows, in fact, that it is primarily the twice cumulated shocks
to the German long-term bond yields which have generated the I(2) stochastic trend.
A significant contribution has been given also by the twice cumulated shocks to the
French yields. This result confirms that the convergence process has been mainly
led by German long-term bond yields and in part by French yields, to which the
other yields have been adjusting. As regards the weights with which the I(2) trend
have influenced the variables, the results have shown that Italian and Spanish yields
were the most influenced, followed, in order, by French and German ones.
4 Conclusions
According to theory, the bond market unification process that we have analysed
would imply a single latent factor, that is a single common stochastic trend as a
driving force underlying the co-movements of yields of the same maturities across
different countries’ market. Analyzing weekly observations of long term bond yields
relative to France, Germany, Italy and Spain within the I(1) model we have detected
some clear signals that the convergence process is more complex than expected
and that, in order to investigate the empirical regularities behind the swings in the
series, we have to use an I(2) model. In fact, analyzing the observations within an
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G. Passamani
I(2) model, we have found evidence of the existence of a common stochastic trend
given mainly by the twice cumulated shocks to the German long-term bond yields,
but also by the twice cumulated shocks to the French ones. Such results indicate
the importance of modelling the smoothing behaviour shown by the series in the
process of convergence using an approach which takes into accounts the curvature
of the co-movements in the series, as well as the level and the slope. As a conclusion,
it’s reasonable to state that the chosen I(2) approach has allowed a better modelling
of the convergence process than within the I(1) model, giving evidence to expected
characteristics that the I(1) model wouldn’t show.
Paper financed by PRIN/MIUR 2007: “Propriet`a macroeconomiche emergenti
in modelli multi-agente: il ruolo della dinamica industriale e della evoluzione della
struttura finanziaria”
The author thanks Refet S. G¨urkaynak, Department of Economics, Bilkent
University, and former Economist, Federal Reserve Board, Division of Monetary
Affairs, for kindly providing data.
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Part VI
Environmental Statistics
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