Defining Benchmark Status: An Application using Euro-Area Bonds This version: 17 September 2003 Peter G. Dunne, Queen's University, Belfast Michael J. Moore, Queen's University, Belfast Richard Portes, London Business School and CEPR An earlier version was circulated as NBER Working Paper 9087 and CEPR Discussion Paper 3490. The paper has benefited from seminar presentations at the London Business School, University of Michigan, the ESRC Money Macro Finance Research Group, the Institute for International Integration Studies, and the Irish Economic Association. It was also presented at the NYU Salomon Center conference on ‘The Euro: Valuation, Hedging and Capital Market Issues’. We are grateful for comments from our discussant, Lasse Pedersen. We have also received very helpful comments from Jim Davidson, David Goldreich, Stephen Hall, Harald Hau, Rich Lyons, Kjell Nyborg, Carol Osler and Kathy Yuan. This paper is part of a research network on ‘The Analysis of International Capital Markets: Understanding Europe’s Role in the Global Economy’, funded by the European Commission under the Research Training Network Programme (Contract No. HPRNŒCTŒ1999Œ00067). We thank Euro- MTS Ltd for providing the data.
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Defining Benchmark Status: An Application using Euro-Area Bonds
This version: 17 September 2003
Peter G. Dunne, Queen's University, Belfast
Michael J. Moore, Queen's University, Belfast
Richard Portes, London Business School and CEPR
An earlier version was circulated as NBER Working Paper 9087 and CEPR Discussion
Paper 3490. The paper has benefited from seminar presentations at the London
Business School, University of Michigan, the ESRC Money Macro Finance Research
Group, the Institute for International Integration Studies, and the Irish Economic
Association. It was also presented at the NYU Salomon Center conference on ‘The
Euro: Valuation, Hedging and Capital Market Issues’. We are grateful for comments
from our discussant, Lasse Pedersen. We have also received very helpful comments
from Jim Davidson, David Goldreich, Stephen Hall, Harald Hau, Rich Lyons, Kjell
Nyborg, Carol Osler and Kathy Yuan. This paper is part of a research network on ‘The
Analysis of International Capital Markets: Understanding Europe’s Role in the Global
Economy’, funded by the European Commission under the Research Training
Network Programme (Contract No. HPRNŒCTŒ1999Œ00067). We thank Euro-
MTS Ltd for providing the data.
Defining Benchmark Status: An Application using Euro-Area Bonds
ABSTRACT
Using a unique data set from the electronic trading platform Euro-MTS, we consider
what is the ‘benchmark’ in the new euro-denominated government bond market.
Consistent with recent theoretical developments we believe that benchmark status
can be associated with characteristics of the price discovery process, and we use the
concept of Irreducibility of Cointegrating Relations among bond yields to identify the
benchmark at each maturity. We show that no one country provides the benchmark
bond at all maturities. The benchmark differs across maturities, and at some
maturities benchmark status is shared by the bonds of more than one country.
Keywords: benchmark, euro government bonds, cointegration JEL: F36, G12, H63 Peter G. Dunne (Queen's University, Belfast) The Queen's University of Belfast School of Management & Economics Belfast BT 7 1NN Northern Ireland e-mail: [email protected] Telephone: +44 (028) 90273310 Fax: +44(028) 90328649 Michael J. Moore (Queen's University, Belfast) The Queen's University of Belfast School of Management & Economics Belfast BT 7 1NN Northern Ireland e-mail: [email protected] Telephone: +44 (028) 90273208 Fax: +44(028) 90328649 Richard Portes (London Business School and CEPR)
Department of Economics London Business School Sussex Place Regent’s Park London NW1 4SA Tel. (+44 20) 7706 6886 Fax ((+44 20) 7724 1598 Email [email protected]
Corresponding author: Richard Portes, Columbia Business School, 822 Uris Hall, 3022 Broadway, New York NY 10027-6902, tel. 212-854-1753, email [email protected]
1
Defining Benchmark Status: An Application using Euro-Area Bonds
Peter G. Dunne (QUB)
Michael J. Moore (QUB)
Richard Portes (LBS, NBER and CEPR)
1. Introduction
The introduction of the euro on 1 January 1999 eliminated exchange risk
between the currencies of participating member states and thereby created the
conditions for a substantially more integrated public debt market in the euro area.
The euro-area member states agreed that from the outset, all new issuance should
be in euro and outstanding stocks of debt should be re-denominated into euro. As a
result, the euro-area debt market is comparable to the US treasuries market both in
terms of size and issuance volume. Unlike in the United States, however, public debt
management in the euro area is decentralised under the responsibility of 12 separate
national agencies.
This decentralised management of the euro-area public debt market is one
reason for cross-country yield spreads. But the evidence of differentiation across
countries has not been thoroughly explored, and one of the contributions of this
paper is to describe patterns in cross-country yield differences. For example, we find
yields are lowest for German bonds; that there is an inner periphery of countries
centred on France for which yields are consistently higher; and that the outer
periphery centred on Italy displays the highest yields.
2
We begin our analysis by discussing why such yield spreads exist. Our main
contribution, however, comes in examining benchmark status. In this decentralised
euro government bond market, there is no official designation of benchmark
securities, nor any established market convention. Indeed, benchmark status is more
or less explicitly contested among countries.
One might ask why this should be so, aside from national pride. What are the
benefits of achieving benchmark status? This leads us to consider the appropriate
definition of ‘benchmark’. If the ‘benchmark’ were simply the security with lowest
yield, the question would answer itself: clearly governments wish to borrow at the
lowest possible yields; and there is an obvious welfare consequence, if foreigners
hold any significant share of domestic government securities.
If indeed lowest yield were all that mattered for benchmark status, then the
German market would provide the benchmark at all maturities (see below). Analysts
who take this view accept that the appropriate underlying criterion for benchmark
status is that this is the security against which others are priced, and they simply
assume that the security with lowest yield takes that role (e.g., Favero et al., 2000,
pp. 25-26). A plausible alternative, however, is to interpret benchmark to mean the
most liquid security1, which is therefore most capable of providing a reference point
for the market. But the Italian market, not the German, is easily the largest and
arguably the most liquid for short-dated bonds; and perhaps the French is most liquid
at medium maturities.
3
Liquidity is to some extent quantifiable, but liquidity alone is unlikely to be a
reliable identifier of benchmark status. For example, Italian bonds are most liquid at
almost all maturities but the Italian long yield is probably too variable to be a good
reference point, or a suitable hedge, for other parts of the market. The characteristic
of being a reference point for the market is something that closely relates to Yuan’s
(2002) definition of a benchmark, as discussed below. We also believe it is possible
to distinguish the benchmark empirically, given that the benchmark is defined this
way. So our approach focuses directly on the price discovery process to reveal
benchmark status (see Hasbrouck, 1995, for a treatment in the context of equity
markets). Indeed, one of the attractions of benchmark status is that benchmark
bonds are held by a wide international base of investors, who often provide an
unofficial market in the benchmark outside normal trading hours. This in turn makes
them more representative of the market.
Yuan’s model employs an exogenously determined benchmark. We expect that
similar attributes would be possessed by an endogenously determined benchmark,
however, and we modify Yuan’s model to fit the Euro-area bond market in this and
other respects. Endogeneity in the emergence of the benchmark is not of central
importance to our identification methodology. If the benchmark bond has benchmark
traits consistent with those outlined by Yuan, then our methodology should be
capable of identifying it as the benchmark.
4
The model of Yuan closely associates benchmark status with the price
discovery process. Once in existence the benchmark security provides an
information externality to the market as a whole because it best represents common
movements of the entire market. Essentially, the benchmark bond is the instrument
to which the prices of other bonds react. On this view, the identification of
benchmark status must emerge from empirical analysis and cannot simply be
asserted or read off the data. A benchmark security concentrates the aggregation of
information and reduces the cost of information acquisition in all markets where a
security is traded against the benchmark.
Since price discovery is central to our definition of benchmark status, we
consider alternative approaches to identifying the price discovery process. Scalia
and Vacca (1999) for example, use Granger-Causality tests to determine whether
price discovery occurs in the cash or futures market in Italian bonds. In the context
of identifying benchmark status, however, we believe that Granger-Causality testing
exhibits significant weaknesses, particularly in the context of high-frequency
transaction data with variable liquidity. We nevertheless begin our empirical analysis
by conducting tests for Granger causality between yields. If a bond yield at a
particular maturity Granger-causes the yields of bonds in other countries at the same
maturity, this suggests that the Granger-causing bond is the benchmark at that
maturity. Despite the simple appeal of this technique and our strenuous efforts to
avoid the worst effects of its weaknesses, we prefer to regard this part of our analysis
as descriptive, and we place more weight on the novel approach we introduce in
section 5.2.
This alternative empirical method exploits the fact that yields are non-stationary
for every country and at every maturity. If there were a unique benchmark at every
maturity, then we would expect that the yields of other bonds would be cointegrated
with that benchmark. Indeed, there should be multiple cointegrating vectors
centering on the benchmark bond. This empirical approach relies on a result, based
on Davidson (1998), that the structural nature of the cointegrating relationship
between a benchmark bond and other bonds can be identified even in the context of
quite a general theoretical framework. We outline this approach in detail in section 5.
5
In the next section, we discuss the structure and development of the market for
euro-area government bonds. Section 3 provides an explicit theoretical framework
within which a benchmark security is defined and we consider the implications of this
framework for the identification of the benchmarks in the euro-denominated
government bond market. Section 4 describes our unique data set. Section 5
presents the novel empirical methodology and analysis and section 6 concludes.
6
2. The market for euro-area government bonds
The euro-area government bond market, at just under USD 3 trillion, is
somewhat larger than that of the United States (Table 1). The largest outstanding
stocks are those of Italy, Germany and France, in that order (Table 2). Turnover has
risen dramatically since 1998 – by a factor of three for France, for example (Figure
1). International participation has also risen rapidly: in the three years from 1997 to
2000, the share of Belgian bonds held by non-residents rose from 29% to 53%
(Galati and Tsatsaronis, 2001); for France, it doubled to reach one-third, which was
also the average for the entire area (ibid. and Blanco, 2001).
McCauley (1999) draws some comparisons between the US municipal bond
market and the euro government bond markets, but there can be no question that the
latter are much more highly integrated. There has been considerable convergence
among countries in the structure and maturities of government debt. The share of
foreign-currency debt has fallen to negligible levels, mainly because that formerly
denominated in other euro-area currencies is now denominated in euros. Each
country is striving to achieve large liquid benchmark-size issues: recent French and
Italian issues have exceeded € 20 bn, putting them at the level of US Treasury
benchmark issues. German issues are in the range of € 10-15 bn, and even the small
countries are now up to € 3-5 bn issue size. Secondary markets have become much
deeper and more efficient (see Favero, et al., 2000).
7
There are still significant impediments to market integration. The single
currency has not brought unification of tax structures, accounting rules, settlement
systems, market conventions, or issuing procedures. On the other hand, a single
electronic trading platform now handles about half of the total volume of secondary
market transactions (see below).
Nor has market integration gone so far as to give identical yields on different
countries’ securities of the same characteristics. Yields have indeed converged. But
there are still significant spreads, and since mid-2000, though not before, all
countries have had positive spreads relative to Germany at all maturities (until very
recently). In our data (see below), for example, the Italian-German yield gap ranges
from 18 bp at the short end to 35 bp at the very long end2. Some observers conclude
that this gives Germany unambiguous status as the benchmark issuer, although
there might have been some multiplicity in the first eighteen months of EMU (Blanco,
2001, p. 14-15, Codogno, et al., 2003).
8
What are the sources of these yield differentials? It is plausible that before
EMU, much of the spread simply reflected exchange-rate risk. Indeed, by comparing
swap rates, Blanco (2001, Sec. 4.1) breaks down the spreads over German yields at
the 10-year maturity between the foreign exchange factor and other factors, which he
identifies with credit (default) risk and microstructure characteristics, in particular
liquidity. He finds that for those countries with wide pre-1999 spreads, the main
component was exchange-rate risk (Table 3). Moreover, taking that factor out,
spreads have in fact widened significantly for all countries since the advent of the
euro. And insofar as bond ratings represent default risk, it seems clear that only part
of these wider spreads is attributable to this factor (in Figure 2, we see substantial
differences in yields between countries in the same risk category). But the
interpretation of the spreads as representing different credit risks and liquidity
characteristics is also problematic, and establishing which of these factors is
dominant is even more difficult (Portes, 2003). The spreads vary over time and along
the yield curve. But credit ratings vary very little indeed over time and typically do not
discriminate across maturities; and we are far from being able to identify time-varying
and maturity-dependent determinants of liquidity.
Whatever the causes of the spreads for other countries over German yields,
the mere fact that they are positive is enough for most observers to conclude that
Germany provides the benchmark all along the yield curve. We shall find that the
dynamic evidence on price discovery suggests a very different view.
9
3. Benchmark securities: a framework
Yuan (2002) formalises the concept of a benchmark security. Adopting her
definition to our context, define a country-specific security as having a yield with the
following factor structure:
(1) % 1.....,fi i i ir r i nβ γ ε= + + =
where is the return on the ith country’s security, irfir is the country-specific risk-free
rate. The risk-free rate can differ across countries because of, for example, political
factors such as the possibility that a country might leave the euro-zone. %γ is euro-
zone wide risk and iβ is country i’s sensitivity to that risk. iε is the country-specific
shock.
As usual, i ∀ = are stationary processes that are independently
distributed normally with mean zero and variance
1.....,iε n
2iσ . But we depart from
convention, including Yuan’s specification, with regard to the systematic risk %γ . We
assume that %γ is an I(1) process3. Consequently, all of the yields are themselves
non-stationary.
At this point, it is worth showing the following result:
Lemma 1. All pairs of country yields { }, 1....,ir i n= are cointegrated.
Proof: For any equation (1) implies that and ir jr
ff
j ji i i
i j i j i j
r rr r jεεβ β β β β β
− = − + −
10
The right hand side is stationary by assumption. The cointegrating vector is 1 1,i jβ β
−
.
Note that the variance of the cointegrating residual is:
22
2Var ji i
i j i
rr2j
j
σσβ β β β
− = +
(2)
We are now in a position to define a benchmark security:
Definition 1 (Yuan): A benchmark security has the following two properties:
(i) it has no sensitivity to country-specific risk,
(ii) it has unit sensitivity to systematic risk.
In our case, systematic risk is the euro-zone risk %γ . The benchmark security can be
constructed as follows. Form the following basket of country-specific securities:
%1 1 1
1 1with 0 and 1
n n nf
b i i i i i ii i i
n n
i i i ii i
r w r w w
w w
β γ ε
ε β
= = =
= =
= + +
→ →
∑ ∑ ∑
∑ ∑where are the weights on each country’s security. In effect, the
benchmark security’s yield, r , is:
1.....,iw i n=
b
(3)
%
1
where
fb b
nfb i
i
r r
r w
γ
=
= +
=∑ fir
It is noteworthy that within this framework there is no explicit role for the level of the
yield. While benchmark status may give rise to lower yields, here it is assumed that
benchmark attributes stem purely from characteristics related to the security’s
information content.
11
Lemma 2. All country yields { }, 1....,ir i n= are pairwise cointegrated with the
benchmark yield r . b
Proof:
From equations (1) and (3),
f
fi ib b
i i
r rr r i
i
εβ β β
− = − +
The right hand side is stationary by assumption. The cointegrating vector is
1 , 1iβ
−
.
Note that the variance of the cointegrating residual is:
2
2Var ib
i i
r r iσβ β
− =
(4)
We are now in a position to state the main result:
Theorem 1:
The variance of the residual error in the cointegrating vector between the yield on
country i’s security and any other country specific security j=1…,n. is always greater
than the variance of the residual error in the cointegrating vector between country j’s
yield and the benchmark yield.
Proof: Compare equations (2) and (4).
The above results have been developed using the property that the
benchmark is a basket of bonds. The concept of a benchmark security as a basket
of bonds is not entirely new. Galati and Tsatsaronis (2001) raise the idea in the
context of euro-area government bonds, only to dismiss it immediately: ‘Market
12
participants, however, are not yet ready to accept a benchmark yield curve made up
of more than one issuer, being wary of the problems posed by small but persistent
technical differences between the issues that complicate hedging and arbitrage
across the maturity spectrum (p. 10).’
The analysis above is predicated on the idea that the benchmark bond is
issued exogenously. In the euro-area bond market, however, this cannot occur. We
argue, instead, that a particular country’s bond emerges endogenously as the
benchmark, at each maturity, with the characteristics outlined in Definition 1.
Whether or not the benchmark is endogenously determined, Yuan’s analysis
regarding its characteristics is likely to hold true. The contest for benchmark status
may itself be worth modelling, but here we restrict attention to the more modest task
of identifying the benchmarks at each maturity. We believe that the empirical
approach we use is capable of identifying the benchmark independent of the nature
of the contest for benchmark status.
In section 5.2, we introduce the methodology for implementing Theorem 1. The
methodology was originally proposed by Davidson (1998) and modified by Barassi,
Caporale and Hall (2000 a, b). Davidson’s original work has direct parallels with the
statistical result used by Yuan to motivate her concept of a benchmark. We return to
the idea of the benchmark as a basket of bonds in section 5.3.
13
4. Data
4.1 Primary data
We have a unique transactions-based data set from Euro-MTS for October and
November of 2000. Since the creation of the euro in 1999, Euro-MTS has emerged
as the principal electronic trading platform for bonds denominated in euros. At the
end of 2000, it handled over 40% of total transactions volume (Galati and
Tsatsaronis, 2001). Government bonds traded on Euro-MTS must have an issue size
of at least € 5 bn. For a discussion of MTS, see Scalia and Vacca (1999).
The full data set consists of all actual transactions. For each transaction, we
have a time stamp, the volume traded, the price at which the trade was conducted
and an indicator showing whether the trade is initiated by the buyer or seller. The
countries represented are Germany, Finland, Portugal, Spain, Austria, Italy, France,
the Netherlands and Belgium: all euro-area countries except Ireland4. Greece joined
the euro-area after the time-period covered by the sample, while the twelfth euro-
area country, Luxembourg, has negligible government debt.
The sample includes all Euro-MTS and country-specific MTS bonds traded on
the electronic platforms. In addition to treasury paper, the data set also includes
French and German mortgage-backed bonds, a European Investment Bank bond,
and a euro-denominated US agency bond (“Freddie-Mac”).
4.2 Derived data
In the analysis below we use the most frequently traded bond on the EuroMTS
platform for each of three countries (Italy, France and Germany) and for each of four
14
maturities. These are short, medium, long and very long. On the EuroMTS platform,
all bonds are grouped into one of these four categories5, as follows:
Maturity Baskets on Euro-MTS
Short 1.25-3.5 years
Medium 3.5-6.5 years
Long 6.5-13.5 years
Very long >13.5 years
Table 2 shows how dominant these three countries are in terms of outstanding
issues of stock. Together they account for over 70% of the market. We found that
the coverage of the data for the other countries was too sparse to get a consistently
clear picture of intra-daily activity. The coverage of our data set for the three
selected countries is set out in Table 4. It is evident that even at the very long
maturity, there is much greater transactions volume for Italy on Euro-MTS than for
either of the other countries (reflecting the fact that the MTS was initially set up to
trade solely Italian bonds). But there is no particular problem of
‘unrepresentativeness’ in our data for the other two countries. For our time-series
analysis, we track only a single security for each country at each maturity, and there
are enough transactions in the most highly traded bonds to give a fully representative
series.
In each case the data are observed twice daily, once in the morning trading
period and once in the afternoon trading period. Our sample covers October and
November of the year 2000. Thus we have 44 trading days and 88 observations for
each bond. This was a consistently active period for the MTS electronic trading
15
platform, and it was a period within which there was increased uncertainty regarding
benchmark status due to the relatively recent nature of the conversion to euro
denomination. More recently the decline of the non-German bond futures markets
may suggest that Germany is once again in the dominant benchmark position, but we
find that German dominance was not at all clear in the period we study. Thus, our
analysis provides some indication of how susceptible German dominance might be to
future threats such as the possibility of a downgrading of German bonds if German
fiscal and macroeconomic conditions were to deteriorate significantly.
The availability and timing of observations is important, especially for causality
testing6. In our case, the transactions for each variable were chosen according to
their closeness in time to (either before or after) the last available transaction in each
period in the least-liquid bond7. Hence, the observations are not necessarily close to
the end of the trading period. This minimises the time gap between observations.
16
Interpolation was done in relatively few cases (never for the Italian) and almost
always involved the use of the most similar bonds from the same country (i.e. similar
in terms of maturity, coupon, liquidity, and the yield gap against the other two
countries). In the case of the long bonds, interpolation of the French benchmark was
sometimes done using the most similar Dutch bond. In the instances where
interpolation was not possible, the previously observed yield was continued forward.
The table below shows how often this was necessary. It is worth pointing out that the
periods of greatest illiquidity were also the periods of least variability, so that our
practice of assuming zero change is not likely to have had large effects on our
results, with one exception. Despite our efforts to avoid the problems arising from
non-trading, the conclusions of our Granger-causality testing are consistent with
bonds of higher liquidity appearing to cause the less liquid bond yields8.
Incidence of Non-trading
German Short 6 French Short 3 German Medium 2 French Medium 0
German Long 1 French Long 3 German Very Long 5 French Very Long 9
4.3 Data Summary
Descriptive statistics for each yield and yield gaps9 are graphically displayed in
the web appendix. The Italian yield is always highest, the German the lowest, with
the French yield in the intermediate position. The French yield is typically closer to
the German than to the Italian yield. The only exception to this is for four days in
early October 2000 during which the French-German yield gap was slightly higher
than the Italian-German yield differential in the long-dated category.
The final set of descriptive statistics anticipates the analysis. For each
maturity, we test each bond and yield gap for stationarity. We use the Dickey-Fuller
test or the Augmented Dickey-Fuller test where necessary. The results are reported
in Table 510.
The outcome of the tests is simple to summarise. In every case, the yield is
non-stationary. The results for the yield gaps, however, are not so clear. This is
reflected in the fact that all of the tests on the yield gaps were carried out first with
just a constant in the specification and then repeated with both a constant and a
trend. For example, at the short end, it is unclear whether the Italian-German or the
Italian-French yield gaps are stationary, whereas the French-German gap appears to
be stationary. The implications of this will be developed in the next section. But
17
Lemma 1 in Section 2 shows that we should not necessarily expect yield gaps to be
stationary.
18
5. Results and analysis
5.1 Granger causality
We begin by examining the flow of causality among the yields at each maturity.
We bypass the issues raised by changes in the term structure by carrying this out for
each maturity separately. In each case, we construct a three-variable vector
autoregression. Using a range of selection criteria the following lag lengths were
selected:11.
VAR lag length
Maturity Number of lags
Short 1
Medium 1
Long 3
Very long 1
The most obvious problem that could arise from data of varying liquidity is that
the most liquid variable will tend to be most up-to-date and appear to Granger-cause
the other variables. The non-synchronous trading model of Lo and Mackinlay (1990)
assumes a single factor model for returns on individual stocks with different
independent non-trading probabilities and different sensitivities to movements in the
common factor. Assuming that stocks are ordered by their non-trading probabilities,
this set-up trivially induces spurious asymmetric cross-autocorrelation that relates to
(i) the product of the factor sensitivities (ii) a nonlinear function of the differential non-
19
trading probabilities and (iii) the ratio of the variance of the common factor to the
product of the individual security standard deviations. In our data we would expect
the factor sensitivities to be near 1. We would expect the ratio of the variance of the
common factor to the product of individual standard deviations to be also near unity.
Under these circumstances, where π is the non-trading probability, the calculation of
the spurious nth order cross-autocorrelation between the ith and jth bond returns can
be performed using;
(1 )(1 )
1i jn n
ij ii j
π πρ π
π π− −
≈−
In our case, we can see that any cross-autocorrelation with the Italian return as the
reference variable would be zero due to a zero Italian non-trading probability
( 0iπ = ). However, the French and German yields should appear to be Granger-
caused by the Italian because these usually have positive non-trading probabilities.
Since the German bonds have the highest non-trading probabilities the Granger-
causality testing should be biased against the German as the benchmark.
The Granger Causality Test results on the whole appear to confirm these
priors12. At the short end no country emerges as benchmark. Non-causality is
rejected in every case: lagged yields of each country affect the yields of one or both
of the other countries. For the medium maturity, where neither the Italian nor the
French bond suffer from the stale price problem, the German bond can be ruled out
as a possible benchmark, but both the Italian and French yields have predictive
power for other countries’ yields. At the long end the Italian bond emerges as a
benchmark and has predictive power for both French and German yields. Finally, for
the very long maturity, as with the medium maturity, only the German bond can be
ruled out as benchmark.
20
These results strongly reject the hypothesis that innovations in German yields
Granger-cause innovations in French and Italian yields, at all maturities. That
interpretation of Germany as the benchmark issuer is not consistent with our data.
We would not wish to place too high a weight on these results, however, partly
because of the weaknesses of the approach in the presence of missing observations,
and more particularly because the analysis of section 2 gives no role for causality in
the definition of a benchmark.
5.2 Cointegration
The Granger-causality analysis is simple but perhaps rather crude. It ignores
long-run relationships. The factor definition of a benchmark in section 3 along with
Lemmas 1 and 2 as well as Theorem 1 suggests that the benchmark should be
identified from an analysis of the cointegration properties of the yield series. Despite
the use of a basket of bonds as benchmark in the analysis of section 3, in this section
we entertain the idea that a single country’s bonds could possess benchmark
characteristics. If a particular country provides the benchmark at a given maturity,
then there should be two cointegrating vectors in the three-variable system of country
yields. For example, if Germany were the benchmark, then the cointegrating vectors
could be13
Italian yield = γGerman yield + nuisance parameters
French yield = δGerman yield + nuisance parameters
.
21
But this approach suffers from an identification problem. Even if we are
satisfied that cointegration vectors along the lines of the above exist, we still cannot
draw any immediate conclusion about the structure of the relationships between
yields such as the identity of the benchmark. The reason for this is that any linear
combination of multiple cointegrating vectors is itself a cointegrating vector. In
particular,
Italian yield = (γ/δ)French yield + nuisance parameters
provides us with a perfectly valid cointegrating vector derived from the above. On the
face of it, any one of the yields can provide the benchmark, and we have made no
progress.
A recent development in non-stationary econometrics due to Davidson (1998)
and developed by Barassi, Caporale and Hall (2000 a,b) [BCH] enables us to
explore the matter further. This involves testing for irreducibility of cointegrating
relations and ranking according to the criterion of minimum variance. The interesting
feature of this method is that it allows us to learn about the structural relationship that
links cointegrated series from the data alone, without imposing any arbitrary
identifying conditions. In this case, the ‘structural’ relationship that we are exploring is
the identity of the benchmark in a set of bond yields.
There is a risk of confusion in the use of the word ‘structure’, because of the
many different uses to which it has been put by different authors. Davidson uses the
term to mean parameters or relations that have a direct economic interpretation and
may therefore satisfy restrictions based on economic theory. It need not mean a
22
relationship that is regime-invariant. The possibility that “incredible assumptions”
(Sims, 1980) need not always be the price of obtaining structural estimates turns out
to be a distinctive feature of models with stochastic trends.
We begin with the concept of an irreducible cointegrating vector.
Definition 2 (Davidson): A set of I(1) variables is called irreducibly cointegrated (IC)
if they are cointegrated, but dropping any of the variables leaves a set that is not
cointegrated.
IC vectors can be divided into two classes: structural and solved. A structural IC
vector is one that has a direct economic interpretation.
Theorem 2 (Davidson). If an IC relation contains a variable which appears in no
other IC relation, it is structural.
The less interesting solved vectors are defined as follows:
Definition 3 (Davidson). A solved vector is a linear combination of structural
vectors from which one or more common variables are eliminated by choice of
offsetting weights such that the included variables are not a superset of any of the
component relations.
A solved vector is an IC vector which is a linear combination of structural IC
vectors. Once an IC relation is found, interest focuses on the problem of
distinguishing between structural and solved forms. Of course, the theoretical model
might answer this question for us, but this would then simply be using the theory to
23
identify the model, so in the absence of overidentifying restrictions we could learn
nothing about the validity of the theory itself. The key issue is whether we can
identify the structure from the data directly.
BCH introduce an extension of Davidson's framework that can be illustrated
concretely with our problem as follows. In our system made up of three I(1)
variables, the French, German and Italian bond yields, consider the case where the
pairs (German yields, French yields) and (German yields, Italian yields) are both
cointegrated. It follows necessarily that the pair (French yields, Italian yields) is also
cointegrated. The cointegrating rank of these three variables is 2, and one of these
three IC relations necessarily is solved from the other two. The problem is that we
cannot know which, without a prior theory. Here is where the BCH extension of
Davidson's methodology shows its effectiveness. In order to detect which of the
cointegrating relations is the solved one and which of the vectors are irreducible and
structural, we calculate the descriptive statistics of each cointegrating relation and
rank these vectors on the basis of the magnitude of the variance of their residual
errors. The structural vectors are identified as the ones corresponding to the lowest
variance. The reason for this is suggested by standard statistical theory and can be
illustrated as follows: Let x, y and z be our cointegrated series and let
y - βx = e1
y - γz = e2
x - δz = e3
be the three irreducible cointegrating relations. Now assume that the structural
relationships are the first two, (y-βx and y-γz), with e1 and e2 being the structural
24
error terms from the first two which are therefore assumed14 to be distributed
independently N(0, 2iσ ), i=1,2. The third equation is just solved from the first two15.
This implies that e3 is a function of e1 and e2, and therefore we expect it to be
distributed N(0,2 21
22σ σ
β+
). If { }2 23 11 then ,Max 2
2β σ σ≤ > σ . The
condition 1 β ≤ holds in every case that we estimate below. Since there exists an
alternative linear combination of the structural relations that could result in the solved
relation with γ as the denominator in the variance expression for the solved relation,
it is worth noting that the condition 1γ ≤ also holds 16 . Therefore cointegrating
relations whose residuals display lower variance should be the structural ones, the
remaining others being just solved cointegrating relations.
Our empirical strategy is therefore as follows. First, we use the Johansen
procedure to identify the number of cointegrating vectors at each maturity in our
three-variable system. Secondly, we use Phillips-Hansen fully modified estimation to
estimate the irreducible cointegrating vectors as recommended by Davidson. Finally
we rank the irreducible cointegrating vectors using the variance ranking criterion of
BCH. From this we identify the structural vectors and therefore the benchmark. The
latter must be the common yield in the two structural irreducible cointegrating
vectors.
The results of the Johansen Procedure and Phillips estimation are shown for
each maturity in Tables 6 to 9.
(i) Johansen Procedure: In Tables 6 , 8 and 9, it is clear that that there are two
cointegrating vectors among the three yields at the short, long and very long
25
maturities. Table 7 provides more ambiguous evidence. For the medium
maturity, there is at least one cointegrating vector using the trace and λ max
tests, but only the latter suggests that there are two cointegrating vectors.
On balance, we conclude that there are two cointegrating vectors at each maturity.
(ii) Phillips-Hansen Estimation:
Short (Table 6): All three pairs are irreducibly cointegrated using standard
ADF tests. Interestingly, the coefficients are statistically significantly less than
unity in each case17.
Medium (Table 7): Two of the pairs are irreducibly cointegrated using
standard ADF tests. This supports the evidence provided by the λ max but
not the trace test above. The remaining pair must be cointegrated as a
consequence. Two out of the three cointegrating vectors displayed slopes
that were significantly less than unity. The third was less than unity but not
significantly so.
Long (Table 8): Two of the pairs are irreducibly cointegrated using standard
ADF tests. From both the Johansen results and arithmetic of multiple
cointegration, the third pair must also be cointegrated. All three pairs have
slopes that are insignificantly different from unity.
Very Long (Table 9): All three pairs are irreducibly cointegrated using
standard ADF tests. In all cases, the coefficients are insignificantly different
from unity.
We conclude that, for each maturity, each pair of yields is cointegrated with a slope
equal to or less than unity18.
26
(iii) BCH minimum variance ranking:
Short: The ranking of the variances of the residuals of the three cointegrating
vectors from smallest to largest is:
Italian-German
Italian-French
French-German
From this we conclude that that the Italian-German and Italian-French pairs
are structural and that the Italian yield provides the benchmark at the short
end.
Medium: The ranking of the variances of the residuals of the three
cointegrating vectors from smallest to largest is:
French-German
French-Italian
Italian-German
On this basis, the French yield is the benchmark at the medium maturity.
Long and Very Long: For both these maturities, the ranking of the variances
of the residuals of the three cointegrating vectors from smallest to largest is:
German-Italian
German-French
French-Italian
Thus the German market provides the benchmarks at both the long and very
long maturities.
The results here contrast sharply with those based on Granger-causality, as
There are similar equations describing the evolution of the other yields.
29
The ECM equation above can be interpreted as follows: construct a portfolio
consisting of a long position in German bonds and an equal short position in Italian
bonds. Call this the first canonical benchmark portfolio. Its return equals the
German/Italian yield gap by construction. The parameter 1λ can be understood as
the loading sensitivity to that portfolio. A similar interpretation also applies to 2λ with
respect to a portfolio that is long in German bonds with an equal short position in
French bonds. Call this the second canonical benchmark portfolio.
In fact, however, we find that the two canonical portfolios constructed above
are not always the benchmark portfolios. Instead, we identify the benchmark
portfolios through estimation using the Phillips-Hansen FMOLS procedure. For
example, at the short maturity, the benchmark portfolios consist of
(i) a portfolio which is long in the Italian bond and (almost in equal measure)
short in the French bond
(ii) a portfolio which is long in the Italian bond and has an almost equal short
position in the German bond
The specific portfolios change depending on the structural relations chosen on
the basis of the cointegration analysis. As shown in Table 10, at the short maturity,
only the Italian-French portfolio (portfolio (i) above) is significant for French yields.
Similarly, only the Italian-German portfolio (portfolio (ii) above) is significant for
German yields. The Italian yield changes are not related to either portfolio, so that
the Italian yield is weakly exogenous - a likely property of a benchmark.
30
At the other maturities things are not as straightforward. Yield changes appear
to react significantly to perturbations in both determining portfolios. While this may
simply reflect complexity in the adjustment of the entire system of yields to
disequilibria, it also suggests that benchmark status could be shared by more than
one country. This is particularly relevant to the medium maturity (Table 11), where
the German yield changes relate significantly only to the determining portfolio that
does not involve the German yield. It could therefore be concluded that the
benchmark is some combination of the Italian and French bonds.
The view that there must be a single benchmark issuer, at least at a given
maturity, is equivalent in our analysis to stipulating that the ‘benchmark portfolios’
enter into the yield change equations in a particularly simple form. In general, this is
not what the data are telling us19. The benchmark portfolios are typically simple, but
not that simple.
31
6. Conclusion
We focus on the meaning of ‘benchmark’ bond in the context of the market for
euro-area government securities, extending the theoretical definition of a benchmark
in Yuan (2002). This market has developed rapidly since the beginning of monetary
union, but it is still not fully integrated, and there is no consensus20 regarding which
securities have benchmark status. We investigate two possible empirical criteria -
using Granger-causality and the Modified Davidson Method which uses
cointegration. We find rather different results with the two methods. The former
generates results which are either inconclusive or lack plausibility. The Modified
Davidson Method gives clear-cut results that are both plausible and congruent with
the theory that we present. But with neither do we find the unambiguous benchmark
status for German securities that would come from a simple focus on the securities
with lowest yield at a given maturity. Our interpretation of the cointegration results in
an error correction framework leads naturally to looking for benchmark portfolios
rather than a single benchmark security. This may be particularly appropriate in this
newly and only partially integrated market.
Clearly more research is needed, and the Euro-MTS database that we use is a
rich source. Meanwhile, however, we believe it is clear from the research reported
here that at least in the euro area, no simple definition of benchmark status will do.
Perhaps the markets are coming to understand this too:
‘German government bonds, long the unrivalled royalty of the European debt market, now find pretenders to the throne. The German government is careful…to protect the benchmark status of its bonds…But all the good intentions…are nothing in the face of the inexorable march of European monetary union. The euro-driven integration of European financial markets is creating vigorous competition to Germany’s long reign as king of the region’s bond markets. “Benchmark status is more contended now than it ever was,” said Adolf Rosenstock, European economist in Frankfurt at Nomura Research…’ (International Herald Tribune, 21 March 2002)
32
References Barassi, M. R., G. M. Caporale, and S. G. Hall, 2000a, Interest Rate Linkages: Identifying Structural Relations, Discussion Paper no. 2000.02, Centre for International Macroeconomics, University of Oxford. Barassi, M. R., G. M. Caporale, and S. G. Hall, 2000b, Irreducibility and Structural Cointegrating Relations: An Application to the G-7 Long Term Interest Rates, Working Paper ICMS4, Imperial College of Science, Technology and Medicine. Blanco, R., 2001, The euro-area government securities markets: recent developments and implications for market functioning, Working Paper no. 0120, Servicio de Estudios, Banco de Espana. Blanco, R., 2002, The euro-area government securities markets: recent developments and implications for market functioning, mimeo, Launching Workshop of the ECB-CFS Research Network on Capital Markets and Financial Integration in Europe, European Central Bank. Codogno, L., C. Favero, and A. Missale, 2003, Yield spreads on EMU government bonds, at www.economic-policy.org/pdfs/Preliminary_drafts/37thPanel_meeting/Codognoetal.pdf, forthcoming in Economic Policy October 2003. Davidson, J., 1998, Structural relations, cointegration and identification: some simple results and their application, Journal of Econometrics 87,87-113. Favero, C., A. Missale, and G. Piga, 2000, EMU and public debt management: one money, one debt?, CEPR Policy Paper No. 3. Galati, G., and K. Tsatsaronis, 2001, The impact of the euro on Europe’s financial markets, Working Paper No. 100, Bank for International Settlements. Engle, R.F., and C. W. J. Granger, 1987, Co-integration and error-correction: representation, estimation, and testing, Econometrica, 55, 1987, 251-76. Hasbrouck, J., One Security, Many Markets: Determining the Location of Price Discovery, Journal of Finance, 50, 4, 1995, 1175-1199 Lo, A., and A. C. MacKinlay, 1990, An Econometric Analysis of Nonsynchronous Trading, Journal of Econometrics, 45, 181-212. McCauley, R., 1999, The Euro and the Liquidity of European Fixed Income Markets, in Part 2.2. of “Market Liquidity: Research Findings and Selected Policy Implications", Committee on the Global Financial System, Bank for International Settlements, Publications No. 11 (May 1999). Portes, R., 2003, Discussion of Codogno et al., forthcoming in Economic Policy October 2003.
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Remolona E. M., 2002, Micro and Macro structures in fixed income markets: The issues at stake in Europe, mimeo, Launching Workshop of the ECB-CFS Research Network on Capital Markets and Financial Integration in Europe, European Central Bank.
Scalia, A., and V. Vacca 1999, Does market transparency matter? a case study, Discussion Paper 359, Banca d’Italia. Sims, C., 1980, Macroeconomics and Reality. Econometrica 48 (1), 1-48. Yuan, K., (2002), The Liquidity Service of Sovereign Bonds, mimeo, University of Michigan Business School, available at http://webuser.bus.umich.edu/kyuan/exter_final.pdf
Johansen Test of Cointegrating Rank. Endogenous Variables: Italian, French and German Yields. Exogenous variables in cointegration space: Drift and trade-type indicators. Unrestricted constant outside cointegration space. Lag length: 1 Effective sample: 2 to 88
Conclusion Both the L-Max and Trace statistics imply that there are two cointegrating vectors. Phillips-Hansen FMOLS. Pairwise cointegrating regressions with nuisance parameters: constant, trend and trade type dummies. Trade type (i) is the trade type of the dependent and trade type (ii) is trade type of the regressor. Std. Errors In brackets & Std. Errors from Unity in the case of the coefficient on the yield regressor. Regressing …. French on German Italian on French Italian on German Yield (It,Ge or Fr) 0.879 (0.034) 0.897 (0.035) 0.848 (0.033) t-test: yield slope = unity 3.46 2.88 4.52 Constant 0.665 (0.175) 0.682 (0.181) 0.979 (0.168) Trend -0.0003 (0.0001) 0.0007(0.0001) -0.0009 (0.0001) Trade Type (i) -0.088 (0.032) -0.063 (0.029) -0.006 (0.028) Trade Type (ii) 0.095 (0.030) 0.018 (0.033) 0.051 (0.030) ADF (Crit.10% -3.5) DF -7.76 DF –5.07 DF -3.97
Conclusion All three pairs are cointegrated. Minimum variance analysis Standard Deviation. 1.22 1.17 1.09
Conclusion Pairs involving Italian Yield have lowest residual variance.
39
Tables 7: Medium Maturity Johansen Test of Cointegrating Rank. Endogenous Variables: Italian, French and German Yields. Exogenous variables in cointegration space: Drift and trade-type indicators. Unrestricted constant outside cointegration space. Lag length: 1 Effective sample: 2 to 88
The L-Max statistic implies 2 cointegrating vectors while the Trace statistics implies 1. Phillips-Hansen FMOLS. Pairwise cointegrating regressions with nuisance parameters: constant, trend and trade type dummies. Trade type (i) is the trade type of the dependent and trade type (ii) is trade type of the regressor. Std. Errors In brackets & Std. Errors from Unity in the case of the coefficient on the yield regressor. Regressing …. French on German French on Italian Italian on German Yield (It,Ge or Fr) 0.855 (0.022) 0.958 (0.028) 0.864 (0.029) t-test: yield slope = unity 6.39 1.43 4.56 Constant 0.814 (0.115) 0.024 (0.155) 0.967 (0.151) Trend -0.0002 (0.0001) 0.0000 (0.0001) -0.0003 (0.0001) Trade Type (i) -0.046 (0.021) -0.060 (0.024) -0.098 (0.027) Trade Type (ii) -0.047 (0.020) 0.079 (0.025) -0.051 (0.027) ADF (Crit.10% -3.5) DF -6.71 ADF(2) -3.73 ADF(1) -2.88
Conclusion All three pairs are cointegrated. Minimum variance analysis
Standard Deviation 0.847 0.949 1.009
Conclusion Pairs involving Italian Yield have lowest residual variance.
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Tables 8: Long Maturity Johansen Test of Cointegrating Rank. Endogenous Variables: Italian, French and German Yields. Exogenous variables in cointegration space: Drift and trade-type indicators. Unrestricted constant outside cointegration space. Lag length: 3 Effective sample: 4 to 88
Conclusion Both the L-Max and Trace statistics imply that there are two cointegrating vectors. Phillips-Hansen FMOLS. Pairwise cointegrating regressions with nuisance parameters: constant, trend and trade type dummies. Trade type (i) is the trade type of the dependent and trade type (ii) is trade type of the regressor. Std. Errors In brackets & Std. Errors from Unity in the case of the coefficient on the yield regressor. Regressing …. German on
French French on Italian German on Italian Yield (Ge or Fr) 0.927 (0.042) 0.977 (0.05) 0.948 (0.04) t-test: yield slope = unity 1.71 0.45 1.27 Constant 0.256 (0.227) -0.033 (0.283) -0.005 (0.225) Trend 0.0000 (0.0001) -0.0004 (0.0001) -0.0003 (0.0001) Trade Type (i) -0.011 (0.030) 0.047 (0.035) -0.022 (0.028) Trade Type (ii) 0.036 (0.029) -0.075 (0.035) 0.009 (0.028) ADF (Crit.10% -3.5) DF -4.50 ADF(1) -4.09 ADF(5) -3.27
Conclusion German-Italian pair fail cointegration test but must be cointegrated since the two other pairs are. Minimum variance analysis Standard Deviation 1.09 1.33 0.98
Conclusion Pairs involving German Yield have lowest residual variance.
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Tables 9: Very-Long Maturity Johansen Test of Cointegrating Rank. Endogenous Variables: Italian, French and German Yields. Exogenous variables in cointegration space: Drift and trade-type indicators. Unrestricted constant outside cointegration space. Lag length: 1 Effective sample: 2 to 88
Conclusion Both the L-Max and Trace statistics imply that there are two cointegrating vectors. Phillips-Hansen FMOLS. Pairwise cointegrating regressions with nuisance parameters: constant, trend and trade type dummies. Trade type (i) is the trade type of the dependent and trade type (ii) is trade type of the regressor. Std. Errors In brackets & Std. Errors from Unity in the case of the coefficient on the yield regressor. Regressing …. German on
French French on Italian German on Italian Yield (It,Ge or Fr) 0.955 (0.028) 1.027 (0.057) 1.006 (0.054) t-test: yield slope = unity 1.56 0.47 0.12 Constant 0.150 (0.163) -0.398 (0.342) -0.372 (0.322) Trend 0.0000 (0.0001) -0.0004 (0.0001) -0.0003 (0.0001) Trade Type (i) -0.003 (0.023) 0.167 (0.043) -0.076 (0.040) Trade Type (ii) 0.032 (0.023) -0.001 (0.043) 0.043 (0.041) ADF (Crit.10% -3.5) DF -6.92 ADF(1) –3.62 DF -3.61
Conclusion All three pairs are cointegrated. Minimum variance analysis Standard Deviation 0.97 1.59 1.46
Conclusion Pairs involving German Yield have lowest residual variance.
42
Table 10 SHORT MATURITY - LOADINGS ON BENCHMARK PORTFOLIOS. Insignificant coefficient estimates were restricted to zero.
Structural cointegration relations (i.e., benchmark portfolios). Portfolio 1: Italian Yield - 0.897 French Yield Portfolio 2: Italian Yield - 0.848 German Yield
Loadings Point estimate (t-statistic) Italian Yield Change on Portfolio 1 0 ( ----) Italian Yield Change on Portfolio 2 0 ( ----)
French Yield Change on Portfolio 1 0.065 (5.31) French Yield Change on Portfolio 2 0 ( ----)
German Yield Change on Portfolio 1 0 ( ----) German Yield Change on Portfolio 2 0.064 (5.18)
Table 11 MEDIUM MATURITY - LOADINGS ON BENCHMARK PORTFOLIOS. Insignificant coefficient estimates were restricted to zero.
Structural cointegration relations (i.e., benchmark portfolios). Portfolio 1: French Yield - 0.855 German Yield Portfolio 2: French Yield - 0.958 Italian Yield
Loadings Point estimate (t-statistic) Italian Yield Change on Portfolio 1 0 ( ----) Italian Yield Change on Portfolio 2 -0.065 (-2.94)
French Yield Change on Portfolio 1 -0.066 (-2.33) French Yield Change on Portfolio 2 -0.083 (-3.18)
German Yield Change on Portfolio 1 0 ( ----) German Yield Change on Portfolio 2 -0.078 (-3.16)
43
Table 12
LONG MATURITY - LOADINGS ON BENCHMARK PORTFOLIOS. Insignificant coefficient estimates were restricted to zero.
Structural cointegration relations (i.e., benchmark portfolios). Portfolio 1: German Yield - 0.967 French Yield Portfolio 2: Italian Yield - 0.927 German Yield
Loadings Point estimate (t-statistic) Italian Yield Change on Portfolio 1 0 ( ----) Italian Yield Change on Portfolio 2 0.028 (2.01)
French Yield Change on Portfolio 1 0.040 (2.26) French Yield Change on Portfolio 2 0.056 (2.94)
German Yield Change on Portfolio 1 0 ( ----) German Yield Change on Portfolio 2 0.050 (3.15)
Table 13
VERY-LONG MATURITY - LOADINGS ON BENCHMARK PORTFOLIOS. Insignificant coefficient estimates were restricted to zero.
Structural cointegration relations (i.e., benchmark portfolios). Portfolio 1: German Yield - 0.955 French Yield Portfolio 2: Italian Yield - 0.967 German Yield
Loadings Point estimate (t-statistic) Italian Yield Change on Portfolio 1 0 ( ----) Italian Yield Change on Portfolio 2 0.021 (2.17)
French Yield Change on Portfolio 1 0.003 (2.14) French Yield Change on Portfolio 2 0.028 (2.61)
German Yield Change on Portfolio 1 -0.039 (-2.46) German Yield Change on Portfolio 2 0.035 (3.19)
44
45
Endnotes A web page which contains a number of more detailed results is referred to in a number of the endnotes below. The address is http://qub-recon.vhost.tibus.com/uploads/Web_Appendix.doc 1 See Blanco (2002). 2 Descriptive statistics are available from the relevant web page listed above. 3 It is essential to make this assumption because bond yields are typically non-atationary. 4 Ireland joined Euro-MTS in June 2002. 5 There is also a fifth category for bills: securities with maturity less than 1.25 years. However, until recently, only Italy was significantly trading such instruments on Euro-MTS. 6 We are grateful to Rich Lyons and Kathy Yuan for pointing out that there may be a “stale price” problem that incorrectly manifests as Granger Causality. This casts further doubt on the usefulness of Granger Causality as a means to identify the benchmark. 7 We use 12.30 as the end of the morning trading period and 4.30 as the end of the afternoon trading period. 8 The Granger causality results are discussed in the next section. 9 Available at the relevant web page listed above. 10 More details of these tests can be obtained from the relevant web address listed above. 11 Details of the lag-length selection can be found on the relevant web page listed above. 12 Granger causality results are summarised in the Table at the end of section 5.2 and further details are available in the web appendix. 13 A strong restriction is that the constant in both cointegrating vectors be unity. This corresponds to two stationary yield gaps. We already know from the discussion in Section 3 that this is problematic. 14 If residuals from the structural vectors are not orthogonal, then it is not clear what ‘structural’ means in this context. It is essential one way or the other to make some assumption about the covariance between the structural relations. However, any assumption other than a zero value makes the application of the irreducible cointegrating vector approach inconclusive.
15 It is easy to show that 2 13 and e ee γδ
β β−
= =
16 In two instances in the very-long maturity the point estimate for what could be referred to as γ is above unity but is not statistically significantly so. 17 To see this, we report a fully modified t-test that tests for the difference between the estimated coefficient and unity. In each of Tables 6 to 9, this is shown in the row labelled “t-test: yield slope = unity”. 18 The Phillips-Hansen approach is a least squares technique so that in small samples, results depend on which variable is selected as the dependent variable. We carried out the above estimation and testing procedure for each pair in each of the two possible variants. We selected the reported result on the basis of the one that had the lowest residual variance. 19 The results for the long and very-long maturities are shown in Tables 12 and 13. 20 Remolona (2002) argues that the swaps market now provides the benchmark yield curve for euro denominated bonds.