Common trends, Cointegration and Competitive
Price Behaviour
John Hunter
Department of Economics and Finance, Brunel University,
Uxbridge, Middlesex, UB8 3PH, England
Simon Burke
Economics Department
Reading University Business School
October 13, 2006
Abstract
The Article considers the impact of a single stochastic trend on the
identi…cation of competitive behaviour. The common trend measures the
cumulated average response of the market to shocks in pricing behaviour.
If the market de…nition is broad then all …rms pricing ought to re‡ect this
competitive urge. More speci…cally, individual prices will adjust in the
long-run to the common trend to survive and this form of error correcting behaviour reveals a set of restrictions that econometrically identify
the cointegrating vectors. Alternatively, non-competitive behaviour is observed either when there is more than one common trend or when a single
price series drives the common trend. The latter case occurs when one of
the price series is weakly exogenous for all the cointegrating vectors.
Key Words: Cointegration, Cointegrating Exogeneity, Common trend,
Competition, Equilibrium Price Adjustment, Stochastic Trend, Weak Exogeneity.
JEL Classi…cation: C32, D18, D40.
1
1
Introduction
In this article we de…ne statistical criteria for determining competitive behaviour from the long-run decomposition of prices. Regulatory authorities and
…rms have exploited tests of stationarity and cointegration to in‡uence regulatory (Hunter and Ioannidis (2001)) and attempt to determine non-competitive
behaviour ((Forni (2004), London Economics (2002)). Here, tests for stationary
relative prices are seen as a special case of cointegration. The existing literature,
addressing this issue through tests of cointegration has either considered small
systems (La Cour and Møllgaard (2002), Hendry and Juselius (2001)) or sets
of binary relations (De Vanya and Wals (1999)). When market prices are su¢ ciently inter-related in the long-run via cointegration, then the market is viewed
as having a broad de…nition or being more competitive. It is our premise that
for the case of multi-price comparisons, testing for stationarity amongst sets of
price indices is best addressed by tests of cointegration, because such tests are
more appropriate with more than two price series and using a systems approach
it becomes feasible to distinguish between competitive and collusive behaviour.
In this article, we generalize the approach outlined by Hendry and Juselius
(2001) to the case of multi-product price comparisons. For a competitive market
with n prices that are all integrated of order one (I(1)), then for a broad market
de…nition n 1 cointegrating relationships are required. The cointegrating relationships can be uniquely identi…ed when all …rms adjust to a single common
trend. This may appear to be counter intuitive as price following is often seen
as de…ning anti-competitive behaviour (Neumann (2001)). However, as is explained by Buccirossi (2006) the observation that prices are independently set
is not necessarily consistent with behaviour that can be objectively described
as being collusive. An important component of the econometric arguments that
have been exploited to show that parallel pricing is linked to a broad market
de…nition dependent on the fact that price series are non-stationary and the
price responses need to be proportional. Here, we demonstrate that competitive behaviour is only required to be proportional in the case where for all
intents and purposes n = 2 as occurs for the conditional model in Hendry and
Juselius (2001) and the sequence of pairwise comparisons associated with the
model analyzed in De Vanya and Wals (1999)). While for n > 2 the observation of long-run parallel pricing could also be consistent with anti-competitive
behaviour.1 In the bivariate case competitive behaviour can often be seen as
being consistent with parallel pricing (Hendry and Juselius (2001)) and this
1 The approach discussed thus far relates only to the interaction of prices, though the
models could be conditioned on other variables. Firstly, this is due to the fact that there is a
signi…cant literature that uses this methodology. Secondly, Forni (2004) explains that other
methods, such of the residual demand approach and any analysis that draws on the calculation
of elasticities may also be ‡awed as it does not account for both sides of the market.
There is clearly some way to go before such an approach might de…ne a test of competitiveness, but the analysis in Forni(2004) and London Economics (2002) study for the Dutch
Telecommunications Regulator have already been used to associate a narrowly de…ned market
with non-competitive behaviour. The current article is a re‡ection of the concerns voiced by
Hunter (2003) as to the way the methodology has been applied.
2
proposition might be appropriately tested by determining whether the natural
logarithm (log) of price proportions are stationary.
Here, n price responses are consistent with competitive behaviour when all
prices have the same order of integration (I(1)), there are n 1 cointegrating
relationships or a single common trend and the common trend is driven by a
combination of shocks to all n prices. The test of cointegration is a primary
test of the proposition that all series are driven by a single common trend and
thus a weighted average of the price shocks of all …rms, but in the multiproduct
case this does not imply parallel pricing. Parallel pricing only arises when n 1
prices respond to a single price and this price is then weakly exogenous for the
vector of cointegrating relationships (Johansen (1992)). In the latter case one
…rm price de…nes the stochastic trend and all …rms respond to the prices set by
that …rm. The price that is weakly exogenous responds only to past values of
that price and more generally to the shocks that apply to that …rms price. In
addition, the I(1) case considered here, can be extended to include variables that
exhibit long-memory, as discussed by Robinson (2006)); the common stochastic
trend is now replaced by a common strong attractor.
Firstly we consider the bivariate case and then the trivariate case. The
trivariate case is an extension of the model analyzed by Hendry and Juselius
(2001) and it is easy to show that the results considered here extend to the
multi-variate case. The structural restrictions associated with a market where
prices strongly interact uniquely identify the cointegrating vectors without further restrictions on the loadings and all prices are strongly inter-related in the
long-run when each …rms pricing behaviour is consistent with long-run correction to a single stochastic trend that excludes none of the other …rms shocks.
The former implies that the cointegrating rank is n 1 and the latter that there
are no weakly exogenous variables in the system.
2
The stochastic trend, Long-run Equilibrium
Price Targeting (LEPT) and Cointegration.
Let us consider a market with n …rms, all …rms are viewed as being competitive
…rm when they all respond to a common stochastic trend and this common trend
is driven by linear combinations of the shocks that apply to all …rms. This is
an equilibrium price target (EPT) when each of the …rms respond in the same
way to this single common trend and the relationship between each …rm price
and this trend de…nes a speci…c set of restrictions on the n 1 cointegrating
relations ( ) and such restrictions are su¢ cient to exactly identify :2
2 Here we re‡ect on the information that derives from the long-run inter-action of prices,
because we believe that in the long-run demand responses are more elastic and that arbitrage
is more likely to force …rms to respond to the forces of competition. However, there are other
empirical methods directed at determining whether markets are competitive. Also see the
discussion of variance in Abrantes-Metz, Froeb, Geweke and Taylor (2005). Hosken, O’Brien,
Sche¤man and Vita (2002) consider studies based on systems of demand equations to analyze
the appropriateness of Horizontal Mergers. And Froeb and Werden (1998) discuss the use
3
Competitive …rms are viewed as correcting their behaviour in response to
some equilibrium price target. The underlying target to which the competitive
…rm responds is an average of all …rms prices that for series that are all I(1) is
de…ned by the non-stationary component of the common trend.
De…nition 1 Let 9 pt where:
pt
t
X
= a0 xt = a0 Cx0 + a0 C(
i
+ )
(1)
i=1
+a0 (
0
)
1
1
X
(I +
0
)i 0 ( i + )
i=0
is a common trend or more generally a common trend plus stationary components that are non-unique. Then Long-run Equilibrium Price Targeting implies
for pit I(d); 8 i = 1; n; then:
pit
where
0
a0 xt
pt = pit
= 1and rank( ) = rank( ) = n
pit
pt
I(0)
1:Therefore:
= pit a0? xt
= (a0 ii
a0 )xt = a0 ( ii
In )xt :
If ( ii In ) = Ri then there are n cointegrating vectors of the form :i = a0 ( ii
In ) that are dependent, when all prices have the same order of integration.
Case 1, bivariate models (Hendry and Juselius (2001), De Vanya and Walls
(1999) and Forni (2004)), then a0 = a01 a02 and:
0
n
=
where R1
=
::1
a0 R1
a0 R2
=
::2
0
1
0
1
0
n
"
After normalization
=
=
; and R2 =
1
a01
a01
a02
a02
1
a02
a01
1
0
a02
a01
1
0
#
of the di¤erence in the Her…ndhal Herschman index to assess welfare gains through mergers
in homogenous product markets. While, Hunter, Ioannidis, Iossa and Skerratt (2001) have
sought to provide measures appropriate to the analysis of consumer harm under conditions of
imperfect information about price and quality.
Of course, in the case of merger, signi…cant resources are often employed by anti-trust
authorities to analyze a small number of cases. Which is the basis of this quest for more
e¤ective techniques as can seen by the signi…cant interest in consumer harm amongst OECD
countries and the European Union, Competition Commission. In this light, the degree of
consumer detriment, as measured on the basis of survey data by the OFT (2000) and at
the level of companies by Pierce-Brown and Graham (2001) consumer detriment, would also
appear to suggest that this is a signi…cant issue.
4
Selecting n
1 = 1 cointegrating vectors from the 0n :
i
h
0
0
1 :
= 1 aa0 2 = 1
2
The cointegrating vectors take the form p1t p2t I(0) either when the price
proportion is tested for stationarity as is suggested by Forni(2004), or from the
cointegrating rank of the bivariate system as suggested by De Vanya and Walls
(1999) or conditional on the price of oil by Juselius and Hendry (2001).
However, such tests of bivariate relationships are not appropriate when there
are more than two price combinations as is the case in Forni(2004), De Vanya
and Walls (1999) and London Economics (2002). This point was addressed by
Hunter (2003) in his response to London Economics (2002). This moves us to
cases where n exceeds 2.
Case 2, The trivariate system with a0 = a01 a02 a03 :
0
n
2
::1
= 4
::2
::3
2
3
2
3
a0 R1
5 = 4 a0 R2 5
a0 Ri3
a02 + a03
a01
= 4
a01
where
2
0
R1 = 4 1
1
0
1
0
3
2
1
0
0 5 ; R2 = 4 0
0
1
3
2
1
0
0 5 and R3 = 4 0
0
1
1
0
1
After normalization
0
n
02
3 And det B6
@4
1
1?
1? + 3?
1?
1? + 2?
2
? 2?
?
2
6
=6
4
a01
a01 +a03
a01
a01 +a02
1
1
a02
a01 +a02
3?
2? + 3?
3?
1? + 3?
2?
1? + 2?
? 3?
2?
2 +
2? + 3?
? 2? + ? 3? + 2?
?
2
? + ? 2? + ? 3?
2 +
? 2? + ? 3? + 2? 3?
?
1
3?
a03
a02 +a03
a03
a01 +a03
a02
a02 +a03
1
2?
2? + 3?
3
a03
a03 5 ;
0
a1 + a02
a02
0
a1 + a03
a02
1
1
31
7C
5A =
3?
2? + 3?
+
2?
2? + 3?
3
73
7
5
2
? + ? 2? + ? 3?
2 +
? 2? + ? 3? + 2?
?
2
? + ? 2? + ? 3?
2 +
? 2? + ? 3? + 2?
?
3?
2? + 3?
3
1
0 5:
1
0
1
0
3?
3?
+
=
= 0:
Therefore the n = 3 potential cointegrating vectors are by de…nition dependent and all the
prices cointegrate with the common trend.
5
Selecting n
0
n
1 cointegrating vectors from the
0
=
"
=
1
21
31
12
1
32
a02
a02 +a03
1
a01
a01 +a03
:
#
a03
a02 +a03
a03
a01 +a03
1
:
This leaves r = n 1 restrictions that exactly identify the cointegrating vectors
(Burke and Hunter, 2005, Chapter 5):
1; 12 + 32 = 1
or
X
f or all i = 1; 2; :::n;
1:
ji =
21
+
=
31
j6=i
Where these are consistent with pit pt I(0) for i = 1; 2:::n, and with a0 = 1.
The above result can be readily generalized to the case where there are more
than three prices and for each additional long-run equation, enough additional
restrictions apply to the cointegrating vectors.
If we consider the special case where a = ? ; then rather than approximating
the common trend, pt = 0? xt and it is possible to remove the initial condition
by sequential de-meaning of each price series in turn (Taylor, 1999). In the case
where a0 = 0? then C = ? ( 0? ? ) 0? ; and we isolate the trend component by
multiplying (1) by 0? where 0? = 0 :
0
? xt
=
=
0
? Cx0
+
t
X
0
? C(
t
X
0
0
x
+
(
? 0
?
i
0
?
+ )+
i=1
i
(
0
)
1
1
X
(I +
0
)i 0 ( i + )
i=0
+ ):
i=1
Setting the initial condition to zero, then for a competitive market there can be
no drift, so = 0 and it follows that the common trend is de…ned by:
0
? xt
0
?
=
t
X
i:
i=1
If we use the results set out above except replacing a0 by 0? then in the
trivariate system with 0? =
after normalization there are
1?
2?
3?
n 1 cointegrating vectors:
0
=
=
1
21
31
12
1
32
1
2?
2? +
1
1?
1? +
3?
6
3?
3?
2? + 3?
3?
1? +
3?
:
and the n 1 restrictions that exactly identify these cointegrating vectors are
the same restrictions as would occur when pt is not exactly equivalent to 0? xt .
It follows that competitive behaviour is rejected when there are less than
n-1 cointegrating vectors as a group of …rms may be following an alternative
common trend and as a result they may not be responsive to the competitive
price. The latter can occur when some of the prices are of another order of
integration, but this would imply a di¤erent cointegrating rank. This leads to
the following result:
Corollary 2 Competitive behaviour requires a similar price response so it follows for LEPT we require pit I(d); 8 i = 1;
; n:
Proof. If there are j prices such that pit
I(d2 ); 8 i = 1;
;j
and d2 > d;then it follows that rank( )
n 1: If d2 = 2; then there are
r1 = j 1 cointegrating vectors j implying rank( ) j 1 when the j prices
satisfy the the restriction associated with LEPT. But when the other n j prices
also satisfy LEPT, then rank( )
n j 1: It follows by combining the two
minimum rank conditions, that rank( ) n j 1 + j 1: Hence, when prices
are of di¤ erent orders of integration they can at best follow two independent
random walks, meaning the market is not competitive.4
It follows from corollary 1 above that LEPT requires all series to be of the
same order of integration otherwise di¤erent market segments may respond to
di¤erent trends as rank( )
n 2 and this violates theorem 1. It follows
from theorem 1, rank( )
n 1 and for LEPT that all prices cointegrate
with the common trend. However, this type of relation is only consistent with
competitive behaviour when the cointegrating relations depend on all prices or
we preclude the case where, by any simple re-ordering, 0? 6= 0 0
:
3?
More speci…cally the identifying cointegrating combination negates the possibility that n 1 prices depend exactly on a single price; this is the case where
one of the prices is long-run weakly exogenous and all prices react to this price.
4 If d = 0; then there are j inter-related prices and the system is separated into two blocks
2
such that:
=
1 ,1
1;2
2 ,1
2 ,2
=
1
0
1
0
2 1
1
2
=
1
1
2
2
0
0
2
0
2
It follows from the di¤erential in the order of integration that there are n j 1 cointegrating
relations amongst the higher order variables that are I(0): Therefore, without loss of generality
n j 1 and for ease of exposition, we set 1;2 = 0 :
2 ,1 = 0; rank( 2 ,2 )
=
1 ,1
0
0
2 ,2
=
1
0
0
1
0
2
0
2
:
If we now consider the cointegrating relations associated with 01 there are j homogeneity
restrictions that apply to these price series and For ease of exposition, we set 1;2 = 0 and
j
X
1 and as a result rank( 1 ,1 ) = j 1: Therefore, LEPT implies that rank( )
ki =
k6=i
n
2:
7
More speci…cally, LEPT relates to a single price leader that does not respond to
any other prices or in the simplest system this price follows a random walk and
all prices respond to this form of price leadership. Hence, LEPT is necessary,
but not su¢ cient for a market to be competitive as we also require none of the
price variables to be weakly exogenous. If a price is weakly exogenous then the
common trend is purely driven by the shocks to that …rms price and do not
respond to the market circumstances of the other …rms.
We will show next that any violation of the condition rank( ) = n 1 will
lead to prices following di¤erent trends, while weak exogeneity (WE), cointegrating exogeneity (CE), or sub-system long-run exogeneity (SSLE) all lead to
a partial or complete failure of competitive pricing.
3
Exogeneity, Rank n
behaviour.
2 and narrow market
Firstly, we consider the case where there are n 1 cointegrating vectors, but a
single price is weakly exogenous for the cointegrating vectors (Johansen (1992)).
Then we will consider further decompositions of the cointegrating vectors associated with CE (Hunter (1992)).
If there are n 1 cointegrating vectors then is an n r matrix of loadings.
It follows from Johansen (1992) that for WE of a variable for the parameters of
interest ( ); then a row of is set to zero. With rank( ) = n 1; then only
one price can be weakly exogenous, otherwise rank( ) < n 1 and there are
fewer than n 1 cointegrating relations and the condition that there is a single
common trend is violated.
If there is a single common trend, a single weakly exogenous variable and
a0 = 0? , then from the condition a 0? = 0; it follows without loss of generality
0 1 and this implies that the nth price de…nes the
that 0? = 0
common trend. More generally, when ai = 0 then the equation driving variable
i contains no long-run components and this means that it follows a generalized
random walk or this series de…nes the stochastic trend solely driven by the
residual mis-pricing associated with that market alone.
h
i
Corollary 3 If rank( ) = rank( ) = n 1 and 0 = A n 01 1 for some
ordering of the pi i = 1; :::n, then this implies a broad market as all prices
interact, but non-competitive behaviour as all price follow pn
Proof. When rank( ) = rank( ) = n 1; then the necessary condition for a
broad market is accepted and an identi…cation of the system occurs by adopting
the normalization rule of Boswijk (1996):
0
In the WE case:
0
In
=
=
h
A
1
b
0
n 1 1
8
:
i
implies that the normalization is unique as it is the only normalization by which
pi for i = 1; :::n is conditioned on an exogenous variable pn : It follows without
loss of generality that under WE, 0? = 0 0 :::
and from the de…3?
nition of a common trend:
= a0 xt =
pt
0 :::
n?
Therefore,
0
xt
0
? xt
2
3
p1t
6 .. 7
4 . 5 = pnt = pn0 +
pnt
In
=
2
6
= 4
b
1
p1t
..
.
pn
1t
t
X
(
n?
3 2
p1t
p1t
6 .. 7 6
4 . 5=4
pnt
pn 1t
3
1 pt
t
X
0
?(
+
ni
+
i
+ )
i=1
n ):
i=1
2
b 1 pt
bn
0
? x0
=
3
b1 pnt
..
.
bn
1 pnt
7
5
7
5
From the de…nition of the common trend pt = pnt . Hence, all prices are driven
by the stochastic behaviour that underlies pnt and as 0 0 :::
this
n?
is without reference to the shocks that underlie the other …rms prices.
If all …rms prices are conditioned on pnt ; then …rm n is the long-run price
leader and should the restriction in theorem 1 apply and pt = pnt ; then equilibrium price targeting implies that:
=
In
1
n 1
and
0
n 1
=
1
1
:5
Therefore, we have a broad market in the sense that …rms follow the common
trend, but when the common trend is driven by a single …rm without reference
to other …rms or more pertinently without reference to the the direct shocks
associated with mis-pricing by these other …rms, then the …rm must hold a
dominant position in the market place or that …rm must de…ne the barometer
to which all other …rms respond. However, a barometer should not normally
behave without reference to the other …rms. It follows, with one price being
weakly exogenous for the parameters of interest, that the nth …rms price can
be viewed as driving all the other …rms prices. This, we would argue is a form
of price leadership as the long-run is conditioned on the behaviour of the nth
…rm price. In this case, under the restrictions associated with LEPT all …rms
respond to those of the nth …rm, but in the long-run the nth …rm does not
5 This is consistent with the restrictions tested by De Vanya and Wals (1999) and by
equation it implies a subset of the restrictions considered by Forni (2004) as stationarity tests
for a broad market de…nition.
9
respond to any of the other …rms prices. Hence, although there are n 1 longrun price relations and satis…es the right restrictions this is not a competitive
case. Hence, for competitive behaviour, we have a further requirement that the
common trend is not de…ned by a single …rms price or that none of the prices
are weakly exogenous for .
Notice that LEPT also precludes the existence of a block triangular form
to the long-run parameter matrix associated with CE (Hunter (2005)) as this
implies a sub-block of equations that do not respond to the other …rms prices.
0
If
=
; then decomposing and , into 1 an n1 r sub-matrix, 2 an
n2 r sub-matrix, 1 an n1 r sub-matrix and 2 an n2 r submatrix, then:
=
1,1
1;2
2,1
2,2
=
1
1
2
2
0
=
0
1
0
2 1
1
1
2
0
2
0
2
:
The the primary condition for CE is 2;1 = 0 and without loss of generality
this implies 2 = [0 : 11 ] and 1 = [ 1;1 : 0]: But this condition imposes
restrictions on
that are not consistent with LEPT as 1;2 is normally not
zeros As according to the CE case the second block of equations do not respond
to behaviour in the …rst block of …rms prices. What is occurring here, is
that some …rms operate without reference to other …rms in the group, or they
cointegrate and all the other …rms long-run price responses do not respond to
them. Hence one …rms prices can be viewed as causing the other prices in the
system and the long-run forecasts of the …rst block of …rms are conditioned on
those of the second block. This is long-run non-causality and this behaviour,
due to the block triangularity, is also re‡ected in the corrections that occur
in the short-run. Hence, the short-run for the …rst n1 …rms responds to the
long-run corrections of all …rms, while the long-run of the the last n2 …rms only
responds to the correction in those …rms alone.
In the case of WE for :1 (…rst block of beta) of x2 , then sub-system WE
(SSWE) implies that a sub-block of …rms may operate as a cartel, while the
remaining …rms operate as a following fringe. The sub-block cartel is noncompetitive, while the other block are ‘innocent followers’. The condition for
SSWE is:
1;2
1;2 (
1
2;2 )
2;2
= 0:
This occurs when either 1;2 = 1;2 ( 2;2 ) 1 2;2 or 1;2 = 0 and 1;2 = 0: This
implies that the long-run associated with the …rst block of equations can be
estimated without reference to the information content of the second block of
equations.
For cases where rank( ) = n 2, this may occur when there is a further
rank de…ciency. For example, where:
:i
=
:j
for some i 6= j: This is matched price following (MPF), but since for LEPT
rank( ) = n 1, then for this type of behaviour to exist rank( ) < n 1.
10
A number of side issues arise from there being at least two common trends.
Firstly, individual prices follow linear combinations of the common trends that
happen to be di¤erent. In this case, one trend may eventually dominate. They
follow the same linear combination (consistent with equilibrium) so there exists
a unique pt ; but rank( ) < n 1. They may also follow di¤erent linear combinations of the common trends that are not consistent with equilibrium as such
divergence of prices implies death or dominance.
4
Conclusion
In this article we considered the conditions required for competitive behaviour.
We …nd that pricing is consistent with competitive behaviour when: i) there are
n 1 cointegrating relationships, ii) the restrictions associated with LEPT are
satis…ed, iii) There does not exist an i = 0 for all i = 1; :::; n. Forni(2004) and
Hendry and Juselius (2001) also require (i) and (iii), but beyond the bivariate
case the restrictions associated with the LEPT are not in general simple parity
conditions. This has the further implication that tests of stationarity in an n
…rm system will not be appropriate.
Although we have concentrated the case where there are stochastic trends,
the same conditions apply when there is drift and more generally for cases where
there is balanced cointegration. It is our conjecture that the restrictions are still
relevant for long memory processes with fractional cointegration. In particular,
Robinson and Yajima (2002) consider the estimation of fractional cointegration
amongst oil prices that turns out to be an n-1 case and Robinson (2006) presents
a very general expression of cointegration for stationary long memory processes
that cointegrate. Hence, the results presented here are not limited to cases
where prices are I(1).
5
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