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DYNAMIC ECONOMETRIC MODELS 
Vol. 11 – Nicolaus Copernicus University – Toruń – 2011 

Sylwester Bejger, Joanna Bruzda†  
Nicolaus Copernicus University in Toruń  

Detection of Collusion Equilibrium in an Industry  
with Application of Wavelet Analysis  

A b s t r a c t. In the present paper an attempt was made to verify the possibilities of the use of 
a marker of structural changes of market price variance in the detection of trade collusion between 
business players. We used the theoretical model of strategic behaviour of trade players with the 
assumption of exogenous and time-constant cartel quota (market shares), which justifies the 
application of a marker for business with specific parameters. The paper contains empirical em-
ployment of a marker for a sequence of average Lysine price on the USA market in 1990–1996. 
Wavelet analysis was applied, for the first time in this context, as the econometric method for the 
detection of structural changes in the variance. 

K e y w o r d s: Explicit and tacit collusion, supergame with a fixed structure of market shares, 
price variance, wavelet analysis  

Introduction  

 In the paper Bejger (2010), on the basis of well-known Lysine1 cartel 
(1990–1996 period) a theoretical model of strategic behaviour of industry play-
ers was constructed as a standard supergame model with a Cournot type stage 
game with additional assumption of exogenous and time-constant cartel quota 
(market shares). For business branches bound by specific parameters, e.g. Ly-
sine manufacturers2 in the above mentioned period, the model may indicate 
certain characteristic structural disorders in the variance of the market price, 
resulting from the likelihood of the development of price war phase, caused by 

                                                 
† The author acknowledges the financial support from the Polish Ministry of Science and 

Higher Education under the grant no. N N111 285135. 
1 Lysine is an α-amino acid that the human body cannot synthesize. Lysine production for an-

imal feed is a major global industry, as Lysine is an important additive to animal feed because it is 
a limiting amino acid when optimizing the growth of certain animals such as pigs and chickens 
for the production of meat. 

2 For a detailed description of Lysine conspiracy see Connor (2001). 



Sylwester Bejger, Joanna Bruzda  156 

a player with no intention of maintaining or establishing collusion due to a too 
low predicted or factual share on the market or a certain market price stiffness 
in the collusion phase in the periods of market contraction. Present paper in-
cludes an attempt of the application of wavelet analysis with the aim of an em-
pirical verification of the correctness of theoretical findings. The verification is 
made possible thanks to the well known history of strategic behaviour of the 
players in the business. The following chapter will introduce the summary of 
the to-date empirical research with the use of variance change marker. Chapter 
2 contains the description of research methodology. Chapter 3 contains empiri-
cal work (in a case-study flavour). In chapter 4 the results of the study were 
presented. Chapter 5 summarizes the whole of the research.  

1.  To-Date Empirical Research with Application of Variance 
Change Marker  

 Assuming that the theoretical model accurately describes strategic behav-
iour in the industry of Lysine manufacturers, the following research hypotheses 
could be put forward: 

 regime changes of the variance of average Lysine prices are possible due to 
a probability of the occurrence of price war phases, 

 in the collusion phase the variance should be lower than in competition 
phase. 

These two specific patterns in variance process could distinguish collusion from 
competition and are usually named markers of collusion (Harrington,  
2005, p. 25). 

With the aim of detecting such variance disorders a wavelet analysis was ap-
plied. So far, such approach has not been used. Basing on theoretical findings, 
one could put forward a hypothesis that, on average, in the collusion phase price 
variance is lower than in the competition phase. One should also expect regime 
variance change while switching from collusion phase to competition phase 
(price war). To-date papers connected with the detection of collusion on the 
basis of the detection of structural changes in the variance embraced the appli-
cation of descriptive statistics methods in the comparison of variance levels in 
the phases of collusion and competition (Abrantes-Metz, Froeb, Geweke, Tay-
lor, 2006), the application of ARCH / GARCH specification in the process of 
market price including the additional 0-1 variable describing the phases of col-
lusion and competition (Bolotova, Connor, Miller, 2008), as well as the applica-
tion of Markov switch model of MS(M)(AR(p))GARCH(p,q) type for the vari-
ance and/or average (constant) of the price process (Bejger, 2009). The two 
latter articles are all the more interesting due to the fact that they refer to the 
cartel of Lysine manufacturers. In the paper (Abrantes-Metz, Froeb, Geweke, 
Taylor, 2006) the cartel of frozen fish suppliers was the research subject.  
The existence and functioning of the cartel was confirmed by the authorities. 



Detection of Collusion Equilibrium in an Industry with Application of Wavelet Analysis  157

In the course of the research it was found that during collusion phase (before the 
discovery of the cartel) market price variance (offered by colluders at bargains) 
was significantly lower than after the fall of the cartel. The hypothesis that justi-
fied the research was the supergame model with SPPE equilibrium. In the arti-
cle (Bolotova, Connor, Miller, 2008) ARCH / GARCH specification was used 
with the aim of examining the disturbances in the variance and the average of 
the Lysine market price process during the phases of competition and collusion. 
The influence was described through 0-1 variables for the phases of cartel and 
competition. A statistically significantly lower average variance was reached in 
the periods of collusion than in the phases of competition. However, it should 
be underlined that the marking off of phases of both types (with the aim of 0-
1variable specification) took place on the basis of a posteriori evidence from the 
proceedings in the case of the cartel. In the same article a cartel of citric acid 
producers was studied, with no particular findings as to the lower variance in 
the collusion phase. The theoretical basis for the marker was SPPE-type equilib-
rium. In the paper (Bejger, 2009) Markov switching model was employed with 
the aim of detecting moments of change in the level of variance and the average 
of Lysine price process. The results confirmed the existence of two variance 
regimes and a high likelihood of the process staying within the lower variance 
regime, especially during the second collusion phase. The study seems of im-
portance in view of the absence of assumptions as to the moments of switch. 
The detection of those moments partly confirmed the history of cartel activity 
known from the trial evidence. 

2.  Wavelet Analysis Methodology 

 Assuming that the theoretical model accurately describes strategic behav-
iour in the business of Lysine manufacturers, one could put forward the follow-
ing research hypotheses: 

 regime changes of the variance of average Lysine prices are possible due to 
different phases of price war, 

 in the collusion phase the variance should be lower than in the competition 
phase. 

With the aim of detecting such variance disorders a wavelet analysis was ap-
plied. So far, such approach has not been used. Comparing it to variance mark-
er-based econometric methods of collusion detection applied before (mentioned 
in chapter 1), it can be noticed that the wavelet analysis makes it possible to 
utilize simple methods of statistical inference, allows the preliminary, graphic, 
estimation of the changes in variance, and as a non-parametric method is not 
burdened with model specification error. In addition, it allows for the indication 
of the scales in case of which a change occurs; this however requires the access 
to long time sequences. 



Sylwester Bejger, Joanna Bruzda  158 

 Wavelet analysis consists in the decomposition of the process into compo-
nents constituting shifted and rescaled versions of the so-called mother wavelet, 

, which is a function with unit energy, fulfilling the so-called admissibility 
condition ( Percival, Walden, 2000, p. 4). Let a vector be given in the form of 

),,,( 110  Nxxx x  where JN 2 . For  and   
(j – decomposition level, t – wavelet coefficient number) we define discrete 
wavelet transformation (DWT) of the vector x: 

  




1

0 ,,
N

n tjntj nxW  , (1) 

where )(, tj  are versions of the basis wavelet shifted by an integer number and 

rescaled on the dyadic scale ,...2,1,2 1   jj
j , i.e.:  

 txx jj
tj   22)( 2/

,  . (2) 

 In the case of the popular Daubechies wavelets for a given j wavelet coeffi-
cients tjW ,  are proportional to differences (of various orders) of weighted aver-

ages over the scale j . For the stochastic process tX  a time-varying wavelet 

variance is defined in the following way: 

)Var(
2

1
)( ,

2
tj

j
jt W


  . (3) 

 Under the assumption that the quantity above does not depend directly on 
time 3, we obtain a decomposition of the variance according to the scale in the 
form (Percival, Walden, 2000, pp. 296-298): 

 









1 1

2
, )()Var(

1

2

1
)Var(

j j
jtj

j
t WX 


. (4) 

Wavelet variance on the level j corresponding to the scale 12  j
j , )(2

j , 

informs about the changeability of fluctuation in cycles contained approximate-

ly in the bracket j2 – 12 j .  

 In the estimation of wavelet variance and wavelet correlation in practice, 
DWT is replaced by its modification in the form of MODWT (maximal overlap 
discrete wavelet transform)4, which does not require handling long ranges being 

                                                 
3 The assumption is fulfilled also for non-stationary processes on condition that the processes 

are integrated of order d, while the applied wavelet filter is sufficient to eliminate the nonstation-
arity,  i.e. it is a Daubechies filter (daublet, symlet or coiflet) of an appropriate length (see Perci-
val, Walden, 2000).  

4 MODWT is preferably pronounced ‘mod WT’ – modified WT see: (Percival, Walden, 2000, 
p. 159). Other names for this kind of transformation are non-decimated wavelet transformation, 
continuous-discrete wavelet transformation or French-derived term algorithme à trous. 

)(

Jj ,,2,1  12,,1,0   jJt 



Detection of Collusion Equilibrium in an Industry with Application of Wavelet Analysis  159

the subsequent powers of the number 2, provides a more effective estimator 

)(2
j  and has invariance properties due to shifts in time. (Percival, Walden, 

2000, pp. 308-310; Gençay and others, 2002, p. 135). The estimator of wavelet 
variance is expressed by means on the following formula: 







1

1

2
,

2 ~
~
1

)(~
N

Lt
tj

j
j

j

W
N

 , (5) 

where  are MODWT coefficients,  is the length of the 

wavelet filter for the scale  (L is the length of the basic wavelet filter), where-

as  is the number of coefficients distorted by the extrapolation 

at the ends of the sample. (1–)-per cent confidence interval for )(2
j  can be 

approximated in the following way: 
5,0

,
~

2
~

)0(ˆ
)(~

2 














j

jW
j

N

f
 , (6) 

where 
2
  is the (1–/2) quantile of the normal distribution, whereas )0(ˆ

,
~

jW
f  is 

an estimate of the spectral density function for squares of wavelet coefficients 
for scale j  at 0. 

2.1. Testing Homogeneity of Variance 

 The attractiveness of wavelet coefficients in testing volatility changes re-
sults from their two vital properties. Firstly, wavelet coefficients are closely tied 
to the changes at various scales and moments in time, and thus they carry in-
formation about the variability of the process. Secondly, (conventional) wavelet 
transformation provides coefficients which can be treated as approximately non-
correlated both in the case of short and long-memory processes, which signifi-
cantly simplifies the inference procedures (comp. Whitcher, 1998, ch. 4.1.2; 
Percival, Walden, 2000, p. 351). Meanwhile, in the estimation of the switch 
moment the employment of MODWT is proposed on the account of eliminating 
subsampling effects, which entails higher precision of an estimate. The ap-
proach put forward in the PhD dissertation of Whitcher (1998) is presented be-
low, and the direct application of the Inclán and Tiao (1994) method to wavelet 
coefficients is proposed. It is obligatory to stress that the methods of wavelet 
detection of changes in variance are discussed primarily in the context of long-
memory processes, but  – because the property of the approximate decorrelation 
is valid also for the short-memory processes – it is possible to apply the method 
in the case of the latter processes as well. 

tjW ,
~

1)1)(12(  LL j
j

j

1
~

 jj LNN



Sylwester Bejger, Joanna Bruzda  160 

 Let }{ ,tjW  be wavelet coefficients of the jth decomposition level. We are 

now interested in testing the following hypothesis: 

)Var()Var()Var(:H
12/,1,,0   j

jj NjLjLj WWW  , 

where jL  is the number of boundary coefficients of the wavelet transform at the  

jth decomposition level (DWT coefficients, the value of which is influenced by 
the extrapolation method at the ends of the sample)5. Furthermore, we assume 
that the wavelet decorrelation is effective what means that the coefficients 

}{ ,tjW  form a second-order Gaussian white noise. Moreover, we assume that the 

length of the wavelet filter is sufficient for the elimination of deterministic 
components, i.e. 0)E( , tjW . The statistics in the test against the alternative 

hypothesis in the form: 

)Var()Var()Var()Var(:H
12/,1,,,1   j

j NjkjkjLj WWWW  , 

where k is the unknown location of the variance change, is based on the normal-
ized cumulated sum of squares (CUSUM for squares): 











12/ 2
,

2
,

j

j

j

N

Lt tj

k

Lt tj

k

W

W
W ,  (7) 

and have the forms: 

A. (Inclán, Tiao, 1994) 
j

j

jJ
j

j

N

Lk
k

Lk

N
WIT 










1

12,,
2 max


;   (8) 

B. (Whitcher, 1998)  
 DDW jN

,max2 ,      (9) 

where:  

 kN

Lk

NLk
WD

j

j

j
j

 





1

1

22/,,
max


,   1
22/,,

max 




 
j

j

j
j

N

Lk
k

NLk
WD


, 

whereas j
j

j LNN  2/  is the number of coefficients distorted by the extrapo-

lation method at the ends of the sample. In the case where jN   amounts to at 

least 128, the following distribution for the test statistics is used (see: Inclán, 
Tiao 1994, p. 923; Whitcher 1998, p. 60). Small sample critical values (9) can 
be found in the works of Whitcher (1998), Whitcher and others (2002). 

 

                                                 
5 For the applied wavelets those values amount to 0 in the case of the Haar wavelet whereas in 

the case of d4 wavelet – 1 for the first decomposition level and 2 for higher levels – compare 
(Percival, Walden, 2000, p. 136). 



Detection of Collusion Equilibrium in an Industry with Application of Wavelet Analysis  161

2.2. Estimating the Location of Variance Change 

 k
~

, for which the appropriate expressions in test statistics reach their max-
imum As mentioned earlier, it is necessary to base the estimation of the location 

of variance change on the coefficients tjW ,
~

 of the non-decimated wavelet trans-

formation. It is then that the index values, indicates the moment of change. Ob-
viously, in such statistics the lower and upper summation limits must be modi-
fied accordingly, in the way so that all non-boundary coefficients are taken into 
account. And thus, the statistic in Inclán and Tiao version now takes the form: 

1

2

1,,12

1
max 





 
j

j

j

j

LN

Lk
k

NLk

LN
WIT


, (10) 

whereas the expressions in statistics (9) transform into: 

 kLN

Lk

NLk
WD

j

j

j

 




 2

2,,1
max


,    1

1, , 2
max ,j

j
j

k L

k N Lk L N
D W

 
  

 


 

then the normalized cumulated sum of squares is in the form: 











1

1

2
,

1

2
,

~

~

N

Lt tj

k

Lt tj

k

j

j

W

W
W .  (11) 

Finally, we receive the moment of change in real time, by applying additional 
corrections due to a phase shift6. 

 In the case in which there is a possibility of the occurrence of multiple 
changes in variance, it is proposed (Inclán, Tiao, 1994) to apply a procedure 
described as the algorithm of Iterated Cumulative Sums of Squares, ICSS, based 
on the method of binary segmentation. The application of such an algorithm in 
the context of wavelet detection was proposed by Whitcher et al. (2000). The 
course of the algorithm can be depicted in the following way: 

 having a succession of non-boundary DWT coefficients: 

12/,1,, ,,,
 j

jj NjLjLj WWW   we assume 12/, 21  j
j NtLt  and compute 

the value of the statistic IT or W for the range ],[ 21 tt . If the statistic points 

to the occurrence of change, we mark the moment k , for which its value 
was determined and we pass on to the next stage. 

 We determine the value of the test statistic to the left of k , i.e. in the range 

]1,[ 1 kt  and to the right of k , i.e. in the interval ],1[ 2tk  . We keep 

                                                 
6 In the case of the filters applied further in the empirical part, the evaluation of the moment 

of change in real time, with the assumption that the time is numbered beginning with 1, is ob-

tained as follows: in the case of the Haar wavelet without any modification of k
~

, while in the 

case of d4 wavelet – by subtracting 1 from k
~

. 



Sylwester Bejger, Joanna Bruzda  162 

finding subsequent points by following the same procedure – we appropri-
ately divide the intervals and determine the values of statistics in the smaller 
ranges of coefficients. The procedure comes to an end when the rejection of 
the zero hypothesis does not occur. 

 We arrange the detected points in the ascending order nkkk ,,, 21   while 

additionally assuming jLk 0  and 12/1 
j

n Nk . For each j = 1, …, n 

we carry out a test in the interval between the points of change adjacent to 

jk , i.e. in the area between 1jk  and 1jk . If a potential change-point is not 

detected again, we reject jk  from the considered set. The procedure is con-

tinued on a new set of points and we ultimately finish when there are no re-
ductions in the number of detected changes. 

 With the aim of selecting the appropriate manner of implementation for the 
presented methods, it is worth discussing their statistical properties. Firstly, 
wavelets with smaller support (shorter wavelet filters) have better localization 
properties – out of the simulation analyses presented in Whitcher’s PhD disser-
tation (1998) we can infer that the localization of the moment of switch is bur-
dened with a bigger mistake for longer wavelet filters. Secondly, wavelet coef-
ficients on the second decomposition level require higher SNR (signal to noise 
ratios), in order to attain the same level of accuracy as on the first level, both in 
terms of the detection of change and its localization – thus it is vital to make use 
of primarily first levels of decomposition. Other conclusions of the simulation 
analyses are as follows: the estimation of the moment of change is lightly biased 
towards the centre of the test (Inclán, Tiao, 1994; Whitcher, 1998), in the test of 
homogeneity of the wavelet variance, the Haar, d4 and la8 wavelets provide 
similar rejection rates in tests for one or two changes in a wide range of value 
for the parameter d of fractional integration (0,05–0,45), especially if small 
sample critical values were used and the test was carried out on low (1–2) de-
composition levels (Whitcher, 1998; Whitcher and others, 2002). Obviously, 
Daubechies wavelets (see Daubechies, 1992; Percival, Walden, 2000) with 
longer support allow to analyse non-stationary processes of a more complicated 
structure but – simultaneously – render a smaller number of useable coeffi-
cients. In the case of the tested sequence we do not analyse higher levels of 
decomposition also for the reason that it would only contain 19 coefficients (so 
there is a slim chance of as many as two moments of change being discovered), 
moreover, wishing to carry out a j-level wavelet analysis without having to 
write in any additional observations we should be equipped with a series of 

length which is a multiple of j2  (the length of our series – 78 – does not allow 
to take the analysis to higher levels without writing in additional observations). 
Summing up, two wavelets will be applied in the analysis: the Haar wavelet and 



Detection of Collusion Equilibrium in an Industry with Application of Wavelet Analysis  163

d4, yielding filters of the length of 2 and 4 respectively7. In the testing for ho-
mogeneity of variance we will use the first decomposition level. 

3.  Empirical Study 

 In the first stage of the analysis graphs were made of the MODWT coeffi-
cients and the wavelet variance in its rolling version. Figure 1 was then marked 
with MODWT coefficients obtained by means of the Haar wavelet on 4 levels 
of decomposition, together with scaling coefficients from the fourth level as 
well as the original time line. The values are shifted on the time axis in the way 
so as to cancel the phase shift.  

Figure 1.  Wavelet, scaling coefficients and the time series. The vertical lines delineate 
the boundary regions 

                                                 
7 The d4 wavelet cannot be written in closed form, while the Haar wavelet is given as: 

)()()( )2/1,0)1,2/1 xxx   11 . A detailed presentation of the level j Haar and d4 wavelet filters 

can be found in Percival and Walden (2000), ch. IV. 

10 20 30 40 50 60 70

0.7

0.8

0.9

1

1.1

1.2

1.3

X

T-1W1

T-2W2

T-2V2

10 20 30 40 50 60 70

0.7

0.8

0.9

1

1.1

1.2

1.3

X

T-1W1

T-2W2

T-4W3

T-4V3

10 20 30 40 50 60 70

0.7

0.8

0.9

1

1.1

1.2

1.3

X

T-1W1

T-2W2

T-4W3

T-8W4

T-8V4



Sylwester Bejger, Joanna Bruzda  164 

 Figures 2 and 3 (next page) depict the wavelet variance in the rolling (local) 
version, computed with small portions of wavelet coefficients, shifted on the 
time axis, together with 95-percent confidence intervals, estimated on the basis 
of MODWT. The calculations omit all boundary coefficients (i.e. the coeffi-
cients the value of which is influence by the extrapolation method at the ends of 
the series). Two moments can be seen: one located around observation 28 and 
the other further off.  

Figure 2.  Rolling wavelet variances with with 95-percent confidence intervals – Haar 
wavelet, window length 30 

 

20 40 60 
0 

0.5 

1 

1.5 x 10
-3

Scale 1 

Wavelet 
variances 

 

20 40 60
0

0.005

0.01

0.015
Scale 2

20 40 60
0

0.02

0.04

0.06

0.08
Scale 4

 

Figure 3.  Rolling wavelet variances with with 95-percent confidence intervals –  
d4 wavelet, window length 30 

 The next stage consisted in testing of structural changes and variances by 
means of wavelet methods and included a direct application of the Inclán and 
Tiao method to DWT coefficients and testing according to Whitcher’s wavelet 
method. The results of both tests are summed up in Tables 1 and 2. 

The occurrence of the subsequent point of change was also verified with the 
application of wavelet d4. This time coefficients with numbers from the interval 
[1, 20] were evaluated. Appropriate statistics were equal: 0.4106 (IT), 0.6290 
(W), pointing to the lack of another switch within the variance. The d4 wavelet 
indicates two moments of change: we infer about the first one at the low signifi-
cance level, i.e. empirical significance level remains below 10%, yet the infer-
ence about the presence of the latter is already made at 1% level. The observa-
tions are made in accordance with the course of rolling wavelet variances.  

20 40 60 0

1

2

3

4
x 10 -3 Scale 1 

Wavelet 
variances 

20 40 60
0

0.005

0.01

0.015
Scale 2

20 40 60
0

0.02

0.04

0.06

0.08
Scale 4

20 40 60 0

0.1

0.2

0.3

0.4
Scale 8 



Detection of Collusion Equilibrium in an Industry with Application of Wavelet Analysis  165

Table 1. Testing for change in variance – the Haar wavelet 

Method and stage of 
analysis 

Statistics No. of the DWT 
coefficient 

(kj) 
 

Location of the 
change in variance 
with the help of the 

MODWT coefficients 
IT - first change-point 1.4144** 22 (k1) 45 
W – first change-point 1.3458** 22 (k1) 45 

IT – second change-point 
searched to the left from 

the first 
0.4627 10 38 

W – second change-point 
searched to the left from 

the first 
0.7356 18 38 

IT – second change-point 
searched to the right from 

the first 
0.5335 27 61 

W – second change-point 
searched to the right from 

the first 
0.8494 34 61 

Note: IT – Inclán and Tiao test; W –Whitcher test; asymptotic critical values: 1.224 (10%), 1.358 (5%), 1.628 
(1%), small-sample critical values (Whitcher, 1998): n = 16 – 1.135 (10%), 1.265 (5%), 1.508 (1%),  n = 32 –  
1.157 (10%), 1.293 (5%), 1.553 (1%); small-sample critical values (Inclán, Tiao, 1994): n = 100 – 1.14 (10%), 
1.27 (5%), 1.52 (1%); numeration of the DWT coefficients: 0, 1,…, 38; location of variance change in real 
time; ‘**’ denotes rejection of the homogeneity of variance at the  5% significance level.  

Table 2. Testing for change in variance – the d4 wavelet 

Method and stage of 
analysis 

Statistics No. of the DWT 
coefficient 

(kj) 
 

Location of the 
change in variance 
with the help of the 

MODWT coefficients 
IT - first change-point 1.1297 23  (k2) 42 
W – first change-point 1.0583 23 42 

IT – second change-point 
searched to the left from 

the first 
1.2241* 20  (k1) 28 

W – second change-point 
searched to the left from 

the first 
1.5256*** 20  (k1) 28 

IT – second change-point 
searched to the right from 

the first 
1.0331 34 63 

W – second change-point 
searched to the right from 

the first 
0.6940 34 63 

Note: IT – Inclán and Tiao test; W –Whitcher test; asymptotic critical values: 1.224 (10%), 1.358 (5%), 1.628 
(1%), small-sample critical values (Whitcher, 1998): n = 16 – 1.135 (10%), 1.265 (5%), 1.508 (1%), n = 32 –  
1.157 (10%), 1.293 (5%), 1.553 (1%); small-sample critical values (Inclán, Tiao, 1994): n = 100 – 1.14 (10%), 
1.27 (5%), 1.52 (1%); numeration of the DWT coefficients: 0, 1,…, 38; location of variance change in real 
time; designation of the coefficients indicating a change in variance is in an increasing order; ‘*’, ‘***’ denote 
rejection of the homogeneity of variance at the 10%, 1% significance level, respectively. 

As far as the moments of switch are concerned, certain ambiguities appear. The 
first change-point is in the interval 42–46 (rather closer to 45–46, as the Haar 



Sylwester Bejger, Joanna Bruzda  166 

wavelet has better localization properties, and moreover  – as it was already 
mentioned – the evaluation of the moment of change is burdened with error 
towards the centre of the sample), while the other one – as it results from the 
test with the use of d4 wavelet as well as the charts – around obser- 
vations 28–29. 

4.  Interpretation of Results  

 In order to better illustrate the received results, the series of average Lysine 
prices was presented in Figure 4, together with observation numbers. 

 
Figure 4.  Average price of Lysine, observation numbers on upper horizontal axis. Data 

from Connor, (2000), appendix A, Table A2 

As results from the study we carried out, the moments of variance change are 
identified around observation 28 – the increase of the variance – and around 
observation 45 – the fall of the process’ variance. This piece of statistical in-
formation is confirmed by the analysis of charts 1 and 2. Basing on theoretical 
motivation of the marker of collusion it is mandatory to make the following 
statements: 

a) until April 1992 no significant change of the variance regime can be found, 
therefore the period cannot be considered a factual phase of the price war, 

b) from April 1992 until August 1993 we can talk about a significant phase of 
the growth of the process variance, which points to the price war. It is the 
period of the announcement by ADM (precisely April 1992) of the propo-
sition to establish the ‘amino acids manufacturer society’, until August 
1993, which is around two months before the Irvine meeting (where repre-
sentatives of two main players, AMD and Ajinomoto had confidentially 
agreed cartel quotas of Lysine supply, as they further witnessed),  

1 7 13 19 25 31 37 43 49 55 61 67 73

0,4

0,5

0,6

0,7

0,8

0,9

1

1,1

1,2

1,3

1,4

Ja
n-

90

M
ay

-9
0

S
ep

-9
0

Ja
n-

91

M
ay

-9
1

S
ep

-9
1

Ja
n-

92

M
ay

-9
2

S
ep

-9
2

Ja
n-

93

M
ay

-9
3

S
ep

-9
3

Ja
n-

94

M
ay

-9
4

S
ep

-9
4

Ja
n-

95

M
ay

-9
5

S
ep

-9
5

Ja
n-

96

M
ay

-9
6

Average price of Lysine (USD/pound)



Detection of Collusion Equilibrium in an Industry with Application of Wavelet Analysis  167

c) from September 1993, therefore exactly a month before the Irvine meeting, 
a meaningful drop of the process variance occurs, which may indicate the 
establishment of collusion and the willingness to maintain it. There might 
be a certain mistake in pinpointing the moment of change or perhaps the 
players had gone into informal agreements before the time they indicated 
in their testimonies. 

 Comparing the results of the wavelet analysis with familiar facts from the 
cartel history (see: Connor, 2000), it is necessary to notice that the analysis 
pointed out significant changes in the process variance. The discovered mo-
ments of change are closely tied to vital facts from the cartel history, therefore 
we can assume that they indicate the change in the players’ price behaviour.  
If we accept the proposed game theory model (Bejger, 2010) as the theoretical 
base for that behaviour, changes in the variance correspond to phases of the 
price war and collusion. In confrontation with the to-date analyses of the work-
ings of the cartel, the present empirical study leads to the following differences 
in the history evaluation: 

 price war phase is delayed by around 12 months, 

 first phase of collusion is not clearly distinguished. 

On the one hand, those differences may be caused by a small sample series, 
which disables wavelet analysis at higher decomposition levels, on the other 
they may point to a history of Lysine industry collusion that is much different 
from the one generally accepted. 

Conclusions 

 Taking as an theoretical motivation the model of strategic behaviour of the 
players in an industry with the assumption of time-fixed cartel quotas (market 
shares) an attempt of explaining the behaviour of the players in the industry of 
Lysine manufacturers in the years 1990 – 1996 was made. The model provides 
theoretical support to the method of the detection of collusion based on the 
analysis of the variance of the market price process, as well. For empirical eval-
uation the marker of variance change was applied for a series of average Lysine 
prices on the USA market in the studied period. Wavelet analysis was em-
ployed, for the first time in such a context, as the econometric method of the 
detection of structural changes in the variance. The proposed method of detec-
tion proved to be very useful, indicating the significant changes of the variance 
regime and precisely detecting the moments of such changes, closely connected 
to the key dates in the history of the cartel. While enumerating the advantages 
of wavelet analysis in the proposed application, it is essential to mention: 

 parsimony of specification – as a non-parametric method it is not burdened 
with specification error of the econometric model, 



Sylwester Bejger, Joanna Bruzda  168 

 simplicity of application – the work indispensable to apply the method to 
the data is minimal. Therefore, the method is quick in application, 

 precise indication of the moment of variance change, without any assump-
tions as to their localization. The method is thus absolutely objective, 

 possibility of preliminary, graphic assessment of the variance changes by 
means of MODWT charts as well as rolling wavelet variances. 

Among the drawbacks to wavelet analysis we can indicate relatively high re-
quirements as to the length of the observation series and the lack of the direct 
link of the method to the structure of equilibrium strategy (as it takes place in 
e.g. the application of Markov switching model of MS-AR-GARCH type). 
Therefore, it can be assessed that wavelet analysis in the proposed application 
may serve as a preliminary detector of the variance changes because its applica-
tion is cheap and quick with accurate findings. Wherever it is theoretically justi-
fied other methods may then be applied with the aim of further verification of 
the hypothesis of collusion in an industry. Our work has a case –study character 
so our conclusions should be verified in further research. Next steps in the study 
will be the modification of the theoretical model for the strategies with different 
penal codes and – on the empirical plane – an attempt to apply the wavelet 
analysis of variance for other price time series in industries susceptible to ex-
plicit or tacit collusion. 

References 

Abrantes-Metz, R., Froeb, L., Geweke, J., Taylor, C. (2006), A variance screen for collusion, 
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Athey, S., Bagwell, K., Sanchirico, C. (2004), Collusion and price rigidity, Review of Economic 
Studies, 71, 317–349. 

Bejger, S. (2010), Collusion and seasonality of market price – a case of fixed market shares, 
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Bejger, S. (2009), Econometric tools for collusion detection, AUNC Ekonomia XXXIX, 125–132. 
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Empirical results from two recent cases, International Journal of Industrial Organization 
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Connor, J. (2001), Our customers are our enemies”: the lysine cartel of 1992–1995, Review of 
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Connor, J. (2000), Archer Daniels Midland: Price-fixer to the World, Staff paper No. 00-11, 
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Daubechies, I. (1992), Ten Lectures on Wavelets, SIAM, Philadelphia. 
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Detection of Collusion Equilibrium in an Industry with Application of Wavelet Analysis  169

Whitcher, B. (1998), Assessing Nonstationary Time Series Using Wavelets, PhD thesis, Universi-
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Detekcja równowagi zmowy w branży z wykorzystaniem analizy falkowej 

Z a r y s  t r e ś c i. Artykuł zawiera empiryczne zastosowanie markera zmian strukturalnych 
wariancji procesu ceny rynkowej dla szeregu cen średnich Lysiny na rynku USA w latach 1990–
1996. Jako metodę ekonometryczną detekcji zmian strukturalnych w wariancji zastosowano, po 
raz pierwszy w tym kontekście, analizę falkową. Metoda ta ma w omawianym zakresie aplikacji 
istotne zalety, takie jak oszczędność w zakresie wymaganych danych statystycznych oraz bardzo 
dobre własności lokalizacyjne w dziedzinie czasu. W pracy wykorzystano model teoretyczny 
zachowań strategicznych graczy w branży  motywujący zastosowanie wymienionego markera. 

S ł o w a  k l u c z o w e: Zmowa jawna i milcząca,  supergra ze stała strukturą udziałów w rynku, 
lysina, wariancja ceny, analiza falkowa.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 

 

 

 

 


