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DYNAMIC ECONOMETRIC MODELS 
Vol. 9 – Nicolaus Copernicus University – Toruń – 2009 

Sylwester Bejger  
Nicholas Copernicus University in Toruń  

  

Econometric Tools for Detection of Collusion Equilibrium  
in the Industry 

A b s t r a c t. The article presents the notion of detection of overt or tacit collusion equilibrium in 
the context of choice of the appropriate econometric method, which is determined by the amount 
of information that the observer possesses. There has been shown one of the collusion markers 
coherent with an equilibrium of the proper model of strategic interaction – the presence of struc-
tural disturbances in the price process variance for phases of collusion and competition. The 
Markov Switching Model with switching of variance regimes has been proposed as a proper 
theoretical method detecting that type of changes without prior knowledge of switching moments. 
In order to verify the effectiveness of the method it has been applied to a series of lysine market 
prices throughout and after termination of its manufacturers’ collusion.  

K e y w o r d s: explicit and tacit collusion, collusive equilibrium, cartel detection, lysine, price 
variance, Markov switching model 

1. Introduction  
 The equilibrium of players’ collusion in the industry may occur as a conse-
quence of players’ strategic interaction of the overt or tacit collusion nature. 
Although these two types of interaction are described by means of different 
models of game theory with various informative assumptions, their equilibriums 
are characterized by similar consequences for the industry, i.e.: 

a) the occurrence players’ market power leading to the loss of social wealth 
(above-normal PCM), 

b)  the limitation of competition and the impediment of the industry develop-
ment . 

 The article presents one of collusion markers resulting from the theoretical 
model of tacit collusion, which is price rigidity in the collusion phase, and pro-
poses the application of the Markov Switching Model of MS-AR-GARCH type 
in order to detect structural changes in market price variance, and thereby to 



Sylwester Bejger 

 

28

verify the presence of the aforementioned marker. The preliminary verification 
of usefulness of the proposed econometric method constitutes a research prob-
lem. For this purpose, in the empirical section there has been applied the pro-
posed model for a series of market prices of cartel players – lysine manufactur-
ers. 

2. Collusion and Its Quantitative Detection  
 Collusion constitutes a serious problem in the market economy. If we take 
into account only overt collusion (mainly in the area of English-speaking report-
ing) between 1990 – 2005, 283 cartels were confirmed (so called hard core 
cartels) with national and global scope of operation, however, it has been esti-
mated that as little as 30% of overt collusion are detected and punished. There 
are no data concerning tacit collusion. The above mentioned 283 cartels influ-
enced sales of 2.1 trillion USD, caused unjustifiable price overcharge of the 
value of 500 bn USD and were punished with pecuniary penalty of the nominal 
value amounting to 25.4 bn USD (Connor, Helmers, 2006).  
 Taking into account the fact how common and harmful such collusion is, it 
seems natural that it should be quickly detected. Unfortunately, although theo-
retical models of overt or tacit collusion are described very well as research 
hypotheses concerning players’ behavior, their empirical verification presents 
great difficulties. It happens mainly due to the fact that the players participating 
in collusion have an advantageous position over the observer in the form of 
private information. Moreover, the resources of public statistics are frequently 
very humble on the disaggregation level of the industry or individual players. 
Thus, it is not difficult to ascertain that econometric methods, that are, on the 
one hand, economical as far as the use of statistical data is concerned, and, on 
the other hand, coherent with the model hypothesis, are particularly valuable. 
Currently known econometric methods of detection can be divided into direct 
and indirect methods: 

a) direct - the assessment of strategy profile in equilibrium that is consistent 
with the assumed collusion model, verification of the hypothesis on con-
formity with theoretical equilibrium, 

b) indirect - measurement and/or identification of market power or detection 
of so called collusion markers (non-competitive behaviors), i.e, certain, 
characteristic for collusion, disorders concerning: 
− the relation between players’ prices and market demand changes, 
− price and market share stability, 
− the relation between players’ prices, 
− investment in production capacity. 



Econometric Tools for Detection of Collusion Equilibrium in the Industry  

 

29

When it comes to statistical data, group a) methods are very demanding, their 
applications are very rare and they are possible only in specific circumstances1. 
Group b) methods are much more common. The following basic methods are 
enumerated in the order of intensity of the use of statistical data: 
− the study of structural changes in price volatility , 
− non-parametric method based on revealed preferences, 
− the Osborne – Pitchik test, 
− the study of asymmetry of price reactions, 
− Hall’s method, 
− the assessment of residual demand elasticity, 
− the Panzar – Rosse method, 
− CPM method.  

3. Price Disorders Characteristic for Collusion  
 One of the most promising collusion markers are markers that are based on 
the analysis of changes in price processes and/or market shares. In accordance 
with known tacit collusion models: 

1. the player’s (players’) price and supply are negatively correlated, the price 
is ahead of the demand cycle, the stochastic process of market price under-
goes changes of the regime type (Green, Porter, 1984; Rotemberg, Salon-
er, 1986; Haltiwanger, Harrington, 1991), 

2. the price process variance is on average lower for collusion phases and may 
undergo changes of the regime type (Athey, Bagwell, Sanchirico, 2004; 
Connor, 2004; Abrantes-Metz, Froeb, Geweke, Taylor, 2006; Bolotova, 
Connor, Miller, 2008),  

The application marker 2 is particularly promising in practice. It is justified by 
the fact that the requirements for this marker concerning data are very little 
(market price is sufficient) and it has clear theoretical justification. Lower price 
volatility in the collusion phase results directly from the equilibrium properties 
of SPPE type (symmetric perfect public equilibrium) of the price super-game 
with standard assumption concerning sufficiently high discount rate. For the 
strategy profile in the equilibrium of this game, players: 
− achieve higher payments than in competitive equilibrium (cartel payments), 
− in the collusion phase, players’ prices are insensitive to costs shocks (play-

ers avoid price changes, even at the cost of effectiveness, not to cause 
switching to penalty phase). 

                                                 
1 For example, see Slade (1992). 



Sylwester Bejger 

 

30

It should be noted that the necessity of using a number of observations includ-
ing both competition and collusion phases and high product homogeneity of the 
analyzed trade is one of the marker’s drawbacks. 

4. Econometric Method Verifying Marker’s Presence  
 The works conducted so far that are connected with collusion detection on 
the basis of the detection of structural changes in variance included the applica-
tion of the descriptive statistics methods for the comparison of variance level in 
collusion and competition phases (Abrantes-Metz, Froeb, Geweke, Taylor, 
2006) and the application of ARCH / GARCH specification for the market price 
process together with additional 0-1 variable describing collusion and competi-
tion phases (Bolotova, Connor, Miller, 2008, further BCM ). 
 This paper suggests using the Markov Switching Model of 
MS(M)(AR(p))GARCH(p,q) type as an econometric method verifying the 
marker for the variance and/or the average (constant) of the price process 2. The 
application of this model has the following advantages:  
− it is theoretically coherent with the structure of equilibrium strategy of the 

super-game model, 
− it allows to model structural changes of process variance directly, without 

the use of additional artificial variables; such modeling is not possible in, 
e.g. ARCH / GARCH specification, 

− it is coherent with informative asymmetry between cartel members and the 
observer. MS(AR)GARCH specification does not require observation 
(knowledge) of the state variable, thus it may be used for actual detection of 
variance regimes and objective determination of switching moments, that is 
the detection of collusion and competition phases. The form of the general 
MS(M)(AR(p))GARCH(p,q) model is a development version of a well-
known MS model (Hamilton, 1989; Hamilton, Susmel, 1994; Krolzig, 1998; 
Stawicki, 2004; Davidson, 2004). The application of the model with re-
gimes in variance refers mainly to high frequency data, such as currency 
rates, rates of return from financial instruments, prices of electric energy3. 

The general form of switching model which has been applied can be written 
down as4: 

                                                 
2 According to Krolzig this type of model could be described as MSI(M)H-AR(q) with 

GARCH(p,q) component, Krolzig (1998). 
3 For example, refer to Fong (1998), Włodarczyk, Zawada (2005, 2007), Kośko, Pietrzak 

(2007). 
4 There are various forms of notation, this one is from Davidson (2004). 



Econometric Tools for Detection of Collusion Equilibrium in the Industry  

 

31

∑
=

− ++=
p

m
tmtmSSt uyy

tt
1

0 ,φα  (1) 

where: 

),1,0.(..~  oraz  2/1 diieehu tttt =  

,
1

2
0 ∑

∞

=
−+=

m
mtmSSt uh

tt
ββ  (2) 

The conditional variance equation (2) uses ARCH(∞) specification which also 
includes models of GARCH (p,q) class. 
In model (1),(2) each parameter may be potentially a random variable switched 
between the values from a finite set of values depending on the actual state of 

tS  where MSt ...,,1= .  

Variable tS  is assumed to be the exogenous, homogeneous Markov process 
with fixed transition probabilities { }ijp  where: 

)Pr( 1 iSjSp ttij === − . 

The probability that the observed process ty  is in the state j in the period t is 
presented by means of the filtering (updating) equation:  

,
)|Pr(),|(

)|Pr(),|(
)|Pr(

11 1

11

−= −

−−

Ω=Ω=

Ω=Ω=
=Ω=
∑ tt

M
i ttt

ttttt
tt

iSiSyf

jSjSyf
jS  (3) 

where tΩ  signifies the entire information available at moment t, and: 

),|Pr()|Pr( 1111 −−=− Ω==Ω= ∑ tt
M
i ijtt iSpjS  (4) 

where transition probability ijp constitutes M(M-1) parameters to be estimated. 

The form of conditional density function of observed variable: 
),,|(. 1−Ω= tt jSf  

requires making an assumption concerning the type of distribution.  
Estimation of model parameters may be obtained through maximum likelihood 
method. For this purpose a likelihood function is used: 

),|Pr(),|(Prlog 11 1
1

−= −
=

Ω=Ω== ∑∑ tt
M
i ttt

T

t
jSjSyfL  (5) 

The maximization of function (5) is conducted by means of a very well known 
method, Expectation Maximization (EM) algorithm (Krolzig, 1998, p. 8). 



Sylwester Bejger 

 

32

5. Empirical Verification  
 The collusion of lysine producers5 was proved in 1996. The test includes 
monthly average lysine prices on the USA market in the period between 01/90 – 
06/966. Within this period, on the basis of collected evidence (Connor, 2001) 
the following phases may be distinguished (Table 1). 

Table 1. The statistics of lysine price (prices per pound) 
Phase  Months 

number 
Average  Standard devia-

tion 
Coefficient of 

variation 
1. Competition (01.90 – 07.92) 31 102.90 16.22 15.8% 

2. Collusion (08.92 – 03.93) 8 90.13 9.83 10.9% 
3. Competition (04.93 – 07.93) 4 70.50 7.72 11% 

4. Collusion (08.93 – 06.95) 23 110.30 8.55 7.8% 
5. Competition (07.95 – 06.96) 12 102.50 9.51 9.3% 

 The purpose of the empirical research is to check to what degree the model 
with the proposed specification may be used to detect changes of the regime 
type in the variance of the process generating data, and thereby detect collusion 
phases. Such verification is possible due to the knowledge of the collusion his-
tory, provided that this case has been correctly determined (i.e., no significant 
evidence was omitted during the trial). 
Initially, there was verified a hypothesis on variances equality for two compara-
ble phases. Table 2 summarizes this step. 

Table 2. The value of statistics for the variance equality test in phase 1 and 4 
Bartlett 7.6064 (0.005) 

Brown-Forsythe 3.9206 (0.053) 
Test F 3.2106 (0.003) 

Note: p-values given in brackets. 

On the basis of the test it can be ascertained that variances in both phases are 
significantly different.  
Next, the properties of the examined series were checked in terms of the distri-
bution characteristics and autocorrelation, stationarity and homoscedasticity of 
residuals. The results of this part of research are included in Table 3. 

                                                 
5 Lysine is an basic amino acid required as a feed component in hog, poultry and fish produc-

tion. 
6 The prices are from Connor, (2000), appendix A, Table A2. 



Econometric Tools for Detection of Collusion Equilibrium in the Industry  

 

33

Table 3. Characteristics of series  

Skewness -0.765  Jarque-Bera 
normality test 

7.634 
(0.021) 

 ADF test -3.627* 
(0.007) 

Kurtosis 3.081  Ljung – Box test 
for levels – Q(5) 

145.260 
(0.000) 

KPSS test 0.160** 

LM test for heteroscedasticity of residuals 9.685 (0.002) 
Note: p- values given in brackets, * value of t statistics (critical values for 1%, 5%, 10% sig. levels – (-3.519), 
(-2.900), (-2.587), ** value of LM statistics (asympt. critical values for 1%, 5%, 10% sig. levels – 0.739, 
0.463, 0.347).  

The series is skewed, the hypothesis of normal distribution was rejected and by 
means of test with different configuration of hypotheses the series stationarity 
was confirmed. After removing autocorrelation, there is explicit heteroscedas-
ticity of a random component, which indicates relations in the variance which 
were not included in the model.  
In the next stage of the research a number of models of 
MS(k)(AR(p))GARCH(p,q) type was constructed and estimated with the use of 
maximum likelihood method. The best results, when it comes to the model 
properties, were achieved for MS(2)(AR(2))GARCH(1;0) specification in the 
form of: 

∑
=

− ++=
2

1
0 ,

m
tmtmSt uyy

t
φα  (6) 

where: 
),1,0.(..~2/1 diieorazehu tttt =  

,2
110 −+= tSt uh

t
ββ  (7) 

.2;1=tS  

This specification assumes controlling the observable price process through 
non-observable stochastic process of state variable st, which is assumed to be a 
homogeneous Markov chain of two states and proper matrix of transition prob-
abilities between states. A constant and unconditional error variance are para-
meter which depends on the regime. Moreover, regardless of the regime, the 
average of a series is described by AR(2) process, whereas, GARCH(1;0) com-
ponent is present in the variance. Table 4 shows estimation results.  



Sylwester Bejger 

 

34

Table 4. Estimation results MS(2)(AR(2))GARCH(1;0) 
Parameter estimation p-value Tests 

p11 0.945 ---- L.og Likelihood -204.670 
p22 0.861 ---- R2 0.961 
φ1 1.290 0 Skewness (residuals) -0.021 
φ2 -0.485 0 Kurtosis (residuals) 2.680 
β1 0.405 0.094 Jarque-Bera test (resi-

duals) 
0.326 (0.849)* 

α01 111.216 0 Ljung-Box test (residuals, 
Q(5)) 

3.206 (0.668) 

α02 83.479 0 Ljung-Box (resi-
duals2,Q(5)) 

4.699 (0.454) 

β01 2.104 
(4.430)** 

---- LM Autocorrelation test 0.042 (0.837) 

β02 3.497 
(12.230)** 

---- LM neglected ARCH test 4.014 (0.547) 

Note: * p-values in brackets, ** variance 

The most essential question is whether the proposed model may serve the as-
sumed objective, i.e. collusion detection. It can be estimated on the basis of 
precision of regime detection. Figure 1 shows the values of the observed varia-
ble and smoothed probabilities for regime 1 (i.e. conditional probabilities of the 
process is in state s1, while taking into account information from the entire sam-
ple) respectively, together with marked collusion phases.  
 

Figure 1. Lysine prices and smoothed probabilities 
 



Econometric Tools for Detection of Collusion Equilibrium in the Industry  

 

35

One can notice significant conformity of changes detected by the model in 
the variance regime and the average with the observed collusion phases (partic-
ularly in the case of phase 2 which is detected almost ideally). The values of the 
average variance of an error term in regime 1 is more than 2.5 times bigger than 
in regime 2 which is consistent with the assumed theoretical hypothesis. Proba-
bilities pii of maintaining in the “competition” and “collusion” states are high, 
which well replicates the structure of profile of the equilibrium supergame strat-
egies. Unfortunately, this model does not detect a regime change for other pe-
riods, which is connected with the fact that both the constant and variance un-
dergo changes. Although the estimation of the value of the constant for the low 
variance regime (collusion phase) is significantly higher than for the competi-
tion regime, which may be confirmed by traditional understanding of price col-
lusion, however, on the basis of regimes, generally speaking, the average price 
level, one cannot unambiguously draw conclusions concerning the type of equi-
librium, without additional statistical information e.g. on demand level. Figure 2 
presents the comparison of price levels and smoothed probabilities for the mod-
el which is not so well fitted (with switching exclusively in variance) but which 
detects particular phases in more unambiguous manner7.  

 

Figure 2. Lysine prices and smoothed probabilities 
 

                                                 
7 Autoregressive component in switching model could bias signal of filtered probabilities if 

corrects other then gaussian disturbances of error term. See Lahiri, Whang, (1994). 



Sylwester Bejger 

 

36

Specification for this model is MS(2)-AR(1), Table 5 presents the estimation of 
parameters. 

Table 5. Estimation results MS(2)-(AR(2)) 
Parameter Estimation p-value Tests 

p11 0.683 ---- L.og Likelihood -236.450 
p22 0.967 ---- R2 0.851 
φ1 0.946 0 Skewness (residuals) -0.115 
α0 104.641 0 Kurtosis (residuals) 2.452 
β01 0.433 

(0.187)** 
 Jarque-Bera test (resi-

duals) 
1.134 (0.567) 

β02 5.911 
(34.939)** 

 Ljung-Box test (residuals, 
Q(5)) 

50.078 ( 0.000) 

   Ljung-Box (resi-
duals2,Q(5)) 

16.650 (0.005) 

   LM Autocorrelation test 11.832 (0.001) 
   LM neglected ARCH test 12.179 (0.032) 

Note: * p-values in brackets, **variance 

 This model regarding the detection of equilibrium types on the basis of va-
riance is closer to the actually observed cartel history. First of all, the average 
length of staying in each regime 1)1( −−= iist pd  is consistent with the history 
as far as proportion is concerned, the probability of switching from the competi-
tion phase to the collusion phase is higher and collusion stability is lower. If we 
compare both models in term of value iip  it turns out that model 2 is more con-
sistent with the collusion history (the average collusion phase is 3.1 month long, 
whereas in the case of model 1 it is 16 months long)  

6. Summary 
 The Markov switching model with the switching component in variance 
and/or GARCH process parameters has unquestionable theoretical advantages 
when it comes collusion detection on the basis of variance changes in the mar-
ket price. On the basis of the empirical research, the correctness of the detection 
may be found at least in respect to variance regimes. The model in specification 
that is more advantageous in terms of process replication in better adjusted to 
data than models used in BCM work (taking into account value of log likelih-
ood). It must be remembered, however, that empirical verification was based on 
one, although unique, but relatively short series of data. At the next stage of 
research, the accepted method should be tested on other empirical series (which 
is difficult due to difficulties in obtaining data) or on the series generated for 
various collusion strategy profiles.  



Econometric Tools for Detection of Collusion Equilibrium in the Industry  

 

37

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Sylwester Bejger 

 

38

Ekonometryczne narzędzia detekcji równowagi zmowy w branży 

Z a r y s t r e ś c i. W artykule przedstawiono problem detekcji równowagi zmowy jawnej lub 
milczącej w kontekście wyboru właściwej metody ekonometrycznej, który determinowany jest 
ilością informacji posiadaną przez obserwatora. Zaprezentowano jeden z markerów zmowy spój-
nych z równowagą właściwego modelu interakcji strategicznej – obecność zaburzeń struktural-
nych w wariancji procesu ceny dla faz zmowy i konkurencji. Jako poprawną teoretycznie metodę 
detekcji tego typu zmian bez wiedzy a-priori o momentach przełączania zaproponowano wyko-
rzystanie przełącznikowego modelu Markowa z przełączaniem reżimów wariancji. W celu wery-
fikacji skuteczności metody aplikowano ją dla szeregu cen rynkowych lysiny w czasie trwania 
i upadku zmowy jej producentów.  

S ł o w a k l u c z o w e: Zmowa jawna i milcząca, równowaga, lysina, wariancja ceny, model 
przełącznikowy Markowa  

 


