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© 2015 Nicolaus Copernicus University. All rights reserved.  
http://www.dem.umk.pl/dem 

D Y N A M I C  E C O N O M E T R I C  M O D E L S  
DOI: http://dx.doi.org/10.12775/DEM.2015.007  Vol. 15 (2015) 111−128 

Submitted October 23, 2015  ISSN (online) 2450-7067 
Accepted December 15, 2015 ISSN (print) 1234-3862 

Sylwester Bejger* 

Testing Parallel Pricing Behavior  
in the Polish Wholesale Fuel Market:  
an ARDL – Bound Testing Approach 

A b s t r a c t. In this study, we investigated whether the observed series of fuel prices can be 
compatible with a specific theoretical model of strategic player interaction. Our primary 
interest is in determining whether a parallel pricing policy, implied by a theoretical model of 
strategic interactions, can be an industry-observed pricing mechanism. Therefore, we first 
calculated various descriptive statistics of the price series to discover any common patterns of 
individual series. Next, we determined whether parallel co-movement of the price levels exist 
using an ARDL – bound testing approach. This study finds that if we restricted our research 
to the described pricing mechanism (IPP pricing based on previous day fundamentals), the 
players will have chosen the levels of price in a parallel mode; this excludes 2007, when 
LOTOS appeared to be the price leader. 

K e y w o r d s: wholesale fuel market, parallel pricing, cointegration. 

J E L Classification: L1, L7, C22. 

Introduction  

 The general objective of this paper was to discover particular, strategic 
patterns in the price behavior of the players in the Polish wholesale fuel 
market in a sample period of 2004 to 2013. This study is shaped as an inter-
section of monitoring and verification (to use the terminology of Harrington, 
2008) in the context of behavioral screening. However, this study is more 
broadly understood because we did not constitute tests for verification of the 
                                                 

* Correspondence to: Sylwester Bejger, Nicolaus Copernicus University, Department of 
Econometrics and Statistics, ul. Gagarina 13A, 87-100 Toruń, Poland, e-mail: sylw@umk.pl. 



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hypothesis of collusion (tacit or overt) existence in the industry on the basis 
of a specific marker1. Instead, we strive to answer the question regarding 
whether the observed series of price levels can be compatible with the 
known model of the strategic interaction of players (may be part of the equi-
librium of a game).  
 On the basis of industry2 research, important factors that influence the 
strategic conduct of the market participants can be enumerated: a duopolistic 
market with high concentration, homogeneous products, high barriers to 
entry, capacity constraints for domestic production, inelastic demand, possi-
ble IPP3 pricing mechanism and major players’ full price transparency.  
 Based on these characteristics, it is possible to restrict the set of adequate 
models of noncooperative games by the specification of the essential ele-
ments of a game model. These specifications include the following: the price 
should be considered as a main strategic variable for the players, there were 
no threats of significant entry to the market in the sample period, and pro-
duction capacities should be treated as exogenous parameters in the game 
period. Through these restrictions, one can isolate a single period game as a 
non-zero-sum, simultaneous moves finite game in pure action spaces. This 
set of assumptions leads to a choice of a standard Bertrand model as a build-
ing block of certain one-shot games; this is enriched with additional ele-
ments to eliminate equilibrium in marginal costs as the sole solution. Models 
based on the well-known Edgeworth model (1925) would be particularly 
well-suited in this instance. Edgeworth’s model modifications have led to 
various models, for example, those by Levitan and Shubik (1972), Kreps and 
Scheinkman (1983), Osborne and Pitchik (1986), Deneckere and Kovenock 
(1992). Because those models are essentially static, one can consider a prop-
er dynamic specification. In this case, a highly popular Maskin and Tirole 
(1988) model and a supergame approach to oligopolistic competition could 
be considered. As a reference work in that topic, one can point to Tirole 
(1998) and Vives (2001). For our empirical research, the works of Brock and 
Scheinkman (1985), Lambson (1987), Green and Porter (1984), Rotemberg 
and Saloner (1986), Dudey (1992), Lu and Wright (2010) could be inspira-
tional because of the choice of price as a strategic variable and because of 
the imposed capacity constraints in certain games. However, analyzing the 
abovementioned models, we can conclude that their structures do not explain 

                                                 
1 Refer to: Harringtopn (2006, 2008) or Aberantes-Metz (2011, 2013), Abrantes-Metz and 

Bajari (2012). 
2 Details of that study will be published in an another paper. 
3 Import Parity Pricing. 



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the very important market’s phenomenon, i.e., IPP pricing mechanism. In 
our view, such a mechanism leads to a concept of a reference point in strate-
gic interactions that could be described as a focal point or ceiling point. 
There is substantial empirical evidence of the use of focal point prices by 
firms. Scherer (1980) found price lining is widespread at the retail level in 
the US. Using data from the US-credit card market during the 1980s, Knittel 
and Stango (2003) show that nonbinding price ceilings, which serve as ex-
ternal focal points, increase the probability that firms engage in tacit collu-
sion. In Faber, Janssen (2011), they investigated the focal point effects of oil 
companies that suggest petrol and diesel prices to their retailers. We wanted 
to utilize the concept then but formulate the question slightly differently: 
what were the strategic implications of IPP treated as the “focal price” for 
player’s daily actions (price levels)? We then decided to construct a game 
theory model of pricing behavior that is suitable for the industry’s parame-
ters4. The main conclusions from our theoretical model are the following:  
 it is strategically possible to use the same (or very close) price levels by 

both players in daily interactions; therefore, regarding the parallel pric-
ing phenomenon, we can state, 

 the common price level for both players should be very close to the IPP 
price level (or a proxy of it), if we assume the ability of players to 
properly calculate it on the basis of commonly known factors. 

Thus, the primary purpose of this article concerns the empirical verification 
of the first of the above implications. We want to determine whether the 
observable data reflects parallel pricing strategic behavior in a market and, 
as an implication, if the rational conduct of the players is coherent with the 
equilibrium strategies of a game theory model. Few studies of broad price 
movements and dependencies in the Polish refining sector in general (Bejger 
and Bruzda, 2002; Miłobędzki, 2008; Leszkiewicz-Kędzior, 2012) exist; in 
addition, no such studies (to our knowledge) are devoted to analyzing the 
pricing behavior of both players in parallel, focused on the strategic interac-
tions among them.  
 The methodological construction of the research first contains the basic 
statistical measures of the empirical distributions of prices used; finally, it 
examines the cointegration between the prices of players and the existence of 
the long-term and short-term relations (if any) for the two series. To investi-
gate the construction, the ARDL – bound testing approach was used because 
it is fairly universal and insensitive to the misspecification of integration 
order. The remainder of this paper is structured as follows. Section 2 briefly 

                                                 
4 Bejger (2015), forthcoming. 



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outlines the data. Section 3 reviews the estimation techniques and contains 
the empirical analysis. Section 4 presents the study’s conclusions. 

2. Data Description  

 The main data set covers wholesale (unregular) daily prices on PKN 
PB95, PKN ON, LOTOS PB95 and LOTOS ON in PLN for cubic meters 
and the 01.01.2004–31.12.2013 sample period. These four time series have 
been obtained from official websites of the players5. We decided to conduct 
research on two slightly different data sets from the basic sample; we examined 
“raw” series (of different lengths) to reveal their individual properties. Next, we 
synchronized these data sets to study cointegration. In the raw data analysis, we 
did not subtract the excise tax and fuel duty because the reactions of price 
levels to the changes in these duties could contain information regarding the 
strategic behavior of players. For the purpose of synchronization of the ob-
servations, we used the dates of observations of PKN’s gasoline price as 
reference. Missing daily observations of LOTOS were replaced by simple 
extrapolation between the values observed immediately before and after; 
therefore, both final series have 2281 observations. Such transformation of 
LOTOS’ data did not harm the statistical properties of original series sub-
stantially (integration order unchanged, descriptive statistics that are closer 
to PKN in values). Additionally, we simplified the names of the synchro-
nized series to LOTOSPB and PKNPB to use them as variable names.  
 Figure 1 shows raw data. There are obvious similarities in the time evo-
lution of the prices of the same type of fuel in both players and in the differ-
ences in the price processes of different fuels. 
 As a preliminary examination, the basic descriptive statistics of the (raw) 
series were calculated. Table 1 contains the results of this step for the indi-
vidual series (in levels)6. From the results, one can observe the similarities of 
moments for the same type of fuels and the strong rejection of the normality 
of empirical distributions in all cases.  
 

                                                 
5 http://www.orlen.pl/PL/DlaBiznesu/HurtoweCenyPaliw; 

http://www.lotos.pl/144/dla_biznesu/hurtowe_ceny_paliw. 
6 As we can observe, the series are of different length. For statistical or econometric re-

search, this difference is an obvious disadvantage; however, we rely strongly on daily obser-
vations such as those that could reveal strategic patterns in player’s behavior.  



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Figure 1. Time series of prices (wholesale, fuels)  

Table 1. Descriptive statistics  

PKN PB95 PKN ON LOTOS PB95 LOTOS ON 
 Mean* 3521.64 3400.70 3530.19 3375.92 

 Median* 3374.00 3184.00 3381.00 3153.00 
 Maximum* 4757.00 4663.00 4768.00 4667.00 
 Minimum* 2425.00 2117.00 2428.00 2121.00 
 Std. Dev.* 573.84 657.40 572.05 664.37 

 Coef. Of var. 16.29% 19.33% 16.20% 19.68% 
 Skewness 0.36 0.45 0.35 0.45 
 Kurtosis 1.98 1.81 1.97 1.87 

 Jarque-Bera 149.24 194.44 149.95 204.37 
 p-value 0.00 0.00 0.00 0.00 

 Number of Obs. 2281 2094 2343 2343 
Note: * – values in PLN.  



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3. Estimation Techniques and Empirical Analysis 

 As stated in the introduction, the main objective for this research was an 
empirical verification of the pricing schema that was implied by the theoreti-
cal model of strategic interaction (the focal price game), parallel pricing 
strategic behavior. 
 At first, the model has to specify what statistical and econometric 
measures can be used to verify such pricing. If one understands movements 
of prices as parallel pricing, Buccirossi (2006) stated, on the basis of a sim-
ple game, that: “concise representation of the degree of price parallelism is 
provided by the correlation between prices”7. A testable proposition (partly 
adequate to our case) is: “If that the market is perturbed only by shocks on 
costs, then: if shocks are perfectly common, the correlation between prices 
equals 1 in both the competitive and the collusive equilibrium”8. Of course, 
such an ascertainment is consistent with intuition of course; however, in our 
opinion, simple correlation is not a sufficient tool for time series of very 
complex structure. We can begin from this point; however, we test other 
measures of similarity of series, as well. However, first, we want to deter-
mine whether the parallel movement could be observed not only in the 
common- comovement of the levels. Parallelism could mean similar patterns 
in the empirical distributions of price changes and price grids, for example. 
We want to research that possibilities at first and move to the examination of 
co-movements next. To limit the size of the analysis, and for data availabil-
ity, we focused solely on gasoline prices. We began with raw data (daily 
observations not paired) to obtain as much information as possible. 

3.1 Statistical Analysis of the Gasoline Price Changes 

 
 To check the movements of the price changes more closely, we calculat-
ed the positive and negative average changes of values (in PLN) and their 
relation to the price level. The results are shown in Table 2.  
 At the beginning of the sample period (2004 and 2005), a higher relative 
average change can be noticed, which could be connected to higher grid 
price levels in that period for LOTOS (Figure 2). 
  

                                                 
7 Buccirossi (2006), p. 92. 
8 Proposition 3, Buccirossi (2006), p. 94. 



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Table 2. Average positive and negative changes of prices by years 

 LOTOS PB95 PKN PB95  

year avg. change 
avg. change as 

% of price 
avg. change 

avg. change as % 
of price 

abs. difference 
of % change 

2004 37.95 1.374% 39.40 1.426% 0.052% 
 –37.55 –1.348% –41.53 –1.488% 0.140% 

2005 38.08 1.280% 46.83 1.575% 0.296% 
 –40.60 –1.315% –55.71 –1.801% 0.486% 

2006 22.18 0.729% 12.04 0.397% 0.332% 
 –21.79 –0.712% –14.32 –0.472% 0.241% 

2007 20.47 0.652% 13.47 0.432% 0.220% 
 –15.05 –0.466% –10.37 –0.322% 0.144% 

2008 18.42 0.552% 13.59 0.410% 0.143% 
 –22.58 –0.709% –18.69 –0.594% 0.115% 

2009 25.47 0.821% 23.80 0.771% 0.049% 
 –20.60 –0.632% –15.02 –0.461% 0.170% 

2010 18.21 0.519% 17.71 0.505% 0.015% 
 –14.90 –0.422% –12.83 –0.363% 0.059% 

2011 20.39 0.508% 19.43 0.484% 0.024% 
 –20.40 –0.505% –19.35 –0.479% 0.026% 

2012 17.05 0.383% 16.33 0.368% 0.015% 
 –22.98 –0.515% –21.59 –0.485% 0.031% 

2013 19.99 0.468% 19.08 0.447% 0.021% 
 –18.88 –0.439% –17.53 –0.408% 0.031% 

Note: Bolded numbers mean the rejection of H0 regarding the equality of a mean positive or negative 
change of prices between LOTOS and PKN in a particular year. 

 

Figure 2.  Empirical distribution of price grids – PKN PB95 (left panel) and LOTOS 
PB95 (right panel) 

 PKN’s grid was more evenly distributed; however, grid 5 was dominant 
in 2004 and 2005, as well. In accordance with Table 2, 2006–2008 were very 



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different than the remainder of the sample because the average changes (of 
both signs) decreased but became significantly different for both players. 
Those changes signal disturbances in the behavior of players in that period. 
As can be observed, the beginning of this change corresponds to LOTOS’ 
introduction of values other than 5 PLN of the price grid and the synchroni-
zation of the announcement of prices. From 2009 to the end of the sample 
average price, the changes became statistically equal; however, we can sub-
jectively (not in the statistical sense) observe that considering relative 
changes, upward movements were closer. The last interesting analysis of raw 
data is contained in Table 3. 

Table 3. Daily distributions of price changes 

     LOTOS PB95 PKN PB95 

 LOTOS 
PB95 

PKN 
PB95 

LOTOS 
PB95 

change 

PKN 
PB95 

change 
up down no 

change 
up down no 

change 

Mon 8 3 0.3% 0.1% 0.3% 0.2% 0.1% 0.2% 0.0% 0.5% 
Tue 432 441 18.6% 19.6% 19.1% 18.0% 17.8% 20.1% 19.1% 16.7% 
Wen 439 446 18.2% 19.5% 17.8% 18.7% 21.7% 17.2% 22.0% 19.8% 
Thur 457 469 19.9% 20.6% 18.0% 21.9% 17.2% 21.0% 20.2% 19.3% 
Frid 461 456 19.8% 19.8% 20.7% 18.9% 18.9% 20.4% 19.2% 22.4% 
Sat 476 462 20.7% 20.3% 21.6% 19.8% 18.1% 21.0% 19.5% 19.8% 
Sun 70 4 2.5% 0.0% 2.5% 2.6% 5.6% 0.1% 0.0% 1.6% 

Note: Columns up, down, no change contain contribution of daily up, down, no change price move-
ments in whole number of movements of a given type. Highest values bolded. 

 Table 3 refers to the daily distributions of price movements of both play-
ers; there are certain differences. However, after checking pairs of empirical 
distributions of the same type (i.e., LOTOS up – PKN up) using the Kolmo-
gorov-Smirnov test, we could not reject the hypothesis of the equality of 
cumulative distributions. Therefore, the days of movements of prices are 
largely the same. 

3.3. Cointegration Analysis  

3.3.1. ARDL – Bounds Testing Procedure 

 With that test, we want to move to the next step of our research, testing 
the common movements of both price series.  
 Preliminarily, we checked the correlation coefficient for both series. As 
we expected, correlations were very strong, with a magnitude of 0.99 in the 
entire sample and not below 0.97 in the sub-samples that encompass each 
year. However, for the possible cointegrated series, we need a more precise 



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evaluation of the co-movement. We were interested in determining whether 
a cointegration exists between the prices of players, and what type of long-
run and short-run relations (if any) exist for the two series. To study these 
relations, we decided to use the ARDL-bound testing approach. The ARDL 
modeling approach was originally introduced by Pesaran and Shin (1999) 
and extended by Pesaran et al. (2001). The ARDL cointegration approach 
has numerous advantages over conventional cointegration testing. The ap-
proach yields consistent estimates of the long-run coefficients that are as-
ymptotically normal irrespective of whether the underlying regressors are 
I(1) or I(0).The approach involves a single-equation set-up, which is simple 
to implement and interpret; different lag-lengths can be assigned to model 
variables. In accordance with Pesaran, Shin (1999), we consider a general 
ARDL(p; q) model: 

∑ ∑ ∗∆ , (1) 

∆ ∆ ∆ ⋯ ∆ .  (2) 

The scalar disturbance, ut in the model (1) is iid(0; ). 
We do not want to impose any integration assumption9.  
The ARDL model used in this study is a version of eq. (1), which could be 
called an unrestricted ECM or, as in Pesaran et al. (2001), a conditional 
ECM. The model is as follows: 

ΔLOTOSPB α ∑ ΔLOTOSPB ∑ ∆
																																								 LOTOSPB , (3) 

where we assume the LOTOS price as “dependent”, in accordance with 
Miłobędzki (2008), when he concluded that PKN is a “price leader” in the 
industry.  
 
3.3.2. Empirical Analysis 
 First, a test for order of integration was conducted. Although the bounds 
test for cointegration allows variables to be a mixture of I(1) and I(0), it is 
important to conduct stationarity tests to ensure that the variables are not all 
I(0) and not I(2). The results reported in Table 4 confirm that all series have 
one unit root, i.e. all are I(1). 
 To exclude the possibility of distinct leader-follower strategic behavior 
in the entire sample, we next tested for causality between the variables. Us-

                                                 
9 The order of integration was preliminary tested (Table 4); however, those were usual 

tests (ADF and KPSS). We did not test the order of integration more thoroughly (in the pres-
ence of breaks in intercept and/or trend, for example). 



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ing the Toda-Yamamoto procedure (Toda and Yamamoto, 1995), the 
Granger causality in both directions between the price series was found. 
Table 5 contains the final Wald test of the Granger non-causality for VAR 
with 13 lags (optimal lag length selected for VAR was 12 + 1 lag to explain 
I(1) in each variable, treated here as exogenous). 

Table 4. ADF Test for unit root 

 Level   1st difference  

Series 
ADF test 
statistics 

p-value* Lag 
ADF test 
statistics 

p-value* Lag 

PKN PB95 0.484 0.819 2 –21.875 0.000 2 
PKN ON 0.845 0.893 2 –24.608 0.000 1 

LOTOS PB95 0.509 0.825 2 –20.218 0.000 3 
LOTOS ON 0.887 0.899 2 –22.859 0.000 2 

Note: ADF Test for H0: series has a unit root. Test's critical value at the 1% level: –2.56607. We con-
ducted the KPSS test for all series with the same results (all series I(1)).*MacKinnon (1996) one-sided 
p-values. 

Table 5. Wald tests for Granger causality 

Dependent variable: LOTOSPB 
Excluded Chi-sq df p-value 
PKNPB  284.433 12  0.000 

All  284.433 12  0.000 

Dependent variable: PKNPB 
Excluded Chi-sq df p-value 

LOTOSPB  105.109 12  0.000 
All  105.109 12  0.000 

 
 The null hypothesis of no causality in both directions must be rejected. 
Regarding the specifications of the model, the lag structure of (3) has been 
defined. The ardlbound: An Eviews add-in by Tarverdi M. Yashar (2014) 
was utilized here. The parameters of the ARDL (1,2) model were subse-
quently estimated. The residuals were tested for a serial correlation next 
because the key assumption in the ARDL / Bounds Testing methodology is 
that the errors of equation (3) must be serially independent. Table 6 summa-
rizes the results of those steps.  
 The estimation results for the ARDL(1,2) specification of choice are 
satisfactory; all coefficients (except one) are statistically significant (at the 
0.05 level), and the residuals are serially independent. Having the parame-
ters’ estimates, we next apply a bounds F-test to Eq (3). The long run rela-
tion between LOTOSPB and PKNPB series was sought. As usually occurs in 



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cointegration testing, the absence of a long-run equilibrium relation between 
the variables was tested for. The hypothesis structure was as follows: 
 H0: 0, 
 H1: . 

Table 6. ARDL (p, q) model selection and estimation 

ARDL Model AIC SC   
ARDL(1,1) 9.2751 9.2852   
ARDL(1,2) 9.2721 9.2847   
ARDL(1,3) 9.2732 9.2883   
Variable Coefficient Std. Error t-Statistic p -value 

C 6.7113 3.2528 2.0632 0.0392 
D(LOTOSPB(–1)) –0.0374 0.0408 –0.9160 0.3598 

D(PKNPB(–1)) 0.3875 0.0408 9.4960 0 
D(PKNPB(–2)) 0.0675 0.0216 3.1240 0.0018 
LOTOSPB(–1) –0.2645 0.0438 –6.0310 0 

PKNPB(–1) 0.2629 0.0438 5.9938 0 
Breusch-Godfrey Serial Correlation LM Test: Durbin-Watson statistic 

F-statistic 0.7081     Prob. F(3,2269) 0.5471 
Obs*R-squared 2.1309     Prob. Chi-Square(3) 0.5457 

Durbin-Watson stat 1.9988    
Note: AIC and SC information criteria for the ARDL preferred structure are in bold. Variables D(*) are 
the first differences of the series of levels. 

A rejection of H0 implies that there is a long-run relation between series.  
The F-statistics in the form presented in Pesaran et al. (2001), p. 297, equa-
tion (21) was used. Because this statistic has non-standard distributions, in 
accordance with the so-called bounds procedure proposed by Pesaran et al. 
(2001), we tested H0 within the conditional ECM (3). The researchers pro-
vided bounds on the critical values for the asymptotic distribution of the F-
statistic for various specifications of ECM for sizes 0.100, 0.050, 0.025 and 
0.010 of the test. The lower bound values assume that the forcing variables 
are purely I(0), and the upper bound values assume that variables are purely 
I(1). If the computed F-statistics fall outside the critical value bounds, a con-
clusive decision results without needing to know the cointegration rank r of 
the variables. However, if the Wald or F-statistic falls within these bounds, 
the inference would be inconclusive. In such circumstances, knowledge of 
the cointegration rank r of the forcing variables is required to proceed fur-
ther10. Additionally, a bounds t-test of the θ0 estimated value was performed: 

                                                 
10 Pesaran et al (2001), p. 299. 



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 H0 : 0,  
 H1 0 .  

Asymptotic critical t-statistic’s values for this test are provided by Pesaran et 
al. (2001, p. 303–304) in a form of similar to the F-statistic’s I(0)–I(1) 
bounds, calculated for various specifications. Inference is similar, either the 
calculated t-value is smaller than the lower bound, which means stationarity 
of the data; a t-value higher than I(1) bound supports the hypothesis of 
a long run relation. Table 7 concludes this examination: 

Table 7. Bounds testing 

test statistic calculated value   
F-statistic 19.6814   
t-statistic –6.0310   

significance level 0.1 0.05 0.01 
F-stat. bounds 4.04; 4.78 4.94; 5.73 6.84; 7.84 
t-stat.bounds –2.57; –2.91 –2.86; –3.22 –3.43; –3.82 

Note: Bound values from Pesaran et al. (2001), p.300, p.303 for unrestricted intercept, no trend specifi-
cation and k = 1. 

 It can be stated that because the value of our F-statistic and t-statistic 
exceeds the upper bound at the 1% significance level, there was evidence of 
a long-run relation between the LOTOSPB and PKNPB. The estimated long-
run multiplier between PKNPB and LOTOSPB is –(0.262933/–0.264599)) = 
0.9937; therefore, in the long run, an increase of 1 unit in PKNPB will lead 
to an increase of 0.9937 units in LOTOSPB.  
After confirming cointegration, there was a possibility to estimate the long-
run relation meaningfully: 

LOTOSPB α α , (4) 

and use the OLS residuals series from the (4) as an error correction term to 
estimate the restricted ECM of the form: 

ΔLOTOSPB α ∑ ΔLOTOSPB ∑ ∆
																																								 , (5) 

where: 
 (6), 

and a0, a1 are OLS estimates of parameters in (6).  
Table 8 contains estimated values of parameters in (5). 
  



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Table 8. Estimation of parameters of restricted ECM  

Variable Coefficient Std. Error t-Statistic p-value   
C 0.3822 0.5224 0.7316 0.4644 

D(LOTOSPB(–1)) –0.0367 0.0408 –0.8988 0.3688 
D(PKNPB(–1)) 0.3866 0.0408 9.4719 0.0000 
D(PKNPB(–2)) 0.0666 0.0216 3.0800 0.0021 

EC(–1) –0.2649 0.0438 –6.0368 0.0000 
Note: Variables D(*) are the first differences of the series in levels. 

It can be observed that the estimated coefficient of the error-correction term, 
ECt-1, is negative, as expected, and significant. The absolute value of this 
coefficient implies that approximately 27% of any disequilibrium between 
the LOTOSPB price level and the PKNPB price level is corrected within one 
day. 

3.4. Preliminary Analysis of Leader-follower Behavior in Subsamples 

 The last problem, which was preliminarily examined, was the leader-
follower behavior of a more complicated nature (as replications of certain 
patterns in price changes and price announcements), which was exhibited in 
the sub-samples. To begin, the sub-samples were examined in the most intui-
tive manner, consistent with the assumed pricing mechanism, i.e., by testing 
the differences between the same day prices of both players, the current price 
of one of the players and the lagged price of the other. Figure 3 provides the 
possibility to visually inspect various series of differences of prices of the 
players: 
 

 
Figure 3.  Absolute values of differences between levels of prices of players and the 

entire sample 

The differences of the same day prices appear to have the smallest mean and 
variance; however, an interesting period (it covers nearly precisely all obser-
vations of 2007) in the series of differences of the PKNPB and lagged 1-day  
  

0

50

100

150

200

04 05 06 07 08 09 10 11 12 13

LOTOSPB minus  PKNPB (abs v alues)

0

50

100

150

200

04 05 06 07 08 09 10 11 12 13

LOTOSPB minus PKNPBt-1 (abs v alues)

0

50

100

150

200

04 05 06 07 08 09 10 11 12 13

LOTOSPB minus PKNPBt+1 (abs v alues)



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Table 9. Price’s differences analysis  

 Descriptive statistics – whole sample 
Descriptive statistics – year 

2007 subsample 
 S1 = yt – xt S2 = yt – xt–1 S3 = xt – yt–1 S1 = yt – xt S3 = xt – yt–1 

Mean 9.749 20.873 19.386 11.530 8.724 
Median 6 16 14 10 7 

Std. Dev. 10.755 19.391 19.853 7.980 6.814 
Skewness 1.992 2.529 2.802 0.341 1.077 
Kurtosis 10.060 17.332 18.119 2.348 4.251 

Test – equality of means E(S1) = E(S2) 
Test test stat value p-value test stat value p-value 
t-test –23.9695 0.0000   

Anova F-test 574.5346 0.0000   
Test – equality of means E(S1) = E(S3) 

t-test –20.4223 0.0000 4.2020 0.0000 
Anova F-test 417.0719 0.0000 17.6569 0.0000 

Test – equality of medians Me(S1) = Me(S2) 
Wilcoxon/Mann-Whitney (tie-

adj.) 
24.3783 0.0000   

Adj. Med. Chi-square 441.7884 0.0000   
van der Waerden 613.5586 0.0000   

Test – equality of medians Me(S1) = Me(S3) 
Wilcoxon/Mann-Whitney  

(tie-adj.) 
21.4293 0.0000 3.8747 0.0001 

Adj. Med. Chi-square 346.6815 0.0000 18.6559 0.0000 
van der Waerden 502.5227 0.0000 11.8360 0.0006 

Test – equality of variances var(S1) = var (S2) 
F-test 3.4100 0.0000   

Bartlett 808.2732 0.0000   
Brown-Forsythe 201.9853 0.0000   

Test – equality of variances var(S1) = var (S3) 
F-test 3.2498 0.0000 1.371368 0.0135 

Bartlett 749.3125 0.0000 6.096009 0.0135 
Brown-Forsythe 282.6512 0.0000 9.518529 0.0021 

Note: Series x – PKNPB in levels, series y – LOTOSPB in levels. All the series in PLN/m3. 

price of LOTOS (rightmost panel)11 can be observed. Descriptive statistics 
and various test statistics are reported in Table 9. These statistics confirm 
that the measures of the average level and dispersion were significantly dif-
ferent in a case of three series, S1, S2 and S3, and that the smallest values of 

                                                 
11 To ensure coherent interpretation, we use differences LOTOS – PKN in every case; 

however, for absolute values, x – yt+1 = y – xt-1  



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the average and dispersion measures can be observed in a case of differences 
between the same day prices in an entire sample. This observance excludes 
the subsample from 2007, when the smallest values of those measures for the 
difference between LOTOS’ day t price and PKN’s day t+1 price were not-
ed. 
 These results lead to the conclusion that, if the research is restricted to 
IPP pricing based on previous day fundamentals, the players will, in fact, 
choose levels of prices in a parallel mode; this excludes 2007, when LOTOS 
appeared to be the price leader12. 

Conclusions 

 To inspect the parallel pricing policy, certain descriptive statistics for the 
raw price series (not paired) were calculated to evaluate how consistent the 
pricing policies were. This study found that for the entire sample, an average 
positive and negative value of prices change were statistically equal (on 
standard 0.05 significance level); however, the mean changes between play-
ers were significantly different. Analysis of the yearly subsamples revealed 
that the years 2006–2008 were highly different than the remainder of the 
sample because the average changes (of both signs) decreased but became 
significantly different for the both players. That finding signals certain dis-
turbances in the behavior of players in that period. The beginning of this 
change corresponded to LOTOS’ introduction of other than 5 PLN values of 
the price grid and the synchronization of the price announcement. From 
2009 to the end of the sample, the average price changes became statistically 
equal. Analyzing the empirical distributions of price grids and the absolute 
values of price changes, other differences in pricing policies were detected. 
However, an examination of the empirical distributions of daily price 
movements has confirmed that the daily price changes policy of both players 
was not significantly different over the entire sample period. Next, the casual 
dependencies and the cointegration of gasoline prices was examined. On the 
basis of the estimated ARDL (1, 2) model, the bound testing procedure of 
cointegration testing was used. F-test and t-test results have confirmed the 
existence of a long-run relation between the wholesale prices of gasoline at 
the 1% significance level. The estimated long-run multiplier between 
PKNPB and LOTOSPB was 0.9937; therefore, in the long run, an increase 

                                                 
12 We have checked the medians and means equality hypothesis in all pairs of the three se-

ries of differences for all the year’s subsamples and have not confirmed the phenomenon of 
2007 in the remaining years.  



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of 1 PLN of PKNPB will lead to an increase of 0.9937 PLN in LOTOSPB. 
The estimation of the regular ECM model provided information regarding 
the short-run adjustments of prices. The absolute value of the estimated coef-
ficient of the error-correction term implies that approximately 27% of any 
disequilibrium between the LOTOSPB price level and the PKNPB price 
level was corrected within one day. Finally, the simple test of the leader-
follower pricing behavior over the subsamples was performed. To check that 
test in a manner consistent with the assumed pricing mechanism, the differ-
ences between the same day prices of both players, the current price of one 
of the players and the lagged price of the other were examined.  
Overall, this study finds that, if we restricted our research to described pric-
ing mechanism (IPP pricing based on previous day fundamentals) players 
really chose the levels of price in a parallel mode, with the exclusion of year 
2007 when LOTOS appeared to be the price leader.  
 In terms of the assessment of the players’ strategic behavior, this study 
confirmed that the publically observable pricing conduct of the players in the 
Polish wholesale fuel market could be consistent with an equilibrium of an 
assumed game theory model. This finding, in turn, implies that the players’ 
observed behavior was coherent with the assumed competition model and 
did not exhibit disturbances (unreasonable conduct), which could reflect 
competition’s distortions. However, it should be emphasized that for a deci-
sive “screening” of conclusions based on the model, the test of the second 
behavioral implication of the theoretical model, i.e., the focal role of the IPP 
price, must be examined. This test and examination will be a subject of the 
subsequent research. 

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Test paralelnych zachowań cenowych na hurtowym rynku  
paliw płynnych w Polsce oparty na podejściu ARDL /bound testing 

Z a r y s  t r e ś c i. Artykuł dotyczy badania, czy obserwowane cen paliw płynnych na rynku 
hurtowym mogą być zgodne z określonym modelem teoretycznym interakcji strategicznych 
graczy. Głównym celem badawczym było określanie, czy paralelne zachowania cenowe, 
implikowane równowagą teoretycznego modelu interakcji strategicznych, mogą być mecha-
nizmem ustalania cen faktycznie obserwowanym na rynku. Aby to stwierdzić wyznaczono 
statystyki opisowe indywidualnych szeregów czasowych cen dla ustalenia podobieństw. 
Następnie zbadano współzależność ruchów cenowych graczy wykorzystując podejście ARDL 
/bound testing. Stwierdzono, że jeśli gracze wykorzystują mechanizm cenowy ceny parytetu 
importowego (przyjęty w modelu teoretycznym), ich zachowania replikują paralelne zacho-
wanie cenowe, za wyjątkiem roku 2007, w którym LOTOS wydaje się być liderem cenowym. 

S ł o w a  k l u c z o w e:hurtowy rynek paliw, zachowania paralelne, kointegracja. 


