**3. Methodology**

The integration analysis was conducted on the monthly prices of four dairy products: fresh milk, butter, Edam cheese, and skimmed milk powder (SMP). The temporal scope of the study covered the period 2001–2021 at the national level, and the period 2013–2021 at the regional level. The time ranges were selected based on available data. The quantitative analysis is based on logarithmic transformations of prices (log-prices) and their first differences (log-returns). Country-level data were obtained from Italian Dairy Economic Consulting (CLAL.IT 2021) and Food and Agriculture Data (FAO 2021). Data at the regional level were obtained from the Czech Statistical Office (2021) and Polish Statistical Office (2021). Country and region abbreviations have been used in the presentation of results: Czech Republic (CZ), Poland (PL), Jihoˇceský kraj (JHC), Jihomoravský kraj (JHM), Karlovarský kraj (KVK), Kraj Vysoˇcina (VYS), Královéhradecký kraj (HKK), Liberecký kraj (LBK), Moravskoslezský kraj (MSK), Olomoucký kraj (OLK), Pardubický kraj (PAK), Plze ˇnský kraj (PLK), Stˇredoˇceský kraj (STC), Zlínský kraj (ZLK), Dolno´sl ˛askie Voivodeship (DOL), Kujawsko-Pomorskie Voivodeship (K-P), Łódzkie Voivodeship (LDZ), Lubelskie Voivodeship (LBL), Lubuskie Voivodeship (LBU), Małopolskie Voivodeship (MLP), Mazowieckie Voivodeship (MAZ), Opolskie Voivodeship (OPO), Podkarpackie Voivodeship (PKR), Podlaskie Voivodeship (PDL), Pomorskie Voivodeship (POM), Sl ˛ ´ askie Voivodeship (SL), Swi ˛ ´ etokrzyskie Voivodeship (SW), Warmi ´nsko-Mazurskie Voivodeship (W-M), Wielkopolskie Voivodeship (WLK), Zachodnio-Pomorskie Voivodeship (Z-P). The locations of Polish and Czech regions are shown in Figure 1.

One of the first analyses was cointegration testing, according to which nonstationary time series are integrated if their linear combination is stationary. Then, we speak of a long-run equilibrium relationship between the price series being studied. Cointegration means that analyzed prices move closely together in the long-run perspective, while in the short-run they may drift apart. For this purpose, the Johansen procedure, which is based on a vector autoregression (VAR) model, was used. The general form of the VAR model is as follows (Neusser 2016):

$$X\_t = \mathbb{C} + \sum\_{i=1}^p A\_i X\_{t-i} + \varepsilon\_{t,i} \tag{1}$$

where *Xt* is the endogenous variable vector, *C* is the constant vector, *Ai* forms the coefficient matrix, and *et* is the white noise vector that is independently and identically distributed with *et* ~ *IID*(0,Σ), where Σ is the positive definite matrix.

**Figure 1.** Map of analyzed regions.

In this work, the price cointegration analysis covered a maximum of two markets (A and B). The VAR model, with an intercept and without other deterministic variables for product prices in markets A and B, can be written as a system of two equations:

$$X\_{At} = \mathfrak{a}\_{A0} + \sum\_{i=1}^{p} \mathfrak{a}\_{Ai} X\_{At-i} + \sum\_{i=1}^{p} \beta\_{Ai} X\_{Bt-i} + \mathfrak{e}\_{At\_{r}} \tag{2}$$

$$X\_{Bt} = \alpha\_{B0} + \sum\_{i=1}^{p} \alpha\_{Bi} X\_{At-i} + \sum\_{i=1}^{p} \beta\_{Bi} X\_{Bt-i} + \varepsilon\_{Bt} \tag{3}$$

where: *α*, *β* are the parameters of the model in the equation of prices on market *A* and prices on market *B*.

Before the analysis of long-run relationships, statistical properties of price series were carried out. Unit root tests are applied to residuals from the cointegrating regression which are used for checking that price series have the same order. If both analyzed price series have the same integration order, then a test for cointegration can be performed. The modified Augmented Dickey–Fuller test (ADF-GLS) and the Phillips–Perron (PP) test were used to evaluate the unit root. In both tests, the null hypothesis was that the time series are nonstationary; the alternative hypothesis was that they are stationary. The ADF-GLS test is a modification of the ADF test suggested by Elliott et al. (1996). In the first step, the *yt* series is trendless and decreased using a generalized least squares method. In the second, the remainders of the equation (*yt*) are used to test the unit root using the ADF equation:

$$
\Delta \widetilde{y}\_t = \rho \widetilde{y}\_{t-1} + \sum\_{i=1}^p \delta\_i \Delta \widetilde{y}\_{t-p} + \varepsilon\_t \tag{4}
$$

where: *ρ* and *δ* are the model coefficients, *εt* is the random component, and *p* is the maximum augmentation lag. The Phillips–Perron unit root test is also a modification of the Dickey–Fuller test. Instead of accounting for autoregressive structure, the PP test corrects for any series correlation and heteroscedasticity in the errors by modifying the Dickey–Fuller test statistics in a non-parametric manner (Phillips and Perron 1988). The lag length for the tests was selected using the Akaike Information Criterion (AIC). Details of the time series tested and verification of the degree of integration are provided in Table A1 in Appendix A. Note that all the time series were integrated at order I(1), except for monthly cheese prices in Poland and the Czech Republic, whose order was I(0).

If the variables are cointegrated, the Equation (1) can be represented in the vector error correction model (VECM) (*p* − 1) (Neusser 2016):

$$
\Delta(X\_t) = \mathbb{C} + \Pi X\_{t-1} + \sum\_{i=1}^{p-1} \tau\_i \Delta(X\_{t-1}) + \varepsilon\_{t\prime} \tag{5}
$$

where: Π = ∑*<sup>p</sup>*−<sup>1</sup> *i*=1 *Ai* − *I* (*I*: identity matrix); Γ*i* = − ∑*<sup>p</sup>*−<sup>1</sup> *j*=*i*+1 *Ai*; Π is the long-run matrix coefficient, Γ*i* is the short-run matrix coefficient. The VECM model (with an unlimited constant) for two series of prices in locations *A* and *B*, assuming that the cointegration vector has the form [1, −1], can be written as a system of the following two equations:

$$
\Delta(X\_{At}) = a\_{A0} + \rho\_A(X\_{At-1} - X\_{Rt-1}) + \sum\_{i=1}^{p-1} a\_{Ai} \Delta X\_{At-i} + \sum\_{i=1}^{p-1} \beta\_{Ai} \Delta X\_{Rt-i} + e\_{At\_i} \tag{6}
$$

$$
\Delta(X\_{Bt}) = a\_{Bt} + \rho\_B(X\_{At-1} - X\_{Bt-1}) + \sum\_{i=1}^{p-1} a\_{Bi} \Delta X\_{At-i} + \sum\_{i=1}^{p-1} \beta\_{Bi} \Delta X\_{Bt-i} + \varepsilon\_{Bt\_r} \tag{7}
$$

where: *ρ* is the parameter of the model in the equation of prices on market *A* and prices on market *B*, with the rest of the markings as in Equations (1)–(3).

One of two tests are used in the Johansen procedure: the trace test (*LRtrace*) or the maximum eigenvalue test (*LRmax*):

$$LR\_{trace}(r) = -(T - p) \sum\_{i=r+1}^{k} n(1 - \lambda\_i),\tag{8}$$

$$LR\_{\max}(r) = -(T - p)\ln(1 - \lambda\_{r+1}),\tag{9}$$

where: *r* is the number of cointegrating relationships, *T* is the sample size, *k* is the number of variables, *λi* is the *i*-th largest canonical correlation, *p* is the maximum augmentation lag. The *LRtrace* tests the null hypothesis of *r* cointegrating vectors against the alternative hypothesis of *n* cointegrating vectors. The *LRmax* tests the null hypothesis of *r* cointegrating vectors against the alternative hypothesis of *r* + 1 cointegrating vectors.

As a result of applying the Johansen test, we may have the following situations: (1) the rank of the matrix Π is equal to 0 and then the Equation (5) is a VAR model for increments of variables in which there is no long-run dependence; (2) the rank of the matrix Π is greater than 0 but less than *r*, then the number of cointegration vectors is equal to this rank; (3) the matrix Π is of full rank, then the series of variables is stationary and, thus, the Equation (5) is a VAR model for the levels of variables.

In the next step, a VAR or VECM model was estimated depending on the results from the Johansen test. In the next step, if significant coefficients from endogenous variables were found by estimation, the Granger causality test and impulse response function (IRF) test were performed. It allowed assessment of the possible direction of price transmission. The Granger causality test detects the causal relationship between the variables being studied. In this test, variable *X* is a cause, in the Granger sense, of variable *Y*, when the values of variable *Y* can be better predicted given the future value of variable *X* than without those values. This test can be described by the following equations (Granger 1969):

$$\mathcal{Y}\_t = \beta\_0 + \sum\_{j=1}^m \beta\_j \mathcal{Y}\_{t-j} + \sum\_{k=l}^n \beta\_k \mathcal{X}\_{t-k} + u\_{t\prime} \tag{10}$$

$$X\_t = \beta\_0 + \sum\_{j=1}^{m} \beta\_j X\_{t-j} + \sum\_{k=l}^{n} \beta\_k Y\_{t-k} + u\_{t\_r} \tag{11}$$

where: *Yt* is the value of variable *Y*; *Xt* is the value of variable *X*; *β* denotes the structural parameters of the model; *t* is the change in time; *ut* is the random component of the model.

The impulse response function indicates how fast a price shock at one price transmits towards another price. It is the response of one price variable to a sudden and temporary change in another price variable.

#### **4. Preliminary Analysis of Polish and Czech Milk and Dairy Markets**

#### *4.1. Trade Exchange*

The degree of integration of separate markets can be characterized using an analysis of changes in trade. The flow of products between different markets expresses the flow of supply and demand impulses that occur between countries/regions (Hamulczuk 2020). Both the Czech Republic and Poland are increasing their trade volume in all dairy products year by year. Milk and dairy products are perishable products with low transport and storage susceptibility. Therefore, they require continuous cold chain maintenance which can limit the transportation distance of these products. However, it is worth noting that advances in logistics have significantly increased the ability to transport milk and dairy products, with low transport and storage vulnerability over much longer distances (Roman 2018).

However, the main trading partners of the Czech Republic and Poland are mostly neighboring countries (Table 4). In addition, the Czech Republic and Poland are also key partners for each other. Czechs imported cheese (25% of import value), butter (24% of value), milk (10% of value), and SMP (9% of value) from Poland. Depending on the product, Poland is the first, second, or, in the worst case, third largest supplier for the Czech Republic in terms of import value. Poland imported the most of the SMP (22% of value), milk (20% of value), and cheese (7% of value) from Czech Republic. The Czech Republic is the second, third, or fourth largest supplier of dairy products in terms of import value.


**Table 3.** Dairy product exports and imports from/to the Czech Republic and of Poland from/to other countries (five best trade partners by value), in %.


**Table 4.** Dairy product exports and imports from/to the Czech Republic and of Poland from/to other countries (five best trade partners by value), in %.

Note: Country codes are based on ISO 3166: AE = the United Arab Emirates, AT = Austria, BE = Belgium, CN = China, CZ = Czech Republic, DE = Germany, DK = Denmark, ES = Spain, FR = France, HU = Hungary, IE = Ireland, IT = Italy, LT = Lithuania, NL = the Netherlands, OTH = Others, PL = Poland, RO = Romania, SA = Saudi Arabia, SK = Slovakia, SW = Sweden, UK = the United Kingdom, ZA = South Africa. Source: own calculation (FAO 2021).

In 2019, Czechs exported the most of the SMP (13% of export value) and cheese (12% of value) to Poland. In the case of Poland, the Czech Republic's share in butter exports amounted to 20% and, at the same time, it was the main foreign recipient of this product. Moreover, 13% of the cheese exported from Poland went to the Czech Republic, which was the second largest trade partner with respect to this product.

Therefore, on the basis of trade flows, it would be reasonable to conclude that these countries are characterized by a long range of linkages, continuously present. Therefore, this means that there are strong grounds with respect to the integration of the two markets.

#### *4.2. The Linkage of Milk Prices between Regions in Poland and the Czech Republic*

The preliminary analysis of market integration can be also conducted based on price analysis. If there are significant price differences between the analyzed markets, then there is a weak integration. Moreover, these differences often increase as distances between the separate markets being analyzed increase (Roman 2020). Average deviations of logarithms of Polish milk prices from Czech prices over the entire period ranged from −9.1% to +3.2%, with an average of −2.0% (Figure 2). In addition, note the sub-period of the largest price deviations occurring between 2002 and 2005, i.e., especially before the accession of both countries to the EU. Over a longer period, milk prices in Poland were only higher than milk prices in the Czech Republic in 2016–2017. This may have been due to changes in the CAP, including the abolition of milk quotas and the period of adjustment to the new milk market situation (Eurostat 2021e). Decreasing differences in milk prices in time are probably a consequence of the influence of various factors, such as the increase in the foreign trade of milk and dairy products between Poland and the Czech Republic. In addition, the change in milk price differences was influenced by a more efficient information flow after both countries joined the EU, as well as by the increasing price integration across EU countries (Benedek et al. 2017; Fousekis 2018).

**Figure 2.** Differences between Czech and Polish milk prices in 2001–2021 (logarithm price). Source: own calculation (Czech Statistical Office 2021; Polish Statistical Office 2021).

#### *4.3. Linking the Prices of Dairy Products between the Polish and Czech Markets*

The final part of the preliminary analysis focuses on the price linkages of dairy products between the Czech Republic and Poland. The average deviations of the logarithms of Polish butter prices from Czech prices over the entire period ranged from −9.2% to +6.4%, with an average of −1.5% (Figure 3a). The largest variation in butter prices occurred between 2002 and 2005, which was similar for milk prices. The average deviations of the logarithms of Polish SMP prices from Czech prices over the entire period ranged from −4.4% to +2.5%, with an average of −0.6% (Figure 3b). Thus, it should be said that the price differences were the smallest for this product. The largest SMP price deviations occurred between 2002 and 2005. The average deviations of the logarithms of Polish Edam cheese prices from Czech prices over the entire period ranged from −4.4% to +2.7%, with an average of −1.6% (Figure 3c). The largest SMP price deviations also occurred between 2002 and 2005.

**Figure 3.** *Cont.*

**Figure 3.** The price difference between dairy products: (**a**) butter, (**b**) SMP, (**c**) Edam, in Poland and the Czech Republic. Source: own calculations (CLAL.IT 2021).

#### **5. Results and Discussion**

#### *5.1. Milk Market Integration of Regions in Poland and the Czech Republic*

Since the milk price series were characterized by first-order I(1) integration, the first step was to perform a Johansen cointegration test. The test was used to verify the long-run relationship between milk price at the national level, then at the regional level. The results of the cointegration test for milk prices at the national level are summarized in Table 5. Note that the statistical values of the tests are greater than their critical values at *p* = 0.05. This means that there is a long-run cointegration relationship between the Czech milk price and the Polish milk price at the national level. Since there was one cointegrating rank in the milk price relationship, the VECM model was used. It can be seen that the coefficient estimates in the long-run equilibrium relationship range from 0.51 to 0.67. The coefficient in the long-run relationship in the model with a limited trend and an unlimited constant is 0.67, which indicates that, in the long-run relationship, a 1% increase/decrease in milk prices in the Czech Republic is reflected by 0.67% increase/decrease in milk prices in Poland. Czech milk prices are an exogenous variable for Polish milk prices, as the only

significant coefficient with deviations from long-run equilibrium (EC) is in the Czech milk price equation. Imbalances due to shocks in the price system are corrected during the month by 6.3% through the Czech Republic's response, and by 2.5% through the Polish price response. Moreover, in the light of the Granger test performed, it can be concluded that future prices in the Czech Republic are a cause, in the Granger sense, of future milk prices in Poland and vice versa. Thus, we identify a two-way causality. Moreover, the reaction of Czech milk prices to Polish milk prices is positive and stable over 8 months (Figure 4). However, the reaction of milk prices in Poland to milk prices in the Czech Republic is shorter and lasts about 3 months.

**Table 5.** Cointegration testing results and selected VECMs statistics for the Czech Republic and Poland's raw milk price series.


Note: r = rank; l = price logarithm; dl = first differences of price logarithms; 1\*l\_CZ Raw milk-0.508\*l\_PL Raw milk which indicates that, in the long-run relationship, 1% increase/decrease in milk prices in the Czech Republic is reflected by a 0.508% increase/decrease in milk prices in Poland; EC (l\_CZ Raw milk) = error correction component for Czech raw milk prices; EC (l\_PL Raw milk) = error correction component for Polish raw milk prices; dl\_CZ Raw milk = > dl\_PL Raw milk means whether future milk prices in the Czech Republic are the cause, in the sense of Granger, of future milk prices in Poland; dl\_PL Raw milk = > dl\_CZ Raw milk means whether future milk prices in Poland are the cause, in the sense of Granger, of future milk prices in the Czech Republic. \*\* *p* < 0.05, \*\*\* *p* < 0.01. Source: own calculation (Czech Statistical Office 2021; Polish Statistical Office 2021).

Results of cointegration testing at the regional level are shown in Table A2 in Appendix A. Cointegration testing for the regions involved a pairwise analysis of each Czech region with each Polish region. In this case, the highest number of long-run relationships was identified for the Czech side: Liberecký kraj (11 long-run linkages), Královéhradecký kraj (8 linkages), and for the Polish side: Warmi ´nsko-Mazurskie Voivodeship (9 linkages) and Podlaskie Voivodeship (8 linkages). In the case of the Czech Republic, these are the regions closest to Poland. The importance of distance was also confirmed by analyzing the correlation of milk prices in each region and the distance between these regions (Figure 5). According to this, as distance increases, the degree of milk price linkage decreases. However, in the case of Poland, the highest number of long-run linkages was obtained by regions farthest from the Czech border; however, in turn, these regions are key from the point of view of Polish milk production. Thus, both the distance and the specialization of the region can be considered as a factor influencing the integration processes of separate markets.

**Figure 4.** Impulse response function between raw milk prices: (**a**) response of the Czech Republic to a shock in Poland; (**b**) response of Poland to a shock in the Czech Republic. Source: own calculation (Czech Statistical Office 2021; Polish Statistical Office 2021).

In the next step, a Granger causality test was performed to examine whether milk prices in Czech regions have predictive power on milk price variation in Polish regions and vice versa (Table 6). There is a one-way (→) or two-way (↔) relationship between Czech and Polish milk prices in the short term. It can be concluded that future milk prices in Poland were the Granger cause of milk prices in all Czech regions. In contrast, for only 50% of the relationship, Czech milk prices were the cause, in the Granger sense, of milk prices in Poland in the analyzed period.

In the light of the Granger causality test, the most exogenous Czech milk prices were in the following regions (Figure 6): Plze ˇnský kraj, Jihomoravský kraj, Moravskoslezský kraj. In the case of milk prices in Poland, the most exogenous prices in the following voivodeships should be recognized: Mazowieckie, Kujawsko-Pomorskie, Wielkopolskie. The positions of Polish voivodeships seem to be justified mainly by the region's specialization in milk production (statistically significant positive correlation between coefficient F (Granger

causality test statistics) and milk production of 0.53. However, in this case, the relationship between the distance of the region and the summed coefficient F was not confirmed for either the Czech Republic or Poland.


**Table 6.** Direction of dependence based on Granger's causality.

Source: own calculation (Czech Statistical Office 2021; Polish Statistical Office 2021).

**Figure 6.** Summary of Granger causality testing results between milk price series in the Czech Republic and Poland: (**a**) Czech regions, (**b**) Polish regions (sum of *F* test statistics). Source: Source: own calculation (Czech Statistical Office 2021; Polish Statistical Office 2021).

#### *5.2. Dairy Products Market Integration of Poland and the Czech Republic*

The final part of the analysis focuses on dairy products. Since the butter and SMP price series were integrated at order I(1), the Johansen cointegration test was performed in the next step. The results of the cointegration test are summarized in Table 7. Note that the statistical values of the tests are greater than their critical values at *p* = 0.05 for SMP prices only. This means that there is a long-run relationship between the price of SMP in the Czech Republic and the price of SMP in Poland. Thus, the results are consistent with the findings of Domagała (2020), who analyzed price relationships over the period 2004–2016. In contrast, there is no long-run relationship between butter prices in these countries.


**Table 7.** Cointegration testing results and selected VAR/VECM statistics for the Czech Republic and Poland's dairy products price series.

\*\* *p* < 0.05, \*\*\* *p* < 0.01. Note: r = rank; l = price logarithm; dl = first differences of price logarithms; 1×l\_CZ SMP-0.907×l\_PL SMP×time which indicates that, in the long-run relationship, 1% increase/decrease in SMP prices in the Czech Republic is reflected by a 0.907% increase/decrease in SMP prices in Poland; EC (l\_CZ SMP) = error correction component for Czech SMP prices; EC (l\_PL Raw milk) = error correction component for Polish SMP prices; dl\_CZ butter = > dl\_PL butter means whether future butter prices in the Czech Republic are the cause, in the sense of Granger, of future butter prices in Poland; dl\_PL butter = > dl\_CZ butter means whether future butter prices in Poland are the cause, in the sense of Granger, of future butter prices in the Czech Republic; the same applies to SMP and EDAM cheese. Source: own calculation (CLAL.IT 2021).

Since there was one cointegrating rank in the SMP price relationship, the VECM model was used. The coefficient in the long-run relationship in the model with a limited trend and an unlimited constant is 0.91; this indicates that, in the long-run relationship, a 1% increase/decrease in SMP prices in the Czech Republic is reflected by 0.91% increase/decrease in SMP prices in Poland. Long-run relationship coefficients close to 1 reflect the validity of LOP in the spatial markets analyzed. Both Czech SMP prices are an exogenous variable for Polish SMP prices, and Polish SMP prices for Czech SMP prices, as the coefficients with deviations from long-run equilibrium (EC) present in the equation of Czech and Polish SMP prices are statistically significant. Imbalances due to shocks in the price system are corrected during the month by 18.1% through the Czech Republic's response, and by 9.9% through the Polish price response. Moreover, in the light of the Granger test performed, it can be concluded that future prices of SMP in Poland are a cause, in the Granger sense, of future SMP prices in the Czech Republic. Thus, this trend continues as confirmed by Domagała's (2020) studies. Moreover, the reaction of Czech SMP prices to Polish SMP prices and vice versa is positive and stable over 5 months (Figure 7).

Since there was no cointegration between the butter price series, a VAR model was estimated. According to the AIC, the minimum lag length in the VAR model was *p* = 3. Since all the endogenous variables of this study are integrated on the first order that is not cointegrated, the VAR (*p* − 1) model is estimated, so the lag length is 2 (*p* − 1 = 2). The R2 value for each VAR model indicates that the overall quality of fit is not satisfactory. For example, about 15% of the variability in Czech butter prices can be explained by Czech and Polish butter prices. Regarding the short-run butter price relationship, there is a two-way result given by the Granger causality test. This means that future butter prices in the Czech Republic are a cause, in the Granger sense, of future butter prices in Poland, and vice versa. Thus, the results of Roman (2018) and Domagała (2021), who examined the price linkage over the period 2012–2016, are confirmed. It should be noted that prior to this year, only butter prices in Poland influenced future butter prices in the Czech Republic. Further evidence of a short-run relationship between butter prices in the analyzed countries can be inferred from the IRF test. The reaction of butter prices in the Czech Republic to butter prices in Poland and vice versa is positive. The impact change occurs after month 2 for Czech butter prices and after month 4 for Polish butter prices.

In the case of cheese, the time series were integrated at order I(0); therefore, no cointegration tests were performed, only the VAR model was estimated. The R2 value for each VAR model indicates that the overall quality of fit is satisfactory. For example, about 89% of the variability in Czech cheese prices can be explained by Czech and Polish Edam prices. As for the short-run butter price relationship, we have a two-way result here as well, following the Granger causality test. This means that future cheese prices in the Czech Republic are a cause, in the Granger sense, of future cheese prices in Poland, and vice versa. In the case of the results from the impulse response analysis, the response of cheese prices in the Czech Republic to cheese prices in Poland, and vice versa, is positive and long-lasting.

**Figure 7.** Impulse response function between butter prices: (**a**) response of Czech Republic to a shock in Poland; (**b**) response of Poland to a shock in the Czech Republic; SMP prices: (**c**) response of Czech Republic to a shock in Poland; (**d**) response of Poland to a shock in the Czech Republic; Edam prices:(**e**) response of Czech Republic to a shock in Poland; (**f**) response of Poland to a shock in the Czech Republic. Source: own calculations (CLAL.IT 2021).
