The nexus between non-fungible tokens (NFTS) and cryptocurrencies

Perron’s [22] study shows that unit root tests become less effective when structural breaks are ignored. These tests lose their validity when the fracture structure is not distinct and smoother transitions are observed [23]. In contrast to conventional unit root tests, the standard ADF unit root test is extended using Fourier functions based on a specific frequency dimension. They could obtain reliable tests regardless of time or breakage pattern by reducing the number of predicted parameters [24].

Since each time series has its characteristics, the number, duration and form of structural breaks in macroeconomic variable series vary. Enders, Becker, and Lee [25] developed the KPSS-type stationarity test by including Fourier functions in the unit root tests. Thus, it is no longer important whether the break occurs suddenly or not in determining the number of breaks by using the Fourier function [26]. The Fourier Kwiatkowski-Phillips-Schmidt-Shin test developed by [25] is independent of the break date and the number of breaks. This test allows cross-section dependence, heterogeneity, and gradual structural changes based on the Fourier stationarity test. When using trigonometric terms for nonlinear structures, this test parallels the unit root tests proposed by [27] and [28].

In the study, the equational structure of the data generation process (DGP) was created as follows:

yt = X ′ t β + Z ′ ty + rt + ɛ t(2)

rt = rt - 1 + μ t(3)

The μt value in the equation 2-3 has an independent structure and is identically distributed with the σu2 variance. ɛt represents stationary error terms. Under the (t) approximation, the Fourier series was constructed as follows:

a ( t ) - α 0 + ∑ k = 1 n α k · sin ( 2 π k t / T ) + ∑ k = 1 n b k cos ( 2 π kt / T ) ; n < T / 2 ,(4)

T and K, respectively, were indicated as the sample size and frequency values in Equations 4 [23].

The FKPSS unit root test compares the null hypothesis, “There is no unit root in the series” to the alternative, “There is a unit root in both the fixed and both fixed and trended form of the time series". The null hypothesis is not rejected if the FKPSS test statistics are lower than the critical table value determined from the simulation result [29].

After the series obtained the stationarity finding, this study examined the long-term relationship between the variables by the Fourier Shin (2016) cointegration test.

Cointegration analysis is defined as the presence of a long-term equilibrium relationship between the variables. After unit root tests determine the series’ stationarity, cointegration analysis is used to investigate the series’ relationship. Along with the cointegration test, which is the test that Engle-Granger [30] brought to the literature, new tests have emerged in the literature that also take into account structural changes. Some of these are Hatemi-J, Gregory and Hansen, and Johensen et al. The main problem seen in these tests is that the number and form of structural change are determined a priori [31].

The problems of cointegration tests were also solved by including Fourier functions in the Equation [32]. [11] and [32] have made significant progress on cointegration in their studies. For example; (11) by including the Fourier equation in the cointegration test, he was able to eliminate the lines that occur due to the number and form of structural breaks in the FKPSS test. FSHIN (cointegration) test is one of the important tests that started to be used in this way.

The Fourier SHIN (FSHIN) cointegration test was introduced to the literature by [11] and is used to test the existence, not the absence, of a long-term relationship between variables in a time series. The FSHIN cointegration test tests cointegration under the null hypothesis. [11], this test is also used as an extended version of the FKPSS stationarity test for cointegration. This test can produce strong results against structural changes such as the FKPSS test. [11] Fourier cointegration FSHIN test is expressed as follows;

y t = d t + x ′ t β + η t , t = 1 , 2 , . , T ,(5)

In the equation is η _t  = y _t  + v_1t, y₀ = 0, y _t  = y_t-1 + u _t , and x _t  = x_t-1 + v_2t. Here u _t represents 0 mean σ u 2 independent variance and similarly distributed error term. y _t shows a random walk process with zero mean [21].

The test statistic for the FSHIN cointegration test is as follows;

CI f m = T - 2 ω ˆ 1 - 2 ∑ t = 1 T S t 2(6)

The S t 2 = ∑ t = 1 T υ ˆ 1 t , in this equation represents the least residual sum of squares obtained from the equation (2.2.1.4). ω ˆ 1 2 shows the consistent estimator of the long-term variance of υ_1t [33].

F statistic is used in the FSHIN cointegration test and if the test statistic is smaller than the critical values of Tsong et al. (2016)’s table, it is decided that there is a cointegration relationship with structural breaks between the variables [34].

After applying the cointegration tests, FMOLS-DOLS tests were used to interpret the coefficients of this relationship. The FMOLS method was developed by [35] to obtain optimal estimates of cointegration regressions. FMOLS and DOLS methods estimate the long-term relationship in variables with the cointegration relationship. These tests can be used when the variables are I(1) or I(0). Unlike the DOLS estimator, the FMOLS method is not sensitive to the number of lag and leading values [36].

The FMOLS estimator is a semi-parametric correction method that considers the internality problem between the explanatory variables and the error term and the autocorrelation between the error terms to avoid the problems caused by the long-run correlation of the cointegrated equation and stochastic shocks [37]. FMOLS’ theoretical background is expressed as follows:

y t = X ′ t β + D ′ 1 t Y 1 + u 1 t , X t = r ′ 21 D 1 t + r ′ 22 D 2 t + ε t(7)

Δ ɛ t = u 2 t(8) where, D t = D 1 t ′ + D 2 t ′ deterministic trend variables and n stochastic variables are determined by the X _t equation system. Ω and Λ long-term covariance matrices are obtained from the error terms u _t =(u_1t,u_2t) in Equations 7 and 8, which are assumed to have zero means and do not contain unit roots. Ω and Λ long-term covariance matrices are obtained as follows:

Λ = ∑ j = 0 ∞ E ( u t , u t - j ) = [ Λ 11 Λ 12 Λ 21 Λ 22 ](9)

Ω = ∑ j = - ∞ ∞ E ( u t , u t - j ) = [ ω 11 ω 12 ω 21 ω 22 ] = Λ + Λ ′ - [ σ 11 σ 12 σ 21 σ 22 ](10)

The FMOLS estimator is obtained using the following Equation (11) after calculating Equations (9) and (10):

θ = [ β ^ γ ] = ( ∑ t = 1 T X t Y t + − T λ 12 + ) = ( ∑ t = 1 T X t X ′ t ) − 1(11)

DOLS, developed by (38) and (39), is calculated as follows:

y t = a 0 + X ′ t β ∑ i = − q r ∂ i Δ X t + i + μ 1 t(12)

Δ denotes the difference operator, ∂ _i the coefficient of the lags and antecedents of the explanatory variables with the first difference.

The variance decomposition test examines the change in one of the endogenous variables by separating them as separate shocks affecting all the endogenous variables. In addition, variance decomposition gives information about the dynamic structure of the system. The purpose of variance decomposition is to reveal the effect of each random shock on the error variance of the prediction for future periods [40].

The dynamic behaviour of each variable in the model’s response to shocks and all other endogenous variables is described by impulse-response functions. Also, it is used to explain the relationship between variables and examine their effects on each other. Impulse-response testing investigates the impact of shocks in the economic analysis of the variables used in the series [41].

4.5 Toda-Yamamoto causality test

This study used causality analysis for determining the relationship between the variables and the direction of this relationship. Toda-Yamamoto [42] developed the Toda-Yamamoto test to investigate Granger causality tests. Based on VAR model estimation, this test takes a straightforward approach to the subject while explaining the causal relationship. This method, proposed by Toda-Yamamoto, is seen as a complementary feature of the technique he developed by [43]. This approach allows causality inferences based on an enhanced VAR model with integrated and cointegrated processes. This method is also more beneficial as it bypasses preliminary tests for unit root tests [44]. In the Toda Yamamoto causality test, which can be used if the variables become stationary due to taking both the level and the first difference, firstly, the dmax (maximum degree of integration) value calculated by the VAR method should be learned. Then, the dmax value of the variable with the highest integrated value is added to the lag length determined in the VAR model. Finally, the causality relationship is determined for the p + dmax delay value in the test [42]. The equation for the model is expressed as follows:

Y t = w ¯ + ∑ i = 1 k α 1 i X t - i + ∑ i = 1 k β 1 i Y t - i + ∑ j = m + 1 d max δ 1 i X t - i + ∑ j = m + 1 d max θ 1 i Y t - i + ɛ 1 t(13)

X t = ∂ + ∑ i = 1 k α 2 i X t - i + ∑ i = 1 k β 2 i Y t - i + ∑ j = m + 1 d max δ 2 i X t - i + ∑ j = m + 1 d max θ 2 i Y t - i + ɛ 2 t(14)

The k term in the equations represents the appropriate lag length. While d_max shows the maximum degree of integration, it reflects that the mean of the ɛ_1t and ɛ_2t error terms is zero on the equation and the covariance matrix is constant [45].

After applying the cointegration tests, FMOLS-DOLS tests were used to interpret the coefficients of this relationship. The FMOLS method was developed by [35] to obtain optimal estimates of cointegration regressions. FMOLS and DOLS methods estimate the long-term relationship in variables with the cointegration relationship. These tests can be used when the variables are I(1) or I(0). Unlike the DOLS estimator, the FMOLS method is not sensitive to the number of lag and leading values [36].

FMOLS and DOLS tests are used to explain how much the independent variables affect the dependent variable with coefficients. The resulting coefficients and the negative/positive nature of these coefficients explain the effect of the variables on each other and whether this effect is positive or negative. According to FMOLS and DOLS test results in Table 5, it was seen that there was a negative and long-term relationship between the variables. For FMOLS results, the lnGasFee variable had the most significant impact on lnSales. This variable was followed by lnETH (1.069) and lnBTC (0.177). According to DOLS results, lnETH (1.048) had the most significant effect on lnSales. This variable was followed by lnGasFee (0.359) and lnBTC (0.188).

Ten-period variance decomposition results for all variables used in the analysis were given in Table 6. This test is used to explain how many of the interactions of the variables in the ten-term process are caused by themselves and how many of them are caused by other variables. A unit change in lnSales was only due to itself in the first period. As seen in Exhibit 6, the effect of a unit change in lnSales was felt in the following periods on lnGasFee, lnBTC and lnETH variables. In the ten lagged periods, the variance of lnSales is 94% due to shocks from itself, 4.9% from lnGasFee, 0.208% from lnBTC, and 0.513% from lnETH.

The results of the Impulse-Response test graphically explain how the response of the variables in ten periods creates an effect and the duration of this effect. Used in conjunction with the Variance decomposition test, this test graphically shows how long it takes to react to shocks and variables.

As seen in Fig. 2, the reaction of lnSales to the shocks coming from itself was insignificant for the first four periods, but then the effect turned positive. While lnSales’s reaction to lnETH was meaningless in the first four periods, the effect became positive afterwards. The response of lnSales to the shocks in lnBTC and lnGasFee did not reflect a theoretical result. lnETH’s reaction to the shocks at lnSales is theoretically meaningless. InETH’s response to the shocks from him was meaningless in the first three periods, but this effect later became positive. While lnETH’s reaction to the shocks in lnBTC and lnGasFee was insignificant in the first six periods, the effect was positive afterwards.

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Fig. 2

Results of Impulse-Response Function Test.

As seen in Fig. 2, the reaction of lnBTC to the lnSales variable was economically significant and positive for the ten periods followed. While the response of lnBTC to the shocks in lnETH was inconsistent in the first five periods, this effect became positive afterwards. The reaction of lnBTC to the shocks coming from itself was economically significant and positive for the ten periods. The response of lnBTC to shocks from lnGasFee was theoretically and statistically insignificant.

The reaction of lnGasFee to the shocks in lnSales, lnBTC and lnGasFee was economically significant and positive for the ten periods followed. InGasFee’s response to the shocks from InETH did not reflect a theoretical result.

Causality analysis is used to explain the existence and direction of the relationship between variables. With the causality test, the significance level (1%, 5%, 10%) of the relationship between the variables was revealed by the analysis.

Table 7 demonstrates that at the percent 10 levels, there was a unidirectional relationship between lnSales and lnGasFee variables. Also, at the percent 5 level, was determined a unidirectional relationship between lnBTC and lnSales variables. It was determined that there is a one-way causality relationship between lnSales and lnETH variables at the 1% significance level.

Bitcoin has the highest volume in the cryptocurrency market and almost all cryptocurrencies in the market are traded with Bitcoin parity. Many of the traders in the cryptocurrency market control their capital with Bitcoin value along with USD. This dominant power of Bitcoin in the market is very effective in the pricing of other cryptocurrencies. It is an expected result that NFT trading within the blockchain ecosystem will also be affected by Bitcoin. Looking at the causality results, it is seen that NFTs (Sales) are in a one-way causality relationship with both gas fees and Ethereum. This causality is most likely related to the use of Ethereum as a main means of payment in the NFT marketplace and gas fees being a significant cost element in NFT trading. However, no causality was observed from gas fees to NFTs. This shows that NFTs sales are not affected by the increase in fees, but NFTs have an impact on fees. Similarly, there is one-way causality relationship from NFTs sales to Ethereum. While NFTs sales affect Ethereum, Ethereum volume does not affect the number of NFTs sales.