Data and Model
The existence of the EKC hypothesis in this study for the 15 highest economic freedom countries is examined, namely Australia, Canada, Denmark, Estonia, Finland, Germany, Latvia, Lithuania, Luxembourg, Netherlands, Norway, Singapore, South Korea, Sweden, and Switzerland, for the period 1996–2018. The ecfo, efr, ren, gdp and gdpsq denote ecological footprint (per capita, global hectares per person), index of economic freedom, renewable energy consumption (per capita, total final consumption, ktoe), per capita GDP (constant 2010 US $) and square of GDP, respectively. The ecfo and efr data are taken from Global Footprint Network database and the Heritage Foundation, respectively. The ren, gdp and gdpsq data are taken from the World Bank database. All data in the study are converted into logarithmic form.
The following model is used to test the existence of the EKC hypothesis:
$$lnecf{o}_{it}={\beta }_{0}+{\beta }_{1}lnef{r}_{it}+{\beta }_{2}lnre{n}_{it}+{\beta }_{3}lngd{p}_{it}+{\beta }_{4}lngdps{q}_{it}+{\epsilon }_{it}$$
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where, \({\beta }_{1}, { \beta }_{2}, {\beta }_{3} and {\beta }_{4}\)are the coefficients of \(lnef{r}_{it}, lnre{n}_{it}, lngd{p}_{it} and lngdps{q}_{it}\), respectively. \({\epsilon }_{it}\) is the error term. To determine the existence of the EKC hypothesis some conditions have to be satisfied. The first condition is the existence of the cointegration relationship and then the coefficient of lngdp should be positive while the coefficient of lngdpsq should be negative. Moreover, these coefficients should be statistically significant in Eq. (1). The turning point formula is as follows:
$${Y}^{*}={e}^{\frac{{\beta }_{3}}{2{\beta }_{4}}}$$
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Methodology
Priori tests
Cross-section dependency is one of the first priori tests used in studies using panel data. For this reason, the first issue addressed in this study is the cross-section dependency. From this point of view, in the study, three different test statistics are calculated, and the cross-section dependence is tested. Along with the variables, the EKC model is also calculated. Cross-sectional dependence of Eq. (1) is tested with the test statistics of Breusch and Pagan (1980).
$$CD=T\sum _{i=1}^{N-1}\sum _{j=i+1}^{N}{\widehat{\rho }}_{ij}^{2}$$
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where \({\widehat{\rho }}_{ij}\) displays the correlation between errors calculated from Eq. (1). The test statistic below has been suggested by Pesaran (2004) due to the calculated test statistic in Eq. (3) might yield deviant results in large samples:
$$C{D}_{LM1}=\sqrt{\frac{1}{N(N-1)}}\sum _{i=1}^{N-1}\sum _{j=i+1}^{N}(T{\widehat{\rho }}_{ij}^{2}-1)$$
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The test statistic is stated above to make some adjustments to the test statistic in Eq. (3). Thus, cross-sectional dependency in large samples has become testable. On the other hand, if the time (T) is smaller than observation (N), then Eq. (4) is adjusted by Pesaran (2004). Afterwards, the obtained test statistic is as follows:
$$C{D}_{LM2}=\sqrt{\frac{2T}{N\left(N-1\right)}}\left(\sum _{i=1}^{N-1}\sum _{j=i+1}^{N}{\widehat{\rho }}_{ij}\right)$$
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For three tests statistics the hypotheses are as follows:
\({H}_{0}:\) there is no cross-sectional dependency
\({H}_{1}:\) there is cross-sectional dependency.
Here, delta tests proposed by Pesaran and Yamagata (2008) are used as homogeneity tests. The test statistic for the first of the delta tests are as follows:
$$\widehat{\varDelta }=\sqrt{N}\left(\frac{{N}^{-1}\tilde{S}-k}{\sqrt{2k}}\right)$$
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Here, S̃ is the changed statistics of Swamy (1970). The Delta adjusted test statistics are as follows:
$${\widehat{\varDelta }}_{adj}=\sqrt{N}\left(\frac{{N}^{-1}\tilde{S}-E({\tilde{z}}_{\widehat{it}}}{\sqrt{var\left({\tilde{z}}_{\widehat{it}}\right)}}\right)E\left({\tilde{z}}_{\widehat{it}}\right)=k, var\left({\tilde{z}}_{\widehat{it}}\right)$$
$$=\frac{2k(T-k-1)}{T+1}$$
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For above test statistics the hypotheses to test slope homogeneity are as follows:
\({H}_{0}:\) there are homogenous slopes
\({H}_{1}:\) there are heterogeneous slopes.
Panel Unit Root Test
Increasing globalization trends in recent years increase the importance of cross-sectional dependency in economic research. In addition, the crises in the world economy and the transformations in the economic structures of countries make structural breaks extremely important. The most important issue for the reliability of structural break unit root tests is that the break dates, numbers and forms can be determined accurately beforehand. The difficulties that may arise here are tried to be overcome by Fourier unit root tests. Because this type of tests allows not only hard breaks but also gradual breaks (soft transitions). During the modeling of the test, it is not necessary to know the breakage form and dates beforehand.
For this reason, the long-run dynamic behaviour of per capita ecological footprint is determined via FPKSS test (Nazlioglu and Karul 2017) in this study. The difference of FPKPSS test from traditional panel unit root methods is that it utilizes Fourier functions to determine structural shifts. Also, unlike the others, the null hypothesis is stationary, while the alternative hypothesis is non-stationary. Furthermore, FPKPSS allows us to smoothly model structural changes and take into account country-specific heterogeneities dealing with cross-sectional dependency. FPKPSS test procedure has explained as follows:
$${y}_{it}={\alpha }_{i}\left(t\right)+{r}_{it}+{\lambda }_{i}{F}_{t}+{\epsilon }_{it}$$
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$${r}_{it}={r}_{it-1}+{u}_{it}$$
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Here, \({r}_{it}\) and \({F}_{t}\) denote a stochastic term, and unobserved common factor,respectively. \({\lambda }_{i}\) denotes the loading weights. The deterministic term of the Eq. (8) as a time-dependent function is denoted by \({\alpha }_{i}\left(t\right)\). \({\epsilon }_{it}\) and \({u}_{it}\) are independent and identically distributed (i.i.d.) across i and over t.
For KPSS stationary test the deterministic component is as follows:
$${\alpha }_{i}\left(t\right)={a}_{i}+{\gamma }_{1i}\text{sin}\left(\frac{2\pi kt}{T}\right)+{\gamma }_{2t}\text{c}\text{o}\text{s}\left(\frac{2\pi kt}{T}\right)$$
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Here, \({\gamma }_{1i}\) indicates the amplitude of the changes, while \({\gamma }_{2i}\) indicates the displacement of the changes. Eq. (10) conjectures the time-varying intersection term, using the nonzero values of \({\gamma }_{1i}\) and \({\gamma }_{2i}\) to detect gradual changes in intersection. Furthermore, fluctuation of the slope of both intercept and time trend is allowed by FPKPSS test. A Fourier expansion has been improved to approach the non-linear trend function either with structural-breaks or another form of non-linearity by Jones and Enders (2014) as follows:
:\({\alpha }_{i}\left(t\right)={a}_{i}+{b}_{i}t+{\gamma }_{1i}\text{sin}\left(\frac{2\pi kt}{T}\right)+{\gamma }_{2t}\text{c}\text{o}\text{s}\left(\frac{2\pi kt}{T}\right)\) (11)
The KPSS procedure of Becker et al. (2006) can estimate the country-specific statistics, with (k) is as follows:
$${\eta }_{i}\left(k\right)=1/{T}^{2}\sum _{t=1}^{T}\frac{{\tilde{S}}_{it}{\left(k\right)}^{2}}{{\tilde{\sigma }}_{\epsilon i}^{2}}$$
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Here, \({\tilde{S}}_{it}\left(k\right)=\sum _{j=1}^{t}{\tilde{\epsilon }}_{ij}\) is a partial sum process obtained from the OLS errors in Eq. (8). Computation of the long-run variance of error term \({\epsilon }_{it}\) is denoted as \({\tilde{\sigma }}_{\epsilon i}^{2}\), which is specified as \({\sigma }_{\epsilon i}^{2}=\underset{T\to \infty }{\text{lim}}{T}^{-1}E\left({S}_{it}^{2}\right)\). The panel statistics \(FP\left(k\right)\), which consists of the average of KPSS-based individual test statistics, can be calculated as follows:
$$FP\left(k\right)=\frac{1}{N}\sum _{i=1}^{N}{\eta }_{i}\left(k\right)$$
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While \(\xi \left(k\right)\) and \(\zeta \left(k\right)\) are the mean and the variance, the final test statistics for panel stationarity test \(FP\left(k\right)\) converges to the standard normal distribution.
$$FZ\left(k\right)=FPKPSS=\sqrt{N}\left(\frac{FP\left(k\right)-\xi \left(k\right)}{\zeta \left(k\right)}\right)\sim N\left(\text{0,1}\right).$$
Panel Cointegration Test:
Panel cointegration test:
While traditional cointegration tests do not consider cross-section dependency, recently developed cointegration tests take this into account. Two test statistics proposed by Westerlund (2008) for the panel cointegration test does consider the cross-sectional dependency:
\(D{H}_{g}=\sum _{i=1}^{n}{\widehat{S}}_{i}{\left({\tilde{\varphi }}_{i}-{\widehat{\varphi }}_{i}\right)}^{2}\sum _{t=2}^{T}{\widehat{e}}_{it-1}^{2}\) (13)
$$D{H}_{p}={\widehat{S}}_{n}{\left(\tilde{\varphi }-\widehat{\varphi }\right)}^{2}\sum _{i=1}^{n}\sum _{t=2}^{T}{\widehat{e}}_{it-1}^{2}$$
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Here, DHg is the panel statistic and obtained by adding the individual terms before they are place together. The average statistic is represented by DHp. Various terms are multiplied and then added together. Thus, this statistical value is created. According to Durbin Hausman test, the null hypothesis indicates the absence of cointegration. Conversely, the alternative hypothesis refers to the existence of cointegration in the panel.