Page 28 - Read Online

Özbek et al. Carbon Footprints 2025, 4, 30 https://dx.doi.org/10.20517/cf.2025.43 Page 7 of 21

will be tested, β , β , β , and β indicate the elasticity coefficients of GDP, GDP2, TO, and URB,
22
23
21
24
respectively. Finally, ε and ε symbolize the error term of models (1) and (2), respectively. Figure 1 shows
2t
1t
the graphs of the variables used in the models.
Methodology
This section of the study focuses on the econometric method used in the analysis. For this purpose, first, the
Augmented Dickey Fuller (ADF) and Phillips-Perron (PP) unit root tests and the Fourier ADF unit root test
are introduced, and then the A-ARDL cointegration technique is discussed. In the following section, Fully
Modified Ordinary Least Square (FMOLS), Dynamic Ordinary Least Square (DOLS), and Canonical
Cointegration Regression (CCR) methods that provide long-term cointegration coefficients are explained.
Finally, the Fourier Bootstrap Toda-Yamamoto (FBTY) Causality technique is presented to investigate the
short-term causality interaction between variables. A summary of the methodology used is given in
Figure 2.

Unit root tests
In the analysis, the ADF unit root test and PP unit root test are used for the unit root research of the
variables examined [55,56] . The PP unit root test has a significant advantage over the ADF unit root test. The
PP test provides robust results in the cases of serial correlation in the considered series, time-sensitive
heteroscedasticity, and time-dependent regime change in the sample period . It is considered important to
[57]
account for structural breaks in the investigation of the unit root process. In this context, Christopoulos and
[58]
León-Ledesma developed a new Fourier-based unit root test that simultaneously considers the nonlinear
fitting process and structural breaks. The new Fourier-based unit root test includes structural breaks by
considering a trigonometric function that allows smooth temporal mean changes, in addition to the linear
fitting process modeled by smooth transition functions. The main advantage of this test is that it does not
consider the factors causing the structural break as external .
[58]

y, a stochastic process, is defined as follows:
t
y = δ(t) + υ (3)
t
t
where δ(t) is the time-varying deterministic component and υ ~ N(0, σ). Christopolous and
t
Leon-Ledesma considered the following Fourier series expansion, which was used by Becker et al. ,
[59]
[58]
[61]
[60]
Becker et al. , and Enders and Lee to investigate the unknown form of δ(t) and the unknown number of
breaks.

where, k is the frequency number of the Fourier Function, t is the trend term, T is the sample size and
π = 3.1416 is defined as. If the appropriate frequency number k were known, Equation (3) could be used to
detect the unknown number of structural breaks. However, since the appropriate frequency number k is not
known, Equation (4) can be expressed as follows by taking into account the Fourier expansion presented by
[62]
Ludlow and Enders , which shows that a single frequency is sufficient. Since the appropriate frequency
number k is unknown, a Fourier-ADF model is estimated for k = 1, 2, ... 5. The lag number p for each k is
selected according to the Schwarz information criterion. The sum of squared residuals of the resulting
model is then calculated for each k. The k value that minimizes this sum is selected as the appropriate
frequency number. This approach models structural breaks not as sharp dates, but as smooth transitions. In
this model, the break date is determined by k, which symbolizes the periodic change in the trend.