The Fourier approach and testing for cointegration in LSTAR model

Abstract

This paper proposes a new logistic smooth transition autoregressive (LSTAR) cointegration test by combining the Fourier function to catch the structural changes that occur over time and the LSTAR model to consider the nonlinearity. Logistic and exponential functions are the main transition functions defining the nonlinearity in STAR models since the exponential smooth transition autoregressive (ESTAR) and LSTAR models can explain the different structures of economic variables. The Fourier approach is a simple and effective way to model the structural changes in time series as an alternative to dummy variables. The most significant advantage of the method is that it does not require prior knowledge about the date, number of breaks, or forms. Monte Carlo simulation results show that the proposed test has good size and power properties for different sample sizes and parameters. The results also revealed that the power performance of the proposed test and the Fourier ESTAR test offered by Güriş and Sedefoğlu (2022) are close to each other. The steps of the Fourier-based tests are illustrated by providing an empirical example of testing the validity of the purchasing power parity (PPP) hypothesis in Türkiye.

1. Introduction

Economic time series are exposed to structural breaks in different forms and numbers to varying dates although many tests have been proposed in the unit root and cointegration literature as if there are no breaks in the series. When we examine the literature for the unit root tests, it appears that the earlier studies including Fuller (1976), Dickey and Fuller (1979, 1981), Nelson and Plosser (1982), Phillips and Perron (1988), Kwiatkowski et al. (1992), Elliot et al. (1996), and Ng-Perron (2001) have pioneered the development of unit root tests. However, the main feature of these tests is that structural breaks are not taken into account in the data-generating process, which will enhance the probability of non-rejection of the false null hypothesis. It is well explained in the seminal paper of Perron (1989) that allows a single break in the series. Later on, studies including Banerjee et al. (1992), Perron and Vogelsang (1992), Zivot and Andrews (1992), Perron (1997), Lee et al. (1997), Lumsdaine and Papell (1997), Bai and Perron (1998), Clemente et al. (1998), Vogelsang and Perron (1998), Ohara (1999), Lee (1999), Bai and Perron (2003), Lee and Strazicich (2003), Sen (2003), Carrion-i Silvestre and Sanso (2007), Papell and Prodan (2007), Narayan and Popp (2010), and Lee and Strazicich (2013) have employed dummy variables to capture one or multiple structural breaks following the paper of Perron (1989). However, these tests were designed to detect potential sharp breaks in series through dummy variables although many time series consist of smooth breaks over time rather than affecting the economy in a specific time. In addition, prior knowledge is needed for the number, date, and form of the breaks when the structural changes are modeled with dummy variables. As such, misspecification of the modeling process of breaks is as severe as ignoring the breaks. On the other hand, the problems seen in modeling structural breaks in linear time series analysis have also been encountered in unit root tests in nonlinear time series analysis including Enders and Granger (1998), Leybourne et al. (1998), Caner and Hansen (2001), Harvey and Mills (2002), Kapetanios et al. (2003), Sollis (2004), Vougas (2006), Pascalau (2007), Chong et al. (2008), Cook and Vougas (2009), Sollis (2009), Cuestas and Garratt (2011), Kılıç (2011), Kruse (2011), Cuestas and Ordonez (2014), Park and Shintani (2016), and Hu and Chen (2016). Nevertheless, several studies have pointed out that the Fourier approach is an alternative approach to dummy variables in terms of capturing breaks well regardless of the number, date, and form of the breaks as detailed in the earlier studies of Gallant (1981), Gallant (1984), Gallant and Souza (1991), Becker et al. (2004), Becker et al. (2006), Enders and Lee (2012a), and Enders and Lee (2012b). The remarkable results obtained by using the Fourier function in unit root tests have paved the way for using this approach in cointegration tests. Tsong et al. (2016) and Banerjee et al. (2017) have pioneered studies that model structural breaks using the Fourier function in cointegration tests. Different from the cointegration tests of Tsong et al. (2016) and Banerjee et al. (2017), Güriş and Sedefoğlu (2022) proposed a cointegration test defining the Fourier function in the deterministic term and accounting for the nonlinearity with the smooth transition autoregressive (STAR) model. Moreover, Güriş and Sedefoğlu (2022) applied grid search instead of the Taylor approximation to overcome the identification problem of the nuisance parameter following the Maki (2010) cointegration test. In this respect, it differs from the Kapetanios et al. (2006) cointegration test which is the first cointegration test in STAR models and ignores breaks.

In this paper, we propose a new cointegration test defining the Fourier function in the deterministic term and considering the logistic transition function in the STAR model. To our knowledge, no cointegration test combines the logistic function for the nonlinearity and the Fourier function for the structural changes. Furthermore, we compare the power performance of the test with the Fourier ESTAR cointegration test proposed by Güriş and Sedefoğlu (2022). Comparing the power performance of the ESTAR and LSTAR type tests which have similar steps but different transition functions will reduce the uncertainty of deciding the function in applied econometrics. In their study, Terasvirta and Anderson (1992) mention that the contraction and expansion periods of the economy have different dynamics in the LSTAR model and the transition in these dynamics is smooth. In the ESTAR model, on the other hand, the contraction and expansion periods of the economy are very similar and only the dynamics of transition between regimes differ. From this point of view, LSTAR and ESTAR models can be used as alternatives to each other to explain different economic structures that enhance the interest in the STAR models in the literature. Additionally, we provide an empirical example of the validity of the purchasing power parity (PPP) hypothesis in Türkiye covering the period of 2010:1 and 2019:12 to demonstrate the steps of the Fourier-based cointegration tests in the STAR model different from the Güriş and Sedefoğlu (2022).

The rest of the paper is organized as follows. Section 2 gives the details of the Fourier LSTAR cointegration test. In Section 3, Monte Carlo simulation results are carried out to compute critical values, size, and power properties of the Fourier LSTAR test. An empirical example of the PPP hypothesis is illustrated in Section 4 and the paper is concluded in Section 5.

2. Fourier LSTAR cointegration test

Nonlinear structure and structural breaks are the two main factors affecting the cointegration testing process since many macroeconomic variables have a wide variety of structural breaks of unknown numbers and forms and given information by the linear time series models may be insufficient for important economic inferences. Results tend to be biased which makes it more difficult to reject an incorrect null hypothesis and find a cointegration relationship when these two factors are ignored. In the traditional unit root and cointegration tests, dummy variables are mostly used to specify the structural changes. However, the usage of dummy variables requires prior knowledge of the number of breaks, dates, and forms of breaks. Employing such a subjective way that requires prior knowledge may lead to misspecification of the model since the actual nature of the breaks is generally unknown and major breaks also sometimes do not exhibit their impacts immediately as noted in Becker et al. (2006) and Tsong et al. (2016). Following Gallant (1981), Gallant (1984), Gallant and Souza (1991), Becker et al. (2004), Becker et al. (2006), Enders and Lee (2012a), and Enders and Lee (2012b) in the unit root literature and Tsong et al. (2016) and Banerjee et al. (2017) in the cointegration literature, Güriş and Sedefoğlu (2022) utilized the Fourier approach as an alternative way to overcome the problems caused by the usage of dummy variables. The approach works quite well for capturing the unknown number of breaks and dates and might catch well both gradual and sharp breaks (Banerjee et al., 2017). Furthermore, the degrees of freedom problem that occurs due to the usage of a large number of dummy variables can be avoided through the Fourier approach (Enders & Lee, 2012a).

The Fourier function $\alpha (t)$ expressed in Eq. (4) is defined in the deterministic term $d_t$ for the models with constant (Eq. (2)) and constant and trend (Eq. (3)) to model the structural changes as follows:

$y_t=d_t+{\beta }^{\prime }{x}_t+e_t,t=1,2,\mathrm{\dots }T,$(1) $d_{2t}=\mu +\alpha (t),$(2) $d_{3t}=\mu +\vartheta t+\alpha (t),$(3) $\alpha (t)={\alpha }_1\sin \left({\displaystyle \frac{2k\pi t}{T}}\right)+{\alpha }_2\cos \left({\displaystyle \frac{2k\pi t}{T}}\right),$(4)

where $y_t$ and $x_t$ are the variables in which the cointegration relation is investigated and are stationary at the first difference as a main assumption of the cointegration analysis. Since the number of independent variables can be more than one, the estimated parameter $\beta$ and observable variable $x_t$ are denoted as ${\beta }^{\prime }=({\beta }_1,{\beta }_2,\mathrm{\dots },{\beta }_m)$ , and $x_t={(x_{1t},x_{2t},\mathrm{\dots },x_{mt})}^{\prime }$ . In the Fourier function defined in Eq. (4), $k$ is the Fourier frequency chosen for the approximation, ${\alpha }_1$ and ${\alpha }_2$ represent the measurement of the amplitude and displacement of the frequency component. $T$ is the sample size, $t$ is a trend term, and $\pi \cong$ 3.1416.

The Fourier LSTAR cointegration test is designed to capture multiple structural breaks in time series that require no prior knowledge regarding the date, number, and forms of breaks through the Fourier function defined in the deterministic term as stated earlier. Given that the true value of the Fourier frequency is unknown, the first step in practice is finding the appropriate frequency $k$ . The grid search procedure is advised by Becker et al. (2006) that we estimate the Eq. (1) through OLS for all possible integer values in the range [1, 5] and the optimum frequency is produced by the model that gives the smallest sum of squared residuals (SSR) in the second step. The reason for practicing with small number frequencies is that higher frequencies are not sufficient to detect breaks in the series depending on the power of the test as highlighted by Becker et al. (2004), Becker et al. (2006), Enders and Lee (2012a), Enders and Lee (2012b), Tsong et al. (2016), Banerjee et al. (2017), and Güriş and Sedefoğlu (2022). In this respect, finding a cointegration relationship between variables becomes more difficult with power loss.

The smooth transition function $G(.)$ is included in the model which is created with residuals obtained by the OLS estimation of Eq. (1) as follows:

$e_t={\varphi }_1{e}_{t-1}+{\varphi }_2{e}_{t-1}G(.)+{\mu }_t,$(5)

where the transition function is considered as a logistic function to model the nonlinearity of the variables and $u_t$ is an error term with a zero mean. In Eq. (5), we assume that ${\varphi }_1=1$ because $e_t$ has a unit root and the tendency to move back to equilibrium is weak when $G(.)=$ 0 and ${\varphi }_1=$ 1.

The nonlinear structure is expressed by the single logistic function as follows:

$G(.)={(1+\exp \{-\gamma (e_{t-1})\})}^{-1}.$(6)

where the parameter $\gamma$ is the smoothness of the function and assumed $\gamma >0$ . To test the null hypothesis of no cointegration, $H_o:\rho =0$ , against the alternative of the presence of cointegration with a STAR adjustment, $H_1:\rho <0$ , we rewrite the Eq. (5) as follows:

$e_t={\varphi }_1{e}_{t-1}+{\varphi }_2{e}_{t-1}{(1+\exp \{-\gamma (e_{t-1})\})}^{-1}+u_t,$(7)

where $e_t$ is predicted from Eq. (1) by applying the OLS estimation method. For the cointegration analysis, the model is rewritten as follows:

$\mathrm{\Delta }{\widehat{e}}_t=\rho {\widehat{e}}_{t-1}{(1+\exp \{-\gamma ({\widehat{e}}_{t-1})\})}^{-1}+{\displaystyle \sum _{j=1}^p}{\psi }_j\mathrm{\Delta }{\widehat{e}}_{t-j}+{ϵ}_t,$(8)

where ${ϵ}_t$ is the stationary error with a zero mean.

3. Data generating process: Monte Carlo simulation study

This section presents the Monte Carlo simulation results for the new cointegration test, providing critical values, size, and power performance. The size and power performance of the test reflects the alfa ( $\alpha$ ) and beta ( $\beta$ ) type errors, respectively. The $\beta$ type error, also known as Type II error, indicates the power of the test when written ( $1-\beta$ ). Thus, we mainly obtain the probability of rejecting the correct null hypothesis and the probability of not rejecting the incorrect null hypothesis with the size and power properties.

The data-generating process begins with the estimation of critical values. Since the nuisance parameter in the transition function $\gamma$ is only defined under the alternative hypothesis, we employ a grid search to overcome the undefined nuisance parameter problem in Eq. (8). In the grid search process, we obtain $t$ statistics of $\rho$ for each value of $\gamma$ for the condition of $\gamma \in [{\gamma }_{\text{min}},{\gamma }_{\text{max}}]$ in which ${\gamma }_{\text{min}}={10}^{-1}{\left(\sum _{t=1}^n{\widehat{e}}_t^2/T\right)}^{-1/2}$ and ${\gamma }_{\text{max}}={10}^3{\left(\sum _{t=1}^n{\widehat{e}}_t^2/T\right)}^{-1/2}$ following Dijk et al. (2002) and Park and Shintani (2016).

The infimum type $t$ statistics are computed as follows:

$\underset{\gamma \in {\mathrm{\Gamma }}_T}{inf}{t}_{\tau }(\gamma )\equiv \underset{\gamma \in {\mathrm{\Gamma }}_T}{inf}{\displaystyle \frac{\widehat{\rho }}{s.e.(\widehat{\rho })}},$(9)

where $\widehat{\rho }$ refers to the OLS estimate of $\rho$ , the standard error of the $\widehat{\rho }$ is shown as $s.e.(\widehat{\rho })$ and random sequences of the parameter space given by the functions of $({\widehat{e}}_1,{\widehat{e}}_2,\mathrm{\dots },{\widehat{e}}_T)$ are symbolized as ${\mathrm{\Gamma }}_T$ in Eq. (9) (Maki, 2010).

To compute the critical values, first, residuals are obtained from Eq. (1) with a constant and a constant and linear trend. Second, the infimum $t$ statistic is obtained from the $t$ statistics calculated for each value of the smoothness parameter $\gamma$ in the defined parameter range. The number of replication is 10,000 for this process. Finally, test statistics are sorted and critical values are presented at significance levels of 1%, 5%, and 10%, respectively. In this process, we assume that the Fourier frequencies are $k=$ 1, 2, 3, 4, 5, sample sizes are $T=$ 250,1000, and the number of independent variables is $n=$ 1, 2, 3. Critical values are presented in Table 1.

Table 1

Critical values of the Fourier LSTAR test

		Case 2: Model with an intercept						Case 3: Model with an intercept and a linear trend
		$T=$ 250			$T=$ 1000			$T=$ 250			$T=$ 1000
$n$	$k$	1%	5%	10%	1%	5%	10%	1%	5%	10%	1%	5%	10%
1	1	$-$ 4.76	$-$ 4.20	$-$ 3.91	$-$ 4.73	$-$ 4.17	$-$ 3.87	$-$ 5.15	$-$ 4.61	$-$ 4.31	$-$ 5.11	$-$ 4.55	$-$ 4.29
	2	$-$ 4.57	$-$ 3.92	$-$ 3.59	$-$ 4.48	$-$ 3.89	$-$ 3.57	$-$ 5.01	$-$ 4.45	$-$ 4.14	$-$ 4.99	$-$ 4.40	$-$ 4.10
	3	$-$ 4.34	$-$ 3.69	$-$ 3.34	$-$ 4.30	$-$ 3.66	$-$ 3.36	$-$ 4.83	$-$ 4.22	$-$ 3.91	$-$ 4.71	$-$ 4.21	$-$ 3.89
	4	$-$ 4.15	$-$ 3.55	$-$ 3.20	$-$ 4.10	$-$ 3.54	$-$ 3.22	$-$ 4.70	$-$ 4.08	$-$ 3.77	$-$ 4.68	$-$ 4.07	$-$ 3.75
	5	$-$ 4.18	$-$ 3.54	$-$ 3.21	$-$ 4.07	$-$ 3.49	$-$ 3.16	$-$ 4.66	$-$ 3.99	$-$ 3.68	$-$ 4.64	$-$ 3.99	$-$ 3.69
2	1	$-$ 5.10	$-$ 4.50	$-$ 4.23	$-$ 5.04	$-$ 4.51	$-$ 4.23	$-$ 5.44	$-$ 4.87	$-$ 4.59	$-$ 5.33	$-$ 4.80	$-$ 4.52
	2	$-$ 5.01	$-$ 4.46	$-$ 4.07	$-$ 4.96	$-$ 4.38	$-$ 4.07	$-$ 5.39	$-$ 4.81	$-$ 4.51	$-$ 5.33	$-$ 4.76	$-$ 4.48
	3	$-$ 4.81	$-$ 4.19	$-$ 3.89	$-$ 4.75	$-$ 4.19	$-$ 3.87	$-$ 5.25	$-$ 4.65	$-$ 4.31	$-$ 5.19	$-$ 4.58	$-$ 4.27
	4	$-$ 4.70	$-$ 4.05	$-$ 3.73	$-$ 4.61	$-$ 4.03	$-$ 3.71	$-$ 5.09	$-$ 4.49	$-$ 4.16	$-$ 5.10	$-$ 4.47	$-$ 4.16
	5	$-$ 4.58	$-$ 3.99	$-$ 3.65	$-$ 4.45	$-$ 3.93	$-$ 3.60	$-$ 4.93	$-$ 4.40	$-$ 4.09	$-$ 4.93	$-$ 4.38	$-$ 4.05
3	1	$-$ 5.35	$-$ 4.83	$-$ 4.53	$-$ 5.34	$-$ 4.74	$-$ 4.48	$-$ 5.67	$-$ 5.13	$-$ 4.85	$-$ 5.62	$-$ 5.07	$-$ 4.80
	2	$-$ 5.36	$-$ 4.79	$-$ 4.48	$-$ 5.28	$-$ 4.70	$-$ 4.42	$-$ 5.74	$-$ 5.13	$-$ 4.81	$-$ 5.57	$-$ 5.03	$-$ 4.77
	3	$-$ 5.22	$-$ 4.64	$-$ 4.30	$-$ 5.18	$-$ 4.55	$-$ 4.26	$-$ 5.52	$-$ 4.98	$-$ 4.67	$-$ 5.50	$-$ 4.90	$-$ 4.62
	4	$-$ 5.11	$-$ 4.50	$-$ 4.15	$-$ 5.03	$-$ 4.44	$-$ 4.10	$-$ 5.51	$-$ 4.88	$-$ 4.54	$-$ 5.37	$-$ 4.81	$-$ 4.51
	5	$-$ 4.96	$-$ 4.37	$-$ 4.06	$-$ 4.96	$-$ 4.35	$-$ 4.03	$-$ 5.47	$-$ 4.78	$-$ 4.44	$-$ 5.26	$-$ 4.71	$-$ 4.41

Note: Critical values of the proposed test are computed by the authors through R programming.

The data-generating process for the size properties of the Fourier LSTAR test is as follows:

$y_t=\mu +{\alpha }_1\sin \left({\displaystyle \frac{2k\pi t}{\tau }}\right)+{\alpha }_2\cos \left({\displaystyle \frac{2k\pi t}{\tau }}\right)+{\beta }_1{x}_t+e_t,$(10) $\mathrm{\Delta }{e}_t=u_t,$(11) $u_t=\alpha u_{t-1}+{ϵ}_{1t},$(12) $\mathrm{\Delta }{x}_t={ϵ}_{2t}$(13) $\left(\begin{array}{c}\hfill {ϵ}_{1t}\hfill \\ \hfill {ϵ}_{2t}\hfill \end{array}\right)\sim \left(0\left[\begin{array}{cc}\hfill {\sigma }_1^2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\sigma }_2^2\hfill \end{array}\right]\right),$(14)

where $\mu =$ 1, ${\beta }_1=$ 2, $\alpha =$ (0.5, 0, $-$ 0.5), ${\sigma }_1^2=$ 1, ${\sigma }_2^2=$ (1, 4), and $k=$ 1, 2, 3, 4, 5. The parameters of the Fourier function, ${\alpha }_1$ and ${\alpha }_2$ , consist of the combination (3, 5), (0, 5), and (3, 0), respectively. The nominal size is 0.05. The size properties of the test are presented in Table 2 for the sample sizes $T=$ 250, 1000. Results indicate that the size performance of the test is close to the nominal level and it is not affected by the change in the autocorrelation coefficient. However, the size has increased without exceeding the range 0.054 and 0.066 with the change in the variance ${\sigma }_2^2$ .

Table 2

Size performance of the Fourier LSTAR test

		${\sigma }_2^2=1,\alpha =0$		${\sigma }_2^2=1,\alpha =0.5$		${\sigma }_2^2=1,\alpha =-0.5$		${\sigma }_2^2=4,\alpha =0$
$k$	$(\alpha ,\beta )$	$T=$ 250	$T=$ 1000	$T=$ 250	$T=$ 1000	$T=$ 250	$T=$ 1000	$T=$ 250	$T=$ 1000
1	(3, 5)	0.051	0.048	0.052	0.047	0.056	0.047	0.062	0.063
	(0, 5)	0.054	0.051	0.054	0.050	0.056	0.049	0.066	0.063
	(3, 0)	0.047	0.048	0.055	0.051	0.052	0.050	0.061	0.057
2	(3, 5)	0.050	0.051	0.051	0.050	0.053	0.053	0.062	0.058
	(0, 5)	0.058	0.051	0.052	0.050	0.053	0.052	0.064	0.059
	(3, 0)	0.052	0.050	0.053	0.047	0.050	0.049	0.065	0.054
3	(3, 5)	0.056	0.053	0.048	0.049	0.048	0.053	0.058	0.058
	(0, 5)	0.049	0.051	0.050	0.054	0.052	0.058	0.058	0.058
	(3, 0)	0.052	0.053	0.048	0.049	0.055	0.050	0.056	0.059

Note: Size properties of the proposed test are computed by the authors through R programming.

The data-generating process for the power properties of the Fourier LSTAR test is as follows:

$y_t=\mu +{\alpha }_1\sin \left({\displaystyle \frac{2k\pi t}{T}}\right)+{\alpha }_2\cos \left({\displaystyle \frac{2k\pi t}{T}}\right)+{\beta }_1{x}_t+e_t,$(15) $\mathrm{\Delta }{e}_t={\rho }_1{e}_{t-1}+{\rho }_2{e}_{t-1}[{(1+\exp \{-\gamma (e_{t-1})\})}^{-1}]+{ϵ}_{1t},$(16) $\mathrm{\Delta }{x}_t={ϵ}_{2t}$(17) $\left(\begin{array}{c}\hfill {ϵ}_{1t}\hfill \\ \hfill {ϵ}_{2t}\hfill \end{array}\right)\sim \left(0\left[\begin{array}{cc}1\hfill & 0\hfill \\ 0\hfill & 1\hfill \end{array}\right]\right),$(18)

where $\mu =$ 1, ${\beta }_1=$ 2, ${\rho }_1=$ 0, $\gamma =$ (0.01, 0.1, 1), ${\rho }_2=$ ( $-$ 0.5, $-$ 0.1), $k=$ 1, 2, 3. The parameters of the Fourier function, ${\alpha }_1$ and ${\alpha }_2$ , consist of the combination (3, 5), (0, 5), and (3, 0), respectively. The proposed test maintains good power for all combinations of $k=$ 1 when $T=$ 1000 and $T=$ 250, as shown in the first part of Table 3. We observe that the power of the test increases as the sample size expands and the test loses power as the Fourier frequency increases. Hence, the test is more likely to catch the breaks gradually through the Fourier function when $k=$ 1 and find a cointegration relation among variables.

Table 3

Power properties with different parameter values

Power properties of the Fourier LSTAR test
${\rho }_1=0,c=0$		$T=$ 250			$T=$ 1000
$({\rho }_2,\gamma )$	$({\alpha }_1,{\alpha }_2)$	$k=$ 1	$k=$ 2	$k=$ 3	$k=$ 1	$k=$ 2	$k=$ 3
( $-$ 0.5, 0.01)	(3, 5)	0.950	0.779	0.633	0.981	0.861	0.870
	(0, 5)	0.948	0.777	0.614	0.982	0.864	0.866
	(3, 0)	0.948	0.786	0.623	0.980	0.861	0.865
( $-$ 0.5, 0.1)	(3, 5)	0.973	0.858	0.630	0.982	0.872	0.868
	(0, 5)	0.976	0.845	0.624	0.983	0.869	0.873
	(3, 0)	0.975	0.850	0.630	0.979	0.873	0.870
( $-$ 0.5, 1)	(3, 5)	0.976	0.854	0.855	0.982	0.865	0.867
	(0, 5)	0.976	0.859	0.855	0.980	0.870	0.875
	(3, 0)	0.976	0.850	0.853	0.986	0.863	0.870
( $-$ 0.1, 0.01)	(3, 5)	0.713	0.486	0.465	0.755	0.525	0.503
	(0, 5)	0.720	0.489	0.462	0.748	0.520	0.499
	(3, 0)	0.710	0.499	0.451	0.748	0.525	0.491
( $-$ 0.1, 0.1)	(3, 5)	0.726	0.516	0.493	0.751	0.528	0.507
	(0, 5)	0.861	0.498	0.487	0.746	0.527	0.498
	(3, 0)	0.733	0.516	0.474	0.749	0.526	0.502
( $-$ 0.1, 1)	(3, 5)	0.727	0.516	0.486	0.742	0.523	0.503
	(0, 5)	0.727	0.514	0.494	0.748	0.530	0.504
	(3, 0)	0.733	0.510	0.489	0.750	0.533	0.501
Power performance comparison of the tests
${\rho }_1=$ 0; $k=$ 1; $({\alpha }_1,{\alpha }_2)=$ (3, 5)			$T=$ 250		$T=$ 1000
	$({\rho }_2,\gamma )$		FESTAR	FLSTAR	FESTAR	FLSTAR
	( $-$ 0.5, 0.01)		0.964	0.950	0.984	0.981
	( $-$ 0.5, 0.1)		0.979	0.973	0.986	0.982
	( $-$ 0.5, 1)		0.979	0.976	0.987	0.982
	( $-$ 0.1, 0.01)		0.745	0.713	0.779	0.755
	( $-$ 0.1, 0.1)		0.752	0.726	0.779	0.751
	( $-$ 0.1, 1)		0.754	0.727	0.785	0.742

Note: Power performance of the proposed test is computed by the authors through R programming.

In the second part of Table 3, we represent the power performance of the Fourier ESTAR and Fourier LSTAR tests for the sample sizes of $T=$ 250, and 1000. The most powerful results are found when $T=$ 1000 for both tests and the power of the tests decreases with the decline in the sample size. When we compare the power performance of the Fourier ESTAR and Fourier LSTAR tests, overall, the power of the Fourier LSTAR test is close to the Fourier ESTAR test. The probability of rejecting the null hypothesis correctly ranges from 74% to 98% out of 10,000 iterations in the Fourier LSTAR test and 77% to 98% out of 10,000 iterations in the Fourier ESTAR test when $T=$ 1000. We can point out that the flexible Fourier approach shows similar behavior in LSTAR and ESTAR models in terms of power properties when we evaluate the results comparatively, which reduces the uncertainty of deciding between LSTAR and ESTAR in real data applications. In other words, the Fourier LSTAR test can be an alternative cointegration test to the Fourier ESTAR test and the previous tests, which successfully modeled nonlinearity but did not consider structural breaks, such as Maki (2010).

4. An application to real data: PPP hypothesis in Türkiye

In the empirical analysis, we test the validity of the PPP hypothesis in Türkiye for the period from 2010:1 to 2019:12 ( $T=$ 120) to demonstrate the steps of the Fourier LSTAR test that we proposed and gave the details in previous sections. As we compare the power performance of the Fourier ESTAR and LSTAR tests, we apply the Fourier ESTAR cointegration test to test the PPP hypothesis along with Fourier LSTAR. The data are extracted from the International Financial Statistics (IFS) and Federal Reserve databases. Series are obtained by considering the following equation before the analysis:

$s_t=\alpha +\beta (p_t-p_t^{*})+e_t,$(19)

where $s_t$ is the logarithm of the nominal dollar exchange rate, $p_t$ is the logarithm of the domestic consumer price index, $p_t^{*}$ is the logarithm of the US consumer price index, and $e_t$ is the error term. According to the PPP hypothesis, finding a cointegration relation between $s_t$ and $(p_t-p_t^{*})$ provides evidence in favor of this hypothesis. The evidence of cointegration relation between these variables is sufficient for the weak form of the PPP hypothesis. The variables demonstrated here are subject to structural breaks for the given period and the suggested cointegration tests with the Fourier function can deal with the breaks at any time of the period. In the analysis, the variables of $s_t$ and $(p_t-p_t^{*})$ are redefined as lter and lcpi, respectively.

For the Fourier ESTAR and Fourier LSTAR cointegration tests, the first step is to decide the optimal Fourier frequency $k$ by estimating the following equation with integer frequencies ranging from 1 to 5:

$y_t=(\mu )+(\vartheta t)+{\alpha }_1\sin \left({\displaystyle \frac{2k\pi t}{T}}\right)+{\alpha }_2\cos \left({\displaystyle \frac{2k\pi t}{T}}\right)+{\beta }^{\prime }{x}_t+e_t\hspace{1em}t=1,2,\mathrm{\dots }T,$(20)

where the model that gives the minimum SSR among the predicted models provides the optimal $k$ value. Equation (20) can be rewritten with the real data variables as follows:

${\text{𝑙𝑡𝑒𝑟}}_t=\mu +{\beta }_1{\text{𝑙𝑐𝑝𝑖}}_t+{\alpha }_1\sin \left({\displaystyle \frac{2k\pi t}{T}}\right)+{\alpha }_2\cos \left({\displaystyle \frac{2k\pi t}{T}}\right)+e_t,$(21)

where $k$ is the optimal frequency and variables are stationary at first difference as the main assumption of cointegration tests.

Residuals are obtained from Eq. (21) as follows:

$e_t={\text{𝑙𝑡𝑒𝑟}}_t-\mu -{\beta }_1{\text{𝑙𝑐𝑝𝑖}}_t-{\alpha }_1\sin \left({\displaystyle \frac{2k\pi t}{T}}\right)-{\alpha }_2\cos \left({\displaystyle \frac{2k\pi t}{T}}\right).$(22)

In the second step, we predict the following equations using the residuals from Eq. (22) to test the cointegration relation in STAR models:

$\mathrm{\Delta }{\widehat{e}}_t=\rho {\widehat{e}}_{t-1}(1-\exp \{-\gamma {\widehat{e}}_{t-1}^2\})+{\displaystyle \sum _{j=1}^p}{\beta }_j\mathrm{\Delta }{\widehat{e}}_{t-j}+v_{1t},$(23) $\mathrm{\Delta }{\widehat{e}}_t=\rho {\widehat{e}}_{t-1}{(1+\exp \{-\gamma (e_{t-1})\})}^{-1}+{\displaystyle \sum _{j=1}^p}{\beta }_j\mathrm{\Delta }{\widehat{e}}_{t-j}+v_{2t},$(24)

where we compute the test statistics of $\rho$ for each possible value of nuisance parameter defined in the exponential and logistic transition function $G(.)$ and choose the minimum value to test the null hypothesis of no cointegration relation against the alternative of cointegration relation with a STAR adjustment. The range of the parameter is as defined in the previous section. In the empirical example, lags of the dependent variable may also be added to the model to avoid the autocorrelation problem.

ADF test statistics for lter and lcpi are $-$ 3.082 and $-$ 2.164 at the level and $-$ 8.034 and $-$ 10.276 at the first difference, respectively. Kapetanios et al. (2003) test statistics are $-$ 0.297 for lter and $-$ 2.373 for lcpi at level. In this case, ADF and Kapetanios et al. (2003) unit root test results indicate that both variables are stationary at I (1). In the first step, Eq. (21) is estimated by the OLS estimation method for different Fourier frequencies ranging from 1 to 5 to define the model that gives minimum SSR. The minimum SSR is obtained with Model 1 at $k=$ 1. Thus, all unknown structural breaks are modeled best with the optimal frequency $k=$ 1. Following, infimum-type test statistics derived from Eqs (23) and (24) are computed to test the cointegration relation through the Fourier ESTAR and Fourier LSTAR cointegration tests in the second step. Critical values of the Fourier LSTAR test at $k=$ 1 are $-$ 4.727, $-$ 4.169, and $-$ 3.873 at the significance level of 1%, 5%, and 10% as presented in Table 1. The critical values of the Fourier ESTAR cointegration test obtained from the paper of Güriş and Sedefoğlu (2022) are $-$ 4.843, $-$ 4.255, and $-$ 3.977 at the significance level of 1%, 5%, and 10%. The infimum $t$ statistics of the coefficient $\rho$ is $-$ 1.147 for the Fourier ESTAR test and $-$ 0.925 for the Fourier LSTAR test. The Akaike Information Criterion (AIC) for both models with the first lag are $-$ 604.847 and $-$ 604.597, respectively. Hence, we can conclude that we cannot reject the null hypothesis of no cointegration relation according to the Fourier ESTAR and Fourier LSTAR test results and the PPP hypothesis is not valid in Türkiye. On the other hand, finding a similar result from both tests is not surprising since the power performance of the tests was close to each other.

5. Conclusion

In this paper, we propose a cointegration test using the logistic transition function in the STAR model and defining the Fourier function in the deterministic term. Simulation results show that the Fourier LSTAR cointegration test is sized correctly except for the small changes caused by the degree of the variance. The size is not affected by the serial correlation of the error term. The test maintains good power properties at different sample sizes and combinations. The good power properties of the test become more pronounced as the sample size gets bigger. Nevertheless, the most potent power properties are obtained at the lowest Fourier frequency similar to the literature. The comparison of the Fourier LSTAR and ESTAR simulation results shows that the proposed test performs well as the Fourier ESTAR test. Therefore, both tests can be used alternatively unless there is a special emphasis on the economic structure or the goal of using prior knowledge. As a result of the empirical example we aimed to reveal the steps of the test, the null hypothesis of no cointegration relation between the variables cannot be rejected by the Fourier ESTAR and Fourier LSTAR tests. Thus, the PPP hypothesis is not valid in Türkiye for the given period. This result supports the simulation results that we obtained from the comparison of the Fourier-based tests. For future work, we note that working with decimal Fourier frequencies to see if the power properties outperform the previous tests and if the test fits the expectation of the theories in the economy can offer exciting findings.