On the performance of the variance ratio unit root tests with flexible Fourier form

Abstract

This article introduces a new unit root test that combines the variance ratio framework with the Flexible Fourier Form under the generalized least squares detrending mechanism. The advantage of the proposed method against its alternatives can be listed as: (1) it suggests a non-parametric procedure that does not require any parametric or semi-parametric model to remove serial correlation in the innovation process; (2) it can reasonably adapt itself to deal with the multiple structural breaks with various functional specifications. In the simulation exercises, we show that the proposed method exhibits satisfactory performance in the size and size-adjusted power analysis.

1. Introduction

After the seminal article of [20], addressing structural breaks in the deterministic component of the time series processes becomes a crucial exercise in unit root testing. [20] tests for the presence of a unit root when there is a single sharp structural break at a known date in the series. More recent studies, such as [14] and [4] focus on the unit root testing in the case that the number and dates of the structural breaks are unknown as well as their functional form. Furthermore, the inclusion of sinusoidal terms, namely the Flexible Fourier Form (hereafter FFF), to the deterministic part of the data generation process (hereafter DGP) emerged as the latest approach to alleviate the challenge appearing with the existence of multiple structural breaks. In this context, [13] state that the Fourier approximation embedded in the FFF captures the deterministic behavior of a series within a desired degree of accuracy, additionally the approximation is well-defined in the sense that it is bounded and is an orthogonal basis which spans the whole domain of the series as well as being a global approximation.

Initially, [1] introduce the FFF approach in order to model structural breaks in the framework of [15] (hereafter KPSS). Subsequently, the FFF approach is employed in the ADF-type unit root testing environment by [9] and an LM-type unit root testing environment by [10]. [19] further extends the unit root tests introduced in [9,10] by allowing fractional frequency components in the FFF rather than restricting them to be positive integers. Furthermore, [23] introduces generalized least squares (hereafter GLS) detrending method of [7] to the FFF approach. The GLS detrending is a popular method in the literature of unit root testing with structural breaks; it has been adapted to treat a single sharp break [21] and multiple sharp breaks [4]. Similarly, [5,6] pair the FFF with OLS detrending mechanism when testing for a unit root in both linear and nonlinear setups.

This article applies the FFF to the variance ratio (hereafter VR) testing procedure. In the literature, there are different VR unit root tests, such as [2,16,17,28]. The common feature of these tests is that test statistics in all of them are based on the ratio of two variances computed from two different transforms of the observed series. However, in our analysis, we utilize the most recent and powerful VR procedure of [17]. A variant of this VR method is introduced to the unit root literature by [2]. Furthermore, [17] modifies the VR unit root tests by utilizing the fractional integration instead of the standard integration operator. [17] shows that this modification in the VR framework improves the size and power properties of the existing unit root tests. In this regard, our objective is to combine the advantages mentioned earlier of the VR method with the FFF method's ability to address structural breaks. We denote this new test as VR-FFF To best of our knowledge, this new procedure is the first and only non-parametric unit root test, which, unlike other FFF tests, does not require lag length selection or bandwidth choice for removing serial correlation in the innovation term. Furthermore, the proposed method looks similar to the Log-periodogram regression framework, which also utilizes Fourier transform. However, the current study is different from this framework since we try to approximate the deterministic term with Fourier functions, while in the log-periodogram framework the Fourier Functions are employed to analyze the stochastic components [22].

Last but not least, we check the validity of the proposed methodology in an empirical exercise. In this exercise, we test the unit root hypothesis in the yield curve spreads with long and short maturity gaps for the US data. We show that the rejection decisions of the VR-FFF procedure for unit root hypothesis are quite different from the findings of the standard unit root tests. While the standard unit root tests without the FFF correction favor the non-stationarity of the US yield curve spreads, our procedure finds that at least short-term interest rate spreads are in general stationary.

The rest of the paper is organized as follows. In Section 2, we introduce the unit root model and discuss the theoretical results of the proposed method. Section 3 investigates the small and large sample properties of the VR test with the FFF type of deterministic component adjustment. In Section 4, we illustrate the empirical application of the proposed method, and Section 5 concludes. The tables for the results of the simulation and empirical exercises are placed in the appendix.

2. Model

Consider the autoregressive unit root model for $\{y_t{\}}_{t=1}^T$ with the deterministic component vector $Z_{j,t}$ as

$$\begin{array}{rl}{y}_t& =Z_{j,t}^{\mathrm{\top }}\theta +x_t,\text{for}j=0,1,2\text{and for}t=1,\dots ,T;\end{array}$$ (1)

$$\begin{array}{rl}{x}_t& =\rho x_{t-1}+\varphi (L)u_t,\end{array}$$ (2)

where the innovation term is $u_t\sim i.i.d.(0,{\sigma }^2)$ with ${\sigma }^2<\mathrm{\infty }$, $\varphi (L)$ is the lag polynomial that is assumed to provides the stationarity conditions for the innovations, the initial condition $y_0$ is an $O_p(1)$ random variable, and j denotes the type of deterministic component and $x_t$ is the stochastic component of $y_t$, which determines the non-stationarity of the observed process. Depending the value of j, three types of the deterministic components are considered. For the sake of simplicity, let us use a single frequency component for now. If j = 0, then $Z_{0,t}=[sin(2\pi {\kappa }_1/T),cos(2\pi {\kappa }_{1}t/T){]}^{\mathrm{\top }}$ and $\theta =[{\alpha }_1,{\beta }_1]$. If j = 1, $Z_{1,t}=[1,sin(2\pi {\kappa }_1/T),cos(2\pi {\kappa }_{1}t/T){]}^{\mathrm{\top }}$, $\theta =[{\delta }_0,{\alpha }_1,{\beta }_1]$ and finally, when j = 2, $Z_{2,t}=[1,t,sin(2\pi {\kappa }_1/T),cos(2\pi {\kappa }_{1}t/T){]}^{\mathrm{\top }}$, $\theta =[{\delta }_0,{\delta }_1,{\alpha }_1,{\beta }_1]$. These specifications can be represented explicitly as it follows:

$$\begin{array}{rl}{y}_t& ={\alpha }_1\sin \left(\frac{2\pi {\kappa }_{1}t}{T}\right)+{\beta }_1\cos \left(\frac{2\pi {\kappa }_{1}t}{T}\right)+x_t,\text{for}j=0;\end{array}$$ (3)

$$\begin{array}{rl}{y}_t& ={\delta }_0+{\alpha }_1\sin \left(\frac{2\pi {\kappa }_{1}t}{T}\right)+{\beta }_1\cos \left(\frac{2\pi {\kappa }_{1}t}{T}\right)+x_t,\text{for}j=1;\end{array}$$ (4)

$$\begin{array}{rl}{y}_t& ={\delta }_0+{\delta }_{1}t+{\alpha }_1\sin \left(\frac{2\pi {\kappa }_{1}t}{T}\right)+{\beta }_1\cos \left(\frac{2\pi {\kappa }_{1}t}{T}\right)+x_t,\text{for}j=2,\end{array}$$ (5)

for $t=1,\dots ,T$. Notice that with Equations (3)–(5), we can control the structural breaks or nonlinearities in the unit root model. Moreover, the parameter ${\kappa }_1$, which takes integer and non-integer values in this study, controls the Fourier frequency. Furthermore, ${\alpha }_1$ and ${\beta }_1$ in (3)–(5) measure, respectively, the amplitude and displacement of the sine and cosine functions in the deterministic term [12] and they are computed at the given frequency. These parameters also controls the displacement of the sinusoidal component of the deterministic term [10]. [1] show that with different values of the parameters ${\kappa }_1$, ${\alpha }_1$ and ${\beta }_1$, the sinusoidal functions in the above representation can capture a wide range of structural break specifications. With the usage of these functions, we can ignore the explicit estimation of the structural break dates and magnitudes.

In the VR procedure, we test the null hypothesis of $H_0:\rho =1$ versus $H_a:|\rho |<1$ based on Equation (2). Due to the nature of the VR procedure, the deterministic term has to be removed before obtaining the test statistic. In the proposed method, we utilize the local GLS detrending approach of [7] to remove the deterministic terms in Equation (1). This approach consists of two stages:

1. In the first stage, we apply the quasi-differencing on the observed series $y_t$ and the deterministic component vector $Z_{j,t}$ for j = 0, 1, 2 and the application is done with the local-to-unity parameter (or local quasi-differencing parameter) ${\overline{c}}_{{\kappa }_1,j}$ as shown below:

   $$\begin{array}{rl}{y}_{{\overline{c}}_{{\kappa }_1,j},t}& =\{\begin{array}{ll}{y}_1& ift=1,\\ y_t-(1+\frac{{\overline{c}}_{{\kappa }_1,j}}{T})y_{t-1}& otherwise;\end{array}\end{array}$$ (6)

   $$\begin{array}{rl}{Z}_{j,{\overline{c}}_{{\kappa }_1,j},t}& =\{\begin{array}{ll}{Z}_{j,1}& ift=1,\\ Z_{j,t}-(1+\frac{{\overline{c}}_{{\kappa }_1,j}}{T})Z_{j,t-1}& otherwise.\end{array}\end{array}$$ (7)

2. Next, we regress $y_{{\overline{c}}_{{\kappa }_1,j},t}$ on $Z_{j,{\overline{c}}_{{\kappa }_1,j},t}$ and obtain the OLS coefficients ${\hat{\theta }}_{{\overline{c}}_{{\kappa }_1,j}}$. The local GLS detrended series are generated as $y_t^{{\overline{c}}_{{\kappa }_1,j}}=y_t-Z_{j,t}^{\mathrm{\top }}{\hat{\theta }}_{{\overline{c}}_{{\kappa }_{!},j}}$.

Although these local GLS detrended series can be used in any unit root test procedure, we focus on [17]'s VR unit root test. To compute the test statistic, the fractional integration operator is utilized. Let $v_t$ be the time series process to which the fractional integration operator is applied. Then, the fractional integration of $v_t$ with the fractional integration parameter $d_1>0$ is defined as ${\mathrm{△}}_{+}^{-d_1}{v}_t=\sum _{j=0}^{t-1}{\pi }_j(d_1)v_{t-j}$, where ${\pi }_j(d)$, for $j=0,\dots ,t-1$, are the fractional binomial coefficients. These coefficients are written as ${\pi }_j(d)=\mathrm{\Gamma }(j+d)/\{\mathrm{\Gamma }(j+1)\mathrm{\Gamma }(d)\}$ [26] and only the past values of $v_t$ with a positive index enter the integration.

After obtaining the fractional transform of the local GLS detreded series, say ${\tilde{y}}_t^{{\overline{c}}_{{\kappa }_1,j}}={\mathrm{△}}_{+}^{-d_1}{y}_t^{{\overline{c}}_{{\kappa }_1,j}}$, we compute [17]'s VR test as the scaled ratio of the variance of the detrended series $y_t^{{\overline{c}}_{{\kappa }_1,j}}$ and the variance of the fractionally integrated series ${\tilde{y}}_t^{{\overline{c}}_{{\kappa }_1,j}}$:

$${\tau }_j(d_1)=T^{2d_1}\frac{\sum _{t=1}^T{\left(y_t^{{\overline{c}}_{{\kappa }_1,j}}\right)}^2}{\sum _{t=1}^T{\left({\tilde{y}}_t^{{\overline{c}}_{{\kappa }_1,j}}\right)}^2}.$$ (8)

Remark 1

Under the assumptions of [17], the asymptotic distribution of the above test statistic is given as:

$${\tau }_j(d_1)=T^{2d_1}\stackrel{}{\to }\frac{{\int }_0^1{B}_{{\overline{c}}_{{\kappa }_1,j}}(s{)}^{2}ds}{{\int }_0^1{B}_{{\overline{c}}_{{\kappa }_1,j},d_1}(s{)}^{2}ds}.$$ (9)

where $B_{{\overline{c}}_{{\kappa }_1,j}}(s)$ is the local GLS detrended version of the standard Brownian motion and $B_{{\overline{c}}_{{\kappa }_1,j},d_1}(s)$ is the local GLS detrended version of fractional Brownian motion with order $1+d_1$. We omit the proof of the specification in Equation (9) because the proof directly follows by combining the results of Theorem 2 in [7], Theorem 1 in [23] and Theorem 4 in [17] with the Continuous Mapping Theorem.

Remark 2

In the [17]'s VR test, we reject the null hypothesis of unit root if the calculated test statistic is larger than the critical value $C{V}_{\xi ,d_1,{\kappa }_1,j}$, where ξ is the significance level. Note that this critical value depends on $d_1$, ${\kappa }_1$ and the type of the deterministic trend. Accordingly, the critical values given in [17] are not applicable.

Remark 3

It is possible to modify the deterministic component in Equation (1) to include multiple values of the Fourier frequencies. For instance, we can combine n different κ values to constitute the FFF. For j = 2, this function and the corresponding coefficients are represented as:

$$\begin{array}{rl}{Z}_{2,t}& ={[1,t,sin\left(2\pi {\kappa }_1/T\right),cos\left(2\pi {\kappa }_{1}t/T\right),\dots ,sin\left(2\pi {\kappa }_n/T\right),cos\left(2\pi {\kappa }_{n}t/T\right)]}^{\mathrm{\top }},\end{array}$$ (10)

$$\begin{array}{rl}\theta & =[{\delta }_0,{\delta }_1,{\alpha }_1,{\beta }_1,\dots ,{\alpha }_n,{\beta }_n].\end{array}$$ (11)

Similarly, one can modify the above FFF for j = 0, 1. Also notice that we can define $\kappa =[{\kappa }_1,\dots ,{\kappa }_n]$ and replace ${\kappa }_1$ with κ in the above construction and the asymptotic analysis of the VR unit root test.

Another issue is that the local quasi-differencing parameter depends on κ and the type of the deterministic trend. Therefore, the value of this parameter is quite different from the values obtained in [17] and [23]. Consequently, we first calculate the values of ${\overline{c}}_{\kappa ,j}$ under different κ and the deterministic component. This value is chosen as the number that generates $50\mathrm{\%}$ power under the data generation $x_t=(1+{\overline{c}}_{\kappa ,j}/T)x_{t-1}+u_t$ where $u_t$ is an i.i.d. random variable. Tables 1 and 2 demonstrate the values of this parameter for different values of κ and j.

Table 1. The values of the local quasi-detrending parameter ${\overline{c}}_{\kappa ,j}$ for the single and multiple Fourier frequencies.

	κ
j	1	2	3	4	5	$[1,2]$	$[1,2,3]$	$[1,2,3,4]$	$[1,2,3,4,5]$
0	−11.45	−9.5	−9.55	−9.7	−9.75	−12.55	−12.55	−13.4	−13.2
1	−15.35	−10.45	−10.1	−10.2	−10.2	−19.5	−22.7	−25.55	−28.35
2	−24.45	−18.4	−16.25	−15.85	−15.75	−32.15	−38.9	−44.95	−50.6

Table 2. The values of the local quasi-detrending parameter ${\overline{c}}_{\kappa ,j}$ for the fractional Fourier frequencies.

	κ
j	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	1.9
0	−10.000	−9.500	−9.750	−10.500	−9.750	−9.750	−9.500	−9.750	−9.750
1	−13.250	−12.250	−12.000	−11.750	−11.750	−11.750	−11.750	−11.500	−11.500
2	−24.750	−24.250	−23.000	−22.000	−21.250	−20.250	−19.750	−19.250	−18.750

Given the local quasi-detrending parameters in Tables 1–2, we simulate the asymptotic distribution and the critical values for the unit root test that we propose. Tables 3 and 4 demonstrate the simulated asymptotic critical values. We generate these values by simulating the associated Brownian and Fractional Brownian motions with the discretization step $(1/1000)$. In the generation procedure, we consider both integer- and fractional-valued κ parameters for our analysis. The latter is investigated by [19].

Table 3. The critical values for the [17]'s VR test at $5\mathrm{\%}$ significance level for integer-valued κ.

	κ
j	1	2	3	4	5	[1,2]	[1,2,3]	[1,2,3,4]	[1,2,3,4,5]
0	1.7281	1.624	1.6131	1.6215	1.6127	1.825	1.8713	1.8991	1.8710
1	1.9432	1.7186	1.6648	1.6475	1.642	2.1203	2.2445	2.3476	2.436
2	2.0963	1.9697	1.8685	1.8413	1.8401	2.2609	2.386	2.4877	2.5675

Table 4. The critical values for the [17]'s VR test at $5\mathrm{\%}$ significance level for fractional-valued κ.

	κ
j	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	1.9
0	1.6672	1.6489	1.643	1.6591	1.6627	1.639	1.6142	1.604	1.6268
1	1.8828	1.837	1.7982	1.7834	1.7646	1.7541	1.758	1.7619	1.7527
2	2.0921	2.0888	2.0772	2.063	2.04	2.0187	1.9993	1.982	1.9684

3. The size and power performance of the unit root tests with the flexible Fourier form

In this section, by using Monte Carlo simulations, we evaluate the small and large sample performances of the VR-FFF test and compare this test with several other well-known unit root tests in the literature. To conduct the simulation exercises, we adopt the following DGP:

$$\begin{array}{rl}{y}_t& =\sum _{i=1}^n[{\alpha }_i\sin \left(\frac{2\pi {\kappa }_{i}t}{T}\right)+{\beta }_i\cos \left(\frac{2\pi {\kappa }_{i}t}{T}\right)]+x_t,t=1,\dots ,T;\end{array}$$ (12)

$$\begin{array}{rl}{x}_t& =\rho x_{t-1}+\varphi (L)u_t,\end{array}$$ (13)

for different values of n, ${\kappa }_i$, ${\alpha }_i$, ${\beta }_i$, ρ and the coefficients in the lag polynomial $\varphi (L)$. We use $n\in \{1,2,3\}$ and ${\kappa }_i\in \{1,2,3\}$ for each $i\in \{1,\dots ,n\}$. For instance, $\kappa =[1,2,3]$ indicates n = 3 along with ${\kappa }_1=1$, ${\kappa }_2=2$ and ${\kappa }_3=3$. To keep the exposition brief under the integer-valued κ case, we only consider $\kappa \in \{1,2,3,[1,2],[1,2,3]\}$ in the size exercise, and $\kappa \in \{3,[1,2,3]\}$ in the size-adjusted power evaluation. Additionally, under the fractional-valued κ case, we utilize $\kappa \in \{1.3,1.5,1.7\}$ in the size adjustment, and $\kappa \in \{1.3,1.7\}$ in the size-adjusted power evaluation. These chosen specifications are the most representative cases in the simulation analysis¹. Moreover, we set ${\alpha }_1=\dots ={\alpha }_n=\alpha$ and ${\beta }_1=\dots ={\beta }_n=\beta$ besides using two different values of α and β by selecting values from the set $\{0,3\}$ following [23].

In this study, we consider T = 50 as the small sample, T = 500 as the medium sample and T = 1000 as the large sample. All simulations are performed with 20, 000 Monte Carlo replications. In all of our simulations, we consider four different scenarios for the serial correlation in the innovation process governed by $\varphi (L)$. These are listed as:

• Model 1 ${ϵ}_t=u_t$.

• Model 2 ${ϵ}_t=-0.5{ϵ}_{t-1}+0.5u_{t-1}+u_t$.

• Model 3 ${ϵ}_t=0.5u_{t-1}+u_t$.

• Model 4 ${ϵ}_t=-0.1{ϵ}_{t-1}+0.2{ϵ}_{t-2}+0.1u_{t-1}-0.2u_{t-2}+u_t$.

where ${ϵ}_t=\varphi (L)u_t$ and $u_t$ is drawn from an $i.i.d(0,1)$. Additionally, we consider three types of deterministic components, which are represented with j. However, we only report the results for j = 1². In the performance comparison of the unit root tests with the FFF, we denote test statistics of the VR test as τ, the Augmented Dickey-Fuller as $AD{F}_t$ and the modified $Z_t$ as $M{Z}_t$: all the tests are modified with the FFF correction. In the VR test, we choose $d_1$ as 0.1, since this number is acknowledged as the optimal choice according to the various studies (see [11,17,18]).

In all the tables, we report the rejection probabilities of the unit root tests. These probability values show how frequently the unit root hypothesis is rejected. In the size exercise, one expects a rejection probability close to the nominal size, which is equal to the significance level of $5\mathrm{\%}$. Furthermore, in the power exercise, one expects a rejection probability higher than the nominal size of $5\mathrm{\%}$.

3.1. The size performance

First, we focus on the size features of the unit root tests, and we set $\rho =1$ to generate the experimental data using the DGP in Equations (12) and (13). The results of the size experiment with the integer-valued κ case are demonstrated in Table 5 for the sample size 50, in Table 6 for the sample size 500 and in Table 7 for the sample size 1000. Furthermore, the results of the size experiment with the fractional-valued κ case are demonstrated in Table 8. Since we use the significance level as $5\mathrm{\%}$, we compare the rejection frquencies

Regardless of the sample size, in the case of single, multiple and fractional frequencies in the FFF, the FFF-VR test outperforms both $AD{F}_t$ and $M{Z}_t$ test as observed in Tables 5–7. The different types of serial correlation in the innovations, the values of κ, α, and β do not undermine this striking result. Furthermore, the VR-FFF test exhibits the ideal rejection probability under the null hypothesis in all cases except the model 3 with T = 50, while the other tests suffer from downward size distortion, especially when T = 50.

When we consider the multiple integer-valued Fourier frequencies in Tables 5–7, some particular circumstances arise. For example, under the serial correlation model 3, $M{Z}_t$ test exhibits less size distortion than the VR-FFF test. We also observe that $AD{F}_t$ is oversized, and $M{Z}_t$ is severely undersized with the serial correlation types 1, 2 and 4, and j = 2 (in the unreported simulation results), while the VR-FFF test achieves the nominal size properties. Additionally, as the sample size increases, all tests exhibit improvement in their size performances.

Note that the usage of the fractional frequencies does not effect the size performance of the VR-FFF test. Moreover, in Table 8, we also observe that the size performance of the $AD{F}_t$ and $M{Z}_t$ tests are similar to those obtained when ${\kappa }_1=1$ and ${\kappa }_1=2$ in Tables 6 and 7. These similarities emerge from the fact that we choose the fractional ${\kappa }_1$ values between 1 and 2.

3.2. The size-adjusted power performance

Because $AD{F}_t$ and $M{Z}_t$ tests are undersized in almost all cases, we apply size adjustment in the power evaluation. In other words, we adopt a size-adjusted power analysis instead of an empirical power exercise³. In addition to the parameters specified in the size exercise, we consider different values for the parameter ρ in this section. The unit root literature considers a range of values for this parameter. In most of the studies, ρ lies between 0.99 and 0.6 (see [10,17]). To keep the exposition brief, we consider two values of ρ in this study. The values are $\rho \in \{0.98,0.80\}$; we employ the case $\rho =0.98$ as the local alternative and $\rho =0.8$ as the stationary alternative. In addition, the size-adjusted power is evaluated by using a proper subset of the parameter values for α, β, and κ in the size exercise. The findings for different sample sizes are presented in Tables 9 and 12.

In Table 9, we observe the small sample size-adjusted power performances of the unit root tests. First, in the single integer-valued frequency case, the VR-FFF test outperforms the other tests under the stationary alternative and all serial correlation specifications. The power of the VR-FFF test at $\rho =0.8$ is almost $25\mathrm{\%}$ higher than the other tests. However, this clear advantage is neutralized when the multiple integer-valued frequency tests are considered. Besides, the findings in this table reveal that the VR-FFF test generates slightly less power compared to the other unit root tests under the local alternative.

Furthermore, Tables 10 and 11 demonstrate improvements in the power performance of the unit root tests when the sample size increases. For instance, the VR-FFF test achieves full power when $\rho =0.8$ in almost any case. However, the other tests exhibit similar power performance after the sample size 500. Finally, as observed in Table 12, introducing the fractional ${\kappa }_1$ values in the testing procedure does not distort the power performance of the considered unit root tests.

4. Empirical application

In this section, we illustrate the empirical practicality of our method through an application. We apply our test on the interest rate spreads, which are the difference between two interest rates with different maturities. In the literature, there are different findings of the stationarity of these spreads. For instance, by using cointegration methods, [3] explicitly state that the interest rate spreads are stationary. Additionally, while [24] rejects the unit root hypothesis for the interest rate spreads between short-term maturities, the author [24] cannot reject non-stationarity of the interest rate spreads between longer-term maturities. Finally, [8] and [25] suggest that the interest rate spreads are stationary when the adjustment toward the long-run equilibrium follows a threshold process. To this end, we check whether our new methodology brings a new perspective on the dynamics of the interest rate spreads.

We consider 18 interest rate spreads, which we have obtained using the Federal Funds Rate, Treasury Bill rate (3 and 6 months), and Treasury Constant Maturity rate (1, 3, 5, and 10 years) series. The descriptions of these spreads are given in Table 13. Additionally, we compute these series by using a dataset that consists of 687 monthly observations between January 1960 and March 2017. Table 14 summarizes the unit root test statistics and p-values of the different tests applied to the spread series. On the one hand, we consider the standard versions of the VR test ( ${\tau }^{\ast }$), modified Z test ( $M{Z}_t^{\ast }$), and the Augmented Dickey and Fuller ( $AD{F}_t^{\ast }$). On the other hand, the FFF counterparts of these tests are denoted without the asterisk in the notation. In all of these tests, we only consider a drift term in the deterministic component, since these series do not exhibit strong trending behavior. We utilize five Fourier frequencies⁴ with $\kappa =[1,2,3,4,5]$ in the FFF adjustment, since we find this specification as the optimal choice according to the criterion shown in [27].

In Table 14, we use the bold font for the p-values if the null is rejected at a $5\mathrm{\%}$ significance level. Moreover, we also use the italics font for the p-values if the rejection decisions of the standard and the FFF version of the same test are different. First, we observe an interesting pattern in the results for the $M{Z}_t^{\ast }$ and $AD{F}_t^{\ast }$ tests, which indicate the absence of the unit root in all the spreads. The only exception to this finding is the result of the $AD{F}_t^{\ast }$ test for the GS10Y-GS5Y series. However, the FFF versions of these tests display contradictory findings. The FFF- $M{Z}_t$ test cannot reject the unit root hypothesis for all the series except GS3Y-TB3M, while the FFF- $AD{F}_t$ rejects the hypothesis for only the TB3M-FFR and GS3Y-TB3M series. For the VR test, we see a slightly different situation. The standard VR test finds a unit root in the TB6M-FFR, but the VR-FFF test concludes that the same series is stationary. This situation is unique in all the test results reported in Table 14. In the rest of the cases, where we find a unit root with the VR-FFF, the standard VR test gives the same result.

Another crucial point is that the standard VR test seems to reject the null hypothesis less frequently than any other test. For example, both the standard and FFF version of the VR test finds that the GS10Y-GS1Y, GS10Y-GS3Y, GS10Y-GS5Y, and GS10Y-TB6M series possess a unit root. This result seems to be the critical difference between the VR-FFF and other FFF-based unit root tests. Furthermore, the FFF-VR test does not seem to favor a unit root in any spreads that contain GS10Y. This is another unique case for GS10Y-GS5Y; all tests except $M{Z}_t^{\ast }$ indicates the series is not stationary around a deterministic term. Lastly, the GS3Y-TB3M is the only series for which all the tests agree upon its stationarity.

We now focus on the graphical results for a few interesting cases. In this context, Figure 1 demonstrates six spread series and their sinusoidal deterministic terms estimated with the FFF. In the first case that we investigate, all the standard tests reject the null hypothesis while their FFF versions does not reject it. The GS5Y-GS3Y spread on the left bottom part of Figure 1 is a typical example of this case. We can attribute this result to the significant nonlinear trend, which is also apparent in the plot of the GS5Y-GS3Y spread. Next, we consider the TB6M-FFR spread on the top left of Figure 1. In this case, the standard VR test does not reject the unit root hypothesis, but the VR-FFF rejects it. Compared to the other graphs, the deterministic part of the TB6M-FFR spread is relatively linear if we ignore the beginning of the sample. In another case, although the GS3Y-FFR series seems to have a relatively linear trend, both the standard VR and FFF-VR indicate that the series is stationary. Moreover, the spreads that contains the FFR and the FFF-VR finds as non-stationary are the series with a long-maturity difference, e.g. GS10Y-FFR and GS5Y-FFR. Finally, the nonlinear trends of the spreads series on the middle left, middle right, and bottom left of Figure 1 seem to follow a similar path.

Figure 1.

Graphs of the selected spreads and their estimated nonlinear trends.

These results imply that the FFF correction changes the rejection decision of the tests since the presence of the ignored structural or smooth breaks in the deterministic component may cause spurious stationarity in the spread data. Although these empirical findings, especially the results of $M{Z}_t^{\ast }$ and $AD{F}_t^{\ast }$ and their FFF counterparts, are controversial to the results in the existing literature (for instance, see [10]), our proposed method found that the spreads with short maturity differences exhibit stationary patterns.

5. Conclusion

The article proposes a unit root test based on the VR procedure of [17] that takes into account possible multiple breaks of an unknown functional form. The test captures the breaks and their functional form via the FFF, which is a method introduced in the unit root literature by [1]. We also incorporate the local GLS detrending proposed by [7] into our FFF-based VR testing procedure.

We show that the resulting unit root test shares similar asymptotic properties as [17], and it possesses a good finite sample size and power features. Although the proposed FFF-VR unit root test has good size properties compared to the alternative tests evaluated, we show that increasing the number of frequency component(s) causes a size distortion for this test. Despite these size distortions, the size features of the VR test is overall more stable than the other unit root tests. Another crucial finding of this study is that the proposed method exhibits better power performance than the other tests in almost all cases. Accordingly, we think that this new testing framework is a good alternative when dealing with unit roots under possible structural breaks in deterministic components.

We also apply the proposed method to the US interest rate spread data. The findings do not, in general, support the stationarity assumption on the interest rate spreads. While we verify the stationarity of the spreads for the shorter maturity, we show the non-stationary patterns in the yield curve spreads for the longer maturity.

Finally, the developed techniques in this article can be utilized to devise a cointegration test under possible structural breaks. Since this extension requires additional theoretical and experimental work, we consider it in a future study. The basic idea in this extension is to combine the VR cointegration test of [18] with the methods used in our study. However, the cointegration framework may cause additional complications, which are currently being investigated in an additional study.

A. Tables.

Table C1. The size performance of the unit root tests when T = 50 and j = 1.

			Model 1			Model 2			Model 3			Model 4
κ	α	β	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$
1	0	0	0.053	0.039	0.006	0.056	0.039	0.006	0.015	0.006	0.015	0.052	0.039	0.005
	0	3	0.052	0.040	0.006	0.054	0.037	0.006	0.014	0.006	0.014	0.057	0.039	0.006
	0	5	0.056	0.040	0.005	0.054	0.041	0.006	0.014	0.006	0.014	0.050	0.038	0.007
	3	0	0.051	0.038	0.005	0.052	0.040	0.006	0.014	0.005	0.014	0.052	0.041	0.006
	3	3	0.053	0.041	0.006	0.051	0.039	0.006	0.014	0.006	0.014	0.052	0.042	0.006
	3	5	0.052	0.040	0.005	0.052	0.038	0.006	0.015	0.006	0.015	0.053	0.041	0.005
	5	0	0.052	0.041	0.005	0.052	0.039	0.006	0.014	0.006	0.013	0.052	0.039	0.006
	5	3	0.053	0.039	0.006	0.053	0.037	0.006	0.014	0.006	0.013	0.053	0.040	0.005
	5	5	0.053	0.038	0.006	0.053	0.039	0.006	0.015	0.006	0.014	0.055	0.042	0.005
2	0	0	0.052	0.020	0.008	0.050	0.021	0.009	0.030	0.005	0.008	0.050	0.019	0.009
	0	3	0.049	0.019	0.009	0.053	0.021	0.008	0.027	0.004	0.008	0.049	0.021	0.008
	0	5	0.054	0.022	0.009	0.051	0.021	0.007	0.026	0.005	0.008	0.049	0.019	0.008
	3	0	0.050	0.022	0.009	0.052	0.022	0.009	0.028	0.006	0.008	0.050	0.020	0.008
	3	3	0.049	0.019	0.008	0.051	0.021	0.009	0.028	0.006	0.008	0.049	0.020	0.008
	3	5	0.053	0.020	0.009	0.048	0.020	0.008	0.031	0.005	0.008	0.052	0.020	0.009
	5	0	0.049	0.022	0.009	0.053	0.022	0.009	0.028	0.005	0.008	0.049	0.019	0.008
	5	3	0.051	0.019	0.008	0.049	0.020	0.008	0.027	0.006	0.007	0.053	0.021	0.009
	5	5	0.050	0.019	0.008	0.049	0.021	0.010	0.028	0.005	0.007	0.054	0.022	0.009
3	0	0	0.051	0.016	0.009	0.051	0.017	0.009	0.029	0.005	0.012	0.051	0.017	0.010
	0	3	0.054	0.018	0.010	0.050	0.014	0.009	0.028	0.005	0.011	0.052	0.017	0.009
	0	5	0.049	0.017	0.010	0.052	0.017	0.009	0.028	0.006	0.013	0.052	0.016	0.010
	3	0	0.051	0.017	0.009	0.050	0.017	0.010	0.029	0.005	0.012	0.053	0.017	0.010
	3	3	0.055	0.018	0.010	0.054	0.016	0.009	0.030	0.005	0.011	0.050	0.016	0.008
	3	5	0.050	0.017	0.009	0.052	0.017	0.009	0.029	0.007	0.012	0.051	0.018	0.010
	5	0	0.053	0.017	0.010	0.051	0.016	0.009	0.027	0.005	0.010	0.050	0.016	0.009
	5	3	0.052	0.018	0.010	0.053	0.017	0.010	0.030	0.006	0.013	0.053	0.018	0.009
	5	5	0.051	0.017	0.009	0.049	0.016	0.010	0.030	0.005	0.012	0.052	0.018	0.009
[1,2]	0	0	0.048	0.057	0.005	0.046	0.055	0.005	0.007	0.004	0.008	0.047	0.058	0.007
	0	3	0.048	0.060	0.006	0.047	0.058	0.005	0.007	0.003	0.009	0.047	0.058	0.006
	0	5	0.048	0.058	0.006	0.047	0.057	0.006	0.007	0.004	0.008	0.045	0.055	0.006
	3	0	0.048	0.059	0.005	0.047	0.058	0.005	0.005	0.003	0.009	0.046	0.055	0.005
	3	3	0.047	0.055	0.006	0.049	0.058	0.006	0.007	0.004	0.008	0.045	0.057	0.006
	3	5	0.046	0.057	0.005	0.047	0.059	0.005	0.005	0.004	0.010	0.046	0.056	0.007
	5	0	0.048	0.058	0.006	0.050	0.061	0.004	0.006	0.004	0.010	0.045	0.057	0.005
	5	3	0.049	0.059	0.005	0.047	0.060	0.005	0.006	0.004	0.009	0.052	0.061	0.005
	5	5	0.047	0.059	0.006	0.048	0.057	0.005	0.006	0.004	0.008	0.046	0.058	0.005
[1,2,3]	0	0	0.050	0.074	0.003	0.052	0.073	0.003	0.004	0.004	0.004	0.052	0.075	0.003
	0	3	0.049	0.076	0.003	0.048	0.072	0.003	0.004	0.005	0.004	0.051	0.071	0.003
	0	5	0.050	0.075	0.003	0.050	0.077	0.003	0.004	0.004	0.004	0.050	0.074	0.002
	3	0	0.050	0.077	0.003	0.052	0.076	0.003	0.004	0.004	0.005	0.051	0.074	0.003
	3	3	0.051	0.073	0.003	0.050	0.074	0.002	0.004	0.005	0.004	0.052	0.077	0.003
	3	5	0.051	0.076	0.002	0.050	0.075	0.003	0.004	0.004	0.003	0.048	0.071	0.003
	5	0	0.051	0.075	0.003	0.051	0.073	0.003	0.003	0.004	0.004	0.053	0.078	0.003
	5	3	0.051	0.076	0.002	0.050	0.074	0.003	0.004	0.004	0.004	0.050	0.074	0.003
	5	5	0.048	0.074	0.003	0.050	0.071	0.002	0.004	0.005	0.004	0.053	0.075	0.003

Table C2. The size performance of the unit root tests when T = 500 and j = 1.

			Model 1			Model 2			Model 3			Model 4
κ	α	β	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$
1	0	0	0.053	0.046	0.042	0.051	0.043	0.039	0.041	0.039	0.046	0.050	0.042	0.038
	0	3	0.051	0.042	0.039	0.052	0.041	0.037	0.043	0.038	0.044	0.052	0.044	0.040
	0	5	0.053	0.042	0.039	0.054	0.043	0.040	0.042	0.037	0.043	0.052	0.044	0.039
	3	0	0.049	0.043	0.039	0.049	0.043	0.039	0.043	0.039	0.045	0.053	0.043	0.040
	3	3	0.052	0.043	0.040	0.052	0.046	0.041	0.043	0.039	0.046	0.052	0.043	0.039
	3	5	0.052	0.045	0.041	0.051	0.045	0.041	0.043	0.039	0.046	0.053	0.044	0.040
	5	0	0.049	0.041	0.037	0.052	0.044	0.040	0.044	0.039	0.046	0.051	0.042	0.038
	5	3	0.050	0.044	0.040	0.051	0.043	0.039	0.042	0.039	0.046	0.051	0.043	0.040
	5	5	0.051	0.043	0.039	0.051	0.044	0.041	0.045	0.040	0.046	0.053	0.043	0.039
2	0	0	0.050	0.042	0.041	0.051	0.042	0.040	0.047	0.038	0.041	0.047	0.040	0.038
	0	3	0.050	0.043	0.041	0.049	0.040	0.038	0.047	0.039	0.042	0.048	0.040	0.038
	0	5	0.051	0.045	0.043	0.048	0.042	0.041	0.048	0.039	0.044	0.051	0.041	0.040
	3	0	0.050	0.043	0.041	0.050	0.041	0.039	0.046	0.038	0.041	0.050	0.040	0.039
	3	3	0.048	0.041	0.040	0.048	0.042	0.040	0.046	0.037	0.041	0.047	0.042	0.040
	3	5	0.051	0.043	0.042	0.050	0.041	0.040	0.045	0.038	0.041	0.050	0.039	0.038
	5	0	0.049	0.040	0.038	0.047	0.040	0.039	0.042	0.036	0.039	0.050	0.042	0.040
	5	3	0.053	0.045	0.042	0.047	0.040	0.038	0.044	0.038	0.041	0.048	0.042	0.041
	5	5	0.047	0.042	0.040	0.051	0.040	0.038	0.042	0.036	0.039	0.048	0.042	0.041
3	0	0	0.049	0.044	0.043	0.047	0.040	0.039	0.043	0.038	0.042	0.049	0.047	0.045
	0	3	0.051	0.045	0.044	0.048	0.044	0.042	0.047	0.043	0.045	0.048	0.044	0.043
	0	5	0.049	0.047	0.046	0.051	0.045	0.044	0.048	0.042	0.045	0.048	0.043	0.042
	3	0	0.048	0.043	0.042	0.049	0.045	0.044	0.045	0.039	0.041	0.048	0.045	0.043
	3	3	0.050	0.044	0.043	0.048	0.045	0.044	0.048	0.043	0.045	0.048	0.044	0.043
	3	5	0.051	0.047	0.045	0.048	0.044	0.043	0.044	0.040	0.043	0.050	0.045	0.045
	5	0	0.049	0.044	0.043	0.050	0.046	0.044	0.043	0.039	0.042	0.047	0.044	0.043
	5	3	0.050	0.047	0.045	0.046	0.043	0.042	0.044	0.042	0.046	0.047	0.044	0.043
	5	5	0.049	0.040	0.040	0.045	0.042	0.041	0.045	0.040	0.043	0.049	0.046	0.044
[1,2]	0	0	0.047	0.037	0.030	0.046	0.035	0.028	0.035	0.027	0.035	0.047	0.037	0.031
	0	3	0.048	0.036	0.029	0.049	0.036	0.028	0.035	0.029	0.037	0.046	0.035	0.029
	0	5	0.046	0.036	0.030	0.048	0.038	0.031	0.032	0.027	0.035	0.050	0.039	0.031
	3	0	0.048	0.036	0.029	0.046	0.036	0.029	0.034	0.030	0.037	0.049	0.039	0.032
	3	3	0.047	0.035	0.028	0.047	0.036	0.029	0.034	0.029	0.036	0.045	0.035	0.028
	3	5	0.048	0.038	0.031	0.046	0.036	0.029	0.033	0.028	0.035	0.048	0.037	0.030
	5	0	0.048	0.039	0.030	0.050	0.040	0.032	0.036	0.032	0.040	0.047	0.037	0.029
	5	3	0.048	0.036	0.029	0.049	0.039	0.032	0.037	0.031	0.038	0.049	0.036	0.030
	5	5	0.051	0.039	0.031	0.050	0.039	0.031	0.035	0.028	0.036	0.051	0.037	0.030
[1,2,3]	0	0	0.048	0.039	0.027	0.047	0.041	0.028	0.033	0.029	0.037	0.049	0.042	0.029
	0	3	0.051	0.041	0.028	0.047	0.038	0.028	0.032	0.029	0.038	0.051	0.044	0.031
	0	5	0.050	0.042	0.031	0.050	0.043	0.031	0.031	0.028	0.036	0.050	0.043	0.031
	3	0	0.049	0.042	0.030	0.050	0.042	0.030	0.031	0.028	0.037	0.047	0.039	0.029
	3	3	0.053	0.041	0.030	0.052	0.042	0.031	0.032	0.029	0.038	0.049	0.041	0.031
	3	5	0.051	0.042	0.029	0.049	0.042	0.031	0.029	0.028	0.034	0.051	0.042	0.031
	5	0	0.050	0.041	0.030	0.051	0.043	0.031	0.033	0.029	0.038	0.051	0.041	0.029
	5	3	0.048	0.040	0.028	0.050	0.041	0.029	0.030	0.026	0.035	0.050	0.040	0.029
	5	5	0.048	0.039	0.028	0.051	0.045	0.032	0.033	0.029	0.037	0.051	0.044	0.031

Table C3. The size performance of the unit root tests when T = 1000 and j = 1.

			Model 1			Model 2			Model 3			Model 4
κ	α	β	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$
1	0	0	0.048	0.044	0.042	0.052	0.045	0.044	0.047	0.043	0.049	0.052	0.045	0.044
	0	3	0.052	0.044	0.042	0.052	0.046	0.045	0.046	0.044	0.048	0.050	0.044	0.042
	0	5	0.052	0.047	0.045	0.051	0.044	0.042	0.043	0.043	0.047	0.051	0.044	0.042
	3	0	0.052	0.046	0.044	0.051	0.043	0.042	0.045	0.044	0.049	0.050	0.043	0.041
	3	3	0.048	0.043	0.041	0.047	0.041	0.040	0.045	0.043	0.048	0.050	0.043	0.042
	3	5	0.049	0.042	0.041	0.048	0.044	0.042	0.045	0.041	0.046	0.050	0.045	0.043
	5	0	0.051	0.045	0.043	0.051	0.045	0.043	0.046	0.042	0.046	0.049	0.042	0.040
	5	3	0.052	0.047	0.045	0.050	0.044	0.042	0.045	0.042	0.046	0.051	0.043	0.042
	5	5	0.053	0.046	0.044	0.049	0.044	0.043	0.045	0.044	0.048	0.048	0.044	0.042
2	0	0	0.048	0.043	0.043	0.048	0.046	0.045	0.047	0.043	0.045	0.052	0.047	0.046
	0	3	0.051	0.045	0.045	0.052	0.047	0.046	0.048	0.044	0.047	0.049	0.044	0.043
	0	5	0.049	0.046	0.045	0.051	0.047	0.047	0.047	0.045	0.047	0.046	0.042	0.042
	3	0	0.048	0.043	0.043	0.047	0.042	0.041	0.047	0.044	0.046	0.050	0.045	0.044
	3	3	0.049	0.046	0.045	0.049	0.043	0.043	0.048	0.043	0.046	0.049	0.045	0.045
	3	5	0.050	0.045	0.044	0.049	0.044	0.044	0.049	0.044	0.047	0.049	0.045	0.044
	5	0	0.051	0.045	0.044	0.051	0.046	0.045	0.049	0.046	0.048	0.048	0.045	0.044
	5	3	0.049	0.047	0.046	0.048	0.044	0.043	0.049	0.045	0.047	0.048	0.044	0.044
	5	5	0.049	0.045	0.044	0.050	0.046	0.046	0.047	0.043	0.045	0.048	0.045	0.044
3	0	0	0.051	0.045	0.044	0.053	0.046	0.046	0.048	0.041	0.042	0.049	0.043	0.042
	0	3	0.052	0.047	0.046	0.050	0.045	0.045	0.049	0.044	0.045	0.052	0.046	0.045
	0	5	0.052	0.046	0.046	0.051	0.045	0.045	0.047	0.041	0.044	0.050	0.043	0.043
	3	0	0.052	0.044	0.043	0.051	0.046	0.045	0.052	0.044	0.046	0.053	0.045	0.045
	3	3	0.051	0.045	0.044	0.052	0.045	0.044	0.049	0.042	0.044	0.049	0.042	0.041
	3	5	0.048	0.042	0.042	0.052	0.042	0.042	0.050	0.044	0.046	0.053	0.046	0.045
	5	0	0.051	0.047	0.046	0.049	0.044	0.044	0.050	0.043	0.045	0.051	0.045	0.045
	5	3	0.054	0.049	0.048	0.052	0.046	0.046	0.047	0.041	0.043	0.051	0.043	0.043
	5	5	0.052	0.044	0.043	0.052	0.046	0.045	0.050	0.044	0.046	0.054	0.047	0.046
[1,2]	0	0	0.048	0.040	0.037	0.048	0.043	0.039	0.040	0.039	0.047	0.045	0.039	0.036
	0	3	0.049	0.044	0.041	0.047	0.041	0.037	0.041	0.042	0.048	0.048	0.041	0.039
	0	5	0.048	0.040	0.038	0.049	0.044	0.039	0.039	0.040	0.047	0.047	0.043	0.040
	3	0	0.048	0.041	0.037	0.048	0.042	0.039	0.039	0.041	0.047	0.049	0.042	0.039
	3	3	0.047	0.042	0.038	0.049	0.044	0.042	0.039	0.041	0.047	0.046	0.041	0.036
	3	5	0.049	0.043	0.039	0.047	0.041	0.038	0.039	0.041	0.048	0.046	0.041	0.038
	5	0	0.049	0.044	0.040	0.047	0.043	0.040	0.041	0.040	0.048	0.045	0.042	0.038
	5	3	0.047	0.042	0.038	0.044	0.039	0.036	0.038	0.039	0.048	0.046	0.041	0.038
	5	5	0.045	0.041	0.038	0.048	0.044	0.040	0.039	0.038	0.046	0.048	0.044	0.041
[1,2,3]	0	0	0.048	0.042	0.036	0.046	0.039	0.034	0.039	0.039	0.048	0.048	0.042	0.036
	0	3	0.049	0.043	0.038	0.049	0.041	0.036	0.038	0.038	0.047	0.052	0.045	0.040
	0	5	0.050	0.043	0.038	0.050	0.040	0.035	0.038	0.038	0.047	0.049	0.043	0.037
	3	0	0.048	0.041	0.036	0.047	0.041	0.036	0.040	0.038	0.049	0.049	0.043	0.037
	3	3	0.049	0.043	0.037	0.049	0.042	0.037	0.038	0.039	0.049	0.048	0.044	0.038
	3	5	0.049	0.044	0.039	0.051	0.041	0.037	0.038	0.038	0.048	0.048	0.041	0.035
	5	0	0.048	0.043	0.038	0.048	0.042	0.037	0.037	0.037	0.049	0.052	0.044	0.038
	5	3	0.049	0.043	0.038	0.052	0.045	0.039	0.038	0.038	0.047	0.045	0.040	0.035
	5	5	0.050	0.043	0.038	0.048	0.042	0.035	0.038	0.039	0.049	0.050	0.043	0.037

Table C4. The size performance of the unit root tests when j = 1 and with fractional κ values.

				Model 1			Model 2			Model 3			Model 4
T	κ	α	β	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$
50	1.3	0	0	0.051	0.030	0.006	0.049	0.030	0.006	0.026	0.006	0.012	0.049	0.032	0.006
		0	3	0.052	0.032	0.007	0.051	0.031	0.006	0.023	0.006	0.011	0.049	0.030	0.006
		0	5	0.051	0.030	0.006	0.051	0.031	0.006	0.025	0.006	0.012	0.050	0.031	0.005
		3	0	0.051	0.031	0.007	0.055	0.033	0.006	0.025	0.006	0.011	0.051	0.029	0.006
		3	3	0.052	0.032	0.007	0.049	0.028	0.005	0.026	0.006	0.011	0.050	0.032	0.007
		3	5	0.047	0.028	0.006	0.051	0.031	0.005	0.025	0.005	0.012	0.049	0.031	0.006
		5	0	0.051	0.031	0.006	0.049	0.030	0.006	0.025	0.006	0.012	0.051	0.031	0.006
		5	3	0.048	0.029	0.006	0.051	0.032	0.006	0.026	0.007	0.012	0.051	0.031	0.005
		5	5	0.051	0.032	0.006	0.051	0.032	0.007	0.025	0.005	0.010	0.054	0.031	0.006
	1.5	0	0	0.050	0.025	0.007	0.049	0.025	0.006	0.028	0.006	0.010	0.050	0.025	0.006
		0	3	0.050	0.027	0.007	0.051	0.026	0.006	0.027	0.006	0.011	0.051	0.025	0.006
		0	5	0.050	0.027	0.007	0.052	0.028	0.007	0.028	0.004	0.010	0.048	0.025	0.006
		3	0	0.051	0.029	0.007	0.050	0.026	0.007	0.027	0.006	0.010	0.049	0.028	0.007
		3	3	0.048	0.025	0.007	0.051	0.025	0.007	0.027	0.005	0.010	0.051	0.028	0.007
		3	5	0.050	0.027	0.007	0.049	0.025	0.007	0.029	0.005	0.009	0.052	0.026	0.006
		5	0	0.051	0.026	0.006	0.054	0.029	0.007	0.029	0.007	0.009	0.051	0.027	0.007
		5	3	0.051	0.026	0.006	0.051	0.028	0.007	0.028	0.005	0.011	0.047	0.024	0.006
		5	5	0.048	0.026	0.006	0.050	0.025	0.007	0.030	0.006	0.010	0.052	0.027	0.006
	1.7	0	0	0.052	0.022	0.007	0.052	0.024	0.006	0.030	0.005	0.009	0.053	0.024	0.006
		0	3	0.047	0.021	0.005	0.050	0.023	0.006	0.030	0.005	0.008	0.050	0.021	0.006
		0	5	0.050	0.021	0.006	0.050	0.021	0.006	0.031	0.005	0.008	0.051	0.024	0.006
		3	0	0.051	0.022	0.006	0.051	0.023	0.006	0.030	0.004	0.008	0.052	0.023	0.007
		3	3	0.052	0.023	0.007	0.052	0.022	0.007	0.028	0.005	0.009	0.049	0.022	0.007
		3	5	0.050	0.023	0.006	0.049	0.021	0.007	0.030	0.005	0.008	0.054	0.024	0.006
		5	0	0.050	0.022	0.007	0.051	0.022	0.006	0.031	0.005	0.009	0.050	0.023	0.006
		5	3	0.052	0.023	0.006	0.052	0.023	0.006	0.030	0.005	0.007	0.051	0.022	0.005
		5	5	0.050	0.021	0.007	0.053	0.022	0.006	0.030	0.005	0.007	0.050	0.022	0.006
1000	1.3	0	0	0.047	0.039	0.038	0.047	0.039	0.038	0.047	0.043	0.046	0.048	0.041	0.040
		0	3	0.047	0.040	0.039	0.049	0.039	0.038	0.045	0.038	0.042	0.046	0.040	0.039
		0	5	0.047	0.040	0.039	0.048	0.039	0.038	0.045	0.038	0.042	0.049	0.040	0.040
		3	0	0.047	0.039	0.038	0.049	0.040	0.039	0.045	0.039	0.042	0.050	0.043	0.042
		3	3	0.044	0.037	0.037	0.045	0.039	0.038	0.044	0.040	0.043	0.048	0.039	0.038
		3	5	0.046	0.040	0.039	0.048	0.039	0.038	0.043	0.037	0.040	0.049	0.041	0.040
		5	0	0.048	0.039	0.038	0.049	0.040	0.039	0.044	0.039	0.042	0.047	0.041	0.040
		5	3	0.047	0.039	0.038	0.047	0.040	0.039	0.045	0.039	0.043	0.048	0.039	0.038
		5	5	0.047	0.040	0.039	0.047	0.041	0.040	0.044	0.039	0.042	0.048	0.041	0.039
	1.5	0	0	0.050	0.036	0.035	0.049	0.035	0.034	0.051	0.038	0.041	0.050	0.036	0.034
		0	3	0.052	0.037	0.036	0.052	0.039	0.038	0.048	0.037	0.040	0.051	0.036	0.035
		0	5	0.049	0.034	0.034	0.050	0.037	0.036	0.049	0.036	0.039	0.051	0.037	0.036
		3	0	0.052	0.036	0.034	0.051	0.038	0.036	0.050	0.037	0.039	0.051	0.038	0.037
		3	3	0.050	0.036	0.035	0.051	0.037	0.036	0.045	0.032	0.036	0.050	0.035	0.034
		3	5	0.051	0.036	0.035	0.049	0.037	0.036	0.044	0.035	0.038	0.051	0.035	0.034
		5	0	0.050	0.035	0.034	0.050	0.038	0.037	0.046	0.031	0.034	0.050	0.037	0.036
		5	3	0.050	0.035	0.034	0.049	0.035	0.034	0.048	0.036	0.038	0.052	0.037	0.036
		5	5	0.051	0.035	0.034	0.050	0.036	0.035	0.050	0.039	0.041	0.050	0.036	0.035
	1.7	0	0	0.050	0.039	0.039	0.052	0.041	0.040	0.052	0.040	0.043	0.053	0.040	0.039
		0	3	0.052	0.040	0.039	0.051	0.042	0.042	0.049	0.040	0.042	0.053	0.038	0.037
		0	5	0.054	0.041	0.040	0.054	0.040	0.040	0.052	0.038	0.041	0.051	0.040	0.039
		3	0	0.053	0.040	0.039	0.052	0.039	0.038	0.051	0.039	0.041	0.053	0.042	0.041
		3	3	0.054	0.039	0.038	0.055	0.041	0.040	0.050	0.039	0.041	0.053	0.040	0.039
		3	5	0.053	0.038	0.037	0.053	0.039	0.037	0.050	0.039	0.042	0.053	0.041	0.040
		5	0	0.051	0.039	0.038	0.052	0.038	0.038	0.050	0.039	0.041	0.055	0.043	0.042
		5	3	0.053	0.041	0.040	0.054	0.040	0.039	0.051	0.037	0.040	0.054	0.042	0.041
		5	5	0.055	0.041	0.040	0.051	0.040	0.039	0.050	0.038	0.041	0.054	0.042	0.041

Table C5. The size-adjusted power performance of the unit root tests when T = 50 and j = 1.

				Model 1			Model 2			Model 3			Model 4
κ	α	β	ρ	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$
3	0	0	0.98	0.074	0.076	0.077	0.076	0.077	0.076	0.071	0.081	0.075	0.071	0.077	0.073
	0	0	0.80	0.464	0.389	0.378	0.456	0.384	0.367	0.436	0.347	0.255	0.451	0.383	0.365
	0	3	0.98	0.071	0.076	0.073	0.071	0.079	0.075	0.070	0.079	0.072	0.073	0.079	0.077
	0	3	0.80	0.453	0.374	0.356	0.466	0.393	0.373	0.444	0.334	0.240	0.457	0.393	0.379
	0	5	0.98	0.078	0.075	0.073	0.072	0.073	0.073	0.068	0.076	0.070	0.071	0.076	0.074
	0	5	0.80	0.469	0.381	0.370	0.454	0.381	0.373	0.428	0.340	0.246	0.446	0.384	0.371
	3	0	0.98	0.071	0.079	0.076	0.072	0.079	0.076	0.074	0.083	0.073	0.069	0.073	0.071
	3	0	0.80	0.462	0.395	0.376	0.462	0.395	0.375	0.442	0.341	0.247	0.453	0.384	0.370
	3	3	0.98	0.066	0.071	0.070	0.071	0.081	0.077	0.071	0.076	0.076	0.076	0.074	0.074
	3	3	0.80	0.443	0.383	0.365	0.441	0.389	0.375	0.431	0.335	0.256	0.460	0.389	0.377
	3	5	0.98	0.072	0.077	0.074	0.070	0.075	0.073	0.071	0.076	0.070	0.068	0.077	0.073
	3	5	0.80	0.466	0.392	0.374	0.454	0.386	0.373	0.438	0.333	0.239	0.462	0.384	0.368
	5	0	0.98	0.068	0.075	0.071	0.073	0.080	0.079	0.073	0.081	0.073	0.075	0.076	0.072
	5	0	0.80	0.451	0.388	0.367	0.458	0.388	0.378	0.451	0.339	0.249	0.469	0.401	0.380
	5	3	0.98	0.072	0.075	0.073	0.069	0.076	0.075	0.063	0.076	0.072	0.069	0.075	0.072
	5	3	0.80	0.460	0.378	0.365	0.452	0.379	0.364	0.419	0.324	0.237	0.457	0.382	0.369
	5	5	0.98	0.071	0.079	0.078	0.073	0.076	0.075	0.066	0.073	0.069	0.073	0.075	0.073
	5	5	0.80	0.466	0.393	0.381	0.463	0.391	0.380	0.424	0.327	0.247	0.461	0.379	0.361
[1,2,3]	0	0	0.98	0.073	0.074	0.072	0.074	0.077	0.076	0.070	0.075	0.074	0.073	0.074	0.070
	0	0	0.80	0.165	0.192	0.181	0.174	0.200	0.188	0.169	0.218	0.196	0.178	0.196	0.179
	0	3	0.98	0.075	0.076	0.076	0.073	0.072	0.071	0.072	0.075	0.074	0.075	0.074	0.074
	0	3	0.80	0.176	0.202	0.198	0.175	0.195	0.186	0.172	0.221	0.199	0.174	0.199	0.194
	0	5	0.98	0.072	0.076	0.075	0.070	0.077	0.076	0.073	0.074	0.070	0.074	0.074	0.071
	0	5	0.80	0.174	0.203	0.189	0.163	0.198	0.187	0.174	0.221	0.193	0.166	0.190	0.173
	3	0	0.98	0.073	0.075	0.076	0.071	0.078	0.075	0.069	0.073	0.070	0.073	0.079	0.077
	3	0	0.80	0.174	0.200	0.189	0.169	0.204	0.186	0.174	0.209	0.187	0.174	0.207	0.194
	3	3	0.98	0.068	0.077	0.079	0.073	0.073	0.074	0.074	0.078	0.076	0.075	0.078	0.075
	3	3	0.80	0.159	0.200	0.192	0.170	0.196	0.186	0.175	0.222	0.196	0.174	0.204	0.185
	3	5	0.98	0.075	0.076	0.075	0.072	0.075	0.077	0.074	0.073	0.071	0.071	0.074	0.071
	3	5	0.80	0.174	0.197	0.189	0.167	0.192	0.192	0.172	0.215	0.195	0.166	0.196	0.179
	5	0	0.98	0.074	0.071	0.076	0.074	0.074	0.076	0.074	0.075	0.072	0.073	0.077	0.075
	5	0	0.80	0.168	0.196	0.190	0.177	0.198	0.195	0.176	0.224	0.202	0.176	0.203	0.194
	5	3	0.98	0.072	0.079	0.073	0.073	0.076	0.076	0.072	0.070	0.068	0.072	0.072	0.070
	5	3	0.80	0.172	0.205	0.192	0.165	0.200	0.195	0.173	0.215	0.190	0.158	0.188	0.183
	5	5	0.98	0.077	0.073	0.074	0.072	0.078	0.076	0.071	0.077	0.071	0.071	0.073	0.073
	5	5	0.80	0.173	0.195	0.190	0.180	0.206	0.189	0.173	0.226	0.192	0.164	0.188	0.174

Table C6. The size-adjusted power performance of the unit root tests when T = 500 and j = 1.

				Model 1			Model 2			Model 3			Model 4
κ	α	β	ρ	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$
3	0	0	0.98	0.482	0.655	0.655	0.505	0.671	0.673	0.499	0.637	0.634	0.484	0.629	0.627
	0	0	0.80	0.963	0.968	0.949	0.964	0.969	0.951	0.994	0.995	0.989	0.964	0.965	0.944
	0	3	0.98	0.482	0.653	0.656	0.492	0.666	0.668	0.486	0.617	0.616	0.494	0.664	0.666
	0	3	0.80	0.962	0.966	0.948	0.964	0.970	0.952	0.993	0.994	0.987	0.962	0.968	0.948
	0	5	0.98	0.493	0.637	0.639	0.476	0.656	0.654	0.482	0.614	0.612	0.490	0.665	0.668
	0	5	0.80	0.965	0.966	0.948	0.961	0.966	0.948	0.994	0.994	0.988	0.963	0.968	0.950
	3	0	0.98	0.490	0.659	0.663	0.489	0.649	0.650	0.491	0.643	0.644	0.502	0.657	0.662
	3	0	0.80	0.963	0.965	0.947	0.964	0.968	0.948	0.993	0.995	0.988	0.963	0.967	0.949
	3	3	0.98	0.489	0.661	0.662	0.489	0.651	0.653	0.474	0.629	0.624	0.492	0.661	0.663
	3	3	0.80	0.963	0.968	0.949	0.964	0.967	0.947	0.993	0.993	0.986	0.965	0.969	0.952
	3	5	0.98	0.480	0.650	0.652	0.495	0.659	0.661	0.501	0.635	0.637	0.484	0.657	0.657
	3	5	0.80	0.963	0.967	0.950	0.964	0.968	0.947	0.993	0.994	0.987	0.963	0.967	0.949
	5	0	0.98	0.488	0.656	0.660	0.477	0.651	0.651	0.491	0.630	0.625	0.499	0.658	0.657
	5	0	0.80	0.961	0.965	0.945	0.963	0.967	0.949	0.993	0.995	0.987	0.963	0.968	0.949
	5	3	0.98	0.479	0.638	0.640	0.497	0.673	0.672	0.496	0.618	0.618	0.497	0.657	0.657
	5	3	0.80	0.961	0.965	0.946	0.962	0.970	0.951	0.994	0.993	0.986	0.965	0.969	0.949
	5	5	0.98	0.488	0.680	0.685	0.506	0.674	0.672	0.490	0.629	0.625	0.488	0.643	0.646
	5	5	0.80	0.964	0.970	0.953	0.965	0.969	0.949	0.993	0.994	0.987	0.964	0.967	0.950
[1,2,3]	0	0	0.98	0.221	0.240	0.242	0.229	0.260	0.263	0.223	0.238	0.235	0.224	0.239	0.243
	0	0	0.80	0.993	0.916	0.946	0.993	0.917	0.948	0.998	0.960	0.975	0.994	0.919	0.951
	0	3	0.98	0.209	0.241	0.243	0.228	0.245	0.246	0.227	0.249	0.237	0.227	0.233	0.229
	0	3	0.80	0.993	0.915	0.946	0.993	0.915	0.947	0.998	0.958	0.973	0.994	0.915	0.946
	0	5	0.98	0.233	0.248	0.252	0.229	0.242	0.246	0.224	0.245	0.240	0.230	0.241	0.242
	0	5	0.80	0.993	0.915	0.946	0.993	0.916	0.948	0.998	0.957	0.973	0.993	0.917	0.950
	3	0	0.98	0.231	0.246	0.251	0.224	0.237	0.238	0.228	0.241	0.235	0.215	0.234	0.236
	3	0	0.80	0.994	0.918	0.950	0.992	0.918	0.949	0.998	0.958	0.974	0.994	0.917	0.948
	3	3	0.98	0.220	0.231	0.227	0.222	0.239	0.240	0.220	0.240	0.241	0.213	0.234	0.234
	3	3	0.80	0.991	0.912	0.946	0.992	0.919	0.950	0.998	0.957	0.973	0.994	0.919	0.949
	3	5	0.98	0.223	0.254	0.250	0.221	0.241	0.245	0.223	0.237	0.234	0.233	0.264	0.266
	3	5	0.80	0.993	0.918	0.947	0.992	0.912	0.944	0.999	0.957	0.972	0.993	0.922	0.951
	5	0	0.98	0.225	0.253	0.254	0.223	0.241	0.241	0.225	0.232	0.229	0.219	0.229	0.229
	5	0	0.80	0.992	0.916	0.947	0.993	0.916	0.948	0.998	0.956	0.971	0.993	0.913	0.946
	5	3	0.98	0.217	0.240	0.240	0.219	0.247	0.252	0.222	0.243	0.241	0.225	0.241	0.240
	5	3	0.80	0.993	0.919	0.949	0.992	0.917	0.947	0.998	0.957	0.973	0.993	0.920	0.949
	5	5	0.98	0.220	0.236	0.235	0.228	0.245	0.249	0.226	0.244	0.240	0.230	0.246	0.249
	5	5	0.80	0.992	0.914	0.947	0.993	0.917	0.947	0.999	0.958	0.973	0.994	0.919	0.949

Table C7. The size-adjusted power performance of the unit root tests when T = 1000 and j = 1.

				Model 1			Model 2			Model 3			Model 4
κ	α	β	ρ	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$
3	0	0	0.98	0.830	0.979	0.981	0.826	0.978	0.979	0.842	0.978	0.978	0.833	0.982	0.983
	0	0	0.80	0.978	0.993	0.985	0.978	0.994	0.987	0.998	1.000	0.999	0.980	0.995	0.988
	0	3	0.98	0.824	0.978	0.979	0.839	0.980	0.980	0.840	0.973	0.974	0.826	0.979	0.980
	0	3	0.80	0.976	0.994	0.984	0.978	0.993	0.985	0.998	1.000	0.999	0.976	0.994	0.985
	0	5	0.98	0.823	0.977	0.979	0.825	0.980	0.981	0.842	0.975	0.975	0.834	0.981	0.982
	0	5	0.80	0.976	0.994	0.986	0.977	0.993	0.984	0.998	1.000	0.999	0.976	0.994	0.986
	3	0	0.98	0.828	0.981	0.982	0.825	0.978	0.979	0.833	0.975	0.975	0.823	0.980	0.981
	3	0	0.80	0.976	0.994	0.986	0.978	0.993	0.986	0.997	1.000	0.998	0.978	0.994	0.986
	3	3	0.98	0.831	0.978	0.980	0.823	0.978	0.979	0.839	0.973	0.975	0.830	0.983	0.983
	3	3	0.80	0.977	0.993	0.986	0.980	0.995	0.987	0.998	1.000	0.999	0.979	0.995	0.988
	3	5	0.98	0.839	0.982	0.983	0.827	0.982	0.983	0.835	0.973	0.975	0.823	0.976	0.977
	3	5	0.80	0.978	0.994	0.987	0.977	0.995	0.986	0.998	1.000	0.999	0.976	0.995	0.985
	5	0	0.98	0.831	0.977	0.978	0.834	0.979	0.980	0.838	0.973	0.974	0.835	0.978	0.978
	5	0	0.80	0.976	0.994	0.985	0.980	0.995	0.987	0.998	1.000	0.998	0.977	0.994	0.986
	5	3	0.98	0.819	0.975	0.976	0.821	0.978	0.980	0.843	0.976	0.976	0.830	0.979	0.981
	5	3	0.80	0.976	0.994	0.985	0.975	0.992	0.985	0.998	1.000	0.999	0.977	0.993	0.985
	5	5	0.98	0.826	0.981	0.982	0.820	0.978	0.979	0.842	0.974	0.974	0.823	0.978	0.978
	5	5	0.80	0.976	0.993	0.984	0.977	0.994	0.986	0.998	1.000	0.999	0.977	0.993	0.985
[1,2,3]	0	0	0.98	0.387	0.445	0.446	0.376	0.429	0.433	0.384	0.419	0.410	0.387	0.435	0.440
	0	0	0.80	0.999	0.957	0.971	1.000	0.954	0.970	1.000	0.988	0.992	1.000	0.956	0.972
	0	3	0.98	0.392	0.441	0.448	0.378	0.426	0.429	0.386	0.422	0.416	0.388	0.430	0.435
	0	3	0.80	1.000	0.960	0.973	0.999	0.952	0.968	1.000	0.989	0.992	0.999	0.954	0.968
	0	5	0.98	0.399	0.442	0.448	0.391	0.430	0.436	0.378	0.394	0.390	0.381	0.436	0.440
	0	5	0.80	1.000	0.959	0.972	1.000	0.956	0.970	1.000	0.987	0.992	1.000	0.954	0.970
	3	0	0.98	0.386	0.433	0.436	0.395	0.452	0.457	0.389	0.409	0.407	0.396	0.447	0.452
	3	0	0.80	0.999	0.956	0.971	1.000	0.956	0.970	1.000	0.987	0.991	1.000	0.954	0.969
	3	3	0.98	0.387	0.439	0.451	0.384	0.444	0.446	0.402	0.437	0.436	0.386	0.441	0.447
	3	3	0.80	1.000	0.955	0.970	0.999	0.955	0.969	1.000	0.989	0.992	0.999	0.952	0.968
	3	5	0.98	0.388	0.436	0.442	0.393	0.452	0.458	0.393	0.423	0.410	0.388	0.441	0.446
	3	5	0.80	1.000	0.954	0.969	1.000	0.960	0.973	1.000	0.989	0.992	1.000	0.954	0.970
	5	0	0.98	0.384	0.434	0.441	0.379	0.428	0.436	0.382	0.410	0.403	0.385	0.444	0.448
	5	0	0.80	1.000	0.954	0.971	1.000	0.953	0.970	1.000	0.988	0.993	0.999	0.955	0.970
	5	3	0.98	0.393	0.430	0.441	0.382	0.421	0.428	0.391	0.431	0.419	0.384	0.444	0.453
	5	3	0.80	1.000	0.951	0.969	0.999	0.951	0.969	1.000	0.988	0.992	0.999	0.955	0.970
	5	5	0.98	0.411	0.443	0.447	0.393	0.437	0.444	0.394	0.416	0.416	0.396	0.446	0.451
	5	5	0.80	1.000	0.955	0.971	1.000	0.956	0.971	1.000	0.987	0.992	1.000	0.956	0.970

Table C8. The size-adjusted power performance of the unit root tests when j = 1 and with fractional κ values.

					Model 1			Model 2			Model 3			Model 4
T	κ	α	β	ρ	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$	τ	$AD{F}_t$	$M{Z}_t$
50	1.3	0	0	0.98	0.075	0.076	0.074	0.075	0.079	0.077	0.071	0.072	0.070	0.073	0.075	0.073
		0	0	0.80	0.392	0.369	0.362	0.383	0.370	0.361	0.392	0.297	0.158	0.391	0.370	0.363
		0	3	0.98	0.079	0.074	0.073	0.074	0.073	0.072	0.075	0.074	0.070	0.075	0.076	0.076
		0	3	0.80	0.390	0.361	0.352	0.395	0.372	0.366	0.396	0.293	0.158	0.394	0.373	0.369
		3	0	0.98	0.079	0.076	0.073	0.077	0.077	0.076	0.075	0.071	0.064	0.070	0.074	0.071
		3	0	0.80	0.391	0.365	0.355	0.384	0.362	0.357	0.399	0.286	0.144	0.389	0.367	0.358
		3	3	0.98	0.072	0.075	0.072	0.070	0.072	0.069	0.071	0.070	0.068	0.074	0.077	0.074
		3	3	0.80	0.383	0.366	0.357	0.385	0.358	0.346	0.395	0.284	0.151	0.394	0.385	0.378
	1.7	0	0	0.98	0.067	0.072	0.072	0.064	0.068	0.067	0.073	0.072	0.063	0.067	0.074	0.073
		0	0	0.80	0.400	0.379	0.368	0.397	0.375	0.362	0.395	0.325	0.220	0.387	0.389	0.376
		0	3	0.98	0.070	0.073	0.071	0.066	0.068	0.066	0.068	0.068	0.061	0.068	0.074	0.072
		0	3	0.80	0.392	0.375	0.368	0.380	0.370	0.358	0.391	0.318	0.213	0.406	0.399	0.390
		3	0	0.98	0.070	0.075	0.073	0.071	0.076	0.076	0.073	0.070	0.066	0.071	0.073	0.070
		3	0	0.80	0.389	0.382	0.368	0.400	0.385	0.376	0.395	0.322	0.224	0.394	0.381	0.370
		3	3	0.98	0.067	0.070	0.069	0.071	0.078	0.075	0.068	0.069	0.066	0.072	0.073	0.070
		3	3	0.80	0.392	0.381	0.371	0.411	0.389	0.377	0.396	0.329	0.224	0.398	0.374	0.364
1000	1.3	0	0	0.98	0.830	0.979	0.981	0.826	0.978	0.979	0.842	0.978	0.978	0.833	0.982	0.983
		0	0	0.80	0.978	0.993	0.985	0.978	0.994	0.987	0.998	1.000	0.999	0.980	0.995	0.988
		0	3	0.98	0.824	0.978	0.979	0.839	0.980	0.980	0.840	0.973	0.974	0.826	0.979	0.980
		0	3	0.80	0.976	0.994	0.984	0.978	0.993	0.985	0.998	1.000	0.999	0.976	0.994	0.985
		3	0	0.98	0.828	0.981	0.982	0.825	0.978	0.979	0.833	0.975	0.975	0.823	0.980	0.981
		3	0	0.80	0.976	0.994	0.986	0.978	0.993	0.986	0.997	1.000	0.998	0.978	0.994	0.986
		3	3	0.98	0.831	0.978	0.980	0.823	0.978	0.979	0.839	0.973	0.975	0.830	0.983	0.983
		3	3	0.80	0.977	0.993	0.986	0.980	0.995	0.987	0.998	1.000	0.999	0.979	0.995	0.988
	1.7	0	0	0.98	0.387	0.445	0.446	0.376	0.429	0.433	0.384	0.419	0.410	0.387	0.435	0.440
		0	0	0.80	0.999	0.957	0.971	1.000	0.954	0.970	1.000	0.988	0.992	1.000	0.956	0.972
		0	3	0.98	0.392	0.441	0.448	0.378	0.426	0.429	0.386	0.422	0.416	0.388	0.430	0.435
		0	3	0.80	1.000	0.960	0.973	0.999	0.952	0.968	1.000	0.989	0.992	0.999	0.954	0.968
		3	0	0.98	0.386	0.433	0.436	0.395	0.452	0.457	0.389	0.409	0.407	0.396	0.447	0.452
		3	0	0.80	0.999	0.956	0.971	1.000	0.956	0.970	1.000	0.987	0.991	1.000	0.954	0.969
		3	3	0.98	0.387	0.439	0.451	0.384	0.444	0.446	0.402	0.437	0.436	0.386	0.441	0.447
		3	3	0.80	1.000	0.955	0.970	0.999	0.955	0.969	1.000	0.989	0.992	0.999	0.952	0.968

Table C9. The list of descriptions for the variables utilized in the empirical analysis.

Series	Description
TB3M-FFR	3 Month Treasury Bill – Federal Funds Rate
TB6M-FFR	6 Month Treasury Bill – Federal Funds Rate
GS10Y-FFR	10 Year Treasury Constant Maturity Rate – Federal Funds Rate
GS5Y-FFR	5 Year Treasury Constant Maturity Rate – Federal Funds Rate
GS3Y-FFR	3 Year Treasury Constant Maturity Rate – Federal Funds Rate
GS1Y-FFR	1 Year Treasury Constant Maturity Rate – Federal Funds Rate
GS10Y-GS1Y	10 Year Treasury Constant Maturity Rate – 1 Year Treasury Constant Maturity Rate
GS10Y-GS3Y	10 Year Treasury Constant Maturity Rate – 3 Year Treasury Constant Maturity Rate
GS10Y-GS5Y	5 Year Treasury Constant Maturity Rate – 5 Year Treasury Constant Maturity Rate
GS10Y TB3M	10 Year Treasury Constant Maturity Rate – 3 Month Treasury Bill
GS10Y-TB6M	10 Year Treasury Constant Maturity Rate – 6 Month Treasury Bill
GS5Y-GS1Y	5 Year Treasury Constant Maturity Rate – 1 Year Treasury Constant Maturity Rate
GS5Y-GS3Y	5 Year Treasury Constant Maturity Rate – 3 Year Treasury Constant Maturity Rate
GS5Y-TB3M	5 Year Treasury Constant Maturity Rate – 3 Month Treasury Bill
GS5Y-TB6M	5 Year Treasury Constant Maturity Rate – 6 Month Treasury Bill
GS3Y-GS1Y	3 Year Treasury Constant Maturity Rate – 1 Year Treasury Constant Maturity Rate
GS3Y-TB3M	3 Year Treasury Constant Maturity Rate – 3 Month Treasury Bill
GS3Y-TB6M	3 Year Treasury Constant Maturity Rate – 6 Month Treasury Bill

Table C10. Results of the unit root tests with standard and flexible Fourier form approaches.

	τ		${\tau }^{\ast }$		$M{Z}_t$		$M{Z}_t^{\ast }$		$AD{F}_t$		$AD{F}_t^{\ast }$
	Test statistic	p-value	Test statistic	p-value	Test statistic	p-value	Test statistic	p-value	Test statistic	p-value	Test statistic	p-value
TB3M-FFR	2.5888	.0013	1.7090	.0462	−5.3612	.0505	−2.9085	.0071	−69.1534	0.0145	−18.8169	.0056
TB6M-FFR	2.5271	.0084	1.6530	.0796	−4.9458	.1034	−3.2724	.0024	−58.2789	.0521	−22.8172	0.0018
GS10Y-FFR	2.3543	.1246	2.1181	.0000	−4.3407	.2014	−3.8599	.0002	−40.2135	.1967	−31.4606	.0002
GS5Y-FFR	2.4189	.0604	2.2393	.0000	−4.6407	.1520	−4.2584	.0000	−46.5320	.1359	−38.8099	.0000
GS3Y-FFR	2.4802	.0228	2.2317	.0000	−4.9249	.1067	−4.4260	.0000	−53.3663	.0811	−42.3891	.0000
GS1Y-FFR	2.5419	.0057	1.8634	.0053	−4.8723	.1142	−3.7770	.0002	−56.1854	.0641	−31.3041	.0002
GS10Y-GS1Y	2.1710	.3405	1.6932	.0547	−3.4299	.3376	−3.6867	.0004	−25.8854	.3556	−25.9590	.0005
GS10Y-GS3Y	2.1372	.3856	1.6099	.1191	−3.2207	.3632	−2.5831	.0197	−22.8586	.3881	−13.4790	.0212
GS10Y-GS5Y	2.1485	.3705	1.5724	.1631	−3.6951	.3018	−2.2178	.0476	−30.2136	0.3110	−10.0575	0.0510
GS10Y-TB3M	2.2740	.2213	1.8410	.0080	−4.7326	.1352	−4.1416	.0002	−47.5124	.1278	−33.5579	.0002
GS10Y-TB6M	2.2047	.3002	1.6937	.0544	−3.6923	.3025	−3.6810	.0004	−29.7461	.3147	−26.3779	0.0005
GS5Y-GS1Y	2.2178	.2850	1.8146	.0121	−3.6848	.3033	−4.5117	.0000	−29.6510	.3156	−37.9814	.0000
GS5Y-GS3Y	2.1694	.3418	1.7152	.0429	−3.5357	.3230	−2.7511	.0119	−26.8969	.3442	−15.6376	.0124
GS5Y-TB3M	2.3717	.1053	2.0347	.0001	−4.6939	.1416	−4.2931	.0000	−48.0973	.1234	−38.8139	.0000
GS5Y-TB6M	2.2801	.2152	1.8102	.0127	−4.1428	.2310	−4.0856	.0002	−37.3722	.2300	−33.4640	.0002
GS3Y-GS1Y	2.2816	.2131	1.9083	.0030	−3.9428	.2669	−4.9108	.0000	−34.5769	.2608	−45.5525	.0000
GS3Y-TB3M	2.4922	.0189	2.1521	.0000	−5.6536	.0246	−5.1228	.0000	−70.0850	.0129	−54.7863	0.0000
GS3Y-TB6M	2.3844	.0915	1.8669	.0049	−4.6488	.1504	−4.0739	.0002	−48.1427	.1230	−34.9391	.0001

Disclosure statement

No potential conflict of interest was reported by the author(s).