- Email: [email protected]
Quantile nonlinear unit root test with covariates and an application to the PPP hypothesis
S0264-9993(19)30722-9
DOI:
https://doi.org/10.1016/j.econmod.2020.01.021
Reference:
ECMODE 5137
To appear in:
Economic Modelling
Received Date: 16 May 2019 Revised Date:
27 August 2019
Accepted Date: 25 January 2020
Please cite this article as: Yang, Y., Zhao, Z., Quantile nonlinear unit root test with covariates and an application to the PPP hypothesis, Economic Modelling (2020), doi: https://doi.org/10.1016/ j.econmod.2020.01.021. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
Quantile Nonlinear Unit Root Test with Covariates and An Application to the PPP Hypothesis Yang Yanga and Zhao Zhao∗b a
School of Finance, Zhongnan University of Economics and Law b School of Economics, Huazhong University of Science and Technology
Abstract In this article, we employ the methods initiated by Hansen (1995) to develop new quantile nonlinear unit root tests with covariates. The limiting distributions of our proposed tests are derived, which are dependent on nuisance parameter reflecting the correlation between the equation error and the covariates. To deal with this inferential difficulty, two alternative procedures based on either consistent estimate of the nuisance parameter or bootstrap implementation of the test are proposed. Monte Carlo simulations show that the proposed tests perform very well in finite samples and large power gains can be achieved by including correlated covariates in the testing equation. The proposed tests are applied to the PPP hypothesis. The empirical results indicate that the real exchange rates are not constant unit root processes.
Keywords: Unit root; Quantile; Covariates; PPP JEL classification: C12; F31
∗ Corresponding
author. E-mail address: [email protected]
1
1
Introduction
Unit root test has been and continues to be a testing problem of great theoretical significance in time series econometrics. Testing the existence of unit roots in economic time series has important implications for the validity of economic theory and the implementation of economic policies (e.g., Nelson and Plosser, 1982). The augmented Dickey-Fuller (ADF) (Dickey and Fuller, 1979, 1981) and Phillips and Perron (PP) (1988) tests are the most widely used unit root tests in empirical studies. However, a lot of literature has documented that the standard linear ADF and PP unit root tests suffer from a large power loss if the true data-generating process involves nonlinear adjustment dynamics (e.g., Michael et al., 1997; Taylor et al., 2001). In fact, the presence of transaction costs or arbitrage boundaries can create nonlinear dynamics in economic variables. In response, more recent studies have developed unit root testing procedures that have power against an alternative of nonlinear stationary process. For example, Caner and Hansen (2001), Bec et al. (2004, 2008), and Kapetanios and Shin (2006) developed nonlinear unit root tests based on threshold autoregressive (TAR) models. On the other hand, Kapetanios et al. (2003) and Kilic (2011) proposed nonlinear unit root tests based on exponential smooth transition autoregressive (ESTAR) models. All of the above unit root tests rely on ordinary least squares (OLS) estimators, which are designed to provide optimal performance under Gaussian conditions. However, it is well known that many financial time series like exchange rates are heavy-tailed (Falk and Wang, 2003). In the absence of Gaussian conditions, traditional tests based on the OLS method can exhibit rather poor power performance (Xiao, 2012). Koenker and Xiao (2004) proposed a new unit root test based on quantile autoregression approach, which is robust to departures from Gaussian conditions. Monte Carlo simulations show their proposed test has good power under non-Gaussian conditions and sacrifices little efficiency in the Gaussian model. Li and Park (2018) extended the approach of Koenker and Xiao (2004) by developing a quantile nonlinear unit root test (hereinafter referred to as the QKSS test) based on a quantile ESTAR model. The QKSS test is robust to non-normal innovation distribution and allows for different and potentially asymmetric adjustment speed at different quantiles. A large body of literature has employed the QKSS test for empirical research (e.g., Bahmani-Oskooee et al., 2018). Although useful, the QKSS test is performed in a univariate framework, ignoring information in related time series. In a seminal work, Hansen (1995) found that including correlated stationary covariates in the unit root testing regression can lead to a more accurate estimate of the autoregressive parameter, and as a result, to enormous power gains. Along this route, Elliott and Jansson (2003), Jansson (2004), Galvao (2009), Tsong (2011) and Westerlund (2015)also applied this technique for developing unit root tests that have more power than their univariate counterparts. Motivated by these studies, this article aims to generalize the QKSS test by introducing stationary covariates into the quantile ESTAR model. 2
Compared with the existing literature, the contributions of this paper are mainly reflected in the following aspects. (1) We employ the methods initiated by Hansen (1995) to develop new quantile nonlinear unit root tests with covariates. The limiting distributions of our proposed tests are derived, which are dependent on nuisance parameter reflecting the correlation between the equation error and the covariates. To deal with this inferential difficulty, two alternative procedures based on either consistent estimate of the nuisance parameter or bootstrap implementation of the test are proposed. (2) This article conducts a series of Monte Carlo simulations, which show that the proposed tests perform very well in finite samples and large power gains can be achieved by including correlated covariates in the testing equation. (3) We apply the new proposed quantile nonlinear unit root tests with covariates to investigate the purchasing power parity (PPP) hypothesis. Compared with previously proposed tests, our proposed tests provide more evidence to support PPP, indicating that the real exchange rates (RERs) are not constant unit root processes. The rest of the paper is organized as follows. In Section 2, we introduce the model and propose the tests. The asymptotic null distributions of the proposed tests are derived. Results of the Monte Carlo simulations are reported in Section 3. In Section 4, we apply our proposed tests to investigate the PPP hypothesis. Section 5 concludes. Proofs are collected in an Appendix.
2 2.1
Quantile Unit Root Test with Covariates in the Nonlinear Framework Model and Assumptions
Following Kapetanios et al. (2003), we consider the univariate time series yt , given by p X 2 ∆yt = γyt−1 1 − exp −θyt−1 + ak ∆yt−k + ut , (1) k=1
2 where t = 1, 2, · · · , T . In this setting, as yt−1 → ±∞, 1 − exp −θyt−1 → 1, 2 and as yt−1 → 0, 1 − exp −θyt−1 → 0, provided that θ > 0. Consequently, the process in (1) has two regimes (i.e., a middle regime when yt−1 is close to zero and an outer regime when |yt−1 | becomes larger) and yt follows a unit root process in the middle regime. We let the error process ut to be related with other stationary covariates. More specifically, ut is defined as ut = B(L)0 xt + et ,
(2)
where xP t is an m-dimentional stationary covariates, L is the lag operator, q1 B(L) = k=−q bk Lk and bk is an m × 1 vector. 2 Combining (1) and (2), we obtain p q1 X X 2 ∆yt = γyt−1 1 − exp −θyt−1 + ak ∆yt−k + b0k xt−k + et . k=1
3
k=−q2
(3)
Denote the information set up to time t as =t−1 = {es , s < t, xt−q1 , · · · , xt+q2 }. Let Qe (τ ) denote the τ -th quantile of et and Q∆yt (τ |=t−1 ) denote the τ -th conditional quantile of ∆yt conditional on =t−1 , then 2 Q∆yt (τ |=t−1 ) = γyt−1 1 − exp −θyt−1 +
p X
ak ∆yt−k +
k=1
q1 X
b0k xt−k + Qe (τ ).
(4)
k=−q2
Remark 1. Li and Park (2018) developed the QKSS test without incorporating any covariates. However, it is rare that the time series yt is observed in isolation. More typically, we observe at least one related time series, say xt . It may be expected that including the related covariates xt will provide useful information to increase the power of the QKSS test. Therefore, we develop new quantile nonlinear unit root tests with covariates based on model (4). The unit root null hypothesis is H0 : θ = 0, and the alternative hypothesis is H1 : θ > 0. Since γ is not identified under the unit root null hypothesis, it is infeasible to test H0 directly (Davies, 1977). To overcome the unidentified parameter problem, a first-order Taylor series expansion of model (4) is used as the following auxiliary regression (see, e.g., Kapetanios et al., 2003; Li and Park, 2018) 3 Q∆yt (τ |=t−1 ) = δyt−1 +
p X
ak ∆yt−k +
k=1
q1 X
b0k xt−k + Qe (τ ).
(5)
k=−q2
For equation (5), the unit root null hypothesis (θ = 0) is equivalent to H0 : 3 δ = 0. Define a0 (τ ) = Qe (τ ), zt = (1, yt−1 , ∆yt−1 , · · · , ∆yt−p , x0t−q1 , · · · , x0t+q2 )0 0 0 and β(τ ) = (a0 (τ ), δ, a1 , · · · , ap , bq1 , · · · , b−q2 )0 . Then we have Q∆yt (τ |=t−1 ) = zt0 β(τ ).
(6)
Estimation of the quantile regression model can be obtained from the following minimization problem ˆ ) = arg min β(τ β
T X
ρτ (∆yt − zt0 β(τ )),
(7)
t=1
where ρτ (u) = u(τ − I(u < 0)) is the check function (Koenker and Bassett, 1978). To derive the asymptotic theory, we impose the following assumptions. Assumption 1 For some ζ > r > 2, A1. xt is covariance stationary and strong mixing with mixing coefficients αm , P∞ 1/r−1/ζ which satisfy m=1 αm < ∞; ζ A2. supt E |xt | < ∞; 4
A3. E [xt−k et ] = 0 for −q2 ≤ k ≤ q1 ; A4. et is an i.i.d. random variable, with mean 0 and variance σ 2
0, where φt = (∆yt−1 , · · · , ∆yt−p , xt−q1 , · · · , xt+q2 ) ; A7. The distribution function of {et }, F , has a continuous Lebesgue density, f , with 0 < f (u) < ∞ on {u : 0 < F (u) < 1}. Remark 2. Assumptions A1 and A2 are conventional weak dependence and moment restrictions, which are standard requirements for Functional Central Limit Theorem. Assumptions A3 and A4 state that the regression error et is orthogonal to the leads and lags of the covariates xt and the lagged differences of yt . These assumptions are imposed to obtain consistent parameter estimates. Assumption A5 implies yt has one unit root at most. Assumption A6 ensures that there is no collinearity among stationary variables in zt . Assumption A7 is a standard requirement to permit the development of an asymptotic theory for quantile regression.
2.2
Quantile Nonlinear Unit Root Test with Covariates
The quantile nonlinear unit root test with covariates is defined as −1 (τ )) 1/2 f (Fd 0 ˆ ), Y−1 Mz Y−1 δ(τ t(τ ) = p τ (1 − τ )
(8)
−1 (τ )) is a consistent estimator of f (F −1 (τ )), Y where f (Fd −1 is a vector of 3 lagged dependent variable (yt−1 ) and Mz is the projection matrix onto the space orthogonal to z = (1, ∆yt−1 , · · · , ∆yt−p , x0t−q1 , ..., x0t+q2 ). Denoting ψτ (u) = τ − I(u < 0) and etτ = ∆yt − zt0 β(τ ), then we have E[ψτ (etτ )|=t−1 ] = 0. The equation (2) can be rewritten as qX 1 −1 0 ˜b0 ∆xt−k , ut = et + B (1) xt + k
(9)
k=−q2
Pk Pq1 where ˜bk = i=−q2 bi if −q2 ≤ k < 0, and ˜bk = − i=k+1 bi if 0 ≤ k < q1 . Using the results in Herrndorf (1984) and Hansen (1992), it can be shown 0 that the partial sums of the vector process (et + B (1) xt , ψτ (etτ )) follow a bivariate invariance principle. Using the results in Phillips and Solo (1992) and PbT rc Pq1 −1 ˜0 p Xiao (2014), we have T −1/2 t=1 { k=−q b ∆xt−k } − → 0. Consequently, we 2 k can show that bT rc
T −1/2
X
(ut , ψτ (etτ ))0 ⇒ (Bu (r), Bψτ (r))0 = BM (0, Σ(τ )),
t=1
where
Σ(τ ) =
σu2 σuψ (τ ) σuψ (τ ) σψ2 (τ ) 5
(10)
is the long run covariance matrix of the bivariate Brownian motion and can be written as Σ0P (τ ) + Σ1 (τ ) + Σ01 (τ ), where Σ0 (τ ) = E[(ut , ψτ (etτ ))0 (ut , ψτ (etτ ))] ∞ and Σ1 (τ ) = s=2 E[(u1 , ψτ (e1τ ))0 (us , ψτ (esτ ))]. Using the above results, we can establish the limiting distribution of t(τ ) in the following theorem. Theorem 1 Under the unit root null hypothesis and Assumptions A1-A7, R1 R1 W dW2 W 1 dW1 p 0 2 + 1 − λ q0 R 1 , (11) t (τ ) ⇒ ξ (τ ) = λ qR 1 1 2 2 W W dr dr 1 1 0 0 R1 where W 1 = W13 − 0 W13 dr, W1 and W2 are the standard Brownian motions and independent of one another, and λ = λ (τ ) =
σuψ (τ ) σuψ (τ ) = p . σu σψ (τ ) σu τ (1 − τ )
(12)
Remark 3. The limiting distribution of t (τ ) has a similar pattern as that of the QKSS test. However, the weight λ in (11) obviously contain information from related covariates xt . The t (τ ) test is a generalization of the QKSS test and, in fact, it has the same asymptotic distribution of the QKSS test when there is no information in the covariates (i.e., B(L) = 0 in equation (2), and ut = et ). Given a consistent estimate of λ, the limiting distribution of t (τ ) can be approximated by a direct simulation, which can be implemented using the following steps: (1) Generate two random standard normal vectors, v1t and v2t , for t = 1, 2, · · · , T , and generate the random walk yt = yt−1 + v1t , with y0 = 0. (2)Approximate t (τ ) by PT PT 3 3 yt−1 − y¯ v1t p yt−1 − y¯ v2t t=2 t=2 2 tˆ(τ ) =λ qP 2 + 1 − λ qPT 2 , T 3 3 y y − y ¯ − y ¯ t−1 t−1 t=2 t=2 PT where y¯ = T −1 t=1 yt3 . (3) Repeat steps 1 to 2 many times and compute the critical values from the corresponding quantiles of tˆ(τ ). Following Hansen (1995), to obtain the asymptotic critical values of t (τ ), we set the sample size T = 1000 and the number of repetition Nc = 60000. The asymptotic critical values of t (τ ) are presented in Table 1 for values of λ2 in steps of 0.1. The critical values for other values of λ2 can be approximated by interpolation. [Table 1 is about here] In order to implement the t (τ ) test and to select the appropriate critical values from Table 1, it is necessary to obtain consistent estimate of the nuisance 6
parameter λ. Therefore, we need to obtain consistent estimates of the variance and covariance parameters σu2 and σuψ according to (12). Following Koenker and Xiao (2004), we use the kernel method to estimate σu2 and σuψ σ ˆu2 =
M X
k
j=−M
j Cuu (j) , M
σ ˆuψ =
M X
k
j=−M
j Cuψ (j) , M
where k(·) is a kernel function, M is the bandwidth, Cuu (j) = T −1 PT and Cuψ (j) = T −1 t=1 u ˆt ψτ (ˆ et+j,τ ).
2.3
PT
t=1
u ˆt u ˆt+j
Bootstrap Implementation of the Test
Theorem 1 shows that the limiting distribution of t(τ ) depends on nuisance parameter reflecting the correlation between the equation error and the covariates. In Section 2.2, we propose testing procedure based on consistent estimate of the nuisance parameter. However, relying on the estimated value of the nuisance parameter would introduce additional source of variability (Chang et al., 2017). In this section, we apply the bootstrap method to the t(τ ) test to deal with the nuisance parameter dependency. Bootstrap methods are widely employed in the unit root testing literature (see, e.g., Park, 2003). The bootstrap implementation of the t(τ ) test is given as follows. Step 1. Let wt = ∆yt , then fit the regression wt =
p X
a ˆk wt−k +
k=1
q1 X
ˆb0 xt−k + eˆt , k
(13)
k=−q2
and obtain estimates a ˆ1 , · · · , a ˆp , ˆb−q2 , · · · , ˆbq1 and the residuals eˆt . Step 2. Fit the lth order autoregression of xt as ˆ 1 xt−1 + · · · + Φ ˆ l xt−l + ηˆt , xt = Φ
(14)
ˆ 1, · · · , Φ ˆ l and the residuals ηˆt . and obtain estimates Φ Step 3. Generate the (1+m)-dimensional bootstrap samples ςt∗ = (e∗t, ηt∗0 )0 by T P resampling from the centered fitted residual vectors ςˆt − T −1 ςˆt , where t=1
ςˆt = (ˆ et , ηˆt0 )0 , and eˆt and ηˆt are the fitted residuals from (13) and (14). Step 4. Generate x∗t from ηt∗ using the fitted autoregression ˆ 1 x∗t−1 + · · · + Φ ˆ l x∗t−l + ηt∗ , x∗t = Φ and then generate wt∗ from e∗t and x∗t using the fitted regression wt∗ =
p X
∗ a ˆk wt−k +
k=1
q1 X k=−q2
7
ˆb0 x∗ + e∗ . k t−k t
∗ Step 5. Generate yt∗ under the unit root null hypothesis yt∗ = yt−1 + wt∗ , and ∗ then compute the bootstrap unit root test statistic t (τ ) using the bootstrap sample yt∗ . Step 6. Repeat steps 3-5 NB times,obtaining NB bootstrap test statistics, deN B P NB . Calculate the bootstrap p-value as NB−1 noted as {t∗ (τ )i }i=1 I (t∗ (τ )i < t(τ )), i=1
where I(.) is the indicator function. It is worth noting that the unit root null must be imposed when generating the bootstrap sample yt∗ in step 5. This is because imposing the unit root null ensures that yt∗ behaves like a unit root process, and therefore causes the bootstrap implementation valid. The asymptotic validity of the bootstrap procedure can be proved using a bootstrap invariance principle which has been established in Park (2002). Remark 4. Li and Park (2018) did not consider the bootstrap implementation of the QKSS test. However, since the t(τ ) test is a generalization of the QKSS test, the bootstrap procedure presented above can be easily extended to a bootstrap QKSS test by ignoring the covariates xt .
2.4
Inference Over a Range of Quantiles
In addition to the t-ratio statistic t(τ ), we may also investigate the unit root property of yt over a range of quantiles τ ∈ Λ, instead of focusing on only a selected quantile. In this section we propose a Kolmogorov-Smirnov (KS) type test and a Cramer-von Mises (CM) type test based on quantile nonliear model for τ ∈ Λ = [τ0 , 1 − τ0 ] for some 0 < τ0 < 1/2. The test statistics for the unit root null hypothesis are defined as follows Z 2 tks = sup |t (τ )| , tcm = t (τ ) dτ. (15) τ ∈Λ τ ∈Λ n−1
In practice, we can calculate t(τ ) at {τi = i/n}i=1 , and then the tks test can be constructed by taking the maximum value over τ ∈ Λ and the tcm test can be computed using numerical integration. The limiting distributions of tks and tcm are given in the following theorem. Theorem 2 Under the unit root null hypothesis and Assumptions A1-A7, Z 2 tks ⇒ sup |ξ (τ )| , tcm ⇒ ξ (τ ) dτ. (16) τ ∈Λ τ ∈Λ
The critical values of tks and tcm can be obtained by the bootstrap implementation similar to that in Section 2.3.
8
3
Monte Carlo Simulations
In this section, we conduct Monte Carlo simulations to investigate the finite sample properties of our proposed tests. The data is generated from the following process 2 ∆yt = γyt−1 1 − exp −θyt−1 + ut , (17) ut 1 σ12 where ∼ i.i.d. (0, Σ) with Σ = . xt σ12 1 Following Galvao (2009), three different distributions for (ut , xt )0 are considered, say standard normal (N (0, 1)), student-t distribution with 2 degrees of freedom (t2 ), and student-t distribution with 3 degrees of freedom (t3 ). We let σ12 vary among {0.3, 0.5, 0.7}. We report the simulation results for the four proposed tests: (1) the t(τ ) test using the critical values in Table 1 at τ = 0.5; (2) the t∗ (τ ) test using the bootstrap procedure in Section 2.3 at τ = 0.5; (3) the tks test with Λ = [0.1, 0.9]; (4) the tcm test with Λ = [0.1, 0.9]. For comparison, we also consider the ADF test proposed by Dickey and Fuller (1979, 1981), the KSS test proposed by Kapetanios et al. (2003), the CKSS test proposed by Tsong (2011), the quantile ADF (QADF) test proposed by Koenker and Xiao (2004), the QKSS test, and the QKS and QCM tests proposed by Li and Park (2018). The QKS and QCM tests are used to examine the unit root hypothesis over a range of quantiles. The number of Monte Carlo replications is 1000 and the number of bootstrap replications is 399. The lag orders for the unit root tests are selected by using the Bayesian information criterion (BIC). For estimation of long-run variance and covariance parameters (σu2 and σuψ ), we use the Quadratic Spectral windows in the kernel estimators. Sample size of T = 100 is examined for 5% significance level1 . We first consider the size performance of the tests and set θ = 0. The size results are reported in Table 2. As can be seen from Table 2, our proposed t(τ ), t∗ (τ ), tks and tcm tests perform well in practice, and their finite sample sizes are close to the nominal 5% level. [Table 2 is about here] Now, we address the finite sample power performance of the tests. We let γ vary among {−0.1, −0.3} and θ vary among {0.01, 0.1}. The power results are reported in Tables 3-5. Compared with the QKSS, QKS and QCM tests, the improvement of power performance of our proposed tests are significant when the covariates are highly correlated with the error (i.e., σ12 is large). For example, when the innovation has N (0, 1) distribution, σ12 = 0.7, γ = −0.3 and θ = 0.1, the power of QKSS, QKS and QCM are 54.1%, 43.7% and 67.8%, respectively. In contrast, the corresponding power of t(τ ), t∗ (τ ), tks and tcm are 86.2%, 82.2%, 85.6% and 94.9%, respectively. Thus, large power gains can be achieved by including correlated covariates in the testing equation. Results in these tables show that our proposed tests are more powerful than the CKSS test 1 The
main results remain unchanged for other values of T
9
in the presence of heavy-tailed disturbances. For example, when the innovation has t2 distribution, σ12 = 0.3, γ = −0.1, and θ = 0.01, the power of CKSS is 21.3%. By contrast, the corresponding power of t(τ ) and t∗ (τ ) are 50.3% and 53.9%, respectively. [Tables 3-5 are about here] The simulation results show that when the correlation between the covariates and the error is low, the power gains of our proposed tests are not significant compared to the QADF and QKSS tests. This result is consistent with that in Hansen (1995). Thus, the choice of covariates is important in empirical studies. In the related literature, economic theory is often used as a guide to select the potential covariates. For example, according to the Fisher hypothesis, inflation is expected to be related with nominal interest rate. Therefore, nominal interest rate can be selected as covariate when testing the unit root hypothesis of inflation. If there are many potential covariates, one strategy is to choose the one which gives us the smallest ρ2 since this covariate provides the most powerful test2 (see, Chang et al., 2017), and another strategy is to use the factor analysis to abstract the useful information among several potential covariates (Lee and Tsong, 2011).
4
Testing the PPP Hypothesis
Testing for PPP has been a major focus of empirical international finance. A necessary condition for validating PPP is that the RER should be a mean reverting process. Despite the theoretical appeal of PPP and the large amount of literature trying to reveal evidence of RER stationarity, studies on the PPP hypothesis remain inconclusive (see, e.g., Taylor and Taylor, 2004; Imbs et al., 2005; Lothian and Taylor, 2008; Zhou and Kutan, 2011). The widespread failure to find support for PPP in the literature using conventional unit root tests may be caused by nonlinear adjustment speed of RER or non-normal innovations. Moreover, including suitable variables related to the RER in the testing equation may improve the power of the unit root test. In this section, we apply the proposed tests to investigate the PPP hypothesis. Specifically, the aim is to test the stationarity property of the RER (qt ) given by qt = st + p∗t − pt ,
(18)
where st is the nominal exchange rate (domestic currency per dollar), p∗t and pt are the foreign and domestic price levels, respectively. All variables are in their logarithmic form. In this paper, we consider testing whether the RER has a unit root for the Eurozone, Japan, the United Kingdom (UK) and Korea. The data employed in this article are monthly data for the period from January of 2000 to December 2 ρ2
2 / σ 2 σ 2 . The smaller the value of ρ2 , the larger the relative contribution of x = σue t u e
to ut .
10
of 2018. The nominal exchange rate st is the end of the monthly value of the domestic currency per dollar. p∗t and pt are the monthly Consumer Price Index (CPI) of the relevant countries. All the data are obtained from Federal Reserve Bank of St. Louis database. The RERs are plotted in Figure 1. [Figure 1 is about here] As a preliminary exercise we compute the descriptive statistics and normality tests for these RERs. The results are reported in Table 6. The formal JarqueBera test rejects normality in the RERs of Eurozone, Japan and Korea at the 1% significance level, supporting the use of quantile regression. [Table 6 is about here] As discussed in Elliott and Pesavento (2006), the choice of covariates is limited only to being sure that they are stationary and correlated with the shocks to the RER. Following Galvao (2009) and Tsong (2011), we use the first difference of the nominal exchange rate as covariate3 . The results of the ADF, KSS, CKSS, QKS, QCM, tks and tcm tests are reported in Table 7, and the results of the QADF, QKSS, t(τ ) and t∗ (τ ) tests are reported in Table 8. [Tables 7-8 are about here] Compared with previously proposed tests, our proposed tests provide more evidence to support PPP. For example, all the previously proposed tests can not reject the unit root null hypothesis for the RER of UK; in contrast, at the 10% significance level the tks and tcm tests can reject the unit root null hypothesis for the RER of UK, and t(τ ) (t∗ (τ )) can reject the unit root null hypothesis for the RER of UK when 0.4 ≤ τ ≤ 0.8 (0.4 ≤ τ ≤ 0.9). Therefore, the empirical results provide strong evidence that the RERs are not constant unit root processes and indicate the superiority of our proposed test. It is interesting to note that, the t(τ ) and t∗ (τ ) tests can reject the unit root null hypothesis for the RER of Korea in the lower quantiles, but can not reject the unit root null for the RER of Korea in the upper quantiles, implying that there is asymmetry in the adjustment speed. Such asymmetries may arise as a result of central bank intervention policies. Since depreciations would have favourable effect on the trade balance, the central bank might want to defend an appreciation of the currency more rigorously than a depreciation, therefore inducing asymmetric adjustment speed.
5
Conclusion
In this article, we employ the methods initiated by Hansen (1995) to develop new quantile nonlinear unit root tests with covariates. The limiting distributions 3 The ADF and KPSS unit root tests confirm that the first difference of the nominal exchange rate is stationary variable.
11
of our proposed tests are derived, which are dependent on nuisance parameter reflecting the correlation between the equation error and the covariates. To deal with this inferential difficulty, two alternative procedures based on either consistent estimate of the nuisance parameter or bootstrap implementation of the test are proposed. We conduct a series of Monte Carlo simulations, which show that the proposed tests perform very well in finite samples and large power gains can be achieved by including correlated covariates in the testing equation. Furthermore, in the non-Gaussian heavy tailed distribution case, our proposed tests are more powerful than the conventional OLS based unit root tests. We apply the new proposed quantile nonlinear unit root tests with covariates to investigate the PPP hypothesis for the Eurozone, Japan, UK and Korea. Compared with previously proposed tests, our proposed tests provide more evidence to support PPP, indicating that the RERs are not constant unit root processes.
Acknowledgments We would like to thank the editors and the two anonymous reviewers for their valuable comments and suggestions. Yang acknowledges research support from the National Natural Science Foundation of China under Grant No. 71903200 and the Fundamental Research Funds for the Central Universities. Zhao acknowledges research support from the National Natural Science Foundation of China under Grant No. 71803055.
Appendix Proof of Theorem 1 The main ideas of the proof are similar to those of Koenker and Xiao (2004) and Li and Park (2018), so we only outline the proofs here. More details can be found in these articles. In order to derive the limiting distribution of t(τ ), we need to derive the ˆ ). Since the comlimiting distributions of the quantile parameter estimates β(τ ˆ ponents in the vector β(τ ) have different convergence rates, we introduce the standardization matrix DT = diag(T 1/2 , T 2 , T 1/2 , · · · , T 1/2 ) and define vˆ = ˆ ) − β(τ )). Then, we have ∆yt − z 0 β(τ ˆ ) = etτ − (D−1 vˆ)0 zt . Thus, the DT (β(τ t T minimization problem in (7) can be rewritten as min v
T X ρτ (etτ − (DT−1 v)0 zt ) − ρτ (etτ ) .
(A1)
t=1
If vˆ is a minimizer of HT (v) =
T P ρτ (etτ − (DT−1 v)0 zt ) − ρτ (etτ ) , then we
t=1
ˆ ) − β(τ )). Following the approach of Knight (1989), it can be have vˆ = DT (β(τ shown that 12
HT (v) = −
T X
(DT−1 v)0 zt ψτ (etτ )
t=1
+
−1 (DT v)0 zt
T Z X t=1
{I(etτ ≤ s) − I(etτ ≤ 0)}ds.
(A2)
0
According the the results in Knight (1989), if HT (v) converges weakly to that of H(v) and H(v) has a unique minimum, then vˆ converges in distribution to the minimizer of H(v). Therefore, in order to derive the limiting distribution ˆ ), we need to derive the asymptotic distribution of the two terms of HT (v) of β(τ in the above equation. Now we consider the first term of HT (v). Under the unit root null hypothesis, we have ∆yt = A(L)−1 ut . Using the BN decomposition and the results in Galvao (2009, proof of lemma 1), we have T −1/2 ybT rc ⇒ A(1)−1 Bu (r). Consequently, it is easy to verify (e.g., Hansen, 1992) T
−2
T X
3 yt−1 ψτ (etτ )
−3
Z
⇒ A(1)
1
Bu3 dBψτ .
(A3)
0
t=1
Because ∆yt and xt are stationary variables, and the regression error et is orthogonal to the leads and lags of the covariates xt and the lagged differences of yt according to Assumptions A3 and A4, we can apply central limit theorem to show that T P −1/2 ∆yt−1 ψτ (etτ ) T t=1 .. . T −1/2 P ∆yt−p ψτ (etτ ) T t=1 (A4) ⇒ Φ, T −1/2 P T xt−q1 ψτ (etτ ) t=1 .. . T −1/2 P T xt+q2 ψτ (etτ ) t=1
where Φ = [Φ1 , · · · , Φp+m(q1 +q2 ) ]0 is a [p+m(q1 +q2 )]-dimensional normal variate with covariance matrix τ (1 − τ )ΩΦ , where the elements of ΩΦ are the elements of the matrix E[φt φ0t ]. Combining (A3) and (A4), we have DT−1
T X
R1 zt ψτ (etτ ) ⇒
t=1
¯u (r) = [1, A(1)−3 Bu3 (r)]0 . where B 13
0
¯u dB τ B ψ Φ
,
(A5)
In the next, we consider the second term of HT (v). Using a similar argument to that of Koenker and Xiao (2004), we have −1 (DT v)0 zt
T Z X t=1
{I(etτ ≤ s) − I(etτ ≤ 0)}ds ⇒
0
1 f (F −1 (τ ))v 0 Ψv, 2
(A6)
R1
¯u B ¯0 B 0 u where Ψ= . 0 ΩΦ Using the results in (A5) and (A6), it can be shown that 0
DT βˆ (τ ) − β (τ ) ⇒
1 f (F −1 (τ ))
R1 0
¯u B ¯u0 B 0
0 ΩΦ
−1 R 1 0
¯u dB τ B ψ Φ
. (A7)
Consequently, using the definition of t(τ ) in (8), Theorem 1 can be proved.
Proof of Theorem 2 Using Theorem 1 and the continuous mapping theorem, Theorem 2 can be proved.
References [1] Mohsen Bahmani-Oskooee, Tsangyao Chang, and Omid Ranjbar. Testing hysteresis effect in u.s. state unemployment: new evidence using a nonlinear quantile unit root test. Applied Economics Letters, 25(4):249–253, 2018. [2] Frederique Bec, Alain Guay, and Emmanuel Guerre. Adaptive consistent unit-root tests based on autoregressive threshold model. Journal of Econometrics, 142(1):94–133, 2008. [3] Frederique Bec, Melika Ben Salem, and Marine Carrasco. Tests for unitroot versus threshold specification with an application to the purchasing power parity relationship. Journal of Business & Economic Statistics, 22(4):382–395, 2004. [4] Mehmet Caner and Bruce E Hansen. Threshold autoregression with a unit root. Econometrica, 69(6):1555–1596, 2001. [5] Yoosoon Chang, Robin C. Sickles, and Wonho Song. Bootstrapping unit root tests with covariates. Econometric Reviews, 36(1-3):136–155, 2017. [6] Robert B Davies. Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika, 64(2):247–254, 1977. [7] David A Dickey and Wayne A Fuller. Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74(366):427–431, 1979. 14
[8] David A Dickey and Wayne A Fuller. Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica, 49(4):1057–1072, 1981. [9] Graham Elliott and Michael Jansson. Testing for unit roots with stationary covariates. Journal of Econometrics, 115(1):75–89, 2003. [10] Graham Elliott and Elena Pesavento. On the failure of purchasing power parity for bilateral exchange rates after 1973. Journal of Money, Credit and Banking, 38(6):1405–1430, 2006. [11] Barry Falk and Chunhsuan Wang. Testing long-run ppp with infinitevariance returns. Journal of Applied Econometrics, 18(4):471–484, 2003. [12] Antonio F Galvao. Unit root quantile autoregression testing using covariates. Journal of Econometrics, 152(2):165–178, 2009. [13] B E Hansen. Rethinking the univariate approach to unit root testing: Using covariates to increase power. Econometric Theory, 11(5):1148–1171, 1995. [14] Bruce E Hansen. Convergence to stochastic integrals for dependent heterogeneous processes. Econometric Theory, 8(04):489–500, 1992. [15] Norbert Herrndorf. A functional central limit theorem for weakly dependent sequences of random variables. Annals of Probability, 12(1):141–153, 1984. [16] Jean Imbs, Haroon Mumtaz, Morten O Ravn, and Helene Rey. Ppp strikes back: Aggregation and the real exchange rate. Quarterly Journal of Economics, 120(1):1–43, 2005. [17] Michael Jansson. Stationarity testing with covariates. Econometric Theory, 20(01):56–94, 2004. [18] George Kapetanios and Yongcheol Shin. Unit root tests in three-regime setar models. Econometrics Journal, 9(2):252–278, 2006. [19] George Kapetanios, Yongcheol Shin, and Andy Snell. Testing for a unit root in the nonlinear star framework. Journal of Econometrics, 112(2):359–379, 2003. [20] Rehim Kilic. Testing for a unit root in a stationary estar process. Econometric Reviews, 30(3):274–302, 2011. [21] Keith Knight. Limit theory for autoregressive-parameter estimates in an infinite-variance random walk. Canadian Journal of Statistics-revue Canadienne De Statistique, 17(3):261–278, 1989. [22] Roger Koenker and Gilbert Bassett. Regression quantiles. Econometrica, 46(1):33–49, 1978.
15
[23] Roger Koenker and Zhijie Xiao. Unit root quantile autoregression inference. Journal of the American Statistical Association, 99(467):775–787, 2004. [24] Cheng Feng Lee and Ching Chuan Tsong. Covariate selection for testing purchasing power parity. Applied Economics, 43(15):1923–1933, 2011. [25] Haiqi Li and Sung Y Park. Testing for a unit root in a nonlinear quantile autoregression framework. Econometric Reviews, 37(8):867–892, 2018. [26] James R Lothian and Mark P Taylor. Real exchange rates over the past two centuries : How important is the harrod-balassa-samuelson effect? The Economic Journal, 118(532):1742–1763, 2008. [27] Panos Michael, A Robert Nobay, and David Peel. Transactions costs and nonlinear adjustment in real exchange rates; an empirical investigation. Journal of Political Economy, 105(4):862–879, 1997. [28] Charles R. Nelson and C Plosser. Trends and random walks in macroeconomic time series: Some evidence and implications. Journal of Monetary Economics, 10(2):139–162, 1982. [29] Joon Y. Park. An invariance principle for sieve bootstrap in time series. Econometric Theory, 18(2):469–490, 2002. [30] Joon Y. Park. Bootstrap unit root tests. Econometrica, 71(6):1845–1895, 2003. [31] Peter C B Phillips and Pierre Perron. Testing for a unit root in time series regression. Biometrika, 75(2):335–346, 1988. [32] Peter C. B. Phillips and Victor Solo. Asymptotics for linear processes. Annals of Statistics, 20(2):971–1001, 1992. [33] Alan M Taylor and Mark P Taylor. The purchasing power parity debate. Journal of Economic Perspectives, 18(4):135–158, 2004. [34] Mark P Taylor, David Peel, and Lucio Sarno. Nonlinear mean-reversion in real exchange rates: Toward a solution to the purchasing power parity puzzles. International Economic Review, 42(4):1015–1042, 2001. [35] Chingchuan Tsong. Testing for a unit root with covariates against nonlinear alternatives. Economic Modelling, 28(3):1226–1234, 2011. [36] Joakim Westerlund. Rethinking the univariate approach to panel unit root testing: Using covariates to resolve the incidental trend problem. Journal of Business & Economic Statistics, 33(3):430–443, 2015. [37] Zhijie Xiao. Robust inference in nonstationary time series models. Journal of Econometrics, 169(2):211–223, 2012.
16
[38] Zhijie Xiao. Unit roots: A selective review of the contributions of peter c. b. phillips. Econometric Theory, 30(04):775–814, 2014. [39] Su Zhou and Ali M Kutan. Is the evidence for ppp reliable? a sustainability examination of the stationarity of real exchange rates. Journal of Banking and Finance, 35(9):2479–2490, 2011.
17
Table 1: Asymptotic critical values of t(τ ) λ
2
1% 5% 10%
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-2.34 -1.65 -1.29
-2.82 -2.14 -1.78
-2.98 -2.33 -1.97
-3.10 -2.46 -2.11
-3.19 -2.56 -2.23
-3.26 -2.65 -2.32
-3.33 -2.72 -2.40
-3.38 -2.79 -2.48
-3.42 -2.85 -2.54
-3.45 -2.89 -2.59
-3.47 -2.92 -2.64
Table 2: Finite sample size of the unit root tests N (0, 1)
t2
t3
σ12
ADF
QADF
KSS
QKSS
CKSS
QKS
QCM
t(τ )
t∗ (τ )
tks
tcm
0.3 0.5 0.7 0.3 0.5 0.7 0.3 0.5 0.7
0.058 0.052 0.052 0.058 0.060 0.059 0.051 0.057 0.061
0.062 0.059 0.057 0.045 0.040 0.042 0.048 0.045 0.053
0.051 0.052 0.048 0.096 0.095 0.097 0.080 0.076 0.078
0.043 0.042 0.048 0.044 0.039 0.038 0.047 0.045 0.040
0.044 0.041 0.049 0.088 0.085 0.078 0.071 0.071 0.066
0.043 0.045 0.043 0.049 0.048 0.042 0.067 0.042 0.045
0.044 0.057 0.051 0.063 0.048 0.045 0.060 0.047 0.049
0.031 0.033 0.030 0.039 0.039 0.030 0.040 0.042 0.025
0.053 0.051 0.043 0.063 0.047 0.054 0.073 0.064 0.074
0.045 0.043 0.039 0.052 0.042 0.046 0.071 0.039 0.052
0.050 0.057 0.053 0.058 0.045 0.054 0.072 0.057 0.066
Table 3: Finite sample power of the unit root tests (N (0, 1)) σ12
γ
θ
ADF
QADF
KSS
QKSS
CKSS
QKS
QCM
t(τ )
t∗ (τ )
tks
tcm
0.3
-0.1
0.01 0.1 0.01 0.1 0.01 0.1 0.01 0.1 0.01 0.1 0.01 0.1
0.088 0.177 0.134 0.642 0.087 0.177 0.139 0.645 0.080 0.180 0.144 0.641
0.090 0.218 0.301 0.591 0.084 0.217 0.303 0.578 0.076 0.202 0.285 0.563
0.083 0.185 0.159 0.652 0.090 0.182 0.152 0.666 0.078 0.186 0.149 0.674
0.051 0.168 0.247 0.527 0.063 0.171 0.253 0.537 0.062 0.172 0.267 0.541
0.101 0.307 0.438 0.836 0.117 0.397 0.551 0.916 0.179 0.591 0.779 0.988
0.077 0.137 0.096 0.442 0.055 0.136 0.097 0.402 0.066 0.142 0.123 0.437
0.079 0.180 0.148 0.653 0.081 0.189 0.156 0.636 0.102 0.194 0.153 0.678
0.062 0.185 0.293 0.574 0.074 0.226 0.354 0.674 0.119 0.347 0.507 0.862
0.100 0.174 0.172 0.480 0.106 0.214 0.214 0.616 0.175 0.376 0.374 0.822
0.075 0.157 0.146 0.485 0.104 0.223 0.182 0.623 0.151 0.351 0.349 0.856
0.107 0.219 0.205 0.720 0.136 0.332 0.282 0.823 0.233 0.527 0.499 0.949
-0.3 0.5
-0.1 -0.3
0.7
-0.1 -0.3
18
Table 4: Finite sample power of the unit root tests (t2 ) σ12
γ
θ
ADF
QADF
KSS
QKSS
CKSS
QKS
QCM
t(τ )
t∗ (τ )
tks
tcm
0.3
-0.1
0.01 0.1 0.01 0.1 0.01 0.1 0.01 0.1 0.01 0.1 0.01 0.1
0.153 0.240 0.521 0.974 0.146 0.249 0.506 0.971 0.151 0.249 0.500 0.970
0.529 0.872 0.961 1.000 0.536 0.877 0.962 0.999 0.534 0.873 0.965 0.999
0.125 0.124 0.521 0.723 0.123 0.127 0.520 0.722 0.129 0.135 0.526 0.708
0.495 0.773 0.954 0.998 0.501 0.779 0.954 0.995 0.492 0.761 0.950 0.995
0.213 0.398 0.780 0.969 0.301 0.526 0.856 0.985 0.460 0.741 0.955 0.997
0.338 0.394 0.701 0.891 0.343 0.402 0.713 0.900 0.336 0.384 0.702 0.879
0.449 0.538 0.826 0.958 0.481 0.542 0.837 0.972 0.486 0.537 0.835 0.962
0.503 0.774 0.964 0.997 0.568 0.838 0.979 0.997 0.693 0.919 0.994 1.000
0.539 0.592 0.837 0.927 0.629 0.691 0.898 0.970 0.742 0.822 0.952 0.984
0.399 0.456 0.745 0.923 0.475 0.584 0.817 0.956 0.605 0.734 0.907 0.982
0.506 0.596 0.868 0.965 0.601 0.699 0.896 0.994 0.734 0.854 0.963 0.992
-0.3 0.5
-0.1 -0.3
0.7
-0.1 -0.3
Table 5: Finite sample power of the unit root tests (t3 ) σ12
γ
θ
ADF
QADF
KSS
QKSS
CKSS
QKS
QCM
t(τ )
t∗ (τ )
tks
tcm
0.3
-0.1
0.01 0.1 0.01 0.1 0.01 0.1 0.01 0.1 0.01 0.1 0.01 0.1
0.126 0.244 0.286 0.924 0.127 0.230 0.282 0.927 0.131 0.217 0.285 0.921
0.252 0.601 0.780 0.975 0.244 0.593 0.784 0.978 0.225 0.567 0.754 0.973
0.126 0.168 0.328 0.747 0.126 0.164 0.328 0.752 0.116 0.160 0.333 0.748
0.235 0.466 0.736 0.946 0.231 0.457 0.736 0.943 0.224 0.443 0.728 0.945
0.178 0.369 0.627 0.932 0.212 0.471 0.742 0.969 0.321 0.675 0.885 0.994
0.156 0.196 0.356 0.736 0.148 0.210 0.383 0.737 0.134 0.201 0.400 0.726
0.237 0.314 0.542 0.896 0.227 0.325 0.553 0.898 0.211 0.322 0.560 0.896
0.251 0.509 0.776 0.961 0.289 0.567 0.832 0.986 0.378 0.711 0.931 0.998
0.279 0.381 0.552 0.828 0.336 0.479 0.653 0.895 0.502 0.639 0.793 0.959
0.180 0.260 0.429 0.803 0.227 0.393 0.568 0.875 0.378 0.560 0.736 0.964
0.286 0.391 0.601 0.930 0.335 0.536 0.701 0.961 0.521 0.696 0.846 0.985
-0.3 0.5
-0.1 -0.3
0.7
-0.1 -0.3
Figure 1: Real bilateral exchange rates with the US dollar
Japan
2000
2010
19
7.3 7.2 7.1 7.0
4.5 4.4 2010
Korea
6.9
4.8 4.6
4.7
0.0 0.1 −0.2 −0.4 2000
UK
−0.6 −0.5 −0.4 −0.3 −0.2
Eurozone
2000
2010
2000
2010
Table 6: Descriptive statistics and normality tests Eurozone Japan UK Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-Bera
-0.198 -0.233 0.122 -0.460 0.139 0.546 2.459 14.110(0.001)***
4.567 4.559 4.819 4.325 0.134 0.045 1.986 9.848(0.007)***
-0.449 -0.447 -0.205 -0.672 0.109 0.074 2.542 2.198(0.333)
Korea 7.052 7.037 7.298 6.879 0.096 0.502 2.748 10.173(0.006)***
Notes: Values in brackets are asymptotic p-values of the Jarque-Bera normality test. *** denotes significance at the 1% level.
Table 7: Results of the ADF, KSS, CKSS, QKS, QCM, tks and tcm tests Eurozone Japan UK Korea ADF KSS CKSS QKS QCM tks tcm
-1.863 -2.028 -0.831 2.332 2.103 1.351 0.540
-1.634 -1.715 -1.498 1.980 1.309 2.736 3.219*
-1.340 -1.479 -1.609 1.514 1.124 3.668** 3.891*
Notes: *** denotes significance at the 1% level.
20
-2.076 -2.645 -2.924*** 2.612 2.789 4.351*** 4.442***
Table 8: Results of the QADF, QKSS, t(τ ) and t∗ (τ ) tests Eurozone
Japan ∗
τ
QADF
QKSS
t(τ )
t (τ )
QADF
QKSS
t(τ )
t∗ (τ )
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-1.310 -1.665 -1.413 -1.091 -0.993 -0.581 -1.037 -0.180 -1.083
-1.380 -1.355 -1.230 -1.131 -0.944 -1.598 -1.601 -1.747 -2.332**
-1.263 -1.143 -1.351* -0.654 0.068 0.191 0.380 0.006 0.246
0.016** 0.013** 0.011** 0.019** 0.128 0.153 0.215 0.129 0.245
-0.060 -1.618 -1.328 -1.028 -0.650 -1.058 -1.080 -1.993 -1.021
-0.061 -1.603 -1.323 -1.048 -0.665 -1.030 -1.095 -1.980 -1.023
-0.229 -1.587* -1.290 -2.021** -2.125** -2.295** -1.427* -2.137** -2.736***
0.148 0.006*** 0.010** 0.000*** 0.001*** 0.001*** 0.008*** 0.000*** 0.001***
UK
Korea
τ
QADF
QKSS
t(τ )
t∗ (τ )
QADF
QKSS
t(τ )
t∗ (τ )
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-1.109 -0.976 -1.568 -0.823 -0.826 -1.571 -0.414 -0.961 -1.657
-1.140 -1.514 -1.480 -0.706 -1.039 -1.185 -0.418 -0.944 -1.185
0.655 0.487 0.360 -1.392* -2.868*** -3.668*** -2.932*** -2.080** -1.253
0.509 0.361 0.291 0.007*** 0.000*** 0.000*** 0.000*** 0.001*** 0.021**
-2.404** -1.959 -2.540** -2.586** -1.287 -1.171 -1.133 -0.043 0.213
-2.518** -2.013* -2.515** -2.612** -1.297 -1.138 -1.151 -0.043 0.220
-2.564*** -4.351*** -2.561*** -2.068** -2.282** -1.294 -0.875 -0.583 -0.304
0.004*** 0.000*** 0.003*** 0.009*** 0.006*** 0.040** 0.135 0.218 0.284
Notes: We report the bootstrap p-value of t∗ (τ ). *,**,*** denote significance at the 10%, 5% and 1% levels, respectively.
21
HIGHLIGHT · This paper develops new quantile nonlinear unit root tests with covariates. · We derive the limiting distributions of the proposed tests. · Monte Carlo simulations show that large power gains can be achieved by including correlated covariates in the testing equation. · An empirical application to the PPP hypothesis, using the proposed tests, is provided.
1