Unit root quantile autoregression testing with smooth structural changes

Finance Research Letters xxx (xxxx) xxx–xxx

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Unit root quantile autoregression testing with smooth structural changes☆ Haiqi Li, Chaowen Zheng

⁎

College of Finance and Statistics, Hunan University, Changsha 410006, China

AR TI CLE I NF O

AB S T R A CT

Keywords: Unit root test Quantile autoregression Flexible fourier form Structural changes

By incorporating the flexible Fourier form into quantile autoregression model, this paper proposes three new unit root test statistics, which are robust to both non-Gaussian condition and structural changes. Since their limiting distributions are non-standard, a bootstrap procedure is developed to calculate their critical values. Monte Carlo simulation results show that, while Koenker and Xiao (2004) tests are quite conservative under various kinds of error distributions and structural changes, the newly proposed tests have good size performance except for a little size distortion occasionally. Moreover, our tests have much higher performance especially when the sample size is small.

JEL classification: C12 C13 C22

1. Introduction Testing unit root hypothesis is of particular interest to economists and finance practitioners since it has important policy implications. For example, the unit root property for the macroeconomic time series is closely related to the persistence of macroeconomic shocks. Nelson and Plosser (1982) investigated the unit root property of 14 U.S. macroeconomic time series and could not reject the null hypothesis of a unit root for any of them. Thus, they advocated that the source of business fluctuations was nonmonetary. Sekioua (2006) studied the unit root property of the forward premium. If the forward premium is a unit root process, it makes the commonly employed market efficiency test inappropriate since it suffers from the spurious regression-type critique. The unit root tests also play fundamental role in checking the validity of financial theories and models, such as the purchasing power parity theory (Ma et al., 2017) and the Fed model (Koivu et al., 2005). However, most widely used unit root tests suffer from low power under the non-Gaussian condition (Koenker and Xiao, 2004; Li and Park, 2016) or when the trend function of the series has structural changes (Perron, 1989), which may induce misleading conclusions and policy implications. To deal with these issues, developing robust unit root test methods is imperative. To improve the power performance under non-Gaussian condition, Koenker and Xiao (2004) developed three tests which are robust to various kinds of error distributions based on the quantile autoregression approach. Galvao (2009) extended their work by introducing stationary covariates and a linear time trend into the model. For power loss caused by structural changes in the deterministic components of the series, Perron (1989) modified the standard Dickey-Fuller (DF) test to allow for a single exogenous structural break at a prespecified change point. For subsequent developments along this line, see Perron (2006) for an excellent survey. Nevertheless, these tests usually depend crucially on the locations, numbers and the type of changes, which makes them inflexible and difficult to implement in practice. Recently, Enders and Lee (2012a) developed a new LM type unit root ☆ This project was supported by the National Natural Science Foundation of China (NSFC) (No.71773026) and the Humanities and Social Sciences Project of Ministry of Education of China (No.17YJA790041). ⁎ Corresponding author. E-mail addresses: [email protected] (H. Li), [email protected] (C. Zheng).

http://dx.doi.org/10.1016/j.frl.2017.10.008 Received 29 June 2017; Received in revised form 26 August 2017; Accepted 11 October 2017 1544-6123/ © 2017 Elsevier Inc. All rights reserved.

Please cite this article as: Li, H., Finance Research Letters (2017), http://dx.doi.org/10.1016/j.frl.2017.10.008

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test by using the flexible Fourier form to account for structural changes. Rodrigues and Taylor (2012) and Enders and Lee (2012b) proposed the corresponding local GLS detrend type and DF type tests. Though power loss in unit root tests due to the departure from Gaussian condition and in the presence of structural changes has been separately studied, they had never been treated simultaneously. Given that most time series in economics and finance have notoriously heavy-tailed behavior and major events such as the global financial crisis could induce a level and/or slope change to the trend function of the time series, it is of great importance to develop new unit root tests which are robust to both non-Gaussian condition and various kinds of structural changes. To the best of our knowledge, this paper is the first in the literature to study unit root tests which are robust to both non-Gaussian condition and structural changes in the trend function of a series. We achieve this by incorporating the flexible Fourier form into the quantile autoregression model and three new unit root test statistics, say, the quantile t-ratio test, the quantile Kolmogorov–Smirnov (QKS) test and the quantile Cramér-von Mises(QCM) tests are developed. Limiting distributions of these test statistics are derived and bootstrap procedures are developed to obtain their finite-sample critical values. Finally, Monte Carlo simulations are conducted to explore the finite sample performance of the newly proposed three tests. 2. Quantile autoregression with a Fourier function Consider the following data generated process (DGP):

yt = α (t ) + γt + et ,

et = ϕet − 1 + μt ,

(1)

where α(t) is a deterministic function of t representing potential level and/or slope changes in the trend function of the series, γt represents the deterministic trend, μt is assumed to be i.i.d with mean 0 and variance σμ2 for simplicity. This model is first suggested by Schmidt and Phillips (1992) to avoid the ambiguous meaning of parameters existing in traditional Dickey–Fuller type unit root test model (also see Galvao, 2009. By employing this model, we can analyze the trending and unit root behavior of the series separately and in particular, we are mainly interested in testing the unit root null hypothesis (i.e., H0: ϕ = 1). However, the functional form of α(t) is usually unknown in practice, which makes any test for H0: ϕ = 1 problematic and might induce misleading conclusion if it is misspecified. It is well documented that an autoregressive process with structural changes is often incorrectly regarded as a unit root process (Perron, 1989). Following Enders and Lee (2012a), we approximate it using the Fourier expansion with only a single frequency as it can often capture the essential characteristics of an unknown functional form:

α (t ) ≅ α 0 + αk sin(2πkt / T ) + βk cos(2πkt / T ).

(2)

Comparing to nonparametric or semi-parametric methods, this method is easy to implement and the convergence rates of parameters are faster. As Enders and Lee (2012a, 2012b) pointed out, this approximation method could help researchers avoid selecting specific endogenous break dates, the number of breaks, and the form of the breaks. The specification problem is transformed to incorporating the appropriate frequency components into the estimating equation. Using this approximation equation and rewriting Model (1), the testing procedure for the unit root null hypothesis could be based on the following model:

yt = ϕy͠ t − 1 + α 0 + αk sin(2πkt / T ) + βk cos(2πkt / T ) + γt + μt ,

(3)

where y͠ t − 1 = yt − 1 − α 0 − αk sin(2πk (t − 1)/ T ) − βk cos(2πk (t − 1)/ T ) − γ (t − 1) . In particular, we are interested in testing unit root property at the τth quantile level of yt:

Q yt (τ yt − 1) = ϕy͠ t − 1 + α 0 + αk sin(2πkt / T ) + βk cos(2πkt / T ) + γt + Fμ−1 (τ ).

(4)

Though there are trigonometric terms, Model (4) is still linear in parameters. Let z t = (y͠ t − 1 , 1, sin(2πkt / T ), cos(2πkt / T ), t )′ , θ (τ ) = (ϕ, α (τ ), αk , βk , γ )′,α (τ ) = α 0 + Fμ−1 (τ ), the estimator θ ̂(τ ) could be obtained by solving the following minimization problem: T

θ ̂(τ ) = argmin ∑ ρτ (yt − θ (τ )′z t ), θ

(5)

t=1

where ρτ (u) = u (τ − I (u < 0)) is the check function. For asymptotic analysis, we impose the following assumptions first. Assumption 1. {μt} are i.i.d random variables with mean 0 and variance σ2 < ∞. Assumption 2. The distribution of {μt}, F(μ), has a continuous density f(μ) with f(μ) > 0 on {μ: 0 < f(μ) < 1}. For simplicity and without loss of generality, the error term μt is assumed to be i.i.d, thus excluding any linear dependence. However, our analysis could be easily extended to more general case where weak dependence is allowed. Identical distribution assumption is needed for the asymptotic analysis of the QKS test and QCM test and also simplifies the bootstrap procedure. Assumption 2 is quite standard in the literature of quantile regression study, see Koenker and Xiao, 2004. Since different components of θ ̂(τ ) have different convergence rates, define diagonal matrix GT = diag (T , T1/2, T1/2, T1/2, T 3/2) . Theorem 1 below gives the asymptotic distribution of θ ̂(τ ) . Theorem 1. Under Assumptions 1 and 2, 2

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GT (θ ̂(τ ) − θ (τ )) ⇒

1

⎡ f (F −1 (τ )) ⎢ ⎣

∫0

−1

1

1

Bμ B′μ⎤ ⎡ ⎥ ⎢ ⎦ ⎣

∫0

Bμ andBψτ

= Bψτ (r ) are standard Brownian motions,ψτ = ψτ (μtτ ) = τ − I (μtτ < 0) and μtτ is the

whereBμ = [Bμ , 1, sin(2πkr ), cos(2πkr ), r ]′ , residual of Model (4).

Proof. See Section 1 of the Supplementary material.

Bμ dBψτ⎤, ⎥ ⎦

(6)

□

In particular, we have the following corollary: Corollary 1. Under Assumptions 1 and 2,

n (ϕ (̂ τ ) − 1) ⇒

1 f

⎡

(F −1 (τ )) ⎢ ⎣

∫0

1

−1

B2μ⎤ ⎡ ⎥ ⎢ ⎦ ⎣

∫0

1

B μ dBψτ⎤, ⎥ ⎦

(7)

whereB μ is the detrended version of Brownian motion Bμ. Remark 1. The derivation of Corollary 1 is quite standard in the unit root test literature. Due to trigonometric terms, B μ is related with the nuisance parameter k. While its exact form is quite complicated, we can proceed without its full knowledge.

3. Inference on unit root null hypothesis 3.1. Inference at a selected quantile level Based on Corollary 1 above, we proposed the following unit root t-ratio test statistic:

t f (τ ) =

 f (F −1 (τ )) τ (1 − τ )

(Y −′ 1 Mx Y−1)1/2 (ϕ ̂ − 1),

(8)

f (F −1 (τ )) is a consistent estimator of f (F −1 (τ )) and the estimating method is quite standard. Y−1 is a vector of lagged dewhere  pendent variables (yt − 1), and Mx is a matrix projected onto the space orthogonal to x t = (1, sin(2πkt /T ), cos(2πkt / T ), t ) . The subscript f is used since there are Fourier terms. Based on Theorem 1 and under the null hypothesis, we can obtain the following theorem. Theorem 2. Under Assumption 1 and 2,

1

⎡ τ (1 − τ ) ⎢ ⎣

t f (τ ) ⇒ t (τ ) =

∫0

1

−1/2

B μ2⎤ ⎥ ⎦

∫0

1

B μ dBψτ .

(9)

Note that, for any given τ, the test statistic is simply the quantile regression counterpart of the one proposed by Enders and Lee (2012b). This asymptotic distribution is nonstandard since Bμ and Bψ are correlated. However, it can be decomposed into a liner combination of two independent parts. Following Koenker and Xiao (2004), we have

∫0

1

B μ dBψτ =

∫0

1

∫0

B μ dBψτ . μ + λμψ (τ )

σμψ (τ )/ σμ2

1

B μ dBμ ,

(10)

Bψτ·μ

and is a Brownian motion independent of Bμ with variance where λμψ (τ ) = Therefore, the asymptotic distribution of tf(τ) can be rewritten as

t f (τ ) =

∫ B μ dBψτ·μ + τ (1 − τ ) ( ∫ B μ2)1/2

λμψ (τ )

1

∫ B μ dBμ

τ (1 − τ ) ( ∫ B μ2)1/2

σψ2·μ

=

σψ2 (τ )

−

2 σμψ (τ )/ σμ2 .

. (11)

could be rewritten as Bμ = σμW1(r), = Let W1(r) and W2(r) be two independent standard Brownian motions, then Bμ and σψ · μW2(r). Moreover, Bμ(r) = σμW1(r), where W1(r) is the detrended version of W1(r) as in Theorem 1. Since for fixed τ, σψ = τ (1 − τ ), then we have

Bψτ·μ

t f (τ ) = ζ

1 ∫0 W 1 (r ) dW1 1 ∫0 W 12 (r ) dr

+

Bψτ·μ

1 − ζ 2 N (0, 1), (12)

where

ζ = ζ (τ ) =

σμψ (τ ) σμ σψ (τ )

=

σμψ (τ ) σμ τ (1 − τ )

.

Thus, the asymptotic distribution of tf(τ) is a convex combination of the Dicker–Fuller type distribution and a standard normal distribution, with the weight determined by the nuisance parameter ζ, which is exactly the correlation coefficient between μt and ψ(μtτ). 3

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3.2. Calculating critical values based on bootstrap method As pointed out by Psaradakis (2001), conventional large-sample approximation to the asymptotic distributions of most unit-root test statistics is inaccurate for certain types of error processes. To improve finite-sample performance of unit root tests in practice, bootstrapped unit root tests were proposed in the literature. For this reason, we develop a bootstrap procedure similar to Smeekes (2013) in order to calculate the critical values of the newly proposed test statistic. The procedure is as follows: Calculate ytd = yt − β′x t , where xt is as defined in Section 3.1 and β is the OLS estimator of {yt} regressed on {xt}. Estimate an AR(1) regression by OLS and calculate the residuals, μtd = ytd − ϕ ŷ td− 1. Resample with replacement from the recentered residuals (μtd − μtd ) to obtain bootstrap errors μt*. Build {ytd * } recursively as ytd * = ytd−*1 + μt* and yt* = ytd * + β * ′x t . Using the bootstrap sample {yt* }, estimate Model (4) and calculate the test statistic t f* (τ ) . Repeat step iii to v B times, obtain bootstrap test statistic t f* (τ ) for b = 1, 2…,B and select the bootstrap critical value c*(τ, α) satisfying P * [t f* (τ ) ≤ c * (τ )] = α . Remark 2. Theoretically, in step (iv), β* could be any constant vector. However, it would be invalid to set β * = β , as this would mean

(i) (ii) (iii) (iv) (v) (vi)

time-varying deterministic trend which is not the case of the original sample (Smeekes, 2013). We recommend setting β* = 0 for simplicity. In the proposed bootstrap procedure, the bootstrap sample {yt* } is generated under the unit root null hypothesis ensuring the validity of the bootstrap method. The asymptotic validity relies on the bootstrap invariance principle which had been well established in Smeekes (2013).

3.3. Inference over a range of quantiles We also proposed two test statistics against the unit root null hypothesis over a range of quantile levels τ ∈ T . We construct a QKS type test and a QCM type tests for τ ∈ T = [τ0, 1 − τ0] with 0 < τ0 < 1. It is important to note that the identical distribution assumption implies that these two tests can be viewed as diagnostic tools for the adequacy of the proposed model. The test statistics are defined as follows:

QKSf ≡ sup t f (τ ) ,

QCMf ≡

τ∈T

∫τ∈T t f (τ )2dτ .

In practice, one may discretize the quantile interval T and compute tf(τ) at each τi ∈ T . Then, the QKSf and QCMf test statistics can be obtained by maximizing over τi ∈ T and using numerical integration, respectively. In addition, their limiting distributions are given by

sup t f (τ ) ⇒ sup t (τ ) , τ∈T

τ∈T

∫τ∈T t f (τ )2dτ ⇒ ∫τ∈T t (τ )2dτ ,

respectively, which can be proved easily using Theorem 2 and the continuous mapping theorem. The critical values of these two tests can be also obtained by the bootstrap procedure similar to the one in Section 3.2 apart from that step (v) and (vi) should be replaced by (v′). Using the bootstrap sample {yt* }, estimate Model (4) at the quantile levels {τi ∈ T }in= 1 and obtain t f* (τi ) . Then QKS f* and QCM f* can be calculated as:

QKS f* = max t f* (τi ) , i∈T

QCM f* =

∑

t f* (τi )2 (τi − τi − 1).

i∈T

* f (I , α ) and (vi′). Repeat step (iii) to (v′) B times, obtain QKS f* and QCM f* for b = 1, 2…,B and select the bootstrap critical values cQKS * f (I , α ) ⎤ = α and P * ⎡QCM f* ≥ cQCM * f (I , α ) ⎤ = α . * f (I , α ) satisfying P * ⎡QKS f* ≥ cQKS cQCM ⎣ ⎦ ⎣ ⎦ 4. Monte Carlo simulation In this section, we conduct the Monte Carlo simulation to study the finite sample performance of the proposed test statistics. Two different cases, say finite sample performance under trigonometric structural changes and other kinds of structural changes in the trend function of the series, are considered. 4.1. Case 1: trigonometric structural changes To implement the Monte Carlo simulation, the data is first generated according to 4

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Table 1 Finite sample performance for Case 1 with estimated k. tf statistic

QKSf statistic

QCMf statistic

t statistic N(0,1)

N(0,1)

t(4)

N(0,1)

t(4)

N(0,1)

t(4)

QKS statistic

QCM statistic

t(4)

N(0,1)

t(4)

N(0,1)

t(4)

0.010 0.010 0.014 0.014 0.002

0.032 0.022 0.014 0.012 0.002

0.019 0.011 0.008 0.003 0.002

0.043 0.023 0.010 0.007 0.017

0.007 0.010 0.013 0.010 0.003

0.021 0.026 0.013 0.011 0.007

0.026 0.018 0.012 0.024 0.018

0.040 0.016 0.032 0.032 0.040

0.030 0.011 0.011 0.018 0.017

0.032 0.022 0.035 0.035 0.017

0.035 0.021 0.021 0.015 0.018

0.038 0.038 0.032 0.037 0.027

0.046 0.028 0.020 0.022 0.030

0.054 0.062 0.050 0.042 0.040

0.028 0.033 0.020 0.028 0.018

0.057 0.033 0.025 0.045 0.031

0.030 0.017 0.026 0.020 0.017

0.048 0.045 0.057 0.052 0.045

0.004 0.008 0.002 0.000 0.010

0.054 0.056 0.032 0.012 0.016

0.003 0.001 0.000 0.001 0.000

0.065 0.022 0.022 0.015 0.035

0.003 0.000 0.004 0.002 0.000

0.038 0.068 0.028 0.013 0.022

0.016 0.018 0.018 0.022 0.010

0.216 0.182 0.172 0.146 0.172

0.021 0.020 0.016 0.007 0.013

0.203 0.155 0.148 0.163 0.087

0.016 0.016 0.011 0.005 0.008

0.222 0.265 0.161 0.155 0.195

0.138 0.144 0.152 0.150 0.150

0.748 0.750 0.730 0.718 0.708

0.115 0.146 0.113 0.110 0.081

0.793 0.735 0.726 0.737 0.730

0.112 0.098 0.152 0.130 0.113

0.797 0.816 0.878 0.823 0.797

T=100

ϕ = 1 (Size)

k k k k k

= = = = =

1 2 3 4 5

0.070 0.056 0.050 0.060 0.076

0.062 0.058 0.072 0.066 0.058

0.078 0.066 0.068 0.060 0.050

0.061 0.072 0.040 0.067 0.042

0.054 0.067 0.045 0.054 0.047

0.084 0.066 0.063 0.048 0.072

k k k k k

= = = = =

1 2 3 4 5

0.072 0.062 0.052 0.054 0.042

0.076 0.078 0.076 0.062 0.072

0.062 0.040 0.043 0.061 0.042

0.056 0.053 0.043 0.057 0.042

0.061 0.071 0.045 0.063 0.048

0.076 0.059 0.052 0.044 0.063

k k k k k

= = = = =

1 2 3 4 5

0.084 0.086 0.064 0.060 0.042

0.092 0.086 0.064 0.060 0.042

0.046 0.038 0.041 0.033 0.040

0.048 0.041 0.036 0.033 0.027

0.080 0.052 0.068 0.051 0.048

0.095 0.071 0.080 0.061 0.073

k k k k k

= = = = =

1 2 3 4 5

0.110 0.122 0.146 0.140 0.150

0.180 0.206 0.240 0.256 0.236

0.117 0.130 0.151 0.158 0.127

0.131 0.268 0.252 0.298 0.226

0.110 0.192 0.145 0.169 0.130

0.198 0.274 0.296 0.232 0.323

k k k k k

= = = = =

1 2 3 4 5

0.238 0.302 0.356 0.356 0.354

0.556 0.584 0.632 0.668 0.648

0.203 0.288 0.348 0.405 0.340

0.356 0.550 0.596 0.751 0.647

0.206 0.356 0.301 0.403 0.341

0.536 0.636 0.728 0.693 0.793

k k k k k

= = = = =

1 2 3 4 5

0.802 0.814 0.920 0.926 0.938

0.984 0.990 0.992 0.992 0.944

0.790 0.812 0.868 0.923 0.930

0.968 0.996 0.998 0.998 0.995

0.872 0.917 0.977 0.975 0.962

0.995 0.997 1.000 1.000 1.000

T=200

T=500

T=100

ϕ = 0.9 (Power)

T=200

T=500

Note: tf, QKSf and QCMf represent the newly proposed test statistics. t, QKS and QCM represent the tests statistics proposed by Koenker and Xiao (2004). N(0, 1) and t(4) denote standard Normal distribution and the t distribution with degree of freedom 4, respectively.

yt = α 0 + αk sin(2πkt / T ) + βk cos(2πkt / T ) + γt + et ,

et = ϕet − 1 + μt ,

with k varying from 1 to 5. ϕ is set to be 1 and 0.9 to evaluate both the size and power performance of the test statistics. αk and βk are simply set as 3 and 5 as in Enders and Lee (2012a); 2012b). α0 and γ are both set as 0. To explore the robustness of the statistics, the error term {μt} is set to follow different distributions, say, Normal(0, 1) and t(4) distributions.1 Three sample sizes (T=100, 500, 1000) are considered and number of repetitions is 1000. Clearly, the test will have better performance if the true value of k in the DGP is known. However, it is usually unknown in practice. To show the robustness and effectiveness of our tests, we consider k estimated from the data.2 This is done by using a twostep procedure. First, for each integer value of k in the interval 1 ≤ k ≤ kmax , estimate the Model (4). Second, choose the estimated value, k ,̂ which minimizes the residual sum of squares across these estimated regression equations. Following Enders and Lee (2012a), we set the maximum frequency at kmax = 5. Results of the three test statistics proposed by Koenker and Xiao (2004) (hereafter KX tests) which do not take the structural changes into consideration are also given for comparison. Their tests are simply denoted as t , QKS and QCM without subscript f. Besides, Bofingeb (1975) bandwidth choice rule is used to estimate the sparsity function f (F −1 (τ )) . The quantile t-ratio statistic is calculated at median and other two type tests are calculated at quantile interval T = [0.1, 0.9]. Table 1 reports the results for our tests and the KX tests. As can be seen, the proposed three test statistics have similar size and power performance which is also the case of the KX tests. For different k, error distributions and sample sizes, our tests show good size performance close to the nominal level 0.05, though a little size distortion could be observed occasionally. Moreover, the KX tests are 1 2

In the Supplementary material, we also report the results for χ2(1) and Log − Normal (0, 1) distributions. In the Supplementary material, results for the case of known k are also reported. The results are quite similar to that of estimated k.

5

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Table 2 Finite sample performance for Case 2. tf statistic

QKSf statistic

QCMf statistic

t statistic N(0,1)

N(0,1)

t(4)

N(0,1)

t(4)

N(0,1)

t(4)

S1 S2 S3 D1 D2 D3

0.088 0.072 0.048 0.084 0.078 0.092

0.084 0.080 0.068 0.092 0.092 0.060

0.076 0.072 0.056 0.094 0.082 0.086

0.096 0.066 0.092 0.094 0.086 0.068

0.078 0.088 0.076 0.080 0.084 0.090

0.078 0.084 0.048 0.078 0.086 0.080

S1 S2 S3 D1 D2 D3

0.088 0.078 0.076 0.090 0.068 0.090

0.094 0.082 0.062 0.090 0.084 0.092

0.094 0.102 0.076 0.078 0.088 0.078

0.074 0.098 0.078 0.086 0.076 0.078

0.088 0.086 0.080 0.076 0.078 0.084

0.076 0.082 0.090 0.088 0.090 0.094

S1 S2 S3 D1 D2 D3

0.084 0.086 0.088 0.093 0.084 0.080

0.093 0.092 0.094 0.072 0.080 0.092

0.090 0.068 0.078 0.070 0.082 0.078

0.094 0.092 0.086 0.090 0.088 0.096

0.084 0.082 0.092 0.088 0.086 0.082

0.088 0.088 0.082 0.090 0.072 0.078

S1 S2 S3 D1 D2 D3

0.092 0.260 0.082 0.098 0.178 0.190

0.162 0.380 0.154 0.214 0.370 0.290

0.088 0.212 0.078 0.102 0.158 0.180

0.122 0.364 0.190 0.186 0.308 0.312

0.078 0.182 0.204 0.086 0.182 0.204

0.184 0.428 0.146 0.218 0.322 0.338

S1 S2 S3 D1 D2 D3

0.132 0.378 0.150 0.192 0.340 0.312

0.442 0.740 0.460 0.520 0.700 0.690

0.148 0.370 0.148 0.180 0.328 0.323

0.424 0.752 0.456 0.462 0.652 0.692

0.124 0.390 0.154 0.152 0.326 0.328

0.472 0.784 0.526 0.558 0.720 0.658

S1 S2 S3 D1 D2 D3

0.536 0.916 0.584 0.574 1.000 0.988

0.954 1.000 0.972 0.970 0.998 0.986

0.550 0.940 0.624 0.598 0.874 0.744

0.978 0.998 0.980 0.720 1.000 0.994

0.966 0.996 0.970 0.970 0.998 0.998

0.978 1.000 0.984 0.970 1.000 0.995

QKS statistic

QCM statistic

t(4)

N(0,1)

t(4)

N(0,1)

t(4)

0.038 0.054 0.028 0.050 0.056 0.034

0.040 0.064 0.058 0.054 0.048 0.058

0.028 0.050 0.022 0.040 0.046 0.066

0.040 0.058 0.028 0.044 0.028 0.026

0.033 0.037 0.034 0.036 0.043 0.031

0.040 0.063 0.038 0.039 0.055 0.034

0.058 0.054 0.034 0.044 0.044 0.036

0.048 0.060 0.046 0.044 0.032 0.036

0.050 0.030 0.030 0.042 0.036 0.026

0.076 0.030 0.042 0.046 0.024 0.034

0.055 0.031 0.036 0.026 0.047 0.048

0.064 0.032 0.043 0.040 0.042 0.036

0.058 0.052 0.050 0.050 0.070 0.044

0.042 0.044 0.044 0.066 0.034 0.068

0.062 0.054 0.062 0.028 0.052 0.026

0.062 0.062 0.038 0.064 0.038 0.054

0.050 0.054 0.052 0.074 0.058 0.038

0.060 0.022 0.058 0.050 0.062 0.022

0.032 0.132 0.030 0.060 0.102 0.106

0.058 0.340 0.104 0.178 0.296 0.254

0.014 0.106 0.014 0.106 0.098 0.178

0.064 0.290 0.122 0.138 0.228 0.174

0.011 0.121 0.018 0.037 0.090 0.087

0.118 0.388 0.128 0.159 0.290 0.213

0.036 0.314 0.026 0.112 0.168 0.158

0.246 0.660 0.258 0.462 0.542 0.520

0.036 0.248 0.032 0.096 0.190 0.138

0.334 0.718 0.236 0.518 0.482 0.472

0.026 0.217 0.048 0.173 0.197 0.242

0.330 0.662 0.274 0.420 0.582 0.534

0.210 0.888 0.188 0.464 1.000 0.418

0.762 1.000 0.758 0.950 0.996 0.930

0.200 0.940 0.184 0.348 0.932 0.386

0.880 0.996 0.758 0.966 0.992 0.950

0.142 0.850 0.202 0.512 0.950 0.478

0.832 0.954 0.910 0.972 1.000 0.936

T=100

ϕ = 1 (Size)

T=200

T=500

T=100

ϕ = 0.9 (Power)

T=200

T=500

Note: tf, QKSf and QCMf represent the newly proposed test statistics. t, QKS and QCM represent the tests statistics proposed by Koenker and Xiao (2004). N(0, 1) and t(4) denote standard Normal distribution and the t distribution with degree of freedom 4, respectively.

conservative especially when the sample size is small. Compared with KX tests, the improvement of power performance of our tests is significant and the improvement becomes larger as the sample size increases. Specifically, the KX tests show low power performance under the Normal(0, 1) distribution even when T = 500 and the powers are under 0.15 for all cases. By contrast, the power performance of our tests are generally over 0.15 and approaching 1 as the sample size increases. Though the power performance of KX tests is better under the t(4) distribution, the powers of the newly proposed tests are generally larger than 0.20. These results suggest that, first, the estimating procedure for k can work well in practice; second, our tests are robust to trigonometric structural changes while the KX tests tend to accept the erroneous null hypothesis too frequently. Moreover, the power performance of our tests also improves with k increasing while the opposite trend could be found for the KX tests. Considering that larger k means more complex structural changes, this results show the merits of our tests further.

4.2. Case 2: various kinds of structural changes To further explore the finite sample performance of our tests, we also consider DGPs with other types of structural changes, say, sharp, exponential smooth transition and logistic smooth transition type structural changes, which are more often encountered in practice. Specifically, we consider DPGs with one single structural change as following:

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Finance Research Letters xxx (xxxx) xxx–xxx

H. Li, C. Zheng

S1: Yt = 8It + et , where It = 1 if t ≤ T /2, S2: yt = 8/[1 − exp (−0.001(t − 0.5T )2)] + et , S3: yt = 8/[1 + exp (0.1(t − 0.5T ))] + et , and DGPs with two structural changes:

D1: Yt = 6(1 − It ) + et , where It = 1 if 0.3T ≤ t ≤ 0.7T , D2: yt = 6/[1 − exp ( −0.001(t − 0.3T )2)] − 6/[1 − exp (−0.001(t − 0.7T )2)] + et , D3: yt = 6/[1 + exp (0.1(t − 0.3T ))] − 6/[1 − exp (−0.1(t − 0.7T )2)] + et , where in all six cases, et = ϕet − 1 + μt . Since there is no linear time trend in these DGPs, a special case of our tests based on Model (4) without liner time trend is used. The frequency term k is estimated from the data using the method proposed above and the results for the KX test are reported for comparison. Table 2 gives the results. Similar size and power performance could be found among three different test statistics both for our and the KX tests. Specifically, our tests are slightly over-sized while the KX tests are slightly conservative at 5% significance level. The slight over-rejection of our tests might be due to the over-fitting problem caused by the approximation of Fourier terms to the unknown structural changes. In contrast, the KX tests tend to accept the null hypothesis too often due to the ignorance of structural changes. It is still the case that the power performance improves as the sample size increases for all tests and they show the highest power for Model S2 and the lowest power for Model S1. Most importantly, our tests have significantly better power performance than KX tests for almost all cases and improve more quickly with sample size. Generally speaking, the improvement of power varies across different sample sizes. While the increase in power performance is generally over 0.05 when T = 100 and over 0.10 when T = 200, most figures are over 0.20 when T = 500 and sometimes, the increase in power can be even as high as 0.8. This power performance results show the robustness of our test to various kinds of structural changes and suggests that they would work well in practice. 5. Conclusion By incorporating the flexible Fourier form into quantile autoregression, this paper contributes to the literature by developing three new unit root test statistics. Since their asymptotic distributions are non-standard, bootstrap procedures are proposed to compute their critical values. Monte Carlo simulation results show that these tests perform well in practice under various kinds of error distributions and structural changes. It can be expected that our tests would provide useful guidance and be widely used by economists and finance practitioners. Supplementary material Supplementary material associated with this article can be found, in the online version, at 10.1016/j.frl.2017.10.008. References Bofingeb, E., 1975. Estimation of a density function using order statistic. Aust. N. Z. J. Stat. 17 (1), 1–7. Enders, W., Lee, J., 2012a. A unit root test using a fourier series to approximate smooth breaks. Oxford Bull. Econ. Stat. 74 (4), 574–599. Enders, W., Lee, J., 2012b. The flexible fourier form and dickey-fuller type unit root tests. Econ. Lett. 117, 196–199. Galvao, A.F., 2009. Unit root quantile autoregression testing using covariates. J. Econom. 152, 165–178. Koenker, R., Xiao, Z., 2004. Unit root quantile autoregression inference. J. Am. Stat. Assoc. 99 (467), 775–787. Koivu, M., Pennanen, T., Ziemba, W.T., 2005. Cointegration analysis of the fed model. Finance Res. Lett. 2, 248–259. Li, H., Park, S., 2016. Testing for a unit root in a nonlinear quantile autoregression framework. Econom. Rev. Forthcoming, https://doi.org/10.1080/00927872.2016. 1178871. Ma, W., Li, H., Park, S.Y., 2017. Empirical conditional quantile test for purchasing power parity: evidence from east asian countries. Int. Rev. Econ. Finance 49, 211–222. Nelson, C., Plosser, C., 1982. Trends and random walks in macroeconmic time series. J. Monet Econ. 10, 139–162. Perron, P., 1989. The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57 (6), 1361–1401. Perron, P., 2006. Econometric Theory vol. 1 of The Palgrave Handbook of Econometrics. Palgrave Macmillan, pp. 278–352. Psaradakis, Z., 2001. Bootstrap tests for an autoregressive unit root in the presence of weakly dependent errors. J. Time Ser. Anal. 22 (5), 577–594. Rodrigues, P., Taylor, A., 2012. The flexible fourier form and local generalised least squares de-trended unit root tests. Oxford Bull. Econ. Stat. 74 (5), 736–759. Schmidt, P., Phillips, P.C.B., 1992. LM tests for a unit root in the presence of deterministic trends. Oxford Bull. Econ. Stat. 54 (3), 257–287. Sekioua, S., 2006. Nonlinear adjustment in the forward premium: evidence from a threshold unit root test. Int. Rev. Econ. Finance 15, 164–183. Smeekes, S., 2013. Detrending bootstrap unit root tests. Econom. Rev. 32 (8), 869–891.

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