Economics Letters 126 (2015) 43–46

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Economics Letters

journal homepage: www.elsevier.com/locate/ecolet

Residual-based test for fractional cointegrationBin Wang a, Man Wang b, Ngai Hang Chan c,∗

a Department of Finance, Yangzhou University, 88 South University Ave., Yangzhou, Jiangsu, 225009, Chinab Department of Finance, Donghua University, 1882 Yan-an Road West, Shanghai, 200051, Chinac Department of Statistics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

h i g h l i g h t s

• We propose a residual-based test for fractional cointegration.• The integration orders can be real-valued and the resulting cointegrating error can be stationary or nonstationary.• The proposed test is simple to implement, has standard asymptotics and does not require a prescribed bandwidth.• The proposed test has better power than the GPH test for unit-root series and has satisfactory sizes when other tests fail.

a r t i c l e i n f o

Article history:Received 4 August 2014Received in revised form30 October 2014Accepted 10 November 2014Available online 17 November 2014

JEL classification:C12C32

Keywords:Fractional cointegrationAsymptotic normalResidual-based testMonte Carlo experiment

a b s t r a c t

By allowing deviations from equilibrium to follow a fractionally integrated process, the notion of frac-tional cointegration analysis encompasses a wide range of mean-reverting behaviors. For fractional coin-tegrations, asymptotic theories have been extensively studied, and numerous empirical studies have beenconducted in finance and economics. But as far as testing for fractional cointegration is concerned, mostof the testing procedures have restrictions on the integration orders of observed time series or integratingerror and some tests involve determination of bandwidth. In this paper, a general fractional cointegrationmodel with the observed series and the cointegrating error being fractional processes is considered, anda residual-based testing procedure for fractional cointegration is proposed. Under some regularity con-ditions, the test statistic has an asymptotic standard normal distribution under the null hypothesis of nofractional cointegration and diverges under the alternatives. This test procedure is easy to implement andworks well in finite samples, as reported in a Monte Carlo experiment.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

In the past two decades, the concept of fractional cointegration,which allows the equilibriumerror to follow a fractional integratedprocess, has received much attention in the finance and econo-metric literature. A partial list of some of the recent developmentsin this area includes Cheung and Lai (1993) and Soofi (1998) whotest the purchasing power parity hypothesis, Baillie and Bollerslev(1994) and Hassler et al. (2006), who investigate the memory ofexchange rates, and Booth and Tse (1995) and Dittmann (2000)who explore the dynamic of interest rate future markets and stockmarket prices, respectively. All of these studies detect evidence offractional cointegration and obtain satisfactory result under the as-sumption that the observations are I(1) processes.

∗ Corresponding author. Tel.: +852 3943 8519.E-mail address: nhchan@sta.cuhk.edu.hk (N.H. Chan).

http://dx.doi.org/10.1016/j.econlet.2014.11.0090165-1765/© 2014 Elsevier B.V. All rights reserved.

Testing for fractional cointegration has subsequently been gen-eralized to fractionally integrated processes. Robinson (2008) pro-poses a test based on the joint local Whittle estimation of allparameters, which rules out the possibility that the two under-lying series have equal integration orders. For time series withequal integration orders, Marinucci and Robinson (2001) proposea Hausman-type test for no cointegration, which involves deter-mination of a bandwidth. Marmol and Velasco (2004) construct aHausman-type test to test for the presence of fractional cointegra-tion, with the additional assumption that the cointegration error isnonstationary.

In this paper, we derive a general cointegration testing proce-dure for two fractionally integrated processes with equal integra-tion orders, which are assumed to be unknown. This test allows thecointegrating error to be fractionally integrated without requiringthem to be nonstationary. The test is shown to be asymptoticallystandard normal under the null hypothesis of no fractional cointe-gration and diverges under the fractional cointegration alternative.

44 B. Wang et al. / Economics Letters 126 (2015) 43–46

The test is residual-based, which was initially suggested by EngleandGranger (1987) and studied by Phillips andOuliaris (1990). Themethodology does not require further specification of the short-run dynamics of the underlying processes because semiparamet-ric estimates of the long-memory parameters and the long-runcovariancematrix are used. Only preliminary estimates of the inte-gration orders of the underlying series and the regression residualare needed. This test is easy to implement and does not require anyuser-chosen number such as bandwidth.

The proposed test requires the integration orders of the ob-served series to be equal, which can be determined by the residual-based test proposed by Hualde (2013) or Wang (2008), both ofwhich are valid irrespective of whether the series are cointegrated.

In the next section, we present the testing procedure andthe asymptotic theory. Section 3 reports the empirical sizes andpowers of our test via a Monte Carlo study. Section 4 concludes.

2. Testing for fractional cointegration

Consider the following bivariate model (yt , xt)′, with primedenoting transposition and t ∈ 0, ±1, ±2, . . .,

yt = βxt + ∆−δϑ1t1(t > 0),

xt = ∆−dϑ2t1(t > 0), (1)

where 1(·) is the indicator function, δ = 1−L, L is the lag operator,d > 1/2, δ ≤ d, and υt = (υ1t , υ2t)

′ is white noise defined inAssumption 1 below. We concentrate on the case that xt and ytare both nonstationary with d > 1/2. The case δ < d indicatesthe existence of fractional cointegration. By Taylor’s expansion,∆−d

=

∞

j=0 πj(−d)Lj, πj(d) =0(j+d)

0(d)0(j+1) for d = 0, −1, −2, . . . ,where 0(·) is the gamma function, taking 0(α) = ∞ for α =

0, −1, −2, . . . , 0(0)/0(0) = 1. Denote the k × k identity matrixby Ik and the Euclidean norm by ∥ · ∥.

Assumption 1. Consider the process υt = A(L)ϵt , A(L) =

∞

j=0 Aj

Lj. Assume that

1.1. det(A(1)) = 0 and

∞

j=1 j∥Aj∥2 < ∞;

1.2. ϵt are i.i.d. vectorswithmean zero, positive definite covariancematrixΩ , and E∥ϵt∥

q < ∞, for some q > max(4, 2/(2d−1));1.3. fii(0) > 0, i = 1, 2, where fij(0) is the (i, j) element of the

spectral density of υt , denoted by f (λ).

Assumption 1 is common because it is satisfied by the usualstationary and invertible autoregressive moving average (ARMA)processes. This assumption is similar to Assumption 1 of Robin-son and Hualde (2003), and Assumptions A–C of Marmol and Ve-lasco (2004), which is a condition for applying the functional limittheorem of Marinucci and Robinson (2000). Assumption 1.1 en-sures that the limiting process of partial sum of υt has nondegen-erate finite-dimensional distributions. Assumption 1.2 along withAssumption 1.3 imply that f (λ) is Lip(γ ), γ > 0, which enablesone to obtain the asymptotic properties of the test statistic, seeTheorem 1 below. A larger d entails weaker moment conditions.If d > 3/4, max(4, 2/(2d − 1)) = 4, then d has no restriction onthe moments of ϵt .

Under Assumption 1, model (1) means that xt and yt are bothtype-II fractionally integrated I(d) processes and a linear combina-tion yt − βxt is I(δ). In this way, the traditional I(1) cointegrationis a special case of model (1) with d = 1 and δ = 0.

Denote the regression residual in (1) by ut = yt − βxt , then utis an I(δ) process. Therefore, testing the hypothesis of no fractionalcointegration between yt and xt against existence of fractionalcointegration can be formulated as H0 : δ = d against H1 : δ < d.

To construct the test statistic, it is important to estimate δ andd precisely, thus we impose the following assumption.

Assumption 2. Under both the null and the alternative hypothe-sis,

2.1. there exists a positive constant K < ∞ and estimates d, δ ofd and δ respectively, such that

|d| + |δ| ≤ K , (2)

and for some η > 0,

d = d + Op(T−η), (3)

δ = δ + Op(T−η); (4)

2.2. f (0)p

→ f (0), wherep

→ stands for convergence in probability.

Assumption 2.1 is the same as Assumption 3 of Robinson andHualde (2003) and Assumption 2 of Hualde and Velasco (2008).Condition (2) is not restrictive if our estimates are optimizers ofthe corresponding functions over compact sets. The parameter dcan be estimated from xt by parametric or semiparametric mem-ory estimates, for example the approximate Gaussian maximumlikelihood estimates proposed by Beran (1995) or Whittle pseudo-maximum likelihood estimation proposed by Velasco and Robin-son (2000), thus condition (3) is easily satisfied.

Condition (4) is more subtle because β is unknown and henceut is unobserved. The estimate of δ requires a proxy ut , which is anestimate of ut . Let β be the Ordinary Least Squares (OLS) or NarrowBand (NB, see Robinson andMarinucci (2001)) estimates ofβ . Thenunder Assumptions 1.2 and 1.3, and some other mild conditions,estimation of the memory parameter of residuals ut = yt − βxtleads to a consistent estimate of δ under thenull and the alternativehypotheses (cf. Velasco (2003), Hualde and Velasco (2008)).

Let F = F(δ, f22(0)) =T−1/2

t ∆δxt(2π f22(0))1/2

be the test for H0 : δ = dagainst the alternative H1 : δ < d.

Theorem 1. Let Assumptions 1 and 2 hold, xt and yt are definedin (1), then F

d→N(0, 1) under H0 and F = Op(T d−δ) under H1,

whered

→ stands for convergence in distribution.

Proof. Since ∆dxt = υ2t , which is an I(0) process, it followsthat F(δ, f22(0)) =

T−1/2 t ∆−(d−δ)υ2t

(2π f22(0))1/2. Under H0, F(δ, f22(0)) =

T−1/2 t υ2t

(2π f22(0))1/2, then F(δ, f22(0))

d→N(0, 1) in view of the functional

limit theory of I(0) process. Under Assumptions 1 and 2, δ andf22(0) are consistent estimates of δ and f22(0). It can be provedthat F(δ, f22(0)) − F = op(1) (see Appendix). Consequently F

d→

N(0, 1). Under H1, ∆δxt is I(d − δ) since xt is I(d) process,then T−1/2 T

t=1 ∆−δxt = Op(T d−δ), and further F(δ, f22(0)) =

Op(T d−δ). Since δ and f22(0) are consistent estimates of δ and f22(0),with Assumptions 1 and 2, it can be proved that F(δ, f22(0))− F =

op(T d−δ) (see Appendix), thus F = Op(T d−δ).

3. Monte Carlo simulations

Monte Carlo experiments are conducted to examine the finitesample performance of the test. Let (yt , xt)′ be generated frommodel (1) with β = 1, υt = (υ1t , υ2t)

′ being a Gaussian whitenoisewith E(vt) = 0, Var(v1t) = Var(v2t) = 1 and Cov(v1t , v2t) =

ρ. The initial values υ1t , υ2t , t ≤ 0 are set to be zero. We considercaseswith ρ = 0 and 0.5, and sample sizes T = 100, 250, and 500.Let d = 0.6, 0.8, 1, 1.2 and for a given d, let δ = d, d−0.2, d−0.4and d−0.6. For a given set of (ρ, d, δ, T ) and β = 1, we obtain theobserved series yt , xt , t = 1, . . . , T , and further testing procedure

B. Wang et al. / Economics Letters 126 (2015) 43–46 45

Table 1Empirical sizes based on different levels d.

α = 0.01 α = 0.05 α = 0.1d/T 100 250 500 100 250 500 100 250 500

ρ = 0

1.2 0.041 0.018 0.012 0.108 0.072 0.065 0.179 0.150 0.1251 0.048 0.021 0.011 0.118 0.093 0.051 0.159 0.137 0.1120.8 0.032 0.020 0.015 0.093 0.081 0.059 0.149 0.124 0.1180.6 0.039 0.017 0.013 0.078 0.056 0.052 0.113 0.105 0.104

ρ = 0.5

1.2 0.043 0.022 0.015 0.103 0.078 0.069 0.189 0.140 0.1301 0.041 0.026 0.014 0.120 0.095 0.061 0.179 0.145 0.1190.8 0.042 0.027 0.019 0.103 0.071 0.053 0.159 0.135 0.1260.6 0.036 0.023 0.015 0.088 0.066 0.058 0.135 0.125 0.108

Table 2Empirical powers based on different d and δ.

α = 0.01 α = 0.05 α = 0.1δ/T 100 250 500 100 250 500 100 250 500

d = 10.8 0.085 0.118 0.218 0.178 0.231 0.354 0.197 0.251 0.4390.6 0.134 0.258 0.497 0.212 0.430 0.732 0.281 0.612 0.8350.4 0.585 0.979 0.989 0.648 0.955 0.992 0.804 0.983 0.999

d = 0.80.6 0.098 0.121 0.231 0.173 0.208 0.387 0.248 0.302 0.4750.4 0.375 0.432 0.521 0.435 0.559 0.638 0.793 0.977 0.9950.2 0.730 0.970 0.998 0.735 0.977 0.998 0.987 0.999 0.985

d = 0.60.4 0.125 0.231 0.650 0.231 0.452 0.765 0.289 0.598 0.8430.2 0.333 0.548 0.689 0.485 0.623 0.718 0.584 0.731 0.8590 0.653 0.955 0.978 0.741 0.967 0.998 0.786 0.971 0.996

is implemented in the following procedure:

Step 1. We estimate d by the method presented in Beran (1995).Let αj(d) =

min(m,j)k=0 (−1)k m!

k!(m−k)!πj−k(−γ ), where d =

m+γ , γ ∈ (−1/2, 1/2), m is an integer, and πj is definedin Section 2. Let et(d, x) =

t−1j=0 aj(d)xt−j, and S(d, x) =T

t=1 e2t (d, x), d is estimated by d = argmind S(d, x). It

follows that d satisfies Assumption 2 as d = d+Op(T−1/2).Step 2. Let f22(0) =

12πT

Tt=1(∆

dxt)2, which is a consistentestimate of f22(0).

Step 3. Compute the OLS estimate β from by β =

t xt ytt x

2t

and

obtain δ from ut = yt − βxt by the same procedure as inStep 1.

In this way, the test statistic F(δ, f22(0)) can be computed. Using10,000 replications, we report the rejection ratio corresponding tonominal sizes α = 0.01, 0.05 and 0.1 respectively.

We first look at the sizes (under H0). Table 1 shows that F isoversized, and all the empirical sizes approach the nominal size αwhen sample size T increases. There is no obvious change in sizewhen d takes different values. The reason is that when H0 hold, theproposed test statistic F depends on the estimation error δ − δ, noton the particular value δ and d.

We also look at the power for δ < d, which is given in Table 2.As expected, the simulated powers tend to 1 as the sample size Tincreases and as the cointegration gap d−δ increases. Although theproposed test allows d to be unknown, for the last experiment weassume d = 1, to compare with the GPH test proposed by Cheungand Lai (1993). The GPH test is a fractional cointegration test onlyfor the case that yt and xt are I(1), thus the residual ut is I(1) and∆ut is I(0) under H0 : d = δ = 1. Table 3 shows that the proposedtest has better powers than the GPH test.

4. Conclusion

Wepropose a residual-based test for no fractional cointegrationagainst fractional cointegration between two integrated time

Table 3Empirical power comparison, d = 1, α = 0.05.

GPH Fδ/T 100 250 500 100 250 500

1 0.095 0.058 0.051 0.103 0.061 0.0530.8 0.165 0.151 0.418 0.189 0.238 0.3710.6 0.291 0.469 0.697 0.341 0.451 0.7790.4 0.699 0.896 0.989 0.723 0.945 0.995

series yt and xt , where the observed series are I(d) processesand the regression residual ut = yt − βxt is an I(δ) process.Our test procedure allows δ and d to be real-valued and includesthe traditional cointegration as a special case. With consistentestimate d and δ obtained from xt and the residual ut = yt − βxt ,we construct a test statistic, which has an asymptotic standardnormal distribution under the null hypothesis of no cointegrationand diverges at the rate of T d−δ under the alternative hypothesisof fractional cointegration. Our test procedure is very simple tocompute and enjoys standard asymptotic properties. Our testshows better power than the customary GPH test for unit-rootcointegration, and performs well in a general framework whereother methods may fail.

Acknowledgments

We would like to thank an anonymous referee for manyhelpful comments. Man Wang’s work is supported in part bythe Fundamental Research Funds for the Central Universities (No.14D210809). Ngai Hang Chan’s research is partially supportedby the HKSAR-RGC-GRF Nos 400313, 400410 and 14300514, andHKSAR-RGC-CRF: CityU8/CRG/12G.

Appendix

Let g(d, zt) = ∆dzt , then g(d, zt) =t−1

i=0 πi(d)zt−i if zt = 0for t ≤ 0 and g(r)(d, zt) =

t−1i=1 π

(r)i (d)zt−i. By Taylor expansion

around d, for a certain constant R to be defined subsequently, we

46 B. Wang et al. / Economics Letters 126 (2015) 43–46

show that

T−1/2T

t=1

∆δxt − T−1/2T

t=1

∆δxt

= T−1/2T

t=1

(g(d − δ; υ2t) − g(d − δ; υ2t))

=1

√T

R−1r=1

(δ − δ)r

r!

Tt=1

g(r)(d − δ; υ2t)

+(δ − δ)R

R!√T

Tt=1

g(R)(d − δ; υ2t)

=

op(1), under H0,

op(T d−δ), under H1,(A.1)

where δ ∈ (min(δ, δ),max(δ, δ)).By some simple calculations, it can be derived that |π

(r)i (0)| ≤

Kr (log(i+1))r−1

i−r+1 , i ≥ r for some constant Kr .UnderH0, by a simplemodification of the proof of (C.8) of Robin-

son and Hualde (2003), Var(T

t=1 g(r)(0, υ2t)) = O(T (log T )2r),

under Assumption 1 and the bound of |π(r)i (0)|. Therefore,T

t=1 g(r)(0, υ2t) = Op(T 1/2(log T )r). From Assumption 2, the first

term of (A.1) is Op(T−η log T ). Next, by the bound of |π(r)i (0)| and

Assumption 2, g(R)(δ − δ; υ2t) = Op(T 1/2) uniformly in T , sothe second term in (A.1) is Op(T 1−Rη). Choose R > (1 + η)/η,then (A.1) is Op(T−η log T ) = op(1). As f22(0) is a consistent es-timate of f22(0), F − F(δ, f22(0)) = op(1). Further, when δ = d,

F(δ, f22(0)) =T−1/2

t υ2t(2π f22(0))1/2

d→N(0, 1), from Theorem 1 of Marin-

ucci and Robinson (2000). Consequently, Fd

→N(0, 1).Under H1, by a modification of the proof of (C.8) of Robinson

and Hualde (2003), it can be shown that Var(T

t=1 g(r)(d −

δ, υ2t)) = O(T 2(d−δ)+1(log T )2r), so (T

t=1 g(r)(d − δ, υ2t)) =

Op(T d−δ+1/2(log T )r), Further the first termof (A.1) isOp(T d−δ−η logT ). As in Lemma C.4 of Robinson and Hualde (2003), Var(g(R)(d −

δ, υ2t)) ≤ Kt−1

i=1 (log i)2Ri2(d−δ)−2 for some constant K . ThenTt=1 g

(R)(d − δ, υ) = Op((log T )RT d−δ) and the second term in

(A.1) is Op((log T )RT−ηR+d−δ). If we choose R > d−δη

, this becomes

Op(T d−δ−η log T ). It follows that (A.1) is op(T d−δ). As f22(0) is aconsistent estimate of f22(0), F − F(δ, f22(0)) = op(T d−δ). Underδ < d, From Assumption 1 and Theorem 1 of Marinucci andRobinson (2000), it follows that F(δ, f22(0)) = Op(T d−δ), whichleads to F = Op(T d−δ). The proof of Theorem 1 completes.

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