国立中山大学：《计量经济学》（英文版） Fractional PP

Testing for a Fractional Unit Root in Time Series Regression Chingnun Lee, Tzu-Hsiang Liao2 and Fu-Shuen Shie Inst. of Economics, National Sun Yat-sen Univ Kaohsiung, Taiwan Dept. of Finance, National Central Univ, Chung-Li, Taiwan Dept. of Finance, National Taiwan Univ, Taipei, Taiwan Abstract This paper presents a nonparametric test of the differencing parameter of a general fractionally integrated process, which allows for weakly dependent and heteroge- neously distributed innovation in the short run dynamics. It is shown that these test statistics derived from standardized ordinary least squared estimator of a simple autoregressive model are consistent and fare well both in terms of power and size The paper ends with two empirical applications Key Words: Unit Root, Fractional Integrated Process, Power JEL classification: C12. C22 August 2004 Corresponding author, Address: Graduate Institute of Economics National Sun Yat-sen University 70 Leng-hai Road Kaohsiung. Taiwan 804 Tel:886-7-5252000ext.5618 ax:886-7-5255611 e-mail: lee_econ @mail nsysu. edu.tw

Testing for a Fractional Unit Root in Time Series Regression Chingnun Lee1 ∗ , Tzu-Hsiang Liao2 and Fu-Shuen Shie3 1 Inst. of Economics, National Sun Yat-sen Univ., Kaohsiung, Taiwan 2Dept. of Finance, National Central Univ., Chung-Li, Taiwan 3Dept. of Finance, National Taiwan Univ., Taipei, Taiwan Abstract This paper presents a nonparametric test of the differencing parameter of a general fractionally integrated process, which allows for weakly dependent and heteroge￾neously distributed innovation in the short run dynamics. It is shown that these test statistics derived from standardized ordinary least squared estimator of a simple autoregressive model are consistent and fare well both in terms of power and size. The paper ends with two empirical applications. Key Words: Unit Root, Fractional Integrated Process, Power. JEL classification: C12, C22. August 2004 ∗Corresponding author, Address: Graduate Institute of Economics National Sun Yat-sen University 70 Leng-hai Road Kaohsiung, Taiwan 804 Tel: 886-7-5252000 ext. 5618 Fax: 886-7-5255611 e-mail: lee econ@mail.nsysu.edu.tw

1 Introduction It has become quite a standard practice in applied work to perform test on whether a variable is integrated or stationary using both the null hypothesis of I(1)and I(o) See Phillips and Xiao(1998) for an updated survey of unit root testing approaches However, by proceeding in this way, it is often found that both the null hypothesis are rejected( for example, see Tsay(2000)), suggesting that many time series are not well represented as either I(1) or I(0). In view of this outcome, the class of fractionally integrated processes, denoted as I(d), where the order of integration d is extended to be any real number, has proved very useful in capturing the persistence of many long-memory processes. This procedure above raises two viewpoints on the issues. First, the power of traditional unit roots test against fractional alternative such as Dickey and Fuller(DF, 1979 and 1981), Phillips and Perron(PP, 1988 and Kwiatkowski, Phillips, Schmidt and Shin(KPSS, 1992). Secondly, the power of various technique of estimation of d both when they are applied to be the test of unit root(d=1) and to the inference of the true value of d The power of traditional unit root tests have been studied by Diebold and Rudebusch(1991), Lee and Schmidt(1996), Kramer(1998), Dolado, Gonzalo and Mayoral (2002) and Lee and Shie(2002). In general, they al show that these unit roots test are consistent when the alternative is a I(d )process but their power turns out to be quite low. In particular, this lack of power has motivated the development of new approach to estimate the order of integration d, of a series directly, where d does not assume a special value such as unity or zero, but is apparently arbitrary To inference about the fractional difference parameter d from an observed process, there are a very large number of rather heterogeneous methods to estimate and/or test d. In parametric methodology, most methods are based on the autore- gressive fractionally-integrated moving average(ARFIMA) process as pioneered by Granger and Joyeux(1980) and Hosking(1981). Fox and Taqqu(1986)construct an asymptotic approximation to the likelihood of an ARFIMa process in the fre- which we mean that other than d, the short-run dynamics have to be specified parametrically to estimation or inference about d

1 Introduction It has become quite a standard practice in applied work to perform test on whether a variable is integrated or stationary using both the null hypothesis of I(1) and I(0). See Phillips and Xiao (1998) for an updated survey of unit root testing approaches. However, by proceeding in this way, it is often found that both the null hypothesis are rejected( for example, see Tsay (2000)), suggesting that many time series are not well represented as either I(1) or I(0). In view of this outcome, the class of fractionally integrated processes, denoted as I(d), where the order of integration d is extended to be any real number, has proved very useful in capturing the persistence of many long-memory processes. This procedure above raises two viewpoints on the issues. First, the power of traditional unit roots test against fractional alternative such as Dickey and Fuller (DF, 1979 and 1981), Phillips and Perron (PP, 1988) and Kwiatkowski, Phillips, Schmidt and Shin (KPSS, 1992). Secondly, the power of various technique of estimation of d both when they are applied to be the test of unit root (d = 1) and to the inference of the true value of d. The power of traditional unit root tests have been studied by Diebold and Rudebusch (1991), Lee and Schmidt (1996), Kramer (1998), Dolado, Gonzalo and Mayoral (2002) and Lee and Shie (2002). In general, they al show that these unit roots test are consistent when the alternative is a I(d) process but their power turns out to be quite low. In particular, this lack of power has motivated the development of new approach to estimate the order of integration d, of a series directly, where d does not assume a special value such as unity or zero, but is apparently arbitrary. To inference about the fractional difference parameter d from an observed process, there are a very large number of rather heterogeneous methods to estimate and/or test d. In parametric methodology, 1 most methods are based on the autore￾gressive fractionally-integrated moving average (ARFIMA) process as pioneered by Granger and Joyeux (1980) and Hosking (1981). Fox and Taqqu (1986) construct an asymptotic approximation to the likelihood of an ARFIMA process in the fre- 1which we mean that other than d, the short-run dynamics have to be specified parametrically to estimation or inference about d. 1

quency domain, Sowell(1992)construct the exact likelihood function of an ARFIMA process in the time domain, and Chung and Baillie(1993 ) propose conditional sums of squared(CSS)estimator for an ARFIMA process Standard maximum likelihood estimator(MLE) inference procedures then apply to these estimates. Besides MLE estimator, Chung and Schmidt(1995) and Mayroal(2001) had proposed a mini- mum distance estimator for the ARFIMa process. On the other hand of parametric methodology, Robinson(1994)and Tanaka(1999) have proposed LM test for d in the frequency and in the time domain, respectively. To implement these tests pirically, a parametric short-run dynamic arMa structure has to be specified Gli-Alana and Robinson(1997) and Gli-Alana(2000) for example To inference d nonparametrically, Geweke and Porter-Hudak(GPH, 1983)sug- gests a regression of the ordinate of the log spectral density on trigonometric function to estimate d. Robinson(1992) consider frequency domain approach to find con sistent estimate of d in the absence of any parameterization of the autocovariance unction. In general the parametric methods present narrower confidence intervals over nonparametric ones. However, a drawback is that the parameters are sensitive to the class of models considered and may be misleading because of misspecification ee Hauser et al.(1999) By allowing the short-run dynamics to be weakly dependent and heteroge- neously distributed, our aim of the present paper is to develop a new nonparametric test for the fractional difference parameter, d for0<d<l, from a generally frac tional integrated process in the time domain. It is shown that this test statistics derived from the simple least squares regression of an simple autoregressive model is easy to compute and is consistent against possible alternative including I(1)process It also fare very well in finite samples, in terms of power and size, when compare to other competing tests The rest of the paper is organized as follows. Section 2 reviews the functional central limit theorem for a quite general fractional integrated process. In Section 3 we define our model. estimators and their limiting distributions. In Section 4.we provide a test statistics for testing the fractional difference parameters. Simulation

quency domain, Sowell (1992) construct the exact likelihood function of an ARFIMA process in the time domain, and Chung and Baillie (1993) propose conditional sums of squared (CSS) estimator for an ARFIMA process. Standard maximum likelihood estimator (MLE) inference procedures then apply to these estimates. Besides MLE estimator, Chung and Schmidt (1995) and Mayroal (2001) had proposed a mini￾mum distance estimator for the ARFIMA process. On the other hand of parametric methodology, Robinson (1994) and Tanaka (1999) have proposed LM test for d in the frequency and in the time domain, respectively. To implement these tests em￾pirically, a parametric short-run dynamic ARMA structure has to be specified. See Gli-Ala˜na and Robinson (1997) and Gli-Ala˜na (2000) for example. To inference d nonparametrically, Geweke and Porter-Hudak (GPH, 1983) sug￾gests a regression of the ordinate of the log spectral density on trigonometric function to estimate d. Robinson (1992) consider frequency domain approach to find con￾sistent estimate of d in the absence of any parameterization of the autocovariance function. In general the parametric methods present narrower confidence intervals over nonparametric ones. However, a drawback is that the parameters are sensitive to the class of models considered and may be misleading because of misspecification. See Hauser et al. (1999). By allowing the short-run dynamics to be weakly dependent and heteroge￾neously distributed, our aim of the present paper is to develop a new nonparametric test for the fractional difference parameter, d for 0 < d ≤ 1, from a generally frac￾tional integrated process in the time domain. It is shown that this test statistics derived from the simple least squares regression of an simple autoregressive model is easy to compute and is consistent against possible alternative including I(1) process. It also fare very well in finite samples, in terms of power and size, when compared to other competing tests. The rest of the paper is organized as follows. Section 2 reviews the functional central limit theorem for a quite general fractional integrated process. In Section 3 we define our model, estimators and their limiting distributions. In Section 4, we provide a test statistics for testing the fractional difference parameters. Simulation 2

evidence of size and power of this test is shown in Section 5. Section 6 discuss some empirical application to the previous test. Finally, Section 7 draws some concluding remarks Proofs of Theorems and Lemma are gathered in Appendix. Fron now on, throughout this paper, the following conventional notation is adopted: L is the lag operator, T( denotes the gamma function, = denotes weak convergence of associated probability measures, denotes convergence in probability, [z] means the largest integer that is smaller than or equal to 2, d=d+l, and we let ei fi denote that ei/fi+ 1 as 2 Preliminaries The class of I(d) process, ut is customarily written in the form innovation process E, the fractional difference of ut, is a stationary and weakly de- pendent process to be specified below. The process ut is covariance stationary and ergodic for -0.52;(c)is covari- ance stationary,and0<2<∞ where a2=lir-sT-1∑1∑=1E(e1s,);(d) is strong mixing with mixing coefficients am that satisfy 2o_1 am-2/7<oo Assumption 1 allow for both temporal dependence and heteroskedasticity in the process of innovations Et. These include all Gaussian and many other stationary finite order ARMA models under very general conditions on the underlying errors 3

evidence of size and power of this test is shown in Section 5. Section 6 discuss some empirical application to the previous test. Finally, Section 7 draws some concluding remarks. Proofs of Theorems and Lemma are gathered in Appendix. Fron now on,throughout this paper, the following conventional notation is adopted: L is the lag operator, Γ(·) denotes the gamma function, ⇒ denotes weak convergence of associated probability measures, p −→ denotes convergence in probability, [z] means the largest integer that is smaller than or equal to z, ˜d = d + 1, and we let ei ∼ fi denote that ei/fi → 1 as i → ∞. 2 Preliminaries The class of I(d) process, ut is customarily written in the form (1 − L) dut = εt , (1) innovation process ε, the fractional difference of ut , is a stationary and weakly de￾pendent process to be specified below. The process ut is covariance stationary and ergodic for −0.5 2; (c) is covari￾ance stationary, and 0 < σ2 ε < ∞ where σ 2 ε = limT→∞ T −1 PT t=1 PT s=1 E(εtεs); (d) is strong mixing with mixing coefficients αm that satisfy P∞ m=1 α 1−2/γ m < ∞. Assumption 1 allow for both temporal dependence and heteroskedasticity in the process of innovations εt . These include all Gaussian and many other stationary finite order ARMA models under very general conditions on the underlying errors. 3

Define the variance of the partial sums of the I(d) process ut by af= Var(>taut then we have the following functional central limit theorem that will be extensive used in our theoretical development below. The results are due to Davidson and De Jong(2000) Lemma 1: Suppose(1-L)ut =Et, -0.5< d <0.5 and Et satisfy Assumption 1, then as 7→∞ a)af w ovaTIt2d, and (b)n1∑→B(r),for∈0,1 Here, Ba(r) is the normalized fractional Brownian motion that is defined by the following stochastic integral: 2 r (r-)B(x)+/(x-a)2-(-)dB(x),(2 T(1+dVi with Va= r(tdr(i+2d+Jo [(1+r)d-rddr)=(+2r(+r(-d and b(r) standard brownian motion This type of fractional Brownian motion is so defined as to make EBa(1)2=1 Fractional Brownian motion differs from standard Brownian motion B(r) in having correlated increments. See Mandelbrot and Van Ness(1968) and Marinucci and Robinson(1999)for additional detail on the fractional Brownian motion Lemma 1(b) is a functional central limit theorem for a generally fractional inte- grated process that could apply to a large class of fractionally integrated process in- cluding the well-known ARFIMA (p, d, q)process. See for example, Davydov(1970) Akonom and Gourieroux(1987), Hosking(1996), Marinucci and Robinson(2000) and Chung(2002) 2The original definition of fractional Brownian motion shown in Sowell(1990)is Ba(r)=1/r(1+ d)Jo(r-r)dB(ar). However, Marinucci and Robinson(1999)show that it require correction by replacing it with the definition of fractional Brownian motion as in(2)

Define the variance of the partial sums of the I(d) process ut by σ 2 T = V ar( PT t=1 ut), then we have the following functional central limit theorem that will be extensive used in our theoretical development below. The results are due to Davidson and De Jong (2000). Lemma 1: Suppose (1 − L) dut = εt , −0.5 < d < 0.5 and εt satisfy Assumption 1, then as T → ∞, (a) σ 2 T ∼ σ 2 εVdT 1+2d , and (b) σ −1 T P[T r] t=1 ut ⇒ Bd(r), for r ∈ [0, 1]. Here, Bd(r) is the normalized fractional Brownian motion that is defined by the following stochastic integral: 2 Bd(r) ≡ 1 Γ(1 + d)V 1 2 d ( Z r 0 (r − x) d dB(x) + Z 0 −∞ [(r − x) d − (−x) d ]dB(x)), (2) with Vd ≡ 1 Γ(1+d) 2 ( 1 1+2d + R ∞ 0 [(1 + τ ) d − τ d ]dτ ) = Γ(1−2d) (1+2d)Γ(1+d)Γ(1−d) and B(r) is the standard Brownian motion. This type of fractional Brownian motion is so defined as to make EBd(1)2 = 1. Fractional Brownian motion differs from standard Brownian motion B(r) in having correlated increments. See Mandelbrot and Van Ness (1968) and Marinucci and Robinson (1999) for additional detail on the fractional Brownian motion. Lemma 1(b) is a functional central limit theorem for a generally fractional inte￾grated process that could apply to a large class of fractionally integrated process in￾cluding the well-known ARFIMA (p,d,q) process. See for example, Davydov (1970), Akonom and Gourieroux (1987), Hosking (1996), Marinucci and Robinson (2000) and Chung (2002). 2The original definition of fractional Brownian motion shown in Sowell (1990) is Bd(r) = 1/Γ(1+ d) R r 0 (r − x) ddB(x). However, Marinucci and Robinson (1999) show that it require correction by replacing it with the definition of fractional Brownian motion as in (2). 4

3 The model and estimators 3.1 A generally nonstationary I(d)process Let yt be a nonstationary fractional difference process generated by There 0.5< d< 1.5, and Et, the fractional difference of yt, satisfy Assumption 1 The process yt can be also represented equivalently as 6=1 and(1-L)u where d=1+d, and.5< d <0.5. Initial condition for (3)are set at t=0 and yo=0. We consider the two least-squares regression equations yt= Byt-1+ it yt=&+ Byt-1+ut, re B and(a, B) are the conventional least-square regression coefficients. We shall be concerned with the limiting distribution of the regression coefficients in(6) and (7)under the hypothesis that the data are generated by 3) or equivalently by(4) and(5). Thus for the null values d= do, it will become B=l, a=0, and d= do Under(4)and(5), sample moments of yt and ut that are useful to derive the Ols estimator are collected in the following lemma Lemma2.AsT→∞,then (a)T l→VoBA(1) T 2-2d 2-1→Va2J[Ba(r)2d (c)T-1-2的→va2[Ba(1)2 T W-1→V=JBar 5

3 The model and estimators 3.1 A generally nonstationary I(d) process Let yt be a nonstationary fractional difference process generated by (1 − L) d˜ yt = εt , (3) where 0.5 < ˜d < 1.5, and εt , the fractional difference of yt , satisfy Assumption 1. The process yt can be also represented equivalently as yt = βyt−1 + ut ; (4) β = 1 and (1 − L) dut = εt , (5) where ˜d = 1 + d, and −0.5 < d < 0.5. Initial condition for (3) are set at t = 0 and y0 = 0. We consider the two least-squares regression equations: yt = βyˆ t−1 + ˆut , (6) yt = ˘α + βy˘ t−1 + ˘ut , (7) where βˆ and (˘α, β˘) are the conventional least-square regression coefficients. We shall be concerned with the limiting distribution of the regression coefficients in (6) and (7) under the hypothesis that the data are generated by (3) or equivalently by (4) and (5). Thus for the null values ˜d = ˜d0, it will become β = 1, α = 0, and d = d0. Under (4) and (5), sample moments of yt and ut that are useful to derive the OLS estimator are collected in the following lemma. Lemma 2. As T → ∞, then (a) T − 1 2 −d P T t=1 ut ⇒ V 1 2 d σεBd(1), (b) T −2−2d P T t=1 y 2 t−1 ⇒ Vdσ 2 ε R 1 0 [Bd(r)]2dr, (c) T −1−2d y 2 T ⇒ Vdσ 2 ε [Bd(1)]2 , (d) T − 3 2 −d P T t=1 yt−1 ⇒ V 1 2 d σε R 1 0 Bd(r)dr, 5

(e)T∑2 -1lt→是Va2[Ba(1)2ifd>0, (g)T1∑v-12-2ifd0 2Vao2Jo[Ba(r)]dr when d 0: and (c)T(B-1) 1{B(1)2-} when d=0 2 o [B(r)]2dr For the regression model(7), then as T-0o when d>0 then ()r+anB①(hB)h-号B(1)B) fo [Ba(r)2dr-lo Ba(r)dr)2 6

(e) T −1 P T t=1 u 2 t p −→ σ 2 u = E(u 2 t ), (f) T −1−2d P T t=1 yt−1ut ⇒ 1 2 Vdσ 2 ε [Bd(1)]2 if d > 0, (g) T −1 P T t=1 yt−1ut p −→ −1 2 σ 2 u if d 0; (b) T 1+2d (βˆ − 1) ⇒ − 1 2 σ 2 u Vdσ 2 ε R 1 0 [Bd(r)]2dr , when d 0, then (d) T 1 2 −dα˘ ⇒ V 1 2 d σεBd(1){ R 1 0 [Bd(r)]2dr − 1 2Bd(1) R 1 0 Bd(r)dr} R 1 0 [Bd(r)]2dr − [ R 1 0 Bd(r)dr] 2 , and 6

()7(-1)=B(1)2-B1)Ba JolBa(r)j2dr-lo Ba(r)dr] when d <0. then ojO Ba(r)dr ()T+→2B()h-[BB (g)m1+4(8-1)→-5VDB()Ph-bB()P when d=0 then o:{B(1)/lB()2d-3B(1)2-:}/bB()b (h)Ta→ and Jo[B(r)2dr-o B(r)dr]2 (i)T( 号{B(1)2-2}-B(1)/B(r)h Jo[B(r)dr-Uo B()dr]2 We first discuss the results from model(6). The convergence rates of (B-1) depend intrinsically on the degree of fractional difference in the ut process. The distribution of Tminl1, 1+2d(Br-1)is therefore called a generalized fractional unit root distribution. This fact is also discussed in Sowell(1990) and Tanaka(1999 Corollary 2. 4) where Et in(3)is assumed to be ii d. and to be infinite order moving average process respectively. It may be easily illustrated that when the innovation process Et is i.i.d. (0, 02), we have g [r(1-2d)/r(1-d)2]02,3 leading to the following simplification of part(b) and(c)of Theorem 1 (+dr(-d. when d<0 1+d ∫Ba(r)2dtr 7(6-1)=B()2-1 Jo[B(r)dr, when d=0 Result( 8)was first given by Sowell(1990)and result(9)was given by Dicke 3See for example Baillie(1996)

(e) T(β˘ − 1) ⇒ 1 2 [Bd(1)]2 − Bd(1) R 1 0 Bd(r)dr R 1 0 [Bd(r)]2dr − [ R 1 0 Bd(r)dr] 2 ; when d < 0, then (f) T 1 2 +dα˘ ⇒ 1 2 σ 2 u R 1 0 Bd(r)dr V 1 2 d σε{ R 1 0 [Bd(r)]2dr − [ R 1 0 Bd(r)dr] 2} , and (g) T 1+2d (β˘ − 1) ⇒ − 1 2 σ 2 u Vdσ 2 ε{ R 1 0 [Bd(r)]2dr − [ R 1 0 Bd(r)dr] 2} ; when d = 0, then (h) T 1 2α˘ ⇒ σε{B(1) R 1 0 [B(r)]2dr − 1 2 {[B(1)]2 − σ 2 u σ2 ε } R 1 0 B(r)dr} R 1 0 [B(r)]2dr − [ R 1 0 B(r)dr] 2 , and (i) T(β˘ − 1) ⇒ 1 2 {[B(1)]2 − σ 2 u σ2 ε } − B(1) R 1 0 B(r)dr R 1 0 [B(r)]2dr − [ R 1 0 B(r)dr] 2 . We first discuss the results from model (6). The convergence rates of (βˆ − 1) depend intrinsically on the degree of fractional difference in the ut process. The distribution of T min[1,1+2d] (βˆ T − 1) is therefore called a generalized fractional unit root distribution. This fact is also discussed in Sowell (1990) and Tanaka (1999, Corollary 2.4) where εt in (3) is assumed to be i.i.d. and to be infinite order moving average process respectively. It may be easily illustrated that when the innovation process εt is i.i.d.(0, σ2 ), we have σ 2 u = [Γ(1 − 2d)/Γ(1 − d) 2 ]σ 2 , 3 leading to the following simplification of part (b) and (c) of Theorem 1: T 1+2d (βˆ − 1) ⇒ − ( 1 2 + d) Γ(1+d) Γ(1−d) R 1 0 [Bd(r)]2dr , when d < 0; (8) and T(βˆ − 1) ⇒ 1 2 {[B(1)]2 − 1} R 1 0 [B(r)]2dr , when d = 0. (9) Result (8) was first given by Sowell (1990) and result (9) was given by Dickey 3See for example Baillie (1996). 7

and Fuller(1979). Theorem 1 therefore extends( 8 )and(9) to the very general case of weakly dependent distributed data after difference-d times It is interesting to note that when d>0, the assumption on Et did not play any role in determining this limiting distribution. It converges to the same distrib ution as that of Sowell(1990) and Tanaka(1999 ) When d<0, the distribution of 1) has the same general form for a very wide class of the innovation process Et. It reduces to be the distribution of Phillips(1987, Theorem 3.1,(c))when d=0 Similar conclusion applies to the results from model (7). The simplification of part (g)of Theorem I when Et is ii d (0, 02)is (+d JoB(r)]2dr-Lo B(r)dr)2 4 Statistical Inference of the fractional differ ence parameter 4. 1 Test for 0.5< d< 1 The limiting distribution of the regression coefficients when -05< d <0(0.5< d 1) given in last section depend upon the nuisance parameter on and a. These distributions are therefore not directly usable for statistical testing. However, since oa and o2 may be consistently estimated and the estimate may be used to construct modified statistics whose limiting distribution are independent of (o2, 02), there exist simple transformation of the test statistics which eliminate the nuisance parameters asymptotically This idea was first developed by Phillips(1987) and Phillips and Perron(1988 in the context of test for a unit root. Here we show how the similar procedure may be extended to apply to test for the fractional difference parameter value in a quite generally fractional integrated process. First due to the ergodicity assumption of ut, consistent estimation of o2 are provided by 02=T-2(yt-yt-12for data

and Fuller (1979). Theorem 1 therefore extends (8) and (9) to the very general case of weakly dependent distributed data after difference-d times. It is interesting to note that when d > 0, the assumption on εt did not play any role in determining this limiting distribution. It converges to the same distrib￾ution as that of Sowell (1990) and Tanaka (1999). When d < 0, the distribution of T 1+2d (βˆ−1) has the same general form for a very wide class of the innovation process εt . It reduces to be the distribution of Phillips (1987, Theorem 3.1, (c)) when d = 0. Similar conclusion applies to the results from model (7). The simplification of part (g) of Theorem 1 when εt is i.i.d.(0, σ2 ) is: T 1+2d (β˘ − 1) ⇒ − ( 1 2 + d) Γ(1+d) Γ(1−d) R 1 0 [B(r)]2dr − [ R 1 0 B(r)dr] 2 . (10) 4 Statistical Inference of the Fractional Differ￾ence Parameter 4.1 Test for 0.5 < ˜d < 1 The limiting distribution of the regression coefficients when −0.5 < d < 0 (0.5 < d < 1) given in last section depend upon the nuisance parameter σ 2 u and σ 2 ε . These distributions are therefore not directly usable for statistical testing. However, since σ 2 u and σ 2 ε may be consistently estimated and the estimate may be used to construct modified statistics whose limiting distribution are independent of (σ 2 u ,σ 2 ε ), there exist simple transformation of the test statistics which eliminate the nuisance parameters asymptotically. This idea was first developed by Phillips (1987) and Phillips and Perron (1988) in the context of test for a unit root. Here we show how the similar procedure may be extended to apply to test for the fractional difference parameter value in a quite generally fractional integrated process. First due to the ergodicity assumption of ut , consistent estimation of σ 2 u are provided by ˜σ 2 u = T −1 PT t=1(yt − yt−1) 2 for data 8

generated by(4)and(5). Since B and(&, B)are consistent by Theorem 1, we may also use a2=T->t(yt-Byt-1)2 and a ∑t=1(y as a consistent estimator of o2 for model(6)and(7), respectively Consistent estimation of o2 can be in the same spirit with that of Phillips and Perron(1988)by the following simple estimator based on truncated sample autocovariance, namely s1=re?+2∑mn∑s- (11) where Et=(1-L)(yt-yt-1)=(1-L)ut and wI=1-T/(L+1).We may also use Et=(1-L)(yt-Byt-1)and Et=(1-L(yt-a-Byt-1) as alternative estimate to Et in the construction of sT. Conditions for the consistency of sti are and ex- plored by Phillips(1987, Theorem 4.2). We now define some simple transformation of conventional test statistics from the regression(6) and(7) which eliminate the nuisance parameter dependencies asymptotically. Specifically, we define Z( T+2a(6-1 z(d)=71+2(3-1) (13) Z(d)is the transformation of the standard estimator T+2d(B-1)and Zu(d) is the transformation of T+2d(B-1). The limiting distribution of Z(d ) and Zu(d )is given bv Theoren2: Assume that l=0(T2), then as t→∞, z(d)→-2VBa(r)dr z(d0)→-1 2 Vdo o Bdo(r)]2dr-[o Bdo(r)dr]2)

generated by (4) and (5). Since βˆ and (˘α,β˘) are consistent by Theorem 1, we may also use ˆσ 2 u = T −1 PT t=1(yt − βyˆ t−1) 2 and ˘σ 2 u = T −1 PT t=1(yt − α˘ − βy˘ t−1) 2 as a consistent estimator of σ 2 u for model (6) and (7), respectively. Consistent estimation of σ 2 ε can be in the same spirit with that of Phillips and Perron (1988) by the following simple estimator based on truncated sample autocovariance, namely: s 2 T l = T −1X T t=1 ε˜ 2 t + 2T −1X l τ=1 wτ l X T t=τ+1 ε˜tε˜t−τ , (11) where ˜εt = (1−L) d (yt −yt−1) = (1−L) dut and wτ l = 1−τ/(l+ 1). We may also use εˆt = (1 − L) d (yt − βyˆ t−1) and ˘εt = (1 − L) d (yt − α˘ − βy˘ t−1) as alternative estimate to ˜εt in the construction of s 2 T l. Conditions for the consistency of s 2 T l are and ex￾plored by Phillips (1987, Theorem 4.2). We now define some simple transformation of conventional test statistics from the regression (6) and (7) which eliminate the nuisance parameter dependencies asymptotically. Specifically, we define Z(d) = s 2 T l σ˜ 2 u T 1+2d (βˆ − 1), (12) and Zµ(d) = s 2 T l σ˜ 2 u T 1+2d (β˘ − 1). (13) Z(d) is the transformation of the standard estimator T 1+2d (βˆ − 1) and Zµ(d) is the transformation of T 1+2d (β˘−1). The limiting distribution of Z(d) and Zµ(d) is given by: Theorem 2: Assume that l = o(T 1 2 ), then as T → ∞, Z(d0) ⇒ − 1 2 1 Vd0 R 1 0 [Bd0 (r)]2dr , and Zµ(d0) ⇒ − 1 2 1 Vd0 ( R 1 0 [Bd0 (r)]2dr − [ R 1 0 Bd0 (r)dr] 2 ) , 9