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ASYMPTOTICS FOR GENERAL FRACTIONALLY INTEGRATED PROCESSES WITH APPLICATIONS TO UNIT ROOT TESTS
Published online by Cambridge University Press: 08 January 2003
Abstract
In this paper, functional limit theorems for general fractional processes are established under quite weak conditions. The results are then used to derive weak convergence of general nonstationary fractionally integrated processes and to characterize unit root distribution in a model with error being a fractional autoregressive moving average process or a nonstationary fractionally integrated process.The authors thank three referees and an associate editor for their detailed reading of this paper and valuable comments, which have led to this much improved version of the paper.
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1. INTRODUCTION
Consider a fractionally integrated autoregressive moving average(autoregressive fractionally integrated moving average) (ARFIMA) process {Xt} defined by
where m ≥ 0 is an integer and d ∈ (−½,½); B is a backshift operator and εt are independently and identically distributed (i.i.d.) random variables with zero mean and finite variance; φ(B) and θ(B) are polynomial functions of B with order p, and q, respectively, and both of them only have roots outside the unit circle, i.e., the ARMA(p,q) process ut is taken to be stationary and invertible. The fractional difference operator (1 − B)γ is defined by its Maclaurin series (by its binomial expansion, if γ is an integer):
If z < 0, Γ(z) is defined by the recursion formula zΓ(z) = Γ(z + 1).
Since model (1.1) was introduced by Granger and Joyeux (1980) and Hosking (1981), it has become very popular in applications. It nests the usual Box–Jenkins autoregressive integrated moving average (ARIMA) model and has an ability to capture both short-term dynamics and a wide variety of low-frequency behavior at the same time. Also, there is considerable evidence on the success of applying ARFIMA models to describe financial data such as forward premiums, interest rate differentials, and inflation rates. Illustrations can be found in survey and review papers of Robinson (1994) and Baillie (1996).
Because of their applications in economics and finance, ARFIMA processes have been studied quite extensively in recent years. In model (1.1), it is well known that the process is stationary and invertible when m = 0 (Hosking, 1981; Odaki, 1993); when m ≥ 1, the process is nonstationary. In particular, when d = 0 and m is an integer, the process becomes a usual unit root process. For estimate of the parameter d + m and other related statistical inferences, because the literature is rather extensive, we here only refer to Hosking (1984), Li and McLeod (1986), Fox and Taqqu (1986), Dahlhaus (1989), Giraitis and Surgailis (1990), Beran (1995), Beran, Bhansali, and Ocker (1998), and Tanaka (1999). For more results, see the references cited in these papers and a review book of Beran (1994).
As to the asymptotics of the ARFIMA processes, Sowell (1990) first derived a result that the partial sum process of a simple fractional process (i.e., m = 0, φ(B) ≡ θ(B) ≡ 1 in model (1.1)) converges weakly to a “type I” fractional Brownian motion1
A definition can be found later in this section. For a correction of Sowell's Theorem 2.2, see Theorem 2.1.1 of Liu (1998).
Motivated by characterizing unit root distribution in a more general model, this paper extends the weak convergence result given by Sowell (1990) to general fractional processes. Instead of assuming that the innovations ut in model (1.1) are an ARMA(p,q) process, we allow it to be a more general linear process. This means that this paper provides a unified treatment for previous studies on weak convergence for summable linear processes and fractional processes. The weak convergence results for summable linear processes can be found in Hannan (1979), Phillips and Solo (1992), and Chan and Tsay (1996). Our results are then used to derive functional limit theorems for general nonstationary fractionally integrated processes and to derive the limit distribution of the least square estimate (LSE) of the coefficient for a AR(1) model when the true coefficient is 1 (i.e., the true model has a unit root) and the error process is a general fractional process or a general nonstationary fractionally integrated process. These results improve essentially the related results in the literature (see Section 3 for more details), and they also provide a unified treatment for unit root tests with the error process being a summable linear process, a fractional process, or a nonstationary fractionally integrated process. The main results of this paper are given under quite weak moment conditions for the innovations εt. For example, weak convergence of general nonstationary fractionally integrated processes and nonstationary unit root distribution are derived whenever the innovations εt have finite second moment. Such a condition is the best possible moment condition in the literature and it is of interest from a theoretical point of view.
This paper is organized as follows. In the next section, we derive weak convergence of general fractional processes without proofs and compare them to related results in the literature. Applications of these results to general nonstationary fractionally integrated processes and testing for unit roots will be presented in Section 3. Finally in the Appendix, we give the proofs of the main theorems in Section 2.
We end this section with some notation. We denote limn→∞ an /bn = 1 by an ∼ bn; C,C1,… are for positive constants, which may take on different values in different places. The expression D[0,1] denotes the space of functions on [0,1] in which all elements are right continuous and have left-hand limits, endowed with the Skorohod topology (see Billingsley, 1968, p. 111). Convergence in distribution and weak convergence of probability measures on D[0,1] are denoted by →d and ⇒, respectively. Finally, we define type I fractional Brownian motions with −½ < d < ½ on D[0,1] as follows:
where W(s) is a standard Brownian motion and
Clearly, Wd(t) is a self-similar Gaussian process with covariance
A more general definition of fractional Brownian motions can be found in Mandelbrot and Van Ness (1968), Samorodnitsky and Taqqu (1994), and Marinucci and Robinson (1999).
2. ASYMPTOTICS OF GENERAL ARFIMA PROCESSES
From here on, we discuss the following general ARFIMA process Xt defined by
where m ≥ 0 is an integer and d ∈ (−½,½); (1 − B)d is defined by (1.2); εj,j = 0,±1,… are i.i.d. random variables with Eε0 = 0, and {ψj,j ≥ 0} is a sequence of real numbers to be specified later.
The two theorems in this section derive results on weak convergence of general stationary fractional processes. They provide a unified treatment for the cases of fractional processes and summable linear processes.
THEOREM 2.1. Let Xj satisfy (2.1) with m = 0 and let ψj,j ≥ 0, satisfy
Assume that Eε02 < ∞. Then, for 0 ≤ d < ½,
where κ2(d) = [bψ2Eε02Γ(1 − 2d)]/(1 + 2d)Γ(1 + d)Γ(1 − d) and Wd(t) is a type I fractional Brownian motion on D[0,1].
If, in addition, E|ε0|(2+δ)/(1+2d) < ∞, where δ > 0, then (2.3) still holds for −½ < d < 0.
For 0 ≤ d < ½, Theorem 2.1 gives the result under the best possible moment condition Eε02 < ∞. If there is a slightly stronger restriction on ψk, the moment condition in Theorem 2.1 for −½ < d < 0 can be weakened to E|ε0|2/(1+2d) < ∞. Explicitly, we have the following theorem.
THEOREM 2.2. Let Xj satisfy (2.1) with m = 0 and let ψj,j ≥ 0, satisfy
Assume that E|ε0|max{22/(1+2d)} < ∞. Then (2.3) holds for d ∈ (−½,½).
The proofs of Theorems 2.1 and 2.2 are given in the Appendix.
If ut is a process satisfying φ(B)ut = θ(B)εt, where polynomials φ(B) and θ(B) with order p and q, respectively, have only roots outside the unit circle, Theorem 3.1.1.1 of Brockwell and Davis (1987, p. 85) implies that
with |ψk| ≤ Ca−k,k ≥ 0, where a > 1 and
. Therefore, the following corollary is a direct consequence of Theorem 2.2.
COROLLARY 2.1. Let Xj satisfy (1.1) with m = 0. If E|ε0|max{22/(1+2d)} < ∞, then (2.3) follows with bψ = θ(1)/φ(1) for d ∈ (−½,½).
We next give some remarks that compare our results to those in the literature.
Remark 2.1. If φ(B) ≡ θ(B) ≡ 1 in model (1.1), Corollary 2.1 reduces to Theorem 2.2 of Sowell (1990), where the author derives (2.3) provided E|εt|r < ∞ for r ≥ max{4,−8d/(1 + 2d)}. If d = 0, then
. In this case κ2(0) = bψ2Eε02 and W0(t) is a standard Brownian motion on D[0,1] . Thus, Hannan's result (1979) becomes a special case of Theorem 2.1. Theorem 3.1.4 of Phillips and Solo (1992) and Theorem 2.2.5 of Chan and Tsay (1996) also give similar results but impose more restrictions on ψj. For further results on weak convergence of linear processes, refer to Wang, Lin, and Gulati (2002).
Remark 2.2. Davydov (1970) and the later work by Gorodetskii (1977), Taqqu (1975), Avram and Taqqu (1987), and Mielniczuk (1997) give results on the functional limit theorem for linear process with square summable weights. Theorems 2.1 and 2.2 derive essentially weak convergence of an infinite weighted function of a linear process, i.e., a process such as
. We note that this infinite weighted function is difficult to rewrite as a linear process with explicit coefficients and therefore the results in the papers cited previously can not be applied directly to our theorems. On the other hand, even in the special case where ψ0 = 1 and ψk = 0,k ≥ 1, Theorems 2.1 and 2.2 are not a direct consequence of the papers cited earlier as their results are held under a higher moment condition for innovations εk, in particular, in the case that −½ < d < 0.
Remark 2.3. For the results on functional limit theorems for fractional processes defined by (2.1) with other dependent innovations instead of a linear process ut, refer to Davidson and Jong (2000) and Wang, Lin, and Gulati (2001).
3. APPLICATIONS
In this section, we apply the main results presented in Section 2 to several well-known examples, namely, to derive functional limit theorems for general nonstationary fractionally integrated processes and fractional unit root distributions. These have been studied by various authors in recent years. As will be seen later, the applications of Theorems 2.1 and 2.2 to the related statistics can lead to better results under weaker conditions.
3.1. General nonstationary fractionally integrated processes
In previous research, the functional limit theorem for general nonstationary fractionally integrated processes has been discussed in a very general framework by Chan and Terrin (1995) under the assumption that ut defined in (2.1) is a class of stationary Gaussian processes. The result of Chan and Terrin (1995) extends the results of Chan and Wei (1988), Parks and Phillips (1988, 1989), and Sims, Stock, and Watson (1990) from the domain of integer m's (i.e., d = 0) to fractional d + m's. More recently, Liu (1998) has derived a functional limit theorem for simple nonstationary fractionally integrated processes (i.e., the process Xt defined by (2.1) with m ≥ 1 and ut = εt) provided E|εt|r < ∞ for r ≥ max{4,−8d/(1 + 2d)}. Theorem 3.1, which follows gives an essential improvement of the results cited previously.
For convenience, we introduce the following conditions.
Condition A. ψj,j ≥ 0 satisfy (2.2) and E|ε0|p < ∞, where p = 2, for 0 ≤ d < ½; p = (2 + δ)/(1 + 2d) < ∞,δ > 0, for −½ < d < 0.
Condition B. ψj,j ≥ 0 satisfy (2.4) and E|ε0|max{22/(1+2d)} < ∞.
THEOREM 3.1. Let Xj satisfy (2.1) with m ≥ 1. Let Condition A or Condition B hold. Then, for 0 ≤ t ≤ 1,
where κ(d), Wd(t) are defined as in Theorem 2.1 and
Proof. We first assume m = 1. Put
By definition of the lag operator, we have that
and
. Now Theorems 2.1 and 2.2 imply that, for 0 ≤ t ≤ 1,
and hence (3.1) holds. By using (3.1) and the continuous mapping theorem, we obtain that
The relations (3.2) and (3.3) thus hold true for m = 1.
For general m ≥ 2, Theorem 3.1 follows by induction, and details are omitted. This completes the proof of Theorem 3.1. █
We next consider another application of our main results.
3.2. Fractional unit root distribution
Let {yt} be a stochastic process generated according to
where y0 = 0 and {Xt} is a sequence of errors. Denote the LSE of α by
. We have that
For the case where model (3.6) has a unit root (i.e., the null hypothesis α = 1 holds), the limit distribution of
was first considered by Dickey and Fuller (1979) under the assumption that Xt are i.i.d. random variables. Since then, considerable attention has been focused on weakening the i.i.d. assumption. Here we only cite Said and Dickey (1984), Phillips (1987), Hall (1989), and Chan and Tsay (1996). In these papers, the unit root distribution is obtained but only for the situation where the error process is a short memory process, such as an ARMA process. For similar results, more references can be found in Phillips and Xiao (1998), where the authors present a survey of unit root theory with an emphasis on testing principles and recent developments.
On weakening the assumption of i.i.d. errors, another important contribution is made by Sowell (1990). By assuming that the error process is a simple fractional process (i.e., ψ0 = 1,ψj = 0, j ≥ 1, and m = 0 in model (2.1)), Sowell (1990) establishes a well-known fractional unit root distribution2
The fractional Brownian motion used as limiting process is insufficiently defined in Sowell (1990). For a correction of Sowell's Theorem 3.1, see Marinucci and Robinson (2000). Also, it can be found in Theorem 3.2 which follows.
In Theorem 3.2, which follows, as an application of Theorems 2.1, 2.2, and 3.1, we derive the limit distribution of
while the error process Xt satisfies (2.1). Under quite weak moment conditions, this result provides a unified treatment of the previously most cited results. In particular, we point out that the limit distribution of
is free to the choice of the weights ψk of the ut in model (2.1) if d + m > 0.
THEOREM 3.2. Let Xt satisfy (2.1) with m ≥ 0 and yt satisfy model (3.6). Let Condition A or Condition B hold.
(a) If m ≥ 1 or m = 0 and 0 < d < ½, then
where Wd,m is defined as in Theorem 3.1.
(b) If m = 0 and d = 0, then
where
.
(c) If m = 0 and d ∈ (−½,0), then
where κ2(d) is defined as in Theorem 2.1 and
Proof. By noting
we can rewrite
as
Because
, where Xj satisfy (2.1), it follows from Theorem 3.1 that
Because of (3.14), it is clear that if m ≥ 1, then
On the other hand, if m = 0 in (2.1), it follows from Lemma 3.3 in the Appendix that {Xt,t ≥ 1} is a stationary linear process with
Because stationary linear processes are ergodic, by using the stationary ergodic theorem (see Stout, 1974, p. 181), we obtain that
In particular, we point out that
. Because (3.16) implies (3.15) when m = 0 and 0 < d < ½, the relation (3.8) follows immediately from (3.11)–(3.13), (3.15), (3.16) and the continuous mapping theorem.
Similarly, the relation (3.9) and (3.10) follow easily from (3.11)–(3.13), (3.16), and the fact that Wd,1(s) = Wd(s) and W0(s) = W(s). This completes the proof of Theorem 3.2. █
APPENDIX: PROOFS OF THEOREMS 2.1 and 2.2
We first give several preliminary lemmas, which are also interesting in their own right.
Let {υj,j = 0,±1,…} be a sequence of random variables, {an,k,k = 0,±1,±2,…} be a triangular array of constants, and
. For reading convenience, we give the following basic assumptions.
Assumption 1. The sequence {υj2} is uniformly integrable.
Assumption 2.
is the σ-field of events generated by {υi, i ≤ j}.
Assumption 3. 0 < An < ∞ for each fixed n ≥ 1, and as n → ∞, An → ∞ and maxk|an,k|/An → 0.
Assumption 4. There exists a positive constant C such that
LEMMA 3.1. Let Assumptions 1–4 hold,
. Then, as n → ∞,
where
.
Proof. Because
, there exists a sequence of positive increasing constants λn such that
For this λn, we rewrite
By a simple calculation, to prove (A.2), it suffices to show that
and
Note that relation (A.4) also implies that
is well defined almost surely.
We next give the proofs of (A.4)–(A.6). It can be easily shown (cf. Chow, 1965) that
By Hölder's inequality, it follows from (A.3) and (A.7) that
This implies (A.4) because An → ∞ from Assumption 3.
To prove (A.5) and (A.6), put
We have that for each fixed n ≥ 1 (recalling that Ynj are well defined),
and similar to (A.7)
Because of (A.9)–(A.11) and Assumptions 1–3, tracing the proof of Lemma 3.1 given in Robinson (1997), (A.5) and (A.6) hold if we prove, as n → ∞,
Because Hölder's inequality implies that for each n ≥ 1,i,j ≥ 0,
elementary calculation shows that
Write
It follows that
In view of Assumptions 3 and 4, we have ηn → ∞ and
Taking account of the following inequality:
maxk|an,k| = An /ηn2, and Assumption 4, we have
Therefore, using (A.13)–(A.15), we obtain that
Now, (A.12) follows immediately from (A.16) and
This also completes the proof of Lemma 3.1. █
LEMMA 3.2. Let ck = Γ(d + k)/Γ(d)Γ(k + 1) for k ≥ 0 and ck = 0 for k < 0, where −½ < d < ½. Then,
for 0 ≤ s < t ≤ 1.
Proof. For the proof of (A.17), see Theorem 1 in Hosking (1981). The proof of (A.18) follows easily from (A.17). By noting Γ(z + 1) = zΓ(z) for all z, we have that for 1 ≤ n ≤ k and d ∈ (−½,½),
which implies (A.19).
To prove (A.20), let ζk,k = 0,±1,±2,…, be i.i.d. N(0,1) random variables and
. Because ck = 0 for k < 0 and hence
clearly we have that for 0 ≤ s < t ≤ 1
By noting that Yj,j ≥ 1 are stationary random variables, using Theorem 2.1 of Sowell (1990), we obtain
where we use the estimate: ([nt] − [ns])/n ∼ t − s. Thus (A.20) follows. This also completes the proof of Lemma 3.2. █
LEMMA 3.3. Let Xt satisfy (2.1) with m = 0 and ψj,j ≥ 0 satisfy (2.2). If Eε02 < ∞, then
Proof of Lemma 3.3. Writing Xt = Ψ(B)εt, we have Ψ(z) = (1 − z)dψ(z), where
, similar to the proof of Theorem 2 part (a) given by Hosking (1981), the power series expansion of Ψ(z) converges for all |z| ≤ 1 when d < ½. Thus, if −½ < d < ½, we have
Now (A.21) follows from the binomial expansion of (1 − z)d (cf. Hosking, 1981).
As is well-known (e.g., see Hosking, 1981), {Zt,t ≥ 1} is a strictly stationary random sequence with 0 mean. It follows from Theorem 3.5.3 of Stout (1974, p. 170) that
has the same properties as Zt. To prove (A.22), let fz(.) be a spectral density of {Zt}. According to Theorem 12.4.1 in Brockwell and Davis (1987, p. 466),
This, together with the second equality of (A.21) and (4.4.3) in Brockwell and Davis (1987, p. 121), shows that {Xt} has a spectral density fX(.) and fX(λ) = |ψ(eiλ)|2fZ(λ), where
. In terms of
, we obtain
If d = 0, the result is obvious since
. The proof of Lemma 3.3 is complete. █
After these preliminaries, we now give the proof of the main results.
Proof of Theorem 2.1. If d = 0, Xt reduces to summable linear processes. In this case, (2.3) follows from Hannan (1979). So, we assume that d ≠ 0 in the sequel. Put
In view of part (b) of Lemma 3.3, Xt is a strictly stationary random sequence with zero mean. Now, using the usual method in the proof of weak convergence for stationary random sequence (cf. Taqqu, 1975, Theorem 2.1), it suffices to show that
(i) for each fixed l ≥ 1 and real constants 0 < t1 ≠ t2 ≠···≠ tl ≤ 1,
where τ1,τ2,…,τl are any real constants and
;
(ii) for some a > 1/(1 + 2d),
We first prove part (i). From part (a) of Lemma 3.3, we have that
and ck are defined as in Lemma 3.2. Let mi = [nti],i = 1,…,l. It follows from (A.25) that
where, by an elementary calculation (recalling ck = 0 if k < 0),
To apply Lemma 3.1 to (A.26), we first show that
In fact, (A.28) follows immediately from Lemma 3.2 (see (A.20)) and
From Lemma 3.2 (i.e., (A.17)–(A.19)), on the other hand, we have that for d ≠ 0,
In view of (A.28)–(A.30), conditions of Lemma 3.1 hold for bn,k defined in (A.26) and (A.27). By applying (A.26), (A.28), and Lemma 3.1, we obtain that
The relation (A.31), together with (A.26) and (A.32), implies (A.23). This completes the proof of part (i).
We next prove part (ii). By applying (A.26) and (A.27) with l = τ1 = t1 = 1, we obtain that
with
. Furthermore, it follows from (A.32) that
Thus the first relation of (A.24) holds.
If 0 < d < ½, the second relation of (A.24) is obvious by letting a = 1. To establish the second relation of (A.24) for −½ < d < 0, we let 2a = (2 + δ)/(1 + 2d). Obviously, we have that a > 1/(1 + 2d) > 1 and E|ε0|2a < ∞ when d < 0. By the Marcinkiewicz–Zygmund inequality (also Burkholder's inequality; see Hall and Heyde, 1980, p. 23) and Hölder's inequality, there exists a constant Ca depending only on a such that for all integers j and s ≤ h,
Because of (A.33) and (A.34), it follows from the Fatou lemma that for all j,
Hence, by Hölder's inequality again, we obtain that
which implies the desired result. This also completes the proof of Theorem 2.1. █
Proof of Theorem 2.2. We only need to consider the case where −½ < d < 0. In this case, because εk are i.i.d. random variables with Eε0 = 0 and E|ε0|2/(1+2d) < ∞, by applying Komlós, Major, and Tusnády (1975, 1976) (also see Csörgõ and Horváth, 1993 Corollary 1.1), on a suitable probability space
we can construct ηk,k = 0,±1,±2,…, which are i.i.d. N(0,Eε02) random variables such that as n → ∞
Let Yt satisfy
where −½ < d < 0 and ψk satisfies (2.4). Because ηk are i.i.d. N(0,Eε02), by applying Theorem 2.1, we have that, for 0 ≤ t ≤ 1,
In view of (A.37) and Theorem 1.4.1 of Billingsley (1968), to prove Theorem 2.2, it suffices to show that
where oP(.) denotes convergence in probability.
By applying part (a) of Lemma 3.3, for any m ≥ 1, we can write
where ck = Γ(d + k)/Γ(d)Γ(k + 1) for k ≥ 0 and ck = 0 for k < 0;
Clearly, (A.38) follows if
We next prove (A.40). Write
By noting c0 = 0 and |ck| ≤ Ckd−1 for k ≥ 1 (see Lemma 3.2), it can be easily shown that for all l ≥ s ≥ 0 and j ≥ 1,
Therefore, it follows that
Similarly, we obtain that
In view of (2.4), (A.41), (A.42), and d < 0, Markov's inequity implies that (A.40) holds for j = 1 and j = 4.
On the other hand, it follows from (A.35) and (A.36) that
This implies that (A.40) holds for j = 2.
We now prove (A.40) for j = 3. For convenience, write
. We have that
Clearly, it follows that
Recalling (A.36), |ck| ≤ Ckd−1, and d < 0, we get that
Similarly, we have
On the other hand, by applying Lemma 3.2, we know that (noting m ≤ k − s)
Therefore, it follows from (A.36) that (recalling d < 0)
In view of (A.43)–(A.47), we have that
This implies (A.40) for j = 3. This also completes the proof of Theorem 2.2. █
References
REFERENCES
Avram, F. &
M.S. Taqqu
(1987)
Noncentral limit theorems and Appell polynomials.
Annals of Probability
15,
767–775.Google Scholar
Baillie, R.T.
(1996)
Long memory processes and fractional integration in econometrics.
Journal of Econometrics
73,
5–59.Google Scholar
Beran, J.
(1994)
Statistics for Long Memory Processes.
New York:
Chapman and Hall.
Beran, J.
(1995)
Maximum likelihood estimation of the differencing parameter for
invertible short and long memory ARIMA models.
Journal of the Royal Statistical Society, Series B
57,
695–672.Google Scholar
Beran, J.,
R.J. Bhansali, &
D. Ocker
(1998)
On unified model selection for stationary and nonstationary short
and long memory autoregressive processes.
Biometrika
85,
921–934.Google Scholar
Billingsley, P.
(1968)
Convergence of Probability Measures.
New York:
Wiley.
Brockwell, P.J. &
R.A. Davis
(1987)
Time Series: Theory and Methods.
New York:
Springer-Verlag.
Chan, N.H. &
N. Terrin
(1995)
Inference for unstable long-memory processes with applications to
fractional unit root autoregressions.
Annals of Statistics
23,
1662–1683.Google Scholar
Chan, N.H. &
B.S. Tsay
(1996)
Asymptotic inference for non-invertible moving-average time series.
Journal of Time Series Analysis
17,
1–17.Google Scholar
Chan, N.H. &
C.Z. Wei
(1988)
Limiting distributions of least squares estimates of unstable
autoregressive processes.
Annals of Statistics
16,
367–401.Google Scholar
Chow, Y.S.
(1965)
Local convergence of martingales and the law of large numbers.
Annals of Mathematics and Statistics
36,
552–558.Google Scholar
Csörgõ, M. &
J. Horváth
(1993)
Weighted Approximation in Probability and Statistics.
Wiley Series in Probability and Mathematical Statistics: Probability
and Statistics.
Chichester:
Wiley.
Dahlhaus, R.
(1989)
Efficient parameter estimation for self-similar processes.
Annals of Statistics
17,
1749–1766.Google Scholar
Davidson, J. &
R.M. Jong
(2000)
The functional central limit theorem and weak convergence to
stochastic integrals, II: Fractionally integrated processes.
Econometric Theory
16,
643–666.Google Scholar
Davydov, Y.A.
(1970)
The invariance principle for stationary processes.
Theory of Probability and Its Applications
15,
487–498.Google Scholar
Dickey, D.A. &
W.A. Fuller
(1979)
Distribution of the estimators for autoregressive time series with
a unit root.
Journal of the American Statistical Association
74,
427–431.Google Scholar
Fox, R. &
M.S. Taqqu
(1986)
Large sample properties of parameter estimates for strongly
dependent stationary Gaussian time series.
Annals of Statistics
14,
517–532.Google Scholar
Giraitis, L. &
D. Surgailis
(1990)
A central limit theorem for quadratic forms in strongly dependent
linear variables and its applications to the asymptotic normality of
Whittle's estimate.
Probability Theory and Related Fields
86,
87–104.Google Scholar
Gorodetskii, V.V.
(1977)
On convergence to semi-stable Gaussian process.
Theory of Probability and Its Applications
22,
498–508.Google Scholar
Granger, C.W.J. &
A. Joyeux
(1980)
An introduction to long memory time series models and fractional
differencing.
Journal of Time Series Analysis
1,
15–29.Google Scholar
Hall, A.R.
(1989)
Testing for a unit root in the presence of moving average errors.
Biometrika
76,
49–56.Google Scholar
Hall, P. &
C.C. Heyde
(1980)
Martingale Limit Theory and its Applications.
New York:
Academic Press.
Hannan, E.J.
(1979)
The central limit theorem for time series regression.
Stochastic Processes and Their Applications
9,
281–289.Google Scholar
Hosking, J.R.M.
(1984)
Modeling persistence in hydrological time series using fractional
differencing.
Water Resources Research
20,
1898–1907.Google Scholar
Kokoszka, P.S. &
M.S. Taqqu
(1995)
Fractional ARIMA with stable innovations.
Stochastic Processes and Their Applications
60,
19–47.Google Scholar
Komlós, J.,
P. Major, &
G. Tusnády
(1975)
An approximation of partial sums of independent R.V.'s and
sample DF, part I.
Zeitschrift für Wahrscheinlichskeitstheorie und Verwandte
Gebiete
32,
111–131.Google Scholar
Komlós, J.,
P. Major, &
G. Tusnády
(1976)
An approximation of partial sums of independent R.V.'s and
sample DF, part II.
Zeitschrift für Wahrscheinlichskeitstheorie und Verwandte
Gebiete
34,
33–58.Google Scholar
Li, W.K. &
A.I. McLeod
(1986)
Fractional time series modeling.
Biometrika
73,
217–221.Google Scholar
Liu, M.
(1998)
Asymptotics of nonstationary fractional integrated series.
Econometric Theory
14,
641–662.Google Scholar
Mandelbrot, B.B. &
J.W. Van Ness
(1968)
Fractional Brownian motions, fractional noises and applications.
SIAM Review
10,
423–437.Google Scholar
Marinucci, D. &
P.M. Robinson
(1999)
Alternative forms of fractional Brownian motion.
Journal of Statistical Planning and Inference
80,
3–13.Google Scholar
Marinucci, D. &
P.M. Robinson
(2000)
Weak convergence of multivariate fractional processes.
Stochastic Processes and Their Applications
86,
103–120.Google Scholar
Mielniczuk, J.
(1997)
Long and short-range dependent sums of infinite-order moving averages
and regression estimation.
Acta Sci. Math. (Szeged).
63,
301–316.Google Scholar
Odaki, M.
(1993)
On the invertibility of fractionally differenced ARIMA processes.
Biometrika
80,
703–709.Google Scholar
Parks, J.Y. &
P.C.B. Phillips
(1988)
Statistical inference in regressions with integrated processes,
part 1.
Econometric Theory
4,
468–497.Google Scholar
Parks, J.Y. &
P.C.B. Phillips
(1989)
Statistical inference in regressions with integrated processes,
part 2.
Econometric Theory
5,
95–131.Google Scholar
Phillips, P.C.B.
(1987)
Time series regression with a unit root.
Econometrica
55,
277–302.Google Scholar
Phillips, P.C.B. &
V. Solo
(1992)
Asymptotics for linear processes.
Annals of Statistics
20,
971–1001.Google Scholar
Phillips, P.C.B. &
Z. Xiao
(1998)
A Primer on Unit Root Testing.
Manuscript.
Robinson, P.M.
(1994)
Time series with strong dependence. In
C.A. Sims (ed.),
Advance in Econometrics, Sixth World Congress.
vol. 1, pp.
47–95.
Cambridge, UK:
Cambridge University Press.
Robinson, P.M.
(1997)
Large sample inference for nonparametric regression with dependent
errors.
Annals of Statistics
25,
2054–2084.Google Scholar
Said, S.E. &
D.A. Dickey
(1984)
Testing for unit roots in autoregressive moving average models of
unknown order.
Biometrika
71,
599–607.Google Scholar
Samorodnitsky, G. &
M.S. Taqqu
(1994)
Stable Non-Gaussian Random Processes.
New York:
Chapman and Hall.
Sims, C.A.,
J.H. Stock, &
M.W. Watson
(1990)
Inference in linear time series models with some unit roots.
Econometrica
58,
113–144.Google Scholar
Sowell, F.B.
(1990)
The fractional unit root distribution.
Econometrica
58,
495–505.CrossRefGoogle Scholar
Stout, W.F.
(1974)
Almost Sure Convergence.
New York:
Academic Press.
Tanaka, K.
(1999)
The nonstationary fractional unit root.
Econometric Theory
15,
549–582.CrossRefGoogle Scholar
Taqqu, M.S.
(1975)
Weak convergence to fractional Brownian motion and to the
Rosenblatt process.
Zeitschrift für Wahrscheinlichskeitstheorie und Verwandte
Gebiete
31,
287–302.CrossRefGoogle Scholar
Wang, Q.,
Y.-X. Lin, &
C.M. Gulati
(2001)
Asymptotics for moving average processes with dependent innovations.
Statistics and Probability Letters
54,
347–356.CrossRefGoogle Scholar
Wang, Q.,
Y.-X. Lin, &
C.M. Gulati
(2002)
The invariance principle for linear processes with applications.
Econometric Theory
18,
119–139.CrossRefGoogle Scholar
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