1. INTRODUCTION
Since the seminal work of Robinson (1995a, 1995b), semiparametric estimation of the long memory parameter of time series has been an active area of research. Let d ∈ (−½,½) be the order of integration. Consider the I(d) process {Xt} defined by
where μ is an unknown mean, B is the backward shift operator, and
is a mean zero, covariance stationary, short memory process. Let fX(·) and fu(·) be the spectral density functions of {Xt} and {ut}, respectively. Then (1) implies
The process {Xt} has long memory if d ∈ (0,½), has short memory if d = 0, and is antipersistent if d ∈ (−½,0). A nonstationary process {Xt} can be defined when d ≥ ½. For example, if
, we can write Xt as the sum of an I(d − 1) process, i.e.,
where X0 is a random variable whose distribution does not depend on t. The case for
can be similarly defined by the repeated use of partial summation (Velasco 1999a, 1999b). An alternative definition of an I(d) process is given by
Equivalently, letting
, k ≥ 1, φ0(d) = 1, we have
Clearly (5) is well defined for any
. Under the formulation (4), the process {Xt} is nonstationary when d ≥ ½ and only asymptotically stationary when d ∈ (−½,½). The main distinction between Type I processes (expressions (1) and (3)) and Type II processes (expression (4)) lies in the presample treatment. These two definitions of I(d) processes could lead to different asymptotic behaviors for various statistics. For example, the normalized partial sum of Xt when d > ½ converges to two distinct fractional Brownian motions (Marinucci and Robinson, 1999a, 1999b). A detailed discussion of their differences can be found in Robinson (2005) and Shimotsu and Phillips (SP hereafter) (2006).
Two popular semiparametric frequency domain approaches to estimate d have been extensively studied in the literature: log periodogram (LP) regression (Geweke and Porter-Hudak, 1983) and local Whittle (LW) estimation (Künsch, 1987). LP regression is easy to compute because it only involves least squares regression, but it is less efficient than LW estimation, which is constructed based on the likelihood principle (Robinson, 1995b). The asymptotic properties of LP regression have been investigated by Robinson (1995a) and Velasco (1999a, 2000) among others for Type I processes and by Kim and Phillips (1999) and Phillips (1999b) for Type II processes. Regarding the asymptotic theory of the LW estimator, Robinson (1995b) and Velasco (1999b) dealt with Type I processes. For Type II processes, see Phillips and Shimotsu (PS hereafter) (2004) and SP (2006). PS (2004) and SP (2006) nearly completed the study of asymptotic properties of the LW estimator for Type II processes in that they characterized the asymptotic distribution for d ∈ [−1,−½) ∪ (−½,½) ∪ (½,M], where M is a fixed constant. Their asymptotic analysis was facilitated by an exact representation of the Fourier transform of Type II fractional processes (Phillips, 1999a). PS (2004) conjectured that their asymptotic results are still valid for Type I processes with different constants in the asymptotic distributions. See the review paper by Moulines and Soulier (2003) for other semiparametric methods of estimating d in the frequency domain.
So far, it seems that most asymptotic analysis of the LW estimator has been limited to linear processes, i.e.,
(cf. Robinson, 1995b; Velasco, 1999b) or
(cf. PS, 2004; SP, 2006), where the ζt are martingale differences with constant conditional variance, i.e.,
Here
. To obtain the asymptotic distribution of the LW estimator, it is often additionally assumed that
, are also constants (Velasco, 1999b; PS, 2004; SP, 2006). As mentioned in Robinson and Henry (1999), it is a drawback not to allow conditional heteroskedasticity, which is an intrinsic feature for autoregressive conditional heteroskedasticity (ARCH) and generalized autoregressive conditional heteroskedasticity (GARCH) models in financial time series. Robinson and Henry (1999) attempted to relax the constant conditional variance condition (6) for the LW estimator in the case of Type I processes when d ∈ (−½,½). For general nonlinear processes, Dalla et al. (2006) proved the consistency of LW estimates and also discussed the convergence rate. Their results indicate that the LW estimator might have a non-Gaussian limit distribution. No distributional theory was provided in their paper.
This paper has two goals. First, we attempt to solve the conjecture posed in PS (2004) for Type I processes when
. Second, we consider the asymptotic distribution of the LW estimator for a wide class of nonlinear processes
where εt are independent and identically distributed (i.i.d.) random variables and F is a measurable function such that ut is well defined. Then ut is a stationary causal ergodic process. The class of processes that (7) represents is huge; see Rosenblatt (1959, 1971), Kallianpur (1981), and Tong (1990, p. 204). As in Wiener (1958), Priestley (1988), and Wu (2005), (7) can be interpreted as a physical system with (…,εt−1,εt) being the input, F being a filter, and ut being the output. Our dependence measures on the process {ut} basically quantify the degree of dependence of outputs on inputs (cf. Remark 2.3 in Section 2). The class (7) includes the linear process
as a special case. It also includes general GARCH (Bollerslev, 1986) and autoregressive moving average–GARCH (ARMA-GARCH) (Ling and Li, 1997) processes. Together with (1), the widely used fractional autoregressive integrated moving average–GARCH (FARIMA-GARCH) model is also within this framework; see Section 4. Our theoretical results confirm the findings from finite-sample simulations in Robinson and Henry (1999), Henry (2001), and Nielsen and Frederiksen (2005) that the LW estimator is robust to conditional heteroskedastic innovations if the bandwidth is appropriately chosen.
The following notation will be used throughout the paper. For a random variable ξ, write
if
and let ∥·∥ = ∥·∥2. For
define projection operators
,
. Let C > 0 denote a generic constant that may vary from line to line. Denote by ⇒ and
convergence in distribution and in probability, respectively. The symbols
,
, and oa.s.(1) signify being bounded in probability, convergence to zero in probability, and convergence in the almost sure sense, respectively. Let N(μ,σ2) be a normal distribution with mean μ and variance σ2.
The paper is organized as follows. Section 2 introduces the LW estimator and some notation and assumptions. Section 3 provides a distributional theory of LW estimates for Type I processes when
. Applications are given in Section 4, and conclusions are made in Section 5. We leave the technical details to the Appendix.
2. LOCAL WHITTLE ESTIMATION
Let
be the imaginary unit. For a process
, define the periodogram
Let λj = 2πj/n,j = 1,…, n, be the Fourier frequencies. Denote the true value of d by d0. Given the observation X1,…, Xn, the LW estimate for d0 is defined as the minimizer of the local objective function R(d), i.e.,
where
Throughout the paper, we assume that m = m(n) satisfies m−1 + m/n → 0, Δ1 and Δ2 satisfy −½ < Δ1 < Δ2 < ∞, and d0 ∈ [Δ1,Δ2].
Let
and
be the covariance function of (ut); let
be the spectral density. Assume throughout the paper that G0 = fu(0) > 0. We make the following assumptions.
Assumption 2.1.
.
Remark 2.1. Assumption 2.1 indicates that ut has short memory. Under this assumption, the spectral density fu is continuous and bounded.
Assumption 2.2.
.
Remark 2.2. The quantity
is closely related to the predictive dependence measure introduced by Wu (2005); see Remark 2.3 for more discussion. Following Wu and Min (2005), Assumption 2.2 is called
dependent with order 1. Assumption 2.2 implies
(cf Lemma A.1 in the Appendix); thus fu(·) is continuously differentiable over [−π,π].
Remark 2.3. Let
be an i.i.d. copy of
and, for k ≥ 0,
, where
. Interpreting (7) as a physical system, Wu (2005) introduced the physical dependence measure
and the predictive dependence measure
, where
. Intuitively, δq(·) quantifies the dependence of uk on ε0 by measuring the distance between uk and its coupled version uk′. For ωq(k), because
is the k-step-ahead predicted mean, ωq(k) measures the contribution of ε0 in predicting future expected values. By Theorem 1 in Wu (2005),
. So Assumption 2.2 is equivalent to
. In many applications physical and predictive dependence measures are easy to use because they are directly related to data-generating mechanisms.
Assumption 2.3.
.
Remark 2.4. Summability conditions on joint cumulants are widely used in spectral analysis. It is an important problem to verify such conditions. For a linear process
with εj being i.i.d., Assumption 2.3. holds if
and
. For nonlinear processes (7), it is satisfied under a geometric moment contraction (GMC) condition with order 4 (Wu and Shao, 2004). The process {ut} satisfies GMC with order α, α > 0, if there exists a C = C(α) and ρ = ρ(α) ∈ (0,1) such that
where un* = F(…,ε−1′,ε0′,ε1,…, εn) is a coupled version of un. Here
is an i.i.d. copy of
. The property (9) indicates that the process {un} forgets its past exponentially fast, and it can be verified for many nonlinear time series models (Wu and Min, 2005; Shao and Wu, 2007). Define the fourth cumulant spectral density
Under Assumption 2.3, f4(·,·,·) is continuous and bounded. Section 4 provides another sufficient condition for the summability of joint cumulants (see Theorem 4.1).
Assumption 2.4. Assume
and
, q > 4.
Remark 2.5. For the linear causal process
, δq(k) = |ak|∥ε0′ − ε0∥q. So Assumption 2.4 holds if
and
. Assumption 2.4 implies Assumption 2.2 because
and
.
Assumption 2.5. For some β ∈ (0,2], fu(λ) = (1 + O(λβ))G0 as λ ↓ 0.
Remark 2.6. Assumption 2.5 is commonly made in the study of local Whittle estimation (PS, 2004; SP, 2006). For the popular FARIMA(p,d,q) or FARIMA-GARCH processes (see Section 4), β = 2. Our Assumptions 2.2 and 2.4 imply the continuous differentiability of fu(·); thus β ≥ 1.
Assumption 2.6. Assume for β > 1,
and for β = 1, (log n)3 = o(m), m3(log m)2/n2 → 0.
Remark 2.7. Previous work by Robinson (1995b) and PS (2004) requires
Note that for β ≥ 1, Assumption 2.6 is stronger than (11). We are unable to relax Assumption 2.6, which is needed for us to apply Corollary 2 in the companion paper in this issue (Wu and Shao, 2007) to obtain asymptotic distributions of
. Under (11), the variance of
dominates its bias. In terms of choosing the bandwidth that minimizes the mean square error, the optimal order for m is n2β/(2β+1) (cf. Henry and Robinson, 1996). On the other hand, Assumption 2.6 does not seem to be overly restrictive: Assumption A4′ in Robinson and Henry (1999) requires m = o(n1/2/log n) in an early attempt to establish a central limit theorem for the LW estimator without imposing (6).
3. MAIN RESULTS
Four cases, d0 ∈ (−½,½), d0 ∈ (½,1),
, and d0 = 1 are dealt with in Sections 3.1, 3.2, 3.3, and 3.4, respectively.
3.1. d0 ∈ (−½,½)
Under the Type I formulation (1), we write
Because IX(λj),j = 1,…, m, which is used in the LW estimation, is invariant to the mean, we can and do assume μ = 0.
PROPOSITION 3.1. For the process (12), let
. Then under Assumption 2.2, we have
and
where
and sj = log(j/m) + 1.
THEOREM 3.1. For the process (12), under Assumptions 2.3–2.6, we have
Robinson (1995b) first obtained the asymptotic distribution of the LW estimator in the setting of linear processes under the assumption that the innovations are martingale differences with constant conditional variance. Robinson and Henry (1999) attempted to relax the latter restriction (see (6)) under the following framework:
where
The widely used ARCH and GARCH models are included in this framework; compare (22) in Section 4. The asymptotic normality was obtained under quite restrictive conditions. For example, they require
,
almost surely and for t ≥ u ≥ v,
In the GARCH case (22), we only need to assume a qth moment condition (q > 4) and allow various forms of conditional heteroskedasticity; see Section 4 for more discussion. Note that the existence of higher order moments requires more restriction on the parameter space in the GARCH type models. On the other hand, our formulation excludes possible long memory in conditional variance, which is included in their framework.
Remark 3.1. The short memory conditions on ut (such as Assumptions 2.2 and 2.4) imply the continuous differentiability of fu(·) over the full band [−π,π], whereas in previous work (see Robinson, 1995b; Robinson and Henry, 1999; Phillips and Shimotsu, 2004), only local smoothness of fu(·) or fX(·) around a neighborhood of zero frequency is imposed. In particular, our assumptions on ut exclude the so-called Gegenbauer process (Gray, Zhang, and Woodward, 1989), in which the spectral density function has a pole at a nonzero frequency (for more discussion, see Phillips and Shimotsu, 2004). These global smoothness assumptions together with the stronger rate condition on m are the prices we pay for allowing nonlinear processes.
3.2. d0 ∈ (½,1)
Recall the definition of Xt in (3). Write
where Aj,n = Φj − Φj−n,
if j ≥ 0 and Φj = 0 if j < 0. Further write
For a complex number z let Re(z) be its real part and z its conjugate. Then
PROPOSITION 3.2. Suppose that Xt is generated by (3) with d0 ∈ (½,1). Assume
and Assumption 2.2; we have
and
.
Remark 3.2. Under Type I formulation, Velasco (1999b) showed that
is consistent for linear processes when d0 ∈ [½,1). For Type II processes, the consistency was established by PS (2004). At d0 = ½, we conjecture that the consistency of
still holds for nonlinear processes.
THEOREM 3.2. Let Xt be defined in (3) with d0 ∈ (½,1).
(i) Under Assumptions 2.3–2.6, we have
where
and U1 and U2 are i.i.d. N(0,1) random variables.
(ii) Let
,
,
,
, and
. Then under Assumption 2.5 and (11), for d0 = ¾, we have
PS (2004) dealt with Type II processes and conjectured that if d0 ∈ [¾,1), then their asymptotic results still hold for Type I processes possibly with different J(d0). Here, we prove the conjecture in the framework of fractionally integrated nonlinear processes. In comparison with the asymptotic distribution in the Type II case, the extra term
in (18) is due to the so-called prehistorical influence. At d0 = ¾, the first two terms in (17) are of the same stochastic magnitude. In this particular case, it seems very hard to obtain asymptotic results for nonlinear processes.
3.3.
THEOREM 3.3. Suppose that {Xt} is generated from (3) with
and
. Then under Assumption 2.2, we have
.
Remark 3.3. Theorem 3.3 confirms the conjecture posed by PS (2004) (see Remark 3.3 therein), which states that
for Type I processes if
. More refined structures exist if d0 = 1; see Section 3.4.
3.4. d0 = 1
THEOREM 3.4. Suppose that {Xt} is generated from (3) with d0 = 1. Assume that
and m = mn → ∞ satisfies
Then under Assumptions 2.3. and 2.5, we have
where W1,W2,W3 are i.i.d. N(0,1).
The condition
in Theorem 3.4 is related to Assumption 2.4 with q = 4. Assumption 2.4 is stronger.
The limit distribution in (20) is equivalent to the form stated in Theorem 4.2 of PS (2004), who obtained the asymptotic distribution of
for Type II processes with d0 = 1. It suggests the interesting dichotomous phenomenon: the asymptotic behaviors in the three cases d0 < 1, d0 = 1, and d0 > 1 are very different. In addition, the condition on the bandwidth m is quite stringent here. Basically we require m = O(n1/6(log n)−4/3).
In summary, for fractional nonlinear processes ((1) and (3)), the LW estimator is consistent when d0 ∈ (−½,½) ∪ (½,1] and is inconsistent when d0 > 1. When d0 ∈ (−½,½) ∪ (½,¾), the asymptotic distribution is normal with asymptotic variance independent of the true value d0. For Type II processes, SP (2000) proposed a modified LW estimator, which basically replaces IX(λj) in R(d) (see (8)) by IX*(λj), where
The modified LW estimator is shown to be consistent for d0 ∈ (0,2) and is asymptotically normally distributed with variance ¼ for
. We would expect that this result still holds for Type I processes in our setting in view of (16), although the boundary case
is hard to handle. We shall not pursue this generalization in this paper.
4. APPLICATIONS
In this section, we shall show that our main technical assumptions, Assumptions 2.3 and 2.4, are satisfied by a number of widely used models in financial time series analysis. The so-called FARIMA(p,d,q)-GARCH(r,s) model (Ling and Li, 1997; Li, Ling, and McAleer, 2002) has been used by Baillie, Chung, and Tieslau (1996), Hauser and Kunst (1998a, 1998b), and Lien and Tse (1999) among others to model both long memory and conditional heteroskedasticity. It admits the following form:
where {εt} are i.i.d. with zero mean and unit variance,
, and
. Assume that all the roots of
and
are outside the unit circle, φp ≠ 0, ψq ≠ 0, and φ(z) and ψ(z) have no common root. Rewrite (21) into the form of (1) with ut = φ(B)−1ψ(B)ζt, where ut is an ARMA-GARCH process. For a GARCH process, the necessary and sufficient conditions for the existence of fourth moments have been investigated by Ling and McAleer (2002a, 2002b). Wu and Min (2005) showed that ζt is GMC(4) (cf. eqn. (9) with α = 4) provided that
. Because an ARMA process with GMC(4) innovations is still GMC(4) (cf. Shao and Wu, 2007, Thm. 5), the process ut is GMC(4) if
. Therefore, our Assumption 2.3 is satisfied because GMC(4) implies the summability of fourth cumulants (cf. Wu and Shao, 2004, Prop. 2).
In the literature, various forms of GARCH have been proposed to model conditional heteroskedasticity. An important class is the asymmetric GARCH(r,s) models (Ding, Granger, and Engle, 1993). Interestingly, for general asymmetric GARCH(r,s) models (which include (22) as a special case), Shao and Wu (2007) showed that they satisfy GMC(q),
under the qth moment condition. Another popular asymmetric GARCH model is the so-called EGARCH(p,q) (exponential GARCH(p,q)) model proposed by Nelson (1991), which admits the following form:
where α0, θ, and γ are constants and
and
are polynomials with zeros outside the unit circle having no common factors. Robinson and Henry (1999) also considered the preceding model in their finite-sample simulation. However, they mentioned that the EGARCH model is not included in their theoretical framework (14). If εt are i.i.d. N(0,1), Min (2004) showed that ζt is GMC(q) for any
.
For other types of nonlinear time series models such as bilinear models (Subba Rao and Gabr, 1984), threshold autoregressive models (Tong, 1990), and signed volatility models (Yao, 2004), the GMC property has been verified by Wu and Min (2005) and Shao and Wu (2007) under certain contraction conditions. Regarding our Assumption 2.4, it holds if the process ut is GMC(q) for some q > 4 (for a rigorous proof see Wu, 2007).
A common assumption in spectral analysis is the summability of joint cumulants up to certain orders (Brillinger, 1975; Rosenblatt, 1985). The following theorem gives a sufficient condition under which the summability of joint cumulants is true. It generalizes Proposition 2 in Wu and Shao (2004).
THEOREM 4.1. Assume
, k ≥ 2,
. Then
provided that
, where δk(m) = ∥um − um′∥k.
Remark 4.1. Because
, (23) holds if
Therefore
, q > 4 implies our Assumptions 2.3, 2.4, and β = 2 in Assumption 2.5.
Example 4.1. ARCH(∞) (Robinson, 1991)
where εt are i.i.d. mean zero random variables having suitable moments. Note that (25) is a special form of (14). Both general ARCH(p) and GARCH(r,s) models fall into this framework when the weights ψj either vanish for j > p or decay exponentially to zero. This property implies the exponential decay rate of the autocorrelation of ζt2. Giraitis, Kokoszka, and Leipus (2000) gave a sufficient condition for the existence of a stationary solution and found that the autocorrelation of ζt2 can decay slowly like a power function, but no long memory structure of ζt2 is allowed. Further development can be found in Zaffaroni (2004). We shall show that our condition can be satisfied with hyperbolically decaying coefficients ψj.
PROPOSITION 4.1. For (25), let
. Assume that
and
, then we have
where ζk′ is the coupled version of ζk as in Remark 2.3.
Example 4.2. LARCH (linear ARCH) (Robinson, 1991)
where a and
are not constrained to be nonnegative. Giraitis, Robinson, and Surgailis (2000) provided a sufficient and necessary condition for the existence of a stationary solution and demonstrated that ζt2 could have long memory autocorrelation unlike the ARCH(∞) case. The following proposition covers part of the short memory case where bj is allowed to have a hyperbolic decay.
PROPOSITION 4.2. For (27), let
. Assume that
and
. Then
5. CONCLUSIONS
This paper presents an asymptotic theory for the LW estimator of a class of fractionally integrated nonlinear processes. The theory we develop here matches the empirical evidence found in finite-sample simulations by Robinson and Henry (1999), Henry (2001), and Nielsen and Frederiksen (2005) that suggest that the LW estimator is robust to conditional heteroskedasticity. Recently, long memory in volatility has received a great deal of attention in the literature (cf. Hurvich, Moulines, and Soulier, 2005, and references therein). Robinson and Henry (1999) showed that the LW estimator of the long memory parameter in the level is unaffected by long memory in volatility. Our framework excludes this interesting case in that the conditional heteroskedastic models included are all of short memory type. However, our framework is general enough to allow various kinds of short memory GARCH models.
Under the current framework, it is certainly worth investigating the asymptotic properties of the LW estimator for Type II processes and also the exact LW estimator (SP, 2005). These topics are beyond the scope of this paper and will be studied in the future.
TECHNICAL APPENDIX AND PROOFS
For convenience of notation, write IXj = IX(λj), Iuj = Iu(λj), IYj = IY(λj), fXj = fX(λj), fuj = fu(λj), fYj = |1 − eiλj|2−2d0fuj,
,
,
, wXj = wX(λj), wuj = wu(λj), and wYj = wY(λj), j = 1,…, m.
Let
and
. Denote
,
, and αj = α(λj). When d0 ∈ (−½,½), it is easy to see that α(·) satisfies the following condition: α(λ) is differentiable in a neighborhood of the origin (0,ε) and also α′(λ) = O(|α(λ)|λ−1) as λ ↓ 0. Let K(w) = (2πn)−1|D(w)|2 be the Fejér kernel. We introduce the following working assumption.
Assumption A.1. fu(·) is differentiable at (0,ε) for some ε > 0 and
The following lemma describes the relationship between γu(k) and
.
LEMMA A.1. For ut in (7),
implies
for q > 0.
Proof of Lemma A.1. Because
and
are orthogonal if j ≠ j′,
Therefore, by the Cauchy–Schwarz inequality,
█
Remark A.1. By Lemma A.1, Assumption 2.2 leads to
, which implies that fu′(·) is continuous on [−π,π]. So Assumption A.1 is satisfied. Because
, we have that
implies
for q > 0.
LEMMA A.2. Suppose that {Xt} is from (1) with d0 ∈ (−½,½). Under Assumptions 2.1 and A.1, the following expressions hold uniformly over 1 ≤ k < j ≤ m = o(n):
Furthermore, if Assumption 2.3 is satisfied, then we have
uniformly over j,k = 1,…, m. Here a ∨ b = max{a,b}, a ∧ b = min{a,b}.
Proof of Lemma A.2. Because the cross spectral density fXu(λ) = (1 − eiλ)−d0fu(λ) when d0 ∈ (−½,½), it is easily seen that (A.1) implies | fXu′(λ)| = O(λ−1−d0) and | fX′(λ)| = O(λ−1−2d0) as λ ↓ 0. Then the lemma is a direct consequence of Robinson (1995a); see Theorem 2 and its proof therein. Regarding (A.3), we have
uniformly over j = 1,…, m in view of Assumption 2.3. and the first assertion. █
Remark A.2. To show (A.2), we only need to use equation (4.1) in the paper by Robinson (1995a), whose proof does not require the rate of convergence of fu(λ) as λ ↓ 0. For the remaining nine statements, our assumptions suffice. See Velasco (1999b) for a similar result when d0 ∈ [½,1).
The next two lemmas (Lemmas A.3 and A.4) are useful for the consistency and the asymptotic normality of
, respectively.
LEMMA A.3. For the process (12) with d0 ∈ (−½,½), suppose Assumption 2.1 holds. Then
Proof of Lemma A.3. Write φk = φk(d0). By (12),
where
. After straightforward calculations, we get
uniformly over j = 1,…, m. The last equality in the preceding expression is due to Lemma 3 of Robinson (1995b) in view of the properties of α(λ). Finally, (A.4) follows from the Cauchy–Schwarz inequality and the fact that
uniformly over j = 1,…, m under Assumption 2.1 (cf. Brockwell and Davis, 1991, Prop. 10.3.1). █
LEMMA A.4. For the process (12) with d0 ∈ (−½,½), suppose Assumptions 2.1, 2.3, and A.1 hold. Then
where C is a generic constant independent of r, m, and n.
Proof of Lemma A.4. The proof is a generalization of the argument in Robinson (1995b, pp. 1648–1651) to the nonlinear case. Let l = 1 + [lfloor ]r1/2 log r[rfloor ]. By Lemma A.3,
. It then suffices to consider l + 1 ≤ j ≤ r. Write
, where
By Lemma A.2, we have
and
Because the summand in a2 is the summand in b2 with j = k, we shall derive the order of the summand in b2 first. After a straightforward calculation (cf. Brillinger, 1975), we can write (2πn)2fuj fuk s(j,k) as
where Π3 = [−π,π]3 and
Under Assumption 2.3, f4(·,·,·) is bounded. Following Robinson (1995b, p. 1649), (A.5) is a sum of three types of components. The first type is
By the Cauchy–Schwarz inequality, Lemma 3 of Robinson (1995b), and the periodicity, (A.6) is bounded in absolute value by Cn2Pj Pk, where
So |(A.6)| ≤ Cn2j−1k−1. A typical second type of component is
Again, by the Cauchy–Schwarz inequality and the periodicity, we have |(A.7)| ≤ Cn2Pj1/2Pk ≤ Cn2j−1/2k−1 because
. It is easily seen that other components of the second type can be bounded either by Cn2j−1/2k−1 or Cn2j−1k−1/2.
We proceed to show that the third type of component is bounded in magnitude by Cn3/2j−1/2k−1/2. An example of the third type of component is
where Π2 = [−π,π]2 and
Then we have |(A.8)| ≤ Cn3/2Pj1/2Pk1/2 ≤ Cn3/2j−1/2k−1/2 by the Cauchy–Schwarz inequality if the following relation holds:
To show (A.9), let c(k1,k2,k3) = cum(u0,uk1,uk2,uk3) and rewrite Gjk(θ,w1) as
By a similar argument as previously, we have
which entails (A.9) under Assumption 2.3.. Finally, we deduce that
The conclusion follows because l = 1 + [lfloor ]r1/2 log r[rfloor ]. █
LEMMA A.5. For the process Xt in (3) with
, under Assumption 2.1,
,
, where C only depends on d0 and CU.
Proof of Lemma A.5. Recall that
. By (3), we have
Fix a δ ∈ (0,1); let hn = δ/n. Then on [0,hn], Gn(λ,d0) ≤ C(nλ)2λ−2d0. On (hn,δ), Gn(λ,d0) ≤ Cλ−2d0 and Gn(λ,d0) ≤ C on [δ,π]. Hence
The constant C only depends on d0 and CU once we fix δ. Thus the conclusion follows because Gn(λ,d0) = Gn(−λ,d0). █
LEMMA A.6. Let ut = F(…,εt−1,εt) and
, k = 1,…, n. Suppose that
and
; then on a richer probability space, there exists a standard Brownian motion
such that
where χ(n) = n−1/4 log(n). Consequently,
where
.
Proof of Lemma A.6. By Theorem 3 and Corollary 5 in Wu (2007), we have (A.10). Because
uniformly over s = 1,…, m, (A.11) follows. █
Remark A.3. The strong approximation (A.11) is used to obtain the asymptotic distribution of
when d0 = 1. A similar result for linear processes has been used by Phillips (1999b) to derive the asymptotic distribution of the estimator from LP regression at d0 = 1.
LEMMA A.7. For (3) with
, assume
, where q > 2/(2d0 − 1) for d0 ∈ (¾,1) and q = 2 for
. Then
where
Proof of Lemma A.7. When
, the result follows from Theorem 2.1 in the paper by Wu and Shao (2006), who proved a functional central limit theorem. When d0 = 1, it follows from Theorem 1 in Hannan (1973). █
Proof of Proposition 3.1. Let ηj = λj2d0IXj /G0. By Theorem 2.1 in Dalla et al. (2006), it suffices to prove that
The former assertion is a direct consequence of Lemma A.3 and the fact that
uniformly over j = 1,…, m under Assumption 2.1 (cf. Brockwell and Davis, 1991, Prop. 10.3.1). For the latter, let τm = [lfloor ]τm[rfloor ]. By Lemma A.3,
We shall adopt a martingale approximation approach to prove
. Let
be stationary martingale differences, and we approximate
by
. By Lemma 4 in Wu and Shao (2007),
, where C is independent of n and λj. Therefore,
where
. Note that
forms martingale differences with respect to
and
under
(see Wu and Shao, 2007, Lem. 4); we have
by Burkholder's inequality (Hall and Heyde, 1980, p. 23) and Lemma 5 in Wu and Shao (2007). Thus the conclusion follows because ∥d0∥2 = 2πG0 (Wu and Shao, 2007) and, by the ergodic theorem, J1n − 1 = oa.s.(1). █
Remark A.4. In Dalla et al. (2006), their d is twice the d used in our paper. Also they set Δ1 = −½ and Δ2 = ½ because they only consider the case d0 ∈ (−½,½). A detailed check of their argument shows that their Theorem 2.1 is still applicable to our case as long as d0 ∈ [Δ1,Δ2].
Proof of Theorem 3.1. By Proposition 3.1, it suffices to show
. Because fXj G0−1λj2d0 = 1 + O(λjβ) holds uniformly over j = 1,…, m under Assumption 2.5 and
under (11), we can write
, where
For I1, by summation by parts, we get from Lemma A.4 that
Let
, and we have
under (11). By Corollary 2 in Wu and Shao (2007), under Assumption 2.4 and (10),
. The conclusion then follows because
. █
Proof of Proposition 3.2. A careful investigation of Theorem 2.1 of Dalla et al. (2006) and its proof suggests that their argument still applies if d0 ∈ (½,1). Let ηj = λj2d0IXj G0−1. It suffices to show (A.13). By Lemma A.5, λj2d0∥Xn − X0∥2 (2πn)−1|1 − eiλj|−2 ≤ Cn2d0−2λj2d0−2 ≤ C, 1 ≤ j ≤ m, where we have applied the fact that |1 − eiλ|−2 = λ−2(1 + O(λ2)) over a neighborhood of λ = 0. Lemma A.3 and the fact that
uniformly over j = 1,…, m (cf. Brockwell and Davis, 1991, Prop. 10.3.1) imply that
for j = 1,…, m. By the Cauchy–Schwarz inequality, the third term in (17) multiplied by λj2d0 is bounded uniformly over j. So
, j = 1,…, m. Next, for any τ ∈ (0,1], let τm = [lfloor ]τm[rfloor ],
By Lemma A.3 and the argument in the proof of Proposition 3.1,
Finally we need to show that
Applying Lemma A.2 to {wYj}, we get
Thus (A.14) holds because
(see Lemma A.5). The conclusion follows. █
Proof of Theorem 3.2. (i). By Proposition 3.2,
. In view of (17), we first show that
Because
(see Lemma A.5), it suffices to show that
Summation by parts yields
, where
, 1 ≤ r ≤ m. Note that |1 − eiλj|−2 = λj−2(1 + O(λj2)) uniformly over j = 1,…, m. By Lemma A.2, we have
Because d0 < 1,
hence (A.16) holds. When d0 ∈ (½,¾), by the argument in the proof of Theorem 3.1. (where the role of λj2d0IXj is replaced by λj2d0−2IYj),
When d0 ∈ (¾,1),
in view of Lemma A.7 and the fact that
.
(ii). Denote
. Recall (15) for the expression of Xn − X0. We first claim that
Let Ω1 = (2π)2d0−2(2d0 − 1)−2(1 − d0), yt = n1/2−d0An−t,nεt = n1/2−d0φn−t(d0)εt, 1 ≤ t ≤ n, and
. By (17) and (A.15), we have
, where
and
The validity of the last equality in the preceding expression follows from Lemma A.4 and the argument in the proof of Theorem 3.1. Under the assumption
, it is easy to see that a(eiλ) is differentiable in a neighborhood of zero and (d/dλ)a(eiλ) = O(λ−1) as λ → 0+. Based on the preceding properties of α(·), Robinson (1995b, p. 1644) showed that
. Therefore
, where z1 = 0,
for t ≥ 2. Here
. Letting Ω2 = Γ(d0)−2(2d0 − 1)−1 and
, we shall show that
where U1, W1, and W2 are i.i.d. N(0,1). Notice that Pn is independent of
; it suffices to show that Pn ⇒ Ω31/2W2 because PS (2004, p. 687) have shown that
. Write
Note that
for all k ≥ n. Then for any ε > 0, the following Lindeberg condition holds:
which entails that K1n [var(K1n)]−1/2 ⇒ N(0,1). Because K1n and K2n are independent of each other, the convergence of Pn follows from the fact that
.
Combining the preceding results, we apply the continuous mapping theorem and get
, where the limit has the same distribution as U1 /2 + J(d0)U22.
It remains to show (A.17). By Beveridge-Nelson decomposition (Phillips and Solo, 1992),
Letting Bj,n = Aj,n − Aj+1,n, then we have
where
and
Thus (A.17) holds, and the proof is completed. █
Remark A.5. As we can see from the proof of Theorem 3.2, the third term of (17) is negligible for all d0 ∈ (½,1). The first term of (17) becomes dominant when d0 ∈ (¾,1), whereas the second term is dominant when d0 ∈ (½,¾). At d0 = ¾, the first two terms have the same stochastic order, and the asymptotic distribution is a mixture. This phenomenon has been observed by PS (2004) in the Type II case, and our result for the Type I case is consistent with their observation. When d0 ∈ (¾,1), Assumption 2.4 can be replaced by
, q > 2/(2d0 − 1); compare Lemma A.7 and Assumption 2.2.
Proof of Theorem 3.3. The proof follows the argument in PS (2004) for the Type II case, Lemmas A.2, A.3, A.5, and A.7. For the details, see Shao and Wu (2005). █
The following lemma is used in proving the asymptotic distribution of
when d0 = 1.
LEMMA A.8. Let the nonnegative random variables ηj = ηj,m, j = 1,…, m satisfy
, 1 ≤ j ≤ m, where C is a finite constant. Suppose that the random variables Qm(x,d),m ≥ 1 satisfy that for any b ∈ (0,1),
,
and for some γ ∈ (0,1),
Then as m → ∞,
Proof of Lemma A.8. Let bm = [lfloor ]bm[rfloor ]; then we have
By (A.19),
. Under (A.20),
as b ↓ 0. The conclusion follows. █
Proof of Theorem 3.4. The following argument is a variant of the one used by Dalla et al. (2006) in their proof of Theorem 2.1. Again, we assume X0 = 0 for the convenience of presentation. Let
and ηj = λj2d0G0−1IXj. Recall (8) for the objective function R(d). Note that R′(d) = 2Tn(d)/Sn(d), where
Fix an ε ∈ (0,¼). On
, we have
which yields
on Ω1n ∩ Ω2n, where
and
as n → ∞. By the mean-value theorem, there exists a d that lies in between
and d0 such that
on Ω1n ∩ Ω2n, where
. Assume that
The proof of (A.21) is given at the end of this proof. Recall (A.11) for the form of ξj,1 ≤ j ≤ m. We apply Lemma A.5 and (17) and get
The last equality in the preceding expression is due to Lemma A.6 and (19). By Theorem 3.3,
. Because
,
where
and
. We shall show that
. By the Cramer–Wold device, it suffices to show that, for any α ∈ (0,1),
where zj = αG0−1|ξj|2 + (1 − α)G0−1/2(ξj e−iλj + ξj eiλj), j = 1,…, m are independent random variables. Note that
,
, and
. Then the convergence (A.22) follows from the obvious Lindeberg condition, i.e., for any δ > 0,
It is easily seen that
is independent of ξj, j = 1,…, m, and thus independent of Zn(1) and Zn(2). Applying the continuous mapping theorem, we get
It remains to show (A.21). To that end, write
Because vj = log(j/m) + 1 + O(log m/m) uniformly over j = 1,…, m (see Robinson, 1995b, Lem. 2) and
(see Lemma A.5), we have
where by (17),
By Lemma A.5,
. With regard to H1n(2), we have
where the variance of the latter term is o(1) by (A.3) of Lemma A.2. That
follows from Lemma A.2 and a similar argument as before; compare the proof of Theorem 3.3.
By Lemmas A.2 and A.5, there exists a finite constant C such that
, j = 1,…, m. Therefore, on Ω1n ∩ Ω2n,
follows from Theorem 3.3 and Lemma A.8 by setting Qm(x,d) = [x2(d−d0) − 1] log(x){log(x) + 1}, d1 = d0 − ε, and d2 = d0 + ε. Thus the conclusion holds. █
Proof of Theorem 4.1. Let J = cum(u0,um1,…,umk−1), where 0 = m0 ≤ m1 ≤ ··· ≤ mk−1. Denote by nl = ml − ml−1, 1 ≤ l ≤ k − 1. Define the random vector Y0 = Y0,l = (um0−ml−1,…,uml−2−ml−1,u0). Further let
be an i.i.d. copy of
, Ωn = (…,εn−1′,εn′) and define ut* = F(Ω0,ε1,…, εt), t ≥ 0. Following Proposition 2 in Wu and Shao (2004), by the stationarity and the additivity of cumulants,
Denote by ζj = ∥uj − uj*∥k and
. Proposition 2 of Wu and Shao (2004) asserts that |Bj| ≤ C1ζml+j−ml−1, where C1 only depends on k and the moments
, 1 ≤ i ≤ k. Therefore, by Proposition 2 of Wu (2007),
where C2 = 18k3/2(k − 1)−1/2C1, C3 = C2 k. Because the preceding expression holds for any l, 1 ≤ l ≤ k − 1, we have J ≤ C3 min1≤l≤k−1 Snl1/2. Finally,
by our assumption. █
Proof of Proposition 4.1. Write (25) into the following form:
Note that
implies
. By Theorem 2.1 of Giraitis, Kokoszka, and Leipus (2000), there exists a unique strictly stationary solution, and
. Denote it by ζt = G(…,εt−1,εt) and let ζt′ = G(…,ε−1,ε0′,ε1,…, εt) for
. Note that
By the Cauchy–Schwarz inequality, we have
, i.e.,
. For
, let
. Thus we get
Let sk := ∥ρk − ρk′∥3, αj = ψj∥ε12∥3; then
, which implies
. It suffices to show that
because it implies
and
where the latter implies the first assertion of (26) in view of Remark 4.1. In the preceding discussion, we have applied the fact that (ζk − ζk′)2 = εk2(σk − σk′)2 ≤ εk2|σk2 − (σk′)2| = |ρk − ρk′|, where σk′ is the coupled version of σk with ε0 replaced by ε0′. Let
and
; then
. Define
and
; then h(z) = s0 /(1 − g(z)). Note that g(1) < 1. Let h(p)(z) be the pth derivative of h(z). A simple calculation shows that h(p)(1), p = 1,2,3 are all finite under
. Thus
. The conclusion follows. █
Proof of Proposition 4.2. Note that
implies
. It follows from Theorem 2.1 of Giraitis, Robinson, and Surgailis (2000) that there exists a strictly stationary solution ζt = G(…,εt−1,εt) and
. By the same argument as in the proof of Proposition 4.1,
. For
,
. The conclusion follows from a similar argument as in the proof of Proposition 4.1, and we omit the details. █
