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Article Excerpt
1. INTRODUCTION
The semiparametric estimation of long memory for weakly stationary univariate series has been studied extensively (see, e.g., Robinson 1994, 1995a,b; Hurvich, Deo, and Brodsky 1998; Moulines and Soulier 1999). Generalizations to the case where additive polynomial trends may be present were considered by Velasco (1999a,b), Hurvich and Chen (2000), and Hurvich, Moulines, and Soulier (2002). All four of these articles use tapering schemes, and the latter two use differencing before tapering.
The idea of differencing to detrend the data followed by tapering to handle difficulties induced by possible noninvertibility was suggested by Hart (1989) in a nonparametric context. In the presence of polynomial trends, adequate differencing will completely annihilate the trends, but if the trend is an arbitrary smooth function, then differencing serves as only an approximate detrending device. Hart (1989) focused on estimation of the autocovariances of the noise process (stochastic component), which was assumed to have short memory, in the presence of a smooth additive nonparametric signal (trend). Here we explore the use of differencing and tapering for estimation of the memory parameter of a long-memory noise process in the presence of a smooth additive nonparametric signal.
There is an existing literature on long memory in the presence of nonpolynomial trends. Kunsch (1986) discussed the difficulty of distinguishing certain monotonic trends from long memory. Hall and Hart (1990), Csorgo and Mielnikzuk (1995a,b), and Deo (1997) discussed the properties of kernel estimators of the mean function in the presence of a long-memory noise. Robinson (1997) discussed this same topic, and also provided a method for estimating the memory parameter in the presence of a nonparametric signal. Our theoretical results here focus only on estimation of the memory parameter of the noise, not on estimation of the signal. Nevertheless, our estimator of the memory parameter of the noise, which does not require any preliminary estimator of the signal, might be useful for estimating standard errors for the signal or in constructing optimal estimators of the signal.
Robinson (1997) showed that the memory parameter of the noise may be estimated consistently from the raw data, even in the presence of an unknown nonparametric signal. He established the [log.sup.1/2](n)-consistency of the estimator, a rate that was sufficient for the purpose at hand, under conditions on the bandwidth that become extremely stringent as the short-memory case is approached. Our procedure, which applies the Gaussian semiparametric estimator (see Robinson 1995b) to the tapered, differenced data, yields [n.sup.2/5-[delta]]-consistency and asymptotic normality for any sufficiently small positive value of [delta], assuming that the process is stationary with a spectral pole at zero frequency.
Section 2 presents the main theoretical results on consistency and asymptotic normality of the proposed estimator. It is assumed in these theorems that an increasing number of low frequencies are trimmed from the objective function of the estimator. Stronger tapers lead to a decrease in the amount of trimming required. In Section 3 we investigate the possibility of using trimming alone, without any differencing or tapering. We find that this direct approach, a trimmed version of the estimator of Robinson (1997), can attain [n.sup.2/5-[delta]]-consistency, but that an excessive amount of trimming is required, so this approach would not be expected to work well in practice even with thousands of observations. Simulation results, presented in Section 4, examine the performance of the proposed estimator and explore the effects of tapering and trimming. Furthermore, a finite-sample correction to the asymptotic variance eventually agrees well with the variances of the estimators as found in our simulations. In our simulation study, we also compare the proposed estimator of the long-memory parameter with the direct estimator considered by Robinson (1997), and finally we study the question of feasible inference for the regression function. We find that the proposed estimator of the long-memory parameter is potentially far less biased than the direct estimator, and consequently the proposed estimator may lead to more accurate inference on the regression function. Section 5 presents an application of our proposed methodology to a series of monthly global temperatures. The article concludes with a mathematical Appendix, containing proofs of theoretical results.
2. ASSUMPTIONS AND MAIN RESULTS
We consider the model
[X.sub.t] = r(t/n) + [[epsilon].sub.t], t = 0,..., n, (1)
where r is a sufficiently smooth function and [epsilon] is a linear process with long-range dependence. Denote
[Y.sub.t] = [X.sub.t] - [X.sub.t-1],
[DELTA]r(t/n) = r(t/n) - r((t - 1)/n), and
[[eta].sub.t] = [[epsilon].sub.t] - [[epsilon].sub.t-1].
This yields
[Y.sub.t] = [DELTA]r(t/n) + [[eta].sub.t], t = 1,..., n.
We assume that the process [eta] is linear with respect to a mean-0 unit variance white noise Z = ([Z.sub.t])[.sub.t[member of]Z],
[[eta].sub.t] = [summation over (j[member of]Z)][a.sub.j][Z.sub.t-j], [summation over (j[member of]Z)][a.sub.j.sup.2] < [infinity]. (2)
We further assume that the spectral density of [eta], denoted by f, can be expressed as
f(x) = |x|[.sup.-2[d.sub.0]]f*(x), (3)
where [d.sub.0] represents the memory parameter of the differenced series and f*(x) satisfies some smoothness condition in the neighborhood of zero frequency (see the assumptions in the statement of Thm. 1). In the sequel we will assume that the original series is stationary with long memory; thus, [d.sub.0] [member of] I [subset] (-1, -1/2).
Let h be a complex-valued function defined on [0, 1]. For any positive integer n, define [H.sub.n] = [[summation].sub.t=1.sup.n] |h(t/n)|[.sup.2]. The tapered discrete Fourier transform (DFT) and the tapered periodogram ordinates of a process [xi] are defined as
[d.sub.[xi],k] = [1/[square root of (2[pi][H.sub.n])]] [n.summation over (t=1)] h(t/n)[[xi].sub.t][e.sup.i[x.sub.k]t] and
[I.sub.[xi],k] = |[d.sub.[xi],k]|[.sup.2], (4)
where [x.sub.k] := 2[pi]k/n (k = 1,..., [(n - 1)/2]) are the Fourier frequencies.
We use the following assumptions on the mean function r and on the taper h. Throughout the article, p is a fixed integer. For our theoretical results, we assume that p [greater than or equal to] 1, whereas in the simulation section we also consider p = 0, that is, no tapering.
(A1) r is a p + 1 times continuously differentiable function on [0, 1].
(A2) h(x) = [[summation].sub.u=0.sup.v][b.sub.u][e.sup.2i[pi]xu] for a nonnegative integer v [greater than or equal to] p and real coefficients [b.sub.u], [less than or equal to] u [less than or equal to] v, such that
[v.summation over (u=0)] [b.sub.u][u.sup.k] = for k = 0,..., p - 1.
A function h that satisfies (A2) has the following important properties:
a. h has at least p - 1 vanishing derivatives at and 1,
[h.sup.(j)](0) = [h.sup.(j)](1) = 0, j = 0,..., p - 1. (5)
b. There exists a constant C such that for all x [member of] [-[pi], [pi]],
|[n.summation over (t=1)]h(t/n)[e.sup.itx]| [less than or equal to] C[n/[(1 + n|x|)[.sup.p+1]]]. (6)
An example of a function h that satisfies (A2) is h(x) = (1 - [e.sup.2i[pi]x])[.sup.p]. This yields the family of tapers introduced by Hurvich and Chen (2000). For this taper, which has v = p, (5) clearly holds, and (6) was proved by Hurvich and Chen (2000). In fact, it is easily seen that (5) implies (6). Chen (2001) has constructed certain tapers that satisfy (A2). These new tapers may have improved efficiency properties compared with the Hurvich-Chen tapers.
The following bound for the tapered DFT of the differenced mean function [d.sub.[DELTA]r,k] is crucial for the derivation of the properties of our estimator. It should be noted that the lemma does not require the specific form for the taper as given in assumption (A2).
Lemma 1. Assume (A1) and let h be a complex-valued p times continuously differentiable function that satisfies (5). Then there exists a constant C such that, for 1 [less than or equal to] k [less than or equal to] n/2,
|[d.sub.[DELTA]r,k]| [less than or equal to] C([k.sup.-p][n.sup.-1/2] + [n.sup.-3/2]). (7)
We now introduce the assumptions on the structure of the process [eta]:
(A3) ([Z.sub.l]) is a fourth-order homoscedastic martingale difference sequence; that is, almost surely,
E[[Z.sub.k]|[F.sub.k-1]] = 0, E[[Z.sub.k.sup.2]|[F.sub.k-1]] = 1, and
E[[Z.sub.k.sup.4]|[F.sub.k-1]] = [[mu].sub.4],
where [F.sub.k] = [sigma]([Z.sub.l], l [less than or equal to] k).
(A4) a(x) := [[summation].sub.j[member of]Z][a.sub.j][e.sup.ijx] can be expressed as a(x) = [x.sup.-[d.sub.0]] X a*(x) (x > 0), where [d.sub.0] [member of] (-1, -1/2) and a* is twice continuously differentiable in a neighborhood [-v, v] of and absolutely integrable on [-[pi], [pi]].
The local Whittle contrast is defined as
[W.sub.m](C, d) = [m.summation over (k=l)]{log(C[x.sub.k.sup.-2d]) + [C.sup.-1][x.sub.k.sup.2d][I.sub.Y,k]}, (8)
where m < n/2 is a bandwidth parameter and l < m is a lower trimming number. Concentrating C out of [W.sub.m] yields the profile likelihood
[^.J.sub.l,m](d) = log([1/[m - l + 1]] [m.summation over (k=l)][k.sup.2d][I.sub.Y,k]) - 2d[[gamma].sub.l,m], (9)
where [[gamma].sub.l,m] = [1/[m-l+1]] [[summation].sub.k=l.sup.m] log(k). We define the local Whittle (or Gaussian semiparametric in the terminology of Robinson 1995b) estimate of [d.sub.0] as
[^.d.sub.n] = arg [min.[d[member of][-1, -1/2]]] [^.J.sub.l,m](d).
Theorem 1. Assume that (A1)-(A4) hold. If l and m are non-decreasing sequences of integers such that l/m [right arrow] and
m/n + n/m[l.sup.1+2p] = O([n.sup.-[zeta]]) (10)
for some [zeta] > 0, then [^.d.sub.n] - [d.sub.0] = [O.sub.P]([n.sup.-[eta]]) for some [eta] > 0.
Remark 1. A suitable choice of the sequences l and m is l = [[n.sup.[delta]/2p]] and m = [[n.sup.1-[delta]]] for some arbitrarily small [delta] > 0.
Theorem 2. Assume (A1)-(A4). Let l and m be nondecreasing sequences of integers such that (10) holds and
[lim.[n[right arrow][infinity]]] {n log(m)/([m.sup.1/2][l.sup.1+2p]) + [m.sup.5]/[n.sup.4] + l/m} = 0. (11)
Then [m.sup.1/2]([^.d.sub.n] - [d.sub.0]) converges weakly to the Gaussian distribution with mean-0 and variance
[[[summation].sub.z=-v.sup.v]([[summation].sub.u=0.sup.v-|z|][b.sub.u][b.sub.u+|z|])[.sup.2]]/[4([[summation].sub.u=0.sup.v][b.sub.u.sup.2])[.sup.2]].
Remark 2. A suitable choice of the sequences l and m is l = [[n.sup.3/(5(1+2p))+[delta]/(4p)]] and m = [[n.sup.4/5-[delta]]] for some arbitrarily small [delta] [member of] (0, 4/5). Hence the number of trimmed lower frequencies is not too high.
Remark 3. For the Hurvich-Chen taper of order p, the asymptotic variance is (4p)!(p!)[.sup.4]/{4(2p)![.sup.4]}.
3. DIRECT ESTIMATION OF THE MEMORY PARAMETER WITH TRIMMING
Robinson (1997) showed that the memory parameter of the original sequence {[X.sub.t]} can be estimated consistently by the standard Gaussian semiparametric or local Whittle estimator. The rate of convergence that Robinson obtained is [log.sup.1/2](n). In this section we prove that, in theory, trimming can improve this rate of convergence. In our context, the memory parameter of {[X.sub.t]} is d* = [d.sub.0] + 1 [member of] (0, 1/2).
In this section only we consider the nontapered periodogram of the nondifferenced data,
[I.sub.X,k] = [1/[2[pi]n]]|[n.summation...
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