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Submitted to the Annals of Statistics arXiv: math.PR/0000000

SUPPLEMENT TO ”GAUSSIAN PSEUDO-MAXIMUM LIKELIHOOD ESTIMATION OF FRACTIONAL TIME SERIES MODELS” By Javier Hualde, Peter M. Robinson Universidad P´ ublica de Navarra and London School of Economics This supplements [4] by providing a Monte Carlo study of ﬁnite sample performance, an application to two empirical time series, and proofs of the four lemmas in [4]. FINITE-SAMPLE PERFORMANCE A Monte Carlo study was conducted in order to throw light on the performance of our estimates in small and moderate samples. We considered several versions of the FARIMA(1, δ 0 , 0) model, which allows for simultaneous variation of both long- and short-range dependence. In (1.2) we have α(L) = (1 − ϕ0 L)−1 , β(L) = 1, for ϕ0 = −0.5, 0, 0.5, while the memory parameter values were δ 0 = −0.6, −0.4, 0, 0.4, 0.6, 1, 1.5, 2, covering values either side of the stationarity and invertibility boundaries, as well as cases with one or two unit roots, and a value between these. We generated xt , t = 1, ..., n, for n = 64, 128, 256 from (1.1), (1.2), using independent N(0, 1) )′ in (1.5) of τ 0 = (δ 0 , ϕ0 )′ , using εt . We computed the estimate τ = (δ, ϕ T =[−3, 3]×[−0.999, 0.999] for each of the 8×3 τ 0 values. This was repeated over 5, 000 independent replications, and Monte Carlo bias and standard deviation computed in each of the 8 × 3 × 3 = 72 cases. From the same data sets we also computed these summary statistics for an estimate which correctly assumes the degree of integer diﬀerencing or aggregating needed to shift the process to the stationarity and invertibility region: we estimated the memory and autoregressive parameter of the appropriately integer differenced or aggregated sequence by the discrete-frequency Whittle pseudo likelihood estimate (i.e. the untapered version of the estimate in [8]) and then added to or subtracted from the former the appropriate integer, denot W )′ . Though τ and τ W are equally ing the resulting estimate τ W = (δ W , ϕ asymptotically eﬃcient, the additional information it employs leads one to expect τ W to be generally more accurate than our τ in ﬁnite samples. Monte Carlo biases of estimates of δ 0 are given in Table 1. Nearly all biases of both estimates are negative, and overall are worst when ϕ0 = 0 and n = 64, though there is considerable improvement with increasing n. 1

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J. HUALDE AND P.M. ROBINSON

The latter phenomenon is mostly repeated, albeit less dramatically, for the other values of ϕ0 , such that for n = 256 absolute bias tends to monotonically increase with ϕ0 . The relative performance of δ and δ W also diﬀers markedly between zero and non-zero ϕ0 . When ϕ0 = ±0.5, δ is more (less) biased in 38 (6) cases out of 48, whereas when ϕ0 = 0 the corresponding scores are 4 (19) out of 24 (though mention must be made of the relatively poor performance here of δ when δ 0 = 2). Otherwise, biases of both estimates are fairly stable across δ 0 . The overall superiority of δ W here might be explained by the fact that it correctly uses the information on the unit length interval in which δ 0 lies. TABLE 1. Bias of estimates of δ0

δ0 -.6 -.6 -.4 -.4 0 0 .4 .4 .6 .6 1 1 1.5 1.5 2 2

ϕ0 n δ δW δ δW δ δW δ δW δ δW δ δW δ δW δ δW

64 -.052 -.019 -.049 -.007 -.056 -.034 -.051 -.021 -.059 -.027 -.071 -.036 -.129 -.028 -.162 -.043

.5 128 -.054 -.043 -.058 -.037 -.059 -.053 -.058 -.054 -.056 -.045 -.066 -.054 -.063 -.038 -.150 -.049

256 -.049 -.043 -.049 -.040 -.050 -.050 -.049 -.044 -.048 -.033 -.050 -.052 -.051 -.040 -.136 -.049

64 -.113 -.244 -.105 -.192 -.106 -.210 -.115 -.245 -.126 -.240 -.115 -.218 -.152 -.265 -.440 -.210

0 128 -.049 -.099 -.047 -.085 -.052 -.098 -.049 -.104 -.052 -.124 -.049 -.098 -.056 -.124 -.246 -.093

256 -.020 -.022 -.021 -.019 -.019 -.026 -.019 -.021 -.019 -.021 -.019 -.029 -.021 -.021 -.120 -.025

64 -.041 -.058 -.037 -.031 -.037 -.052 -.039 -.048 -.039 -.037 -.046 -.057 -.119 -.043 -.098 -.052

-.5 128 -.018 -.015 -.018 -.006 -.020 -.019 -.018 -.015 -.018 -.005 -.016 -.017 -.038 -.010 -.017 -.015

256 -.009 -.005 -.010 .000 -.008 -.008 -.008 -.004 -.008 .005 -.008 -.008 -.011 -.002 -.009 -.009

More surprising are the Monte Carlo standard deviations of estimates of δ 0 , displayed in Table 2. Again, for both estimates ϕ0 = 0 (overspeciﬁcation) with n = 64 is a bad case, there is improvement with increasing n, standard deviations tend to increase with ϕ0 for large n, and there is reasonable stability across δ 0 . However, with the notable exception of the 9 cases when τ 0 = (1.5, −0.5)′ , τ 0 = (2, 0)′ , τ 0 = (2, 0.5)′ for n ≥ 128, and τ 0 = (2, −0.5)′ for n = 64, δ is consistently the more precise, in 63 out of 72 cases.

GAUSSIAN ESTIMATION OF FRACTIONAL TIME SERIES MODELS

TABLE 2. Standard deviation of estimates of δ 0

δ0 -.6 -.6 -.4 -.4 0 0 .4 .4 .6 .6 1 1 1.5 1.5 2 2

ϕ0 n δ δW δ δW δ δW δ δW δ δW δ δW δ δW δ δW

64 .252 .299 .256 .297 .255 .293 .257 .303 .255 .310 .255 .292 .259 .306 .271 .290

.5 128 .203 .236 .209 .239 .207 .235 .204 .238 .207 .251 .212 .233 .208 .243 .242 .232

256 .165 .189 .165 .188 .165 .186 .166 .190 .165 .198 .166 .187 .166 .192 .220 .187

64 .282 .417 .286 .408 .284 .406 .295 .424 .305 .441 .286 .410 .333 .441 .433 .404

0 128 .176 .286 .176 .282 .179 .284 .177 .294 .185 .337 .179 .283 .192 .317 .376 .281

256 .099 .137 .100 .135 .099 .135 .097 .136 .102 .169 .102 .145 .103 .141 .291 .128

64 .150 .246 .148 .217 .151 .219 .151 .222 .152 .233 .172 .222 .328 .230 .298 .222

-.5 128 .091 .107 .093 .110 .092 .106 .091 .105 .091 .113 .091 .106 .173 .108 .097 .104

256 .062 .068 .062 .069 .062 .067 .062 .067 .062 .070 .061 .067 .076 .068 .061 .067

TABLE 3. Bias of estimates of ϕ0

δ0 -.6 -.6 -.4 -.4 0 0 .4 .4 .6 .6 1 1 1.5 1.5 2 2

ϕ0 n ϕ W ϕ ϕ W ϕ ϕ ϕW ϕ W ϕ ϕ W ϕ ϕ W ϕ ϕ W ϕ ϕ W ϕ

64 -.001 -.034 -.003 -.039 .004 -.016 -.003 -.032 .006 -.040 .016 -.017 .076 -.032 .107 -.010

.5 128 .022 .007 .025 .004 .025 .015 .026 .016 .022 -.004 .035 .018 .030 -.002 .115 .015

256 .031 .021 .030 .019 .032 .028 .031 .023 .030 .006 .031 .029 .032 .016 .112 .025

64 .095 .213 .087 .166 .088 .177 .095 .215 .106 .221 .094 .181 .133 .240 .429 .176

0 128 .043 .091 .039 .077 .043 .084 .043 .095 .044 .122 .043 .087 .049 .118 .245 .083

256 .017 .022 .017 .017 .016 .022 .016 .020 .017 .027 .016 .025 .018 .022 .118 .020

64 .039 .064 .036 .049 .036 .054 .037 .054 .035 .057 .042 .057 .128 .056 .101 .054

-.5 128 .018 .022 .017 .018 .017 .020 .017 .020 .016 .021 .016 .020 .039 .021 .017 .019

256 .009 .010 .008 .007 .008 .009 .008 .008 .008 .008 .007 .009 .011 .009 .008 .010

3

4

J. HUALDE AND P.M. ROBINSON

In Table 3, we compare the estimates of ϕ0 in terms of bias. There are is the more similar overall patterns to those in Table 1, but now, while ϕ biased when ϕ0 = 0.5 (in 18 against 6 cases, the latter being ones when is superior (in 37 against n = 64 and δ 0 ≤ 1), for both ϕ0 = 0 and −0.5 ϕ 18 cases, the latter mostly being ones when τ 0 = (1.5, −0.5)′ , (2, 0)′ . The Monte Carlo standard deviations of estimates of ϕ0 , in Table 4, show a clearly dominating, though for broadly similar picture to Table 2, with ϕ is even more imprecise. ϕ0 = 0 and n = 64 ϕ TABLE 4. Standard deviation of estimates of ϕ0

δ0 -.6 -.6 -.4 -.4 0 0 .4 .4 .6 .6 1 1 1.5 1.5 2 2

ϕ0 n ϕ W ϕ ϕ W ϕ ϕ W ϕ ϕ W ϕ ϕ ϕW ϕ W ϕ ϕ W ϕ ϕ W ϕ

64 .255 .273 .258 .275 .257 .268 .260 .280 .257 .288 .255 .268 .247 .280 .238 .267

.5 128 .207 .226 .211 .228 .208 .224 .208 .227 .209 .243 .210 .223 .209 .234 .214 .222

256 .167 .185 .166 .184 .166 .181 .167 .185 .167 .195 .166 .181 .166 .187 .191 .183

64 .300 .406 .303 .389 .300 .387 .311 .413 .319 .416 .304 .389 .348 .424 .468 .387

0 128 .197 .291 .195 .283 .200 .285 .199 .297 .206 .333 .199 .285 .212 .326 .411 .282

256 .117 .147 .118 .145 .117 .146 .115 .146 .121 .176 .120 .154 .120 .152 .311 .140

64 .159 .230 .156 .198 .154 .202 .156 .205 .156 .219 .177 .207 .382 .212 .346 .204

-.5 128 .100 .105 .098 .103 .097 .102 .098 .102 .097 .106 .096 .101 .201 .104 .104 .098

256 .068 .069 .067 .067 .066 .068 .065 .066 .068 .068 .065 .066 .086 .067 .066 .068

With the aim of providing a clearer picture of the pattern of estimates with respect to variations in τ 0 , we plot in Figure 1 the Monte Carlo root mean square error of δ, δW , for n = 128, as a function of 15 values of δ 0 (those in the initial choice plus -0.2, 0.2, 0.8, 1.2, 1.4, 1.6, 1.8) for ϕ0 = −0.5, 0, 0.5. In each of the plots, the thick line corresponds to results for δ, whereas the thin one records results for δ W . As anticipated, the best results are for ϕ0 = −0.5, δ being superior to δ W , except in the region between δ 0 = 1.4 and 2. For δ W , results are worst for ϕ0 = 0, when δ clearly dominates (except when δ 0 = 2), whereas when ϕ0 = 0.5, both estimates behave similarly, although δ is slightly superior overall.

GAUSSIAN ESTIMATION OF FRACTIONAL TIME SERIES MODELS

5

FIGURE 1. Root Mean Square Error of estimates of δ 0

, ϕ W (recorded Finally, in Figure 2 we plot corresponding results for ϕ in the thick and thin lines, respectively). The pattern is in all cases almost identical to that in Figure 1. FIGURE 2. Root Mean Square Error of estimates of ϕ0

EMPIRICAL EXAMPLE We now report an empirical application to US quarterly income and consumption data 1947Q1-1981Q2 (n = 138), which was previously analyzed

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J. HUALDE AND P.M. ROBINSON

by [3], for example. By means of traditional testing procedures [3] found evidence of a unit root in both series, and the semiparametric fractional approach of [5] tended to support this conclusion. Our analysis did not. We determined θ (s; ϕ) from the data, our approach permitting comparison among competing parametric models. This was achieved by ﬁrst obtaining a preliminary estimate of δ0 , which was used to ﬁlter the series to have, approximately, short memory, and then employing the model choice procedure √ of [1] to select p1 and p2 . For this purpose we cannot use a n−consistent parametric estimate of δ 0 (for example, one based on a FARIMA(0, δ 0 , 0)) because under-speciﬁcation of p1 or p2 , or over-speciﬁcation of both, results in inconsistent estimation of δ 0 . Instead, we employed a semiparametric estimate of δ 0 , which converges more slowly but does not require short memory speciﬁcation and is thus more robust. In addition, we examine the issue of truncation, which is inherent to model (1.1), and arises because the model reﬂects the data start-time: given a sample xt , t = 1, ..., n, the ﬁrst observation of the ﬁltered sequence ∆d {xt 1 (t > 0)} equals the unﬁltered x1 , the second is a linear combination of x1 , x2 , and so on. We check stability with respect to omitting from the analysis l initial observations of the ﬁltered series. We look ﬁrst at the income series. We computed the local Whittle or semiparametric Gaussian estimate (see e.g. [6]) on ﬁrst-diﬀerenced observations ∆xt , followed by adding back 1 (an alternative semiparametric estimate, which is valid also under nonstationarity, and thus avoids the initial ﬁrstdiﬀerencing, was proposed and justiﬁed by [7]). In order to reﬂect possible sensitivity to choice of bandwidth m (the number of low Fourier frequencies employed) and because the choice of m only indirectly aﬀects the ﬁnal outcome, rather than employing an optimal, data-dependent m, we tried three diﬀerent values, m = 8, 17, 34, obtaining estimates δ = 1.107, 1.017, 1.084, respectively. Using these δ, the ﬁltered ∆δ {xt 1 (t > 0)} were generated, and from their simple and partial correlograms we identiﬁed in the spirit of [1] the parametric model θ (s; ϕ). For the various estimates of δ 0 , the methodol −1 ogy suggested that θ (s; ϕ0 ) = 1 − ϕ0 s10 might be adequate. We report our estimates of δ 0 , ϕ0 in Table 5, along with t-ratios (denoted by tδ , tϕ ) corresponding to the null hypotheses H0 : δ 0 = 1, H0 : ϕ0 = 0, where denominators are corresponding elements n nof the 2-dimensional square ma )′ . trix ∂εt (τ )/∂τ (∂εt (τ )/∂τ )′ / ε2 (τ ), where τ = (δ, ϕ t=10+l t=10+l t For l > 2, the corresponding null hypotheses are in all cases rejected at 1% signiﬁcance level, thus casting doubt on the unit root hypothesis.

GAUSSIAN ESTIMATION OF FRACTIONAL TIME SERIES MODELS

7

TABLE 5. Parameter estimates for the income series

l δ tδ ϕ tϕ

1 1.12 2.29 .204 2.55

2 1.14 2.54 .257 3.43

3 1.15 2.62 .236 3.02

4 1.15 2.62 .235 3.01

5 1.15 2.62 .235 3.01

6 1.15 2.67 .233 3.01

7 1.15 2.66 .247 3.18

8 1.15 2.64 .249 3.13

9 1.15 2.66 .245 3.07

10 1.16 2.68 .242 3.03

For the consumption series results are provided in Table 6. As before, we computed three diﬀerent δ = 0.855, 0.976, 1.127, for m = 8, 17, 34, respectively. We again identiﬁed θ (s; ϕ) based on the corresponding residuals, but now the greater variation of the δ, leads to two diﬀerent speci(1) ﬁcations, namely θ (s; ϕ0 ) = (1 − ϕ0 s)−1 (suggested by δ = 0.855) and (8) 8 −1 θ (s; ϕ0 ) = (1 − ϕ0 s ) (suggested by δ = 0.976, 1.127). Given the discrepancy, we let the two short run models compete in our parametric spec(1) (8) iﬁcation, setting θ (s; ϕ0 ) = (1 − ϕ0 s − ϕ0 s8 )−1 , obtaining parametric (1) (8) , ϕ . As before, t-ratios for identical null hypotheses are estimates δ, ϕ (1) provided, supporting clearly the speciﬁcation with ϕ0 = 0, a unit root being again strongly rejected. TABLE 6. Parameter estimates for the consumption series

l δ

tδ (1) ϕ tϕ(1) (8) ϕ tϕ(8)

1 1.07 2.33 -.016 -.178 -.164 -2.06

2 1.10 2.74 -.054 -.600 -.196 -2.50

3 1.11 2.78 -.068 -.750 -.213 -2.62

4 1.11 2.76 -.072 -.785 -.220 -2.67

5 1.12 2.70 -.074 -.807 -.221 -2.68

6 1.12 2.63 -.075 -.809 -.223 -2.64

7 1.15 2.79 -.092 -1.01 -.225 -2.77

PROOFS OF LEMMAS IN SECTION 5 PROOF OF LEMMA 1. Clearly, (0.1)

cj (τ ) =

j

k=0

φk (ϕ) aj−k ,

8 1.14 2.65 -.064 -.674 -.220 -2.70

9 1.15 2.72 -.054 -.570 -.240 -3.01

10 1.15 2.63 -.041 -.423 -.233 -2.89

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J. HUALDE AND P.M. ROBINSON

writing aj = aj (δ0 − δ), so that for any δ ∈ I, by Stirling’s approximation sup |cj (τ )| ≤ K

ϕ∈Ψ

≤ K (0.2)

j−1

(j − k)δ0 −δ−1 sup |φk (ϕ)|

(j − k)δ0 −δ−1 sup |φk (ϕ)|

k=0 [j/2]

ϕ∈Ψ

ϕ∈Ψ

k=0 j−1

+K

k=[j/2]

(j − k)δ0 −δ−1 sup |φk (ϕ)| . ϕ∈Ψ

(0.2) is bounded by Kj

δ0 −δ−1

∞

k

−1−ς

+ Kj

j−1

−1−ς

k=1

k=[j/2]

(j − k)δ0 −δ−1 = O j max(δ0 −δ−1,−1−ς) ,

because ς > 1/2 and the second sum is O j δ0 −δ if δ < δ 0 , O (log j) if δ = δ 0 , and O (1) if δ > δ 0 . The proof of (5.2) is almost identical on noting cj+1 −cj = φj+1 (ϕ)+

j

k=1

PROOF OF LEMMA 2. From (5.1), (0.1) εt (τ ∗ ) =

t−1

j=0

aj

t−j−1

φk (ϕ0 ) ut−j−k =

t−1

aj εt−j + vt (δ) ,

j=0

k=0

where (0.3)

vt (δ) = −

t−1

aj

j=0

∞

φk (ϕ0 ) ut−j−k .

k=t−j

Thus ∞ sup |vt (δ)| ≤ K j κ−1 φk (ϕ0 ) ut−j−k . k=t−j δ 0 −κ≤δ<δ 0 − 12 +η j=1 t

Now



V ar 

∞

k=t−j

φk (ϕ) (aj+1−k − aj−k ) , aj+1 −aj = O j δ0 −δ−2 .



φk (ϕ0 ) ut−j−k  ≤ K

∞

k=t−j

φ2k (ϕ0 ) ≤ K (t − j)−1−2ς .

GAUSSIAN ESTIMATION OF FRACTIONAL TIME SERIES MODELS

9

Thus 

|vt (δ)| = Op 

sup δ 0 −κ≤δ<δ0 − 12 +η

t−1

j=1

j κ−1 (t − j)



−1/2−ς 

= Op tκ−1 ,

as in the proof of Lemma 1, noting that 1 + ς > 3/2. Finally, by (0.3) vt (δ 0 ) = −

∞

φk (ϕ0 ) ut−j−k = Op t−1/2−ς ,

k=t

by previous arguments. PROOF OF LEMMA 3. Since εt (τ ) = ξ (L; ϕ) εt (τ ∗ ), following similar steps as in [2] (p.346),

wε(τ ) (λ) = ξ n−1 eiλ ; ϕ wε(τ ∗ ) (λ) + Un (λ; τ ) , where ξ n−1 (z; ϕ) =

n−1 j=0

ξ j (ϕ) z j and 1

Un (λ; τ ) = −n− 2

n−1

ξ k (ϕ) eikλ

k=1

n

εt (τ ∗ ) eitλ ,

t=n−k+1

so that (5.3) holds with n n 2 2 iλj iλj Vn (τ ) = |Un (λj ; τ )|2 ξ n−1 e ; ϕ − ξ e ; ϕ Iε(τ ∗ ) (λj ) + j=1

(0.4)

+2 Re

 n  

 

ξ n−1 eiλj ; ϕ wε(τ ∗ ) (λj ) Un (−λj ; τ ) .

j=1



The third term of (0.4) is −

n n−1 n n 2 n−1 ξ k (ϕ) ξ l (ϕ) εt (τ ∗ ) εs (τ ∗ ) ei(k+t−l−s)λj n k=0 t=1 l=1 s=n−l+1 j=1

= −2

n−1 n−1

n+min(k−l,0)

ξ k (ϕ) ξ l (ϕ)

k=1 l=1

εs+l−k (τ ∗ ) εs (τ ∗ ) ,

s=n−l+1

where by Lemma 2 (0.5)

j=1

∗

εs (τ ) =

s−1 j=0

aj εs−j + vs (δ) .

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J. HUALDE AND P.M. ROBINSON

By summation by parts, for s ≥ 2, the ﬁrst term on the right of (0.5) is as−1

s−1 j=0

εs−j −

so that

s−1 E sup a ε j s−j ≤ δ 0 −κ≤δ<δ 0 − 12 +η j=0

It can be readily shown that V ar in j, s, V ar

j

ε k=0 s−k

s−2 j=0

(aj+1 − aj )

j

εs−k ,

k=0

s−1 sup |as−1 | E εs−j j=0 δ0 −κ≤δ<δ 0 − 12 +η

j + sup |aj+1 − aj | E εs−k . 1 k=0 j=0 δ 0 −κ≤δ<δ 0 − 2 +η s−2

s−1 j=0

εs−j

= O (s), whereas, uniformly

= O (j), so that

  s−1 s−2 1 3 sδ− 2 + E sup aj εs−j ≤ K sup j δ− 2  1 1 δ 0 −κ≤δ<δ 0 − 2 +η j=0 δ 0 −κ≤δ<δ 0 − 2 +η j=1

≤ K log s1 (κ = 1/2) + sκ−1/2 1 (κ > 1/2) ,

(0.6)

whereas by Lemma 2 E

sup δ0 −κ≤δ<δ 0 − 12 +η

Then since

|vs (δ)| ≤ Ksκ−1 .

n+min(k−l,0) ∗ ∗ E sup εs+l−k (τ ) εs (τ ) δ 0 −κ≤δ<δ 0 − 12 +η s=n−l+1 2

≤ Kl log n1 (κ = 1/2) + nκ−1/2 1 (κ > 1/2)

+ Kl,

we have

n−1 n−1 n+min(k−l,0) ∗ ∗ E sup ξ (ϕ) ξ (ϕ) ε (τ ) ε (τ ) s s+l−k k l δ0 −κ≤δ<δ 0 − 12 +η k=1 l=1 s=n−l+1 ϕ∈Ψ

≤ K log2 n1 (κ = 1/2) + n2κ−1 1 (κ > 1/2) sup

2

2κ−1

≤ K log n1 (κ = 1/2) + n

∞

ϕ∈Ψ k=0

1 (κ > 1/2) .

|ξ k (ϕ)|

∞ l=0

l |ξ l (ϕ)|

11

GAUSSIAN ESTIMATION OF FRACTIONAL TIME SERIES MODELS

Following similar steps to previous ones, it is immediate to show that n

sup

δ 0 −κ≤δ<δ0 − 12 +η j=1 ϕ∈Ψ

|Un (λj ; τ )|2 = Op log2 n1 (κ = 1/2) + n2κ−1 1 (κ > 1/2) .

Finally n 2 2 iλj iλj sup ξ e − ξ e I ∗ (λ ) j ε(τ ) n−1 δ 0 −κ≤δ<δ 0 − 12 +η j=1 ϕ∈Ψ

≤

sup λ∈[−π,π] ϕ∈Ψ

2 2 iλ iλ ξ n−1 e − ξ e

sup δ 0 −κ≤δ<δ0 − 12 +η

n

ε2t (τ ∗ ) .

t=1

By previous results sup

n

δ 0 −κ≤δ<δ 0 − 12 +η t=1

ε2t (τ ∗ ) = Op n log2 n1 (κ = 1/2) + n2κ 1 (κ > 1/2) ,

and noting that sup λ∈[−π,π] ϕ∈Ψ

2 2 −ς iλ iλ ξ = o (1) , n−1 e ; ϕ − ξ e ; ϕ = O n

the ﬁrst term on the right of (0.4) is of smaller order, to conclude the proof. PROOF OF LEMMA 4. The proof is very similar to that of Lemma 1. The only point worth mentioning is the calculation of the order of magnitude of ∂ (aj+1 (c) − aj (c)) /∂c and ∂ 2 (aj+1 (c) − aj (c)) /∂c2 . First, ∂ (aj+1 (c) − aj (c)) /∂c is (0.7)

ψ (j + 1 + c) aj+1 (c) − ψ (j + c) aj (c) − ψ (c) (aj+1 (c) − aj (c)) .

The third term in (0.7) is O j c−2 , whereas since

and

|ψ (j + 1 + c) − ψ (j + c)| ≤ K ψ′ (j + c) ≤ K (j + 1)−1 , |ψ (j + c)| ≤ K log (j + 1) ,

then (aj+1 (c) − aj (c)) ψ (j + 1 + c) + aj (c) (ψ (j + 1 + c) − ψ (j + c))

12

J. HUALDE AND P.M. ROBINSON

is O j c−2 log j . Thus (0.7) is O j c−2 log j . Second, it can be shown that ∂2 (aj+1 (c) − aj (c)) ∂c2 ∂aj+1 (c) ∂aj (c) = ψ (j + 1 + c) − ψ (j + c) + o j c−2 log2 j ∂c ∂c c−2 2 = O j log j ,

by similar steps to those in the treatment of (0.7). References. [1] Box, G.E.P. and Jenkins, G.M. (1971). Time series analysis. Forecasting and control. Holden-Day, San Francisco. [2] Brockwell, P.J. and Davis, R.A. (1990). Time series: theory and methods. SpringerVerlag, New York. [3] Engle, R.F. and Granger, C.W.J. (1987). Cointegration and error correction: representation, estimation and testing. Econometrica 55, 251-276. [4] Hualde, J. and Robinson P.M. (2011). Gaussian pseudo-maximum likelihood estimation of fractional time series models. [5] Marinucci, D. and Robinson P.M. (2001). Semiparametric fractional cointegration analysis. Journal of Econometrics 105, 225-247. [6] Robinson, P.M. (1995). Gaussian semiparametric estimation of long-range dependence. Annals of Statistics 23, 1630-1661. [7] Shimotsu, K. and Phillips, P.C.B. (2005). Exact local Whittle estimation of fractional integration. Annals of Statistics 33, 1890-1933. [8] Velasco, C. and Robinson, P.M. (2000). Whittle pseudo-maximum likelihood estimation for nonstationary time series. Journal of the American Statistical Association 95, 1229—1243. J. Hualde Departamento de Econom´ıa ´ blica de Navarra Universidad Pu Campus Arrosad´ıa 31006 Pamplona Spain E-mail: [email protected]

P.M. Robinson Department of Economics London School of Economics Houghton Street London WC2A 2AE United Kingdom E-mail: [email protected]