# Two estimators of the long-run variance: Beyond short memory

**ABSTRACT** This paper deals with the estimation of the long-run variance of a stationary sequence. We extend the usual Bartlett-kernel heteroskedasticity and autocorrelation consistent (HAC) estimator to deal with long memory and antipersistence. We then derive asymptotic expansions for this estimator and the memory and autocorrelation consistent (MAC) estimator introduced by Robinson [Robinson, P. M., 2005. Robust covariance matrix estimation: HAC estimates with long memory/antipersistence correction. Econometric Theory 21, 171–180]. We offer a theoretical explanation for the sensitivity of HAC to the bandwidth choice, a feature which has been observed in the special case of short memory. Using these analytical results, we determine the MSE-optimal bandwidth rates for each estimator. We analyze by simulations the finite-sample performance of HAC and MAC estimators, and the coverage probabilities for the studentized sample mean, giving practical recommendations for the choice of bandwidths.

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**ABSTRACT:**We construct a two-sample test for comparison of long memory parameters based on ratios of two rescaled variance (V/S) statistics studied in Giraitis et al. [L. Giraitis, R. Leipus, A. Philippe, A test for stationarity versus trends and unit roots for a wide class of dependent errors, Econometric Theory 21 (2006) 989–1029]. The two samples have the same length and can be mutually independent or dependent. In the latter case, the test statistic is modified to make it asymptotically free of the long-run correlation coefficient between the samples. To diminish the sensitivity of the test on the choice of the bandwidth parameter, an adaptive formula for the bandwidth parameter is derived using the asymptotic expansion in Abadir et al. [K. Abadir, W. Distaso, L. Giraitis, Two estimators of the long-run variance: beyond short memory, Journal of Econometrics 150 (2009) 56–70]. A simulation study shows that the above choice of bandwidth leads to a good size of our comparison test for most values of fractional and ARMA parameters of the simulated series.Journal of Multivariate Analysis 10/2010; · 1.06 Impact Factor - SourceAvailable from: Donatas Surgailis[Show abstract] [Hide abstract]

**ABSTRACT:**Nous proposons un test pour comparer le paramètre de longue mémoire de deux processus éventuellement corrélés. Le test est construit à partir de la statistique V/S basée sur deux estimations de la variance asymptotique des sommes partielles. Nous établissons la consistance asymptotique du test. Des simulations illustrent les performances du test sur des petits échantillons et sa sensibilité au paramètre de type fenêtre de la statistique V/S. A partir d'un développement asymptotique de la statistique V/S, nous obtenons un critère adaptatif pour le choix de ce paramètre.42èmes Journées de Statistique. 01/2010;

Page 1

Electronic copy available at: http://ssrn.com/abstract=1984844

Two estimators of the long-run variance: beyond short

memory

Karim M. Abadir∗

Imperial College Business School, Imperial College London, London SW7 2AZ, UK

Walter Distaso

Imperial College Business School, Imperial College London, London SW7 2AZ, UK

Liudas Giraitis

Department of Economics, Queen Mary, University of London, London E14 NS, UK

January 19, 2009

Abstract

This paper deals with the estimation of the long-run variance of a sta-

tionary sequence. We extend the usual Bartlett-kernel heteroskedasticity and

autocorrelation consistent (HAC) estimator to deal with long memory and an-

tipersistence. We then derive asymptotic expansions for this estimator and the

memory and autocorrelation consistent (MAC) estimator introduced by Robin-

son (2005). We offer a theoretical explanation for the sensitivity of HAC to the

bandwidth choice, a feature which has been observed in the special case of short

memory. Using these analytical results, we determine the MSE-optimal band-

width rates for each estimator. We analyze by simulations the finite-sample

performance of HAC and MAC estimators, and the coverage probabilities for

the studentized sample mean, giving practical recommendations for the choice

of bandwidths.

JEL Classification: C22, C14.

Keywords: long-run variance, long memory, heteroskedasticity and autocorrelation

consistent (HAC) estimator, memory and autocorrelation consistent (MAC) estima-

tor.

∗Corresponding author. E-mail address: k.m.abadir@imperial.ac.uk.

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Electronic copy available at: http://ssrn.com/abstract=1984844

1 Introduction and setup

In empirical studies, it is now standard practice to produce robust estimates of stan-

dard errors (SEs). Popular references in econometrics for such procedures include

White (1980), Newey and West (1987), Andrews and Monahan (1992). In statis-

tics, the literature goes further back to Jowett (1955) and Hannan (1957). These

procedures for estimating covariance matrices account for heteroskedasticity and

autocorrelation of unknown form, for short memory models.

There is now an increasing body of evidence suggesting the existence of long

memory in macroeconomic and financial series; e.g. see Diebold and Rudebusch

(1989), Baillie and Bollerslev (1994), Gil-Alaña and Robinson (1997), Chambers

(1998), Cavaliere (2001), Abadir and Talmain (2002). It is therefore of interest to

adapt the most popular of these procedures, the Bartlett-kernel heteroskedastic-

ity and autocorrelation consistent (HAC) estimator, to account for the possibility

of long memory and antipersistence. In addition to HAC, we study the alterna-

tive memory and autocorrelation consistent (MAC) estimator recently introduced

by Robinson (2005). He established the consistency of his MAC estimator of the

covariance matrix, leaving open the issue of its higher-order expansion.

Our first contribution is to derive second order expansions for HAC and MAC

in the univariate case, reducing the problem to the estimation of a scalar (the long

run variance) instead of estimating the covariance matrix. Our derivations give

an insight into the more difficult multivariate case and provide the first step in

understanding this problem.

The second contribution of this paper is to provide a theoretical explanation for

the sensitivity of HAC estimators to the choice of bandwidth, a feature that has

been widely observed in the special case of short memory. Our results show that

the HAC estimator is sensitive because the minimum-MSE bandwidth depends on

the persistence in the series. The theoretical part of this paper explains where the

problem comes from and gives some practical advice for selecting the bandwidth.

We also show that, on the other hand, the MAC estimator is more robust to the

bandwidth selection, since its asymptotic properties are not affected by long memory

or antipersistence.

The final theoretical contribution of this paper is to obtain the distribution

of the estimated normalized spectrum at the origin, by virtue of its link to the

long-run variance. The distribution is Gaussian for MAC, but the one for HAC is

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Gaussian only if the long memory is below some threshold. In the case of short

memory, HAC is the usual Bartlett-kernel estimator of the spectral density at zero

frequency, and its bias and asymptotic distribution are well investigated in the

literature. The asymptotic results for the HAC estimator provide the background

for the development of kernel estimation of a spectral density under long memory

and antipersistence.

The plan of the paper is as follows. In Sections 2 and 3, we derive the bias

and asymptotic expansions for both types of estimators, allowing us to describe the

limiting distributions as well as the asymptotic MSEs. This enables us to determine

the rate of the MSE-optimal bandwidth for each estimator. Section 4 investigates

by simulations the finite-sample performance of HAC and MAC estimators, and

coverage probabilities for the studentized sample mean, giving practical recommen-

dations for the choice of bandwidths. Section 5 concludes. The derivations are given

in the Appendix.

We now detail the setting for our paper. Let {}∈Zbe a stationary sequence

with unknown mean := E(). Let the spectral density of {} be denoted by

() and defined over || ≤ . Suppose that it has the property

() = 0||−2+ (||−2) as → 0(1.1)

where || 12 and 0 0.

ARIMA(): when and are finite; but see Abadir and Taylor (1999) for

Special cases include stationary and invertible

identification issues when or are allowed to be infinite. We shall call the

memory parameter of {}; with = 0 indicating short memory, 0 12 long

memory, and −12 0 antipersistence.

To conduct inference on , define the sample mean¯ := −1P

satisfies

var(12−¯ ) = −1−2

−

As → ∞, we can use assumption (1.1) and a change of variable of integration to

get the convergence

Z∞

where we have the continuous function

⎧

⎩

3

=1 which

Z

µsin(2)

sin(2)

¶2

()d

var(12−¯ ) → 2

:= 0

−∞

µsin(2)

2

¶2

||−2d = 0()(1.2)

() :=

⎨

2Γ(1−2)sin()

(1+2)

if 6= 0

if = 02

(1.3)

Page 4

We notice from (1.2) that 2

is just a scaling of 0by the function (), so in the

usual short memory case of = 0 we get

2

= 2(0)and0= (0)

In general, the problem of the estimation of the long-run variance 2

related to the estimation of and 0≡ lim→0||2() appearing in (1.1). The

HAC and MAC procedures mentioned at the start of this section hinge on the

is closely

estimation of the long run variance 2

.

We will consider the behaviour of the estimators under two alternative sets of

assumptions. The first one is stronger than the second one. It allows the derivation

of asymptotic expansions and the resulting investigation of MSE-optimal bandwidth

rates. The second one is sufficient to establish the consistency of the estimators

for a wide class of stationary sequences. It allows the use of estimates of for

robust SEs for¯ . The second type of conditions are very weak, so they yield only

consistency and are not sufficient to obtain other asymptotic results. 2 The first set

of assumptions is common for HAC and MAC:

Assumption L. {} is a linear sequence

= +

∞

X

=0

− ∈ Z

whereP∞

zero mean and unit variance. Moreover, the spectral density () of {} has the

property

() = ||−2()

where ∈ (−1212) and (·) is a continuous bounded function such that () =

0(1 + (||2)) as → 0 and 0= (0) 0.

Let b 2

t := 12−(¯ − )

b

For HAC, the second type of assumptions (to establish consistency) is:

=02

∞, is a real number and {} are i.i.d. random variables with

(1.4)

be a consistent estimator of 2

the sample mean¯ satisfies

. Under condition (1.4), the t-ratio for

→ N(01) → ∞(1.5)

so that a consistent HAC or MAC estimator of 2

allows inference on .

Assumption M. {} is a fourth order stationary process such that, for some

∈ (−1212) and 6= 0,

∼ 2−1if 6= 0

∞

X

=−∞

|| ∞ if = 0

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Page 5

where := cov(+); and

∞

X

=−∞

|()| ≤ if 0 sup

X

=−

|()| ≤ 2if ≥ 0

where denotes a generic constant and () is a fourth-order cumulant defined

by () := E(+++)−(−+−+−). In addition, if

0, then () ≤ ||−2 ∈ [−].

For MAC, the second type of assumptions differs from Assumption M and is

straightforward to discuss at the end of Section 3.

2Asymptotic properties of HAC-type estimators

In this section, we first adapt the HAC estimator to allow for long memory and an-

tipersistence, introducing two HAC-type estimators. Then, we analyze their prop-

erties under Assumption L that {} is a linear process, presenting limiting distrib-

utions and asymptotic expansions for the estimators. To the best of our knowledge,

the asymptotic normality of the HAC estimator was investigated in the literature

only in the short memory case of = 0 and under the assumption that E(4

) ∞.

Our Theorem 2.1(a) will require for {} the existence of only a moment of order

2+ (for some 0), which is a new result in the field. It also shows that, under the

strong persistence 14 12, the asymptotic distribution will be non-Gaussian.

Finally, we show that Assumption M guarantees consistency (but not necessarily

the other properties) of the estimators.

Let

e := −1

−

X

=1

(− E())(+− E()) 0 ≤

be the sample autocovariances of {} centered around E(), and

:= −1

−

X

=1

(−¯ )(+−¯ ) 0 ≤

the sample autocovariances of {} centered around the sample mean¯ .

Define

e 2

() := −1−2

X

=1

e |−|= −2(e 0+ 2

X

=1

(1 − )e )(2.1)

5

Page 6

which uses a known (or correctly hypothesized) E(), and

¯ 2

() := −1−2

X

=1

|−|= −2(0+ 2

X

=1

(1 − )) (2.2)

where the mean is estimated unrestrictedly, and assume that the bandwidth para-

meter satisfies

→ ∞ = (1−) (2.3)

for some 0.The difference between the stochastic expansions of the two

estimators will reveal just how much is the impact of estimating E(). The

asymptotically-optimal choice of will arise from the first theorem below. To make

() operational, we can employ any estimatorb that is consistent at the

and two such estimators ofb will be discussed later in Section 3.

of this section, we need to assume that the coefficients decay as

e 2

() and ¯ 2

rate of log or faster, calculating e 2

We start by making Assumption L. In addition, to establish the main theorem

(b) and ¯ 2

(b). This is a very weak condition,

= −1+(1 + (−1)) 6= 0 if 6= 0;

∞

X

=0

= 0 if 0; and(2.4)

∞

X

=

|| = (−2) if = 0(2.5)

Such additional requirements are satisfied, for example, by ∼ ARIMA()

where ∈ (−1212). We now derive asymptotic expansions for the estimators

e 2

:=1

2

−∞

(b) and ¯ 2

(b), where the bias will be expressed in terms of

()

sin2(2)1||≤−0||−2

Z∞

µ

(2)2

¶

d(2.6)

In the case of −12 14, these HAC estimators have Gaussian limit distri-

butions. However, if 14 12, then the limit can be written in terms of a

random variable given by the double Itô-Wiener integral

Z00

where (d) is a standard Gaussian complex measure ((−d) is the conjugate of

(d)) with mean zero and variance E(|(d)|2) = d. The limit variable () has

a (non-Gaussian) Rosenblatt distribution and is well-defined when 14 12.

The symbolR00

6

() :=

R2

ei(1+2)− 1

i(1+ 2)

|1|−|2|−(d1)(d2)(2.7)

R2indicates that one does not integrate on the diagonals 1= ±2.

Page 7

Theorem 2.1. Supppose that {} satisfies Assumption L and (2.4)—(2.5), and that

b is an estimator of such that

(b − )log = (1)

(a) If −12 14 and E(||2+) ∞, for some 0, then, as → ∞,

(2.8)

e 2

(b) − 2

= ()12+ −1−2 + (()12) + (−1−2)(2.9)

and

¯ 2

(b) − 2

= ()12+ −1−2 + (()12) + (−1−2)(2.10)

where

→ N(02

Z∞

),

2

:= 82

0

0

µsin(2)

2

¶4

−4d =

⎧

⎩

⎨

162

0

2(21+4−1)

Γ(4+4)cos(2)if 6= 0

if = 0

1622

03 = 44

3

(2.11)

and it is understood that lim→−14(21+4− 1)cos(2) = log4.

(b) If 14 12, E¡4

then

¢ ∞ and () in (1.4) has bounded derivative,

e 2

(b) − 2

= ()1−2e+ −1−2 + (()1−2) + (−1−2)

X

(b) has the property

¯ 2

(2.12)

where

e:= −2

=1

(2

− E(2

))

→ 20();

whereas ¯ 2

(b) − e 2

(b) =

³

()1−2´

) ∞, the MSEs of HAC-type esti-

(2.13)

Under the additional assumption that E(4

mators exist and are minimized asymptotically by

∝

(

1(3+4)−12 14

14 12

12−

(2.14)

where ∝ denotes proportionality. We now list other comments and implications

arising from Theorem 2.1:

Remark 2.1. Since E() = E(e) = 0, the asymptotic bias of the estimators is

given by −1−2. It tends to zero as (hence ) tends to infinity.

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Page 8

Remark 2.2. When −12 14, the convergence ¯ 2

implies that 2

can be consistently estimated by replacing 0by ¯ 2

(b)

−→ 2

(b)(b) in (2.11).

= ()0

Remark 2.3. If 14, then estimates with known and estimated mean have the

same asymptotic properties. However, if 14, then the rate of convergence of

the sample mean to is rather slow, and replacing by¯ leads to an additional

term in the limiting distribution of the HAC estimator whose consistency is nev-

ertheless unaffected. In the context of hypothesis testing about the mean , one

can estimate the long run variance by treating as unknown and estimating it by

the sample mean. Alternatively, one can compute the long-run variance under the

null hypothesis, treating as known. This will improve the size but may have an

adverse effect on the finite-sample power of tests based on HAC estimators.

Remark 2.4. As a general rule, convergence in distribution does not necessarily

imply a corresponding convergence for moments such as the MSE. However, our

proofs are based on 2expansions for which this implication holds if we make the

additional assumption that E(4

) ∞, hence our stated results for the asymptotic

bias and variance. Note that for the validity of asymptotic expansions (2.9)—(2.10),

only 2 + moments of {} are needed.

Remark 2.5. If {} is a nonlinear process, then Theorem 2.1 might not hold.

For example, the nonlinear transformation = eof a linear process {} will, in

general, increase the bias of estimators. Therefore, the optimal minimizing the

MSE might also change in this case.

Relaxing Assumption L, we obtain the following concistency result.

Theorem 2.2. Suppose that → ∞, = (12), that Assumption M holds, and

thatb − = (1log). Then,

¯ 2

→ 2

(b)

e 2

(b)

−→ 2

as → ∞(2.15)

3Robinson’s MAC estimator

In this section, we derive the asymptotic properties of Robinson’s MAC estimator

of 2

= ()0, where () is given by (1.3). We shall show that the asymptotic

properties of the MAC estimator do not depend on the memory parameter , and its

asymptotic distribution is always Gaussian. Hence, it is more robust than HAC to

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Page 9

the bandwidth selection in practice, something that will be illustrated numerically

in the subsequent section. Define

b 2

() := ()b()

X

where

b() := −1

=1

2

()

is a consistent estimator of 0,

() := (2)−1

¯¯¯¯¯

X

=1

ei

¯¯¯¯¯

2

is the periodogram, = 2 are the Fourier frequencies, and the bandwidth

parameter satisfies → ∞ and = ((log)2).

This estimator has a number of features. First, it does not require estimation

of the unknown mean E() since the periodogram is self-centring at the Fourier

frequencies . Contrast this with HAC estimators; see also Remark 2.3. Second,

as the following theorem will show, the bias and asymptotic distribution of the

estimator do not depend on ∈ (−1212), and the asymptotic distribution is

always Gaussian.

In addition to Assumption L, we will need the condition that () :=

P∞

d

d() = (|()|)

=0eisatisfies

as → 0+(3.1)

in order to derive the CLT in the following theorem.

Theorem 3.1. Suppose that {} satisfies Assumption L with E(4

Assume thatb is an estimator of such thatb − = (1log). Then

b 2

) ∞ and (3.1).

(b) − 2

= −122

+ 2(b − )(log)2

(1 + (1))(3.2)

+(()2) + (−12)

where

→ N(01)(3.3)

The parameter can be estimated, for example, by the local Whittle estimator

b := argmin∈ [−1212]()

9

Page 10

which minimizes the objective function

() := log

⎛

⎝1

X

=1

2()

⎞

⎠−2

X

=1

log

with bandwidth parameter such that → ∞ and = ((log)2). We

use the notation for the bandwidth of the local Whittle estimator, stressing

that it can be set to values that can differ from the bandwidth used in b 2

Theorem 3.1

√(b − )

For the estimation of , the log-periodogram estimator can be used as an alternative

to the local Whittle estimator; see Robinson (1995a).

(b), whenb is the local Whittle estimator. Let

a CLT to analyze the decline of the MSE as increases. Under Assumption L,

E((b − )2) = (−1

b 2

by (3.2). Since (3.5) is derived using an 2approximation, a more detailed analysis

shows that the MSE is ¡−1+ (log)2−1

optimal bandwidth is therefore the one taking that grows at the maximal rate of

45.

.

If = (45), then Robinson (1995b) showed that under the assumptions of

→ N(014) (3.4)

We now turn to the MSE of b 2

= (45) and = (45), since we only need a consistency rate rather than

). Therefore,

³

(b) − 2

´2

=

µ1

+(log)2

¶

(3.5)

¢, hence decreasing in . The MSE-

In general, without recourse to Assumption L, the consistency of Robinson’s

MAC estimator follows immediately fromb

assume Gaussianity or linearity of {}; see Dalla et al. (2006) and Abadir et al.

(2007). For example, if = 0 and (1.4) holds, then such consistency follows under

the assumptionP∞

4Simulation results

→ andb(b)

→ 0. The estimators

b andb(b) are consistent under very weak general assumptions, which do not

=−∞|()| ≤ ; see Corollary 1 of Dalla et al. (2006).

The objective of this section is to illustrate the asymptotic results for the HAC and

(b) and b 2

MAC estimators ¯ 2

(b), to examine their finite-sample performance, and

to give advice on how to choose the bandwidth parameters in practical applications.

10

Page 11

We focus on the MSE because the primary use of these estimators is the consis-

tent estimation of the long-run variance 2

used in various statistics; e.g. in the

denominator b of HAC and MAC robust t-ratios

t := 12−(¯ − )

b

→ N(01) → ∞(4.1)

For this reason, we also consider the closeness of HAC and MAC robust t-ratios to

their limiting normal distributions; see Velasco and Robinson (2001) for expansions

relating to t-ratios using smoothed autocovariance estimates for (0). We study

the coverage probabilities (CPs) of 95% asymptotic confidence intervals (CIs) for ,

considering how the choice of bandwidths affects the closeness of CPs to the nominal

95% level based on the limiting normal distribution of the t-ratio.

We let {} be a linear Gaussian ARIMA(10) process with unit standard

deviation, for different values of (AR parameter) and . We link to 2

, the object

of our analysis, by means of (1.2)—(1.3). Throughout the simulation exercise, the

number of replications is 5,000. We consider three sample sizes = 2505001000

and we estimate the parameter using the local Whittle estimatorˆ with bandwidth

=¥065¦. We do not report the results for =¥05¦¥08¦because they

Table 1 contains the MSE of the HAC estimator ¯ 2

are dominated by =¥065¦.

values of the bandwidth . The minimum-MSE value for each and is highlighted

(b) calculated for different

by shaded gray boxes. The results for these optima are so scattered across the table,

that in practice it will be difficult to achieve them.

Table 2 reports the MSEs of the HAC estimator ¯ 2

(b) when is chosen according

to the asymptotically-optimal rule (2.14). It gives MSEs comparable to the optimal

MSEs of Table 1, except when and are simultaneously large. In this case, the

cost in terms of the MSE can be substantial.

Table 3 contains the MSE of the MAC estimator b 2

that resulted from (3.5): almost all the optima are for = (45) and, in the four

(b) calculated for different

values of the bandwidth . It reveals the accuracy of the simple bandwidth rule

exceptions (shaded boxes), there is little loss in nevertheless sticking to = (45).

Both Tables 2 and 3 show that the MSEs of HAC and MAC estimators usually

increase when || or || increase.

Tables 4 and 5 report CPs for using, respectively, the HAC estimator ¯ 2

(b)

with chosen by the rule (2.14) and the MAC estimator with various bandwidths

. HAC and MAC estimators gives comparable CPs, which are slightly better

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Page 12

for MAC. CPs approach the nominal 95% level as sample size increases. They are

close to the 95% level except when → 05 or when becomes negative. The

bandwidth =¥08¦tends to give better CPs for MAC, and this is in line with

the recommendations of Table 3.

Because of the specificity of MC studies to the generating process that is used, it

is recommended in practice that the user tries also bandwidths that are smaller than

the maximum allowable =¥08¦which we recommended. This could be used to

check the stability of the estimator as varies near its (unknown) optimal value.

For example, data that are not generated by a linear process (such as ARIMA)

require smaller bandwidths like¥07¦; see Dalla et al. (2006).

5Concluding Remarks

In this paper, the properties of two alternative types of estimators of the long-run

variance have been derived. The first one is an extension of the widely used Bartlett-

kernel HAC estimator, while the second one is the frequency-based MAC estimator

suggested by Robinson (2005). We give guidance on how to choose the bandwidths

in practice, for each estimator. The calculation of both estimators is numerically

straightforward, and allows for the possibility of long-memory or antipersistence in

the data.

Our theoretical results explain that the HAC estimator is sensitive to the se-

lection of the bandwidth , since the order of minimizing the MSE depends on

the extent of the memory in the series. This problem often complicates bandwidth

selection in applied work. The MAC estimator is more robust to the choice of the

bandwidth, which does not depend on the memory. The simulation study confirms

this analytical finding.

On the other hand, the paper does not provide a theory of deriving optimal

estimators, e.g. under MSE-optimality or closeness to normality of the Studentized

t-ratio for . We have studied two types of estimators without establishing whether

or not they are dominated by others, but the asymptotic normality of the MAC

estimator for ∈ (−1212) is an encouraging sign, and so is the good simulation

performance of the two estimators.

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Page 13

Appendix

AProofs of the theorems, auxiliary lemmas and propo-

sitions

There are four subsections. The first proves the results relating to the theorems of

Section 2, while the second proves the theorem of Section 3. For the first theorem,

we need lemmas that are derived in the third subsection, and propositions that are

obtained in the fourth one. We require these auxiliary results here, but they can

also be of use beyond our paper.

Throughout this section, we take ∼ to mean that → 1 as → ∞.

A.1Proof of Theorems 2.1 and 2.2

proof of Theorem 2.1. By definitions (2.1), (2.2), and (2.8)

e 2

(b) = e 2

(b) and ¯ 2

()(1 + (1))and¯ 2

(b) = ¯ 2

()(1 + (1))

Condition (2.8) and asymptotic results derived for ¯ 2

(b) by e 2

Also, observe that

Z

where

() and ¯ 2

() below then allows

us to replace e 2

() and ¯ 2

() in the statement of the theorem

without altering the expansions, so we will prove the theorem for e 2

e =

() and ¯ 2

().

−

ei()d=

Z

−

ei¯()d 0 ≤

() := (2)−1¯¯¯

are the corresponding periodograms. Therefore,

X

=1

ei(− E())

¯¯¯

2

¯() := (2)−1¯¯¯

X

=1

ei(−¯ )

¯¯¯

2

e 2

¯ 2

() =

Z

−

()()d (A.1)

and

() =

Z

−

()¯()d (A.2)

where

() := −1−2¯¯¯

X

=1

ei¯¯¯

2= −1−2

µsin(2)

sin(2)

¶2

(A.3)

13

Page 14

is the renormalized Fejér kernel.

By (A.1) and (A.2), we can write ¯ 2

() = e 2

()(¯() − ())d

() + , where

:=

Z

−

(A.4)

In Lemma A.4, we will show that E(||) ≤ (()1−2+ ()). Hence,

¯ 2

() = e 2

() + (()1−2) (A.5)

If −12 14, then ()1−2= (()12), and we can write (A.5) as

³Z

where

:= ()12³

By Proposition A.1,

−→ N(02

sition A.2,

Z

which proves (2.9) and (2.10).

¯ 2

() − 2

= ()12+

−

()()d − 2

´

+ (()12)

e 2

() −

Z

−

()()d

´

), where 2

is given by (2.11), whereas by Propo-

−

()()d = 2

+ −1−2 + (−1−2)

In the case 14 12, write

e 2

() − 2

=¡e 2

() − E¡e 2

()¢¢+ E¡e 2

()¢− 2

= −1−2+(−1−2)+

Proposition A.3 derives the asymptotic bias E¡e 2

totic behavior

()¢−2

(()1−2) and shows that the stochastic term exhibits the nonstandard asymp-

()1−2¡e 2

Thus, the term on the left-hand side above can be approximated by the normalized

sum −2P

Gaussian limit distribution. These relations imply (2.12) and (2.13).

Proof of Theorem 2.2. The conditionb− = (1log) allows us to prove

(2.15) was shown in Giraitis et al. (2003, Theorem 3.1). For 0,

() − E¡e 2

()¢¢= −2

X

=1

(2

− E(2

)) + (1)

−→ 20()

=1(2

− E(2

)) of strongly dependent variables 2

which has a non-

the theorem for e 2

() and ¯ 2

() instead of e 2

1 −||

(b) and ¯ 2

³

(b). For ≥ 0, convergence

2¯ 2

=

X

||

³

´

e +

X

||

1 −||

´

=: 1+ 2

14

Page 15

where

e

= −1

−||

X

=1

(− )(+||− )

´

=(− ). It suffices to show that

=

³

1 −||

(¯ − )2− −1(¯ − )(1−||+ ||+1)

= E(), and :=P

−21

→ 2

and−22

−→ 0 (A.6)

The verification of the relations −22

→ 0 and −2E(1) → 2

is the same as

in Giraitis et al. (2003).

To prove the convergence (A.6), it remains to check that E((1− E(1))2) =

(4) We have E((1− E(1))2) ≤ || + 0, where

:=

X

|||0|

³

1 −||

´³

1 −|0|

´

−2

−||

X

=1

−|0|

X

0=1

(00)

(00) := −0−0+||−|0|+ −0−|0|−0+||

and

0

:= −2

X

X

|||0|≤

−||

X

∞

X

=1

−|0|

X

0=1

|(||0− 0− + |0|)|

≤ −2X

||≤

=−∞|()| ∞ for 0, −12 0 and = ().

To work out , write = + 0where

=1

00=−∞

|(||00)| ≤ −1= (4)

by the assumptionP∞

:=

X

|||0|

³

1 −||

´³

1 −|0|

´

−2

∞

X

=−∞

−|0|

X

0=1

(00)

whereas 0can be bounded by

|0

| ≤ −2

X

|||0|

X

−|| ≤0

X

0=1

|(00)|

We split summation over into three regions: − || ≤ ≤ , , and ≤ 0. In

the case of − || ≤ ≤ , the order of this part of the sum is straightforward.

15

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