band spectral regression with trending data

8/7/2019 Band spectral regression with trending data

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BAND SPECTRAL REGRESSION

WITH TRENDING DATA

BY

DEAN CORBAE, SAM OULIARIS

and PETER C. B. PHILLIPS

COWLES FOUNDATION PAPER NO. 1039

COWLES FOUNDATION FOR RESEARCH IN ECONOMICS

YALE UNIVERSITY

Box 208281

New Haven, Connecticut 06520-8281

2002

http://cowles.econ.yale.edu/


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Econometrica, Vol. 70, No. 3 (May, 2002), 10671109

BAND SPECTRAL REGRESSION WITH TRENDING DATA

By Dean Corbae, Sam Ouliaris, and Peter C. B. Phillips1

Band spectral regression with both deterministic and stochastic trends is considered. It

is shown that trend removal by regression in the time domain prior to band spectral regres-

sion can lead to biased and inconsistent estimates in models with frequency dependent

coefficients. Both semiparametric and nonparametric regression formulations are consid-

ered, the latter including general systems of two-sided distributed lags such as those aris-

ing in lead and lag regressions. The bias problem arises through omitted variables and is

avoided by careful specification of the regression equation. Trend removal in the frequency

domain is shown to be a convenient option in practice. An asymptotic theory is developedand the two cases of stationary data and cointegrated nonstationary data are compared.

In the latter case, a levels and differences regression formulation is shown to be useful in

estimating the frequency response function at nonzero as well as zero frequencies.

Keywords: Band spectral regression, deterministic and stochastic trends, discrete

Fourier transform, distributed lag, integrated process, leads and lags regression, nonsta-

tionary time series, two-sided spectral BN decomposition.

1 introduction

Hannans (1963a, b) band-spectrum regression procedure is a usefulregression device that has been adopted in some applied econometric work,notably Engle (1974), where there are latent variables that may be regarded asfrequency dependent (like permanent and transitory income) and where there isreason to expect that the relationship between the variables may depend on fre-quency. More recently, band spectral regression has been used to estimate coin-tegrating relations, which describe low frequency or long run relations betweeneconomic variables. In particular, Phillips (1991a) showed how frequency domainregressions that concentrate on a band around the origin are capable of pro-

ducing asymptotically efficient estimates of cointegrating vectors. Interestingly,in that case the spectral methods are used with nonstationary integrated timeseries in precisely the same way as they were originally designed for stationaryseries, on which there is now a large literature (see, inter alia, Hannan (1970 Ch.VII), Hannan and Thomson (1971), Hannan and Robinson (1973), and Robinson(1991)). Related methods were used in Choi and Phillips (1993) to construct fre-quency domain tests for unit roots. In all of this work, the capacity of frequencydomain techniques to deal nonparametrically with short memory components in

1 Our thanks go to a co-editor and three referees for helpful comments. An early draft was written

in the summer of 1994. The first complete version of the paper was written over the period 1 January1 July 1997 while Phillips was visiting the University of Auckland. Lemmas AD are due to Phillips

and are taken from other work to make this paper self contained. Phillips thanks the NSF for research

support under Grant Nos. SBR 94-22922, SBR 97-30295, and SES 0092509.

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1068 d. corbae, s. ouliaris, and p. c. b. phillips

the data is exploited. Robinson (1991), who allowed for band dependent modelformulations, showed how, in such semiparametric models, data based methods

can be used to determine the smoothing parameter that plays a central role inthe treatment of the nonparametric component. Other data based methods havebeen used directly in the study of cointegrating relations and testing problems byXiao and Phillips (1998, 1999).

To the extent that relations between variables may be band dependent, it isalso reasonable to expect that causal relations between variables may vary accord-ing to frequency. Extending work by Geweke (1982), both Hosoya (1991) andGranger and Lin (1995) have studied causal links from this perspective and con-structed measures of causality that are frequency dependent. Most recently, fre-

quency band methods have been suggested for the estimation of models with longmemory components (e.g., Marinucci (1998), Robinson and Marinucci (1998),Kim and Phillips (1999)), where salient features of the variables are naturallyevident in frequency domain formulations.

This paper studies some properties of frequency band regression in the pres-ence of both deterministic and stochastic trends, a common feature in economictime series applications. In such cases, a seemingly minor issue relates to themanner of deterministic trend removal. In particular, should the deterministictrends be eliminated by regression in the time domain prior to the use of thefrequency domain regression or not? Since spectral regression procedures were

originally developed for stationary time series and since the removal of deter-ministic trends by least squares regression is well known to be asymptoticallyefficient under weak conditions (Grenander and Rosenblatt (1957, p. 244)), itmay seem natural to perform the trend removal in the time domain prior to theuse of spectral methods. Indeed, this is recommended in Hannan (1963a, p. 30;1970, pp. 443444), even though the development of spectral regression thereand in Hannan (1970, Ch. VII.4) allows for regressors like deterministic trendsthat are susceptible to a generalized harmonic analysis, satisfying the so-calledGrenander conditions (Grenander and Rosenblatt (1957, p. 233), Hannan (1970,

p. 77)). Theorem 11 of Hannan (1970, p. 444) confirms that such time domaindetrending followed by spectral regression is asymptotically efficient in modelswhere all the variables are trend stationary and where the model coefficients donot vary with frequency.

In time domain regression, the FrischWaugh (1933) theorem assures invari-ance of the regression coefficients to prior trend removal or to the inclusionof trends in the regression itself. Such invariance is often taken for granted inempirical work. However, as will be shown here, invariance does not alwaysapply for band-spectrum regression when one switches from time domain to fre-quency domain regressions. In particular, detrending by removing deterministiccomponents in the time domain and then applying band-spectrum regression isnot necessarily equivalent to detrending in the frequency domain and applyingband-spectrum regression. The reason is that a band dependent spectral modelis similar in form to a linear regression with a structural change in the coeffi-cient vector. So, when there are deterministic trends in the model, the original


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band spectral regression 1069

deterministic regressors also need to be augmented by regressors that are rele-vant to the change period (here, the frequency band where the change occurs).

Thus, the seemingly innocuous matter of detrending can have some nontrivialconsequences in practice. In particular, detrending in the time domain yields esti-mates that can be biased in finite samples, and, in the case of nonstationary data,inconsistent. The appropriate procedure is to take account of the augmentedregression equation prior to detrending and this can be readily accomplished byincluding the deterministic variables explicitly in the frequency domain regres-sion. These issues are relevant whenever band spectral methods are applied,including those cases that involve cointegrating regressions. In the latter case,a levels and differences regression formulation is shown to be appropriate anduseful in estimating the frequency response function at nonzero as well as zerofrequencies.

The present paper provides an asymptotic analysis of frequency domainregression with trending data for the two cases of stationary and cointegratednonstationary data. We consider both semiparametric and fully nonparametricformulations of the regression model, in both cases allowing the model coeffi-cients to vary with frequency. In dealing with the nonstationary case, the paperintroduces some new methods for obtaining a limit theory for discrete Fouriertransforms (dfts) of integrated time series and makes extensive use of two sidedBN decompositions, which lead to levels and differences model formulations.

This asymptotic theory simplifies some earlier results given in Phillips (1991a)and shows that the dfts of an I(1) process are spatially (i.e., frequencywise)dependent across all the fundamental frequencies, even in the limit as the sam-ple size n , due to leakage from the zero frequency. This leakage is strongenough to ensure that smoothed periodogram estimates of the spectrum awayfrom the origin are inconsistent at all frequencies = 0. The techniques andresults given here should be useful in other regression contexts where these quan-tities arise. Some extensions of the methods to cases of data with long rangedependence are provided in Phillips (1999).

Most of our notation is standard: [a] signifies the largest integer not exceedinga > signifies positive definiteness when applied to matrices, a is the complexconjugate transpose of the matrix a, a is a matrix norm of a, a is the MoorePenrose inverse of a, Pa = aaaa is the orthogonal projector onto the rangeofa, L20 1 is the space of square integrable functions on [0, 1], 1[A] is the indi-

cator of A,d signifies asymptotically distributed as and d and p are used to

denote weak convergence of the associated probability measures and convergencein probability, respectively, as the sample size, n ; I(1) signifies an integratedprocess of order one, BM denotes a vector Brownian motion with covariance

matrix , and we write integrals like 10 Brdr as 10 B, or simply B if thereis no ambiguity over limits; MN0 G signifies a mixed normal distribution withmatrix mixing variate G, Nc 0 G signifies a complex normal distribution withcovariance matrix G, the discrete Fourier transform (dft) of at t = 1 n iswritten wa = 1/

nn

t=1 ateit , and s = 2s/n s = 0 1 n 1 are

the fundamental frequencies.


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2 models and estimation preliminaries

Let xt

t=

1 n be an observed time series generated by

xt = 2zt + xt(1)

where zt is a p + 1-dimensional deterministic sequence and xt is a zero meank-dimensional time series. The series xt therefore has both a deterministiccomponent involving the sequence zt and a stochastic (latent) component xt . Indeveloping our theory it will be convenient to allow for both stationary and non-stationary xt . Accordingly, we introduce the following two alternative assump-tions. The mechanism linking the observed variables xt , unobserved disturbances

t , and a dependent variable yt is made explicit later.

Assumption 1: t = t xt is a jointly stationary time series with Wold repre-sentation t =

j=0 Cjtj, where t = iid 0 with finite fourth moments and the

coefficients Cj satisfy

j=0 j12 Cj < . Partitioned conformably with t, the spec-

tral density matrix f of t is

f =

f 00 fxx

with ffxx > 0 .Assumption 2: xt is an I(1) process satisfying xt = vt, initialized at t = 0

by any Op(1) random variable. The shocks t = t vt satisfy Assumption 1 withspectral density f partitioned as

f =

f 00 fvv

with ffvv > 0

.

Assumptions 1 and 2 suffice for partial sums of t to satisfy the functional law

n1/2nr

t=1 td Br = BM, a vector Brownian motion of dimension k + 1

with covariance matrix = 2f0 (e.g., Phillips and Solo (1992, Theorem3.4)). The vector process B and matrix can be partitioned conformably with as B = B Bx and

=

00 xx

where xx > 0, so that xt is a full rank vector I(1) process. In all cases, xt is takento be independent oft . This amounts to a strict exogeneity assumption when weintroduce a generating mechanism for a dependent variable yt . The assumptionis standard for consistent estimation in the stationary case (as in Hannan (1963a,b)). In the nonstationary case, it is not required for the consistent estimation of


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(low frequency) cointegrating regression components. However, at bands awayfrom the origin, some version of incoherency between the regressors and errors

is required for consistency and asserting incoherency enables us to examine thebias and inconsistency effects of detrending in such regressions, just as in thecase of stationary xt .

We make the following assumptions concerning the deterministic sequence zt .We confine our treatment to time polynomials and set zt = 1 t tp, themost common case in practice, although we anticipate that our conclusions aboutbias and inconsistency will apply more generally, for instance to trigonometricpolynomials, after some appropriate extensions of our dft formulae in Lemma Aand the limit results in Lemma C below. Let n = diag1 n np, and definedt =

1

n zt . Then

dnr = 1n znr ur = 1 r rp(2)

uniformly in r 0 1. The limit functions ur are linearly independent inL20 1 and n

1n

t=1 dtdt

10

uu > 0. Now let Z (respectively, D) be the npobservation matrix of the nonstochastic regressors zt (respectively, standardizedregressors dt), PZ be the projector onto the range of Z, and QZ = In PZ be theresidual projection matrix (respectively, PD and I PD). Clearly, PZ = PD andQZ

=QD.

The Nonparametric Model

The type of model we have in mind for the stochastic component of the dataallows for the regression coefficient to vary across frequency bands. In the timedomain, suppose a (latent) dependent variable yt is related to xt and t accordingto a (possibly two-sided) distributed lag

y

t =

j= j xtj+ t = L xt + t(3)where the transfer function of the filter, b = ei =j= jeij, is assumedto converge for all . Under stationarity (Assumption 1), the crossspectrum fyx between yt and xt satisfies

fyx = bfxx(4)

whose complex conjugate transpose gives the complex regression coefficient

= b = fxx1fxy(5)

As argued in Hannan (1963a, b), spectral methods provide a natural mechanismfor focusing attention on a particular frequency, or band of frequencies, and forperforming estimation of .


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Next, suppose that the coefficients j in (3) are such that L has a valid two-sided spectral BN decomposition (see part (b) of Lemma D in the Appendix),

viz.

L = ei+ eiLeiL 1(6)

where

L =

j=jL

j j =

k=j+1

keik j 0

j

k= keik j < 0with

j= j2 < . Using (6) in (3) we have

yt = Lxt + t(7)= b + eiLeiL 1xt + t= bxt + eiL 1at + t

in which

at

= e

iL

xt. Then, setting

=s in (7) and taking dfts at fre-

quency s we obtain the following frequency domain form of (3):

wys = bswxs + ws +1n

eis as 0 as n

(8)

For stationary xt at is also stationary and (8) can be written as

wys = bswxs + ws + Op

1n

(9)

giving an approximate frequency domain version of (3).The case of integrated xt (Assumption 2) can be handled in a similar way.

Here it is useful to apply to (3) the two-sided BN decomposition at unity (part(a)of Lemma D), viz.

L = 1+ LL 1 where(10)

L =

j=jL

j j =

k=j+1

k j 0

j

k=k j < 0

so that (3) can be interpreted as

yt = 1xt + t Lvt = 1xt + t say(11)


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which is a cointegrating equation between yt and xt with cointegrating vector1

1 and composite error

t

=t

Lvt . Setting t

=Lvt and taking

the dft of (11), we obtain

wys = 1wxs+ ws ws(12)= bswxs + ws ws+ 1 bswxs(13)

The final two terms of (13) are important, revealing that the approximation (9)fails in the nonstationary case and has an error of specification that arises fromomitted variables.

Assuming it is valid, the spectral BN decomposition of L at frequency s is

L = eis +s e

is Leis L 1

wheres is constructed as in Lemma D(b). Then

ws = eis wvs +1n

eis vs 0 vs n

(14)

= eis wvs + Op1n

where vs t =s e

is Lvt is stationary. It follows that (12) has the form

wys = 1wxs eis wvs + ws + Op

1n

(15)

giving an approximate frequency domain version of the cointegrated model thatapplies for all frequencies. Since vt = xt, it is apparent from (15) that if data on

yt and

xt were available, the equation could be estimated in the frequency domain

by band spectral regression of wys on wxs and wxs for frequencies scentered around some frequency . For s centered around = 0, this procedurewas suggested originally in Phillips (1991b) and Phillips and Loretan (1991),where it is called augmented frequency domain regression, although in that workdeterministic trends were excluded. In the general case where s is centered on = 0, we can recover the spectral coefficient = b from the coefficients1 and eis = eis in (15) using the fact that

b = ei = 1+ eiei 1(16)

Causal economic relationships are often formulated as in (3) with one-sidedrelations where j = 0 for j < 0, but two-sided relations are not excluded. Theyare well known to occur in situations where there is feedback in both directionsbetween the variables y and x. A notable recent example arises in the simplifi-cation of cointegrating systems to single equation formulations where two-sided


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dynamic regressions like (11) arise and where they have been used in regres-sion analysis to obtain efficient estimates of the cointegrating vector (Phillips and

Loretan (1991), Saikonnen (1991), Stock and Watson (1993)).The mechanism linking the observed dependent variable yt to zt and yt (and,

hence, xt, and t) is now given by

yt = zt1 + yt(17)

where yt follows (3) in the stationary case with xt t satisfying Assumption 1,and (11) in the cointegrated case with xt t satisfying Assumption 2. Theobserved series yt and xt therefore have deterministic trends and stochastic com-ponents that are linked by a system of distributed lags with stationary errors.

The natural statistical approach would appear to be deterministic trend removalby regression of yt xt on zt, followed by an analysis of the system of laggeddependencies between the residuals from these regressions. The latter can beconducted in the frequency domain where the coefficient of interest is the trans-fer function of the filter b. This coefficient will vary according to frequency(unless j = 0 for j= 0) and can be estimated by nonparametric regression tech-niques. One of the aims of this paper is to examine the finite sample and asymp-totic performance of this natural procedure against that of a procedure thatseeks to perform the regression analysis in the frequency domain directly on the

observed quantities yt xt zt rather than on detrended data.

A Fixed Band Regression Model

A simple situation of interest occurs when the response function b is con-stant, let us say on two fixed frequency bands. This case captures the essentialfeatures of the problem that we intend to discuss, is easy to analyze, and will beconsidered at some length here and in the asymptotic theory. It has the advan-tage that b is real and parametric and can be estimated at parametric rateswhen the bands are of fixed positive length. This case is analogous to a regression

model with a structural break and that analogy helps to explain why time and fre-quency domain detrending have different effects. The nonparametric model canbe interpreted as a limiting case in which one of the bands shrinks to a particularfrequency as the sample size increases. To promote the analogy with a structuralbreak model we will develop a conventional regression model formulation forthe data.

To this end, we postulate the dual bands A = 0 0 and cA = ,00 for some given frequency 0 > 0 and define the frequency depen-dent coefficient vector

b = A1 A+ Ac 1 cA(18)

where A is a k-vector of parameters pertinent to the band A, and Ac is thecorresponding k-vector of parameters pertinent to the bandcA. This formulationof b enables us to separate low frequency 0 from high frequency


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> 0 responses in a regression context. The coefficients j in the Fourierrepresentation, b

=j= je

ij, of (18) are

j =1

2

beij dw =

A0

+ Ac

0 j= 0

sin j0j

A Ac j= 0(19)

so that the filter in (3) is symmetric and two-sided. This is an interesting casewhere the coefficients j decay slowly and do not satisfy the sufficient condition

j

12 j < given in Lemma D for the validity of the BN decomposition of

L=j= Lj. Nevertheless, the BN decomposition of L is still valid inthis case, as shown in Remark (i) following Lemma D in the Appendix.

While data for yt can be generated by (3) and (19) by truncating the filter, analternate frequency domain approach that uses (18) is possible. This mechanismhas the advantage of revealing the impact of the shift in (18) in terms of a stan-dard regression model with a structural change. Like xt, the dependent variableyt is assumed to have both deterministic and stochastic components. Its stochas-tic component yt is generated from xt and t by means of a triangular arrayformulation which we explain as follows.

Let X = x1 xn be the n

k matrix of observations of the exogenous

regressors xt , let U be the n n unitary matrix with jk element e2ijk/n/n,and let W = E1nU, where

E1n =

0 1In1 0

is an n n permutation matrix that reorders the rows of U so that the lastrow becomes the first and succeeding rows are simply advanced by one position.Then, WW = In, and the matrix W

X has kth row given by the dft wxs

withs

=k

1.

We introduce the following matrices that are used to provide a frequencydependent structure to the model for yt . Define:

A = n n selector matrix that zeroes out frequencies in WX that are notrelevant to the primary band of interest, say A. Then, A

cW= IAW extractsthe residual frequencies over cA. Note that A

cA = AAc = 0. = WAW and c = WAcW = I .

Then, for each n, y = y1 yn is generated in the frequency domain in dftform by the system

AWy = AWXA + AW(20)AcWy = AcWXAc + AcW(21)

where = 1 n. Equations (20) and (21) generate observations in thefrequency domain that correspond to the model (8) in the nonparametric case


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with the final term of (8) omitted. In the stationary case, this omitted term isOpn

12 as seen in (9), so the difference between the generating mechanisms

is negligible asymptotically. In the nonstationary case, however, the omitted termis nonnegligible, as is apparent from (13) and (15). In consequence, the two gen-erating mechanisms differ in the nonstationary case and this leads to a differencein the asymptotic theory between the two models that will figure in our discus-sion in Section 5.

Adding (20) and (21) and multiplying by W gives

y = XA + c XAc +(22)which is the time domain generating mechanism for

y. By construction, the matri-

ces and c have elements that depend on n, and it follows that yt t =1 n has a triangular array structure, although we do not emphasize this byusing an additional subscript and we will have no need of triangular array limittheory in our asymptotic development.

In (22), t xt satisfies Assumption 1 in the stationary case and t x

t sat-

isfies Assumption 2 in the nonstationary case. Thus, (22) is a variable (banddependent) coefficient time series regression with exogenous regressors and sta-tionary errors under Assumption 1, and a cointegrating regression with exoge-nous regressors and band dependent coefficients under Assumption 2. Note that

model (22) is semiparametric: parametric in the regression coefficients A andAc and nonparametric in the regression errors .

We now suppose that the observed series yt has deterministic components likext in (1) and is related to the unobserved component yt by means of

y = Z1 + y(23)

Using (22) and (23), it follows that the observed data satisfy the model

y = Z1 2A + XA + cZ2A Ac cXA Ac + (24)

or, equivalently,

y = Z1 2Ac + Z2Ac A + XAc XAc A+ (25)

or

y = Z1 2A + cZ1 2Ac + XA + cXAc + (26)

These models extend (22) to cases where the observed data contain deterministictrends. Observe that the trend regressors z

tnow appear in the model with band

dependent coefficients, just like the exogenous regressors xt .The formulations (24)(26) make it clear that detrending the data in the time

domain is not a simple matter of applying the projection matrix QZ, as mightbe expected immediately from (1) and (23). In fact, correct trend removal isaccomplished by the use of the operator QV = I PV, where V = ZcZ or,


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equivalently, in view of (26) V = ZcZ. Methods that rely on prefilteringby means of QZ do not fully remove the trends and this can have important

consequences, like biased and inconsistent estimates of the regression coefficientsA and Ac . Of course, when A = In the coefficient is invariant across bandsA = say, we have = In and c is null, so that V = Z and usual detrendingby QZ is appropriate. In that case (26) reduces to

y = Z1 2+ X + Put another way, conventional detrending by QZ implicitly assumes that there isno variation in the coefficient across frequency bands. If there are such variations,then correct detrending needs to take into account the effects of such variations

on the trends, as in (26). Otherwise, there will be misspecification bias fromomitted variables in the deterministic detrending.

Fixed Band Regressions and Misspecification Bias

A simple illustration with the Hannan (1963a) inefficient band-spectrumregression estimator shows the effects of such misspecification. This estimatorcan be constructed for the band A and then has the form

A =X Q

Z

QZ

X1X QZ

QZ

y(27)

with a corresponding formula for Ac , the estimator over the band cA. In form-

ing A and Ac , the data are filtered by a trend removal regression via the projec-tion QZ before performing the band-spectrum regression. This procedure followsHannans (1963a, b) recommendation for dealing with deterministic trends and,as we have discussed in the introduction, is the conventional approach in thiscontext. Using (24) and (27), we find

A = A + X QZ QZX1XQZ QZcZ2A Ac (28) cXA Ac +

= A X QZ QZX1XQZ QZc XA Ac = A X QZ QZ X1X QZ QZc XA Ac

and the corresponding formula for Ac is

Ac = Ac X QZcQZ X1X QZcQZXAc A (29)It follows that

EAX = A X QZ QZX1X QZ QZc XA Ac (30)and

EAc X = Ac XQZcQZX1XQZcQZXAc A


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Hence, band-spectrum regression yields biased estimates of the coefficients whenA

=Ac and trend removal regression is conducted via the projection QZ.

To some extent, the problem with A is a finite sample one. In fact, for sta-tionary xt , the bias disappears as n . But, as we shall see in Section 5, whenxt is integrated the bias in Ac does not always disappear in the limit.

Narrow Band Regressions

In the nonparametric case we estimate in (5) at a particular frequency ,rather than over a fixed discrete band. In that case the band spectral estimatorhas the same form as (27) but the selector matrix A = A (respectively, = )chooses frequencies in a shrinking band about . We write the estimator as

= X QZQZX1X QZQZy(31)From the form of (15) it is apparent that the narrow band regression leading

to (31) omits the term involving the dft, wvs, of the differences vt = xt. Sothis narrow band regression also involves omitted variable misspecification.

An alternate nonparametric procedure is to use an augmented narrow bandregression in which the differences are included. To fix ideas, we detrend thedata using QZ and then perform a band spectral regression of the form

wyzs=

a1

wxzs+

a2

wxzs+

residual(32)

for frequencies s in a shrinking band around . In equation (32) the affix z onthe subscript (e.g., in wxzs) signifies that the data have been detrended beforetaking dfts. This approach is analogous to the augmented spectral regressionapproach of Phillips and Loretan (1991) for estimating a cointegrating vector.However, the narrow band regression (32) now applies for frequencies thatare nonzero as well as zero and prior detrending of the data has occurred in thetime domain. Define X = XX and a = a1 a2. Then

a

=X QZQZX

1X QZQZy(33)

In view of the relation (16), an estimate of = ei may be recovered froma using the linear combination

=

a1 ei 1a2 = 0a10 = 0

(34)

The asymptotic theory for the narrow band estimates and is developedin Section 5.

3 dft recursion formulae

Here we develop some useful formulae for dfts of the deterministic sequencezt. The following lemma gives general recursion formulae for the quantityWks =

nt=1 t

keis t . The case for s = 0 is well known, of course, but the resultfor s = 0 appears to be new.


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Lemma A: (a) For s = 0,

Wks =n

t=1tk = 1

k + 1nk+1 k + 11 B1nk +k + 12 B2nk1+

k + 13

B3n

k2 + +

k + 1k

Bk1n

where Bj are the Bernoulli numbers.(b) For s = 0, Wks satisfies the recursion

Wk

s

=

nkeis

eis 1 +1

eis 1

k

j=1kj1jWkjs(35)with initialization

W0s =n

t=1eis t =

n s = 00 s = 0(36)

Using (35), the standardized quantities Wks = 1n

nt=1

tn

k

eis t satisfy therecursion

Wks =1n

eis

eis 1 +1

eis 1k

j=1

1

nj

kj

1jWkjs(37)

and then the dft of dt is simply

wds =1n

nt=1

dteis t = W0s Wps

The main cases of interest are low order polynomials, where explicit expres-

sions for the discrete Fourier transforms Wks are easily obtained fromLemma A. Thus, when k = 0 1 2 we have

W0s =

n s = 00 s = 0(38)

W1s =

n + 12n1/2

s = 01

n1/2

eis

eis 1 s

=0

(39)

W2s =

n + 12n + 1

6n3/2 s = 0

1

n1/2eis

eis 1 2

n3/2eis

eis 12 s = 0(40)


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In case (38), it is apparent that eliminating the zero frequency will demeanthe data and leave the model unchanged for s

=0. Then, cZ

=0 and so

QZc = c Pzc = 0, and QZ = QV. It follows that XQZ QZc X = 0in (30) and therefore A is unbiased in this case of simple data demeaning.

On the other hand, when zt = 1 t and dt = 1t/n, it follows from (38) and(39) that

wds = 1

n

nt=1

dteis t =

n

n + 12n1/2

s = 0

0

1

n1/2eis

eis 1 s = 0

(41)

and the second component of wd

s has nonzero elements for s

=0. Hence,

cZ = 0 and QZ = QV, so that A is biased in this case. The same appliesfor higher order trends.

Frequencies in the band cA satisfy s > 0 > 0. It therefore follows fromLemma A and (37) that Wks = On

12 uniformly for s cA. Hence,

wds = On1/2 for all s cA(42)and, thus, cD = WAcWD has elements that are of On1/2.

4 frequency domain detrending

We consider first the fixed band regression model (22) and (23). The problemof the omitted variable bias evident in (28) can be dealt with either in the fre-quency domain or in the time domain. In the time domain, one simply detrendsusing the projection operator QV = I PV, where V = ZcZ, as is appar-ent from the form of (26). In the frequency domain, the alternative is to leavethe detrending until the regression is performed in the frequency domain.

To do frequency domain detrending, we simply apply the discrete Fouriertransform operator W to (24) and then perform the band spectrum regression.The transformed model is

Wy = WZ1 2A +W cZ2A Ac + WXA W cXA Ac + W

= WZ1 2A +AcWZ2A Ac + WXA AcWXA Ac + W

The resulting band spectral estimator for the band A is equivalent to a regres-sion on

AWy = AWZ1 2A + AWXA + AW(43)since AAc = 0, and therefore this estimator has the form

fA = X WAQAWZAWX1X WAQAWZAWy(44)

= A + X WAQAWZAWX1X WAQAWZAW


16/44


Clearly, EfAX = A, and the estimator is unbiased. A similar result holds for

the corresponding estimator

fAc of Ac .

In this frequency domain approach to detrending, the so called FrischWaugh(1933) theorem clearly holds, i.e., the regression coefficient

fA on the variable

AWX in (43) is invariant to whether the regressor AWZ is included in the regres-sion or whether all the data have been previously detrended in the frequencydomain by regression on AWZ.

Following (32), the natural narrow band approach is to use an augmentedregression model that includes the dft of the trend in the regression, viz.

wys = c1wxs + c2wxs+ c3wzs+ residual(45)

for frequencies s centered on . This narrow band regression leads to the estimate

f =

c1 ei 1c2 = 0c10 = 0

(46)

similar to (34).

5 asymptotic theory

We derive a limit distribution theory for the detrended band spectral regres-sion estimates and consider what happens to the bias as n . We start byintroducing some notation and making the framework for the limit theory moreprecise.

We start with the parametric case where there are the two discrete bandsA and Ac . Let na = #s A and nc = #s Ac be the number offundamental frequencies in the bands A and Ac . It is convenient to subdivide into subbands j of equal width (say, /J) that center on frequenciesj = j/J j= J+ 1 J 1. Let m = #s j and suppose that Ja ofthese bands lie in A. It follows that n and na can be approximated by n

=2mJ

and na = 2mJa, respectively.In the nonparametric case, we focus on a single frequency and consider a

shrinking band of width /J centered on . Again, we let m = #s .The following condition will be useful in the development of the asymptotics

and will be taken to hold throughout the remainder of the paper. Additionalrequirements will be stated as needed.

Assumption 3: (a) na/n and nc/n 1 for some fixed number 0 1 as n .

(b) m J , and J /n 0 as n .For the bias in A to vanish asymptotically, the deviation that depends on the

term

XQZ QZX1XQZ QZ

c XA Ac


17/44


in (28) needs to disappear as n . We will distinguish the two cases of sta-tionary and nonstationary (integrated)

xt corresponding to Assumptions 1 and 2

in the following discussion.

51 The Stationary Case

Here, the bias in A will disappear whenX QZ QZX

n

1XQZ QZc Xn

p 0(47)

A similar requirement, obtained by interchanging and c in (47), holds forthe bias in Ac . We have the following result.

Theorem 1 (Semiparametric Case): If xt and t are zero mean, stationary,and ergodic time series satisfying Assumption 1, and yt is generated by (22), bandspectral regression with detrending in the time domain or in the frequency domainis consistent for both A and Ac . The common limit distribution of A and

fA is

given by

nA AnfA A

d

N 0 Vawhere

Va =A

fxxd

12A

fxxfd

A

fxxd

1(48)

with an analogous result for Ac and fAc .

In the stationary case, therefore, the bias in band spectral regression from timedomain detrending disappears as n and there is no difference between thetwo bands A and

cA in terms of the limit theory. It is therefore irrelevant

whether the main focus of interest is high or low frequency regression. The formof the asymptotic covariance matrix Va is a band spectral version of the familiarsandwich formula for the robust covariance matrix in least squares regression.Va can be estimated by replacing the spectra in the above formula with cor-responding consistent estimates and averaging over the band A. For the fullband case where A = , the matrix Va is the well known formula for theasymptotic covariance matrix of the least squares regression estimator in a timeseries regression (c.f. Hannan (1970, p. 426)). A formula related to Va was givenin Hannan (1963b, equation (16)) for band spectral estimates in the context ofmodels with measurement error.

A similar result holds in the nonparametric case, but with a different conver-gence rate.


18/44


Theorem 1 (Nonparametric Case): If xt and t are zero mean, stationary,and ergodic time series satisfying Assumption 1, and

yt is generated by (3), non-

parametric band spectral regression with detrending in the time domain or in thefrequency domain is consistent for = b. The common limit distribution of and

f is given by

m

mf

d Nc0 V

where

V = ffxx1

52 The Nonstationary Case

Here, the distinction between low frequency regression and regression at otherfrequencies becomes important. In the band regression model (22), there areonly two bands A and

cA. Over A, which includes the zero frequency, the

estimator is known to be n-consistent when A = Ac (see Phillips (1991a)) sincein that case, the regression equation is a conventional cointegrating relation.When A = Ac , the same result continues to hold over the band A, as shownin Theorem 2 below. In this case, the bias in (28) disappears when

X QZ QZX

n2

1X QZ QZc Xn2

p 0(49)

Note that in (49) the moment matrices are standardized by n2, because the datanonstationarity is manifest in bands like A that include the zero frequency. Overfrequency bands like cA that exclude the zero frequency the rate of convergence

of the moment matrices is slower and the bias in Ac will disappear when

X QZcQZXn

1X QZcQZXn

p 0(50)Our starting point in the I 1 case is to provide some limit theory for dis-

crete Fourier transforms of I 1 and detrended I 1 processes. The followingtwo lemmas do this and provide a limit theory for periodogram averages of suchprocesses. The remainder of the limit theory then follows in a fairly straightfor-ward way from these lemmas. To make the derivations simpler we confine ourattention here to the case of a linear trend.

Lemma B: Let xt be an I 1 process satisfying Assumption 2. Then, the discreteFourier transform ofxt for s = 0 is given by

wxs =1

1 eis wvs eis

1 eisxn x0

n1/2(51)


19/44


Equation (51) shows that the discrete Fourier transforms of an I 1 process are

not asymptotically independent across fundamental frequencies. Indeed, they are

frequency-wise dependent by virtue of the component n1/2xn, which produces acommon leakage into all frequencies s = 0, even in the limit as n . As thenext lemma shows, this leakage is strong enough to ensure that smoothed peri-

odogram estimates of the spectral density fxx = 1ei2fvv are inconsis-tent at frequencies cA. Lemma C(f) shows that the leakage is still manifestwhen the data are first detrended in the time domain. On the other hand, it is

apparent from (41) that we can write (51) as

wxs = 11 eis wvs+ w tn sxn x0(52)

from which it is clear that frequency domain detrending (i.e., using residuals from

regressions of the frequency domain data on wt/ns or wds) will remove the

second term of (51) and thereby eliminate the common leakage from the low

frequency.

Lemma C: Let xt be an I 1 process satisfying Assumption 2 and dt = 1t/n.Define xdt = xt dtDD1D X and let wxd be the discrete Fourier trans-form of xdt . Then, as n :

(a) n2

sA wxswxs d 1

0BxB

x

(b) n3/2

sA wxswds d 1

0Bxu

(c) n2

sA wxdswxds d 1

0BxuB

xu

(d) n1/2

scA

wxswds d 1

2Bx1

cA

eif1/1 eid

(e) n1scA wxswxs dcA fxx+ 121/1ei2Bx1Bx1d(f) n1

scA wxdswxds

d

cA

fxx+21gBxgBx

d

(g) n1/2

scA wxdswds d 1

2

cA

gBxf1 d

(h)

scA wdswds 1

2

cA

f1f1 d

(i) n1

sA wdswds 1

0uu

(j) n2

s0 wxswxs d 1

0BxB

x

(k) n2s0 wxdswxdsd

1

0BxuB

xu

(l) 1n s0 wdswds 10 uu(m) n3/2

s0 wxswds

d 10

Bxu

(n) n/m

s wdswds f1f1 = 0

(o)

nm1

s wxswds d ei/1 eiB1f1 = 0


20/44


where fxx = 1 ei2fvv, f1 = 0 ei/ei 1 for = 0, and

Bxur = Bxr 10

Bxu1

0uu1ur

and

gBx =ei

1 ei Bx1 +1

0Bxu

1

0uu1

f1

Joint convergence applies in (a)(o).

With this result in hand, the I 1 band regression model can be analyzed. Thelimit theory for time domain detrended band spectral regression is as follows.

Theorem 2: Suppose t xt satisfies Assumption 2 and yt is generated by (22).Then

nA Ad1

0BxuB

xu

110

BxudB

(53)

=MN0

1

0B

xuB

xu1

2f

0and

Acd Ac 1Ac A(54)

where

=

cA

2fxx+ gBxgBx d

and

=

cA

gBxf1 d

10

uu1 1

0uBx

(55)

So, when the regressors are I 1, band spectral regression is inconsistent infrequency bands that exclude the origin when detrending is performed in the timedomain prior to frequency domain regression. The bias is random and is linearin the differential, Ac

A, between the coefficients in the frequency bands. In

the case of a linear trend zt = t, the limit function f1 = ei/ei 1 and thenthe bias depends on

gBx =ei

1 ei

Bx1

10

Bxr

10

r21


21/44


When fxx = 1/21 ei2 (i.e., when vt is iid(0, 1)), a simple calculationreveals that the probability limit (54) of Ac simplifies to

Ac + A Ac

10

Bxr 1

0r21

Bx11

0r211

0Bxr

1 +

Bx1 1

0Bxr

10

r212 (56)

So, in this case, the bias is not dependent on the width of the band Ac .The following theorem gives the corresponding results for the nonparametric

estimators and .

Theorem 2: Suppose t xt satisfies Assumption 2 and yt is generated by (11)where L and L both have valid BN decompositions.

(a) For = 0,

n0 0d1

0BxuB

xu

110

BxudB 1

0BxudB

x + x

1

(57)

where 0 = 1 and x =

j= Ev0vj. For = 0,

d + 21 fxx 1

eiei 1(58)where = ei = b,

= 2fxx+ gBxgBxand fxx = 1 ei2fvv.

(b) For = 0,

n0 0d 1

0BxuB

xu

1 10

BxudB

For

=0,

m dNc

0

f

fxx

Remarks: (i) In Theorem 2(a) it is apparent that the limit distribution of0 has a second order bias term involving1

0Bxu dB

x + x

1

which is nonzero except when

1=

0, a circumstance that arises when the filterL is symmetric (see Remark (ii) following Lemma D in the Appendix) as itis in the case (19). In that case, the limit (57) reduces to the same mixed normaldistribution as (53). Part (a) shows also that the estimator is inconsistent atall frequencies = 0 except when ei = 0. The bias in 0 and inconsistencyin are due to omitted variable misspecification in the regression.


22/44


(ii) Part (b) of Theorem 2 shows that the estimator based on the aug-mented regression (32) is consistent for both

=0, and

=0. The same mixed

normal distribution as (53) applies when = 0. When = 0 the limit distributionis the same as in the stationary case given in Theorem 1 .

The following results give the limit theory for the frequency domain detrendedestimators in the fixed band and narrow band cases for nonstationary data.

Theorem 3: Under the conditions of Theorem 2

n

fA A

d

1

0BxuB

xu

1

1

0Bxu dB

(59)

and

n

fAc Ac

d N0

cA

fxxd

12

cA

fxxfd

(60)

cA

fxxd

1

In fixed band regression models there is usually some advantage to be gainedby averaging over the band and using weighted regression, as shown in Hannans(1963a, b) original work. Efficient regression is based on a weighted band spec-tral regression that uses a preliminary regression to obtain estimates of the equa-tion errors and a corresponding estimate of the error spectrum, say f, thatis uniformly consistent so that sup f f p 0 (e.g., Hannan (1970,p. 488)). When such weighted regression is performed with frequency domaindetrending, the resulting estimates have optimal properties for both the nonsta-tionary component and the stationary component. For A, the limit distributionis the same as (59) and is optimal in the sense of Phillips (1991b). For

Ac ,

the limit distribution of the efficient estimate is normal with variance matrix2

cA

fxxf1 d1 and therefore attains the usual efficiency bound in

time series regression (e.g., Hannan (1970, eqn. (3.4), p. 427)) adjusted here forband limited regression (Hannan (1963, eqn. (16))). Details are omitted and areavailable in an earlier version of the present paper.

Theorem 3: Under the conditions of Theorem 2, at = 0

nf0 0 d 10 BxuBxu1

10 Bxu dBand for = 0

m

f d Nc0 ffxx1(61)


23/44


So, fA and

f0 are consistent and have the same mixed normal limit distribution

as that of

A in (53). The limit distribution makes asymptotic inference about A

and 0 straightforward, using conventional regression Wald tests adjusted in theusual fashion so that a consistent estimate of the spectrum of t is used (basedon regression residuals) in place of a variance estimate.

The frequency domain detrended estimators fAc and

f are also consistent,

unlike the time detrended estimator = 0 in the nonstationary case. Thelimit distribution of

fAc is the same as it is in the case of trend stationary regres-

sors (Theorem 1) and has the same

n rate. The nonparametric estimator f is

m consistent and is asymptotically equivalent to the augmented regression esti-mator . As far as the model (11) is concerned, it is therefore apparent from

Theorems 2 and 3 that if the correct augmented regression model is used inestimation, it does not matter asymptotically whether detrending is done in thetime domain or the frequency domain.

6 conclusion

It is natural to eliminate deterministic trends in the time domain by simple leastsquares regression because the GrenanderRosenblatt (1957, p. 244) theoremshows that such regression is asymptotically efficient when the time series aretrend stationary (although this conclusion does not hold when there are stochas-tic as well as deterministic trendssee Phillips and Lee (1996)). In a similar way,it seems natural to eliminate deterministic trends in band spectral regressions bydetrending in the time domain prior to the use of spectral methods because thesemethods were originally intended for application to stationary time series. How-ever, this paper shows that such time domain detrending will lead to biased coef-ficient estimates in models where the coefficients are frequency dependent. Inmodels that have both deterministic and stochastic trends, time domain detrend-ing can lead to inconsistent estimates of the coefficients at frequency bands awayfrom the origin. The inconsistency, which arises from omitted variable effects,

can be substantial and has been confirmed in simulations that are not reportedhere (Corbae, Ouliaris, and Phillips (1997)).

The bias and inconsistency arise from omitted variable misspecification andare managed by use of an appropriate augmented regression model. The situa-tion is analogous to a structural break model, but here the coefficients changeacross frequency rather than over time. An alternate approach that is suitable inpractice is to model the data and run regressions, including detrending regres-sions, in the frequency domain. In effect, discrete Fourier transforms of all thevariables in the model, including the deterministic trends, are taken and bandregression is performed. When nonparametric estimation is being conducted, thesame principle is employed but one uses a shrinking band that is local to a par-ticular frequency. In the nonstationary case, it turns out to be particularly impor-tant to specify the model in terms of levels and differences as in (15) leading tothe fitted regressions (32) and (45). An estimate of the frequency response coef-ficient at a particular frequency is then recovered from a linear combination of


24/44


the coefficients in the regression as in (34) and (46). This approach, which canbe regarded as a frequency domain version of leads and lags dynamic regression,

provides a convenient single equation method of estimating a long run relation-ship in the presence of deterministic trends and short run dynamics.

Dept. of Economics, University of Texas at Austin, Austin, TX 78712, U.S.A.;

[email protected]; http://www.eco.utexas.edu/corbaeSchool of Business, National University of Singapore, Singapore 117591; and

Research Dept., IMF, Washington DC, U.S.A.; [email protected]

andCowles Foundation for Research in Economics, Yale University, Box 208281,

New Haven, CT 06520-8281, U.S.A.; and University of Auckland and University ofYork; [email protected]; http://Korora.econ.yale.edu

Manuscript received September, 1997; final revision received June, 2001.

APPENDIX

Lemma D (Two Sided BN Decompositions): If CL = j= cjLj and j= j 12 cj < then:

(a) CL = C1+ CL1 L, whereCL =

j= cjL

j, with

cj =

k=j+1

ck j 0

j

k=ck j < 0

(b) CL = Ceiw + Cweiw Leiw L 1 where CwL =j= cwjLj with

cwj

=

k=j+1

ckeikw j 0

j

k=cke

ikw j < 0

Proof of Lemma D: The proof is along the same lines as the proof in Phillips and Solo (1992,

Lemma 2.1) of the one sided BN decomposition.

Remarks: (i) The condition

j=

j 12 cj < (62)

is sufficient for (a) and (b) and ensures that j= c2j < but it is not necessary for the latter. Forinstance, the series L =j= jLj, whose coefficients j are given by (19), fails (62) but still hasa valid BN decomposition with

j=

2j < . To see this, let cj = eijx/j and then j is a constant

times the imaginary part ofcj. Define Sn =n

j=1 eijx = eix /eix 1einx 1. Partial summation gives

Pn =n

j=1

eijx

j= 1

nSn +

eix

eix 1n

j=2

eij1x 1jj1


25/44


It follows that

Pn Pm =Sn

n Sm

m +eix

eix 1n

j=m+1 eij1x

1

jj 1 and letting n we have

cm =

j=m+1

eijx

j= Sm

m+ e

ix

eix 1

j=m+1

eij1x 1jj 1 = O

1

m

with a similar result for cm. We deduce that

cm2 < . Hence, L =

j= jLj with j

given by (19) has a valid BN decomposition with

j= 2j <

(ii) The case of a symmetric filter with cj = cj for j = 0 is an important specialization, whichincludes the series with coefficients j given by (19). In this case, cm =

j=m cj =

j=m cj =

cm + cm. Then, C1 =j=1cj + cj + c0 =j=1cj cj cj +j=1 cj = 0. If CL has a valid BNdecomposition, we deduce thatCL = C1 +CL1 L2 CL =

j=cjLj(63)

where

cj=

k=j+1

ck j 0

j

k=ck j < 0

A sufficient condition for the validity of (63) is

j= j12 cj < or

j= j

32 cj < in terms

of the original coefficients.

Proof of Lemma A: Part (a) is well known (e.g., Gradshteyn and Ryzhik (1965, formula

0.121)). For part (b) we use partial summation to give

n

t=1tkeis t

=

nt=1

tk

eis t +

nt=1

t 1keis t

=

n

t=1k

j=1k

jtkj1jeis t +n

t=1t 1keis t1eis 1or

nk =n

t=1

kj=1

k

j

tkj1j

eis t +

nt=1

t 1keis t1eis 1Using this formula in the identity

nt=1

tkeis t

eis 1= nkeis 1+ n

t=1t 1keis t1eis 1

we get n

t=1tkeis t

eis 1= nkeis 1+ nk n

t=1

kj=1

k

j

tkj1j

eis t

= nkeis +k

j=1

k

j

1j

n

t=1tkjeis t


26/44


giving the recursion

Wks = nkeis

eis 1 +1

eis 1k

j=1kj1jWkjs

which holds for s = 1 n (i.e., eis = 1).The initial condition

W0s =n

t=1eis t =

n s = 00 s = 0

follows by elementary calculation. For higher order trends with k = 1 2 3 we get

W1s =nn +1

2

s

=0

neis

eis 1 s = 0

W2s =

nn +12n+1

6 s = 0

neis

eis 1

n 2eis 1

s = 0

and

W3s =nn +1

2 2

s=

0

neis

eis 1

n2 3n 1 1

eis 1 + 6 1eis 12

s = 0

which lead to the expressions for Wks k = 0 1 2, that are given in Section 2 following Lemma A.

Proof of Theorem 1: This follows standard lines and is omitted (see Corbae, Ouliaris, and

Phillips (1997)).

Proof of Theorem 1: First, observe that

= m1 X QZ QZ X1m1 X QZ QZ y(64)Next, from (8) and (9) we have

wys = bs wxs + ws +1n

eis as 0 as n

(65)

= bwxs + ws +Op

1n

(66)

in the stationary case. We now find that

m1

X QZQZ

X = m1

s wxs wxs

+ op1 p 2fxx (67)

and using (66) we have

m1 X QZQZy = m1 X y + op1(68)= m1

s wxs wxs

b+ m1 s

wxs ws +op1


27/44


Then, from (64), (67), and (68) we have the expression

m = m1 s

wxs wxs 1m1/2

s wxs ws

+op1(69)The family ws s are known to satisfy a central limit theorem (with limit Nc0 2f)for dfts of stationary processes and are independently distributed2 as n . It follows from thisresult and (67) that

m1/2

s wxs ws

d Nc0 22ffxx(70)

which leads to the stated limit theorem for . A similar argument gives the result for f.

Proof of Lemma B: Take dfts of the equation xt = t , giving

wxS = n1/2n

t=1xt1e

is t +wvs = eiswxs n1/2

eis nxn x0

+wvs Then, transposing and solving yields the stated formula

wxs =1

1 eis wvs eis

1 eisxn x0

n1/2(71)

Proof of Lemma C: Part (a) This result may be proved as in Phillips (1991a). A new and

substantially simpler proof uses part (e) and is as follows.

n2

s Awxs wxs

= n2 s

wxs wxs n2

s cA

wxs wxs

= n2n

t=1xt x

t +Opn1

d1

0BxB

x

where the error magnitude in the second line follows directly from part (e) below.

Part (b) First note from (41) that wd0 = nn + 1/2n1/2, so that n1/2wd0 1 12 =f0 = 10 u, andwds

=

01

n1/2eis

eis 1

= 1

nf1s

s = 0

It follows that as n

wds =

0

n

2s i1 +o1

(72)

2 Hannan (1973, Theorem 3) showed that a finite collection ofws satisfy a central limit theorem

and are independent. Here the collection s has m members and is asymptotically infinite.Phillips (2000, Theorem 3.2) showed that an asymptotically infinite collection of ws satisfy acentral limit theorem and are asymptotically independent for frequencies s in the neighborhood of

the origin provided the number of frequencies m = on 12 + 1p where Et p < for some p > 2,i.e., provided the number of frequencies does not go to infinity too fast. This result can be extended

to frequency bands away from the origin, although a proof was not given in that paper.


28/44


a formula that holds for both s fixed and for s with s/n 0. On the other hand, whens = 0 as n we have

wds = 1

n

0

ei

ei 1+o 1

n

= 1n

f1 + o 1

n

(73)

Write the summation over A as follows:

s A =

s

s Ac . Then

n3/2

s Awxs wds

= n3/2 s

wxs wds n3/2

s Acwxs wds

= n3/2 s

wxs wds n2

s Acwxs f1s

First,

n2

s Acwxs f1s

= n2 s Ac

11 eis wvs

eis

1 eisxn x0

n1/2

f1s

= Opn1

Second,

n3/2

s wxs wds

= n2n

t=1xt

s

eis t wds

=n3/2

n

t=1 xt dt = n1n

t=1xt

ndt

d1

0Bxrur

dr

It follows that

n3/2

s Awxs wds

d1

0Bxrur

dr

Part (c) First, observe that

n1/2

x

dnr

d

B

xr

ur

1

0uu

1

1

0uB

x =B

xur

Then

1

n2

s A

wxd s wxds

d1

0BxuB

xu

as in part (a).

Part (d) From Lemma B we have

wxs =1

1 eis wvs eis

1 eisxn x0

n1/2

As in the proof of part (b), we have n1/2xnd Bx1, and, if s = 0 as n ,

wxs d 1

1 ei Nc0 2fvv ei

1 ei N 0 2fvv 0

= Nc

0 2fvv + fvv0

1 ei2


29/44


It follows that

n1/2 s cA wxs wds = n1 s cA wxs f1s = n1

s cA

1

1 eis

ws f1s n1 xn x0

n1/2

s cA

eis

1 eis

f1s

= term I + term II say.

Since the w s are asymptotically independent Nc0 2fvv s , term Ip 0. For term II, since

x0 = Op1, we have

n1xn x0

n1/2 s cA

eis

1 eis f1s

1

2

xn

n1/2

2

n s cAeis

1 eis f1s

d 12

Bx1

cA

eif1

1 ei d

so that

term IId 1

2

Bx1

cA

eif1

1 ei d

(74)

giving part (d).

Part (e) From Lemma B we get

n1

s cA

wxs wxs d n1

s cA

1

1 eis 2

wvs wvs + xn

n1/2xn

n1/2

= 12J

JajJ

1

m

s j

1

1 eis 2

wvs wvs + xn

n1/2xn

n1/2

+op1

= 12J

JajJ

1

1 eij2 2fvv j + n1

s cA

1

1 eis 2xn

n1/2xn

n1/2+op1

d

cAfvv

1

ei

2

d+ 12 cA

1

1

ei

2

d Bx1Bx1

as required. In the penultimate line above, 2fvvj = 1/m

s j w s ws and fvv

fvv p 0.Part (f) Note that

wxds = wxs wds n1DD1n1D X(75)

= 11 eis wvs

eis

1 eisxn x0

n1/2n1/2wds n1DD1n3/2D X

d 11 ei Nc0 2f

ei

1 ei Bx1 f1

10

uu11

0uBx

= 1

1 ei Nc0 2f gBx say

when s = 0 as n . Here

gBx =ei

1 ei Bx1 +1

0Bxu

1

0uu1

f1


30/44


and the complex normal variate Nc0 2f in (75) is independent of the Brownian motion Bx.If j = j/J is the midpoint of the band j and j as n , then

2fxxd j =1

m

s j

wxds wxds d 2f 1 ei2 + gBxgBx

= 2fxx + gBxgBx

We deduce that

n1

s cA

wxd s wxds = 1

2J

JajJ

1

m

s j

wxds wxds + op1

=1

2J JajJ 2 fxxd j +op1d

cA

fxx + 21gBxgBx

d

which gives part (f).

Part (g) Observe that

n1/2

s cA

wxds wds = n1

s cA

wxd s f1s + op1

d

n1 s cA gs Bxf1s

d

1

2

2

n

s cA

gs Bxf1s

d 12

cA

gBxf1 d

as stated.

Part (h) From (73) we have

s cA

wds wds 1

n s cA

f1s f1s 1

2 cA

f1f1 d

as required.

Part (i) Using part (h), we get

n1

s Awds wds

= n1 s

wds wds +Op

n1

= n1n

t=1dt

s

eis t wds +Op

n1

= n1n

t=1dt d

t +Opn1

1

0urur dr

giving the stated result.


31/44


Part (j) Using the representation of wxs in Lemma B, the fact that s > J when s 0, and proceeding as in part (a), we find that

n2

s 0wxs wxs

= n2 s

s 0

wxs wxs

+op1

= n2 s

wxs wxs

+Op

nsJ

1

s2

wvs wvs

+ xnn1/2

xnn1/2

=

n2n

t=1xt x

t

+ op1

d1

0BxB

x

Part (k) In the same way as parts (j) and (c) we find that

n2

s 0wxd s wxds

= n2n

t=1xdt x

dt + op1

d1

0BxuB

xu

Part (l) From part (b) we have n1/2wd0 f0 =

1 1

2

= 10

u and, for s = 0 with s/n 0,we have wds

= 0 n/2si1 + o1 from (72). Thus, for = 0, we find that1

n

s 0

wd s wds = 1

nwd0wd0

+ 1n

s 00

wds wds

= 1n

wd0wd0 +0 0

0 2 1

2

2ms=1

1

s2

f0f0 +

0 0

0 112

=1

0uu

Part (m) As in part (b) we find that

n3/2

s 0wxs wds

= n3/2 s

wxs wds n3/2

s 0wxs wds

= n3/2

nt=1 x

t dt op1d1

0Bxrur

dr

since

n3/2

s 0wxs wds

= Op

1

nm

s 0

wxs

= Op

1

nm s 0

wvs eis xnx0n1 eis

= op1

Part (n) For = 0, using (73) we obtainn

m

s

wds wd s 1

m

s

f1s f1s f1f1


32/44


Part (o) For = 0, using (73) we obtain

nm s wxs wds = 1m s wxs f1s +o1= 1

m

s

wvs eis xnx0n

1 eis

f1s

+ op1

d ei

1 ei B1f1

Finally, joint weak convergence in (a)(o) applies because the component elements jointly converge

and one may apply the continuous mapping theorem in a routine fashion. In particular, Assumptions 1

and 2 ensure thatn1/2

nrt=1

t wv

d Br with = Nc0 2fvv

and where is independent ofBr for all = 0. The required quantities in (a)(k) are functionalsof these elements and smoothed periodogram ordinates (like 1/m

s j wvs wvs

for j as in the case of part (f)) that converge in probability to constants.

Proof of Theorem 2: Note that

A

A

= X

QZ QZ X1X

QZ QZ c

XA Ac We consider the limiting behavior of n2 X QZ QZ X and n1 X QZ QZc X. Define xdt = xt dt D

D1D X. Since xt is an I 1 process and satisfies an invariance principle when standardizedby n1/2, we have

n1/2xdnrd Bxr ur

10

uu1 1

0uBx = Bxur say(76)

Write the discrete Fourier transform of xdt as wxd s and then from Lemma C(c) we have

n2 X QZ QZ X = 1n2 s A wxds wxd s

d

10 BxuBxu(77)Since

10

BxuBxu > 0 (see Phillips and Hansen (1990)), n

2 X QZ QZ X has a positive definite limitas n .

Next, decompose n1 X QZ QZ c X as follows:X QZ QZ c X

n=

X QZ WAWPZ WAcWXn

X QDWAWPDWAcWX

n(78)

=X PDW

AWPDWAcW

X

nX WAWPDW

AcW

X

n

= term A term B

Take each of these terms in turn. Factor term A as follows and consider each factor separately. Write

X PD WAWPDWAcWXn

= X PD WAWD

n3/2

DD

n

1DWAcWX

n1/2

(79)


33/44


The first factor is

n1/2 X PDWAWDn = n1/2 X Dn DDn 1

DWAWD

n(80)

d1

0Bxu

1

0uu11

0uu

=1

0Bxu

in view of (76) and Lemma C(i). The third factor is the conjugate transpose of

n1/2 X WAcWD = n1/2 s cA

wxs wds (81)

d 12

Bx1

cAeif1

1

ei

d

from Lemma C(d). The limit of term A now follows by combining (81) and (80) and using joint weakconvergence:

X PDWAWPDWAcWXn

= n1/2 X PDWAWD

n

DD

n

1DWAcW n1/2 X(82)

d1

0Bxu

1

0uu1

1

2

cA

eif1

1 ei d

Bx1

Next consider term B of (78). Using Lemma C, we obtain

X WAWPDWAcWXn

(83)

= 1n

n1/2

s A

wxs wds

DD

n

1n1/2

s cA

wds wxs

d1

0Bxu

1

0uu1

1

2

cA

eif1

1 ei dBx1

Combining (82) and (83) in (78) we find3

n1 X QZ QZ c X = op1(84)As for the limit distribution of A, we have

nA A = X QZ QZ X

n2

1 X QZ QZ n

(85)

X QZ QZ X

n2

1 X QZ QZc Xn

A Ac

From Lemma C(c) and (77) above, we have

n2 X QZ QZ X d 10 BxuBxu = G(86)

3 By changing the probability space (on which the random sequence xt is defined), we can ensurethat both terms tend almost surely to the same random variable (see, for example, Theorem 4,

page 47 of Shorack and Wellner (1986)). Then, in the original space the difference of the terms tends

in distribution to zero giving the stated result.


34/44


and from (84)

X QZ QZ Xn2

1 X QZ QZ c Xn

p 0(87)Next,

X QZ QZ n

= n1 s A

wxds wds = 1

n

s A

wxd s ws (88)

1

n32 s A

wxds wds

DD

n 1

D

n

Then, using (76) and proceeding as in the proof of Lemma C(a) above, we find

n1

s Awxd s ws

= n1 s

wxd s ws n1

s Acwxds ws

(89)

= n1n

t=1xdt t op1

d1

0BxudB

Further, from Lemma C(b), (i) and Assumption 2, we obtain

1

n32

s A

wxd s wds

DD

n

1Dn

d1

0Bxuu

1

0uu1

0u dB

= 0(90)

since1

0Bxuu

= 0. Combining (88), (89), and (90), we deduce that

X QZ QZ n

d1

0Bxu dB(91)

The limit distribution (91) is a mixture normal distribution with mixing matrix variate1

0BxuB

xu.

It now follows from (86) and (91) that

X QZ QZ Xn2

1 X QZ QZ n

d1

0BxuB

xu

110

BxudB

(92)

MN

0 2f0

10

BxuBxu

1

Using (85), (87), and (92), we deduce that

nA Ad1

0BxuB

xu

110

BxudB

which gives the stated result.


35/44


For the limit of Ac , we need to examine the asymptotic behavior of the bias term in (29), whichdepends on the matrix quotient n1

X QZ

cQZ

X1n1

X QZ

cQZ

X. Take each of these fac-

tors in turn. First,X QZ cQZ Xn

=X QDcQD X

n= n1

s cA

wxd s wxds

where wxds = wxs wds n1DD1n1D X. From Lemma C(f) we have

n1

s cA

wxds wxds d

cA

fxx + 21gBxgBx

d(93)

which is a positive definite limit. Next,

n1 X QZ cQZX= n1 X QZcPZ X = n1 X QDcPDX=

n1/2

s cA

wxd s wds

n1DD1

n3/2

s Awds wxs

From Lemma C(g) and (b), we obtain

n1

X QZ

cQZX

d

12 cA

gBxf1 d

1

0uu

1

1

0uBx(94)

It follows from (93) and (94) and joint weak convergence that the asymptotic bias term for Acinvolves

n1 X QZcQZ X1n1 X QZcQZ Xd

cA

2fxx + gBxgBx

d

1

cA

gBxf1 d

10

uu1 1

0uBx

establishing the stated result.

Proof of Theorem 2: Part (a) From (64) and (11) we have

= 1+X QZQZ X1X QZ QZ(95)

Since t is a strictly stationary and ergodic sequence with mean zero and satisfies a central limittheorem, n1DD1n1D = Opn1/2, and so

t zt n1ZZ1n1Z = t zt 1n n1DD1n1Dp t (96)

Then, since L has a valid BN decomposition,

n1 X QZ QZ = n1 s

wxds ws +op1(97)

= n1 s

wxds ws n1

s wxd s wvs

eis +op1


36/44


When = 0, we find as in (89) and Theorem 3.1 of Phillips (1991a) that

n1

s 0wxds ws

d

10 Bxu dB(98)and

n1

s 0wxds wvs

d1

0Bxu dB

x +x(99)

where x =

j= Ev0vj. We deduce that

n1

X QZQZ

d

1

0BxudB

1

0BxudB

x + x

1(100)

Next, as in Lemma C(c) we find that

n2 X QZ QZ X d 10

BxuBxu(101)

Combining (100) and (101), we obtain

n0 1d1

0BxuB

xu

110

Bxu dB 1

0BxudB

x + x

1

Now consider the case where = 0. First we have

m1 X QZQZ X = m1 s

wxds wxds

and in a similar fashion to (93) we find that

m1

s wxd s wxds

d 2fxx +gBxgBx = say(102)

Next, from (15) we have

wys = 1wxs eis wvs +ws + Op1n(103)

Now

wyd s = wys wds n1DD1n1Dy(104)

and, in view of the cointegrating relation (11), we have

n12 ynr d 1Bxr = Byr(105)

and it follows as in part (f) of Lemma C that

wyd s wys f1s 10

uu110

uBx1 +op1(106)However, from the proof of Lemma C(f) we also have

wxds = wxs f1s

10

uu11

0uBx

+ op1(107)


37/44


We deduce from (103), (106), and (107) that

wyds wxs 1 f1s 10 uu110 uBx1+ ws (108)

wvs eis + op1= wxds 1 wvs eis +ws + op1

Then

m1 X QZQZy = m1 s

wxd s wyds (109)

m1

s wxd s wxds

1

m1 s

wxds wvs ei+ op1

Using (51) we find that

m1

s wxd s wvs

= m1 s

wxs wvs + op1(110)

= m1 s

1

1 eis wvs wvs +op1

p2

1

eifvv

It follows from (95), (102), (109), and (110) that

d 1 2

1 ei 1 fvve

i = 1+ 21 ei2 1 fvve

iei 1

The true coefficient is

= b = ei = 1 + eiei 1

so we have

d +2

1

ei

2

1 fvv 1eiei 1

which gives the stated result since fxx = 1 ei2fvv.Part (b) The estimator is derived from the augmented spectral regression (32) and the relation

(34). In (32), yt and xt are first detrended in the time domain and then dfts are taken of the detrendeddata and differences of the detrended data. Since wxds wvs +op1, (108) can be written as

wyds = wxds 1 wxds eis + ws +op1(111)

First, take = 0. In this case, the regressors wxds and wxds in (111) are asymptoticallyorthogonal after appropriate scaling because

m1/2n1

s 0wxd s wvs

= Opm1/2(112)

in view of (99). It then follows from (98), Lemma C(k), (112), and (111) that

n0 1d1

0BxuB

xu

110

BxudB

as required.


38/44


Now take the case = 0. From part (f) of Lemma C and (52) we have

wxzs = wxds =w

v

s

1 eis eis

1 eisx

n x0

n1/2 n3/2 XDn1DD1n1/2wds = wvs

1 eis + w tn s xn x0 n3/2 X Dn1DD1n1/2wds

= wvs 1 eis + Ann

1/2wds

Here, n1/2wds = Op1 from (42) and

An =

0xn x0

n

n3/2

X Dn1DD1 = Op1

Then (111) is

wyd s =

wvs

1 eis +Ann1/2wds

1 wvs eis +ws +op1(113)

= wvs

1

1 eis eis

+wds n1/2An +ws +op1

The augmented narrow band regression (32) around frequency can therefore be written as

wyzs = a1wvs

1

eis

+Ann1/2wds + a2wvs + op1 + residual(114)

which is asymptotically equivalent to the regression

wyzs = wvs b1 + n1/2wds b2 + residual(115)

with

b1 =a1

1 ei + a2 b2 = a1An

In view of (113), (115), and the asymptotic orthogonality of the regressors in (115) (i.e., m1

s wvs n1/2wds

p 0), we find that

b1 p1

1 ei ei =

1 ei

and

m

b1

1 ei

d Nc

0

2f

2fvv

We deduce that

= a1 + 1 eia2 = b11 ei p

and

m

d Nc0 ffxx



39/44


Proof of Theorem 3: Note that

n

f

A

A=

n2X WAQAWZ

AWX1n1X WAQAWZ

AW

= n2X QV QVX1n1X QV QV= n2 X QV QVX1n1 X QV QV= n2 X Q Z Q Z X1n1 X Q Z Q Z

Next,

X Q Z = X X P Z = X X ZZ Z1Z=

X

X DD D1D

and so

X Q Z Q Z X = X Q Z Q Z X = XX X DD D1DX which is

s Awxs wxs

s Awxs wds

s Awds wds

1

s Awds wxs

Now, as in Lemma C(a), (b) and (i), we have

n2

s Awxs wxs

d1

0BxB

x

n3/2

s Awxs wds

dd1

0Bxu

n1

s Awds wds

1

0uu

Thus,

n2 X Q Z Q Z X d 10

BxBx

10

Bxu1

0uu11

0f0B

x

=1

0BxuB

xu(116)

Next, consider the limit behavior of

n1 X Q Z Q Z = n1X X DD D1D = n1

s Awxs ws

n3/2

s Awxs wd s

n1 s A

wds wds 1n1/2 s A

wds ws As in the proof of (91),

n1

s Awxs ws

d1

0Bx dB


40/44


and

n1/2 s A wds ws d

1

0

u dB

Thus,

n1 X Q Z Q Z d 10

Bx dB 1

0Bxu

1

0uu1 1

0udB =

10

Bxu dB

It follows that

nfA A

d1

0BxuB

xu

110

Bxu dB

MN

0

10

BxuBxu

12f0

giving the stated result for the band A.

For the band cA, we have

n

f

Ac Ac = n1X WAcQAc WZ AcWX1n1/2X WAcQAc WZ AcW(117)= n1 X Qc Z cQc Z X1n1/2 X Qc Z cQc Z

As above, X Qc ZcQc Z X = X Qc ZccQc Z X= X c X X cDD cDDc X =

s cAwxs wxs

s cA

wxs wds

s cA

wds wds 1

s cA

wds wxs

From the above expression and Lemma C(d), (e), and (h) we deduce that

n1 X Qc Z cQc Z X(118)d

cA

fxx +

1

2

1

1 ei2 Bx1Bx1

d

1

2Bx1

cA

eif1

1 ei d

1

2 cA f1f1 d

1

2 cA f1ei

1 ei

dBx1

=

cA

fxx +

1

2

1

1 ei2 Bx1Bx1

d 12

Bx1Bx1

cA

eif1

1 ei d

cA

f1f1 d

cA

f1ei

1 ei d

Next, observe that

cA

eif1

1 ei d

cA

f1f1 d

f1

is the L2cA projection of the function e

i1

ei1 onto the space spanned by f1. When

the deterministic variable zt includes a linear time trend, we know from (39) that the vector f1includes the function ei1 ei1 as one of its components. Hence, in this case we have

cA

ei1 ei1f1 d

cA

f1f1 d

f1 = ei1 ei1(119)

for cA. It follows that (118) is simply

cA

fxx d.


41/44


Proceeding with the proof, the second factor of (117) decomposes as

n1/2 X Qc Z cQc Z = n1/2 s cA

wxs ws

n1/2

s cA

wxs wds

s cA

wds wds

s cA

wds ws

Using (119) and the independence of xt and t we find

n1/2 X Qc Z cQc Z d n1/2 s cA

wxs ws

12

Bx1

cA

ei

f11 ei d 12 c

A

f1f1 d

n1/2 s cA

f1s ws

d n1/2 s cA

wxs

1

2Bx1

cA

eif1

1 ei d

1

2

cA

f1f1 d

f1s

ws

= n1/2 s cA

wxs Bx1 eis1 eis ws = n1/2

s cA

1

1 eis w s eis

1 eisxn x0

n1/2 Bx1

eis

1 eis

ws

= n1/2 s cA

1

1 eis w s

ws +op1

d

N0

2

n s cA1

1 ei

s 2

w s w s

fs d N

0 2

cA

fxx fd

We deduce that

n

f

Ac Ac d N

0

cA

fxx d

12

cA

fxx fd

cA

fxx d

1


Proof of Theorem 3: Part (a) First note that

wxs = 2wzs +wxs = 2nwds + wxs with a similar expression for wy s . The regression (45) is equivalent to the following regressionwith frequency domain detrended data:

wfyds = c1wfxd s + c2wfxds + residual(120)


42/44


where

wfxds = wxs wds s

wds wds 1 s

wds wxs = wxs wds

s

wds wds 1

s wds wxs

= wfxd s say

with a similar expression for wfyds = wfyd s , and

w

f

xds

= wxs

wds s wds wds

1

s wds wxs = wvs + op1

For = 0, as in the proof of Theorem 2, the regressors in (120) are asymptotically orthogonalbecause

n3/2

s 0w

fxds wvs

= Opn1/2

Next, using Lemma C(j)(m), we find as in (116) above that

n2

s 0w

fxds w

fxds

d 10

BxuBxu

and as in (89) we get

n1

s 0w

fxds ws

d1

0BxudB

It follows using (15) and these two results that

nf 0 d 10

BxuBxu110

BxudBNext consider the = 0 case. First we simplify the regression equation (120):

wfyd s = c1wfxds + c2wfxds + residual(121)

= c1wfxds + c2wvs + op1 + residual

Using (73) and Lemma C(n)(o) we obtain

wf

xds = wxs wds s

wds wds 1

s

wds wxs (122)= wvs

1 eis eis

1 eisxn x0

n1/2

1n

f1

m

nf1f1

mn

f1ei

1 ei B1


43/44


= wvs

1 eis eis

1 eisxn x0

n1/2 e

i

1 ei B1

= wvs 1 eis +op1

So the regression (121) is asymptotically equivalent to

wfyds

=

wvs

1 eis +op1

c1 + wvs + op1c2 + residual(123)

From (122) and (15) we deduce, as in Theorem 2, that

bf1 =

c11 ei + c2p p

1

1 ei ei =

1 ei

and

m

b1

1 ei

d Nc

0

f

fvv

It follows that

f = c1 + 1 eic2 = b11 ei p

and

m

f

d Nc0

f

fxx REFERENCES

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Journal of Econometrics, 59, 263286.

Corbae, D., S. Ouliaris, and P. C. B. Phillips (1997): Band Spectral Regression with Trending

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Engle, R. (1974): Band Spectrum Regression, International Economic Review, 15, 111.Frisch, R., and F. V. Waugh (1993): Partial Time Series Regression as Compared with Individual

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Geweke, J. (1982): Measurement of Linear Dependence and Feedback Between Multiple TimeSeries, Journal of the American Statistical Association, 77, 304313.

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Granger, C. W. J., and J.-L Lin (1995): Causality in the Long Run, Econometric Theory, 11,530536.

Grenander, U., and M. Rosenblatt (1957): Statistical Analysis of Stationary Time Series.

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(1973): Central Limit Theorems for Time Series Regression, Zeitschrift fr Wahrschein-

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Hannan, E. J., and P. J. Thomson (1971): Spectral Inference over Narrow Bands, Journal of

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Hosoya, Y. (1991): The Decomposition and Measurement of the Interdependence between

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