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Discrete Fourier Transforms of Fractional Processes with Econometric Applications*

Peter C. B. Phillips (Yale University, New Haven, CT, USA; University of Auckland, Auckland, New Zealand; Singapore Management University, Singapore; and University of Southampton, Southampton, UK)

Essays in Honor of Joon Y. Park: Econometric Theory

ISBN: 978-1-83753-209-4, eISBN: 978-1-83753-208-7

ISSN: 0731-9053

Publication date: 24 April 2023

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Abstract

The discrete Fourier transform (dft) of a fractional process is studied. An exact representation of the dft is given in terms of the component data, leading to the frequency domain form of the model for a fractional process. This representation is particularly useful in analyzing the asymptotic behavior of the dft and periodogram in the nonstationary case when the memory parameter d≥12. Various asymptotic approximations are established including some new hypergeometric function representations that are of independent interest. It is shown that smoothed periodogram spectral estimates remain consistent for frequencies away from the origin in the nonstationary case provided the memory parameter d < 1. When d = 1, the spectral estimates are inconsistent and converge weakly to random variates. Applications of the theory to log periodogram regression and local Whittle estimation of the memory parameter are discussed and some modified versions of these procedures are suggested for nonstationary cases.

Keywords

Citation

Phillips, P.C.B. (2023), "Discrete Fourier Transforms of Fractional Processes with Econometric Applications*", Chang, Y., Lee, S. and Miller, J.I. (Ed.) Essays in Honor of Joon Y. Park: Econometric Theory (Advances in Econometrics, Vol. 45A), Emerald Publishing Limited, Leeds, pp. 3-71. https://doi.org/10.1108/S0731-90532023000045A001

Publisher

:

Emerald Publishing Limited

Copyright © 2023 Peter C. B. Phillips

1. Introduction

Studies of nonstationary time series over the last four decades have produced a vast body of knowledge that has transformed the conduct of empirical research in economics. The impact of this research is now manifest in empirical work throughout the social and business sciences. A catalyst supporting these developments was the widespread recognition that real world processes in society, economics, and politics are influenced in fundamental ways by advances in technology, firm investments, and individual human decision-making. These processes are rarely, if ever, stationary. Inevitably they evolve in uncertain ways over time, reflecting the arrival of new shocks to the system, some of which have persistent effects. Recognizing this reality led to an understanding that methods of data analysis need to account for the fact that the way in which memory is carried in the data differs in a fundamental manner among stationary, near-stationary, and nonstationary processes.

Acknowledgment of the importance of this distinction is evident in early researches of statisticians and economists at the turn of the twentieth century (Hooker, 1901; Pearson & Elderton, 1923; Yule, 1926) on nonsense correlations1 and the work of the mathematician Bachelier (1900) on speculative prices, which introduced the notion of a stochastic process. Methods began to emerge later that provided probabilistic underpinnings and foundations for statistical inference with data that demonstrated long range memory or dependence (Granger & Joyeux, 1980; Hosking, 1981; Hurst, 1951; 1956; Mandelbrot & Van Ness, 1968) and various types of random wandering behavior over time. In economics in the 1980s, advances in the use of function space limit theory were made that enabled the full trajectory features of nonstationary data to be reflected in regression asymptotics, leading to new understanding of such regressions, including both cointegrating and spurious regressions, and new methods of testing and inference for analyzing nonstationary data (Durlauf & Phillips, 1988; Phillips, 1986b; 1987; 1988; Phillips & Durlauf, 1986).

Joon Park played a big part in these developments, starting with his doctoral dissertation research and early research at Yale (Park & Phillips, 1988; 1989) and a sustained series of subsequent works that have helped to push out the envelope of econometric methodology for linear, nonlinear, and continuous time methods of analysis with nonstationary data. Many of these works have been jointly conducted with the present author in a longstanding collaboration that has been as pleasurable and special an academic fellowship as much as it has enriched this field of research.

My contribution to this symposium of works honoring Joon Park relates to his research on nonstationary processes and focuses on some of the defining properties of long range-dependent time series. The present work has a history reaching back more than two decades and it is hoped that a good part of its value is retained amidst the considerable body of work that has emerged since the original version of the paper (Phillips, 1999a) was written. The first contribution of this chapter is to provide an exact representation of the dft of a fractional process, which enables asymptotic analysis of its behavior and various functionals such as the periodogram in the nonstationary case when the memory parameter d≥12. The methods reveal that smoothed periodogram spectral estimates remain consistent for frequencies away from the origin in the nonstationary case provided the memory parameter d < 1. When d = 1, the spectral estimates are inconsistent and converge weakly to random variates. Some useful applications of this theory are given for log periodogram regression and local Whittle estimation of the memory parameter in nonstationary cases. For an advanced textbook treatment of long memory processes, readers are referred to Surgailis et al. (2012).

The plan of this chapter is as follows. Various preliminaries are given in the following Section 2. Some useful new decompositions and representations in the frequency domain are developed in Section 3 that extend related decompositions in the time domain. Section 4 develops asymptotic approximations for dfts involving special functions that help to simplify representations and enable development of limit theory for dfts of fractional processes in nonstationary cases. These results extend earlier work on the limit theory of dfts of stationary processes to the fractional case. For higher levels of dependence, when d = 1, the leakage from the zero frequency becomes dominant and affects the limit theory at all frequencies, so that dfts are spatially correlated across frequency asymptotically, quite unlike the stationary case. Section 5 provides some applications of the results to spectral estimation and to semiparametric estimation of the memory parameter. Particular attention in the latter case is given to log periodogram regression and local Whittle estimation. Some modified versions of these procedures are suggested which conveniently extend their range of applicability to the nonstationary case. Final remarks on long memory and autoregressive approaches to nonstationarity close out Section 5. Proofs and technical results are in the Appendix in Section 6. A notational summary is given at the end of this chapter in Section 7.

A final word of introduction. While our focus is on the case where d∈12,1, the methods introduced here are applicable when d>1, and in modified form when |d|<12. A particularly useful approach is to combine the exact representation (3.7) that applies when d = 1 with results for fractional d to produce valid representations for the d>1 case. The remarks and results in paragraphs 3.6–3.8 indicate some of these possibilities.

2. Preliminaries

We consider the fractional process Xt generated by the model

(2.1)(1−L)dXt=ut, t=0,1, …

Our interest is primarily in the case where Xt is nonstationary and d≥12, so in (2.1) we work from a given initial date t = 0, set uj = 0 for all j ≤ 0, and assume that ut(t > 0) is stationary with 0 mean and continuous spectrum fu(λ)>0. This formulation corresponds to a Type II fractional process (Davidson & Hashimzade, 2009; Marinucci & Robinson, 1999). Expanding the binomial in (2.1) gives the form

(2.2)∑k=0t(−d)kk!Xt−k=ut,

where

(d)k=Γ(d+k)Γ(d)=(d)(d+1) … (d+k−1)

is Pochhammer’s symbol for the forward factorial function and Γ(·) is the gamma function. When d is a positive integer, the series in (2.2) terminates, giving the usual formulae for the model (2.1) in terms of differences and higher order differences of Xt. An alternate form for Xt is obtained by inversion of (2.1), giving

(2.3)Xt=(1−L)−dut=∑k=0t(d)kk!ut−k.

Throughout this chapter it will be convenient to assume that the stationary component ut in (2.1) is a linear process of the form

(2.4)ut=C(L)εt=∑j=0∞cjεt−j, ∑j=0∞jcj<∞, C1≠0,

for all t and with εt=iid0,σ2 with finite fourth moments. The summability condition in (2.4) is satisfied by a wide class of parametric and nonparametric models for ut, enables the use of the techniques in Phillips and Solo (1992), and ensures that partial sums of ut satisfy a functional central limit theorem, which will be needed later.

Under (2.4), the spectrum is fu(λ)=σ22π∑j=0∞cjeijλ2 and fu(0)=σ22πC(1)2>0.2 In view of (2.1), it is natural to define

(2.5)fx(λ)=|1−eiλ|−2dfu(λ).

The function fx(λ) gives the spectrum of Xt when it exists and Xt is stationary (i.e., for d<12 and under infinite past initialization of Xt in (2.3)) and is the analog of the spectrum in the nonstationary case when d≥12 even though it is not integrable. In that case, Solo (1992) gave a formal justification of fx(λ) as a spectrum in terms of the limit of the expectation of the periodogram. Taking logarithms of (2.5) produces the equation

(2.6)ln(fx(λ))=−2dln(|1−eiλ|)+ln(fu(λ)),

which motivates a linear log periodogram regression for the estimation of d, in which fx(λ) is replaced by periodogram ordinates Ix(λ) evaluated at the fundamental frequencies λs=2πsn, s=0,1,…,n−1 Here, Ia(λs)=wa(λs)wa(λs)*, wa(λs) is the dft, wa(λs)=12πn∑t=1nateitλs of a time series at, and w* is the complex conjugate of w. With this substitution (2.6) becomes

(2.7)lnIxλs=−2dln1−eiλs+lnfuλs+Uλs,

where Uλs=lnIxλs/fxλs. By virtue of the continuity of fu, fuλs is effectively constant for frequencies in a shrinking band around the origin, suggesting a linear least squares regression of lnIxλs on ln1−eiλs over frequencies s=1,…,m (with m a truncation number). The method has undoubted appeal, is easy to perform in practice and has been commonly employed in applications. However, (2.6) is a moment condition, not a data generating mechanism, and the analysis of this regression estimator is complicated by the difficulty of characterizing the asymptotic behavior of the dft wx(λs), which is the central element in determining the properties of the regression residual Uλs in (2.7).

An important contribution by Künsch (1986) showed that, for fractional processes like (2.1), wx(λs) has quite different statistical properties from the corresponding dft, wu(λs), of the stationary process ut for frequencies in the immediate neighborhood of the origin. In particular, for λs=2πsn→0, with s fixed as n→∞ the dft ordinates are asymptotically correlated, not uncorrelated. Analyses by Robinson (1995b) and Hurvich et al. (1998) for Gaussian ut have provided an asymptotic theory in the stationary case, thereby placing log periodogram regression on a rigorous footing. More recent work has dealt with nonstationary cases where d≥12 (Kim & Phillips, 2006; Phillips, 2007; Velasco, 1999). Another semiparametric estimation procedure, suggested by Künsch (1987), is the Gaussian estimator which maximizes a local version of the Whittle likelihood, which is known to have a smaller variance than log periodogram regression in the stationary case (Robinson, 1995a). This estimator also relies on the behavior of wx(λs) for frequencies in the vicinity of the origin. More recent work on Whittle estimation has focused on nonstationary cases where d≥12 (Abadir et al., 2007; Phillips, 2014; Phillips & Shimotsu, 2004; Shao, 2010; Shimotsu & Phillips, 2005; 2006; Velasco & Robinson, 2000) and cases of noise contaminated data (Sun & Phillips, 2003) such as in the estimation of the Fisher equation (Sun & Phillips, 2004).

This chapter provides new methods for studying the asymptotic behavior of wx(λs) for nonstationary values of d. The approach relies on an exact representation of wx(λs) in terms of the dft wu(λs) and certain residual components. This representation aids in the analysis of the properties of wx(λs) and, thereby, in the study of log periodogram regression and local Whittle estimation. The representation also provides a frequency domain version of the data generating mechanism (2.1) above. As such, it is useful in motivating some alternative approaches to inference about d that are proposed here and which have been explored in subsequent work that has appeared since the first version of this chapter circulated in 1999.

3. Frequency Domain Decompositions

It is convenient to manipulate the operator 1−Ld in (2.1), with its polynomial expansion (2.2), in a form that more readily accommodates dfts. This can be done algebraically, as in Phillips and Solo (1992), by expanding the polynomial operator about its value at the complex exponential eiλ, leading to the following decomposition.

3.1. Lemma

Define the fractional operator expansion DnL;d=∑k=0n−dkk!Lk. Then

(3.1)DnL;d=Dneiλ;d+D˜nλe−iλL;de−iλL−1,

where D˜nλe−iλL;d=∑p=0n−1d˜λpe−ipλLp and d˜λp=∑k=p+1n−dkk!eikλ.

The representation (3.1) is an immediate consequence of formula (32) in Phillips and Solo (1992) and can be obtained by straightforward algebraic manipulation. No summability conditions are required here for its validity since it is a finite sum. However, the value of d does affect the order of the terms in this expansion and, consequently, the order of magnitude of these terms when n→∞, a fact that does affect subsequent theory. Additionally, when λ depends on n, the order of these terms is affected and this too needs to be accounted for in the asymptotic theory. Much of this chapter is devoted to this accounting to assist in characterizing the limit behavior of the dft wxλ=12πn∑t=1nXteitλ.

Using the operator (3.1), we may write the model (2.1) in the following form for all t≤n

(3.2)ut=DnL;dXt=Dneiλ;dXt+D˜nλe−iλL;de−iλL−1Xt

Taking dfts of the left and right sides of (3.2) now yields an exact expression for wxλ in terms of wuλ. The result is stated as follows.

3.2. Theorem

(3.3)wuλ=wxλDneiλ;d+12πnX˜λ0(d)−einλX˜λn(d)

where Dneiλ;d=∑k=0n−dkk!eikλ,

X˜λt(d)=D˜nλe−iλL;dXt=∑p=0n−1d˜λpe−ipλXt−p,

and

.(3.4)D˜nλe−iλL;d=∑p=0n−1d˜λpe−ipλLp, with  d˜λp=∑k=p+1n−dkk!eikλ

3.3. Remark

Equation (3.3) provides an exact representation of wxλ in terms of wuλ and a residual component involving n−12X˜λn(d). Explicitly,

(3.5)wxλ=Dneiλ;d−1wuλ−12πnDneiλ;d−1X˜λ0(d)−einλX˜λn(d).

In fact, (3.3) or (3.5) may be interpreted as a frequency domain version of the original model (2.1). In terms of periodogram ordinates, we have the corresponding equation

(3.6)Ixλs=wxλs2=Dneiλs;d−1wuλs−12πnX˜λ0(d)−einλsX˜λsn(d)2=Dneiλs;d−2Iuλs−2Re12πnX˜λs0(d)−X˜λsn(d)wuλs*+12πnX˜λs0(d)−X˜λsn(d)2,

which may be interpreted as the data generating mechanism for the ordinates Ixλs that are used in a log periodogram regression. Equation (3.6) reveals the model that is implicit in (2.7). To the extent that Dneiλs;d−2 can be replaced by 1−eiλs−2d and the component n−12X˜λsn(d) is small enough to be neglected, (3.6) and (2.5) might seem to suggest that Uλs=lnIxλs/fxλs will behave like the corresponding functional, logIuλs/fuλs, of the errors in (2.1). However, as will become apparent in our analysis, the residual component n−12X˜λsn(d) in (3.5) and (3.6) cannot be neglected, in general, and its importance grows as d increases.

3.4. Remark

When d=1, the forward factorial (−d)k=0 for all k>1, so that series involving these coefficients terminate at k = 1. In this case Dneiλ;1=1−eiλ, d˜λ0=−eiλ, X˜λ0(1)=−eiλX0, and X˜λn(1)=−eiλXn. Equation (3.3) then reduces to the simple form

(3.7)wuλ=1−eiλwxλ+eiλ2πneinλXn−X0,

an expression obtained by the author in earlier work and used in Corbae et al. (2002, Lemma B). In this case, it is apparent that n−12X˜λsn(d)=eiλsn−12Xn=Op(1) for all λs. Thus, in the unit root case, the residual correction term n−12X˜λsn(d) definitely matters, plays a role in the asymptotic behavior of wxλs at all frequencies and thereby affects the asymptotic theory of estimators of d like those arising from log periodogram regression and local Whittle estimation. Indeed, in those cases the author has shown in other works (Phillips, 2007; Phillips & Shimotsu, 2004) that these estimators have limiting mixed normal distributions rather than normal distributions when d = 1.

3.5. Remark

When ut = 0 for t≤0, in (2.1), it follows that Xt = 0 for t≤0 and hence X˜λ0(d)=0. In this event, expression (3.3) becomes

(3.8)wuλ=wxλDneiλ;d−einλ2πnD˜nλe−iλL;dXn=wxλDneiλ;d−einλ2πnX˜λn(d),

or, in the unit root case,

(3.9)wuλ=1−eiλwxλ+eiλ2πneinλXn,

in place of (3.7). Since these initial conditions are assumed in (2.1), and since the effect of relaxing them will usually be apparent, we will henceforth use (3.8) in place of (3.3).

3.6. Remark

Another useful representation for the dft of Xt can be obtained by combining the representation (3.8) with the unit root decomposition (3.9). It is especially useful when d>1. Write (2.1) as

(3.10)1−LXt=1−L1−dut:=zt

so that Xt=∑j=1tzj+X0. Then, taking dfts in (3.10), we first apply (3.9) to write wxλs in terms of wzλs and then use (3.8) to reduce wzλs in terms of wuλs and a correction term. The outcome is formalized in the following theorem.

3.7. Theorem

If Xt follows (2.1), then

(3.11)wxλ1−eiλ=wzλ−eiλeiλnXn2πn=Dneiλ;fwuλ−eiλn2πnU˜λnf−eiλeiλnXn2πn

where f=1−d,

(3.13)U˜λnf=D˜nλe−iλL;fun=∑p=0n−1f˜λpe−ipλun−p, and f˜λp=∑k=p+1n−fkk!eikλ.

3.8. Remark

Some further decomposition beyond (3.11) and (3.12) is possible. As in Phillips and Solo (1992), we can decompose the operator CL that appears in ut=C(L)εt as

CL=Ceiλ+C˜λe−iλLe−iλL−1, C˜λL=∑j=0∞c˜jλLj, c˜jλ=e−iλj∑k=j+1∞ckeiλk,

where ∑j=0∞|c˜jλ|<∞ in view of the summability condition on cj in (2.4). Then,

(3.14)ut=CLεt=Ceiλεt+e−iλελt−1−ελt,

is a valid decomposition of ut into the iid component Ceiλεt and a stationary error that telescopes under the dft operation, with ελt=C˜λe−iλLεt=∑j=0∞c˜jλe−iλjεt−j. In particular,

wuλ=Ceiλwελ+12πnελ0−einλελn=Ceiλwελ+Op1n.

Using this representation in (3.12) we get

(3.15)wxλ1−eiλ=Dneiλ;fCeiλwελ−eiλn2πnU˜λnf−eiλeiλnXn2πn+Dneiλ;f×Op1n.

Additionally, zt in (3.10) can be written as

(3.16)zt=Ceiλ1−Lfεt+1−Lfe−iλL−1ελt.

Set ηt=1−Lfεt, ηλt=1−Lfελt in (3.16) and take dfts, giving

(3.17)wzλ=Ceiλwηλ+12πnηλ0−einληλn=Ceiλwηλ+Op1n,

since ηλt is stationary with finite variance for all d∈12,32 because then f<12. (Note that ηλt=ελt when d=1). Next write

(3.18)ηt=1−Lfεt=DnL;f+RnL;fεt

with

RnL;f=∑k=n+1∞−fkk!Lk,

and note that

εnt:=RnL;fεt=Op1n12+f.

Applying (3.3) to the dft wηλ calculated from (3.18) we have

(3.19)wηλ=wελDneiλ;f+12πnε˜λ0(f)−einλε˜λn(f)+wnελ,

with

(3.20)ε˜λn(f)=∑p=0n−1f˜λpe−ipλεn−p,  f˜λp=∑k=p+1n−fkk!eikλ,

and

wnελ=12πn∑t=1nεnteiλt.

Now wnελ=Opn−f because

(3.21)Ewnελwnελ*=12πn∑t=1n∑s=1nei2πnt−sEεntεns=12πn∑t=1n∑s=1nOn−1−2f=On−2f.

Using (3.19) and (3.21) in (3.17) we get

(3.22)wzλ=CeiλDneiλ;fwελ+12πnε˜λ0(f)−einλε˜λn(f)+Op1n+Op1nf.

Then, combining (3.22) with the unit root decomposition (3.11) leads to the representation

(3.23)wxλ1−eiλ=CeiλDneiλ;fwελ−eiλXn2πn+12πnCeiλε˜λ0(f)−einλε˜λn(f)+Op1nf

This representation holds uniformly over λ and is likely to be most useful when λ=λs=2πsn→0 and s→∞.

3.9. Remark

The representations (3.8), (3.11), and (3.12) hold for all fundamental frequencies λs=2πsn. They are helpful in providing asymptotic representations of wxλs. In such expansions, it is useful to allow for situations where s→∞ as well as n→∞. In some cases, as in spectral density estimation at some frequency ϕ≠0, we want the expansion rate of s to be the same as n, so that we can accommodate λs→ϕ as n→∞. In other cases, as in log periodogram and Gaussian semiparametric regression, interest centers on frequencies λs in the vicinity of the origin, so then we consider cases where s is fixed or s→∞ and sn→0 as n→∞.. The following section gives results that are helpful in the determination of the asymptotic form of these representations as n→∞ under these various conditions.

4. Asymptotic Approximations

4.1. Component Approximations

We start with the sinusoidal polynomial Dneiλ;d=∑k=0n−dkk!eikλ that appears in the decomposition (3.1) and Theorems 3.2 and 3.7. The series can be summed in terms of hypergeometric functions and the asymptotic form taken as n→∞ depends on λ. The behavior is described in the following lemma.

4.2. Lemma3

Suppose d > 0 and is noninteger. Then

(4.1)Dneiλ;d=1−eiλd−ei(n+1)λ−dn+1n+1! 2F1n+1−d,1;n+2;eiλ,

and, for cos(λ)<12,

(4.2)Dneiλ;d=1−eiλd+ei(n+1)λeiλ−1−dn+1n+1! 2F11+d,1;n+2;eiλeiλ−1.

The following asymptotic representations hold:

  • (a)

    For fixed λ≠0

    Dneiλ;d=1−eiλd−1Γ−dn1+deinλ1−eiλ1+O1n.

  • (b)

    For λ=λs=2πisn→0 and s→∞ as n→∞

    Dneiλs;d=1−eiλsd+12πi1Γ−dnds1+O1s+O1n1+d.

  • (c)

    For λ=λs=2πisn→0 and s fixed as n→∞

    Dneiλs;d=1Γ1−dnd 1F11,1−d;−2πis+O1n1+d.

  • (d)

    For λ = 0

    Dn1;d=1Γ1−d1nd1+O1n.

In the above formulae,  1F1a,b;z and  2F1a,b,c;z denote the confluent hypergeometric function and the hypergeometric function, respectively.

From part (d), it follows that Dn1;d differs from 0 by a term of On−d. From part (c), the same also applies to Dneiλs;d when s is fixed and λs=2πisn→0. Of course, in the event that d is a positive integer, we have the following terminating polynomials

Dn1;d=∑k=0n−dkk!=∑k=0d−dkk!=∑k=0ddk−1k=1−1d=0,

and

Dneiλs;d=∑k=0n−dkeiλskk!=∑k=0ddk−eiλsk=1−eiλsd

in this case.

Our next focus of interest is the correction term in (3.8) that involves X˜λn(d). We are especially interested in deriving an asymptotic approximation to X˜λn(d) at the fundamental frequencies λs. As in Lemma 3.1, the asymptotic behavior of X˜λs,n(d) is sensitive to the value of s in λs=2πsn. In particular, when d∈12,1, the asymptotic form of X˜λs,n(d) differs, depending on whether s is fixed or whether s→∞ as n→∞. In the latter case, n−12X˜λs,n(d)=op1, while in the former n−12X˜λs,n(d)=Op1. On the other hand, when d=1 n−12X˜λs,n(d)=Op1 for all s≠0. The results are given in the following theorem.

4.3. Theorem

Suppose d∈12,1. Then

  • (a)

    For fixed λ≠0 as n→∞,

    X˜λ,n(d)n=−eiλ1−eiλ1−dXnn+op1n1−d=Op1n1−d.

  • (b)

    For λ=λs=2πsn→0 and snα→∞, as n→∞, for some α∈12,1

    X˜λs,n(d)n=−eiλs1−eiλs1−dXnn+op1s1−d=−eiλs−2πis1−dXnnd−12+op1s1−d=Op1s1−d.

  • (c)

    For λ=λs=2πsn and s fixed, as n→∞,

    X˜λsn(d)n= 1F11,1−d;−2πisΓ1−d∫01e2πisrXn,d(r)dr−1Γ1−d∫01 1F11,1−d;−2πisrr−dXn,d(1−r)dr+Op1n1−d=Op1,

    where Xn,d(r)=Xnrnd−12.

  • (d)

    When d=1, the equation

    X˜λ,n(1)n=−eiλXnn=Op1

    holds for λ fixed, or λ=λs=2πsn→0 with s→∞ or λs=2πsn→0 with s fixed.

In parts (a) and (b) of Theorem 4.3 the leading term in the asymptotic approximation of n−12X˜λ,n(d) is the same and so, although the error order of magnitude differs, we may write

X˜λ,n(d)n=−eiλ1−eiλ1−dXnn+opeiλ1−eiλ1−dXnn,

for both these cases. Further, the leading term of n−12X˜λ,n(d) is Op1n1−d for fixed λ≠0, is Op1s1−d for λs=2πsn→0 and snα→∞, and is Op(1) for λs=2πsn→0 with s fixed. Thus, the correction term n−12X˜λ,n(d) is nonnegligible in a region around the origin when d∈12,1. The asymptotic form of n−12X˜λ,n(d) in that case (i.e., case (c), with λs=2πsn, and s fixed) is more complicated than the other cases and it involves hypergeometric series. The representation given in case (c) actually includes s=0, for which we have the simpler form

(4.3)X˜λ0n(d)n=1Γ1−d∫01Xn,d(r)dr−1Γ1−d∫01r−dXn,d(1−r)dr+Op1n1−d.

When d=1, the formula given in (d) is exact, as follows directly from (3.9).

Finally, we look at the correction term U˜λnf that appears in (3.12). We concentrate on the interesting case where λ is in the vicinity of the origin and give the result corresponding to part (c) of Theorem 4.3.

4.4. Theorem

Suppose d∈12,32 and f=1−d. Then, for λ=λs=2πsn and s fixed, as n→∞

(4.4)U˜λsn(f)2πn=12π1Γ1−fnf 1F11,1−f;−2πis∫01e−2πisrdXn1−r−∫01r−f 1F11,1−f;−2πisrdXn1−r+Op1n,

where Xn(r)=n−12∑t=0nrut. When f=0, U˜λsn(0)=0.

4.5. Approximations for wx(λ)

Evaluating (3.8) at λs, we have

wxλs=Dneiλs;d−1wuλs+12πnX˜λsn(d).

We use Lemma 4.2 and Theorem 4.3 to obtain explicit expressions for wxλs in terms of wuλs and a correction term. When d=1, the following exact form comes directly from (3.9)

(4.5)wxλs=1−eiλs−1wuλs−eiλs1−eiλsXn2πn,

and holds for all s=1,2,…. When d∈12,1, it is convenient to separate the following three cases:

  • (a)

    Case λs→ϕ≠0

Here, from Lemma 4.2 we have

Dneiλs;d=1−eiλsd−1Γ−dn1+deinλs1−eiλs1+O1n=1−eiλsd+O1n1+d,

uniformly for λs∈Bϕ=ϕ−πM,ϕ+πM where M→∞ as n→∞. Similarly, from Theorem 4.3,

X˜λs,n(d)n=−eiλs1−eiλs1−dXnn+op1n1−d

uniformly for λs∈Bϕ. It follows that

(4.6)wxλs=1−eiλs−dwuλs−eiλs1−eiλsXn2πn+op1n1−d,

uniformly for λs∈Bϕ.

  • (b)

    Case λs=2πisn→0 and s→∞

From Lemma 4.2 (b) when s→∞ as n→∞

Dneiλs;d=1−eiλsd+1Γ−dnd12πis1+O1s+O1n1+d.

And from Theorem 4.3 (b) with snα→∞ for some α∈12,1, as n→∞,

X˜λs,n(d)n=−eiλs1−eiλs1−dXnn+op1s1−d.

It follows that if sn+nαs→0 as n→∞, for some α∈12,1, then

(4.7)wxλs=1−eiλs−dwuλs−eiλs1−eiλsXn2πn+op1−eiλs−ds1−d.

Observe that the first two terms of (4.6) and (4.7) are the same. Although the order of magnitude of the error differs in the two cases, we may write

(4.8)wxλs=1−eiλs−dwuλs−eiλs1−eiλsXn2πn+opeiλs1−eiλsXnn

for both these cases, and (4.8) is valid for all λs=2πsn with nαs→0.

  • (c)

    Case λs=2πisn→0 and s fixed

From Lemma 4.2 (c) when s is fixed as n→∞, we have

(4.9)Dneiλs;d=1Γ1−dnd 1F11,1−d;−2πis+O1n1+d,

and it follows that

1ndwxλs=1nd1Γ1−dnd 1F11,1−d;−2πis+O1n1+d−1×wuλs+12πnX˜λsn(d),

giving

(4.10)wxλsnd=Γ1−d 1F11,1−d;−2πiswuλs+12πnX˜λsn(d)+Op1n.

Further, from Theorem 4.3 (c),

X˜λsn(d)n= 1F11,1−d;−2πisΓ1−d∫01e2πisrXn,d(r)dr−1Γ1−d∫01 1F11,1−d;−2πisrr−dXn,d(1−r)dr+Op1n1−d,

so that

(4.11)1ndwxλs=Γ1−d 1F11,1−d;−2πiswuλs+12π∫01e2πisrXn,d(r)dr−2π−12 1F11,1−d;−2πis∫01 1F11,1−d;−2πisrr−dXn,d(1−r)dr+Op1n1−d.

Unlike (4.6) and (4.8), the term

(4.12)eiλs1−eiλsXn2πn

does not figure directly in (4.11). In fact, as the alternate representation in the next section shows, the term (4.12) is absorbed into the series expression in (4.11), so it is still present and figures in the leading term of the dft wxλs when s is fixed.

  • (c)

    Case λs=2πisn→0 and s fixed: alternate form.

Theorem 3.7 gives

(4.13)wxλs1−eiλs=Dneiλs;fwuλs−12πnU˜λsnf−eiλsXn2πn,

with f=1−d, Lemma 4.2 (c) gives

Dneiλs;f=1Γ1−fnf 1F11,1−f;−2πis+O1n1+f,

and Theorem 4.4 gives

U˜λsn(f)2πn=12π1Γ1−fnf 1F11,1−f;−2πis∫01e−2πisrdXn1−r−∫01r−f 1F11,1−f;−2πisrdXn1−r+Op1n.

Also,

wuλs=12πn∑t=1ne2πsitnut=12π∑k=1ne2πsin−knun−kn=12π∫01e−2πisrdXn1−r+Op1n.

Combining these last three representations in (4.13), we get

wxλs1−eiλs=1Γ1−fnf 1F11,1−f;−2πis12π∫01e−2πisrdXn1−r+Op1n−12π1Γ1−fnf 1F11,1−f;−2πis∫01e−2πisrdXn1−r+12π1Γ1−fnf∫01r−f 1F11,1−f;−2πisrdXn1−r−eiλsXn2πn+Op1n=12π1Γ1−fnf∫01r−f 1F11,1−f;−2πisrdXn1−r−eiλsXn2πn+Op1n,

leading to

(4.14)1ndwxλs=12π1Γ1−fn1−eiλs∫01r−f 1F11,1−f;−2πisrdXn1−r−12πeiλsn1−eiλsXnnd−12+Op1nd−12,

which shows how (4.12) continues to play a role in the leading term of wxλs.

4.6. Limit Theory

Under (2.4), partial sums of ut satisfy the functional law

(4.15)Xn(r)=1n∑t=0nrut→dB(r),

where B is Brownian motion with variance ω2=σ2C(1)2, for example, Phillips and Solo (1992). There is a corresponding functional law for suitably standardized elements of the time series Xt.. Akonom and Gouriéroux (1987) showed such a functional law for n12−dXt when the components ut follow a stationary ARMA process and the following simply extends their result to the linear process ut.

4.7. Lemma

For ut satisfying (2.4) and with εt iid 0,σ2 and Eεtp<∞ for p>max1d−12,2,,

(4.16)Xn,d(r)=Xnrnd−12→dBd−1(r)=1Γd∫0rr−sd−1dB(s),

a fractional Brownian motion where B(s) is Brownian motion with variance ω2.

Like Xt, the fractional Brownian motion Bd−1(r) is initialized at the origin, and therefore has nonstationary increments, in contrast to the other fractional process

(4.17)WH(r)=1CH∫−∞∞r−s+H−12−−s+H−12dB(s), H=d−12,CH=12H+∫0∞1+sH−12−sH−122ds12, 0<H<1

introduced by Mandelbrot and Van Ness (1968) and studied by Samorodnitsky and Taqqu (2017) in this form. Both processes reduce to Brownian motion for special cases of the parameters, viz., d = 1 for (4.16), and H=12 for (4.17).

These functional laws enable us to get limit representations of the correction term n−12X˜λsn(d). The case where s is fixed as n→∞ is especially interesting, the other two cases following immediately from (4.16) and the respective expressions (4.6) and (4.7).

4.8. Lemma

For λs=2πisn→0 and s fixed

(4.18)X˜λsn(d)n→d1Γ1−d∫01e2πisrBd−1(r)dr1F11,1−d;−2πis−∫01e2πisrdBr.

The next result gives formulae for the stochastic Fourier integral ∫0re2πsiqdB(q) that appears in (4.18) and (when s = 0) for the constituent Brownian motion B in terms of the fractional Brownian motion Bd−1.

4.9. Theorem

For fixed integer s

(4.19)∫0re−2πsi(r−q)dB(q)=1Γ1−d∫0r 1F11,1−d;−2πis(r−q)r−q−dBd−1(q)dq,

and, in the special case where s=0,

(4.20)Br=1Γ1−d∫0rr−q−dBd−1(q)dq.

The equality (4.20) is the inverse (integral) transform of the fractional Brownian motion Bd−1(r). In effect, the right side of (4.20) is the (1−d)’th fractional integral of the d−1’th fractional derivative of Brownian motion. Formula (4.19) extends this representation to the case s≠0. When r=1, (4.19) becomes

∫01e2πsiqdB(q)=1Γ1−d∫01 1F11,1−d;−2πis(1−q)1−q−dBd−1(q)dq.

4.10. Theorem

Suppose d∈12,1. The following limit results apply.

  • (a)

    Let ϕ>0 and suppose λsj∈Bϕ=ϕ−π2M,ϕ+π2M for a finite set of distinct integers sj (j=1,…,J). When M→∞ as n→∞, the family {wx(λsj)}j=1J are asymptotically independently distributed as complex normal Nc0,fxϕ, where fxϕ=1−eiϕ−2dfuϕ.

  • (b)

    Let {sj}j=1J be distinct integers with 0<l<sj<L for each j and with Ln+nαl→0 as n→∞, for some α∈12,1. The family {(λsj)dwx(λsj)}j=1J are asymptotically independently distributed as Nc0,fu0.

  • (c)

    Let {sj}j=1J be a finite set of distinct positive integers which are fixed as n→∞. Then, for each j

    (4.21)1ndwxλsj→d12π∫01e2πisjrBd−1rdr,

    where Bd−1 is the fractional Brownian motion given in (4.16). Joint convergence also applies.

    When d=1, the following limits apply.

  • (d)

    Let ϕ>0 and suppose λsj∈Bϕ=ϕ−πM,ϕ+πM for a finite set of distinct integers sj (j=1,…,J). When M→∞ as n→∞, the family {wx(λsj)}j=1J are asymptotically distributed as

    (4.22)11−eiϕξj−eiϕ1−eiϕηj=1J,

    where the {ξj}j=1J are iid Nc0,fuϕ and are independent of

    (4.23)η=B(1)2π,

    where B is Brownian motion with variance ω2.

  • (e)

    Let {sj}j=1J be a finite set of distinct positive integers for which sjn→0 as n→∞. The family {λsjwx(λsj)}j=1J are asymptotically distributed as

    (4.24)iξj−η,

    where ξj and η are as in (4.22) and (4.23).

  • (f)

    When sj is fixed as n→∞, the ξj in (e) have the representation

    (4.25)ξj=12π∫01e2πisjrdBr,

    and

    (4.26)1nwxλsj→d12π∫01e2πisjrBrdr,

    which also holds for sj=0.

Parts (a) and (d) show that Hannan (1973) result for the limit theory of dfts of stationary processes extends to fractional processes at frequencies removed from the origin when d∈12,1 but not when d=1. In the latter case, the leakage from the zero frequency is so substantial that it affects the limit theory of the dft at all frequencies, although the limit distribution is still normal. Moreover, as is apparent from the form of (4.22), the limit variates are spatially correlated across frequency by virtue of the presence of the random component η, through which the leakage is transmitted.

Part (b) shows that, when d∈12,1, a version of Hannan’s result applies to the scaled transforms sjndwxλsj in a (distant) vicinity of the origin where λsj=2πsjin→0 but nαsj→0 as n→∞, for some α∈12,1. However, when d=1, the scaled transforms sjnwxλsj are asymptotically dependent across frequency.

Part (c) shows that in the immediate vicinity of the origin (i.e., for λsj=2πsjin→0 with sj fixed), the n−dwx(λsj) are asymptotically dependent for d∈12,1 and each converges weakly to an integral functional of fractional Brownian motion that involves the integer sj. In earlier work, Akonom and Gouriéroux (1987) gave (4.21) in the case of ARMA ut. An alternate expression for (4.21), which relates to (4.14) is

1ndwxλsj→d12πΓ1+d∫01 1F11,1+d;−2πisjrrddB1−r

and can be obtained from the formula

∫01e2πisrBd−1rdr=1Γ1+d∫01 1F11,1+d;−2πisrrddB1−r,

which is proved in Lemma E in the Technical Appendix and Proofs.

The methods in the proof of Theorem 4.10 are used in (Phillips, 2007, Theorem 3.2) to extend existing theory showing the asymptotic independence of a finite collection of dfts of stationary time series (Hannan, 1973) to collections of a small (i.e., with less than sample size) infinity of dfts at Fourier frequencies.

5. Statistical Applications

5.1. Spectrum Estimation for Fractional Processes

The limit theory in Section 4.6 is useful in obtaining the asymptotic behavior of spectral estimates for fractional processes. We give some results for smoothed periodogram estimates for frequencies at the origin and away from the origin. The former are of interest in procedures that are used to estimate the memory parameter d. The latter reveal any leakage from low to high frequencies that occurs in spectrum estimation.

For frequencies away from the origin such as ϕ≠0, the usual smoothed periodogram estimator of fxϕ is given by

(5.1)f^xxϕ=1m∑λs∈Bϕwxλswxλs*,

where Bmϕ=ϕ−π2M,ϕ+π2M and M is the bandwidth parameter that determines the number of frequencies m=#λs∈Bmϕ=[n/2M] used in the smoothing. At the zero frequency ϕ=0, we consider a one-sided average of m periodogram ordinates at the origin

(5.2)f^xx0=1m∑s=0m−1wxλswxλs*.

The following theorem gives the asymptotic behavior of f^xxϕ for these two cases and for d∈(12,1) and d=1.

5.2. Theorem

  • (a)

    For ϕ≠0 and 12<d<1

    f^xxϕ→p fxϕ=fuϕ1−eiϕ2d.

  • (b)

    For ϕ≠0 and d = 1

    f^xxϕ→d fxϕ+12π1−eiϕ−2B12.

  • (c)

    For 12<d<1 and m such that mnα→∞ with α≥12d

    mn2df^xx0→d 12π∫01Bd−1r2dr.

  • (d)

    For d = 1 and m such that mn→∞

    mn2f^xx0→d 12π∫01Br2dr.

According to part (a), spectral estimates like f^xxϕ at frequencies removed from the origin are consistent for fxϕ=1−eiϕ−2dfuϕ provided d<1. When d=1, the estimate is inconsistent and converges weakly to a random quantity. In this case, the leakage from low frequency behavior is strong enough to persist in the limit at all frequencies ϕ>0. Part (d) was given in earlier work by Phillips (1991), where it was shown to be useful in analyzing regression in the frequency domain with integrated time series. A new and simpler derivation is given here based on the decomposition (3.9). Part (c) can be expected to be useful in similar regression contexts with fractional processes.

5.3. Semiparametric Estimation of d

We indicate some potential applications of the above theory for the estimation of the memory parameter d in (2.1). This is a large subject which goes beyond the scope of this chapter and for which theoretical development was undertaken after the original version of this chapter was completed in 1999. The main references will be reported in the following discussion. The presentation here focuses on the new ideas that led into these developments and not the technical details.

Concordant with the nonparametric approach, our concern is with the case where little is known about the short memory component ut of (2.1) and its spectrum fu(λ) is treated nonparametrically. In both log periodogram estimation and local Whittle estimation, this is accomplished by working with the dft wxλs of the data Xt over a set of m Fourier frequencies λs=2πsn:s=1,…,m that shrink slowly to origin as the sample size n→∞ by virtue of a condition on m of the type mn→0. It has been suggested that, in view of the asymptotic correlation of the ordinates in the vicinity of the origin (Künsch, 1986), it may be useful to trim this set of frequencies away from the origin and restrict attention to λs=2πsn:s=l,…,m where l is a trimming number that satisfies l→∞ and mlogml→0 (Robinson, 1995b), although it is now known that this trimming is not necessary (Hurvich et al., 1998).

From (4.7) we know that for d∈12,1 the dft wxλs

(5.3)wxλs=1−eiλs−dwuλs−eiλs1−eiλsXn2πn+op1−eiλs−ds1−d,

when sn+nαs→0 as n→∞, for some α∈12,1. The asymptotic behavior of wxλs is dominated by the first two terms of (5.3), and as d→1 the importance of the second term in (5.3), which is Op(nd/s), rivals that of the first term, which is Op(nd/sd). Apparently, therefore, it would seem desirable to correct the dft wxλs for the effects of leakage in semiparametric estimation of d simply by adding the correction term supplied by the known form of the expansion (5.3). For log periodogram regression this amounts to using the quantity

(5.4)vxλs=wxλs+eiλs1−eiλsXn2πn

in place of wxλs in the regression. Thus, in place of the usual least squares regression (over s=1,…,m)

lnIxλs=c^−d^ln1−eiλs2+error

that is inspired by the form of the moment relation (2.6) in the frequency domain, the argument above suggests the linear least squares regression

(5.5)lnIvλs=c˜−d˜ln1−eiλs2+error,

in which the periodogram ordinates, Ixλs, are replaced by Ivλs=vxλsvxλs*. We call this procedure modified log periodogram regression. This replacement is inspired by (5.3), which approximates the data generating process of the dft wxλs over the relevant set of frequencies as m→∞ in the regression. In place of the “regression model”:

lnIxλs=c−dln1−eiλs2+uλs,

with c=lnfu0 and

uλs=lnIxλs/fxλs+lnfuλs/fu0,

as in (2.7), we now have from (5.3)

Ivλs=1−eiλs−dwuλs+opnds2=1−eiλs−2dIuλs1+1−eiλsdwuλs−1opnds=1−eiλs−2dIuλs1+op1s1−d2,

which leads to the new regression model

(5.6)lnIvλs=c−dln1−eiλs2+aλs,

with

(5.7)aλs=lnIuλs/fuλs+lnfuλs/fu0+Op1s1−d.

This relationship holds for frequencies λs satisfying sn+nαs→0 as n→∞, in view of (5.3).

The new regression (5.5) seems likely to be most useful in cases where nonstationarity is suspected. Note, however, that when d<12, the correction term in (5.4) is op(1) when ns→0, so that use of (5.5) can also be expected to be satisfactory in the stationary case. When d=1, the correction is exact for all frequencies, as is clear from (3.9). In that case, therefore, (5.6) is an exact regression relation whose error is given by

(5.8)aλs=lnIuλs/fuλs+lnfuλs/fu0.

It is then a relatively straightforward matter to show that the modified log periodogram estimator has the following limit theory

(5.9)md˜−d→dN0,π224,

that is the same limit distribution as the log periodogram estimator in the stationary case (Hurvich et al., 1998; Robinson, 1995b). By contrast, the usual log periodogram estimator d^ has a mixed normal limit theory when d=1,, as shown in Phillips (1999b, 2007). The mixed normal limit arises in this case because of the presence of the term (2π)−12eiλsn−12Xn in (3.9) which is Op(1) and does not vanish as n→∞. Moreover, the usual log periodogram estimator d^ is inconsistent and converges in probability to unity when d∈(1,2) as shown in Kim and Phillips (2006), which makes use of some of the present methods.

The modified regression (5.5) appears to be even more useful in the nonstationary case when d>1. In that case, the usual estimator d^ is inconsistent, and d^→p1,, a fact that can be established using the expansions obtained in Sections 2 and 3, whereas d˜ is consistent and has the same limit distribution as that shown in (5.9). Further analysis of this modified log periodogram estimator, together with an empirical application to the Nelson-Plosser data (Nelson & Plosser, 1982), was given in Kim and Phillips (2003).

The intuition leading to the modified regression (5.5) can also be employed in the case of the local Whittle estimator Künsch (1987) and Robinson (1995a). We will not go into details here. Suffice to remark that we would simply replace Iv(λs;d) in the extremum estimation problem (5.16)–(5.18) given below by Iv(λs), which can be computed from vx(λs) as in (5.4). The resulting estimator is a modified local Whittle estimator, and, like the modified log periodogram regression estimator in (5.5), its asymptotic properties can be expected to be the same for stationary and nonstationary values of the memory parameter, including those for which d>1.

Our theory also suggests some other possibilities. In particular, we may build on the idea noted above that (5.6) gives an exact relationship when d = 1 with error (5.8). Indeed, the decomposition (3.8) implies the following exact relation between the transforms wxλs and wuλs

wxλs=Dneiλs;d−1wuλs+12πnX˜λn(d).

Define the new transform

(5.10)vxλs;d=wxλs−Dneiλs;d−112πnX˜λn(d),

which is dependent on the memory parameter d and for which the equation

(5.11)vxλs;d=Dneiλs;d−1wuλs

holds exactly. Extending the ideas that led to (5.6) above, we have the exact periodogram relation

(5.12)lnIvλs;d=c+lnDneiλs;d−2+aλs,

with Ivλs;d=vxλs;dvxλs;d*, and

aλs=lnIuλs/fuλs+lnfuλs/fu0,

just as in (5.8). In place of linear least squares regression, it is now possible to apply nonlinear regression directly to the regression model (5.12). Let Ysd=lnIvλs;d, and As=lnDneiλs;d−2. Then, nonlinear regression leads to the following extremum estimator

d#=argmin dQmd,

where

Qmd=1m∑s=1mYsd−Ysd¯−dAs−As¯Ysd−Ysd¯−dAs−As¯*,

and As¯=m−1∑s=1mAs, Ys(d)¯=m−1∑s=1mYs(d). The advantage of d# is that it is the natural estimator of d that emerges from the exact formulation of the regression model in the frequency domain, that is, (5.12). Its disadvantage is that it is more complicated to compute than the conventional log periodogram regression estimator d^ and the modified estimator d˜, neither of which require numerical methods. Some simplifications in computation can be obtained by using some of the approximations developed in Sections 3 and 4.

Finally, we remark that the exact relationship (5.11) can be used to obtain an exact form of local Whittle estimator under Gaussian assumptions about ut. The local Whittle likelihood suggested by Künsch (1987) and studied by Robinson (1995a) has the form

(5.13)KmG,d=1m∑s=1mlogGλs−2d+λs2dGIxλs,

and is minimized jointly with respect to the parameters (G,d), where G0=fu(0) is the true value of G. The (negative) Whittle likelihood (e.g., Hannan & Deistler, 2012, pp. 224–225) based on frequencies up to λm and up to scale multiplication is

(5.14)∑s=1mlogfuλs+∑s=1mIuλsfuλs.

The objective function (5.13) is derived from (5.14) by using the approximate relationship

wxλs∼1−eiλs−dwuλs∼−λs−dwuλs,

or

Iuλs∼λs2dIxλs,

to transform (5.14) to be data dependent, in conjunction with the local approximation fuλs∼G0. We may now proceed to transform (5.14) using the exact relationship between wuλs and wxλs that is given by (5.11) and (5.10). We get

12∑s=1mlogDneiλs;d−2fuλs+12∑s=1mDneiλs;d2Ivλs;dfuλs,

and this leads directly to the following “exact” version of the local Whittle likelihood

(5.15)LmG,d=1m∑s=1mlogDneiλs;d−2G+Dneiλs;d2GIvλs;d.

The new estimates are obtained from the joint minimization

G**,d**=argmin d,GLmG,d.

Concentrating out G, we find that d** satisfies

(5.16)d**=argmin dRmd,

with

(5.17)Rmd=logG**d−21m∑j=1mlogDneiλs;d,

where

(5.18)G**d=1m∑j=1mDneiλs;d2Ivλs;d.

The estimator d** would seem to offer an attractive semiparametric procedure because it is based on likelihood principles and involves the exact data generating mechanism for the dft. This procedure is more computationally intensive than the usual Whittle estimator but no impediment to practical use. A full analytic investigation of the exact local Whittle estimator was conducted and reported in Shimotsu and Phillips (2005) showing that the same asymptotic properties of the local Whittle estimator apply to the exact local Whittle estimator over a full range of stationary and nonstationary values of the memory parameter d. This approach enables consistent estimation of d and the construction of valid confidence intervals for d for both stationary and nonstationary long memory time series. The procedure has proved popular in empirical research. Further work on nonstationarity-extended Whittle estimation has been done by Abadir et al. (2007) and Shao (2010).

5.4. Final Remarks

Fractional processes conveniently embody in a single memory parameter d an index that measures the extent of long range dependence in an observed time series. When a nonparametric formulation is employed for the innovations that drive the observed process, a great deal of model generality is achieved. Integer values of d include integrated processes and the special value d=12 provides a simple boundary between stationary and nonstationary cases. This flexibility has enabled a fundamental extension of the simple ARIMA models popularized in the 1970s wherein variate differencing became a common method of dealing with nonstationarity. The flexibility of long memory also enriched the concept of cointegration by allowing for fractional possibilities in long run equilibrium errors, thereby narrowing the differential (between variables and errors) that distinguishes a cointegrating relationship among observable integrated time series. In view of this generality, semiparametric methods and frequency domain methods such as those used in the present work have been found to be very useful in estimation, inference, and asymptotic analysis of long memory systems.

In spite of the generality that long range dependence brings to empirical analysis, it is worth remembering that some important cases are not included in its orbit. Explosive and mildly explosive time series are prime examples that have particular relevance in economics and finance where exuberance and speculation are not uncommon in real estate and financial asset markets. A simple autoregressive time series with an explosive root is not rendered stationary by differencing or fractional differencing, just as differentiating an exponential function produces a derivative that simply reproduces the exponential. Parameterizations of nonstationarity using simple autoregressive coefficients and the tests that are so enabled by such formulations therefore offer possibilities that are not encompassed in the notion of long range dependence. While autoregressions and long memory systems provide a dual parametric source of unit root dynamics, these parameterizations deliver alternative departures from unit roots that help enrich our capacity to model different types of nonstationary time series behavior and evolution.

6. Technical Appendix and Proofs

6.1. Preliminary Results

We provide some technical lemmas that are useful throughout this chapter. Lemmas A and B provide results on binomial coefficients and hypergeometric functions that are either standard (e.g., Erdélyi, 1953) or follow from standard results. We give them here to facilitate our own derivations and to make this chapter more accessible. Lemmas C and D provide some more specific results on sinusoidal polynomials and hypergeometric functions of sinusoids that are immediately relevant to formulae in this chapter. Lemma E gives a useful inverse transform of fractional Brownian motion, an inverse transform for a hypergeometric series of fractional Brownian motion and some useful relationships between certain integral functionals of fractional Brownian motion and Brownian motion. Lemma F provides a new asymptotic expansion for hypergeometric series that allows for increasing coefficients as well as an argument that tends to unity. The expansion should be useful in other work with hypergeometric series.

6.1.1. Lemma A

  • (a)

    dk=−1k−dkk!.

  • (b)

    p+aj=j+apajap, aj+k=(a)j(a+j)k.

  • (c)

    ∑k=0n−dkk!=1−dnn!1d≠0,1,..+∑k=0d−dkk!1(d=0,1,..).

  • (d)

    Γn+αΓn+β=nα−β1+O1n.

Proof

Part (a) is immediate from the definition

dk=d!d−k!k!=dd−1 … d−k+1k!=−1k−d … −d+k−1k!=−1k−dkk!.

The second formula in Part (b) is immediate from the definition of the forward factorial. The first formula in Part (b) follows from

p+aj=Γp+a+jΓp+a=Γp+a+jΓj+aΓj+a/ΓaΓp+a/Γa=j+apajap.

For part (c), we write the sum as a terminating hypergeometric function, and use Lemma B (a) and (c) to obtain

∑k=0n−dkk!=−dnn! 2F1−n,1;d−n+1;1=−dnn!ΓdΓd−n+1Γd+1Γd−n=Γ−d+nΓ−dn!d−nd=Γ−d+n+1Γ−d+1n!=1−dnn!,

for d≠0,1,2,.., while for d=0,1,.. the sum ∑k=0n−dkk! simply terminates at k=d.

Part (d) is a standard result that follows from the Stirling approximation, for example, Erdélyi (1953, p. 47).

6.1.2. Lemma B

In the following formulae,  2F1(a,b,c;z)=∑k=0∞akbkk!ckzk is the hypergeometric function.

  • (a)

    ∑k=0n−dkk!zk=−dnn!zn 2F1−n,1;d−n+1;z−11d≠0,1,..+∑k=0d−dkzkk!1(d=0,1,..)

  • (b)

    ∑t=m+1∞−dtt!zt=zm+1−dm+1m+1! 2F1m+1−d,1;m+2;z.

  • (c)

     2F1(a,b,c;1)=ΓcΓc−a−b/Γc−aΓc−b for Rec−a−b>0 and c≠0,−1,−2,…..

  • (d)

    If |z|<1 and |z/(z−1)|<1

    (6.1) 2F1(a,b;c;z)=(1−z)−a 2F1(a,c−b;c;z/(z−1)),

    the right hand side giving an analytic continuation of the hypergeometric function to the half-plane Re(z)<12.

  • (e)

    ∑k=0n−dke−iλkk!=1−dne−iλnn! 2F1(−n,1;1−d;1−eiλ)1d≠0,1,..+∑k=0d−dke−iλkk!1(d=0,1,..)

  • (f)

    If Re(c)>Re(b)>0

    (6.2) 2F1(a,b;c;z)=ΓcΓbΓc−b∫01tb−11−tc−b−11−tz−adt,

    which gives an analytic continuation of  2F1(a,b;c;z) to the entire z plane cut along 1,∞, that is, to all z for which arg(1−z)<π.

Proof

Part (a) is given in Erdélyi (1953, pp. 87 and 101) in terms of binomial coefficients. Using the form given there and Lemma A (a), we have for d≠0,1,…

∑k=0n−dkk!zk=∑k=0ndk−zk=dn−zn 2F1−n,1;d−n+1;z−1=−dnn!zn 2F1−n,1;d−n+1;z−1.

When d=0,1,.. the sum simply terminates at k = d and the stated result follows.

For part (b) we have

(6.3)∑k=m+1∞−dkk!xk=xm+1∑k=0∞−dm+1+km+1+k!xk=xm+1∑k=0∞Γm+1+k−dΓ−dΓm+2+kxk=xm+1∑k=0∞m+1−dkm+2kΓm+1−dΓ−dΓm+2xk=xm+1Γm+1−dΓ−dΓm+2∑k=0∞m+1−dkm+2kk!xk=xm+1Γm+1−dΓ−dΓm+2 2F1m+1−d,1;m+2;x=xm+1−dm+1m+1! 2F1m+1−d,1;m+2;x.

The hypergeometric function  2F1(a,b,c;z)=∑k=0∞akbkk!ckzk is absolutely convergent for all z≤1 when Rea+b−c<0 (Erdélyi, 1953, p. 57). Hence, the series in (6.3) converges absolutely for all z≤1 when d>0.

Part (c) is a well-known summation formula (Erdélyi, 1953, p. 61). Part (d) is Euler’s formula (Erdélyi, 1953, pp. 64 & 105). The series for  2F1(a,b,c;z) converges absolutely for all |z|<1 and converges absolutely for |z|=1 when Rec−a−b>0 Erdélyi, 1953, p. 57). The series for  2F1(a,c−b;c;z/(z−1)) converges for |z/(z−1)|<1. Since the latter inequality holds for all z for which  Re(z)<12, it follows that the right side of (6.1) gives the analytic continuation of  2F1(a,b;c;z) to the half-plane Re(z)<12 (Erdélyi, 1953, p. 64).

Part (e) is obtained by direct calculation. Using (a), we proceed as follows for the case d≠0,1,…:

(6.4)(6.5)∑k=0n−dke−iλkk!=−dnn!e−iλn 2F1−n,1;d−n+1;eiλ=−dne−iλnn!∑j=0n−nj1j1+eiλ−1jj!d−n+1j=−dne−iλnn!∑j=0n−njd−n+1j∑q=0jjqeiλ−1q=−dne−iλnn!∑j=0n−njd−n+1j∑q=0jj!j−q!q!eiλ−1q=−dne−iλnn!∑q=0n1q!eiλ−1q∑j=qn−njj!d−n+1jj−q!=−dne−iλnn!∑q=0n1q!eiλ−1q∑s=0n−q−ns+qs+q!d−n+1s+qs!

Since −nq+s=−nq−n+qs, and d−n+1s+q=d−n+1qd−n+1+qs from Lemma A (b), (6.5) becomes

(6.6)−dne−iλnn!∑q=0n−nqd−n+1qeiλ−1q∑s=0n−qq−nsq+1sd−n+1+qss!=−dne−iλnn!∑q=0n−nqd−n+1qeiλ−1q 2F1q−n,q+1;q−n+d+1;1.

In this expression, the  2F1 series terminates, so Lemma B (c) holds and (6.6) sums to

−dne−iλnn!∑q=0n−nqd−n+1qeiλ−1qΓq−n+d+1Γd−qΓd+1Γd−n=−dne−iλnn!∑q=0n−nq1qq!eiλ−1qΓd−n+1Γd−qΓd+1Γ−n+d=d−nd−dne−iλnn!∑q=0n−nq1qq!eiλ−1qΓd−qΓd=1−dne−iλnn!∑q=0n−nq1qq!eiλ−1q1d−1d−2…(d−q)=1−dne−iλnn!∑q=0n−nq1qq!eiλ−1q−1q1−dq=1−dne−iλnn! 2F1(−n,1,1−d;1−eiλ),

giving the stated result for the case d≠0,1,…. The result for d=0,1,.. follows immediately because the series terminates at k=d. An alternative and more direct proof of the result makes use in (6.4) of the fact that

(6.7) 2F1−n,1;d−n+1;eiλ=(d−n)n(d−n+1)n 2F1−n,1;1−d;1−eiλ

employing the linear transformation formula  2F1−m,b;c;z=(c−b)m(c)m 2F1−m,b;b−c−m;1−z for terminating hypergeometric series – see Olver et al. (2010, Formula 15.8.7, p. 390).

Part (f) is a standard result (Erdélyi, 1953, p. 59).

6.1.3. Lemma C

Assume d≠0,1,... Then:

  • (a)

    For fixed λ≠0 as n→∞

    ∑k=n+1∞−dkk!eiλk=O1n1+d.

  • (b)

    For λs=2πsn→0 and s→∞ as n→∞

    ∑k=n+1∞−dkk!eiλsk=−12πi1Γ−dnds1+O1s+O1n1+d.

  • (c)

    For λs=2πsn→0 and s fixed as n→∞

    ∑k=n+1∞−dkk!eiλsk=O1nd.

Proof

Using Lemma B (b), Lemma A (d), and Lemma F (b), given below, we get

(6.8)∑k=n+1∞−dkk!eiλk=eiλn+1Γn+1−dΓ−dΓn+2 2F1n+1−d,1;n+2,eiλ=eiλ(n+1)1Γ−dn1+d1+O1n11−eiλ1+O1n=O1n1+d,

giving part (a). For λ=λs=2πsn→0 and s→∞ as n→∞ we have, using Lemma F (a),

(6.9)eiλsn+1Γn+1−dΓ−dΓn+2 2F1n+1−d,1;n+2;eiλs=eiλsΓ−dn1+d1+O1n 2F1n+1−d,1;n+2;eiλs=eiλsΓ−dn1+d11−eiλs∑j=0k−11+dj1jj!12πis1+Osnj+O12πis1+Osnk1+O1n=1Γ−dn1+deiλs1−eiλs1+O1s+O1n=1Γ−dn1+deiλs1−eiλs1+O1s+O1n1+d=−1Γ−dnd1+Osn2πsi1+O1s+O1n1+d=−12πi1Γ−dnds1+O1s+O1n1+d,

giving part (b). Finally, for s fixed as n→∞, we have

∑k=n+1∞−dkk!eiλsk=O∑k=n+1∞1k1+d=O1nd

giving part (c).

6.1.4. Lemma D

Assume d≠1,2,…, let r∈0,1 and let λs=2πsn→0 with s fixed as n→∞. Then:

(6.10) 2F1(−nr,1,1−d;1−eiλs)= 1F11,1−d;2πisr+On−1,
(6.11) 2F1(−nr,1,1−d;e−iλs−1)= 1F11,1−d;2πisr+On−1,

and for nonnegative integer p≤n

(6.12) 2F1(−p,1,1−d;1−e−iλs)= 1F11,1−d;−2πispn+Op−1.

Proof

The same argument gives both results (6.10) and (6.11). We prove (6.10).

(6.13) 2F1(−nr,1,1−d;1−eiλs)=∑j=0nr−nrj1−dj−2πisn+On−2j=∑j=0nr1j−nrj−nrj1−djj!2πisr+On−1j=∑j=0∞1j1−djj!2πisrj+On−1−∑j=nr+1∞1j1−djj!2πisrj= 1F11,1−d;2πisr+On−1,

because

∑j=N+1∞1jxj1−djj!=xN+1∑k=0∞xk1−dk+N+1=xN+1Γ1−d−1∑k=0∞xkΓk+2+N−d=xN+1Γ1−dΓN+2−d∑k=0∞xk1k2+N−dkk!=xN+1Γ1−deN+1−d2πN+2−dN+1−d∑k=0∞xk1k2+N−dkk!1+O1N=O1NN−δ,

for all δ>0 and all finite x. Line (6.13) above follows because, for 1≤j≤nr,

1−−nrj−nrj=1−11−1nr…1−j−1nr≤1−1−j−1nrj=Oj2nr,

and

(6.14)1n∑j=0nr1jj21−djj!2πisrj=O1n∑j=0nr3j1−djj!2πisrj=O1n 1F13,1−d,2πisr=O1n,

since the  1F1 function is everywhere convergent.

Next, for (6.12) we have

(6.15) 2F1−p,1,1−d;1−e−iλs=∑j=0p−pj1−dj2πisn+On−2j=1+−p1−d2πisn+On−2+−p−p+11−d22πisn+On−22+…+−pp1−dp2πisn+On−2p=1+−11−d2πisr+On−1+−1−1+Op−11−d22πispn+On−12+…+−1+Op−1p1−dp2πispn+On−1p=∑j=0p1j1+Ojpj1−djj!−2πispn+On−1j=∑j=0∞1j1−djj!−2πispnj+Op−1++On−1−∑j=p+1∞1j1−djj!−2πispnj= 1F11,1−d;−2πispn+On−1+Op−1,

giving (6.12). Again, line (6.15) above follows because

1p∑j=0p1jj21−djj!−2πispnj=O1p∑j=0p3j1−djj!−2πispnj=O1p.

6.1.5. Lemma E

  • (a)

    For j=1,2,…

    Γj+1−d−1∫0rr−sj−dBd−1sds=Γj−1∫q=0rr−qj−1B(q)dq

    and for j=0,1,2,…

    Γj+1−d−1∫0rr−sj−dBd−1sds=Γj+1−1∫q=0rr−qjdB(q).

  • (b)

    Γ1−d−1∫0r 1F11,1−d;−2πisr−qr−q−dBd−1qdq=∫q=0re−2πisr−qdBq.

  • (c)

    1Γ1+d∫01 1F11,1+d;−2πisrrddB1−r=1Γ1−f−2πsi∫01r−f 1F11,1−f;−2πisrdB1−r+1−2πsiBd−11.

  • (d)

    1Γ1+d∫01 1F11,1+d;−2πisrrddB1−r=∫01e2πisrBd−1rdr.

In the above formulae, B is Brownian motion with variance ω2 and Bd−1(r)=1Γd∫0rr−sd−1dB(s) is a fractional Brownian motion initialized at the origin, as in Lemma 3.4.

Proof

To prove part (a) we use an operator approach with D=ddx and allow for fractional powers of D with a Weyl integral interpretation (see Lovoie et al., 1976; Phillips et al., 1986a) for the approach used here). The operator eqD is treated at the translation operator, so that eqDf(x)=f(x+q). Setting Bd−1s=0 for all s≤0 we have

(6.16)1Γj+1−d∫0rr−sj−dBd−1sds=1Γj+1−d∫q=0∞qj−dBd−1r−qdq=1Γj+1−d∫q=0∞qj−de−qDBd−1rdq=Dd−j−1Bd−1x|x=r=Dd−j−1D1−dBx|x=r=D−jBx|x=r=Γj−1∫q=0rqj−1Br−qdq=Γj−1∫q=0rr−qj−1Bqdq,

giving the first of the stated results and, consequently,

∫0rr−sj−dBd−1sds=Γ1−d1−djΓj∫q=0rr−qj−1Bqdq.

To obtain the second form of the result we use integration by parts to give

(6.17)Γj−1∫q=0rr−qj−1Bqdq=j−1Γj−1∫q=0rr−qjdB(q)=Γj+1−1∫q=0rr−qjdB(q).

Combining (6.16) and (6.17), we have

1Γj+1−d∫0rr−sj−dBd−1sds=Γj+1−1∫q=0rr−qjdB(q)

which holds also when j=0 giving the inverse relation

(6.18)1Γ1−d∫0rr−s−dBd−1sds=B(r),

(see Theorem 4.9). An alternate weak convergence proof of (6.18) is given in the proof of Theorem 4.9 below and, from this result, (6.17) can alternatively be obtained by subsequent integration.

To prove part (b) we proceed as follows:

1Γ1−d∫0r 1F11,1−d;−2πisr−qr−q−dBd−1qdq=1Γ1−d∑j=0∞1j−1j1−djj!∫0r2πisr−qjr−q−dBd−1qdq=1Γ1−d∑j=0∞−2πisj1−dj∫0rr−qj−dBd−1qdq=∑j=0∞−2πisj1−dj1−djΓj∫q=0rr−qj−1Bqdq=∑j=0∞−2πisjΓj∫q=0rr−qj−1Bqdq=∑j=0∞−2πisjj!∫q=0rr−qjdBq=∫q=0re−2πisr−qdBq,

using (6.17) in the penultimate line. This proves part (b).

To prove part (c), we expand the  1F1 function on the right side of the formula and use

Bd−1(1)=1Γd∫011−sd−1dB(s)=−1Γd∫01rd−1dB(1−r),

to get

1Γ1−f−2πsi∫01r−f 1F11,1−f;−2πisrdB1−r+1−2πsiBd−11=1Γ1−f−2πsi∑j=0∞1j−2πsijj!1−fj∫01rj−fdB1−r−1−2πsi1Γ1−f∫01r−fdB(1−r)=1Γ1−f−2πsi∑j=1∞1j−2πsijj!1−fj∫01rj−fdB1−r=∑j=1∞−2πsij−1Γj+1−f∫01rj−fdB1−r=∑k=0∞−2πsikΓk+1+d∫01rk+ddB1−r=1Γ1+d∑k=0∞1k−2πsikk!1+dk∫01rk+ddB1−r=1Γ1+d∫01 1F11,1+d;−2πisrrddB1−r,

giving the stated result.

To prove part (d) we use the exponential expansion for e2πisr in the integral on the right side, giving

(6.19)∫01e2πisrBd−1rdr=∫01e2πis(1−r)Bd−11−rdr=∫01e−2πisrBd−11−rdr=∑j=0∞−2πsijj!∫01rjBd−11−rdr=∑j=0∞−2πsijj!∫011−rjBd−1rdr.

From part (a) we have

Γj+1−d−1∫0rr−sj−dBd−1sds=Γj+1−1∫q=0rr−qjdB(q),

and setting k=j−d and r = 1 gives the formula

Γk+1−1∫011−skBd−1sds=Γk+d+1−1∫q=011−qk+ddB(q),

or

(6.20)Γk+1−1∫01skBd−11−sds=Γk+d+1−1∫q=01qk+ddB(1−q).

Using (6.20) in (6.19) we get

∫01e2πisrBd−1rdr=∑j=0∞−2πsijj!∫01rjBd−11−rdr=∑j=0∞−2πsijj!Γj+1Γj+d+1∫01qj+ddB(1−q)=∑j=0∞−2πsijj!1jΓj+d+1∫01qj+ddB(1−q)=1Γd+1∫01∑j=0∞−2πsiqjj!1j1+djqddB(1−q)=1Γd+1∫01 1F11,1+d;−2πisqqddB(1−q),

giving the stated result.

6.1.6. Lemma F

Let α and β be constants for which Re(β),Re(β−α)>0. The following asymptotic expansions to some given order k hold

  • (a)

    Let λs=2πsn. If sn→0 as n→∞ and s→∞, then

     2F1α,n−β;n;eiλs=(1−eiλs)−α∑j=0k−1(α)j(β)jj!12πis1+Osnj+O12πis1+Osnk=(1−eiλs)−α∑j=0k−1(α)j(β)jj!12πis1+Osnj+O1sk.

  • (b)

    Let λ≠0 be fixed as n→∞. Then

     2F1α,n−β;n;eiλ=(1−eiλ)−α∑j=0k−1(α)j(β)jj!1neiλeiλ−11+O1nj+O1nk.

  • (c)

    Let λs=2πsn. If sn+nsp→0 as n,s,p→∞, then

     2F1α,p−β;p;eiλs=(1−eiλs)−α∑j=0k−1(α)j(β)jj!n2πisp1+Osnj+On2πisp1+Osnk.

Proof

Since Re(β−α)>0 the series for  2F1α,n−β;n;eiλs converges absolutely for all λs. Using (6.1) from Lemma B (d), we write

(6.21) 2F1α,n−β;n;eiλs=(1−eiλs)−α 2F1α,β;n;eiλseiλs−1,

where the right side has a convergent series representation for suitable λs, viz., when |eiλs/(eiλs−1)|<1, or cos(λs)<12. Although the domain of convergence of the series on the right side series is restricted, the right hand side has a valid asymptotic expansion for large n that applies to all λs as we shall now show.

First observe that as n,s→∞ with sn→0, the complex quantity

(6.22)Zns=eiλseiλs−1=n2πis1+Osn=n2πis1+o1

lies inside the plane cut along 1,∞, that is, |arg(1−Zns)|<π. Hence, we may use the analytic continuation of the right hand side of (6.21) based on the following integral representation (Erdélyi, 1953, p. 59; Lemma B(f)):

(6.23) 2F1β,α;n;Zns=ΓnΓαΓn−α∫01tα−11−tn−α−11−tZns−βdt.

An asymptotic series that is valid even for |Zns|>1 for large n may now be obtained using a method due to MacRobert (see Erdélyi, 1953, p. 76) as follows. Expand the last binomial factor in (6.23) in MacLaurin’s expansion up to k terms with remainder as

1−tZns−β=∑j=0k−1(β)jj!tZnsj+(β)kk!tZnsk∫01k1−qk−11−qtZns−β−kdq.

Now scale this expansion by ΓnΓαΓn−αtα−11−tn−α−1 and integrate term by term, using the formula

ΓnΓαΓn−α∫01tα+j−11−tn−α−1dt=ΓnΓαΓn−αΓα+jΓn−αΓn+j=(α)j(n)j.

This leads to

(6.24) 2F1β,α;n;Zns=∑j=0k−1(α)j(β)j(n)jj!Znsj+Rkn=∑j=0k−1(α)j(β)j(n)jj!n2πis1+Osnj+Rkn=∑j=0k−1(α)j(β)jj!12πis1+Osn1+On−1j+Rkn=∑j=0k−1(α)j(β)jj!12πis1+Osnj+Rkn,

where

Rkn=(β)kk!Bα,n−α∫01tα−11−tn−α−1tZnsk∫01k1−qk−11−qtZns−β−kdqdt=k(α)k(β)kn2πis1+Osnk(n)kk!Bα+k,n−α×∫01tα+k−11−tn−α−1∫011−qk−11−qtn2πis1+Osn−β−kdqdt,

since the beta function factors as follows

1Bα,n−α=ΓnΓαΓn−α=Γα+kΓnΓαΓn+kΓn+kΓα+kΓn−α=(α)k(n)kBα+k,n−α.

In view of (6.22) there exists a constant c > 0 for which  Im(Zns)≥c. Then, for any given β and k, there exists an M, independent of n and s, such that

supt,q∈0,11−qtZns−β−k<M.

Then,

Rkn≤Mk(α)k(β)kn2πis1+Osnk(n)kk!Bα+k,n−α∫01tα+k−11−tn−α−1∫011−qk−1dq=Mk(α)k(β)kn2πis1+Osnk(n)kk!Bα+k,n−αBα+k,n−αBk,1=Mk(α)k(β)kn2πis1+Osnk(n)kk!ΓkΓk+1=M(α)k(β)k12πis1+Osn1+O1nkk!=M(α)k(β)kk!12πis1+Osnk,

so that Rkn has the same order of magnitude as the first neglected term in the expansion (6.24). Thus, (6.24) is a valid asymptotic expansion of the form

 2F1β,α;n;n2πis1+Osn=∑j=0k−1(α)j(β)jj!12πis1+Osnj+O12πis1+Osnk,

giving the required result for part (a). Part (b) follows in an identical manner using

Z=eiλeiλ−1

in place of Zns.

To prove part (c) we proceed as in the proof of part (a), setting Zns=eiλseiλs−1 as in (6.22). Then

 2F1β,α;p;Zns=∑j=0k−1(α)j(β)j(p)jj!Znsj+Rknp=∑j=0k−1(α)j(β)j(p)jj!n2πis1+Osnj+Rknp=∑j=0k−1(α)j(β)jj!n2πisp1+Osn1+Op−1j+Rknp=∑j=0k−1(α)j(β)jj!n2πisp1+Osnj+Rknp,

since psn→∞. The remainder is

Rknp=(β)kk!Bα,p−α∫01tα−11−tp−α−1tZnsk∫01k1−qk−11−qtZns−β−kdqdt=k(α)k(β)kn2πis1+Osnk(p)kk!Bα+k,p−α×∫01tα+k−11−tp−α−1∫011−qk−11−qtn2πis1+Osn−β−kdqdt.

As in the case of Rkn, we have

Rknp≤M(α)k(β)kk!n2πisp1+Osn1+Op−1k=M(α)k(β)kk!n2πisp1+Osnk,

again since psn→∞. Thus, Rknp has the same order as the first neglected term in the series and we get the asymptotic expansion

 2F1β,α;p;eiλseiλs−1=∑j=0k−1(α)j(β)jj!n2πisp1+Osnj+On2πisp1+Osnk,

which leads to the stated result.

6.2. Proofs of Main Lemmas and Theorems

6.2.1. Proof of Lemma 3.1

See Phillips and Solo (1992, formula (32)).

6.2.2. Proof of Theorem 3.2

From (3.2) we have the following alternate form for the model (2.1) for all t≤n

(6.25)ut=1−LdXt=DnL;dXt=Dneiλ;dXt+D˜nλe−iλL;de−iλL−1Xt.

Observe that

(6.26)D˜nλe−iλL;de−iλL−1Xt=e−iλL−1X˜λt=e−iλX˜λt−1(d)−X˜λt(d),

where X˜λt(d)=D˜nλe−iλL;dXt=∑p=0n−1d˜λpe−ipλXt−p. Since the right side of (6.26) is a telescoping Fourier sum, taking dfts of (6.26) leaves us with 12πnX˜λ0(d)−einλX˜λn(d). It follows that when we take dfts of expression (6.25) we have

(6.27)Dneiλ;dwxλs+12πnX˜λ0(d)−einλX˜λn(d)=wuλ,

giving the required formula (3.3).

6.2.3. Proof of Theorem 3.7

Equation (3.11) follows immediately from the definition 1−LXt=zt and (3.9). Equation (3.12) follows by applying (3.8) to zt=1−L1−dut.

6.2.4. Proof of Lemma 4.2

Using the hypergeometric series representation from Lemma B (b), and the asymptotic expansion in Lemma A (d), we have for d > 0

(6.28)Dneiλ;d=∑k=0n−dkk!eikλ=∑k=0∞−∑k=n+1∞−dkk!eikλ=1−eiλd−ei(n+1)λΓn+1−dΓ−dn+1! 2F1n+1−d,1;n+2;eiλ=1−eiλd−ei(n+1)λΓ−dn1+d1+O1n 2F1n+1−d,1;n+2;eiλ,

giving (4.1). Formula (4.2) follows immediately from Lemma B (d), noting that |eiλ/(eiλ−1)|<1 when 2cos(λ)<1.

Next, using Lemma F (b), we have for fixed λ≠0,

(6.29) 2F1n+1−d,1;n+2;eiλ=(1−eiλ)−11+O1n.

It follows from (6.28) and (6.29) that as n→∞ and for fixed λ≠0

Dneiλ;d=1−eiλd−1Γ−dn1+dei(n+1)λ1−eiλ1+O1n,

giving part (a).

When λs=2πisn→0 as n→∞ and s→∞,, we proceed as follows. Using Lemma F (a) in the hypergeometric factor in the second term of (6.28), we have

(6.30) 2F1n+1−d,1;n+2;eiλs=11−eiλs∑j=0k−11+dj1jj!12πis1+Osnj+O12πis1+Osnk.

Then, as in the argument leading to (6.9), the second term of (6.28) admits the following valid asymptotic expansion for λ=λs→0 as n→∞ and s→∞:

(6.31)eiλsΓ−dn1+d1+O1n 2F1n+1−d,1;n+2;eiλs=−12πi1Γ−dnds1+O1s+O1n1+d,

and so from (6.28) and (6.31) we get

Dneiλs;d=1−eiλsd+12πi1Γ−dnds1+O1s+O1n1+d,

giving part (b). The result can also be shown directly by noting from Lemma C (b) that

Dneiλs;d=1−eiλsd−∑k=n+1∞(−d)kk!eikλs=1−eiλsd+12πi1Γ(−d)nds1+O1s+O1n1+d.

For part (c), we start by using the following summation formula from Lemma B (e)

∑k=0n−dkeiλskk!=1−dneiλsnn! 2F1(−n,1,1−d;1−e−iλs).

Since s is fixed, we have from Lemma D (6.12) with p = n

 2F1(−n,1,1−d;1−e−iλs)= 1F11,1−d;−2πis+On−1.

It follows that

(6.32)∑k=0n−dkeiλskk!=1−dneiλsnn! 1F11,1−d;−2πis+On−1=1−dnn! 1F11,1−d;−2πis+O1n1+d,

and, then, for fixed s as n→∞, we have

(6.33)Dneiλs;d=∑k=0n−dkeiλskk!=1Γ1−dnd 1F11,1−d;−2πis+O1n1+d,

as required for part (c).

Part (d) follows as a special case of formula (6.33) with s=0. We also get the result directly from Lemma A (c), viz.,

Dn1;d=∑k=0n−1−dkk!=1−dn−1n−1!=1Γ1−d1nd1+O1n.

It follows that Dn1;d differs from zero by a term of On−d.

6.2.5. Proof of Theorem 4.3

Parts (a) and (b). We write X˜λn(d) as the sum of two components, the first involving L + 1 components, with 1<L<n and where the choice of L will be discussed below. We have:

X˜λn(d)=D˜nλe−iλL;dXn=∑p=0n−1d˜λpe−ipλXn−p=∑p=0n−1∑k=p+1n−dkk!eikλe−ipλXn−p=∑p=0L∑k=p+1n−dkk!eikλe−ipλXn−p+∑p=L+1n−1∑k=p+1n−dkk!eikλe−ipλXn−p.

Then

(6.34)X˜λn(d)n=1n1−d∑p=0L∑k=p+1n−dkk!eikλe−ipλXn−pnd−12+1n1−d∑p=L+1n−1∑k=p+1n−dkk!eikλe−ipλXn−pnd−12.

Next, look at the sinusoidal sum ∑k=p+1n−dkk!eikλ that appears in (6.34). We use the truncated binomial series formula from Lemma B (b) in this sum, giving

(6.35)∑k=p+1n−dkk!eiλk=∑k=p+1∞−dkk!eiλk−∑k=n+1∞−dkk!eiλk=eiλp+1−dp+1p+1! 2F11+p−d,1;p+2,eiλ−eiλn+1−dn+1n+1! 2F1n+1−d,1;n+2,eiλ.

For large n and fixed λ≠0, we have, using Lemma C (a),

(6.36)∑k=n+1∞−dkk!eiλk=O1n1+d,

while for λ=λs=2πisn→0 and s→∞ as n→∞ we have from Lemma C (b)

(6.37)∑k=n+1∞−dkk!eiλsk=−1Γ−dnd12πis1+O1s+O1n1+d.

So, neglecting the second term of (6.35) in view of (6.37), we get

(6.38)∑t=p+1n−dtt!eiλst=eiλsp+1Γp+1−dΓ−dp+1! 2F11+p−d,1;p+2,eiλs+O1nds

for all s→∞, as n→∞. Finally, for s fixed as n→∞, we have from Lemma C (c)

∑k=n+1∞−dkk!eiλsk=O1nd,

so that (6.38) also holds with s fixed.

Using (6.38), we deduce that

(6.39)1n1−d∑p=0L∑k=p+1n−dkk!eikλse−ipλs=eiλs1n1−d∑p=0L−dp+1p+1! 2F11+p−d,1;p+2,eiλs+OLns=eiλs1n1−d∑p=0∞−dp+1p+1! 2F11+p−d,1;p+2,eiλs−eiλs1n1−d∑p=L+1∞−dp+1p+1! 2F11+p−d,1;p+2,eiλs+OLns.

Now

(6.40)∑p=0∞−dp+1p+1! 2F11+p−d,1;p+2,eiλ=∑p=0∞−dp+1p+1!∑k=0∞1+p−dk1kk!p+2keiλk=∑k=0∞∑p=0∞−dp+1p+1!1+p−dkp+2keiλk=∑k=0∞∑p=0∞−dp+1p+1!1−d+kp2p1−dpk+2p1−dk2keiλk=∑k=0∞∑p=0∞−dp+1p+1!1−d+kp2p1−dpk+2p1−dk2keiλk.

Next, since 2p=(p+1)! and

−dp+1=Γ1−d+pΓ(−d)=−dΓ1−d+pΓ(1−d)=−d1−dp

we have

(6.41)∑p=0∞−dp+1p+1!1−d+kp2p1−dpk+2p=−d∑p=0∞1−d+kpk+2p=−d∑p=0∞1−d+kp1pk+2pp!=−d 2F1k+1−d,1;k+2;1=−dΓk+2ΓdΓk+1Γ1+d=−k+1,

where the explicit representation in the last line follows by the summation formula of Lemma B (c). Using (6.41) in (6.40) we get

(6.42)∑k=0∞∑p=0∞−dp+1p+1!1−d+kp2p1−dpk+2p1−dk2keiλk=−∑k=0∞k+11−dk2keiλk=−∑k=0∞1−dkk!eiλk=−11−eiλ1−d.

Thus,

(6.43)∑p=0∞−dp+1p+1! 2F11+p−d,1;p+2,eiλs=∑k=0∞∑p=0∞−dp+1p+1!1−d+kp2p1−dpk+2p1−dk2keiλsk=−11−eiλs1−d.

Next, using Lemma F (c) we find that for sn+nLs→0 (which holds under the conditions on s and L that are given below),

(6.44)∑p=L+1∞−dp+1p+1! 2F11+p−d,1;p+2,eiλs=O∑p=L+1∞−dp+1p+1!11−eiλs1+Onsp=O11−eiλs∑p=L+1∞1p1+d1+Op−11+Onsp=O1Ld11−eiλs.

It follows from (6.39), (6.43), and (6.44) that

(6.45)1n1−d∑p=0L∑k=p+1n−dkk!eikλe−ipλs=−1n1−deiλs1−eiλs1−d+OLns+1n1−d∑p=L+1∞−dp+1p+1! 2F11+p−d,1;p+2,eiλs=−1n1−deiλs1−eiλs1−d+OLns+OndLd1s.

The first term in (6.45) is O1s1−d and dominates the second term. The first term also dominates the third term when nLs→0, which will be the case when snα→∞, as n→∞, for some α∈0,1 and L=n1−α and when d<1. (Note that for s fixed the last term of (6.45) does matter, and this distinguishes the s fixed case, which will be considered below in the proof of part (c).) Hence, when n→∞, λs→0, and snα→∞ (with L chosen as L=n1−α), we have

(6.46)1n1−d∑p=0L∑k=p+1n−dkk!eikλse−ipλsXn−pnd−12=1n1−d∑p=0L∑k=p+1n−dkk!eikλse−ipλsXnnd−12+op1
(6.47)=−1n1−deiλs1−eiλs1−dXnnd−12+op1s1−d=−eiλs1−eiλs1−dXnn+op1s1−d.

Line (6.46) above is justified by a separate argument, which we now develop. We use the fact, from Lemma 4.7, that n12−dXn−p=Op(1) and p≤L=n1−α. We proceed as follows. Select K=n1−η→∞ with 0<η<α (we will place a further condition on η below). Then, LK+Kn→0 and we may write (for large n)

(6.48)Xn−pnd−12=1nd−12∑j=0n−pdjj!un−p−j=1nd−12∑j=K+1n−pdjj!un−p−j+1nd−12∑j=0Kdjj!un−p−j=∑j=K+1n−p1jn1−dun−p−jn1+Op1K+Knd−121Kd−12∑j=0Kdjj!un−p−j=∑j=K+1n−p1jn1−dun−p−jn+Op1K+OpKnd−12=∑j=K+1n−p1j+pn1−dun−p−jnj+pj1−d+Op1K+OpKnd−12=∑k=K+p+1n1kn1−dun−knkk−p1−d+Op1K+OpKnd−12=∑k=K+p+1n1kn1−dun−kn−1+Opkd−1+∑k=1K+p1kK+p1−dun−kK+p=+Oppnd−12∑k=K+p+1n1k2−dun−k+Op1K+OpKnd−12.

Observe that for any δ>0, ∑k=1∞1k1+δun−k converges almost surely since ∑k=1∞1k1+δE|un−k|<∞. Then,

E∑k=K+p+1n1k2−dun−k≤∑k=K+p+1n1k2−dEun−k≤∑k=K+p+1∞1k2−dEun−k≤1K1−d−δ∑k=K+p+1∞1k1+δEun−k=o1K1−d+δ,

and so

∑k=K+p+1n1k2−dun−k=op1K1−d−δ.

It follows that

pnd−12∑k=K+p+1n1k2−dun−k=oppnd−121n(1−η)(1−d−δ)=opLn12−η(1−d−δ)−δ=opnnα−η(1−d−δ)−δ

uniformly for p≤L. For K=n1−η and with η satisfying

0<η<minα,α−12−δ1−d−δ,

and choosing δ such that 0<δ<α−12, we have

(6.49)pnd−12∑k=K+p+1n1k2−dun−k=op1,

uniformly for p≤L.

Using (6.49), we find that (6.48) can be written as

Xn−pnd−12=1nd−12∑k=0ndkk!un−k+op1+OpK+pnd−12+op1+Op1K+OpKnd−12=Xnnd−12+OpKnd−12+Op1K+op(1)=Xnnd−12+op(1),

uniformly for p≤L=n1−α with α>12, thereby establishing (6.46).

When n→∞ with fixed λ≠0, we have, in view of the use of (6.36) rather than (6.37) in the above arguments, the same expression but with an opn−(1−d) error. Specifically,

(6.50)1n1−d∑p=0L∑k=p+1n−dkk!eikλe−ipλsXn−pnd−12=−1n1−deiλ1−eiλ1−dXnnd−12+op1n1−d+O1nndLd11−eiλ+O1n1−d1nd=−1n1−deiλ1−eiλ1−dXnnd−12+op1n1−d.

In both cases the dominant approximation is given by the first term and we can write

1n1−d∑p=0L∑k=p+1n−dkk!eikλe−ipλXn−pnd−12=−eiλ1−eiλ1−dXnn+opeiλ1−eiλ1−dXnn.

It remains to show that we may neglect the second term of (6.34). Using Lemma C (b), Lemma 4.7, (6.38), and Lemma F (c), we have, when n→∞, λs→0, snα→∞, and L=n1−α

(6.51)1n1−d∑p=L+1n−1∑k=p+1n−dkk!eikλse−ipλsXn−pnd−12=1n1−d∑p=L+1n−1eiλs(p+1)−dp+1p+1! 2F11+p−d,1;p+2,eiλs+O1ndse−ipλsXn−pnd−12=eiλsn1−d∑p=L+1n−1−dp+1p+1! 2F11+p−d,1;p+2,eiλsXn−pnd−12+Op1s=Oeiλs1−eiλs1n1−d∑p=L+1n−1−dp+1p+1!1+OnspXn−pnd−12+Op1s=OpndLds+Op1s,

which is op1s1−d since nLs→0.

For the case of fixed λ≠0 and with L=n1−α we get

(6.52)1n1−d∑p=L+1n−1∑k=p+1n−dkk!eikλse−ipλsXn−pnd−12=Op1n1−d∑p=L+1n−11p1+dXn−pnd−12=Op1n1−dLd=Op1n1−αd=op1n1−d

In both cases (6.51) and (6.52), the order is smaller than the leading term of (6.47) and (6.50), respectively. Hence, for both fixed λ≠0 and λs→0 and snα→∞ as n→∞, we have

X˜λn(d)n=1n1−d∑p=0n∑k=p+1n−dkk!eikλe−ipλsXn−pnd−12=−eiλ1−eiλ1−dXnn+opeiλ1−eiλ1−dXnn,

giving the required results.

Part (c). Our interest is in

X˜λsn(d)n=1n1−d∑p=0n−1∑k=p+1n−dkk!eikλse−ipλsXn−pnd−12.

From Lemma B (e) we have

(6.53)∑k=0m−dkeiλskk!=1−dmeiλsmm! 2F1(−m,1,1−d;1−e−iλs).

Since s is fixed, 1−e−iλs=2πisn+On−2 and using Lemma D and (6.53) we get

(6.54)∑k=0n−dkeiλskk!=1−dnn! 1F11,1−d;−2πis+O1n1+d.

Using (6.53) with m = p and Lemma D again we obtain

(6.55)∑k=0p−dkeiλskk!=1−dpeiλspp! 2F1(−p,1,1−d;1−e−iλs)=1−dpeiλspp! 1F11,1−d;−2πispn+O1p1+d.

Now n12−dXn−p=Op1, uniformly in p≤n, so that

1n1−d∑p=0n∑k=p+1n−dkk!eikλse−ipλsXn−pnd−12=1n1−d∑p=0n∑k=p+1n−dkk!eikλse−ipλsOp1.

Using (6.54) and (6.55) and noting that ∑p=0np−1−d=O1, we have

1n1−d∑p=0n∑k=p+1n−dkk!eikλse−ipλsXn−pnd−12=1n1−d∑p=0n1−dn 1F11,1−d;−2πisn!−1−dpeiλsp 1F11,1−d;−2πispnp!e−ipλsXn−pnd−12+Op1n1−d.

Next observe that, since s is fixed as n→∞,

1n1−d1−dnn!∑p=0ne−ipλsXn−pnd−12=1Γ1−d1n∑p=0ne−ipλsXn−pnd−12+Op1n=1Γ1−d1n∑p=0ne−2πispnXn−pnd−12+Op1n=1Γ1−d∫01e−2πisrXn,d(1−r)dr+Op1n=1Γ1−d∫01e2πisrXn,d(r)dr+Op1n.

Further,

1n1−d∑p=0n1−dp 1F11,1−d;−2πispnp!Xn−pnd−12=1Γ1−d1n1−d∑p=1n 1F11,1−d;−2πispnpdXn−pnd−12+Op1n1−d=1Γ1−d1n∑p=1n 1F11,1−d;−2πispnpndXn−pnd−12+Op1n1−d=1Γ1−d∫01 1F11,1−d;−2πisrr−dXn,d1−rdr+Op1n1−d.

We deduce that

1n1−d∑p=0n∑k=p+1n−dkk!eikλse−ipλsXn−pnd−12=1n1−d∑p=0n1−dn 1F11,1−d;−2πisn!−1−dpeiλsp 1F11,1−d;−2πisrp!e−ipλsXn−pnd−12+Op1n1−d=1Γ1−d∫01e−2πisr1F11,1−d;−2πisXn,d(1−r)dr−∫01 1F11,1−d;−2πisrr−dXn,d(1−r)dr+Op1n1−d= 1F11,1−d;−2πisΓ1−d∫01e2πisrXn,d(r)dr−1Γ1−d∫01 1F11,1−d;−2πisrr−dXn,d(1−r)dr+Op1n1−d,

giving the stated result.

Part (d). When d = 1 the series expression for n−12X˜λn(d) terminates because −dk=0 for all k>1, so that only the term involving p = 0 is retained. We then have

X˜λn(1)n=−eiλXnn,

which holds for all λ.

6.2.6. Proof of Theorem 4.4

By definition, zt=1−L1−dut=1−Lfut , and from Theorem 3.7 we have

 wxλ1−eiλ=wzλ−eiλXn2πn=Dneiλ;fwuλ−eiλn2πnU˜λnf−eiλXn2πn,

where

U˜λnf=D˜nλe−iλL;fun=∑p=0n−1f˜λpe−ipλun−p, and f˜λp=∑k=p+1n−fkk!eikλ.

Now, as in Lemma B (e), we have

U˜λsn(f)2πn=12πn∑p=0n−1f˜λspe−ipλsun−p=12π∑p=0n−1∑k=p+1n−fkk!eikλse−ipλsun−pn=12π∑p=0n1−fneiλsnn! 2F1−n,1,1−f;1−e−iλs−1−fpeiλspp! 2F1(−p,1,1−f;1−e−iλs)e−ipλsun−pn.

As in the proof of Theorem 4.3 and using the fact that ∑p=1np−1−fun−p=Op1 as n→∞, we proceed as follows

U˜λsn(f)2πn=12π∑p=0n1−fn 1F11,1−f;−2πisn!−1−fpeiλsp 1F11,1−f;−2πispnp!+O1p1+fe−ipλsun−pn=12π∑p=0n1−fn 1F11,1−f;−2πisn!−1−fpeiλsp 1F11,1−f;−2πispnp!e−ipλsun−pn+Op1n=12π1−fn 1F11,1−f;−2πisn!∑p=0ne−ipλsun−pn−12π∑p=0n1−fp 1F11,1−f;−2πispnp!un−pn+Op1n=12π1−fn 1F11,1−f;−2πisn!∫01e−2πisrdXn1−r−12π1Γ1−f∑p=0n1pf1+O1p 1F11,1−f;−2πispnun−pn+Op1n=12π1−fn 1F11,1−f;−2πisn!∫01e−2πisrdXn1−r−12π1Γ1−fnf∑p=0n1pnf 1F11,1−f;−2πispnun−pn+Op1n=12π1−fn 1F11,1−f;−2πisn!∫01e−2πisrdXn1−r−12π1Γ1−fnf∫01r−f 1F11,1−f;−2πisrdXn1−r+Op1n=12π1Γ1−fnf 1F11,1−f;−2πis∫01e−2πisrdXn1−r−∫01r−f 1F11,1−f;−2πisrdXn1−r+Op1n.

So we have

U˜λsn(f)2πn=12π1Γ1−fnf 1F11,1−f;−2πis∫01e−2πisrdXn1−r−∫01r−f 1F11,1−f;−2πisrdXn1−r+Op1n,

as required. Note that when f=0, we get

 1F11,1;−2πis=e−2πsi=1,  1F11,1;−2πisr=e−2πisr,

and U˜λsn(0)=0.

6.2.7. Proof of Lemma 4.7

Akonom and Gouriéroux (1987) prove the result when ut follows a stationary and invertible ARMA process. Using the device in Phillips and Solo (1992), we write

,ut=CLεt=C1εt+ε˜t−1−ε˜t

where ε˜t=C˜Lεt=∑j=0∞c˜jεt−j and c˜j=∑k=j+1∞ck. Under (2.4), ε˜t is stationary with mean zero and finite variance σ2∑j=0∞c˜j2. Then

Xt=1−L−dut=C11−L−dεt−1−L1−dε˜t.

Now for 12<d≤1, ξt=1−L1−dε˜t is stationary with mean zero and finite variance, so that n12−dξnr→p0.. On the other hand, Xtε=1−L−dεt is a fractional process constructed from iid0,σ2 innovations with E|εt|p<∞, and so from Akonom and Gouriéroux (1987)

Xn,dε(r)=1nd−12Xnrε→dσΓd∫0rr−sd−1dW(s).

It follows that

Xn,d(r)=1nd−12Xnr→dBd−1(r)=σC1Γd∫0rr−sd−1dW(s)=1Γd∫0rr−sd−1dB(s),

as stated.

6.2.8. Proof of Lemma 4.8

By Theorem 4.3 (c), Lemma 4.7 and the continuous mapping theorem we have

(6.56)X˜λn(d)n= 1F11,1−d;−2πisΓ1−d∫01e2πisrXn,d(r)dr−1Γ1−d∫01 1F11,1−d;−2πisrr−dXn,d(1−r)dr+Op1n1−d→d 1F11,1−d;−2πisΓ1−d∫01e−2πisrBd−1(1−r)dr−1Γ1−d∫01 1F11,1−d;−2πisrr−dBd−1(1−r)dr.

In the above, we can replace Xn,d(r) by a continuous polygonal version up to an op(1) error uniformly over r∈[0,1]. The continuous mapping theorem then applies since the mapping f↦∫01 r−df(1−r)dr is continuous when d < 1 for all continuous functions f, and since the confluent hypergeometric function  1F1a,c;x is an entire function of x.

Now observe from Lemma E that

Γ1−d−1∫01 1F11,1−d;−2πis1−q1−q−dBd−1qdq=∫q=01e−2πis1−qdBq.

It follows that (6.56) is

(6.57) 1F11,1−d;−2πisΓ1−d∫01e−2πisrB