Elsevier

Journal of Econometrics

Volume 130, Issue 2, February 2006, Pages 209-233
Journal of Econometrics

Local Whittle estimation of fractional integration and some of its variants

https://doi.org/10.1016/j.jeconom.2004.09.014Get rights and content

Abstract

Asymptotic properties of the local Whittle estimator are studied under a model of fractional integration that provides a uniform representation of the data for any value of d. For d(-12,12), the estimator is shown to be consistent and asymptotically normally distributed. For d[-1,-12), the estimator converges either to the true parameter value or to zero, depending on the number of frequencies used in estimation. This behavior manifests itself as a positive bias in finite samples and the inconsistency affects the asymptotic properties of the ‘differencing/adding-one-back’ estimator that has been suggested for use. The relationship between several variants of the LW estimator are discussed. It is shown that the LW estimator, the ‘differencing/adding-one-back’ estimator and the modified LW estimator (Cowles Foundation Discussion Paper no. 1265, Yale University, 2000) can all be viewed as approximations to an exact LW estimator (Cowles Foundation Discussion Paper no. 1367, Yale University, 2003) that uses an exact expression for the model in frequency domain form. The approximations are valid over specific domains of d, like the stationary region (-12,12) and the nonstationary region (12,32), whereas the exact LW estimator is consistent and asymptotically N(0,14) for all values of d.

Introduction

Fractional integration and the study of so-called I(d) processes has recently attracted a good deal of attention amongst theorists and empirical researchers. In applied econometric work, I(d) processes with d>0 have been found to provide good empirical models for certain financial time series and volatility measures, as well as certain macroeconomic time series like inflation and interest rates. Fractional processes accommodate temporal dependence in a time series that is intermediate in form between short-memory series (the so-called I(0) processes) and unit root time series (I(1) processes). Fractional models encompass both stationary and nonstationary processes depending on the value of the memory parameter and include both I(0) and I(1) processes as limiting cases when the memory parameter takes on the values zero and unity. For these reasons, fractional integration is attractive to empirical researchers, providing some liberation from the classical dichotomy of I(0) and I(1) processes. Growing evidence in applied work indicates that fractionally integrated processes can describe certain long range characteristics of economic data rather well, including the volatility of financial asset returns, forward exchange market premia, interest rate differentials, and inflation rates.1

The memory parameter, d, plays a central role in the definition of fractional integration and is often the focus of empirical interest. When -12<d<12, the process has a stationary representation. Under stationarity, two commonly used semiparametric estimators (log periodogram (LP) regression, local Whittle (LW) estimator) are known to be consistent and asymptotically normally distributed (Robinson, 1995a, Robinson, 1995b). However, these estimators are also known to exhibit nonstandard behavior when d>12. Although they are consistent for d(12,1] and asymptotically normally distributed for d(12,34), the LW estimator has a nonnormal limit distribution for d[34,1], and for d>1 they both converge to unity in probability and are inconsistent (Kim and Phillips, 1999, Phillips, 1999b, Phillips and Shimotsu, 2004; Velasco, 1999a, Velasco, 1999b). To avoid inconsistency and the unreliable basis for inference when d may be larger than 12, a simple procedure is to estimate d by taking first differences of the data, estimating d-1, and adding back one to the estimate. This ‘differencing + add-one-back’ procedure has been used in practical work for estimating the value of d in the economic data. However, as many economists and econometricians have argued, economic time series may be trend stationary, i.e. I(d) with d[0,12) around a linear time trend. If this is true, then taking a first difference of a time series reduces it to I(d) with d[-1,-12). Little is known about the properties of these estimators in this case (and hence this ‘difference + add-one-back’ procedure), apart from some Monte Carlo simulation results that suggest the possibility of biased estimation (Hurvich and Ray, 1995) for this range of d values. As we show here, however, the procedure may lead to inconsistent estimation of d.

Fractional integration is often defined implicitly in terms of a process that becomes stationary or weakly dependent after the application of the filter (1-L)d. To write down a fractionally integrated process explicitly as a function of weakly dependent innovations, there appear to be two main approaches. One approach (“type I”) seeks to invert the filter (1-L)d, expand it, and express the resulting process as an infinite order moving average of the innovations. This approach corresponds to the model used in Robinson, 1995a, Robinson, 1995b and gives a stationary process when d<12. Although stationarity of the resulting process facilitates the analysis substantially, when d>12 this approach breaks down because the infinite order moving average does not converge. In that case, the approach is modified by modeling the process as the partial sum of the component I(d-1) (or I(d-2)) process and expanding (1-L)1-d or (1-L)2-d in terms of innovations (e.g., Velasco, 1999a, Velasco, 1999b). This approach has the advantage that the integer differenced process is stationary, but the disadvantage that it introduces a discontinuity in the data generating mechanism at the fractional points d=12,32,. Moreover, in practical work with economic data there are no inherent restrictions on the value of d, and this modeling approach seems awkward and may be problematic, especially because the value of d is unknown before estimation and the degree of integer differencing to achieve stationarity must simply be guessed.

To circumvent these problems, an alternate (“type II”) model of fractional integration that is designed to cover a wider range of d has been used by some authors, including Robinson (1994), Phillips (1999a), Tanaka (1999) and Robinson and Marinucci (2001). This alternate model initializes the process at t=0 by setting the past innovations to zero and expresses the data as a finite order moving average. This formulation gives a valid representation of the data for all values of d and provides an attractive alternative for modeling data in a uniform way, irrespective of the order of integration, although it is only asymptotically stationary if d<12. It has been used in earlier work (Kim and Phillips, 1999, Phillips and Shimotsu, 2004) to examine the statistical properties of the LP and LW semiparametric estimators under nonstationarity (d>12). However, the statistical properties of these semiparametric estimators under this alternate model of fractional integration with d<12 are not yet established.

The present paper has three main objectives. First, it shows that the LW estimator is consistent and asymptotically normally distributed for d(-12,12) under the alternate model of fractional integration. Therefore, the LW estimator is indeed a valid estimator for both models of fractional integration. Second, it shows that, for d[-1,-12), this estimator converges either to d0, the true parameter value, or to 0 depending on the number of frequencies used in estimation. So the estimator may or may not be consistent when the data is strongly antipersistent or overdifferenced prior to estimation. In finite samples the inconsistency manifests itself as a positive bias whose magnitude depends on the value of d and the number of frequencies included in the objective function. These results complement Phillips and Shimotsu (2004), which used the same model of fractional integration as the present paper and studied the asymptotic properties of the LW estimator for d>1/2, largely completing the study of the statistical properties of the LW estimator under the alternate model of fractional integration. Third, the paper discusses the relationship between the LW estimator, a modified LW estimator (Phillips, 1999a, Shimotsu and Phillips, 2000), and the exact LW estimator (Shimotsu and Phillips, 2003). The modified LW estimator was proposed as a possible solution in the nonstationary environment (d>12), and the exact LW estimator is based on an exact frequency domain transformation of fractional integration and has been shown to be consistent and asymptotically normally distributed for any value of d. It turns out that both the LW estimator and the modified LW estimator can be viewed as approximants of the exact LW estimator for certain values of d. The former provides a good approximation for d(-12,12), while the latter provides a good approximation for d(12,32).

The technical approach taken in the present paper draws on an exact representation and approximation theory for the discrete Fourier transform (dft) of fractionally integrated processes. This theory, developed in Phillips (1999a) and Phillips and Shimotsu (2004), provides a uniform apparatus for analyzing the asymptotic behavior of the discrete Fourier transform of fractionally integrated time series and provides the key element in developing the theory in the present paper.

The remainder of the paper is organized as follows. Section 2 introduces the model used in this paper. Consistency of the LW estimator for d(-12,12) and its probability limit for d[-1,-12) are demonstrated in Section 3. Section 4 demonstrates asymptotic normality. Section 5 reports some simulation results. The relationship between the three estimators mentioned above is discussed in Section 6. Section 7 discusses the alternative representations of I(d) processes. Proofs are collected together in Appendix A.

Section snippets

Preliminaries: a model of fractional integration

We consider the fractional process Xt generated by the model(1-L)d(Xt-X0)=utI{t1},t=0,±1,,where X0 is a random variable with a certain fixed distribution and ut is stationary with zero mean and spectral density fu(λ). Expanding the binomial in (1) gives the formk=0t(-d)kk!(Xt-k-X0)=utI{t1},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) terminates, giving the

LW estimation: consistency and inconsistency for -1d<1/2

The LW estimator was originally developed by Künsch (1987) and Robinson (1995b) under the assumption that Xt is stationary and its spectral density behaves like Gλ-2d as λ0+. It starts with the frequency domain Gaussian likelihood function localized to the vicinity of the origin:Qm(G,d)=1mj=1mlog(Gλj-2d)+λj2dGIx(λj),where m is some integer less than n controlling the number of frequencies included in the local likelihood. The LW procedure estimates G and d by minimizing Qm(G,d), so that (G^,d^

LW estimation: asymptotic distribution

We introduce some further assumptions that are used in the results in this section.

Assumption 1

For some β(0,2], fu(λ)=G0(1+O(λβ));G0(0,)asλ0+.

Assumption 2

In a neighborhood (0,δ) of the origin, C(eiλ) is differentiable and ddλC(eiλ)=O(λ-1)asλ0+.

Assumption 3

Assumption 3 holds and alsoE(εt3|Ft-1)=μ3,E(εt4|Ft-1)=μ4a.s.,t=0,±1,,for finite constants μ3 and μ4.

Assumption 4

As n, 1m+m1+2β(logm)2n2β0.

Assumption 5

Uniformly in k=0,1, jkγj=O((log(k+1))-4),jkcj=O((log(k+1))-4),where γjEutut+j and cj is defined in Assumption 3.

Assumptions 14 are

Simulations

This section reports some simulations that were conducted to examine the finite sample performance of the LW estimator. We generate I(d) processes according to (3) with X0=0 and utiidN(0,1). The bias, standard deviation, and mean squared error (MSE) were computed using 10,000 replications.

Table 1 shows the simulation results for n=500, m=n0.65=56 and different values of d. For d[-0.4,0.4], the bias is small and the standard deviations changes little for different values of d, corroborating

Likelihood-based semiparametric estimation of d: LW, modified LW, and exact LW procedures

As mentioned in Section 3, the local Whittle likelihood was originally suggested under the stationarity assumption on Xt. When Xt is generated by model (1) and hence is nonstationary, we can reinterpret the LW likelihood function as an approximation of the Whittle likelihood function of ut, the weakly dependent innovation, transformed to be data dependent. The Whittle likelihood function of ut taken locally over frequencies λ1 to λm and up to scale multiplication isj=1mlogfu(λj)+j=1mIu(λj)fu(λ

Different characterizations of I(d) processes

Two main approaches to defining an I(d) process have been used in the literature to date. They are by no means exhaustive. The first, which is used in Hosking (1981) among others, is to assume d<12 and define the observed process Xt as an infinite order moving average of short-memory innovations, viz.Xt=X0+(1-L)-dut=k=0(d)kk!ut-k,where ut is a weakly dependent stationary process. Xt is stationary when d<12. This approach is extended by partial summation methods to produce I(d) series with d[1

Acknowledgements

An earlier version of this paper was presented at the Cowles Foundation Econometrics Conference “New Developments in Time Series Econometrics”, October 23–24, 1999 and circulated under the title “Modified Local Whittle Estimation of the Memory Parameter in the Nonstationary Case”. Shimotsu thanks the Cowles Foundation for support under a Cowles prize and the Sloan Foundation for Fellowship support. Phillips thanks the NSF for research support under Grant nos. SBR97-30295 & SES 00-92509.

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