A Test Against Spurious Long Memory

This paper proposes a test statistic for the null hypothesis that a given time series is a stationary long memory process against the alternative hypothesis that it is of short memory, a¤ected by regime change or a smoothly varying trend. The proposed test is in the frequency domain and explores the derivatives of the pro…led local Whittle likelihood function in a degenerating neighborhood of the origin. The assumptions adopted are the same as in Robinson (1995b), which allow for non-Gaussianity. The resulting null limiting distribution is nuisance parameter free and can be easily simulated. The test is straightforward to implement: no kernel smoothing is required and no estimation of nuisance parameters is necessary. Also, there is no need to specify the number or locations of the di¤erent regimes that occur under the alternative hypothesis. Monte Carlo simulation shows that the test has good size, even for relatively small sample sizes, and that it has excellent power against alternatives of interest. JEL Classi…cation Number: C12, C22. Keywords: structural change, trend, fractional integration, semiparametric, frequency domain estimates. Department of Economics, Boston University, 270 Bay State Rd., Boston, MA, 02215 ([email protected]). I wish to thank Pierre Perron for useful suggestions, and Adam McCloskey and Yohei Yamamoto for detailed comments on a previous draft that improved the presentation. 1 Introduction There has long been an interest in long memory models, especially regarding applications to …nancial time series. Formally, a scalar process is said to have long memory if its spectral density at frequency is proportional to 2d as approaches zero, where 0 < d < 1 is the memory parameter. Long memory models provide a middle ground between short memory and unit root speci…cations, hence allowing for more ‡exibility in modelling the persistence of a time series. Estimation theory for these models has been well worked out. Among others, Fox and Taqqu (1986) studied parametric long memory models and Geweke and PorterHudak (1983), Robinson (1995a, b), and Hurvich et al. (2005) considered semiparametric models. The latter literature has been very in‡uential because it does not require speci…c assumptions on the process generating the di¤erence of order d of the series, hence minimizing the impact of misspeci…cation in the spectrum at frequencies distant from zero. Applications to macroeconomics and …nance are numerous. For example, Ding et al. (1993) argued that stock returns volatility is well described by a long memory process. Andersen et al. (2003) used a fractionally integrated (vector) autoregressive model to forecast realized volatility for the Deutschmark/Dollar and Yen/Dollar spot exchange rates. Also see Andersen et al. (2001) and Deo et al. (2006), among others, for other interesting applications. Two important features of a long memory process are that the spectral density of the process at the origin is unbounded and that its autocorrelation function decays at a slow hyperbolic rate at long lags. However, these features can also be present for a short memory process a¤ected by regime change or a smoothly varying trend, leading to so-called spurious long memory. Such phenomena have been widely documented, see Diebold and Inoue (2001), Gourieroux and Jasiak (2001), Lobato and Savin (1998), Mikosh and St1⁄4 aric1⁄4 a (2004), Parke (1999), and Teverosovky and Taqqu (1997). Whether the observed long memory characteristic is true or spurious is of substantial empirical importance. For example, Taylor (2000) showed that the long memory assumption has a signi…cant impact on the term structure of implied volatilities. Ohanissian et al. (2004) showed that when the true data generating process (DGP) is of spurious long memory, using either a short memory model or a true long memory model leads to severe underpricing of call options. Meanwhile, when the DGP is of true long memory, using a short memory model or a spurious long memory model leads to general overpricing of call options. Such concerns have been raised in other literatures as well, see Klemeš(1974) on hydrology, Mills (2007) on northern hemisphere temperatures, and Karagiannis et al. (2004) on internet tra¢ c. Unfortunately, it has proven rather di¢ cult 1 to distinguish between true and spurious long memory, partly because the commonly used tests for regime switching or structural change are biased toward over rejection (of the null hypothesis of no change) when the process is indeed fractionally integrated. This paper proposes a test statistic to distinguish between true and spurious long memory. The proposed test is in the frequency domain. The motivation for it is that the spectral domain properties of the aforementioned processes di¤er when di¤erent frequency bands local to zero are examined. Speci…cally, if a process is of short memory with level shifts, then the e¤ect of nonstationarity is visible only within a narrow range of j = O(n ), where n is the sample size and j = 2 j=n (j = 1; :::; [n=2]) are the Fourier frequencies. More importantly, within this range the mean of the periodogram is proportional to n 2 j ; implying a memory parameter of d = 1. Outside this range, the short memory component dominates and the periodogram is basically ‡at. This is in contrast to a true long memory process, where the spectral density is proportional to 2d j uniformly for j = o(1): The above result was documented by Perron and Qu (2008) via the study of a simple mean plus noise model with the mean component speci…ed as a compound binomial process. Now, if a process is of short memory with a smoothly varying trend, this paper obtains a result showing that the behavior of the periodogram is similar to that with level shifts. The above …ndings suggest constructing a test statistic using a process of weighted periodograms, where the weights are chosen to exploit the non-uniform behavior of the periodogram under the alternative hypothesis. This leads to a Kolmogorov-Smirnov type test based on derivatives of the pro…led local Whittle likelihood function evaluated sequentially in a degenerating neighborhood of the origin. The test is very simple to implement: no kernel smoothing is required and no estimation of nuisance parameters is necessary. Also, there is no need to specify the number or locations of the di¤erent regimes that occur under the alternative hypothesis. It is also quite general in the sense that the null hypothesis only requires specifying the process semiparametrically. The assumptions used are the same as in Robinson (1995b), which allow for non-Gaussianity. The null limiting distribution is nuisance parameter free and can be easily simulated. Monte Carlo simulation shows the test has good size, even for relatively small sample sizes, and has excellent power against alternatives of interest. This paper is closely related to an interesting article by Ohanissian et al. (2008), although the construction of the test and its underlying assumptions are very di¤erent. Ohanissian et al. (2008) explored the idea that temporal aggregation does not change the order of fractional integration (see Chambers, 1998). They showed that for a Gaussian long memory process, the Geweke Porter-Hudak (GPH) estimates obtained from di¤erent temporal aggregates are 2 jointly normally distributed, while for spurious long memory, the estimates depend on the level of aggregation. However, their result requires Gaussianity, which is di¢ cult to relax. Also, the assumption they place on the bandwidth of the GPH estimates is very stringent and requires a very large sample size for the test to be applicable. Our test does not require Gaussianity; it is based on an estimator that is more e¢ cient than the GPH estimator and can be applied to sample sizes that are typical in economics and …nance. This paper is also related to Berkes et al. (2006), who considered CUSUM procedures, and Sibbertsen and Venetis (2003), who proposed a test based on the di¤erence between the standard GPH estimator of d and a version of the estimator based on the tapered periodogram. From a methodological perspective, the idea of using integrated periodograms as the basis for speci…cation testing can be traced back to Grenander and Rosenblatt (1953, 1957) and Bartlett (1955). See Priestley (1983) for a review of the literature covering short memory processes. The literature concerning long memory processes is relatively sparse. Ibragimov (1963) provided an early seminal contribution. Recently, Kokoszka and Mikosch (1997) derived a functional central limit theorem for the integrated periodogram of a long memory process with …nite or in…nite variance. They only considered parametric models and the parameters were assumed to be known. Nielsen (2004) considered semiparametric models for multivariate long memory processes and proved the weak convergence of the integrated periodogram. His results greatly facilitate our subsequent analysis. However, he also assumed the memory parameter is known and his result only applies to d 2 (0; 1=4): In this paper, we obtain a weak convergence result for weighted periodograms that involve estimated parameters in a semiparametric setting. The result relies on a uniform weak law of large numbers, a functional central limit theorem, and a Taylor approximation that holds uniformly for parameters of interest. The remainder of this paper is organized as follows. Section 2 discusses spectral domain properties of processes with true and spurious long memory. Section 3 introduces the test statistic. Section 4 presents the assumptions used and the limiting distribution under the null hypothesis. Section 5 includes simulations, for …nite sample properties. Section 6 concludes. The proofs are contained in the Appendices, with Appendix A containing the proof of the main results and Appendix B some auxiliary lemmas. The following notation is used throughout the paper. jzj denotes the modulus of a complex number z. The imaginary unit is denoted by i. For a real number x, [x] denotes the largest integer less than or equal to x. The subscript 0 indicates the true value of a parameter. The symbols “)”and “!”signify weak convergence under Skorohold topology 3 and convergence in probability. And Op( ) and op( ) are the usual symbols for orders of convergence. 2 Spectral domain properties of processes with true and spurious long memory Let xt (t = 1; 2; :::; n) be a scalar process with n denoting the sample size. The periodogram of xt evaluated at frequency j = 2 j=n (j = 1; 2; :::; [n=2]) is given by Ix( j) = 1 2 n n X t=1 xt exp(i jt) 2 : (1) In this section, we will consider the spectral domain properties of xt for frequencies lying in a degenerating neighborhood of the origin. Special attention will be paid to the periodogram (1), which is the building block for various long memory parameter estimators and the test statistic proposed in the next section. First consider true long memory processes. Recall that xt is said to have long memory if its spectral density f( ) satis…es f( ) G 2d as ! 0+; (2) where the symbol “ ”means that the ratio of expressions on the left and right sides of the symbol tends to unity, G 2 (0;1), and d > 0. In the following analysis, we will assume d 2 (0; 1=2), which is arguably the case of most practical interest. A special case of a process satisfying (2) that has been widely applied in economics and …nance is the ARFIMA(p; d; q) process, introduced by Granger and Joyeux (1980) and Hosking (1981): A(L) (1 L) xt = B(L)"t; (3) where A(L) = 1 a1L ::: apL, B(L) = 1 + b1L + ::: + bqL, and "t is a white noise process with mean zero and variance ". This process has a spectral density that satis…es f( ) 2 " 2 jB(1)j jA(1)j 2d as ! 0 + : The statistical properties of the periodogram of a long memory process have been extensively studied, see Robinson (1995a, b) and Lahiri (2003). The result that is central to our analysis is Theorem 2 of Robinson (1995a), which states that under fairly mild conditions, when xt satis…es (2), E Ix( j) f( j) ! 1 for all j such that j !1 as n!1 but j=n! 0. 4 This uniform behavior of Ix( j) is the very property that allows us to test against spurious long memory. Now consider short memory processes contaminated by level shifts. In this case, the periodogram also diverges at the zero frequency and the autocovariance function also exhibits a slow rate of decay, see Klemeš(1974), Diebold and Inoue (2001), and Gourieroux and Jasiak (2001). More closely related to the current paper are the …ndings of Perron and Qu (2008). They considered the following simple process involving random level shifts for some series xt: xt = vt + ut; (4) ut = ut 1 + t t; where vt is a stationary short memory process (e.g., an ARMA process), t i:i:d: (0; 2 ); and t is a Bernoulli random variable that takes value 1 with probability pn and 0 otherwise, i.e., t i:i:d: B(1; pn). They used pn = p=n with 0 < p <1 to model rare shifts so that as n increases, the expected number of shifts remains bounded. The components t; t; and vt are assumed to be mutually independent. Then, the periodogram of xt can be decomposed into the following three components: Ix( j) = 1 2 n n X t=1 vt exp(i jt) 2 + 1 2 n n X t=1 ut exp(i jt) 2 + 2 2 n n X