Wavelet semi-parametric inference for long memory in volatility in the presence of a trend

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1 Journal of Statistical Computation and Simulation ISSN: (Print) (Online) Journal homepage: Wavelet semi-parametric inference for long memory in volatility in the presence of a trend Agnieszka Jach & Piotr Kokoszka To cite this article: Agnieszka Jach & Piotr Kokoszka (2017): Wavelet semi-parametric inference for long memory in volatility in the presence of a trend, Journal of Statistical Computation and Simulation, DOI: 1080/ To link to this article: Published online: 01 Jan Submit your article to this journal Article views: 3 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at Download by: [Colorado State University] Date: 09 January 2017, At: 08:05

2 JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION, Wavelet semi-parametric inference for long memory in volatility in the presence of a trend Agnieszka Jach a and Piotr Kokoszka b a Department of Finance and Statistics, Hanken School of Economics, Helsinki, Finland; b Department of Statistics, Colorado State University, Fort Collins, CO, USA ABSTRACT Risk of investing in a financial asset is quantified by functionals of squared returns. Discrete time stochastic volatility (SV) models impose a convenient and practically relevant time series dependence structure on the log-squared returns. Different long-term risk characteristics are postulated by short-memory SV and long-memory SV models. It is therefore important to test which of these two alternatives is suitable for a specific asset. Most standard tests are confounded by deterministic trends. This paper introduces a new, wavelet-based, test of the null hypothesis of short versus long memory in volatility which is robust to deterministic trends. In finite samples, the test performs better than currently available tests which are based on the Fourier transform. ARTICLE HISTORY Received 5 January 2016 Accepted 11 December 2016 KEYWORDS Deterministic trend; long memory; stochastic volatility; wavelets AMS SUBJECT CLASSIFICATION 62M10; 42C40 1. Introduction The long-memory stochastic volatility (LMSV) model [1,2]forreturns{r t } is defined as r t = exp(y t /2)e t, (1) where the multiplicative errors {e t } are mean-zero iid with unit variance. The assumption of long memory is introduced through the spectrum of {Y t }, S Y (f ) = 1 e i2πf 2d S Y (f ), f, (2) where the function S Y (f ) is assumed to be bounded and bounded away from zero. The parameter d is called the memory parameter. If d <, the process {Y t } is stationary. If d = 0, it is said to have short memory; if d > 0, long memory. Ourfocusisonthevolatilityseries X t = log(r 2 t ) = μ + Y t + η t, (3) where μ = E[log(e 2 t )]andη t = log(e 2 t ) E[log(e2 t )]. The volatility thus has a constant-plus-signalplus-noise decomposition, and has the same memory parameter d as the process {Y t }.Thepaperis concerned with inference on d if the process {X t } defined by Equation (3) is not directly observable, but is contaminated by a trend m t, i.e. a realization of {X t + m t } is observed. The above problem, in many model settings, has received a great deal of attention over the last decade, the review of the most relevant research is presented in Section 2. The main point is that CONTACT Agnieszka Jach agnieszka.jach@hanken.fi 2017 Informa UK Limited, trading as Taylor & Francis Group

3 2 A. JACH AND P. KOKOSZKA estimating or testing pertaining to the parameter d canbeseverelybiasedifthereisatrend.thisis true not just for the LMSV model, but for all long memory models, and, in fact, most time series models. The objective of this paper is to derive and assess in finite samples a new test of d = 0. Testing is performed in the wavelet domain, which is a natural approach if the effect of the trend m t is to be mitigated. The wavelet transform cancels polynomial trends, and so, locally, most regular trends. The paper is organized as follows. After reviewing in Section 2 related recent research, we introduce in Section 3 the requisite wavelet transform background and adapt the approach of Hurvich et al. [3]toderiveaWaveletLocalWhittle(WLW)estimatorofd in trend-contaminated model (3). Using this estimator as a starting point, we derive in Section 4 a GLR chi-square test of d = 0vs.d > 0 (inthepresenceofatrend).insection5, we evaluate the performance of the test in a Monte Carlo study and compare it to tests based on the Fourier transform. For illustration, we apply the test to exchange rate and stock index data. 2. Previous research Several methods of the estimation of d in model (3) exist. Semi-parametric estimators include various modifications of two popular frequency estimators: of the log-periodogram estimator [4,5], and of the local Whittle (LW) or Gaussian semi-parametric estimator [6,7]. Modified estimation schemes can be found in [3, 8 13]. When estimating the parameter d in the log-squared returns, these authors take into account the presence of the noise sequence {η t } and the short-memory dynamics of the signal {Y t } in Equation (3). Without the adjustments for these two factors, the memory parameter tends to be underestimated (see, for example, [14]). An important issue, which can hinder statistical inference about the memory parameter d is the presence of a deterministic trend in log-squared returns. If a trend is present, a short-memory volatility can be spuriously classified as long-memory [15 18]. This practically relevant feature has been recently addressed by Iacone [19], and in a more general setting, by McClosky and Perron [20]. Both papers propose to trim lower frequencies when using log-periodogram or LW estimator, but theprocessestheyconsiderarefreeofthenoiseterm{η t }. An estimation procedure that compensates for the possible presence of three sources of bias: the noise, the short-memory and the deterministic trend is proposed by McClosky [21], who employs a fully parametric approach (with trimming). In the semi-parametric context, this three-fold issue has been considered by Hou and Perron [22], who proposed a modified (frequency) LW estimator of the long-memory parameter in model (3). Moulines et al. [23] established consistency and rate optimality of the WLW estimator if {X t } is a linear process and asymptotic normality if {X t } is Gaussian. Roueff and Taqqu [24]showedasymptotic normality of the WLW estimator if {X t } is a linear process. A review and comparison to other estimators are presented in [25]. We conjecturethat the asymptotictheorycan be developed for the LMSV modelaswell,butthisisadifficulttask,evenassumingthat{y t } is a linear process. Our goal is to provide an attractive tool set for robust estimation of and testing for long memory in trend-contaminated volatility, and to validate its usefulness and superiority via computational methods. In our approach, the other two sources of bias, the noise and the short memory, are adjusted for by adapting the procedure of Hurvich et al. [3] and Hurvich and Ray [26], who derived the LW estimator (see Section 3). The WLW estimator we work with is automatically invariant to polynomial trends of low order. In contrast, frequency-based estimators require tapering or trimming to be robust to deterministic trends [27]. 3. WLW estimator WebeginthissectionwithabriefdescriptionoftheLWestimatorofHurvichetal.[3] followedby a review of the wavelet transform. Once these two preliminary steps are accomplished, we derive the new WLW estimator on which the test developed in Section 4 is based.

4 JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 3 Local Whittle estimator in the LMSV setting. Frequency-based LW estimator of Hurvich et al. [3] is obtained by optimizing the (empirical) Whittle criterion function that includes an additional term to account for the contribution of the noise sequence {η t },aswellasthatof{y t }, to the low-frequency behaviour of the spectral density of {X t }.Specifically,thespectraldensityof{X t } is represented as S X (f ) = 1 e i2πf 2d S X (f ),andmodelledas S X (f ) = S Y (f ) {}}{ 1 e i2πf 2d S Y (f ) +σ 2 η 1 e i2πf 2d G(1 + h(d, θ, f )), (4) where G isapositiveconstantandthefunctionhisselectedbytheusertoallowforaflexiblemodelling of S X (f ) via G(1 + h(d, θ, f )) near 0. The idea behind such treatment is that G(1 + h(d, θ, f )) will concurrently capture the effect of {η t } and the short-memory dynamics of {Y t } on the spectrum of {X t } in the neighbourhood of 0. The parameter θ can be regarded as a nuisance parameter. Hurvich et al. [3] proposed parametrization h(d, θ, f ) = θ(2πf ) 2d (P1 parametrization of that paper) as, under appropriate assumptions, when f 0, we have S X (f ) (2πf ) 2d G(1 + h(d, θ, f )),with G = S Y (0) and θ = σ 2 η S Y (0). Once we observe a discrete realization X 1, X 2,..., X n of the log-squared returns, the Whittle criterion function over periodogram ordinates x = [x 1, x 2,..., x m ], x j = 2jπ/n, j = 1, 2,..., m < n/2 close to 0 is defined as C(d, θ) = log 1 m m xj 2d I j 1 + θxj 2d + 1 m j=1 m j=1 log(xj 2d (1 + θxj 2d )). (5) We discard the highest frequencies as only those close to 0 are relevant in the estimation of d.the periodogram at x j is I j = d j 2 and d j = (1/ 2πn) n t=1 X t e itx j.thelwestimatorof(d, θ) is the minimum of this function, that is min (d,θ) D T C(d, θ), for some admissible sets D and T. Discrete wavelet transform. Now, we review some basic facts about the discrete wavelet transform (DWT), which one needs to understand our approach. A systematic account of the DWT can be found in [28]. The DWT of a discrete realization of a time series X = [X 0, X 1,..., X n 1 ] produces a set of J 0 + 1vectorsj = 1, 2,..., J 0 J = [log 2 (n)]. The first J 0 vectors, whose lengths decrease by a factor of 2, contain wavelet coefficients W j = [W j,0,, W j,1,..., W j,nj 1], n j = 2 J j.theseareassociatedwith time scales between 2 j and 2 j+1.thelastvectorv j of length n J0 contains the scaling coefficients which are associated with averages over time scales greater than 2 J 0+1. Both sets of coefficients are localized in time and frequency and are obtained through a linear transformation of the data. More specifically, the wavelet coefficients are defined as L j 1 W j,k = h j,l X 2 j (k+1) l 1modn, L j = (2 j 1)(L 1) + 1, l=0 where {h j,l } is the jth-level DWT wavelet filter, which is a high-pass filter with nominal pass band [1/2 j+1,1/2 j ], while the scaling coefficients are L j 1 V j,k = g j,l X 2 j (k+1) l 1modn, l=0

5 4 A. JACH AND P. KOKOSZKA where {g j,l } is the jth-level DWT scaling filter. This filter is a low-pass filter with nominal pass band [0, 1/2 j+1 ]. In both cases L is the length of the underlying wavelet filter (for example, Daubechies D(L) or least asymmetric LA(L) filter, see Section 4.8 of Percival and Walden [28]). All coefficients are arranged in a column vector W = [W 1, W 2,..., W J, V J ]andcanbewrittenasw = WX,where the elements of the matrix W are based on the wavelet and scaling filters. The transfer function for the level j wavelet filter is j 2 ( ) H j,l (f ) = e i2π(lj 1 1)f H 1,L (2 j 1 1 f ) H 1,L 2 2 k f k=0 and its squared gain function is H j,l (f ) = H j,l (f ) 2.HereH 1,L ( ) is the Fourier transform of the wavelet filter. From the definition of the DWT coefficients, we see that the large scales/low frequencies are suitable for long-memory parameter estimation and thus our focus will be on large indices j.let w j,k, j = 1, 2,..., J, k = 0, 1,..., M j 1, M j = N j L j denote the nonboundary DWT coefficients of the realization from the process (3), where L j = min([(l 2)(1 2 j )], n j ) (see Section 4.11 of Percival and Walden [28] fordetailsonboundary coefficients). Because the DWT acts like a decorrelating transform, the coefficients {w j,k } can be regarded as a weakly correlated sequence, even though the observations themselves are highly correlated. Here, we will assume that for a fixed j the sequence {w j,k } has autocovariances s j,τ, τ = 0, 1,..., M j τ. Thes j,τ depends, among other things, on the spectrum of the underlying process and is given by Percival and Walden [28] where s j,τ (d, θ, G) = 1/2 1/2 e i2πτf S j (f ) df, 2j 1 S j (f ) = 2 j H j,l (2 j (f + k))s X (2 j (f + k)). k=0 In the parametric estimation, the spectrum S X ( ) is fully specified for all j s, while in the semiparametric setting, in which we wish to estimate the memory parameter, we focus on the frequencies close to 0 and S X ( ) is not completely specified. WLW estimator in the LMSV setting. We now proceed to the derivation of the WLW estimator in volatility. Using approximation (4) for S X near the zero frequency (equivalently, for large scales), we write 1/2 s j,τ (d, θ, G) = G 1/2 = Gc j,τ (d, θ). 2j 1 e i2πτf 2 j k=0 H j,l (2 j (f + k)) 1 + h(d, θ,2 j (f + k)) (2π2 j (f + k)) 2d df Assuming Gaussian white noise model for the w j,k forarangeofscalesj = j 0, j 0 + 1,..., J 0 J, (j j 0 for j 0 as n ), we can derive the estimators of parameters d and θ based on the pseudo-maximum likelihood approach. We have w j,k iid N(0, Gc j,0 (d, θ)).twicethenegativelog

6 JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 5 likelihood, 2logf (w; d, θ, G), is given as (the summation extends over j = j 0,..., J 0 ) 2logf (w; d, θ, G) = M log(2πg) + j = M log(2πg) + j M j log(c j,0 (d, θ))+ j M j log(c j,0 (d, θ))+ j M j 1 k=0 w 2 j,k c j,0 (d, θ)g R j c j,0 (d, θ)g, where R j = M j 1 k=0 w 2 j,k and M = j M j. Minimizing this expression with respect to G yields the maximum likelihood estimate of G as a function of the remaining parameters: Ĝ(d, θ) = 1 M j R j c j,0 (d, θ). (6) Replacing G by Ĝ(d, θ) in 2logf (w; d, θ, G) (and dropping w) yields a function of d, andθ only, namely 2logf (d, θ) = M(log(2π)+ 1) + M log(ĝ(d, θ))+ j M j log(c j,0 (d, θ)). Minimizing the wavelet Whittle criterion function E(d, θ) = log(ĝ(d, θ))+ 1 M j log(c j,0 (d, θ)), M with respect to d and θ (the term c j,0 (d, θ) is obtained by numerical integration) gives the WLW estimators (ˆd, ˆθ),thatis, (ˆd, ˆθ) = j min E(d, θ), (7) (d,θ) D T foradmissibleintervalsd ( (, ) to ensure stationarity and invertibility of {Y t })andt ( (0, ) to include positive values). 4. Robust test for long memory in volatility In this section we develop a test to distinguish a weakly dependent volatility with a trend from a volatility with long memory, possibly with a trend. Our test falls into the broad category of Generalized Likelihood Ratio (GLR) tests. This approach is different form the previously proposed approaches which are based on asymptotic distributions of estimators of d. The testing problem is as follows: Null hypothesis: The observations U 0, U 1,..., U n 1 follow the model U t = X t {}}{ μ + Y t + η t +m t, (8) where X t is given by Equation (3) with d = 0and{m t } is a deterministic polynomial. Alternative hypothesis: The observations U 0, U 1,..., U n 1 follow model (8), where X t is given by Equation (3) with d > 0.

7 6 A. JACH AND P. KOKOSZKA The trend function {m t } need not be the same under the null and the alternative. To focus attention, the above testing problem can be equivalently stated as H 0 : d = 0 versus H 1 : d > 0, keeping in mind that the observations follow model (8). WenowderivethewaveletdomainGLRtest.WeapplytheDWTtothedataU 0, U 1,..., U n 1 of length n = 2 J. Next we carry out the WLW estimation of Section 3. Letβ = (d, θ) be the vector of parameters, d [0, ), θ (0, ), and introduce the parameter spaces 0 ={β : d = 0} and ={β : d 0}. Minimizing the 2logf (w; β),wherew are the wavelet coefficients of U 0, U 1,..., U n 1,over 0 leads to estimators ˆβ 0 and minimizing over to ˆβ,where Thus the GLR test statistic is ˆβ 0 = (0, ˆθ 0 ) and ˆβ = (ˆd, ˆθ). λ(w) = max{f (w; β) : β 0} max{f (w; β) : β } = f (w; ˆβ 0 ) f (w; ˆβ). Under H 0, 2logλ(w) is approximated by the chi-square distribution with 1 degree of freedom. Therefore, the size α asymptotic GLR test rejects H 0 if 2logλ(w) >χ1 2(α), whereχ 1 2 (α) is the upper α quantile of the chi-square distribution with 1 degree of freedom. We have that 2logλ(w) = M(E(0, ˆθ 0 ) E(ˆd, ˆθ)) (the term M(log(2π)+ 1) cancels out). Spectral domain tests. We compare our test to that of Hou and Perron [22] (Section 5 of that paper), which is based on the following modification of the LW profile function (5), J(d, θ, ϑ) = log 1 m xj 2d I j m 1 + θxj 2d + ϑxj 2d m log(xj 2d (1 + θxj 2d + ϑxj 2d 2 /n)). (9) /n m j=1 The nuisance parameter, ϑ, is included to control for the influence of the deterministic (and other) trends.whenthisadjustmentisomitted,oneobtainstheestimatorofhurvichetal.[3]. Hou and Perron [22] establish asymptotic properties of their estimator with only one nuisance parameter, ϑ, but not both θ and ϑ. Nevertheless, we will use the asymptotic normality of the former to produce the (asymptotic) standard normal test statistic z ModLW = (d ModLW d 0 )/ 1/4m, whichusestheestimator of the memory parameter d ModLW obtained from minimizing the profile function (9). The size α asymptotic normal test rejects H 0 : d = d 0 versus H 1 : d > d 0 if z ModLW > z(α),wherez(α) is the upper α quantile of the standard normal distribution. Additionally, we consider two (heuristic) adjustments of the LW estimator of Hurvich et al. [3], one with tapering and one with trimming to control for the presence of a trend. In the first case, we optimize profile function (5) with periodogram I j at x j replaced by a tapered periodogram I tap,p j of Hurvich and Chen [27], I tap,p j = 1 2π n k=1 h k 2p t=1 j=1 2 n h p t D t e itx j, where p is the taper s order, {D t } is obtained by differencing {X t } p times, and ( ( )) i2π(t ) h t = 1 exp, t = 1, 2,..., n. n We denote the corresponding estimator by d TapLW. The asymptotic theory for the tapered LW estimator is not available, so we use the results for the tapered Gaussian semiparametric estimator

8 JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 7 (S X (f ) parametrized as a constant) of Hurvich and Chen [27]. This leads to the asymptotic standard normal test statistic z TapLW = d TapLW d 0 p /4m, p = πɣ2 (2p 1)Ɣ 2 ((4p 3)/2) Ɣ 4 ((2p 1)/2)Ɣ(4p 3), where Ɣ( ) is the gamma function. In case of trimming, similarly to [19], we optimize the LW profile function(5) over the reduced number of Fourier frequencies j = l, l + 1,..., m instead of j = 1, 2,..., m. This leads to the standard normal test statistic z TrimLW = d TrimLW d 0 1/4S(l, m), where, following Iacone [19], we employ trimming-related correction term, S(l, m), inthedenominator. The correction is given by S(l, m) = m νj 2, j=l ν j = log(j) 1 m l + 1 m log(k). k=l The size α asymptotic normal test rejects H 0 : d = d 0 versus H 1 : d > d 0 if z TrimLW > z(α). 5. Finite-sample evaluation of the wavelet-domain test Theobjectiveofthissectionistoevaluatethefinite-sampleperformanceofthetestintroducedin Section 4. In the setting of the stochastic volatility model, it is important to take into account the effectofthenoise{η t } in Equation (3). As the ratio of this noise to the signal {Y t } increases, a test of d = 0 will be increasingly less likely to reject H 0. As in the setting of linear processes, the robustness of the test to the short-memory autocorrelation structure and to deterministic trends must be assessed as well. The set-up related to the data-generating process involves sample sizes n = 2 J for J {10, 11, 12, 13}, (n {1024, 2048, 4096, 8192}) and values of noise-to-signal ratio (nsr), nsr = ση 2/S Y (0) {1, 5, 10} (the last two were used by Hurvich et al. [3] andthevalueof1isincludedfor comparison). We use R = 500 replications. The Monte Carlo error in the size study at the 5% nominal level is approximately 5(0.95)/500 1%. The shocks {e t } are iid standard normal, and the signal {Y t } follows a Gaussian FARIMA(1,d,0) process with φ {0,,..., }, d {0,,..., }. Thespectrumof{Y t } is S FARIMA(1,d,0) (f ) = σz 2(1/(1 + φ2 2φ cos(2πf ))) 1 exp( i2πf ) 2d,whereσZ 2 isthevarianceofthegaussianinnovations, which is set in such a way that the selected values of nsr = ση 2/σ Z 2/(1 + φ2 2φ)are attained, that is, we put σz 2 = σ η 2(1 + φ2 2φ)/nsr. If {e t } are standard normal, then η t has a log-chi-square distribution with 1 degree of freedom and thus has variance ση 2 = π 2 /2. If {e t } follow a Student distribution with ν being degrees of freedom (normalized to a unit variance), then ση 2 = ψ (ν/2) + π 2 /2, where ψ ( ) is the derivative of digamma function. We consider two types of a deterministic trend function m t : fractional function of Iacone [19], m t,frac = 10t 5 (scaledby10ratherthanby2asinthatpaper), polynomial function m t,poly = 20(t n/2) 2 /n 2. The ranges of the trends are (1, 10) and (0, 5), respectively. We perform additional computations with the above settings for {e t } following a Student distribution with 3 degrees of freedom (standardized to a unit variance). In Figures A1 and A2, we show examples of realizations from model {X t + m t,frac } and their corresponding sample autocorrelation functions.

9 8 A. JACH AND P. KOKOSZKA Now we describe the estimation/testing procedure. The wavelet estimator ˆd uses the lower cut-off scale j 0 = [log 2 (n 1 β )], β {,, }, that matches the number of Fourier frequencies m = [n β ] used with the frequency estimators. The upper level J 0 is the highest level available (uncontaminated by the boundary coefficients). Least asymmetric wavelet filter of length 8 is utilized, which corresponds to the taper order p = 3.Underthealternativehypothesis,thenumericaloptimization is performed on the set (d, θ) ( 1, 0.99) (001, 10 9 ) with the starting value of d = d GPH, where d GPH is the regression estimate of Geweke and Porter-Hudak [4]andstartingvalueofθ = 5. Under the null, the optimization is carried only with respect to θ (001, 10 9 ) and d = 0(no starting value required). The performance of the new test is measured with reference to the normal tests based on the adjusted LW estimators. The optimization of these is done with the same specifications as for the wavelet estimator, except for the tapered LW estimator for which d ( (p + 1) +, ) (the data are differenced p times prior to tapering, as stipulated by Hurvich and Chen [27])and we also set m = [5n β ] as in that paper. Parameter ϑ in the LW estimator of Hou and Perron [22] is optimized like θ, while the lower and upper trimming cut-offs in d TrimLW are l = [1 + n (β ) ]andm = [n β ], which are obtained by adapting formulas suggested by Iacone [19]. For all tests we compute the empirical size and power, that is, the fraction of rejections of H 0 in R = 500 replications. Because some of the tests have empirical size different from the nominal one, we compute size-adjusted power as prescribed by [29], Section 3. The adjustment relies on using the critical value which makes the estimated size as close as possible to nominal. All tests were evaluated on the same replications from the process {X t + m t }. Most results of our extensive simulation study are tabulated in the appendix (with normally distributed {e t }, while those with heavy-tailed shocks are available upon request). A close examination Estimator ModLW TapLW TrimLW WLW Empirical size nsr = 1 nsr = 5 nsr = 10 α= α=5 φ Figure 1. Size, fractional trend. Empirical size of the wavelet and frequency tests (H 0 : d = 0versusH 1 : d > 0) based on R = 500 replications of {X t + m t,frac } of length n = 2 11 as a function of φ, the AR parameter of the Gaussian FARIMA(1,d,0) model for {Y t }, {e t } N(0, 1). The nominal level 5% (bottom) and 10% (top) indicated by the horizontal line. Bandwidth parameter β =.

10 JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 9 of Tables A1 A4 (sample sizes 2 10,2 11,2 12,2 13, respectively) reveals that the larger bandwidths lead to oversized tests, hence we focus on the smallest bandwidth parameter β = and a moderate sample size of n = Selected results are presented in form of graphs to make a quick comparison easy. Figures 1 and 2 show the empirical size of the chi-square (WLW) and normal tests for the nominal levels of 5 and 10 percent as a function of φ, for three values of the nsr. Fractional trend has a particularly detrimental effect on the empirical size of the tests based on trimmed and modified versions of the LW estimator, they are very seriously oversized. By contrast the new WLW test holds the empirical size very well. For the polynomial trend, the wavelet- and the modified-lw-based tests have empirical size closer to the target value, compared to the tapered-lw- and trimmed-lw-based tests, which are oversized, especially the latter. The small bandwidth (β =, hence j 0 = 4) is particularly important when nsr = 1. For such bandwidth, test based on wavelet estimator remains reasonable for φ when nsr = 1andthatbasedonModLWforφ, with the latter being more conservative. As the AR coefficient φ approaches 1, the time series model approaches a nonstationary model. This is why we see increasing rejection rates as φ gets closer to 1. Figures 3 and 4 show the size-adjusted empirical power of the chi-square and normal tests for the nominal level α = 5 as a function of d > 0. In these multi-panel figures, different rows correspond to different values of φ, while columns refer to the nsr. For modified and tapered LW and wavelet estimators, the power curves have the desired shape the power increases with d as expected, as the data provides more evidence in favour of the alternative hypothesis. All three estimators have low power for larger values of nsr. ModLW-based and wavelet tests are more powerful than the TapLW-based test. Normal test based on trimmed LW estimator has no size-adjusted power. This is due to its inflated size. Estimator ModLW TapLW TrimLW WLW Empirical size nsr = 1 nsr = 5 nsr = 10 α= α=5 φ Figure 2. Size, polynomial trend. Empirical size of the wavelet and frequency tests (H 0 : d = 0versusH 1 : d > 0) based on R = 500 replications of {X t + m t,poly } of length n = 2 11 as a function of φ, the AR parameter of the Gaussian FARIMA(1,d,0) model for {Y t }, {e t } N(0, 1). The nominal level 5% (bottom) and 10% (top) indicated by the horizontal line. Bandwidth parameter β =.

11 10 A. JACH AND P. KOKOSZKA Regarding the sample size effect, by comparing Tables A1 A4 (empirical size) and Tables A5 A8 (size-adjusted empirical power), we see that the results improve with increasing n. Summary of the simulation study. Our GLR test based on the WLW estimator which uses the lower cut-off scale j 0 = [log 2 (n 1 β )]withβ = hastheempiricalsizeclosesttothenominalsize.using β = or β = leads to poorer empirical size for all four tests, but our GLR test still dominates the other three tests (with β>). The size-adjusted power of the new test is larger than that of the tests that over-reject under H 0. Some frequency domain tests have empirical size close to 100 % in certain scenarios, which makes them practically unusable. The good performance of our test comes from its robustness to trends achieved by formulating the procedure in the wavelet domain, and from more effective decorrelation in the wavelet domain than in the spectral domain Application to index and FX returns We conclude this section by applying all tests to return data. We consider daily (log) returns on S&P500 index and on USD/GBP exchange rate ending on the last day of Nov 2015, both of length Inbothcaseswecalculatethevolatilityseries(logsofsquaredreturns).Thereturnsandthevolatility series are displayed in Figure 5. Estimation and testing set-up is the same as in the Monte Carlo experiment. In Table 1 we report the test statistics and p-values of all testing procedures. All tests reject the null hypothesis (at the 5% level) of weakly correlated volatility with a deterministic trend for the S&P500 data set. For the USD/GBP exchange rate data set, only tests based on modified and trimmed LW estimators do. This brief illustration provides another piece of empirical evidence that long memory is an important feature of volatility of returns on speculative assets, even after deterministic trends have been accounted for. Estimator ModLW TapLW TrimLW WLW Empirical power nsr = 1 α=5 nsr = 5 α=5 nsr = 10 α=5 φ= φ= φ=0 d Figure 3. Power, fractional trend. Empirical size-adjusted power of the wavelet and frequency tests (H 0 : d = 0versusH 1 : d > 0) based on R = 500 replications of {X t + m t,frac } of length n = 2 11 asafunctionofd. Different rows correspond to different values of φ, the AR parameter of the Gaussian FARIMA(1,d,0) model for {Y t }, {e t } N(0, 1). Bandwidth parameter β =.

12 JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 11 Estimator ModLW TapLW TrimLW WLW Empirical power nsr = 1 α=5 nsr = 5 α=5 nsr = 10 α=5 φ= φ= φ=0 d Figure 4. Power, polynomial trend. Empirical size-adjusted power of the wavelet and frequency tests (H 0 : d = 0versusH 1 : d > 0) based on R = 500 replications of {X t + m t,poly } of length n = 2 11 asafunctionofd. Different rows correspond to different values of φ, the AR parameter of the Gaussian FARIMA(1,d,0) model for {Y t }, {e t } N(0, 1). Bandwidth parameter β =. 0 S&P500 USD/GBP Return Volatility Date Figure 5. Returns and log-volatility of S&P500 index and of USD/GBP exchange rate.

13 12 A. JACH AND P. KOKOSZKA Table 1. Test statistics and p-values (in parentheses) of the wavelet and frequency tests (H 0 : d = 0versusH 1 : d > 0) applied to, possibly trend-contaminated, volatility of real data of length n = Bandwidth parameter β =. Estimator Data ModLW TapLW TrimLW WLW S&P (< 01) (360) (< 01) (< 01) USD/GBP (< 01) (581) (335) (935) Disclosure statement No potential conflict of interest was reported by the authors. Funding A. Jach was supported by grant Jose Castillejo JC of Spanish Ministry of Education. References [1] Breidt FJ, Crato N, Lima P. The detection and estimation of long memory in stochastic volatility. J. Econom. 1998;3: [2] Harvey A. Long-memory in stochastic volatility. In: Knight J, Satchell SE, editors. Forecasting volatility in financial markets. London: Butterwoth-Heinemann; p [3] Hurvich CM, Moulines E, Soulier P. Estimating long memory in volatility. Econometrica. 2005;73: [4] Geweke J, Porter-Hudak S. The estimation and application of long memory time series models. J Time Ser Anal. 1983;4: [5] Robinson PM. Log-periodogram regression of time series with long range dependence. Ann Stat. 1995;23: [6] Künsch H. Discrimination between monotonic trends and long-range dependence. J Appl Probab. 1986;23: [7] Robinson PM. Gaussian semiparametric estimation of long range dependence. Ann Stat. 1995;23: [8] Andrews DWK, Sun Y. Adaptive local polynomial Whittle estimation of long-range dependence. Econometrica. 2004;72: [9] Arteche J. Semiparametric estimation in perturbed long memory series. Comput Stat Data Anal. 2006;51: [10] Arteche J. Semiparametric inference in correlated long memory signal plus noise models. Econom Rev. 2012;31: [11] Deo RS, Hurvich CM. On the log periodogram regression estimator of the memory parameter in long memory stochastic volatility models. Econom Theory. 2001;17: [12] Frederiksen P, Nielsen FS, Nielsen MO. Local polynomial Whittle estimation of perturbed fractional processes. J. Econom. 2012;167: [13] Sun Y, Phillips PCB. Nonlinear log-periodogram regression for perturbed fractional processes. J Econom. 2003;115: [14] Haldrup N, Nielsen MO. Estimation of fractional integration in the presence of data noise. Comput Stat Data Anal. 2007;51: [15] Baek C, Pipiras V. Statistical tests for a single change in mean against long-range dependence. J Time Ser Anal. 2012;33: [16] Berkes I, Horváth L, Kokoszka P, et al. On discriminating between long-range dependence and changes in mean. Ann Stat. 2006;34: [17] Bhattacharya RN, Gupta VK, Waymire E. The Hurst effect under trends. J Appl Probab. 1983;20: [18] Mikosch T, Stărică C. Nonstationarities in financial time series, the long-range dependence, and the Igarch effects. Rev Econ Stat. 2004;86: [19] Iacone F. Local Whittle estimation of the memory parameter in presence of deterministic components. J Time Ser Anal. 2010;31: [20] McClosky A, Perron P. Memory parameter estimation in the presence of level shifts and deterministic trends. Econom Theory. 2013;29: [21] McClosky A. Estimation of the long-memory stochastic volatility model parameters that is robust to level shifts and deterministic trends. J Time Ser Anal. 2013;34:

14 JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 13 [22] Hou J, Perron P. Modified local Whittle estimator for long memory processes in the presence of low frequency (and other) contaminations. J Econom. 2014;182: [23] Moulines E, Roueff F, Taqqu MS. A wavelet Whittle estimator of the memory parameter of a nonstationary Gaussian time series. Ann Stat. 2008;36: [24] Roueff F, Taqqu MS. Asymptotic normality of wavelet estimators of the memory parameter for linear processes. J Time Ser Anal. 2008;30: [25] Fay G, Moulines E, Roueff F, et al. Estimators of long-memory: Fourier versus wavelets. J Econom. 2009;151: [26] Hurvich CM, Ray B. The local Whittle estimator of long-memory stochastic volatility. J Financ Econom. 2003;1: [27] Hurvich CM, Chen WW. An efficient taper for potentially overdifferenced long-memory time series. J Time Ser Anal. 2001;21: [28] Percival DB, Walden AT. Wavelet methods for time series analysis. Cambridge: Cambridge University Press; [29] Lloyd CJ. Estimating test power adjusted for size. J Statist Comput Simul. 2005;75: Appendix 5 X t + m t d=0 d= t Figure A1. Examples of realizations from a model {X t + m t,frac } of length n = 2 11 with nsr = 5 and φ =, the AR parameter of Gaussian FARIMA(1,d,0) model for {Y t }, {e t } N(0, 1).

15 14 A. JACH AND P. KOKOSZKA 0 ACF d=0 d= Lag Figure A2. Sample autocorrelation functions of realizations from a process {X t + m t,frac } of length n = 2 11 with nsr = 5 and φ =, the AR parameter of Gaussian FARIMA(1,d,0) model for {Y t }, {e t } N(0, 1).

16 JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 15 Table A1. Empirical size of the wavelet and frequency tests (H 0 : d = 0versusH 1 : d > 0) at the 5% nominal level based on R = 500 replications of {X t + m t,frac } and {X t + m t,poly } of length n = 2 10 for selected values of φ, the AR parameter of Gaussian FARIMA(1,d,0) model for {Y t }, {e t } N(0, 1). Empirical size, fractional trend d φ nsr β ModLW TapLW TrimLW WLW Empirical size, polynomial trend ModLW TapLW TrimLW WLW

17 16 A. JACH AND P. KOKOSZKA Table A2. Empirical size of the wavelet and frequency tests (H 0 : d = 0versusH 1 : d > 0) at the 5% nominal level based on R = 500 replications of {X t + m t,frac } and {X t + m t,poly } of length n = 2 11 for selected values of φ, the AR parameter of the Gaussian FARIMA(1,d,0) model for {Y t }, {e t } N(0, 1). Empirical size, fractional trend d φ nsr β ModLW TapLW TrimLW WLW Empirical size, polynomial trend ModLW TapLW TrimLW WLW

18 JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 17 Table A3. Empirical size of the wavelet and frequency tests (H 0 : d = 0versusH 1 : d > 0) at the 5% nominal level based on R = 500 replications of {X t + m t,frac } and {X t + m t,poly } of length n = 2 12 for selected values of φ, the AR parameter of the Gaussian FARIMA(1,d,0) model for {Y t }, {e t } N(0, 1). Empirical size, fractional trend d φ nsr β ModLW TapLW TrimLW WLW Empirical size, polynomial trend ModLW TapLW TrimLW WLW

19 18 A. JACH AND P. KOKOSZKA Table A4. Empirical size of the wavelet and frequency tests (H 0 : d = 0versusH 1 : d > 0) at the 5% nominal level based on R = 500 replications of {X t + m t,frac } and {X t + m t,poly } of length n = 2 13 for selected values of φ, the AR parameter of the Gaussian FARIMA(1,d,0) model for {Y t }, {e t } N(0, 1). Empirical size, fractional trend d φ nsr β ModLW TapLW TrimLW WLW Empirical size, polynomial trend ModLW TapLW TrimLW WLW

20 JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 19 Table A5. Empirical size-adjusted power of the wavelet and frequency tests (H 0 : d = 0versusH 1 : d > 0) at the 5% nominal level based on R = 500 replications of {X t + m t,frac } and {X t + m t,poly } of length n = 2 10 for selected values of d and φ, the AR parameter of the Gaussian FARIMA(1,d,0) model for {Y t }, {e t } N(0, 1). Empirical power, fractional trend d φ nsr β ModLW TapLW TrimLW WLW Empirical power, polynomial trend ModLW TapLW TrimLW WLW

21 20 A. JACH AND P. KOKOSZKA Table A6. Empirical size-adjusted power of the wavelet and frequency tests (H 0 : d = 0versusH 1 : d > 0) at the 5% nominal level based on R = 500 replications of {X t + m t,frac } and {X t + m t,poly } of length n = 2 11 for selected values of d and φ, the AR parameter of Gaussian FARIMA(1,d,0) model for {Y t }, {e t } N(0, 1). Empirical power, fractional trend d φ nsr β ModLW TapLW TrimLW WLW Empirical power, polynomial trend ModLW TapLW TrimLW WLW

22 JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 21 Table A7. Empirical size-adjusted power of the wavelet and frequency tests (H 0 : d = 0versusH 1 : d > 0) at the 5% nominal level based on R = 500 replications of {X t + m t,frac } and {X t + m t,poly } of length n = 2 12 for selected values of d and φ, the AR parameter of Gaussian FARIMA(1,d,0) model for {Y t }, {e t } N(0, 1). Empirical power, fractional trend d φ nsr β ModLW TapLW TrimLW WLW Empirical power, polynomial trend ModLW TapLW TrimLW WLW

23 22 A. JACH AND P. KOKOSZKA Table A8. Empirical size-adjusted power of the wavelet and frequency tests (H 0 : d = 0versusH 1 : d > 0) at the 5% nominal level based on R = 500 replications of {X t + m t,frac } and {X t + m t,poly } of length n = 2 13 for selected values of d and φ, the AR parameter of Gaussian FARIMA(1,d,0) model for {Y t }, {e t } N(0, 1). Empirical power, fractional trend d φ nsr β ModLW TapLW TrimLW WLW Empirical power, polynomial trend ModLW TapLW TrimLW WLW