Testing the Fractionally Integrated Hypothesis using M Estimation

Transcription

1 esting the Fractionally Integrated Hyothesis using M Estimation Matei Demetrescu a, Paulo M.M. Rodrigues b and Antonio Rubia c a Christian-Albrechts-University of Kiel b Banco de Portugal and Nova School of Business and Economics, Universidade Nova de Lisboa c University of Alicante Work in Progress. his version: December 8, 207 Abstract his aer develos a class of tests for fractional integration in the time domain based on M estimation. his testing strategy o ers more robust roerties against non-gaussian errors than least squares (LS) or other estimation rinciles. We discuss the asymtotic roerties of the new tests under fairly general assumtions, and for di erent estimation aroaches based on direct otimization of the M lossfunction and on iterated k-ste and reweighted LS numeric algorithms. Monte Carlo exerimentation shows that M tests for fractional integration exhibit emirical size close to the nominal level in nite samles and enhanced ower in relation to alternative rocedures, such as LS and even quantile regression, when innovations are drawn from heavy-tailed distributions. An alication to daily volatility of several stock market indices, roxied by log absolute returns and log high-low rice ranges, shows the emirical erformance of the new tests. Keywords: Fractional integration, M estimation, heavy-tails, long memory, volatility. JEL: C2, C22.

2 Introduction Most unit-root and cointegration tests build on least squares (LS) estimation. his rincile ensures e ciency under Gaussian conditions. In ractice, however, macroeconomic and nancial variables are usually driven by heavy-tailed distributions. In this context, LS-based tests remain asymtotically valid under aroriate conditions, but are no longer e cient. Suitable alternatives, which either accommodate the true likelihood of the data or ensure robust roerties against deviations from normality, can exhibit imroved ower. his consideration becomes articularly relevant in stochastic-trend detection and ersistence analysis because the relative losses in ower from using ine cient tests tend to be greater under nonstationarity; see, among others, Rothenberg and Stock (997) and Georgiev, Rodrigues and aylor (207). his concern is extensible to fractional integrated models (Beran, 994; Haldru and Nielsen, 2003; olvi, 2003), which generalize the unit-root setting and rovide a convenient form to describe long-range deendence; see Baillie (996) and Robinson (2003) for reviews. While the unit root literature has suggested alternative rocedures to deal with non-gaussian errors (see, among others, Lucas, 993; Cambell and Dufour, 995; Phillis, 995; Herce, 996; Breitung and Gourieroux, 997; Rothenberg and Stock, 997; Hasan and Koenker, 997; Wright 2000; Ling and McAleer, 2004; Koenker and Xiao 2004; and Galvao, 2009), there have been little attemts to develo robust tests for fractional integration. o the best of our knowledge, only the tests in Delgado and Velasco (2005), based on signed residuals, and Hassler, Rodrigues and Rubia (206), based on quantile regression (QR), ensure a form of robustness in the estimation of the fractional arameter in a general context characterized by either stationary or nonstationary dynamics with errors drawn from a heavy-tailed distribution. In this aer, we roose a class of tests for fractional integration in the time domain under M estimation; see Huber (98) and Amemiya (985) for a review of this toic. his framework is fairly general and encomasses di erent estimation techniques includ- Beran (994) rooses an aroximated maximum likelihood estimator based on the autoregressive reresentation of a stationary ARFIMA model which belongs to a class of M-estimators; see also Agostinelli and Bisaglia (200). Li and Li (2008) discuss the asymtotic roerties of LAD estimators of stationary ARFIMA models in a Lalace quasi-maximum likelihood estimation setting. 2

3 ing, among others, LS, QR and maximum likelihood (ML) estimation. he main interest is on the class of weighting functions that reduce the in uence of large observations and, therefore, lead to robust roerties against heavy-tailed distributions. his analysis extends the tests for fractional integration in Breitung and Hassler (2002), Demetrescu, Kuzin and Hassler (2008) and Hassler et al. (2009) to a non-gaussian framework. We discuss the asymtotic roerties of the new tests under fairly general conditions on the data generating rocess, and for di erent estimation strategies based on direct otimization of the M loss-function and on iterated k-ste and reweighted LS numeric algorithms. As in the LS context, the null asymtotic distribution of these tests is shown to be standard normal and indeendent of the value of the fractional integration arameter. Monte Carlo exerimentation shows that M tests for fractional integration exhibit emirical size close to the nominal level in nite samles and enhanced ower in relation to alternative rocedures, such as LS and even QR, when innovations are drawn from heavy-tailed distributions. he emirical section addresses the long-run dynamics of di erent volatility measures of stock market indices in develoed and emerging markets. In articular, we consider daily observations of log absolute returns and log rice range estimates (the log-transform of the sread between the highest and lowest log-asset rices over the day) in the eriod Absolute returns and related transformations are well-known roxies of volatility. Similarly, the high-low range is a highly e cient roxy of volatility as it builds on the entire intraday rice ath rather than on closing rices; see, among others, Alizadeh, Brandt and Diebold (2002) and references therein. Both measures exhibit the distinctive attern of long-range ersistence in the autocorrelation function that characterizes fractional integration, but they are drawn from statistical distributions with di erent roerties. Whereas log absolute returns are highly non-normal, the distribution of logrange estimates is aroximately Gaussian; see Alizadeh et al. (2002). Consequently, the use of di erent estimation techniques with comlementary roerties in terms of ef- ciency/robustness given the distribution of the data seems naturally motivated in this context. Given these series, we rovide formal insight on the long-term dynamics of volatility by constructing con dence intervals of the fractional arameter using a large battery of alternative estimation rinciles. hese include regression-based tests for long-memory 3

4 based on LS, QR and M estimation, the sign test in Delgado and Velasco (2005), and the frequency-domain local Whittle estimator in Shimotsu and Phillis (2005). Consistent with revious evidence, the overall results from this analysis inoint that market return volatility is driven by fractional integration. he analysis of log-range volatility estimates systematically indicates a stronger degree of ersistence than that based on log absolute returns. Because the former is widely considered as a more e cient volatility roxy (see, e.g., Andersen and Bollerslev 998; Alizadeh et al. 2002; Brandt and Diebold 2006), the related evidence is more reliable. In this analysis, M estimation roduces estimates which are not markedly di erent from those based on LS, but which nevertheless tend to exhibit smaller amlitude (i.e., smaller arameter uncertainty). his evidence, which comletely agrees with the exerimental results reorted in the Monte Carlo section, suggests that M-based testing could imrove the emirical e ciency of LS-based inference, with relative gains that deend on the extent of non-normality of the data. In ractice, these gains come at the exense of little incremental comutational cost because iterated algorithms used in M estimation do not require numeric otimization or comlex oerations. M-based inference, therefore, reresents a valuable alternative (or comlement) since it rovides signi cant re nements over LS even in a quasi-gaussian context. he remainder of the aer is organized as follows. Section 2 outlines the general fractional integration context analyzed in the aer. Section 3 details the asymtotic behavior of the test statistics. Section 4 reorts Monte Carlo simulation results on the small-samle erformance of the tests. Section 5 discusses the emirical alication. Finally, Section 6 summarizes and concludes. All roofs of the main theoretical statements are collected in a technical aendix. In what follows, ) and! denote weak convergence and convergence in robability, resectively, of a sequence of random elements when the samle length is allowed to diverge. he terms o () and O () reresent a sequence of random numbers converging to zero in robability and bounded in robability, resectively. 4

5 2 esting for fractional integration 2. he data generating rocess Assume that the observable time series, fy t g t= ; is generated as, ( L) d+ y t = " t I(t > ) () where L denotes the lag oerator, d + is a real value usually referred to as the longmemory or fractional integration arameter, I() denotes the indicator function, and f" t g is a covariance stationary and invertible noise rocess. According to this seci cation, fy t g in () is generally said to be a (ye-ii) fractionally integrated rocess of order d + ; which shall be referred to as F I(d + ) in the sequel. he fy t g variable is driven by a unit root rocess when d + =, and by a weakly-stationary rocess when d + = 0. Non-integer values in the range (0; ) give rise to long-range deendence characterized by hyerbolically-decaying imulse resonse functions (Hassler and Kokoska, 200), o ering an intermediate case between the characteristic exonential decay of short memory and the in nite ersistence of unit-root rocesses. In contrast to most of the existing tests, model () does not require that d + lies in the ( 0; 5; 0:5) interval in which fy t g is stationary and invertible. his outstanding roerty rovides a considerable degree of generality in our analysis. For a real-valued d; our main urose is to address whether fy t g is F I (d) or, equivalently, to test the null hyothesis H 0 : = 0: his hyothesis is tested against the two-sided alternative H : 6= 0, noting that one-sided alternatives are also a straightforward ossibility in this context. he following assumtion lays out the roerties of f" t g in () and comletes the characterization of the data generating rocess (DGP) considered in this aer. Assumtion. he error rocess f" t g in () is characterized as A (L) " t = e t ; with P A (L) = j= a jl j having all roots outside the unit root circle, and fe t ; F t g is a strictly stationary and ergodic Martingale Di erence Sequence (MDS) with E(e t jf t ) = 0; E(e 2 t jf t ) = 2, and E (je t j 2+ ) < for some > 2=3; where F t = fe s : s tg denotes the - eld generated by fe s ; s tg : 5

6 Remark. Assumtion allows for AR() short-run dynamics. When the innovations fe t g are i.i.d., () is often referred to as an ARFIMA(; d+; 0) rocess. For large values of, this can be seen as a truncated aroximation of an ARFIMA( ; d + ; q ) model with nite 0 and q > 0: Assumtion allows fe t g to be a conditionally homoskedastic MDS, a generalization of the i.i.d. setting usually considered in the fractional integration literature; see, e.g., Hualde and Robinson (2007). In this literature it is also customary to assume L 4 -bounded innovations, but a natural motivation for M estimators is that they can downweigh the in uence of innovations drawn from a heavy-tailed distribution for which lower-order moments may not exist. Assumtion only requires innovations to be L 2+2=3 -bounded. 2.2 Regression-based tests for fractional integration Given the fy t g series and the real value d, de ne the stochastic rocess and with f j (d)g t j=0 " t;d := ( L) d + y Xt t = j (d) y t j ; (2) j=0 Xt x t ;d := j " t j;d ; t = 2; :::; (3) j= characterized by the truncated series of olynomial coe cients in the binomial exansion ( L) d := P j=0 j (d) L j, namely, 0 (d) := ; and j (d) := j d j j (d) ; j : (4) As discussed in Breitung and Hassler (2002) and Demeterscu et al. (2008), testing the null hyothesis that d is the order of integration of fy t g in (); or H 0 : = 0; is equivalent to testing H 0 : = 0 in the LS auxiliary regression " t;d = x t ;d + X j " t j;d + v t ; (5) j= because this characterization holds exactly with := 0; j := a j ; and v t := e t under H 0 : = 0 and Assumtion. his is the result of the alication of the Lagrange Multilier (LM) rincile; see also anaka (999). Under local alternatives of the form H : = c= with a xed c 6= 0, it can be shown that = c= + O ( ) and that the regression disturbances fv t g have a fractionally integrated noise comonent. As 6

7 a result, the heterogenous behavior of and the di erent stochastic roerties of the random disturbance rovide a sound statistical basis to identify the order of fractional integration in fy t g. Desite the aarent theoretical simlicity of this framework, the fact that x t ;d converges in mean square sense to x t ;d := P j= j " t j;d under the null hyothesis and Assumtion, with x t ;d being a stationary linear rocess with non-absolutely summable coe cients, is a source of major technical di culties for the asymtotic analysis in this context. Demetrescu et al. (2008) and Hassler et al. (2009) derive the asymtotic theory of the fractional integration tests under LS estimation of the set of arameters := (; ; :::; ) 0, showing -consistency and asymtotic normality under fairly general conditions. As a result, H 0 : = 0 can be tested by means of a standard t-ratio, or measurable transformations such as its squares. If Assumtion is strengthened to require " t iidn (0; 2 ), the seci c harmonic weighting uon which x t ;d is constructed in (3) ensures e cient testing, and the squared t-statistic for H 0 : = 0 is asymtotically equivalent to the LM test for H 0 : = 0 under ML; see also Robinson (994) and anaka (999). If Assumtion holds with non-gaussian innovations, the regression-based aroach still ensures asymtotically correct nominal size, indeendently of the underlying distribution of fe t g. However, LS estimation is no longer e cient, and alternative rocedures may render more owerful tests. Hassler et al. (206) discuss the asymtotic theory for QR estimators, showing that median-based tests can largely outerform LS-based inference when innovations are driven by heavy-tailed distributions. Similarly, LS is known to be highly sensitive to extreme values, which may lead to arameter bias and wrong inference even in large samles. hese considerations rovide a natural motivation for develoing robust tests based on alternative estimation rinciles, such as M estimation. 7

8 3 M estimation 3. heoretical setu We now roceed to the asymtotic analysis of M-based tests for fractional integration. o this end, consider the following auxiliary regression: X " t;d = + x t ;d + j " t j;d + u t ; t = + ; :::; ; (6) j= or, in vector notation, " t;d = 0 x t ;d + u t (7) with := (; ; ; :::; ) 0, x t ;d := (; x t ;d ; " t ;d; :::; " t ;d ) 0 and fu t g denoting a random disturbance. Although the similitudes with (5) are straightforward, there are meaningful di erences. In the LS context, the restriction = 0 follows directly from model () and, hence, there is no need for an intercet. In contrast, in M regressions it is necessary to include this additional term because it can generally be di erent from zero. Seci cally, its the theoretical value, denoted, is determined by samle-deendent features of the data and the choice of the () function that characterizes M otimization. 2 Consequently, under the null hyothesis, it follows readily that u t := e t, and hence Var(u t ) = 2 : he M estimator of in (6), denoted b M ; can generally be de ned as the solution of the otimization roblem: min 2 Q () := min 2 t=+ "t;d 0 x t ;d b where () is a measurable function, b is a reliminary consistent estimate of the scale of the residuals, and 2 R +2 denotes the arameter sace. When = 0, then = 0 holds true in (6) ; indeendently of the choice of the () function, and therefore the null hyothesis that fy t g is F I (d) can be addressed by testing the restriction H 0 : = 0 on the solution of (8) ; with the validity of this rocedure relying once more on the LM rincile under quasi-ml estimation; see Aendix A for a discussion. he articular choice of the 2 his term corresond to the M-measure of location of the error term; see, for instance, Huber (98). Note, for instance, that in the QR setting in Hassler et al. (206), the theoretical value of the intercet corresonds to the -th conditional quantile of fe t g, which is dictated by the distribution of innovations and the (arbitrary) choice of the quantile in the estimation. (8) 8

9 () function is in general driven by e ciency, robustness, or comutational issues and characterizes the roerties of the resultant M estimator. For instance, (r=b) = r 2 leads to the LS estimation in Demetrescu et al. (2008). Also, (r=b) = r( I(r < 0)); with 2 (0; ) ; leads to the QR estimator in Hassler et al. (206), and (r=b) = ln f " (r) ; with f " (r) denoting a di erentiable density function, leads to the ML estimator. 3.2 Asymtotic distribution of the M estimator In this section, we characterize the existence, consistency, and rst-order asymtotic distribution of the solution of min 2 Q () for a class of () functions. his is the basis to construct tests for fractional integration. o this end, we rst introduce further notation and additional conditions that, together with Assumtion, conform the set of su cient conditions in our analysis. hus, de ne the variables, u t;s () := " t;d 0 x t ;d ; and u t;s () := " t;d s 0 x t ;d s for all 2 ; where s > 0 is a generic scale factor and x t ;d := ; x t ;d ; " 0 t ;d; :::; " t ;d ; with x t ;d := P j= j " t j;d. Similarly, given the theoretical value, formally de ned in Assumtion 2 below, de ne e t;s := (e t e t;s. (9) ) =s; noting that under H 0 : = 0, u t;s () := Assumtion 2. : R! R is a measurable function satisfying the following conditions: i) (r) is twice di erentiable; ii) (r) (r) =@r is bounded; iii) 0 (r) (r) =@r is rst-order Lischitz continuous and bounded; iv) E ( (e t; ) jf t ) = 0, E( 0 (e t; ) jf t ) > 0 almost surely, where is the unique, real-valued solution of mine et c jft c2 ; v) E ( 0 (e t; ) e t jf t ) and E ( 0 (e t; ) jf t ) are constant. Assumtion 3. he scale estimator b satis es b = o =4 : Assumtion ; where = is a comact subset of R +2 ; with 0 := ( ; 0 0) 0 and 0 := (0; a ; :::; a ) 0 denoting the true value of when H 0 : = 0 and the revious assumtions hold true. Remark 2. Assumtion 2 is slightly more general than related conditions in the extant literature; see, for instance, Lucas (995). 9 In this literature, it is often assumed

10 that (r) is bounded, but this restriction is not essential and can be disensed since E j u t; () j <, here imlied under Assumtion. Condition i) requires twice di erentiability, which can be weakened by simly requiring Lischitz-continuity without a ecting consistency, but i) lays a role in deriving the limiting distributions; see Aendix B for details. he boundedness and smoothness conditions in ii) and iii) aim to reduce the in uence of large observations. Condition iv) ensures that 0 is the unique solution of min 2 E u t; () under the null hyothesis. Condition v) is a technical restriction that ensures the asymtotic negligibility of certain remaining terms. It holds true, for instance, when fe t g is an i.i.d. rocess, but it may hold under more general conditions as well. Assumtion 3 requires b to be consistent at a rate greater than =4, which could be obtained from the residuals of any reliminary -consistent estimate of ; see heorem 3 below. Finally, Assumtion 4 is standard in this framework. he following heorems characterize the existence, consistency, and asymtotic null distribution of b M under H 0 : = 0 and Assumtions to 4. Detailed roofs of all these theoretical statements are rovided in Aendix B. heorem. Let fy t g t= be a samle generated according to (). Consider the otimization roblem min 2 Q () as characterized in (8) ; with f" t;dg and x t ;d generated as in (2) and (3), resectively. Under the null hyothesis, H 0 : = 0; and Assumtions to 4, there exists a random vector b M which solves min 2 Q () such that, b M! 0 (0) with 0 := ( ; 0; a ; :::; a ) 0 denoting the vector of true arameters. heorem 2. For constants K and such that K > 0 and 3=8 < < =2, de ne := f 2 : jj 0 jj Kg and let e M be the solution of min Q () : hen, 2 under the null hyothesis, H 0 : = 0; and Assumtions -4, it follows that: where := 2 A B A em ; A := E 0 (e t; ) x t 0 ) N (0; ) () ;d x0 t ;d, B := E 2 (e t; ) x t ;d x0 t ;d : 0

11 Remark 3. When the function is strictly convex such that is monotonically increasing, Q () is a strictly convex function with a unique minimum attainable in, so b M exists and is unique. Unfortunately, strict convexity is not comatible with the class of - functions tyically used in related literature and, hence, we do not imose this restriction. As a result, min 2 Q () may resent multile solutions corresonding to local minima, a well-known ractical concern in M estimation; see Amemiya (985). heorem ensures the existence of a correct solution which converges in robability to the true arameter vector. In order to characterize the asymtotic distribution of this solution, heorem 2 considers the comact ball in a local neighborhood of 0 : he bounds on are chosen to ensure that any -consistent reliminary estimator of belongs to with robability one. Since Q () is continuous on and is comact, the objective function takes its minimum value in, which ensures the existence of a local solution, denoted e M ; which necessarily corresonds to the global minimum in large samles. Consequently, heorem 2 characterizes the asymtotic behavior of the consistent solution in heorem, showing that b M with zero mean and covariance matrix. Remark 4. is -consistent and asymtotically normal distributed Consistency holds if twice di erentiability in Assumtion 2i) is relaced by Lischitz continuity, a more general condition; see Lemma A in Aendix B. his allows us to extend consistency to the QR context as a corollary of heorem 2. is characterized by (r=b) = r( QR I(r < 0)), 2 (0; ) ; i.e., a Lischitz-continuous function not di erentiable at the origin. Hence, the QR test in Hassler et al. (206) derived under i.i.d. innovations is shown to generate consistent estimates under the more general conditions considered here. Since Hassler et al. (206) require L 2+ -bounded innovations for some > 0; and Assumtion requires 2=3; this roerty may seem to come at the cost of a slight strengthening of the moment condition. Neverthelss, this requirement is due to scaling the residuals under in Assumtion 2 which, unlike the check function used for QR, is not homogenous. Consequently, for the seci c urose of showing consistency in QR, Assumtion could be weakened to simly require > 0 as in Hassler et al. (206); see Aendix B.

12 3.3 Asymtotic distribution of iterated estimators Alternatively to the direct otimization of Q (), a consistent M estimator of can be obtained using iterated numerical methods. hese build on a reliminary -consistent estimate, say b (0) ; which can be obtained from the otimization of a strictly convex function. his reliminary estimate is then iterated in a numeric algorithm without engaging in further otimization, obtaining an estimator which can be shown to be asymtotically distributed as b M ; see, for examle, Kreiss (985), Welsh and Ronchetti (2002), and Ling and Li (2003). In this section, we rst roose a suitable methodology to determine b (0), and then characterize the asymtotic null distribution of two alternative estimators building on iterated algorithms, namely, the k-iterated Newton-Rahson and the Iterated Reweighted Least Squares (IRLS) algorithms. heorem 3. Let b (0) be a -consistent estimate of : = (; ; :::; ) 0 under Assumtion. Denote be (0)t := " t;d b 0 (0)z t ;d and let b2 (0) be the samle variance of be (0)t. P Furthermore, let b (0) := arg min c2 t=+ be(0)t c and b (0) := b (0) ; b (0) 0 0 : Under Assumtions to 4 and H 0 : = 0, it then follows that b (0) = 0 + O =2 and b (0) = o =4 : Remark 5. he simlest method to construct a -consistent estimate of under Assumtion is LS, but other methods are ossible as well. Given the resulting residuals, it is straightforward to construct a consistent estimate of satisfying Assumtion 3. he simlest alternative is the standard deviation of residuals. Other alternatives building on absolute residuals, such as transformation of the mean absolute deviation, may render this roerty as well. Finally, given the regression residuals be (0)t, a reliminary estimate of arises by solving the M equation. Otimization at this stage only involves a single arameter and, hence, grid-search methods are highly e ective in ensuring convergence to the global minimum. As a result, b (0) := b (0) ; b (0) 0 0 can be seen as a two-stage consistent reliminary estimator of 0. b (0) 2

13 3.3. Iterated Newton-Rahson estimators Given b (0) and b (0), the one-ste M estimator of based on the Newton-Rahson algorithm, denoted b NR ; is determined as b NR = b (0) " b (0) t=+ 0 u t;b (0) () x t ;dx 0 t ;d # = b (0) " t=+ # u t;b (0) () x t ;d = b (0) (2) see Lehmann (983, heorems 3. and 4.2 of Chater 6). his rocedure can be iterated a nite k number of times, leading to the k-ste Newton-Rahson M estimator of. In ractice, the most common choice is k =, since a single iteration starting from a -consistent re-estimate su ces to ensure the asymtotic roerties of M estimators. he following result formally roves this statement in our context and characterizes the asymtotic distribution of b NR under the set of assumtions considered. heorem 4. Let b (0) be a reliminary estimator of such that b(0) 0 = O () and b (0) = o =4 under H 0 : = 0 and Assumtions to 4. Denote b NR as the M estimator of based on a one-ste iteration of the Newton-Rahson algorithm as de ned in (2). hen, under the set of conditions considered, bnr 0 ) N (0; ) : (3) Remark 6. heorem 4 shows that iterated estimators from the Newton-Rahson algorithm, based on a reliminary -consistent estimate, have the same asymtotic null distribution as the consistent solution from min 2 Q () : he null asymtotic distribution of b NR is not a ected by the distribution of the reliminary estimate b (0) and, remarkably, even a single iteration su ces to roduce an estimator which is asymtotically equivalent to that obtained from the direct numerical otimization of Q () : Furthermore, because a single iteration roduces a -consistent estimator of, heorem 4 alies trivially on any of the subsequent iterations, thereby characterizing the distribution of k-iterated Newton-Rahson estimators for any k when building on the estimates of the revious iteration. 3

14 3.3.2 Iterated Reweighted Least Squares (IRLS) he k-ste Newton-Rahson algorithm involves the comutation of the Hessian. Alternative methods which do not rely uon this estimation may result more attractive. Among these alternatives, the IRLS estimator is robably the most common numerical technique imlemented in ractice. his rocedure exloits the analogy between the rst-order condition of the otimization roblem and the equation vector that characterizes a simle Weighted Least Squares (WLS) roblem. In articular, if we denote! (r t ) := (r t ) =r t ; setting! (0) := 0; the rst-order condition equation system P t=+ u t;b () x t ;d = 0 that characterizes M estimators can be rewritten as P t=+! u t;b () u t;b () x t ;d = 0: As a result, b M := min 2 Q () admits an imlicit WLS-tye reresentation of the form: t=+! u t;b () x t ;dx 0 t ;d! t=+! u t;b () x t ;d" t;d : (4) his roerty suggests an iterated algorithm to aroximate the solution in the otimization roblem by recursive methods using (4). In articular, starting from a reliminary estimate b (0) ; one obtains the scaled residuals u b(0) t;b and the corresonding weights! t u b(0) t;b and, hence, a new estimator, say b IRLS;() ; by direct comutation of (4). his rocedure is then reeated a number of times until convergence, leading to the IRLS M estimator of, which we shall denote as b IRLS. heorem 5 below characterizes the asymtotic distribution of this estimator uon additional regularity conditions. heorem 5. Let b (0) be a reliminary estimate of satisfying heorem 3, and denote b IRLS as the M estimator of based on k iterations of the IRLS algorithm. Under the null hyothesis, Assumtions to 4, and when i) (r) is odd such that (0) = 0 and 00 (0) is nite, ii)! (r) is Lischitz continuous, and iii) the eigenvalues of D := I C A are smaller than unity, with C := lim! P t=+! t (e t; ) x t ;d x0 t ;d ; it follows that: when k is allowed to diverge. birls 0 ) N (0; ) 4

15 Remark 7. he condition that is odd is standard in this literature; e.g., Welsh and Ronchetti (2002). In contrast to the Newton-Rahson algorithm, the IRLS method generally yields a numerical aroximation of the true global solution. When C = A, the term D = 0 and the aroximation of the asymtotic distribution of the test is exact for any nite number of iterations. his may be the case, for instance, if has a constant, non-trivial third derivative. More generally, C may di er from A, but if the di erence is small enough because the eigenvalues of D are smaller than unity, the asymtotic null distribution of b IRLS is the same as b M when the number of iterations is allowed to diverge. It can be seen from the roof (see Aendix B) that the k-ste IRLS estimator, as diverges, has a di erent limiting distribution for each k. In this context, condition iii) in heorem 5 guarantees numerical convergence of this sequence of aroximations for a nite, large enough value of k. his is the case, for instance, when using Huber s function. 3.4 esting for fractional integration Building on any of the revious estimates of and on a consistent estimate of the asymtotic covariance matrix, a test for the order of fractional integration of fy t g can readily be imlemented through a standard t-statistic. he asymtotic distribution of this test is formally stated in the following heorem. heorem 6. Let b n M 2 S; S := bm ; b NR ; b o IRLS; ; be the M estimator of such that bm 0 = O () holds true under the resective conditions outlined in heorems 2, 4 and 5. Given bu t := " t;d b 0 Mx t u) 2, and the matrices A := P 0 bu t t=+ b M x t ;d, de ne b2 M := P t=+ (bu t ;d x0 t ;d, B := P and := b 2 M A B A ; it then follows that b2 M and, consequently,! : 2 bu t t=+ b M x t ;d x0 t ;d ;! 2 ; A! A ; B! B ; Hence, under the null hyothesis, H 0 : = 0; and the remaining assumtions considered; t M := b M b!22 = ) N (0; ) where b M and b! 22 denote the second element in b M and in the diagonal of ; resectively. 5

16 Remark 8. heorem 6 states the null distribution of the t-statistic for H 0 : = 0: It also gives a theoretical basis for the construction of con dence intervals that include the true value of the fractional arameter with 00 ( ) % asymtotic coverage by inverting the non-rejection region of t M ; see Hassler et al. (206). More seci cally, let t M (l) denote the value of t M when testing H 0 : y t F I (l) for an arbitrary l. For a closed interval R, de ne D = fs 2 : Pr [Z jt M (s)j] g with 2 (0; ) ; and Z the standard normal variate, i.e., the subset of for which the null hyothesis cannot be rejected at the signi cance level. From heorem 6, it follows that if D is in the interior of, then the robability of the true order of integration being within D is at least ( hus, a con dence interval can be constructed through a grid-search rocess in ; which is comutationally feasible because the fractional arameter tiically lies in the interval (0; ). Finally, the following theorem characterizes the asymtotic distribution of t M under sequences of local alternatives in a neighbourhood of the null hyothesis and, hence, comletes the theoretical discussion. It is shown that M-based tests for fractional integration exhibit non-trivial ower against such alternatives, so deartures from the null hyothesis will be detected with increasing robability. heorem 7. Consider a sequence of local alternatives of the form H : = c= for some nite c 6= 0: Under the assumtions of heorem 6, it then follows under the alternative hyothesis that: c t M ) N ;!22 where! 22 is the second element in the diagonal of : Consequently, Pr jt M j > z is an increasing sequence on jcj, with z standard normal distribution for the % nominal size level. in ). =2 =2 denoting the two-sided critical value of the 4 Monte Carlo analysis In this section, we analyze the nite-samle roerties of the M tests for fractional integration by means of Monte Carlo simulation. We consider two alternative -functions widely used in related literature, namely, the so-called Huber and Bisquare (or Biweight) 6

17 h -functions de ned, resectively, as (r) = 0:5r 2 I (jrj c H ) + h with c H = :345, and (r) = c 2B6 r k c H jrj k 2 2 i I (jrj > k) 2 i 3 I (jrj c B ) + c2 B 6 I (jrj > c B ) with c B = 4:685. Since IRLS is the most common aroach used in ractice, we reort results based on this algorithm, noting that results based on Newton-Rahson estimation are similar. In the emirical imlementation of the IRLS algorithm, we rst use LS to estimate a reliminary value of the sloe arameters in the augmented regression, and then infer the scatter of the residuals using the Mean Absolute Deviation, as usual in this literature. We iterate until convergence setting max k = 00. he t-statistics from the Huber and Bisquare -functions are denoted as t H M and tb M ; resectively. For benchmarking uroses, we consider the LS-based test statistics with standard errors comuted for i.i.d. innovations as in Breitung and Hassler (2000), denoted t LS, and using Eicker-White s correction against heteroskedasticity as in Demetrescu et al. (2008), denoted t HC LS. In addition, we comute the QR t-test in Hassler et al. (206) from the QR estimation of (6) at the 50th quantile, denoted t QR ; using Powell s robust standard errors with a Gaussian kernel and bandwidth arameter 0:3 min fb u ; IQR u g =5 ; where b u and IQR u denote the samle standard deviation and the samle interquartile range of the residuals of the regression, resectively. discussed reviously, both LS and QR can be seen as articular cases of the generalized M estimation framework, exhibiting di erent roerties because of the di erent choices of the weighting function imlemented. LS-based inference is exected to rovide more owerful results under normality, whereas QR-based inference can lead to enhanced ower in relation to LS under deartures from normality, as shown exerimentally in Hassler et al. (206). Finally, we also consider the sign test in Delgado and Velasco (2005), denoted DV. his test is based on the same harmonic weighting structure that characterizes the regression-based tests discussed in this aer, but has the outstanding roerty of being formally valid even if E (e 2 t ) = : On the other hand, it requires the median of e t to be zero, i.e., requires symmetric errors, which in ractice may result excessively restrictive, but which holds true in our exerimental analysis. he DV test statistic is asymtotically distributed as a standard normal distribution under the null hyothesis and, given S t;d := sign (" t;d ) ; can be comuted as DV = r 6 2 X j= j t=j+ As S t;d S t j;d (5) 7

18 4. Short-run dynamics In the rst exeriment, we consider a DGP given by ( a L)( L) + y t = e t ; t = ; :::; ; where a 2 f0; 0:5g ; fe t g are i.i.d. innovations drawn from a Student-t distribution with v 2 f2; 3; 000g degrees of freedom, and 2 f250; 500g. he case v = 000 corresonds closely to the Gaussian distribution, whereas v 2 f2; 3g are characterized by heavy-tailed distributions, having in nite variance when v = 2. As in Breitung and Hassler (2002) and Hassler et al. (206), under the null hyothesis we test for a unit root, namely, H 0 : y t F I(); noting that the true order of integration is given by d 0 = + with 2 f 0:3; :::; 0:; 0; 0:; :::; 0:3g. We then comute the di erent test stastistics against a twosided alternative at the 5% signi cance level and analyze the average rejection frequencies given 5; 000 relications of the exeriment. he case = 0 determines the emirical size of the tests, while values 6= 0 characterize the nite-samle ower behaviour. Finally, the autoregressive coe cients a = 0 and a = 0:5 allow us to analyze the erformance of the tests under errors driven by i.i.d. innovations and short-run deendence, resectively. In the latter case, LS- and M-based tests are comuted from auxiliary regressions augmented with one lag of the deendent variable, while the DV test is comuted on the residuals of a rst-order autoregression. [Insert able around here] able reorts the rejection frequencies in the i.i.d. exeriment (a = 0). Under the null hyothesis, all regression-based tests exhibit aroximately correct size with rejection frequencies close to 5%. LS-based estimation tends to yield more stable results than QR in small samles, since the latter requires numerical otimization, but these di erences tend to disaear quickly as the samle size increases. For v = 2 and = 500, t LS and t QR su er size deartures in this exeriment, which is not surrising because errors have in nite variance and this ossibility is ruled out in their theoretical derivations. In contrast, the remaining tests, including t HC LS, exhibit good size erformance in this context, even though only DV is theoretically ensured to exhibit correct nominal size (asymtotically) when errors have in nite variance. Under the alternative hyotheses, the Gaussian environment v = 000 rovides the conditions for the otimality of t LS, which generally outerforms the alternative tests. 8

19 he di erences are fairly small under M estimation and tend to be more substantial for the QR and DV tests. When v = 2 or v = 3, LS-based tests tend to become conservative in relation to QR- and M-based tests. In this context, t LS is no longer e cient, and the Monte Carlo analysis con rms that M-estimation leads to large gains in relative ower. For instance, for = 250; v = 3 and = 6:22%, resectively, whereas that of t H M and tb M 0:; the ower of t LS and t HC LS is 54:00% and is 73:62% and 72:96%, resectively. he di erences in ower between the two IRLS tests are small and tend to disaear as the samle length increases. he QR test exhibits enhanced roerties under heavy-tailed distributions, as reorted in Hassler et al. (206), but the ower of this test tends to be dominated by M testing. For instance, for = 250; v = 3 and = 0:; t QR has ower of 64:0%. he DV test dislays considerably smaller ower in comarison to the other tests, which is not surrising because sign tests are fairly robust, but known to exhibit reduced ower in small samles. he ability to reject the false null increases with the samle length. In the case of M-based tests, this is formally exected from heorem 7. For instance, for = 500; v = 3 and = 0:, the ower of t LS and t HC LS is 82:24% and 83:64%, showing sizeable increments. Similarly, the ower of the M tests, t H M and tb M ; is 96:04% and 95:74%, resectively, while the ower of t QR is 90:98%. All these tests exhibit good ower, and the di erences tend to disaear in large samles, but M-based tests clearly outerform the other alternatives under heavy-tailed innovations. Interestingly, ower exhibits an asymmetric attern such that the relative gains tend to be much larger when < 0, a feature noted in revious literature (see e.g. Hassler et al., 206). his attern is data-deendent and, for instance, tends to disaear as v aroaches 2, for which ower exhibits a more symmetric behavior. [Insert able 2 around here] able 2 reorts the rejection frequencies when the DGP is driven by weakly-deendent errors with a = 0:5. Under the null hyothesis all tests, and articularly the DV test, dislay nite samle size deartures as a consequence of augmentation. hese deartures are more evident for = 250; and with Gaussian errors. he size distortion caused by augmentation is a small-samle feature and, it is almost comletely eliminated when the samle length increases to = 500. For the DV test, the overall distortion still remains sizeable with = 500. Finally, under the alternative hyotheses, all tests dislay 9

20 signi cant ower reductions in relation to the i.i.d. context, a well-known feature caused by augmentation. Nevertheless, ower increases as jj and/or the samle size increases, showing that these distortions are a nite-samle result. In this context, the ower of all tests is characterized by a strong asymmetric attern such that when < 0 alternatives are easier detected. As in the i.i.d. case, the ower of the M- and QR-based tests largely imroves in relation to the LS-based alternatives as the degree of excess kurtosis increases, articularly. when < 0. In summary, the overall icture that emerges from this exeriment suorts the asymtotic theory discussed in heorems 6 and 7, showing that M-based tests are well-suited in nite-samles and exhibit aroximately correct emirical size. Short-run deendence can be handled successfully through augmentation, articularly in large samles, and M estimation can rovide imroved erformance over LS- and even QR-based alternatives when the data is driven by non-gaussian distributions. 4.2 MDS with time-varying volatility he second exeriment addresses the emirical size and ower of M tests when errors exhibit time-varying deendence in conditional volatility. Assumtion does not formally allow for this ossibility, but our interest is motivated by the emirical relevance of this attern in nancial variables samled at high frequencies. As in the revious exeriment, data are generated from ( L) + y t = " t ; where " t := t t ; 2 t characterizes the GARCH dynamics, and t are i.i.d. innovations drawn from a Studentt distribution with v 2 f2; 3; 000g degrees of freedom. In the exeriment, we consider two GARCH rocesses, namely, 2 t = 0:0 + 0:0" 2 t + 0:60 2 t (GARCH-A) and 2 t = 0:0 + 0:05" 2 t + 0:90 2 t (GARCH-B), characterizing di erent degrees of ersistence in volatility: Since time-varying volatility is a distinctive attern of high-frequency data, the samle length is tyically large and often sans thousands of obsevations (see, for instance, the emirical analysis in Section 5). In order to acknowledge this emirical feature, we set = 000 in this exeriment. As in the revious exeriment, LS- and M-based tests are comuted from augmented regressions with one lag of the deendent variable. [Insert able 3 around here] 20

21 able 3 resents the rejection frequencies when the volatility dynamics is generated by the GARCH-A and GARCH B rocesses. he icture that emerges from this analysis is remarkably similar to that discussed reviously. Under the null hyothesis, emirical sizes are close to the nominal level in all cases, with the degree of ersistence in volatility laying little role. For M-based estimators, size ranges between 5.28% and 6.26%. Under the alternative hyothesis, the emirical rejection frequencies of all tests are very similar regardless of whether GARCH-A or GARCH-B is considered. Remarkably, and as also observed in ables and 2, deartures from normalilty lead the M-tests tend dislay imroved ower erformance, even in relation to QR-based tests. 5 Emirical analysis: volatility of nancial assets Volatility modeling has taken a redominant osition in nancial econometrics. In this section, we analyze the long-term dynamics of two alternative daily volatility measures. On the one hand, we consider log absolute returns. Functions of absolute returns, such as log-ower transformations, are the most common roxy of the conditional variability of nancial assets and exhibit autocorrelations with hyerbolic rates of decay (e.g., Bollerslev and Wright 2000) and a considerable degree of non-normality; see, e.g., Brand and Jones (2006) and references therein. On the other hand, we consider the log transform of the high-low range rice range, a simle, yet highly e ective emirical roxy of volatility. his measure dislays the characteristic attern of strong ersistence of volatility measures, but in contrast to log absolute returns, its distribution is close to be normal. his framework oses an interesting scenario to emirically comare the erformance of M tests in relation to LS. We comute both measures on daily rices of stock market indices in develoed and emerging markets over the eriod 0/0/2000 to 3/2/206. hese include indices for the U.S. (SP500 and NASDAQ Comosite), France (CAC40), Germany (DAX30), Jaan (NIKKEI250), Sain (IBEX35), U.K. (FSE00), Brazil (BOVESPA), Hong Kong (HANG SENG), Argentina (MERVAL), and Mexico (MXX). 3 For both volatility measures, able 4 reorts standard descritive statistics (mean, standard deviation, skewness, 3 he data used to comute these measures (maximum, minimum and closing rices) are available from commercial data roviders and can be obtained freely from Yahoo Finance. 2

22 and kurtosis) as well as the Ljung-Box Q-test statistic for absence of serial correlation u to the rst 00 lags, and the Jarque-Bera test statistic for normality. he main features of this analysis shall be commented in greater detail below. [Insert able 4 around here] Given these series, we construct 95% con dence intervals for the fractional arameter by inverting the non-rejection regions of the following test statistics: the LS-based test in Demetrescu et al. (2008) comuted with robust standard errors to (conditional) heteroskedasticity, denoted t HC LS ; the QR test in Hassler et al. (206) comuted at the median with robust standard errors, as described in the Monte Carlo section, denoted t QR ; the Kolmogorov-Smirnov joint QR test to simultaneously address the null hyothesis at the quantiles [0:; 0:9], as described in Hassler et al. (206), denoted KS; and, nally, the M-based tests comuted from IRLS and one-ste Newton-Rahson algorithms, denoted IRLS and NR, resectively, with Huber and Bisquare weighting -functions, denoted with subscrits H and B, resectively (e.g., IRLS H and IRLS B ). hese algorithms are iterated starting from the LS estimation, setting the initial value of the intercet equal to zero and allowing the iterative rocedure to freely determine all the arameters involved. In the imlementation of these tests, we follow two common methodological aroaches. First, in order to account for a likely non-zero deterministic mean in the level of the volatility measures, we use the demeaning rocess described in Robinson (994); see also Demetrescu et al. (2008) and Hassler et al. (206). In articular, model () can be generalized by setting y t = + ( L) d " t I(t > ) with 6= 0: Under H 0 : = 0; ( L) d + y t = ( L) d + + " ti(t > ), so can be estimated consistently under Assumtion from the regression of d +y t := P t j=0 j (d) y t j on b t;d := P t j=0 j (d) ; t = 2; :::; ; with f j (d)g as de ned in (4) : he residuals of this regression corresond to f" t g and, therefore, it su ces to rede ne (2) as " t;d := d +y t bb t;d, with b denoting the estimate of, to remove the e ects of the deterministic mean. We determine b from a LS regression in all cases, noting that alternative estimation methods (e.g., M or QR estimation) may be valid as well. Secondly, the auxiliary regressions are augmented based on Schwert s rule, h i.e., := 4 (=00) =4i ; see Demetrescu et al. (2008) for a discussion on the convenience of this rocedure. Finally, together with these tests, we imlement the DV sign test and the Exact Local Whittle (ELW) estimator in Shimotsu and Phillis (2005). In order to account 22

23 for short-run deendence in the DV test, this test is comuted on the residuals of an AR(). he 95% con dence intervals are then constructed by inverting the emirical nonrejection regions of this test. he con dence intervals from the ELW are constructed from oint-estimates based on estimation with bandwidth arameter [ 0:6 ], using asymtotic standard errors, and building on the asymtotic normality of this estimator. 5. Log absolute returns For illustrative uroses, Figure dislays the time-series dynamics, the samle histogram (confronted with the theoretical normal distribution), and the samle autocorrelation function (with asymtotic 95% con dance bands) of the log absolute-valued returns of the SP500 index. As exected, log absolute returns exhibit a attern of slow-decaying autocorrelations that suggests long-range ersistence. Furthermore, as discussed reviously, these series are highly non-normal owing to the occurrence of extreme returns that cause excess kurtosis and skewness, even after alying the log-transform. For instance, the log absolute return of the SP500 index has samle skewness and kurtosis of :09 and 5:27, resectively, so normality is strongly rejected according to the Jarque-Bera test owing to extreme observations in the left tail of the distribution. he remaining series are characterized by similar characteristics; see able 4, Panel A. [Insert Figure around here] able 5 reorts the 95% con dence interval estimates for the fractional arameter from the di erent testing rocedures. In this analysis, the DV test always rejects the null hyothesis, failing to rovide reasonable estimates, so we do not reort results for this statistic. his evidence should not be surrising in view that log absolute returns are strongly left-skewed and that the DV test builds on the critical assumtion that the underlying distribution is symmetric. As a result, the test is largely biased towards overrejection, with rejection signalling that (at least) one of the key assumtions that de ne the DGP under the null is not suorted by the data. [Insert able 5 around here] In contrast, the results from the alternative time- and frequency-domain tests suggest 23

24 strong evidence of fractional integration, with the hyotheses of F I(0) or F I() dynamics being largely rejected in all cases. Excluding MERVAL, all con dence intervals include d = 0:4, considered as the characteristic value of the long-memory arameter in emirical studies involving daily transformations of squared returns or realized volatility; see, for instance, Bollerslev and Wright (2000) and Andersen et al. (200, 2003). For summarizing uroses, the column labelled Intersection in able 5 shows the set of values for which the null hyothesis cannot be rejected by any of the regression-based tests, i.e., a core set of values for which there is methodological agreement. his region is essentially formed by values smaller than the cut-o level d < 0:5, essentially suggesting that log absolute returns are driven by a stationary long-memory rocess. he results from local Whittle estimation agree with this evidence. Particularizing on M estimates, there are minor di erences attending to the choice of the weighting function or the iterative algorithm. Con dence intervals are not markedly di erent from those based on LS estimation, which is not surrising since the emirical analysis involves a fairly large samle and LS-based inference is consistent and should not be a ected critically by the distribution of the data rovided regularity conditions. As discussed reviously, however, LS is not e cient in a non-gaussian context, and it is worth noting that the amlitudes of the M con dence intervals tend to be smaller than their LS counterarts in all cases. As shown in the Monte Carlo section, this evidence is consistent with relative gains in ower erformance. Similarly, M con dence intervals are smaller than their QR counterart, which, again, agrees with the nite-samle erformance exhibitted by M and QR tests under heavy-tailed innovations in the Monte Carlo analysis. In order to give a sense of the relative size of these reductions, the last two columns in able 5 reort the average relative change in the amlitude of the con dence intervals when moving from LS to M estimation with IRLS and NR algorithms, resectively. 4 Relative changes are determined as RC := (A M A LS ) =A LS, with A M and A LS denoting the amlitude of the M- and LS-based con dence interval, resectively. For ease of resentation, we reort the average values of RC based on the estimates from the two di erent weighting functions. For instance, for the log absolute return of the SP500, the 4 Note that this is an intuitive way to araise the di erences in the estimation. We do not ursue a formal, statistical analysis to determine if, for instance, reductions are statistically signi cant. 24