Robust testing of time trend and mean with unknown integration order errors

Transcription

1 Robust testing of time trend and mean with unknown integration order errors Jiawen Xu y Shanghai University of Finance and Economics and Key Laboratory of Mathematical Economics Pierre Perron z Boston University March 3, 23; Revised January 3, 24 Abstract We provide tests to perform inference on the coe cients of a linear trend assuming the noise to be a fractionally integrated process with memory parameter d 2 ( :5; :5) by applying a quasi-gls procedure using d-di erences of the data. Doing so, the error term is short memory, the asymptotic distribution of the OLS estimators applied to quasi-di erenced data and their t-statistics are una ected by the value of d and standard procedures have a limit normal distribution. No truncation or pre-test is needed given the continuity with respect to d. o have feasible tests, we use the Exact Local Whittle estimator of Shimotsu (2), valid for processes with a linear trend. he nite sample size and of the tests are investigated via simulations. We also provide a comparison with the tests of Perron and Yabu (29) valid for a noise component that is I() or I(). he results are encouraging in that our test is valid under more general conditions, yet has similar as those that apply to the dichotomous cases with d either or. JEL Classi cation: C22 Keywords: fractional integration, long-memory, linear time trend, inference, con- dence intervals, quasi-gls procedure. We are grateful to two referees and the editor for useful comments. y School of Economics, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai, China 2433, and Key Laboratory of Mathematical Economics (SUFE), Ministry of Education, Shanghai 2433, China (xu.jiawen@mail.shufe.edu.cn). z Corresponding author: Department of Economics, Boston University, 27 Bay State Rd., Boston MA 225 (perron@bu.edu).

2 Introduction Many time series are well captured by a deterministic linear trend. With a logarithmic transformation, the slope of the trend function represents the average growth rate of the time series, a quantity of substantial interest. o be more precise, consider the following model for the time series process y t : y t = + 2 t + u t ; () where u t are the deviations from the trend. he parameter 2 is of primary interest. If 2 =, then tests about pertains to the mean of the time series. Hypothesis testing on the slope of the trend function is important for many reasons. First, assessing whether a trend is present is of direct interest in many applications. Second, the correct speci cation of the trend function is important in other testing problems, such as assessing the nature of the noise component u t. hird, tests for hypotheses about the values of and 2 allow constructing con dence intervals via inversions. here is a large literature on issues pertaining to inference about the slope of a linear trend function, most related to the case where the noise component is stationary, i.e., integrated of order zero, I(). A classic result due to Grenander and Rosenblatt (957) states that the estimate of 2 obtained from a simple least-squares regression of the form () is asymptotically as e cient as that obtained from a Generalized Least Squares (GLS) regression when the process for u t is correctly speci ed. However, when u t has an autoregressive unit root, i.e., integrated of order one, I(), the estimate of the mean of the rst-di erenced series is e cient in large samples. Several papers tackled the issue of constructing tests and con dence intervals for the parameter 2 when it is not known a priori if u t is I() or I(). Sun and Pantula (999) proposed a pre-test method which rst applies a test of the unit root hypothesis and then chooses the critical value to be used according to the outcome of the test. Since the probability of using the critical values from the I() case does not converge to zero when the errors are I(), the simulations reported accordingly show that substantial size distortions remain. Canjels and Watson (997) considered various Feasible GLS methods. heir analysis is, however, restricted to the cases where u t is either I() or the autoregressive root is local to one. hey do not allow I() processes and, moreover, their method yields con dence intervals that are substantially conservative with common sample sizes. Roy et al. (24) considered a test based on a one-step Gauss Newton regression but its limit distribution is not the same in the I() and I() cases (see Perron and Yabu, 22). Vogelsang (998), Bunzel and Vogelsang (25) and Harvey et al. (27) proposed tests valid with either I() or I() errors. heir

3 approach, however, uses randomly scaled versions of tests for trends so that in nite samples the good properties of such tests are lost, at least to some extent. Perron and Yabu (29) considered a Feasible Quasi GLS approach that uses a supere cient estimate of the sum of the autoregressive parameters when =. he estimate of is the OLS estimate obtained from an autoregression applied to detrended data and is truncated to take a value when the estimate is in a neighborhood of. his makes the estimate super-e cient when = and implies that inference on the slope parameter can be performed using the standard normal or chi-square distribution whether = or jj <. Much of the literature focused on u t being I() or I(), special cases of fractionally integrated, I(d), processes with memory parameter d. Since d can take any real value (within some interval), a long-memory process extends the classical dichotomy of I() and I() processes. Our aim is to provide tests to perform inference on the coe cients of a linear trend function assuming the noise component to be an I(d) process with d 2 ( :5; :5). he methodology is similar to that in Perron and Yabu (29) and applies a quasi-gls procedure using d-di erences of the data. he error term is then short memory and the asymptotic distribution of the OLS estimators of ( ; 2 ) and their t-statistics are una ected by the value of d and standard OLS procedures can be applied with the limit normal distribution. No truncation or pre-test is needed given the continuity with respect to d. o make our procedure feasible, we need an estimator of d valid with a tted linear time trend and for a wide range of d. After experimenting with various possible estimators, we opted to use the Exact Local Whittle (ELW) estimator of Shimotsu (2) who extended Shimotsu and Phillips (25) to cover processes with a linear trend. It is valid for values of d in the range ( :5; :5) and yields tests with good nite sample properties. A related paper is Iacone and al. (23) who proposed a test for a break in the slope of a linear time trend when the order of integration is unknown, whose methodology is similar to ours. Also, Abadir, Distaso and Giraitis (2) considered an I(d) model with trend and cycles and derived the asymptotic distribution of the OLS estimate of the parameter of the slope of the trend. his note is organized as follows. Section 2 describes the model and the test statistics, and Section 3 the estimate of d used to have feasible tests. Section 4 presents simulation results about the size and of the tests in nite samples. We provide a comparison with the tests of Perron and Yabu (29) valid when u t is either I() or I(). he results are encouraging in the sense that our test is valid under much more general conditions, yet has similar as those that apply only to the dichotomous cases with d either or. Section 5 provides brief conclusions and a mathematical appendix some technical derivations. 2

4 2 he model and test statistics he data-generating process is assumed to be: y t = + 2 t + u t (2) for t = ; :::;, with u t a fractionally integrated process satisfying the following assumptions. Assumption : he process u t is generated by d u t = ( L) d u t = " t {ft g, where d is the fractional di erence operator and {fag is the indicator function of the event A. Also, " t is a linear short memory process generated by " t = A(L)v t = P j= A jv t j with A() 2 >, P l= lja lj <, v t i:i:d: (; 2 v) and Ejv t j q < with q > max(4; 2=(3 2d)). Assumption 2: f " () is bounded for 2 [; ]; f " () G 2 (; ) as! + and, for some 2 (; 2], f " () = G ( + O( ); in a neighborhood (; ) of the origin, A(e i ) is di erentiable and (d=d)a(e i ) = O( ) as! +. Assumptions -2 are mostly from Shimotsu (2) and allow the estimate of d to be consistent and asymptotically normally distributed. Assumption strengthens some of his conditions in order to have a functional central limit theorem for the partial sums of the u t. he stated conditions for this to hold follow Marinucci and Robinson (2). Applying a d-di erencing transformation, the DGP can be written as: y d t d y t = d {ft g + 2 d t{ft g + d u t {ft g; (t = ; :::; ) Note that d u t = " t and d y = y : We also de ne X t = [; t] and Xt d d X t = [ d {ft g; d t{ft g] with d X = [; ]. Hence, the GLS transformed regression is: y d t = X d t + " t ; (t = ; :::; ) o obtain a feasible regression, we need to replace d by some consistent estimate ^d to be discussed in the next section. he tests will then be based on the regression y ^d t = X ^d t + u ^d t ; (t = ; :::; ) (3) where u ^d = ^du t ft g. Let ^ = (X ^d X ^d) X ^d y ^d denote the OLS estimator of [ ; 2 ], where X ^d = [X ^d ; :::; X ^d ] and y ^d = [y ^d ; :::; y ^d ]. he test statistic on the time trend coe cient 2 for H : 2 = 2 against H : 2 6=, is constructed as the usual t-statistic: t^2 = R(^ )=[^ 2 R(X ^d X ^d) R ] =2 3

5 where R = [ ], = ( ; 2) and ^ 2 is a consistent estimator of long-run variance 2 = P j= (j) where (j) = E(" t" t j ). Similarly, the test statistic on the constant term for H : = can also be constructed as usual with: t^ = R (^ )=[^ 2 R (X ^d X ^d) R ] =2 where R = [ ]. he following theorem provides the limit distribution of the test statistics. heorem Let fy t g be generated by (2) under Assumptions -2. Suppose that we have estimates ^d and ^ 2 such that ^d d = O p ( ) for some > and ^ 2 2 = o p (). hen, a) under H : 2 = 2, t^2! d N(; ) for any d 2 ( :5; :5); b) under H : =, t^! d N(; ) for any d 2 ( :5; :5). A consistent estimate of 2 is readily available. Popular estimates are weighted sums of autocovariances of the form ^ 2 = ^()+2 P j= (j; m)^(j), where ^(j) = P t=j+ u ^d t u ^d t j with u ^d t the OLS residuals from the regression (3) and () a kernel function with bandwidth m. In the simulations below, we use the Bartlett kernel and Andrews (99) data dependent method for selecting the bandwidth based on an AR() approximation. he choice of an appropriate estimate of d is more delicate and discussed in the next section. 3 Estimate of d he Exact Local-Whittle (ELW) estimation procedure for the order of fractional integration of a process was studied by Shimotsu and Phillips (25). It was subsequently extended by Shimotsu (2) to cover the case with an unknown trend function, a needed feature in our context. It is also valid under a wide range of possible values for d including values greater than. Accordingly, we shall adopt it as the estimator of d to be used in constructing our test statistics. o describe the estimation procedure, de ne the discrete Fourier transform and the periodogram of y t evaluated at the fundamental frequencies as! y ( j ) = p 2 X t= y t exp(it j ); I y ( j ) = j! y ( j )j 2 for j = (2j= ), j = ; :::;. he ELW estimator of d is the minimizer of Q m (G; d) = m X m j= [log(g 2d j ) + G I d y( j )]: Concentrating Q m (G; d) with respect to G, the objective function is R(d) = log ^G(d) 2d(m ) P m j= log( j), where ^G(d) = m P m j= I y( d j ) and, within a pre-speci ed range 4

6 to be de ned below, the ELW estimator is d e = arg min d2[ ; 2 ] R(d). Shimotsu (2) extended the ELW estimation procedure to cover an unknown linear time trend via a twostep procedure applied to detrended data. he rst step detrends the data by an OLS regression of y t on (; t) with the residuals denoted ^y t. he modi ed objective function is then: R F (d) = log ^G F (d) 2d m X m X m j= log( j); ^GF (d) = m j= I d (^y '(d))( j ) where '(d) = ( w(d))^y with w(d) a twice continuous di erentiable weight function such that w(d) = for d =2 and w(d) = for d 3=4. As recommended by Shimotsu (2), w(d) = (=2)[ + cos(4d)] for d 2 [=2; 3=4]. A two-step procedure is applied to ensure the global consistency of the estimate. In the rst step, one uses the tapered local Whittle estimator of Velasco (999) denoted ^d, which is p m-consistent and invariant to a linear trend for d 2 ( =2; 5=2). he second step estimator involves the following modi cation: ^d ELW = ^d RF ( ^d )=R F ( ^d ) (4) where RF ( ^d ) and R F ( ^d ) are the rst and second derivatives of R F (d). Following Shimotsu (2), we use max[rf ( ^d ); 2] to improve the nite sample properties. he nal estimator, denoted ^d ELW, is obtained iterating (4). Now consider the following assumption. Assumption 3: a) (=2) < < 2 (7=4); b) as!, m + m +2 (log m) m log!, for any >, with as de ned in Assumption 2. From Shimotsu (2), under Assumptions A-A3, p m( ^d ELW d)! d N(; =4). Hence, if our test statistics are constructed using ^d ELW, heorem continues to hold. 4 Simulation results In this section, we consider the size and of the test for the slope of the trend via t^2 simulations, using, replications throughout (the results for the test for the mean are t^ qualitatively similar for the range d 2 ( :5; :5)). he data are generated by (2) with u t an ARF IMA(p; d; q) of the form ( L) d u t = " t {ft g with A(L)" t = B(L)e t, where A(L) = a L ::: a p L p and B(L) = +b L+:::+b q L q are the autoregressive and moving average lag polynomials, respectively, and e t i:i:d N(; ). Assumptions A and A2 are satis ed if the roots of A(L) = and B(L) = are outside the unit circle. In all cases, we set = 2 = under the null hypothesis without loss of generality. Also, the estimate ^d ELW is constructed with m = :65. We consider two-sided tests at the 5% signi cance level and for 5

7 d = or, the results are compared to those obtained with the two versions of the Perron and Yabu (29) tests, t F S S (MU) or tf (UB), which use di erent autoregressive estimates before applying the truncation (MU stands for Median Unbiased and UB for Upper Biased). We start with the case of pure fractional processes with A(L) = B(L) =. We consider the range d 2 [ :4; :4] and = 5; and 2. he results, presented in able, show that the exact sizes of the test t^2 are close to the nominal size in all cases. On the other hand, t F S S (MU) and tf (UB) show substantial size distortions unless d = ;. When d is negative the tests are very conservative, while when < d <, the tests are liberal. he liberal size distortions are especially pronounced when d = :4. he functions for a two-sided test of 2 = are presented in Figure for = 5. Given the size distortions of the Perron and Yabu (29) tests when d is di erent from and, we include them only for the case d = (we return below to the case d = ). When d =, t F S S (MU) and tf (UB) have higher, as expected. his due to the fact that the Perron and Yabu (29) tests apply a truncation to when the autoregressive parameter is in a neighborhood of leading to a smaller bias when d =. However, the di erences are not large and decrease as increases (from unreported simulations). As expected, the of is highest when d is t^2 small with the decreasing monotonically as d increases (note the di erent scaling on the horizontal axis). able 2 presents results about the size of the tests for processes with short-run dynamics of the autoregressive form with an AR() so that A(L) = al with d =, cases for which the Perron and Yabu tests were designed. We consider values of a ranging from to :95. he results show that the exact size remains close to the nominal 5% level, unless a is close to, in which case the exact size of is below nominal size. It is well known that t^2 in the presence of a short-run component that has strong correlation, most estimates of d are biased. Accordingly, it is of some comfort to see that our test retains decent size and exhibits no liberal size distortions. he functions for a two-sided test of 2 = are presented in Figure 2 for = 5. When a =, :3 or :5, all tests have essentially the same. When a = :7 or :9, the Perron and Yabu tests have slightly higher. When a = :95, t^2 has much higher, despite being conservative, unless the alternative is close to the null value. We next consider the size and of the tests using ve di erent DGPs used in Qu (2), which were motivated by nancial applications of interest. hese are given by: DGP. ARFIMA(; d; ): ( a L)( L) :4 " t = e t, where a = :4 and :4: DGP 2. ARFIMA(; d; ): ( L) :4 " t = ( + b L)e t, where b = :4 and :4: 6

8 DGP 3. ARFIMA(2; d; ): ( a L)( a 2 L)( L) :4 " t = e t, with a = :3, a 2 = :5: DGP 4. " t = z t + t, where ( L) :4 z t = e t and t i:i:d N(; var(z t )): DGP 5. ( L) :4 " t = t with t = t e t ; 2 t = + : 2 t + :85 2 t : In all cases, e t i:i:d: N(; ). DGPs -3 are di erent cases of ARFIMA processes, DGP 4 is a fractionally integrated process with measurement errors and DGP 5 is a GARCH process. Note that DGPs 4 and 5 do not satisfy the conditions of Assumptions -2. We nevertheless include them to assess the robustness of the results given that conditional heteroskedasticity and measurement errors are prevalent features of many time series. Given the size distortions of the Perron and Yabu (29) tests when d is di erent from or, we only present results for the test. t^2 able 3 presents the exact sizes of the tests. In all cases, the exact size of is near t^2 5%, except for DGPs 2 and 5 for which the test has slight liberal size distortions when = 5, which decrease as increases. he functions of the test for = 5 are presented in Figure 3. In all cases, increases rapidly to as 2 deviates from, with the exception perhaps of the case with GARCH errors. Comparing the results across DGPs, decreases when additional short-run dynamics is present. he e ect of measurement errors on the is minor. 5 Conclusion We provided tests to perform inference on the coe cients of a linear trend function assuming the noise to be a fractionally integrated process with memory parameter in the interval ( :5; :5). he results are encouraging in the sense that our test is valid under much more general conditions, yet has similar to the Perron and Yabu (29) tests that apply only to the dichotomous cases with d either or. When d is di erent from or, its exact size is close to the nominal size and is good. Our procedure provides a useful tool for inference about the coe cients of a linear trend under general conditions on the noise component. hough we assumed the errors to follow a ype II long-memory process, we conjecture that our results remain valid with a ype I process as de ned by Marinucci and Robinson (999). First, as Shimotsu (2) argues, his results remain valid for both types of processes. Also, the conditions for a functional central limit theorem for ype I processes are very similar, see e.g., Wang et al. (23) and could be slightly modi ed accordingly. 7

9 References Abadir, K. M., Distaso, W. and Giraitis, L. (2) An I(d) model with trend and cycles. Journal of Econometrics 63, Andrews, D. W. K. (99) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, Bunzel, H. and Vogelsang,. J. (25) Powerful trend function tests that are robust to strong serial correlation with an application to the Prebish-Singer hypothesis. Journal of Business and Economic Statistics 23, Canjels, E. and Watson, M. W. (997) Estimating deterministic trends in the presence of serially correlated errors. Review of Economics and Statistics 79, Grenander, U. and Rosenblatt, M. (957) Statistical Analysis of Stationary ime Series, New York: John Wiley. Harvey, D. I., Leybourne, S. J. and aylor, A. M. R. (27) A simple, robust and ful test of the trend hypothesis. Journal of Econometrics 4, Iacone, F., Leybourne, S. J. and aylor, A. M. R. (23) esting for a break in trend when the order of integration is unknown. Journal of Econometrics 76, 345. Marinucci, D. and Robinson, P. M. (999) Alternative forms of fractional Brownian motion. Journal of Statistical Planning and Inference 8, -22. Marinucci, D. and Robinson, P. M. (2) Weak convergence of multivariate fractional Brownian motion. Stochastic Processes and their Applications 86, 3-2. Perron, P. and Yabu,. (29) Estimating deterministic trends with an integrated or stationary noise component. Journal of Econometrics 5, Perron, P. and Yabu,. (22) esting for trend in the presence of autoregressive error: a comment. Journal of the American Statistical Association 7, 844. Qu, Z. (2) A test against spurious long memory. Journal of Business & Economic Statistics 29, Robinson, P. M. (25) E ciency improvements in inference on stationary and nonstationary fractional time series. Annals of Statistics 33, Robinson, P. M. and Iacone, F. (25) Cointegration in fractional systems with deterministic trends. Journal of Econometrics 29, Roy, A., Falk, B. and Fuller, W. A. (24) esting for trend in the presence of autoregressive errors. Journal of the American Statistical Association 99, Shimotsu, K. (2) Exact local Whittle estimation of fractional integration with unknown mean and time trend. Econometric heory 26,

10 Shimotsu, K. and Phillips, P. C. B. (25) Exact local Whittle estimation of fractional integration. Annals of Statistics 33, Sun, H. and Pantula, S. G. (999) esting for trends in correlated data. Statistics and Probability Letters 4, Velasco, C. (999) Gaussian semiparametric estimation of non-stationary time series. Journal of ime Series Analysis 2, Vogelsang,. J. (998) rend function hypothesis testing in the presence of serial correlation. Econometrica 66, Wang, Q., Lin, Y.-X. and Gulati, C. M. (23) Asymptotics for general fractionally integrated processes with applications to unit root tests. Econometric heory 9,

11 Appendix Proof of heorem : We start by assuming that d and 2 are known and then show that the results remain the same under the condition stated. Consider rst part (a). Let K = diagf =2 d ; 3=2 d g, X d = [X d ; :::; X d], Xt d = [ ;t ;t ], i;t = d t i {ft g for i = f; g and " = [" ; :::; " ] then: t^2 = R(^ ) R(K = Xd X d K ) (K Xd ") [ 2 R(X d X d ) R ] =2 [ 2 R(K Xd X d K ) R ] =2 Lemma A. () for :5 < d < :5: t^2 d! RC L=[RC R ] =2 A, where 2 C = 4 f ( d)} 2 ( 2d) ( d) (2 d)(2 2d) ( d) (2 d)(2 2d) { (2 d)} 2 (3 2d) 3 2 5, L = 4 R ( d) (2 d) r d dw (r) R r d dw (r) 3 5 and A = p 3 (2) for :5 d < :5: 2d (2 2d) Z r d dw (r) ( 2d) Z r d dw (r) = R( K e Xd X d K e ) ( K e Xd ") d t^2 [ 2 ( K e Xd X d K e )! R C e L e ] =2 [ 2 RC e R ] =2 = C =2 22 L 2 = p 3 2d Z r d dw (r) A 2 where e K = diagf; 3=2 d g with C 22 and L 2 the relevant sub-matrices of C and L: Proof: From Lemma of Robinson (25), as t!, for d 2 (; ), d {ft g = ( d) t d + O(t ) and d t{ft g = (2 d) 2 t d + O(). For d 2 (; :5); d t{ft g = (2 d) t d + O(t ). From (A.34) of Robinson and Iacone (25), for any r 2 (; ], we have a) for d 2 (; :5): d d {f[r ] g! ( d) r d ; b) for d 2 (; :5), d d [r ]{f[r ] g! (2 d) r d. he facts that K Xd X d K! C and K Xd "! L are proved in (A.36) of Robinson and Iacone (25). For part (2), the result follows given that, when :5 d < :5; ek Xd X d e K! 2 4 C 22 Lemma A.2 A and A 2 have a N(; ) distribution: 3 5 and K e Xd "! L e L 2 A-

12 p Note that A P 2 can be approximated by the sum of functions of normal random variables 3 2d d 3=2 j= j d e j, with e j i:i:d: N(; ) so that P j= j d e j is N(; P j= j2( d) ). According to Faulhaber s formula, P j= jp = [(B+ ) p+ B p+ ]=(p+), with B the Bernoulli number. Also, P j= j2( d) ' 3 2d =(3 2d), so that p 3 2d P d 3=2 j= j d e j is N(; ). Similarly, A can be approximated by (2 2d) p X p 3 2d d 3=2 j d e j 3 2d( 2d) d =2 j= X j d e j : he rst term is N(; (2 2d) 2 ) and the second is N(; (3 2d)( 2d)). he covariance of the two terms is (3 2d)( 2d), so that A is N(; ), since (2 2d) 2 + (3 2d)( 2d) 2(3 2d)( 2d) =. For part (b), we have! d R t^ C L=[ 2 R C R] =2 B, where j= B = p 2d[(2 2d) Now, B can be approximated by Z Z r d dw (r) (3 2d) r d dw (r)]: (2 2d) p 2d d =2 P j= j d e j p 2d(3 2d) d 3=2 P j= j d e j : he rst term is N(; (2 2d) 2 ) and the second is N(; (3 2d)( 2d)): heir covariance is (3 2d)( 2d), so that B is N(; ), since (2 2d) 2 +(3 2d)( 2d) 2(3 2d)( 2d) =. It remains to show that the results remain the same with estimates of d and 2. he fact that the results remain the same when using a consistent estimate of 2 is trivial, hence we concentrate on using an estimate of d. We need to show that if ^d d = O p ( ) for any > ; then (a) for jdj < :5; and K X ^d X ^dk K Xd X d K d! (A.) K X ^d u ^d K Xd " d! (A.2) where X ^d = [X ^d ; :::; X ^d ], u ^d = [u ^d ; :::; u ^d ], X ^d t = [^ ;t ^ ;t ], u ^d t = ^du t {ft g, and ^ i;t = ^dt i {ft g for i = f; g; (b) for :5 d < :5; and ek X ^d X ^d K e ek Xd X d K e d! (A.3) Note that from Iacone et al. (23, A.22), ek X ^d u ^d e K Xd " d! (A.4) ^ i;t i;t = o p (t i d ) (A.5) A-2

13 if ^d d = O p ( ) for some >. Consider rst the case jdj < :5. We need to show that X 2d i j ( for i; j = f; g, or equivalently, that t= ^ i;t^ j;t X i;t j;t )! d t= X 2d i j [ (^ i;t i;t ) j;t + t= X i;t (^ j;t j;t ) + t= X (^ i;t i;t )(^ j;t j;t )] t= d! (A.6) We have X X X 2d i j (^ i;t i;t ) j;t 2d i j [ (^ i;t i;t ) 2 t= t= t= 2 j;t] =2 (A.7) Using (A.5) and the fact that j i;t j Ct i d for d 2 ( :5; :5) from Lemma of Robinson (25), (A.7) is o p ( 2d i j+i d+=2+j d+=2 ) = o p (). Using similar arguments, the other terms in (A.6) are o p (), which establishes (A.). For d 2 ( :5; :5), we want to show that X d 3=2 ^ ;t u ^d t= X d 3=2 ;t " d t! t= or, equivalently, X d 3=2 [ (^ ;t ;t )" t + t= X ;t (u ^d t= " t ) + X t= (^ ;t ;t )(u ^d " t )] d! According to (A.29) and (A.3) in Iacone et al. (23), X d 3=2 ;t (u ^d t= " t ) = d 3=2 X t= + d 3=2 X BX ;t r! ( ^d d) r g (r) (" t ; ) t= r= ;t B! ( ^d d) B g (B) (" t ; e d d) where jd e dj j ^d dj, g (r) (" t ; v) = P t s= a(r) s (v)" t s and a (r) s (v) r (v) s =@v r. Also j P t= g(r) (" t ; )j = O p ( =2 ) and j ;t+ ;t j Ct d, hence X X d 3=2 r ;t g (r) (" t ; ) d 3=2 r j ;t+ ;t jj t= t= s=t = o p ( d 3=2 r d+=2+ ) = o p () A-3 X g (r) (" s ; )j

14 X d 3=2 B ;t g (B) (" t ; d e X d) d 3=2 B [ t= 2 ;t t= t= for B > =. According to Lemma 4 of Robinson (25), with (r) i;t ^ i;t i;t = BX r= = (ln ) r d t i {ft g and e (B) i;t X X d 3=2 (^ ;t ;t )" t = d 3=2 and t= d 3=2 r X t= t= X (g (B) (" t ; d e d)) 2 ] =2 = O p ( d 3=2 B+ d+ 2 + ) = O p ( B ) = o p () r! ( ^d d) r (r) i;t + B! ( ^d d) B e (B) i;t = (ln ) B ed t i {ft g. Hence, BX " t r= (r) d 3=2 r ;t " t r! ( ^d d) r (r) ;t + d 3=2 X t= j (r) ;t+ (r) ;t jj X t= X " s j = o p () s=t (A.8) " t B! ( ^d d) B e (B) ;t since j P s=t " sj = O p ( =2 ) and j (r) ;t+ (r) ;t j = O((ln ) r t d ). Also, P d 3=2 B t= " te (B) ;t = o p () using similar arguments. Hence, P d 3=2 t= (^ ;t ;t )" t = o p (). Similarly, we can sow that P d 3=2 t= (^ ;t ;t )(u ^d " t ) = o p (). It remains to show that for d 2 ( :5; :5) or, equivalently, d =2 [ From (A.8), X (^ ;t ;t )" t + t= X d =2 ^ ;t u ^d t= X ;t (u ^d t= X X d =2 (^ ;t ;t )" t = d =2 since, t= d =2 r X t= t= BX " t r= " t (r) ;t d =2 r j (r) ;t jj X d =2 ;t " d t! " t ) + t= X t= (^ ;t r! ( ^d d) r (r) ;t + d =2 ;t )(u ^d X t= " t )] d! (A.9) " t B! ( ^d d) B e (B) ;t = o p () X " s j = O p ( d =2 r d+=2 ) = o p () and j (r) ;t j = O((ln t) r (t d + (ln t) t )). Using similar arguments, the other two terms in (A.9) are also o p (). his completes the proof of (A.)-(A.4). s= A-4

15 able : Finite Sample Size; Pure Fractional Processes. d ELW MU UB ELW MU UB ELW MU UB able 2: Finite Sample Size; AR() Processes with d=. AR ELW MU UB ELW MU UB ELW MU UB able 3 Finite Sample Sizes; DGP -5 with d=.4 DGP- DGP- DGP-2 DGP-2 DGP-3 DGP-4 DGP-5 AR=.4 AR=-.4 MA=.4 MA=-.4 AR=.3, Measurement AR2=.5 error GARCH

16 Figure : Unadjusted for pure fractional processes.2 d=.4 ELW.2 d= x 3 d= d= d= d= ELW MU UB Figure 2: Unadjusted for AR() processes.2 a=.2 a= ELW MU UB a=.5 a= a=.9 a=

17 Figure 3: Unadjusted for DGP -5 AR=.4,d=.4 AR=.4,d= ELW MA=.4,d= MA=.4,d= AR=.3,AR2=.5,d= measurement error GARCH