Inference on a Structural Break in Trend with Fractionally Integrated Errors

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1 Inference on a Structural Break in rend with Fractionally Integrated Errors Seong Yeon Chang y Xiamen University Pierre Perron z Boston University April, ; Revised: September, 5 Abstract Perron and Zhu (5) established the consistency, rate of convergence and the limiting distributions of parameter estimates in a linear time trend with a change in slope with or without a concurrent change in level. hey considered the dichotomous cases whereby the errors are shortmemory stationary, I(), or have an autoregressive unit root, I(). We extend their analysis to cover the more general case of fractionally integrated errors for values of d in the interval ( :5, :5) excluding the boundary case :5. Our theoretical results uncover some interesting features. For example, when a concurrent level shift is allowed, the rate of convergence of the estimate of the break date is the same for all values of d in the interval ( :5; :5). his feature is linked to the contamination induced by allowing a level shift, previously discussed by Perron and Zhu (5). In all other cases, the rate of convergence is monotonically decreasing as d increases. We also provide results about the so-called spurious break issue. Simulation experiments are provided to illustrate some of the theoretical results. JEL Classi cation: C, C8, C Keywords: Fractionally integrated process, Linear trend, Segmented trend, Spurious break, Structural change We thank the editor Robert aylor, the co-editor, two anonymous referees, Dukpa Kim and Zhongjun Qu for constructive comments and suggestions. his paper is a revised version of parts of Seong Yeon Chang s Ph.D. Dissertation at Boston University (Chang, ). y Wang Yanan Institute for Studies in Economics, MOE Key Laboratory of Econometrics, Fujian Key Laboratory of Statistical Science, Xiamen University, Xiamen, Fujian 5, China (sychang@xmu.edu.cn) z Department of Economics, Boston University, 7 Bay State Rd., Boston, MA 5 (perron@bu.edu)

2 Introduction Economic relationships are often subject to structural changes. Hence, testing for a structural break and estimating the break date have been important topics in both economics and statistics; see Perron () for a review. o test for a structural break, or instability of the parameters, important contributions include Andrews (99) and Andrews and Ploberger (99). Bai (99, 997) showed that the break date can be estimated consistently by minimizing the sum of squared residuals (SSR) from the unrestricted model and derived the limiting distribution of the estimate of the break date, which can be applied to constructing con dence intervals for the true break date. Bai and Perron (998, ) considered statistical inference related to multiple structural changes under general conditions. In the literature, most of the work assumed that the regressors and the errors are short-memory stationary processes. Structural breaks in trend regressors and non-stationary processes are also important from a practical perspective. Perron (989) showed that the Dickey and Fuller (979) type unit root test is biased in favor of a non-rejection of the unit root null hypothesis when the process is trend stationary with a structural break in slope. Work related to changes in trend include the following. Feder (975) considered estimating the joint points of polynomial type segmented regressions. Bai (997) and Bai and Perron (998) provided inference results with trending regressors. In order to obtain the limiting distribution, the trending regressors are assumed to be a function of t=, say g(t= ), with the sample size. Deng and Perron () analyzed the consequences of specifying the trend function in scaled form when a structural break is involved. Bai et al. (998) analyzed the limiting distribution of the estimated break date for multivariate time series with a change in slope. Chu and White (99) suggested a testing procedure for a change in a trend function with short-memory stationary errors. Perron (99) and Vogelsang (997) considered testing a structural break in trend when the errors are either short-memory stationary, I(), or having an autoregressive unit root, I(). Vogelsang (999) devised a test whose limiting distribution does not change depending on whether the noise component is I() or I(). Recently, Perron and Yabu (9) considered testing for structural changes in the trend function of a time series without any prior knowledge about whether the errors are I() or I(). heir testing procedure adopts a quasi-feasible generalized least squares (GLS) approach that uses a super-e cient estimate of the sum of the autoregressive parameters when =. Harvey et al. (9) proposed a GLS-based trend break test that is asymptotically size robust with I() or I() errors. Sayginsoy and Vogelsang () (SV, henceforth) suggested xed-b asymptotics-based slope change tests with either I() or I() errors. Since the limiting null distributions of the tests vary

3 depending on the structure of the noise component, i.e., I() or I(), a scaling factor approach to align the two distributions for a given signi cant level has been adopted. Yang and Vogelsang () applied the xed-b theory to a sup-lm type test in order to test a level shift. Interestingly, they found that there is a bandwidth such that the xed-b asymptotic critical value is the same for both I() and I() errors. With respect to the problem of estimating the break date of the change in the slope of a linear trend with or without a concurrent level shift, Perron and Zhu (5) (PZ, henceforth) analyzed the consistency, rate of convergence and the limiting distributions of the parameter estimates when the errors are either I() or I(). he results of PZ and Perron and Yabu (9) have been used in Kim and Perron (9) to provide unit root tests with improved power that allow for a change in the trend function under both the null and alternative hypotheses. Fractionally integrated processes have been popular in the economics and statistics literature, in particular following the introduction of the ARFIMA processes by Granger and Joyeux (98) and Hosking (98). Kuan and Hsu (998) considered a change in mean model and established the consistency and the rate of convergence of the least square estimate of the break date when the errors are fractionally integrated; see also Lavielle and Moulines (). hey found that the convergence rate depends on the order of integration d. Moreover, when no such change in mean is present, the estimate of the break date obtained by minimizing the sum of squared residuals indicates a spurious break date when d (; :5). Hsu and Kuan (8) showed that the least square estimate of the break date in a mean change model is not consistent when the errors are fractionally integrated with d (:5; :5), and that the spurious feature also occurs. Gil-Alana (8) executed a set of Monte Carlo simulations to con rm that both the order of fractional integration and the break date can be estimated simultaneously by minimizing the SSR considering a range of grid values for d and the break date. In the context of testing for a structural change in the framework of fractionally integrated processes, the following work are relevant. Shao () proposed a simple testing procedure to test for a level shift in a stationary long memory time series based on the self-normalization idea of Shao (). More recently, Iacone et al. (a) proposed a robust test for a slope change in trend when the order of fractional integration d in the errors is located in an interval [; :5) excluding the boundary case.5. Iacone et al. (b) considered the same problem, but the testing procedure is based on xed-b asymptotics developed by SV. By developing d -adaptive critical values, the proposed test is (asymptotically) size controlled and improves power when d = compared to the test of SV. Iacone et al. () analyzed a change in mean model and suggested a sup-wald test based on xed-b asymptotics. he main contribution of this paper is to extend PZ s analysis to cover the more general case of fractionally integrated errors for values of d in the interval ( :5; :5) excluding the boundary

4 case :5. We establish the consistency, rate of convergence, and the limiting distributions of the parameter estimates in models when the trend function exhibits a slope change with or without a concurrent change in level. Our theoretical results uncover some interesting features. First, when a concurrent level shift is allowed, the rate of convergence of the estimate of the break fraction is the same for all values of d in the interval ( :5; :5). his feature is linked to the contamination induced by allowing a level shift, previously discussed by PZ. In all other cases, the rate of convergence is monotonically decreasing as d increases. Second, the coe cient of the slope change can be estimated consistently for all d ( :5; :5) [ (:5; :5) while the level shift coe cient is asymptotically unidenti ed for d (:5; :5). We extend Hsu and Kuan s (8) result to the slope change model with a concurrent level shift. hird, we also provide results about the so-called spurious break issue. For d (; :5)[(:5; :5), it is very likely to estimate a spurious break when there is no break in the data generating process. Simulation experiments are provided to illustrate some of the theoretical results in the paper. he structure of the paper is as follows. In section, we review fractionally integrated processes, fractional Brownian motion and useful related functional central limit theorems. Section presents the models, the assumptions and a key inequality used throughout the proofs. Section provides the main contributions related to the limit properties of the estimates: consistency (Section.), rate of convergence (Section.), limit distributions of the estimate of the break date (Section.) and limit distributions of the estimates of the other parameters (Section.). he problem of the possibility of a spurious break is discussed in Section 5. Section provides brief concluding remarks. All technical derivations are relegated to an online Supplementary Material. Fractionally Integrated Processes and Functional Central Limit heorem In this section, we brie y de ne fractionally integrated processes and review results to be used in subsequent developments. We follow the notation of Wang et al. () and Robinson (5). De ne rst a = X j (a)l j ; j (a) = j= (j + a) (a) (j + ) where L is the lag operator, = L is the di erence operator and is the Gamma function with (a) = for a = ; ; : : :, and ()= () =. Let f t ; t = ; ; : : :g be a zero-mean short-memory covariance stationary process, with spectral density that is bounded and bounded away from zero. For d ( :5; :5), t = d t ; t = ; ; : : : ; () ()

5 is covariance stationary and invertible for d > :5. he truncated version of t is de ned as # t = t t ; t = ; ; : : : ; () where A is the indicator function for the event A. For an integer m, u t = m # t ; t = ; ; : : : () is called a type I I(m + d) process. A zero-mean short-memory covariance stationary process t can be represented as a one-sided moving average: X t = j t j ; t = ; ; : : : ; (5) j= where =, P j= j <, and t, t = ; ; : : : are independent and identically distributed (i.i.d.) random variables with mean zero. Let D[; ] be the space of functions on [; ] which are right continuous and have left limits, equipped with the Skorohod topology. Let ) denote weak convergence in distribution under the Skorohod topology and p! convergence in probability. Denote by [a] the integer part of a R. he order of integration is d = m + d with m N. Wang et al. () derived a functional central limit theorem (FCL) for m which includes the non-stationary cases. condition is required. o consider general non-stationary fractional processes, the following Condition A: j ; j in (5) satisfy P j= j= d j j j < and () P j= j =. Also, E(j j maxf;=(+d)g ) <. We summarize their results insofar as they will be relevant for subsequent derivations. Lemma (Wang et al.,, heorem.) Let u t satisfy () with m = and assume Condition A holds. hen, for d ( :5; :5), [ r] X (d) =+d u t ) B d (r); () t= where (d) = f () ( d)e( )g=f( + d) ( + d) ( d)g and B d() is a type I fractional Brownian motion on D[; ], i.e., Z B d (t) = [(t s) d ( s) d ]db(s) + (d + ) with B() a standard Brownian motion. Z t (t s) d db(s) ; Lemma (Wang et al.,, heorem.) Let u t satisfy () with m = and assume Condition A holds. hen, for d ( :5; :5), a) [(d) =+d ] u [ r] ) B d (r), b) [(d) =+d+ ] P [ r] t= u t ) R r B d(s)ds, c) [ (d) (d+) ] P [ r] t= u t ) R r [B d(s)] ds.

6 he Models We consider the series of interest y t as consisting of a systematic part f t and a random component u t, namely, y t = f t + u t. For the noise component u t, the following two assumptions hold. Assumption A: u t is a type I I(m + d) process de ned by ()-(5). Assumption A: he conditions of Lemmas and are satis ed. For the systematic part f t, we consider two cases. he rst, labeled Model I, speci es that f t is a rst-order linear trend with a single change in slope. In this case, the trend is joined at the time of break and there is no concurrent level shift. he second, labeled Model II, speci es that f t is a rst-order linear trend with a concurrent break in both intercept and slope. Let = = denote a generic break fraction with a postulated break date, whose true value is. Model I (Joint Broken rend): he deterministic component f t is speci ed as f t = + t + b B t ; where B t is a dummy variable for the slope change de ned by B t = t if t > and otherwise. Hence, the slope coe cient changes from to + b at time. Note that the trend function is continuous at the time point, hence the labeling of a joint broken trend. Model II (Local Disjoint Broken rend): he deterministic component is speci ed by f t = + t + b C t + b B t ; where C t is a dummy variable for the level shift de ned by C t = if t > and otherwise. At the break date, there is a slope change with a concurrent level shift. he magnitude of the level shift is b, which is asymptotically negligible compared to the level of the series +, hence the labeling of a local disjoint broken trend. In matrix notation, the models de ned above can be speci ed as Y = X + U, where Y = [y ; : : : ; y ], U = [u ; : : : ; u ], X = [x( ) ; : : : ; x( ) ], with x( ) t = [ t B t ] and = [ b ], for Model I, while for Model II, x( ) t = [ t C t B t ] and = [ b b ]. Note that the matrix X depends on the candidate break date. 5

7 Remark In the literature, the following high level assumption on the regressors is standard. D [P [ ] t= x( ) t x( ) t]d p! Q() uniformly in [; ] for some D, where Q() is a positive semi-de nite, symmetric, and an absolutely continuous, monotonically increasing function of. We do not introduce this assumption explicitly because it automatically holds for both Models I and II. Later, this high level assumption with assumptions A and A are used to obtain asymptotic results for the so-called spurious break issue. he break date can be estimated by using a global least-squares criterion: ^ = arg min Y (I P )Y where P is the matrix that projects on the range space of X, i.e., P = X (X X ) X and = [; ( ) ], < < =. With X ^ constructed using the estimate ^, the OLS estimate of is ^ = (X ^ X ^ ) X ^ Y and the SSR, for an estimated break fraction ^ = ^ =, is S(^) = X ^u t = t= X t= y t x( ^ ) t^ = Y (I P ^ )Y where P ^ is the projection matrix associated with X ^. We denote the true value of each parameter with superscript : = [ b ] in Model I, = [ b b ] in Model II,, and = =. Hence, the data generating process (DGP) is speci ed as h Y = X + U = x( ) ::: x( ) i + U: (7) hroughout, we assume that there is at least a change in slope as stated in the following assumption. Assumption A: b = and (; ) for some (; =). his assumption is required to ensure that there is a break in slope and that the pre and post break samples are asymptotically large enough to obtain consistent estimates of the unknown coe cients. his is a standard assumption needed to derive any useful asymptotic result. As in PZ, a key inequality plays a crucial role in proving the asymptotic results. By construction, we have for all, S(^) S( ), or equivalently, Y (I P ^ )Y Y (I P )Y. Using (7), this inequality can be written as Y (P P ^ )Y, or equivalently, ( X + U )(P P ^ )(X + U) = X (P P ^ )X + X (P P ^ )U + U (P P ^ )U = (X X ^ ) (I P ^ )(X X ^ ) + (X X ^ ) (I P ^ )U + U (P P ^ )U ( ^X ^X) + ( ^X ^U) + ( ^U ^U) (8) where we use the fact that X P = X and X (I P ) =.

8 Asymptotic Results We consider in turn the consistency, rate of convergence and limit distributions of the estimates, concentrating on the estimate of the break fraction.. Consistency We show that ^ is a consistent estimate of when the errors are fractionally integrated with parameter d ( :5; :5) [ (:5; :5). he idea behind the proof is the following. Unless ^ p!, the rst term in (8) would asymptotically dominate the others since it is positive provided the event f = ^ g does not hold for all, which occurs with probability one. It means that the key inequality does not hold if ^ does not converge to in probability, which leads to the desired contradiction. he following theorem states the consistency result. heorem Under Assumptions A-A, in Models I and II, ^ p!, 8d ( :5; :5) [ (:5; :5).. Rate of Convergence he following theorem shows that the rate of the convergence of the estimate of the break fraction, ^, depends on the order of fractional integration d. It also di ers across the two models being faster with no concurrent level shift. heorem Under Assumptions A-A, for every d ( :5; :5): ) For Model I: 8 < ^ O p ( =+d ) if m = = : O p ( =+d ) if m =, ) For Model II: 8 < ^ O p ( ) if m = = : O p ( =+d ) if m =. heorem implies that the rate of convergence is slower when allowing for a concurrent level shift, even if none is present, for d ( :5; :5). It is, however, the same when d (:5; :5). hese results accord with those from PZ who considered I() and I() processes. For Models I and II with I() errors, ^ = O p ( = ). On the other hand, for Model I with I() errors, ^ = O p ( = ) and for Model II with I() errors, ^ = O p ( ). PZ presented an intuitive explanation for the change in convergence rate induced by introducing a level shift. Brie y, a random deviation from a deterministic trend function is subject to be captured as if it were a level shift. Hence, it can have an e ect on the precision of the estimate. 7

9 he results show that the rate of convergence is linearly decreasing as d increases for all models except Model II for d ( of convergence is the same for all d ( :5; :5). he result for this latter case is quite interesting as the rate :5; :5). he explanation for this feature is again related to the contamination induced by allowing a concurrent level shift, which implies added noise. If the process is stationary, d ( convergence invariant to d. :5; :5), this added noise dominates and renders the rate of If the process is non-stationary, d (:5; :5), the noise is small compared to the signal and we are back essentially to the case with no concurrent level shift.. he Limiting Distribution of the Estimate of the Break Date Given results about the consistency and the rate of convergence of the estimate of the break fraction ^, we can now consider its limiting distribution. he results are stated in the following theorem. heorem Under Assumptions A-A, we have for every d ( :5; :5): ) For Model I: a) if m =, = d (^ ) ) (d)=[ ( ) b ], b) if m =, = d (^ ) ) (d) R B d (r)dr =[ ( ) b ], where = Z db d (r) + Z db d (r) ( ) Z rdb d (r) ( Z ) ( (r ) )db d (r); and Z B d (r)dr = Z B d (r)dr + ( Z ) ( (r ) Z B d (r)dr )B d (r)dr: ( ) Z rb d (r)dr ) For Model II: a) if m =, de ne a stochastic process S (v) on the set of integers as follows: S () =, S (v) = S (v) for v < and S (v) = S (v) for v >, with X X S (v) = ( b + b k) ( b + b k)u k; v = ; ; : : : ; S (v) = k=v+ vx ( b + b k) + k= k= k=v+ vx ( b + b k)u k; v = ; ; : : : : If u t is strictly stationary with a continuous distribution, (^ ) ) arg min v S (v). b) If m =, 8

10 de ne = = = = Z B d (r)dr; ; ; B d ( ); Z Z = [(r Z Z Z rb d (r)dr; B d (r)dr ; r )=( ) ]db d (r); B d (r)dr; Z (r [(r )(r )=( ) ]db d (r); = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) f( ) +g ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) )B d (r)dr ; ; 7 5 ( ) f( ) +g ( ) ( ) ( ) ( ) ( ) ( ) f( ) +g f( ) ( ) + g ( ) ( ) ( ) ( ) ( ) ( ) Also de ne Z (v) as: Z () =, Z (v) = Z (v) for v < and Z (v) = Z (v) for v >, with Z (v) = ( b ) jvj = + v (d) b + v(d) [ ]; v < ; Z (v) = ( b ) jvj = + v (d) b + v(d) [ ]; v > : hen, = d (^ ) ) arg min v Z (v). : 7 5 heorem implies that the limiting distributions have interesting qualitative di erences across models. First, in Model I, even though the magnitude of the break is xed, the limiting distributions of the estimate of the break fraction do not depend on the structure of the error process, except via (d) which is required to properly scale the distribution. his feature contrasts with results for stationary regressors. Bai (997), among many others, showed that the limiting distribution of the estimate of the break fraction depends on the exact distributions of both the regressors and the errors in linear regression models with stationary regressors. o avoid this issue, the so-called shrinking shift framework was introduced, whereby the magnitude of break decreases as the sample 9

11 size increases. heorem, however, shows that we do not have to rely on such a shrinking shift framework to obtain the limiting distributions with a joint-segmented trend. Second, in Model II, the limiting distributions are functions of a two-sided random process in which many nuisance parameters are involved. In particular, when d ( :5; :5), the limiting distribution depends on the exact distributions of the errors. On the other hand, for d (:5; :5), the limiting distribution does not depend on the exact distribution of the errors. Hence, a con dence interval for the break date can be formed by estimating the nuisance parameters consistently and simulating the various functionals of the fractional Brownian motions. hird, comparing Models I and II, we nd that including a level shift component as a regressor, even if irrelevant, has an important e ect on the asymptotic distributions. For illustrative purpose, assume that the DGP does not have a level shift, i.e., b =. While Model I does not allow a level shift, Model II introduces a dummy variable C t to incorporate an irrelevant level shift. From heorem, we know that the rate of convergence of the estimated break fraction is slower in Model II when d ( :5; :5). Furthermore, the asymptotic distributions are di erent across models. In Section, we provide simulation experiments to further analyze the implications of incorporating a level shift component.. he Limiting Distribution of the Estimates of the Other Parameters We turn to the limiting distribution of the other parameter estimates in the models, that is (^ ; ^ ; ^ b ) for Model I, and (^ ; ^ b ; ^ ; ^ b ) for Model II. heorem Under assumption A-A, the following results hold for all d ( Model I: = d (^ ) = d (^ ) 7 5 ) a if m = ; = d (^ b b ) = d (^ ) = d (^ ) 7 5 ) a if m = ; = d (^ b b ) where a = ( +) ( +) ( ) ( ) ( ) ( +) ( +) ( +) ( ) ( ) ( ) ( ) ( +) ( ) ( ) ( ) ( ) ( ) ( ) 7 5 ; :5; :5). ) For

12 and = (d) = (d) Z Z ( ) r+r ( ) ( ) ( ) r+r ( ) ( ) ( +r) ( ) ( )r +( ) r ( ) ( ) f ( ) gr f ( ) gr ( ) ( ) ( ) fr r ( ) g ( ) 7 5 db d(r) + Z 7 5 db d(r) + ( + r) ( ) + r+r Z + r fr 7 5 db d(r) C A ; fr (+ )r+ g ( +)fr (+ )r+ g 7 5 db d(r) C A : ( + )r + g ) For Model II: = d (^ ) R db d(r) = d (^ R ) = d (^ b b ) = d b ( ^ ) ) (d) rdb d(r) R 7 db d(r) R = d (^ b b ) (r )db d (r) if m = : Hence, ^ b is asymptotically unidenti ed and ^ b b ) b arg min S (), as de ned in heorem ; = d (^ ) R B d(r)dr = d (^ R ) = d (^ b b ) = d b ( ^ ) ) (d) rb d(r)dr R 7 B if m = : d(r)dr R = d (^ b b ) (r )B d (r)dr his implies that ^ b is asymptotically unidenti ed because = d [(^ b b ) b ( ^ )] ) +, where and are random variables de ned in heorem. Note that except for the unidenti ed intercept shift ^ b, the other parameters, (^ ; ^ ; ^ b ), have the same stochastic order for Models I and II. As noted in PZ, the exact model speci cation does not matter if one wants to make asymptotic inference on these parameters. 5 Spurious Break In this section, we consider the properties of the least square estimate of a structural break date when no break is present in the data generating process. Nunes et al. (995) and Bai (998) showed that the least square estimate of the break date can lead to a spurious break date when

13 the error is an I() process, in the sense that the estimate will not gather around either end of the sample. Kuan and Hsu (998) considered a change in mean model for a fractionally integrated process with d ( :5; :5) and showed that a spurious break can be estimated if d (; :5). Hsu and Kuan (8) con rmed the possibility of estimating a spurious mean break if the series is a non-stationary fractionally integrated process, i.e., d (:5; :5). Here, we consider the issue of spurious breaks in the context of Model II with a disjoint-segmented trend. he data generating process is speci ed as: y t = + t + u t ; with u t = m # t for t = ; ; ; : : : as de ned in (). Let S( ) denote the sum of squared residuals related to a generic break date, that is, S( ) = Y (I P )Y. We have that P ^ = arg min S( ) = arg min fs( ) t= u t g because P t= u t is independent of. If no structural change is allowed, then! X M ( ) S( ) = X t= x tu t! t= u t X t= x tx t! X t=! x tu t X t= + x tu t X t= + where x t = (; t). Let M ( ) be the normalized version of M ( ), that is, M ( ) (d+m) M ( )! X = (d+m) x tu t D D t= + X t= + x tu t A D X x tx t D X t= +! D x tx t D A (d+m) x tx t A X t= x tu t! X t= + X t= + x tu t A x tu t A where D = diagf = ; = g and m f; g. Note that ^ = arg max M ( ) = arg max M ( ) because the normalization (d+m) does not depend on. We start with the case d ( :5; :5) with m =. If assumptions A-A hold, we have M ( ) ) M () (d) G() Q() G() + [G() G()] [Q() Q()] [G() G()] (9) where G() = (B d (); R rdb d(r)), G() G() = (B d () B d (); R rdb d(r)), Q() = A ; Q() Q() ( )= A : = = ( )= ( )=

14 aqqu (977) showed that for d ( logarithms: for some positive constant c, lim sup t! :5; :5), B d (t); t R satis es the following law of iterated B d (t) = almost surely (a:s:): (c t +d log log t) = Since B d (t) is self-similar with self-similarity parameter :5 + d, for any c > it satis es, B d (t) d = c (:5+d ) B d (c t), where d = denotes equality in distribution. Applying the law of iterated logarithms to B d (=t) and self-similarity, we have hen, for d ( :5; ], lim sup! lim sup t! B d () p B d (t) = a:s: (c t +d log log(=t)) = q = lim sup c d log log(=) =! It is easy to verify that B d () B d () is also a fractional Brownian motion B d (s) with s =. For d ( :5; ], lim sup! On the other hand, for d (; :5), lim sup! B d () B d () p = lim sup s! B d () p = lim sup! B d (s) p s = a:s: B d () B d () p = a:s: Since the above are almost sure limits, we can de ne M () and M () as the almost sure limit of M () as! ;, respectively. Hence, with probability, ( Z M () = M () = (d) Z ) B d() rdb d (r) B d () + rdb d (r) : heorem 5 Under assumptions A-A, i) for d ( :5; ], lim sup! M () = lim sup! M () = a:s:; ii) for d (; :5), there exist some (; ) s.t. M () = M () < M () a.s. heorem 5 implies that no spurious break is estimated if the order of fractional integration is a value in ( :5; ]. It is not the case for d (; :5) as M () = M () is stochastically bounded while M () can be arbitrarily large with s close to either ends. Now, consider the possibility of estimating a spurious break date when the errors are nonstationary, i.e., d (:5; :5). non-stationary fractional process with a deterministic trend. a:s: Here, heorem in Bai (998) is generalized to incorporate a Proposition Under assumptions A-A, for d (:5; :5), sup (;) M () = O p ().

15 Proposition implies that M () is stochastically bounded even when! or! for d (:5; :5). Of interest is the limit behavior of M () when gets closer to either or. From (9), M () = (d) G() Q() G() using the fact that G() Q() G()! as!. he latter follows from the fact that (d) G() Q() G() is the limit of the rst term in M ( ). hen, (d+)! X x tu t X x tx t! X! [ ] X x tu t (d+) (d+) u t t= t= t= t= and (d+) P [ ] t= u t p! as!. hus, M () = lim! M (), thereby M () is continuous at =. Similarly, M () = (d) G() Q() G() is de ned as the limit of M () as!. heorem Under assumptions A-A, for d (:5; :5), with probability, M () = M () < M () for every < <. heorem implies that M () cannot attain a maximum at zero or one almost surely. Simulation Experiments In this section, we provide simulation experiments to illustrate various theoretical results. We rst assess whether the asymptotic distributions are good approximations to the nite sample distributions. We highlight the bimodality of the distribution induced by an irrelevant level shift included in the regression. We also illustrate the spurious break problem.. Finite Sample and Limiting Distributions We start with simulations showing that the nite sample distributions of various estimates are well approximated by their asymptotic distributions. Of interest are three estimates: ^ (break date), ^ b (slope change), and ^ b (level shift). hroughout, we use, replications and two sample sizes = and 8. Whenever the asymptotic distributions are non-normal, we use simulations of the fractional Brownian motion and estimates of the various parameters to evaluate the probability density function using a kernel-based method applied to the simulated realizations. We rst consider the following DGP: y t = x( ) t + u t = + t + b B t + u t ; () where u t = m ( t t ), t = d t, t i:i:d:n(; ) for t = ; ; : : :, m f; g, and B t = (t ) t>[ ] with = [ ]. We set the various parameters at the following values: = :5, = :7, = :, b = :, = : for stationary case, and = : for

16 non-stationary case. he con gurations are the same as those in PZ, chosen to obtain distributions that easily reveal the main features of interest. Using DGP (), we consider two regression models: the joint broken trend (Model I) and the local disjoint broken trend (Model II). Figure presents the nite sample and asymptotic probability density function (pdf) of the normalized estimates of the break date and the slope change when the order of fractional integration d = : and = :. For Model I, the normalized estimate of the break date is given by = d ( ^ ) and the normalized estimate of the slope change is = d (^ b b ). Simulation results pertaining to Model I are in the top panels. he results reveal that the nite sample distribution is well approximated by the asymptotic distribution for both estimates. For Model II, the normalized estimates ( ^ ) and = d (^ b b ) are considered and the results are presented in the bottom panels. Since we set b =, Model II incorporates an irrelevant level shift. We nd that the nite sample distribution of the estimate of the break date is clearly bimodal. Furthermore, the asymptotic distribution is a good approximation to the nite sample distribution when = 8 but less so when =. For the slope change, when =, the nite sample distribution is right-skewed. However, as the sample size increases, the nite sample distribution approaches the limiting distribution. Figure presents a similar set of results for the non-stationary case. he DGP is still () but with d = : and = :. For both Models I and II, the normalized statistics are = d ( ^ ) and = d (^ b b ). When the regression from Model I is used (which is well speci ed), the asymptotic distribution is a good approximation to the nite sample distribution for these two parameters. On the other hand, when the regression from Model II is used (which introduces an irrelevant level shift regressor), the asymptotic distribution of the estimated break fraction exhibits a minor bimodal pattern that is not present in the nite sample distribution when = or = 8. However, increasing to, the nite sample distribution indeed also exhibits a slight bimodal pattern and the approximation is satisfactory. For the slope change, the asymptotic approximation is still good. Figure presents the nite sample and asymptotic distribution of the estimate of the level shift ^ b from the regression Model II. he DGP is given by () where d = : and = : in panel (a), while d = : and = : in panel (b). he values of the other parameters remain as stated above. In panel (a), with d = :, the distributions are clearly bimodal and the approximation is quite satisfactory with = 8. We explain the feature in detail below. When the errors are non-stationary with d = :, we only plot in panel (b) the nite sample distributions, which show little changes between = and 8. Remark In heorem, we show that introducing a level shift regressor reduces the rate of con- 5

17 vergence of the estimate of the break fraction when the order of fractional integration d is in ( :5; :5) and that the rate of convergence is invariant in Model II. Furthermore, it induces bimodality in the distribution of this estimate (both in nite samples and in the limit). PZ (Section 5) provided an intuitive explanation for this phenomenon. Since the level shift regressor C t can categorize random departure from the trend line around the true break date as a level shift, this induces increased randomness in the estimate of the break date. his is referred to as a contamination. In the proof of heorem.(), we show that ^ b b = b ( ^ ) + o p() when d ( :5; :5). Accordingly, since ^ is contaminated by the level shift regressor C t, it also has an e ect on the estimate ^ b, referred to as a feedback e ect. Because of the feedback e ect, the true parameter of the level shift b cannot be identi ed. Figures and 5 consider the case with a genuine level shift, i.e., b =. he DGP is y t = x( ) t + u t = + t + b C t + b B t + u t ; () where C t = t>[ ]. We consider the regression from Model II to estimate the parameters. In Figure, we present the results pertaining to the case where the errors are stationary with d = : and = :. We make a comparison between the nite sample distribution and the limiting distribution derived in heorem. In panel (a), b = : and the asymptotic distribution shows strong bimodality (the right mode being more important). Moreover, the asymptotic distribution is a good approximation to the nite sample distribution when the sample size is large, = 8, but less so with =. In panel (b), b = : and the left mode clearly dominates. he asymptotic distribution approximates the nite sample distribution better compared to the case where b = :. In panel (c), we plot a set of the asymptotic distributions changing the value of b. As the absolute value of b increases, one mode dominates the other and is more centered around. his implies that a large level shift is helpful in identifying the true break date. b Figure 5 presents similar results for non-stationary errors with d = : and = :. We set = : in panel (a), : in panel (b), and :5 in panel (c). o understand the implications of the results in Figure 5, rst note that the level shift parameter b does not appear in the limiting distribution of the estimated break date in Model II for m =, i.e., when d (:5; :5). As the sample size increases, the magnitude of the level shift b is relatively small compared to the level of the trend function. In the limit, the level shift e ect is concealed by random variations in the non-stationary errors. Hence, the asymptotic distribution is an appropriate approximation when the magnitude of the level shift b is relatively small. he adequacy of the approximation decreases as b increases. For a large level shift, we can expect that b would a ect the limiting distribution. PZ suggested to use an asymptotic expansion under this circumstance (see their heorem 5).

18 . Spurious Break We consider simulation experiments to illustrate the issue of a potential spurious break. he data generating process is speci ed by y t = + t + u t ; where u t = m ( t t ), t = d t, t i:i:d:n(; ) for t = ; ; : : : and m f; g. Without loss of generality, we set = = and consider d f :; :; :7; :g. he sample sizes used are = and. For each value of d, the results are obtained from, replications. We consider estimating the date of a structural break using Model II (locally disjoint broken trend). Figure (a) presents histograms of the estimates ^ when =. For d = :, the estimates are concentrated at the two end points ( and ) indicating that the estimate of the break date is consistent and no spurious break feature is present, consistent with heorem 5. For d = :, the histogram of the estimate ^ spreads out across all admissible break dates with the exception of the end points. For d f:7; :g, the estimates of the break date ^ tend to cluster near the middle of the sample, which falsely indicates that there is a break in the sample. Figure (b) presents histograms of the estimate ^ with =. With this larger sample, the estimates often occur near the boundaries, though there is no mass at or very near or with d f:; :7; :g. Hence, the theoretical results are supported by the simulations. hese results reinforce the feature discussed in the literature to the e ect that structural change and long memory imply similar features in the data, and it is di cult to distinguish one from the other at least in small samples. his suggests the importance of implementing a proper testing procedure for a structural break which should be robust to any a priori unknown order of integration. Recently, Harvey et al. (9) and Perron and Yabu (9) suggested testing procedures for a structural change in trend function designed to be robust to I() or I() errors. Iacone et al. () presented a sup-wald type test for a change in the slope of a trend function which is robust across fractional values of the order of integration. 7 Conclusion In this paper, we establish the consistency, rate of convergence and limit distributions of parameter estimates in models where the trend function experiences a slope change at some unknown date, with or without a concurrent level shift, when the errors are fractionally integrated processes with the order of fractional integration d ( :5; :5)[(:5; :5). It is worth noting that introducing a level shift has a crucial e ect on the asymptotic results. Our theoretical results uncover some interesting features. First, when a concurrent level shift is allowed, the rate of convergence of the estimate of 7

19 the break date is slower, and it is the same for all values of d ( :5; :5). his feature is linked to the contamination induced by allowing a level shift. In all other cases, the rate of convergence is monotonically decreasing as d increases. Second, the level shift coe cient b is asymptotically unidenti ed when the errors are non-stationary fractionally integrated processes, i.e., d (:5; :5) while the slope change coe cient b can be estimated consistently for all d ( :5; :5)[(:5; :5). hird, we also provide results about the so-called spurious break issue and show that it cannot occur in the limit when d in the interval ( :5; ]. Lastly, via the simulation experiments, we con rm that an irrelevant level shift induces bimodality in the distribution of the break date estimates, but a relevant level shift improves the precision of the estimate. he results in this paper can be useful for subsequent work. For instance, Lobato and Velasco (7) considered e cient Wald test for a unit root against a fractionally integrated process with unknown order. However, their procedure does not allow a break under both the null and alternative hypotheses. Accordingly, an interesting avenue would be to extend the Kim and Perron (9) unit root testing procedure that allows a structural change in the trend function under both the null and alternative hypotheses. Just as the results of Perron and Zhu (5) and Perron and Yabu (9) were useful to achieve this task, one could use our results and those of Iacone et al. () to extend the test of Lobato and Velasco (7). his is currently the object of ongoing research. Notes. he restriction that d = :5 is standard in the long memory literature (e.g. Iacone et al., ). anaka (999) showed that the case with d = :5 needs to be treated separately from the case with d = :5.. o generate one dimensional fractional Brownian motion B d (t) on t [; ], we use the MALAB code hurst.m of Kroese and Botev () that applies the Fast Fourier ransform (FF) to a circulant covariance matrix.. For a set of statistics fx i g i=;:::;n, the pdf at a value x is estimated by ^g(x) = (Nh x ) P n i= K((x x i )=h x ) where K() is a kernel function and h x is the bandwidth. We use the standard normal kernel and n = ;. As mentioned in PZ, the cross-validation method for choosing the optimal bandwidth does not work well because the estimates of the break date are discrete integers. As a rule of thumb, the bandwidth is set to h x = :^ x where ^ x is the estimated standard deviation of the sample statistics fx i g i=;:::;n. 8

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23 .. Normalized Break Date Estimate in Model I (m=,d=.) Limiting = =8.. Normalized Slope Change Estimate in Model I (m=,d=.) Limiting = = (a) (b).5. Normalized Break Date Estimate in Model II (m=,d=.) Limiting = =8.. Normalized Slope Change Estimate in Model II (m=,d=.) Limiting = = (c) (d) Figure : Finite sample and asymptotic distributions in Models I and II with d =.: µ b =. he statistics are normalized as follows: / d ( ˆ ) for the break date in Model I (panel (a)) but ˆ in Model II (panel (c)); and / d ( ˆβ b βb ) for the slope change in both models (panels (b) and (d)). he finite sample distributions are obtained using u t = ζ t t, ζ t = d η t and η t i.i.d.n(, σ ) with, replications. he values of the parameters are set to λ =.5, µ =.7, β =., β b =., σ =.. Because the limiting distributions are non-standard, we use 5, simulated values to construct the pdf.

24 .5 Normalized Break Date Estimate in Model I (m=,d=.) Limiting = =8 5 Normalized Slope Change Estimate in Model I (m=,d=.) Limiting = = (a) (b).5 Normalized Break Date Estimate in Model II (m=,d=.) Limiting = =8 = 5 Normalized Slope Change Estimate in Model II (m=,d=.) Limiting = =8 = (c) (d) Figure : Finite sample and asymptotic distributions in Models I and II with d =.: µ b =. he statistics are normalized as follows: / d ( ˆ ) for both models (panels (a) and (c)); and / d ( ˆβ b βb ) for the slope change in both models (panels (b) and (d)). he finite sample distributions are obtained using u t = ζ t t, ζ t = d η t and η t i.i.d.n(, σ ) with, replications. he values of the parameters are set to λ =.5, µ =.7, β =., β b =., σ =.. Because the limiting distributions are non-standard, we use 5, simulated values to construct the pdf.

25 .. Unidentified Level Shift in Model II (m=,d=.) Limiting = =8 5 Unidentified Level Shift in Model II (m=,d=.) = = (a) (b) Figure : Unidentified level shift in Model II: µ b =. he statistics are normalized as follows: ˆµ b µ b for d =. (panel (a)); and / d (ˆµ b µ b ) for d =. (panel (b)). he finite sample distributions are obtained using u t = ζ t t, ζ t = d η t and η t i.i.d.n(, σ ) with, replications. he values of the parameters are set to λ =.5, µ =.7, β =., β b =.. Moreover, σ =. if d =. and σ =. if d =.. Because the limiting distributions are non-standard, we use 5, simulated values to construct the pdf in panel (a).

26 ..5 Normalized Break Date Estimate in Model II (m=,d=.,µ b =.) Limiting = = Normalized Break Date Estimate in Model II (m=,d=.,µ b =.) Limiting = = (a) (b)..5. Asymptotic Distributions in Model II with Varying µ b (m=,d=.) µ b =. µ b =. µ b = µ b =. µ b =.... (c) Figure : Finite sample and asymptotic distributions in Model II with d =.: µ b. he statistic for the break date is normalized as ˆ. he finite sample distributions are obtained using u t = ζ t t, ζ t = d η t and η t i.i.d.n(, σ ) with, replications. he values of the parameters are set to λ =.5, µ =.7, β =., β b =., σ =.. Because the limiting distributions are non-standard, we use 5, simulated values to construct the pdf. In Panel (a), where µ b =., we compare the finite sample distributions for = and 8 against the limiting distribution; in Panel (b), we let µ b =.; in Panel (c), we compare the limiting distributions of ˆ varying µ b.