Inference on a Structural Break in Trend with Fractionally Integrated Errors

Transcription

1 Inference on a Structural Break in rend with Fractionally Integrated Errors Seongyeon Chang Boston University Pierre Perron y Boston University November, 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 short-memory, 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 in the paper. JEL Classi cation: C. Keywords: linear trend. long-memory, segmented trend, structural change, spurious break, Department of Economics, Boston University, Bay State Rd., Boston, MA 5 (seongch@bu.edu) y Department of Economics, Boston University, 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, 99) 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 construct con dence intervals (CIs) 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 stationary. Structural breaks in trend regressors and non-stationary processes are also important from a practical perspective. Perron (989) showed that the Dickey and Fuller (99) 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. 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 short-memory, 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 and show that it cannot occur for all values of d in the interval ( :5; :5). Simulation experiments are provided to illustrate the theoretical results in the paper. Work related to changes in trend include the following. Feder (95) considered estimating the joint points of polynomial type segmented regressions. Bai (99) 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

3 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 non-stationary type series with a change in slope. Chu and White (99) suggested a testing procedure for a change in a trend function with stationary errors. Perron (99) and Vogelsang (99) considered testing a structural break in trend when the errors are either stationary or have a unit root. Vogelsang (999) devised a test whose limiting distribution does not change depending on whether the noise component is stationary or integrated. 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 stationary or integrated. heir testing procedure adopts a quasi-feasible generalized least squares approach that uses a super-e cient estimate of the sum of the autoregressive parameters when =. Harvey et al. (9) proposed a generalized least squares (GLS)-based trend break test that is asymptotically size robust with I() and I() errors. 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 showed 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 supports 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 all possible grids on d and break dates. More recently, Iacone et al. () provided a sup-wald type test for a structural change in trend when the order of integration in the errors is unknown a priori. 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

4 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 and an appendix contains all technical derivations. 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 Marinucci and Robinson (999) and Wang et al. (). De nition An autoregressive fractionally integrated moving average (ARFIMA) process u t is de ned as ( L) d+m u t = t and t = (L)" t () for t = ; ; : : :, where m is an integer and d ( :5; :5); L is the lag operator such that Lu t = u t and " t are independently and identically distributed (i.i.d.) random variables with zero mean and nite variance. Using this notation, note that the order of integration is d = m + d. Wang et al. () derived the invariance principle for m which includes the non-stationary cases. summarize their results insofar as they will be relevant for subsequent derivations. Lemma (Wang et al.,, heorem.): Let u t satisfy () with m = and let f j ; j g satisfy j j j < and b j= Assume that E <. hen, for d < :5, (d) =+d j= We j = () [ r] u j ) W d (r); r [; ]; () j= where (d) = b ( d)e ( + d) ( + d) ( d) ()

5 and W d (r) is a ype I fractional Brownian motion on D[; ]. Also, if Ej j (+c)=(+d) <, where c >, then () still holds for d ( :5; ). Lemma (Wang et al.,, heorem.): Let u t satisfy () with m = and let f j ; j g satisfy j = d j j j < and b j= Assume that Ej j maxf;=(+d)g <. hen, () holds for d ( :5; :5). j= j = : (5) Corollary (Wang et al.,, Corollary.): Let u j satisfy () with m =. If Ej j maxf;=(+d)g <, then () follows with b = ()=() for d ( :5; :5). In order to consider general non-stationary fractionally integrated processes, two additional conditions are required. Condition : f j ; j g satisfy (), and Ej" j p < where p =, for d [; :5); p = ( + c)=( + d) <, c > for d ( :5; ). Condition : f j ; j g satisfy (5), and Ej" j maxf;=(+d)g <. Lemma (Wang et al.,, heorem.): Let u j satisfy () with m. Let Conditions and hold. hen, for r, (d) =+d+m u [ r] ) W d;m (r); (d) =+d+m (d) (d+m) [ r] u j ) j= [ r] u j ) j= Z r Z rm Z r Z r : : : W d (r )dr dr : : : dr m ; [W d;m (s)] ds Marinucci and Robinson (999) de ned type I fractional Brownian motions with d ( :5; :5) on D[; ] as follows: W d (t) = Z [(t s) d ( s) d ]dw (d) + A(d) where W (s) is a standard Brownian motion and A(d) = d + + Z Z t [( + s) d s d ] ds = : (t s) d dw (s); hey explained the di erence between ype I and ype II fractional Brownian motion and showed how those two types are related.

6 where 8 < W d (r) if m = W d;m (r) = : R rm R r W d (r )dr dr dr m if m : R r We shall be interested in the case m =, in which case (d) =+d+ (d) (d+) he Models and Assumptions (d) ) W d(r); () =+d [ r] u j ) j= [ r] u j ) j= Z r Z r W d (s)ds; [W d (s)] ds: he series of interest, y t, is assumed to consist of some systematic part f t and a random component u t, namely, y t = f t + u t : For the noise component u t, we assume Eu t = and that the following two assumptions hold. Assumption A: For d ( :5; :5) [ (:5; :5), u t = d t t> where d ( L) d, with L being a lag operator such that L t = t and d = P j= jl j with j (j + d )=[ (d ) (j + )], where () denotes the Gamma function. Assumption A: t I(). More speci cally, t is such that P = [ r] t= t ) b W (r) where b = lim! E( P t= t) exists and is strictly positive. Here ) denotes weak convergence in distribution (under the Skorohod topology) and W () is the standard Wiener process. Remark here are many sets of su cient conditions to guarantee that the invariance principle in Assumption A holds. For example, it holds when t is a linear process such that t = P j= c j" t j with P j= jjc jj <, where f" t ; F t g is a martingale di erence sequence with F t the ltration to which " t is adapted. For the systematic part f t, we consider two cases studied in PZ with I() and I() errors. he rst, labelled Model I, speci es that f t is a rst-order linear trend with a single change 5

7 in slope. In this case, the trend is joined at the time of break and there is no concurrent level shift. he second, labelled Model II, speci es that f t is a rst-order linear trend with a concurrent break in both intercept and slope. Let = = denote the break fraction with a postulated break date. Model I: Joint Broken rend with Fractionally Integrated Errors: 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 8 < if t ; B t = : t if t > : Hence, the slope coe cient changes from to + b at time. Note that the trend function is continuous at the time point, hence the labelling of a joint broken trend. Model II: Local Disjoint Broken rend with Fractionally Integrated Errors: he deterministic component f t is speci ed by f t = + t + b C t + b B t ; (8) where C t is a dummy variable for the level shift de ned by 8 < if t ; C t = : if t > : At the break date, there are concurrent slope and level shifts. he magnitude of the level shift is b, which is asymptotically negligible compared to the level of the series +, hence the labelling of a local disjoint broken trend. hroughout, we assume that there is at least a change in slope as stated in the following assumption. Let the true break date be denoted by and let the break fraction be = =. Assumption A: b = and (; ).

8 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. By construction, the true break date increases in the sample size. In matrix notation, the Data Generating Processes can be speci ed as follows Y = + U (9) where Y = [y ; : : : ; y ], U = [u ; : : : ; u ], k = [x(k) ; : : : ; x(k) ], with x(k) t = [ t B t ] and = [ b ], for Model I, while for Model II, x(k) t = [ t C t B t ] and = [ b b ]. Note that the matrix depends on the candidate break date, while depends on the true break date. he parameters are estimated using a global leastsquares criterion. he estimate of the break date is ^ = arg min Y (I P )Y where P is the matrix that projects on the range space of, i.e., P = ( ). With ^ constructed using the estimate of the break date ^, the OLS estimate of is ^ = ( ^ ^ ) ^ Y and the resulting sum of squared residuals is, for an estimated break fraction ^ = ^ =, S(^) = ^u t = t= t= y t x( ^ ) t^ = Y (I P ^ )Y where P ^ is the projection matrix associated with ^. he limiting distributions of ^ and ^ have been derived by PZ for the dichotomous case with either I() and I() errors. In what follows, our aim is the following. First, we show that the break fraction can be estimated consistently by minimizing the sum of squared residuals when the errors are fractionally integrated. Second, we derive the limit distributions of the estimates. hird, we show that a spurious break phenomenon can occur even in the case of a break in a linear time trend when the errors are fractionally integrated.. A Key Inequality As in PZ, a key inequality will play a crucial role in proving the limit results. First, by construction, we have for all, S(^) S( )

9 or equivalently, Y (I P ^ )Y Y (I P )Y: Using (9), this inequality can be rewritten as or equivalently, Y (P P ^ )Y ( + U )(P ^ P ^ )( + U) = (P P ^ ) + (P P ^ )U + U (P P ^ )U = ( ^ ) (I P ^ )( ^ ) + ( ^ ) (I P ^ )U + U (P P ^ )U () + (U) + (UU) () where we have made use of the fact that P = it is straightforward to show that and (I P ) =. Moreover, arg min [SSR( )] = arg min [SSR( ) SSR( )] = arg min [ ( ) (I P )( ) + ( ) (I P )U + U (P P )U]: We will make use of this result later to derive the rate of convergence of ^ = ^ =. Asymptotic Results We consider in turn the consistency, rate of convergence and limit distribution of the estimates, concentrating on the estimate of the break date.. 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 (non-negative) term in () would asymptotically dominate the others. It means that the key inequality does not hold if ^ does not converge to, which leads to the desired contradiction. We start with the following theorem. 8

10 heorem De ne () = ( ) (I P )( ) (U) = ( ) (I P )U (UU) = U (P P )U: Under the Assumptions A-A, the following results hold uniformly over all generic [; ( ) ] for some arbitrary small such that [; ]. First, for d ( :5; :5), we have for Model I: i) if m = : () = j j O( ) (U) = j jo p ( =+d ) (UU) = j jo p ( +d ); ii) if m = : () = j j O( ) (U) = j jo p ( =+d ) (UU) = j jo p ( +d ): For Model II: i) if m = : () = j j O() (U) = j j =+d O p () (UU) = j j =+d O p ( =+d ): ii) if m = () = j j O() (U) = j j O p ( =+d ) (UU) = j jo p ( +d ): Note that () is always positive because it is quadratic and (I P ) is positive semi-de nite. Given the results in heorem, unless ^! p, then () = O( ), (U) = O p ( =+d ), and (UU) = O p ( d ) with m =. Similarly, () = O( ), (U) = O p ( 5=+d ), and (UU) = O p ( +d ) with m =. Hence, for large and d ( :5; :5), with 9

11 some probability, the positive term () dominates the other two terms (U) and (UU) such that inequality () will not hold with probability. Hence, we have a contradiction since the inequality () holds by construction. herefore, we can conclude that ^! p, as stated in the following theorem. heorem Under Assumptions A-A, in Model I-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, when 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 Model I and II with I() process, ^ = O p ( = ). On the other hand, for Model I with I() process, ^ = O p ( = ) and for Model II with I() process, ^ = O p ( ). PZ (5) 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. he results show that the rate of convergence is linearly decreasing as d increases for all models except Model II for d ( :5; :5). he result for this latter case is quite interesting as the rate of convergence is the same for all d ( :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 ( :5; :5), this added noise dominates and renders the rate of convergence invariant to d. If the process

12 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 convergence 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 heorem. heorem Under Assumptions A-A, we have for every d ( :5; :5): ) For Model I: a) if m =, b) if m =, where (d) is de ned by (), = d (^ ) ) (d) ( ) b ; = d (^ ) ) (d) R Wd (r)dr ( ) b = Z dw d (r) + Z ( ) (r ( ) Z dw d (r) )dw d (r); ( ) Z rdw d (r) and Z W d (r)dr = Z Z W d (r)dr + W d (r)dr Z ( ) (r )W d (r)dr: ( ) ( ) Z rw d (r)dr ) For Model II: a) if m =, de ne a stochastic process S (m) on the set of integers as follows: S () =, S (m) = S (m) for m < and S (m) = S (m) for m >, with S (m) = ( b + b k) ( b + b k)u k ; m = ; ; : : : ; S (m) = k=m+ m ( b + b k) + k= k= k=m+ m ( b + b k)u k ; m = ; ; : : : :

13 If u t is strictly stationary with a continuous distribution, S is a two-sided random walk with drift, and d (^ ) ) arg min m S (m). b) if m =, de ne = = = = Z W d (r)dr; ; ; W d ( ); Z Z [(r Z Z Z rw d (r)dr; W d (r)dr ; r )=( ) ]dw d (r); W d (r)dr; [(r )(r )=( ) ]dw d (r); = = Z ( ) (r ( ) ( ) ( ) ( +) ( ) ( ) 5 ( ) ( +) ( ) ( ) ( +) ( + ) ( ) ( ) )W d (r)dr ; Also de ne Z (v) as follows: Z () =, Z (v) = Z (v) for v < and Z (v) = Z (v) for v >, with : 5 Z (v) = ( b ) jvj = + v (d) b + v [ ]; v < ; Z (v) = ( b ) jvj = + v (d) b + v [ ]; v > : hen, = d (^ ) ) arg min v Z (v).. he limiting distribution of other parameters We turn to the limiting distribution of the other parameters in the models, that is (^ ; ^ ; ^ b ) for Model I, and (^ ; ^ b ; ^ ; ^ b ) for Model II.

14 heorem 5 Under assumption A-A, the following results hold for all d ( :5; :5). ) For Model I: = d (^ ) = d (^ ) 5 ) a if m = ; = d (^ b b) = d (^ ) = d (^ ) 5 ) a if m = ; = d (^ b b) where and a = = (d) = (d) Z ( + )= ( + )= = ( ) ( + )= ( + )= ( + )= ( ) 5 ; = ( ) ( + )= ( ) = ( ) Z + r r + r r ( ) ( +r) ( )r +( ) r ( )r ( )r ( ) (r r ) 5 dw d(r) + 5 dw d(r) + Z Z ( + r) + r+r fr + r fr (+ )r+ g ( +)fr (+ )r+ g 5 dw d(r) C A ; 5 dw d(r) C A : ( + )r + g ) For Model II: = d (^ R ) dw d(r) = d (^ ) R = d (^ b b ) = d b ( ^ ) (d) rdw d(r) R ) 5 dw d (r) R 5 = d (^ b b) (r )dw d (r) if m = ;

15 = d (^ R ) W d(r)dr = d (^ ) R = d (^ b b ) = d b ( ^ ) (d) rw d(r)dr R ) 5 W d (r)dr R 5 = d (^ b b) (r )W d (r)dr his implies that ^ b is asymptotically unidenti ed. if m = : Note that except for the unidenti ed intercept shift ^ b, the other parameters, (^ ; ^ ; ^ b ), have the same stochastic order for Model 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 structural break is present in the data generating process. Nunes et al. (995) and Bai (998) showed that the least square estimator of the break date can lead to a spurious break date when the errors are integrated, in the sense that the estimate will not converge to 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 I with a joint-segmented trend. he DGP is speci ed as follows; for t = ; : : : ; ; y t = + t + u t ; () and ( L) d u t t> = t () where t i:i:d:n(; ). When estimating a one-break model in slope using Model I, the regression for a candidate break date is y t = + t + B t + u t ; t = ; : : : ; : he so-called spurious break problem has been analyzed in the segmented regression model (see, e.g., Nunes et al, 995). However, we take a global least squares approach. Hence, the theoretical derivations are di erent. In matrix form, the DGP is Y = + U;

16 and Model I can be written as: Y = + U: he OLS estimate of is ^ = ( ) Y, the OLS residuals are ^U = [I ( ) ]Y; and the sum of squared residuals is given by ^U ^U = Y [I ( ) ]Y: It is straightforward to show that ^ = arg min = arg max ^U ^U = arg min f ^U ^U U Ug U ( ) U = arg max M ( ): Let M ( ) be the normalized version of M ( ), that is, M ( ) (d+m) M ( ) = (d+m) U D = (D = D = ) D = U (d+m) where D = diagf; ; g and m f; g. In order to derive the asymptotic distribution of M ( ), we need the following conditions which are similar to those of Nunes et al. (995) and Bai (998). Condition S here exists a diagonal matrix D such that D = D = Q(), uniformly in (; ) where Q() is assumed to be a positive de nite matrix for all >. Condition S For some, = D = U! p G(), where G() is a stochastic process having continuous sample paths. Conditions S and S hold with Assumptions A and A. Note that ^ = arg max M ( ) = arg max M ( ) because the normalization factor (d+m) does not depend on. If Conditions S and S hold, we have M ( ) p! M () G() Q()G(): It can be shown that (see, Bai, 99) ^ d! arg max (;) M (). Hence, the estimate of the break fraction ^ is a random variable with support in (; ). Note that this is true for all 5 p!

17 d ( :5; :5)[(:5; :5), which generalizes the results for I() and I() processes in Nunes et al. (995) and Bai (998). Below, we show that M ()! as! f; g if d ( :5; :5). aqqu (9) showed that for d ( :5; :5), the fractional Brownian motion W d (t); t R satis es the following law of iterated logarithms: lim sup t! W d (t) (ct +d log log t) = = a:s: for some positive constant c. Since W d (t) is self-similar with self-similarity parameter :5+d, for any c > it satis es, W d (t) d = c (:5+d) W d (ct), where d = denotes equality in distribution. Applying the law of iterated logarithms to W d (=t) and self-similarity, we have lim sup t! W d (t) (ct +d log log(=t)) = = a:s: hen, for d ( :5; ], lim sup! W d () p q = lim sup c d log log(=) =! a:s: It is easy to verify that W d () s =. For d ( :5; ], W d () is also a fractional Brownian motion W d (s) with lim sup! W d () W d () p = lim sup s! W d (s) p s = a:s: Of interest is the behavior of M () when gets closer to either or. heorem Under Assumption A and A: ) for any d ( :5; ], lim sup! M () = lim sup! M () = a:s:; ) for any d (; :5), lim sup! M () = a:s:. heorem implies that no spurious break is estimated if the order of fractional integration is a value in ( :5; :5). Proposition For de ned in Condition S, assume d with d (:5; :5). hen, sup (;) M () = O p (). he proof is similar to that of heorem in Bai (998) with Lemma and is omitted. Proposition implies that M () is stochastically bounded for d (:5; :5). In the following theorem, we show that it is not possible that ^! f; g in the limit. Note that M ()

18 and M () are de ned without the dummy variable for a slope change B t in the model. After some algebra, we have Z M () = M () = (d)[ Z W d (r)dr Z W d (r)dr Z rw d (r)dr + rw d (r)dr ]: heorem Under Assumption A and A, for any d (:5; :5), M () = M () < M (), for every (; ): heorem implies that the maximum value of M () cannot be located at or and the value that maximizes M () on any subset of [; ] is bounded away from or since M () is not a constant process. Hemce, the spurious break feature applies when d (:5; :5). 5. Monte Carlo Experiments 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 ; ( L) d u t t> = t ; t i:i:d:n(; ): for t = ; : : : ;. Without loss of generality, we set = = and we consider d f :; :; :8; :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 either Model I (joint-segmented trend) or Model II (locally disjoint broken trend). Figure presents the results pertaining to Model I. Figure (a) presents histograms of the estimates ^ when =. For d f :; :g, 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. For d f:8; :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 estimates ^ 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:8; :g. Hence, the theoretical results are supported by the simulations.

19 he corresponding results for Model II are presented in Figure. Interestingly, in this case a spurious break occurs for all positive values of d even when =. Hence, it appears that simply introducing an irrelevant level shift can alter the results towards having a spurious break. More work is needed to theoretically assess this feature. 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. hese tests are useful to avoid the spurious break problem. Conclusion his paper considered the consistency, rate of convergence and limit distribution of the estimate of a break date in the slope of a linear trend function, with or without a concurrent level shift, when the errors are fractionally integrated with d ( :5; :5) [ (:5; :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 allowed 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 and show that it cannot occur in the limit when d in the interval ( :5; :5). Simulation experiments illustrate this theoretical result. he results in this paper can be useful for subsequent work. For instance, Lobato and Velasco () considered e cient Wald test of 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 (). his is currently the object of ongoing research. 8

20 Appendix We consider the proofs of heorems - for Models I and II separately, for ease of exposition. Note rst that all limit statements are taken as!. A. Results for Model I Model I can be represented in matrix notation as Y = + U i = h t B b 5 + U where Y = (y ; : : : ; y ), U = (u ; : : : ; u ), = (x( ) ; : : : ; x( ) ) with x( ) t = (; t; B t ) and = ( ; ; b ). Note that the matrix depends on the candidate value of the break date. In the proof, we only consider the case >. It is straightforward to apply the same arguments to the case where <. For >, let 8 if t >< and for =, let With this notation, we can write t e b (t) = if < t < >: if t : 8 < if t e b (t) = b (t) = : if < t : ( ) = b ( )e b Note that e b ([ r]) converges to a continuous function f eb (r) over [; ] de ned by, for >, 8 if r >< f eb (r) = r if < r < >: if r ; and by, for =, 8 < if r f eb (r) = f b (r) = : if < r : 9

21 Pertaining to the proof of heorem, we rst consider the term (). We have () = ( ) (I P )( ) = ( ) be b(i P )e b where the second equality holds because the rst two columns of ( ) are zeros by construction. Note that e b(i P )e b is the sum of squared residuals from a regression e b on [ t B ]. De ne S = e b(i P )e b : Next, consider the continuous time least-squares projection of the function f eb (r) on [ r f B (r)], where f B (r) = (r ) r. Let [^ ^ ^] denote the estimates of the coe cients and let S denote the resulting SSR. From the de nition of a Riemann integral, S! S, where S = Z f eb (r) ^ ^r ^fb (r) dr: Suppose that ^ = ^ =. It is easy to show that S > from the de nition of f eb (r) and f B (r). Otherwise, we have S Z minf; g Z minf; g f eb (r) ^ ^r ^fb (r) dr = (^ + ^r) dr > where the equality holds because of the de nition on f eb (r) and f B (r) and the fact that both and are bounded away from zero. Hence, < S < and S = O( ). Accordingly, Next, we consider the term (U). We have () = ( ) bo( ): (U) = ( ) (I P )U = b ( )e b(i P )U De ne f e eb (r) as the projection residuals from a least-squares regression of f eb (r) on [ r f B (r)]. By the properties of orthogonal projections and the result for (), we have Z ef eb (r)dr = Z f eb (r) ^ ^r ^fb (r) dr = and Z [ f e eb (r)] dr = S = O() uniformly over all. By the functional central limit theorem (FCL) and the continuous mapping theorem, (d+=) e b(i Z P k )U ) (d) ef eb (r)dw d (r) if m = :

22 Similarly, Z (d+=) e b(i P k )U ) (d) ef eb (r)w d (r)dr if m = ; where (d) = [b ( d)e" ]=( + d) ( + d) ( d). We deduce that E[ Z ef eb (r)dw d (r)] = and E[ Z ef eb (r)w d (r)dr] = ; Z V ar[ ef eb (r)dw d (r)] = Z Z = d Z Z ef eb (r) e f eb (u)e[dw d (u)dw d (r)] where d = d(d + =) and H is a Banach space, and V ar[ Z ef eb (r)w d (r)dr] = = = Z Z Z Z Z Z r = O p (); if m = ju rj d e feb (r) e f eb (u)dudr = k e f eb k H; if m = ef eb (r) e f eb (u)e[w d (r)w d (u)]dudr ef eb (r) e f eb (u) (jujd+ + jrj d+ ju rj d+ )dudr ef eb (r) e f eb (u)(juj d+ + jrj d+ ju rj d+ )dudr uniformly over all [; ]. herefore, R f e eb (r)w d (r)dr = O p () and 8 < O e p ( d+= ) if m = ; b(i P )U = : O p ( d+= ) if m = : Hence, we have 8 < b ( )O p ( d+= ) if m = ; (U) = : b ( )O p ( d+= ) if m = : Finally, we consider the term (UU). De ne D = diag( d+= ; d+= ; d+= ) with d ( :5; :5). We have (UU) = U (P P )U = Uf ( ) ( ) gu = U( )D [D D ] D U + U D [D D ] D [ ] D [D D ] D U + U D [D D ] D ( ) U:

23 Applying the FCL for d ( :5; :5) and m =, (d+=) (d+=) u t ) (d)w d (); t= tu t ) (d)[w d () t= Also, from Lemma with m =. (d+=) (d+5=) In addition, it is easy to show that t= + t= + Z u t ) (d) t= tu t ) (d) t= t= + We next consider each term in (UU). (t )! (t )t! (t )! W d (r)dr] = (d) Z Z Z Z Z W d (r)dr; rw d (r)dr: (r ) dr; (r )rdr; (r )dr:. D D and D D are O( d ) uniformly in. Z rdw d (r):. When m =, D U and D U are O p () uniformly in, and P (d+=) t= u t (d)w d () D U = (d+=) P t= tu t (d+=) P t= + (t )u t 5 ) (d) R rdw d(r) (d) R (r When m =, D U and D U are O p ( ) uniformly in, and D U = (d+=) P t= u t (d+5=) P t= tu t (d+5=) P t= + (t )u t 5 ) 5 : )dw d(r) (d) R W d(r)dr (d) R rw d(r)dr (d) R (r 5 : )W d(r)dr

24 . U ( )D. It su ces to consider the third column of ( the rst two columns are zeros. We have ) because (d+=+m) U (B B ) = (d+=+m) (t )u t + (d+=+m) ( ) + + = j jo p () for m f; g:. D [ ]D. As noted earlier, it su ces to consider the terms in which B and B are involved. Hence, we have B B B B = j jo( ) B t B t = j jo( ) B B = j jo( ) D [ ]D = j jo( (+d) ); for m f; g: Based on the results -, 8 < j jo p ( +d ) if m = (UU) = : j jo p ( +d ) if m = his completes the proof of heorem for Model I. A.. Proof of Consistency (heorem ) From the proof of heorem, we know that for Model I, if m =, and, if m =, ( ^ ^) = ( ( ^ ^U) = b ( ( ^U ^U) = j ( ^ ^) = ( ( ^ ^U) = b ( ( ^U ^U) = j ^ ) bo( ) ^ )O p ( =+d ) ^ jo p ( +d ): ^ ) bo( ) ^ )O p ( =+d ) ^ jo p ( +d ) We consider the proof for m = (the proof for m = is similar). Suppose that ^ 9 p. hen, the results above imply that ( ^ ^) = O( ), ( ^ ^U) = O p ( =+d ), and ( ^U ^U) = O p ( d ) for d ( :5; :5). herefore, for su ciently large, the term ( ^ ^) dominates the others with some probability. It implies that the key inequality ( ^ ^)+( ^ ^U)+( ^U ^U) cannot hold with probability. Since this inequality is valid for all, we have a contradiction. Hence, we can conclude that ^! p. u t

25 A.. Rate of Convergence (heorem ) Consider the set V () = f : j j < ; 8 > g: From the consistency of ^ in heorem, Pr ^ V ()! as!. Hence, it su ces to consider the behavior of S( ) for all V (). Consider another set V c () de ned by V c () = f : j j < and j j > C =+d+m ; 8 > ; 8d ( :5; :5); m = f; gg: Note that V c () V (). Since S( ^ ) S( ) with probability, we can claim that = V c () by showing that for each >, there exists a constant C > such that Pr min fs( ) V c() S( )g < (A.) Equation (A.) implies that a minimum cannot be obtained in the set V c () and that j j C =+d+m must hold with an arbitrary large probability. Equation (A.) is equivalent to Pr min f() + (U) + (UU)g V c() < Based on results derived in heorem, we can apply the following normalizations. If m =, then If m =, then () j j = j j bo( ) > C =+d bo( ) =+d j j =+d =+d (U) j j = j j b O p ( =+d ) = O =+d j j =+d p () (UU) j j = j jo p ( +d ) = o =+d j j =+d p (): () j j = j j bo( ) > C =+d bo( ) =+d j j =+d =+d (U) j j = j j b O p ( =+d ) = O =+d j j =+d p () (UU) j j = j jo p ( +d ) = o =+d j j =+d p (): = ac + o() = ac + o() where a is a positive constant. Here, we simply use the fact that j j < and j j > C =+d+m in V c (). herefore, Equation (A.) is satis ed for all > if we choose a su ciently large C >.

26 A.. Limiting Distribution of the estimate of the break date Consider rst the case with m =. De ne the set D(C) = f : j j < C =+d g, for some positive number C, and m = = d j j. We are interested in the stochastic orders of the other parameters. We analyze arg min D(C)[SSR( ) SSR( )]: For D(C), we have j j = O( =+d ). Hence, () = j j O( ) = O( +d ), (U) = j jo p ( =+d ) = O p ( +d ) and (UU) = j jo p ( +d ) = O p ( =+d ). hen, arg min D(C)[SSR( ) SSR( )] = arg min D(C)[() + (U) + (UU)]= +d = arg min D(C)[()= +d + (U)= +d + o p ()]; hence we only need to consider the rst two terms. Note that on the set D(C), j j = O( =+d ) for d ( :5; :5). Using this fact, we can derive the following results that will subsequently be applied: = ( ) = d D D = = = ( ) ( + )= 5 + o() ( ) = ( ) ( + )= ( ) = a + o(); and the inverse is d (D D ) = a + o() with ( + )= ( + )= = ( ) We have a = ( + )= ( + )= ( + )= ( ) 5 = ( ) ( + )= ( ) = ( ) () = b(b B ) (I P )(B B ) = bf(b B ) (B B ) he second term in () is such that (B B ) D (D D ) D (B B )g (B B ) D = j j = d =+d e b D i = m h ( ) + o() 5

27 where m = = d j j. Using the results above, h d (B B ) D (D D ) = m Hence, d (B B ) ( ) (B ( ) B ) = and d (B B ) (B B ) = d j j e be b = m e be b = ( )m + o() i ( ) ( ) + o() ( ) (A.) m + o() Combining (A.) and (A.), we obtain ( d (B B ) ) (I P )(B B ) = m + o() (A.) Now, We have, (U) = ( ) (I P )U = (B B ) (I P )U: d (B B ) U = j j = d = d e bu = m (d) Z ( r=)w d (r)dr + o p (); and d (B B ) ( ) U = d (B B ) D (D D ) D U = (B B ) D d (D D ) D U; " # D U = = d u t = d tu t = d (t )u t t= t= t= + = = d u t 5= d tu t 5= d (t )u t + o p () 5 : t= t= t= +

28 Hence, for d ( :5; :5) and m =, we have d (B B ) (I P )U Z Z = (d)f W d (r)dr + W d (r)dr Z ( ) ( ) rw d (r)dr ( ) Z = (d) b m Wd (r)dr + o p (); Z (r )W d (r)drg b m + o p () where Wd (r) is the residuals function from a continuous time least-squares regression of W d (r) on f; r; (r ) r> g. herefore, m = arg min m D(C)[() d + (U) d + o p ()] Z = arg min m D(C)[m ( ) b + (d)m b Wd (r)dr] + o p () by the continuous mapping theorem. Note that the objective function does not change if <. We can conclude that m = = d j ^ j ) (d) R W d (r)dr ( ) b : Next, consider the case with m =. De ne m = =+d j j for this case. Note that = d e bu ) (d) R dw d (r). For (), we have the same results as for m =. For (U), we have: (U) = b (B B ) (I P )U = = d b m e b(i P )U = = d b m e = d b b m e b D (D D ) D U Z = b m (d) dw d (r) R ( ) ( ) dw d(r) R ( ) rdw d(r) R 55 + o p() (r )dw d (r) Z Z + r r ( + r) = b m (d) dw d (r) + dw d (r) + o p () ( ) b m (d) + o p ():

29 For (UU), we know that U is an I(d) process with d ( :5; :5). It is easy to show that U ( )D = j jo p ( ), D U = O p (), and D D = O p ( d ). Hence, (UU) = j jo p ( d ) which is dominated by (U) asymptotically. he optimal m is therefore given by A.. m = =+d (^ ) ) (d) b ( ) : Limit Distributions of the Other Parameters he OLS estimates of the regression coe cients is Hence, First, for m =, ) ^ = ( ^ ^ ) ^ Y = ( ^ ^ ) ^ + ( ^ ^ ) ^ U = + D (D ^ ^ D ) D ^ ( ^ ) + D (D ^ ^ D ) D ^ U: D (^ ) = (D ^ ^ D ) [D ^ ( ^ ) + D (d) b ( ) b = (d) : Z D ^ ( ^ ) + D = D ^ b j je b + D ^ U ^ U = D ^ b j j = d e b =+d + D = m b =+d D ^ e b + D 5 + (d) ( ) R (r + r r + r r ( ) ( +r) Since d (D D ) p! a, 5 dw d(r) + ^ U R dw d(r) R rdw d(r) Z 5 )dw d (r) d D (^ ) ) a : 8 ( + r) ^ U + r+r + r ^ U]: 5 dw d(r) C A

30 Second, for m =, D (^ ) = (D ^ ^ D ) [ D ^ ( ^ ) + D hen, we have D ^ ( ^ ) + D D ^ ^ b j je b + D = D ^ b j j = d e b =+d + D = m b =+d D ^ e b + D Z ) (d) Wd (d)dr = (d) : Z (+ ) ( ) ( )r +( ) r ( )r ( )r ( ) (r r ) ^ U 5 + (d) R (r 5 dw d(r) + ^ U R W d(r)dr R rw d(r)dr 5 )W d dr(r) Z fr (+ )r+ g ( +)fr (+ )r+ g fr ^ U ^ U]: 5 dw d(r) C A ( + )r + g herefore, d D (^ ) ) a : A. Results for Model II We now consider results for Model II. he proofs of the consistency is similar to that for Model I. In any event, the relevant bound will be derived in the proof of the limit distribution. A.. Consistency (heorem ) From heorem, for m = : ( ^ ^) = ( ( ^ ^U) = b ( ( ^U ^U) = j ^ ) bo() ^ ) =+d O p () ^ j =+d O p ( =+d ); and for m = : ( ^ ^) = ( ( ^ ^U) = b ( ( ^U ^U) = j ^ ) bo() ^ ) O p ( =+d ) ^ jo p ( +d ): 9

31 he proof of consistency is similar to that for Model I. Suppose that ^ p 9. hen, with m =, ( ^ ^) = O( ), ( ^ ^U) = O p ( 5=+d ) and ( ^U ^U) = O p ( +d ) for all d ( :5; :5). Hence, with some positive probability, ( ^ ^) dominate the other two terms, so that this result cannot be compatible with the key inequality (). Hence, we have a contradiction and conclude that ^ p!. A.. Rate of Convergence (heorem ) We consider the set ev c () = f : j j < and j j > C m(d+=) ; 8 > ; 8d ( :5; :5); m = f; gg: Given the results in heorem, if m = : () j j = j j bo() = j =+d j j =+d j = d bo() > C = d bo() = ac + o(); (U) j j = j j =+d b O p () = O =+d j j =+d p (); (UU) j j =+d = j j =+d O p ( =+d ) j j =+d = o p (); and if m = : for 8d ( () j j = j j bo() =+d j j > C =+d bo() =+d =+d (U) j j = j j b O p ( =+d ) = O =+d j j =+d p (); (UU) j j = j jo p ( +d ) = o =+d j j =+d p () = ac + o(); :5; :5) where a is a positive constant. It is easy to show that! P r min fs( ) S( V e )g c() for any > if we choose a su ciently large C >. his completes the proof. A.. < Limit Distribution of the Estimate of the Break Date Given the results in heorem, we work on the set D (C) = f : j j < d Cg if m = and D (C) = f : j j < =+d Cg if m =, for some positive C. In other words, for = =, j j = O p ( +d ) with m = and j j = O p ( =+d ) with m =. In matrix notation, Model II can be expressed as Y = + U

32 with i = h t C B where = (; : : : ; ), t = (; ; : : : ; ), C = (C ; : : : ; C ), B = (B ; : : : ; B ) and = [ b b ]. For >, 8 < if t C C = : otherwise; and 8 < t B B ( if t )C = : otherwise: When <, and 8 < if t C C = : otherwise; 8 < (t B B ( ) if t )C = : otherwise: We shall use the following notations. For >, g ( ) = t= + [ b + b (t )]; and for <, h ( ) = g ( ) = t= + [ b + b (t )] t= + [ b + b (t )]; h ( ) = t= + [ b + b (t )]

33 We rst consider the term (). Noting that ( )(I P )C =, we have () = ( ) (I P )( ) = [(C C ) b + (B B ( )C ) b )] (I P ) = [(C C ) b + (B B ( )C ) b )] t= + [ b + b (t )] [ b + b (t )]x( ) td (D D ) D t= + where D = diag( =+d ; =+d ; =+d ; =+d ). Note that for >, [ b + b (t )]x( ) td t= + = d t= + [ b + b (t )][ = = = d t = ] t= + [ b + b (t )][ t= ] = = d [ b + b k][ (k + )= ] k= t= + x( ) t [ b + b (t )] = = d g [ = ] + = d [ b + b k][ k= ] = d jg j[ = ] + jg j = d j j [ ] = O p (jg j = d ): where the last step follows from the fact that j j=! p on both D (C) and D (C). Also, (D D ) = O p ( d ): Hence, the second term in () is such that k= ( ) P ( ) = O p (g ) = o p (h )

34 because j j = O p ( +d ) if m = and j j = O p ( =+d ) if m = where d ( :5; :5). herefore, 8 < h + o p (h ) if > () = : h + o p (h ) if his implies that () = j j O() since b is xed. Consider now the term (U). For m =, we have (U) = ( ) (I P )U = t= + [ b + b (t )]u t [ [ b + b (t )]x( ) td ](D D ) D t= + We consider each term of this expression.. When <, let u t = u + v k. hen, = d t= + [ b + b (t )]u t = = d. d D D = + o(), where = k= [ b + b k]u +k U = = d g u + = d [ b + b k]v k = g (d)w d ( ) + o p (g ): ( ) ( ) ( ) ( ) ( +) ( ) ( ) ; 5 k=

35 and = ( ) ( ) ( ) (+ ) ( ) ( ) ( ) (+ ) ( ) ( ) : 5. D U ) (d), where R W d(r)dr R = rw d(r)dr R W d (r)dr 5 R (r )W d (r)dr using integration by parts.. When <, = R = d [ b + b (t )] D t= + = d R ( r)dw d(r) R r dw d (r) R ( )dw d (r) + R ( t= + [ b + b (t )][ t= ] r)dw d (r) ( ) dw d (r) + R ( ) ( ) dw d (r) = d g [ ] + d [ b + b k][ k= ] k= = d g [ ] + o p (g d ): Combining the results -, we obtain that (U) = =+d fg (d)w d ( ) g [ ] (d) + o p ()g = =+d g (d)fw d ( ) [ ] + o p ()g = =+d g (d) + o p ( =+d g ): 5 After some algebra, we have = W d ( ) [ ] = Z r r dw d (r):

36 We can show that when >, = d [ b + b (t )]u t = g (d)w d ( ) + o p (g ) t= + and = d [ b + b (t )]x( ) td = d g [ ] + o p (g d ): t= + Hence, where = (U) = =+d g (d) + o p ( =+d g ) Z hese results imply that 8 < (U) = =+d (d) : and We nally consider the term (UU). We have We rst have [(r )(r )=( ) ]dw d (r): (UU) = U (P P )U g if < + o p () g if > (U) = j j O p ( =+d ): = U ( )D (D D ) D U + U D (D D ) D [ ] D (D D ) D U + U D (D D ) D ( ) U Z = d U (C C ) = (d)( ) W d (r)dr + o p (); = d U (B B ) = (d)( ) Z rw d (r)dr + o p (): Hence, U ( )D = ( )[(d) + o p ()] 5

37 where = [ R W d (r)dr R rw d (r)dr]. For the second term in (UU), we have D [ ]D = ( ) d f with Hence, f = ( ) : 5 +d U D (D D ) D [ ] D (D D ) D U = ( ) +d (d)[ + o p ()] where = f = Collecting the results above, we have ( ) ( +) ( ) ( ) ( +) ( + ) ( ) ( ) : 5 his implies that with m =, (UU) = ( ) +d (d)[ + o p ()]: (UU) = j jo p ( +d ): De ne m = ( ) = d. It is easy to show that both h and h are asymptotically equivalent to =+d ( b ) jm j = and both g and g are asymptotically equivalent to +d m b=, therefore = d () = bjm j = + o p (); 8 < (d)m = d (U) = b + o p () if m > : (d)m b + o p () if m < = d (UU) = m (d) [ ] + o p ():

38 De ne Z (v; ; b ; (d)) as follows: Z () =, Z (v) = Z (v) for v < and Z (v) = Z (v) for v >, with Z (v; ; b ; (d)) = ( b ) jvj = + v (d) b + v(d) [ ] + o p (); Z (v; ; b ; (d)) = ( b ) jvj = + v (d) b + v(d) [ ] + o p (): By the continuous mapping theorem, we have m ( ^ ) = d ) arg min v Z (v; ; ; ): Now, consider the case with d ( :5; :5) and m =. he following argument applies to the set D (C) = f : j j < d Cg and accordingly we have j j = O p ( +d ) for = =. As in the case with m =, 8 < h + o p (h ) if < () = : h + o p (h ) if > : If >, (U) = t= + [ b + b (t )]u t f [ b + b (t )]x( ) td g(d D ) D U: t= + We next consider each term of (U).. t= + [ b + b (t )]u t = = k= k= [ b + b k]u k+ b u k+ + b k= ku k+ = O p (j j =+d ):

39 . [ b + b (t )]x( ) td t= + = = d t= + [ b + b (t )][ t= ] = = d [ b + b k][ (k + )= ] k= = = d [ b + b k][ k= ] + = d [ b + b k][ ] k= = O p (j j = d ): k=. (D D ) = O p ( d ):. D U = O p (): Since we search in a set for which j j < d C for some C > and j j = O p ( +d ), ( ^ ) P U is dominated by ( ^ ) U asymptotically. Hence, (U) = j j =+d O p (): We can derive the results for > in a similar way. In sum, 8 P >< t= +[ b + b (t )]u t + o p () if > (U) = if = >: P t= + [ b + b (t )]u t + o p () if < : Next, consider the term (UU). We have = d U (C C ) = = d maxf ; g t=minf ; g+ u t = = d j j =+d j j = d = = d j j =+d O p (): maxf ; g t=minf ; g+ u t 8

40 and = d U (B B ) = ( = d U B = d U B ) = j jo p (): Hence, U ( )D = j j =+d O p ( = d ): hen following the same arguments as for Model I, we have (UU) = j j =+d O p ( = d )O p ( d )O p () = j j =+d O p ( =+d ): Following Bai (99), we de ne a stochastic process S () on the set of integers as follows: 8 S >< () if < S () = if = >: S () if > with S () = S () = k=+ ( b + b k) ( b + b k) k= k=+ ( b + b k)u k ; = ; ; : : : ; ( b + b k)u k ; = ; ; : : : : k= Under the assumption that u t is strictly stationary and has a continuous distribution, the rest of the proof is similar to that of Bai (99, p.59) and, hence omitted. A.. Limit Distributions of the Other Parameters As for Model I, we use the facts that and where D (^ ) = (D ^ ^ D ) [D ^ ( ^ ) + D d (D ^ ^ D ) = + o() = ( ) ( ) ( ) ( ) ( +) ( ) ( ) 9 : 5 ^ U];

41 Hence, we obtain d D (^ ) R dw d(r) ( = b j ^ ) R j = d + (d) rdw d(r) R + o p () 5 dw d (r) C ( ) R 5A (r )dw d (r) R dw d(r) R = b j ^ j = d + (d) rdw d(r) R + o p (): 5 dw d (r) R 5 (r )dw d (r) Note that the limiting distribution of ^ b depends on that of j ^ to show that, when m =, j. Similarly, it is easy d D (^ ) R W d(r)dr R = b j ^ j = d + (d) rw d(r)dr R + o p (): 5 W d (r)dr R 5 (r )W d (r)dr Proof of heorem : Consider rst the case with d ( we have :5; :5). After some algebra, M ( ) p! M () = (d)[ + [W d()] ( + ) + + ( ( + ) Z W d () rdw d (r) Z Z ( + ) ) W d() (r )dw d (r) ( ) Z Z rdw d (r) + (r )dw ( ) d (r) ]: Z rdw d (r) (r )dw d (r)

42 We can write R (r )dw d(r) = W d () + W d () + R rdw d(r). hen M () = (d)[ + [W d()] ( + ) Z W d () rdw d (r) ( ) W Wd () W d () d() p + = ( Z ( + ) Wd () W d () + rdw ( ) = d (r) p Z ( + ) Z rdw d (r) ( ) + ( ) W d () + W d () + ( + ) rdw d (r) + Z Z ) W d() rdw d (r) rdw d (r) ]: Z rdw d (r) Since the last term is quadratic, it dominates the other terms if it diverges. Note that ( ) = ( ) W d () + W d () + Z rdw d (r) W d() W d () p + ( ) = Z rdw d (r) : (A.) By applying the law of iterated logarithms for a fractional Brownian motion, we can show that lim sup! M () = a:s: for d ( :5; ]. Furthermore, note that ( ) = ( ) W d () + W d () + Z rdw d (r) W d() W d () + ( ) Using the iterated law of logarithms for d (; :5), we obtain Z rdw d (r) ; lim sup! W d () W d () = lim sup s! W d (s) s = a:s:: Hence, this shows that lim sup! M () = a:s: for d (; :5). On the other hand, when!, we can apply the law of iterated logarithms to the quadratic term only for d ( :5; ] because the order of is not su cient for the law of iterated logarithms to hold with d (; :5). Proof of heorem : Consider the case with d (:5; :5). With the functional central