New Distribution Theory for the Estimation of Structural Break Point in Mean

Transcription

1 New Distribution Teory for te Estimation of Structural Break Point in Mean Liang Jiang Singapore Management University Xiaou Wang Te Cinese University of Hong Kong Jun Yu Singapore Management University September, 6 Abstract Based on te Girsanov teorem, tis paper obtains te exact distribution of te maximum likeliood estimator of structural break point in a continuous time model. Te exact distribution is asymmetric and tri-modal, indicating tat te estimator is biased. Tese two properties are also found in te finite sample distribution of te least squares LS estimator of structural break point in te discrete time model, suggesting te classical long-span asymptotic teory is inadequate. Te paper ten builds a continuous time approximation to te discrete time model and develops an in-fill asymptotic teory for te LS estimator. Te in-fill asymptotic distribution is asymmetric and tri-modal and delivers good approximations to te finite sample distribution. To reduce te bias in te estimation of bot te continuous time and te discrete time models, a simulation-based metod based on te indirect estimation IE approac is proposed. Monte Carlo studies sow tat IE acieves substantial bias reductions. JEL classification: C; C46 Keywords: Structural break, Bias reduction, Indirect estimation, Exact distribution, In-fill asymptotics We would like to tank te editors and 3 referees, participants of many conferences and seminars for teir constructive comments tat ave significantly improved te paper. Tis researc was supported by te Singapore Ministry of Education MOE Academic Researc Fund Tier 3 grant wit te MOE s offi cial grant number MOE3-T3--9. Corresponding autor: Jun Yu, Scool of Economics and Lee Kong Cian Scool of Business, Singapore Management University; yujun@smu.edu.sg.

2 Introduction Statistical inference of structural breaks as received a great deal of attention bot in te econometrics and in te statistics literature over te last several decades. Tremendous efforts ave been made in developing te asymptotic teory for te estimation of te fractional structural break point te absolute structural break point divided by te total sample size, including te consistency, te rate of convergence, and te limiting distribution; see, for example, Yao 987 and Bai 994, 997b, among oters. Asymptotic teory as been developed under te long-span asymptotic sceme under wic te time span of data is assumed to infinity. Tis long-span asymptotic distribution is te distribution of te location of te extremum of a two-sided Brownian motion wit triangular drift over te interval, +. It is symmetric wit te true break point being te unique mode, indicating tat te estimators ave no asymptotic bias. Interestingly and rater surprisingly, ow well te asymptotic distribution works in finite sample is largely unknown. Focusing on simple models wit a sift in mean, tis paper systematically investigates te performance of te long-span asymptotic distribution, te exact distributional properties, and te bias problem in te estimation of te structural break point. To te best of our knowledge, our study is te first systematic analysis of te exact distribution teory in te literature. Our paper makes several contributions to te literature. First, by using te Girsanov teorem, we develop te exact distribution of te maximum likeliood ML estimator of te structural break point in a continuous time model, assuming tat a continuous record over a finite time span is available. It is sown tat te exact distribution is asymmetric wen te true break point is not in te middle of te sample. Moreover, te exact distribution as trimodality wen te signal-to-noise ratio te break size over te standard deviation of te error term is not very large, regardless of te location of te true break point. Asymmetry togeter wit trimodality makes te ML estimator biased and suggests tat te long-span asymptotic distribution does not conform to te exact distribution. It is also found tat upward downward bias is obtained wen te fractional structural break point is smaller larger tan 5%, and te furter te fractional structural break point away from 5%, te larger te bias. Second, te properties of asymmetry and trimodality are found to be sared by te finite sample distribution of te LS estimator of te structural break point in te discrete time model, suggesting a substantial bias in te LS estimator and te inadequacy of te long-span asymptotic distribution in finite sample approximations. To better ap-

3 proximate te finite sample distribution, we consider a continuous time approximation to te discrete time model wit a structural break in mean and develop an in-fill asymptotic teory for te LS estimator. Te developed in-fill asymptotic distribution retains te properties of asymmetry and trimodality, and, ence, provides better approximations tan te long-span asymptotic distribution. Te in-fill asymptotic sceme leads to a break size of a smaller order tan tat assumed in Bai 994. It is tis important difference in te break size tat leads to a different asymptotic distribution. Tird, an indirect estimation IE procedure is proposed to reduce bias in te estimation of te structural break point. One standard metod for bias reduction is to obtain an analytical form to approximate te bias and ten bias-correct te original estimator via te analytical approac as in Kendall 954 and Yu. However, it is diffi cult to use te analytical approac ere, as te bias formula is diffi cult to obtain analytically. Te primary advantage of IE lies in its merit in calibrating te binding function via simulations and avoiding te need to obtain an analytical expression for te bias function. It is sown tat IE, witout using te analytical form of te bias, acieves substantial bias reduction. Te in-fill asymptotic treatment is not new in te literature. Recently, Yu 4 and Zou and Yu 5 demonstrated tat te in-fill asymptotic distribution provides better approximations to te finite sample distribution tan te long-span asymptotic distribution in persistent autoregressive models. To te best of our knowledge, it is te first time in te literature of structural breaks tat te in-fill asymptotic distribution is derived. As in Yu 4 and Zou and Yu 5, we also find tat te in-fill asymptotic distribution conforms better to te finite sample distribution tan te longspan counterpart. Te rest of te paper is organized as follows. Section gives a brief review of te literature and provides te motivations of te paper. Section 3 develops te exact distribution of te ML estimator of structural break point in a continuous time model. Section 4 establises a continuous time approximation to te discrete time model previously considered in te literature and develops te in-fill asymptotic teory for te LS estimator under different settings. Te IE procedure and its applications in te continuous time and te discrete time models wit a structural break are introduced in Section 5. In Section 6, we provide simulation results and compare te finite sample performance of IE wit tat of te traditional estimation metods and of Pillips 987 and Perron 99 developed te in-fill asymptotic distributions of te LS estimator of te autoregressive parameter. Barndorff-Nielsen and Separd 4 developed te in-fill asymptotic distribution of te LS estimator in regression models. 3

4 oter simulation-based metods. Section 7 concludes. All proofs are contained in te Appendix. Literature Review and Motivations Te literature on estimating structural break points is too extensive to review. partial list of contributions include Hinkley 97, Hawkins et al. 986, Yao 987, Bai 994, 995, 997a, 997b, Bai and Perron 998 and Bai et al In tese studies, large sample teories for different estimators under various model settings are establised. A simplified model considered in Hinkley 97 is µ + ɛ t if t k Y t =, t =,..., T, µ + δ + ɛ t if t > k were T denotes te number of observations, ɛ t is a sequence of independent and identically distributed i.i.d. random variables wit E ɛ t = and V ar ɛ t = σ. Let k denotes te break point wit true value k. Te condition of k < T is assumed to ensure tat one break appens. Te fractional break point is defined as τ = k/t wit true value τ = k /T. Constant µ measures te mean of Y t before break and δ is te break size. Let te probability density function pdf of Y t be fy t, µ for t k and fy t, µ + δ for t > k. Under te assumption tat te functional form of f, and te parameters µ and δ are all known, te ML estimator of k is defined as k } T kml,t = arg max log fy t, µ + log fy t, µ + δ. k=,...,t t=k+ Te corresponding estimator of τ is τ ML,T = k ML,T /T. Yao 987 developed a longspan limiting distribution under te sceme of T followed by δ wic takes te form of δ I µ kml, k d arg max u, A W u u }, 3 were I µ is te Fiser information of te density function fy, µ, W u is a twosided Brownian motion wic will be defined below, and d denotes convergence in distribution. Te closed-form expressions for te pdf and te cumulative distribution function cdf of te long-span limiting distribution were derived in Yao

5 Density.5.45 argmax W u u / N, u Figure : Te pdfs of arg max u, W u u } and a standard normal distribution. et al. For te same model as in Equation wit unknown parameters µ and δ, Hawkins 986 and Bai 994 studied te long-span asymptotic beavior of te LS estimator of te break point. Te LS estimator takes te form of were S k = kls,t = arg min k=,...,t k T Yt Y k + t=k+ } S k = arg max [Vk Y t ] }, 4 k=,...,t Y t Y k wit Y k Y k being te sample mean of te first k last T k observations and [V k Y t ] T T k. = Y T k Y k Te corresponding estimator of τ is τ LS,T = k LS,T /T. Hawkins et al. 986 sowed tat T α τ LS,T τ p for any α < /, were p denotes convergence in probability. Bai 994 improved te rate of convergence by sowing tat τ LS,T τ = O p T δ. In addition, by letting te break size depend on T denoted by δ T, and assuming tat δ T wit T δ T log T as T, Bai 994 derived an asymptotic distribution as T δ T /σ ˆτ LS,T τ arg d max u, W u u }, 5 wic is te same as in 3. Tis long-span asymptotic distribution in 5 is widely used as an approximation to te finite sample distribution for models wit a small break. Note tat wen ɛ t is normally distributed, te Fiser information I µ in Equation 3 is σ. In tis case, te asymptotic teory for τ ML,T in Yao 987 is exactly te same as tat for τ LS,T in Bai 994. However, Bai s results were obtained witout assuming Gaussian errors, and, ence, an invariance principle applies. Figure plots te pdf of te limiting distribution given in 3 and 5. For te purpose of comparison, te pdf of a standard normal distribution is also plotted. It 5

6 Density finite sample distribution argmax Wu u / Figure : Te pdf of te finite sample distribution of T δ Tσ τ LS,T τ wen T =, δ T =., σ = and τ =.3 in Model and te pdf of arg max W u u }. u, can be seen tat, relative to te standard normal distribution, te limiting distribution obtained in te literature as muc fatter tails and a muc iger peak. More importantly, te limiting distribution as a unique mode at te origin and is symmetric about it, suggesting tat bot te ML estimator and te LS estimator ave no asymptotic bias, no matter wat te true value of te structural break point is. Unfortunately, te long-span asymptotic distribution developed in te literature does not perform well in many empirically relevant cases. To see tis problem, in Figure we plot te pdf of te long-span asymptotic distribution listed in 5 and te finite sample distribution of T δ σ τ LS,T τ wen T =, δ =., σ = and τ =.3 in Model. Te finite sample distribution is obtained from simulated data. It is clear tat te two distributions are very different from eac oter. Tree striking distinctions can be found. First, te finite sample distribution is asymmetric, wereas te long-span asymptotic distribution is symmetric. Second, te finite sample distribution displays trimodality wile te long-span asymptotic distribution as a unique mode. Tird, te finite sample distribution indicates tat te LS estimator τ LS,T is seriously biased. Te simulation result sows tat te bias is.74, wic is about 57% of te true value. In contrast, tere is no bias suggested by te long-span asymptotic distribution. It is tis inadequacy of te long-span asymptotic distribution for approximating te finite sample distribution tat motivates us to develop an alternative distribution teory for te estimation of te structural break point. 6

7 3 A Continuous Time Model In tis section we focus our attention on a continuous time model wit a structural break in te drift function. Te model considered ere is δ dxt = µ + [t>τ ] dt + σdbt, 6 were t [, ], [t>τ ] is an indicator function, µ, δ, and τ are constants wit δ / being te break size, σ is anoter constant capturing te noise level, and Bt denotes a standard Brownian motion. Te condition of τ [α, β] wit < α < β < is assumed to ensure tat one break appens during te time interval,. We furter assume tat a continuous record is available and all parameters are known except for τ. Wit a continuous record, assuming a more complex structure for σ suc as a time varying diffusion will not cange te analysis because te diffusion function can be estimated by quadratic variation witout estimation error. Te continuous time diffusion model is a natural coice to study te asymmetry of te sample information before and after te break point. Tis is because it is well-known in te continuous time literature tat te longer te time span over wic a continuous record is available, te more information tat te continuous record contains about te parameters in te drift function; see Pillips and Yu 9a, 9b. Hence, wen τ /, te amount of information contained by observations over te time interval [, τ ] is different from tat over te time interval [τ, ]. Tis difference is captured by te asymmetry in te lengt of te time span before and after te break point. Terefore, te exact distribution of te ML estimator of te structural break point is expected to be asymmetric. For any τ, we can obtain te exact log-likeliood function of Model 6 via te Girsanov Teorem as log Lτ = log dp τ dp B = σ µ + δ [t>τ] dxt µ + δ } [t>τ] dt, were P τ is te probability measure corresponding to Model 6 wit τ being replaced by τ for any τ, and P B is te probability measure corresponding to Bt. Tis leads to te ML estimator of τ as τ ML = arg max log Lτ. 7 τ, See also Pillips and Yu 9b for a recent usage of te Girsanov Teorem in estimating continuous time models. 7

8 Following te literature, we now define a two-sided Brownian motion as W u = Bτ Bτ u if u W u = W u = Bτ Bτ + u if u >, 8 were W s = Bτ Bτ s and W s = Bτ Bτ + s are two independent Brownian motions composed by increments of te standard Brownian motion B before and after τ, respectively. Teorem 3. reports te exact distribution of τ ML. Teorem 3. Consider Model 6 wit a continuous record being available. For te ML estimator τ ML defined in 7, a wen is a constant, we ave te exact distribution as δ τ ML τ = d arg max σ u τ σ δ, τ σ δ W u u } ; 9 b wen, te break size δ /, we ave te small- distribution as δ d τ ML τ arg σ max u, W u u }, were W u is te two-sided Brownian motion defined in 8, and d = denotes equivalence in distribution. Part a of Teorem 3. gives te exact distribution of τ ML wen a continuous record over a finite time span is available. It is different from te long-span limiting distribution developed in te literature as in 5 in two obvious aspects. First, te limiting distribution in 5 corresponds to te location of te extremum of W u u over te interval of,. As te interval is symmetric about zero, te limiting distribution is symmetric too. However, te exact distribution in 9 corresponds to te interval of τ δ σ, τ δ, wic depends on te true value of te fractional σ break point τ. Only wen te true break point is exactly in te middle of te sample, i.e., τ = /, does te interval become δ σ /, δ σ /, being symmetric about te origin. In tis case te exact distribution is symmetric. However, if τ is not /, te interval and ence te exact distribution will be asymmetric, indicating tat τ ML is biased. It is easy to see tat te exact distribution in 9 suggests upward bias wen τ < / and downward bias wen τ > /, and te furter τ away from /, te larger te bias. In addition, te signal-to-noise ratio δ contributes to te degree of σ asymmetry of te interval, and, ence, affects te exact distribution and te magnitude 8

9 Density Density Density Figure 3: Te density of τ ML τ given in Equation 9 wen τ =.4,.5,.6 te left, middle and rigt panel respectively and te signal-to-noise ratio δ is. σɛ of bias. Tese findings are confirmed by te simulation results reported in Figures 3-4, in wic we plot te density functions of te exact distribution wen τ =.4,.5,.6 te left, middle and rigt panel respectively, = and δ being and 4, respectively. σ Second, te interval to locate te argmax in te exact distribution in 9 is always bounded. Wereas, te interval to locate te argmax in te long-span limiting distribution in 5 is unbounded. Suc a difference as an implication for te modality of te distribution. As sown in Figure, te long-span limiting distribution as a unique mode at te origin. Wereas, Figures 3-4 sows tat te exact distribution displays trimodality. One mode is at te origin. Te oter two modes are at te two boundary points, τ δ σ and τ δ σ. Wen τ = /, te two modes at te boundary points ave te same eigt. Wen τ /, te two modes at te boundary points do not ave te same eigt. From te comparison of Figures 3-4, we can also find tat te modes at te two boundary points are iger wen te signal-to-noise ratio is smaller. As a result, for te case were τ /, te exact distribution is more skewed and leads to a larger bias of τ ML wen te signal-to-noise ratio is smaller. Te mode at te origin is well expected. Tis is because te drift term and te random term in W u u are u and W u = O p u, respectively. Wen u is large, te negative drift term dominates te random term. As a result, te probability for W u u to reac te maximum at a large value of u sould be small, and decreasing as u getting larger. In te mean time, because of te randomness in W u, it is still possible for W u u to reac te maximum at any large value of u. Tis also explains te sape of te long-span limiting distribution in 5 as apparent in Figure. Wen te interval of te exact distribution in 9 is bounded wit a comparatively small value of δ, u takes small values even at te boundary points. Tis means tat σ te negative drift term becomes less dominant, ence, it is more likely for W u u 9

10 to reac te maximum in te neigboroods of te two boundary points. To explain wy tere are two modes at te two boundary points, take te rigt boundary point τ δ σ as an example. Being a mode at tis boundary point means tat it is more likely for W u u to reac te maximum at τ δ σ tan at any point arbitrarily close to but strictly less tan τ δ σ. Given te randomness of W u, te probability for W u u to reac te maximum in any small left neigborood of τ δ σ is nonzero. Conditional on te event tat W u u reaces te maximum in a small left neigborood, for τ δ σ to be te arg max point, te value of W u u at τ δ σ only needs to be larger tan te value of W u u at te points smaller tan τ δ σ. However, for any interior point to be te arg max, we ave to compare te value of W u u at tis interior point wit tat at bot sides of tis interior point. Terefore, τ δ σ is more likely to be te arg max of W u u tan any interior point. Similar arguments apply to te oter boundary point, τ δ. σ Wen te signal-to-noise ratio δ gets smaller, te values of te two boundary points σ become smaller too. Hence, te probabilities of W u u reacing its maximum in te neigboroods of te two boundary points get larger, leading to larger values of te modes at te two boundary points. Similar arguments explain te reason wy te boundary point closer to te origin as a larger mode tan te oter boundary point. Moreover, wen δ is very small, te lengt of te interval over wic W u u is σ maximized is very small. In tis case, te negative drift term is stocastically dominated by te random term in W u u. Tis explains wy te origin may not be te igest mode wen te signal-to-noise ratio is very small, as apparent in Figure 3. Part b of Teorem 3. sows tat te asymptotic distribution of τ ML wen is te same as te long-span asymptotic distribution developed in Yao 986 and Bai 994. Te same small- asymptotic distribution is also obtained in Ibragimov and Has minskii Continuous Time Approximation to Discrete Time Models Motivated by te findings in te exact distribution in te continuous time model, in tis section we first build a continuous time approximation to te discrete time structural break model widely studied in te literature. Ten, we develop te in-fill asymptotic teory for te LS estimator of te break point, and sow tat te in-fill asymptotic distribution provides better approximations to te finite sample distribution tan te

11 Density Density Density Figure 4: Te density of τ ML τ given in Equation 9 wen τ =.4,.5,.6 te left, middle and rigt panel respectively and te signal-to-noise ratio δ is 4. σɛ long-span asymptotic distribution developed in te literature. Consider te continuous time process X t defined in 6. We now assume tat te observations are only available at discrete time points, say at T equally spaced points t} T, were is te sampling interval and T = / is te sample size. For simplicity, we assume te structural break point T τ to be an integer, denoted by k. Let X t } T denote te discrete time observations. Ten, te exact discretization of te continuous time process defined in 6 can be written as µ + ɛ t for t =,, k, X t X t = µ + δ / + ɛ t for t = k +,, T, were ɛ t iid N, σ. Letting Z t = X t X t /, we ave Z t = µ + ɛ t if t k, µ + δ / + ɛ t if t > k. It can be seen tat, wenever is fixed, te discrete time model in Equation is te same as te one studied in Yao 987 and Bai 994 given in Equation wit ɛ t being normally distributed and te sift in mean being δ = δ /. We now develop te asymptotic teory of te LS estimator of τ = k /T under te in-fill asymptotic sceme were wit a fixed time span T =. Clearly, if, te sample size T. In te limit of, a continuous record is available. As it can be seen clearly in te proofs in Appendix, te development of te in-fill asymptotic teory does not require te assumption of Gaussian errors. Terefore, an invariance principle applies. Moreover, te in-fill asymptotic teory continues to old wen µ in Model is replaced wit µ. In oter words, making te means of Z t before and after

12 break to be around a constant different from zero, instead of converging to zero wen as required in Model, would not cange te in-fill asymptotics developed in te section. Wit a fixed, te in-fill asymptotic sceme implies tat te break size δ / goes to zero at te rate of / T. Tis rate is faster tan tat assumed in Bai 994. Tis key difference makes our in-fill asymptotic teory different from te long-span asymptotic teory developed in Bai 994. Wen µ and δ / are known, te in-fill asymptotic distribution is sown to be te same as te exact distribution of te ML estimator wen a continuous record is available, as given in Part a of Teorem 3.. Wen µ and δ / are unknown, we derive an in-fill asymptotic distribution wic is asymmetric if τ /, and as trimodality. In bot cases, simulation results sow tat te in-fill asymptotic distribution provides better approximations to te finite sample distribution. We also consider te in-fill asymptotic sceme wit and δ /. In tis case te break size goes to zero but at a rate slower tan / T. It is sown tat te in-fill asymptotic distribution wit is te same as te long-span asymptotic distribution obtained in Yao 987 and Bai 994. generalize and connect naturally wit tose in te literature. 4. In-fill asymptotics wen only τ is unknown Wen µ and δ / are known, te LS estimator of k is defined as k Z t µ T + kls,t = arg min k=,...,t = arg max k=,...,t δ / k t=k+ Z t µ Hence, our setup and results Z t µ + δ / } + δ / k/ }. Te corresponding estimator of τ is τ LS,T = k LS,T /T. Wen te errors in Model are normally distributed, te LS estimators of k and τ are identical to te ML estimators as defined in Yao 987. Compared to Yao s long-span asymptotic distribution, te in-fill asymptotic distribution given in Part a of Teorem 4. provides an alternative asymptotic approximation to te finite sample distribution of τ LS,T. Part b of te teorem connects our in-fill asymptotics to Yao s long-span asymptotics. Teorem 4. Consider Model wit known µ and δ /. Denote te LS estimator τ LS,T = k LS,T /T wit k LS,T defined in. Ten,

13 a wen wit a fixed, we ave te in-fill asymptotic distribution as δ d T τ LS,T τ arg σ u max τ σ δ, τ σ δ W u u } ; b wen and simultaneously wit δ /, we ave te small- in-fill asymptotic distribution as δ d T τ LS,T τ arg σ max u, were W u is te two-sided Brownian motion defined in 8. W u u }, δ Remark 4. Note tat T = / implies T σ = δ / σ. Hence, te in-fill asymptotic distribution of τ LS,T in Teorem 4. is te same as te exact distribution of τ ML obtained in Teorem 3.. δ Remark 4. Wen wit a fixed, T σ = δ / σ is a constant. In tis case, Part a of Teorem 4. sows tat τ LS,T is inconsistent and k LS,T k diverges at te rate of T. Wen and simultaneously wit δ /, te break size srinks to zero but at a rate slower tan / δ T. In tis case, T σ and τ LS,T becomes consistent as sown by Part b of Teorem 4.. Moreover, te small- in-fill asymptotic distribution obtained in Part b of Teorem 4. is te same as te long-span asymptotic distribution obtained in Bai 994. Clearly, by relaxing te assumption of Bai, we get te same asymptotic distribution. Remark 4.3 Te proof of Teorem 4. does not depend on te assumption of Gaussian errors. Terefore, an invariance principle applies to te in-fill asymptotics. Moreover, te proof of Teorem 4. can be easily extended to te case were te errors in Model follow a weakly stationary process wit a long-run variance [a ]. In tis case, te results in Teorem 4. still old but wit σ being replaced by [a ]. δ Figure 5 plots te finite sample distribution of T σ τ LS,T τ wen τ =.3,.5,.7 te left, middle and rigt panel respectively obtained from simulations, te density of te in-fill asymptotic distribution given in Part a of Teorem 4. and te density of te long-span limiting distribution given in Yao 987. Te data are simulated from Model wit µ =, δ =, =, σ = and = /. So te break size is δ =.. Te experiment is replicated, times to obtain te density. Te first part of Table reports te finite sample bias of τ LS,T, te bias 3

14 Density Density Density Table : Te table sows te finite sample bias of ˆτ LS,T, te bias from te in-fill asymptotic distribution, and te bias from te long-span asymptotic distribution. Tese tree kinds of bias are denoted by FS, IF, LS, respectively, wen τ is te only unknown parameter; and are denoted by FS, IF, LS, respectively, wen more parameters are unknown. Te number of replications is,. δ σ τ FS IF LS FS IF LS δ Figure 5: Te pdf of T σ τ LS,T τ wen τ =.3,.5,.7 te left, middle and rigt panel respectively and δ =. Te blue solid line is te finite sample σ distribution wen T = ; te black broken line is te density given in Part a of Teorem 4.; and te red dotted line is te long-span limiting distribution in Yao

15 implied by te in-fill asymptotic distribution in Part a of Teorem 4., and te bias implied by te long-span limiting distribution in Yao 987, for te cases were te signal-to-noise ratio δ =, 4, 6, respectively. σ Several features are apparent in Figure 5 and te first part of Table. First, te finite sample distribution is not symmetric about wen τ /. In particular, if τ is smaller larger tan /, te density is positively negatively skewed, indicating an upward downward bias in τ LS,T. Te bias is 3% above te true value wen τ =.3 wic is substantial. Second, te finite sample distribution as trimodality. Te origin is one of te tree modes and te two boundary points, τ δ and τ δ are te oter two. σ σ, Tird and most importantly, te in-fill asymptotic distribution given in Part a of Teorem 4. sares te two important features of te finite sample distribution, namely, asymmetry and trimodality, and captures te finite sample bias very well. Not surprisingly, it provides muc better approximations to te finite sample distribution tan te long-span asymptotic distribution. Fourt, as revealed by te first part of Table, as te signal-to-noise ratio δ increases, te magnitude of asymmetry σ in te finite sample distribution decreases, and, ence, te finite sample bias becomes smaller. Tis property is also well captured by te in-fill asymptotics. 4. In-fill asymptotics wit more unknown parameters Wen µ and δ / are unknown, te means before and after te break point ave to be estimated. As in Bai 994, te LS estimator of te break point is now defined as, k } T kls,t = arg min Zt Z k + Z t Z k = arg max [Vk Z t ] }, k=,...,t k t=k+ were Z k Z k is te sample mean of te first k last T k observations and [V k Z t ] = T T k T Z k Z k. Similarly, τ LS,T = k LS,T /T. Teorem 4. Consider Model wit unknown parameters of µ and δ /. For te LS estimator τ LS,T = k LS,T /T wit k LS,T defined in, a wen wit a fixed, we ave te in-fill asymptotic distribution as wit δ d δ [ ] T τ LS,T τ arg max B u, 3 σ σ u τ, τ B u = B τ u B τ + u τ τ +u δ τ for u u σ B τ u B τ + u τ τ u δ for u >, σ τ +u 5

16 were B s is a standard Brownian motion and B s B B s; b wen and simultaneously wit δ /, we ave te small- in-fill asymptotic distribution as δ d T τ LS,T τ arg σ max u, were W u is te two-sided Brownian motion defined in 8. W u u }, Remark 4.4 Te in-fill asymptotic distribution reported in Part a of Teorem 4. is new to te literature. Wen τ /, te interval τ, τ is asymmetric about zero and, not surprisingly, te in-fill asymptotic distribution is asymmetric too. Wen τ = /, te interval becomes symmetric, and we ave B / u B / + u B u = B / u B / + u /+u δ / u / u δ /+u for u σ, for u > σ wic is symmetrically distributed about zero. As a result, te distribution in Part a of Teorem 4. is symmetric about zero wen τ = /. In practice, one needs to estimate τ and te signal-to-noise ratio δ and ten insert te estimated values into te σ in-fill asymptotic distribution reported in Part a of Teorem 4. for te purpose of making statistical inference. Remark 4.5 By using te Beveridge-Nelson decomposition and te functional central limit teory for serially dependent processes, Teorem 4. can be extended to te case were te errors in Model follow a weakly stationary process wit a long-run variance [a ]. In tis case, te results in Teorem 4. still apply wit σ being replaced by [a ]. δ Figures 6 and 7 plot te finite sample distribution of T σ τ LS,T τ, obtained from simulated data, wen τ =.3,.5,.7 te left, middle and rigt panel respectively, te density of te in-fill asymptotic distribution given in Part a of Teorem 4. and te density of te long-span limiting distribution given in Bai 994. Te data are simulated from Model wit µ =, δ =, =, σ = and = / and so te break size is δ =. in Figure 6. Figure 7 corresponds to δ = 4 and so te break size is δ =.4. Te experiment is replicated, times. Te finite sample bias of τ LS,T, te bias implied by te in-fill asymptotic distribution, and te bias implied by te long-span limiting distribution are reported in te second part of Table. 6

17 Density Density Density δ Figure 6: Te pdf of T σ τ LS τ wen τ =.3,.5,.7 te left, middle and rigt panel respectively and δ =. Te blue solid line is te finite sample distribution σ wen T = ; te black broken line is te density given in Part a of Teorem 4.; and te red dotted line is te long-span limiting distribution in Bai 994. Several features are apparent in Figures 6-7 and te second part of Table. First, te finite sample distribution is asymmetric about wen τ /, and, ence, τ LS,T is biased. In particular, if τ is less greater tan /, te density is positively negatively skewed, leading to an upward downward bias in τ LS,T. Te bias is more tan 5% of te true value if τ =.3, wic is very substantial. Second, te finite sample distribution is not as concentrated around zero as suggested by te long-span limiting distribution. Te finite sample distribution as trimodality. Te origin is one of te tree modes and te two boundary points, δ σ τ and δ σ τ, are te oter two. Te peak at te origin can be smaller tan tose at te boundary points wen δ σ is small. Tird and most importantly, te in-fill asymptotic distribution given in Part a of Teorem 4. as trimodality, and is asymmetric about zero wen τ /. It provides better approximations to te finite sample distribution tan te long-span limiting distribution. Comparing two parts in Table, it can be seen tat wen oter parameters are unknown, te bias in τ LS,T increases. In spite of te increased bias in τ LS,T, it can be seen from te second part of Table tat te in-fill asymptotic distribution also captures te finite sample bias very well. 5 Bias Correction via IE Indirect estimation IE is a simulation-based metod, first introduced by Smit 993, Gouriéroux et al. 993, and Gallant and Taucen 996. Tis metod is particularly useful for estimating parameters of a model were moments and likeliood function are 7

18 Density Density Density δ Figure 7: Te pdf of T σ τ LS τ wen τ =.3,.5,.7 te left, middle and rigt panel respectively and δ = 4. Te blue solid line is te finite sample distribution σ wen T = ; te black broken line is te density given in Part a of Teorem 4.; and te red dotted line is te long-span limiting distribution in Bai 994. diffi cult to calculate, but te model is easy to simulate. It uses an auxiliary model to capture aspects of te data upon wic to base te estimation. Te parameters of te auxiliary model can be estimated using eiter te observed data or te data simulated from te true model. Ten, IE estimates are obtained by minimizing te distance between te two sets of parameter estimates. Typically, one cooses an auxiliary model tat is amenable to estimate and well approximates te true model at te same time. To improve finite sample properties of te original estimator, McKinnon and Smit 998 and Gouriéroux et al. developed an IE procedure, were te auxiliary model is cosen to be te true model. In tis section, we apply tis IE procedure to do bias correction in estimating τ and k. It is important to obtain te bias function via simulations because te bias formula and te bias expansion of te ML and LS estimators studied in tis paper are diffi cult to obtain. Te same IE procedure was also used to do bias correction in continuous time models by Pillips and Yu 9a, c and in dynamic panel data models by Gouriéroux et al.. Te application of IE for estimating te structural break point proceeds as follows. Given a parameter θ say τ, we simulate data ỹθ = ỹ, s ỹ, s..., ỹt s } from te true model, suc as, Equation 6 or, were s =,..., S, and S is te number of simulated pats. Note tat T in ỹθ sould be cosen as te same number of te actual data under analysis so tat te bias of te original estimator from te actual observations can be calibrated by simulated data. IE ten matces te estimate from te actual data wit tat from te simulated data. To be specific, let ˆθ T be an estimator of θ from te actual data and θ s T θ be te estimator of θ based on te st simulated 8

19 pat for some fixed θ. Te IE estimator is ten defined as ˆθIE,T,S = arg min θ Θ ˆθ T S θs T θ S, 4 were is some finite-dimensional distance metric and Θ is te compact parameter space. Wen S, it is expected tat S θ s p S s= T θ E θ s T θ := b T θ, were b T θ is known as te binding function. Ten te IE estimator becomes ˆθ IE,T = arg min θ Θ ˆθT b T θ. Gouriéroux et al sowed tat if b T θ is an affi ne function in θ for any T, ˆθ IE,T is exactly mean-unbiased. Wen te auxiliary model is identical to te true model and ˆθ T is consistent, Gouriéroux et al gives non primitive conditions for te second order bias corrections by ˆθ IE,T. Arvanitis and Demos 4 provided more primitive conditions to ensure te validity of moment expansions and te second order bias correction by ˆθ IE,T. In our setup, if τ is te only unknown parameter, we can easily obtain τ IE,T based on Equation 4. And te IE estimator of k can be obtained as ˆk IE,T = τ IE,T T. Let te corresponding binding function be b T k = b T τ T. Since τ ML in te continuous time model and τ LS,T in te discrete time model are consistent wen, we can establis te second order bias correction by te IE estimator under some regularity conditions. To derive te asymptotic distribution of te IE estimator, one needs to verify tat te binding function is asymptotically locally relatively equicontinuous Pillips,. If te binding function is indeed asymptotically locally relatively equicontinuous and lim T E τ T = τ were τ T is eiter τ ML or τ LS,T, te Delta metod can be applied to te original estimator τ ML and τ LS,T and te asymptotic teory including te rate of convergence and te limiting distribution sould be te same as tat of te original estimator. Unfortunately, since te pdf of τ T is unknown analytically, finding te binding function is only possible numerically. As a result, calculating te derivative of te binding function and verifying asymptotically locally relative equicontinuity of s= te binding function are very diffi cult, if not impossible. Wen is fixed, if te binding function is invertible, tat is, τ IE,T = b T τ T, one may informally apply te Delta metod to study te effi ciency of te indirect estimator as Var τ IE,T bt τ τ Var τ T. Hence, te effi ciency loss or gain is measured by b T τ. If τ b T τ τ <, τ IE,T as a bigger variance tan τ T. However, if b T τ τ >, τ IE,T will ave a smaller variance tan τ T. As bot te simulation results and te large sample teory suggest tat τ is over estimated wen τ < / and is under estimated wen τ > /, te binding function is expected to be flatter tan te 45 degrees line. As a result, τ IE,T is expected to lose some effi ciency compared to τ T. 9

20 As suggested by a referee, we also consider oter simulation-based metods to do bias correction, and compare teir performance to te IE approac. One alternative bias correction metod is te so-called median unbiased estimator denoted by τ MU,T as in Andrews 993 wic is obtained by replacing te sample mean in Equation 4 wit te sample median. As te finite sample distribution of τ T is asymmetric, median migt be able to better measure te location tan te mean. Wen te binding function is invertible and monotonic, τ MU,T is exactly median unbiased. Anoter bias correction metod is te bootstrap metod of Efron 979. Hall 99 sowed tat te parametric bootstrap metod is an effective metod for bias correction. Te idea of parametric bootstrap is to generate many bootstrap sample pats, eac of wic aving te same structure as te estimated pat from te initial estimation, and ten to obtain a new estimate from eac bootstrap sample pat by applying te same estimation procedure, denoted as τ s T τ T for s =,..., S. Let τ T τ T = S S s= τ s T τ T. Ten, te bias of τ T wen τ = τ T is approximated by τ T τ T τ T, and, ence, te bootstrap estimator is defined as τ BS,T = τ T τ T τ T τ T = τ T τ T τ T. Many oter simulation-based metods and teir comparisons are discussed in Forneron and Ng 5. In Model wit unknown parameters oter tan τ, using Equation 4 to obtain τ IE,T and te IE estimators of oter parameters simultaneously will be numerically very time consuming. Tis is because te binding function now becomes a system of multivariate functions and as to be computed via simulations for combinations of some cosen values of all parameters. Given tat τ is te parameter of interest, we propose a way to reduce te computational cost in calculating te binding function b T τ. First, it as been sown in te subsection 4. tat te developed in-fill asymptotic distribution well approximates te finite sample distribution. We terefore suggest to approximate te binding function b T τ by its limit under te in-fill asymptotic sceme, wic is [ ] bτ = E τ + arg max u τ, τ B u were B u is defined as in 3. To reduce te dimensionality of te binding function, note tat te in-fill asymptotic distribution of τ LS,T given in 3 depends on te signal-to-noise ratio δ /σ as a wole, not on te break size δ / and te standard variance σ individually. We ence propose to replace δ /σ in B u wit its LS estimate, and treat it as known wen B u and bτ are simulated.

21 6 Monte Carlo Results In tis section, we design tree Monte Carlo experiments to examine te bias of te ML estimator of τ in te continuous time model 6 and te LS estimator of k in te discrete time model, and compare teir performance to te estimators from IE and oter simulation-based bias-correction metods. In te first experiment, data are generated from Model 6, wit µ =, σ =, =, δ =, 4, 6, τ =.3,.5,.7, dbt iid N, and =. For eac combination of δ and τ, we obtain te ML estimate of τ from 7 and several biased corrected estimates of τ wit S =,. 3 Table reports te bias, te standard error, and te root mean squared errors RMSE of te ML estimator, te IE estimator, te median unbiased MU estimator, and te parametric bootstrap PB estimator, obtained from, replications. Some observations can be obtained from te table. First, wen τ =.5, te ML estimator does not ave any noticeable bias in all cases. However, wen τ.5, te ML estimator suffers from a bias problem. For example, wen τ =.3 and δ =, te bias is.9, wic is about 3% of te true value. Tis is σ very substantial. In general, te bias becomes larger wen τ is furter away from.5, or wen te signal-to-noise ratio gets smaller. To te best of our knowledge, suc a bias as not been discussed in te literature. Second, in all cases wen τ.5, te IE approac substantially reduces te bias. For example, wen δ σ = and τ = 7%, IE removes about two tirds of te bias in te ML estimator. Tird, te bias reduction by IE comes wit a cost of a iger variance, wic causes te RMSE of te IE estimator sligtly iger tan its ML counterpart. Finally. compared wit IE, te MU estimator is less effective for bias reduction but is more effi cient in terms of variance. In terms of RMSE, te MU estimator performs better. Tis finding is consistent wit wat was reported in Tables 7-8 of Pillips and Yu 9a for a continuous time model. However, compared wit IE, te PB estimator performs similarly in terms of bias reduction but increases te variance more in almost all cases. In te second experiment, data are generated from Model, wit µ =, σ =, iid δ =.,.4,.6, τ =.3,.5,.7, ɛ t N,, T =. 4 For eac combination of δ and τ, we obtain te LS estimate of k from and several biased corrected estimates of k wit S =,. Table 3 reports te bias, te standard error, and te RMSE of te LS estimator, defined in, and te tree simulation-based estimators, obtained 3 We also try oter values for H, suc as H =, and 5,. Te results are almost uncanged. 4 We also try oter values for T in te second and te tird experiments, suc as T = 8 and. Te results remain qualitatively uncanged.

22 Table : Monte Carlo comparisons of bias and RMSE of te ML estimator, te MU estimator, te PB estimator, and te IE estimator for te continuous time model 6. Te number of simulated pats is set to be,. Te number of replications is set at,. Case Bias Standard Error RMSE δ σ τ ML MU PB IE ML MU PB IE ML MU PB IE Table 3: Monte Carlo comparisons of bias and RMSE of te LS estimator, te MU estimator, te PB estimator, and IE estimator for te discrete time model wen only τ is unknown. Te sample size is set to be T =. Te number of simulated pats is set to be,. Te number of replications is set at,. Case Bias Standard Error RMSE δ σ τ k LS MU PB IE LS MU PB IE LS MU PB IE

23 Binding Function.8 45 degree line Delta =. Delta =.4 Delta = True value for tau Figure 8: Binding functions of te LS estimator for discrete time model wit T = from, replications. Te conclusions drawn from Table 3 are nearly identical to tose from Table. To understand wy IE increases te variance relative to te original estimator in tese two experiments, we plot te binding function in te second experiment in Figure 8. In Figure 8 we also plot te 45 degrees line for te purpose of comparison. Several conclusions can be made. First, every binding function passes troug te 45 degrees line wen τ =.5, suggesting tat no bias exists wen τ =.5. Second, te binding functions are flatter tan te 45 degrees line in all cases, explaining wy te variance of te IE estimator is larger tan tat of te ML estimator. Te smaller te signal-tonoise ratio, te flatter te binding function and ence te bigger te loss in effi ciency. Tird, no binding function is exactly a straigt line. Nonlinearity can be found near te two boundary points. Consequently, according to Gouriéroux et al., te IE estimator is not exactly mean unbiased. Altoug not plotted, te binding function in te first experiment sares te same caracteristics. In te tird experiment, data are generated from Model, wit µ =, σ =, δ =.,.4,.6, τ =.3,.5,.7, ɛ t iid N,, T =. Different from te second experiment, all te parameters, including τ, are assumed to be unknown. For eac combination of δ and τ σ, we obtain te LS estimate of k from and te indirect estimate of k wit S =,. Table 4 reports te bias, te standard error, and te RMSE of k LS,T defined in and k IE,T proposed in te end of Section 5, obtained from, replications. Several conclusions can be made from Table 4. First, wen τ =.5, te LS estimator does not ave any noticeable bias. However, wen τ.5, te LS estimator suffers from a severe bias problem. In general, te bias becomes larger wen τ is furter away from.5 or wen te signal-to-noise ratio gets smaller. Second, 3

24 Table 4: Monte Carlo comparisons of bias, standard error and RMSE of te LS estimator and te IE estimator wen more parameters oter tan τ are unknown. Te sample size is set to be T =. Te number of simulated pats is set at, for indirect estimation. Te number of replications is set at,. Case Bias Standard Error RMSE δ τ σ k LS IE LS IE LS IE IE can reduce te bias in all cases. For example, wen δ =.6 and τ σ =.7, IE removes about 59% of te bias of te LS estimator. Moreover, te variance of te IE estimator is comparable to tat of te LS estimator. Overall, te RMSE of te IE estimator is similar to its LS counterpart. Unfortunately, analyzing te beavior of te binding function ere is complicated for two reasons. First, we replace te signal-tonoise ratio δ /σ by its LS estimator wic inevitably canges te curvature of te binding function in obtaining te IE estimator. Second, in general te binding function is a system of functions tat depend on all unknown parameters. 7 Conclusions Tis paper is concerned about te in-fill asymptotic approximation to te exact distribution in te estimation of structural break point in mean. We find tat te exact distributions of te traditional estimators of structural break point are often asymmetric and ave trimodality bot in te continuous time model and in te discrete time model. It is also found tat te traditional estimators are biased. Unfortunately, te literature on structural breaks as always focused te attention on developing asymptotic teory wit a time span being assumed to go to infinity. Te long-span limiting distribution developed in te literature is symmetric and as te true break point as te unique mode. As a result, it provides poor approximations to te exact distribution in many empirically relevant cases. In tis paper we address te finite sample problem in several aspects. First, we 4

25 derive te exact distribution of te ML estimator of te structural break point in a continuous time model wen a continuous record is available. It is sown tat te exact distribution as trimodality, regardless of te location of te break. Wen te true break point is in te middle of te sample, te exact distribution is symmetric. However, wen te true break point occurs earlier later tan te middle of te sample, te exact distribution is skewed to te rigt left, leading to a positive negative bias in te ML estimator. In a discrete time model wit a break in mean, we continue to find te trimodality and asymmetry in te finite sample distribution of te LS estimator of te structural break point. To better approximate te finite sample distribution, we deviate from te literature by considering a continuous time approximation to te discrete time model and developing an in-fill asymptotic teory. For te discrete time model wit te break point being te only unknown parameter, te in-fill asymptotic distribution is te same as te exact distribution in te continuous time model. For te discrete time model wit more unknown parameters, te in-fill asymptotic distribution is new to te literature. We sow tat tis distribution as trimodality and is asymmetric wen te true break point is not in te middle of te sample and te in-fill asymptotic distribution better approximates te finite sample distribution tan te long-span limiting distribution developed in te literature. Given tat te exact distribution suggests a substantial bias in te ML/LS estimators, to reduce te bias, we propose to use te IE tecnique to estimate te break point. Indirect estimation inerits te asymptotic properties of te original estimator but reduces te finite sample bias. Monte Carlo results sow tat te IE procedure is effective in reducing te bias in te commonly used break point estimators. Te models considered in tis paper are very simple in nature. Also, te estimators considered are based on te full sample. Real time and ence subsample estimators tend to ave more serious finite sample problems. Furter studies on developing better approximations to te finite sample distribution for more realistic models and real time estimators are needed. Appendix 5