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On the asymptotic behavior of bubble date estimators

In this study, we extend the three-regime bubble model of Pang et al. (2021) to allow the forth regime followed by the unit root process after recovery. We provide the asymptotic and finite sample justification of the consistency of the collapse date estimator in the two-regime AR(1) model. The consistency allows us to split the sample before and after the date of collapse and to consider the estimation of the date of exuberation and date of recovery separately. We have also found that the limiting behavior of the recovery date varies depending on the extent of explosiveness and recovering.


Introduction
The estimation of the dates of regime changes has a long history in the econometric literature (see, e.g., Casini and Perron (2019) for a recent review). In this study, we address the issue of estimating the break dates in explosive bubble model. Such a bubble model is often generated (see Phillips et al. (2011)) as a unit root process followed by explosive process, which in turn followed by a unit root regime (with possible stationary recovering regime after the collapse of the bubble). For estimation of bubble dates, Phillips et al. (2015a,b) (PSY hereinafter) proposed the recursive algorithm based on the right-tailed ADF test. Generally speaking, the origination of the bubble is taken at the date for which the test statistic begins to exceed the critical value, and the date of collapse is taken at the date for which the test statistic subsequently falls below the critical value (see Skrobotov (2021) for review).
However, as demonstrated in Harvey et al. (2017), the bubble dates estimates are more accurate if they are obtained by minimizing the sum of the squared residuals over all possible dates. For the model with only two regimes, Chong (2001) investigated consistency and the limiting distribution of the break date estimator for the AR(1) model with a break in the coefficient. The break date estimator is based on minimizing the sum of the squared residuals (SSR) over all possible break dates. The results depend on the direction of the coefficient change. Chong (2001) considered three cases, both regimes are stationary, the first regime is a unit root regime, and the second regime is stationary and vice versa. Pang et al. (2018) extended the result of Chong (2001), allowing the AR(1) coefficient to be dependent on the sample size. They also considered the case with change in persistence from a unit root to moderately explosive and vice versa. Pang et al. (2021) (PDC hereinafter) considered the model with three regimes that reflect the explosive financial bubble model: the first regime is a unit root regime, followed by an explosive regime and then by a stationary collapsing regime. PDC investigated the sample splitting strategy: The first break date is estimated for the two regime AR(1) model. Results indicate that the break date estimator is consistent for the date of collapse (i.e., the second break). Then, one could consider the sample before the date of collapse and investigate the asymptotic behavior of the date of origination of the bubble -the results are the same as that in Pang et al. (2018). From a different perspective, the estimation procedure used by PDC is the same as that in Chong (1995) and Bai (1997), in which multiple breaks are estimated one at a time in the regime-wise stationary model with level shifts.
In this study, we extend the results of PDC by allowing the forth unit root regime. The advantage of this extension is that we can investigate not only the emerging and collapsing dates of the bubble but also the recovering date to the normal market. As a result, we can totally investigate the abnormal market behavior. In our procedure, we first prove the consistency of the collapse date estimator by minimizing the sum of the squared residuals using the two-regime model, allowing nonstationary volatility. Second, as the estimated break date is consistent with the collapsing date, one could split the sample at the estimated break date and consider the estimator of the market recovery date (i.e., the date of change from stationary collapsing regime to unit root regime) using the subsample after the break date detected in the first step. 1 Third, we allow a weak dependence in the innovation errors.
Our approach closely resembles that of Harvey et al. (2017)  It should be noted that our approach can be easily extended to the multiple bubbles context. Harvey et al. (2020) proposed to initially identify the bubble regimes based on PSY approach and then obtain more precise bubble date estimates during each bubble episode.
Subsequently, one can use our approach in the second step.
The remainder of this paper is organized as follows. Section 2 formulates the model and assumptions. In Section 3, we define the main procedure and provide the limiting behaviors of the break date estimators. The model with serially correlated shocks is considered in Section 4. The finite sample performance of the estimated break dates is demonstrated in Section 5, and the empirical example is given in Section 6. Section 7 concludes the paper. All proofs are collected in the appendix.

Model and Assumptions
Let us consider the following bubble's emerging and collapsing model for t = 1, 2, . . . , T : where y 0 = o p (T 1/2 ), c 0 ≥ 0, η 0 > 1/2, φ a := 1 + c a /T a with c a > 0 and 0 < a < 1, φ b := 1 − c b /T b with c b > 0 and 0 < b < 1, c 1 ≥ 0, and η 1 > 1/2. In this model, the process evolves up to time k e as a unit root process with a possibly positive drift shrinking to zero, and then y t behaves mildly explosive and next starts collapsing at t = k c + 1. Thereafter, the adjustment (collapsing and recovering) period lasts up to k r and then returns to a unit root process. We denote the mildly explosive regime with φ a := 1 + c a /T a and the mildly stationary regime with φ b := 1 − c b /T b for ease of exposition. However, we can also consider these AR(1) parameters in more general forms, such as φ a := 1 + c a /h T and φ b : where h T + T /h T → ∞ and k T + T /k T → ∞. The break points k e , k c , and k r are unknown, and the corresponding break fractions are defined as τ e := k e /T , τ c := k c /T , and τ r := k r /T , respectively. Note that PDC considered a model up to k r . Thus, our model is an extension of theirs to include the normal market behavior. Note that the result on the estimator of k r is new to the existing literature.
For model (1), we make the following assumption.
Assumption 1 implies that break dates are distinct and not quite close each other, which is typically assumed in the existing literature. Assumption 2 is general enough to allow for nonstationary unconditional volatility.
Assumption 2 is used mainly for the estimation of the collapsing date k c and the market recovering date k r for a < b. For the estimation of the emerging date of a bubble k e and the last break point k r for a > b, we impose the following stronger assumption: Under Assumption 3, the FCLT in (2) becomes simpler and is given by This result is used for deriving the limiting distributions of the estimators of k e and k r for a > b in Theorems 2(iii) and 3(ii), respectively.

Individual Estimation of Break Dates
Following PDC, we estimate the model with a one-time break. Note that model (1) can be expressed as For a given 1 ≤ k ≤ T − 1, we denote the sum of the squared residuals as where m t= is abbreviated just as m and φ a (k/T ) = Defining the break point estimator ask := arg min k SSR(k/T ), we have the following theorem.
Theorem 1 is the same as Theorem 3(a) in PDC and implies that, even if the last regime with a unit root process is additionally included in the sample period, we can consistently estimate the bubble collapsing date. This may not be a surprising result because the explosive behavior in the second regime is quite different from those in the other regimes. Therefore, even if we estimate only a one time break date, the estimator is consistent with the collapsing date of the bubble. We also note that we allow nonstationary volatility in {ε t }, general enough to apply our method to practical analyses. It might be possible to estimate the break dates by minimizing the weighted sum of the squared residuals using the volatility estimated by Harvey et al. (2022), but this is beyond the scope of this paper.
The consistency of the collapsing date is intuitively explained as follows. As given in the proof of Lemma 2, during the emergence of the bubble in the second regime, the process can be expressed as T weakly converges to a (time-transformed) Brownian motion. Therefore, the process evolves monotonically at a geometric rate (noting that φ a > 1) and attains its peak at k = k c . Conversely, the process collapses at a geometrically decaying rate (noting that φ b < 1) because it can be expressed as for k c + 1 ≤ k ≤ k r , where the first term on the right hand side can be shown to dominate the second term, at least, when k is relatively close to k c . These geometric explosive and collapsing behaviors help us identify the collapsing date k r consistently.
After we obtained the consistent estimatork of k c , we decompose the whole sample into two sub-samples for t = 1, . . . ,k andk + 1, . . . , T and estimate k e and k r from the first and second sub-samples, respectively. Becausek is a consistent estimator of k c , we can treat k c as a known break point. Thus, we use k c , instead ofk, in the following.
Estimation of k e is based on the minimization of the sum of the squared residuals using the first sub-sample, and the estimator is defined aŝ k e := arg min τ e ≤k/T ≤τe SSR 1 (k/T ) where 0 < τ e < τ e <τ e < τ c and The corresponding break fraction estimator is defined byτ e :=k e /T .
The consistency of theτ e has been already established in Theorem 1.3 of Pang et al.
(2018) and stated in the next theorem.
On the other hand, for the estimation of k r , we minimize the sum of the squared residuals using the second sub-sample, and the estimator is defined aŝ k r := arg min τ r ≤k/T ≤τr where τ c < τ r < τ r <τ r < 1 and The corresponding break fraction estimator is defined byτ r :=k r /T .
To investigate the asymptotic property ofk r , we must distinguish the two cases: a < b and a > b. In the case of a < b, the explosive behavior is faster than the collapsing (recovering) speed, and we continue to make Assumption 2. On the other hand, to derive the limiting distribution, we need to impose stronger restrictions on the shocks given by Assumption 3 in the case of a > b, in which the process recovers relatively faster than the evolution of the bubble.
The following theorem reveals that the convergence order ofτ r depends on whether a < b or a > b.
Theorem 3 Suppose that Assumptions 1 and 3 hold. (i) When a < b, for s ≥ 0 and B 1 (·) and B 2 (·) being two independent standard Brownian motions on R + .
Remark 1 Evidently, Theorems 2 and 3 hold under Assumptions 1 and 2, except for the derivation of the limiting distributions. That is, the same convergence orders of the break date estimators will be obtained with independent and heteroskedastic errors. We made Assumption 3 to derive the limiting distributions in Theorems 2(iii) and 3(ii), which are the same as those obtained in the existing literature.
From Theorem 3, we can observe that the break date from the collapsing (recovering) regime to the normal market can be consistently estimated if a < b; that is, the explosive speed of the process is faster than the collapsing speed. In this case, as given in (6), the process decays at geometric rate for k c + 1 ≤ k ≤ k r (because the first term of (6) dominates the second one) and reaches at That is, the explosive effect induced by φ a remains in the process during the whole collapsing (recovering) regime, after which the process evolves as → ∞ for any α > 0. This monotonically geometrical decay during the collapsing regime followed by the asymptotically flat behavior makesk r consistent.
On the other hand, when the collapsing speed is faster (a > b), the effect of the explosive component in y k in the recovering regime diminishes to zero as k approaches k r because → 0 when a > b, as can be observed in (7). That is, the initial value effect by y kc in the recovering regime gradually disappears and the process behaves as if it started from the small initial value. Therefore, we obtain the same result as Theorem 1 of Pang et al. (2018).
The asymptotically different convergence orders in Theorem 3 reflect the finite sample performance ofk r , as presented in Section 5.
Remark 2 It is possible to consider the case where a = b. As is seen from the expansion given by (7), the case where (τ c − τ e )c a > (τ r − τ c )c b will have the same result as the case of a < b and the reverse relation corresponds to the case of a > b. We can also consider the knife edge case in which (τ c − τ e )c a = (τ r − τ c )c b . In this case, the asymptotic behavior of depends on the higher order expansion of the logarithms as well as the values of a = b. Because this knife edge case will make our analysis complicated, we do not pursue this case and concentrate on the two cases where a < b or a > b to clearly state our main results.

Serially Correlated Case
In this section, we extend model (1) with {ε t } being serially correlated. We consider the case where {ε t } is a linear process given by For the innovations {v t }, we make the following assumption.
Linear process (8) can be expressed as, by the BN decomposition, and ψ = ∞ j=0 ψ j . It is also well known that the summability condition in (8) implies ∞ =0 |ψ | < ∞. In this case, we have the following corollary.
Corollary 1 Suppose that we estimate k c , k e , and k r as in the previous section and that Assumptions 1 and 4 holds. Then, we obtain the same results as given in Theorems 1, 2, and 3(i), while Theorem 3(ii) holds with the limiting distribution replaced by. ( where ψ * depends on whether v < 0 or v ≥ 0 and is defined in Appendix C. By Corollary 1, we can estimate the three break dates by using the misspecified AR(1) model with the same convergence orders of the estimators as before, although we may expect that the finite sample accuracy of the estimators would be improved by correctly specifying the correlation structure by considering the ADF type regression with an additional lag structure.

Monte-Carlo Simulations
In this section, we examine the performance of the estimates of the bubble regimes dates in finite samples. Our purpose is to demonstrate how the finite sample properties of the estimated break dates reflect the asymptotic results obtained in the previous section.
The Monte-Carlo simulations reported in this section are based on the series generated by (1) with y 0 = 0 and {ε t } ∼ IIDN (0, 1). Data were generated from this DGP for samples of T = (400, 800) with 50, 000 replications. 2 We set the drift terms in the first and fourth regimes to c 0 T −η 0 = 1/800 and c 1 T −η 0 = 1/800, respectively, following PDC. In this experiment, we focus on two cases. In the first case, the explosive coefficient φ a changes with a fixed φ b ; φ b is set to 0.96 while φ a takes values among (1.01, 1.05, 1.09). In the second case, the collapsing coefficient φ b varies with a fixed φ a ; φ a is set to 1.05 while φ b varies among (0.98, 0.96, 0.94).
We set the localising parameters c a = c b = 1; then, the values of a and b are uniquely determined based on the definitions given by φ a = 1 + 1/T a and φ b = 1 − 1/T b . For the dates of bubble regimes, we use (τ e , τ c , τ r ) to be equal to (0.4,0.6,0.7). 3 Further, in the minimization of SSR(k/T ), SSR 1 (k/T ), and SSR 2 (k/T ), we excluded the first and last 5% observations from the permissible break date k. For example, when estimating k r based on SSR 2 (k/T ), the permissible break date k ranges fromk c + 0.05T + 1 to 0.95T . If the break date estimatek c exceeds 0.95T , then we cannot estimate k r ; we do not include such a case in any bins of the histogram and thus the sum of the heights of the bins does not necessarily equal one fork r in some cases. Similarly, we cannot estimatek e when k c < 0.05T . In the following, we investigate onlyk c andk r because, oncek c is obtained, the estimate of k e in the four regime model is the same as that in the three regime model, which has been already investigated by PDC.  which corresponds to the case of a > b, we can observe that the accuracy ofk r deteriorates, while for a < b (φ a = 1.05 and φ a = 1.09) the estimate becomes more accurate. In particular, 2 All simulations were programmed in R with rnorm random number generator. 3 To compare the estimator of kc with that of PDC, we also set (τe, τc, τr) to be equal to (0.4,0.6,1), that is, without the last unit root regime (the normal market behavior after the collapse and recovering). The results are entirely similar and omitted for brevity.
for the large sample size of T = 800, the frequency of selecting the true break date is very close to 100%.
In the case where φ a is fixed at 1.05 and φ b varies, we can observe the good performance of the estimatek r even for the case of a > b (φ a = 1.05, φ b = 0.94) as shown in Figures 4(e) and 4(f). Reflecting the result in Theorem 3, the estimatek r is most accurate when φ a = 1.05 and φ b = 0.98; the frequency of selecting the true break date is approximately 75% and 100% for T = 400 and 800, respectively.
We next consider the case where the explosive and collapsing regimes are shorter such as (τ e , τ c , τ r ) = (0.5, 0.55, 0.6), which is relatively close to NASDAQ data in empirical appli- We also experiment with shorter minimum separation periods; we exclude the first and last 1% samples from the permissible break date k. The results provided in the appendix demonstrate that there is virtually no significant difference between two trimming parameters by comparing Figures 1-4 with D.5-D.8.
For σ 1 /σ 0 = 1/3, with which the volatility in the first half of the sample is higher than in the later sample,k c andk r perform similarly to the case of the constant unconditional volatility; the frequency of correctly selecting the true collapsing date k c is relatively high, whereas the performance ofk r is not satisfactory when φ a = 1.01 and φ b = 0.96 ).
In the case where the low volatility regime shifts to the high volatility one at the middle of the sample (σ 1 /σ 0 = 3), it becomes more difficult to select the true collapsing and recovering dates than in the case of σ 1 /σ 0 = 1/3, in particular, when φ a = 1.01 and φ b = 0.96. The possible reason is that both k c and k r are in the volatile regime when σ 1 /σ 0 = 3 and thus it would be difficult to distinguish between the shifts in the parameters and the large shocks.
However, the performance of the estimates improves for the larger values of φ a and/or as T gets larger ).

Empirical Application
In this section, we demonstrate the application of the bubble dates estimation method for the four regime model investigated in the previous sections to two time series data sets. It is worth noticing that even though we may detect emergence and collapse of a bubble by some tests, we are not sure whether the final observation available for estimation is included in the collapsing regime or the unit root regime. Therefore, before estimating the break dates, we need to determine whether the three or four regime model is appropriate. which is February 2000, as depicted in Figure 5. Next, by splitting the whole sample at the estimated collapsing date, the date of origination of the bubble is detected at August 1998 from the first subsample, while the date of recovery is estimated at September 2001 from the seconod subsample. As is observed in Figure 5, our method can detect the explosive and collapsing behavior very well.
The second example is an application of our method to the logarithm of the U.S. house price index from January 1991 to December 2012, provided by the Federal Housing Finance Agency, adjusted by the consumer price index. 5 Again, because the BIC selects the four regime model (BIC 2 , BIC 3 , and BIC 4 equal −2809.591, −2895.905, and −2918.223, respectively), we first estimate the collapsing date, which is estimated at November 2006. As is seen in Figure 6, the explosive behavior becomes mild around early in 2006 but the collapsing date is estimated just before the series staring crashing. By splitting the whole sample at the estimated collapsing date, the emergence date of the explosive behavior is estimated at September 1997, while the recovering date is at May 2011. These estimated dates are consistent with the visual inspection and the proposed method works very well.

Concluding Remarks
In this paper, we considered the four regime bubble model and investigated the break date estimators using the sample splitting approach; the break dates are estimated one at a time.
We showed that the break date estimated initially is consistent with the collapsing date of the bubble. We used the second subsample after the first estimated break date to estimate the break date returning to the normal market. The results revealed that this estimator is consistent with the recovering date if the explosive speed of the process during the bubble period is faster than the collapsing (recovering) speed, whereas in the case where the explosive speed is relatively slower, only the corresponding break fraction estimator is consistent. We need further investigation when the two speeds are the same.
Although we considered the case where the bubble occurred only once, we can extend our approach to the multiple bubble model in conjunction with the approach used by Harvey et al. (2020). In this case, we first identify the bubble regimes based on the PSY approach and then implement our method to each estimated regime. In Appendix A, we provide the proof of theorems. All the lemmas related to these proofs are given in Appendix B, while Appendix C gives the proof in the case of the serially correlated errors. The additional figures are collected in Appendix D. In the appendices, we denote C a generic constant that may differ all over and

A. Proofs of Theorems
Let M T be some positive and increasing sequence such that M T → ∞ and M T /T min(a,b) → 0.
We use the following relations repeatedly: for any values of α 1 , α 2 and α 3 > 0. These relations hold because We also note that, in the proofs, the orders of the terms of our interest may depend on in some places, which diverges to infinity when a < b, because This implies that the following relations similar to (A.1) hold: for any values of α 1 , α 2 , α 3 , and α 4 > 0.
Proof of Theorem 1: To prove the consistency ofk, we first show thatk − k c = O p (1).
Similarly to Chong (2001), we prove this by the contradiction argument. Suppose thatk − k c is not O p (1), which implies that, for some increasing sequence M T such that M T → ∞ and holds for some 0 < α ≤ 1. We shall prove that (A.2) does not occur as T → ∞. For this purpose, we decompose the whole sample into five subsamples such that D 0T = {k : To derive the asymptotic order of SSR(τ c ), we decompose it into To investigate the order of ϕ 1T , we can observe from (B.3) and (B.9) that and thus, the second term in the numerator of ϕ 1T dominates the first term. Because from (B.2) and (B.7), the second term in the denominator of ϕ 1T dominates the first term.
Therefore, we can observe that Using this result, we have Similarly, for S 2T , we can observe that 3),(B.7) and (B.9), we can observe that Then, we have from (B.7) and (B.9).
Using this result, we have Similarly, we have Combining (A.5), (A.6), (A.8), and (A.9), we can observe that We next show that (A.3) holds for i = 1, . . . , 4. Let us first consider the case where it is sufficient to show that the last term diverges to infinity faster than SSR(τ c ) uniformly

The order of each term of the denominator becomes
from (B.32) and (B.37). Then, in the numerator of ϕ T , the third term dominates the others ). Similarly, the order of the denominator of ϕ T is T k+1 uniformly over k ∈ D 1T from (B.2), (B.7), (B.19), (B.32), and (B.37). Therefore, we have uniformly over k ∈ D 1T , where we note that the dominant term is positive and thus (A.11) diverges to infinity. This implies that which is strictly faster than (A.10). Therefore, (A.3) is proved for i = 1.
Allowing for an abuse of notation, we repeatedly redefine ϕ T in the following. Similarly, we can observe that kr kc+1 The order of the first term of the numerator becomes uniformly over k ∈ D 2T from (B.11), (B.22), (B.34), and (B.38), while the third term is from (B.32) and (B.37). For the second term, note that ; thus, using (B.7) and (B.8), the following relation holds.
Thus, in the numerator of ϕ T , the second term, which is always positive, dominates the others, and its order is bounded from below and above, as in (A.12). On the other hand, the order of the denominator of ϕ T is bounded below by T k+1 from (B.19) and bounded above by from (B.7), (B.19), (B.32), and (B.37). By combining (A.12), (A.13), and (A.14), we can observe that ϕ T is asymptotically positive and Using this result, we obtain which is strictly faster than (A.10). Therefore, (A.3) is proved for i = 2.
For k ∈ D 3T , we have Then, we can observe that The order of the first term of the numerator becomes uniformly over k ∈ D 3T from (B.3), (B.9), and (B.23), while the second term is using (B.19) and (B.20), the following relation holds.
Thus, in the numerator of ϕ T , the third term, which is always negative, dominates the others, and its order is bounded from below and above, as in (A.17). On the other hand, the order of the denominator of ϕ T is bounded below by from (B.7) and bounded above by Using this result, we obtain uniformly over k ∈ D 3T from (B.7), (B.9), and (A.20), where the inequality holds because the third term asymptotically dominates the others. This implies that which is strictly faster than (A.10). Therefore, (A.3) is proved for i = 3.
For k ∈ D 4T , we have We can see that The order of each term of the numerator becomes ). On the other hand, the order of the denominator is given by uniformly over k ∈ D 4T from (B.7) and (B.9). This implies that which is strictly faster than (A.10). Therefore, (A.3) is proved for i = 4.
As we establishedk − k c = O p (1), there exists a positive integer M for any given such that P (|k − k c | > M ) < for all sufficiently large T . In the same manner as Pang et al.
(2021), we can observe that We show that the probabilities on the right-hand side go to 0. Note that the diverging order of SSR(k c ) has been already obtained in (A.10), which is strictly smaller than ; thus, we can ignore SSR(τ c ) divided by the scaling term.
We first focus on SSR(τ c − m/T ). Let k = (τ c − m/T )T = k c − m. Then, allowing an We investigate each of S it for i = 1, . . . , 5.
For S 1T , we can observe that which is obtained by using (B.2), (B.3), (B.7), and (B.9) because m is a fixed integer value ; thus, S 1T divided by the scaling term is negligible.

Thus,
; thus, S 4T divided by the scaling term is negligible.
To summarize the result, we have, for a given positive integer value of m, which implies that A similar result is obtained for SSR(τ c + m/T ). Let k = (τ c + m/T )T = k c + m. Then, in exactly the same manner, we can show that Thus, for a given positive integer value of m, does not hold for any 0 < α ≤ 1 where M T → ∞ and M T /T → 0 For this purpose, we decompose the range of the permissible break dates into three sub-samples such that we will show that, for i = 1 and 2, We first consider the case where k ∈ E 1T . Following (B.2) in Chong (2001), we have Thus, as explained in Appendix C on page 122 of Chong (2001), it is sufficient to show that Using (B.32) and (B.34), we have, for a < b, while using (B.21) and (B.24), To derive the order of A 3 , we expand it, as given in Appendix C of Chong (2001), using Lemmas 3 and 4, where the second terms in the two maxima are evaluated as (1) by (B.26) and (B.23), and by (B.24), (B.32), and (B.34), respectively. Then, we obtained (A.24) for i = 1.
In the case of k ∈ E 2T , we have, following (B.4) in Chong (2001), Thus, as explained in Appendix C on page 124 of Chong (2001), it is sufficient to show that Using (B.19) and (B.22), we have, for a < b, while using (B.33) and (B.35), To derive the order of A 6 , we expand it, as given in Appendix C of Chong (2001), using Lemmas 3 and 4, where the second terms in the two maxima are evaluated as by (B.40) and (B.36), respectively. Then, we obtained (A.24) for i = 2.
As we establishedk r − k r = O p (1), there exists a positive integer M for any given such that P (|k r − k r | > M ) < for all sufficiently large T . In the same manner as PDC, we have From expression (A.25) with k replaced by k r − m, we can show that the second term of (A.25) dominates the first and third terms using Lemmas 3 and 4. By focusing the second term, we can observe that, for a given fixed m, using (B.1) and (B.26), which implies that Similarly, from expression (A.28) with k replaced by k r + m, we can show that the second term of (A.28) dominates the first and third terms using Lemmas 3 and 4. By focusing on the second term, we can observe that, for a given fixed m, using (B.1) and (B.26) and thus, for a given positive integer value of m, In case (ii), we first prove (A.23) with M T replaced by M bT does not hold, where M bT → ∞ and T b+ /M bT → 0 for a given > 0. That is, for any given > 0, the objective function cannot be minimized outside the T b+ neighborhood of k r . Let, for a given , where we suppress the dependence on for notational simplicity. Then, as in case (i), it is sufficient to prove that for i = 1 and 2 when a > b.
In exactly the same manner as in case (i), the difference in the sums of the squared residuals is expressed as (A.25) for k ∈ E b1T and (A.28) for k ∈ E b2T . Thus, it is sufficient , and A b6 are defined as A 2 , A 3 , A 5 , and A 6 with E iT replaced by E biT for i = 1 and 2, respectively.
uniformly over k ∈ E b1T , where the second equality holds by (B.37) and because the term in the parentheses uniformly converges in probability to σ 2 /(2c b ), as shown in the proof of Lemma 5, while the last equality is obtained by noting that For the numerator of the second term, we have uniformly over k ∈ E b1T , where the second equality holds by (B.38) and because the term in the parentheses is uniformly o p (1), as shown in the proof of Lemma 5. Therefore, we can observe that the second term in the second maximum in (A.27) is O p (1) uniformly over E b1T .

Summarizing these results, we can observe that
For the first maximum in (A.29), the first term is O p (1) as shown above for A b3 , while for the second term, and the numerator is uniformly O p (T 2 ) by (B.39). Thus, the second term is O p (1) uniformly over k ∈ E b2T . Summarizing these results, we can observe that To derive the limiting distribution of (1 − φ b )T (τ r − τ r ), let k = k r + vT b where v is a finite constant, and consider the O(T b ) order neighborhood of k r . We first derive the limiting distribution when v < 0 (k = k r − |v|T b ) using (A.25), in which Λ T (k/T ) can be shown to be o p (1) using the same technique as before. Because time is measured in 1/T b units rather than 1/T in this case, we note that by the FCLT, where B 1 (·) is a standard Brownian motion on R + . Then, we can observe that Using these results, each term in (A.25) becomes, by noting that T k+1 y 2 t−1 = T kr+1 y 2 t−1 (1+ o p (1)) in this case, where we note that E[B 2 c b (s)] = 1/(2c b ) for any s ≥ 0 in the last equality. Then, we have we obtain the limiting distribution given in the theorem.
Similarly, for k = k r + vT b with v ≥ 0, it can be shown that Λ T (k/T ) = o p (1) in (A.28).

Because the last two terms in the second braces can be expressed as
the limiting distribution can be expressed as for v ≥ 0. Then, we obtain the limiting distribution for v ≥ 0 given in the theorem.

(B.15)
We thus obtain kc kc−M T +1 (B.9) is obtained by noting that Similarly, by using (B.6), we can observe that Again, by the HRI for any fixed value of N > 0, ) uniformly over k e + 1 ≤ k ≤ k c . Combining this with (B.16), we have (B.10).
Lemma 3 Under Assumption 2, for k c + 1 ≤ k ≤ k r , Proof of Lemma 3: For k c + 1 ≤ k ≤ k r , y k is expressed as In particular, when k = k r , (B.26) becomes Note that the dominant order depends on the behavior of φ kc−ke From the definition of φ b , we can observe that the first term on the right-hand side of (B.27) ) (when k is finite) and at most O p (T b+1 φ 2(kc−ke) a ) (when k = k r ). For the third term, we can observe that Therefore, the first term on the right hand side of (B.27) dominates the third term for all k c + 1 ≤ k ≤ k r . We can also see that the former dominates the second term by Cauchy- The order of the first term of (B.28) is given by where the second last relation holds because by applying (B.14). For the third term of (B.28), we have which is the smaller order than the first term when a < b. Because the second term of (B.28) is shown to be dominated by the first term using Cauchy-Schwarz inequality, we have (B.21).
which implies that the first term on the right-hand side of (B.30) is On the other hand, the variance of the second term becomes Then, the second term on the right-hand side of (B.30) is O p (T (b+1)/2 ) and dominated by the first term. We then have (B.22).
Similarly to (B.22), we have, for (B.23), The first term is at least O p (y kc ) = O p (T 1/2 φ (kc−kr) a ) whereas, by the HRI, and thus, the first term is dominant for all k c + 1 ≤ k ≤ k r . In addition, we have, again using the HRI, Then, because y kc = O p (T 1/2 φ Lemma 4 For k r + 1 ≤ k ≤ T , Proof of Lemma 4: We first note that for k r + 1 ≤ k ≤ T , In particular, we have y T = y kr + c 1 (T − k r )/T η 1 + T j=kr+1 ε j ; thus, we obtain (B.31) using (B.17).
When a < b, the first term on the right-hand side of (B.40) dominates the second and third terms uniformly over k r + 1 ≤ k ≤ T and thus using (B.17), ).
When a > b, (B.39) is obtained similarly by the FCLT.
Lemma 5 Under Assumption 3, we have for a > b, Proof of Lemma 5: We first note that, from (B.26), we can observe that for a > b. Thus, to prove (B.41), it is sufficient to show that We first show that the denominator converges to σ 2 /(2c b ) uniformly over k ∈ E b1T . Note (B.45) The first term on the right-hand side of (B.45) is further decomposed into by changing the order of the summations. Note that can be shown to be a decreasing function of k − k c for all sufficiently large T , we have, using the HRI, On the other hand, D 2 is decomposed into Clearly, the first term in the brackets divided by k r − k goes to σ 2 , while the third term is For the second term, by applying the HRI in the reverse order, we have and thus the second term divided by k r − k uniformly converges in probability to zero.
Similarly, the fourth term in the brackets divided by k r − k is show to be o p (1) uniformly by applying the HRI in the reverse order. As a result, uniformly over k ∈ E b1T . By (B.47) and (B.48), for the first term on the right hand side of (B.45), we have Next, we will show that the second term on the right-hand side of (B.45) divided by converges in probability to 0 uniformly. By changing the order of the summations, we can observe that The first term on the right-hand side of (B.49) becomes where the last relation is obtained by changing the order of the summations. By letting the first term in the brackets can be expressed as Noting that φ kc− −1 b ξ 1kc+ forms a martingale difference sequence and that where the first inequality follows by the HRI by noting that φ is a decreasing function of k − k c for all sufficiently large T and the second . This implies that the first term in the brackets in (B.50) divided by k r − k converges in probability to zero uniformly over k ∈ E b1T . By a similar argument, the second term divided by k r − k is shown to be o p (1).
As a result, for the first term on the right hand side of (B.49), we have We next investigate the second term of (B.49). Note that We first show that the first term on the right hand side of the brackets divided by k r − k is uniformly o p (1) outside of the T 2b neighborhood of k r , and then refinement will be performed.
Let M b1T → ∞ with T 2b /M b1T → 0. Then, we can observe that Because we obtained (B.52), we focus on the rage of k given by k The first term becomes, by noting that and thus by the HRI, This implies that max kr−CT 2b ≤k≤kr−M b2T we can show that, by using the HRI in the reverse order, On the other hand, from the definitions of V t and W t , we can easily show that and thus W kr−C 2 T 2b −2 W kr−C 2 T 2b /M bT = o p (1). Therefore, by combining these results with Next, we focus on the rage of k given by k r − C 3 T 3b/2 ≤ k ≤ k r − M b3T for a sufficiently large C 3 , where M b3T → ∞ and T 5b/4 /M b3T → 0. Then, by using the same argument, we can observe that (B.53) holds with M b2T replaced by M b3T . By repeating the same argument, we can observe that where M bT → ∞ and T b+ /M bT → 0 for a given > 0.
On the other hand, the order of the second term of (B.51) is shown to be o p (1) because Therefore, for the second term on the right hand side of (B.49), The third term on the right hand side of (B.49) becomes We express the first term in the brackets as Becauseξ 1t is a martingale difference sequence with the filtration in the reserve order and A similar result can be obtained for the second term in the brackets, and thus, for the third term on the right-hand side of (B.49), uniformly over k ∈ E 1T . By combining these result, we can observe that the denominator of (B.44) converges to σ 2 /(2c b ) uniformly over k ∈ E b1T . The numerator of (B.44) is expressed as, by changing the order of the summations, The first term on the right-hand side has the same structure as the first term in the brackets of (B.51), and we have shown (B.54). On the other hand, the second term on the right-hand side is essentially the same as the first term in the brackets in (B.55), which, divided by k r − k, is o p (1) uniformly. Therefore, the numerator of (B.44) is o p (1) uniformly and thus we obtain (B.44).
To prove (B.42), note that (B.56) From (B.40), we can observe that, for a > b, y k ∼ a k j=kc+1 u j uniformly over k ∈ E b2T because y kr = O p (T b/2 ) by (B.17) for a > b. Thus, the denominator of (B.56) becomes The tedious but direct calculation shows that where the o(1) term is uniform over k ∈ E b2T . In this case, the following relation holds: uniformly over k ∈ E b2T . We next investigate the numerator of (B.56). Note that From (B.40), we have By noting that for a > b, Therefore, we obtain (B.42) .

C. Proofs in the Serially Correlated Case
In this appendix, we first show that Lemmas 1-4 hold for model (8) under Assumption 4. As a result, we obtain the same result as Theorem 1.
It is well known in the econometric literature that Lemma 1 holds for a linear process with (B.1) replaced by 1 √ T y ke ⇒ ψW (τ e ) and we omit the details.
To prove (B.6) in Lemma 2, from (B.13) it is sufficient to prove that The first term of (C.1) is O p (T a/2 ) uniformly as proved in (B.6), while the second term can be expanded as The first term is O p (1) while the third terms are o p (T a/2 ) because φ −(k−ke) a < 1 and max k |ṽ k | = o p (T a/2 ) as proved in Proposition A3 in Phillips and Magdalinos (2007). Similarly, the second term is Thus, we obtain (C.2).
For (B.10), we note that 3) The first term in the parentheses is O p (T a/2 φ (kc−ke) a ) uniformly as proved in Appendix B, whereas the second term of (C.3) becomes Because max t |ṽ t | = o p (T a/2 ), we can see that the second term of (C.3) is o p (T a/2 φ (kc−ke) a ) uniformly and thus (B.10) holds.
In exactly the same manner, Lemmas 3 and 4 hold for the model with serially correlated shocks using the BN decomposition given by (9).
Theorem 2 in the serially correlate case is obtained in Pang et al. (2018).
For the proof ofk r , let k = k r + vT b for a given v and we proceed with the proof in the same way as Theorem 3 but we need to deal with the serially correlated shocks carefully.
For I, we have, as derived in the proof of Theorem 3, that the first and third terms in the curly braces are o p (T b ), we have, by the WLLN, These results yield Similarly, it can be shown that II and III are o p (1). For IV , we have, by the WLLN, These results imply that Similarly, it is shown using (C.4) that These results yield for v < 0. We obtain the result in the theorem by defining ψ * := 1 −ψ.
Then, it can be shown that On the other hand, we can observe that Combining these results, we have for v ≥ 0. By defining ψ * := 1 + (1 − ∞ j=0 ψ 2 j /ψ 2 )/(c bB