A Modified Confidence Set for the S Title Date in Linear Regression Models.

Transcription

1 A Modified Confidence Set for the S Title Date in Linear Regression Models Author(s) Yamamoto, Yohei Citation Issue Date Type Technical Report Text Version publisher URL Right Hitotsubashi University Repository

2 Graduate School of Economics, Hitotsubashi University Discussion Paper Series No A Modified Confidence Set for the Structural Break Date in Linear Regression Models Yohei Yamamoto May 24

3 A Modied Condence Set for the Structural Break Date in Linear Regression Models Yohei Yamamoto y Hitotsubashi University May 7, 24 Abstract Elliott and Muller (27) (EM) provides a method to construct a condence set for the structural break date by inverting a locally best test statistic. Previous studies show that the EM method produces a set with an accurate coverage ratio even for a small break, however, the set is often overly lengthy. This study proposes a simple modication to rehabilitate their method. Following the literature, we provide an asymptotic justication for the modied method under a nonlocal asymptotic framework. A Monte Carlo simulation shows that like the original method, the modied method exhibits a coverage ratio that is very close to the nominal level. More importantly, it achieves a much shorter condence set. Hence, when the break is small, the modied method serves as a better alternative to Bai's (997) condence set. We apply these methods to a small level shift in post-98s Japanese ination data. JEL Classication Number: C2, C38 Keywords: condence set, coverage ratio, nonlocal asymptotics, heteroskedasticity and autocorrelation consistent covariance The author acknowledges nancial support for this work from the MEXT Grants-in-Aid for Scientic Research No y Hitotsubashi University, Department of Economics, 2- Naka, Kunitachi, Tokyo, Japan (yohei.yamamoto@econ.hit-u.ac.jp).

4 Introduction This study provides a simple modication of the condence set that was proposed by Elliott and Muller (27) (EM hereafter) for a single structural break date in linear regression models. EM proposed a method of inverting a test statistic that species the null hypothesis of the break occurring at a certain date and the alternative hypothesis of it occurring at another date. The idea is to implement such a test for each possible break date and arrive at the collection of dates where the test fails to reject the null hypothesis; this collection of dates is considered a condence set. Their method has at least a strong theoretical advantage in the sense that it is based on the locally best test and is considered asymptotically ecient under certain conditions, including the break magnitude shrinking to zero at a fast rate. More importantly, their simulation study suggests that this method exhibits very good coverage properties across a wide range of break magnitudes in nite samples. Especially for small breaks where the standard structural break test is marginally signicant or insignicant, the existing method of Bai (997) is known to exhibit undercoverage, whereas the EM method still achieves a coverage ratio that is very close to the nominal level. However, previous studies show that the EM condence set is often much longer than the one computed by other methods. For example, in their Table 8 (say, the case of r = :5 and d = 6), the coverage ratio of Bai (997) is a few percentage points lower, whereas that given by the EM method is very close to the nominal level. As a cost of this advantage, the average length of the EM method is about four times that of Bai's method. More recently, Chang and Perron (23) investigated the same problem by extending the comparison to the set of linear models including lagged dependent variables. They show that EM's condence set indeed has a better coverage ratio than other methods but often becomes so long that it includes the whole sample period. More interestingly, they discuss a theoretical base for this phenomenon by using a nonlocal asymptotic framework in which the break does not shrink to zero as the sample size increases. Perron and Yamamoto (23) and Yamamoto and Tanaka (23) recently showed that the same technique is useful for analyzing the widely known nonmonotonic power problem in structural break tests. (See also Crainiceanu and Vogelsang 27 and Perron 99, 26 for details of the nonmonotonic power problem.) Following this literature, we provide an asymptotic justication of our condence set under the nonlocal asymptotic framework with a xed break magnitude. The results show Eo and Morley (23) recently proposed a method that inverts the likelihood ratio test statistic. A comparison with this method is beyond the scope of this paper.

5 that under the null hypothesis, the modied and original tests are asymptotically equivalent. Under the alternative hypothesis, the modied test is an increasing function in the break magnitude, but the original test is not. If serial correlation in the errors is accounted for by the standard heteroskedasticity and autocorrelation consistent (HAC) correction, then the original test may become a decreasing function. This asymptotic property predicts a higher power of the modied test thus it excludes false time points from the condence set more eciently in nite samples. Our Monte Carlo simulation shows that the modied method gives a very good coverage ratio, similar to the original method. More importantly, the average length of the modied method is much smaller than that of the original method over a wide range of break magnitudes. This improvement is signicant whether or not the HAC correction is conducted. This analysis has a clear practical implication that the modied condence set serves as a good alternative to Bai's method when the break is small. If the break is not small, Bai's method will achieve good coverage with a shorter length than ours and that of the original method on average, so is still recommended. The remainder of this paper is structured as follows. Section 2 introduces the model and assumptions. Section 3 explains the EM method to construct the condence set for a single structural break date. Section 4 proposes a modication of the EM method and investigates its asymptotic property using a nonlocal asymptotic framework. Section 5 provides the results of a Monte Carlo simulation to compare nite sample properties of the new and existing methods. Section 6 illustrates the usefulness of our approach by using an empirical example of post-98s Japanese ination data and section 7 presents a conclusion. Throughout the paper, the symbols \!" p and \)" denote convergence in probability and convergence in distribution. 2 Model and assumptions Consider the linear regression model y t = I(t Tb )Xt + I(t > Tb )Xt 2 + Zt + u t ; t = ; ; T; () where y t is a scalar variable, X t is a k vector of regressors with breaking coecients and 2 at a date t = Tb and Z t is a p vector of regressors with time-invariant coecients. I() is the indicator function. The error term u t has mean zero. The break magnitude is denoted by = 2. Our goal is to compute a condence set for the unknown break date Tb at a specied condence level. To this end, we introduce the following assumptions. 2

6 Assumption i) Tb = [ T ], where < < : ii) T P =2 [st ] t= X tu t ) =2 W (s) for s and T P =2 [st ] X t=t tu b + t ) =2 2 (W (s) W (r )) for s where and 2 are symmetric and positive denite k k long-run variance (LRV) matrices and W () is a k standard Wiener process. iii) Let W t = [X t ; Z t ]. The matrices j P j t= W twt, j P T t=t j+ W twt, j P Tb W t=tb j+ twt, and j P Tb +j W t=t tw b + t have minimum eigenvalues bounded away from zero in probability for all j p + k. T iv) sup P =2 [st ] s t= Z tu t = Op (): v) T P [st ] t= W twt p! sq ; uniformly in s and T P [st ] W t=t tw p b + t! (s )Q 2 ; uniformly in s, where Q and Q 2 are full rank. Parts i), ii), iv), and v) follow Condition of EM. Part iii) follows Bai (997) A3 and enables to estimate the break date by ^T b = arg min T b 2[T;( )T ] SSR(T b ); (2) where SSR(T b ) is the sum of squared residuals of the regression of y t on fi(t T b )X t, I(t > T b )X t, Z t g. The quantity in (2) is the small trimming value ( > ). We know that such estimator satises consistency for the ratio ^T b =T p! as discussed in Bai (997) under these assumptions. (See also Lemma in the Appendix.) These assumptions also enable us to use the condence set of Bai (997) in the form of [ ^T b [c (+)=2 g] ; ^T b [c ( )=2 g] + ] where c is the percentile of the suggested random variable and g = ^ ^^=(^ ^Q^) 2. 3 Elliott and Muller's (27) condence set Earlier studies show that Bai's condence set may exhibit undercoverage when the break is small. To overcome this problem, EM proposes an alternative method that inverts the test statistic U T (T m ) specifying the null hypothesis of a break occurring at t = T m and the alternative hypothesis of one occurring at t 6= T m P U T (T m ) = Tm 2 Tm P t t= s= v s +(T T m ) 2 P T t=t m+ P t s= v s P t s=t m+ v s 2 P t s=t v m+ s ; 3

7 where v t = X t u t and its LRVs j (j = ; 2) are replaced by their estimates, respectively, as in the following steps.. For any T m = p + 2k + ; : : : ; T p 2k, compute the least squares regression of fy t g T t= on fi(t T m)x t ; I(t > T m )X t ; Z t g T t=. Call the residuals ^u t. 2. Construct f^v t g T t= = fx t^u t g T t=. 3. Compute the LRV estimators ^ and ^ 2 of f^v t g Tm t= and f^v tg T t=t m+, respectively2. 4. Compute U T (T m ) with v t replaced by ^v t and and 2 by ^ and ^ 2, respectively. 5. Include T m in the level condence set when U T (T m ) is less than the critical values provided by EM and exclude it otherwise. In nite samples, this method indeed achieves a coverage ratio that is very close to the nominal level even when the break is small. However, it is also shown that this method often gives an overly lengthy condence set. For example, in Table 8 of EM (say, the case of = :5 and d = 6), the coverage ratio of Bai (997) is a few percentage points lower than, whereas the EM method is very close to, the nominal level. However, as a cost of this advantage, the average length of the EM method is about four times that of Bai's method. More recently, Chang and Perron (23) investigated the same problem by extending the comparison to linear models including lagged dependent variables. They show that EM's condence set indeed has a better coverage ratio than other methods but often becomes so long that it includes the whole sample period. 4 A modied method This section proposes a simple modication to overcome the drawback of the EM condence set. The basic idea is to estimate the LRV by using the residuals obtained under the alternative hypothesis, that is, the residuals assuming a break at an unknown point t 6= T m. To do so, we use the unknown break date estimation of Bai (997) that minimizes the sum of squared residuals over possible unknown break dates. If the true break date is consistently estimated, such residuals are asymptotically free from the presence of a break. These residuals are used to estimate the LRV to construct the test. However, the other part of the test 2 An attractive choice is to use the automatic bandwidth estimators of Andrews(99) or Andrews and Monahan (992). If it is known that = 2, the it is advisable to rely instead on a single LRV estimator based on fv t g T t=. This footnote also applies to the step 3 of the modied procedure in the next section. 4

8 is constructed as suggested by EM assuming the break at T m. Hence, under the alternative hypothesis of T m 6= Tb, the true break is present in these residuals, which makes the test statistic large. The following steps describe the algorithm.. Estimate a break date ^T b by minimizing the sum of squares in the regression, that is, (2). 2. For any T m = p + 2k + ; : : : ; T p 2k, do the following: (a) Compute the least squares regression of fy t g T t= on fi(t T m)x t ; I(t > T m )X t ; I(t > ^T o T b )X t ; Z t n t Call the residuals eu t. Note that for T m such that T m ^Tb < k, omit I(t > ^T o T b )X t t= from the regressors. Construct fev t g T t= = fx teu t g T t=. (b) Compute the least squares regression of fy t g T t= on fi(t T m)x t ; I(t > T m )X t ; Z t g T t=. Call the residuals ^u t. Construct f^v t g T t= = fx t^u t g T t=. 3. Compute the LRV estimators e and e 2 of fev t g Tm t= and fev tg T t=t m+, respectively. 4. Compute U T (T m ) with v t replaced by ^v t. Replace and 2 by e and e 2, respectively. Call it V T (T m ). 5. Include T m in the level condence set when V T (T m ) is less than the critical values provided by EM and exclude it otherwise. Intuitively, this modication only changes the LRV estimate to another consistent estimate, with which the consistency holds under assumption of the break magnitude that shrinks to zero or not. Therefore, EM's Proposition 3 is still valid and the asymptotic critical values are invariant under the null hypothesis. The following theorem is obtained. Theorem Suppose Assumption holds. a) Let v t be independent and identically distributed (i.i.d.) processes and j (j = ; 2) estimated under the i.i.d. assumption or b) let v t be strictly stationary processes and j (j = ; 2) estimated using the HAC estimator. Then, U T (Tb ) and V T (Tb ) are asymptotically equivalent. More importantly, we derive the asymptotic behavior of the tests under the alternative hypothesis. To do so, we follow the literature and use a nonlocal asymptotic framework that assumes the break magnitude to be xed. 5

9 Theorem 2 Suppose Assumption holds. a) Let v t be i.i.d. processes and j (j = ; 2) estimated under the i.i.d. assumption. Then, for T m 6= T b, T U T (T m ) = O p (); T V T (T m ) = O p (kk 2 ): b) Let v t be strictly stationary processes and j (j = ; 2) estimated using the HAC estimator of Andrews(99) with the bandwidth chosen by an AR() approximation. Then, for T m 6= T b, T = U T (T m ) = O p (); T = V T (T m ) = O p (kk 2 ); where = 3 when the Bartlett kernel is used and = 5 when the Quadratic Spectral (or Parzen or Tukey-Hanning) kernel is used. Theorem 2 gives clear guidance for predicting the behavior of the tests U T and V T under the alternative hypothesis. What is important is that the original test U T is a nonincreasing function in kk. On the other hand, the modied test V T is increasing in kk whether or not the HAC correction is conducted. This result predicts that in nite samples the modied test has a higher power and excludes false time points t 6= T b eciently. from the condence set more Remark If the regressor is a constant, then Condition A in the Appendix is satised. In this case, the selection of bandwidth for constructing the standard nonparametric HAC covariance estimate is aected by the break magnitude, even asymptotically. Suppose that we use the popular Andrews' (99) AR() approximation. Then, the AR() coecient estimate ^ approaches one as the break magnitude increases. Given that the rule is ^m / (T ) =, where = 4^ 2 =( ^) 4 and = 5 in the case of the Quadratic Spectral kernel, and ^ is the OLS estimate for the AR() coecient of ^v t, the bandwidth is ^m = O p (kk 4= T = ). This changes the above results for case b) to T = U T (T m ) = O p (kk 4= ); so that U T is indeed a decreasing function in kk. This result closely follows Chang and Perron's (23) Theorem and applies to the other kernels proposed in Andrews (99). 6

10 5 Monte Carlo simulation This section investigates the nite sample property of the proposed condence set as a comparison of the two existing methods. To this end, we replicate EM's simulation design and consider the following data generating processes: Model : y t = + dt =2 I(t > [ T ]) + u t ; where u t i:i:d:n(; ): Model 2: y t = + dt =2 I(t > [ T ]) + u t ; u t = ( + I(t > [ T ]) " t ; where " t i:i:d:n(; ): Model 3: y t = + dt =2 I(t > [ T ]) + u t ; u t = :3u t + " t ; where " t i:i:d:n(; ): Model 4: y t = + dt =2 I(t > [ T ]) + u t ; u t = " t :3" t ; where " t i:i:d:n(; 2:4): Model 5: y t = + x t + dt =2 x t I(t > [ T ]) + u t ; x t = :5x t + t ; where t i:i:d:n(::75): Model 6: y t = + x t + dt =2 x t I(t > [ T ]) + u t ; x t = :5x t + t ; u t = " t jx t j ; 7

11 where t i:i:d:n(::75) and " t i:i:d:n(; :333): In every model, we consider d = [4; 8; 2; 6] and = [:5; :35; :2] and set T =. The following condence sets are considered i) EM's method assuming equal LRVs in the two regimes (denoted by U :eq ), ii) EM's method assuming unequal LRVs (U: neq ), iii) the modied method assuming equal LRVs (V :eq ), iv) the modied method assuming unequal LRVs (V :neq ), v) Bai's method assuming equal LRVs (Bai :heq ), and vi) Bai's method assuming unequal LRVs (Bai :hneq ). We report the coverage ratio and average length of the condence set evaluated by 3, replications in Tables through 6 for Models through 6, respectively. These tables exactly correspond to those reported by EM. Since there is no serial correlation in the errors in Models, 2, 5, and 6, White's (98) heteroskedasticity robust covariance is used. For Models 3 and 4, we use the HAC covariance of Andrews(99) using Quadratic Spectral kernel with the bandwidth selected by an AR() approximation of a prewhitened series of Andrews and Monahan (992) to account for serial correlation in the errors 3. We preliminarily point out that the U and Bai condence sets in Tables through 6 almost completely replicate EM's results. There is a minor dierence, because the extent to which Bai undercovers the nominal level in the case of d = 4 is somewhat smaller in our simulation than in EM's simulation. However, we can conrm that the EM method gives a very good coverage ratio, whereas Bai shows undercoverage when the break is small (d = 4 and 8). In addition, the average length of U is more than Bai in every case as pointed out by earlier studies. The main ndings of this simulation are summarized by the following two points. First, the modied method V gives a very good coverage ratio and is very similar to the original method U. We see a slight undercoverage only in DGP3 with a very small break (d = 4), however, the deviation is minor and rare. Furthermore, it quickly achieves the correct nominal level when the break is as large as d = 8. Second and more importantly, the average length of V is much smaller than that of U over the range of break magnitudes. However, as the theory suggests, the advantage of V over U is intensied when the break gets larger and/or the HAC correction is conducted. For example, the dierence in average lengths is not very large in the case of d = 4 in Model 2. However, the average length of V is ve times less than that of U in the case of d = 6 in Model 3. This conrms Theorem 2. We also nd that in all cases the average length of V falls between U and Bai. This 3 In practice, this information is not available and serial correlation in the errors is to be tested prior to form a condence set. From this point of view, we also implemented the test for which the HAC correction is applied only when the pre-test rejects. However, the results are very close to what are reported here so is not separately reported. 8

12 implies that as long as the break is not small, Bai's method gives good coverage as well as the shortest condence set, so it is recommended. When the break is small, Bai's method still gives a short condence set, but at the risk of undercoverage. Hence, the modied condence set proposed in this paper must be an attractive alternative method. 6 Empirical illustration : Level shift in the post-98s Japanese ination rate After a long recession, Japanese ination dynamics have now been attracting much attention. As is common to most developed countries, Japan experienced high ination period in the 97s, recording two-digit annual ination rates in terms of the core consumer price index (CPI). The inationary pressure came down as late as the 98s and ination became fairly stable since the middle of that decade. Of interest is that Japanese economy experienced a strong deationary pressure in the late 99s (See De Veirman 29 and the references therein). The source of this deation is a controversial issue and detailed investigation is out of the scope of this paper. However, identifying the dates of level shifts should help one understand the major cause. We use the quarterly annualized Japanese ination rate Y t = 4(log(P t ) log(p t )), where P t is the core CPI (seasonally adjusted). We take the sample of a relatively stable ination rate from 985Q through 23Q4; the series is presented in Figure. We are interested in determining whether there is a break in the mean of Y t, and if so, when it occurred. To see this, we rst apply the SupF structural change test for the parameter in the regression model Y t = + u t. We use the HAC covariance of Andrews(99) using Quadratic Spectral kernel with the bandwidth chosen by an AR() approximation. The SupF test gives a value of 8., suggesting that there is a break at % signicance level. We then use the sequential method of Bai and Perron (998) to see if there is a second signicant break. We nd that the SupF (2j) test is 3.96 and is insignicant even at the % level. We conclude that there is one break in the mean of ination during this period. We now compute the condence intervals by using the EM method (U), the modied method (V ), and Bai's method (Bai). Figure 2 illustrates the 9% condence set in black and the 95% condence set in black and gray. The 9% condence set of the original EM method covers the period from 993Q2 through 2Q3, however, the 95% set includes almost the entire sample period and is thus not very informative. However, Bai gives a set centered at 998Q and the 9% and 95% sets span from from 995Q through 2Q2 and 993Q4 through 22Q3, respectively. Our method provides a set that covers a time period somewhat earlier than Bai from 993Q3 through 2Q2 for 9% and 993Q through 9

13 2Q4 for 95%. The lengths of V and Bai are very similar or V is somewhat shorter in this example. Given that the break is small and Bai's method may be subject to undercoverage, we prioritize the result obtained by our modied method, followed by Bai and U in that order. This example clearly illustrates the usefulness of the proposed method. 7 Conclusion This paper provides a simple modication of the condence set of the single break date in linear regression models proposed by Elliott and Muller (27). The method involves a step that estimates an unknown break point to obtain the residuals under the alternative hypothesis. These are then used to construct the LRV estimate for the test statistic. Following the literature, an asymptotic justication of the proposed method is provided using the nonlocal xed break asymptotic framework. Our Monte Carlo simulation shows that the modied method gives an equally correct coverage ratio but a signicantly shorter condence set than the original approach. This method is a good alternative to the condence set proposed by Bai (997) when the break is small, because it is known that the Bai's method may result in undercoverage in such cases.

14 and Appendix : Proof of Theorems Consider the LRV estimators ^ = ^R() + 2 P T l= (l; ^m) ^R(l) (A.) e = e R() + 2 P T l= (l; em) e R(l), (A.2) where ^R(l) and R(l) e for l = ; ; are the sample autocovariance estimates of f^v t g T t= and fev t g T t= of order l, respectively. (; ) is a kernel function and ^m and em are the bandwidths. We let ^m and em be selected by the Andrew's (99) data dependent method with an AR() approximation, however, this particular choice does not aect the qualitative results of this paper. For simplicity, we consider the model with p = since this part does not aect the nal result. Lemma : Suppose is a xed parameter and Assumption holds. Let ^T b be obtained by (2). Then, ^T b T b = O p(kk 2 ). where Proof of Lemma : See Bai (997), Proposition. Proof of Theorem : We consider the case of T m = Tb 8. < X t u t X t Xt(^ ) for t = ; ; Tb ^v t = ; : X t u t X t Xt(^ 2 2 ) for t = Tb + ; ; T; ^ = [ P X tx t] [ P X ty t ] ; = + [ P X txt] [ P X tu t ] ; ^ 2 = [ P 2 X txt] [ P 2 X ty t ] ; = 2 + [ P 2 X tx t] [ P 2 X tu t ] ; so that ^ = o p () and ^ 2 2 = o p () under Assumption ii), respectively. Note that the subscripts and 2 of the summation symbol denote the regimes dened by t 2 [; Tb ] and t 2 [Tb + ; T ] for and 2, respectively. The sample autocovariance of ^v t in the pre- T m (= Tb ) period is ^R (l) = (T b l) P X tu t u t l X t l +(^ ) (T b l) P X tx t X t lx t l ^ +(^ ) (T b l) P X tx t u t l X t l + (T b l) P X tu t X t lx t l ^ ; = I + II + IIIa + IIIb:

15 Term I converges in probability to R (l) under the stationary assumption. II; IIIa and IIIb are o p () since ^ = o p (). Hence, ^R (l)! p R (l). Similarly, ^R 2 (l) = (T Tb l) P 2 X tu t u t l Xt l +(^ 2 2 ) (T Tb l) P ^2 2 X tx t Xt lx t l 2 +(^ 2 2 ) (T Tb l) P 2 X tx t u t l Xt l + (T Tb l) P 2 X ^2 tu t Xt lx t l 2 : Given that ^ 2 2 = o p (), ^R 2 (l)! p R 2 (l) for l = ; ; 2;. Next, suppose we use the modied method. Without loss of generality, let ^T b Tb. Then, 8 X >< t u t X t Xt( e ) for t = ; ; ^T b ; ev t = X t u t X t Xt( e 2 ) for t = ^T b + ; ; Tb ; >: X t u t X t Xt( e 3 2 ) for t = Tb + ; ; T; where e = [ P X tx t] [ P X ty t ] ; = + [ P X txt] [ P X tu t ] ; e 2 = [ P 2 X txt] [ P 2 X ty t ] ; = + [ P 2 X txt] [ P 2 X tu t ] ; e 3 = [ P 3 X txt] [ P 3 X ty t ] ; = 2 + [ P 3 X tx t] [ P 3 X tu t ] : Note that the subscripts, 2, and 3 of the summation symbol denote the regimes t 2 [; ^T b ]; t 2 [ ^T b + ; T b ]; and t 2 [T b + ; T ], respectively. It is straightforward to show that e = o p () and e 3 2 = o p (). In addition, to derive the order of e 2, it is reminded that the number of terms in P 2 is O p(kk 2 ) by Lemma so that e 2 = [ P 2 X tx t] [ P 2 X tu t ] ; = [O p (kk 2 ) O p ()] = o p (): [O p (kk 2 ) o p ()]; 2

16 The sample autocovariances of ev t in the pre- and post- T m periods are er (l) = (Tb l) P ;2 X tu t u t l Xt l +( e ) (Tb l) P X tx t Xt lx t l e +( e 2 ) (Tb l) P 2 X tx t Xt lx t l 2 e +( e ) (Tb l) P X tx t u t l Xt l + (Tb l) P X tu t Xt lx t l e +( e 2 ) (Tb l) P 2 X tx t u t l Xt l + (Tb l) P 2 X tu t Xt lx t l 2 e ; = I + II + III + IV a + IV b + V a + V b: We can show that term II = o p ()O p ()o p () = o p (), IV a and IV b are o p ()O p () = o p (). Terms III, V a, and V b are O p (T ) o p () = o p (T p ). Hence, R e (l)! R (l). Similarly, er 2 (l) = (T T b l) P 3 X tu t u t l X t l +( e 3 2 ) (T T b l) P 3 X tx t X t lx t l e 3 2 +( e 3 2 ) (T T b l) P 3 X tx t u t l X t l + (T T b l) P 3 X tu t X t lx t l ( e 3 2 ); so that given e 3 2 = o p () the terms except for the rst one are o p (). Hence, e R 2 (l) p! R 2 (l). We now consider the covariance matrix. For part i) (the i.i.d. case), j = R j () and the estimators are constructed by ^ j = ^R j () and e j = e R j (). The above results show that ^ j p! j and e j p! j and Proposition 3 of EM gives the nal results. For part ii), the AR() coecient to construct m in f^v t g j and fev t g j is consistently estimated since R j () and R j () are consistently estimated. This means ^ p! and j j < by stationary assumption. Thus, we obtain ^ j p! j and e j p! j at the same rate. Finally, Proposition 3 of EM gives the nal results. We now move on to the proof under the alternative hypothesis. Lemma 2 is a preliminary result to derive Theorem 2. Let = lim T m =T. Lemma 2: Suppose that the LRVs j (j = ; 2) are known. Then, for T m 6= T b, T U T (T m ) p! j (; ) Q j j Q j ; 3

17 with j = if T b < T m and j = 2 if T b > T m, where (; ) = 2 ( ) ; 2(; ) = ( ) 2 ( ) 2 3( ) 3 : Proof of Lemma 2: The test statistic in this case is 2 T T U T (T m ) = T P T m t= T P t s= X s^u s T P t s= X s^u s {z } T m + T T T m A 2 T P T t=t T P t m+ s=t X m+ s^u s 2 T P t s=t X m+ s^u s : {z } B We rst consider the case of Tb < T m. The residuals are 8 u >< s + X s X s^ for s = ; ; Tb ; ^u s = u s + X s 2 X s^ for s = Tb + ; ; T m; >: u s + X s 2 X s^2 for s = T m + ; T; where ^ = = PTm PTm t= X txt PTm t= X ty t ; t= X PT txt b t= X t Xt + P T m t=tb + X txt 2 + P T m t= X tu t ;! p + 2 ; by Assumption v). It is straightforward to show ^ p 2! 2 so that B! p. This is because the residuals are consistent estimates of the true errors in the second regime and T P t s= X s^u s = o p (). Hence, we consider term A. It can be separated into two terms before and after Tb such that A = T P Tb t= T P t s= X s^u s T P t s= X s^u s +T P T m t=t b + For t Tb, the component in A has a limit T P t s= X s^u s T P t s= X s^u s : T P t s= X s^u s = T P t s= X su s + T P t s= X sxs T P t s= X sxs^ ;! p + rq rq + 2 ; = r Q ; 4

18 and for t > T b ; T P t s= X s^u s = T P T b s= X s u s + T P T b s= X s X s T P T b s= X s X s^ so that +T P t s=t b + X su s + T P t s=t b + X sx s 2! p + Q + (r )Q 2 rq + = r Q ; R A! ( p 2 r2 ) 2 2 Q Q dr + R 2 r 2 ( ) 3 = ( ) Q Q = 2 ( ) Q Q : 3 4 Q Q ; T P t s=tb + X sxs^ ; 2 ; 2 Q Q dr; For the case of Tb > T m, we follow the same process to reach the stated result (however, term B dominates now, instead of term A). Hence, we have omitted this part of proof to avoid the repetition. Condition A: Let L T be the maximum autoregressive lag order of v t to be used for bandwidth selection. Then, the following hold: sup sup T P [rt ] t= X tx tx t l Xt l rm l = op (); l2[;::;l T ] r2(;] with a positive denite matrix M l and M M! as kk!. T b Proof of Theorem 2: We consider the case of T m 6= Tb. Without loss of generality, let < T m. Then, 8 X >< t u t X t Xt(^ ) for t = ; ; Tb ; ^v t = X t u t X t Xt(^ 2 ) for t = Tb + ; ; T m; >: X t u t X t Xt(^ 2 2 ) for t = T m + ; ; T; 5

19 where ^ = [ P 2 X tx t] = [ P 2 X tx t] [ P 2 X ty t ] ; [ P X tx t + P 2 X tx t 2 + P 2 X tu t ] ;! p + 2 ; ^ 2 = [ P 3 X txt] [ P 3 X ty t ] ; = 2 + [ P 3 X tx t] [ P 3 X tu t ] p! 2 ; by using Assumption v). Note that the subscripts, 2, and 3 of the summation symbol denote the regimes t 2 [; T b ]; t 2 [T b + ; T m]; and t 2 [T m + ; T ], respectively.the sample autocovariance of ^v t in the pre- T m period is ^R (l) = (T m l) P 2 X tu t u t l X t l +(^ ) (T m l) P X tx t X t lx t l ^ +(^ 2 ) (T m l) P ^ 2 X tx t Xt lx t l 2 +(^ ) (T m l) P X tx t u t l Xt l + (T m l) P X ^ tu t Xt lx t l +(^ 2 ) (T m l) P 2 X tx t u t l X t l + (T m l) P 2 X tu t X t lx t l ^ 2 ; = I + II + III + IV a + IV b + V a + V b: (A.3) Again, I converges in probability to R (l). It is shown that II and III are O p (kk 2 ) since p lim(^ ) = ( ) and p lim(^ 2 ) =. Similarly, IV and V are O p (kk). Hence, ^R (l) = R (l) + O p (kk 2 ). The sample autocovariance in the post-t m period is, ^R 2 (l) = (T T m l) P 3 X tu t u t l Xt l +(^ 2 2 ) (T T m l) P ^2 3 X tx t Xt lx t l 2 +(^ 2 2 ) (T T m l) P 3 X tx t u t l Xt l + (T T m l) P 3 X ^2 tu t Xt p lx t l 2! R2 (l): given ^ p 2! 2. 6

20 We now consider the modied method. We also assume ^T b Tb without loss of generality. Then, 8 X t u t X t Xt( e ) for t = ; ; ^T b ; >< X t u t X t Xt( e 2 ) for t = ^T b + ; ; Tb ev t = ; X t u t X t Xt( e 2 2 ) for t = Tb + ; ; T m; >: X t u t X t Xt( e 3 2 ) for t = T m + ; ; T; where e = [ P X txt] [ P X ty t ] ; = + [ P X txt] [ P X p tu t ]! ; e 2 = [ P 23 X txt] [ P 23 X ty t ] ; 2 3 = 6 4 P 2 X tx t {z } =O p(kk 2 ) e 3 = [ P 3 X tx t] = 2 + [ P 3 X tx t] + P 3 X txt 7 5 [ P 3 X ty t ] ; 2 [ P 3 X tu t ] 6 4 P 2 X tx t {z } + P 3 X tx t 2 + P 23 X 7 tu t 5! p 2 ; =O p(kk 2 ) p! 2 : Note that the subscripts, 2, 3, and 4 of the summation symbol denote the regimes t 2 [; ^T b ]; t 2 [ ^T b + ; T b ]; t 2 [T b + ; T m], and t 2 [T m + ; T ], respectively ( 4 appears later). The sample autocovariance of ev t in the pre- T m period is 3 7

21 er (l) = (T m l) P 23 X tu t u t l Xt l +( e ) (Tm l) P {z } X tx t Xt e lx t l {z } =o p() =O p() +( e 2 ) (Tm l) P {z } 2 X tx t Xt e2 lx t l {z } =O p(kk) =O p(t kk 2 ) +( e 2 2 ) (Tm l) P {z } 3 X tx t Xt e2 lx t l {z } 2 =o p() =O p() +( e + (T m ) (T m l) P X tx t u t l Xt l l) P X tu t Xt lx t l e +( e 2 + (T m ) (T m l) P 2 X tx t u t l Xt l l) P 2 X tu t Xt lx t l 2 e +( e 2 2 ) (T m l) P 3 X tx t u t l X t l + (T m l) P 3 X tu t X t lx t l e 2 2 ; = I + II + III + IV + V a + V b + V Ia + V Ib + V IIa + V IIb: Term I converges in probability to R (l) and II; IV, V a, V b, V IIa, and V IIb are o p (). Term III is O p (T ). Terms V Ia and V Ib are O p (T kk ). Therefore, e R (l) p! R (l). Further, er 2 (l) = (T T m l) P 4 X tu t u t l X t l +( e 3 2 ) (T T m l) P 4 X tx t X t lx t l e 3 2 +( e 3 2 ) (T T m l) P 4 X tx t u t l X t l + (T T m l) P 4 X tu t X t lx t l ( e 3 2 ) p! R2 (l); since e 3 p! 2. We now consider the LRV. For both methods, the LRV estimate in the rst regime dominates that in the second, with respect to the break magnitude. (This is without loss of generality, because the LRV estimate in the second regime dominates in turn if we assume T m < T b.) For part a), the estimators in the rst regime are ^ = ^R () = R () + O p (kk 2 ) and e = e R () = R () + o p (). Hence, we can use Lemma 2 by replacing the denominator to obtain the results of U T and V T, respectively. We move on to part b). For V T, the AR() coecients in fev t g j are consistently estimated, because R j () and R j () are consistently estimated as shown above. Hence, Andrews' (99) 8

22 method is appropriately applied to obtain e p j! j. We use Lemma 2 by replacing the denominator and obtain the result. For U T, we invoke the HAC covariance estimator (A.), where ^m is chosen by Andrews' (99) AR() approximation so that ^m _ (T ) = where = 4^ 2 =( ^) 4 or = 4^ 2 =( ^ 2 ) 2 with ^ = ^R ()= ^R (). Because the terms that dominate in ^R () and ^R () are II and III, respectively, in (A.3), ^! p where j j < and ^m = O p (T = ) if Condition A is not satised. We combine the fact P T l= (l; m) = O(m) and obtain the following: ^ = + O p (kk 2 ) O( ^m); = + O p (kk 2 T = ): Finally, replacing 2 in Lemma 2 by this quantity gives the result for U T. We also prove the statement in Remark. If Condition A is satised, then ^ = ^R () ^R () as kk!. In this case, the bandwidth inates as the break magnitude increases. If we use the Quadratic Spectral kernel, then ^m = O p (kk 4= T = ). Hence, ^ = + O p (kk 2 ) O( ^m); = + O p (kk 2+4= T = ): Replacing 2 in Lemma 2 by this quantity conrms the statement in Remark. p! 9

23 References Andrews, D.W.K. (99). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, Andrews, D.W.K., and Monahan, J. (992). An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica 6, Bai, J. (997). Estimation of a change point in multiple regressions. Review of Economics and Statistics 79, Bai, J., and Perron, P. (998). Estimating and testing linear models with multiple structural changes. Econometrica 66, Chang, S., and Perron, P. (23). A comparison of alternative methods to construct condence intervals for the estimate of a break date in linear regression models. Working Paper Boston University. Crainiceanu, C., and Vogelsang, T. (27). Nonmonotonic power for tests of mean shift in a time series. Journal of Statistical Computation and Simulation 77, De Veirmann, E. (29). What makes the output-ination trade-o change? The absence of accelerating deation in Japan. Journal of Money, Credit and Banking 4, 7-4. Elliott, G., and Muller, U.K. (27). Condence sets for the date of a single break in linear time series regressions. Journal of Econometrics 4, Eo, Y., and Morley, J. (23). Likelihood-based condence sets for the timing of structural breaks. Unpublished Working Paper, University of Sydney. Perron, P. (99). A test for changes in a polynomial trend function for a dynamic time series. Econometric Research Program Research Memorandum No. 363, Princeton University. Perron, P. (26). Dealing with structural breaks, Patterson, K. Ed. Palgrave Handbook of Econometrics, Vol. : Econometric Theory. Palgrave Macmillan pp Perron, P., and Yamamoto, Y. (23). On the usefulness or lack thereof of optimality criteria for structural change tests. forthcoming in Econometric Reviews. Yamamoto, Y., and Tanaka, S. (23). Testing for factor loading structural change under common breaks. Discussion Paper, Graduate School of Economics, Hitotsubashi University,

24 Table : Empirical coverage ratio and average length of condence set : Model coverage ratio average length λ=.5 U.eq U.neq V.eq V.neq Bai.het Bai.hneq λ=.35 U.eq U.neq V.eq V.neq Bai.het Bai.hneq λ=.2 U.eq U.neq V.eq V.neq Bai.het Bai.hneq Table 2: Empirical coverage ratio and average length of condence set : Model 2 coverage ratio average length λ=.5 U.eq U.neq V.eq V.neq Bai.het Bai.hneq λ=.35 U.eq U.neq V.eq V.neq Bai.het Bai.hneq λ=.2 U.eq U.neq V.eq V.neq Bai.het Bai.hneq

25 Table 3: Empirical coverage ratio and average length of condence set : Model 3 coverage ratio average length λ=.5 U.eq U.neq V.eq V.neq Bai.het Bai.hneq λ=.35 U.eq U.neq V.eq V.neq Bai.het Bai.hneq λ=.2 U.eq U.neq V.eq V.neq Bai.het Bai.hneq Table 4: Empirical coverage ratio and average length of condence set : Model 4 coverage ratio average length λ=.5 U.eq U.neq V.eq V.neq Bai.het Bai.hneq λ=.35 U.eq U.neq V.eq V.neq Bai.het Bai.hneq λ=.2 U.eq U.neq V.eq V.neq Bai.het Bai.hneq

26 Table 5: Empirical coverage ratio and average length of condence set : Model 5 coverage ratio average length λ=.5 U.eq U.neq V.eq V.neq Bai.het Bai.hneq λ=.35 U.eq U.neq V.eq V.neq Bai.het Bai.hneq λ=.2 U.eq U.neq V.eq V.neq Bai.het Bai.hneq Table 6: Empirical coverage ratio and average length of condence set : Model 6 coverage ratio average length λ=.5 U.eq U.neq V.eq V.neq Bai.het Bai.hneq λ=.35 U.eq U.neq V.eq V.neq Bai.het Bai.hneq λ=.2 U.eq U.neq V.eq V.neq Bai.het Bai.hneq

27 Figure Japanese ination rate (core CPI) Figure 2 Condence set of a level shift in Japanese ination rate U V Bai Note: The 9% confidence sets are in black.the 95% confidence sets are in black and gray. 24