Fractional Unit Root Tests Allowing for a Structural Change in Trend

Fractional Unit Root Tests Allowing for a Structural Change in Trend under Both the Null and Alternative Hypotheses Abstract This paper considers testing procedures for the null hypothesis of a unit root process against the alternative of a fractional process, called a fractional unit root test. We extend the Lagrange Multiplier (LM) tests of Robinson (1994) and Tanaka (1999), which are locally best invariant and uniformly most powerful. We generalize the LM tests to allow for a slope change in trend with or without a concurrent level shift under both the null and alternative hypotheses. We show that the limit distribution of the proposed LM tests is standard normal. Finite sample simulation experiments show the tests to have good size and power. As an empirical analysis, we apply the tests to the Consumer Price Indices of the G7 countries. 1 Introduction Non-stationarity in economic time series is a pervasive feature. In order to carry proper inference, it is important to …nd the exact features that lead to this non-stationarity. A unit root process is a well known example of non-stationary processes and testing for a unit root against stationarity has been a topic of substantial interest from both theoretical and empirical perspectives. Perron (1989), however, showed that the Dickey and Fuller (1979) type unit root test is biased in favor of a non-rejection of the unit root null hypothesis when the process is trend stationary with a structural change in slope. Perron (1989, 1990) proposed testing procedures in which a structural break is allowed under both the null and alternative hypotheses. Later, Christiano (1992) and Zivot and Andrews (1992) criticized the assumption that the date of the structural break is known a priori. In succeeding research, Zivot and Andrews (1992), Perron (1997), and Vogelsang and Perron (1998) treated the break date as unknown and proposed testing procedures for a unit root. In much work, especially that of Zivot and Andrews (1992), it was common to allow for a structural break only under the alternative hypothesis, not under the null hypothesis of a unit root. This is very restrictive and can lead to misleading results. Recent advances in testing for and estimating a structural break in a trend function have made possible the development of unit root tests that allow for a change in trend under both the null and alternative hypotheses. Perron and Zhu (2005) established the consistency, rate of convergence, and limiting distribution of the parameter estimates when there is a break in a trend function with or without a concurrent level shift. Perron and Yabu (2009) suggested a testing procedure for structural changes in the trend function of a time series without any prior knowledge of whether the noise component is stationary or has an autoregressive unit root. Building on this work, Kim and Perron (2009) proposed unit root testing procedures which allow for a structural change under both the null and alternative hypotheses. See also Carrion-i-Silvestre, Kim, and Perron (2009) for an extension to the case with multiple changes. Fractional processes with the order of integration d > 0:5 are non-stationary as well. Standard unit root tests often reject the null hypothesis when the true process is fractionally integrated with d 2 (0:5; 1). This can lead to the misleading conclusion that the process of interest is stationary. This motivated researchers to introduce unit root tests which are powerful against the alternative hypothesis of a fractional process. Robinson (1991, 1994) proposed a Lagrange Multiplier (LM) test of a unit root process against a fractional process, i.e., I(1) vs. I(d) with d 2 [0; 1), in the frequency domain. Tanaka (1999) suggested a LM test in the time domain and showed that it is locally best invariant and uniformly most powerful. Dolado, Gonzalo, and Mayoral (2002) introduced a Wald type unit root test 1 against the alternative of fractional integration. This test is based on the Dickey and Fuller (1979) type test using an auxiliary regression with a consistent estimate of the integration order. Lobato and Velasco (2007) established a Wald type test which is more e¢ cient and is asymptotically equivalent to the LM test. Recently, Cho, Amsler, and Schmidt (2015) suggested combining the test of Kwiatkowski, Phillips, Schmidt, and Shin (1992) and a unit root test to test the null of integer integration, i.e., I(0) or I(1) against the alternative of fractional integration, i.e., I(d); d 2 (0; 1). In this line of work, the process of interest has been limited to either a random walk or a purely fractional process. Lobato and Velasco (2007) considered short-run dynamics in the process. Dolado, Gonzalo, and Mayoral (2008) extended the work of Dolado et al. (2002) and Lobato and Velasco (2007) to incorporate some deterministic components, for instance, a constant and a linear trend function. Our main contribution is to extend the LM test for a fractional unit root to allow for a structural change in a trend function under both the null and alternative hypotheses. This extension has some advantages: (i) it imposes a symmetric treatment of the nature of the deterministic trend under both the null and alternative hypotheses, (ii) it is not required to distinguish long memory from structural change; (iii) the power of fractional unit root tests can be substantially improved when a break is actually present. We consider linear trend models in which a structural change in slope occurs with or without a concurrent level shift. The rest of this paper is organized as follows. In Section 2, we …rst introduce fractional processes and the Lagrange Multiplier test of Tanaka (1999) along with preliminary results to be used subsequently. In Section 3, the LM tests are generalized to allow for a structural break in trend under both the null and alternative hypotheses. Extensions to processes with short-run dynamics are discussed in Section 4. The results of simulation experiments about the size and power of the tests are presented in Section 5. As an empirical application, we test for a fractional unit root in the Consumer Price Indices (CPI) of the G7 countries in Section 6. Concluding remarks are in Section 7. All mathematical proofs are relegated to an Appendix. 2 Lagrange Multiplier Test For an integer d = 1; 2; : : :, the operator d = (1 L)d denotes the di¤erencing operator with the usual lag operator L, i.e., LXt Xt 1 , Xt = Xt Xt 1 , Xt 2Xt 1 + Xt 2 , and so on. For a non-integer real number d > d = (1 L)d is de…ned by means of the binomial expansion d 1 1 X X d k = ( L) = k k=0 k=0 2 k (d)L k ; 2 Xt = (1 2L + L2 )Xt = 1, the di¤erence operator where d k d! d (d 1) = k!(d k)! k (k 1) k (d) with d k ( 1)k = k Y s s=1 ( ) the gamma function, so that k (d) (x + 1), x = 0; 1; : : :, and for k = 1; 2; (x 1) (x 1) = = (x 1) 1 s (x = d = k d 1 k (d (k d) ; (k + 1) ( d) and , 0 < x < 1, k) (x k + 1) ; 2 1 0 (d) = 1. Recall that x! k) is de…ned as (x (x) = k). To de…ne a fractional process, we use the notation of Robinson (2005). Let f t ; t = 0; 1; : : :g be a short-memory zero-mean covariance stationary process, with spectral density that is bounded and bounded away from zero. For d 2 ( 0:5; 0:5), t = d t; is covariance stationary and invertible. The truncated version of # t = (1) t = 0; 1; : : : ; t 1t 1 ; t is de…ned as where 1A is the indicator function for the event A. For an integer m ut = (2) t = 0; 1; : : : ; m # t ; 0, (3) t = 0; 1; : : : is called a type I I(m + d) process. Let D[0; 1] be the space of functions on [0; 1] which are p right continuous and have left limits, equipped with the Skorohod topology. Let ! denote d convergence in probability and ! convergence in distribution. Denote by [a] the integer part of a 2 R. The order of integration is d0 = d + m with d 2 ( 0:5; 0:5) and m 2 f0; 1g.1 Remark 1 Marinucci and Robinson (1999) de…ned type I and type II fractional Brownian motions with d 2 ( 0:5; 0:5) on D[0; 1], respectively as follows: Z 0 Z t 1 d d [(t s) ( s) ]dB(s) + (t BI (t) = (d + 1) 1 0 Z t 1 BII (t) = (t s)dB(s) ; (d + 1) 0 s)dB(s) ; where B( ) denotes the standard Brownian motion. Furthermore, Robinson (2005) and Davidson and Hashimzade (2009) pointed out that asymptotic results vary depending on the de…nition of fractional Brownian motions considered, which requires one to design simulation experiments in accordance with the particular type used. 1 The restriction that d0 6= 0:5 is standard in the long memory literature. Tanaka (1999) showed that the case with d0 = 0:5 needs to be treated separately from the case with d0 6= 0:5. 3 Now we consider a fractional unit root test. Under the null hypothesis, fut g is a unit root process, that is, H0 : d0 = 1, i.e., d = 0 and m = 1. The alternative hypothesis can be either one sided (H1 : d0 > 1 or H1 : d0 < 1) or two sided (H1 : d0 6= 1). Robinson (1994) and Tanaka (1999) considered the Lagrange Multiplier test in the frequency and time domain, respectively. It is well known that the LM test is locally best invariant. Further, Tanaka (1999) showed that the LM test is locally uniformly most powerful invariant because it achieves the power envelope of all the invariant tests against local alternatives. The test statistic suggested in Tanaka (1999) is LM = where uals k P = ( Tj=k+1 uj k uj )= PT j=1 ( p T r T 1 6 X1 2 k k=1 k; uj )2 is the kth order autocorrelation of the resid- ut . Local alternatives to the null hypothesis are often considered in the literature, with the integration order de…ned as d0 = 1 + T 1=2 with …xed. We state the limiting distribution of the LM test under local alternatives as it will be relevant for subsequent derivations. Lemma 1 (Theorem 3.1 in Tanaka, 1999) Under the assumption that ut is generated p d 2 =6; 1). by (3) with d0 = 1 + T 1=2 and …xed, it holds that, as T ! 1, LM ! N ( 3 Deterministic Components Allowing for a Structural Change In this section, we extend the LM test for a fractional unit root to allow for a structural change in trend with or without a concurrent level shift. We consider the time series of interest yt as consisting of a deterministic component (ft ) and fractionally integrated errors. The data-generating process (DGP) is speci…ed as yt = ft + ut : For ut , we impose E(ut ) = 0 and the following assumption. Assumption 1 ut is a type I I(m + d) process which is de…ned in (1)-(3). Moreover, the short-memory zero-mean covariance stationary process is assumed to be independent and t identically distributed (i:i:d:) with zero mean and …nite variance. The i:i:d: assumption on the short-memory process t will be relaxed later to allow for short-run dynamics. The unit root null hypothesis corresponds to the case with m = 1 and d = 0, which implies that ut is a weighted sum of 4 t. 3.1 Change in Mean We …rst consider the case where yt experiences a level shift at an unknown time Tb . The DGP is speci…ed as yt = 1 + b Ct + ut ; b 6= 0; (4) where Ct is a dummy variable for a level shift de…ned by: 8 < 0 if t Tb ; Ct = : 1 if t > Tb ; where Tb = [T b] is the true break date with the corresponding true break fraction b 2 (0; 1). Theorem 1 (Change in Mean). Under Assumption 1, suppose that the process fyt g is generated under the null hypothesis of (4). Consider the Lagrange Multiplier test LMM de…ned by: LMM = p T r 6 2 PT t=1 ( log yt ) yt : PT 2 t=1 ( yt ) d Under the null hypothesis H0 : d0 = 1, it holds that, as T ! 1, LMM ! N (0; 1). Theorem 1 implies that Tanaka’s (1999) LM test is robust to the presence of a level shift. In the following subsection, we consider the LM test in the context of trending series. 3.2 Slope and Intercept Change in Trending Series We now introduce a deterministic time trend in the models. We follow the notation in Kim and Perron (2009) (henceforth KP) from which we will use some relevant results. The DGPs are speci…ed as 1. Model A0: (Deterministic time trend without a structural change) yt = 1 + 1t + ut ; (5) + 1t + ut ; (6) b Bt + ut ; (7) 2. Model A1: (Level shift) yt = 1 + b Ct 1 + 1t 3. Model A2: (Joint broken trend) yt = 5 + where Bt is a dummy variable for a slope change in trend given by 8 < 0 if t Tb ; Bt = : t T if t > T ; b b 4. Model A3: (Locally disjoint broken trend) yt = 1 + b Ct + 1t + b Bt + ut : Following KP, we can rewrite Models A1-A3 as follows: 0 yt = z(Tb )0t + ut = zt;1 where zt;1 = (1; t)0 , z(Tb )t;2 = ( 1 ; 1 )0 , 8 > > C > < t ; = Bt > > > : (C ; B )0 t t 1 2 = + z(Tb )0t;2 1 8 > > > < 2 + ut ; for Model A1 b b > > > : ( ; b for Model A2 0 b) for Model A3: In matrix notation, the models de…ned previously can be speci…ed as Y = ZTb + U , where = ( 01 ; Y = [y1 ; : : : ; yT ]0 , ZTb = [z(Tb )1 ; : : : ; z(Tb )T ]0 , 0 0 2) , and U = [u1 ; : : : ; uT ]0 . Consider …rst Model A0 where no structural change is allowed. By taking …rst di¤erences, we can rewrite (5) as follows: yt = 1 + ut = The ordinary least squares (OLS) estimate of under both H0 and H1 .2 We de…ne g yt = yt 1 1 + 1 d0 (8) t 1t 1 : PT 1 yt , which is consistent is ^ 1 = T t=1 ^ 1 , the OLS residuals from the regression model (8). Theorem 2 (Linear Trend). Under Assumption 1, suppose that the process fyt g is gener- ated under the null hypothesis of (5). Consider the Lagrange Multiplier test LMT de…ned by: r PT log g yt g yt p t=1 6 LMT = T : 2 2 PT g yt t=1 d Under the null hypothesis H0 : d0 = 1, it holds that, as T ! 1, LMT ! N (0; 1). Under H0 , ^ 1 is a T 1=2 -consistent estimator of the slope coe¢ cient 1 . Hosking (1996) considered a stationary ARFIMA(p; d; q) process fyt g and showed the weak convergence of the sample mean for d 2 ( 0:5; 0:5). It is not di¢ cult to generalize the result to the case where d 2 (0:5; 1), for which ^ 1 is a T 3=2 d -consistent estimator. 2 6 In what follows, the aim is to devise Lagrange Multiplier tests allowing for a slope change in trend with or without a concurrent level shift. The following assumption is essential to that e¤ect. Assumption 2 b 6= 0 and b 2 ( ;1 ) for some 2 (0; 1=2). Assumption 2 ensures that there is a single slope change in trend and that the pre and post break samples are not asymptotically negligible, which is a standard assumption needed to derive useful asymptotic results. Model A1 (level shift only) will be revisited later. The break date can be estimated by using a global least-squares criterion: T^b = arg minT1 2 Y 0 (I PT1 )Y; where PT1 is the matrix that projects on the range space of ZT1 , i.e., PT1 = ZT1 (ZT0 1 ZT1 ) 1 ZT0 1 and = [ T; (1 )T ], 0 < < 1=2. Note that ZT1 is the same as ZTb except replacing Tb with a generic break date T1 . Perron and Zhu (2005) (henceforth, PZ) established the consistency, rate of convergence, and limiting distribution of parameter estimates when the error is an I(1) process. With ZT^b constructed using the estimate T^b , the OLS estimate of is ^ = (Z 0 Z ^ ) 1 Z 0 Y and the resulting sum of squared residuals is, for an estimated break T^b Tb T^b fraction ^ s = T^b =T (the subscript s refers to the fact that we consider a static regression; a dynamic regression with lagged dependent variables will be considered later): S( ^ s ) = T X u^2t = t=1 T X z(T^b )0t ^ yt 2 = Y 0 (I PT^b )Y; t=1 where PT^b is the projection matrix associated with XT^b . The rate of convergence of ^ s for 1=2 Models A2 and A3 is ^ s ) with I(1) errors (see Theorem 3 in PZ). Chang b = Op (T and Perron (2016) derived the consistency and rate of convergence of ^ s when the noise component is a fractional process with the di¤erencing parameter d0 2 ( 0:5; 0:5)[(0:5; 1:5). 1=2+d Speci…cally, for Models A2 and A3, ^ s ) if m = 1 and d 2 ( 0:5; 0:5). With b = Op (T ^ ^ ^ the consistent estimates ( s ; ^ ; ^ ; ; ), we can construct the detrended process fe yt g, and 1 b 1 b the Lagrange Multiplier test statistic LMT;^s is given by LMT;^s = p T r 6 2 PT t=1 ( log yet ) yet : PT et )2 t=1 ( y The convergence rate of the estimate ^ s is not fast enough to guarantee that LMT;^s has the standard normal limit under H0 . KP faced a similar issue in dealing with unit root tests. 7 They introduced a heuristic explanation of the issue involved, which we brie‡y review. Let { ^ = T^b =T denote an estimate of the break fraction such that ^ ) for some b = Op (T { 0. The detrended series fe yt g is given by ^ zY = M ^ z Z(Tb ) + M ^ zU = M ^ z Z(Tb )2 Ye = M 2 ^ z U; +M (9) where Z(Tb ) and Z(Tb )2 are matrices stacking fz(Tb )0t g and fz(Tb )0t;2 g, respectively, and ^ z = I P^z = I Z(T^b )[Z(T^b )0 Z(T^b )] Z(T^b )0 . It is obvious that the idempotent matrix M ^ z Z(Tb )2 2 = 0 only if the true break date is used in M ^ z . In …nite samples, ^ 6= b in M ^ z Z(Tb )2 2 will not be zero. It turns out that a fast rate of convergence general, thereby M ^ z Z(Tb )2 2 on the Lagrange for the estimate of the break date is needed for the e¤ect of M Multiplier test to become negligible asymptotically. The following proposition provides a d su¢ cient condition under which LMT;^ ! N (0; 1) under H0 . Proposition 1 Suppose that the process fyt g is generated under the null hypothesis of Model d A2 or A3, and that Assumptions 1-2 hold. Then, it holds that, as T ! 1, LMT;^ ! N (0; 1) 1=2 if ^ ). b = op (T Proposition 1 implies that the estimate of the break fraction should converge at a rate faster than T 1=2 . As shown above, ^ s does not satisfy this condition. Hence, we need to consider alternative ways to accelerate the rate of convergence of the estimate of the break fraction. KP suggested two possible approaches. The …rst is based on minimizing the sum of squared residuals (SSR) of a dynamic regression model. This method is similar to that in Hatanaka and Yamada (1999). The relevant dynamic regressions are speci…ed as follows: 1 D(Tb )t + + z(Tb )0t + ut ; yt = y t 1 + yt = y t 1 + D(Tb )t + z(Tb )0t + ut ; 2 1t Tb (Model A2) (Models A1 and A3) (10) where D(Tb )t = 1 for t = Tb + 1 and 0 otherwise. We obtain, under the null hypothesis, an estimate of the break fraction ^ d which has a faster rate of convergence such that ^ d = Op (T 1 ) for Models A2 and A3 (see Proposition 1 in KP). Let LMT;^d denote the Lagrange Multiplier test statistic with ^ d replacing ^ s . It is worth noting that, as discussed by Hatanaka and Yamada (1999) and KP, the estimate ^ d has a negative bias in …nite b samples especially for Model A3. As we shall see, this will a¤ect the …nite sample properties of the tests. The second approach is to use a trimmed data set using a window whose length depends on the sample size and which contains the estimated break date. The trimmed series then 8 consists of the original one with the data points in the window excluded. KP showed that the rate of convergence of ^ s can be increased with the trimmed data set. Suppose that a the estimate of the break fraction satis…es ^ ) for some 0 < a < 1 and the b = Op (T trimming window has length 2w(T ) with w(T ) c1 T , c1 > 0 and 1< a< < 0. With this speci…cation, the length of the window is negligible in the limit compared to the sample size T , but is still large enough to include the true break date asymptotically. Following KP’s suggestion, one proceeds as follows: Estimate the break fraction ^ s from the original data set and form a window that ranges from Tl T ( ^ s w(T )) to Th T ( ^ s + w(T )). A trimmed data set is constructed by removing the original data from Tl + 1 to Th and then shifting down the data after the window by D(T ) = yTh yTl . After the trimming and connecting procedures, we now have a new series fyt g, for t = 1; : : : ; T ( T 2w(T )T ), de…ned by: 8 < y t yt = : y if t t+Th Tl D(T ) Tl ; if t > Tl : Test the null hypothesis H0 : d0 = 1 using Tl as the break date, i.e., ^ tr = Tl =T . The Lagrange Multiplier test statistic is then given by r PT p yet ) yet 6 t=1 ( log LMT;^tr = T PT 2 et )2 t=1 ( y where yet is the detrended version of yt using the estimate of the break date Tl (or break fraction ^ tr ). Remark 2 If the window contains either end of the data, then the process fyt g turns out to be Model A0 (no structural break), and the statistic in Theorem 2 should be applied to the trimmed data fyt g. The trimmed process fyt g will satisfy the properties of Model A2 regardless of the speci- …cation of the original data fyt g, which implies that we can use a common limit distribution. The following proposition states the limiting distribution of the Lagrange Multiplier test based on the trimmed data, which is the same as would be obtained if the break date was known in Model A2. 9 Proposition 2 Suppose that the process fyt g is generated under the null hypothesis of Model d A2 or A3, and that Assumptions 1-2 hold. Then, it holds that, as T ! 1, LMT;^tr ! N (0; 1). As shown in KP, under the null hypothesis of a unit root, the estimate of the break fraction ^ tr = Tl =T converges in probability to the true break fraction at some rate greater than T . Hence, the su¢ cient condition in Proposition 1 is satis…ed, so that the proof of Proposition 2 is trivial and omitted. b is, In concluding this section, we consider the case where there is a change in mean, that is, 6= 0 as in Models A1 and A3. In Model A1, we assume that there is a level shift only, that b 6= 0 and b = 0. Under the null hypothesis, a stochastic trend generated by the I(1) error process tends to dominate a level shift. Hence, we cannot estimate the break fraction consistently because the magnitude of the level shift is asymptotically negligible. In …nite samples, we can ignore the level shift if the magnitude of the break is small. Then Model A1 b can be treated as Model A0, and we can follow the testing procedure pertaining to Theorem 2. However, a loss of power is inevitable if a large change in mean is ignored. On the other hand, the level shift can be speci…ed as an increasing function of the sample size, i.e., b = c2 T 1=2+ for some c2 > 0 and > 0. As addressed in Harvey, Leybourne, and Newbold (2001), PZ, and KP, this speci…cation provides better approximations of the properties of the tests in …nite samples when the level shifts are not very small. The models with b = c2 T 1=2+ are labeled as Models A1b and A3b, respectively. Proposition 3 Suppose that the process fyt g is generated under the null hypothesis of Model A1b or A3b. Then, LMT;^ diverges as T ! 1. Although the rate of convergence of the estimate of the break fraction is faster than in the case of a change in slope (see Proposition 7 in KP), Proposition 3 states that the LM tests cannot obtain the standard normal limiting distribution. Hence, the LM test LMT;^ , using the critical values from the standard normal distribution, su¤ers from some liberal size distortions even when j b j is large.3 3.3 Using a Pre-test for a Break in Slope The results of Theorem 2 and Proposition 2 show that the limit distribution of the test is the same whether there is a break in slope introduced as a regressor or not, even when the DGP speci…es that no break is present. Hence, unlike the case of testing for a unit root 3 Simulation results related to this issue are available upon request. 10 as in KP, theoretically there is no need to carry a pre-test to improve the power of the test. However, Chang and Perron (2016) considered Models A2 and A3 with fractionally integrated errors and showed that the so-called spurious break issue occurs with the order of fractional integration d0 2 (0; 0:5) [ (0:5; 1:5). This extended the results on Nunes, Kuan, and Newbold (1995) who considered the unit root case. This means that under both the null and alternative hypotheses, if a break in slope is not present and one is allowed in the regression, the …tted model will with large probability suggest the presence of a break. This could have an e¤ect on both the size and power of the test. On the one hand, the slope change regressor may induce added liberal size distortions in …nite samples because of the over…t. On the other hand, since when no break exists in the DGP it is a super‡uous regressor, power maybe be reduced. Hence, it may be the case that in …nite samples it is bene…cial to use a pre-test for a change in slope and try to choose between models (5) and (7). Since a test for a change in mean will be inconsistent, there is no point in trying to distinguish between models (5) and (6) or between model (4) and the corresponding one without the change in mean. Iacone, Leybourne, and Taylor (2013) suggested a sup-Wald type test (SW) for Model A2. In particular, it is robust to any order of fractional integration d0 located in an interval [0; 1:5) excluding the boundary case 0.5. More precisely, given their recommended choice for the bandwidth when constructing the local Whittle estimate of d0 , their test is consistent for values of d0 in the interval [0; 1:32], though we believe the proof can be modi…ed to allow the interval [0; 1:5). It follows the generalized least squares approach to construct the test statistic for a structural change in trend by taking d0 -di¤erences from the data. To make the test feasible, the fully extended local whittle (FELW) estimator d^F ELW of Abadir, Distaso, and Giratis (2007) is considered. While the FELW estimator is constructed under the null hypothesis of no structural change, Iacone et al. (2013) showed that it also satis…es the necessary condition for consistency even with a local break in trend. Since the true break date is unknown a priori, the …nal statistic SW uses the sup functional of Andrews (1993) across all admissible break dates. This test is asymptotically size controlled for all d0 ’s in the prescribed range. Using this pre-test, we can then de…ne the alternative estimate of the break fraction e = ^ 1SW> where is the critical value for the SW test with a nominal size p%. Given that SW is a consistent test, plimT !1 e = b if ^ is a consistent estimate of b . If there is no break in the DGP, we can expect that p% of the estimates e’s are nonzero. In order to obtain a consistent estimate of b under the null of no structural break, we assume that the critical value is a function of the sample size T . Since SW = Op (T % ); % > 0 with a local break, let = cT % for 0 < < %. This speci…cation introduced in KP is useful 11 because it does not have any e¤ect on the consistency of the test SW and does guarantee that plimT !1 e = 0 when no break is present. Hence, based on the consistency of e, it is recommended to use LMT if e = 0 and LM ^ if e 6= 0. The LM test statistics with T; the pre-test are denoted by LMpT;^ , LMpT;^ , and LMpT;^ . Whether using a pre-test is s tr d bene…cial will be assessed later via simulation experiments about the size and power of the tests. 4 Short-Run Dynamics We now relax Assumption 1 to introduce short-run dynamics in the noise component. A zero-mean short-memory covariance stationary process t can be represented as a one-sided P P 2 (1) t = 0; 1; : : : ; where 0 = 1, 1 moving average: t = 1 j=0 j < 1, j=0 j t j ; P1 j=0 j , and f t , t = 0; 1; : : :g are i.i.d. random variables with mean zero. A special case of interest is an autoregressive moving average (ARMA(p; q)) process given by (L) t = (L) t . In order to implement the Lagrange Multiplier test, we …rst estimate the parameters = ( 1 ; : : : ; p ; 1 ; : : : ; q ) consistently. Then under the null hypothesis, we can construct ^t = ^ (L)^(L) 1 (1 L)d0 u^t where u^t is the OLS residuals from the model considered, whereas ^ (L) and ^(L) are estimated from (L)(1 L)d0 u^t = (L) t , using d0 = 1. With short-run dynamics in the noise component, we consider the following test statistic, ^ = LM T 1 p X 1 T ^k ; k k=1 where ^k is the kth order autocorrelation of residuals ^1 ; : : : ; ^T . Tanaka (1999) derived an important result related to this statistic when no break is present. Lemma 2 (Tanaka, 1999, Theorem 3.3) Under local alternatives, that is, d0 = 1 + d ^ ! T 1=2 with …xed, it holds that, as T ! 1, LM N ( ! 2 ; ! 2 ), where 2 !2 = 6 ( 1; : : : ; i = 1 X 1 j=1 1 p; 1; : : : ; q ) j gj i ; i = ( 1; : : : ; 1 X 1 j=i j j 0 p; 1; : : : ; q ) ; hj i ; with gj and hj the coe¢ cients of L in the expansion of 1= (L) and 1= (L), respectively, and the Fisher information matrix for Remark 3 Note that ! 2 < 2 and . =6, hence comparing Lemmas 1 and 2, the LM test has lower local asymptotic power in the presence of short-run dynamics of any kind. As will be shown via simulations, the loss in power can be substantial. It remains, nevertheless, inevitable. 12 d ^ !! With the maximum likelihood estimate ! ^ , we show that LM=^ N (0; 1) as T ! 1 under the null hypothesis. In particular, when p = 1 or q = 1, it is easy to compute ! ^ . Since vj = &vj 1 + j in both cases, gj = & j , 1 = 1 X 1 j=1 j &j 1 1 log(1 & = &); 1 and =1 & 2: Hence, we have 2 !2 = &2 1 &2 (log(1 &))2 ; 6 and ! ^ can be computed using ^& . All these results remain valid for all trending models with a break considered. The relevant correction needed is a simple scaling by ! ^ so that the test becomes LMT;^ LMT;^ =^ !. Proposition 4 Suppose that the process fyt g is generated under the null hypothesis of Model A2 or A3, and that Assumptions 1-2 hold with t being an ARMA(p,q) process. Then, it d 1=2 holds that, as T ! 1, LM ^ ! N (0; 1) if ^ ) and ! ^ ! = op (1). b = op (T T; The su¢ cient conditions in Proposition 4 follow from Lemma 2 and Proposition 1, hence the proof is omitted. The …nite sample performance of LM ^ with ^ 2 f ^ s ; ^ d ; ^ tr g allowing T; for a structural break under both the null and alternative hypotheses will be examined in the next section. 5 Simulation Experiments In this section, we present results from simulation experiments to illustrate the various theoretical results. Throughout the simulations, the true break fraction is set to The DGP is speci…ed as yt = ft + ut where 8 > > 1 + 1t > > > > < C b t ft = > > > b Bt > > > : C + B b and ut = 1 # t = 1 t 1t 1 , t b t b = 0:5. for Model A0; for Model A1; for Model A2; for Model A3; t = 0; 1; 2; : : :, where t is a short-memory zero-mean covariance stationary process that will be speci…ed below. We set some parameters as follows: 1 = 1:72, 1 = 0:03, b = 1, and b = 1. The con…gurations are the same as those in PZ, chosen to obtain distributions that easily reveal the main features of interest. In all cases, the results are obtained via 10,000 replications. Also, 5% nominal size tests are considered. 13 First, to illustrate the e¤ect of a structural break on the power of the fractional unit root test, we consider two di¤erent models when a structural change in slope is allowed in the DGP; (i) Model A0 (which ignores a relevant slope change) and (ii) Model A2 (which is well speci…ed). The results are provided in Table 1. It is clear that the power of LMT is much lower than that of LMT;^d , which supports the fact that a structural break in the DGP should be allowed when testing for a fractional unit root. Tables 2-5 present the rejection probabilities of the tests LMT and LMT;^ at the 5% signi…cance level when t i:i:d: N (0; 1). In Table 2, no structural change is allowed in the DGP (Model A0), i.e., yt = 1 + 1t + ut . The size of LMT is well controlled, which is 0.05 and 0.06 with the sample sizes T = 150; 500, respectively. Table 3 reports the results for Model A1. The break fraction is not estimated consistently because the level shift is negligible compared to the stochastic trend induced by the I(1) errors. Hence, LMT;^s and LMT;^d su¤er from severe size distortion while LMT maintains size close to the nominal level 5%. Table 4 presents the results pertaining to Model A2. We also consider the test based on trimmed data, LMT;^tr . The test LMT;^d is size controlled while the others show minor size distortion. However, the power of LMT;^d is always lower than that of the other two tests. Table 5 presents the results pertaining to Model A3. Here, we set b = 0 to consider the e¤ect of an irrelevant level shift. Notice that LMT;^s exhibits liberal size distortion and LMT;^d also shows considerable size distortion. As Chang and Perron (2016) noted, the estimate of the break date shows a pattern of bi-modality when an irrelevant level shift is introduced. This phenomenon is referred to the “contamination”e¤ect because the irrelevant level shift can make the estimate of the true break date less precise. By construction, the contamination e¤ect is marginal on LMT;^tr , whose exact size is 6.7% when T = 500. In Tables 6-8, we provide simulation results when the errors have short-run dynamics, i.e., (1 L) t = t, t i:i:d: N (0; 1). We set the value of the autoregressive (AR) parameter at 2 f 0:5; 0; 0:3; 0:6; 0:8g. When = 0, we can compare the loss of power caused by allowing for dynamics when none is present. The other parameters remain unchanged. Table 6 reports the size and power of the Lagrange Multiplier tests pertaining to Model A1, LMT . It is well size controlled with less persistent AR parameters 2 f0; 0:3g, but it is very conservative with a higher AR coe¢ cient 2 f0:6; 0:8g while it shows liberal size distortions with = 0:5. We …nd some interesting features in terms of power. First, power is higher when the AR parameter Second, as is negative (in part due to the liberal size distortions). becomes positive and large, power shrinks considerably. In particular, the loss of power is substantial when the AR parameter has increased from 0.6 to 0.8. This implies that a su¢ ciently large time span is needed to distinguish fractional integration from weak 14 dependence. Comparing Table 3 with Table 6, for the = 0 case, it is obvious that power is substantially lower when an irrelevant AR parameter is introduced. This result suggests that selecting the number of lags in the noise component is crucial to obtain good power. Lastly, with a persistent AR parameter = 0:8, the LM tests have non-monotonic power, that is, power does not increase when the order of integration d0 moves away from the null of a unit root. As also discussed in Lobato and Velasco (2007), it is di¢ cult to distinguish fractional integration from a highly persistent stationary short-memory process. Table 7 reports the results pertaining to Model A2. They show similar patterns as for Model A1. It is noticeable that LMT;^d performs well in terms of size across all cases while its power is always lower than that of the other tests. Table 8 presents the results for Model A3. LMT;^d has size distortion even with negative and less persistent AR parameters. This happens because when using the dynamic regression to estimate the break fraction, the estimate ^ d is negatively biased for Model A3. Among the three LM statistics, LMT;^tr based on the trimmed data performs well in …nite samples. The size of LMT;^tr is well controlled across various values of while LMT;^s and LMT;^d show liberal size distortions. Moreover, the power loss of LMT;^tr is minor relative to the other tests. Hence, LMT;^tr and LMT;^tr are the recommended tests in practice. 5.1 The Size and Power When a Pre-test is Used In Figures 1-4, we present the size and power of the LM tests as the slope change parameter ( b ) changes in Models A2 and A3 with and without the pre-test being used. As a pre-test, we use the SW test of Iacone et al. (2013) at the nominal 5% level. We only consider the version of the LM statistics based on the trimmed estimate of the break fraction, denoted LMT;^tr and LMpT;^ when no short-run dynamics is allowed and by LMT;^tr and LMT;p^ tr tr when an AR(1) structure is allowed. To assess the extent of the di¤erences in the size distortion and power, we also report the infeasible LM test based on the true value of break fraction, denoted LMT; b and LMT; b . The results are presented in Figure 1 for Model A2 (no short-run dynamics), Figure 2 for Model A2, and in Figures 3 and 4 for Model A2 and A3 with short-run dynamics of the form (1 0:3L) t = t, t i:i:d: N (0; 1). For LMpT;^ tr and LMT;p^ , the trimming window contains 6 observations (the simulation results are not tr sensitive to the length of the window). The magnitude of break in the slope of the trend b varies from -4 to 4 in increments of 0.2. We set b = 0 for Model A3. The sample sizes are T = 150, 300 and the number of replications is 10,000 for each value of the parameters. One sided tests against the alternative hypothesis H1 : d0 < 1 are constructed at the nominal 5% level. For power, d0 is set to 0.8. 15 The results for Model A2, presented in Figure 1, show …rst that the version without the pre-test exhibits some liberal size distortions when b is near 0, which reduce when T increase, though remain noticeable even with T = 300. On the other hand, the version with the pre-test exhibits conservative size-distortions when b is near but not equal to 0, which again reduce but remain noticeable when T = 300. The most drastic di¤erences occur when considering the power of the tests. The power of the version without the pre-test is slightly below but near to the power of the version with the true break fraction when T = 150 for all values of b. When T = 300, the power functions are nearly the same. Things are very di¤erent when the version with the pre-test is used. When b is near but not equal to zero the power reduces drastically creating pronounced power valleys. This reduction in power alleviates somewhat when T = 300 but remains important. This is due to the fact that for low values of b the SW test of Iacone et al. (2013) is not powerful enough so that a change in slope regressor is not included. Yet, the magnitude of the change in slope is big enough to induce a considerable loss in power. This is akin to the problem faced by the Kim and Perron (2009) test in the context of testing for a unit root. The results for Model A3, presented in Figure 2, show a similar picture. This is also the case when considering the tests LMT;^tr and LMT;p^ tr with serial correlation in the DGP (Figures 3 and 4), though here the size distortions of both tests are somewhat higher, liberal for LMT;^tr and conservative for LMT;p^ . The power losses of LMT;p^ is severe, especially tr tr when T = 150. Based on the simulation results, it is recommended to use the LMT;^tr or LMT;^tr tests without the pre-test. In our view, the reduction in power when using a pre-test considerably outweigh the di¤erences in size distortions. The SW test of Iacone et al. (2013) is nevertheless still useful to assess the presence of large breaks. 6 An Empirical Application We analyze the aggregate price indices of the G7 countries. Monthly seasonally adjusted CPI series were obtained from the OECD Main Economic Indicators. All series are analyzed with a logarithm transformation and are plotted in Figure 5 where the vertical line is the break date estimated by minimizing the sum of squared residuals from Model A2. The results are presented in Table 9. We only consider Model A2 and use the test for a slope change of Iacone et al. (2013). Based on the simulation results, we recommend using the LM tests with ^ tr . We present results with and without short-run dynamics. When dynamics is allowed, an AR(1) speci…cation is used. The time span is from January 1969 to December 2007 (T = 468). First, the augmented Dickey-Fuller type test (ADF) cannot 16 reject the null of a unit root against the alternative of trend stationarity for all G7 countries. On the other hand, the SW test of Iacone et al. (2013) detects a change in the slope of the trend. Allowing for a structural change in trend, the fractional unit root tests LMT;^tr and LMT;^tr lead to a rejection of the unit root in favor of fractional integration. Speci…cally, the test results state that the order of fractional integration is greater than unity for all G7 countries. This result is compatible with that in Gil-Alana (2008) where he estimated the order of fractional integration for the U.S. CPI and found that the con…dence intervals were located above unity. Hassler and Wolters (1995) considered the in‡ation rates for various countries and found that the order of fractional integration was located in an interval (0; 0:5), that is, the in‡ation rate is a long-memory process. 7 Conclusion We established testing procedures for a fractional unit root allowing for a structural change under both the null and alternative hypotheses. Following Robinson’s (1994), Tanaka (1999) derived a Lagrange Multiplier test in the time domain, and Dolado et al. (2002, 2008) and Lobato and Velasco (2007) considered Wald-type tests for a unit root null hypothesis against fractional integration. Although Dolado et al. (2008) introduced deterministic components, the case with a structural break in trend has not been considered in the literature. In contrast to the large amount of work related to testing the null hypothesis of long-memory against the alternative of stationarity with level shifts, and vice versa, work related to a fractional unit root test allowing for a structural break in trend is more scarce. To the best of our knowledge, this paper is the …rst that addresses testing for a fractional process allowing a structural break under both the null and alternative hypotheses. Fractional unit root tests allowing for a structural break under both the null and alternative hypotheses have some desirable features: (i) given that economic variables are often subject to structural changes, our approach imposes a symmetric treatment of the change under both the null and alternative hypotheses; (ii) it is not required to distinguish long memory from structural change; (iii) the power of fractional unit root tests can be substantially improved when a break is actually present. Under some conditions, the proposed LM test statistics have the standard normal limit under the null hypothesis. Simulation experiments con…rmed that the tests have good size and power. Hence, we believe that our procedures o¤er useful complements to existing tests and should be used in practical applications. 17 Appendix Proof of Theorem 1. The DGP is speci…ed by (4), that is, for t = 1; : : : ; T , yt = 1 + b Ct + ut ; b 6= 0: Under H0 , take …rst di¤erences of yt and de…ne D(Tb )t = 1 if t = Tb + 1 and 0 otherwise. Then, T 1 T X 2 ( yt ) = T 1 t=1 =T =T 1 T X ( b D(Tb )t + t=1 T X t=1 T X 1 2 t +2 T X 2 t) b D(Tb )t t t=1 2 t +2 b Tb +1 2 b + t=1 =T 1 T X 2 t + T X 2 2 b D(Tb )t t=1 ! ! + op (1); t=1 and T 1 T X j=2 ( log j 1 T X X 1 1 yj ) yj = T ( b D(Tb )j k j=2 k=1 j 1 T X X 1 1 =T k j=2 k=1 =T 1 j 1 T X X 1 j=2 k=1 k k b D(Tb )j k 2 b D(Tb )j k j 1 T X X 1 1 +T k j=2 k=1 j k A1 + A2 + A 3 + A4 : ! + + ! ! j k) j 1 X 1 k=1 k ( b D(Tb )j + j k ! D(Tb )j + T j) ( b D(Tb )j + 1 j 1 T X X 1 j=2 k=1 j 1 T X X 1 1 b D(Tb )j + T k j=2 k=1 k j k j) b D(Tb )j k ! j It is easy to show that A1 = 0. For A2 , we have 0 12 0 10 1 T (Tb +1) T (Tb +1) T (Tb +1) X X X 1 1 1 A@ 2 1 @ 2 A A (A2 )2 = 2b 2 @ T1 +1+k T1 +1+k b 2 2 T k T k k=1 k=1 k=1 0 1 0 1 T (Tb +1) T (Tb +1) 2 2 X X 1 @ 2 2 2 A = 2b @ 1 A 1 = Op (T 1 ) = op (1); T1 +1+k T1 +1+k b 2 6 T 6 T k=1 T k=1 18 ! j where the …rst inequality follows from the Cauchy-Schwarz inequality, and the second inP 2 equality holds because 1 = 2 =6 (see Tanaka, 1999). Because t is a short-memory k=1 k PT (T +1) 2 zero-mean covariance stationary process, it is straightforward to show that T 1 k=1 b T1 +1+k = Op (1). By the continuous mapping theorem, jA2 j = op (1), which implies that A2 = op (1). Similarly, 1 (A3 )2 = 2b 2 T 2 2 b Tb X 1 k=1 Tb X 1 T 6 Tb +1 k k !2 2 Tb +1 k k=1 ! Tb X 1 k2 k=1 2 1 b 2 T ! Tb X 2 Tb +1 k k=1 ! 1 = op (1): T Hence, LMM = p T r 6 T 1 PT 2 1 T Pj 1 1 k=1 k j k j=2 PT j=2 2 j + op (1) j = + op (1) p r T T 6 X1 2 k k=1 k + op (1): d The result in Lemma 1 and Slutsky’s theorem imply that LM ! N (0; 1). Proof of Theorem 2. Under H0 : d0 = 1, yt = 1 + ut = 1 + t 1t 1 . De…ne g yt = yt ^ 1 . First, consider the denominator of LMT . Under H0 , conditioning on y0 = 0, T 1 T X t=1 g yt 2 =T 1 T X ^ yt 2 1 =T 1 t=1 =( 1 ^ )2 + 2( 1 1 ^ )T 1 1 T X t=1 T X t+T 1 t=1 = Op (T 1 ) + op (1) + T 1 T X 2 t 1 =T =T j 1 T X X 1 1 =T k j=2 k=1 19 yj 1 T X 2 t T X 2 t + op (1); t=1 P 1=2 where we use the fact that ^ 1 ), and T 1 Tt=1 1 = Op (T large numbers. Second, the numerator of LMT is given by ! j 1 T T X X X 1 ^ T 1 log g yj g yj = T 1 yj k g yj k j=2 j=2 k=1 j 1 T X X 1 k j=2 k=1 t t=1 t=1 1 2 ^ + 1 1 k ^ ^1 + 1 p t ! j k ! 0 by the weak law of ^ yj ! 1 1 ^1 + j = ^ 1 2 1 T 1 j 1 T X X 1 k j=2 k=1 + ^ 1 1 where we use the expansion i = 1; 2; 3. 1 B2 = 1 B3 = 1 ^ ^ ^ 2 1 1 1 1 ^ 1 j 1 T X X 1 1 T k j=2 k=1 = B1 + B2 + B3 + T B1 + j k ! j 1 T X X 1 k j=2 k=1 1 T 1 j=2 k k=1 j 1 T X X 1 1 +T k j=2 k=1 ! j k ! j k j ! j j; = L + 21 L2 + 31 L3 + log j 1 T X X 1 . We show that Bi = op (1) for T (log T + + T ) = Op (T 1 ) (log T + + T ) = op (1); 1 0 T T X1 1 X A = Op (T 1=2 )op (1) = op (1); @ T 1 j k j=k+1 k=1 0 1 T X1 1 TXk @ A = Op (T 1=2 )op (1) = op (1); T 1 j k T 1 j=1 k=1 where is the Euler-Mascheroni constant and T 1=(2T ) which approaches 0 as T ! 1. The results for B2 and B3 follow from the arguments used in the proof of Theorem 1 for the terms A2 and A3 . Hence, T 1 T X j 1 T X X 1 1 g g yj yj = T k j=2 k=1 log j=2 Then, under H0 , T 1 j=2 LMT = T T = This completes the proof. PT 1 PT j=2 T Pj 1 1 k=1 k j k PT 1 1 j 2 t=1 t + op (1) Pj 1 1 j k=1 k j k PT t=1 2 t j k ! j + op (1): + op (1) + op (1): Proof of Propositions 1 and 3. When the estimate of the break date is consistent, the proof is trivial and omitted. We here focus on having a consistent estimate of the break fraction at some rate T { for 0 < { 1: Speci…cally, suppose that the estimate of the break { fraction ^ satis…es that ^ ) for 0 < { 1. For all models A1-A3, we have b = op (T a detrended sequence fe yt g based on the OLS method in PZ. The Lagrange Multiplier test 20 statistic is given by: r PT r PT Pt 1 1 p p yet k ( log y e ) y e 6 6 t t k=1 t=2 t=2 LMT;^ = T = T PT PT k 2 2 2 et ) et )2 t=1 ( y t=1 ( y yet : We show that the stochastic orders of terms associated with the deterministic time trend are smaller than those of terms associated with the error process. We can write (9) as: f+ U e . Since LM ^ is a functional of M T; to consider the inner product of Ye 0 Ye , that is, and Ye = ^ z Z(Tb ) + M ^ zU = M ^ z Z(Tb )2 Ye = M Ye 0 Ye = 2 ^ zU = M f+ U e; +M Ye and subvectors of f0 M f+ 2 M f0 U e+ M Ye , it su¢ ces e0 U e: U Note that we only need to check the stochastic order of Ye 0 Ye because the lag of order k is e0 U e controlled to be small relative to the sample size T . We want to show that the term U e0 U e = Op (T ) uniformly over dominates the others. It is straightforward to show that U all admissible break dates Tb 2 f T; (1 )T g for some 2 (0; 1=2) in all models. When 0 f M f has a smaller order of magnitude compared to that of U e0 U e , so does M f U e M f0 M f cannot be by the Cauchy-Schwartz inequality. Further, the order of magnitude of M f0 M f. The order of magnitude of M f0 M f is Op (T 2 2{ ) for Models greater than that of M A2 and A3, which implies that the break fraction should be estimated consistently at some f0 M f rate greater than T 1=2 . On the other hand, for Model A1b, the stochastic order of M is Op (T 1+2 ). Hence, the orders of magnitude of terms associated with the deterministic trend are greater than those of the terms associated with the error process, thereby the LM statistic diverges as T ! 1. 21