Bootstrap Procedures for Detecting Multiple Persistence Shifts

Bootstrap Procedures for Detecting Multiple Persistence Shifts in Heteroskedastic Time Series Abstract This paper proposes new bootstrap procedures for detecting multiple persistence shifts in a time series driven by nonstationary volatility. The assumed volatility process can accommodate discrete breaks, smooth transition variation as well as trending volatility. We develop wild bootstrap sup-Wald tests of the null hypothesis that the process is either stationary [I(0)] or has a unit root [I(1)] throughout the sample. We also propose a sequential procedure to estimate the number of persistence breaks based on ordering the regime-speci…c bootstrap p-values. The asymptotic validity of the advocated procedures is established both under the null of stability and a variety of persistence change alternatives. A comparison with existing tests that assume homoskedasticity illustrates the …nite sample improvements o¤ered by our methods. An application to OECD in‡ation rates highlights the empirical relevance of the proposed approach and weakens the case for persistence change relative to existing procedures. 1 Introduction Distinguishing among di¤erent forms of nonstationarity has been a topic of long-standing interest in time series analysis; e.g., distinguishing a deterministic trend from a unit root process (I(1), a stochastic trend) and infrequent shifts in trend versus unit root (Perron, 1989). Also, tests for changes in the deterministic trend are sensitive to the nature of the stochastic component, stationary [I(0)] or I(1) (Harvey et al., 2009, Perron and Yabu, 2009). An important related problem concerns inference about the conditional mean in the presence of nonstationarity in variance. Structural changes in variance have been extensively documented for macroeconomic and …nancial time series; e.g., Sensier and van Dijk (2004), Perron and Yamamoto (2020). The non-robustness of unit root tests to nonstationary volatility was established by Cavaliere (2005) and Cavaliere and Taylor (2007, 2008a, 2009). A smaller literature addressed the problem of discriminating between changes in the conditional mean and nonstationary volatility. Hansen (2000) shows that standard structural change tests do not have the correct size with nonstationary variance. Pitarakis (2004), Perron and Yamamoto (2019) and Xu (2015) document the extent of size distortions and power losses for various tests. Perron, Yamamoto and Zhou (2020) develop likelihood ratio tests of the joint hypothesis of changes in coe¢ cients and error variance in a linear regression model. Given the importance of allowing for nonstationary volatility, this paper deals with the problem of testing for structural changes in the persistence of a time series in this context; i.e., changes involving switches between unit root I(1) and stationary I(0) processes and changes that preserve the I(0) properties across regimes. Most procedures available are based on a global homoskedasticity assumption; e.g., Kim (2000), Busetti and Taylor (2004), Harvey et al. (2006) for a single break and Bai and Perron (1998, BP henceforth), Leybourne et al. (2007), Kejriwal et al. (2013, KPZ henceforth) for multiple breaks as well as Kejriwal (2019) for procedures to determine the number of breaks. Cavaliere and Taylor (2008b, CT henceforth) develop bootstrap tests robust to nonstationary volatility based on the ratio of partial sums of demeaned (or detrended) residuals. Their procedure assumes that the process is I(0) under the null hypothesis of stability, a single break under the alternative with either a I(1)-I(0) or I(0)-I(1) shift but not an I(0)-I(0) shift and a stable trend function. We provide a comprehensive treatment of issues related to testing for changes in persistence with heteroskedastic errors. Our approach is general and allows: (1) an I(1) or I(0) null hypothesis; (2) multiple changes with unknown number and timing; (3) changes of the form I(1)-I(0); I(0)-I(1) and I(0)-I(0), without prior knowledge of the speci…c form; 1 (4) disentangling persistence shifts from shifts in the trend function. The assumed volatility process is general and accommodates breaks, smooth transitions and trending volatility. We develop sup-Wald tests based on a wild bootstrap scheme that have accurate size and satisfactory power in …nite samples. We also propose a sequential method to estimate the number of persistence breaks. Extensive simulation experiments are provided to assess the …nite sample properties of the methods suggested, including comparisons with existing tests. Our proposed methods can be applied to study a wide range of important empirical issues. We comment here on three potential applications, one explored in detail later. The …rst concerns the issue of in‡ation persistence that plays a key role in the formulation and evaluation of quantitative macroeconomic models (see, e.g., Korenok et al., 2010). The Lucas Critique suggests that the parameters of reduced form speci…cations depend implicitly on agents’expectations of the policy process and are unlikely to remain stable as policymakers change their behavior, if agents are forward looking. An empirical …nding of high and stable persistence in such a context can potentially be interpreted either in terms of the presence of a strong backward looking component in the dynamics of in‡ation induced through, say indexation or rule-of-thumb behavior on the part of the price setters, or in terms of historical policy shifts being of relatively modest magnitude. Our approach enables a robust treatment of in‡ation dynamics that allows disentangling breaks in mean and persistence allowing for changing volatility and thereby provides reliable guidance on whether changes in persistence is a feature that a reasonable macroeconomic model should be able to replicate; see Section 8 for the analysis, references and further discussions. A second application concerns climate change. The anthropogenic theory of climate change postulates that human activity increases emissions of radiatively active gases relative to natural sources and sinks. It can do so in a way that changes global biogeochemical cycles thereby increasing the persistence of radiative forcing and surface temperature. Using data over 1500-2011, Dergiades et al. (2016) …nd evidence supporting persistence change [I(0) to I(1)] for both series based on the single break tests of Busetti and Taylor (2004) and Harvey et al. (2006). The long time span, however, covers the three Industrial Revolutions (the steam engine, electricity and mass production, and digital technology) suggesting that the single break assumption may be unduly restrictive. Further, they assume homoskedasticity in contrast to evidence favoring heteroskedasticity in both series (see Cavaliere et al., 2018, for global CO2 emissions and Chang et al., 2020, for temperature). Our approach o¤ers methods to comprehensively evaluate this hypothesis owing to its ability to accommodate multiple persistence breaks, non-stationary volatility, and broken trends (Estrada and Perron, 2017). 2 A third possible application is to predictive regression. Predicting a low persistence I(0) variable like stock returns using a highly persistent predictor, say the dividend-price ratio (D/P), involves a (nearly) unbalanced regression that can potentially explain its sporadic predictive power in practice. Lettau and Nieuwerburgh (2008) argue that D/P is well represented by a regime-wise stationary process and use demeaned residuals obtained from the BP procedure to construct a new predictor that is shown to deliver superior predictive performance (see Verdickt et al., 2019, for a more recent related contribution). In contrast, Park (2010) argues, based on single break homoskedasticity-based persistence change tests (Harvey et al., 2006), that D/P is better approximated by a process that switches between I(1) and I(0) regimes and consequently has strong predictive power in the I(0) regime but not in the I(1) regime due to the unbalanced regression problem. Given that neither of these studies allow for nonstationary volatility in D/P or returns (Johannes et al., 2014), their estimation and inference results can be potentially misleading. The generality a¤orded by our approach can be fruitfully employed in this context to distinguish between the mean shift and persistence change alternatives for the D/P process, demarcate the I(1) and I(0) regimes, and assess regime-wise predictability accordingly. The paper is organized as follows. Section 2 describes the models and the assumptions and Section 3 the testing procedures. The large sample e¤ects of nonstationary volatility on these persistence change tests are studied in Section 4. The proposed bootstrap tests are presented in Section 5. Section 6 discusses extensions of the procedures to deal with deterministic trends as well as disentangling shifts in persistence from shifts in the trend function. Section 7 provides a summary of the Monte Carlo evidence, Section 8 presents an application to OECD in‡ation rates and Section 9 concludes. An online supplement contains all proofs, detailed simulation results and additional empirical results. 2 The Persistence Change Model We start with a univariate time series yt generated by the following AR(p) model: yt = ht = + ut ; ut = uTi0 1 + ht (t = Ti0 1 + 1; :::; Ti0 ; i = 1; :::; m + 1) Pp 1 i ht 1 + j=1 ij ht j + et (hTi0 1 = ::: = hTi0 1 p+1 = 0) i (1) 0 with the convention T00 = 0 and Tm+1 = T , where T is the sample size. The process is therefore subject to m breaks or m + 1 regimes with break dates (T10 ; :::; Tm0 ). Both the break dates and the number of breaks are assumed unknown. The autoregressive order p is 3 assumed to be …nite. This Data Generating Process (DGP) was considered by Leybourne et al. (2007) and Kejriwal (2019) and is designed to ensure that adjacent I(1) and I(0) regimes join up at the breakpoints thereby avoiding the problem of spurious jumps to zero in ut . While we assume a common lag length p across regimes, regime-speci…c lag lengths can be accommodated interpreting p as the maximum lag length across the (m + 1) regimes. Our analysis is based on the following assumptions: Assumption A: 1) Ti0 = T 0i ; where 0 < 01 < ::: < part of its argument; 2) All roots of the polynomial outside the unit circle; 3) et = t "t , i (L) 0 m =1 < 1 and b:c denotes the integer i1 L i2 L 2 ::: i;p 1 L p 1 lie where f"t g is an i.i.d. sequence with zero mean and unit variance and f t g is a strictly positive non-stochastic sequence, also supt E("4+ ) < t 1 for some > 0; 4) For some strictly positive deterministic sequence faT g; f t g satis…es aT 1 bT sc = g(s); s 2 [0; 1], where g(:) is a strictly positive, non-stochastic function with a …nite number of discontinuities satisfying a uniform …rst-order Lipschitz condition except at the points of discontinuity. Assumption A1 is standard and dictates the asymptotic framework adopted so that each segment increases proportionately with T . A2 speci…es at most one unit root in each regime and precludes explosive regimes. A3 speci…es that the stochastic process for fet g is deter- mined by the time-varying volatilities f t g (e.g., Xu, 2008). In contrast to CT who make a mixing-type assumption on the errors thereby allowing for moving average processes and conditional heteroskedasticity, our analysis is based on the stronger assumption of a …nite order autoregressive process with i.i.d. innovations. We do, however, demonstrate the robustness of our procedures to moving average errors through simulations (see Section 7). A4 is the key assumption, which allows fyt g to be generated by a wide class of nonstationary heteroskedastic errors; e.g., single or multiple volatility breaks, linearly trending volatility, piece-wise linear trends in variance and smooth transition shifts satisfy A4 with aT = 1 with a particular choice of g(:). Models with explosive deterministic volatility are allowed specifying aT appropriately; see Cavaliere and Taylor (2008a, 2009). The function g(:) is assumed to be non-stochastic to enable simpli…cation of the theoretical analysis. Hence, A4 rules out nonstationary autoregressive stochastic volatility (SV) models (Hansen,1995), SV models with jumps (Georgiev, 2008, Perron and Qu, 2010), “nonstationary nonlinear heteroskedastic” models with stochastically trending volatility, and near-integrated GARCH models. This assumption can be weakened to allow sequences f t g and f"t g that are stochastically independent and interpreting the results as holding conditional on a given realization of g(:). In order to accommodate I(0) preserving persistence changes as in the framework of Bai 4 and Perron (1998), we also consider the following data generating process for yt : yt = i + ut ; ut = i ut 1 + Pp 1 j=1 ij ut for t = Ti0 1 + 1; ; :::; Ti0 ; i = 1; :::; m + 1, with u0 = ::: = u Assumption A are assumed to hold for (2) as well. 3 j (2) + et p+1 = 0. The conditions in Testing Procedures This section describes the testing procedures which form the basis of our proposed bootstrap tests. These procedures are not new and were developed in KPZ and BP. We …rst consider the KPZ tests that specify the null hypothesis of an I(1) process throughout the sample (no (1) change), H0 : i = 1, versus the alternative that the process switches between being I(1) and I(0). The following two models are considered depending on whether the initial regime contains a unit root or not: Model 1a: Model 1b: i i = 1 in odd regimes and j i j < 1 in even regimes; = 1 in even regimes and j i j < 1 in odd regimes. We next review the test statistics designed to detect a speci…ed number of breaks and outline the procedures when the number of breaks is not speci…ed. (1) Tests for a Speci…ed Number of Breaks. To test the null hypothesis H0 : i = 1 for all i in (1), the regression used is yt = ci + ( with ci = (1 i )[uTi0 1 + i] i 1)yt 1 + Pp 1 j=1 j yt j (3) + et (1) and et the residuals. Under H0 ; ci = 0 for all i, which is imposed. The true lag order p is assumed known but can be estimated using standard information criteria such as the AIC or BIC. The coe¢ cients of the lagged di¤erences in (3) are not allowed to change since, as argued in KPZ, the goal is to direct power against changes in the persistence parameter i. A joint test on all parameters would not be particularly infor- mative in this context given the di¢ culty in interpreting a rejection. As shown in KPZ, the test does not have much power against pure changes in short-run dynamics but is powerful when there is a change in both persistence and these dynamics. In fact, the simulation experiments conducted in Section 7 for assessing power with serially correlated errors consider data generating processes that involve changes in both persistence and short-run dynamics in their autoregressive representation. Consider …rst the Wald test that applies when the alternative involves a …xed value m = k of changes. Denote a candidate vector of break fractions by 5 = ( 1 ; :::; k) and the (1) (1) alternative hypotheses corresponding to Models 1a and 1b as Ha;k and Hb;k ; respectively. The corresponding tests are for d = a; b: F1d ( ; k) = (T k 1 (1) ij)(SSR0 (1) (1) where j = 1 if k is even and j = 0 if k is odd, i = b. Here, (1) SSR0 (4) SSR1d;k )=[(k + 1 + ij)SSR1d;k ] 1 if d = a and i = 1 if d = (1) is the sum of squared residuals (SSR) under H0 (1) while SSR1a;k and (1) SSR1b;k denote, respectively, the SSR from estimating (3) under the restrictions imposed by Models 1a-1b. For some small positive number , we de…ne the set k = f : j i+1 ij ; 1 ; k 1 g. The sup-Wald tests are then F1a (k) = sup 2 k F1a ( ; k) and F1b (k) = sup 2 k F1b ( ; k). When the initial regime is unknown, the relevant test statistic is W1 (k) = max[F1a (k); F1b (k)]. The stable I(0) null can be tested using the BP procedure. (0) This amounts to testing H0 : ci = c; i = ; for all i with j j < 1 in (3). The relevant (0) alternative hypothesis is H1;k : 1 = 6 6 ::: 6= k+1 , j i j < 1 for all i, so that the time series 2 = (0) is regimewise-I(0) under H1;k . The BP test for a …xed number m = k of changes is given by G1 ( ; k) = [T (0) (0) 2(k + 1)](SSR0 (0) where SSR0 denotes the SSR under H0 is G1 (k) = sup 2 k (0) (0) (5) SSR1;k )=[kSSR1;k ] (0) and SSR1;k the unrestricted SSR. The BP test G1 ( ; k). To control asymptotic size when the process is either I(1) or (1) (0) I(0) under the null hypothesis, KPZ proposed a joint test. Let H0 = H0 [ H0 . The test for H0 is H(k; ) = min [W1 (k), [cvw;k ( )=cvg;k ( )]G1 (k)], where cvw;k ( ) and cvg;k ( ) are the critical values of the statistics W1 (k) and G1 (k), respectively, for some signi…cance level . Computing G1 (:) and W1 (:) is done using the dynamic programming algorithms of Bai and Perron (2003) and Perron and Qu (2006), respectively. Tests when the Number of Breaks is Unknown. With the number of breaks unknown up to an upper bound A; KPZ proposed the following test statistic to detect processes alternating between I(1) and I(0) regimes: W max1 = max1 k A W1 (k). Similarly, to detect I(0)- preserving changes, the BP test is U Dmax1 = max1 k A G1 (k). To achieve correct size under H0 ; KPZ also suggested the test Hmax1 ( ) = min [W max1 ; [cvw;max ( )=cvg;max ( )]U Dmax1 ], where, cvw;max ( ) and cvg;max ( ) are the critical values of W max1 and U Dmax1 , respectively. The decision rule is to reject H0 if H max1 ( ) > cvw;max ( ). 4 The Large Sample E¤ects of Nonstationary Volatility We consider the large sample behavior of the KPZ and BP tests in the presence of nonstationary volatility as speci…ed in A(3-4). Theorems 1 and 2 below show that the null limiting 6 distributions of the tests are not pivotal and depend on the sample path of the volatility process g(:); hence the tests do not have the correct asymptotic size unless g(:) is a constant. Rr Rr As a matter of notation, for r 2 [0; 1], let ge(r) = ( 0 g(s)2 )1=2 , Bg;1 (r) = ge(1) 1 0 g(s)dB1 (s) Rr and Bg;2 (r) = ge(1) 1 0 g(s)2 dB2 (s). The process Bg;1 (s) is Gaussian with zero mean and variance (s) = ge(s)2 =e g (1)2 so that Bg;1 (:) is a variance-transformed Brownian motion with directing process ; see Davidson (1994, Section 29.4) and Cavaliere (2005). w (1) 0 ( ; k) de…ned by Theorem 1 Under Assumptions A and H0 , F1d ( ; k) ! F1d 2 i+1 i+1 6 P 6 1 6 k + 1 + ij n= 1 i 6 4 2 k+ij 2 (2n+ 2 ) ( [fBg;1 (2n+ 2 ) ( )g2 fBg;1 2n+ i+1 2 i 4 + 1 2n+ i+1 2 R 2n+ i 2 2n+ i+1 2 2n+ i 2 1 2n+ i 2 1 2 g e(1) 1 )g 2 fe g( (2n+ i+1 2 ) (r)]2 dr [Bg;1 fBg;1 ( ) 2n+ i+1 2 )2 g e( 2n+ i+1 2 Bg;1 ( 2n+ i 2 1 )g 2n+ i 2 2 2 1 ) g] 2 3 7 7 7 7 5 where j = 1 if k is even and j = 0 if k is odd, i = 1 if d = a and i = 1 if d = b. w w 0 0 0 (k); F1b (k)]; W max1 = Also, F1d (k) ! sup 2 k F1d ( ; k) (d = a; b), W1 (k) ! max[F1a max1 w k A W1 (k) ! max1 k A 0 0 (k); F1b (k)]g. fmax[F1a w (0) Theorem 2 Under Assumption A and H0 , G1 ( ; k) ! G01 ( ; k), de…ned by " # 2 2 2 2 k 1 P fe g ( ) B ( ) g e ( ) B ( )g f n Bg;1 ( n+1 ) B ( )g n g;2 n+1 n+1 g;2 n n+1 g;1 n + 2g 2 fe 2 k n=1 ( ) g e ( ) e ( ) g ( ) g e ( n )2 g n n+1 n+1 n n n+1 n+1 w w Also, G1 (k) ! sup 2 k G01 ( ; k) and U Dmax1 ! max1 k A fsup 2 k G01 ( ; k)g. The non-robustness of G1 (1) to shifts in the marginal distribution of the regressors was shown in Hansen (2000). The absence of large sample invariance of the KPZ and BP tests to unconditional heteroskedasticity continues to hold for the heteroskedasticity-robust versions of these tests; see Georgiev et al. (2018, Remark 12). Unreported simulations did not reveal any advantage of the robust versions in the presence of unconditional heteroskedasticity. (0) Hence, we focus on non-robust version for simplicity. Note also that under H0 , the KPZ (1) tests diverge to positive in…nity while under H0 ; the BP tests have incorrect asymptotic size even when conditional homoskedasticity holds. It can be shown that these properties continue to hold under Assumptions A. Monte Carlo evidence indicates that the extent of size distortions in …nite samples can be considerable (see Supplement B, Table B-1). 7 5 The Wild Bootstrap Versions of the Tests We now propose wild bootstrap versions of the tests and establish their asymptotic validity under Assumption A. Unlike the standard residual bootstrap, the wild bootstrap procedure (Liu, 1988) can mimic the pattern of heteroskedasticity in the errors. We also show that the bootstrap KPZ and BP test statistics are consistent under the relevant alternatives. With the direction of persistence change typically unknown, our subsequent analysis will only consider the recommended W1 (:), W max1 , G1 (:) and U Dmax1 tests. Since the null hypothesis H0 involves both I(1) and I(0) processes, the algorithm is based on generating two kinds of bootstrap samples, one for each case, conditional on the data fyt gTt=1 : The I(1) (resp., I(0)) bootstrap samples are used to approximate the …nite sample distribution of the KPZ (resp., BP) tests. For reasons discussed below, our proposed bootstrap scheme is not recursive as in Xu (2008) for the stationary autoregressive model. Denote by fvt ; t = 1; :::; T g a sequence of i.i.d. random variables with zero mean, unit variance and uniformly bounded fourth moments (i.e., supt E("4+ ) < 1 for some > 0) that are independent of fyt gTt=1 . t We now describe the algorithms to generate the bootstrap samples. PlT I(1) Bootstrap Samples: 1) Estimate the regression yt = j=1 j yt j + et (t = lT + 2; :::; T ) where lT is chosen using the BIC. Denote the estimates by (lT ; 1 ; :::; lT ) PlT and construct the residuals et = yt j=1 j yt j (t = lT + 2; :::; T ); 2) Obtain the (1) bootstrap residuals et = et vt (t = lT + 2; :::; T ); 3) Generate the bootstrap sample as (1) (1) (1) (1) follows: yt = yt 1 + et (t = lT + 2; :::; T ), yt = yt (t = 1; :::; lT + 1); 4) Construct the (1) bootstrap versions of the W1 (:) and W max1 statistics using fyt gTt=1 based on a regression (1) that does not include lagged …rst di¤erences of yt ; 5) Repeat steps (2)-(4) B times to approximate the bootstrap distribution of the statistics. I(0) Bootstrap Samples: The algorithm is the same except that in step (1) the regression PT PelT is yt = c+ yt 1 + lj=1 et = yt e c e yt 1 j yt j +et and the residuals are e j=1 ej yt j ; (0) (0) (0) also in step (3), we generate yt = et = eet vt (t = e lT + 2; :::; T ), yt = 0 (t = 1; :::; e lT + 1). For the I(1) scheme, we do not introduce …rst-di¤erences in step (4) to avoid explosive or multiple unit roots, as in Cavaliere and Taylor (2008a). The I(0) bootstrap scheme is non-recursive since we do not “add back” the conditional mean component based on the (0) parameter estimates. Rather, the bootstrap samples fyt g have constant (zero) mean and are serially independent, conditional on the data. Using a recursive scheme leads to tests with lower power when the data contain an I(1) segment since the estimated persistence 8 parameter converges to 1 at rate T so that the recursive bootstrap samples are e¤ectively drawn from an autoregressive process with a root close to unity. This feature contributes to an increase in the bootstrap critical values (relative to the non-recursive bootstrap) with an adverse e¤ect on power; see Gulesserian and Kejriwal (2014) in the context of stationarity testing based on the sieve bootstrap in the homoskedastic case. Simulations suggest notable power gains from using the non-recursive form of the wild bootstrap for alternatives that involve switches between I(1) and I(0) regimes (see Supplement C). Note that, in both cases, step (4) constructs the bootstrap statistics from an AR(1) speci…cation. The bootstrap residuals are then serially independent conditional on the data; hence, no need to control for serial correlation through lagged di¤erences. Unreported simulations showed that an AR(1) bootstrap speci…cation resulted in improved …nite sample properties (size and power). Denote the bootstrap analogues of W1 (k), W max1 , G1 (k) and U Dmax1 by W1 (k), W max1 , G1 (k) and U Dmax1 , respectively, with associated p-values pk;W1 , pW max , pk;G1 and pU Dmax , omitting the dependence on T for ease of notation. Similarly, for a given signi…cance level , denote the bootstrap critical values by cvw;k ( ), cvw;max ( ), cvg;k ( ) and cvg;max ( ). Our proposed statistics are the “hybrid”tests H (k; ) = min[W1 (k), [cvw;k ( )=cvg;k ( )]G1 (k)] and Hmax1 ( ) = min[W max1 ; [cvw;max ( )=cvg;max ( )]U Dmax1 ]. The following results state that (a) the wild bootstrap versions of the tests can successfully replicate the …rst order asymptotic distribution of the original tests; (b) for a given signi…cance level , the statistics H1 (k; ) and Hmax1 ( ) have asymptotic size at most ; (c) the test statistics are consistent when k = m changes in persistence are present. w Theorem 3 Under Assumption A with “!p ” denoting weak convergence in probability unw (1) w 0 0 der the bootstrap measure: a) under H0 : (i) W1 (k) !p max[F1a (k); F1b (k)], W max1 !p w w 0 0 max1 k A fmax[F1a (k); F1b (k)]g ; (ii) pk;W1 ! U [0; 1], pW max ! U [0; 1], a uniform distribuw w (0) tion. Under H0 , (i) G1 (k) !p sup 2 k G01 ( ; k); U Dmax1 !p max1 k A fsup 2 k G01 ( ; k)g; w w (ii) pk;G1 ! U [0; 1], pU Dmax ! U [0; 1]; b) Under H0 ; limT !1 P (H (k; ) > cvw;k ( )) and P (Hmax1 ( ) > cvw;max ( )) p p ; c) With p 0 2 m (1) (1) (0) , then, under Ha;m ; Hb;m and H1;m , p we have pm;W1 ! 0, pW max ! 0, pm;G1 ! 0, pU Dmax ! 0. 5.1 Estimating the Number of Breaks A bootstrap procedure can be devised to estimate the number of breaks based on a sequential test of l versus l + 1 breaks, following Kejriwal (2019) who assumed conditional homoskedasticity. Heteroskedasticity precludes using critical values obtained using the full sample when 9 testing stability in each segment. Hence, we propose a new bootstrap sequential procedure. We …rst apply a sequential test of the null hypothesis of l ( 1) breaks against the alternative of (l + 1) breaks. We partition the sample into (l + 1) segments using the l estimated break dates (T^1 ; :::; T^l ) obtained by minimizing the unrestricted SSR. The one break KPZ and BP statistics are then applied to all estimated (l + 1) regimes with the statistics denoted (i) (i) by W1 (1) and G1 (1), respectively, for i = 1; :::; l + 1. The parameter estimates in the (l + 1) regimes are used to generate the regime-speci…c I(1) and I(0) bootstrap samples, (i) (i) which are used to compute the bootstrap p-values of the statistics W1 (1) and G1 (1), de;(i) ;(i) noted by p1;W1 and p1;G1 . For a given signi…cance level ; we reject the null of l breaks in favor ;(i) ;(i) of (l + 1) breaks if min1 i l+1 fpi g < l+1 (decision rule), where pi = maxfp1;W1 ; p1;G1 g and (1 )1=(l+1) . As shown in Supplement A, this decision rule has asymptotic size l+1 = 1 at most under the null hypothesis of l breaks. The steps to implement the sequential pro- cedure are the following: 1) test the null of no break (H0 ) against the alternative of at least one break. For a given signi…cance level ; reject H0 if pmax = maxfpW max ; pU Dmax g < and conclude in favor of at least one break; otherwise stop and the number of breaks selected is 0; 2) Upon a rejection, use the decision rule with l = 1 to determine if there is more than one break. Repeat by increasing l sequentially until the test fails to reject the null hypothesis of no additional break; 3) The estimate m ^ is obtained as the total number of rejections obtained from steps 1 and 2. The probability of selecting the true number of breaks is then at least ) in large samples as stated in the following theorem. (1 Theorem 4 Under the conditions of Theorem 3, limT !1 P (m ^ = m) 6 1 : Extensions This section discusses extensions to deal with: (1) the presence of deterministic trends; (2) distinguishing between a pure trend shifts process from one exhibiting shifts in persistence. Deterministic Trends. We consider an extension of (1) that includes the possibility of m breaks in the deterministic trend, so that: yt = 0 + 0t + Pm j=1 j DUjt + Pm j=1 j DTjt + ut (t = Ti0 1 + 1; :::; Ti0 ) (6) for i = 1; :::; m + 1, with ut as de…ned by (1), where DUjt = I(t > Tj0 ), DTjt = I(t > Tj0 )(t and Tj0 ), j = 1; :::; m. We de…ne m = f( 1 ; :::; m) ; j j+1 jj 0 j = Tj0 =T and for some generic break date " (j = 1; :::; m 10 1); 1 "; m 1 j = Tj =T "g, for some small " > 0. The DGP (and regression used) can be expressed as y t = c i + bi t + with ci = (1 i )f 0 + Pi 1 j=1 j ). Pi 1 j=1 ( j i yt 1 0 j Tj ) + Pp 1 j=1 yt ij + uTi0 1 g + ( i j (7) + et Pp 1 j=1 Pi 1 ij )f 0 + j=1 j g, bi = (1) e 0 : ci = c; i = 1 for H (1 KPZ proposed tests of the null hypothesis i )( 0 + e 0(1) ; bi = 0 for all i so that the process follows a stable unit root all i. Note that under H process with possible drift. Again, two models are consider depending on whether the initial regime is trend or di¤erence stationary. In accordance with the notation in Section 3, the test statistics are denoted by F2a ( ; k); F2b ( ; k); W2 (k) and W max2 . The null hypothesis of e 0(0) : ci = c; bi = b; i = for all i where j j < 1 a stable trend stationary process is given H and the test for a …xed number of m = k changes and known break dates is G2 ( ; k) = (0) (0) g (0) g (0) denotes the SSR under H e 0(0) , i.e., [T 3(k + 1)](SSR SSR2;k )=[kSSR2;k ], where SSR 0 0 obtained from OLS estimation of (7) subject to the restrictions ci = c; bi = b; i = for all i, (0) SSR2;k and denotes the unrestricted SSR. Since, in general the break dates are unknown, the test statistic is de…ned as G2 (k) = sup 2 m G2 ( ; k). When the number of breaks is unknown, the relevant test statistic is U Dmax2 = max1 k A G2 (k). The limit distributions of G2 (:) and U Dmax2 under homoskedastic errors are derived in Kejriwal (2019). Under Assumption A, the above test statistics are not asymptotically pivotal and depend on the sample path of f t g. We propose the following bootstrap algorithm that enables asymptotically valid inference. As in Section 5, we generate both I(1) and I(0) bootstrap samples to ensure e0 = H e 0(1) [ H e 0(0) . The algorithms that the procedure has correct asymptotic size under H are exactly the same except that for the I(1) (resp., I(0)) case a constant (resp., a time trend) is added included in the autoregression in step (1). The bootstrap analogues of W2 (k); W max2 ; G2 (k) and U Dmax2 and the associated p-values are obtained as described in Section 5. The sequential procedure outlined in Section 5.1 is accordingly modi…ed. The following result states the large sample validity of the proposed procedures. Theorem 5 Under Assumption A, and using the tests and bootstrap procedures described e0. above: Theorem 3 holds with H0 replaced by H Disentangling Trend and Persistence Shifts. An important feature is that the statis- tics test the null hypothesis that the persistence parameters and those of the trend function are jointly stable. Hence, they can have power against processes driven by pure trend shifts with no change in persistence. To distinguish between trend and persistence shifts, we can adapt the three-step approach of Kejriwal (2019) for the homoskedastic case to the present 11 context. Consider …rst the non-trending case. The …rst step entails determining the number of breaks (m) e using the sequential procedure described in Section 5.1 and the associated breakpoint estimates (T^1 ; :::; T^m^ ) obtained from the unrestricted model that allows all parameters including those of the lagged …rst di¤erences to change across regimes. Second, using the estimated breakpoints, the Wald statistic for testing the null hypothesis of stable I(0) persistence is constructed [i.e., constancy of i over all i] while allowing all other parameters to vary across the (m ^ + 1) regimes. To account for nonstationary volatility, the Wald statistic is computed using a heteroskedasticity-robust estimator of the variance-covariance matrix (cf., Phillips and Xu, 2006). Third, the null hypothesis of stable I(0) persistence is rejected if the Wald statistic is signi…cant using the critical value from a 2 (m) ^ distribution. Otherwise, the null is not rejected and we conclude in favor of a model with pure level shifts. The trending case is more complex since the process can be either I(1) (with a possibly time-varying drift) or I(0) (around a broken deterministic trend). As above, we develop tests separately for the I(1) and I(0) null and use the intersection of the critical regions of the two tests. The three-step approach is implemented as follows. First, estimate the number of breaks (say m) applying our proposed sequential procedure and breakpoints from the unrestricted speci…cation. Second, compute the Wald statistic (using heteroskedasticity robust standard errors) to test the null of constant persistence allowing the parameters of the trend and lagged di¤erences to change at the estimated breakpoints. In the I(0) case, the statistic has a limiting 2 (m), with m used to obtain critical values. In the I(1) case, apply a second wild bootstrap scheme based on residuals estimated under (3) allowing the constant to change across regimes at the estimated breakpoints. The I(1) bootstrap samples are obtained from a DGP that now includes the estimated regime-speci…c drift in step (3) of the bootstrap algorithm. The bootstrap distribution and critical values of the Wald statistic can then be approximated using simulations. Finally, the null hypothesis of stable [I(1) or I(0)] persistence is rejected if the I(0) and I(1) Wald statistics are both signi…cant. 7 Summary of the Simulations This section summarizes the results of simulation experiments designed to assess the …nite sample performance of our procedures and to provide a comparison with existing approaches. The full set of results is available in Supplements B-C. Following CT, we consider three speci…cations for the volatility process: (i) single discrete break; (ii) deterministically trending volatility; (iii) near-integrated stochastic volatility. Three types of error structures are considered: i.i.d., AR(1) and MA(1). While our theory does not formally allow for moving 12 average processes, we nevertheless include this case in our simulations as a robustness check. The wild bootstrap is implemented using a two point distribution, i.e., vt 2 f 1; 1g with equal probability. The trimming is set at = :15, T 2 f200; 400g and 1000 replications are used. We report results for the non-trending case only (those for the trending case are qualitatively similar). The lag length in the KPZ and BP procedures is selected using BIC with maximal value set to …ve. We report the performance of the tests H (k; ); k = 1; 2 and Hmax1 ( ) = maxfH (1; ); H (2; )g as well as their non-robust (homoskedasticity-based) asymptotic counterparts H(k; ); k = 1; 2 and Hmax1 ( ) = maxfH(1; ); H(2; )g. The ratio-based bootstrap tests of CT are designed to test the I(0) null hypothesis while our tests allow the process to be either I(1) or I(0) under the null. Further, while our tests are based on a …nite order autoregressive model, the CT tests are non-parametric and based on a mixing-type assumption for the innovations. Given that conducting a full and fair comparison of tests with di¤erent underlying models and null hypotheses is not possible, we did not include the CT tests in our analysis. The main …ndings are summarized as follows: Finite Sample Size. The asymptotic tests are considerably oversized indicating their lack of robustness to nonstationary volatility, consistent with the large sample results in Section 4. In contrast, the proposed bootstrap tests are robust to I(1) or I(0) processes maintaining empirical size close to the nominal 5% level across the di¤erent volatility speci…cations. The same is generally true for the di¤erent error structures considered. Finite Sample Power. We consider DGPs with one and two breaks involving switches between I(1) and I(0) regimes as well as between I(0)-preserving regimes. In terms of sizeadjusted power, the bootstrap tests are broadly comparable to their asymptotic counterparts, with neither class of tests uniformly dominating the other. The e¤ect of underspecifying the number of breaks can be seen by comparing the power of H (1) and H (2) for DGPs with two breaks, where the former is generally less powerful than the latter, though not in all cases. The Hmax1 test often has power close to that of the more powerful test amongst H (1) and H (2); highlighting the practical advantage of using Hmax1 to detect the presence of at least one break. Further, the proposed tests have substantial power against I(0)-preserving breaks, a feature that distinguishes these tests from most existing persistence change tests (e.g., the ratio-based tests) that are designed to detect switches between I(1) and I(0) regimes. Finally, the proposed tests are generally more powerful with deterministic rather than stochastic volatility. Number of Breaks. The sequential procedure is generally reliable in selecting the number 13 of breaks in the stable and single break cases. Its performance deteriorates in the two breaks case when the probability of underestimation can be non-negligible. For instance, with an abrupt increase in volatility, the breakpoint estimate used to partition the sample is typically close to the second true breakpoint, so that the …rst segment includes an I(0) to I(1) break while the second is I(0). Whether a second break is selected depends on the power of the single break test in the …rst segment, which is relatively low. A notable improvement is observed as the magnitude of the volatility shift decreases and/or the shift occurs near the second persistence break in the increasing volatility case and near the …rst break otherwise. Disentangling Trend and Persistence Shifts. We consider DGPs with a single break in persistence in addition to a DGP that involves a pure mean shift and apply the procedure proposed in Section 6 to compute the probabilities of selecting the true model for each for these DGPs. The performance is generally satisfactory and improves as T increases. 8 Empirical Application This section undertakes a detailed empirical examination of the nature of in‡ation persistence for a set of OECD countries. Our analysis sheds light on whether persistent, though stable, in‡ation should be regarded as a metric for evaluating macroeconomic models or if persistence varies across monetary policy regimes depending on the relative importance accorded to in‡ation in the monetary authority’s objective function. While early empirical studies (e.g., Cogley and Sargent, 2001) on the stability of in‡ation persistence assumed constant volatility, subsequent work recognized the importance of allowing for time-varying volatility. A substantial literature uses Bayesian methods in a time-varying parameter VAR framework with stochastic volatility to study the stability of the persistence of the in‡ationgap, de…ned as the deviation of in‡ation from its target level. These studies typically assume that the target level of in‡ation evolves as a driftless random walk (see, e.g., Cogley et al., 2010).1 Our proposed procedures complement this literature by examining the persistence properties of in‡ation itself without restricting it to be I(1) a priori as well as obviating the need to specify prior distributions. Our approach also improves upon existing frequentist analyses on the topic that either assume homoskedasticity or restrict the nature of the null and alternative hypotheses allowed and thereby potentially overstate/understate the aggressiveness of the monetary policy stance towards combating in‡ation. For instance, Bataa et 1 An exception is Eo (2016) who considers the persistence of in‡ation itself within a Markov-switching framework with the noise driven by normally distributed innovations. 14 al. (2014) used the BP approach to identify breaks in the mean, dynamics and volatility of in‡ation assuming it is an I(0) process under the null hypothesis of stability and regime-wise I(0) in the presence of breaks, thereby ruling out I(1) regimes. Noriega et al. (2013) and Kejriwal (2019) allowed for unit roots but assumed homoskedasticity. Our empirical investigation is based on monthly CPI in‡ation data for nineteen OECD countries as used in Noriega et al. (2013) and Kejriwal (2019), thereby facilitating a direct comparison with these studies. The data span the period 1960:1-2008:6 (T = 582), except for Germany and Korea where the starting point is 1960:2. The in‡ation rates are seasonally unadjusted and computed as it = 1200(ln Pt ln Pt 1 ), with Pt the CPI at time t.2 The main results are reported in Table 1. The analysis proceeds in six steps. First, we apply the sequential algorithm to estimate the number of breaks m ^ [column (2)] with A = 5; = :15 and = :10. Second, conditional on m, ^ the breakpoint estimates are obtained by minimizing the unrestricted SSR [column (3)]. Third, to distinguish persistence shifts from pure mean shifts, we conduct Wald tests (at the 10% level) of the null that it is subject to m ^ mean shifts against the alternative of m ^ mean and persistence shifts [column (4)]. Here, a heteroskedasticity robust standard error estimate is used to construct the statistics although a wild bootstrap approach could also be used (the results were nearly identical). Fourth, based on the selected model, the largest (across regimes) estimated sum of the autoregressive parameters is computed [column (5)] along with equal-tailed 90% con…dence intervals [column (6)] based on the procedure of Andrews and Guggenberger (2014, AG henceforth) uniformly valid over the stationary and non-stationary regions as robust to conditional (though not unconditional) heteroskedasticity. We use the BIC to select the number of lags within each regime with a maximum of 12 lags. Fifth, the results of the CT procedure are included to highlight the di¤erences in terms of model selection [columns (7) and (8)]. Sixth, unit root tests allowing for nonstationary volatility (Cavaliere and Taylor, 2009) are presented as a robustness check on the model selection [column (9)]. We now turn to a discussion of the results. Evidence of at least one break (m ^ > 0) is obtained for seven countries, of which two (Austria, Korea) favor an I(0) process with a single mean shift. The AG interval estimates are consistent with the presence of at least one I(1) segment in fourteen countries of which three are subject to at least one persistence break. Next, we provide a comparison between our results and those from the CT procedure. For the latter, we …rst apply their test K4 (at the 10% level with 15% trimming) designed to detect 2 We prefer to use seasonally unadjusted rates since commonly used adjustment procedures can have adverse e¤ects on the power of structural change tests (see Ghysels and Perron, 1993). 15 a single persistence change [I(1)-I(0) or I(0)-I(1)]. Upon a rejection, the p-values of their K1 and K10 tests (designed to detect the I(0)-I(1) and I(0)-I(1) alternatives, respectively) are computed and the direction of persistence change determined by the smaller of the two p-values. If both p-values are (near)-zero, the evidence is not conclusive. Comparing columns (7)-(8) shows that the procedures agree only for Luxembourg and Netherlands and point to di¤erent models for all other countries. The CT approach is inconclusive in seven cases. Further, for the eleven cases where the proposed approach decides in favor of a pure I(1) process, the CT procedure suggests a break in persistence, consistent with the fact that it is designed to test the null hypothesis of a stable I(0) process. In the two cases for which we select a pure mean shift process, the CT approach again points to a persistence break, again potentially explained by the non-robustness of their approach. Column (9) supplements our analysis with unit root tests applied to the regime with the largest estimate of the sum of the autoregressive coe¢ cients based on the selected model in column (7). We report the p-values of the wild bootstrap ADF test proposed by Cavaliere and Taylor (2009) which is robust to nonstationary volatility. The lag length in the ADF regression was selected using the Modi…ed Akaike Information Criterion (MAIC) of Ng and Perron (2001) with the maximum lag set at 12(T =100)1=4 . The …ndings match the model selection outcomes in column (7) for fourteen of the nineteen countries, indicating a fair degree of consistency between the two approaches. Table 2 presents the regime-wise estimates for the countries that are subject to at least one persistence break. Interestingly, for four of the …ve countries with persistence breaks, the …rst break corresponds to an increase in persistence that occurs between the early and mid 70s, a period often described as one of “the Great In‡ation”and commonly believed to be associated with both a high level and high degree of persistence. In contrast, for France and Germany which experience two persistence breaks, the second break is associated with a persistence decline occurring in the 80s and 90s. To justify the importance of allowing for nonstationary volatility, Figure D-1 in Supplement D plots the volatility estimates obtained by …tting a nonparametric regression to the squared residuals obtained by estimating the model selected in column (7) of Table 1. As suggested by Xu and Phillips (2008), a Gaussian kernel is used with the bandwidth chosen by cross validation, searching over bandwidths hi = ci T 0:4 (i = 1; :::; 4) with fc1 ; :::; c4 g = f:25; :4; :6; :75g. The estimates show considerable variation over time with di¤erent patterns across countries. While a smooth trend suggests itself for some countries (e.g., France and Norway), more irregular movements are observed for others (e.g., Belgium, UK, USA). A similar overall picture is obtained if one plots the estimated 16 variance pro…le (Figure D-2) as suggested by Cavaliere and Taylor (2007) indicating that the nonstationary behavior of the sample volatility paths is a key feature of the in‡ation data, which if ignored, might lead to misleading inferential results. Finally, to evaluate the impact of nonstationary volatility on persistence change, it is useful to compare our results with the asymptotic sequential procedure of Kejriwal (2019) which assumes homoskedasticity. Using the same dataset, Kejriwal (2019) concludes in favor of a persistence change model for six additional countries (Canada, Finland, Greece, Japan, UK, USA) all of which are found to be pure I(1) processes according to our analysis that accounts for nonstationary volatility. Interestingly, Kejriwal’s analysis for USA suggests a shift from a high persistence I(0) regime to a low persistence I(0) regime, consistent with the view (e.g., Sims, 2001) that the case for unstable persistence is weakened once allowance is made for shifts in the variance of the innovations.3 Thus, evidence for persistence shifts obtained using more restrictive methods might overstate the role of monetary policy, e.g., a purported signi…cant decline in persistence may be attributed to a more aggressive stance taken by the monetary authority towards in‡ation. 9 Conclusion We proposed wild bootstrap sup-Wald tests to detect persistence change in a time series with nonstationary volatility. The set of alternative hypotheses considered include processes exhibiting switches between I(1) and I(0) regimes or that remain I(0) in each regime. The performance of the methods suggested was shown to be reliable in …nite samples and to have better properties than existing tests. An application to in‡ation rates further illustrates the usefulness of the proposed approach in practice. 3 Using the same dataset, Noriega et al. (2013) applies the doubly recursive test of Leybourne et al. (2007) to conclude in favor of a stable I(0) process for 75% of the OECD countries analyzed here. 17 Table 1: Break selection in OECD in‡ation rates C o u ntry m ^ B re a k D a te s P u re M e a n S h ifts L a rg e st A R su m 90% B a n d S e le c te d M o d e l (H ) S e le c te d M o d e l (K) A D F p-va lu e (1 ) (2 ) (3 ) (4 ) (5 ) (6 ) (7 ) (8 ) (9 ) A u stria 1 1 9 6 9 :7 Ye s .6 5 [.4 7 , .8 7 ] I(0 ) w ith o n e m e a n sh ift I(0 )-I(1 ) .1 6 B e lg iu m 1 1 9 7 4 :9 No .8 2 [.7 1 , 1 .0 5 ] I(0 )-I(1 ) I(1 )-I(0 ) .1 1 C anada 0 - No .8 8 [.8 1 , 1 .0 4 ] I(1 ) I(1 )-I(0 ) .1 4 F in la n d 0 - No .8 7 [.7 8 , 1 .0 6 ] I(1 ) I(0 )-I(1 ) .2 7 .1 0 Fra n c e 2 1 9 7 3 :1 ; 1 9 8 3 :4 No .6 5 [.5 1 , .8 9 ] I(0 )-I(0 )-I(0 ) I(0 )-I(1 )/ I(1 )-I(0 ) G e rm a ny 2 1 9 7 0 :9 ; 1 9 9 3 :7 No .5 2 [.4 2 , .6 4 ] I(0 )-I(0 )-I(0 ) I(0 )-I(1 ) .0 0 G re e c e 0 - No .8 1 [.6 4 , 1 .1 2 ] I(1 ) I(0 )-I(1 )/ I(1 )-I(0 ) .1 0 Ita ly 2 1 9 7 2 :6 ; 1 9 7 9 :1 0 No .8 8 [.8 2 , 1 .0 2 ] I(0 )-I(0 )-I(1 ) I(0 )-I(1 )/ I(1 )-I(0 ) .0 7 Japan 0 - No .8 4 [.6 6 , 1 .1 8 ] I(1 ) I(0 )-I(1 ) .3 1 K o re a 1 1 9 8 1 :9 Ye s .1 4 [.0 0 , .2 9 ] I(0 ) w ith o n e m e a n sh ift I(0 )-I(1 )/ I(1 )-I(0 ) .0 0 L u x e m b o u rg 1 1 9 9 9 :1 No .8 7 [.7 6 , 1 .1 0 ] I(1 )-I(0 ) I(1 )-I(0 ) .5 1 N e th e rla n d s 0 - No .0 4 [-.0 3 , .1 2 ] I(0 ) I(0 ) .0 0 N o rw ay 0 - No .7 8 [.6 6 , 1 .0 2 ] I(1 ) I(0 )-I(1 )/ I(1 )-I(0 ) .0 8 P o rtu g a l 0 - No .8 3 [.7 1 , 1 .0 8 ] I(1 ) I(0 )-I(1 )/ I(1 )-I(0 ) .1 6 S p a in 0 - No .8 8 [.7 7 , 1 .0 9 ] I(1 ) I(0 )-I(1 )/ I(1 )-I(0 ) .3 5 Sweden 0 - No .8 2 [.7 1 , 1 .0 5 ] I(1 ) I(0 )-I(1 ) .1 0 S w itz e rla n d 0 - No .8 2 [.6 7 , 1 .1 1 ] I(1 ) I(0 )-I(1 ) .0 7 UK 0 - No .8 8 [.7 6 , 1 .1 1 ] I(1 ) I(0 )-I(1 ) .2 0 USA 0 - No .9 0 [.8 3 , 1 .0 5 ] I(1 ) I(1 )-I(0 ) .1 6 T h is ta b le re p o rts th e e m p iric a l re su lts b a se d o n m o nth ly O E C D in ‡a tio n ra te s d a ta ove r 1 9 6 0 :1 -2 0 0 8 :6 . C o lu m n (1 ): th e c o u ntry n a m e ; c o lu m n (2 ): th e e stim a te m ^ o b ta in e d fro m a p p ly in g th e se q u e ntia l a lg o rith m o f S e c tio n 5 .1 w ith = :10 a n d A = 5; c o lu m n (3 ): th e e stim a te d b re a k d a te s o b ta in e d by m in im iz in g th e u n re stric te d su m o f sq u a re d re sid u a ls w ith m ^ b re a k s; c o lu m n (4 ): th e o u tc o m e o f th e te st fo r th e nu ll hy p o th e sis o f p u re m e a n sh ifts; c o lu m n (5 ): th e O L S e stim a te o f th e la rg e st su m o f th e A R c o e ¢ c ie nts a c ro ss th e e stim a te d re g im e s; c o lu m n (6 ): A n d re w s a n d G u g g e nb e rg e r’s (2 0 1 4 ) 9 0 % c o n …d e n c e b a n d fo r th e la rg e st su m o f th e A R c o e ¢ c ie nts; c o lu m n (7 ): th e m o d e l se le c te d by th e se q u e ntia l a lg o rith m ; c o lu m n (8 ): th e m o d e l se le c te d by th e C T p ro c e d u re ; c o lu m n (9 ): th e p-va lu e o f th e w ild b o o tstra p A D F te st o f C ava lie re a n d Tay lo r (2 0 0 9 ). Table 2: Regime-wise estimates of the persistence parameters for the OECD countries C o u ntry m ^ R e g im e A R su m 90% B a n d A D F p-va lu e B e lg iu m 1 1 st .5 7 [.4 4 , .7 5 ] .0 0 2nd .8 2 [.7 1 , 1 .0 5 ] .1 1 1 st .1 0 [-.1 2 , .3 8 ] .0 0 2nd .6 5 [.5 1 , .8 9 ] .1 0 3 rd .6 3 [.5 1 , .7 9 ] .0 0 1 st .2 1 [.1 0 , .3 6 ] .0 0 2nd .5 2 [.4 2 , .6 4 ] .0 0 3 rd .0 6 [-.5 1 , 1 .1 5 ] .0 8 1 st .0 1 [-.1 3 , .1 9 ] .0 0 .0 0 Fra n c e G e rm a ny Ita ly L u x e m b o u rg 2 2 2 1 2nd .5 1 [.3 6 , .7 3 ] 3 rd .8 8 [.8 2 , 1 .0 2 ] .0 7 1 st .8 7 [.7 6 , .1 .1 0 ] .5 1 2nd -.1 3 [-.8 0 , 1 .1 7 ] .0 1