Testing for a change in persistence in the presence of non-stationary volatility

Transcription

1 Testing for a change in persistence in the presence of non-stationary volatility by Giuseppe Cavaliere and A. M. Robert Taylor Granger Centre Discussion Paper No. 6/4

2 Testing for a Change in Persistence in the Presence of Non-stationary Volatility Giuseppe Cavaliere a and A.M. Robert Taylor b a Department of Statistical Sciences, University of Bologna b School of Economics, University of Nottingham July 29, 26 Abstract In this paper e consider tests for the null of trend-) stationarity against the alternative of a change in persistence at some knon or unknon) point in the observed sample, either from I) to I) behaviour or vice versa, of, inter alia, Kim 2). We sho that in circumstances here the innovation process displays non-stationary unconditional volatility of a very general form, hich includes single and multiple volatility breaks as special cases, the ratiobased statistics used to test for persistence change do not have pivotal limiting null distributions. Numerical evidence suggests that this can cause severe oversizing in the tests. In practice it may therefore be hard to discriminate beteen persistence change processes and processes ith constant persistence but hich display time-varying unconditional volatility. We solve the identified inference problem by proposing ild bootstrap-based implementations of the tests. Monte Carlo evidence suggests that the bootstrap tests perform ell in finite samples. An empirical application to a variety of measures of U.S. price inflation data is provided. Keyords: Persistence change; non-stationary volatility; ild bootstrap. JEL Classification: C22. Introduction Recently, both applied economists and econometricians have questioned hether, rather than simply being either I) or I), series might experience a change in persistence We are grateful to Peter Robinson, Jiti Gao and participants at the TSEFR conference held in Perth, 29th June to st July, 26, for helpful comments on an earlier version of this paper. This paper extends upon earlier research in the authors orking paper Testing for a Change in Persistence in the Presence of a Volatility Shift. Correspondence to: Robert Taylor, School of Economics, University of Nottingham, University Park, Nottingham, NG7 2RD, U.K. Robert.Taylor@nottingham.ac.uk

3 beteen separate I) and I) regimes. There is no a relatively large body of evidence on changes of this kind in macroeconomic and financial time series; see, inter alia, Kim 2), Busetti and Taylor 24) [BT], and Leybourne et al. 23), and the citations therein. Commensurately, a number of procedures designed to test against changing persistence have been suggested in the literature. The most popular of these are the ratio-based persistence change tests of, inter alia, Kim 2), Kim, J. et al. 22) and BT, inter alia, hich e focus on in this paper. These test the null hypothesis that a series is a constant I) process against the alternative that it displays a change in persistence from I) to I), or vice versa. The persistence change tests proposed in the literature are all based on the maintained assumption that, both under the null hypothesis of no change in persistence and the alternatives of I)-I) or I)-I), the time series of interest displays stable unconditional) volatility. This assumption contrasts ith a groing body of recent empirical evidence hich documents that many of the main macro-economic and financial variables across developed countries are characterized by the existence of significant non-stationarity in unconditional volatility, in particular, single and multiple possible smooth transition) breaks in volatility and/or broken) trending volatility; see, inter alia, Busetti and Taylor 23), Sensier and van Dijk 24), Kim and Nelson 999), McConnell and Perez Quiros 2), and the references therein. Sensier and van Dijk 24), for example, find that over 8% of the real and price variables in the Stock and Watson 999) data-set reject the null hypothesis of constant unconditional innovation variance. Considerable evidence against the constancy of unconditional variances in stock market returns and exchange-rate data has also been reported; see, inter alia, Loretan and Phillips 994). Hansen 995) also notes that empirical applications of autoregressive stochastic volatility [SV] models to financial data generally estimate the dominant root in the SV process to be close to one, such that volatility is non-stationary. It has recently been demonstrated that both conventional unit root and stationarity tests suffer from potentially large size distortions in the presence of non-stationary unconditional volatility; cf., Kim, T.-H. et al. 22), Busetti and Taylor 23), Cavaliere 24a,b) and Cavaliere and Taylor 25,26). These findings cast doubt over the reliability of the inferences from persistence change tests hen applied to series hich are subject to non-stationary volatility effects. For instance, a rejection of the null hypothesis of no change in persistence by these tests might in fact be attributable to a structural break in the unconditional volatility process rather than a true change in persistence, making these events hard to distinguish beteen in practice. In this paper e address this issue formally by examining the behaviour of persistence tests under a class of non-stationary unconditional volatility processes hich includes smooth volatility changes, multiple volatility shifts and trending volatility, among other things. In Section 2 the model of persistence change hich e focus on ill be outlined. This model extends that previously considered in the literature by alloing not only for a change in persistence in the series but also for non-stationarity in the unconditional volatility process hich may be present under the constant I) null hypothesis or under the persistence change alternative. In doing so, rather than assuming a specific 2

4 parametric model for the volatility dynamics, e do not impose any constraint on the volatility dynamics, apart from the requirement that the unconditional) variance is bounded, deterministic and displays a countable number of jumps. In Section 3 e provide a brief revie of the ratio-based persistence change test statistics of Kim 2), Kim, J. et al. 22) and BT. In Section 4 e derive the large sample null distributions of these statistics against processes hich display non-stationary volatility. Section 6 uses Monte Carlo methods to explore the effects of a variety of nonstationary volatility processes, including single and multiple breaks in volatility and near-integrated autoregressive stochastic volatility, on the finite sample size and poer properties of the persistence change tests. In most of these cases the size properties of the persistence change tests are found to be highly unreliable. Consequently, in Section 5 e propose ild bootstrap-based versions of the tests of Section 3. These are shon to solve the identified inference problem, providing asymptotically pivotal inference under the class of volatility processes considered here, and, in Section 6, to perform ell in finite samples. In section 7 e report an application of the persistence change tests of section 3 and their bootstrap counterparts from section 5 to U.S. price inflation rate series from the Stock and Watson 25) database. Section 8 concludes. Proofs of our main results are placed in a mathematical appendix. Throughout the paper e ill use the notation: C := C[, ] to denote the space of continuous processes on [, ], and D := D[, ] the space of right continuous ith left limit càdlàg) processes on [, ]; to denote eak convergence in the space D endoed ith the Skorohod metric, p convergence in probability and p eak convergence in probability Giné and Zinn, 99), in each case as the sample size diverges; to denote the integer part of its argument; I ) to denote the indicator function, and x := y y =: x ) to mean that x is defined by y. Reference to a variable being O p T k ) is taken throughout to hold in its strict sense, meaning that the variable is not o p T k ). Finally, given to processes X, Y on [, ], for any s [a, b] [, ] e define P X Y s; a, b) := b Y r) X a r) b X r) X a r) dr) X s), Q X Y s; a, b) := b dy r) X ) b a r) X r) X s a r) dr X r) dr, a P XY s; a, b) := Y s) P X Y s; a, b), and Q X Y s; a, b) := Y s) Q XY s; a, b). 2 The Persistence Change Model Generalising Kim 2,p.99), inter alia, consider the null hypothesis, denoted H, that the scalar time-series process y t is formed as the sum of a purely deterministic component, d t, and a short memory I)) component hich displays a time-varying unconditional volatility process; that is, y t = d t + z t,, t =,..., T ) d t = x tβ 2) z t, = σ t ε t 3) 3

5 This DGP generalizes that of Kim 2,p.99), reducing to Kim s model only here the process displays constant unconditional volatility; that is, σ t = σ, t =,..., T. In hat follos e ill assume that the folloing conditions hold on σ t, ε t and d t in )-3): Assumption V. The term {σ t } satisfies σ st = ω s), here ω ) D is a nonstochastic function ith a finite number of points of discontinuity; moreover, ω ) > and satisfies a uniform) first-order Lipschitz condition except at the points of discontinuity. Assumption E. {ε t } is a zero-mean, unit variance, strictly stationary mixing process ith E ε t p < for some p > 2 and ith mixing coefficients {α m } satisfying m= α2/r /p) m < for some r 2, 4], r p. The long run variance λ 2 ε := k= Eε tε t+k ) is strictly positive. As is standard, e refer to {ε t } as an I) process. Assumption X. x t is a k +) deterministic vector ith x t =, all t, and satisfying the condition that there exists a scaling matrix δ T and a bounded pieceise continuous function F ) on [, ] such that δ T x T x ) uniformly on [, ], and here, for all Λ, Λ = [ l, u ] the compact subset of [, ] used in section 3 belo, x s) x s) ds and x s) x s) ds are both positive definite. Under Assumption V, z t, := σ t ε t is heteroskedastic; hoever, z t, is still short memory in the sense that its scaled partial sums admit a functional central limit theorem see the proof of Lemma ) and e shall therefore refer to such processes as I) throughout the paper. Observe, that {y t } in ) is therefore also I) and heteroskedastic. Assumption V requires the variance process only to be non-stochastic, bounded and to display a countable number of jumps and therefore allos for an extremely ide class of possible volatility processes. Models of single or multiple variance shifts satisfy Assumption V ith ω ) pieceise constant. For example, the function ω s) := σ + σ σ ) I s m) gives the single break model ith a variance shift at time mt, < m <. If ω ) 2 is an affine function, then the unconditional variance of the errors displays a linear trend. Pieceise affine functions are also permitted, alloing for variances hich follo a broken trend. Moreover, smooth transition variance shifts also satisfy Assumption V: e.g., the function ω s) 2 := σ 2 + σ 2 σ)s 2 s), S s) = + exp γ s m))), hich corresponds to a smooth logistic) transition from σ 2 to σ 2 ith transition midpoint mt and speed of transition controlled by γ. The case of constant unconditional volatility here σ t = σ, for all t, clearly satisfies Assumption V ith ωs) = σ. Remark. The assumption that the volatility function ω ) is non-stochastic allos for a considerable simplification of the theoretical set-up. Hoever, e conjecture that, under suitable memory and moment conditions, this assumption can be eakened to allo for cases here the innovations {e t } and ω ) are stochastically independent. Indeed, in such cases if the exogenous) volatility process ω ) has sample paths satisfying Assumption A 3, then the results presented in the paper can be thought of as holding conditional on a given realization of ω ). The conditioning argument used here in the 4

6 context of the volatility function serves the same purpose as the exogeneity assumption used by Perron 989,pp.387-8) to permit stochastic changes in the trend function. In the exogenous) SV frameork, Markov-sitching variances obtain by assuming that ω ) is a strictly positive, continuous-time Markov chain ith a finite number of states, hile non-stationary autoregressive SV models obtain for ω s) = h J s)), J ) a diffusion process in D and h ) a strictly positive continuous function; see Hansen 995). Remark 2. Assumption E imposes the familiar strong mixing conditions of, inter alia, Phillips and Perron 988, p.336). If ω ) is non-constant then {z t, } is an unconditionally heteroskedastic process. Conditional heteroskedasticity is also permitted through Assumption E; see, e.g., Hansen 992). The strict stationarity assumption is made ithout loss of generality and may be eakened to allo for eak heterogeneity of the errors, as in, e.g., Phillips 987), although here one ould need to explicitly assume that λ ε is strictly positive. Moreover, the results presented in this paper are not edded to the mixing aspect of Assumption E, and remain valid provided the partial sum processes involved in the construction of the statistics admit a functional central limit theorem. An important further example satisfying this condition is the linear process assumption of, inter alia, Phillips and Solo 992). Remark 3. The conditions placed on the vector x t in Assumption X are based on the mild regularity conditions of Phillips and Xiao 998). A leading example satisfying these conditions is given by the k-th order polynomial trend, x t =, t,..., t k ). Furthermore, the broken intercept and broken intercept and trend functions considered in, for example, Busetti and Harvey 2) are also permitted. Notice that, since the first element of x t is fixed at unity throughout, model ) alays contains an intercept. Folloing Kim 2) e consider to alternative hypotheses: the first, denoted H, is that y t displays a change in persistence from I) to I) behaviour at time t = T ], hile the second, H, is that there is a change in persistence from I) to I) behaviour at time t = T. Both may be expressed conveniently ithin a generalization of the persistence change data generating process DGP) of Kim 2,p.) y t = d t + z t,, t =,..., T,, ) 4) y t = d t + z t,2, t = T +,..., T. 5) The I)-I) persistence change alternative is obtained under the alternative hile the I)-I) alternative is given by H : z t,2 = z t,2 + σ t ε t 6) z t, = σ t u t z T,2 = z T, H : z t, = z t, + σ t ε t 7) z t,2 = σ t u t + z T,. An I) series is defined to be one formed from the accumulation of an I) series. 5

7 Both 6) and 7) embody end-effect corrections, as are also used in Banerjee et al. 992,p.278) and BT, hich ensure that a given realization of the process ill not display a spurious sharp jump in level at the break point. Under both H and H e require Assumptions V and X to hold on σ t and x t, respectively. Furthermore, e require that both ε t and u t are I), as stated in the folloing assumption. Assumption E. Both {ε t } and {u t } satisfy Assumption E ith strictly positive long-run variances, denoted by λ 2 ε and λ 2 u, respectively. Remark 4. Again, notice that under both H and H, 4)-5) reduces to the corresponding persistence change model in Kim 2) only here σ t = σ, t =,...T. 3 Persistence Change Tests Kim 2), Kim, J. et al. 22) and BT, develop tests hich reject the constant I) null H ) in favour of the I)-I) change alternative H ), based on the ratio statistic K) := T T ) 2 T t=+ S t )) 2 2 t= Ŝt)) 8) 2 here t t S t ) := ε i,, Ŝ t ) := ˆε i, 9) i=+ here, in order to obtain exact invariance to β the vector of parameters characterising d t ), ˆε t, are the residuals from the OLS regression of y t on x t, for t =,...,. Similarly, ε t, are the OLS residuals from regressing y t on x t for t = +,..., T. 2 Where the true potential) changepoint,, is knon the null of no persistence change is rejected for large values of K ). Hoever, in the more realistic case here is unknon, Kim 2), Kim, J. et al. 22) and BT consider three statistics based on the sequence of statistics {K), Λ}, here Λ = [ l, u ] is a closed subset of, ). These are: K := K 2 := T max Ks/T ) s { l T,..., ut } ut s= l T K 3 := ln T i= Ks/T ) ) ut s= l T exp Ks/T )) 2, 2 When constructing the sub-sample residuals, ˆε t, and ε t,, if any of the elements of x t, other than the first, are constant throughout the sub-sample they must be omitted from x t, in accordance ith the requirement that both x s) x s) ds and x s) x s) ds must be positive definite. 6

8 here T u T l T +. The first of these, after Andres 993), takes the maximum over the sequence, the second uses Hansen s 99) mean score statistic, and the third Andres and Ploberger s 994) mean-exponential statistic. In each case the null is rejected for large values of these statistics. In order to test H against the I)-I) change DGP H ), BT propose further tests based on the sequence of reciprocals of K), Λ; precisely, K := K 2 := T max Ks/T ) s { l T,..., ut } ut s= l T K 3 := ln T Ks/T ) ) ut s= l T exp 2 Ks/T ) ), and, in order to test against an unknon direction of change that is, either a change from I) to I) or vice versa), they also propose K 4 := maxk, K ), K 5 := maxk 2, K 2), K 6 := maxk 3, K 3). Representations for and critical values from the limiting distributions of the foregoing statistics under the null hypothesis )-3) in the constant unconditional volatility case, σ t = σ, for all t, are given in Kim, J. et al. 22) and BT. Crucially, they sho that these representations do not depend on the long run variance of {ε t }, λ 2 ε, even though neither the numerator nor the denominator of K) of 8) is scaled by a long run variance estimator. Although the original ratio-based tests of Kim 2), Kim, J. et al. 22) and BT are based on statistics here no variance estimator is employed, Leybourne and Taylor 24) have recently discussed tests based on statistics here the numerator and denominator of 8) are scaled by appropriate sub-sample long run variance estimators. Precisely, they consider replacing K) of 8), for each Λ, by the modified standardized) statistic K ) := ˆλ 2 m T, K) 2) λ 2 m T, here, folloing Kiatkoski et al. 992) [KPSS], ˆλ 2 m T, : = ˆε 2 t, + 2 k j/m T ) t= j= T λ 2 m T, : = T t=+ ε 2 t, + t=j+ T 2 T j= ˆε t, ˆε t j, k j/m T ) T t=j++ ε t, ε t j,, 7

9 ith k ) any suitable kernel function see Assumption K belo), are long run variance estimators applied to the first and last T sample observations respectively. The various tests for a change in persistence occurring at an unknon date are then constructed as above, replacing K) by K ) throughout. With an obvious notation e denote these statistics as Kj, j =,..., 6 and K j, j =,..., 3. The limiting null distribution of each of these statistics coincides ith that of the corresponding unstandardized statistic. Using the Bartlett kernel function k j/m T ) = ω B j/m T ), ω B x) := x) I x ) Leybourne and Taylor 24) find significant improvements in the finite sample size properties of the tests based on K ) in the presence of eak dependence in {ε t }. The bandidth parameter m T used in K ) is not required to gro to infinity as the sample size diverges to obtain pivotal limiting distributions. Indeed, Leybourne and Taylor 24) find that setting m T = or m T = 2 i.e., the sub-sample long run variance estimators contain either zero or one lagged covariance terms) provides a useful pragmatic balance beteen re-dressing the finite size problems of the tests under eakly dependence yet keeping poer losses, relative to the un-standardized tests, hen there is persistence change relatively small. 4 The Effects of Non-stationary Volatility In this section e derive the asymptotic distribution of existing persistence change tests of section 3 in the presence of time-varying unconditional variances satisfying Assumption V. First, in section 4., in order to assess hether the presence of non-stationary volatility might be confused ith a change in persistence, e derive representations for the asymptotic null) distributions of the persistence change tests under H. Second, in section 4.2, e analyze to hat extent non-stationary volatility affects the poer properties consistency) of the tests by analysing their large sample behaviour under H and H. In hat follos, to key processes ill play a fundamental role. The first is given by the folloing function in C: η s) := hich e term the variance profile. The second is the process ) s ωr) 2 dr ωr) 2 dr ; 3) B ω s) := s ωr)dbr) ωr)2 dr ) /2 hich, up to a scaling factor, is the diffusion solving the stochastic differential equation, db ω s) = ω r) db r), B ) a standard Bronian motion. 8

10 Remark 5. The variance profile satisfies η s) = s under constant unconditional volatility, hile it deviates from s if σ t is non-constant. Under Assumption V, the square of the denominator of 3), say ω 2 := ωr)2 dr, is the limit of T T t= σ2 t, hich may therefore be interpreted as the asymptotic) average unconditional) variance. Remark 6. Since B ω is Gaussian, has independent increments and unconditional variance EB ω s) 2 ) = η s), B ω is a time-change Bronian motion; see Cavaliere 24b) and Cavaliere and Taylor 26) for further discussion on such process. 4. Asymptotic Size Theorem provides representations for the limiting null distributions of the persistence change tests of Kim 2), Kim, J. et al. 22) and BT under non-stationary volatility satisfying Assumption V. Initially, e assume that the potential persistence change date is specified a priori. Theorem Suppose that {y t } is generated according to the DGP ) 3) under Assumptions V, E and X. Then, for any Λ, K ) of 8) satisfies K ) L ω ) := ) 2 B ω s, ) 2 ds 2 ˆB ω s, ) 2 ds here B ω s, ) := Q X B ω s;, ) B ω ) and ˆB ω s, ) := Q X B ω s;, ) are the residuals from the non-orthogonal Hilbert projections of B ω s) on the space spanned by x s), s [, ] and s [, ], respectively. Remark 7. The key implication of Theorem is that under non-stationary volatility the persistence change tests of section 3 do not have their usual asymptotic null distributions. Rather, their distributions depend on the sample path of the volatility process, ω ). Remark 8. In the special case ω ) = σ, B ω ) reduces to a standard Bronian motion and the above asymptotic distributions reduce to those given in Kim 2), Kim, J. et al. 22) and BT. We no derive the asymptotic null distributions of the tests hen the variance standardization of Leybourne and Taylor 24) is employed. To that end, e make the folloing assumption regarding the bandidth, m T, and kernel function, k ). Assumption K de Jong, 2). K ) For all x R, k x) and k x) = k x); k) = ; k x) is continuous at and for almost all x R; k x) dx < ; k x) l x), here l x) is a non-increasing function such that x l x) dx < ; K 2 ) m T as T, and m T = o T γ ), γ /2 /r, here r is given in E. Remark 9. Notice that under Assumption K, the bandidth parameter, m T, is assumed to increase as the sample size increases. This requirement is, hoever, not 9

11 strictly necessary and most of the results given in this paper continue to hold if m T = O ). In such cases, ˆλ 2 m T, and λ 2 m T, no longer consistently estimate the long run variance, even in the homoskedastic case. Consistent estimation of the long run variance is, hoever, not required to obtain asymptotically) similar tests under H. Theorem 2 Under the conditions of Theorem and provided that Assumption K also holds, then for any Λ, K ) of 2) satisfies K ) κ ω ) L ω ) =: L ω ) 4) here κ ω ) := [η)/ η))] is the ratio of the asymptotic average volatilities in the first and second sub-samples. Remark. As Theorem 2 demonstrates, the standardization suggested in Leybourne and Taylor 24) introduces the additional term κ ω ) into the asymptotic null distributions of the statistics, relative to those for the un-standardized statistics in Theorem. This term depends on the time-path of the volatility process, and equals unity if and only if the asymptotic average volatilities are equal in the first and second sub-samples. Notice, hoever, that κ ω ) does not depend on the long run variance λ 2 ε. Remark. As in Remark 8, in the special case here ω ) = σ, B ω ) reduces to the standard Bronian motion B ) and κ ω ) =, and, hence, the representation in 4) reduces to that given in Kim 2), Kim, J. et al. 22) and BT. Remark 2. Interestingly, in the special case of a single break in volatility occurring at time ε T, it can be shon that K ε ) L ε ), hich is therefore independent of the break in volatility. Hence, under these circumstances, a test based on K ε ) ould be correctly sized in the limit. In Theorem 3 e no generalize the foregoing results to the case of an unspecified persistence change date. Theorem 3 Under the conditions of Theorem, and defining a := u l ), K 3 sup L ω ) =: K,, K sup L ω ) =: K, K Λ u K 2 a { u ln a exp l 2 L ω ))d l } L ω ) d =: K 2,, K 2 =: K 3,, K 3 a ln Λ u L ω ) d =: K 2, { l u a exp } 2 L ω ))d =: K 3, hile K 4 maxk,, K, ), K 5 maxk 2,, K 2, ), and K 6 maxk 3,, K 3, ). Moreover, if Assumption K also holds, K 3 K 2 K sup Λ u a { u ln a exp l 2 L ω ))d L ω ) =: K,, K l } L ω ) d =: K2,, K 2 =: K 3,, K 3 sup a ln Λ u l L ω ) =: K, l L ω ) d =: K 2, { a u l exp } 2 L ω ))d =: K 3,

12 hile K 4 maxk,, K, ), K 5 maxk 2,, K 2, ), and K 6 maxk 3,, K 3, ). Remark 3. Notice that, even under the conditions of Remark 2, Kj, j =,..., 6, and K i, i =,..., 3, ill not have pivotal limiting null distributions because the asymptotic) invariance to the break in that case occurs only at = ε. 4.2 Consistency We no turn to an analysis of the consistency properties of the persistence change tests of section 3 under non-stationary volatility satisfying Assumption V. In sections 4.2. and e derive the large sample distributions of the basic and standardized statistics, respectively, of section 3, together ith the consistency rates of the associated tests, under the persistence change model H ; recall from Section 2 that this model corresponds to a change in persistence from I) to I) behaviour occurring at time T for some, ). Results for the tests under H are briefly discussed in section H : ratio tests We first analyze the behaviour of a test based on K ) in the folloing theorem, here the folloing notation is used: Bω ) := B ω ) I ), B ω ) := B ω and B ω ) := B ω. Theorem 4 Suppose that {y t } is generated according to the DGP 4)-5) under H of 6) and Assumptions V, E and X. Then, for < < <, K ) of 8) satisfies hile, for < <, K ) 2 Q x B ω s;, ) B ω ) ) 2 ds ) 2 Q x B ω s;, )) 2 ds 5) T 2 K ) 2 λ 2 ε ) 2 λ 2 u Q x B ω s;, ) B ω ) ) 2 ds ˆB ω s, ) 2 ds. 6) For the tests based on an unknon persistence change date, e have the folloing corollary of Theorem 4: Corollary Under the conditions of Theorem 4, provided [, ] [ l, u ], K i, i =,..., 6, are of O p T 2 ). Conversely K i, i =,..., 3, are of O p ). As can be seen from the results in Theorem 4, a persistence change test based on K ) ill be consistent at rate O p T 2 ) provided, as ill the tests based on the K i, i =,..., 6, statistics provided Λ i.e. provided the persistence changepoint is included in the search set). These are the same rates of consistency as hold for these

13 tests in the constant unconditional volatility case; see BT. Hoever, since all of these statistics scaled by T 2 ) have distributions hich depend upon the dynamics of the volatility process, it is anticipated that the finite sample poer of the associated tests ill depend on the time-series behaviour of the underlying volatility process. Notice also that although not consistent under H, the behaviour of tests based on the K ) statistic for > and on K i, i =,..., 3, ill also depend on the volatility process H : standardized ratio tests We no derive the large sample properties of the standardized persistence change tests of Leybourne and Taylor 24) under H. As discussed in section 3, these require a choice of the bandidth parameter, m T, hich, as ould be expected, affects the consistency rate under the alternative. This result is formalized in Theorem 5. Theorem 5 Let the conditions of Theorem 4 hold and let Assumption K hold. Then, for < < <, K ) of 2) satisfies K ) hile, for < <, m T T K ) P x B ω s;, ) ) 2 ds P x B ω s;, )) 2 ds η ) ) + k Q x B ω s;, ) B ω ) ) 2 ds Q x B ω s;, )) 2 ds Q x B ω s;, ) B ω ) ) 2 ds P x B ωs;, )) 2 ds ˆB ω s, ) 2 ds 7) 8) here k ) is the kernel function defined in Assumption K. For the case of an unknon persistence change date, e therefore have the folloing corollary: Corollary 2 Under the conditions of Theorem 5, provided [, ] [ l, u ], Ki, i =,..., 6, are of O p T/m T ) under Assumption K. Conversely K i, i =,..., 3, are of O p ). As ith the results in section 4.2., the standardized persistence change statistics have limiting distributions hich depend on the underlying volatility process, so that again the volatility process is anticipated to impact on the finite sample behaviour of the tests. Moreover, the rate of consistency of tests based on K ) is also sloed don, relative to those based on K ), since, under H, K ) is of O p T/m T ), provided. Again, these are the same rates of consistency as apply to these tests in the constant unconditional volatility case; see Leybourne and Taylor 24). Remark 4. It can be shon that Leybourne and Taylor s 24) suggestion of m T = yields tests, Ki, i =,..., 6, hich are consistent at rate O p T ), provided Λ. This result holds for any finite integer value of m T. 2

14 4.2.3 Results under H Under the alternative of a change from I) to I) behaviour at time T, that is H, a very similar analysis omitted in the interests of brevity) to that given above under H shos that for, K) [K ) ] is of O p T 2 ) [O p T/m T )], hile for <, K) and K ) are both of O p ). Consequently, if the intersection of the intervals [, ] and Λ is non-empty then K j, j =,..., 3, and K k, k = 4,..., 6, [K j, j =,..., 3, and Kk, k = 4,..., 6] are each of O pt 2 ) [O p T/m T )], but are otherise of O p ), hile the K j and Kj j =,..., 3, are each of O p ) for all Λ. As ith the results under H, the limiting distributions of all of these statistics scaled here appropriate) can be shon to depend on the dynamics of the underlying volatility process. 5 Bootstrap Persistence Change Tests In order to overcome the inference problems identified above ith the persistence change tests of section 3, in this section e propose bootstrap versions of these tests. We demonstrate that in the presence of volatility satisfying Assumption V the bootstrap tests provide asymptotically pivotal inference under H. We also derive their consistency properties under H and H. In order to account for x t, the test builds on Hansen s 2) heteroskedastic fixed regressor ild) bootstrap; see also Cavaliere and Taylor 25). Our bootstrap tests for both the knon and unknon changepoint cases are outlined in section 5.. Their large sample size and poer properties are established in sections 5.2 and 5.3 respectively. 5. The Bootstrap Algorithm The first stage of the bootstrap algorithm is to compute the full sample residuals, say ε t, obtained by regressing y t on x t for t =,..., T. A bootstrap sample is then generated as y b t := ε t t, t =,..., T, 9) ith { t } T t= an independent N, ) sequence. No, let ε b t, be defined as the residuals obtained from the OLS projection of y b t on x t for t = T +,..., T ; similarly, let ˆε b t, be defined as the residuals obtained from the OLS projection of y b t on x t for t =,..., T. The bootstrap analogue of K ) of 8) is then given by the statistic K b ) := T ) 2 T t ) 2 t=+ i=+ εb i, 2 t= t i= ˆεb i,) 2 2) hich corresponds to the statistic in 8) except that it is constructed from the pseudoresiduals ε b t, and ˆε b t, rather than the residuals based on the original time series, ε t, and ˆε t,, respectively. The associated bootstrap p-value is then given by p b T ) := 3

15 G b T K ) ; ), here Gb T ; ) denotes the cumulative distribution function cdf) of K b ). In the case of Leybourne and Taylor s 24) standardized version of K ), K ) of 2), the bootstrap p-value is given by p b T ) := G b T K ) ; ), here G b T ; ) denotes the cdf of the bootstrap statistic K b ) = ˆλ b 2 m b T, ˇλ b 2 m b T, K b ) here ˆλ b 2 b 2 and ˇλ are long run variance estimators of the same form as used m b T, m b T, in 2), ith bandidth m b T, applied to, respectively, the first and last T observations from the bootstrap sample, yt, b t =,..., T. Where the potential) changepoint is knon, the foregoing quantities are evaluated at =. Where the potential persistence change point is not specified a priori e form the corresponding bootstrap equivalents of the K j and Kj, j =,..., 6, and K j and K j, j =,..., 3, tests of section 3. For brevity, but ithout loss of generality, e shall confine our discussion to the K and K tests. Exactly the same reasoning extends straightforardly to the other tests in an obvious ay. The bootstrap analogue of K of ) is constructed as K b := max K b s/t ), s { l T,..., ut } ith the associated bootstrap p-value given by p b,t := Gb,T K ), here G b,t ) denotes the cdf of K b. The bootstrap version of the K test is constructed in a similar manner. Specifically, the bootstrap analogue of K of ) is given by K b := max K b s/t ) s { l T,..., ut } ith associated p-value p b,t := G b,t K ), here G b,t ) denotes the cdf of K b The bootstrap analogues of the K j and Kj, j = 2,..., 6, and K j and K j, j =,..., 3, statistics ill be denoted similarly as Kj b and Kj b, j = 2,..., 6, and K j b and K j b, j =,..., 3, respectively. Remark 5. In practice the cdfs G b T ; ), G b T ; ), Gb,T ) and G b,t ) ill be unknon. Hoever, they can be approximated in the usual ay. Taking the K statistic to illustrate the procedure is as follos. Generate N conditionally independent bootstrap statistics, K,i, b i =,..., N, computed as above but from yi,t b := ε t i,t, t =,..., T ith {{ i,t } T t=} N i= a doubly independent N, ) sequence. The simulated bootstrap p-value is then given by p b,t := N N i= I ) K,i b K. By standard arguments, see e.g. Hansen 996), p b,t is consistent for pb,t as N. Remark 6. As is ell knon in the ild bootstrap literature see Davidson and Flachaire, 2, for a revie) in certain cases better bootstrap accuracy can be obtained by replacing the Gaussian distribution used for generating the pseudo-data t in 9) 4.

16 by an asymmetric distribution ith E t ) =, E t 2 ) = and E t 3 ) = Liu, 988). A ell knon example of this is Mammen s 993) to-point distribution: P t =.5 5 )) = )/ 5 =: π, P t = )) = π. We found no discernible differences beteen the finite sample properties of the bootstrap persistence tests based on the Gaussian or Mammen s distribution. 5.. Asymptotic Size The next to theorems establish that in the presence of volatility satisfying Assumption V, the bootstrap p-values defined above are asymptotically pivotal and uniformly distributed and, hence, that the associated bootstrap tests are correctly sized for samples of sufficiently large dimension. Theorem 6 Under the conditions of Theorem : i) for all Λ, K b ) p L ω ), and p b T ) U[, ], a uniform distribution on [, ]; ii) K b p K, and p b,t U[, ]. Turning to the studentized bootstrap statistics, K b ) and K b, provided e make the additional assumption that ε t has finite fourth moments, the folloing results hold under H. Theorem 7 Under the conditions of Theorem 2, and if E ε 4 t ) < and m b T /T /2 as T, then: i) for all Λ, K b ) p L ω ) and p b T ) U[, ], and ii) p K, and p b U[, ]. K b,t Remark 7. Theorems 6-7 sho that as the sample size diverges, the bootstrap statistics, K b ), K, b K b ) and K b, have the same null distribution as those of the original statistics, K ), K, K ) and K, respectively, and, hence, that the associated bootstrap p-values are uniformly distributed under the null hypothesis, leading to tests ith asymptotically) correct size. These results hold for any volatility process satisfying Assumption V. Remark 8. In relation to the bootstrap K b ) and K b ) statistics, it is orth noting that m b T can either be fixed or diverge at rate ot /2 ). Moreover, m b T needs not equal the bandidth parameter, m T, used to compute the original statistic, K ) Consistency Rates We no consider the behaviour of the bootstrap tests of section 5. under the I)- I) persistence change alternative, H. We ill demonstrate that the bootstrap tests attain exactly the same rates of consistency as given for the corresponding standard test from section 4.2. Formal statements of the asymptotic distribution of the bootstrap statistics under H are provided in the appendix. 5

17 Theorem 8 Under the conditions of Theorem 4, for < <, K b ) = O p ) and K b ) = O p ). Consequently, provided, p b T p ). Moreover, provided [, ] [ l, u ], p b p,t. Theorem 9 Under the conditions of Theorem 5, and if E ε 4 t ) < and m b T /T /2 as T, then for < <, K b ) = O p ) and K b ) = O p ). Consequently, provided, p b T ) p ; furthermore, provided [, ] [ l, u ], p b p,t. Remark 9. An important consequence of the results in Theorems 8 and 9 is that, as ith their standard counterparts, the bootstrap K b ) and K b ) tests are consistent at rates O p T 2 ) and O p T/m T ), respectively, provided. This is the case because hile the bootstrap K b ) and K b ) statistics are both of O p ) for all, the K ) and K ) statistics diverge at rates O p T 2 ) and O p T/m T ), respectively, provided ; cf. Theorems 4 and 5. Similarly, the bootstrap Ki b and Ki b, i =,..., 6, tests are also consistent at rates O p T 2 ) and O p T/m T ), respectively, provided Λ. Notice, moreover, that these results hold irrespective of the choice of the bootstrap bandidth parameter, m b T. Remark 2. Observe from A.4) and A.7) in the proof of Theorems 8 and 9 that the limiting distributions of the bootstrap statistics under H again depend on the behaviour of the underlying volatility process through ω ) of 3). Hoever, it is important to note that these distributions are not the same as those obtained under H cf. Theorems 6 and 7) nor do they coincide ith those of the scaled) standard tests under H cf. Theorems 4 and 5). The asymptotic theory therefore predicts that the finite sample poer properties of the standard and corresponding bootstrap tests ill not, in general, coincide. Remark 2. Under H the bootstrap statistics all remain of O p ) for all and, hence, the bootstrap tests ill all have same rates of consistency as noted in section For example, bootstrap implementations of the K j, j =, 2, 3 and K i, i = 4, 5, 6 tests ill therefore all be consistent at rate O p T 2 ). Full details are omitted in the interests of brevity but are available on request. 6 Numerical Results In this section e use Monte Carlo simulation methods to compare the finite sample size and poer properties of the K, K, K 4, K, K and K4 persistence change tests of section 3, the tests being run at the nominal asymptotic) 5% level using the critical values from BT Table, p.38), ith their bootstrap counterparts of section 5, based on de-meaned x t = ) data, for a variety of volatility processes. 3 The finite sample size and poer properties of the tests are discussed in sections 6. and 6.2 respectively. 3 Results for the other persistence change tests discussed in this paper and for tests based on de-trended data are qualitatively similar and are available on request. 6

18 As is typical e take the search set Λ to be [.2,.8]. Results are reported for samples of size T = and 2, ith all experiments conducted using, replications and the rndkmn random number generator of Gauss 5.. All bootstrap tests used N = 4 bootstrap replications; cf. Remark 5. For the standardized ratio tests e set m T = thereby yielding OLS sub-sample variance estimators), as suggested by Leybourne and Taylor 24), and, accordingly, e also set a bandidth of m b T = in their bootstrap counterparts. Results are reported for the folloing models for σ t : Model. Single volatility shift): σ t = σ + σ σ )I t ε T ), ith ε =.5. Model 2. Trending volatility): Volatility follos a linear trend, beteen σ for t = and σ for t = T ; that is, σ t = σ + σ σ) t ), t =,..., T. T Model 3. Exponential near-) integrated stochastic volatility): Folloing Hansen 995, p.6), the volatility process is generated as σ t = σ exp νb 2 t/ T ) here b t is generated according to the first-order autoregression, b t = c/t )b t +k t, t =,..., T, ith k t NIID, ) and b =. Without loss of generality, e set σ = in all cases. For Model e let the ratio δ := σ/σ vary among {, /3, 3} notice that δ = yields a benchmark constant volatility process) so that both positive δ < ) and negative δ > ) breaks in volatility are alloed. For Model 2 e let δ := σ/σ take values among {/3, 3} so that both positively and negatively trending volatilities are generated. For Model 3 e consider ν = 5 and vary c among {, } Size Properties Table reports the empirical rejection frequencies sizes), for the K, K, K 4, K, K and K 4 tests for data generated according to the null DGP no persistence change) )-3) ith β = ithout loss of generality) and σ t generated according to the models detailed above. The innovation process {ε t } as generated according to the ARM A, ) design, ε t = φ ε t + v t θv t, v t NIID, ) ith φ, θ) {, ),.5, ),,.5)}, thereby alloing for IID, AR) and MA) innovations. Corresponding size results for the bootstrap counterpart tests are reported in Table 2. Consider first Model, the case of a single break in volatility. Where δ the results in Table highlight the presence of large size distortions in the basic persistence change tests. For δ = /3 the K test for a change in persistence from I) to I) is severely over-sized hen δ = /3 and severely under-sized hen δ = 3. The reverse 4 For each model other combinations of parameter values ere also considered, but these qualitatively add little to the reported results. 7

19 pattern holds for the K test for a change from I) to I). The K 4 test for either direction of change is severely over-sized for both δ = /3 and δ = 3. These size distortions vary slightly ith φ and θ, ith sizes increased decreased), relative to φ = θ =, hen φ > θ > ): this pattern is also observed under Models 2 and 3. The studentized K, K and K4 tests appear much better behaved avoiding the large over-size problems that are seen ith the basic tests hen δ. It should, of course, be stressed that these statistics do not have pivotal limiting null distributions cf. Theorems 2 and 3) and so hile the distortions are modest for the models considered here this should not be expected to necessarily hold in general. The studentised tests also appear somehat less dependent on φ and θ than the basic tests. Turning to Table 2, it is seen that the bootstrap tests also generally avoid the size distortions seen in the basic tests under the non-stationary volatility models considered and appear to deliver further improvements relative to the size properties of studentized tests, as should be expected; cf. Theorems 6 and 7. Tables 5 about here. The results for Model 2 in Tables and 2 suggest that in general, linear trending volatility has a loer impact on the size of the standard tests than abrupt changes, for a given value of δ, although here under-sizing occurs it tends to be slightly orse than under Model. The basic conclusions dran for the relative performance of the various tests for Model above appears germane here also. Finally for Model 3, severe over-sizing is again seen in the basic persistence change tests hich is greatest, other things equal, for the case of c =. The studentized tests again behave better but still significantly over-sized for c =. The bootstrap tests again appear to deliver a further improvement overall. Overall, across the volatility models considered, the bootstrap K b, K b and K4 b tests deliver the best size control among the tests considered in the presence of both non-stationary volatility and serially correlated innovations. 6.2 Poer Properties Tables 3 and 4 report the empirical rejection frequencies poers) and size-adjusted poers respectively, for the K, K, K 4, K, K and K4 tests for data generated according to the I) to I) sitching AR) DGP, here y t = ρ t y t + z t,, t =,..., T z t, = σ t ε t, ε t NIID, ) ρ t = {.8, t =,..., T., t = T +,..., T The persistence change-point is varied among {.25,.5,.75}, for the same set of models for σ t as considered in section 6.. Results for the corresponding bootstrap 8

20 tests are reported in Table 5. Recall that under H the K and K tests and their bootstrap analogues are not consistent. 5 For the case of homoskedastic errors, that is Model ith δ =, there tends to be a drop in empirical poer in using the bootstrap analogues of the basic K and K 4 tests, although in all but the case of =.75 these losses are generally quite modest. Consequently, in general, our bootstrap procedure does not seem to cause significant poer losses hen unnecessary. In contrast, significant poer losses are seen throughout in using the studentized K and K4 tests and their bootstrap analogues, hich display considerably loer poer than both the basic and bootstrapped basic tests under homoskedasticity. This ranking also holds true, in general, under the nonstationary volatility models considered. The effect of non-stationary volatility on poer is mixed and depends on hether e consider ra or size-adjusted poer for the basic tests, recalling that in some scenarios these ere heavily over-sized and in others badly under-sized. Different volatility models also have different impacts on the poer rankings beteen the various tests, as predicted by the large sample theory; cf. Theorems 4 and 5 and Remark 2. For example, under Model 3 the size-adjusted poer of the basic tests is much loer than for their bootstrap equivalents, hile under Models and 2 the opposite tends to be the case. Taking both size and poer results into consideration, e recommend the use of the bootstrap K, b K b and K4 b tests. Although these do not control size quite as ell as the bootstrap studentised K b, K b and K4 b tests in the presence of serially correlated innovations, they do not suffer the large poer losses associated ith the latter and do not require the additional assumption of finite fourth moments in {ε t }; cf. Theorem 7. 7 Application to U.S. Inflation Data In this section e apply, for Λ = [.2,.8] as in section 6, the K i, K j, i =,..., 6, j =,..., 3, and the studentised ratio tests Ki, K j, i =,..., 6, j =,..., 3, together ith the corresponding bootstrap tests to the monthly U.S. price inflation series from Stock and Watson 25). Specifically, e consider tenty series of inflation rates, measured as the first difference of the logarithm of the relevant monthly seasonally adjusted) price indices/deflators. The data are identified by the same reference codes as given on page 47 of Appendix A of Stock and Watson 25). The sample period used for all series as 967:-23:2. The series are graphed in Figure. In order to assess the time-series behaviour of volatility in these series e also graph in Figure 2 Cavaliere and Taylor s 26, section 4.) estimate of the variance profile, ωs) of 3), for each series. For almost all of the series the estimated variance profile shos substantial deviations from the 45 line hich pertains to a constant variance process; cf. Remark 5. Typically these patterns 5 Results for a corresponding I)-I) sitching AR) DGP ere also computed and gave qualitatively similar conclusions. These results are available on request. 9

21 are consistent ith the presence of multiple breaks in variance. For some series the breaks appear to follo relatively abrupt transition paths e.g. PWIMSA and PU83), hile for others e.g. PSM99Q and PUCD) the transition path tends to be sloer, consistent ith smooth-transition breaks. The estimated variance profile for PU85 follos a relatively smooth arc above the 45 line, consistent ith negatively trending volatility, or possibly a single relatively slo) smooth-transition variance break. Figures 2 and Tables 6 7 about here Tables 6 and 7 reports the outcome of the persistence change statistics for these data. All of the statistics ere computed on de-meaned data. For each outcome to bootstrap p-values are reported. The first, denoted p hom, is obtained from a standard bootstrap and the second, denoted p het, from using the ild bootstrap method of section 5. The standard bootstrap as implemented exactly as detailed in section 5 except that the bootstrap sample in 9) as replaced by yt b := t, t =,..., T. Finally, for the standardized ratio tests e set m T = m b T =, as in section 6. Consider first the results for the K 4 statistic hich does not assume a knon direction of persistence change a priori), in Table 6. Using the homoskedastic bootstrap p values it is seen that the null hypothesis of no persistence change can be rejected for 5 of the 2 series at the % level and for 8 of the 2 series at the 5% level. Hoever, using the heteroskedastic bootstrap p values reduces this to out of 2 significant at the % level and 5 out of 2 significant at the 5% level. In no case is the estimated p-value smaller for the heteroskedastic bootstrap-based tests. The most striking difference beteen the homoskedastic and heteroskedastic-based bootstraps is for PU84 here the former yields a significant outcome at the 5% level hile the latter is only just significant at the 5% level. Of the series here the K4 b test rejects the null hypothesis at the % level, namely PUNEW, PU83, PUCD, PUS, PUXHS, PUXM, GMDC, GMDCD, GMDCN and GMDCS, a comparison of the heteroskedastic) p- values for the outcomes of the K and K statistics are suggestive of I)-I) changes for PU83, PUCD and GMDC, and I)-I) changes for PUNEW, PUXHS, PUXM, GMDCD, GMDCN and GMDCS. The results for the tests based on K 6 are very similar to those discussed above for K 4, hile those for K 5 are again similar although both tests tend to be less significant in general. Specifically, K 6 yields 5 and 7 and 5) out of 2 significant rejections at the % and 5% levels level, respectively, based on the homoskedastic heteroskedastic) bootstrap, hile K 5 yields and 2 7 and 2) out of 2 significant rejections at the % and 5% levels level, respectively, using the homoskedastic heteroskedastic) bootstrap. Turning to the results for the studentised statistic in Table 7, a further reduction in the number of significant outcomes relative to those in Table 6 is seen. For example, the K4 statistic yields 3 and 6 7 and 3) out of 2 significant rejections at the % and 5% levels level, respectively, based on the homoskedastic heteroskedastic) bootstrap. The studentised K4 b test rejects the null hypothesis at the % level for PU85, PUCD, GMDC, GMDCD, GMDCN and GMDCS, and a comparison of the heteroskedastic) p-values for the outcomes of the K and K statistics are suggestive 2