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Harvey, David I. and Leybourne, Stehen J. and Taylor, A.M. Robert (04) On infimum Dickey Fuller unit root tests allowing for a trend break under the null. Comutational Statistics & Data Analysis, 78.

1 Harvey, David I. and Leybourne, Stehen J. and Taylor, A.M. Robert (04) On infimum Dickey Fuller unit root tests allowing for a trend break under the null. Comutational Statistics & Data Analysis, ISSN Access from the University of Nottingham reository: htt://erints.nottingham.ac.uk/33//zar.df Coyright and reuse: The Nottingham eprints service makes this work by researchers of the University of Nottingham available oen access under the following conditions. This article is made available under the University of Nottingham End User licence and may be reused according to the conditions of the licence. For more details see: htt://erints.nottingham.ac.uk/end_user_agreement.df A note on versions: The version resented here may differ from the ublished version or from the version of record. If you wish to cite this item you are advised to consult the ublisher s version. Please see the reository url above for details on accessing the ublished version and note that access may require a subscrition. For more information, lease contact erints@nottingham.ac.uk

2 On Infimum Dickey-Fuller Unit Root Tests Allowing for a Trend Break Under The Null David I. Harvey, Stehen J. Leybourne and A.M. Robert Taylor Granger Centre for Time Series Econometrics and School of Economics, University of Nottingham October 0 Abstract Trend breaks aear to be revalent in macroeconomic time series. Consequently, to avoid the catastrohic imact that unmodelled trend breaks have on ower it is standard emirical ractice to emloy unit root tests which allow for such effects. A oularly alied aroach is the infimum ADF-tye test. Its aeal has endured with ractitioners desite results which show that the infimum ADF statistic diverges to as the samle size diverges, with the consequence that the test has an asymtotic size of unity when a break in trend is resent under the unit root null hyothesis. The result for additive outlier-tye breaks in trend (but not intercet) is refined and shows that divergence to occurs only when the true break fraction is smaller than /3. An alternative testing strategy based on the maximum of the original infimum statistic and the corresonding statistic constructed using the time-reversed samle data is considered. Keywords: Unit root test; trend break; Minimum Dickey-Fuller test. Address corresondence to: David Harvey, School of Economics, University of Nottingham, Nottingham NG7 RD, UK. Tel.: +44 (0) Fax: +44 (0) dave.harvey@nottingham.ac.uk

3 Introduction Macroeconomic series aear to often be characterized by broken trend functions; see, inter alia, Stock and Watson (99,999,005) and Perron and Zhu (005). In a seminal aer, Perron (989) shows that failure to account for trend breaks resent in the data results in unit root tests with zero ower, even asymtotically. Consequently, when testing for a unit root it has become a matter of regular ractice to allow for this kind of deterministic structural change. While Perron (989) initially treated the location of the break as known, subsequent aroaches have focused on the case where the break location is unknown and is chosen via a data-deendent method; see, inter alia, Zivot and Andrews (99) [ZA], Banerjee et al. (99) and Perron (997); see also Pitarakis (0). Of these endogenised aroaches, the testing methodology roosed by ZA has been widely used by ractitioners (for a recent examle, see Alexeev and Maynard, 0). The aroach suggested by ZA is to calculate the ADF t-ratio-tye statistic of Perron (989) for all candidate break oints within a trimmed range and to then form a test based on the infimum (most negative) of this sequence of statistics. This infimum test is simle to comute and, by selecting the statistic within the sequence which gives most weight to the alternative, follows an established aroach to such roblems in econometrics. A significant drawback with the infimum aroach, however, is that it is redicated on the maintained hyothesis that no break in trend occurs under the unit root null hyothesis. This assumtion needs to be made in order for the infimum statistic to have a ivotal limiting null distribution. Investigating what haens when this maintained assumtion does not hold, results resented in Vogelsang and Perron (998) show that, for both sudden additive outlier () breaks and slowly evolving innovational outlier (IO) breaks, when a trend break of fixed non-zero magnitude occurs under the unit root null, so the infimum statistics diverge to as the samle size diverges and, hence, cause the tests to have asymtotic size of unity. In this aer we revisit this issue, focusing on -tye breaks in the trend (but not intercet). Our rimary contribution is to refine the theoretical results given in Vogelsang and Perron (998), showing that the divergence of the infimum statistic to occurs only when the true break fraction, τ 0 say, is smaller than /3. We also briefly consider an alternative testing strategy based on the maximum of the original infimum statistic and the corresonding statistic constructed using the time-reversed samle data. We find that such an aroach aears to offer considerable imrovements in finite samle size control relative to the original test, while retaining attractive ower roerties. In the following denotes the integer art of its argument, and denote weak convergence and convergence in robability, resectively, in each case as the samle size diverges to +, x := y ( x =: y ) indicates that x is defined by y (y is defined by x), and ( ) denotes the indicator function.

4 The Model and the Infimum Unit Root Test We consider a univariate time series {y t } generated by the DGP, y t = µ + βt + γdt t (τ 0 ) + u t, t =,..., T, () u t = ρu t + ε t, t =,..., T () where, for a generic fraction τ, DT t (τ) := (t > τt )(t τt ) in (), and τ 0 is an (unknown) utative trend break fraction, with associated break magnitude γ. The break fraction is assumed to be such that τ 0 Λ, where Λ := [τ L, τ U ] with 0 < τ L < τ U < ; the fractions τ L and τ U reresenting trimming arameters. In (), {u t } is an unobserved mean zero stochastic rocess, initialised such that u = o (T / ). Also, for simlicity of exosition, we will assume that ε t in () is an indeendent and identically distributed sequence with mean zero, variance σ ε and finite fourth moment. The theoretical results given in the aer would continue to hold under a more general weak deendence assumtion rovided the Dickey-Fuller-tye unit root regression in (3) below was augmented with k lags of the deendent variable and where k satisfies the usual condition that /k + k 3 /T 0 as T. We examine the roblem of testing the unit root null hyothesis H 0 : ρ =, against the alternative, H : ρ <, without assuming knowledge of where, or indeed if, the trend break occurs in the DGP. Let û t denote the residual from fitting the OLS regression of y t on z t := [, t, DT t (τ)] (we suress the deendence of û t and any associated OLS estimators on τ for convenience of notation), i.e. û t := y t ˆµ ˆβt ˆγDT t (τ) and let tˆφ (τ) denote the Dickey-Fuller t-ratio for testing φ = 0 in the fitted OLS regression û t = ˆφû t + ˆε t (3) that is tˆφ (τ) := T t= û tû t ˆσ ε T t= û t with ˆσ ε := (T ) T t= ˆε t. Then the infimum ZA-tye rocedure we consider is based on the statistic ZA := inf τ Λ tˆφ (τ). 3 Limiting Behaviour of ZA In order to derive the large samle behaviour of the ZA statistic we must first evaluate the limiting behaviour of the tˆφ (τ) statistic for τ τ 0. This is rovided in Theorem. Theorem Let y t be generated according to () and (), with γ = κσ ε, κ 0. Then for τ τ 0 T / tˆφ (τ) κ {N(, τ 0, τ) N(0, τ 0, τ) } κ { + κ M(τ 0, τ)} 0 N(r, τ 0, τ) dr

5 with N(r, τ 0, τ) := I r τ 0 (r τ 0 ) P P r I r τ P 3 (r τ), M(τ 0, τ) := L(τ 0, τ) {N(, τ 0, τ) N(0, τ 0, τ) } 0 N(r, τ, 0, τ) dr L(τ 0, τ) := ( τ 0 ) + P + ( τ)p3 ( τ 0 )P ( τ 0 I τ τ 0 (τ τ 0 ))P 3 + ( τ)p P 3 and P P P 3 := 3 ( τ) ( τ) (+τ) ( τ) ( τ) (+τ) ( τ) 3 3 ( τ 0 ) ( τ 0 ) (+τ 0 ) { τ 0 I τ τ 0 (τ τ 0 )} {+τ 0 3τ+4I τ τ 0 (τ τ 0 )} where I y x := (y > x). Notice that when τ = τ 0, we find that [P, P, P 3 ] = [0, 0, ] and N(r, τ 0, τ 0 ) = 0; consequently, the limit of T / tˆφ (τ 0 ) is undefined. We know, however, from Kim and Perron (009) that tˆφ (τ) has a well-defined limit distribution under H 0 when τ is within an o(t / ) neighbourhood of τ 0, thus tˆφ (τ) = O () here, and we will therefore consider the limit of T / tˆφ (τ 0 ) to be zero. For τ τ 0, Theorem imlies that tˆφ (τ) = O (T / ). Now since ZA takes the minimum value of tˆφ (τ) across all τ Λ, the ertinent issue is the sign of the limit of T / tˆφ (τ). More secifically, if, given a break at time τ 0, T / tˆφ (τ) is ositive for all τ Λ, then, since tˆφ (τ) + for all τ τ 0, ZA would not be minimised over this roblem region, but rather for a value of τ within a shrinking neighbourhood of τ 0. On the other hand, if T / tˆφ (τ) is negative for any τ Λ, then tˆφ (τ) for some τ τ 0, and consequently ZA also, resulting in unit asymtotic size. We now therefore examine the sign of the limit of T / tˆφ (τ) as a function of τ 0 and τ. The sign of the limit of T / tˆφ (τ) is determined by the sign of N(, τ 0, τ) N(0, τ 0, τ) as the other terms in the limit are unambiguously ositive (as is clear from the Proof of Theorem, + κ M(τ 0, τ) is the limit of ˆσ ε/σ ε and is therefore ositive). Next note that we can write N(0, τ 0, τ) = P, N(, τ 0, τ) = ( τ 0 ) P P P 3 ( τ) and so N(, τ 0, τ) N(0, τ 0, τ) = {( τ 0 ) P P P 3 ( τ)} P. First consider the case where τ < τ 0. Here, we find (uon simlification) P P P 3 = ( τ 0 ) (τ 0 τ) ( τ) 3( τ 0) (τ 0 τ) τ( τ) ( τ 0 ) (3τ 0 τ ττ 0 ) τ( τ) 3 yielding N(, τ 0, τ) N(0, τ 0, τ) = j τ,τ 0 h τ,τ 0 (4) 3

6 where j τ,τ 0 := ( τ 0 ) (τ 0 τ) ( τ + τ 0 τ) /4 ( τ) 4, h τ,τ 0 := τ 0 ( τ) ( τ 0 ) with j τ,τ 0 always ositive when τ < τ 0. Now, the function h τ,τ 0 is, for a given τ 0, monotonically decreasing in τ, and since τ L τ < τ 0 it is bounded by [τ 0 ( τ L ) ( τ 0 ), (τ 0 )( τ 0 )). We then find that For τ 0 < /3 h τ,τ 0 < 0 for all τ L τ < τ 0, { < 0 for τ L τ < 3τ 0 τ For /3 τ 0 < / h 0 τ,τ 0, 0 for 3τ 0 τ 0 τ < τ 0 For τ 0 / h τ,τ 0 > 0 for all τ L τ < τ 0. Next, when τ > τ 0 we have P P P 3 = τ 0 (τ τ 0 )( τ+τ 0 +ττ 0 ) τ (τ τ 0 )(τ τ 0 +ττ 0 3ττ 0 ) τ 3 τ 0 ( 3τ+τ 0+ττ 0 ) τ 3 (τ ) giving N(, τ 0, τ) N(0, τ 0, τ) = k τ,τ 0 l τ,τ 0 (5) where k τ,τ 0 := τ 0 (τ τ 0 ) (τ τ 0 ) /4τ 4, l τ,τ 0 := τ 0 τ( τ 0 ) with k τ,τ 0 always ositive when τ > τ 0. The function l τ,τ 0 is, for a given τ 0, monotonically decreasing in τ and since τ 0 < τ τ U it is bounded by (τ 0 (τ 0 ), τ 0 τ U ( τ 0 )]. Then, For τ 0 < / l τ,τ 0 < 0 for all τ 0 < τ τ U, < 0 for τ 0 τ < τ 0 ( τ For / τ 0 < /3 l 0 ) τ,τ 0 τ 0 for 0 ( τ 0 ) τ τ U For τ 0 /3 l τ,τ 0 > 0 for all τ 0 < τ τ U., Drawing on the above results for (4) when τ < τ 0 and (5) when τ > τ 0, we can then write For τ 0 < /3 lim(t / tˆφ (τ)) < 0 for all τ L τ τ U, { < 0 for some τ For /3 τ 0 < /3 lim(t / L τ τ U tˆφ (τ)), 0 for some τ L τ τ U For τ 0 /3 lim(t / tˆφ (τ)) > 0 for all τ L τ τ U. 4

$We therefore find that, under H 0, ZA will suriously reject with robability aroaching one in the limit rovided the true break fraction τ 0 lies below /3.$

7 To further illustrate the behaviour of T / tˆφ (τ), Figure dislays the regions in (τ, τ 0 ) sace where lim(t / tˆφ (τ)) is ositive/negative. Translating this behaviour into that of ZA, which locates the minimum of tˆφ (τ) across all τ Λ, we consequently obtain that { τ 0 < /3 ZA O () τ 0 /3. We therefore find that, under H 0, ZA will suriously reject with robability aroaching one in the limit rovided the true break fraction τ 0 lies below /3. It will, however, not suriously reject with robability one in the limit if τ 0 is /3 or above. It is this second finding that refines the result resented in Vogelsang and Perron (998), since the finding that surious rejections of the null occur with robability one in the limit is here shown to deend on where the true break is located. (Note that, in contrast, the innovational outlier version of the statistic diverges to for all τ 0, as stated by Vogelsang and Perron (998).) Of course, in an emirical setting where uncertainty exists as to the resence or location of a break, one is unlikely to have any confidence that the utative break will lie in the region τ 0 [/3, τ U ], and so the fundamental roblem raised by Vogelsang and Perron (998) of otentially serious over-sizing ersists in the ractical alication of ZA. lim(t / tˆφ(τ)) < 0 lim(t / tˆφ(τ)) > 0 Figure. Sign of lim(t /tˆφ(τ)) 4 Use of Time-Reversed Data In order to consider how the asymtotic over-size roblem region of a ZA -tye test might be reduced, we now consider the following imrovisation. If we time-reverse the data, i.e. consider {y T t+ } T t= in lace of {y t } T t=, then any break aearing in the first half of the original samle {y t} is translated into one occurring in the second half of {y T t+ }. Thus alication of ZA to {y T t+ }, which we denote by ZA, will deliver a test which does not suriously reject in the limit with robability one when a break occurs in the first third of {y t }. Of course, since in ractice we have no information regarding 5

8 the location of a break, ZA does not achieve robustness, since here a break in the last two-thirds of {y t } would cause surious rejection of the null by ZA. However, if we consider the maximum of ZA and ZA (cf. Leybourne, 995, in the context of unit root testing without allowance for a break in trend); that is, ZA max := max(za, ZA ) the ZA roblem of suriously rejecting the null with robability aroaching one when τ 0 [τ L, /3] restricted to the region τ 0 (/3, /3). Under H 0 and for the case γ = 0, it is straight- is for ZA max forward to show that where ZA max ( ) max inf Z(τ), inf Z (τ) τ Λ τ Λ Z(τ) := K(, τ) K(0, τ), Z (τ) := K (, τ) K (0, τ) 0 K(r, τ) dr 0 K (r, τ) dr with K(r, τ) and K (r, τ) the continuous time residual rocesses from the rojections of W (r), and W ( r), resectively, onto the sace sanned by {, r, (r τ)i r τ }, where W (r) is the Brownian motion rocess defined by T / rt t= ε t σ ε W (r). Table reorts asymtotic nominal 0.0, 0.05 and 0.0 level critical values for ZA max for a selection of commonly used trimming arameters. The critical values were obtained by direct simulation of (), aroximating the Wiener rocesses in the limiting functionals using N IID(0, ) random variates, with the integrals aroximated by normalized sums of,000 stes. Unreorted simulations, available from the authors on request, suggest that (i) ZA max dislays good finite samle size control for all τ 0 values, unless a large magnitude break occurs at τ 0 (/3, /3) in an extremely large samle, and (ii) ZA max has attractive ower roerties under H, regardless of whether or not a break is actually resent. Given that substantial over-size is only seen to occur for series that are unreresentative of those encountered in tyical economic alications, a ragmatic case could be made for using ZA max, although it is difficult to fully justify such an aroach, given that the test still has asymtotic size aroaching one when τ 0 (/3, /3). () Table. Asymtotic ξ-level critical values for the ZA max test [τ L, τ U ] ξ = 0.0 ξ = 0.05 ξ = 0.0 [0.05, 0.95] [0.0, 0.90] [0.5, 0.85] [0.0, 0.80] Acknowledgment Financial suort rovided by the Economic and Social Research Council of the United Kingdom under research grant RES is gratefully acknowledged by the authors.

9 References Alexeev, V. and Maynard, A. (0). Localized level crossing random walk test robust to the resence of structural breaks. Comutational Statistics and Data Analysis 5, Banerjee, A., Lumsdaine, R. and Stock, J. (99). Recursive and sequential tests of the unit root and trend break hyotheses: theory and international evidence. Journal of Business and Economics Statistics 0, Kim, D. and Perron, P. (009). Unit root tests allowing for a break in the trend function at an unknown time under both the null and alternative hyotheses. Journal of Econometrics 48, -3. Leybourne, S.J. (995). Testing for unit roots using forward and reverse Dickey-Fuller regressions. Oxford Bulletin of Economics and Statistics 57, Perron, P. (989). The great crash, the oil rice shock, and the unit root hyothesis. Econometrica 57, Perron, P. (997). Further evidence of breaking trend functions in macroeconomic variables. Journal of Econometrics 80, Perron, P. and Zhu, X. (005). Structural breaks with deterministic and stochastic trends. Journal of Econometrics 9, 5-9. Pitarakis, J.-Y. (0). A joint test for structural stability and a unit root in autoregressions. Comutational Statistics and Data Analysis, forthcoming. Stock, J.H. and Watson, M.W. (99). Evidence on structural instability in macroeconomic time series relations. Journal of Business and Economic Statistics 4, -30. Stock, J.H. and Watson, M.W. (999). A comarison of linear and nonlinear univariate models for forecasting macroeconomic time series. Engle, R.F. and White, H. (eds.), Cointegration, Causality and Forecasting: A Festschrift in Honour of Clive W.J. Granger, Oxford: Oxford University Press, -44. Stock, J. and Watson, M.W. (005). Imlications of dynamic factor analysis for VAR models, NBER Working Paer #47. Vogelsang, T.J. and Perron, P. (998). Additional tests for a unit root allowing the ossibility of breaks in the trend function. International Economic Review 39, Zivot, E. and Andrews, D.W.K. (99). Further evidence on the great crash, the oil-rice shock, and the unit-root hyothesis. Journal of Business and Economic Statistics 0,

10 Aendix: Proof of Theorem In what follows we can set µ = β = 0 without loss of generality. When τ τ 0 we have T T t= y t = T T t= u t + κσ ε T T t=τ 0 T + (t τ 0T ) κσ ε ( τ 0 ) /, T 3 T t= ty t = T 3 T t= tu t + κσ ε T 3 T t=τ 0 T + t(t τ 0T ) κσ ε ( τ 0 ) ( + τ 0 ) /, T 3 T t=τt + (t τt )y t = T 3 T t=τt + (t τt )u t + κσ ε T 3 T t=τt + (t τt )DT t(τ 0 ) κσ ε [{ τ 0 I τ τ 0 (τ τ 0 )} { + τ 0 3τ + 4I τ τ 0 (τ τ 0 )}]/. Examining the limit behaviour of the residuals û t we obtain T ˆµ ˆβ = ˆγ giving We also require T T T T t= t T T t=τt + (t τt ) T T t= t T 3 T t= t T 3 T t=τt + t(t τt ) (t τt ) T 3 T t(t τt ) T (t τt ) t=τt + T T t= y t T 3 T t= ty t T 3 T t=τt + (t τt )y t 3 ( τ) ( τ) (+τ) = : κσ ε P P P 3 ( τ) ( τ) (+τ) ( τ) 3 3 t=τt + 3 T t=τt + κσ ε( τ 0 ) κσ ε( τ 0 ) (+τ 0 ) κσ ε{ τ 0 I τ τ 0 (τ τ 0 )} {+τ 0 3τ+4I τ τ 0 (τ τ 0 )} T û rt = T y rt T ˆµ T ˆβ rt T ˆγI r τ ( rt τt ) = T u rt + κσ ε I r τ 0 (r τ 0 ) T ˆµ ˆβr ˆγI r τ (r τ) û t = y t ˆβ ˆγDU t (τ) κσ ε {I r τ 0 (r τ 0 ) P P r I r τ P 3 (r τ)} =: κσ ε N(r, τ 0, τ). = κσ ε DU t (τ 0 ) + u t ˆβ ˆγDU t (τ) T t= û t = κ σ ε(t τ 0 T ) + T t= u t + (T )ˆβ + ˆγ (T τt ) T T t= û t +κσ ε T t=τ 0 T + u t κσ ε (T τ 0 T )ˆβ κσ εˆγ T t= DU t(τ 0 )DU t (τ) ˆβ T t= u t ˆγ T t=τt + u t + ˆβˆγ(T τt ) σ ε[ + κ {( τ 0 ) + P + ( τ)p3 ( τ 0 )P ( τ 0 I τ τ 0 (τ τ 0 ))P 3 +( τ)p P 3 }] =: σ ε{ + κ L(τ 0, τ)} 8

11 and T ˆφ = T û T T û T T t= û t T 3 T t= û t N(, τ 0, τ) N(0, τ 0, τ) 0 N(r, τ 0, τ) dr. Now we can establish the limit of ˆσ ε ˆσ ε = T T t= ˆε t = T T t= û t + T ˆφ T 3 T t= û t T ˆφT T t= û tû t [ ] σ ε{ + κ N(, τ 0, τ) N(0, τ 0, τ) L(τ 0, τ)} + 0 N(r, τ κ σ 0, τ) ε N(r, τ 0, τ) dr dr 0 [ ] N(, τ 0, τ) N(0, τ 0, τ) 0 N(r, τ {κ σ εn(, τ 0, τ) κ σ εn(0, τ 0, τ) } 0, τ) dr [ ]) = σ ε ( + κ L(τ 0, τ) {N(, τ 0, τ) N(0, τ 0, τ) } 0 N(r, τ =: σ ε{ + κ M(τ 0, τ)}. 0, τ) dr It then follows using standard theory that T / tˆφ = T û T T û T T ˆσ εt 3 T t= û t t= û t κ {N(, τ 0, τ) N(0, τ 0, τ) } κ { + κ M(τ 0, τ)}. 0 N(r, τ 0, τ) dr 9

Johan Lyhagen Department of Information Science, Uppsala University. Abstract

Johan Lyhagen Department of Information Science, Uppsala University. Abstract Why not use standard anel unit root test for testing PPP Johan Lyhagen Deartment of Information Science, Usala University Abstract In this aer we show the consequences of alying a anel unit root test that