To conclude we suggest two areas of **possible** further research. First, in this paper we have focused attention on classical **testing** methods, rather than Bayesian approaches to **unit** **root** **testing** or other model selection based methods, which might also be fruit- fully explored. For further general discussion on these alternative approaches we direct the interested reader to, inter alia, Phillips (1991a,b), Phillips and Ploberger (1994) and Hansen (2007a,2007b). Second, our analysis allows for the possibility of a single **break** in **trend**. Extending the ideas in this paper to the case of multiple (deterministi- cally occurring) **trend** breaks should be feasible using similar computational methods to those used in Carrion-i-Silvestre et al. (2007). For any more than two breaks, how- ever, this procedure becomes problematic and a useful alternative is to consider the case where the breaks may be generated by an auxiliary stochastic component; impor- tant preliminary work in this area allowing for stochastic level shifts is considered in Cavaliere and Georgiev (2007).

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their analysis to the corresponding tests based on the M **testing** principle. We then demon- strate that HHLT’s modified **break** fraction estimator retains the same rates of consistency in both the **trend** **break** and no **trend** **break** cases in the **presence** of non-stationary volatility as were stated in HHLT for the case of constant unconditional volatility. We also show that in both the **trend** **break** and no **trend** **break** cases, the **unit** **root** test proposed in HHLT, based around this modified estimator, has a non-pivotal limiting distribution with its form depending on the underlying volatility process. The same is shown to be true of the corre- sponding M-type tests. The impact of a one-time change in volatility - including the case where this occurs simultaneously with a **break** in **trend** - on the asymptotic properties of these statistics is explored numerically through Monte Carlo simulations. In section 4 we propose a wild bootstrap-based implementation of the HHLT procedure. We demonstrate that the wild bootstrap analogue of the HHLT statistic replicates the first-order asymptotic null distribution of the standard HHLT statistic, such that the corresponding bootstrap tests are asymptotically valid, in the **presence** of non-stationary volatility. The same is shown to be true for the corresponding M-type tests. Simulation evidence presented in Section 5 suggests that the proposed bootstrap tests perform well in small samples. Concluding remarks are offered in section 6. Proofs are collected in an Appendix.

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Given the apparent prevalence of deterministic breaks in **trend** observed in macroeconomic time series data, it is now common practice to allow for such structural change when conducting **unit** **root** tests. Initial work by Perron (1989) assumed the location of a potential single **trend** **break** to be known, but more recent approaches have focused on the case where the **possible** **break** occurs at an unknown point in the sample; see, inter alia, Zivot and Andrews (1992) [ZA], Banerjee et al. (1992), Perron (1997) and Perron and Rodríguez (2003) [PR]. An important issue surrounding such procedures is that there is also an underlying problem of uncertainty as to whether **trend** breaks exist in the data or not. To illustrate the point, when a single **trend** **break** is known to be present, the test based on PR’s local GLS detrended ADF statistic which allows for a **trend** **break** is (near) asymptotically e¢ cient. This holds provided the **break** point is known, or can be dated endogenously with su¢ cient precision. However, when a **trend** **break** does not occur the PR test is not asymptotically e¢ cient, the redundant **trend** **break** regressor compromising power. Moreover, the asymptotic critical values for the PR test based on an estimated **break** point di¤er markedly according to whether a **trend** **break** occurs or not. In response to this problem, Kim and Perron (2009), Carrion-i-Silvestre et al. (2009) [CKP] and Harris et al. (2009) [HHLT] focused on developing **testing** procedures which utilize auxiliary statistics to detect the **presence** of **trend** **break**(s) occurring at unknown point(s) in the sample, and then use the outcome of the detection step to indicate whether or not the **unit** **root** test employed should include **trend** **break**(s) in the deterministic speci…cation. Assuming the **trend** **break** magnitudes to be …xed (independent of sample size), CKP and HHLT show their methods achieve asymptotically e¢ cient **unit** **root** inference in both the no **trend** **break** and **trend** **break** environments. Crucially they assume the **trend** **break** magnitude(s) to be …xed, which renders the **trend** **break** pre-tests used in these procedures consistent against breaks of …xed magnitude and so the correct **unit** **root** test variant (either allowing for **trend** breaks or not) is applied in large samples. However, in …nite samples the pre-tests will not provide perfect discrimination; i.e., some degree of uncertainty will necessarily exist in …nite samples as to whether breaks are present or not. As a result, the asymptotic properties of these procedures contrast sharply with the …nite sample simulations reported in CKP and HHLT which show the **presence** of pronounced “valleys” in the …nite sample power functions (mapped as functions of the **break** magnitudes), such that power is initially high for very small breaks, then decreases as the **break** magnitudes increase, before increasing again.

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It should be emphasized that, because any SV function L(n) possesses asymptotic order o( √ n), the OLS estimators, ˆ α and ˆ β, cannot be consistently estimated in the model (5). This result contrasts with the case where the simple **trend** t is employed. Considering models with an SV regressor, we therefore remark that the existence of a **unit** **root** leads to a meaningless regression and that **testing** for a **unit** **root** is indispensable.

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structural breaks in the deterministic **trend** of real wages (1973) and deflator (1965). Results in the Monte Carlo section show that the test loses some power in the pres- ence of positive autocorrelation for sample sizes below 200. Notwithstanding, the test still has enough power to reject the null hypothesis in all but four cases. Furthermore, the combined results of P97 and CS06 tests for the series, industrial production, em- ployment, GNP deflator, wages, real wages and money stock, can be interpreted and reconciled as follows. For all these series, the P97 test does not reject the null hypoth- esis of **unit** **root**, whereas the CS06 test does reject the null; the CS06 test rejects the null because one or more of the constraints related to the slope or the slope shift are not met, and not necessarily because of the absence of a **unit** **root**. These results imply the **presence** of a **unit** **root** and the absence of a drift/drift and shift, among others. Since our test also rejects the null hypothesis, we can conclude that all these series contain both, a deterministic and a stochastic **trend**. Moreover, besides the deterministic **trend**, our test shows that the GNP deflator and real wages also have a structural **break** in the deterministic rate of growth. The application of our test further refines the results of those of P 97 and CS06 tests. For example, the model Θ 5,1 (λ) of CS06 tests under

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In this paper we contribute to two separate literatures. Our principal contribution is made to the literature on **break** fraction estimation. Here we investigate the properties of a class of weighted residual sum of squares estimators for the location of a level **break** in time series whose shocks display non-stationary volatility (permanent changes in unconditional volatility). This class contains the ordinary least squares (OLS) and weighted least squares (WLS) estimators, the latter based on the true volatility process. For fixed magnitude breaks we show that the estimator attains the same consistency rate under non-stationary volatility as under homoskedasticity. We also provide local limiting distribution theory for the estimator when the **break** magnitude is either local-to-zero at some rate in the sample size or exactly zero. The former includes the Pitman drift rate which is shown via Monte Carlo experiments to predict well the key features of the finite sample behaviour of the OLS estimator and a feasible version of the WLS estimator based on an adaptive estimate of the volatility path of the shocks. The simulations highlight the importance of the **break** location, **break** magnitude, and the form of non-stationary volatility for the finite sample performance of these estimators, and show that the feasible WLS estimator can deliver significant improvements over the OLS estimator in certain heteroskedastic environments. We also contribute to the **unit** **root** **testing** literature. We demonstrate how the results in the first part of the paper can be applied, by using level **break** fraction estimators on the first differences of the data, when **testing** for a **unit** **root** in the **presence** of **trend** breaks and/or non-stationary volatility. In practice it will be unknown whether a **trend** **break** is present and so we also discuss methods to select between the **break** and no **break** cases, considering both standard information criteria and feasible weighted information criteria based on our adaptive volatility estimator. Simulation evidence suggests that the use of these feasible weighted estimators and information criteria can deliver **unit** **root** tests with significantly improved finite sample behaviour under heteroskedasticity relative to their unweighted counterparts.

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The use of cointegration has reached the rank of a standard econometric tool since the seminal work of Engle and Granger (1987). This method has the great advantage that nonstationary **unit** **root** process can be examined without differencing. The existence of a cointegration relations leads to the interpretation that there is stationary equilibrium between nonstationary time series, which is attractive for empirical research in economics and especially in finance. In this view, it seems to be natural, that the concept of frac- tional cointegration has attracted more attention than it can be seen as a generalization of the standard cointegration concept. The fractional methodology has the great advan- tage, that the integration order is not stucked to an integer digit anymore and opens up a very wide range of modeling empirical specifics like long range dependencies. This leads to the interpretation of fractional integrated processes for long memory property as the coefficients of an infinite moving average representation is decaying in a hyperbolic way. It should be regarded, that for most time series, the hypothesis d = 1 can’t be rejected even if fractional integration is taken into account. The concept is more interesting to model cointegration error processes as it shows the persistence of exogenous shocks to a system of time series. The possibility of fractional integrated errors can also be seen as a reason for rejecting cointegration in classical approaches.

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of a change occurs instantaneously, its effect most likely evolves gradu- ally over a period of transition. To account for that effect Perron (1989a) proposed the so-called innovational outlier model, which assumes that “the economy responds to a shock to the **trend** function the same way as it reacts to any other shock”, Perron (1989a, p.1380). This amounts to adding a step dummy variable to the augmented Dickey-Fuller regression instead of ap- plying the ADF test after removing all deterministics. Leybourne, Newbold and Vougas (1998) considered **unit** **root** **testing** in the **presence** of more gen- eral deterministic smooth transition functions, see also Lin and Ter¨asvirta (1994), however without providing asymptotic theory. Limiting results un- der smooth transitions have been established in Saikkonen and L¨ utkepohl (2001) for known breakpoint, and in Saikkonen and L¨ utkepohl (2002) in case the date of the **break** is not known a priori. For a comparison of related tests see also Lanne, L¨ utkepohl and Saikkonen (2002).

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been able to apply this **unit** **root** **testing** framework in judging the stationarity of the US FX rates for twenty-two (22) different currencies, cutting across America, Europe, Asia-Pacific and Southern Africa using three different data frequencies – daily, weekly and monthly were used in the study, with the duration of the data capturing significant periods of financial crisis and/or some other peculiar events. These events caused some level(s) of shifts, which resulted in structural breaks in the **trend** pattern of the series. A similar feat was observed in the preliminary analysis for the three different frequencies, whereby the FX rates revealed the **presence** of heteroscedasticity among residuals and implied that all the FX series exhibited ARCH effect at higher lag. Consequently, our findings indicated the appropriateness of adapting a parsimonious GARCH process in the residuals, in contrast to the white noise disturbance assumption. Also, with significant **trend** estimates for both the OLS regression (**Trend**) and the regression with the inclusion of dummies for the structural breaks (Trend1), the importance of the inclusion of a **trend** term in the model for FX rates cannot be overemphasized.

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Perron (1989) analyzes the properties of classical **unit** **root** tests in the **presence** of a structural change. There he is able to show that ignoring a structural **break** in the **trend** function leads to a remarkable reduction of power of **unit** **root** tests. For this reason Perron develops a Dickey-Fuller-(DF-)type **unit** **root** test which explicitly accounts for a **break** with known **break** date. He distinguishes two **testing** approaches, that di¤er in their assumptions about the adjustment process towards the new equilibrium after a shock. The additive outlier (AO) model assumes an instantaneous adjustment, whereas in the innovational outlier (IO) model the adjustment takes place gradually.

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Recently there has been much interest in **testing** for a **unit** **root** in a time series that has a **trend** **break**. Leybourne, Mills and Newbold (1998) study the behavior of standard **unit** **root** tests when the data generating process contains a **trend** **break** not accounted for by the fitted model. For data consisting of a random walk with a shift in level, they report empirical sizes less than the nominal level when the shift is not too near the beginning of the series. However, if the shift is near the beginning of the series, they report too many rejections of the **unit** **root** null hypothesis. Thus a **unit** **root** process with an early level shift is too often declared stationary. Leybourne et al (1998) call this the ‘converse Perron phenomenon’ in contrast to the well known ‘Perron phenomenon’ investigated by Perron (1989).

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→ τ ∗ i whenever β 3,i 6 = 0 at the same rate as b τ → p τ ∗ in the single **break** case considered above. For Model A, it would seem likely that the same parallel with the single **break** case would hold, but formally Chang and Perron (2016) only consider the case of a single **break** in **trend**. For both Models A and B one would also need to formally establish that analogous uniformity arguments to those made in the proof of Theorem 1 can also be made in those cases where β 3,i = 0. Remark 15. Although based on different models, it is nonetheless worth noting an important differ- ence between the large sample results in Theorem 1 and those which hold for autoregressive **unit** **root** tests and stationarity tests which allow for the possibility of **trend** **break**(s). The limiting distributions of these, under both the null and the relevant local alternatives, depend on the number of **trend** breaks fitted, the number of breaks present in the data and the locations of these; see, for example, Perron and Rodr´ıguez (2003) in the context of **unit** **root** tests, and Busetti and Harvey (2001,2003) in the context of stationarity tests. Moreover, their asymptotic local power functions depend on the number of **trend** breaks fitted, decreasing the more breaks are fitted, other things equal. This is not the case in our setting where, as the results in Theorem 1 demonstrate, the limiting distribution of our feasible LM ( b τ ) statistic is independent of any nuisance parameters arising from the deterministic kernel under both the null hypothesis and local alternatives. However, it is important to emphasise that this is an asymptotic result and so it will be important to investigate how well this asymptotic prediction holds up in finite samples. This we will investigate by Monte Carlo simulation methods in section 4. Remark 16. Consider the case where an observed time series x t satisfies the DGP

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The **presence** of structural change in time series can lead to erroneous inference and cause **unit** **root** tests, e.g. the Dickey-Fuller test, to be biased. Therefore, we are interested in discovering and **testing** for any **possible** structural changes in our data. The rst step is the visual inspection of each country's REER development over time in a time series plot, as can be seen in Figure 1 through 10. Through visual inspection of the graphs it is easy to spot sudden changes in the **trend** of some of the data. The time-series graphs show that the REER has been quite volatile over time in all countries. For most, if not all, countries, it is also easy to draw the conclusion that we have structural breaks in the data set. We also test for the **presence** of structural change using the supF-test Andrews (1993).

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What all of the aforementioned **unit** **root** test procedures have in common is that they treat the location of the **break** as unrestricted, other than making various arbitrary assumptions to exclude a common proportion of **break** dates at the beginning and end of the sample period (so-called trim- ming). However, it is often the case that a practitioner will have some degree of con dence as to the approximate location of a putative **break**, despite not knowing it precisely. Andrews (1993), in the context of **testing** for general structural instability, introduces this possibility motivated by two sets of examples: (i) where a political or institutional event has occurred during a de ned time-frame (e.g. a war) but it is unknown exactly when any change-point takes e ect; (ii) where an event occurs at a known date but its e ect is either anticipated or occurs after a delay. In each case, an analyst has information on the approximate timing of any **break**, but remains unsure over its exact date and its magnitude, or indeed its **presence** at all.

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Preliminary results suggest that this may be the case, but more extensive numerical experiments are required for a confident conclusion. The second direction is studying **trend**-and-**break** time series. Since breaks of three types are **possible** in this case (a change in intercept, a change in slope of the **trend** function, and the combination of both changes), it is interesting how the nonlinear-test statistic differs across these types. Judging from the fact that the asymptotic distributions of the Perron statistic for Θ = 0 and Θ = 1 coincide with the Dickey-Fuller distribution associated with regression having the intercept and **trend**, the nonlinear test statistic may be expected to have the latter (limiting) distribution for all three types of the **break**.

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In situations such as these, where inference depends on the choice of deterministic speci…cation, a practitioner uncertain about the **presence** or otherwise of a **trend** must choose which test to employ. A risk-averse strategy might be to always employ **trend**- invariant **unit** **root** tests. However, Marsh (2007) shows that, in the case of standard DF tests where data is generated by a linear AR process, the Fisher information for a test statistic invariant to a linear **trend** is zero at the **unit** **root**. Consequently, when a **trend** is absent, the power of a **unit** **root** test that is invariant to a **trend** will be compromised relative to the power of an appropriate demeaned but not detrended test statistic. Harvey et al. (2009) show that these power losses can be substantial, therefore opting to always use the **trend**-invariant test is a costly strategy. Conversely when a **trend** is present, the power of a DF test that is demeaned but not detrended is shown to decrease as the magnitude of the **trend** increases. Motivated by these considerations, in this paper we not only examine the power performance of a union of rejections based on demeaned tests and a union of rejections based on demeaned and detrended tests, but also consider a union of rejections based on all four **possible** tests (i.e. demeaned DF and KSS tests, and demeaned and detrended DF and KSS tests). This union procedure is shown to achieve attractive power levels across all settings of the **trend** coe¢ cient.

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We can summarise the results in this subsection by observing that lag order selection based on MAIC has a negative impact on the finite sample power of the resulting wild bootstrap ADF **unit** **root** test if nonstationary volatility is present, with the extent of this effect depending on the specific volatility model. Based on our results, we recommend the use of the RSMAIC lag selection criterion for selecting the lag length in the context of ADF **unit** **root** **testing**, given its greater degree of robustness to nonstationary volatility than the standard MAIC lag selection criterion, and the resulting higher finite sample power which is achievable when using RSMAIC over MAIC. These power gains are most strongly seen for single **break** in volatility models. Moreover, under homoskedasticity we found almost no differences in power between the **unit** **root** tests which use RSMAIC and MAIC to select the lag order. Under all of the volatility and ARMA models considered the finite sample size properties of the **unit** **root** tests based on MAIC and RSMAIC were virtually identical. As such we believe it provides a reliable practical alternative to MAIC.

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Several researchers including Perron (1989, 1990), Rappoport and Rechlin (1989), Zivot and Andrews (1992), Lumsdaine and Papell, and Bai and Perron (1998) have recognized alternative **trend** specifications in **testing** for the **unit** **root** hypothesis. This strand of literature has focused on models with segmented line trends; and single or multiple breaks (Vougas, 2006). Yet, another strand of literature has developed **unit** **root** tests where the alternative hypothesis is that of stationarity around a smoothly changing **trend**. Leybourne, Newbold and Vougos (1996, 1998) (LNV, hereafter) and Sollis (2004) used logistic **trend** functions 3 that allow for a smooth **break** in the deterministic **trend** of the data. Bierens (1997) modeled nonlinear **trend** using Chebyshev polynomials, while Becker et al. (2006) used trigonometric functions (via means of Fourier transformations) to model **possible** gradual breaks in the data generating process. The use of either Chebyshev polynomials or trigonometric functions might be problematic, because there is no unique way of choosing the order of polynomials or the frequency components for the trigonometric functions. However, in the case of logistic **trend** functions the parameters of interest in the gradually changing **trend** function may be estimated using a convenient nonlinear estimation algorithm. By the same token, smooth transition regression (STR) models have also been proved to capture gradual structural breaks quite well (e.g., Granger and Teräsvirta, 1994; Lin and Teräsvirta, 1994; Greenaway et al. 1997). Moreover, the STR type **trend** modeling can incorporate broken or unbroken **trend** lines, thereby allowing for gradual as well as abrupt **break** (Vougas, 2006). Along these lines, the STR type of **trend** modeling can also be seen as a generalization of the first strand of **trend** modeling. Due

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Table 2. present the result for the **unit** **root** test, where ADF and PP test has the null hypothesis that the series is integrated of order 1, while KPSS null hypothesis is that the series is stationary. We find that for euro/dollar, euro/£, euro/CHF, euro/yen we reject the **unit**-**root** hypothesis based on the ADF test, PP test find that the returns don’t have a **unit**-**root**, the KPSS accept the stationary of all series.

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