Bootstrap tests for time varying cointegration

ABSTRACT This article proposes wild and the independent and identically distibuted (i.i.d.) parametric bootstrap implementations of the time-varying cointegration test of Bierens and Martins (2010). The bootstrap statistics and the original likelihood ratio test share the same first-order asymptotic null distribution. Monte Carlo results suggest that the bootstrap approximation to the finite-sample distribution is very accurate, in particular for the wild bootstrap case. The tests are applied to study the purchasing power parity hypothesis for twelve Organisation for Economic Cooperation and Development (OECD) countries and we only find evidence of a constant long-term equilibrium for the U.S.–U.K. relationship.


Introduction
Structural change is of key importance in economics and econometrics, especially for cointegration analysis, as it normally involves long term historical trends, which, consequently, are likely to display breaks in their equilibrium relationship. In a recent article, Bierens and Martins (2010) proposed a vector error correction model in which the cointegration vectors change smoothly over time and, from that model, a likelihood ratio test for standard time-invariant cointegration. Despite the simplicity of the test, the asymptotic chi-square distribution appears to be a poor approximation to the relevant nitesample distribution. In particular, the test falsely indicates the existence of time-varying cointegration too o en.
To address this problem, we propose in this article a bootstrap algorithm for obtaining critical values and show that this alternative approach does lead to test procedures that have, approximately, the correct size. As in many other estimation and inference contexts, in our case the bootstrap distribution is also an accurate approximation to the nite-sample one under the null hypothesis of constant cointegration vectors, as initially presented by Johansen (1988Johansen ( , 1991Johansen ( , 1995. It has extensively been shown in the literature that bootstrap methods provide higher order asymptotic re nements and, thus, better results in bias reduction, con dence interval construction, and hypothesis testing in nite-samples, even in the cases where analytical results are known. See, for example, Jeong and Maddala (1993); Horowitz (2001), as well as Li and Maddala (1996) and Härdle et al. (2001) surveys, the last two in the time series context. As an example in the structural breaks literature, Diebold and Chen (1996) demonstrate the bene ts of bootstraping the "supremum" tests of Andrews (1993).
The likelihood ratio bootstrap tests we suggest follow very closely those already proposed in the literature for testing and determining the cointegration rank in Vector Autoregressive Model (VAR). Swensen (2006Swensen ( , 2009 consider an independent and identically distubuted (i.i.d.) bootstrap version of the pseudo Likelihood Ratio (LR) trace test statistic whereas Cavaliere et al. (2010a) advocate the usage of the wild bootstrap instead as it provides better results under conditional heteroskedasticity when compared to the i.i.d. procedure. Moreover, Cavaliere et al. (2012) propose a bootstrap scheme that improves Swensen's approach to create bootstraped data once the VECM is estimated under the null hypothesis and not under both hypothesis as in Swensen (2006). 1 The contents of the article are as follows. In Section 2, we review the time-varying Vector Error Correction Model (VECM) and the original LR test for time-invariant cointegration and introduce the wild and i.i.d. parametric bootstrap versions of the pseudo LR test, showing their consistency because the three share the same chi-square asymptotic distribution. Section 3 sets out the designs of the Monte Carlo experiments and suggests that the bootstrap approximation to the nite-sample distribution is very accurate, especially for the wild bootstrap. Finally, Section 4 discusses the purchasing power parity hypothesis in the context of time-varying cointegration and reports results from an application of the methodology to international prices and nominal exchange rates.

The time-varying cointegration model
As in Bierens and Martins (2010), consider the time-varying VECM(p) with a dri , where Y t ∈ R k , ε t ∈ R k , µ, α, , and the Ŵ j 's are xed coe cients, the β t 's are time-varying (TV) k × r matrices of cointegrating vectors, and T is the number of observations. 2 The initial values Y t , t = 0, −1, . . . , −p+1, are assumed to be xed. The LR test is de ned for the null hypothesis of standard timeinvariant (TI) cointegration, β t = β for all t, against time-varying cointegration (TVC) in which the cointegrating relationship varies smoothly over time, maintaining the number of cointegration relations as equal to r < k. In this test setup, there must exist a xed certain number of cointegration relations, r > 0. Time-varying cointegrating systems do not necessarily generate time-varying cointegration spaces. The conditions for the existence of time-invariant or time-varying cointegration spaces can be found at Martins and Gabriel (2013). Assuming that the function of discrete time β t is smooth (see Bierens and Martins, 2010 for details), it can be written as for some xed m < T − 1, where the orthonormal Chebyshev time polynomials P i,T (t) are de ned by P 0,T (t) = 1, P i,T (t) = √ 2 cos (iπ (t − 0.5) /T), t = 1, 2, . . . , T, i = 1, 2, 3, . . . , m, and ξ i,T = 1 T T t=1 β t P i,T (t) are unknown k × r matrices. Similar to Johansen (1988Johansen ( , 1991Johansen ( , 1995, model (1) can be speci ed more conveniently as  Giersbergen (1996), Harris and Judge (1998), and Mantalos and Shukur (2001) focus on the properties of the bootstrap by means of Monte Carlo simulations. For bootstrapping methods in standard unit root testing, see, for example, Park (2003) and Paparoditis and Politis (2003). 2 In this Bierens and Martins (2010) baseline model, there is a drift in Y t which is denoted by µ. This does not cover all the usual leading cases for the deterministic components within standard cointegration test frameworks. In particular, we do not consider the cases of the intercept being absorbed into the cointegration relation and the existence of a linear trend. Accounting for a trend is not straightforward in time-varying cointegration as pointed out by Bierens and Martins (2010).
. Under the null hypothesis, ξ ′ = β ′ , 0 r,k.m , where β is the standard k × r matrix of TI cointegrating vectors, so that then αξ ′ Y (m) . Therefore, given m and r, the LR test takes the form where 1 > λ m,1 ≥ λ m,2 ≥ · · · ≥ λ m,r ≥ · · · ≥ λ m,(m+1)k are the ordered solutions of the generalized The λ 0,j 's are similarly de ned by imposing m = 0 (standard cointegration). Assuming that the errors are i.i.d. Gaussian, i.e., ε t ∼ i.i.d. N k [0, ] , Bierens and Martins (2010) show that, given m ≥ 1 and r ≥ 1, under the null hypothesis of standard cointegration, LR tvc m,T is asymptotically χ 2 mkr distributed. They also concluded that, for small T and large m, the test su ers from size distortions and tends to overreject the correct null hypothesis of standard cointegration. Given this result, we propose in the next Section two bootstrap versions of LR tvc m,T along the lines of Cavaliere et al. (2010aCavaliere et al. ( , 2012 and Swensen (2006).
Following the seminal work by Johansen (1988Johansen ( , 1991Johansen ( , 1995, Bierens and Martins (2010) take the normal distribution for ε t and from that derive the exact likelihood function and LR statistic. Actually, imposing normality is not necessary for deriving the asymptotic distribution of LR tvc m,T under the null hypothesis. By simply assuming i.i.d. errors, a straightforward but tedious exercise would be to show that the pseudo LR tvc m,T statistic is also asymptotically χ 2 mkr distributed when based on the pseudo Gaussian likelihood function. 3 In terms of the bootstrap approach, we relax the original assumptions by taking those at Cavaliere et al. (2010a) under our context of standard cointegration Assumption 1. (a) For m = 0, the usual conditions related to the characteristic roots and the nonsingularity of matrix α ′ I k − Ŵ 1 − · · · − Ŵ p−1 β are taken to be true. (b) The innovations {ε t } form a martingale di erence sequence with respect to the ltration F t , where F t−1 ⊆ F t for t = ..., −1, 0, 1, 2, . . . , satisfying (b1) the global homoskedasticity condition and (b2) E ε t 4 ≤ K < ∞.

Remark .
Here, {ε t } is serially uncorrelated and possibly conditionally heteroskedastic. Instead, Cavaliere et al. (2012) and Swensen (2006) assume i.i.d. innovations. See Cavaliere et al. (2010a) for more details about the underlying di erences and the functional central limit theorem and the law of large numbers that apply to martingale di erence sequences.

Bootstrap tests
The parametric bootstrap versions of the TVC test statistic we consider are the wild bootstrap and the i.i.d. bootstrap. They both are implementations to TVC of the Cavaliere et al. (2010a, 2012) (wild andi.i.d.) and Swensen (2006) (i.i.d.) bootstrap procedures for testing the cointegration rank. Next, we de ne the bootstrap procedures using the unrestricted residuals (and µ, Ŵ) as in Cavaliere et al. (2010a) and Swensen (2006), and at the end of this section we de ne the alternative procedure based upon the restricted residuals (and µ, Ŵ) as in Cavaliere et al. (2012). As in Cavaliere et al. (2010a), we construct the pseudo Gaussian likelihood function and the pseudo LR test by assuming i.i.d. Gaussian errors. For the unrestricted TVC model (3), m > 0, let µ, Ŵ denote the pseudo maximum likelihood (ML) estimate of (µ, Ŵ) and ε t , t = p + 1, . . . , T, the pseudo ML residuals. Moreover, let α, β denote the restricted pseudo ML estimates of (α, β) under the null hypothesis of standard cointegration, m = 0. The bootstrap algorithms are described as follows.

Wild Bootstrap Test for TVC Using the Unrestricted Residuals
recursively from the equation with initial values Y b t = Y t , t = 1, . . .
Fi h, for a signi cance level δ, reject the null hypothesis of standard cointegration if p b m,T < δ. As pointed out by Cavaliere et al. (2010a), an alternative could be drawing from Eq. (10) but without the estimate of the deterministic part, µ, and setting initial values to zero, Swensen (2006), but also  for a discussion on why the rank statistic is invariant with respect to µ and recall that Bierens and Martins (2010) showed that LR tvc m,T has the same distribution whether or not there is an intercept in the model. Swensen's i.i.d. Bootstrap Test for TVC. This is the same as the wild bootstrap approach, except that the bootstrap pseudo-disturbances ε b t , t = p + 1, . . . , T, are drawn randomly with replacement from the residuals ε t . Following the same argument as in Cavaliere et al. (2010a), obtain instead the centered if drawing from Eq. (10) without the intercept term.
The number of bootstrap pseudo-samples B must be "large enough" since p b m,T converges to p b m,T as B increases. In their numerical experiments, Cavaliere et al. (2010aCavaliere et al. ( , 2012 set B = 399, and Swensen (2006) considers B = 5, 000. In the next theorem we show that the rst-order asymptotic distributions of the bootrastrap TVC test statistics and the Bierens and Martins (2010) original TVC test statistic are the same. Moreover, in the next Section, we provide, by means of simulations, evidence that the bootstrap approximation to the nite sample distribution seems to be much more accurate, as opposite to the asymptotic approximation. Hence, we consider the wild and i.i.d. bootstrap versions of the TVC test statistic as valid ones to test for standard cointegration and recommend using their critical values to ensure correct 5% type-I errors, say.
The proof of Theorem 1 is presented in a similar way as in Cavaliere et al. (2010aCavaliere et al. ( , 2012 and Swensen (2006). Note that, under the null hypothesis, our model is the same as in Cavaliere et al. (2010aCavaliere et al. ( , 2012 and Swensen (2006) and the assumptions as in Cavaliere et al. (2010a). We also consider the wild and the i.i.d. bootstraps, but the di erence is that, in our context, we test against TVC and x r. Theorem 1. Let LR wb m,T and LR sb m,T denote the LR b m,T bootstrap test statistics for the wild and i.i.d. cases, respectively. Given m ≥ 1 and r ≥ 1, under the null hypothesis of standard cointegration and Assumption 1, the bootstrap LR statistics LR wb m,T and LR sb m,T de ned above are asymptotically χ 2 mkr distributed, as T → ∞.
Proof. For the sake of simplicity, and as in Cavaliere et al. (2010a), we take the no-dri case, µ = 0, for which Bierens and Martins (2010) showed that LR tvc m,T is also asymptotically χ 2 mkr distributed. To avoid confusion with the "wb/sb" notation, we denote any of the two bootstrap probabilities by P * and the expectation under P * by E * , and we do the same, with the bootstrap quantities where for all η > 0, P * max where W (u) , u ∈ [0, 1] is a k−dimensional Wiener process with covariance matrix . Contrary to Cavaliere et al. (2010aCavaliere et al. ( , 2012 and Swensen (2006), Ŵ and ε t are obtained from the TVC model (3 ) and, as a result, our bootstrap samples With the following lemmas, we show that the TVC test statistic applied to the bootstraped sample, LR b m,T , is also asymptotically χ 2 mkr distributed, as T → ∞. The lemmas are as those at Bierens and Martins (2010) and invoking the functional central limit theorem and the law of large numbers that apply to martingale di erence sequences instead (Brown, 1991, andHeyde, 1972), and where it is shown that the original limiting results still hold. 1 where W (u) is a k-variate standard Wiener process, Proof. Let u ∈ [0, 1] , x j, h = 1, . . . , m, and let l = 1, . . . , p − 1. For the rst result, note that From Bierens (1994, Lemma 9.6.3, p. 200), where the last term vanishes because for all η > 0, From (12) and the continuous mapping theorem (CMT), the rst term converges in distribution to via integration by parts (see Bierens and Martins, 2010). Then, due to convergence in probability of C (see Bierens and Martins (2010) for the limiting properties of the pseudo MLE C and the Proof of Lemma S2 in the Supplement to Swensen (2006), available at Econometrica's website, for why the CMT still holds under P * ), and the result follows.
For the second result, where M b 0 is the nonrandom k × k matrix that de nes the limiting variance term (see Bierens and Martins, 2010, for the equality). On the other hand, Then, and the second result follows. For the last one, again, by Bierens (1994, Lemma 9.6.3), and the result holds given that (see Bierens and Martins, 2010, for the integration by parts).

Lemma 2.
Let α ⊥ and β ⊥ be the orthogonal complements of α and β, respectively. The following quantities have the same limiting laws as in Bierens and Martins (2010): Proof. It follows from Lemma 1, given the de nition of S b,(m) ij,T , i, j = 1, 2 (see above) and the sequence of arguments in Bierens and Martins (2010). Notice that these results are related to Lemma A6 at Cavaliere et al. (2010a) and Lemma S2 at Swensen (2006) (for i.i.d. errors) where in their case m = 0. In our test, m being greater than zero implies deriving the limiting laws of processes that involve the Chebishev polynomials. That is provided previously by Lemma 1.

Lemma 3.
Under the null hypothesis of standard cointegration and Assumption 1 and if η > 0, in probability, where · is the Euclidean norm and converge in probability to constants 1 > λ 1 ≥ · · · ≥ λ r > 0, which do not depend on m. Thus, these probability limits are the same as in the standard TI cointegration case: where ρ m,1 ≥ ρ m,2 ≥ .... ≥ ρ m,k−r are the k − r largest solutions of the generalized eigenvalue problem and with V an r.m × (k − r) random matrix with i.i.d. N[0, 1] elements.
Proof. It follows from applying the previous results to the correspondent lemmas at Bierens and Martins (2010) (A7, A8, and A9). Just as in Bierens and Martins (2010), we need to de ne 11,T ξ ⊥,T from converging to a singular matrix (see Andersson et al., 1983). See Theorems 3 and 4 at Cavaliere et al. (2010a) and Proposition 1 at Swensen (2006) (i.i.d.) for the case of m = 0.
Proof of Theorem 1. Following the same route as in Bierens and Martins (2010), the χ 2 mkr distribution of the bootstrap TVC test statistic follows from a combination of gaussian and χ 2 processes, as shown by Johansen (1995), once we apply the Taylor expansion of Johansen (1988) around the pseudo MLE to the pseudo LR statistic. Recall that the exact ML function (Bierens and Martins, 2010) is the same as the pseudo ML function once we assume Gaussian disturbances. To that end, and given the previous lemmas, we state that the pseudo ML estimator of ξ with a bootstrapped sample, . . , r, are the eigenvectors associated with the r largest eigenvalues λ b m,i , has the same limiting distribution as the one in Bierens and Martins (2010). More speci cally, for the m is a k.m × r matrix, and under the null hypothesis of standard cointegration, where W α is an r-variate standard Wiener process, V α is a k.m × r matrix with independent N[0, 1] distributed elements, and V α , W α , and W k−r,m are independent. The result can be shown by generalizing Lemma A6 of Cavaliere et al. (2010a) and Lemma S2 of Swensen (2006) to the case of m > 0. That can be done given our Lemma 1 that involves the Chebishev polynomials. See also page 9 of the Supplementary Material to Swensen's, (2006) article for why the continuous mapping theorem still holds under P * .

Bootstrap Test for TVC Using the Restricted Residuals.
In the context of the LR cointegration rank test and associated sequential rank determination procedure of Johansen (1995), Cavaliere et al. (2012) show that important gains can be achieved once the residuals (and µ, Ŵ) are obtained from the cointegration model under the null hypothesis. In particular, the bootstrap sequential rank determination procedure consistently determines the cointegration rank, in the sense that the probability of choosing a rank smaller than the true one will converge to zero. As mentioned before, in our test for time varying cointegration, the cointegration rank, r, is xed and assumed to be known, and therefore, the "sequential rank determination procedure" is not relevant in our framework. Still, we see no reason why using restricted residuals should not work as well for bootstrap TVC testing and, thus, suggest it as an alternative to the unrestricted procedure de ned above.
Let all estimated quantities be exclusively obtained from the standard time invariant cointegration model, m = 0 : α, β, µ, Ŵ and ε t , t = 1, . . . , T. Following Cavaliere et al. (2012), this alternative Wild and i.i.d. bootstrap procedure is the same as the one above, but (i) the bootstrap pseudo-disturbances ε b t are obtained from the restricted residuals ε t ; and (ii) the bootstrap sample △Y b procedures using the restricted residuals are not suitable for models with r > 1 under conditionally heteroskedastic errors and, thus, alternative methods need to be proposed in the literature.

Monte carlo study
In this section we illustrate the merits of the Bootstrap TVC tests by assessing their nite-sample size performance through numerical simulations. We consider three cointegration models that follow closely the literature and combine distinct values for the number of cointegrating vectors, the number of variables in the system and the VAR lag order.

The designs
The data generating processes (DGP's) correspond to the standard cointegrated VAR's (m = 0) presented by Bierens and Martins (2010) (denote it by BM), Johansen (2002) and Swensen (2006) (JS), and Engle and Yoo (1987) (EY). In the BM model, there is a single cointegrating relationship driving a bivariate system and where p = 2; the JS model is instead of dimension 3 and p = 1; and, relatively to JS, the EY model assumes that there are two cointegrating vectors. There are no deterministic components and the models parameters are as follows: The errors ε t are independent, Gaussian, and with diagonal covariance matrix = I k , except for EY where = 100I k . As in Cavaliere et al. (2010a), we also consider a fatter-tails case of independent errors ε t that follow a t -distribution with ve degrees of freedom and the conditional heteroskedasticity case N (0, 1), independent across i, and h it = 1 + 0.3ε 2 it−1 + 0.65h it−1 , t = 1, . . . , T. These three distinct speci cations for the errors ε t are consistent with our Assumption 1.
The initial values are set equal to zero and the rst y observations are dropped. All experiments are based on 10,000 replicas and consider samples of size T = 50, 100 and 200. The original and bootstrap TVC tests are calculated for a wide range of Chebyshev time polynomials, with a maximum number of T/10 of them: m = 1, 2, 5 for T = 50, m = 1, 2, 5, 10 for T = 100, and m = 1, 2, 5, 10, 20 for T = 200. Following Cavaliere et al. (2010aCavaliere et al. ( , 2012, for each replica, the number of bootstrap pseudosamples B equals 399. For each procedure, we impose a nominal test size of 5%.

The results
We seek to estimate the probability of rejecting the null hypothesis of TI cointegration when it is in fact true, for each of the three TVC tests described above and to observe how those results vary with m and T, besides p, r and k which change across DGP's. Tables 1 (BM model), 2 (JS model), and 3 (EY model) report the nite-sample test-levels for each distributional approximation, where tvc denotes the original TVC test against the chi-squared distribution and wb and sb refer to the wild and i.i.d. TVC tests, respectively, against the bootstrap distribution. The results based on the bootstrap procedure using the unrestricted residuals (and Ŵ) are presented at the right of label "UR" and those using the restricted ones at the right of "R. " The results for tvc under t-distributed and heteroskedastic errors must be somehow read in careful under the original assumption of normality at Bierens and Martins (2010), although we conjecture that the result hold true when relaxing this distributional restriction. The examination of the empirical properties of the original test illustrates the severity of the size distortions, especially for small T and large m, regardless of the DGP under consideration. For large m, the empirical size approaches the nominal one at the expense of a (much) higher number of observations. In contrast, the bootstrap tests provide in general near-exact levels for any combination of T or m. This fact is robust to any of the considered model speci cations except for the conditional heteroskedasticity case with large m where real sizes of the i.i.d. bootstrap test are equal to about 10% or above. Regarding this particular setup, only the wild bootstrap test showed very reasonable results in general. This is somehow consistent with the ndings at Cavaliere et al. (2010a). n/a n/a n/a n/a ---200 UR 9.7 7.0 8.6 12.6 6.5 10.0 19.9 7.2 13.0 41.0 8.5 18.8 88.1 10.6 29.1 R n/a n/a n/a n/a n/a n/a stands for "not available. " In terms of the unrestricted and restricted approaches, we observe that the empirical size of the latter is always smaller than the former. For most cases, this implies having an empirical size (even) closer to the nominal 5% for the restricted procedure. But on the other hand, the Wild bootstrap under the restricted approach fails to reject the null too o en for m large and models JS and EY. Noticeably, the restricted procedure seems not to work for the EY model under conditionally heteroskedastic errors (contrary to models BM and JS, r = 2 at the EY model): the largest solution of the generalized eigenvalue problem under the null and/or under the alternative, λ 0,1 , λ m,1 , is systematically larger than one for the simulated data, thus invalidating the calculation of the test statistic (recall that it must hold true that 1 > λ m,1 ≥ λ 0,1 ).
Hence, we advocate the usage of the bootstrap versions of the TVC test, namely the wild bootstrap using the unrestricted residuals, since these seem to be the only techniques that work well globally when testing for standard cointegration against TVC.
In a recent paper,  extend Swensen (2006) and Cavaliere et al. (2010aCavaliere et al. ( , 2012) by adapting Killian's (1998) bootstrap-a er-bootstrap (BaB) framework to their bootstrap-based LR cointegration rank tests. More speci cally, bias-corrected VECM parameter estimates, obtained from the bootstrap replications, are used instead to generate the pseudo-data and calculate the test statistic. Because Cavaliere, Taylor, and Trenkler's standard cointegration model is the same as ours, we also implemented their algorithm to the TVC unrestricted bootstrap procedures in order to assess whether small sample gains can be obtained. The TVC BaB algorithm is the same as the one previously described with the following di erence: The estimated Ŵ ′ j s used to obtain the bootstrap sample △Y b t T t=1 in step two, and consequently the LR test statistic in step three, are replaced by the bootstrap bias-corrected estimates. That is, where Ŵ b j is the estimate based on the b − th bootstrap sample. According to the Monte Carlo results for model BM, there seems not to exist gains by implementing the BaB procedure. In general, the empirical level increases slightly in both wild and i.i.d. bootstrap schemes and for all sample sizes. The only exceptions are as follows: wb with m = 1 and T = 50 (drops from 5.9 to 5.7) and sb with m = 10 and T = 100 (a drop from 7.2 to 6.8).

Reassessing TV PPP
In Bierens and Martins (2010), the original TVC test statistic was applied to the purchasing power parity (PPP) hypothesis: In its absolute form, it means that the same bundle of goods, measured in real terms, should have the same value across countries. By taking the U.S.A. as the domestic country, Bierens and Martins (2010) concluded that price indices and nominal exchange rates are cointegrated, but in a TV fashion. Given the size distortions of the test and the accurate distribution approximation of the bootstrap versions of the test, in this section, we reevaluate the PPP hypothesis using the same data as in Bierens and Martins (2010), but now computing the wild and i.i.d. bootstrap TVC test statistics as well, for both unrestricted (UR) and restricted (R) approaches.

PPP in the context of TVC
The literature on the PPP hypothesis is recognizably vast and several reviews have been proposed (see, for example, Taylor and Taylor (2004). In our notation, Y t = ln S f t , ln P n t , ln P f t ′ , where P n t and P f t are the price indices in the domestic and foreign economies, respectively, and S f t is the nominal exchange rate in home currency per unit of the foreign currency. Taking the symmetry and proportionality restrictions of the absolute version of PPP, β ′ = (1, −1, 1) , is not expected to hold in empirical work due to several aspects namely measurement errors of the price indices. Hence, the traditional empirical strategy assumes β to be unknown and estimates the deviation series from PPP, e t = β ′ Y t , under the Engel and Granger (1987) or Johansen's methodology, β = (1, β 2 , β 3 ) . Under the assumption of no transactions costs, PPP requires that e t follows a stationary process.
Whether it be the existence of transaction costs, nontradable goods, or market structures with imperfect competition, it is highly unlikely that the equilibrium parity condition holds in its traditional representation. Due to the presence of such market frictions or measurement errors of the price indices in equilibrium models of real exchange rate determination, which may imply a nonlinear adjustment process in the PPP relationship, we test for the single constant cointegration hypothesis against our TVC speci cation. The short run deviations from the PPP due to shocks in the system are measured by e t = β ′ t Y t , where β t is an unknown deterministic function of time that is approximated by β t (m) = m i=0 ξ i P i,T (t) , where the ξ i 's are the Fourier coe cients. Under the standard PPP hypothesis (timeinvariant cointegration), ξ i = 0 for all i = 1, . . . , m, for a xed m.
The way to departure from the traditional Engle and Granger and Johansen's approaches might not be consensual. To put it simple, depending on the underlying model speci cation and assumptions and the properties of the data, one may t the PPP theory within an I(1) or I(2) framework; assume a linear or nonlinear type of cointegration model; and impose a set of coe cients that are either xed or timevarying (threshold cointegration, smooth transition, markov-switching, and so forth).
For example, Falk and Wang (2003) considered the PPP relationship within an I(1) framework whereas Johansen et al. (2010) and Frydman et al. (2008) argue that it ts instead within the I(2) framework and thus making the standard approach possibly misleading. Based on rational expectations hypothesis sticky-price monetary models, the I(1) approach assumes that nominal exchange rates and relative good prices are unit-root processes, while the real exchange rate is stationary (or a near-I(1) process). Instead, Johansen et al. (2010) question these monetary models, specifying an I(2) model with piecewise linear trends where the change in real exchange rate is stationary but highly persistent and apply it to German-USA data in the period 1975-1999. 4 On the other hand, Hong and Phillips (2010) propose a RESET-type test for linear against nonlinear cointegration and applied it to the PPP theory using U.K.-U.S.A., Mexico-U.S.A., Canada-U.S.A., and Japan-U.S.A. data from 1971 to 2004. They found little evidence for a linear relationship, except for the Mexico-U.S.A. case.
An important branch of the PPP literature assumes a nonlinear adjustment process in the xed PPP cointegration relationship. It is argued that due to the presence of transactions costs, the deviations from the PPP e t = β ′ Y t is a nonlinear process that can very well be characterized in terms of a smooth transition autoregressive model (ESTAR model). In this type of models, regime changes occur gradually rather than suddenly and the speed of adjustment varies with the extent of the deviation from parity. Typically, the deviations from the PPP are obtained (i.e., estimated, e t = β ′ Y t ) using the Engle and Granger (see Michael et al., 1997) or the Johansen's cointegration method (see Baum et al, 2001). The results provide strong evidence of mean-reverting behavior for PPP deviations and against the linear framework.
It is known that testing for a linear speci cation with time varying coe cients against a nonlinear model with xed parameters, or selecting the best out of the two, is not an easy task. Once we believe in Clive W. J. Granger's assertion that "any non-linear model can be approximated by a time-varying parameter linear model" (Granger, 2008), we cannot reject a priori the relevance of a speci cation such as the one we are considering in the article.
As it just so happens with the nonlinear adjustment speci cation, our model also assumes PPP cointegration in a nonstandard fashion, including the smooth transitions. In particular, the TVECM is able to empirically assess the strongest assumption in PPP theory: The single cointegration vector being of the form β = (1, −1, 1) and, correspondingly, the real exchange rate a stationary process. This absolute version of the PPP hypothesis occurs if the null hypothesis of our TVC test is not rejected and, furthermore, the restrictions are also not rejected. The bootstrap TVC test is shown to be a "good" statistical tool to see if those changes around a constant β are signi cant or not. Cheung and Lai (1993) claim that, due to transaction costs and measurement errors in prices, if e t is stationary and (β 2 < 0, β 3 > 0) , PPP holds. The Appendix to Bierens and Martins' article includes the plots of the estimated β t . There, one can see that, in general, β t seems to uctuate around (1, −1, 1) and, in particular, β t2 < 0, β t3 > 0 for most t. From an economic point of view, this means that the presence of market frictions and/or measurement errors of the price indices is the cause of time-varying adjustments on the relative importance of each variable in Y t (nominal exchange rates and prices) to guarantee stationary PPP deviations. 5 That is, contrary to the former two-stage approach (obtain e t = β ′ Y t and then t a STAR model to it), at the TVECM, the PPP equilibrium is directly restored via the cointegration vector, β t .

Empirical results
The data we use to illustrate the usefulness of the bootstrap TVC tests in empirical work is the same as in Falk and Wang (2003), downloaded from the Journal of Applied Econometrics data archives website. For this particular dataset, they put the PPP relationship within an I(1) framework. 6 The U.S.A. bilateral relationship of study is with the U.K., Japan, Canada, France, Italy, Germany, Belgium, Denmark, the Netherlands, Norway, Spain, and Sweden. The data are monthly comprising 324 observations and cover the period from January 1973 to December 1999.
In this empirical application where a dri is included, k = 3, and r = 1, we consider m ranging from 1 to 32 = ⌊T/10⌋ . For the bootstrap tests, B = 399 and the initial values Y b t , t = −p + 1, . . . , 0, are set equal to the rst observation in the sample, Y 1 . Just as in Bierens and Martins (2010), the admissible values for the lag order include p = 1, 6, 10, and 18. Based on the Hannan-Quinn information criterion

Conclusion
In this article we have considered two alternative bootstrap algorithms to the time-varying cointegration test proposed by Bierens and Martins (2010), based on a VECM speci cation where the cointegration vector changes smoothly over time. The original likelihood ratio test and its wild and i.i.d. bootstrap versions have the same rst-order asymptotic distribution under the null hypothesis of standard/timeinvariant cointegration. According to some extensive Monte Carlo simulations, and contrary to what happens with the original test statistic, the bootstrap procedures did not show severe size distortions. That is, the bootstrap approximation to the nite-sample distribution can be considered very accurate, especially for the wild bootstrap case. We have applied the tests to the purchasing power parity hypothesis of international prices and nominal exchange rates with the U.S. as the home economy, and found evidence of standard cointegration in the U.S.A.-U.K. bilateral relationship and time-varying cointegration in the remaining eleven cases. The simplicity of application of the bootstrap TVC tests and their good performance in nite-samples make the procedures discussed in this article a valuable tool when addressing the possibility for smooth time-transitions of the equilibrium relationship in several other examples of cointegrated variables. It is important to notice that the LR test setup is conditional on the existence of cointegration. An interesting topic that deserves further attention is how to test for "spurious" regression in our time-varying framework. The work by Park and Hahn (1999) in singleequation time-varying cointegration can be helpful in this respect.