Estudos e Documentos de Trabalho. Working Papers


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1 Estudos e Doumentos de rabalho Working Papers 9 9 HE FLEXIBLE FOURIER FORM AND LOCAL GLS DERENDED UNI ROO ESS Paulo M. M. Rodrigues A. M. Robert aylor September 9 he analyses, opinions and findings of these papers represent the views of the authors, they are not neessarily those of the Bano de Portugal or the Eurosystem. Please address orrespondene to Paulo M.M. Rodrigues Eonomis and Researh Department Bano de Portugal, Av. Almirante Reis no. 7, 5 Lisboa, Portugal; el.: ,
2 BANCO DE PORUGAL Edition Eonomis and Researh Department Av. Almirante Reis, 76 th 5 Lisboa Prepress and Distribution Administrative Servies Department Doumentation, Editing and Museum Division Editing and Publishing Unit Av. Almirante Reis, 7 nd 5 Lisboa Printing Administrative Servies Department Logistis Division Lisbon, September 9 Number of opies 7 ISBN ISSN 877 Legal Deposit No 3664/83
3 he Flexible Fourier Form and Loal GLS Detrended Unit Root ests Paulo M.M. Rodrigues a and A.M. Robert aylor b a Bano de Portugal and Faulty of Eonomis, Universidade Nova de Lisboa b Granger Centre for ime Series Eonometris, University of Nottingham September 9 Abstrat In two reent papers Enders and Lee (8) and Beker et al. (6) provide Lagrange multiplier and OLS detrended unit root tests, and stationarity tests, respetively, whih inorporate a Fourier approximation element in the deterministi omponent. Suh an approah an prove useful in providing robustness against a variety of breaks in the deterministi trend funtion of unknown form and number. In this paper, we generalise the unit root testing proedure based on loal GLS detrending proposed by Elliott, Rothenberg and Stok (996) to allow for a Fourier approximation to the unknown deterministi omponent in the same way. We show that although the resulting unit root tests possess good nite sample size and power properties, their limit null distributions are unde ned. Keywords: Loal GLS detrending; exible Fourier approximation; trend breaks; unit root. JEL lassi ations: C, C We thank Walter Enders and Junsoo Lee for providing us with a opy of their paper together with their Gauss ode. Paulo Rodrigues gratefully aknowledges nanial support from POCI/FEDER (grant ref. PDC/ECO/64595/6). Correspondene to: Robert aylor, Shool of Eonomis, University of Nottingham, Nottingham, NG7 RD, U.K.
4 Introdution he di ulties inherent in testing for a unit root in a time series whih is subjet to strutural breaks in its deterministi trend funtion are well doumented in the eonometri time series literature. Sine the seminal work of Perron (989), a large literature has developed around providing unit root test proedures whih aount for suh breaks; see Perron (6) for a reent review. Initial researh onsidered the presene of at most one break in the data generation proess [DGP] while more reent researh has foused on the possibility of multiple possible breaks in the level and/or the trend; see, in partiular, the loal generalised least squares [GLS] detrended unit root tests of CarrioniSilvestre, Kim and Perron (9). he performane of extant unit root tests depends ruially on the estimated break loation(s) and on the assumed maximum number of breaks; see, inter alia, Enders and Lee (8), Beker, Enders and Hurn (4) and Beker, Enders and Lee (6). It has been observed by, among others, Gallant (98), Davies (987), Beker, Enders and Hurn (4) and Harvey, Leybourne and Xiao (8) that a Fourier approximation an, to any desired degree of auray, apture the behaviour of a deterministi trend funtion of unknown form, even if the funtion itself is aperiodi. his result has been reently employed by Enders and Lee (8) who generalise the Shmidt and Phillips (99) and Shmidt and Lee (99) Lagrange multiplier [LM] type unit root test (whih employ rst di erene [FD] detrending) together with the ordinary least squares [OLS] detrended DikeyFuller [DF] (979) unit root tests through the introdution of a Fourier approximation to the deterministi trend omponent. hey show the resulting tests to be robust against a large variety of possible break mehanisms in the deterministi trend funtion. Beker, Enders and Lee (6) provide stationarity tests ug the same framework and show that the resulting Kwiatkowski, Phillips, Shmidt and Shin (99) [KPSS] type tests display good size and power properties in the presene of a variety of strutural break designs. An empirially attrative feature of these proedures is that there is no need to assume either that the potential break dates or the number of breaks are known to the pratitioner, a priori. he simpliity with whih this approximation an be implemented is also an important advantage of this approah relative to existing methods whih require numerially involved searhing proedures and are in any ase operationally infeasible if the pratitioner wishes to allow for more than two putative breaks; see CarrioniSilvestre, Kim and Perron (9). Our objetive in this paper is to apply the exible Fourier form to the loal GLS unit root testing proedure of Elliott, Rothenberg and Stok [ERS] (996) and to ompare this with the orresponding DF and LM based unit root tests of Enders and Lee (8). It is known that loal GLS detrending an yield unit root tests whih are onsiderably more powerful than their OLS and FD detrended ounterparts; see, in partiular, ERS for the onstant and linear trend ases, Perron and Rodríguez (3) for the ase of a gle break in level/trend, and CarrioniSilvestre, Kim and Perron (9) for the ase of multiple level/trend breaks. In this paper we demonstrate that these power gains arry over, at least in nite samples, when ug loal GLS detrended unit roots based around the exible Fourier form. he paper is organised as follows. Setion introdues the loal GLS detrended unit root tests whih inorporate the exible Fourier form and brie y outlines the orresponding LM and OLS tests of Enders and Lee (8). In setion 3 we provide nite sample ritial values for the GLS detrended tests and ompare the nite sample size and power properties of these tests with the orresponding OLS detrended and LM tests. Large sample properties
5 of the loal GLS detrended tests are disussed in setion 4. Setion 5 onludes. Proofs are ontained in a mathematial appendix. In what follows we use the notation x := y ( x =: y ) to indiate that x is de ned by y (y is de ned by x), and ) to denote weak onvergene. he Model and Unit Roots ests. he Flexible Fourier Model Consider data generated aording to following DGP: y t = + t os + x t ; t = ; :::; () x t = x t + u t () where it is assumed, for the present, that u t iid(; ) and that the starting value, x ; is an O p () random variable. he Fourier frequeny,, is taken to be a xed value. Our interest in this paper lies in testing the unit null hypothesis, H : =, against the stationary alternative, H : jj <, in (). Remark.: he deterministi kernel onsidered in () inludes a linear time trend, but we may also onsider the ase where only a onstant and the two Fourier terms are onsidered; i.e., the ase where = in (). his will be referred to as the onstant ase in what follows, while the more general ase where 6= will be termed the linear trend ase. Remark.: he model in ()() ontains a gle Fourier frequeny. We fous our attention on this model, based on the observations made in Enders and Lee (8) and Beker, Enders and Lee (6) that a gle Fourier frequeny an mimi a large variety of breaks in the deterministi trend funtion. Enders and Lee (8) note that, for any desired level of auray, a more general Fourier expansion of the form f t;n () := + P` i= i it + P`, where ` < =, and with < < < `, ould be onsidered. i= i os it However, Enders and Lee (8) argue against the use of many Fourier frequeny omponents beause it an lead to problems of over tting. Remark.3: Equation () an be rewritten as, y t = z t+f t () ' + x t ; where z t := (; t) and := ( ; ) (or, in the onstant ase z t := and := ), f t () := ( ; os ) ; and ' := (' ; ' ), or in vetor notation as y = Z + f()' + x where Z := (z ; :::; z ) and f() := (f () ; :::; f () ) are matries (Z is a vetor in the onstant ase) and y and x are vetors. 3
6 . GLS Detrended Unit Root ests In this setion, we extend the loal GLS detrending approah of ERS to the problem of testing for a unit root within the ontext of ()(). his is ahieved through a twostep proedure. In the rst step we estimate the OLS regression of y := y ; y ( + )y ; :::; y ( + )y (3) onto V := v ; v ( + )v ; :::; v ( + )v (4) where v t := (zt; f t () ), to obtain an estimate of the parameter vetor := ( ; ' ). Denote this estimate by b := (b ; b' ). he value of the loal GLS detrending parameter,, depends on the form of the deterministi omponent in (); ERS suggest ug = 7 for the onstant ase and = 3:5 for the linear trend ase. In the seond step we run the DFtype unit root test regression on the loal GLS detrended series, yt := y t zt b f t () b', t = ; :::; ; that is, ompute the tstatisti for = in the regression equation yt = yt + u t : (5) We denote the resulting statisti as t ERS f, = ; ; where indiates that the statisti is omputed for the onstant ase, z t =, and that the statisti is omputed for the linear trend ase, z t = (; t). In what follows, where generi statements are being made whih apply in both the onstant and linear trend ases, we will omit the supersript. Remark.4: Enders and Lee (8) extend the LM type unit root tests of Shmidt and Phillips (99) and Shmidt and Lee (99) to this ontext. In the rst step of this testing proedure, the parameters of the deterministi variables (onstant, time trend and Fourier terms) are estimated under the null hypothesis, i.e., y t = z t + f t () + x t where z t := ; := ( ; 3 ) and f t () := ; os. he estimated oe ients, ~ j, j = ; :::; 3, from this regression are then used to onstrut the FD detrended series: yt LM := y t zt e f t() ; e t = ; :::; where e := ( e ; e ); e := y e e e 3 os, and e := ( ~ ; ~ 3 ). he seond step then involves estimating the auxiliary regression y t = z t # + f t () # + y LM t + u t (6) to obtain the regression tstatisti for = in (6), t LM f say. Remark.5: Enders and Lee (8) also onsider OLS detrended DFtype statistis for testing H again H in ()(). In this ase, the appropriate DFtype regression is given by y t = v t! + y t + v t (7) 4
7 where v t := (z t; f t()). he OLS detrended DFtype statisti is then given by the regression tstatisti for = in (7), say t DF f ; where the nomenlature = ; has the same meaning as outlined for t ERS f above. Notie that this proedure is asymptotially equivalent to the twostep proedure where H is tested ug the regression tstatisti for = in y t = y t + v t (8) where y t := y t z t f t() ', are the OLS detrended data from regresg y t onto v t ( and ' being the resulting OLS estimates of and ' respetively), t = ; :::;. Remark.6: It is straightforward to show that all of the three unit root statistis disussed above, namely t ERS f, t LM f and t DF f are exat invariant with respet to the parameters haraterig the deterministi trend funtion in ()(). he three statistis di er purely in the manner in whih this invariane is ahieved; i.e., through the detrending method they employ. Remark.7: We have assumed thus far that u t in () is serially unorrelated. Short run dynamis in the u t proess an be handled in the usual way by augmenting test regressions (5), (6) and (7), with su ient lags of the dependent variable to orret for the serial orrelation present; see, inter alia, ERS, Chang and Park () and Ng and Perron (). 3 Finite Sample Simulations In this setion we provide nite sample ritial values for the unit root tests outlined in the previous setion, together with an investigation of their relative nite sample size and power properties. 3. Finite Sample Critial Values able below, presents a seletion of nite sample ritial values for the t ERS f, t DF f and t LM f unit root tests from setion. he ritial values provided are valid for the onstant () and linear trend () ases. For the ERS type test statistis we followed Elliott et al. s (996) suggestion and set the loal GLS detrending parameter to = 7 in the onstant ase and = 3:5 in the linear trend ase. he reported ritial values were omputed by Monte Carlo from the random walk proess x t = x t + u t ; with u t NIID(; ). Without loss of generality, we set x =, the three tests all being exat similar with respet to x. he test regressions used for eah proedure were those desribed in the previous setion; i.e., (5), (6) and (7) for t ERS f, t LM f and t DF f, respetively. Critial values are reported for (; ; 3; 4; 5) and (; ; ). All of the simulations reported in this paper were programmed in Gauss 9. ug Monte Carlo repliations. able about here Although these ritial values are generated assuming a known value of the Fourier frequeny parameter,, they an also be used as an approximation to the nite sample ritial values in ases where the value of is unknown but has been estimated. As Beker et al. 5
8 (6,p.39) argue In most instanes with highly persistent maroeonomi data, ug the value k = or k = should be su ient to apture the important breaks in the data. However, there are irumstanes where the researher may want to selet some frequeny other than k = or k =. Hene... we onsider is to selet k for ug a ompletely datadriven method. o that end, Davies (987) shows that a onsistent estimate of an be obtained by minimig the residual sum of squares resulting from estimating a sequene of regressions of the form given in () over a suitable grid of values of. An interesting feature that an be observed in the results in able is that as the Fourier frequeny parameter inreases, so the ritial values of the unit root tests whih inlude the Fourier regressors appear to onverge, other things being equal, towards the ritial values for the unit roots tests that omit the Fourier terms (i.e., the unit root tests of DF, ERS and Shmidt and Phillips, 99). his result an be attributed to the asymptoti orthogonality that exists between the elements of the frequeny zero deterministi regressors in z t and the Fourier terms in f t () in ases where =, < < :5, suh that the Fourier terms are loated at the harmoni frequeny pair (; ) whih is bounded away from zero and therefore have no impat on the distribution of the unit root tests in the limit. his is, of ourse, a purely nite sample e et beause! in (), as!. 3. Finite Sample Size and Power of the ests 3.. Conventional Unit Roots ests Before looking at the nite sample size and power properties of the t ERS f, t LM f and t DF f unit root tests from setion, we rst investigate the impliations for the orresponding onventional unit root tests of ERS, Shmidt and Phillips (99) and DF, omputed ug a deterministi kernel whih inludes a onstant only or a onstant and a time trend, but whih do not take aount of the Fourier terms in (). With an obvious notation we denote these tests by t ERS, respetively. o that end, we generate data from the DGP y t = + os + x t (9) x t = x t + u t ; u t NIID(; ); t = ; :::; (), t LM and t DF with x N(; ), independent of u t. he autoregressive parameter is de ned as := + : able reports results for = whih orresponds to the null hypothesis, H, while able 3 reports orresponding results for = 5 whih orresponds to the alternative hypothesis, H. he other parameters are varied aording to (; ; 3; 4; 5); (; 3) and (; 5). his orresponds to the simulation design used in Enders and Lee (8). We report results for samples of length = and = ables 3 about here he results in able demonstrate that under the unit root null hypothesis all of the onventional tests beome undersized, in many ases very severely so, in the presene of k in the notation of Beker et al. (6) is equivalent to in our notation. his does not, however, imply that the inlusion of the Fourier terms in the test regression is unneessary in these ases. As will be seen in the next setion, where onventional unit root tests are evaluated in this ontext the omission of these Fourier terms has severe impliations for the nite sample properties of the tests. 6
9 negleted Fourier terms 3. In general the undersizing is marginally worse, other things being equal, for t ERS and t LM than for t DF, with the degree of undersizing seen in all three tests beoming inreagly severe as and/or beome larger. Other things being equal, the size distortions are worse the greater is and the smaller is the sample size. Notie that in small samples as inreases so the Fourier terms present in the DGP move further away from the zero frequeny and, hene, the impat of these negleted deterministi terms (i.e. the lak of similarity of the test statistis) beomes inreagly pronouned. Regarding the empirial power of the proedures, we observe that when no Fourier terms are present in the DGP the ERS test presents the best power performane followed by the LM type test. he DF is the test with the lowest power of the three. Where (negleted) Fourier terms are present in the DGP we see from the results in able 3 that all of the onventional tests show atastrophi losses in power relative to the ase where no Fourier terms are present. Indeed in the majority of reported ases all three tests display rejetion frequenies below the nominal 5% level. 3.. ests with Known In order to evaluate the nite sample power properties of the t ERS f, t DF f and t LM f unit root tests we generate data from (9)(), again with x N(; ), independent of u t, and the autoregressive parameter set as := + but now for ( 5; ; 5; ): Given the exat invariane of t ERS f, t DF f and t LM f to and when is known we may set = =, without loss of generality. able 4 about here able 4 presents the nite sample power results for the three tests. It is lear from the results in able 4 that the loal GLS detrended unit root test, t ERS f, proposed in this paper enjoys signi ant power gains over both the OLS and FD detrended tests, t DF f and t LM f respetively. It is lear that the OLS detrended test onsistently displays the lowest power among the three tests while the loal GLS detrended test onsistently displays the highest power among the three tests ests with Unknown Following the disussion in setion 3.., we now turn to an evaluation of the nite sample size and power properties of the t ERS f, t DF f and t LM f tests in the ase where is taken to be unknown and is estimated from the data. Data are again generated from (9)() for := + with ( 5; ; 5; ). Beause the tests whih are based on an estimate of are no longer exat invariant to parameters of the Fourier terms (they are, however, asymptotially invariant to these parameters) we generated the Fourier terms for (; ; 3; 4; 5) with, in eah ase, (; 3) and (; 5). In order to make the tests operational we must rst estimate the true but unknown Fourier frequeny parameter,. his is done ug the approah of Davies (987). Following Beker 3 Corresponding experiments with a onstant only were also omputed but gave qualitatively similar results to those presented for the onstant and linear trend ase in able and are therefore omitted. hese results an be obtained from the authors on request. 7
10 et at. (6, p.39) we estimate the regression equation kt kt y t = + t os + x t : () for eah integer value of k in the interval k k max. he estimated value, b, is then given by the value of k whih minimises the residual sum of squares aross these estimated regression equations. Following the arguments given in Beker et al. (6, p. 39) we set the maximum frequeny at k max = 5. he small sample behaviour of this estimator is explored in detail in setion 3 of Beker et al. (6) and is shown to perform well in pratie. he t ERS f of., t DF f and t LM f tests are then alulated as before taking b as if it were the true value ables 5 6 about here ables 5 ( = ) and able 6 ( = ) present the empirial rejetion frequenies of the resulting t ERS f, t DF f and t LM f tests for the unknown ase. Although some small size distortions are observed, when 6= and/or 6=, the results are qualitatively very similar to those reported in able 3 for the known ase, suggesting that the estimation proedure for works well in pratie, at least from the perspetive of maintaining the size and power properties of the resulting unit root tests relative to the known ase. 4 Asymptoti Results In this setion, we show that the loal GLS detrended unit root test statisti, t ERS f, from (5) is asymptotially infeasible. his is established by showing in heorem that the rst stage loal GLS detrending regression of y onto V is unde ned in the limit due to the asymptoti gularity of the assoiated (saled) Gram matrix, V V. heorem Let fy t g be generated aording to ()() under the onditions stated in setion. hen under H : =, and as! p b = V V = V x 3 3 x ) x 5 () + + W 3 where x := x ; x ( + )x ; :::; x ( + )x, R W :=( )W () W (r)dr R rw (r)dr, with W (r) a standard Brownian motion, and where is the value of the loal GLS detrending parameter used. Remark 4.: Note that () orresponds to the limit when z t = (; t) is used in detrending the data. For the onstant only ase, z t =, it is straightforward to show, ug results from 3 3 x the proof of heorem, that for this ase the limit redues to x 8
11 Remark 4.: It is immediately seen from () that V V is asymptotially rank de ient, onverging to a (nonrandom) matrix with rank two (for the onstant only ase of Remark 4. the Gram matrix is asymptotially rank de ient with rank one). Consequently, the statisti t ERS f ; = ;, of (5) is asymptotially unde ned under the unit root null hypothesis. Remark 4.3: If, instead of ug loal GLS detrending, the parameters of the deterministis are estimated by OLS detrending from regresg y t onto v t, where v t is as de ned in setion (as is done in the DFtype test of Enders and Lee, 8, or with the KPSS test of Beker et al. 6  see Remark.5), the asymptoti gularity problem observed in heorem 4. is not enountered, e here the orresponding quantity V V, where V := [v ; v ; :::; v ] is nongular in the limit in both the onstant and linear trend ases; see Beker et al. (6) for further details. Remark 4.4: he results in heorem show that the Fourier regressors and os are asymptotially ollinear with the onstant term when subjeted to the loal GLS transformation in (4). An alternative to the full loal GLS detrending approah outlined in setion might then be to apply the loal GLS detrending stage in the rst step only to the elements of z t, and to then inlude the Fourier terms diretly in the seond step regression. hat is, to ompute the tstatisti for =, thy BRID say, in the regression equation y t = v t! + y t + v t where y t := y t zt with the estimated parameter vetor from regresg y onto z := z ; z ( + )z ; :::; z ( + )z. Although t HY BRID an be shown (available from the authors on request) to have a pivotal and wellde ned limiting null distribution its exat distribution, unlike the t ERS f, t DF f and t LM f tests, depends on the nuisane parameters haraterig the Fourier terms (arig from the fat that the rst stage loal GLS regression is misspei ed). In unreported Monte Carlo simulations we found thy BRID to behave very poorly, both in terms of size and power in small samples. 5 Conlusions In this paper, we generalise the DikeyFullertype unit root testing proedure based on loal GLS detrending proposed by Elliott, Rothenberg and Stok (996) to inorporate a Fourier approximation to the unknown deterministi omponent in the same was as is done for the orresponding OLS and FD detrended DikeyFullertype unit root tests of Enders and Lee (8). We show that although the resulting unit root tests possess good nite sample size and power properties when ompared to the OLS and FD detrended tests of Enders and Lee (8), their limit null distributions are unde ned. Referenes [] Bai, J. and P. Perron (998). Estimating and testing linear models with multiple strutural hanges. Eonometria 66,
12 [] Beker, R., W. Enders and S. Hurn (4). A general test for time dependene in parameters. Journal of Applied Eonometris 9, [3] Beker, R., W. Enders and J. Lee (6). A stationarity test in the presene of an unknown number of smooth breaks, Journal of ime Series Analysis 7, [4] Bierens, H.J. (994). opis in Advaned Eonometris. Cambridge University Press. [5] Bierens, H.J. (997). esting the unit root with drift hypothesis against nonlinear trend stationarity, with an appliation to the US prie level and interest rate. Journal of Eonometris 8, [6] CarrioniSilvestre, J.L., D. Kim and P. Perron (9). GLSbased unit root tests with multiple strutural breaks both under the null and the alternative hypotheses. Forthoming in Eonometri heory. [7] Chang, Y. and J.Y. Park (). On the asymptotis of ADF tests for unit roots. Eonometri Reviews, [8] Davies, R.B. (987). Hypothesis testing when a nuisane parameter is present only under the alternative. Biometrika 74, [9] Dikey, D. and W. Fuller (979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the Amerian Statistial Assoiation 74, [] Elliott, G.,.J. Rothenberg and J.H. Stok (996). E ient tests for an autoregressive unit root. Eonometria 64, [] Enders, W. and J. Lee (8). he exible Fourier form and testing for unit roots: an example of the term struture of interest rates. Working Paper, Department of Eonomis, Finane and Legal Studies, University of Alabama. [] Gallant, R. (98). On the basis in exible funtional form and an essentially unbiased form: the exible Fourier form. Journal of Eonometris 5, [3] Harvey, D. I., S.J. Leybourne and B. Xiao (8). A powerful test for linearity when the order of integration is unknown. Studies in Nonlinear Dynamis and Eonometris, Issue 3, Artile [4] Kwaitowski, D., P.C.B. Phillips, P. Shmidt, P, and Y. Shin (99). esting the null hypothesis of stationarity against the null hypothesis of a unit root. Journal of Eonometris 54, [5] Ng S. and P. Perron (). Lag length seletion and the onstrution of unit root tests with good size and power. Eonometria 69, [6] Perron, P. (989) he Great Crash, the Oil Prie Shok, and the Unit Root Hypothesis. Eonometria 57, [7] Perron, P. (6) Dealing with Strutural Breaks. In Palgrave Handbook of Eonometris, Vol. : Eonometri heory, Patterson, K., and.c. Mills (eds.), Palgrave Mamillan, 7835
13 [8] Perron, P. and G. Rodríguez (3). GLS detrending, e ient unit root tests and strutural hange. Journal of Eonometris 5, 7. [9] Phillips, P.C.B. and P. Perron (988). esting for a unit root in time series regression. Biometrika 75, [] Shmidt, P. and J. Lee (99). A modi ation of the ShmidtPhillips unit root test. Eonomis Letters 36, [] Shmidt, P. and P.C.B. Phillips (99). LM tests for a unit root in the presene of deterministi trends. Oxford Bulletin of Eonomis and Statistis 54,
14 Appendix Proof of heorem he saled loal GLS estimator of an be written as: p b = V V = V x : he olumns of V := (V ; ; V ; ; V 3; V 4; ) are given by: V ; := ( + )e ; V ; := V 3; := os V 4; := + ; ; os +( + )e ; (A.) where is a vetor of ones, e is a vetor with rst element equal to one and all others equal to zero and is a vetor suh that := (; ; :::; ), and := := os := os := os os ; ; ; os ; os ; :::; ; :::; ; :::; os ; :::; os : he following Lemma details the large sample behaviour of the saled produts involved in (A.). he joint onvergene results in (A.)(A.6) of Lemma A., together with appliations of the ontinuous mapping theorem, are su ient to establish the stated result in (). Lemma A. Let the onditions of heorem hold. hen, as!, and V ;V ;! ; V ;V ;! ; V 4;V 4;! ; V ;V ;! ; V ;V 4;! ; V 4;V ;! ; = V ;x ) x ; = V ;x ) ; = V 4;x ) ( )W () Z V 3;V 3;! ; V 4;V 3;! V ;V 3;! ; = V 3;x ) x ; Z W (r)dr rw (r)dr (A.) (A.3) (A.4) (A.5) (A.6) where V := (V ; ; V ; ; V 3; V 4; ), is suh that V := N V with N = diag( = ; = ; = ; ):
15 Proof of Lemma A. Ug the saling matrix := diag = ; = ; = ; = results for the main diagonal elements of V V. we obtain the following limit V ;V ; = ( + )e ( + )e = ( + ) e e ( + ) e + = + = + o(): V ;V ; = = X t= kt X + t= X kt = o(); t= k (t ) k (t ) where we have used the result from Enders and Lee (8) that kt V 3;V 3; = = os X t= os +( + )e kt X os X + t= t= os k (t ) +( + ) = + o(); os k os kt : os os +( + )e kt k (t ) os + ( + )e os ug the result from Enders and Lee (8) that os kt + + V 4;V 4; = = k + + ( + )e os kt. = o(): urning to the o diagonal elements of the symmetri matrix V V, we have that: 3
16 V ;V ; = ( + )e k = ( + X kt ) t= k X k (t ) ( + ) + t= k = + o() = o() where we have used the identity k V ;V 3; = ( + )e os k = ( + ) os k k : os +( + )e X kt os t= X k (t ) + os + ( + ) ( + ) t= k = ( + ) os + k X kt t= ( + ) + X k (t ) os + ( + ) t= k = os + o() = + o() where we have used the identity os k V ;V 4; = = os k os k : = ( + )e + = ( + ) = = o(): = ( + ) k os 4
17 V ;V 3; = os os +( + )e = os os os + os + ( + )e ( + )e k = os os + + ( + ) X k (t) k (t ) k = + + ( + ) t= k = + o() = o(): V ;V 4; = = + = = + = = + X k (t) X! k (t ) = = (t ) (t ) t= t= = k X kt X k (t ) 5= (t ) os 5= (t ) = o(): t= t= V 3;V 4; = = = = os = os +( + )e + X kt (t ) os ( + ) + = ( + ) k 5= t= X kt (t ) t= 5= X! k (t ) (t ) os t= X k (t ) (t ) os = o(): t= urning nally to the numerator, V x ; in (A.), noting that x = x ; x we observe that: x ; :::; x x ; p V ;x = ( + )e x = ( + )x x X t= x t + X x t = x + o p (): (A.7) 5 t=
18 p V ;x = and x k X kt = x + t= X k (t ) x t t= k = x + k X kt os t= X k (t ) u t + t= k = x + o p () = o p () : p V 3;x = x t x t u t X t= os os +( + )e x k X kt = os x + os t= X k (t ) os x t t= x t k X kt os x t t= k (t ) x t x t x t x t + ( + )x (A.8) = x + o p (): (A.9) Remark A.: We have used results from Bierens (994, Lemma 9.6.3) in establishing (A.8) and (A.9). hese results state that, X F t= t u t = F ()S() Z f(r)s (r)dr where f(r) = F (r). Consequently, for F t = kt it follows that f( t and if F t = os kt then f( t ) = kt (k) : As a result, p X t= kt os u t = W () + k Z (kr) W (r)dr kt ) = (k) os and p X t= kt u t = (k) W () k Z os (kr) W (r)dr : 6
19 Finally, to establish the result in (A.6) observe that p V 4;x = p + x = p x + X x t t= X t= x t + X (t )x t t=! X (t )x t t= and, hene, p V 4;x = ) = t= W () X x t Z 3= X x t + t= W (r)dr + 3= t= Z X (t )x t 5= t= Z rdw (r) rw (r)dr =: W: X (t )x t + o p () 7
20 heorem Considering the DGP in ()(), under the null hypothesis, H : = ; and assuming that is xed, it follows as! that, p (b ) = ) Z Z = Z (f ()' + x ) (x + ' ) W (A.) (A.) where W =( )W () R W (r)dr R rw (r)dr; = x + ' and = W : ++ 3 Note that if ' = ; () orresponds to the results obtained by Elliott et al. (996). heorem 3 Considering the DGP in ()(), under the null hypothesis, H : = ; and assuming that is xed, it follows as! that the limits of the salled parameter estimates, b = (b' ; b' ; b ); obtained from test regression (5) i.e., b = V t V t V t y t where V t = (f t () ; y t ) and = diag n p ; p ; o ; will be the following, b ) 4 R 3 (r) dr w R 3 os (r) dr w 3 5 w 3 w 3 w 33 4 D D D where w 3 = R (r) W (r)dr R r (r) dr; w 3 = R R r os (r) dr; w 33 = R W (r) dr R D = rw (r)dr+ 3 ; D = W () + R (r) W (r)dr and D 3 = h R W (r)dw (r) os (r) W (r)dr () W () R os ( R R rdw (r) W (r)dr As an be observed from heorem 4., the limit distributions of the test statistis will only depend on k, the frequeny used in the Fourier approximation. Note that although in the rst step the limit of b is a funtion of x and the unknown oe ient ' the limit distribution of the estimators omputed in the seond step are free of this nuisane parameter. Proof of heorem he proof of theroem 3. follows along similar lines as the proof of heorem 3.. Hene, onsider rst the limit results for the parameter estimates of the deterministi omponent (just a onstant or a onstant and a time trend) estimated in the rst step. We onsider the more general quasidi erened (QD) deterministi kernel whih inludes a onstant and a time trend, Z = (Z ; ; Z ; ) (A.) where Z ; = ( + )e Z ; = + ; ; 8
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