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 DE-RENDED 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.: , [email protected]

2 BANCO DE PORUGAL Edition Eonomis and Researh Department Av. Almirante Reis, 7-6 th 5- Lisboa Pre-press 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 87-7 Legal Deposit No 3664/83

3 he Flexible Fourier Form and Loal GLS De-trended 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 de-trended 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 de-trending 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 de-trending; 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. [email protected]

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] de-trended unit root tests of Carrion-i-Silvestre, 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] de-trending) together with the ordinary least squares [OLS] de-trended Dikey-Fuller [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 Carrion-i-Silvestre, 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 de-trending an yield unit root tests whih are onsiderably more powerful than their OLS and FD de-trended 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 Carrion-i-Silvestre, 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 de-trended unit roots based around the exible Fourier form. he paper is organised as follows. Setion introdues the loal GLS de-trended 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 de-trended tests and ompare the nite sample size and power properties of these tests with the orresponding OLS de-trended and LM tests. Large sample properties

5 of the loal GLS de-trended 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 re-written 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 De-trended Unit Root ests In this setion, we extend the loal GLS de-trending approah of ERS to the problem of testing for a unit root within the ontext of ()-(). his is ahieved through a two-step 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 de-trending 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 DF-type unit root test regression on the loal GLS detrended series, yt := y t zt b f t () b', t = ; :::; ; that is, ompute the t-statisti 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 de-trended 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 t-statisti for = in (6), t LM f say. Remark.5: Enders and Lee (8) also onsider OLS de-trended DF-type statistis for testing H again H in ()-(). In this ase, the appropriate DF-type regression is given by y t = v t! + y t + v t (7) 4

7 where v t := (z t; f t()). he OLS de-trended DF-type statisti is then given by the regression t-statisti 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 two-step proedure where H is tested ug the regression t-statisti for = in y t = y t + v t (8) where y t := y t z t f t() ', are the OLS de-trended 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 de-trending 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 de-trending 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 data-driven 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 under-sized, 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 under-sizing is marginally worse, other things being equal, for t ERS and t LM than for t DF, with the degree of under-sizing 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 de-trended unit root test, t ERS f, proposed in this paper enjoys signi ant power gains over both the OLS and FD de-trended tests, t DF f and t LM f respetively. It is lear that the OLS de-trended test onsistently displays the lowest power among the three tests while the loal GLS de-trended 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 de-trended 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 de-trending 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 de-trending parameter used. Remark 4.: Note that () orresponds to the limit when z t = (; t) is used in de-trending 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 (non-random) 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 de-trending, the parameters of the deterministis are estimated by OLS de-trending from regresg y t onto v t, where v t is as de ned in setion (as is done in the DF-type 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 non-gular 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 de-trending approah outlined in setion might then be to apply the loal GLS de-trending 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 t-statisti 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 well-de 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 mis-spei 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 Dikey-Fuller-type unit root testing proedure based on loal GLS de-trending 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 de-trended Dikey-Fuller-type 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 de-trended 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] Carrion-i-Silvestre, J.L., D. Kim and P. Perron (9). GLS-based 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, 78-35

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 Shmidt-Phillips 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 quasi-di erened (QD) deterministi kernel whih inludes a onstant and a time trend, Z = (Z ; ; Z ; ) (A.) where Z ; = ( + )e Z ; = + ; ; 8

21 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 =(; ; :::; ) : Similarly as in the proof of heorem 3., de ne the diagonal matrix N = diag = ; and onsider Z = Z N : Consequently, the following Lemma an be stated. Lemma A. Considering the DGP ()-() under the null hypothesis, H assumption A (k xed), it follows as! that, and : = ; and Z ;Z ;! (A.3) Z ;Z ;! (A.4) Z ;Z ;! (A.5) = Z ;x ) x (A.6) Z Z = Z ;x ) ( )W () W (r)dr rw (r)dr (A.7) = Z ;Z ;! ; = Z ;Z os;! ; = Z ;Z ;! (A.8) = Z ;Z os;! : (A.9) As in the ase of Lemma A., also these results are useful to haraterize the limit results when only a onstant is onsidered in (A.). hus, under joint onvergene, the results in (A.3) - (A.9) provide the neessary limits to obtain the asymptoti results for the rst step estimators. Proof of Lemma A.. Consider rst the limit results for the denominator of (A.) i.e., the results in (A.3) - (A.5). Saling the elements of Z Z by = diag = ; = we observe that the limits of Z ;Z ; ; Z ;Z ; ; and Z ;Z ; are equivalent to those of V ;V ; ; V 4;V 4; and V ;V 4; provided in (A.7), (A.7) and (A.7), respetively. Regarding the numerator, i.e. the results in (A.6)-(A.9), onsider = Z (f ()' + x ) = = Z f ()' + Z x = : For proof of the results we analyse these two terms separately. Consider rst Z x =. Sine x = x ; x x ; :::; x x = ( + )x e + x x ; it follows that the results for p Z ;x ; and p Z ;x orrespond to those of p V ;x and p V 4;x presented in (A.7) and (??). Regarding = Z f ()', note that = Z f ()' = = " Z ;Z ; Z ;Z ; Z ;Z os; Z ;Z os; # ' ' 9

22 where f () = (Z ; ; Z os; ) ; Z ; = Z os; = os ; os +( + )e ; and the vetors ; ; os and os are as previously de ned in (A.) - (A.), respetively. Hene, the elements of = Z f ()' will have the following limit results: = Z ;Z ; = ( + )e = ( + ) = X X (t ) os + t= t= + o() = o(): (A.) Note that!! as! : hus, for xed we observe that as! ; Z ;Z = ;! : Furthermore, = Z ;Z os; = ( + )e = ( + ) os os os +( + )e X os t= X (t ) + os t= = ( + os ) os + ( + ) + X (t ) os t= = os + ( + ) ( + ) X t= ( + ) os + ( + ) + o() = + o(): (A.) hus, for xed we observe that as! ; = Z ;Z os;! : Moreover,

23 = Z ;Z ; = = = = = 5= = o(); + (t) X (t ) t= X (t ) os t= herefore, = Z ;Z ;!. Finally, = Z ;Z os; = = = = os = = ( + ) 5= 5= X! (t ) (t ) t= X (t ) (t ) t= os +( + )e + ( + ) + X (t ) os t= X! (t ) (t ) os t= X 5= (t ) t= X (t ) (t ) os t= = o() (A.3) and = Z ;Z os;! : Hene, the results in (A.) - (A.3) omplete the proof of Lemma A.. Proof of heorem 3 In order to derive the limit results of the seond step estimators the following test regression with no augmentation is used, y t = f t () ' + y t + u t : Considering = (' ; ' ; ) and V t = (f t () ; yt estimators, n o where = diag p ; p ;. b = X t= V t V t ) ; we look at the limit results of the salled! X t= V t y t (A.)

24 X t= hus, onsider rst the denominator, V t V t = = = X t= f t () f t () 3= X t= X t= 3= y t f t() X t= X t= os 3 X f t ()yt t= X 7 5 t= yt X t= y t 3= X os t= X t= os os the following Lemma with the neessary limit results an be stated. 3= 3= X t= X t= y t Lemma A.3 Considering the DGP ()-() under the null hypothesis, H assumption A (k xed), it follows as! that, ii) iv) v) i) X t= iii) 3= 3= X t= t= X! t= os! X os! t= yt ) Z Z X os yt ) vi) X t= y t Z Z (r) dr os (r) dr y t os y t X t= yt : = ; and Z (r) W (r)dr r (r) dr Z os (r) W (r)dr r os (r) dr Z Z ) W (r) dr rw (r)dr + 3 : where = W ++ 3 and W =( )W () R W (r)dr R rw (r)dr: Proof of Lemma A.3 Noting that yt = y t Z t b ; where y t = + +3 os +xt and Z t = (; ) : Consequently, yt = y t Z t b = + 3 os + x t (b ; ) : Moreover,

25 yt = y t Z t b = + 3 os hus, regarding the result in iv) it follows that, + x t Z t (b ) : X 3= yt X (t ) (t ) = 3= + 3 os + x t Z t (b ) X X = 3= x t 3= Z t (b ) + o() X X p = 3= x t Z t (b ) + o() Z Z ) (r) W (r)dr r (r) dr (A.4) P Note that the result 3= xt ) R (r) W (r)dr follows from Bierens (997, Lemma A.5). Similarly, for v), we observe that, X os 3= yt X (t ) (t ) = os 3= + 3 os + x t Z t (b ) Z Z ) os (r) W (r)dr r os (r) dr: (A.5) Note that in order to prove (A.4) and (A.5) the following results proved quite useful. X (t ) t= = X (t ) os t= = X 4t os os t= = X 4t os os t= os X 4t = os t= (t ) os + os 3 4t X 4t t=

26 and thus, Sine, os t= X 4t os! t= X 4t! ; t= X (t )! : Furthermore, Sine, X (t ) os t= = X 4t + = t= X t= 4 t os + ( os 4 t ) : and X t= 4 t os! it follows that X t= ( os 4 ) t = =! X (t ) os! : t= Finally, with respet to vi), X y t = X (t ) + 3 os = X (xt Z t (b )) + o() (t ) + x t Z t (b ) = X x (b ) X t xt Z t + (b ) X Z t Z t + o() Z Z ) W (r) dr rw (r)dr + 3 : 4

27 his last result follows e the squares and ross produts of the e and oe terms are all of at most order O( ): Regarding the limits of Vt yt note that, P p 3 6 t= y t 4 y 7 t 5 V t y t = = 6 4 p P t= p P t= p P t= os P t= y t y t + 3 os + xt (b ; ) os + 3 os + xt (b ; ) P t= y t + 3 os + xt (b ; ) : Lemma A.4 Considering the DGP ()-() under the null hypothesis, H : = ; and k xed, it follows as! that, p p X t= X t= os yt ) () W () yt ) W () + Z Z os (r) W (r)dr (r) W (r)dr X Z Z yt yt ) W (r)dw (r) t= Proof of Lemma A.4 Following Enders and Lee (4), we an establish that = os and from whih it follows that, Vt yt 6 = 4 Furthermore, e P simpli es to Vt yt 6 = 4 os p P t= p P t= P t= y t t= P t= y t x t = rdw (r) (xt (b ; )) + o() os (xt (b ; )) + o() P = t= p P t= p P t= os + 3 os + xt (b ; ) : os = when is an intenger, this still xt + o() xt + o() p (b; ) 3= P t= y t 5 + o p() : (A.6)

28 Making use of the result in Bierens (994, Lemma 9.6.3) - refered to earlier in the proof of theorem 3. - we obtain, p X t= os u t ) W () + Z (r) W (r)dr and p X t= u t ) () W () Z os (r) W (r)dr : Regarding the last element of the vetor (A.6), reall that (t ) (t ) yt = + 3 os + x t Z t (b ) ; from whih we observe that 3= X yt = t= 3= t= Z ) X (x t Z t (b )) W (r)dr hus, Similarly, p (b; ) 3= X Z yt ) t= X yt x t = t= W (r)dr : X (x t Z t (b )) x t t= Z Z ) W (r)dw (r) rdw (r) : 6

29 able : Critial Values for t ERS f ; t DF f and t LM f Unit Root ests t ERS f t ERS f t DF f t DF f t LM f Note: he results in bold orrespond to the ritial values of the unit root test proedures as originally proposed, by Elliott, et al. (996, able I, p.85), Fuller (996, able.a., p.64) and Shmidt and Phillips (99, able IA, p.64), i.e. with no Fourier terms. he ritial values for the test statistis t ERS f and t DF f were omputed from test regressions with Fourier terms and a onstant only, whereas the ritial values for the test statistis t ERS f, t DF f and t LM f were omputed from test regressions with Fourier terms, a onstant and a time trend..,.5 and. orrespond the %, 5% and % perentiles, respetively

30 able : Empirial Size of Conventional ERS (t ERS ), DF (t DF ) and LM ype (t LM ) Unit Root ests = = t ERS t DF t LM t ERS t DF t LM Note: All results are based on a 5% nominal signi ane level. Critial values used for the t ERS ; t DF and t LM unit root tests were obtained from Elliott, et al. (996, able I, p.85), Fuller (996, able.a., p.64) and Shmidt and Phillips (99, p.64).

31 able 3: Empirial Power of Conventional ERS (t ERS ), DF (t DF ) and LM ype (t LM ) Unit Root ests = = t ERS t DF t LM t ERS t DF t LM See note under able.

32 able 4: Empirial power of Fourier ERS, DF and LM ype ests (Known ) t ERS f t LM f t DF f

33 able 5: Empirial Size and Power of Fourier ERS, DF and LM ype ests (Estimated ) t ERS f t LM f t DF f

34 able 6: Empirial Size and Power of Fourier ERS, DF and LM ype ests (Estimated ) t ERS f t LM f t DF f

35 WORKING PAPERS / UNEMPLOYMEN DURAION: COMPEING AND DEFECIVE RISKS John. Addison, Pedro Portugal / HE ESIMAION OF RISK PREMIUM IMPLICI IN OIL PRICES Jorge Barros Luís 3/ EVALUAING CORE INFLAION INDICAORS Carlos Robalo Marques, Pedro Duarte Neves, Luís Morais Sarmento 4/ LABOR MARKES AND KALEIDOSCOPIC COMPARAIVE ADVANAGE Daniel A. raça 5/ WHY SHOULD CENRAL BANKS AVOID HE USE OF HE UNDERLYING INFLAION INDICAOR? Carlos Robalo Marques, Pedro Duarte Neves, Afonso Gonçalves da Silva 6/ USING HE ASYMMERIC RIMMED MEAN AS A CORE INFLAION INDICAOR Carlos Robalo Marques, João Mahado Mota / HE SURVIVAL OF NEW DOMESIC AND FOREIGN OWNED FIRMS José Mata, Pedro Portugal / GAPS AND RIANGLES Bernardino Adão, Isabel Correia, Pedro eles 3/ A NEW REPRESENAION FOR HE FOREIGN CURRENCY RISK PREMIUM Bernardino Adão, Fátima Silva 4/ ENRY MISAKES WIH SRAEGIC PRICING Bernardino Adão 5/ FINANCING IN HE EUROSYSEM: FIXED VERSUS VARIABLE RAE ENDERS Margarida Catalão-Lopes 6/ AGGREGAION, PERSISENCE AND VOLAILIY IN A MACROMODEL Karim Abadir, Gabriel almain 7/ SOME FACS ABOU HE CYCLICAL CONVERGENCE IN HE EURO ZONE Frederio Belo 8/ ENURE, BUSINESS CYCLE AND HE WAGE-SEING PROCESS Leandro Arozamena, Mário Centeno 9/ USING HE FIRS PRINCIPAL COMPONEN AS A CORE INFLAION INDICAOR José Ferreira Mahado, Carlos Robalo Marques, Pedro Duarte Neves, Afonso Gonçalves da Silva / IDENIFICAION WIH AVERAGED DAA AND IMPLICAIONS FOR HEDONIC REGRESSION SUDIES José A.F. Mahado, João M.C. Santos Silva Bano de Portugal Working Papers i

36 / QUANILE REGRESSION ANALYSIS OF RANSIION DAA José A.F. Mahado, Pedro Portugal / SHOULD WE DISINGUISH BEWEEN SAIC AND DYNAMIC LONG RUN EQUILIBRIUM IN ERROR CORRECION MODELS? Susana Botas, Carlos Robalo Marques 3/ MODELLING AYLOR RULE UNCERAINY Fernando Martins, José A. F. Mahado, Paulo Soares Esteves 4/ PAERNS OF ENRY, POS-ENRY GROWH AND SURVIVAL: A COMPARISON BEWEEN DOMESIC AND FOREIGN OWNED FIRMS José Mata, Pedro Portugal 5/ BUSINESS CYCLES: CYCLICAL COMOVEMEN WIHIN HE EUROPEAN UNION IN HE PERIOD A FREQUENCY DOMAIN APPROACH João Valle e Azevedo 6/ AN AR, NO A SCIENCE? CENRAL BANK MANAGEMEN IN PORUGAL UNDER HE GOLD SANDARD, Jaime Reis 7/ MERGE OR CONCENRAE? SOME INSIGHS FOR ANIRUS POLICY Margarida Catalão-Lopes 8/ DISENANGLING HE MINIMUM WAGE PUZZLE: ANALYSIS OF WORKER ACCESSIONS AND SEPARAIONS FROM A LONGIUDINAL MACHED EMPLOYER-EMPLOYEE DAA SE Pedro Portugal, Ana Rute Cardoso 9/ HE MACH QUALIY GAINS FROM UNEMPLOYMEN INSURANCE Mário Centeno / HEDONIC PRICES INDEXES FOR NEW PASSENGER CARS IN PORUGAL (997-) Hugo J. Reis, J.M.C. Santos Silva / HE ANALYSIS OF SEASONAL REURN ANOMALIES IN HE PORUGUESE SOCK MARKE Miguel Balbina, Nuno C. Martins / DOES MONEY GRANGER CAUSE INFLAION IN HE EURO AREA? Carlos Robalo Marques, Joaquim Pina 3/ INSIUIONS AND ECONOMIC DEVELOPMEN: HOW SRONG IS HE RELAION? iago V.de V. Cavalanti, Álvaro A. Novo 3 /3 FOUNDING CONDIIONS AND HE SURVIVAL OF NEW FIRMS P.A. Geroski, José Mata, Pedro Portugal /3 HE IMING AND PROBABILIY OF FDI: AN APPLICAION O HE UNIED SAES MULINAIONAL ENERPRISES José Brandão de Brito, Felipa de Mello Sampayo 3/3 OPIMAL FISCAL AND MONEARY POLICY: EQUIVALENCE RESULS Isabel Correia, Juan Pablo Niolini, Pedro eles Bano de Portugal Working Papers ii

37 4/3 FORECASING EURO AREA AGGREGAES WIH BAYESIAN VAR AND VECM MODELS Riardo Mourinho Félix, Luís C. Nunes 5/3 CONAGIOUS CURRENCY CRISES: A SPAIAL PROBI APPROACH Álvaro Novo 6/3 HE DISRIBUION OF LIQUIDIY IN A MONEARY UNION WIH DIFFEREN PORFOLIO RIGIDIIES Nuno Alves 7/3 COINCIDEN AND LEADING INDICAORS FOR HE EURO AREA: A FREQUENCY BAND APPROACH António Rua, Luís C. Nunes 8/3 WHY DO FIRMS USE FIXED-ERM CONRACS? José Varejão, Pedro Portugal 9/3 NONLINEARIIES OVER HE BUSINESS CYCLE: AN APPLICAION OF HE SMOOH RANSIION AUOREGRESSIVE MODEL O CHARACERIZE GDP DYNAMICS FOR HE EURO-AREA AND PORUGAL Franiso Craveiro Dias /3 WAGES AND HE RISK OF DISPLACEMEN Anabela Carneiro, Pedro Portugal /3 SIX WAYS O LEAVE UNEMPLOYMEN Pedro Portugal, John. Addison /3 EMPLOYMEN DYNAMICS AND HE SRUCURE OF LABOR ADJUSMEN COSS José Varejão, Pedro Portugal 3/3 HE MONEARY RANSMISSION MECHANISM: IS I RELEVAN FOR POLICY? Bernardino Adão, Isabel Correia, Pedro eles 4/3 HE IMPAC OF INERES-RAE SUBSIDIES ON LONG-ERM HOUSEHOLD DEB: EVIDENCE FROM A LARGE PROGRAM Nuno C. Martins, Ernesto Villanueva 5/3 HE CAREERS OF OP MANAGERS AND FIRM OPENNESS: INERNAL VERSUS EXERNAL LABOUR MARKES Franiso Lima, Mário Centeno 6/3 RACKING GROWH AND HE BUSINESS CYCLE: A SOCHASIC COMMON CYCLE MODEL FOR HE EURO AREA João Valle e Azevedo, Siem Jan Koopman, António Rua 7/3 CORRUPION, CREDI MARKE IMPERFECIONS, AND ECONOMIC DEVELOPMEN António R. Antunes, iago V. Cavalanti 8/3 BARGAINED WAGES, WAGE DRIF AND HE DESIGN OF HE WAGE SEING SYSEM Ana Rute Cardoso, Pedro Portugal 9/3 UNCERAINY AND RISK ANALYSIS OF MACROECONOMIC FORECASS: FAN CHARS REVISIED Álvaro Novo, Maximiano Pinheiro Bano de Portugal Working Papers iii

38 4 /4 HOW DOES HE UNEMPLOYMEN INSURANCE SYSEM SHAPE HE IME PROFILE OF JOBLESS DURAION? John. Addison, Pedro Portugal /4 REAL EXCHANGE RAE AND HUMAN CAPIAL IN HE EMPIRICS OF ECONOMIC GROWH Delfim Gomes Neto 3/4 ON HE USE OF HE FIRS PRINCIPAL COMPONEN AS A CORE INFLAION INDICAOR José Ramos Maria 4/4 OIL PRICES ASSUMPIONS IN MACROECONOMIC FORECASS: SHOULD WE FOLLOW FUURES MARKE EXPECAIONS? Carlos Coimbra, Paulo Soares Esteves 5/4 SYLISED FEAURES OF PRICE SEING BEHAVIOUR IN PORUGAL: 99- Mónia Dias, Daniel Dias, Pedro D. Neves 6/4 A FLEXIBLE VIEW ON PRICES Nuno Alves 7/4 ON HE FISHER-KONIECZNY INDEX OF PRICE CHANGES SYNCHRONIZAION D.A. Dias, C. Robalo Marques, P.D. Neves, J.M.C. Santos Silva 8/4 INFLAION PERSISENCE: FACS OR AREFACS? Carlos Robalo Marques 9/4 WORKERS FLOWS AND REAL WAGE CYCLICALIY Anabela Carneiro, Pedro Portugal /4 MACHING WORKERS O JOBS IN HE FAS LANE: HE OPERAION OF FIXED-ERM CONRACS José Varejão, Pedro Portugal /4 HE LOCAIONAL DEERMINANS OF HE U.S. MULINAIONALS ACIVIIES José Brandão de Brito, Felipa Mello Sampayo /4 KEY ELASICIIES IN JOB SEARCH HEORY: INERNAIONAL EVIDENCE John. Addison, Mário Centeno, Pedro Portugal 3/4 RESERVAION WAGES, SEARCH DURAION AND ACCEPED WAGES IN EUROPE John. Addison, Mário Centeno, Pedro Portugal 4/4 HE MONEARY RANSMISSION N HE US AND HE EURO AREA: COMMON FEAURES AND COMMON FRICIONS Nuno Alves 5/4 NOMINAL WAGE INERIA IN GENERAL EQUILIBRIUM MODELS Nuno Alves 6/4 MONEARY POLICY IN A CURRENCY UNION WIH NAIONAL PRICE ASYMMERIES Sandra Gomes 7/4 NEOCLASSICAL INVESMEN WIH MORAL HAZARD João Ejarque 8/4 MONEARY POLICY WIH SAE CONINGEN INERES RAES Bernardino Adão, Isabel Correia, Pedro eles Bano de Portugal Working Papers iv

39 9/4 MONEARY POLICY WIH SINGLE INSRUMEN FEEDBACK RULES Bernardino Adão, Isabel Correia, Pedro eles /4 ACOUNING FOR HE HIDDEN ECONOMY: BARRIERS O LAGALIY AND LEGAL FAILURES António R. Antunes, iago V. Cavalanti 5 /5 SEAM: A SMALL-SCALE EURO AREA MODEL WIH FORWARD-LOOKING ELEMENS José Brandão de Brito, Rita Duarte /5 FORECASING INFLAION HROUGH A BOOM-UP APPROACH: HE PORUGUESE CASE Cláudia Duarte, António Rua 3/5 USING MEAN REVERSION AS A MEASURE OF PERSISENCE Daniel Dias, Carlos Robalo Marques 4/5 HOUSEHOLD WEALH IN PORUGAL: 98-4 Fátima Cardoso, Vanda Geraldes da Cunha 5/5 ANALYSIS OF DELINQUEN FIRMS USING MULI-SAE RANSIIONS António Antunes 6/5 PRICE SEING IN HE AREA: SOME SYLIZED FACS FROM INDIVIDUAL CONSUMER PRICE DAA Emmanuel Dhyne, Luis J. Álvarez, Hervé Le Bihan, Giovanni Veronese, Daniel Dias, Johannes Hoffmann, Niole Jonker, Patrik Lünnemann, Fabio Rumler, Jouko Vilmunen 7/5 INERMEDIAION COSS, INVESOR PROECION AND ECONOMIC DEVELOPMEN António Antunes, iago Cavalanti, Anne Villamil 8/5 IME OR SAE DEPENDEN PRICE SEING RULES? EVIDENCE FROM PORUGUESE MICRO DAA Daniel Dias, Carlos Robalo Marques, João Santos Silva 9/5 BUSINESS CYCLE A A SECORAL LEVEL: HE PORUGUESE CASE Hugo Reis /5 HE PRICING BEHAVIOUR OF FIRMS IN HE EURO AREA: NEW SURVEY EVIDENCE S. Fabiani, M. Druant, I. Hernando, C. Kwapil, B. Landau, C. Loupias, F. Martins,. Mathä, R. Sabbatini, H. Stahl, A. Stokman /5 CONSUMPION AXES AND REDISRIBUION Isabel Correia /5 UNIQUE EQUILIBRIUM WIH SINGLE MONEARY INSRUMEN RULES Bernardino Adão, Isabel Correia, Pedro eles 3/5 A MACROECONOMIC SRUCURAL MODEL FOR HE PORUGUESE ECONOMY Riardo Mourinho Félix 4/5 HE EFFECS OF A GOVERNMEN EXPENDIURES SHOCK Bernardino Adão, José Brandão de Brito 5/5 MARKE INEGRAION IN HE GOLDEN PERIPHERY HE LISBON/LONDON EXCHANGE, Rui Pedro Esteves, Jaime Reis, Fabiano Ferramosa 6 /6 HE EFFECS OF A ECHNOLOGY SHOCK IN HE EURO AREA Nuno Alves, José Brandão de Brito, Sandra Gomes, João Sousa Bano de Portugal Working Papers v

40 / HE RANSMISSION OF MONEARY AND ECHNOLOGY SHOCKS IN HE EURO AREA Nuno Alves, José Brandão de Brito, Sandra Gomes, João Sousa 3/6 MEASURING HE IMPORANCE OF HE UNIFORM NONSYNCHRONIZAION HYPOHESIS Daniel Dias, Carlos Robalo Marques, João Santos Silva 4/6 HE PRICE SEING BEHAVIOUR OF PORUGUESE FIRMS EVIDENCE FROM SURVEY DAA Fernando Martins 5/6 SICKY PRICES IN HE EURO AREA: A SUMMARY OF NEW MICRO EVIDENCE L. J. Álvarez, E. Dhyne, M. Hoeberihts, C. Kwapil, H. Le Bihan, P. Lünnemann, F. Martins, R. Sabbatini, H. Stahl, P. Vermeulen and J. Vilmunen 6/6 NOMINAL DEB AS A BURDEN ON MONEARY POLICY Javier Díaz-Giménez, Giorgia Giovannetti, Ramon Marimon, Pedro eles 7/6 A DISAGGREGAED FRAMEWORK FOR HE ANALYSIS OF SRUCURAL DEVELOPMENS IN PUBLIC FINANCES Jana Kremer, Cláudia Rodrigues Braz, eunis Brosens, Geert Langenus, Sandro Momigliano, Mikko Spolander 8/6 IDENIFYING ASSE PRICE BOOMS AND BUSS WIH QUANILE REGRESSIONS José A. F. Mahado, João Sousa 9/6 EXCESS BURDEN AND HE COS OF INEFFICIENCY IN PUBLIC SERVICES PROVISION António Afonso, Vítor Gaspar /6 MARKE POWER, DISMISSAL HREA AND REN SHARING: HE ROLE OF INSIDER AND OUSIDER FORCES IN WAGE BARGAINING Anabela Carneiro, Pedro Portugal /6 MEASURING EXPOR COMPEIIVENESS: REVISIING HE EFFECIVE EXCHANGE RAE WEIGHS FOR HE EURO AREA COUNRIES Paulo Soares Esteves, Carolina Reis /6 HE IMPAC OF UNEMPLOYMEN INSURANCE GENEROSIY ON MACH QUALIY DISRIBUION Mário Centeno, Alvaro A. Novo 3/6 U.S. UNEMPLOYMEN DURAION: HAS LONG BECOME LONGER OR SHOR BECOME SHORER? José A.F. Mahado, Pedro Portugal e Juliana Guimarães 4/6 EARNINGS LOSSES OF DISPLACED WORKERS: EVIDENCE FROM A MACHED EMPLOYER-EMPLOYEE DAA SE Anabela Carneiro, Pedro Portugal 5/6 COMPUING GENERAL EQUILIBRIUM MODELS WIH OCCUPAIONAL CHOICE AND FINANCIAL FRICIONS António Antunes, iago Cavalanti, Anne Villamil 6/6 ON HE RELEVANCE OF EXCHANGE RAE REGIMES FOR SABILIZAION POLICY Bernardino Adao, Isabel Correia, Pedro eles 7/6 AN INPU-OUPU ANALYSIS: LINKAGES VS LEAKAGES Hugo Reis, António Rua 7 /7 RELAIVE EXPOR SRUCURES AND VERICAL SPECIALIZAION: A SIMPLE CROSS-COUNRY INDEX João Amador, Sónia Cabral, José Ramos Maria Bano de Portugal Working Papers vi

41 /7 HE FORWARD PREMIUM OF EURO INERES RAES Sónia Costa, Ana Beatriz Galvão 3/7 ADJUSING O HE EURO Gabriel Fagan, Vítor Gaspar 4/7 SPAIAL AND EMPORAL AGGREGAION IN HE ESIMAION OF LABOR DEMAND FUNCIONS José Varejão, Pedro Portugal 5/7 PRICE SEING IN HE EURO AREA: SOME SYLISED FACS FROM INDIVIDUAL PRODUCER PRICE DAA Philip Vermeulen, Daniel Dias, Maarten Dosshe, Erwan Gautier, Ignaio Hernando, Roberto Sabbatini, Harald Stahl 6/7 A SOCHASIC FRONIER ANALYSIS OF SECONDARY EDUCAION OUPU IN PORUGAL Manuel Coutinho Pereira, Sara Moreira 7/7 CREDI RISK DRIVERS: EVALUAING HE CONRIBUION OF FIRM LEVEL INFORMAION AND OF MACROECONOMIC DYNAMICS Diana Bonfim 8/7 CHARACERISICS OF HE PORUGUESE ECONOMIC GROWH: WHA HAS BEEN MISSING? João Amador, Carlos Coimbra 9/7 OAL FACOR PRODUCIVIY GROWH IN HE G7 COUNRIES: DIFFEREN OR ALIKE? João Amador, Carlos Coimbra /7 IDENIFYING UNEMPLOYMEN INSURANCE INCOME EFFECS WIH A QUASI-NAURAL EXPERIMEN Mário Centeno, Alvaro A. Novo /7 HOW DO DIFFEREN ENILEMENS O UNEMPLOYMEN BENEFIS AFFEC HE RANSIIONS FROM UNEMPLOYMEN INO EMPLOYMEN John. Addison, Pedro Portugal /7 INERPREAION OF HE EFFECS OF FILERING INEGRAED IME SERIES João Valle e Azevedo 3/7 EXAC LIMI OF HE EXPECED PERIODOGRAM IN HE UNI-ROO CASE João Valle e Azevedo 4/7 INERNAIONAL RADE PAERNS OVER HE LAS FOUR DECADES: HOW DOES PORUGAL COMPARE WIH OHER COHESION COUNRIES? João Amador, Sónia Cabral, José Ramos Maria 5/7 INFLAION (MIS)PERCEPIONS IN HE EURO AREA Franiso Dias, Cláudia Duarte, António Rua 6/7 LABOR ADJUSMEN COSS IN A PANEL OF ESABLISHMENS: A SRUCURAL APPROACH João Miguel Ejarque, Pedro Portugal 7/7 A MULIVARIAE BAND-PASS FILER João Valle e Azevedo 8/7 AN OPEN ECONOMY MODEL OF HE EURO AREA AND HE US Nuno Alves, Sandra Gomes, João Sousa 9/7 IS IME RIPE FOR PRICE LEVEL PAH SABILIY? Vitor Gaspar, Frank Smets, David Vestin Bano de Portugal Working Papers vii

42 /7 IS HE EURO AREA M3 ABANDONING US? Nuno Alves, Carlos Robalo Marques, João Sousa /7 DO LABOR MARKE POLICIES AFFEC EMPLOYMEN COMPOSIION? LESSONS FROM EUROPEAN COUNRIES António Antunes, Mário Centeno 8 /8 HE DEERMINANS OF PORUGUESE BANKS CAPIAL BUFFERS Miguel Bouinha /8 DO RESERVAION WAGES REALLY DECLINE? SOME INERNAIONAL EVIDENCE ON HE DEERMINANS OF RESERVAION WAGES John. Addison, Mário Centeno, Pedro Portugal 3/8 UNEMPLOYMEN BENEFIS AND RESERVAION WAGES: KEY ELASICIIES FROM A SRIPPED-DOWN JOB SEARCH APPROACH John. Addison, Mário Centeno, Pedro Portugal 4/8 HE EFFECS OF LOW-COS COUNRIES ON PORUGUESE MANUFACURING IMPOR PRICES Fátima Cardoso, Paulo Soares Esteves 5/8 WHA IS BEHIND HE RECEN EVOLUION OF PORUGUESE ERMS OF RADE? Fátima Cardoso, Paulo Soares Esteves 6/8 EVALUAING JOB SEARCH PROGRAMS FOR OLD AND YOUNG INDIVIDUALS: HEEROGENEOUS IMPAC ON UNEMPLOYMEN DURAION Luis Centeno, Mário Centeno, Álvaro A. Novo 7/8 FORECASING USING ARGEED DIFFUSION INDEXES Franiso Dias, Maximiano Pinheiro, António Rua 8/8 SAISICAL ARBIRAGE WIH DEFAUL AND COLLAERAL José Fajardo, Ana Laerda 9/8 DEERMINING HE NUMBER OF FACORS IN APPROXIMAE FACOR MODELS WIH GLOBAL AND GROUP-SPECIFIC FACORS Franiso Dias, Maximiano Pinheiro, António Rua /8 VERICAL SPECIALIZAION ACROSS HE WORLD: A RELAIVE MEASURE João Amador, Sónia Cabral /8 INERNAIONAL FRAGMENAION OF PRODUCION IN HE PORUGUESE ECONOMY: WHA DO DIFFEREN MEASURES ELL US? João Amador, Sónia Cabral /8 IMPAC OF HE RECEN REFORM OF HE PORUGUESE PUBLIC EMPLOYEES PENSION SYSEM Maria Manuel Campos, Manuel Coutinho Pereira 3/8 EMPIRICAL EVIDENCE ON HE BEHAVIOR AND SABILIZING ROLE OF FISCAL AND MONEARY POLICIES IN HE US Manuel Coutinho Pereira 4/8 IMPAC ON WELFARE OF COUNRY HEEROGENEIY IN A CURRENCY UNION Carla Soares 5/8 WAGE AND PRICE DYNAMICS IN PORUGAL Carlos Robalo Marques Bano de Portugal Working Papers viii

43 6/8 IMPROVING COMPEIION IN HE NON-RADABLE GOODS AND LABOUR MARKES: HE PORUGUESE CASE Vanda Almeida, Gabriela Castro, Riardo Mourinho Félix 7/8 PRODUC AND DESINAION MIX IN EXPOR MARKES João Amador, Lua David Opromolla 8/8 FORECASING INVESMEN: A FISHING CONES USING SURVEY DAA José Ramos Maria, Sara Serra 9/8 APPROXIMAING AND FORECASING MACROECONOMIC SIGNALS IN REAL-IME João Valle e Azevedo /8 A HEORY OF ENRY AND EXI INO EXPORS MARKES Alfonso A. Irarrazabal, Lua David Opromolla /8 ON HE UNCERAINY AND RISKS OF MACROECONOMIC FORECASS: COMBINING JUDGEMENS WIH SAMPLE AND MODEL INFORMAION Maximiano Pinheiro, Paulo Soares Esteves /8 ANALYSIS OF HE PREDICORS OF DEFAUL FOR PORUGUESE FIRMS Ana I. Laerda, Russ A. Moro 3/8 INFLAION EXPECAIONS IN HE EURO AREA: ARE CONSUMERS RAIONAL? Franiso Dias, Cláudia Duarte, António Rua 9 /9 AN ASSESSMEN OF COMPEIION IN HE PORUGUESE BANKING SYSEM IN HE 99-4 PERIOD Miguel Bouinha, Nuno Ribeiro /9 FINIE SAMPLE PERFORMANCE OF FREQUENCY AND IME DOMAIN ESS FOR SEASONAL FRACIONAL INEGRAION Paulo M. M. Rodrigues, Antonio Rubia, João Valle e Azevedo 3/9 HE MONEARY RANSMISSION MECHANISM FOR A SMALL OPEN ECONOMY IN A MONEARY UNION Bernardino Adão 4/9 INERNAIONAL COMOVEMEN OF SOCK MARKE REURNS: A WAVELE ANALYSIS António Rua, Luís C. Nunes 5/9 HE INERES RAE PASS-HROUGH OF HE PORUGUESE BANKING SYSEM: CHARACERIZAION AND DEERMINANS Paula Antão 6/9 ELUSIVE COUNER-CYCLICALIY AND DELIBERAE OPPORUNISM? FISCAL POLICY FROM PLANS O FINAL OUCOMES Álvaro M. Pina 7/9 LOCAL IDENIFICAION IN DSGE MODELS Nikolay Iskrev 8/9 CREDI RISK AND CAPIAL REQUIREMENS FOR HE PORUGUESE BANKING SYSEM Paula Antão, Ana Laerda 9/9 A SIMPLE FEASIBLE ALERNAIVE PROCEDURE O ESIMAE MODELS WIH HIGH-DIMENSIONAL FIXED EFFECS Paulo Guimarães, Pedro Portugal Bano de Portugal Working Papers ix

44 /9 REAL WAGES AND HE BUSINESS CYCLE: ACCOUNING FOR WORKER AND FIRM HEEROGENEIY Anabela Carneiro, Paulo Guimarães, Pedro Portugal /9 DOUBLE COVERAGE AND DEMAND FOR HEALH CARE: EVIDENCE FROM QUANILE REGRESSION Sara Moreira, Pedro Pita Barros (to be published) /9 HE NUMBER OF BANK RELAIONSHIPS, BORROWING COSS AND BANK COMPEIION Diana Bonfim, Qinglei Dai, Franeso Frano 3/9 DYNAMIC FACOR MODELS WIH JAGGED EDGE PANEL DAA: AKING ON BOARD HE DYNAMICS OF HE IDIOSYNCRAIC COMPONENS Maximiano Pinheiro, António Rua, Franiso Dias 4/9 BAYESIAN ESIMAION OF A DSGE MODEL FOR HE PORUGUESE ECONOMY Vanda Almeida 5/9 HE DYNAMIC EFFECS OF SHOCKS O WAGES AND PRICES IN HE UNIED SAES AND HE EURO AREA Rita Duarte, Carlos Robalo Marques 6/9 MONEY IS AN EXPERIENCE GOOD: COMPEIION AND RUS IN HE PRIVAE PROVISION OF MONEY Ramon Marimon, Juan Pablo Niolini, Pedro eles 7/9 MONEARY POLICY AND HE FINANCING OF FIRMS Fiorella De Fiore, Pedro eles, Oreste ristani 8/9 HOW ARE FIRMS WAGES AND PRICES LINKED: SURVEY EVIDENCE IN EUROPE Martine Druant, Silvia Fabiani, Gabor Kezdi, Ana Lamo, Fernando Martins, Roberto Sabbatini 9/9 HE FLEXIBLE FOURIER FORM AND LOCAL GLS DE-RENDED UNI ROO ESS Paulo M. M. Rodrigues, A. M. Robert aylor Bano de Portugal Working Papers x