Estudos e Documentos de Trabalho. Working Papers


 Laurence Lambert
 2 years ago
 Views:
Transcription
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
BANCO DE PORTUGAL Economic Research Department
BANCO DE PORTUGAL Economic Research Department ON THE FISHERKONIECZNY INDEX OF PRICE CHANGES SYNCHRONIZATION D.A Dias, C.Robalo Marques P.D.Neves, J.M.C.Santos Silva WP 704 June 2004 The analyses, opinions
More informationEstudos e Documentos de Trabalho. Working Papers
Estudos e Documentos de Trabalho Working Papers 25 2009 ARE ATM/POS DATA RELEVANT WHEN NOWCASTING PRIVATE CONSUMPTION? Paulo Soares Esteves November 2009 The analyses, opinions and findings of these papers
More informationEstudos e Documentos de Trabalho. Working Papers
Estudos e Documentos de Trabalho Working Papers 4 2008 THE EFFECTS OF LOWCOST COUNTRIES ON PORTUGUESE MANUFACTURING IMPORT PRICES Fátima Cardoso Paulo Soares Esteves March 2008 The analyses, opinions
More informationEstudos e Documentos de Trabalho. Working Papers
Estudos e Documentos de Trabalho Working Papers 5 2008 WHAT IS BEHIND THE RECENT EVOLUTION OF PORTUGUESE TERMS OF TRADE? Fátima Cardoso Paulo Soares Esteves March 2008 The analyses, opinions and findings
More informationEstudos e Documentos de Trabalho. Working Papers
Estudos e Documentos de Trabalho Working Papers 2 2009 FINITE SAMPLE PERFORMANCE OF FREQUENCY AND TIME DOMAIN TESTS FOR SEASONAL FRACTIONAL INTEGRATION Paulo M. M. Rodrigues Antonio Rubia João Valle e
More informationEstudos e Documentos de Trabalho. Working Papers
Estudos e Documentos de Trabalho Working Papers 12 2006 THE IMPACT OF UNEMPLOYMENT INSURANCE GENEROSITY ON MATCH QUALITY DISTRIBUTION Mário Centeno Alvaro A. Novo Junho 2006 The analyses, opinions and
More informationEstudos e Documentos de Trabalho. Working Papers
Estudos e Documentos de Trabalho Working Papers 1 2010 MEASURING COMOVEMENT IN THE TIMEFREQUENCY SPACE António Rua January 2010 The analyses, opinions and findings of these papers represent the views
More informationEstudos e Documentos de Trabalho. Working Papers
Estudos e Documentos de Trabalho Working Papers 7 2008 FORECASTING USING TARGETED DIFFUSION INDEXES Francisco Dias Maximiano Pinheiro António Rua May 2008 The analyses, opinions and findings of these papers
More informationEstudos e Documentos de Trabalho. Working Papers
Estudos e Documentos de Trabalho Working Papers 11 2007 HOW DO DIFFERENT ENTITLEMENTS TO UNEMPLOYMENT BENEFITS AFFECT THE TRANSITIONS FROM UNEMPLOYMENT INTO EMPLOYMENT? John T. Addison Pedro Portugal August
More informationRestricted Least Squares, Hypothesis Testing, and Prediction in the Classical Linear Regression Model
Restrited Least Squares, Hypothesis Testing, and Predition in the Classial Linear Regression Model A. Introdution and assumptions The lassial linear regression model an be written as (1) or (2) where xt
More informationThe Real Business Cycle paradigm. The RBC model emphasizes supply (technology) disturbances as the main source of
Prof. Harris Dellas Advanced Macroeconomics Winer 2001/01 The Real Business Cycle paradigm The RBC model emphasizes supply (echnology) disurbances as he main source of macroeconomic flucuaions in a world
More informationEstudos e Documentos de Trabalho. Working Papers
Estudos e Documentos de Trabalho Working Papers 4 2009 INTERNATIONAL COMOVEMENT OF STOCK MARKET RETURNS: A WAVELET ANALYSIS António Rua Luís C. Nunes March 2009 The analyses, opinions and findings of these
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationThe Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas
The Greek financial crisis: growing imbalances and sovereign spreads Heaher D. Gibson, Sephan G. Hall and George S. Tavlas The enry The enry of Greece ino he Eurozone in 2001 produced a dividend in he
More informationEstudos e Documentos de Trabalho. Working Papers
Estudos e Documentos de Trabalho Working Papers 3 2008 UNEMPLOYMENT BENEFITS AND RESERVATION WAGES: KEY ELASTICITIES FROM A STRIPPEDDOWN JOB SEARCH APPROACH John T. Addison Mário Centeno Pedro Portugal
More informationVector Autoregressions (VARs): Operational Perspectives
Vecor Auoregressions (VARs): Operaional Perspecives Primary Source: Sock, James H., and Mark W. Wason, Vecor Auoregressions, Journal of Economic Perspecives, Vol. 15 No. 4 (Fall 2001), 101115. Macroeconomericians
More informationBALANCE OF PAYMENTS. First quarter 2008. Balance of payments
BALANCE OF PAYMENTS DATE: 20080530 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se
More informationCapacity at Unsignalized TwoStage Priority Intersections
Capaity at Unsignalized TwoStage Priority Intersetions by Werner Brilon and Ning Wu Abstrat The subjet of this paper is the apaity of minorstreet traffi movements aross major divided fourlane roadways
More informationA Holistic Method for Selecting Web Services in Design of Composite Applications
A Holisti Method for Seleting Web Servies in Design of Composite Appliations Mārtiņš Bonders, Jānis Grabis Institute of Information Tehnology, Riga Tehnial University, 1 Kalu Street, Riga, LV 1658, Latvia,
More informationPhillips Curve. Macroeconomics Cunningham
Phillips Curve Macroeconomics Cunningham Original Phillips Curve A. W. Phillips (1958), The Relaion Beween Unemploymen and he Rae of Change of Money Wage Raes in he Unied Kingdom, 18611957, Economica.
More informationChapter 9 Bond Prices and Yield
Chaper 9 Bond Prices and Yield Deb Classes: Paymen ype A securiy obligaing issuer o pay ineress and principal o he holder on specified daes, Coupon rae or ineres rae, e.g. 4%, 5 3/4%, ec. Face, par value
More information5. Hypothesis Tests About the Mean of a Normal Population When σ 2 is Not Known
Statistial Inferene II: he Priniples of Interval Estimation and Hypothesis esting 5. Hypothesis ests About the Mean of a Normal Population When σ is Not Known In Setion 3 we used the tdistribution as
More informationEstudos e Documentos de Trabalho. Working Papers
Estudos e Documentos de Trabalho Working Papers 9 27 TOTAL FACTOR PRODUCTIVITY GROWTH IN THE G7 COUNTRIES: DIFFERENT OR ALIKE? João Amador Carlos Coimbra April 27 The analyses, opinions and findings of
More informationEstudos e Documentos de Trabalho. Working Papers
Estudos e Documentos de Trabalho Working Papers 15 7 INFLATION (MIS)PERCEPTIONS IN THE EURO AREA Francisco Dias Cláudia Duarte António Rua October 7 The analyses, opinions and findings of these papers
More informationWage Bargaining, Job Matching, and the Great Depression
Wage Bargaining, Job Maching, and he Grea Depression Chris Reicher UC San Diego May 2008 Wage Bargaining, Job Maching, and he Grea Depression January 2008 Oupu, employmen (% of rend): 19231941 110% 100%
More informationChapter 1 Microeconomics of Consumer Theory
Chapter 1 Miroeonomis of Consumer Theory The two broad ategories of deisionmakers in an eonomy are onsumers and firms. Eah individual in eah of these groups makes its deisions in order to ahieve some
More informationChapter 6: Business Valuation (Income Approach)
Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationChapter 6 A N ovel Solution Of Linear Congruenes Proeedings NCUR IX. (1995), Vol. II, pp. 708{712 Jerey F. Gold Department of Mathematis, Department of Physis University of Utah Salt Lake City, Utah 84112
More informationLecture 5. Gaussian and GaussJordan elimination
International College of Eonomis and Finane (State University Higher Shool of Eonomis) Letures on Linear Algebra by Vladimir Chernyak, Leture. Gaussian and GaussJordan elimination To be read to the musi
More informationINVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS
INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,
More informationEstimating TimeVarying Equity Risk Premium The Japanese Stock Market 19802012
Norhfield Asia Research Seminar Hong Kong, November 19, 2013 Esimaing TimeVarying Equiy Risk Premium The Japanese Sock Marke 19802012 Ibboson Associaes Japan Presiden Kasunari Yamaguchi, PhD/CFA/CMA
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationExponential Maps and Symmetric Transformations in ClusterSpin. YouGang Feng
Exponential Maps and Symmetri Transformations in ClusterSpin System for Lattie Ising Models YouGang Feng Department of Basi Sienes, College of Siene, Guizhou University, Caijia Guan, Guiyang, 550003
More informationChapter 5 Single Phase Systems
Chapter 5 Single Phase Systems Chemial engineering alulations rely heavily on the availability of physial properties of materials. There are three ommon methods used to find these properties. These inlude
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationMinimum LM Unit Root Test with One Structural Break. Junsoo Lee Department of Economics University of Alabama
Minimum LM Unit Root Test with One Structural Break Junsoo Lee Department of Economics University of Alabama Mark C. Strazicich Department of Economics Appalachian State University December 16, 2004 Abstract
More informationCointegration Analysis of Exchange Rate in Foreign Exchange Market
Coinegraion Analysis of Exchange Rae in Foreign Exchange Marke Wang Jian, Wang Shuli School of Economics, Wuhan Universiy of Technology, P.R.China, 430074 Absrac: This paper educed ha he series of exchange
More informationImpact of Debt on Primary Deficit and GSDP Gap in Odisha: Empirical Evidences
S.R. No. 002 10/2015/CEFT Impac of Deb on Primary Defici and GSDP Gap in Odisha: Empirical Evidences 1. Inroducion The excessive pressure of public expendiure over is revenue receip is financed hrough
More informationWeighting Methods in Survey Sampling
Setion on Survey Researh Methods JSM 01 Weighting Methods in Survey Sampling Chiaohih Chang Ferry Butar Butar Abstrat It is said that a welldesigned survey an best prevent nonresponse. However, no matter
More informationPrice elasticity of demand for crude oil: estimates for 23 countries
Price elasiciy of demand for crude oil: esimaes for 23 counries John C.B. Cooper Absrac This paper uses a muliple regression model derived from an adapaion of Nerlove s parial adjusmen model o esimae boh
More informationAn empirical analysis about forecasting Tmall airconditioning sales using time series model Yan Xia
An empirical analysis abou forecasing Tmall aircondiioning sales using ime series model Yan Xia Deparmen of Mahemaics, Ocean Universiy of China, China Absrac Time series model is a hospo in he research
More informationThe Interest Rate Risk of Mortgage Loan Portfolio of Banks
The Ineres Rae Risk of Morgage Loan Porfolio of Banks A Case Sudy of he Hong Kong Marke Jim Wong Hong Kong Moneary Auhoriy Paper presened a he Exper Forum on Advanced Techniques on Sress Tesing: Applicaions
More informationA Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)
A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke
More informationPerformance Center Overview. Performance Center Overview 1
Performance Cener Overview Performance Cener Overview 1 ODJFS Performance Cener ce Cener New Performance Cener Model Performance Cener Projec Meeings Performance Cener Execuive Meeings Performance Cener
More informationAn Iterated Beam Search Algorithm for Scheduling Television Commercials. Mesut Yavuz. Shenandoah University
0080569 An Iterated Beam Searh Algorithm for Sheduling Television Commerials Mesut Yavuz Shenandoah University The Harry F. Byrd, Jr. Shool of Business Winhester, Virginia, U.S.A. myavuz@su.edu POMS 19
More information5.2 The Master Theorem
170 CHAPTER 5. RECURSION AND RECURRENCES 5.2 The Master Theorem Master Theorem In the last setion, we saw three different kinds of behavior for reurrenes of the form at (n/2) + n These behaviors depended
More informationUser s Guide VISFIT: a computer tool for the measurement of intrinsic viscosities
File:UserVisfit_2.do User s Guide VISFIT: a omputer tool for the measurement of intrinsi visosities Version 2.a, September 2003 From: Multiple Linear LeastSquares Fits with a Common Interept: Determination
More informationChannel Assignment Strategies for Cellular Phone Systems
Channel Assignment Strategies for Cellular Phone Systems Wei Liu Yiping Han Hang Yu Zhejiang University Hangzhou, P. R. China Contat: wliu5@ie.uhk.edu.hk 000 Mathematial Contest in Modeling (MCM) Meritorious
More informationFrom (2) follows, if z0 = 0, then z = vt, thus a2 =?va (2.3) Then 2:3 beomes z0 = z (z? vt) (2.4) t0 = bt + b2z Consider the onsequenes of (3). A ligh
Chapter 2 Lorentz Transformations 2. Elementary Considerations We assume we have two oordinate systems S and S0 with oordinates x; y; z; t and x0; y0; z0; t0, respetively. Physial events an be measured
More informationClassical Electromagnetic Doppler Effect Redefined. Copyright 2014 Joseph A. Rybczyk
Classial Eletromagneti Doppler Effet Redefined Copyright 04 Joseph A. Rybzyk Abstrat The lassial Doppler Effet formula for eletromagneti waves is redefined to agree with the fundamental sientifi priniples
More informationDSPI DSPI DSPI DSPI
DSPI DSPI DSPI DSPI Digital Signal Proessing I (879) Fall Semester, 005 IIR FILER DESIG EXAMPLE hese notes summarize the design proedure for IIR filters as disussed in lass on ovember. Introdution:
More informationThe B.E. Journal of Macroeconomics
The B.E. Journal of Maroeonomis Contributions Volume 7, Issue 2007 Artile 9 Nominal Debt Dynamis, Credit Constraints and Monetary Poliy Liam Graham Stephen Wright University College London, liam.graham@ul.a.uk
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationMACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR
MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR The firs experimenal publicaion, which summarised pas and expeced fuure developmen of basic economic indicaors, was published by he Minisry
More informationDeterminants of Bank Longterm Lending Behavior in the Central African Economic and Monetary Community (CEMAC)
Review of Economics & Finance Submied on 05/Jan./2012 Aricle ID: 1923752920120210708 Consan, Fouopi Djiogap and Augusin Ngomsi Deerminans of Bank Longerm Lending Behavior in he Cenral African Economic
More informationAggregate Output. Aggregate Output. Topics. Aggregate Output. Aggregate Output. Aggregate Output
Topics (Sandard Measure) GDP vs GPI discussion Macroeconomic Variables (Unemploymen and Inflaion Rae) (naional income and produc accouns, or NIPA) Gross Domesic Produc (GDP) The value of he final goods
More informationMobile Broadband Rollout Business Case: Risk Analyses of the Forecast Uncertainties
ISF 2009, Hong Kong, 224 June 2009 Mobile Broadband Rollou Business Case: Risk Analyses of he Forecas Uncerainies Nils Krisian Elnegaard, Telenor R&I Agenda Moivaion Modelling long erm forecass for MBB
More informationSURVEYING THE RELATIONSHIP BETWEEN STOCK MARKET MAKER AND LIQUIDITY IN TEHRAN STOCK EXCHANGE COMPANIES
Inernaional Journal of Accouning Research Vol., No. 7, 4 SURVEYING THE RELATIONSHIP BETWEEN STOCK MARKET MAKER AND LIQUIDITY IN TEHRAN STOCK EXCHANGE COMPANIES Mohammad Ebrahimi Erdi, Dr. Azim Aslani,
More informationSupply chain coordination; A Game Theory approach
aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 upply hain oordination; A Game Theory approah JeanClaude Hennet x and Yasemin Arda xx x LI CNRUMR 668 Université
More informationMeasuring the Effects of Exchange Rate Changes on Investment. in Australian Manufacturing Industry
Measuring he Effecs of Exchange Rae Changes on Invesmen in Ausralian Manufacuring Indusry Robyn Swif Economics and Business Saisics Deparmen of Accouning, Finance and Economics Griffih Universiy Nahan
More informationWorking Paper Monetary aggregates, financial intermediate and the business cycle
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Hong, Hao Working
More informationAn integrated optimization model of a Closed Loop Supply Chain under uncertainty
ISSN 18166075 (Print), 18180523 (Online) Journal of System and Management Sienes Vol. 2 (2012) No. 3, pp. 917 An integrated optimization model of a Closed Loop Supply Chain under unertainty Xiaoxia
More informationLecture Note on the Real Exchange Rate
Lecure Noe on he Real Exchange Rae Barry W. Ickes Fall 2004 0.1 Inroducion The real exchange rae is he criical variable (along wih he rae of ineres) in deermining he capial accoun. As we shall see, his
More information1.3 Complex Numbers; Quadratic Equations in the Complex Number System*
04 CHAPTER Equations and Inequalities Explaining Conepts: Disussion and Writing 7. Whih of the following pairs of equations are equivalent? Explain. x 2 9; x 3 (b) x 29; x 3 () x  2x  22 x  2 2 ; x
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationcos t sin t sin t cos t
Exerise 7 Suppose that t 0 0andthat os t sin t At sin t os t Compute Bt t As ds,andshowthata and B ommute 0 Exerise 8 Suppose A is the oeffiient matrix of the ompanion equation Y AY assoiated with the
More informationINTEREST RATE PASSTHROUGH IN COLOMBIA: A MICROBANKING PERSPECTIVE* Rocío Betancourt Hernando Vargas Norberto Rodríguez**
INTEREST RATE PASSTHROUGH IN COOMBIA: A MICROBANKING PERSPECTIVE* Rocío Beancour Hernando Vargas Norbero Rodríguez** Bogoá, Sepember 2006 *The opinions expressed in his paper are hose of he auhors and
More informationIatrogenic Specification Error: A Cautionary Tale of Cleaning Data
DISCUSSION PAPER SERIES IZA DP No. 1093 Iatrogeni Speifiation Error: A Cautionary Tale of Cleaning Data Christopher R. Bollinger Amitabh Chandra Marh 2004 Forshungsinstitut zur Zukunft der Arbeit Institute
More informationThe Determinants of Trade Credit: Vietnam Experience
Proceedings of he Second AsiaPacific Conference on Global Business, Economics, Finance and Social Sciences (AP15Vienam Conference) ISBN: 9781634158336 Danang, Vienam, 1012 July 2015 Paper ID: V536
More information) ( )( ) ( ) ( )( ) ( ) ( ) (1)
OPEN CHANNEL FLOW Open hannel flow is haraterized by a surfae in ontat with a gas phase, allowing the fluid to take on shapes and undergo behavior that is impossible in a pipe or other filled onduit. Examples
More informationLECTURE: SOCIAL SECURITY HILARY HOYNES UC DAVIS EC230 OUTLINE OF LECTURE:
LECTURE: SOCIAL SECURITY HILARY HOYNES UC DAVIS EC230 OUTLINE OF LECTURE: 1. Inroducion and definiions 2. Insiuional Deails in Social Securiy 3. Social Securiy and Redisribuion 4. Jusificaion for Governmen
More informationBALANCE OF PAYMENTS AND FINANCIAL MA REPORT 2015. All officiell statistik finns på: www.scb.se Statistikservice: tfn 08506 948 01
RKET BALANCE OF PAYMENTS AND FINANCIAL MA REPORT 2015 All officiell saisik finns på: www.scb.se Saisikservice: fn 08506 948 01 All official saisics can be found a: www.scb.se Saisics service, phone +46
More informationWORKFLOW CONTROLFLOW PATTERNS A Revised View
WORKFLOW CONTROLFLOW PATTERNS A Revised View Nik Russell 1, Arthur H.M. ter Hofstede 1, 1 BPM Group, Queensland University of Tehnology GPO Box 2434, Brisbane QLD 4001, Australia {n.russell,a.terhofstede}@qut.edu.au
More informationBANCO DE PORTUGAL Economics Research Department
BANCO DE PORTUGAL Economics Research Deparmen Why should Cenral Banks avoid he use of he underlying inflaion indicaor? Carlos Robalo Marques Pedro Duare Neves Afonso Gonçalves da Silva WP 500 Augus 2000
More informationChapter 5 lecture Notes
Lebanese Amerian University From the SeletedWorks of Ghassan Dibeh 04 Chapter 5 leture Notes Ghassan Dibeh, Lebanese Amerian University Available at: http://works.bepress.om/ghassan_dibeh/00/ Chapter
More informationHow much depreciation of the US dollar for sustainability of the current accounts?
How much depreciaion of he US dollar for susainabiliy of he curren accouns? Eiji Ogawa and Taeshi Kudo Firs version: May 27, 2004 This version: June 6, 2004 This paper is prepared for a conference of he
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationII.1. Debt reduction and fiscal multipliers. dbt da dpbal da dg. bal
Quarerly Repor on he Euro Area 3/202 II.. Deb reducion and fiscal mulipliers The deerioraion of public finances in he firs years of he crisis has led mos Member Saes o adop sizeable consolidaion packages.
More informationSymmetric Subgame Perfect Equilibria in Resource Allocation
Proeedings of the TwentySixth AAAI Conferene on Artifiial Intelligene Symmetri Subgame Perfet Equilibria in Resoure Alloation Ludek Cigler Eole Polytehnique Fédérale de Lausanne Artifiial Intelligene
More informationworking papers Paulo Soares Esteves October 2011
working papers 29 2011 DIRECT VS BOTTOMUP APPROACH WHEN FORECASTING GDP: RECONCILING LITERATURE RESULTS WITH INSTITUTIONAL PRACTICE Paulo Soares Esteves October 2011 The analyses, opinions and findings
More informationWhy Do Real and Nominal. InventorySales Ratios Have Different Trends?
Why Do Real and Nominal InvenorySales Raios Have Differen Trends? By Valerie A. Ramey Professor of Economics Deparmen of Economics Universiy of California, San Diego and Research Associae Naional Bureau
More informationThe Influence of Iran's Entrance into the WTO on Major Indexes of Tehran Stock Exchange
2013, TexRoad Publicaion ISSN: 20904274 Journal of Applied Environmenal and Biological Sciences www.exroad.com The Influence of Iran's Enrance ino he WTO on Major Indexes of Tehran Sock Exchange Darush
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationInvestigation of the effect of the degree of openness of the economy on real effective exchange rate Volatility: case study (the Iran economy)
saqartvelos mecnierebata erovnuli akademiis moambe,. 9, #2, 2015 BULLETIN OF THE GEORGIAN NATIONAL ACADEMY OF SCIENCES, vols. 9, no. 2, 2015 Economy Invesigaion of he effec of he degree of openness of
More informationKONSTANTĪNS BEŅKOVSKIS IS THERE A BANK LENDING CHANNEL OF MONETARY POLICY IN LATVIA? EVIDENCE FROM BANK LEVEL DATA
ISBN 9984 676 20 X KONSTANTĪNS BEŅKOVSKIS IS THERE A BANK LENDING CHANNEL OF MONETARY POLICY IN LATVIA? EVIDENCE FROM BANK LEVEL DATA 2008 WORKING PAPER Lavias Banka, 2008 This source is o be indicaed
More informationNew Classical Economics. Graduate Macroeconomics I ECON 309 Cunningham
New Classical Economics Graduae Macroeconomics I ECON 309 Cunningham New Classical Economics 1. Acceps model of GE wih no imperfecions. 2. Prices are perfecly flexible, and all markes are permanenly cleared
More informationDeterminants of Capital Structure: Comparison of Empirical Evidence from the Use of Different Estimators
Serrasqueiro and Nunes, Inernaional Journal of Applied Economics, 5(1), 1429 14 Deerminans of Capial Srucure: Comparison of Empirical Evidence from he Use of Differen Esimaors Zélia Serrasqueiro * and
More informationOil Price Fluctuations and Firm Performance in an Emerging Market: Assessing Volatility and Asymmetric Effect
Journal of Economics, Business and Managemen, Vol., No. 4, November 203 Oil Price Flucuaions and Firm Performance in an Emerging Marke: Assessing Volailiy and Asymmeric Effec Hawai Janor, Aisyah AbdulRahman,
More informationCausal Relationship between MacroEconomic Indicators and Stock Market in India
Asian Journal of Finance & Accouning Causal Relaionship beween MacroEconomic Indicaors and Sock Marke in India Dr. Naliniprava ripahy Associae Professor (Finance), Indian Insiue of Managemen Shillong
More informationSebastián Bravo López
Transfinite Turing mahines Sebastián Bravo López 1 Introdution With the rise of omputers with high omputational power the idea of developing more powerful models of omputation has appeared. Suppose that
More informationPresent Value Methodology
Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer
More informationRisk Modelling of Collateralised Lending
Risk Modelling of Collaeralised Lending Dae: 4112008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies
More informationAnalysis of tax effects on consolidated household/government debts of a nation in a monetary union under classical dichotomy
MPRA Munich Personal RePEc Archive Analysis of ax effecs on consolidaed household/governmen debs of a naion in a moneary union under classical dichoomy Minseong Kim 8 April 016 Online a hps://mpra.ub.unimuenchen.de/71016/
More informationWhat is a swap? A swap is a contract between two counterparties who agree to exchange a stream of payments over an agreed period of several years.
Currency swaps Wha is a swap? A swap is a conrac beween wo counerparies who agree o exchange a sream of paymens over an agreed period of several years. Types of swap equiy swaps (or equiyindexlinked
More informationStatic Fairness Criteria in Telecommunications
Teknillinen Korkeakoulu ERIKOISTYÖ Teknillisen fysiikan koulutusohjelma 92002 Mat208 Sovelletun matematiikan erikoistyöt Stati Fairness Criteria in Teleommuniations Vesa Timonen, email: vesatimonen@hutfi
More informationBrazil through the eyes of CHORINHO
Brazil hrough he eyes of CHORINHO Fabio Kanczuk FEA/USP Dep of Economics, Universiy of São Paulo April 204 Absrac CHORINHO, a medium scale DSGE model used in he financial secor o inform invesmen decisions,
More informationThe Quantum Theory of Radiation
A. Einstein, Physikalishe Zeitshrift 18, 121 1917 The Quantum Theory of Radiation A. Einstein (Reeived Marh, 1917) The formal similarity of the spetral distribution urve of temperature radiation to Maxwell
More informationWorking Paper Deriving the Taylor principle when the central bank supplies money
eonstor www.eonstor.eu Der OpenAessPublikationsserver der ZBW LeibnizInformationszentrum Wirtshaft The Open Aess Publiation Server of the ZBW Leibniz Information Centre for Eonomis Davies, Ceri; Gillman,
More informationAppendix D Flexibility Factor/Margin of Choice Desktop Research
Appendix D Flexibiliy Facor/Margin of Choice Deskop Research Cheshire Eas Council Cheshire Eas Employmen Land Review Conens D1 Flexibiliy Facor/Margin of Choice Deskop Research 2 Final Ocober 2012 \\GLOBAL.ARUP.COM\EUROPE\MANCHESTER\JOBS\200000\22348900\4
More information