Choice of Spectral Density Estimator in Ng-Perron Test: A Comparative Analysis

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1 Inernaional Economeric Review (IER) Choice of Specral Densiy Esimaor in Ng-Perron Tes: A Comparaive Analysis Muhammad Irfan Malik and Aiq-ur-Rehman Inernaional Islamic Universiy Islamabad and Inernaional Islamic Universiy Islamabad ABSTRACT Ng and Perron (1) designed a uni roo es, which incorporaes he properies of DF- GLS and Phillips Perron es. Ng and Perron claim ha he es performs excepionally well especially in he presence of a negaive moving average. However, he performance of he es depends heavily on he choice of he specral densiy esimaors used in he consrucion of he es. Various esimaors for specral densiy exis in he lieraure; each have a crucial impac on he oupu of es, however here is no clariy on which of hese esimaors gives he opimal size and power properies. This sudy aims o evaluae he performance of he Ng-Perron for differen choices of specral densiy esimaors in he presence of a negaive and posiive moving average using Mone Carlo simulaions. The resuls for large samples show ha: (a) in he presence of a posiive moving average, esing wih he kernel based esimaor gives good effecive power and no size disorion, and (b) in he presence of a negaive moving average, he auoregressive esimaor gives beer effecive power, however, huge size disorion is observed in several specificaions of he daa-generaing process. Key words: Ng-Perron Tes, Mone Carlo, Specral Densiy, Uni Roo Tesing JEL Classificaions: C1, C15, C63 1. INTRODUCTION Uni roo esing is a well-known and one of mos debaed issues in economerics. There are los of economic and economeric implicaions of he exisence of a uni roo in ime series daa, including he incidence of spurious regression (Aiq-ur-Rehman, 11; Libanio, 5). Due o is imporance, many ess and esing procedures were developed for esing for a uni roo. However, he size and power properies of uni roo ess have always been subjec o debae. In many economic ime series models, errors may have heerogeneiy and emporal dependence of unknown forms. This is he main source of size and power disorion of uni roo ess. In order o draw more accurae inferences from esimaes of parameers, consrucing uni roo ess based on long run variance (LRV) esimaes has become imporan. LRV esimaes ake serial correlaion and heerogeneiy ino accoun. The key o consrucing an LRV is o esimae he specral densiy (SD hereafer) a zero frequency. There are wo main ypes of SD esimaors: (1) auoregressive esimaor of specral densiy, (2) kernel based esimaor of specral densiy. However, lieraure does no provide any informaion abou he relaive performance of hese esimaors of specral densiy. Many of exising ess for uni roo, including he Ng-Perron es, use an esimaor of specral densiy a Muhammad Irfan Malik, PhD. Scholar (Economerics), Inernaional Insiue of Islamic Economics, Inernaional Islamic Universiy Islamabad, ( irfanmalik@oulook.com), Tel: Aiq-ur-Rehman, Assisan Professor (Economerics), Inernaional Insiue of Islamic Economics, Inernaional Islamic Universiy Islamabad. 51

2 Malik and Rehman-Choice of Specral Densiy Esimaor in Ng-Perron Tes: A Comparaive Analysis zero frequency. Ng and Perron (1) have developed a new sui of ess, which according o hem ouperforms he oher ess, especially in case of a negaive moving average process. The oupu of he es is also affeced criically by he choice of specral densiy esimaor, and he lieraure does no provide any guide in his regard. Ng and Perron (1) do no discuss he effec of he choice of specral densiy esimaor and hus leave praciioners wihou guidance regarding he choice of an esimaor of specral densiy. This sudy aims o invesigae he properies of Ng-Perron es for differen choices of SD esimaors using Mone Carlo simulaions. We examine he size disorion and effecive power of he es, wih boh auoregressive (AR) esimaor and kernel based (KB) esimaors of specral densiy, in he presence of a negaive and posiive moving average. The remainder of he paper is organized as follows: In Secion 2 we discuss he Ng-Perron es and various esimaors of specral densiy. Secion 3 consiss of our Mone Carlo design. Secion 4 explains he resuls. Secion 5 provides deails of deecing he sign of a moving average. Secion 6 presens some concluding remarks. 2. EFFECT OF SPECTRAL DENSITY ESTIMATOR ON OUTPUT OF NG-PERRON TEST: A REAL DATA ILLUSTRATION Like oher ess, he oupu of Ng-Perron es depends crucially on he choice of he specral densiy esimaor, and he final decision may be quie conradicory for wo differen choices of he densiy. This fac is illusraed below wih he help of a real daa example. We apply he Ng-Perron es on log GDP of UK from Table 2.1 provides he oupus of Ng-Perron es wih boh esimaors of specral densiy esimaors. Specral Densiy Esimaor Ng-Perron ess Wih Drif Wih Drif and Trend AR Esimaor -8.25* * KB esimaor (Parzen Kernel) 1.54* -3.81* Criical Value 5% Criical Value Table 2.1 Oupu of Ng-Perron es wih AR and KB esimaor for log UK GDP daa Noes: * 5% level of significance According o he resuls in Table 2.1, he Ng-Perron es saisics is below he criical value for he auoregressive esimaor of specral densiy; hence, he uni roo hypohesis should be considered rejeced. On he oher hand, for Kernel based esimaor of specral densiy, he Ng- Perron es saisics is far above he criical value; hus, he null of uni roo could no be rejeced even a a loose significance level. Therefore, he person applying he uni roo es may be confused in he choice of resul. In response o his ambiguiy, we designed our sudy o compare he size and power properies of Ng-Perron es so ha a praciioner may ge some guidance on he selecion of an opimal specral densiy esimaor. 3. COMPUTATION OF NG-PERRON TEST In his secion we discuss he Ng-Perron uni roo es and differen esimaors of specral densiy a zero frequency. 52

3 Inernaional Economeric Review (IER) 3.1. Tes Saisics Dufour and King (1991) and Ellio e al. (1996) found ha local GLS derending of he daa yields significan power gains. Phillips and Perron (1987) found ha use of SD could improve he performance of he es. Ng and Perron (1) combine GLS derending wih SD o design a new es. The proposed es consiss of a suie of four ess, namely MZ a, MZ, MSB, and MPT. The four es saisics proposed by Ng-Perron are: 1 T ~ 2 ( y ) fˆ() MZ a 2k MZ MZ MSB a * 1/ 2 ˆ() k MSB f c k ct ( ~ y ) fˆ() MPT 2 1 c k (1 c) T ( ~ y ) fˆ() 2 when d when d 1 where represen drif and drif and rend in DGP, and indicaes he esimae of specral densiy a frequency zero Specral Densiy a Frequency zero -. The symbol The specral densiy a frequency zero represen he heeroskedasiciy and auocorrelaed correced (HAC) sandard error. There are many ways o esimae he specral densiy, which can be divided ino wo ypes: (a) auoregressive specral densiy, and (b) kernel based specral densiy, which can be furher subdivided ino four ypes. This hierarchy is summarized in he following Figure 3.1: Figure 3.1 Summary of Specral Densiy Hierarchy SD Esimaor Auoregressive Kernel Based Parzen Kernel Barle Kernel Quadraic Specral Kernel Tucky-Hanning Kernel The compuaional deails of hese esimaors are as under: 53

4 Malik and Rehman-Choice of Specral Densiy Esimaor in Ng-Perron Tes: A Comparaive Analysis Auoregressive (AR) Esimaor of Specral Densiy Auoregressive esimaor of specral densiy was proposed by Sock (199; see also Sock, 1994; Perron and Ng, 1998). This esimaor, based on he esimaion of parameric model, is idenical o he equaion of he ADF es equaion. Afer having GLS derending series esimae he regression equaion given below: ~ y ~ y Auoregressive esimaor of specral densiy is: 2 ˆ f ˆ() (1 ˆ(1)) where (1) = and. T ~ y ˆ 1 ll (3.1) l1 (1) is he sum of coefficiens of lags of. Here and represen he variance of residuals ( ) from he equaion (3.3) Kernel Based (KB) Esimaor of Specral Densiy Non parameric kernel based esimaor of specral densiy was proposed by Phillips (1987) and hen resrucured by Phillips and Perron (1988). Kernel based esimaor of specral densiy is he weighed sum of auo covariance, in which weighs are decided by he kernel and bandwidh parameer. Esimaing he equaion using GLS derended series, ~ y ~ (3.2) The kernel based esimaor given as: y 1 T l 2 fˆ() ˆ( j) K( j / l) (3.3) j( T l) T ˆ ˆ 1 ˆ( j) T j j where l is bandwidh parameer, which ac as a runcaion lag in he covariance weighing, and K is he kernel funcion, which can be esimaed in muliple ways lised below. (j) is j h order auo covariance of residual from equaion (3.2). For he esimaion of he kernel esimaor of specral densiy we consider he following kernels: 1. Barle Kernel j 2. Parzen Kernel 54

5 Inernaional Economeric Review (IER) 3. Quadraic Specral Kernel 4. Tukey-Hanning Kernel where x = 1991). for all kernels. Asympoically, all of hese kernels are equivalen (Andrews, The compuaional deails are given below. Ng and Perron poin ou ha hese four ess are equivalen in erms of size and power. Throughou our discussion, MZ a is aken as represenaive of hese four. 4. MONTE CARLO EXPERIMENT In order o compare he performance of Ng-Perron es wih AR and KB esimaor of specral densiy, we perform exensive Mone Carlo experimens, which is given in Figure 4.2 below. Figure 4.2 Flow Char of Mone Carlo Experimen Selec DGP Chose Auoregressive and Moving Average Coefficien Generae Series Apply Ng-Perron es wih Auoregressive Esimaor Apply Ng-Perron es wih Kernel Based Esimaor Parzen Kernel Barle Kernel Quadraic Specral Kernel Tucky-Hanning Kernel Compue Size/ Power Compue Size/ Power Compare Resuls 55

6 Malik and Rehman-Choice of Specral Densiy Esimaor in Ng-Perron Tes: A Comparaive Analysis Every sep of he above menioned Mone Carlo experimen is summarized as under: 4.1. Daa Generaing Process The following forms of he daa generaing process were used o conduc he Mone Carlo experimen: DGP-I y = α + u u = ρu 1 + δe 1 + e, DGP-II y = α + + u u = ρu 1 + δe 1 + e, DGP-I resembles an ARMA process wih an inercep bu no rend, whereas DGP-II resembles ARMA wih drif and rend, where =,,,T Auoregressive Coefficien Seing he auoregressive coefficien ρ = 1 will generae a uni series, which could be used o compue he power of Ng-Perron es, whereas seing Rho < 1 generaes saionary series, which can be used o compue he power of he es. The following values of Rho were used for he Mone Carlo experimen:.99,.98,.95,.9,.85,.8, and Moving Average Coefficien The aim of his sudy was o evaluae he performance of he Ng-Perron es boh for posiive and negaive moving average processes. The following values were used in he experimen: -.8, -.6, -.4, -.,,.2,.4,.6, and Calculaing Size and Size Disorion Ng and Perron provide a se of asympoic criical values for heir es. The es saisics calculaed on he series generaed under he null were compared wih hese criical values in order o calculae he acual size of he es. The size disorion is he difference beween he acual size and nominal level of significance Calculaing Power and Effecive Power The power of he es was compued by applying uni roo ess o series generaed wih a saionary roo. The probabiliy of rejecion of he null is he power of he es. However, for several daa generaing processes, heavy size disorion was observed. Since i is no reasonable o compare he power of wo ess wih differen sizes, we have used he effecive power of he ess for comparison. The effecive power was calculaed as follows: Effecive Power for a DGP = Acual Power a Rho < 1 Acual Size for Rho = MONTE CARLO RESULTS This secion illusraes he equivalence of KB esimaors. There are four choices of kernels in his sudy whose compuaional deails are given in Secion 3. The figures below summarize he size and power of Ng-Perron es for differen choice of kernels. The figures show ha he power curves remains same of various choices of kernels. The experimen was repeaed for a var y DGP a d m a u b a d. A a y a k does no significanly affec he size and power of es, herefore here is no need of 56

7 Perecnage Inernaional Economeric Review (IER) summarizing he simulaions for all four kernels. Only one of hese kernels will be sufficien o observe he behavior of he remaining ones. Figure 5.3 The Size and Power of MZ a when Sample Size is 15 and δ =.4 wih DGP-I Size Power for Rho= Lag Truncaion Barle Parzen Lag Truncaion Barle Parzen Quadraic-Specral Tucky-Hanning Quadraic-Specral Tucky-Hanning Figure 5.4 The Size and Power of MZ a when Sample Size is 8, δ = -.4 wih DGP-I Barle Size Quadraic-Specral Lag Truncaion Parzen Tucky-Hanning For he comparison of size and power of he es wih AR esimaor and KB esimaor, we use he Parzen kernel as a represenaive for hese four kernels Effecive Power versus Power Our resuls indicae ha he size of he es is no sable, rendering comparison of he power meaningless. For a more meaningful comparison, we compare he disorion in size and he effecive power of he es. Size disorion is he difference beween he observed size and heoreical size (here 5%) of he es; effecive power is defined as he difference beween he empirical power and empirical size of he es Performance of Tes wih DGP-I Boh KB and AR esimaors of specral densiy are equivalen mahemaically a zero lag lengh/ lag runcaion for any daa generaing process. We discuss he performance of he es Barle Power for Rho= Quadraic-Specral Lag Truncaion Parzen Tucky-Hanning

8 Malik and Rehman-Choice of Specral Densiy Esimaor in Ng-Perron Tes: A Comparaive Analysis wih KB and AR esimaors of specral densiy a nonzero lag lengh/ lag runcaion. Our resul shows ha he es has very low effecive power in a small sample size wih boh esimaors. Figures 3 and 4 depic he size disorion and he effecive power of he es wih a posiive value of moving average coefficien. Figure 5.5 The Size Disorion and Effecive Power Ng-Perron Tes wih AR and KB Esimaor when MA =.2 for DGP-I Size Disorion a sample size 4 and AR(25) 1 KB(4) KB(25) AR(4) Figure 5.6 The Size Disorion and Effecive Power Ng-Perron Tes wih AR and KB Esimaor when MA =.6 for DGP-I Size Disorion a sample size 4 and AR(25) 1 KB(4) KB(25) AR(4) According o he figures above, he disorion in size and effecive power of he es increases wih lag lengh when we use an AR esimaor of specral densiy; on he oher hand, when using a KB esimaor, he effecive power of he es improves wih large lag runcaion wihou any disorion in he size of he es. Therefore i could be deduced ha in he case of a posiive moving average, he KB esimaor ouperforms he AR esimaor. The behavior of he effecive power and disorion remains similar, for experimens wih differen values of MA and auoregressive parameers. A differen picure emerges when we have a negaive moving average in he daa generaing process. Lag lengh selecion has significan consequences on he performance of he es wih an AR esimaor. As eviden in Figures 5 and 6, we observed ha for a weaker negaive 58 Effecive Power for Rho=.95 a sample size 4 and AR(25) KB(25) AR(4) Effecive Power for Rho=.95 a sample size 4 and 25 AR(25) KB(25) AR(4) KB(4) KB(4)

9 Inernaional Economeric Review (IER) moving average srucure, he performance of he es was similar for boh esimaors of SD in large samples. As he negaive moving average srucure becomes sronger, disorion is high when using he effecive power, demonsraing non-monoonic behavior and a decreasing KB esimaor, regardless of he choice of he runcaion lag and sample size. On he oher hand, he size disorion wih an AR esimaor is smaller and reduces o zero when lag lengh is 5. The effecive power shows non-monoonic behavior and sars decreasing afer reaching is maximum value. Effec power is maximized when he lag lengh/ lag runcaion is 2 and sars decreasing a a higher lag. Figure 5.7 The Size Disorion and Effecive Power Ng-Perron Tes wih AR and KB Esimaor when MA =. for DGP-I Size Disorion a sample size 4 and AR(4) KB(4) KB(25) AR(25) Effecive Power for Rho=.95 a sample size 4 and KB(4) AR(4) KB(25) AR(25) Figure 5.8 The Size Disorion and Effecive Power Ng-Perron Tes wih AR and KB Esimaor when MA =.6 for DGP-I Size Disorion a sample size 4 and KB(4) KB(25) AR(4) AR(25) Effecive Power for Rho=.95 a sample size 4 and KB(25) AR(4) AR(25) KB(4) 59

10 Malik and Rehman-Choice of Specral Densiy Esimaor in Ng-Perron Tes: A Comparaive Analysis 5.4. Performance of Tes wih DGP-II In his par of he discussion we sudy he effecive power and size disorion when he daa generaing process consiss of a drif as well as a ime rend. Figures 5.9 and 5.1, given below, depic he performance of he es when we have a posiive moving average srucure in DGP. Like he resuls for DGP-I, we observed ha he KB esimaor is a beer choice for he Ng-Perron es when here is a posiive moving average in DGP wih nonzero lag runcaions. The size disorion of he es is very small wih he KB esimaor even in small samples. On he oher hand, he AR esimaor gives huge size disorion for he small sample sizes, which could be as high as 6% in some cases. The effecive power of he wo esimaors as shown in he righ panel of Figure 5.9 is same for he smaller as well as for he larger sample size. Therefore, he esimaor wih a smaller size disorion should be preferred. Thus he KB esimaor is preferred if here is a posiive moving average. This conclusion maches wih wha we conclude for DGP-I in size is high wih AR esimaor wih deerminisic par consiss boh drif and ime rend a he same ime. Figure 5.9 The Size Disorion and Effecive Power Ng-Perron Tes wih AR and KB Esimaor when MA =. for DGP-II Size Disorion a sample size 4 and AR(4) AR(25) 1 KB(25) KB(4) 4 5 Effecive Power for Rho=.9 a sample size 4 and 25 AR(25) AR(4) KB(4) KB(25) Figures 5.9 and 5.1 show he disorion in size and effecive power of es for he AR and KB esimaors a differen lag lengh/ lag runcaions. The Mone Carle experimen resuls are similar o DGP-1 for a negaive moving average. In he large sample wih a srong negaive moving average srucure size disorion is very high wih boh esimaors a zero lag lengh/ lag runcaion. There is a sharp decrease in he size disorion, approaching zero a lag 5 for he AR esimaor wih effecive power a nearly 77%. On he oher hand, he es wih he KB esimaor has a very low effecive power i.e. maximum 35% power, and disorion in size is well above 4% for any sample size in he presence of a srong negaive moving average. Figures 5.11 and 5.12 shows ha Mone Carlo he resuls obained for he negaive moving average also suppor our previous finding ha he AR esimaor is a beer opion in he presence of a negaive moving average. Based on our Mone Carlo resuls for he Ng-Perron es, we come o he conclusion ha he naure of a moving average is imporan for he selecion of an esimaor for specral densiy. Poor selecion of an esimaor may lead o incorrec inferences abou he exisence of a uni roo. 6

11 Inernaional Economeric Review (IER) Figure 5.1 The Size Disorion and Effecive Power Ng-Perron Tes wih AR and KB Esimaor when MA =.6 for DGP-II Size Disorion a sample size 4 and AR(4) AR(25) 1 KB(25) KB(4) Effecive Power for Rho=.9 a sample size 4 and 25 AR(25) KB(25) AR(4) KB(4) 4 5 Figure 5.11 The Size Disorion and Effecive Power Ng-Perron Tes wih AR and KB Esimaor when MA =. for DGP-II Size Disorion a sample size 4 and AR(4) KB(25) AR(25) KB(4) Effecive Power for Rho=.9 a sample size 4 and KB(25) AR(25) AR(4) KB(4) Figure 5.12 The Size Disorion and Effecive Power Ng-Perron Tes wih AR and KB Esimaor when MA =.6 for DGP-II Size Disorion a sample size 4 and AR(4) KB(25) AR(25) KB(4) Effecive Power for Rho=.9 a sample size 4 and KB(25) AR(25) AR(4) KB(4)

12 Malik and Rehman-Choice of Specral Densiy Esimaor in Ng-Perron Tes: A Comparaive Analysis 6. SUMMARY AND CONCLUSION In his research we have evaluaed he performance of he Ng-Perron es for he following choices of specral densiy esimaors: 1. Auoregressive esimaor 2. Kernel based esimaor wih a. Barle Kernel b. Parzen Kernel c. Quadraic Specral Kernel d. Tukey-Hanning Kernel T mu a xp m a d a va y DGP a d a d a g pa am values. The simulaion resuls reveal ha he kernel based esimaor wih differen kernels resul in similar sizes and powers even in small samples. Hence we conclude ha he choice of he kernel does no make any difference. Furher analysis reveals ha if a daa generaing process conains a posiive moving average, he kernel based esimaor performs beer, and for negaive moving average, an au g v ma p m b. I p d DPG au g v ma, ug z d u, a p d a g DGP KB esimaor no size disorion was observed. REFERENCES Andrews, D. W. (1991). Heeroskedasiciy and Auocorrelaion Consisen Covariance Marix esimaion. Economerica, 59 (3), Aiq-ur-Rehman (11). Impac of Model Specificaion Decisions on Performance of Uni Roo Tess. Inernaional Economeric Review, 3 (2), Dufour, J. M. and M. L. King (1991). Opimal Invarian Tess for he Auocorrelaion Coefficien in Linear Regressions wih Saionary or Nonsaionary Errors. Journal of Economerics, 47 (1), Ellio, G., T. J. Rohenberg and J. H. Sock (1996). Efficien Tess for an Auoregressive Uni Roo. Economerica, 64 (4), Libanio, G. A. (5). Uni Roos in Macroeconomic Time Series: Theory, Implicaions, and Evidence. Nova Economia Belo Horizone, 15 (3), Ng, S. and P. Perron (1). Lag lengh Selecion and he Consrucion of Uni Roo Tess wih Good Size and Power. Economerica, 69 (6), Perron, P. and S. Ng (1998). An Auoregressive Specral Densiy Esimaor a Frequency Zero for Nonsaionary Tess. Economeric Theory, 14, Phillips, P. C. (1987). Time Series Regression wih a Uni Roo. Economerica, 55 (2),

13 Inernaional Economeric Review (IER) Phillips, P. C. and P. Perron (1988). Tesing for a uni roo in ime series regression. Biomerika, 75 (2), Sock, J. H. (199). A Class of Tess for Inegraion and Coinegraion. Manuscrip, Harvard Universiy. Sock, J. H. (1994). Uni Roos, Srucural Breaks and Trends. Handbook of Economerics, 4,