CESIS Elecronic Working Paper Series Paper No. 214 The Properies of Procedures Dealing wih Uncerainy abou Inercep and Deerminisic Trend in Uni Roo Tesing R. Sco Hacker* and Abdulnasser Haemi-J** *Jönköping Inernaional Business School, **UAE Universiy February 2010 The Royal Insiue of Technology Cenre of Excellence for Science and Innovaion Sudies (CESIS) hp://www.cesis.se - 1 -
The Properies of Procedures Dealing wih Uncerainy abou Inercep and Deerminisic Trend in Uni Roo Tesing R. Sco Hacker Jönköping Inernaional Business School Jönköping Universiy P.O. Box 1026, SE-551 11, Jönköping, Sweden Sco.Hacker@ihh.hj.se Telephone: +46 36 10 17 50, Fax + 46 36 10 17 50 Abdulnasser Haemi-J UAE Universiy P.O. Box: 17555, AL AIN, UAE E-mail: AHaemi@uaeu.ac.ae Tel: +971-3-713 3213 Absrac The classic Dickey-Fuller uni-roo es can be applied using hree differen equaions, depending upon he inclusion of a consan and/or a ime rend in he regression equaion. This paper invesigaes he size and power properies of a uni-roo esing sraegy oulined in Enders (2004), which allows for repeaed esing of he uni roo wih he hree equaions depending on he significance of various parameers in he equaions. This sraegy is similar o sraegies suggesed by ohers for uni roo esing. Our Mone Carlo simulaion experimens show ha serious mass significance problems prevail when using he sraegy suggesed by Enders. Excluding he possibiliy of unrealisic oucomes and using a priori informaion on wheher here is a rend in he underlying ime series, as suggesed by Elder and Kennedy (2001), reduces he mass significance problem for he uni roo es and improves power for ha es. Subsequen esing for wheher a rend exiss is seriously affeced by esing for he uni roo firs, however. Key words: Uni Roos, Deerminisic Componens, Model Selecion Running ile: Uni Roo Tesing wih Equaion Uncerainy JEL classificaion: C30-2 -
Inroducion Uni roo esing is one of he mos common procedures in modern ime series analysis. This has arisen since deerminaion of uni roo saus is a prerequisie o figure ou wheher correlaion beween variables in a regression is spurious or wheher coinegraion exiss. 1 The earlies and one of he simples uni roo ess used is he Dickey and Fuller (1979) es, which is based on one of he hree regression equaions below Δy = 1 + ε (1) by by Δy = a + 1 + ε (2) by Δy = a + c + 1 + ε (3) where y is he variable being esed for uni roo, is ime, ε is whie noise, a, b, and c are parameric consans, and he firs difference operaor is represened by Δ. 2 The null hypohesis of uni roo in his formulaion is expressed as a zero resricion on b. Noe ha equaions (1) and (2) are simply resriced forms of equaion (3). The uni roo es is onesided and he disribuion of he es saisic is non-sandard under he null, and differs depending upon which equaion is used. Dickey and Fuller (1979) have provided he special criical values for a finie se of observaions 3 and MacKinnon (1991) has offered a means for deermining he special criical values more generally. Unforunaely, one ofen does no know which of he hree equaions is appropriae for esing. Some auhors have recommended sequenial esing sraegies o deermine uni roos under such circumsances. The ime series economerics exbook by Enders (2004) for example presens a sequenial esing sraegy, which is repeaed in he applied economerics exbook by Aseriou and Hall (2007). Some auhors, including Enders (2004), have recommended sequenial esing sraegies o deermine uni roos under such circumsances. These sraegies ypically sar wih esing he uni roo using equaion (3), and depending 1 The issue of spurious regression was illusraed by Granger and Newbold (1974) as heir simulaion showed ha a regression of variables wih uni roos produced significan correlaion even if he variables are independen. This poin was also proved analyically by Phillips (1986). The issue of coinegraon was brough up by Granger (1981) and ess for coinegraion were developed by Engle and Granger (1987), Phillips (1987), Johansson (1988), and Johansson and Juselius (1990), among ohers. 2 According o Said and Dickey (1984), he es equaion should be augmened wih lags of Δ y if auocorrelaion exiss for he error erm ε. 3 Dickey and Fuller (1981) show ha he disribuion of he F es is nonsandard when a uni roo is included in he null hypohesis. They provide new criical values for he F es under such circumsances. - 3 -
upon he resul of ha es and ohers, allow consideraion of esing using he more resriced equaions (1) and (2). Such echniques are likely o suffer from he problem of mass significance due o he repeaed esing. The purpose of his paper is o evaluae he precision of inference based on he sequenial esing echniques of Enders (2004) and show he effec of a priori eliminaion of possible oucomes based upon argumens by Elder and Kennedy in heir 2001 aricle Tesing for Uni Roos: Wha Should Sudens Be Taugh?. To achieve his purpose, we conduc Mone Carlo experimens on he sequenial esing echnique pu forward by Enders (2004), and he uni roo esing sraegy suggesed by Elder and Kennedy (2001). The res of his paper is organised in he following way. Nex secion describes he sequenial esing sraegy oulined by Enders (2004) and evaluaes his sraegy. Secion 3 presens he uni roo esing sraegy wih prior resricions as suggesed by Elder and Kennedy (2001), and his is also evaluaed. The las secion concludes he paper. I. The Sequenial Uni Roo Tesing Sraegy of Enders (2004) Of he hree previous equaions, equaion (3) is he mos general wih equaions (1) and (2) nesed in i. Since each of hese hree equaions has one of wo uni roo sauses a uni roo exis or i does no we can consider six possible models as shown in Table 1. These models are presened based on differen resricions imposed on he parameers of he underlying daa generaing process. The main goal is o find ou wheher y has a uni roo or no. To achieve his i is ofen crucial o appropriaely include or no include he inercep and/or ime rend erm in he uni-roo es equaion. To deal wih he lack of informaion of wheher an inercep or ime rend should be included, Enders (2004) provides a muliple-sep sequenial sraegy for going abou he esing for uni roos. He aribues his mehodology as being a modificaion of one suggesed by Dolado, Jenkinson, and Sosvilla-Rivero (1990). Elder and Kennedy (2001) lis oher sources wih similar recommendaions: Perron (1988), Holden and Perman (1994), and Aya and Burridge (2000). The Enders sraegy is shown in Figure 1 for a Dickey-Fuller (DF) environmen. 4 Δy 4 Formally, Enders presened his sraegy in an augmened DF environmen wih various lags of as addiional explanaory variables o handle possible auocorrelaion in he error erms, bu his modificaion is suppressed in his paper for simpliciy. The laer simulaions will have no inheren auocorrelaion in he error - 4 -
Table 1. Definiions of Models Based on he General Equaion Δy = a + by 1 + c + ε Model Model Model Model Model Model (1) (2) (3) (4) (5) (6) a = 0 a = 0 a 0 a 0 a 0 a 0 b = 0 b < 0 b = 0 b < 0 b = 0 b < 0 c = 0 c = 0 c = 0 c = 0 c 0 c 0 Uni roo, no inercep, no ime rend Saionary around zero equilibrium Uni roo wih drif Saionary around nonzero consan equilibrium Uni roo wih inercep and ime rend Deerminisic rend, rend saionary Enders does no specify he concluding model as is done in Figure 1. He jus provides he conclusion of wheher here is a uni roo or no. The concluding model noed in he figure is he curren auhors inerpreaion of he implied model. The following is also done in his paper o complee he model-selecion inerpreaion. Toward he op of he figure one conclusion is Decide no uni roo (model (2), (4) or (6)). In ha case, which of hese hree models is ulimaely concluded is deermined by sandard -saisic esing. If c = 0 can be rejeced, hen we conclude model (6); if i canno be rejeced hen we esimae Δy = by 1 + a + ε and es wheher a = 0 can be rejeced, wih an affirmaive indicaing model (4) and a negaive answer indicaing model (2). Likewise, furher down near he middle of he figure one conclusion is Decide no uni roo ((model (2) or (4)). In ha case, which of he wo models is concluded is deermined by sandard -saisic esing: if a = 0 can be rejeced, hen we conclude model (4); if i canno be rejeced hen we conclude model (2). erms, so his seems reasonable. Also, o be fair o Enders, he warns ha no procedure can be expeced o work well if i used in a compleely mechanical fashion. Ploing he daa is usually an imporan indicaor of he presence of deerminisic regressors. (p. 214) - 5 -
Figure 1. Enders Sraegy Esimae Δy = a + by 1 + c + ε Can b = 0 be rejeced using DF criical values? Yes Decide no uni roo (model (2), (4) or (6)) No Can b = 0 and c = 0 be rejeced using DF criical values? No Yes Can b = 0 be rejeced using a normal disribuion? No Yes Decide uni roo (model 5) Decide no uni roo (model 6) Esimae Δy = a + by 1 + ε Can b = 0 be rejeced using DF criical values? Yes Decide no uni roo ((model (2) or (4)) No Can a = 0 and b = 0 be rejeced using DF criical values? Yes No Can b = 0 be rejeced using a normal disribuion? No Yes Decide here is a uni roo (model 3) Decide no uni roo (model 4) Esimae Δy = by 1 + ε Can b = 0 be rejeced using DF criical values? Yes Decide has no uni roo (model (2)) No Decide here is a uni roo (model (1)) Noes: DF sands for Dickey-Fuller. Each quesion abou parameers is answered based on he las esimaed equaion before he quesion. - 6 -
To make an evaluaion of he Enders (2004) sraegy, we conduc some Mone Carlo simulaions using a program developed for GAUSS. The design of hese simulaions is as follows. Fify observaions are generaed according o parameers consisen wih models (1), (3), or (5), i.e. he models wih a uni roo, using an error erm drawn independenly from a sandard normal disribuion. The Enders sraegy wih he model-choice exensions noed previously is hen employed o deermine wheher a uni roo exiss or no and he implied model given he resuls. This experimen is performed 5000 imes and he percen of imes he null hypohesis of a uni roo is rejeced is repored in Table 2 and he percen of imes each model is chosen is repored in Table 3. The nominal significance level indicaed (10%, 5%, or 1%) is applied on every hypohesis es performed. Table 2. Frequency of Rejecing Uni Roo When There Is a Uni Roo Based on he General Equaion Δy = a + by 1 + c + ε ; Using Enders Sraegy Nominal a= 0, b = 0, a = 0.25, b = 0, a = 1, b = 0, a = 1, b = 0, Significance c = 0: c = 0: c = 0: c =0.4: Level rue model is rue model is rue model is rue model is Model (1) Model (3) Model (3) Model (5) 10% 23.0% 15.3% 10.5% 0.1% 5% 11.5% 7.6% 5.1% 0.0% 1% 2.6% 2.9% 1.2% 0.0% - 7 -
Table 3. Percenage Choosing Various Models, Based on he General Equaion Δy = a + by 1 + c + ε 3(i). Model Chosen, given rue model is (1); a = 0, b = 0, c = 0 Nominal Model Model Model Model Model Model Significance (1) (2) (3) (4) (5) (6) Level 10% 73.5 % 8.7 % 1.7 % 14.1 % 1.8 % 0.3 % 5% 86.1 % 4.9 % 1.3 % 6.8 % 1.1 % 0.0 % 1% 97.0 % 0.9 % 0.2 % 1.7% 0.2 % 0.0 % 3(ii). Model Chosen, given rue model is (3); a = 0.25, b = 0, c = 0 Nominal Model Model Model Model Model Model Significance (1) (2) (3) (4) (5) (6) Level 10% 63.4 % 1.8 % 19.5 % 11.1 % 1.8 % 2.4 % 5% 76.8 % 1.2 % 14.4 % 5.7 % 1.2 % 0.7 % 1% 92.7 % 0.4 % 5.1 % 1.5 % 0.4 % 0.1 % 3(iii). Model Chosen, given rue model is (3); a = 1, b = 0, c = 0 Nominal Model Model Model Model Model Model Significance (1) (2) (3) (4) (5) (6) Level 10% 0.0 % 0.0 % 87.7 % 0.4 % 1.8 % 10.1 % 5% 0.0 % 0.0 % 93.7 % 0.3 % 1.2 % 4.9 % 1% 0.0 % 0.0 % 98.6 % 0.1 % 0.2 % 1.2 % 3(iv). Model Chosen, given rue model is (5); a = 1, b = 0, c = 0.4 Nominal Model Model Model Model Model Model Significance (1) (2) (3) (4) (5) (6) Level 10% 0.0 % 0.0 % 0.0 % 0.0 % 99.9 % 0.1 % 5% 0.0 % 0.0 % 0.0 % 0.0 % 100.0 % 0.0 % 1% 0.0 % 0.0 % 0.0 % 0.0 % 100.0 % 0.0 % - 8 -
The resuls of he simulaion experimens, presened in Tables 2 and 3, may be inerpreed as follows: Wih no inercep and no ime rend, he frequency of concluding a uni roo when here is acually a uni roo is oo low (acual size is oo high relaive o nominal size). This ensues because he Enders mehodology allows for rejecion of he null hypohesis of he uni roo a various seps, so mass significance becomes very problemaic. As Table 3 indicaes, model choices in his siuaion end o be spread over all possible models, wih model (4) as he main alernaive followed by model (2). Wihou a ime rend bu wih a weak drif erm, here is sill over-rejecion of he null hypohesis of a uni roo, alhough no as much as when here is no drif erm. Wihou a ime rend bu wih a srong drif erm, here are less ess ha are relevan in he Enders mehodology esing model (1) versus model (2) is no done since a = 0 is always rejeced. This is visible in Table 3 also, wih models (1) and (2) never chosen when a = 1, b = 0, and c = 0. As a resul, he mass significance problem is reduced, and he acual sizes are closer o he nominal sizes, alhough he acual sizes are oo high. Model (6) is he main incorrecly chosen alernaive, and he percenage choice of ha model closely maches he nominal sizes. Wih a ime rend and a uni roo, he size suddenly becomes oo low, almos always failing o rejec he null hypohesis of a uni roo. This comes abou because his simple Dickey-Fuller es is no sufficien for esing uni roos when here is boh a uni roo and a ime rend. An addiional regressor such as 2 would be needed o es for a uni roo under such circumsances since he number of deerminisic regressors needs o be a leas as numerous as he number deerminisic componens (Harris and Sollis (2003), p. 45). 5 5 Sraegy S1 in Aya and Burridge (2002) includes a 2 regressor in he firs uni-roo es of a sraegy similar o ha of Enders, bu a he cos of more mass significance difficulies and oo-frequen spurious idenificaion of a quadraic rend when a srong linear rend exiss. - 9 -
II. Uni Roo Tesing Sraegy wih Prior Limiaions Elder and Kennedy (2001) have criicized mehods like Enders mehod based on he following argumens: hey are double esing and riple esing for uni roos (he mass significance problem), hey allow for unrealisic oucomes, and hey do no ake advanage of prior knowledge of ime series growh. The problem of mass significance has already been demonsraed by he simulaion resuls ha we have jus presened. Cuing down on possible models based on removing unrealisic oucomes and using prior knowledge abou ime series growh provides a way o deal wih he mass significance problem. Elder and Kennedy (2001) claim ha model (5) should no be allowed due o is explosive naure, 6 and model (2) should no be allowed since a saionary process around an equilibrium of exacly zero is unlikely. When here is no prior knowledge abou growh in he variable, only models (1), (3), (4), and (6) should be allowed. However, if we have a good reason o hink here is a ime rend or a rend creaed by a drif erm we can narrow down our choices furher o models (3) and (6) only. If insead we have a good reason o hink here is no ime rend or a rend creaed by a drif erm, we can narrow down our choices furher o models (1) and (4). How he Enders sraegy is modified by Elder and Kennedy s suggesions (referred o as he Elder and Kennedy sraegy) when here is no prior knowledge of he variable s growh is lised below. 7 Elder and Kennedy Sraegy, No Prior Knowledge of Growh in Variable A. Esimae he equaion Δy = a + by + c + ε, and es wheher b = 0 can be rejeced 1 using DF criical values. If i can be rejeced, conclude no uni roo, and if no, conclude here is a uni roo. 8 6 By explosive is mean he series has a rae of change ha is ever increasing or ever decreasing. Elder and Kennedy (2001) more widely criicize his model as unrealisic for economic ime series, wih explosiveness being one reason why i is unrealisic. Elder and Kennedy refer o Perron (1988) and Holden and Perman (1994) on discussing he issue of he unrealisic naure of his model. 7 Again, he presenaion is in a Dickey-Fuller environmen raher han an augmened Dickey-Fuller one, maching he presenaion in Elder and Kennedy. 8 Elder and Kennedy accep ha some double esing for uni roos could be appropriae a his poin o improve power. The problem of mass significance is reinroduced wih ha double esing, however. - 10 -
B. If b = 0 can be rejeced in sep A, use sandard esing o deermine wheher c = 0 can be rejeced (i.e. conclude model 6) or no (i.e. conclude model 4). C. If b = 0 canno be rejeced in sep A, esimae he equaion Δ y = a + ε, and es wheher a = 0 can be rejeced using sandard -saisic esing. If i can be rejeced, conclude model (3), if no, conclude model (1). If nonzero growh in he y variable is known a priori, he Elder and Kennedy sraegy becomes he same as sep A above, wih he conclusion of no uni roo implying model (6) and he conclusion of uni roo implying model (3) (neiher sep B nor sep C need be done). If zero growh in he y variable is known a priori, he Elder and Kennedy sraegy becomes he same as sep A above wih c excluded in esimaion. The conclusion of no uni roo would hen imply model (4) and he conclusion of a uni roo implying model (1) (neiher sep B nor sep C need be done). The issue of wha consiues appropriae prior knowledge may no be clear however. Some variables for heoreical reasons have growh or no, and ha cerainly consiues prior knowledge. However, if he growh (or no) of a variable is deermined prior o esing solely by looking a he daa or previous similar daa, hen ha eyeball es arguably should be considered par of he esing procedure and again could lead o a mass significance problem afer being followed by oher ess. Examining he Elder and Kennedy sraegy wih no prior knowledge of growh, we can see ha hey avoid he issue of mass significance on he uni roo es by avoiding muliple esing of he uni roo; deerminaion of wheher a uni roo or no exiss is based enirely on one es. A second es afer ha is suggesed by Elder and Kennedy only o deermine wheher growh or no is involved along wih he saionariy or nonsaionariy deermined by he uni roo es. Due o his srucure in heir sraegy, hey compleely conrol for size in heir uni roo es he acual size for ha es should be very close o he nominal size, and simulaions we have done have indicaed ha is rue. 9 9 Anoher sraegy wih good size properies under similar condiions is sraegy S3 of Aya and Burridge (2000), which includes pre-esing for he linear rend using Vogelsang s (1998) -PS1 saisic, which is invarian o he uni roo, followed by a single uni-roo es appropriae given he resuls of he rend es. Tha - 11 -
The power of he uni roo es of course depends on he rue parameer values and he associaed issue of which alernaive siuaion saionary around a nonzero consan or rend saionary is he rue one, and he associaed rue parameer values. If saionariy around a nonzero consan (model 4) is he rue model, hen he power funcion has he ypical shape, wih small magniudes of b resuling in power close o he size, and successively larger magniudes (more negaive) of b resuling in successively higher power. This is demonsraed in he simulaion resuls of Figure 2 when no prior knowledge of growh is used and when correc prior knowledge ha here is no growh is used. The figure also shows ha power improvemen is possible from uilizing prior correc knowledge abou non-growh when saionariy around a nonzero consan is rue, confirming he saemen on his maer by Elder and Kennedy. Figure 2. Power Funcion When DGP is Δy 1 = by 1 + + ε Using Elder and Kennedy Sraegy wih No Prior Knowledge of Growh and 5% Nominal Size for All Tesing. 1 Power 0.8 0.6 0.4 0.2 Absence of growh assumed and used No knowledge of growh used -1-0.5 0 b 0 If rend saionariy (model 6) is he rue model, he power funcion represening he likelihood of acceping saionariy correcly does no differ wheher or no we use correc a priori informaion on growh in he Elder and Kennedy mehod. This is rue since he equaion esimaed for he uni roo es would be he same regardless of he a priori informaion; he esimaed equaion would include ime as an explanaory variable along wih a consan regardless. However, he power funcion under such circumsances can have an sraegy was found o have good size properies for he uni roo es, bu he power properies for ha es were no impressive. - 12 -
unusual shape (rising, falling, and rising again wih higher magniudes of b) as demonsraed in Figure 3. This is perhaps aribuable o he fac ha for values of b near 0, here is slow convergence so he variable can ake on aribues ha seem like hose of a variable generaed by he excluded model (model (5)), in which here is nonsaionariy around a rend leading o explosiveness. Figure 3. Power Funcion When DGP is Δy = 1 + by + + ε Using Elder and Kennedy 1 Sraegy wih No Prior Knowledge of Growh and 5% Nominal Size for Tesing. 1.2 1 0.8 Power 0.6 0.4 0.2-1 -0.8-0.6-0.4-0.2 0 b 0 The Elder and Kennedy sraegy using no prior knowledge on growh is admirable in is conrol of size on he uni roo es. However, hose auhors also sugges a second es (sep B or C) as a possibiliy for hose ineresed in wha is he appropriae model o conclude upon. A his poin he legiimacy of he size of he second es becomes quesionable due o he prior esing for he uni roo. In Table 5 we presen for various daa generaing processes (DGP) he simulaed size or power on he firs es and second es in he Elder and Kennedy sraegy when here is no prior knowledge of growh in he variable. The firs es is he uni roo es. The second es is eiher he es for a drif if a uni roo is no rejeced in he firs es, or he es for a deerminisic ime rend if a uni roo is rejeced in he firs es. The able also shows he frequency of choosing he correc DGP srucure (uni roo no drif, uni roo wih drif, saionary around nonzero consan, or rend saionary). Each cell in he able presens hree - 13 -
numbers. The firs number in each case is he simulaed value when a nominal size of 10% is used on all ess. The second number is he corresponding value when a nominal size of 5% is used on all ess, and he hird number is he corresponding value when a nominal size of 1% is used on all ess. Table 5. Size and Power Properies for he Firs and Second Tess of he Elder and Kennedy Tesing Sraegy wih No Prior Knowledge of Growh True daa generaing process, Size Power Size Power Frequency wih ε ~ N(0,1) 1 s es 1 s es 2 nd es 2 nd es Choosing rue DGP srucure (i) Δy 9.8% - 19.9% - 72.3% = ε 4.9% 9.9% 85.7% 0.8% 1.7% 97.5% (ii) Δy 9.6% - - 100.0% 90.4% = 1 + ε 4.5% 100.0% 95.5% 1.0% 100.0% 99.0% (iii) Δy 10.3% - - 68.2% 61.2% = 0. 25 + ε 5.1% 53.7% 51.0% 0.9% 26.6% 26.4% (iv) Δy - 96.9% 0% - 96.9% = 0.5y 1 + 1+ ε 90.3% 0% 90.3% 62.7% 0% 62.7% (v) Δy - 11.4% 56.1% - 5.0% = 0.05y 1 + 2 + ε 6.8% 41.2% 4.0% 1.7% 17.6% 1.4% (vi) Δy - 96.4% - 100.0% 96.4% = 0.5y 1 + 1+ 0. 4 + ε 89.9% 100.0% 89.9% 60.3% 100.0% 60.3% (vii) Δy - 47.4% - 100.0% 47.4% = 0.1y 1 + 1+ 0. 4 + ε 32.5% 100.0% 32.5% 12.1% 100.0% 12.1% (viii) Δy - 96.6% - 97.7% 94.4% = 0.5y 1 + 1+ 0. 2 + ε 90.1% 63.9% 57.6% 60.5% 4.3% 2.6% (ix) Δy - 96.9% - 1.4% 1.4% = 0.5y 1 + 1+ 0. 1 + ε 90.4% 0.0% 0.0% 62.7% 0.0% 0.0% Noes: a. The firs es is he uni roo es. The second es is eiher he es for a drif if a uni roo is no rejeced in he firs es, or he es for a deerminisic ime rend if a uni roo is rejeced in he firs es. b. The hree numbers in each cell are, in order, he resuls when he nominal size of 10%, 5%, or 1% is used. c. The size and power found on second es are calculaed as he frequencies respecively of rejecing he rue null hypohesis and of acceping he rue alernaive hypohesis for hose siuaions in which saionariy saus was chosen correcly. - 14 -
Wih he rue DGP in case (i) in he able here is a uni roo wih no drif, so he simulaions are providing informaion on size on he firs es (he uni roo es) and size on he second es (he es for a drif erm given here is a uni roo). The resuls indicae he acual size maches he nominal size well on he firs es, bu he second es has acual size oo high compared o he nominal level. The frequency of choosing he rue DGP srucure appears quie good, beween 72% and 98%. Wih he rue DGP in (ii) and (iii), here is a uni roo wih drif, so he simulaions are providing informaion on size on he firs es (he uni roo es) and power on he second es (he es for a drif erm given here is a uni roo). Again, he size on he firs es is wha we expec for each of hese rue DGPs. The power is varying on he second es, bu no in an unexpeced way: when he drif erm is srong as in case (ii), he power is 100%, and when i is weak as in case (iii), he power is noably weak. The frequency of choosing he correc model is srong when he power is srong as in case (ii) and is weakened by he weak power in case (iii). Wih he rue DGP in case (iv) here is saionariy around a nonzero consan, so he simulaions are providing informaion on power on he firs es (he uni roo es) and size on he second es (he es for a deerminisic rend wih an oherwise saionary process). The power on he firs es wih hese parameers seems good, bu he mos surprising aspec here is he acual size on he second es is zero for all hree nominal size levels considered. Because of his, he frequency of choosing he rue DGP srucure is exacly equal o he power on he firs es. Case (v) is he same as case (iv) bu wih a very low rae of convergence for he saionary process (he coefficien on y -1 is very low). Under hese circumsances he rue DGP is geing close o a random walk wih drif, making he disincion difficul beween he rue DGP srucure (saionary around a nonzero consan) and a DGP srucure wih drif-induced growh. However, since a DGP srucure wih drif-induced growh is no available as an opion afer he uni roo has been rejeced, he second es will misake near drif-induced growh for ime-rend induced growh more ofen in case (v) han in case (iv). This explains he high size values found in case (v) for he second es while in case (iv) hey were all zero. Wih he rue DGP in case (vi) here is a rend saionary process, so he simulaions are providing informaion on power on he firs es (he uni roo es) and power on he second es (he es for a deerminisic rend wih an oherwise saionary process). Here we see - 15 -
srong power in boh cases due o he srong convergence parameer and he srong coefficien on he ime variable. The frequency of choosing he correc model is high and equal o he power on he firs es since he power on he second es is 100%. Cases (vii), (viii), and (ix) are aleraions of case (vi) on one parameer each. The changed parameer is made o be lower so we can examine siuaions where he rue model is more likely no o be chosen. In case (vii) he convergence parameer is lowered in magniude from -0.5 o -0.1, resuling in much lower power on he firs es while he power on he second es remains a 100%. In case (viii) he coefficien on he ime variable is reduce from 0.4 o 0.2 in comparison o case (vi). The power on he firs es is no affeced much by his change, bu he power on he second es has gone down. Ineresingly magniude of he drop in he power on he second es varies from lile wih 10% nominal size o very much wih 1% nominal size. This paern is refleced in how he frequency of choosing he rue DGP srucure is reduced. Finally, in case (ix) he coefficien parameer for he ime rend is reduced from 0.4 o 0.1 in comparison o case (vi). The paern of changes observed beween cases (vi) and (viii) are generally repeaed beween cases (vi) and (ix), alhough sronger in magniude. In case (ix) here is very lile power on he second es wih 10% nominal size, and virually zero power on ha es wih 5% or 1% nominal size. - 16 -
III. Conclusions One objecive of his paper has been o evaluae a uni roo model selecion sraegy suggesed by Enders (2004) via Mone Carlo experimens. Our simulaion resuls indicae serious mass significance problem if his sraegy is used in conducing ess for uni roos. Our simulaion resuls also indicae ha uilizing prior resricions o remove non-credible models, as suggesed by Elder and Kennedy (2001), is excepionally helpful, if no crucial, o have Dickey-Fuller uni roo esing have empirical sizes close o heir nominal counerpars. Since he Enders (2004) sraegy does no uilize such prior resricions, is use can be misleading as our simulaions show. We also invesigae he size and power properies of he Elder and Kennedy (2001) uni roo esing sraegy when here is no knowledge of he growh saus of he examined variable. Our simulaions indicae ha afer he uni roo saus has been deermined, he acual size of he subsequen es suggesed by hose auhors (deermining saionariy around a nonzero consan versus rend saionariy, or random walk versus random walk wih drif) is rarely close o he nominal size, wih here being he disinc possibiliy ha he acual size will be exremely far from he nominal size when esing for a ime rend afer saionariy has been deermined. The simulaions also indicae ha when rend saionariy is he rue model, he es for inclusion of a ime rend afer he uni roo has been rejeced is more robus in is power o a low coefficien for he ime variable when higher nominal size levels are used for boh he uni roo es and he rend es (e.g. here is more robusness a he 10% significance level han a he 1% significance level). Acknowledgemens A version of his paper was presened a a conference eniled The Coinegraed VAR Model: Mehods and Applicaions, Copenhagen, 18-20 June 2006. The auhors would like o hank he paricipans, especially David Hendry, Soren Johansen, and Kaarina Juselius, for heir useful commens and suppor. The usual disclaimer applies however. - 17 -
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Phillips, P. (1986) Undersanding Spurious Regressions in Economerics. Journal of Economerics, 33, 311-40. Phillips, P. (1987) Time Series Regression wih a Uni Roo, Economerica, 55(2): 277-301. Said, S. and Dickey, D. (1984) Tesing for Uni Roos in Auoregressive-Moving Average Models wih Unknown Order. Biomerica, 71, 599-607. - 19 -