A Poin Opimal es for he Null of Near Inegraion A. Aznar and M. I. Ayuda Universiy of Zaragoza he objecive of his paper is o derive a poin opimal es for he null hypohesis of near inegraion (PONI-es). We show ha he limiing disribuion of he procedure, under he null hypohesis, is Sandard Normal. In he las secion we provide Mone- Carlo simulaion resuls o illusrae he finie sample performance of he es. Key Words: Near inegraion, Poin opimal, Mone-Carlo, Uni roo. JEL classificaions: C, C5, C Deparameno de Análisis Económico. Universiy of Zaragoza. Gran Vía -4. 55- Zaragoza. Spain. email:aaznar@.unizar.es. Phone: 349767689. Fax: 3497676996.
.- Inroducion Deermining wheher a paricular ime series is rend saionary or has a uni roo is a key aspec of any specificaion process for ime series models. A huge effor has been made in he las hiry years or so o develop esing procedures o deermine he sochasic characer of he series. wo broad caegories of procedures can be disinguished: uni-roo ess and saionary ess. he uni-roo ess consider he simple null hypohesis ha he observed series is difference saionary agains a composie alernaive ha specifies ha he series is rend saionary. he null hypohesis of he saionary ess saes ha he sochasic par of he series is saionary while he alernaive specifies ha i has a uni roo. here is a vas lieraure dedicaed o developing procedures o es for he null hypohesis of a uni roo. (See Haldrup and Jansson (6) for a recen survey of his lieraure). Useful references for wha follows in his paper are Ellio, Rohenberg and Sock (996), and Ng and Perron (). In he firs of hese, Ellio e al. propose a new family of esing procedures based on he Neyman-Pearson principles o defining he power envelope for uni roo ess. Furhermore, Ellio e al. propose a new mehod, he GLS mehod, o deal wih he deerminisic erms. Ng and Perron (), compare differen uni roo ess paying a special aenion o he deerminaion of he number of lags ha, as is well known, is a key aspec of any parameric approach. For he saionary ess, he null hypohesis specifies ha he sochasic par of he series is saionary while, under he alernaive, i has a uni roo. If, following he Dickey and Fuller (979) approach, we consider a firs order auoregressive - AR() - process, as he esing model, hen he null hypohesis specifies ha he auoregressive parameer is smaller han one while he alernaive hypohesis says ha he parameer
equals one. Hence, i can be seen ha he null hypohesis is composie while he alernaive is simple. Kwiakowski, Phillips, Schmid, and Shin (99), sae ha all procedures ha adop a null hypohesis ha is composie..."suffer from he lack of a plausible model in which he null of saionariy is naurally framed as a parameric resricion". For his reason, hey propose using a reparameerizaion in which he ime series is decomposed ino he sum of a deerminisic erm, a pure random walk, and a saionary process. he null hypohesis of rend saionariy corresponds o he hypohesis ha he variance of he random walk equals zero. (See Leybourne and McCabe (994, 999), Lanne and Saikkonen (3) and Jansson (5)). In hese papers i can be seen ha he model based on his decomposiion is second order equivalen in momens o an ARIMA(,,) model and hen, he null hypohesis is ha he moving-average parameer is one. In his paper, we derive a esing procedure o es he null hypohesis of local o uni roo saionary agains he uni roo alernaive. We consider a model similar o ha of Müller (5) and Harris e al. (7) and, using he Neyman-Pearson principles, we derive a parameric poin opimal es afer selecing a paricular value of he null hypohesis. We show ha, under he null hypohesis, he es saisic, asympoically, follows a Sandard Normal disribuion and ha, under he uni roo alernaive, i is consisen. Firs, hese resuls are derived assuming ha he model has no deerminisic erms and, hen a consan and a ime linear rend are included. he simulaion resuls we repor in he las secion show ha he performance of he esing procedure developed in he paper is reasonably good. he res of he paper is organized as follows. In Secion, we commen on he sochasic characerisics of he model ha we assume as he Daa Generaing Process (DGP) and we derive some preliminary resuls. he es saisic and is limiing 3
behaviour are examined in Secion 3. In Secion 4 we assess he performance of he es wih finie samples reporing he resuls from a Mone-Carlo simulaion sudy. he main conclusions are summarised in Secion 5..- Models and Some Preliminary Resuls Consider he following daa generaing process (DGP) for an observed ime series, y y d u () u u... p u p () where is i.i.d. N(,. he deerminisic erm, d, is specified as d x where and x = ( ). For he case wih no deerminisic erms, ()-() can be wrien as Δy y Δy... p Δy p (3) p where = i and j = i. We assume ha = c, c and we wish o es i= p i= j+ he null hypohesis of local o uni roo saionary agains he uni roo alernaive. Following Müller (5) and Harris e al. (7), we sae hese as H : c c agains H c= where c specifies he minimal amoun of mean reversion under : he saionary null hypohesis. Noe ha, in his paper, near inegraion is defined by assuming ha is p ( O ) whereas in oher papers in he area, see, for example, he cied papers of Müller (5) and Harris e al. (7), i is assumed ha is O ( ). his is a key aspec of he proposal made in his paper. In order o explain he moivaion of his 4
change, consider ha u in () follows an AR() process ha we wrie as: c = + ε where = +, c < wih f ( ) eiher or f( ) u u. Alernaively, we can wrie j u ε j j= =. Define z = uε ; i is well known ha z is a maringale difference sequence when j= <. (See Example 7.5 of Hamilon (994)). Now, j j c c since = + exp f( ) f( ) j j we can wrie j as j= j= c exp f( ) j. If f ( ) =, hen c exp j is a sequence such ha he smalles value is c e for j. However, when f ( ) =, here exiss a such ha, for any finie ε, c exp < ε for all >. his implies ha if f ( ) =, he condiion does no hold and he Ornsein-Uhlenbeck process is required o derive he limiing j= < j disribuions. When f ( ) =, he condiion holds, and he limiing disribuions can be derived by using a sandard Gaussian framework, because uε is a maringale difference sequence. Le us now derive he log-likelihood funcion corresponding o he process generaed by (3). Le ( ) be denoed by. As can be seen in Hamilon (994), condiional on p iniial values, he log-likelihood funcion for he complee sample is: Δ ( y y Δy... p Δy p+ ) p+ l( ) = k (4) σ where values. k is a consan ha depends on he parameers of he model and on p iniial 5
Consider a paricular value of under he null hypohesis,. According o he Neyman-Pearson Lemma, when we are esing H : = agains he alernaive, H :, = he bes esing procedure is he es ha rejecs he null hypohesis when l() l( ) < log h= h (5) where h is a consan beween zero and one. Using (4), he lef hand side of (5) can be wrien as l() l( ) = ( Δy Δy... p y p ) σ Δ + p+ Δ ( y y Δy... p Δy p+ ) p+ + (6) Now, since Δy y Δy p Δ y p+ = (... ) p p+ ( Δy Δy... Δ y ) + y y ( Δy Δy... Δy ) p p+ (7) we can wrie (6) as l() l( ) = y y ( Δy Δy... p Δy p+ ) (8) σ p+ p+ Hence, he inequaliy in (5) is equivalen o y ( Δy Δy... p Δy p+ ) ( y ) p+ p+ h σ > σ ( y ) σ ( y ) p+ p+ (9) 3.-es Procedure and is limiing behaviour Using (3), he lef hand side erm of (9) can be wrien as 6
y y p+ p+ + y σ y p+ p+ ε σ( ) ( ) () Since, as we commened before, y ε is a maringale difference sequence, he second erm of (), asympoically, follows a sandard Normal disribuion. Le ˆ,..., ˆ p be consisen esimaors under he null hypohesis, so ha * * ( ˆ ),...,( ˆ ) * * * * p p are / O ). hen we have: p ( * * * y ˆ ˆ Δy Δy p Δy p+ y Δy Δy p Δy p+ p+ p+ (... ) (... ) () where signifies asympoic equivalence. he saisic we propose o es he null hypohesis is: ˆ ˆ Δ Δ p Δ p+ p+ p+ σ ( y ) σ p+ y ( y y... y ) c( y / ) PONI = + () where σ is he OLS esimaor of σ using (3). In heorem we derive he limiing disribuion of he PONI es. heorem. If he daa are generaed by (3), wih ε being i.i.d. N(, hen, under he null hypohesis ha <, he PONI saisic, asympoically, follows a Sandard Normal disribuion. Proof: Using (), he firs erm on he righ hand side of he PONI es is asympoically equivalen o () and he resuls follows because / / * y y y c p+ p p + + = = y σ σ p+ σ ( ) 7
saionary is Now, he criical region we propose o es he null hypohesis ha he series is PONI N (3) Where N ε is he criical poin of a Sandard Normal disribuion for a chosen nominal size,. Le us nex derive he limiing behaviour of he es saisic when he daa are generaed by he alernaive hypohesis ha he series has a uni roo. heorem. When he daa are generaed assuming ha c = hen he PONI es is consisen. Proof: Under H we have PONI ε p+ p+ ( y ) σ p+ y c( y / ) = + σ (4) Now, since, as can be seen in Proposiion 7.3 of Hamilon (994), y and y are, respecively, ( ) Op and O ( p ), he firs erm of (4) converges o a well defined limi because we can wrie p p y y p p y / y / On he oher hand, we have c p y / c p y / 8
I is clearly seen ha his expression, under he alernaive hypohesis, does no converge as he sample size increases. When one considers models wih deerminisic erms, an appropriae derending procedure of he daa is needed. wo approaches have been proposed in he lieraure. In he firs, he derended series is he vecor of OLS residuals of he regression of he original series on a consan (Model ) or on a consan and a ime linear rend (Model 3); see, for example, Xiao (). he second approach is he GLS derending procedure developed in Ellio e al. (996) and Ng and Perron (). his mehod can be described as follows. For any series, x, of lengh and any consan c, define c x = ( x, Δx c x,..., Δx c x ). hen, he wo parameers of γ are esimaed by using he regression of y, is given by c y on c x. Le ˆ γ be his esimaor. he GLS derended series, y = y xγ ˆ Afer carrying ou some Mone-Carlo simulaion experimens, we concluded ha he second approach is more efficien in erms of he rade-off beween size and power achieved by he PONI es. 4.- Mone Carlo experimens In his secion, we provide Mone Carlo simulaion resuls o illusrae he finie sample performance of he PONI es. We also compare he finie sample performance of our es o ha of he modified KPSS es (MKPSS) for near inegraion of Harris e al. (7). We consider he following DGP: y = μ + δ+ u, =,,...,, u = u + ε, =,...,, ε ~ iidn(,) 9
We have resuls for hree differen models depending on he form adoped by he deerminisic erms: Model : when μ = and δ = ; Model : when δ = and Model 3: when μ and δ. he values considered for are =.9,.93,.95,.97,.98, If daa are generaed using Model, he PONI saisic we use is: PONI y p+ = + / σ y p+ Δy c y / p+ σ / where σ is he maximum-likelihood esimaor of he variance of he perurbaion in he following regression Δ y = βy + u; c is saed as.5 so ha a heoreical size of 5% is guaraneed for a value of he null parameer,, equal o.5 given. For example, a 5% significance level corresponds o =.95 when =. For model, he modified KPSS es of Harris e al. (7) is he sandard KPSS saisic consruced as: MKPSS = = i= ˆ ωc c where rci, = y ρc, y. In he experimens we consider ρ c, = =.95, so ha c is differen for each, ( c =.5 for = 5, c = 5 for = and c = 5 for = 5), in order o assure a 5% significance level for =.95 when is big. long-run variance esimaor of he form: r ci, ˆc ω is any sandard c= c, + ( j/ l) c, j, c, j= rc, rc, j j= = j+ ˆ ω ˆ γ λ ˆ γ ˆ γ,
where λ(.) is a kernel funcion (QS = Quadraic specral, B = Barle) and l is he bandwidh parameer ( M =, and M = *in( /) /4 and in(.) is he ineger par of (.). As Harris e al. (7) say, he MKPSS is relaed o he prewhiened esimaed long-run variance suggesed by Sul, Phillips and Choi (5). he difference is ha our AR() filering uses ( c ).95 = in boh he numeraor and he denominaor of he es. 5% criical values of MKPSS are known o be.656 for Model,.463 for Model and.46 for Model 3. hese asympoic criical values can be found in Kwiakowski e al. (99) for Models and 3, and Hobijn e al. (998) for Model. If daa are generaed using Model or 3, hen he es saisic is he same afer eliminaing he deerminisic erms from y using he procedure menioned in Secion 3. For Model 3, we propose o use c =.75 in order o achieve an empirical significance level close o 5% when =.95 and =. In he experimens we have considered μ = for Model, and μ =, δ = for Model 3. We have sudied he behaviour of he ess for hree differen sample sizes, = 5,, 5, and all he experimens are based on, replicaions. he simulaion resuls are presened in he following hree ables, one for each of he hree models. Since he disribuion of he PONI es depends on he iniial value of he auoregressive process, in order o assess he robusness of he es procedure, he resuls are repored for four differen iniial valuesε =,3,6,9, which, approximaely, represen,, and 3 imes he sandard deviaion of he auoregressive process for =.95 and for a variance of ε equal o one. (See Müller and Ellio (3) for a reamen of he relevance of iniial values).
Each able gives he rejecion rae of he PONI and he modified KPSS ess corresponding o each sample size and for each value of he auoregressive parameer for a 5% nominal significance level. Noe ha when =, he rejecion rae is he power of he corresponding es. Wih respec o he performance of he PONI es, wo main conclusions can be exraced from he resuls presened in able for Model. Firs, he resuls are quie robus o he four iniial values considered. Secondly, he empirical size corresponding.5 o values of equal o or smaller han is smaller han he heoreical 5%. his implies ha he PONI procedure guaranees a 5% empirical level for values of closer o, as he sample size grows. Noe, in his respec, ha when = 5 he empirical size corresponding o =.98 is.6. When we compare he performance of he PONI es wih ha of he modified KPSS es, we find ha, for moderae size samples, say =, he rade-off beween size and power of he wo ess is very similar. However, when he sample size grows, say = 5, hen he PONI es clearly ouperforms he modified KPSS es. For values close o he alernaive hypohesis, he rejecion rae corresponding o he PONI es is much smaller han ha of he modified KPSS es. he resuls corresponding o Model, repored in able, are very similar o hose jus commened for Model. he only difference is ha he power is lower. his endency o a lower discriminaory power is accenuaed when we consider a model wih a consan and a ime linear rend. In order o achieve a given combinaion beween size and power, a larger sample size is required. For small samples, say = 5, empirical size and power are very close.
hese resuls can be visualized by examining Figure (Model ), Figure (Model ) and Figure 3 (Model 3). Figures 4 and 5 inform us on how robus he wo procedures are wih respec o changes in he iniial values of he auoregressive process. 3
able: Model MODEL ε = ε = 3 PONI MKPSS PONI MKPSS QS B QS B M M M M M M M M M 5.9.......... 5.93.3.....3.... 5.95.9.6.6.6.6.9.6.6.6.6 5.97.3.6.5.7.5.4.7.5.7.6 5.98.34.5.3.5.3.35.6.4.6.5 5.59.47.44.48.45.6.5.48.5.49.9...........93...........95.3.5.3.5.3.3.5.3.6.3.97.6.....6.....98.3.3.6.3..3.33.7.33..67.63.36.63.44.69.65.4.65.47 5.9.......... 5.93.......... 5.95..5.4.5.4..5.4.5.5 5.97..5.4.5.5..5.4.5.5 5.98.6.45.5.45.8.6.45.6.45.9 5.9.94.73.94.77.9.94.74.94.78 ε = 6 ε = 9 PONI MKPSS PONI MKPSS QS B QS B M M M M M M M M M 5.9.......... 5.93.3.....3.... 5.95.9.7.6.7.6..7.6.7.6 5.97.6.9.7.9.8.9.... 5.98.39.3.8.3.9.45.35.33.36.34 5.7.6.58.6.59.8.73.7.74.7.9...........93...........95.3.6.3.6.4.3.6.4.6.4.97.7....3.8.4.3.4.5.98.33.36..36.4.36.4.5.4.9.74.7.5.7.57.8.79.64.79.69 5.9.......... 5.93.......... 5.95..5.4.5.4..5.5.5.5 5.97..5.4.5.6..5.5.6.6 5.98.7.46.6.46.3.6.47.8.48.3 5.9.95.76.95.8.93.96.8.96.8 4
able : Model MODEL ε = ε = 3 PONI MKPSS PONI MKPSS QS B QS B M M M M M M M M M 5.9.3.....3.... 5.93.7.4.3.3.4.8.3.4.4.4 5.95..6.6.7.6.3.7.6.7.6 5.97.8......... 5.98......3.3..4.3 5.36.4.3.4.3.36.4.4.5.4.9...........93.4..3...5..3...95..6.5.6.4..6.5.6.5.97..3.9.4...4..5..98.8.9...3.9..3..4.5.9.9.9..5.9.9.9. 5.9.......... 5.93..5........ 5.95...5.5.5..5.5.5.5 5.97.8.9.5.7.7..7.6.7.7 5.98.9.5.7.48.3..48.8.49.3 5.86.96.74.9.79.86.9.74.9.78 ε = 6 ε = 9 PONI MKPSS PONI MKPSS QS B QS B M M M M M M M M M 5.9.3.....3.... 5.93.9.4.4.4.4..4.4.4.4 5.95.5.7.7.7.7.9.8.7.8.8 5.97.3.3..3..9.4.3.4.4 5.98.7.5.4.5.5.3.7.6.8.7 5.36.5.4.5.5.36.5.4.5.5.9...........93.6..3...8..3...95.4.6.6.6.5.9.7.6.7.6.97.7.6..6..35.8.3.8.3.98.34.3.6.3.7.43.6.9.7..5.8..9..5.8..8. 5.9.......... 5.93.......... 5.95.4.5.5.5.5.7.5.5.5.5 5.97.5.7.6.7.8.5.8.7.8.8 5.98.3.5.8.5.3.44.5.3.5.33 5.86.9.74.9.78.86.9.73.9.77 5
able 3: Model 3 MODEL 3 ε = ε = 3 PONI MKPSS PONI MKPSS QS B QS B M M M M M M M M M 5.9..3.3.3.3..3.3.3.3 5.93..5.5.6.5..6.5.6.5 5.95.3.8.7.8.7.3.8.7.8.8 5.97.4..8..9.4..9.. 5.98.5..9...5.... 5.6.....6.....9...7.....7..3.93...8.3.3..3.8...95.4.6.9.6.6.4.6..6.6.97.9....9......98.3.3..3..63.4.3.4..8.6.3.6..8.6.5.6.3 5.9.......... 5.93.......... 5.95..6.5.6.5..6.5.6.5 5.97.7.3.7.3.9.8.3.8.33. 5.98.4.54.3.54.35.6.54.3.55.35 5.74.85.64.85.69.74.84.63.84.69 ε = 6 ε = 9 PONI MKPSS PONI MKPSS QS B QS B M M M M M M M M M 5.9..3.3.3.3..3.3.4.3 5.93..6.6.6.6..6.6.7.6 5.95.3.8.8.9.8.3.9.9..9 5.97.4.....4.3..3.3 5.98.5...3..5.5.4.5.5 5.6.5.4.5.4.6.7.7.8.7.9...7.....7...93..3.8.3.3..3.9.3 4.95.5.7..7.6.6.8..8.7.97..3.4.3...5.6.5.3.98.4.6.5.6.3.4.8.8.8.5.8.8.7.8 5.8....8 5.9.......... 5.93......... 5.95..6.6.6.5..6.6.6.6 5.97..33.8.34..6.34.9.35. 5.98.3.55.3.56.37.36.57.34.57.38 5.74.83.63.84.68.74.83.6.83.67 6
Figure: Model MODEL = 5,7,6,5,4,3 PONI M M,,,,9,93,95,97,98, MODEL =,8,7,6,5,4 PONI M M,3,,,9,93,95,97,98 MODELO = 5,9,8,7,6,5 PONI M M,4,3,,,9,93,95,97,98 7
Figure : Model MODEL = 5,4,35,3,5, PONI M M,5,,5,9,93,95,97,98 MODEL =,6,5,4,3 PONI M M,,,9,93,95,97,98 MODEL = 5,9,8,7,6,5,4 PONI M M,3,,,9,93,95,97,98 8
Figure 3: Model 3 MODEL 3 = 5,,,8,6 PONI M M,4,,9,93,95,97,98 MODEL 3 =,,8,6,4,, PONI M M,8,6,4,,9,93,95,97,98 MODEL 3 = 5,9,8,7,6,5 PONI M M,4,3,,,9,93,95,97,98 9
Figure 4: PONI, Iniial condiions MODEL = PONI,9,8,7,6,5,4 ε= ε=3 ε=6 ε=9,3,,,9,93,95,97,98 Figure 5: KPSS, Iniial condiions MODEL = MKPSS,9,8,7,6,5,4 ε= ε=3 ε=6 ε=9,3,,,9,93,95,97,98
5.- Conclusions In his paper, we have proposed a new procedure o es for he null hypohesis of near inegraion agains he alernaive of a uni roo. Using he Neyman-Pearson Lemma, we have firs derived he es saisic and hen we have shown ha, under he null hypohesis of local o uni roo saionary, which specifies he auoregressive parameer as c /, he limiing disribuion of he es procedure is he Sandard Normal. We have also shown ha, when he daa are generaed by he alernaive hypohesis ha he ime series has a uni roo, he es saisic has no finie limi as he sample size grows. Finally, examining he simulaion resuls repored in Secion 4, i can be said ha he performance of he new es is quie good and ha, when he sample size is large, i ouperforms he modified KPSS es recenly proposed in Harris e al. (7). Acknowledgemens We hank M.. Aparicio, A. Novales and M. Salvador for helpful commens on earlier drafs. he auhors wish o hank he Spanish Deparmen of Educaion-DGICY projec SEJ 6-996 for financial suppor. References Dickey, D. A. and W. A. Fuller, 979, Disribuion of he esimaors for auoregressive ime series wih a uni roo. Journal of he American Saisical Associaion 74, 47-43. Ellio, G.,. J. Rohenberg and J. H. Sock, 996, Efficien ess for an auoregressive uni roo. Economerica 64, 83-836. Haldrup, N. and M. Jansson, 6, Improving size and power in uni roo esing. In Palgrave Handbook of Economerics, Vol., Economeric heory, Chaper 7.. C.
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