A complementary test for ADF test with an application to the exchange rates returns

Transcription

1 MPRA Munich Personal RePEc Archive A complemenary es for ADF es wih an applicaion o he exchange raes reurns Venus Khim-Sen Liew and Sie-Hoe Lau and Siew-Eng Ling 005 Online a hp://mpra.ub.uni-muenchen.de/518/ MPRA Paper No. 518, posed 19. Ocober 006

2 A Complemenary Tes for ADF Tes wih An Applicaion o he Exchange Raes Reurns Venus Khim-Sen Liew Labuan School of Inernaional Business and Finance, Universii Malaysia Sabah Sie-Hoe Lau Faculy of Informaion Technology and Quaniasive Science, Universii Teknologi MARA, Sarawak Campus Siew-Eng Ling Faculy of Informaion Technology and Quaniasive Science, Universii Teknologi MARA, Sarawak Campus Absrac This sudy shows ha augmened Dickey-Fuller (ADF) es failed o deec covariance nonsaionary series. Supporive of Ahamada (004), his sudy finds ha he cumulaive sums of squares procedure in Inclán and Tiao (1994) is useful o complemen he ADF es. As illusraion, he ADF es indicaes ha here is no uni roo in he reurns of Japanese yen/us dollar, Briish pound/ US dollar and Swiss franc/us. However, he complemenary es reveals ha each of hese reurns conains heerogeneous variance. To sum, i can be concluded ha hese exchange rae reurns are covariance nonsaionary alhough here is no uni roo. 1

3 A Complemenary Tes for ADF Tes wih An Applicaion o he Exchange Raes Reurns 1. Inroducion A basic requiremen for ime series modelling is ha he series under sudy mus be weakly saionary, i.e. i has consan mean and covariance. Numerous saionary ess have been developed in he pas o es for saionariy and he popularly applied ess include he augmened Dickey-Fuller (ADF) es (Fuller 1976, Dickey and Fuller 1979), Phillips-Perron (PP) es (Phillips 1987, Phillips and Perron 1988) and Kwiakowski- Phillips-Schmid-Shin (KPSS) es (Kwiakowski e al. 199). Laely, Ahamada (004) demonsraes via a simulaion exercise ha KPSS es fails o deec a form of nonsaionariy due o a shif in he uncondiional variance. They poined ou ha he nonrejecion of he null hypohesis of no uni roo in he KPSS es does no neccesarily imply he saionariy of he daa, as here is a possibiliy ha he daa may exhibi heerogeneous uncondiional variance. The auhor furher proposed a complemenary es o complee he KPSS esing procedure and he complemenary es was shown o be useful deecing he nonsaionary covariance of he daily reurns of US dollar/euro exchange rae, in which he KPSS es has failed o do so. Given he surprising defec in one of he mos powerful saionary es, i is ineresing o find ou wheher he mos commomly uilised ADF es is robus agains nonsaionary covariance. As such, he his simulaion sudy is conduced o examine wheher he ADF es is able o deec nonsaionary covariance. Besides, he performance of he

4 complemenary es as proposed in Ahamada (004) in correcly idenifying simulaed series of nonsaionary covariance is also scruinized in his simulaion sudy. To preview our findings, he curren sudy discovers ha he ADF es has idenified he simulaed nonsaionary covariance as saionary series wih a uni probabiliy. Similar finding is observed in he DF es, which is included in his simulaion sudy for comparison purpose. On he oher hand, using he complemenary es as proposed in Ahamada (004), nonsaionary covariance has been correcly idenified in almos all cases. Hence, his sudy proposes he use of his complemenary es in he case of ADF es o deec nonsaionary covariance if ADF es suggess no uni roo in he series of ineres. In his regards, he curren sudy simulaes and repors he criical values of his complemenary es. In addiion, his sudy applies he same complemenary es in he case of ADF (hereafer referred as complemenary ADF es) o he reurns of few US dollar based exchange rae series of some developed counries o illusrae he usefulness of his complemenary ADF es. The remainder of sudy is srucured as follows: Secion discusses he complemenary ADF es. Secion 3 explains he simulaion process and presens he resuls of sudy. Secion 4 illusraes he usefulness of he complemenary ADF es using emperical daa. Finally, Secion 5 concludes his sudy. 3

5 . The Complemenary ADF Tes Ahamada (004) wisely ailored he cumulaive sum of square (CSS) procedure in Inclán and Tiao (1994) o formulae a complemenary es for he KPSS esing procedure (hereafer, complemenary KPSS es). This useful es is easily applied and ineresed readers may refer o Ahamada (004) 1. In he vein of Ahamada (004), his sudy exends he applicaion of he same CSS procedure in he case of ADF, yielding o he socalled complemenary ADF es. Consider he following ime series { y }, which is saionary around he level r 0 : y = 0 + ε, 1,..., T r =, (1) where ε is independen and idenically disribued (i.i.d.) wih a zero mean and consan variance, denoed ε ~ i.i.d.(0, σ ε ). The saionariy of { y } may be esed by he augmened Dickey-Fuller (ADF) es 3 : 1 Available a hp:// For compaibiliy, he curren sudy follows closely he definiions and noaions in Ahamada (004). 3 ADF is he improved version of Dickey-Fuller (DF) es of he framework y = y 1 + ω, where ω ~ i.i.d. (0, σ ω ). Here, he null hypohesis of =1 (uni roo) is esed agains he alernaive hypohesis of < 1 (no uni roo). 4

6 p y = y + βi y i + η i= 1 1, () where η ~ i.i.d.(0, σ η ), p is he auoregressive lag lengh large enough o eliminae possible serial correlaion in η and is he coefficien of ineres. Convenionally, if = 0, he series conains a uni roo implying nonsaionary, whereas if < 0, here is no uni roo implying saionariy. In he ADF es, he null hypohesis of uni roo, i.e. ADF H 0 : = 0 is esed agains he alernaive hypohesis of no uni roo, i.e. ADF H A : < 0 using he es of individual significance. I is obvious ha under he generaing mechanism in (1) wih ε ~ i.i.d.(0, σ ε ), in () equals 0, hereby convenionally one may conclude ha { y } is saionary. The concern of his sudy is wheher or no he ADF es is robus agains heerogeneous variance process i.e. E( ε ) = σ σ ε. In his regard, a simulaion sudy has been conduced and we will see shorly ha ADF es had idenified nonsaionary covariance series as saionary process 4. A complemenary es for ADF es is herefore needed o differeniae compleely saionary process (mean and covariance saionary) from mean saionary bu covariance nonsaionary process. As in Ahamada (004), he curren sudy uilises he supremum T D saisic proposed in Inclán and Tiao (1994), defined as 5 : / K 4 Alhough sriking, he resuls come as no surprise as Ahamada (004) has already shown similar failure of he mos powerful uni roo es. 5 Wih he pruden adapaion of Ahamada (004). 5

7 τ = max T / D k = 1,..., T K (3) where D k k Ck k =, C k = e, k = 1,..., T. C T T = 1 e in urn is he ordinary leas squares (OLS) residuals from regressing { y } on a consan as in (1). Under he null hypohesis of e is independen and idenically disribued wih zero mean and homogeous variance, i.e. C H 0 : e ~ i.i.d. (0, given by one of he sup{ W 0 }, where σ e ), Ahamada (004) showed ha he limiing disribuion of τ is 0 W is a sandard Brownian Bridge. I is noed here ha he above assumpion is also valid and herefore he disribuion of sup{ W 0 } given by Billingsley (1968) is applicable in he curren case 6 : Pr 0 { sup W b} k k b = 1+ ( 1) e, b > 0 (4) k = 1 where Pr{A } denoes he probabiliy of even A occurs and b is he criical value. Based on simulaion exercises done by Inclán and Tiao (1994), he asympoic 10%, 5% and 1% criical values for τ are corresponding 1.4, and See proof of Proposiion 1 in Ahamada (004) and proof of Theorem 1 in Inclán and Tiao (1994). 7 Inclán and Tiao (1994) esimaed hese criical values from replicaions of T independen N(0,1) observaions. Using his specificaion, he simulaed criical values obained in he curren sudy for T = are raher close o heirs, i.e. 1.5, and 1.613, in he same order. As for differen specificaions of variance, hese values do no vary subsanially, see Appendix 1 for more simulaed criical values for τ. 6

8 Wih he availabiliy of his complemenary ADF es, we may now conduc a complee ADF es by carrying ou he following wo-sep procedure 8 : Firs, apply he ADF es. If he null hypohesis is no rejeced, hen we may conclude ha he daa is nonsaionary, i.e. i conains a uni roo. If he null hypohesis is rejeced, here is no uni roo bu a shif in he variance is possible. For his case, we sugges o apply he complemenary ADF es. If he τ saisic fails o rejec he null hypohesis, hen we have enough saisical evidence o conclude ha here is a complee covariance saionariy. Oherwise, he daa have variance shif and he process is no covariance saionary alhough here is no uni roo. 3. Simulaion Procedures and Resuls Consider he following daa-generaing processes (DGP) specified in Ahamada (004): DGP : x = ε, (5) H 0 where ε ~N(0,1) for = 1,, 00; and DGP A ' H : y ε =, (6) 8 The null and alernaive hypohesis of KPSS es is he reverse of ADF es, see Ahamada (004) for complemenary KPSS es. 7

9 where ' ε ~N(0,1) for = 1,, 100 and ' ε ~N(0,1.5) for = 101,, 00. Noe ha he series { x } is saionary around he level 0.01 bu { y } is nonsaionary as he variance varies. The esimaed rejecion rae of he null hypohesis of nonsaionary a 1%, 5% and 10% level for boh series for 1000 replicaions of each DGP is given in Table 1. TABLE 1. Rejecion Rae of he Null Hypohesis of Nonsaionary Series DF Tes ADF Tes a Complemenary Tes 10 5% 1% 10% 5% 1% 10 5% 1% x } { { y } Noe: a Resuls repored are for p= 4. Similar resuls (no shown) are obained wih oher specificaions of p. Table 1 shows ha boh he DF and ADF es correcly rejec he null hypohesis of uni roo (implying saionariy) in he { x } series, whereas he performance of he complemenary es is well close o he nominal levels. On he oher hand, boh he DF and ADF es errorneously rejec he null hypohesis of uni roo in he nonsaionary { y } series. Noneheless, he complemenary es is able o correcly idenify he nonsaionary variance and he performance is again as good as nominal levels. Thus, he complemenary es has good size and power of es, bu he DF and ADF have only saisfacory size of es. 8

10 4. Illusraions of Complemenary ADF Tes To demonsrae he poenial usefulness of he complemenary ADF es, his sudy applies i o he he reurns of hree US dollar based nominal exchange rae series of developed counries, namely he Japanese yen, Briish pound and Swiss franc. Quarerly daa of hese nominal bilaeral exchange raes covering 1957Q1 o 004Q1 (amouning o 188 usable observaions) are obained from he Inernaional Financial Saisics. The reurns of hese series compued from X = log( S / S 1) where S is Japanese yen/us dollar, Briish pound/ US dollar or Swiss franc/ US dollar are ploed in Figure 1. I is seen from Figure 1 ha hese reurns series are raher saionary around he level 0 bu here is obviously a shif in variance in all cases. Based on he formal DF and ADF ess, in which he resuls are summarised in Table, he null of uni roo has been rejeced a 1% significance level in all cases. However, as argued earlier, his finding does no auomaically implies saionariy since he homogeneiy condiion of variance is ye o be deermined. In his respec, furher applicaion of he complemenary es is obligaory o complee he ADF esing procedure and he resuls are also given in Table. In line wih our earlier observaion (eye-inspecion), srong evidence of heeroscasic variance in all reurns series are given by he complemenary es. Thus, we may conclude ha while here is no uni roo in all he reurns series under sudy, hey are acually covariance nonsaionary. Our resuls are supporive of Ahamada (004), which repors similar 9

11 finding on he daily reurns of US dollar/euro exchange rae by he complemenary KPSS es FIGURE 1. The exchange rae reurns Japanse yen/us dollar Q1 003 Q1 001 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Briish pound/us dollar Q1 003 Q1 001 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Swiss franc/us dollar Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q1 001 Q

12 11

13 TABLE. DF, ADF and complemenary ess resuls wih simulaed criical values Exchange Rae DF ADF Complemenary Tes Yen/US dollar * * 3.98 * Pound/US dollar * *.01 * Swiss Franc/US dollar * *.81 * Simulaed Criical Values a 1% % % Noe: a Esimaed from 1000 replicaions of 188 independen N(0,1) observaions. Aserisk (*) denoes significan a 1% level. 5. Conclusion This sudy demonsraes hrough a simulaion sudy ha he mos commonly applied ADF es failed o deec covariance nonsaionary series. This finding is no surprising as Ahamada (004) has already shown ha he KPSS es, one of he mos powerful saionary es has similar deficiency. Following Ahamada (004), his sudy uilises he cumulaive sums of squares in Inclán and Tiao (1994) o form a complemenary es for he ADF es. Simulaion resuls show ha his complemenary es has he desired good size and power of es, bu no he ADF es. Hence, a wo-sep esing procedure saring from he ADF es and ending wih he complemenary es is essenial for a complee saionary es. This sudy considers he reurns of Japanese yen/us dollar, Briish pound/ US dollar and Swiss franc/us dollar for illusraion of his wo-sep procedure. The ADF es indicaes ha here is no uni roo in hese reurns. However, he complemenary es idenifies ha each of hese reurns conains a shif in variance. Summing up boh es 1

14 resuls, i is concluded ha hese exchange rae reurns are covariance nonsaionary alhough here is no uni roo. References Ahamada, I. (004) A complemenary es for he KPSS es wih an applicaion o he US dollar/euro exchange rae Economic Bullein 3(4), 1 5. Billingsley, P. (1968) Convergence of Probabiliy Measures, John-Wiley: New York. Dickey, D. (1976) Inroducion o Saisical Time Series, Wiley: New York. Dickey, D. and W. A. Fuller (1979) Disribuion of he Esimaors for ime series regressios wih a uni roo Journal of he American Saisical Associaion 74, Inclán, C. and G.C. Tiao (1994) Use of cumulaive sums of squares for rerospecive deecion of changes of variance Journal of American Saisical Associaion 89, Phillpis, P.C.B. (1987) Time series regression wih a uni roo Economerica 55, Phillips, P. C. B. and P. Perron (1988) Tesing for a uni roo in ime series regressions Biomerika 65, Kwiakowski, D., P.C.B. Phillips, P. Schmid and Y. Shin (199) Tesing he null hypohesis of saionariy agains he alernaive of uni roo Journal of Economerics 54,

15 APPENDIX 1 TABLE 3. Criical values of τ saisic for various sample size, T. Sample size, T Criical values 10% 5% 1% Noe: Esimaed from series ha are replicaed from independen random errors wih N(0,1) disribuion. Each series conains T usable observaions. TABLE 4. Criical values of τ saisic for various residuals variance, σ ε. σ Criical values ε 10% 5% 1% Noe: Esimaed from series ha are replicaed from independen random errors wih N(0, σ ) disribuion. Each series conains usable observaions. ε 14