Cointegration and the joint confirmation hypothesis

Transcription

1 Cointegrtion nd the joint confirmtion hypothesis Vsco J. Gbriel,b, * Deprtment of Economics, University of Minho, Gultr, , Brog, Portugl b Deprtment of Economics, Birkbeck College, Oxford, UK Abstrct In this pper, the discussion concerning the joint use of unit root nd sttionrity tests is extended to the cse of cointegrtion. Criticl vlues for testing the joint confirmtion hypothesis of no cointegrtion re computed nd smll Monte Crlo experiment evlutes the reltive performnce of this pproch. Keywords: Cointegrtion; Joint confirmtion hypothesis; Monte Crlo simultions JEL clssifiction: C1; C15; C 1. Introduction The issues of unit roots nd cointegrtion hve generted vst literture in the pst few yers. More recently, it hs been rgued tht confirmtory nlysis (i.e. pplying unit root tests in conjunction with sttionrity tests) my in some cses led to better description of the series, improving upon the seprte use of ech type of test (see, for exmple, Amno nd Vn Norden (199) nd the discussion in Mddl nd Kim (1998)). If the two pproches give consistent results, i.e. there is n cceptnce nd rejection of the nulls, one my conclude whether given series is sttionry or not. On the other hnd, if both tests either reject or ccept their respective null hypotheses, the results re inconclusive. Some prcticl spects concerning the joint use of unit root nd sttionrity tests hve been ddressed by Chremz nd Syczewsk (1998) nd Crrion et l. (001). The first uthors suggest tht, insted of conventionl individul criticl vlues for ech type of test, one should use symmetric criticl power vlues. These correspond to the probbility with which the two types of tests mke *Tel.: ; fx: E-mil ddress: vjgbriel@eeg.uminho.pt (V.J. Gbriel).

2 wrong decision when both cumultive mrginl distributions re equl. Chremz nd Syczewsk (1998), using Monte Crlo methods, tbulted the new criticl vlues needed for the joint confirmtion hypothesis (JCH) of sttionrity when the ugmented Dickey Fuller (ADF) nd the Kwitkowski et l. (199), KPSS henceforth) tests re to be used. However, this pproch depends on the prmeteriztion of the utocorreltion in the errors. Hence, Crrion et l. (001) recommend tht the JCH of unit root should be tested insted, providing new set of criticl vlues. In this pper, we study the ppliction of this methodology to cointegrtion testing. Following Chremz nd Syczewsk (1998) nd Crrion et l. (001), we show how the testing procedure my be implemented nd the relted criticl vlues obtined for tests with null hypothesis of no cointegrtion, s well s null of cointegrtion. We ddress the cses where Engle Grnger s ADF nd Phillips Ouliris Z nd Zt tests re used in conjunction with the KPSS-type test for the null hypothesis of cointegrtion developed by McCbe et l. (1997) (see Gbriel (001) for comprtive study of the properties of null of cointegrtion tests). Furthermore, the ppliction of the joint confirmtion procedure is ssessed by mens of set of Monte Crlo experiments, estblishing some comprisons with the seprte use of ech type of test. This is of gret interest, since joint testing will be n lterntive pproch only if it is ble to produce better results thn individul testing. The pper proceeds s follows. The next section estblishes the nottion for the JCH in the context of cointegrtion, while Section 3 presents the criticl vlues for the JCH of no cointegrtion. The Monte Crlo study is undertken in Section 4 nd Section 5 concludes.. Joint confirmtion hypothesis nd cointegrtion The first step in order to implement the joint use of null of no cointegrtion nd null of cointegrtion tests is to decide whether one wishes to test the JCH of cointegrtion or no cointegrtion. A simple cointegrted model is generlly formulted s yt5 x9t b 1 u t, (1) where y is sclr I(1) process nd x is vector I(1) process of dimension k. The vribles y nd x t t t t re sid to be cointegrted if ut is I(0), wheres if ut is I(1) there is no long run equilibrium reltionship between yt nd x t. A common prmeteriztion for the error process is to ssume tht ut is n utoregressive process 1 v, v n.i.d.(0, s ), with uru, 1 in the cse of cointegrtion nd r 5 1 when there is no ut5 rut1 t t v cointegrtion. Another possibility is to consider tht under the hypothesis of no cointegrtion the disturbnce ut my be decomposed into the sum of rndom wlk nd sttionry component, u 5 g 1, () t t t t t1 t 0 t h t t h t where the rndom wlk is g 5 g 1h, with g 5 0 nd h distributed s i.i.d.(0, s ), while the sttionry prt is distributed s i.i.d.(0, s ) nd is ssumed independent of h. Cointegrtion stems from this formultion when s 5 0, so tht g 5 0 nd no longer is rndom wlk. Note tht in this cse the i.i.d. ssumption of the errors (ut5 t) is not very relistic, since in empiricl pplictions we should expect some degree of seril correltion. Thus, we my relx this ssumption nd ssume tht t t1 t t z 5 p 1 z, z being i.i.d.(0, s ). If one chooses to test the JCH of cointegrtion (mening I(0) errors), the criticl vlues would

3 lwys depend on the vlue of the utoregressive prmeter of the error term, be it r if we specify the null hypothesis of cointegrtion s H 0: uru, 1, or p if H 0: s h 5 0, llowing for utocorreltion in t. It would involve extensive tbultions for few prticulr vlues of r (or p), very likely to be different from the ctul, unknown vlue in the empiricl sitution the resercher is deling with. Note tht this is similr problem to tht pointed out by Crrion et l. (001) for the univrite cse. Therefore, wy to circumvent this obstcle is to specify the JCH of no cointegrtion (i.e. the residuls hve unit root). We closely follow the nottion of Chremz nd Syczewsk (1998) nd Crrion et l. (001) by defining the probbility of joint confirmtion (PJC) of no cointegrtion s ` ` D K EE f D,K(z D, z K;Q, TuH 0, H 1)dzD dzk5 PJC. (3) z PJC z PJC D K Here, z j ( j 5 D, K) represents the test sttistics (in which we mintin the originl nottion), D for the ADF t-sttistic nd K for the KPSS cointegrtion version of McCbe et l. (1997). The vector of DGP PJC prmeters is denoted s Q, T is the smple size, f is the joint density function, while z D,K j re the criticl vlues from the joint distribution for given PJC significnce level. As discussed in the bove mentioned ppers, for ech PJC significnce level the number of possible criticl vlues is infinite. However, if we impose the restriction tht the mrginl probbilities (MPr) should be equl, then there PJC PJC is unique pir (z, z ) stisfying D K ` ` D K E f D(z D; Q, TuH 0 )dzd5e f K(z K; Q, TuH 1 )dzk5mpr. (4) z PJC z PJC D K This restriction mens tht the probbility of deciding wrongly when pplying ech sttistic is equl, tht is, when the ADF sttistic does not reject the null of no cointegrtion (type II error) nd the PJC PJC KPSS-type test rejects true null of cointegrtion (type I error). Such pirs (z, z D K ) re dubbed symmetric criticl power vlues (SCPV). Therefore, we find cointegrtion t PJC significnce level PJC PJC if the joint ADF KPSS sttistic is in the intervl h(`, z ), (0, z D K )j, wheres the converse sitution leds to non-rejection of the JCH of no cointegrtion. In principle, this strtegy voids prioritizing either the cointegrtion or no cointegrtion hypotheses, lthough the prcticl implictions my turn out to be different, s the simultion results in Crrion et l. (001) show. We will return to this below. Also note tht we my lso consider the JCH with other pirs of tests, chnging the nottion conformbly. In fct, we will lso consider the joint ppliction of the Phillips Ouliris Z nd Zt tests, nd KPSS-type test. In the next section, criticl vlues for these cses re presented. 3. Criticl vlues for the JCH of no cointegrtion As known, criticl vlues for cointegrtion testing depend not only on the number of regressors k, but lso on the deterministic components tht my be present in the cointegrtion spce. We will restrict our ttention to single eqution models with single cointegrtion vector. Generlizing (1) s

4 yt5 1 dt 1 x9t b 1 u t, (5) where t denotes time trend, we consider three cses: no constnt ( 5 d 5 0), constnt with no trend 1 ( ± 0, d 5 0) nd the model with trend component ( ± 0, d ± 0), up to three regressors. Since we re considering the JCH of no cointegrtion, then ut5 ut1 1 v t (r 5 1), vt is ssumed to be n.i.d.(0, 1) nd u We lso set 5 1 nd d 5 1 for the relevnt cses. After generting n 5 50,000 replictions for smple sizes T 5 50, 100 nd 50, pirs of ADF KPSS, Z KPSS nd Z t KPSS tests re computed. Using OLS, n pproprite lg length for the ADF test is obtined with t-test downwrd selection procedure, by setting the mximum lg equl to 6 nd then testing downwrd until significnt lst lg is found, t the 5% level. Concerning Z nd Z t, the long run vrince is estimted by mens of prewhitened qudrtic spectrl kernel with n utomticlly selected bndwidth estimtor, using first-order utoregression s prewhitening filter, s recommended by Andrews nd Monhn (199). As for the KPSS cointegrtion sttistic, we use Sikkonen s (1991) dynmic lest squres estimtor nd filter the residuls with n ARIMA( p, 1, 1) model, then using the vrince estimtor suggested by Leybourne nd McCbe (1999) (see McCbe et l. (1997) nd Gbriel (001) for more detils on the computtion of the sttistic). Agin, we follow the methodology of Chremz nd Syczewsk (1998) nd Crrion et l. (001) to obtin the criticl vlues. Thus, the n pirs of observtions re sorted ccording to the ADF (or Z-type) test nd then 50 frctiles re computed. For ech of these ADF (Z-type) frctiles, nother 50 frctiles were obtined for the KPSS sttistic, which mens tht we get tble of empiricl joint frequencies. After cumulting these frequencies nd thus obtining the joint distribution function, we my tbulte criticl vlues for the desired significnce levels. These re shown in Tble 1. The computer routine to obtin these criticl vlues ws written in GAUSS nd is n dpttion of the progrm used by Chremz nd Syczewsk (1998). Since the ADF nd Zt shre the sme (mrginl) symptotic distribution nd given tht the results obtined in the simultions for these two tests re prcticlly the sme, we only show the criticl vlues for the ADF test. 4. Monte Crlo experiment In order to ssess the performnce of the JCH of no cointegrtion in terms of clssifying the model s cointegrted or not, we devised set of Monte Crlo simultions. The DGP is similr to the one in Crrion et l. (001) nd is the sme s tht of the previous section, lthough the errors re llowed to follow n ARMA(1, 1) process of the form u 5 ru 1 v 1uv, (6) t t1 t t1 where r tkes the vlues h0.5, 0.9, 1j nd u 5 h 0.8, 0j. For simplicity, we only consider model with single regressor nd constnt term, setting the smple size s T nd 50, computing 500 replictions. 1 Criticl vlues for k 5 4 nd 5 were lso computed nd re vilble upon request. Avilble upon request.

5 Tble 1 Criticl vlues PJC No constnt Constnt Trend ADF MLS Z ADF MLS Z ADF MLS Z T k k k T k k k T k k k The results from this simultion exercise re shown in Tbles nd 3 for T nd T 5 50, 3 respectively. We considered different testing pproches. Firstly, computing ech test individully nd using the respective mrginl distributions (i.e. the stndrd criticl vlues), we guge the proportion of times tht the tests clssify given DGP s being cointegrted (line C) or not cointegrted (line NC), t the 5% level of significnce. This corresponds to the usul power-size nlysis. Secondly, nd still resorting to the 5% criticl vlues from the mrginl distributions, we count the frequency reliztion of the DGP is clssified s cointegrted or not in the following wy: (i) if tests for the null of no cointegrtion (ADF, Z nd Z t) reject their null nd the KPSS null of cointegrtion test does not, the process is considered to be cointegrted (C); (ii) if tests for the null of 3 This lterntive ws not considered in Crrion et l. (001) for the univrite cse.

6 Tble Monte Crlo results for ADF, Z nd KPSS tests (T 5 100) (r, u ) ADF Z KPSS D K Z K JU(D K) JU(Z K) JS(D K) JS(Z K) (0.5, 0) C NC Inc. A Inc. B (0.5, 0.8) C NC Inc. A Inc. B (0.9, 0) C NC Inc. A Inc. B (0.9, 0.8) C NC Inc. A Inc. B (1, 0) C NC Inc. A Inc. B (1, 0.8) C NC Inc. A Inc. B no cointegrtion do not reject their null nd the null of cointegrtion test does, the process is considered not to be cointegrted (NC); (iii) if both types of tests reject their nulls (inconclusive type A) or do not reject the respective nulls (inconclusive type B), no conclusion is chieved. These joint tests re lbeled s D K for ADF nd KPSS tests nd Z K for Z nd KPSS tests. Finlly, similr exercise is crried out, this time using the 5% criticl vlues from the joint distribution s displyed in Tble 1, with the tests denoted s JU(D K) nd JU(Z K). From the nlysis of Tbles nd 3, we observe tht testing the JCH of no cointegrtion with JU(D K) nd JU(Z K) leds to very smll number of correct decisions when the errors re sttionry. This is lso the cse for joint testing with stndrd criticl vlues. Indeed, most of the times n inconclusive response is obtined, nmely rejections by both tests (type A inconclusive nswers). Moreover, the results do not seem to improve for lrger smple sizes, when we compre Tbles nd 3. On the other hnd, when the DGP is truly non-cointegrted, the JCH pproch with JU(D K) is generlly the most ccurte in delivering the correct nswer, except when negtive MA component is present. In this sitution, JU(D K) still performs better thn D K, lthough the JU(Z K) version is gretly ffected by this error structure. Overll, these results re in ccordnce with the simultions for univrite testing in Crrion et l. (001), lthough with much poorer performnce.

7 Tble 3 Monte Crlo results for ADF, Z nd KPSS tests (T 5 50) (r, u ) ADF Z KPSS D K Z K JU(D K) JU(Z K) JS(D K) JS(Z K) (0.5, 0) C NC Inc. A Inc. B (0.5, 0.8) C NC Inc. A Inc. B (0.9, 0) C NC Inc. A Inc.B (0.9, 0.8) C NC Inc. A Inc. B (1, 0) C NC Inc. A Inc. B (1, 0.8) C NC Inc. A Inc. B Compring this performnce with tht of individul tests, we see tht the ltter hve much more relible behviour in terms of providing the correct decision, both when there is cointegrtion nd when there is not. The performnce of the KPSS cointegrtion test should be highlighted, given its reltive robustness to seril correltion nd most especilly to the introduction of negtive MA components in the errors. In fct, the performnce of ADF nd Z tests, s well s tht of joint tests, seems to suffer gret del with negtive MA error structure, which confirms previous results in the 4 literture. On the other hnd, if we stick to prticulr (individul) test, we will not get inconclusive nswers, s it hppens with the JCH methodology. Given these results, it would lso be interesting to investigte wht the outcome would be if one tested the JCH of cointegrtion. As explined erlier, there is the problem with the criticl vlues depending on the degree of correltion of the errors. However, the resercher could choose n intermedite, though non-optiml, pproximtion by fixing r t n empiriclly plusible vlue nd use the corresponding criticl vlues. Such vlue could be r , which is lso recommended nd tbulted by Chremz nd Syczewsk (1998). Of course, if the true r is lrger thn 0.75, the criticl 4 This could eventully be overcome by using the procedure of Ng nd Perron (1998), for exmple.

8 vlues would be too conservtive, while the converse would led to overrejecting the JCH of cointegrtion. Nevertheless, despite the rbitrriness of such choice, this seems firly relistic wy to proceed. Therefore, dpting the methodology discussed in Sections nd 3 to the JCH of cointegrtion, we 5 computed the 5% criticl vlues for the DGP in this section nd evluted its performnce using the sme set of simultion experiences. The results re lso displyed in Tbles nd 3, under the columns JS(D K) nd JS(Z K). We observe tht this strtegy clerly improves upon tht of JCH of no cointegrtion, since lot more correct decisions re chieved when the DGP is cointegrted. However, this behviour is not sustined symptoticlly, s the results for T 5 50 re in generl worse. On the other hnd, the bility to detect non-cointegrted models improves with the smple size nd ttins very resonble levels. Still, this pproch does not seem to bet the conventionl one, with individul testing. 5. Concluding remrks In this pper, we extended the joint confirmtion hypothesis pproch to the context of cointegrtion. Following Chremz nd Syczewsk (1998) nd Crrion et l. (001), we tbulted criticl vlues for the JCH of no cointegrtion. However, our subsequent Monte Crlo simultions, despite its limittions, question the usefulness of such methodology, s they led us to conclude tht it seems preferble to use the stndrd individul testing pproch, which consistently gve better (or t lest s good) results. Indeed, the joint ppliction of different types of tests my obscure, rther thn clrify, the process of deciding whether given model is cointegrted or not. In prticulr, testing the JCH of no cointegrtion with the criticl vlues derived here is to be voided, s it minly leds to inconclusive nswers when the DGP is truly cointegrted. By reversing the JCH to be tested (tht is, cointegrtion), slightly better results re chieved. Further reserch is required, however, s there re issues tht should be ddressed. For exmple, it would be interesting to chrcterize nd compre the behviour of both types of tests (for the null of cointegrtion nd no cointegrtion) under different types of DGPs. This line of reserch is lredy under study. Acknowledgements I wish to thnk Wojciech Chremz for kindly mking his computer code vilble. I m lso indebted to Josep Lluis Crrion-i-Silvestre, Ron Smith, Hris Psrdkis nd Luis Mrtins for helpful comments, lthough retining responsibility for ny remining errors. Finncil support from the Fundço pr Cienci ˆ e Tecnologi is grtefully cknowledged. 5 These re.48, nd (T5100) nd 4.065, nd (T550) for the ADF, Z nd KPSS tests, respectively, for ( r, u )5(0.75, 0). A more extensive tbultion is vilble upon request.

9 References Amno, R., Vn Norden, S., 199. Unit-Root Tests nd the Burden of Proof. Bnk of Cnd. Andrews, D.W.K., Monhn, J.C., 199. An improved heteroskedsticity nd utocorreltion consistent covrince mtrix estimtor. Econometric 60, Crrion, J.L.I., Snso, A.I., Ortuno, M.A., 001. Unit root nd sttionrity tests wedding. Economics Letters 70, 1 8. Chremz, W.W., Syczewsk, E.M., Joint ppliction of the Dickey Fuller nd KPSS tests. Economics Letters 61, Gbriel, V.J., 001. Tests for the null hypothesis of cointegrtion: Monte Crlo comprison, Mnuscript. Kwitkowski, D., Phillips, P.C.B., Schmidt, P., Shin, Y., 199. Testing the null hypothesis of sttionrity ginst the lterntive of unit root. Journl of Econometrics 54, Leybourne, S.J., McCbe, B.P.M., Modified sttionrity tests with dt-dependent model-selection rules. Journl of Business nd Economic Sttistics 17, Mddl, G.S., Kim, I.M., Unit Roots, Cointegrtion nd Structurl Chnge. Cmbridge University Press, Cmbridge. McCbe, B.P.M., Leybourne, S.J., Shin, Y., A prmetric pproch to testing for the null of cointegrtion. Journl of Time Series Anlysis 18 (4), Ng, S., Perron, P., An utoregressive spectrl density estimtor t frequency zero for nonsttionrity tests. Econometric Theory 14, Sikkonen, P., Asymptoticlly efficient estimtion of cointegrting regressions. Econometric Theory 7, 1 1.