GENERALISED WALD TYPE TESTS OF NONLINEAR RESTRICTIONS. Zaka Ratsimalahelo

Transcription

1 GENERALISED WALD TYPE TESTS OF NONLINEAR RESTRICTIONS Zaka Ratsmalahelo Unversty of Franche-Comté, U.F.R. Scence Economque, 45D, av. e l Observatore, Besançon - France Abstract: Ths paper proposes a generalse Wal type tests to test the hypothess of nonlnear restrctons. We crcumvent the problem of sngularty of the covarance matrx assocate wth the usual Wal test by proposng a generalse nverse proceure, an an alternatve smple proceure whch can be approxmate by a sutable ch-square strbuton. New threshol values are erve to estmate the rank of the covarance matrx. We also propose an operatonal proceure to test the hypothess of the Granger non-causalty n contegrate systems. Copyrght 005 IFAC. Key wors: Wal tests; Matrx perturbaton theory; Nonlnear restrctons; Spectral ecomposton; Covarance matrx egenerate; Granger causalty.. INTRODUCTION Analyses of economc ata often ental the testng of hypotheses that mply complex nonlnear restrctons on subsets of parameters, an t s thus esrable to employ econometrc methos that are flexble both n terms of ther applcatons an mplementaton for typcal ata sets. In ths paper we are ntereste n testng the null hypothess H n the followng form 0 H0 : g( θ ) = 0 () where the vector parameter θ Θ a k-mensonal k compact subspace of an g ) s a mappng k l from to contnuously fferentable functons of θ. In econometrcs lterature the Wal tests are commonly use to test ths null hypothess snce the generalty of ts formulaton affors the testng of several nterestng economc hypotheses whch mght (θ present formable ffcultes for other proceures (Sargan 980; Gregory an Veall, 985, 986). Moreover, t s well known that n a lnear regresson moel wth normally strbute errors, the Wal statstc for a set of lnear restrctons s a monotonc transformaton of the lkelhoo rato (LR) test statstcs. Uner regularty contons, the null asymptotc strbuton s ch-square strbuton, an ths s the strbuton that one usually uses to carry out hypothess tests. On the other han, uner the nonregularty, Wal tests fal to have lmtng chsquare strbuton n general (Anrew 987). For example, n vector autoregressve (VAR) processes, f the process s statonary, the multvarate least squares (LS) estmator of the coeffcents has a nonsngular asymptotc strbuton whereas the strbuton becomes sngular f some varables are ntegrate or contegrate. Another example s the Granger non-causalty test. As shown n Toa an Phllps (993) when there s a contegratng relatonshp, n general the Wal statstc of the Granger non-causalty test has a non-

2 stanar lmtng strbuton, epenng on nusance parameters. To overcome ths problem, Lutkepohl an Bura (997) use a generalze nverse of the asymptotc covarance matrx an they showe that the Wal test has an asymptotcally stanar strbuton. However, the serous problem s how etect the egeneracy or the rank of asymptotc covarance matrx. It s well known that the rank of matrx s equal to the number of nonzero egenvalues. Lukepohl an Bura (997) use an a-hoc threshol to etermne f the egenvalues are zero or not. Unfortunately ther choce of a threshol s more approxmate. The values of the threshol o not appear to be base on any explct analytcal expressons. An obvous way of crcumventng ths ffculty woul be to etermne the strbuton of the egenvalues. The purpose of ths paper s twofol. Frst, uner nonregularty contons, we mofy the usual Wal test to ensure that the null asymptotc strbuton s a ch-square strbuton. Seconly we propose new threshol values to test f the egenvalues are sgnfcantly fferent from zero or not. Ths permts to etermne the rank of the covarance matrx. The paper s organse as follows. Secton evelops the general expresson for the tests statstcs. Secton 3 proposes new threshol values for the egenvalues. The tests of Granger non-causalty are propose n secton 4. Secton 5 gves conclung remarks. n() g θ N(0, ) where Σ = G() θ ΩG()' θ wth Σ () g() θ G() θ = s the matrx of frst θ ' orer partal ervatves of the restrctons. To establsh the lmtng strbuton of the stanar Wal test, we assume the followng two regularty contons. C. The matrx of frst-orer partal ervatves of the restrctons G () θ s of full rank for all θ n the parameter space. C. The asymptotc covarance matrx Ω s non sngular. Let Σ be a consstent estmator of Σ, obtane by replacng G an Ω by ther consstent estmator ( θ () θ G ) an Ω respectvely. Hence uner the two regularty contons C an C, the Wal statstc for testng () s gven by θ (3) W = ng( θ)' Σ g( ) an W s strbute as ch-square wth k egrees of freeom on H. 0. WALD TESTS Now, when the covarance matrx Σ s sngular,.e. at least one of these regularty contons (C an In ths secton, we wll evelop general expressons C) s not satsfe, the Wal statstc may not have for Wal test statstcs for nonlnear restrctons. an asymptotc ch-square strbuton. The sngularty of Σ comes from three possble stuatons. Frst the Let θ be an estmator of θ base on a sample of matrx of frst-orer partal ervatves of the sze n. We make the followng assumpton restrctons, G() θ has reuce rank. Seconly the regarng the estmator θ. asymptotc covarance Ω s a matrx egenerate an / Assumpton () θ = θ OP( n ); () nv ar( θ ) = Ωn ; () n( θ θ) N l (0, Ω), where Ω = lm Ω n s fnte non zero but possbly n sngular; an the symbol represents weak convergence of the assocate probablty measure. (v) a consstent estmator Ω of Ω s avalable. The asymptotc normalty of the null hypothess θ mples that uner fnally both matrces G (θ) an Ω are sngular. () θ If the functon g nvolves proucts of the elements of θ then the matrx of frst-orer partal ervatves G( θ) s lkely to have a reuce rank over part of the parameter space. Such functons are relatvely common n tme seres analyss. For nstance, mpulse responses an relate quanttes of nterest n a vector autoregressve (VAR) analyss nvolve proucts of the VAR coeffcents. Smlarly, such functons come up n analyzng mult-step causalty n VAR moels (Lutkepohl an Bura, 997); or Granger causalty n VARMA moels (Lutkepohl, 99) an n testng of restrctons on the levels of parameters of a contegrate VAR process.

3 The sngularty of the asymptotc covarance matrx of the estmator parameters occurs for nstance n VAR processes. If the process s nonstatonary,.e. some varables are ntegrate or contegrate, the multvarate least squares (LS) estmator of the coeffcents has a sngular asymptotc strbuton. Ths sngularty problem has also been note or scusse n the context of the long-run mpact matrx, see (Johansen, 995). In that case, t s usual practce to resort to a generalze nverse proceure when we have nverte a sngular matrx. That s, we have uner the null hypothess, W = ng( θ)' Σ g( θ ) Σ (4) where s the Moore-Penrose generalze nverse of a matrx Σ. The Wal statstcs stll have an asymptotc ch-square strbuton uner an s equal to the rank of Σ, see (Anrews, 987). Unfortunately, ths latter conton s not always easy to verfy n practce. atonal conton, that s the rank of Σ To overcome ths ffculty, the Moore-Penrose generalze nverse can be obtane by usng the spectral ecomposton of Σ. Dagonalsng Σ as Q' Λ Q, where Λ (r r) s the agonal matrx of the postve egenvalues: Λ = ag( λ,..., λr ) wth Q λ... λ λ r 0 an the columns of, η ( =,...,r) are the egenvectors corresponng to the postve egenvalues. Hence the generalse Wal statstcs W = n g( θ)' Q ' Λ Q g( θ) (5) s asymptotcally ch-square wth r egrees of freeom on H 0, an r = r ank( Σ). In some nstances we wll nee to conser the case where the null hypothess H 0 hols only approxmately. The alternatve hypothess wll then be consere H : g ( θ) = γ/ n. (6) When H hols, The Wal statstc W satsfes W χ (, r ζ) (7) where χ (, r ζ) s the noncentral ch-square strbuton wth r egrees of freeom, an r = r ank( Σ) an noncentralty parameter ζ = γ' Q' Λ Qγ. (8) Ths test statstc has fewer egrees of freeom than the usual Wal test snce r = rank( Σ) k. Hence t has mprove power, n partcular f superfluous restrctons are remove. Gallant (977) fns n a small sample Monte Carlo nvestgaton that power avantages may n fact result from takng nto account fewer restrctons. The generalse Wal statstc s compare wth a crtcal value raw from a central ch-square strbuton wth r egrees of freeom. Inee the etermnaton of the egrees of freeom s a serous problem. The next secton proposes a soluton to ths problem. 3. NEW THRESHOLD VALUES FOR THE EIGENVALUES However the problem assocate wth ths test statstc s the etermnaton of the number of nonzero egenvalues of the matrx Σ, that s the rank of Σ. We suggest a sequence pretest proceure. We evelop the test proceures for H ' : λ = 0 (9) 0 where the λ are the egenvalues of the matrx Σ. To ece f the egenvalues are nonzero or not, Lutkepohl an Bura (997) use a threshol c whch epens on fferent factors such as the sample sze n an the value of the largest egenvalue λ of Σ. Then λ λ s nonzero f the egenvalue of the estmator Σ of Σ s greater than c. In other wors, for our purposes one rejects the null hypothess f λ > c. Unfortunately ther choce of a threshol c s more approxmate, the values of the threshol o not appear to be base on any explct analytcal expressons but are selecte on an a hoc bass. Consequently, t wll not be sure to take all restrctons whch correspon to nonzero egenvalues of the covarance matrx Σ. Thus the rank estmate may be larger than the true rank. We shall propose an approprate threshol level crteron. The approach conssts to erve the statstcal propertes of λ. To ths en, we use some results of the matrx perturbaton theory.

4 Let us conser the matrx perturbaton theory εb Σ Σ Σ = Σ εb (0) where, wth small, s the error matrx or the matrx perturbaton. These results of the matrx perturbaton theory ncate how much the egenvalues an egenvectors of the matrx Σ can ffer from those of Σ for small. Assume that the estmator Σ of Σ s r oot n consstent an has a lmtng normal strbuton, then the perturbaton term εb s of orer / OP( n ) an can be seen as a zero mean Gaussan ranom matrx. Hence the egenvalues λ of are asymptotcally normal, that s n( λ λ) N(0, σ ), see (Ratsmalahelo, 000, 003) ε Σ Havng fully efne the statstcal propertes of λ, we shall propose two alternatve test statstcs to ece whether egenvalue shoul be eclare zero or not. () Frst, let σ be a consstent estmator of σ, accorng to the asymptotc strbuton of λ, we reject the null hypothess H ' 0 f λ > z α σ / ε n () where z α represents the percentle of the stanar normal strbuton. () Seconly, uner H n q =, the test statstc ' 0 λ () σ s a ch-square ranom varable wth one egree of freeom. We reject H ' 0 f q > χ α where χ α s the crtcal value of the ch-square strbuton. We have efne two values of the threshol aequately. To assess the test statstcs, approxmate p values are compute wth reference to stanar normal strbuton an ch-square strbuton respectvely. In the next secton we wll gve an example where testng problem occurs an we wll show how t can be solve wth the metho propose n the prevous sectons. 4 GRANGER NON-CAUSALITY TEST In general, Wal tests for Granger non-causalty n vector autoregressve (VAR) process are known to have non-stanar asymptotc propertes for contegrate systems. The test base upon the nonstanar strbuton s very ffcult, f not mpossble to use n practce. In ths secton, we propose a proceure for conuctng Granger non-causalty tests that are base on methoology evelopng n sectons an 3. We wll show that the Granger non-causalty tests have an asymptotcally stanar strbuton. Conser m-vector process { generate by vector autoregressve (VAR) moel of orer p, y t y t = µ A(L)y t- Φ D t εt m } t =,,,T (3) where y = { y,..., y }', µ s the constant vector, t t mt p AL ( ) = AL = s the backwar shft operator, Φ s the m k coeffcent matrx, an ε = { ε,..., ε }' s a zero mean t t mt nepenent whte nose process wth covarance matrx Ψ an, for =,...,r E τ ε t < for some τ > 0. The etermnstc terms Dt can contan a lnear tme, seasonal ummes, nterventon ummes, or other regressors that we conser fxe an non-stochastc. It wll be convenent to wrte the process (3) n a vector error correcton (VEC) form : p y = µ Π y Γ y Φ D ε t t t t = where Π = = for =,..., p. p A I, L an Γ = A p j= (4) By portonng y t n ( h) an ( h = m h) wth h mensonal subvectors y t an y t an A matrces portone accorngly, then y oes t not Granger-cause the y f the followng hypothess t t

5 s true. H : A = 0 =,...,r 0 for (5) Let use efne y t = Lhy wth L h = [ Ih,0] an y t = Lh y wth Lh = [0, Ih] an I h s the entty matrx of rank h. Let A = [ A, A,..., A p ] an α = vec( A) where vec(.) enotes the vectorzaton operator that stacks the columns of argument matrx. The null hypothess that y t oes not Granger-cause the y t can be wrtten as or equvalently H : L AL ' = 0,..., ) 0 h h ( = p (6) H0 : L hal ' = 0 or R α = 0 (7) where R = L h L an L = I p Lh. Let α = vec( A) where A s the least squares (LS) estmator of A n equaton (3) then followng Theorem.3 of Phllps (998), we have / T ( α α) N(0, Ωα) where the asymptotc covarance matrx Ωα (8) s fnte non zero but possbly sngular. Consequently, we have uner the null hypothess of non-causalty n equaton (7) / T R α N(0, RΩ α R') (9) If R Ω α R' s of full rank then a stanar Wal test statstc, for H 0 n levels VAR s format, has a chsquare strbuton wth egrees of freeom equal to the number of restrctons as T grows: W = T ( Rα)'( RΩαR') ( Rα) χv (0) However, RΩ α R' can be egenerate, as s well known, because Ω α s sngular. When R Ω α R' s egenerate, the Wal statstc has an asymptotcally non-stanar strbuton, an cannot be easly teste (Toa an Phllps, 993, 994). Thus, the levels VAR s moel has not been use n practce n possbly contegrate systems. To overcome ths problem, we mofy the Wal statstc n the followng. The generalse Wal test of the null hypothess of non-causalty n equaton (7): W = T ( Rα)'( RΩ R') ( Rα) () has an asymptotc ch-square strbuton wth r egrees of freeom, an r = r ank( RΩ α R' ). Here (RΩ α R') s the Moore-Penrose generalze nverse of a matrx R R'. Ω α The testng proceure epens upon how we etect the rank of R Ω α R'. In orer to etect egeneracy or the rank of ts matrx, we use the spectral ecomposton of an apply the testng proceure evelope n secton 3. α R R' Ω α 5. CONCLUDING REMARKS In ths paper, we have propose a generalse Wal type tests whch guarantees the asymptotc ch-square strbuton n the case where the stanar Wal statstc fals to have ts lmtng ch-square strbuton ue to nonregularty contons. Snce the lmtng strbuton of the Wal statstc for testng nonlnear restrctons epens substantally on the sngularty or not of the asymptotc covarance matrx, the etermnaton of ts rank s of great mportance n ths context. It s well known that the rank of the matrx s equal to the number of the egenvalues nonzero. The present paper proves new threshol values for the egenvalues whch permt to ece f the egenvalues are nonzero or not. We have also propose an operatonal proceure to test the hypothess of the Granger non-causalty n contegrate systems. It crcumvents the problem of possble egeneracy of the varance-covarance matrx assocate wth the usual Wal test statstc. The mofcaton of the Wal tests consere here proves general solutons to the nonsngularty problem. It can be use n statstcal sgnal processng. REFERENCES Anrews, D.W.K. (987). "Asymptotc results for generalze Wal tests". Econometrc Theory, 3, Dolao, J., J; an Lutkepohl, H. (996) Makng Wal Tests work for contegrate VAR systems, Econometrcs Revews, 5,

6 Gallant, A.R. (977). "Testng a nonlnear regresson specfcaton: A nonregular case". Journal of Amercan Statstcal Assocaton, 7, Gregory, A.W. an M. R. Veall (985). "Formulatng Wal tests of nonlnear restrctons", Econometrca, Gregory, A.W. an M. R. Veall, (986). "Wal tests of Common Factor Restrctons". Economcs Letters,, Johansen, S. (995). Lkelhoo-Base Inference n contegrate Vector Autoregressve Moels, Oxfor Unversty Press, Oxfor. Lutkepohl, H. (99). Introucton to multple tme seres analyss. Sprnger, Berln. Lutkepohl, H. an Bura (997). "Mofe Wal tests uner nonregular contons". Journal of Econometrcs, Paruolo, P. (997) Asymptotc Inference on Movng Average Impact Matrx n Contegrate I() VAR Systems, Econometrc Theory, 3, Phllps, P. C. B. (998): Impulse Response an Forecast Error Varance Asymptotcs n Nonstatonary VARs, Journal of Econometrcs, 83, -56. Ratsmalahelo, Z. (000). Rank Test Base on Matrx Perturbaton Theory. Workng paper, Unversty of Franche-Comté, U.F.R Scence-Economque. A secon verson (00). Ratsmalahelo, Z. (003): Specfcaton of VARMA Echelon Form Moels, n Neck R. (es) on Moellng an Control of Economc Systems, pp 9-37, Elsever, Pergamon Press. Sargan, J. D. (980). Some Tests of Dynamc Specfcaton for a sngle Equaton Econometrca, 48, Toa H. Y. an P.C. B. Phllps (993): Vector Autoregresson an Causalty, Econometrca, 59, Toa H. Y. an P.C. B. Phllps (994): Vector Autoregresson an Causalty: A Theoretcal Overvew an Smulaton Stuy, Econometrc Revews, 3,