A Wald test of restrictions on the cointegrating space based on Johansen s estimator

Transcription

1 Economcs Letters 59 (998) A Wald test of restrctons on the contegratng space based on Johansen s estmator James Davdson* Cardff Busness School, Colum Drve, Cardff CF 3EU, UK Receved 3 July 997; accepted January 998 Abstract A test s derved of the hypothess that the contegratng space of a collecton of I() varables contans a vector subject to a set of lnear restrctons. Applcatons to the problem of testng for rreducble contegratng relatons, and also structural hypotheses, are dscussed. 998 Elsever Scence S.A. Keywords: Contegraton; Irreducble contegratng vector; Johansen; Vector error correcton; Wald test JEL classfcaton: C3. Introducton Ths artcle derves a test of restrctons on the contegratng vectors estmated by the Johansen (988), (99) maxmum lkelhood estmator. Let the DGP for the I() process x t (m 3 ) be wrtten as k Dx 5OGDx ab9x u, (.) t j tj tk t j5 for t 5,...,T, where a and b are m 3 s matrces of rank s (assumed known). Recall that n Johansen s procedure b, ˆ the MLE of b, s calculated as the set of egenvectors correspondng to the s largest egenvalues of S9k S Sk wth respect to S kk, where n the usual notaton, S, Skk and Sk are the moment matrces formed from the resduals from regressng Dxt and x tk, respectvely, onto the ˆ ˆ ˆ ˆ Dx. Accordngly, b s normalsed to satsfy b9s b 5 I and b9s9 S S ˆ b 5 D, ˆ where Dˆ tj kk s k k s the dagonal matrx formed from the s largest egenvalues. As s well known, these vectors span the contegratng space, but have no nterpretaton as structural economc parameters. Johansen and Juselus (99, Secton 5.) show how to construct Wald tests for hypotheses of the form Hb 5, where H s a known p 3 m matrx of constants, usng the result shown n Johansen (99) that T(bˆ b ) s locally asymptotcally mxed normal (LAMN) under standard assumptons. *Tel.: ; fax: ; e-mal: davdsonje@cardff.ac.uk / 98/ $ Elsever Scence S.A. All rghts reserved. PII: S65-765(98)4-

2 84 J. Davdson / Economcs Letters 59 (998) Whle ths type of hypothess s nvarant to the normalsaton snce all the columns of b satsfy the same restrctons under the null, t s not wdely applcable. By contrast, the hypothess we consder here s that the contegratng space contans a vector whch satsfes restrctons H. Ths may be expressed as H:'a(s 3 ) such that Hba 5. (.) The chef motvaton of ths approach s to gve drect tests of structural hypotheses on the contegratng relatonshps of the model. As a byproduct, estmates of dentfed structural parameters can be solved from b. ˆ The test s descrbed n Secton, and the applcatons to structural modellng are brefly outlned n Secton 3.. The test Assume p $ s, snce otherwse H s trvally true. Further, assume that the vector a specfed n (.) s unque up to a normalsaton. (Both of these assumptons wll be motvated n Secton 3.) Then, under H the matrx b9h9hb (s 3 s) has rank of s, ts smallest egenvalue s, and a equals the egenvector correspondng to ths smple egenvalue, up to a normalsaton. On the other hand, f H s false the mnmal egenvalue s postve. H can therefore be restated as Hba 5 where a s the egenvector correspondng to the smallest egenvalue of b9h9hb. The vector a so defned s a contnuous functon of b, dfferentable to all orders. If aˆ denotes the egenvector correspondng to the smallest egenvalue of b9h9hb, ˆ ˆ then aˆ s consstent for a. Snce T(bˆ b ) 5 O p(), we may wrte, when H s true THba ˆˆ 5 TH(bˆ b )a THb(aˆ a) o (). (.) p To derve the asymptotc dstrbuton of ths vector we can use the fact (see Magnus and Neudecker, 988, Chapt. 8, Th. 7) that the dfferental wth respect to b of an egenvector of b9h9hb, correspondng to a smple egenvalue l, has the form da 5 P(db9H9Hb b9h9hdb )a, (.) where P denotes the Moore Penrose nverse of the sngular matrx lis b9h9hb. Lettng h9 the th row of H, we can wrte denote h9 bda 5 a9b9h9hdbpb9h h9bpb9h9hdba 5 (h9bp ^ a9b9h9h a9 ^ h9bpb9h9h)vec db, and hence hd(ba) 9 5 hdba 9 h9 bda 5 (a ^ h Pb9h ^ H9Hba a ^ H9HbPb9h )9 Vec db ; k9 Vec db (say). The p-vector n (.) s therefore approxmated lnearly, element by element, by

3 J. Davdson / Economcs Letters 59 (998) Th9ba ˆˆ 5 Tk9 Vec(bˆ b ) o (), 5,...,p. (.5) p The vectors k (sm 3 ) are contnuous functons of b. Under H, (.5) can be treated as exact f k s evaluated at a pont on the lne segment jonng ˆ b and b, say b *. Whether evaluated at ˆ b or at b *, consstency of ˆ b for b mples that the k converge to fxed lmts at b as T `. Accordng to Theorem 5. of Johansen (99), T Vec(bˆ b )(sm 3 ) s LAMN under standard assumptons, and the lmtng condtonal covarance matrx s consstently estmated by ˆ ˆ ˆˆ ˆ ˆ ˆ s b b T(D I ) ^ M nn9m ; TA ^ B, (.6) where the dentty defnes Aˆ and B, ˆ ˆ n (m 3 (m s)) s the matrx of the egenvectors correspondng to the m s smallest egenvalues of S9 S S wth respect to S (.e., those not ncluded n ˆ k k kk b ), and ˆ ˆ ˆ ˆ M 5 I b(b9b ) b9. ˆ Johansen s covarance matrx formula s gven for ˆ b m b normalsed by a matrx ˆ ˆ ˆ c, or n other words for bc 5 b(c9b ) such that c9b s a matrx of full rank (Johansen, 99, Secton 5). Here, we choose c 5 b(b9b ), so that bc 5 b. The formula s also derved for the general case where b s subject to restrctons of the form b 5 Hw where H s a known matrx and w s a matrx of parameters, but we consder the case H 5 I m, so that ths matrx dsappears from the formula. Wth these substtutons, and the replacement of b by the consstent estmate b, ˆ (.6) corresponds to Johansen s expresson (5.6). For the case of the model wth determnstc trends constraned to zero, the correspondng verson of Johansen s expresson (5.8) s used. Combnng (.6) wth (.5) evaluated at the estmate b, ˆ the asymptotc condtonal covarance matrx of THba ˆˆ s consstently estmated by the matrx TV ˆ ( p 3 p) where (omttng hats for clarty) the typcal element of Vˆ s Under H vj 5 a9aa.h9 Bhj h9 bpaa.a9b9h9hbhj a9aa.h9 BH9HbPb9h j a9apb9h j.h9 BH9Hba h9 bpapbh j.a9bh9hbh9hba h9 bpa.a9b9h9hbh9hbpb9h j a9aa.h9bpb9h9hbhj a9apb9h j.h9bpb9h9hbh9hba a9aa.h9 bpb9h9hbh9hbpb9h j. (.7) the statstc ˆ ˆ ˆ T a9bh9v ˆ Hbaˆ (.8) s asymptotcally ch-squared wth p degrees of freedom, provded Vˆ s of rank p. The latter requrement presents a dffculty when p. m s, snce the matrx Bˆ 5 Mˆ nn9m ˆˆ ˆ b b n (.6) s of rank m s. Note that the dstrbuton of T Vec(bˆ b ) s sngular, of rank s(m s), n vew of the s restrctons to whch ˆ b s subject. Ths dffculty can be overcome by employng the Moore Penrose nverse of the varance matrx. Dagonalsng Vˆ as Q9LQ, where L (r 3 r) s the dagonal matrx of the postve egenvalues, wth r 5 mnh p,m sj, the test statstc T a9b9h9q ˆ ˆ 9L QHba ˆˆ s asymptotcally ch-squared wth r degrees of freedom on H. The mplcaton of havng p. m s (.9) Note that n ths model the matrx specfed s random even asymptotcally. It can be called the covarance matrx because test statstcs formed from t n the conventonal manner have the standard asymptotc dstrbutons under H. See Johansen (99) for further detals.

4 86 J. Davdson / Economcs Letters 59 (998) restrctons s that we are unable to test them all ndependently because of the way restrctons are mposed on b for estmaton. We can however test the m s ndependent restrctons represented by QH, whch n general wll hold only f (.) s true. 3. Applcatons Consder the case of zero (excluson) restrctons. After sutable re-orderng, let H 5 [,I p] for s # p # m. Ths corresponds to the hypothess that the contegratng space contans a vector wth only m p nonzero elements, and equvalently that the frst m p elements of xt form a contegrated subset. Moreover, the contegratng vector for ths subset s drectly estmated by the vector bˆ5 Gba ˆˆ ((m p) 3 ), (3.) where G 5 [I,], and bˆ mp s LAMN wth asymptotc condtonal covarance matrx estmated by lettng G replace H n (.7). The assumptons of Secton can be convenently motvated wth reference to ths case. Note that f we have s lnearly ndependent contegratng vectors for m varables, we can always construct a contegratng vector contanng s zeros, by formng approprate lnear combnatons. The test statstc correspondng to ths case s dentcally zero by constructon. Next, consder the case where b9h9hb has more than one zero egenvalue. Then, there exst two or more lnearly ndependent contegratng vectors for the same subset of m p varables. By the precedng argument, ths means there exsts a contegratng vector wth more than p zero elements, and hence a more restrctve hypothess than that under test s also true. So whle the dervaton of the null dstrbuton s not vald for multple zero egenvalues, we note that there exsts a test statstc computed for at least p restrctons whch does have the stated dstrbuton. On heurstc grounds we should expect the test sze n the case of multple zeros to be f anythng lower than the nomnal sze, so that the test should perform correctly n practce. Monte Carlo experments support ths conjecture. The phenomenon of a contegratng subset, n the context of a structural model of contegraton, s analysed n Davdson (994) and Davdson (997). The latter paper utlses the present test to mplement the algorthm MINIMAL, whch determnes all the rreducble contegratng subsets of a data set (those from whch no varable can be dropped wthout losng contegraton) by an exhaustve sequence of excluson tests. It s shown that an dentfed structural contegratng relaton s always rreducble. Moreover, LAMN estmates of the structural parameters are obtaned drectly from the Johansen matrx by formula (3.), wthout the need for nstruments or teratve methods. A test of long run structural restrctons can always be cast n the form of (.). Thus, suppose that the economc model generatng the data accordng to (.) has structural parameters aand b. In (.) b 5 b d and a 5 a d, where d (s 3 s) s an arbtrary nonsngular matrx to whch Johansen s procedure assgns a value dfferent from I, n general. Pror restrctons on a and b are requred to s dentfy the remanng elements from b and a, accordng to the usual rank condton. The structural Ths assumes that no further restrctons on the vector are accepted. A vector whch s not rreducble s ether not structural, or not dentfed. Also, note that an rreducble contegratng relaton s not necessarly structural. See the above-cted papers for detals.

5 J. Davdson / Economcs Letters 59 (998) hypothess that b (a column of b, labelled wthout loss of generalty) s subject to p lnear restrctons can be wrtten as Hb 5. But f b s dentfed (.e., such that the restrctons are overdentfyng) t has the representaton b 5 ba, where a s unque up to normalsaton. In other words, the structural hypothess always has the representaton (.), n whch the unqueness assumpton of Secton must hold. Fnally, we note that Pesaran and Shn (994) have constructed lkelhood rato tests of structural restrctons on b by estmatng (.) subject to the constrants, n effect settng d 5 Is by mposton of dentfyng restrctons. Ths provdes an alternatve to the present procedure, but numercal optmsaton s requred to compute the constraned estmates, whereas our procedure requres only the estmaton of the standard Johansen model. (Gauss code to compute the statstcs descrbed n ths paper s avalable from the author.) Acknowledgements I thank, wthout mplcatng, Jan Magnus for hs advce. References Davdson, J., 994. Identfyng contegratng regressons by the rank condton. Oxford Bulletn of Economcs and Statstcs 56 (), 3 8. Davdson, J., 997. Structural relatons, contegraton and dentfcaton: Some smple results and ther applcaton. Journal of Econometrcs (forthcomng). Johansen, S., 988. Statstcal analyss of contegraton vectors. Journal of Economc Dynamcs and Control, Johansen, S., 99. Estmaton and hypothess testng of contegraton vectors n gaussan vector autoregressve models. Econometrca 59 (6), Johansen, S., Juselus, K., 99. Maxmum lkelhood estmaton and nference on contegraton, wth applcatons to the demand for money. Oxford Bulletn of Economcs and Statstcs 5, 69. Magnus, J.R., Neudecker, H., 988. Matrx Dfferental Calculus wth Applcatons n Statstcs and Econometrcs. Wley, Chchester. Pesaran, M.H., Shn, Y., 994. Long-run structural modellng. Workng Paper, Department of Appled Economcs, Unversty of Cambrdge (September).