Frequency Granger Causality Test in Cointegration System by Wavelet Analysis

Transcription

1 Proceedngs of the 1th WSEAS Internatona Confenrence on APPLIED MATHEMATICS, Daas, Texas, USA, November 1-3, 6 8 Frequency Granger Causaty Test n Contegraton System by Waveet Anayss Yuan Ja-ha, Zhao Chang-hong, Xong Mn-peng Schoo of Busness Admnstraton North Chna Eectrc Power Unversty Changpng Dstrct Beng 16 Chna Abstract:- Frequency doman Granger causaty test s an mportant fed n economcs however by now no smpe toos exst for the cacuaton. In the paper, we propose a nove method based on waveet anayss to test the mpact of specfc frequency fuctuaton of seres n estabshng the contegraton and Granger causty among seres. The method s rather tme doman orented and can rey on the standard procedure of error correcton mode (Johansen & Juseus 199; Johansen 1995). A smpe case study for eectrcty consumpton and economc growth n Chna s reported n the paper to vadate our method. Key words: Granger causty; Frequency doman; Waveet anayss; Emprca study waveet anayss s a nd of ocazed tme-frequency sgna anayss wth fxed wndow 1 Introducton Granger causaty test has become a rather standard technque n economc research(johansen & Juseus 199; Johansen 1995). As to tme doman anayss, the error correcton mode has become a prototype too. Concernng frequency doman anayss, Yao and Hosoya () has proposed a method based on Wad test to test one-way Granger effect for contegrated vector tme seres. However, though mathemetcay perfect, the method proposed by Yao and Hosoya s very compcated n computaton and can not rey on exstng mature computaton toobox such as Evews ect. So the appcaton of the method s rare. In the paper, we are to propose a nove method for frequency Granger causaty test. In genera sense, our method s rather tme doman orented. So t s possbe to rey on exstng toobox to reaze the comuputaton of the method, whch gves t an adavantage for economst not famar wth compex computaton toobox to use t wthout dffcuty. Tme-frequency Waveet Anayss of Tme seres.1 Dscrete waveet anayss and aterabe form. Suppose that ψ () t L ( R) (square ntegraabe rea number space) wth ts Fourer transformaton beng ψ ( ω ). When ψ ( ω ) satsfes the admssabty condton ψω ( ) dω < (1) R ω ψ () t s caed as waveet base or mother functon. Breang functon f (t) downng n any L ( R) space on the transformaton of mother waveet s caed contnous waveet transform, whch s expressed as 1 t τ WTf ( a, τ ) =< f ( t), ψ a, τ ( t) >= f t dt a ( ) ψ ( ) R a () where a s scae factor and τ s transaton factor. When anaysng sgnas of a non-statonary nature, t s often benefca to be abe to acqure a correaton between the tme and frequency domans of a sgna. The Fourer transform, provdes

2 Proceedngs of the 1th WSEAS Internatona Confenrence on APPLIED MATHEMATICS, Daas, Texas, USA, November 1-3, 6 9 nformaton about the frequency doman, however tme ocased nformaton s essentay ost n the process. The probem wth ths s the nabty to assocate features n the frequency doman wth ther ocaton n tme, as an ateraton n the frequency spectrum w resut n changes throughout the tme doman. In contrast to the Fourer transform, the waveet transform aows exceptona ocasaton n both the tme doman va transatons of the mother waveet, and n the scae (frequency) doman va datons. Ths s why we use waveet anayss for pretreat tme seres before testng Granger causty because the seres to be deat wth s non-statonary. In practce, for convenece of cacuaton t s needed to dscretze the contnous waveet. Generay, t s to dscretze the scae factor a and transaton factor τ. To dscretze a and τ, assumng that a > 1 and b, then a m m = a = nba b, thereby transformng contnous waveet nto dscrete one m / m ψ, ( x) = a ψ ( a x nb ) m, n Z (3) m n In dgtzaton reazton, dscrete waveet transformaton s competed by famous Maat agorthms, whch s c d as = = c c h g (4) the reconstructon agorthm can be expressed + 1 c = c h + d h where { h } ( Z ) Z (5) are the coeffecents of ow-pass fter and { } are the ( Z ) coeffecents of hgh-pass fter. s the +1 c approxmaton of and s the detament g Z d c +1 c of.. Mut-resouton anayss by mut-resouton anayss, the sgna n every resouton s the sum of an approxmate sgna and deta sgna. When contnousy addng the deta sgna nto approxmaton sgna,the orgna sgna s reconstruted. Mut-resouton anayss s a seres of cose V subspace { } condtons: Concordant Z meets wth the foowng monotoy:... V V V V V Gradua perfectbty: ={}, Z V = L ( R) Fex reguarty: Transaton nvarabty: f ( t n) V Z f ( t) f ( t) f ( t) for a n Z Exstence of orthogonaty: φ V, {φ ( t n)} n Z s the orthogonaty base of V V = span{ φ ( t n)}, φ( t n) φ( t m) dt = δ m, n R (6) where condton of exstence of orthogonaty can be reaxed to exstence of Reze base. Accordng to the propertes of mut-resouton anayss, the equaton beow can be satsed = φ( t ) = h[ ] φ (t ) = h( ) φ(t ) = (7) whch ndcates the reatonshp between two ordna scaes. Its frequency verson s ω 1 ω Φ( ω) = H ( e ) Φ( ) (8) where H ( e ω transformaton of h[n]. ) s the dscrete fourer

3 Proceedngs of the 1th WSEAS Internatona Confenrence on APPLIED MATHEMATICS, Daas, Texas, USA, November 1-3, 6 1 Athough Fourer anayss and waveet anayss can both dst the wave components of seres n dfferent frequences, Fourer anayss s good at deang wth statonary seres. For non-statonary seres, the waveet s better than Fourer. And aso, waveet s we n ocazed tme-frequency unte anayss. 3 Frequency Granger Causaty Test Based on Waveet Anayss 3.1 Granger causaty test n contegrated system When both seres are ntegrated of the same order, we can proceed to examne for the presence of contegraton. The Johansen Maxmum ehood procedures are used for the test (Johansen and Juseus, 199). Any ong-tem contegratng reatonshp found between the seres w contrbute an addtona error-correcton term to the ECM. The Johansen procedure s a vector autogressve (VAR) based test on restrcton mposed by contegraton n the unrestrcted VAR. The nu hypothess n consderaton s Ho, that there are a dfferent number of contegraton reatonshp, aganst H1, that a seres n the VAR are statonary. The ECM used n ths paper s specfed as foows: yt = α + β ECM + α () y t t α () x + ε t t xt = α + β ECM + α () y t 1 1 t α () x + ε 1 t 1t (9) (1) Where Yt, Xt are the dfferences n the varabes that capture ther short-run dsturbances, ε, ε are the seray uncorreated error terms, and 1t t ECTt-1 s the error-correcton term (ECT), whch s derved from the ong-run contegraton reatonshp and measures the magntude of the past dsequbrum. In each equaton, change n the endogenous varabe s caused not ony by ther ags, but aso by the prevous perod s dsequbrum n eve,.e. ECTt-1. Gven such a specfcaton, the presence of short and ong-run causaty coud be tested. Consder Eq.(1), f the estmated coeffcents on agged vaues of Xt are statstcay sgnfcant, then the mpcaton s that Xt Granger causes Yt n the short-run. On the other hand, ong-run causaty can be found by testng the sgnfcance of the past dsequbrum term. The nu hypothess of the F test s: Hα ( ) =, = 1, p, and the statstcs s ( RSS R RSS ) / J F = ~ F( J, T K) (11) RSS /( T K) 3. Frequency Granger test based on waveet anayss In reequency doman tme seres can be seen as the sum of specfc frequnecy components. So the Granger causaty among seres s the resut of the sum of dfferent frequency components. Because of the dfference of fuctuaton n dfferent frequency components, the nfucence of dfferent on the Granger causaty s dfferent aso. So the natura dea s to remove specfc frequency component n the seres frsty, then to test whether the Granger causaty s changed n the reconstructed seres. The decomposton and reconstructon of waveet anayss provdes such the too. For exampe, consder the Granger causaty of two seres, eectrcty consumpton and economc growth. If by deetng specfc frequency component n eectrcty consumpton, for exampe four year/cyce component, the unetera Granger cause runnng from eectrcty consumpton to economc growth s not sgnfcant, t s evdent that the four year/cyce component of eectrcty consumpton s mportant n estabshng the Granger cause from eectrcty consumpton to economc growth. Otherwse, the frequency component s of no nfuence to the

4 Proceedngs of the 1th WSEAS Internatona Confenrence on APPLIED MATHEMATICS, Daas, Texas, USA, November 1-3, 6 11 Granger cause. On the other hand, f by deetng the specfc frequency component of both seres, the batera Granger cause of the two seres changes dramatcay, t s evdent that ths frequency wave has sgnfcant nfuence on the Granger cause of the seres. Fnay, f by deetng the same component n two seres, the contegraton reatonshp of the two seres does not exst anymore, t means that the specfc component s rghty the man component to estabsh the ong-run equbrum for the two seres. 4 Case Study: Eectrcty Consumpton and Economc Growth n Chna In ths secton, we use the test of Granger cause between eectrcty consumpton and economc growth n Chna to vadate our dea. In the paper,waveet haar s choosen as the mother waveet of the anayss and accordng to the requrement the scae of resouton s determned as four. Thereby, the hgh frequency component of the owest eve corresponds to wave of two year/cyce, and four year, eght year, sxteen year n turn. Because waveet anayss s senstve to the choose of mother waveet, we adopt bor and rbo waveet aso to anayze the frequency characterstc of the seres, amost same resuts are obtaned. Sampe the eectrcty consumpton and economc growth(gross domestc product GDP) data for anayss, the tme-frequency characterstc of the two seres are shown n fgure 1 and. We are to deetng the two year, four year, eght year and sxteen year/cyce component of the seres and reconstruct them. Then we proceed to test the Granger cause of the reconstructed seres accordng to the dea expaned n secton 3. accordng our research n (Yuan Jaha, Wang Jng, Hu Zhaoguang 6), there exsts contegraton between eectrcty consumpton and GDP n Chna durng 1978 to 4 and there exsts untera Granger cause runnng from eectrcty consumpton to GDP. We mae three battery of tests n such way: 1) deetng the two year, four year, eght year and sxteen year/cyce component of eectrcty consumpton and reconstruct t, then test Granger cause wth untreated GDP seres; ) deetng the two year, four year, eght year and sxteen year/cyce component of GDP and reconstruct t, then test Granger cause wth untreated eectrcty consumpton seres; 3) deetng the two year, four year, eght year and sxteen year/cyce component of two seres and reconstruct them, then test Granger cause wth reconstructed seres. The resuts of the test are reported n tabe 1 to tabe 3. The cacuaton s done n exstng toos: Evews for the Granger causaty test and MATLAB WAVELET TOOLBOX for waveet anayss. From tabe 1, we can see that by deetng the two year, four year and eght (and above) year/cyce wave component of eectrcty consumpton, the Granger cause runnng from eectrcty consumpton to GDP become nsgfcant, whch mpes that, the fuctuaton of eectrcty consumpton, be t hgh frequency or ow frequency, has sgnfcant nfuence on GDP. From tabe, we can see that by deetng the two year, four year and eght (and above) year/cyce wave component of GDP, except for four year/cyce component, the Granger cause s unchanged, whch mpes that, the nfuence of GDP to eectrcty consumpton, many wors n ong-run. From tabe 3, t s evdent that the ong run fuctuaton components of the two seres (eght year and above /cyce) pay eadng roe n estabshng ther ong run reatonshp. Especay, deetng the eght and sxteen year cyce components of the two seres eads to the vanshng of contegraton between them. To sum up, the fuctuaton of eectrcty

5 Proceedngs of the 1th WSEAS Internatona Confenrence on APPLIED MATHEMATICS, Daas, Texas, USA, November 1-3, 6 1 consumpton, be t short term or ong term, has sgnfcant mpact on GDP; that of GDP, ony ong run ow frequency component has sgnfcant mpact on eectrcty consumpton, whch s evdence that the busness cyce s correated wth fuctuaton n eectrcty consumpton; the contegraton reatonshp s many estabshed by the ow frequency components of the two seres, whch s n ne wth the concuson of (Wang Jng, Yu En-ha, Yuan Ja-ha 6). Fg 1 the tme and frequency characterstc of ogarthm seres of GDP Fg the tme and frequency characterstc of ogarthm seres of eectrcty consumpton Tab 1-test resuts of Granger cause runnng from reconstructed eectrcty consumpton seres to GDP Deetng component from eectrcty consumpton Short-run Granger cause Long-run Granger cause year /cyce nsgnfcant nsgnfcant 4 year/cyce nsgnfcant nsgnfcant 8 year and above/cyce nsgnfcant nsgnfcant Tab -test resuts of Granger cause runnng from reconstructed GDP seres to eectrcty consumpton Deetng component from GDP Short-run Granger cause Long-run Granger cause year /cyce nsgnfcant sgnfcant 4 year/cyce sgnfcant sgnfcant 8 year and above/cyce nsgnfcant sgnfcant

6 Proceedngs of the 1th WSEAS Internatona Confenrence on APPLIED MATHEMATICS, Daas, Texas, USA, November 1-3, 6 13 Tab 3-test resuts of Granger cause wth reconstructed GDP and eectrcty consumpton seres Deetng component from GDP and eectrcty consumpton Short-run Granger cause Long-run Granger cause 8 year /cyce component Eectrcty consumpton to GDP nsgnfcant nsgnfcant GDP to eectrcty consumpton nsgnfcant sgnfcant 8 year and above /cyce components Eectrcty consumpton to GDP nsgnfcant nsgnfcant GDP to eectrcty consumpton nsgnfcant nsgnfcant 5 Concudng Remars In the paper, we propose a nove method based on waveet anayss to test the mpact of specfc frequency fuctuaton of seres n estabshng the contegraton and Granger causty among seres. The method s rather tme doman orented and can rey on the standard procedure of error correcton mode. A case to test the Granger cause between eectrcty consumpton and economc growth n Chna s nvestgated to vadate our method. References: [1] Feng Y. Hosoya Y. Inference on one-way effect and evdence n Japanese macroeconomc data. Journa of Econometrcs,, 98():5-55. [] Johansen, S. (1995), Lehood-Based Inference n Contegrated Vector Autoregressve Modes (Oxford Unvesty Press: Oxford) [3] S. Johansen and K. Juseus. Maxmum Lehood Estmaton and Inference on Contegraton wth Appcatons to the demand for money [J]. Oxford Bueton of Ecomomse and Statstcs, 199, 5, [4] Wang Jng, Yu En-ha, Yuan Ja-ha. Research of the Reatonshp Between Eectrcty Consumpton and Busness Cyce n Chna. WSEAS Transacton on Systems, Oct. 6, 5(1): [5] Yuan Jaha, Wang Jng,Hu Zhaoguang Contegraton and Co-feature Anayss on Eectrcty Consumpton and Economc Growth n Chna WSEAS Transacton on Systems, June 6, 5(6): [6] Yuan Jaha, Wang Jng, Hu Zhaoguang.Eectrcty consumpton and economc growth n Chna: contegraton and co-feature anayss.5th WSEAS Int. Conf. on Apped Computer Scence (ACOS '6). Hangzhou Chna, Apr 6