Granger Causality Test: A Useful Descriptive Tool for Time Series Data
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1 Ieraioal OPEN ACCESS Joural Of Moder Egieerig Research (IJMER) Grager Causaliy Tes: A Useful Descripive Tool for Time Series Daa OGUNTADE, E. S 1 ; OLANREWAJU, S. O 2., OJENII, J.A. 3 1, 2 (Deparme of Saisics, Uiversiy of Abuja, P.M.B.117, Abuja. Nigeria) 3 (Deparme of Cyber-Securiy Sciece, Federal Uiversiy of Techology, Mia) Absrac: Ierdepedecy of oe or more variables o he oher has bee i he exisece over log ime whe i was discovered ha oe variable has o move or regress oward aoher followig he work doe by Galo (1886); Pearso & Lee (193); Kedall & Suar, (1961); Johso ad DiNardo, (1997); Gujarai, (24) ec. I was i he ligh of his depedecy over ime he researcher uses Grager Causaliy as a effecive ool i ime series Predicive causaliy usig Nigeria GDP ad Moey Supply o kow he ype of causaliy i exisece i he wo ime series variables uder cosideraio ad which oe ca saisically predics he oher. The research work aimed a esig for aure of causaliy bewee GDP ad moey supply for Federal Republic of Nigeria for he period of hiry years usig he daa sourced from Ceral Bak of Nigeria Saisical Bullei. Afer observig he various codiios of Grager causaliy es such as esurig saioariy i he variables uder cosideraio; addig eough umber of lags i he prescribed model before esimaio as Grager causaliy es is sesiive o he umber of lags iroduced i he model; ad as well as assumig he disurbace erms i he various models are ucorrelaed, he resul of he aalysis idicaes a bilaeral relaioship bewee Nigeria GDP ad Moey Supply. I implies Nigeria GDP Grager causes moey Supply ad vice versa. Based o he resul of his sudy, boh Nigeria GDP ad moey Supply ca be successfully model usig Vecor Auoregressive Model sice chages i oe variable has a sigifica effec o he oher variable. Keywords: Bilaeral Causaliy, Time Series, VARMA Model, Saioariy, Wold represeaio GDP, Moey Supply, I. Iroducio Two or more variables ca be ierdepedece o oe aoher if he occurrece of oe causes he oher o ake place or vice versa, he oe ca alk of uidirecioal causaliy ad feedback causaliy. Chages i he firs variable precede chages i he oher. Tha is oe Grager causes aoher kowig fully ha he prese wih is lagged values ca oly predic he fuure bu he fuure cao predic he pas. A saisical relaioship i iself cao logically imply causaio, o ascribe causaliy; oe mus appeal o a priori or heoreical cosideraios (Gujarai, 24 pp23). A saisical relaioship, however srog ad however suggesive, ca ever esablish causal coecio: our ideas of causaio mus o from mere saisics bu ulimaely from some heory or oher (Kedall & Suar, 1961). II. Model Descripio ad Noaios 2.1 Overview of ARMA Models A se of repeaed observaios of he same variable such as Sock Reurs, GDP, Moey Supply, Ieres Raes ec each oe beig recorded a a specific ime are ermed Time Series. These se of variables i heir saioariy form ca be modelled usig ARMA model espoused i Box Jekis &Reisel, (1994).ARMA model of order (p,q) ca be viewed as liear filer from of digial sigal processig perspecive,( Fagge ad Peixia,(211). Cosider he liear combiaio of he lagged variables i he equaio (1) give below p y, 1 (1) j j j j j j where E( ) ; { } q E s 2, > s IJMER ISSN: Vol. 4 Iss. 5 May
2 are shocks or iovaio series ad ha E( y ) E( y ) for all ad s s E( y y ) E( y y ) for all ad s j s sj Grager Causaliy Tes: A Useful Descripive Tool for Time Series Daa The leer E is use o deoe he expecaio operaor. The ARMA (p,q) i equaio (1) is a combiaio of he auoregressive srucure of he residues (movig average MA) ad a liear relaioship bewee he value prediced by he model a ime ad he pas values of he ime series(auoregressive AR). Cosrucio of ARMA model requires four ieraive seps which are made explici i Box, Jekis, & Reisel,(1994); Hamilo,(1994); Salau, (23); Oguade,(21); Oguade ad Ogufidiimi (213) ec. 2.2 Rus Tes for Saioariy ARMA models ca be used direcly if he ime series is wide sese saioary, (Fagge & Peixia, 211).The saioary assumpio of ay give ime series eed be checked for proper ideificaio ad esimaio of he model. This is achieved by Rus Tes for radomess. The saioariy assumpio is rejeced if Asymp. Sig.5.Oherwise ime series should be differeced or rasformed usig various ime series rasformaio echiques uil saioariy is achieved. 2.3 Grager Causaliy Tes If variable coais useful iformaio for predicig variable, he causes.tha is is Grager / Predicive causaliy of.the causes is deoed as ( ) ad causes is deoed as ( ) where he arrow pois o he direcio of Causaliy. The Grager Causaliy Tes assumes ha he iformaio releva o he predicio of variables ad is coaied i he ime series daa o hese variables. The bilaeral causaliy are relaed by he followig regressio models i i j ij 1 i1 j1 (1) i i j ij 2 i1 j1 (2) Where 1 ad 2 are orhogoal disurbaces 1 From he regressio equaios, we es he hypohesis H ha lagged erms do o belog i he regressio ( j ). Tha is do o cause usig F saisic give below wih m ad ( k) degrees of j1 freedom ( RSSR RSSUR ) / m F RSS /( k) Also, UR 2 H : i versus i1 H 2 : 1 i1 i is also esed for he model above Noe: rejecio of H a cerai criical value of F implies oe variable Grager causes aoher See Gujarai (24) ad Cochrae (25) 2.4 VAR as a ime series Ecoomeric Model Mulivariae causaliy amog depede or respose variables is made possible hrough Vecor Auoregressive echique.causaliy, Coiegraio ad VAR as ime Series Ecoomerics erms are powerful ools i esimaio ad predicio of Vecor Auoregressive Movig Average Models which akes is roo from he popular Box Jekis mehodology. See for example Brockwell ad Davis, (199); Gujarai, (24); Cochrae, (25) (3) IJMER ISSN: Vol. 4 Iss. 5 May
3 Cosider he VAR give i (1) o Grager causes if Tha is if VAR i (1) is equivale o j Grager Causaliy Tes: A Useful Descripive Tool for Time Series Daa Grager causes i i i j 1 i1 j1 if helps o forecas, give pas. Bu, (3) i i j ij 2 i1 j1 Equaio (1) i marix oaios becomes a b i 1 c d j 2 I La Lb 1 Lc Ld 2 a ( L) b ( L) 1 c ( L) d ( L) 2 does Iverig he auoregressive represeaio, we have (5) 1 d ( L) b ( L) 1 (5) a ( L) d ( L) b ( L) c ( L) c ( L) a ( L) 2 Equaio (5) is movig average represeaio called Bivariae Wold represeaio. From (5) does o Grager cause Iff b * ( L), or if he auoregressive marix lag polyomial is lower riagle. Thus, does o Grager cause if ad oly if he Wold movig average marix lag polyomial is lower riagular. Tha is does o Grager cause if ad oly if s bivariae Wold represeaio is he same as is uivariae Wold represeaio. The projecio of o pas ad is he same as projecio of o he pas, ad ha is a fucio of is shocks oly ad does o respod o shocks while is a fucio of boh shocks ad shocks. Deails of hese ad ohers are available i Cochra, (25). III. Numerical Example 3.1 Daa Descripio The daa for his research paper were exraced from he bullei published by he Ceral Bak of Nigeria (CBN) (as show i figure 1).The daa o GDP ad Moey supply were used i he predicig mulivariae model. The prelimiary rasformaios were o lef ou as reds were discovered i he Figure 1 (4) Fig1:Time Plo of Nigeria GDP ad Moey Supply IJMER ISSN: Vol. 4 Iss. 5 May
4 3.2 Empirical Resuls Table 3.1 Null Ui Roo Tes for GDP Hypohesis: D(GDP,2) has a ui roo Exogeous: Cosa Lag Legh: 7 (Auomaic - based o SIC, maxlag=12) Grager Causaliy Tes: A Useful Descripive Tool for Time Series Daa -Saisic Prob.* Augmeed Dickey-Fuller es saisic Tes criical values: 1% level % level % level *MacKio (1996) oe-sided p-values. Table 3.2 Null Ui Roo Tes for MS Null Hypohesis: D(moey supply,2) has a ui roo Exogeous: Cosa Lag Legh: 12 (Auomaic - based o SIC, maxlag=12) -Saisic Prob.* Augmeed Dickey-Fuller es saisic Tes criical values: 1% level % level % level *MacKio (1996) oe-sided p-values. Table 3.3 Grager Causaliy Tess Pairwise Grager Causaliy Tess Dae: 1/17/13 Time: 5:4 Sample: 1981Q1 21Q4 Lags: 2 Null Hypohesis: Obs F-Saisic Prob. GDP does o Grager Cause Moey supply E-16 moey supply does o Grager Cause GDP E-15 Table 3.4: Oucome of Ui Roo Tess ADF TEST F P-VALUE DECISION GDP SIGNIFICANT AT 5% MS SIGNIFICANT AT 5% 3.2.1: Resul of Saioariy Tes Boh GDP ad moey supply are o saioary i heir level form bu he desired level of saioariy was achieved afer secod differece wih sigifica ADF values of ad i absolue value respecively. We rejec he Null hypohesis of presece of ui roos i boh cases a 5% as he P values i able 3.1 above are sigifica. Table 3.5: Resuls of Causaliy Tes DIRECTION OF CAUSALIT F DECISION MSGDP REJECT GDPMS REJECT IJMER ISSN: Vol. 4 Iss. 5 May
5 Grager Causaliy Tes: A Useful Descripive Tool for Time Series Daa 3.2.2: Discussio of Resuls The resul suggess ha he direcio of causaliy is bi-direcio i aure, sice he esimaed f values are sigifica a 5% level of sigifica; he criical F values are ad respecively as elicied i Table 3.2 The grager causaliy es uder he ull hypoheses (H o ) GDP does o grager cause moey supply ad vice-versa are saisically sigifica, which implies ha here is bilaeral/feedback causaliy bewee GDP ad moey supply. The sigifica value of he es i he able above Implies he se of coefficies of GDP ad Moey Supply are saisically ad sigificaly differe from zero i boh regressios. Chages i eiher GDP or moey supply causes chages i he oher variable ad hece chages i he ecoomic growh ad developme of Nigeria. Ivesig more moey io he ecoomy leads o icrease GDP. REFERENCES [1.] Box, G.E.P., Jekis, G.N. ad Reisel G.C. (1994): Time Series Aalysis: Forecasig ad Corol, 3h Ediio, Preice-Hall,New Jersey. [2.] Brockwell, P ad Davis R.A. (199); Time Series: Theory ad Mehods. New ork: Spriger-Verlag. [3.] Cochrae, J.H. (25) Time series for Macroecoomics ad Fiace [4.] Chicago: Sprig pub. [5.] Fagge, L. &Peixia, L. (211) AR MA Model for Predicig he Number of Oubreaks of Newcasle diseases Durig he Moh i IEEE [6.] Galo F. (1886) Family Likeess i Saure i Proceedigs of Royal Sociey, Lodo.Vol4:42-72 [7.] Gujarai, D.N. (24): Basic Ecoomerics, 4 h Ed. New ork: Taa Graw Hill Publishig Co. Ld. [8.] Kedall, M.G ad Suar, A. (1961): The Advaced Theory of Saisics. New ork: Charles Griffi Publishers. [9.] Hamilo, J.D. (1994): Time Series Aalysis. Priceo: Priceo Uiversiy Press. [1.] Oguade, E.S. (21) Time Series Modellig of Malaria Pademic i Nigeria i Global Joural of Mahemaics ad Saisics. Vol.2 (1): [11.] Oguade, E.S. ad Ogufidiimi, F O (213): O Recursive ad Numerical Forecass: A Ieresig Characerizaio of he Behaviour of Time Series i Ieraioal Joural of Numerical Mahemaics. Vol.7 (2): [12.] Pearso K ad Lee A. (193) O he Laws of Iheriace i Biomerika.Vol2: IJMER ISSN: Vol. 4 Iss. 5 May
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