An Overview of Methods for Testing Short- and Long-Run Equilibrium with Time Series Data: Cointegration and Error Correction Mechanism

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1 ISSN (prin) Medierranean Journal of Social Sciences Published by MCSER-CEMAS-Sapienza Universiy of Rome An Overview of Mehods for Tesing Shor- and Long-Run Equilibrium wih Time Series Daa: Coinegraion and Error Correcion Mechanism Doi: 0.590/mjss.203.v4n4p5 Absrac Ferdinand Niyimbanira Lecurer in Economics Faculy of Managemen Sciences Vaal Universiy of Technology Privae Bag X02, Vanderbijlpark, Tel: Time series daa are usually wih a naural emporal ordering. This makes ime series analysis disinc from oher common daa analysis problems, in which here is no naural ordering of he observaions. This paper is abou he mehods used in economics when analyzing ime series daa. I shows he seps of esing for saionariy where uni roo ess in which he Dickey-Fuller and Augmened Dickey-Fuller ess are discussed. For coinegraion, Durbin-Wason, Engle-Granger and Augmened Engle-Granger ess are presened sep by sep. This paper does no end wih he analysis of coinegraion ess only, bu i proceeds o error correcion mehods which is usually used o make adjusmens in a dependen variable which depends no on he level of some explanaory variable, bu o he exen o which an explanaory variable deviaes from an equilibrium relaionship wih he dependen variable and half-life formula is inroduced o show how long i may ake for readjusmen o equilibrium. Wih all discussions, policy implicaions and suggesions for fuure research are made in he paper. Key Words: Coinegraion, Error Correcion Mechanism, Time Series and Uni Roo. Inroducion The purpose of his paper is o presen he mehodology used in economics when one is conducing a research wih ime series daa. Researchers can analyse ime series daa erroneously by formulaing a radiional regression model o represen he behaviour of daa bu no pay oo much aenion o he specificaion of he dynamic srucure of he ime series. One also needs o worry abou simulaneiy bias and auocorrelaed errors. Time series daa are assumed by economericians o be non-saionary (Kennedy, 996). In oher words, ime series daa do no have a fixed saionary mean. Therefore, running a regression on non-saionary daa may give misleading values of R 2, DW and saisics; and his leads o he incorrec conclusion ha a meaningful relaionship exiss among he regression variables, when i does no (Kennedy, 996, p.263). To solve his problem of spurious resuls, one uses he mehod of coinegraion. Indeed, he mehod of coinegraion is used o esimae he long-run equilibrium, while he error correcion mechanism (ECM) should be applied o esimae he shor-run dynamics of he model. Afer collecing daa, specifying research model (mahemaical and economeric model) one needs o proceed o he hypohesis esing. According o Hawkins and Weber (980, p.45), a ime series is a sequence of observaions aken on some process ha varies over ime. This ype of daa poses many challenges o researchers, especially economericians. One may ask why? The key problem is beween daa being saionary and non-saionary. Mos empirical work based on ime series daa assumes ha he underlying ime series daa is saionary, or is mean and variance do no flucuae sysemaically over ime (Gujarai, 2003, p.26). However, i is known ha many macroeconomic ime series daa are nonsaionary (Hill, Griffihs and Judge, 200). Mos economic ime series are generally inegraed of order one I() and become saionary only afer aking firs differences. One may ask wha he problem wih non-saionary daa is. The answer is ha when ime series are used in a regression model he resuls may spuriously indicae a significan relaionship when here is none (Hill e al., 200, p.340). To check for saionariy, here are differen ess one can use. 5

2 ISSN (prin) Medierranean Journal of Social Sciences Published by MCSER-CEMAS-Sapienza Universiy of Rome 2. Checking for Saionariy I is advisable, as a firs sep, o plo he daa under sudy before he researcher pursues formal ess o check variables for saionariy. This gives an iniial idea abou he likely naure of he ime series (Gujarai 2003, p.807). I is preferable o use naural logarihms when ploing regression variables in order o show heir growh. However, i should be noed ha looking a ime series plos alone is no enough o ell wheher a series is saionary or non-saionary. The auocorrelaion funcion a lag k, denoed by k, is defined as he covariance a lag k divided by he variance. A Plo of k versus k is known as he sample or populaion correlogram (Gujarai, 2003, p.808). One denoes k as he lag lengh when compuing he sample auocorrelaion funcion. According o Shumway and Soffer (2000, p. 26), he auocorrelaion funcion has a sampling disribuion, under complee independence, which allows us o assess wheher he daa comes from a compleely random or whie series or wheher correlaions are saisically significan a some lags. Hence, if a ime series is saionary, he auocorrelaion coefficien a various lags will remain around zero and decline quickly, while in a non-saionary ime series he auocorrelaion coefficien sars a a high value and declines very slowly owards zero as he lag lenghens (Gujarai, 2003, p.80-8). The populaion correlogram is defined as follows: k 0 k [] = covariance a lag k / sample variance. If 0 will be equal o. However, only an esimaion of he sample auocorrelaion funcion k can be worked ou. According o Gujarai (2003), his requires one firs o compue he sample covariance a lag k, ŷ k ; and he sample variance, ŷ 0, and are represened by he following equaions: ^ yˆ yˆ k 0 ( y y)( y k y) n y y ( ) 2 n. [3] [2] Where n represens sample size and y sands for he sample mean. Therefore, he sample auocorrelaion funcion a lag k is: ˆ ˆ y k k yˆ0. [4] The above saemen should be for he correlogram of some of he variables may reveal hemselves o be nonsaionary, while ohers show a subjecive probabiliy ha he daa series may be saionary. Uni Roo Tes When discussing saionary and non-saionary ime series, an alernaive es, which has recenly become popular, is known as he uni roo es. This es is imporan as i helps o avoid he problem of spurious regression. In defense of his poin, Harris (995, p.27) wries ha if a variable conains a uni roo hen i is non-saionary and unless i combines wih oher non-saionary series o form a saionary coinegraion relaionship, hen regression involving he series can falsely imply he exisence of a meaningful economic relaionship. Tesing for he presence of uni roos is no sraighforward, bu he easies way o inroduce his idea is o consider he following equaion: 52

3 ISSN (prin) Medierranean Journal of Social Sciences Published by MCSER-CEMAS-Sapienza Universiy of Rome y y u. [5] And u in he equaion is he sochasic error erm or whie noise error erm. There are several ways of esing for he presence of uni roo. This sudy uses he Dickey-Fuller (DF) and Augmened Dickey-Fuller (ADF) ess for esing he null hypohesis ha a series does conain a uni roo or is non-saionary. Boh he DF and ADF approaches are developed from equaion 5. One needs o consider hese ess in more deail, by developing 5. Dickey-Fuller Tes (DF) If a series is differenced d imes, for example, before i becomes saionary, hen i is said o be inegraed of order d, and is denoed I(d). If a series Y is I(d) and Y is non-saionary bu is saionary where = y y- (Cuhberson, e al., 995, p.30), an appropriae es for saionariy has been suggesed by Dickey and Fuller (Hill, e al., 200, p.344). According o Gujarai (2003, p.84), he following equaions can be used for such a es: y y y y u y ( ) y y u u. [6] Where - and -difference operaor. In his scenario one is esing he null hypohesis if, ha is, here is a uni roo. In oher words, he ime series under consideraion is non-saionary if he null hypohesis is rue. Moreover, if uni roo is ignored equaion 6 is esimaed, hen i can be shown ha he disribuion of he ordinary leas square s (OLS) esimae of is no cenred a and he corresponding saisic does no have a suden s disribuion and herefore he usual es for does no apply (Ramanahan, 995, p.553). Insead of a es, hree forms of he (au) es are used (Gujarai, 2003, p.85). Pracically, he Dickey-Fuller es is applied o regressions using equaions 7 o 9 as follows: y y u ; [7] y y u; [8] y 2 y u. [9] Where is he ime or rend variable. The difference beween 7 and he oher wo equaions lies in he inclusion of he ) and a rend erm. Equaion 7 is he formula for a random walk, 8 he random walk wih drif, while 9 represens he random walk wih drif around a sochasic rend. If he c he DF criical value he null hypohesis is no rejeced. Thus he ime series is non-saionary. Gujarai (995, p.79) sresses ue relaive o he criical value is generally an indicaion of saionariy. Thus one does no fail o rejec he null hypohesis of non-saionary in his insance. Augmened Dickey-Fuller Tes (ADF) This es suggess ha he au saisic mus ake larger negaive values han usual in order for he null hypohesis 0), a uni roo or non-saionary process o be rejeced in favour of he alernaive ha is, which indicaes a saionary process. To preclude he possibiliy ha he error erm in one of he above equaions (under DF), are 53

4 ISSN (prin) Medierranean Journal of Social Sciences Published by MCSER-CEMAS-Sapienza Universiy of Rome auocorrelaed, some addiional erms are included. These addiional erms are usually he lagged values of he dependen variables (Hill e al., 200, p.344). An imporan assumpion of he DF es, according o Gujarai (2003, p.88), is ha he error erms are independenly and idenically disribued, while he ADF es adjuss he DF es o ake care of possible serial correlaion in he error erms by adding he lagged and differenced erms of he regressand. If he error erm is found o be auocorrelaed under he Dickey-Fuller es, he Augmened Dickey-Fuller es (ADF), which is a es ha includes addiional lagged erms, is used. In his case, he ADF equaion is: y y 2 m i i y. [0] Where y ( y y2), y ( y2 y3) and for he lag lengh. The ADF es is comparable o he simple DF es, bu he sligh difference is ha he firs involves adding an unknown number of lagged firs differences of he dependen variable o capure auocorrelaion in omied variables ha would oherwise ener he error erm. However, as emphasized by Harris (995, p.34), i is also very imporan o selec he appropriae lag lengh; oo few lags may resul in over-rejecing he null hypohesis when i is rue while oo many lags may reduce he power of he es. One should make sure ha he sample size is enough wih a high probabiliy of obaining accurae resuls. This concurs wih Keller and Warrack (2003) and Mann (2004), who confirmed ha he resuls from a sample size equal o or greaer han, 30 make more sense han he ones from a small sample size (< 30). 3.Tesing for Coinegraion The heory of coinegraion was developed in he 980s and 90s by several researchers such as Engle and Granger (987), Johansen (988) and Engle and Yoo (987) and ohers. Similarly, Robinson and Marinucci (2003, p.334) reconfirm ha coinegraion analysis has been developed as a major heme of ime series economerics and generaed much applied ineres, promping considerable mehodological and heoreical developmens during he 990s. Therefore he coinegraion mehod has become a useful economeric ool (Johansen and Juselius, 990, p.92). According o Harris (995, p.22), if a series mus be differenced d imes before i becomes saionary, hen i conains d uni roos and is said o be inegraed of order d, denoed I(d). Bu he quesion o be asked is why are observed ime series inegraed? Granger and Newbold (974, p.5) reply:...variables are inegraed eiher because hey are driven by oher inegraed variables, or because he dynamic processes generaing hem conain auoregressive roos of uniy; in oher words, uni roos may be found in eiher he marginal or condiional subsysems, or, of course, boh. In he case where residuals are expressed, as a linear combinaion of he variables which are all I(), his linear combinaion will iself be I(), bu i would be desirable o obain residuals ha are I(0). This can only be achieved if he variables are coinegraed (Brook, 2002). Engle-Granger and Augmened Engle-Granger Tes The Engle-Granger es is one of he mehods ha are used when he daa available are hough o be non-saionary and possibly coinegraed. As a rule, non-saionary ime series should no be used in regression models, o avoid he problem of spurious regression (Hill e al., 200, p.346). If ime series daa are I() or non-saionary, hen we esimae he coinegraing regression using ordinary leas squares. However, i is no possible o perform any inferences on he coefficien esimaes from he usual regression. One can only esimae he parameer values afer making sure ha he residuals of he coinegraing regression are I(0), and if so hen one can proceed o he nex sep, which is he error correcion mechanism (ECM). If he residuals are I(), one canno use he esimaed sandard errors and he associaed values of he esimaed coefficiens (Gujarai, 995, p.727), bu a model conaining only firs differences should be esimaed (Brooks, 2002). The differen orders of inegraion imply a hidden assumpion of he error erm being nonsaionary. An imporan poin wih his esing mehod is ha if wo individual I() variables are co-inegraed, when a linear combinaion of boh variables is I(0), hen heir enry ino he esimaing equaion will no creae spurious resuls (Kennedy, 998, p.228). To avoid he problem of a meaningless regression, he Engle-Granger es is used. From his model he residuals are esimaed and a uni roo es is uilized o find ou wheher variables co-inegrae. This deermines wheher or no here is a long-run relaionship beween hem. If his es does no give a saisfacory resul, he Augmened Engle-Granger (AEG) es is used. However, he difference beween Engle-Granger and AEG is o run a 54

5 ISSN (prin) Medierranean Journal of Social Sciences Published by MCSER-CEMAS-Sapienza Universiy of Rome coinegraion regression, by esimaing he Augmened Dickey-Fuller regression, bu wih he AEG he lagged values of he residuals are applied (Gujarai, 995). The lesson o be reained from using he Engle-Granger es is ha one mus be aware of he fac ha i does no prove ha here is really a long-run relaionship. According o Charemza and Deadman (993, p.57), a srong belief in a long-run equilibrium relaionship beween he variables mus be suppored by relevan economic heory. Coinegraion Regression Durbin-Wason (CRDW) An alernaive, easy and a quicker mehod of finding ou wheher dependen and independen variables are coinegraed is he Durbin-Wason es, whose criical values are firs inroduced by Sagan and Bhargava (983). Charemza and Deadman (993, p.53) poin ou ha he disribuion of he coinegraion regression Durbin-Wason es is no fully invesigaed and is criical values are no known. Based on simulaions formed from 00 observaions each, Gujarai (995, p.728) noes ha he, 5 and 0 per cen criical values of d (no DW) o es he null hypohesis ha d = 0 are 0.5, and 0.322, respecively. Therefore he alernaive hypohesis of coinegraion will be rejeced if he compued d value is smaller han, say, a he 5 per cen level and if i is greaer han he criical value, he null hypohesis is acceped, which means ha he variables are coinegraed. I should be remembered ha he power of a coinegraion regression es depends posiively on he goodness of fi of he ordinary leas squares esimae of he long-run relaionship of he specified model. From his, Banerjee e al. (986) propose a simple rule of humb for a quicker evaluaion of he coinegraion hypohesis: ha if compued d value for he residuals is smaller han he coefficien of deerminaion (R 2 ) he apparen significance of a saisic relaionship is likely o be false. This is an indicaion ha he model has a problem of auocorrelaion. If he Durbin-Wason value is above R 2, here is a higher probabiliy ha coinegraion needs invesigaion. 4.Error Correcion or Equilibrium Correcion Mechanism (ECM) The error correcion model was iniially used by Sargan (984), Hendry and Anderson (977) and Davidson e al. (978) o make adjusmens in a dependen variable which depends no on he level of some explanaory variable, bu o he exen o which an explanaory variable deviaes from an equilibrium relaionship wih he dependen variable. In oher words, if here is coinegraion beween variables and here is a possibiliy ha in he shor-run here may be disequilibrium one uses his model. Therefore, o correc his disequilibrium, an error correcion mechanism hopefully pushes he model back owards he long-run equilibrium (Engle and Granger, 987: 25). The error correcion model hus plays an imporan role, in ha i is a force ha pulls he error back oward zero as should be he case when moving back owards equilibrium. The error correcion model is simply a linear ransformaion of he auoregressive-disribued lag model. One may ask wha is disinguishing feaure is. The difference in he error correcion modelling is ha parameers describe he exen of shor-run adjusmen o equilibrium are immediaely provided by he regression (Benerjee e al., 993, p.5). Therefore, in pracice, he error correcion erm, which is nohing more han he lagged residuals from he levels regression, Uˆ and is preferable o oher regression mehods. The error correcion model can be esimaed for more han ^ wo variables. During periods of disequilibrium, U is non-zero and measures he disance real money demand is away from equilibrium during ime. Thus an esimae of he coefficien of Uˆ will provide informaion on he speed of adjusmen back o equilibrium (Harris, 995, p.24). Is sric definiion is ha i measures he proporion of las period s equilibrium error ha is correced for (Brooks, 2002, p.39). A large coefficien of error erm close o negaive one implies a quick adjusmen, while a small value close o zero suggess ha an adjusmen o he long-run seady-sae is slow. This makes he equilibrium correcion model formulaion aracive, because i immediaely provides he parameer describing he rae of adjusmen from disequilibrium in he shor-run (Ericsson and Sharma, 996, p.26). The conclusion is made from he sign and value of coefficien of error erm. Wih he error coefficien, one can use half-life formula o indicae how long i may ake o re-adjus o equilibrium. This can be also checked by a Mone Carlo simulaion of same equaion wih 2000 repeaed random samples (Niyimbanira, 202) and i gives he same resuls wih Johansen Coinegraion es. 5. Conclusion This paper discusses problems encounered in ime series daa and how o overcome hem. I shows ha regression wih 55

6 ISSN (prin) Medierranean Journal of Social Sciences Published by MCSER-CEMAS-Sapienza Universiy of Rome non-saionary series is generally biased and inconsisen. In oher words, regressing one non-saionary series on one anoher is likely o yield spurious resuls. However, his paper explains how a coinegraion analysis allows one o conduc an economeric analysis using non-saionary variables. According o Harris (995, p.25), failure o esablish coinegraion ofen leads o spurious regressions which do no reflec long-run economic relaionships bu, raher, reflec he common rends conained in mos non-saionary ime series. This paper presens he seps which should be followed: checking for saionariy, esing for uni roos when esing for he order of inegraion of he residuals from he coinegraion regression, using he Dickey-Fuller (DF) es and he augmened Dickey-Fuller (ADF) es. In erms of coinegraion esing, his paper focuses on Engle-Granger and Augmened Engle-Granger ess for he long-run relaionship beween dependan variable and is explanaory variables. For he shor-run relaionship, he error correcion mechanism is used. Therefore here is a useful and meaningful link beween he long- and shor-run approaches o economeric modelling. Furher sudies could be conduced by examining usage of half-life formula in economics when one uses ime series daa. References Banerjee, A., Dolado, J., Hendry, D. and Smih, G. (986). Exploring Equilibrium Relaionships in Economerics hrough Saic Models: Some Mone Carlo Evidence. Oxford Bullein of Economics and Saisics, Vol. 48, No. 3, pp Banerjee, A., Dolado, J., Galbraih, J. and Hendry, D. F. (993). Coinegraion, Error-Correcion, and Economeric Analysis of Nonsaionary Daa: Advanced ex in Economerics. New York, Oxford Universiy Press. Brooks, C. (2002). Inroducory Economerics for Finance. Cambridge, Cambridge Universiy Press. Charemza, W. W. and Deadman, D. F. (993). New Direcion in Economeric Pracice: General o Specific Modelling, Coinegraion and Vecor Auoregression. Brookfield, Edward Elgar Publishing Limied. Cuhberson, K, Hall, S. G and Taylor, P. M. (995). Applied Ecomerics Techniques. Ann Arbor, The Universiy of Michigan Press. Engle, R. F. and Granger, C.W.J. (987). Co-inegraion and Error Correcions: Represenaion, Esimaion and Tesing. Economerica, 55(2), Engle, R. F. And Yoo, B. S. (987), Forecasing and Tesing in Coinegraed Sysems. Journal of Economerics, 35, Ericsson, N. R. and Sharma, S. (996). Broad Money Demand and Financial Liberalisaion in Greece. Inernaional Discussion Paper 559, Board of Governors of he Federal Reserve Sysem (U.S), -5 Granger, C. W. J. and Newbold, P. (974). Spurious Regressions in Economerics. Journal of Economerics, 2, -20. Gujarai, D. (995). Basic Economerics. 3rd Ediion. New York, McGraw Hill. Gujarai, D. (2003). Basic Economerics. 4 h Ediion. New York, McGraw Hill. Harris, R. (995). Using Coinegraion Analysis in Economeric Modelling. London, Prenice Hall/Harveser Wheasheaf. Hawkins, C. and Weber, J. (980). Saisical Analysis: Applicaion o Business and Economics. New York, Harper & Row. Hendry, D.F. and G.J. Anderson (977). Tesing dynamic specificaion in small simulaneous sysems: An applicaion o a model of Building Sociey behaviour in he Unied Kingdom. In M.D. Inriligaor (Ed.), Froniers in Quaniaive Economics, 3, Hill, R. C, Griffihs, W. E. and Judge, G. G. (200). Undergraduae Economerics. 2 nd Ediion. America, John Wiley & Sons, Inc. Johansen, S. (988). Saisical Analysis of Coinegraing Vecors. Journal of Economic Dynamics and Conrol, 2, Johansen, S. and Juselius, K. (990). Maximum Likelihood Esimaion and Inference on Coinegraion: wih Applicaion o hedemand for Money. Oxford Bullein of Economics and Saisics,52, Kennedy, P. (996). A Guide o Economerics. 3rd Ediion. Massachuses, Blackwell Publishers Inc. Kennedy, P. (998). A Guide o Economerics. 4 h Ediion. Massachuses, Blackwell Publishers Inc. Niyimbanira, F. (202). Real Demand for Money in Souh Africa: An Economeric Analysis. Saarbrücken, Lamber Academic Publishing. Ramanahan, R. (995). Inroducory Economerics: wih Applicaions. 3 rd Ediion. New York, The Dryden Press. Sargan, J. D. (984). Wages and Prices in he Unied Kingdom: A sudy in Economeric Mehodology. Originally Published in 964 and Reproduced in Wallis K.F and Hundry D. F eds., Quaniaive Economics and Economeric Analysis, Oxford, Basil Blackwell. Sargan, J. D. and Bhargava, A. S. (983). Tesing Residuals from Leas Squares Regression for Being Generaed by he Gaussian Random Walk. Economerica, 5, Shumway, R. and Soffer, D. S. (2000). Time Series Analysis and Is Applicaions. Springer Texs in Saisics. New York, Springer- Verlag New York, Inc. 56