# Cointegration: The Engle and Granger approach

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1 Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be non-saionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require he exisence of a combinaion of he variables o be saionary. Oherwise, any deviaion from equilibrium will no be emporary. Consider he equilibrium demand for money equaion (all variables excep he ineres rae are in logs): m - p = y r m money demand; p price level; y real income; r ineres rae. Economic heory requires and <0. We wan he deviaion o be saionary. This will ensure ha he variables do no wander off from he equilibrium pah. So we need rules concerning linear combinaion of inegraed series. a) x I(0) a + b x I(0) x I(1) a + b x I(1) b) x I(0) and y I(0) a x + b y I(0) c) x I(0) and y I(1) a x + b y I(1) i.e. I(1) is a dominan propery. d) Generally if x I(1) and y I(1) hen ax + by I(1). Bu under cerain condiions he linear combinaion may be I(0). We hen say ha x and y are coinegraed. I.e. we have a saionary equilibrium relaionship. i.e. z = a x + by I(0) ; e) Adding or subracing a consan from a coinegraing equaion does no aler is properies. 1

2 Definiion The componens of a (k1) vecor, y, are said o be coinegraed of order d, b, denoed, y ~CI(d, b), if (i) all he componens of he vecor y are I(d), ha is, hey need d differences o induce saionariy, and (ii) here exiss a vecor ( 0) z ' y ~ I ( d b). The vecor is called he coinegraing vecor. Usually we consider he case wih d=b=1. so ha This is an imporan resul as any arbirary linear combinaion of I(1) series will be I(1) (unless he series are coinegraed). Coinegraing combinaions are equilibria. So i is imporan o be able o discover and model hese relaionships. An alernaive approach o he analysis of long-run (equilibrium) relaionship would be o analyse he relaionships beween he differences of he series, i.e. among I(0) series. However, his approach is only concerned wih shor-run movemens, while i hrows useful long-run informaion. 2

3 Spurious versus coinegraing relaionships The spurious regression problem Compleely unrelaed ime series may appear o be relaed using convenional esing procedures. Suppose y =y -1 +v where v iid N(0, 2 v) (1) and x =x -1 + where iid N(0, 2 ) (2) wih v and independen, i.e. E(v s )=0 for all and s. When =1, y and x are random walks: y =y -1 +v (3) x =x -1 + If we run a regression beween y and x : y =x +u (4) despie lack of causal relaionship, we are likely o find a significan -raio for he null H 0 :=0. A simulaion sudy would show ha Pr(rejec H 0 using 5% level es) >75%. The problem is ha he -es of =0 is no N(0,1) even asympoically. The sandard asympoic disribuion heory does no apply when variables have uni roo. So, i is imporan o discriminae beween wo siuaions: 1. Spurious regressions. Apparenly significan relaionship beween unrelaed series. 2. Genuine relaionships which arise when he ime series are coinegraed. 3

4 Coinegraing regressions and Granger represenaion heorem Here we are primarily concerned wih esing for coinegraion in a sysem of k=2, I(1) variables, in which case here is a mos r=1 coinegraing relaionship. Granger, 1983, Co-inegraed Variables and Error-Correcing Models, Unpublished discussion paper, 83-13, Universiy of California, San Diego. Engle and Granger, 1987, Coinegraion and Error Correcion: Represenaion, Esimaion and Tesing, Economerica, If a se of variables are coinegraed, hen here exiss a valid error correcion represenaion of he daa, and viceversa. If y and x are boh I(1) and have a long run relaionship, here mus be some force which pulls he equilibrium error back o zero. Engle and Granger (1987, Economerica) recommend a wo-sep procedure for coinegraion analysis. (i) Esimae he long-run (equilibrium) equaion: y 0 1x u (5) The OLS residuals from (5) are a measure of disequilibrium: û y ˆ ˆ 0 1x A es of coinegraion is a es of wheher uˆ is saionary. This is deermined by ADF ess on he residuals, wih he MacKinnon (1991) criical values adjused for he number of variables (which MacKinnon denoes as n). If coinegraion holds, he OLS esimaor of (5) is said o be super-consisen. Implicaions: as T (i) here is no need o include I(0) variables in he coinegraing equaion. NOTE: The -raios from equaion (5) are no inerpreable, as i is a long-run equaion, and herefore will have serial correlaion (due o misspecified dynamics) as well as omied variable problems, and as such he disribuion of he -raio is no known. 4

5 The radiional diagnosic ess from (5) are unimporan as he only imporan quesion is he saionariy or oherwise of he residuals. (ii) Second sep: esimae he Error Correcion Model y y x uˆ 0 j j h h 1 j1 h0 by OLS as his equaion has only I(0) variables, sandard hypohesis esing using - raios and diagnosic esing of he error erm is appropriae. The adjusmen coefficien mus be negaive. Special case: y y x (y ˆ ˆ x ) ECM describes how y and x behave in he shor run consisen wih a long run coinegraing relaionship. 5

6 Dynamic approach o ECM and coinegraion The esimaes from OLS in he saic equaion (equaion 5), alhough consisen, can be subsanially biased in small samples, parly due o serial correlaion in he residuals. The bias can be reduced by allowing for some dynamics. In sage (i) we can esimae, wih OLS, an ADL model : y = + 0 x + 1 x -1 + y -1 + (6) and solve for he long run equaion The residuals from (6) y x u uˆ y x are a measure of disequilibrium and a es of coinegraion is a es of wheher uˆ is saionary. As an alernaive o he wo-sep Engle and Granger procedure, he ECM model can be esimaed using he residuals from (6). If coinegraion holds, he OLS esimaor of (6) are super-consisen. 6

7 Balanced regressions Consider again, as an example, he equilibrium demand for money equaion (all variables excep he ineres rae are in logs): m = y r where his ime m real money demand; y real income; r ineres rae. If m is I(1), wha are he necessary condiions for a balanced regression? (i) (ii) (iii) eiher or boh y and r are I(1); if jus y or r is I(1) hen he oher mus be I(0); a balance could also be achieved, in principle, if y and r were of he same bu higher order of inegraion, for example I(2), bu hey coinegraed on a pairwise basis o be I(1) (Paerson, p. 441). 7

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