Testing for Cointegration in Misspecified Systems A Monte Carlo Study of Size Distortions

Transcription

1 Tesing for Coinegraion in Misspecified Sysems A Mone Carlo Sudy of Size Disorions Pär Öserholm * Augus 2003 Absrac When dealing wih ime series ha are inegraed of order one, he concep of coinegraion becomes crucial for he specificaion of a model. Using he bes available ess, one can reduce he probabiliy of esimaing economeric models ha are misspecified. This paper invesigaes he small sample performance of four well-known coinegraion ess when a sysem has been misspecified by leaving ou one relevan explanaory variable from a sysem wih one coinegraing vecor. In a Mone Carlo sudy, he size disorions of he Augmened Engle-Granger (Engle and Granger, 1987), Johansen s (1988) imum eigenvalue, Johansen s (1991) and he Boswijk (1989) Wald ess are examined. The Johansen es adjused by he finie sample correcion of Reinsel and Ahn (1988) is found o have he mos robus performance when lag lengh in he es equaions is chosen according o radiional informaion crieria. JEL Classificaion: C12, C15 Keywords: Coinegraion, Tess, Mone Carlo I am graeful o Michael Jansson, Per Jansson and Rolf Larsson and seminar paricipans a Uppsala Universiy for valuable commens on his paper. Financial suppor from Sparbankernas Forskningssifelse and Jan Wallander s and Tom Hedelius foundaion is graefully acknowledged. * Deparmen of Economics, Uppsala Universiy, Box 513, S Uppsala, par.oserholm@nek.uu.se

2 1. Inroducion In empirical work i is no uncommon ha he esimaed models are inspired by, raher han derived from, heory. This may generae several problems since here may be a number of conflicing heories in he area so ha exacly which variables o include in he esimaion could be less han obvious. Needless o say, using his approach, e.g. in aheoreical vecor auoregressions (VARs), i is possible ha we could end up wih a misspecified model. The omission of relevan explanaory variables is known o generae a number of problems. For example, i may be he reason why we ge unexpeced resuls such as he price puzzle when sudying moneary policy in VARs. The price puzzle is he common observaion ha in a VAR of oupu, prices, ineres raes, money and possibly some oher variables, conracionary shocks of moneary policy lead o persisen price increases. Sims (1992) argues his could be due o a misspecificaion of he model, such as omiing a leading indicaor used by he cenral bank. When dealing wih variables ha are inegraed of order one, I(1), we have more concerns han usual abou omied variables since he choice of variables o include in a sysem will affec possible coinegraing relaions. Since he inroducion of he erm coinegraion by Granger (1981), and furher developmen by Engle and Granger (1987), several ways of esing for he presence of coinegraion have been proposed. If, afer applying a es, we reach he conclusion ha here is no coinegraion, we say ha he variables have no long run equilibrium relaionship. No coinegraion migh lead us o esimae a VAR in differences. However, if he variables are coinegraed, misakenly esimaing he VAR in differences means no jus hrowing away informaion i is misspecified. 1 Conversely, if we ac as if we have coinegraion when here is none, he model will also suffer from misspecificaion and we have o consider unpleasan problems such as spurious regression. 2 Hence, geing he properies of he sysem righ is an imporan maer in order o ge esimaion and inference as correc as we possibly can. 1 See for insance Engle and Granger (1987). 2 See for insance Phillips (1986). 2

3 The purpose of his paper is o invesigae he behaviour of four differen coinegraion ess in small samples when a relevan explanaory variable has been omied from a sysem. The quesion o be answered is how likely i is ha he differen coinegraion ess he Augmened Engle-Granger (Engle and Granger, 1987), Johansen s (1988) imum eigenvalue, Johansen s (1991) and he Boswijk (1989) Wald ess reach he correc conclusion of no coinegraion when we have a sysem where he variables are relaed bu he model has been misspecified. There is a fairly exensive lieraure on he subjec of coinegraion ess and heir behaviour regarding size and power under differen circumsances, such as Banerjee e al (1986), Haug (1996), Bewley and Yang (1998) and Pesaveno (2000). This sudy furher clarifies he problems and advanages of well-known coinegraion ess by invesigaing heir size disorions in a new and empirically reasonable siuaion. The focus on he four included ess is based on he fac ha he Augmened Engle-Granger es and Johansen s ess are by far he mos frequenly used ess in empirical macroeconomics whils he very good resuls of he Boswijk Wald es in a number of Mone Carlo sudies recommend i. Oher ess ha have been shown o have good properies in some aspecs, such as he ess proposed by Sock and Wason (1988), Hansen (1990), Bewley and Yang (1995) are no addressed. Though i would be ineresing o know he properies of hose ess, i is beyond he scope of his paper. The paper is organised as follows. Secion wo describes he coinegraion ess o be considered and heir properies in some previous sudies. In secion hree a Mone Carlo experimen is performed and he resuls are discussed. Secion four empirically applies he ess considered in he paper o real macro daa and, finally, secion five concludes. 2. Tesing for coinegraion 2.1 Four coinegraion ess Among he four ess o be considered in his sudy, we will firs look a he Augmened Engle-Granger (AEG) es. Iniially a saic OLS regression of he form in equaion (1) is run. 3

4 y = a + β x + υ (1) The residuals from his regression are hen esed for he presence of a uni roo using an Augmened Dickey-Fuller es (Said and Dickey, 1984), as shown in regression (2) and he es saisic is simply he -saisic on ρˆ as given in (3). f + γ i ˆ 1 υ i i= 1 ˆ υ = ρυˆ + (2) AEG = ˆ ρ ˆ (3) ˆ ρ If he es saisic, which follows a non-sandard disribuion, is small enough he null hypohesis of a uni roo is rejeced and we conclude ha we have found a coinegraing relaionship. Second, consider wo differen ess based on he mehodology developed by Johansen (1988). Consider a VAR of order p, as given by equaion (4). y + = A1 y 1 + L+ A py p ε (4) where y is a nx1 vecor of non-saionary I(1) variables and ε is a nx1 vecor of innovaions. We can rewrie he VAR as p Γ i y i + i= 1 y = Πy ε (5) where p Π = A i= 1 i I p and Γ = A (6) i j= i+ 1 j 4

5 If he coefficien marix Π has reduced rank r<n, hen here exis nxr marices α and β each wih rank r such ha Π = αβ and β y is saionary. r is he number of coinegraing relaionships, he elemens of α are known as he adjusmen parameers in he vecor error correcion model and each column of β is a coinegraing vecor. If Π has full rank all variables are saionary. I can be shown ha for given r, he imum likelihood esimaor of β defines he combinaion of y 1 ha yields he r larges canonical correlaions of wih y afer correcing for lagged differences and deerminisic variables when presen. 3 Johansen proposes wo differen likelihood raio ess o es he significance of hese canonical correlaions and hereby he reduced rank of he Π marix: he es and imum eigenvalue es. These are shown respecively in equaions (7) and (8). y 1 J = T n i= r+ 1 ( ln 1 ˆ λ ) (7) ( 1 ˆ ) = T ln r+ 1 i J λ (8) where is he i:h larges canonical correlaion. The saisic,, ess he null λˆi hypohesis of he number of coinegraing vecors being less han or equal o r agains he alernaive hypohesis of r+1 or more. The imum eigenvalue saisic,, ess he null hypohesis of he number of coinegraing vecors being less han or equal o r agains he alernaive hypohesis of r+1. If he es saisic is large enough we rejec he null for he alernaive. Noe ha neiher of he es saisics for hese likelihood raio ess follows chi square disribuions. Asympoic criical values can be found in Johansen and Juselius (1990) and are given by mos economeric sofware packages. Cheung and Lai (1993), however, show ha here is over-rejecion of he null hypohesis of no coinegraion when he es saisics are compared o he asympoic criical values. One way of dealing wih his problem is o apply he finie sample correcion proposed by Reinsel and Ahn (1988) in which he es saisic is adjused by a facor of ( T np) / T J J and hen compared o he asympoic criical values. Cheung and 3 For a deailed descripion of he procedure, see for insance Johansen (1995). 5

6 Lai (1993) find ha his mehod performs well, even if here appears o be some bias, and he mehod will be used for finie sample correcions in his paper. Finally, we will look a a Wald-ype es in an error correcion model, an approach suggesed by Boswijk (1989). This es is a mulivariae generalizaion of a model used in Banerjee e al (1986). Consider he error correcion model given in equaion (9) below. y m 0 x x j= 1 ( y θ x ) + ( ϕ j y j + δ j j = δ λ ) + ξ (9) This could equivalenly be wrien as equaion (10). y m ( ϕ j y j + δ j j ) + = δ ξ (10) 0 x + π z 1 + x j= 1 where z = ( y, x ) (11) π = λ ( 1, θ ) (12) If λ in equaion (9) is zero, we have no error correcion mechanism and no coinegraion. Looking a he definiion of π we can conclude ha λ = 0 implies π = 0, and hus we have a way of esing for coinegraion. The es is, as previously menioned, of Wald-ype and he es saisic is given by equaion (13). 1 [ ˆ( πˆ )] πˆ W = πˆ V (13) where is he OLS esimaor of and V ˆ πˆ is he esimaed OLS covariance marix. πˆ π ( ) If he null is rejeced we have found a coinegraing relaionship. The disribuion of he es saisic is, however, no chi squared bu raher a generalizaion of he squared Dickey-Fuller -saisic. 6

7 2.2 Previous sudies Previous simulaion sudies have invesigaed he power and size of hese ess for various daa generaing processes (DGPs). In an early sudy, Banerjee e al (1986) compared he Coinegraing regression Durbin-Wason es (Sargan and Bhargava, 1983) o a -es on he error correcion erm in a dynamic model. The laer es is, as previously menioned, he predecessor of he Boswijk Wald es. Using a simple DGP wih zero or one coinegraing vecor hey found ha he -es was more powerful han he Coinegraing regression Durbin-Wason es, bu slighly oversized a he five percen level. In mos of he laer simulaion sudies, he AEG es has been a frequen gues. Using he same DGP as Banerjee e al (1986), Kremers e al (1992) compared he AEG es o he above menioned -es and found he AEG es o be less powerful. Boswijk and Franses (1992) compared he AEG, he Boswijk Wald and he Johansen imum eigenvalue ess for wo differen DGPs. They found he Boswijk Wald es o ouperform he ohers in erms of size and power. Furhermore, he AEG es urned ou o perform badly, wih low power and large size disorions, for one of he DGPs an ARMA model wih explanaory variables. A comprehensive sudy where nine differen ess boh single equaion and sysem were compared, was conduced by Haug (1996). For a simple DGP wih zero or one coinegraing vecor, he Sock and Wason (1988) and Phillips and Ouliaris (1990) Pˆz ess were found o perform bes in erms of power when he regressors were endogenous. Wih exogenous regressors, he Phillips and Ouliaris (1990) Ẑ α es performed bes. In he sudy i was also found ha he AEG es and Hansen s (1992) L c es showed he overall leas size disorions. A general observaion in he sudy is ha single equaion ess have smaller size disorions, bu also have lower power han sysem based ess. The recommendaion by Haug based on he sudy is o use he Sock and Wason and AEG ess as a combinaion. Worh poining ou is ha he in empirical work less used Sock and Wason es was preferred over he widely used Johansen imum eigenvalue es and ha he Boswijk Wald es was no considered a all in he sudy. 7

8 Focusing on sysem ess only, Bewley and Yang (1998) compared he Sock and Wason es o he Johansen imum eigenvalue es and he Bewley and Yang (1995) es. For DGPs wih zero o wo coinegraing vecors, hey were unable o find any es ha dominaed over a wide range of parameers. In general, hough, he Sock and Wason es and Johansen imum eigenvalue es were more powerful han he Bewley and Yang es. However, size disorions were found o be severe for he Sock and Wason es in some cases. The sudies referred o so far have all been pure Mone Carlo sudies. Pesaveno (2000) on he oher hand compares, among ohers, he AEG es, he Johansen imum eigenvalue es and he Boswijk Wald es in a sudy invesigaing properies boh analyically and in large and small samples. Using a DGP wih one coinegraing vecor, he overall conclusion is ha he Boswijk Wald es performs beer han he oher ess in erm of power and no worse in erm of size disorions. Wih hese resuls from previous sudies in mind, we now urn o he Mone Carlo simulaions in his paper. 3. A Mone Carlo experimen Iniially, a sysem wih one coinegraing vecor is generaed. The Phillips (1991) riangular represenaion of he sysem is given in equaions (14) and (15) below. y = α x + (14) x = η (15) 2 where is a kx1 vecor, α = 1 L 1, ~ NID 0,, η ~ NID 0, Σ, x ( ) ( 1 L 1 ) Σ and ( ) = diag ηk ( ) z E η = 0 z. ( ) The nex sep is esimaion of equaions (1), (5) and (10) and performing he relaed ess. When hese equaions are esimaed, however, he model is inenionally incorrecly specified; from he x vecor, one variable,, is excluded. The exclusion x k, 8

9 of x k, urns he sysem ino one wihou coinegraion and accordingly we wan he coinegraion ess o reach his conclusion wih a probabiliy of one minus he chosen significance level. If we do no reach his conclusion, we will end up wih a misspecified model and esimaion of equaions such as (1) will be spurious regressions, jus like Granger and Newbold s (1974) regressions wih independen random walks in heir seminal aricle on he opic. Given he daa generaing process in equaions (14) and (15) which has one coinegraing vecor, we hroughou he sudy assume ha we are ineresed in finding ou wheher here is zero or one coinegraing vecor. The performance of he ess is evaluaed boh by seing he lag lengh in equaion (2) o f = ( ), in equaion (5) o p = ( ) and in equaion (10) o m = ( as well as deermining lag lengh based on he Akaike (1974) and Schwarz (1978) crieria. In he experimen he following parameers are also varied: he sysem size is se o ) k +1 = ( ) ; he sample size is se o = ( ) T ; he sandard deviaion of he error erm of he omied variable is se o = ( ), as is he sandard deviaion of he error erm of he dependen variable,. For each combinaion of parameers, replicaions are performed. The Malab programming language is used for simulaion and he rouine NDN generaes pseudo-random normal innovaions. 3.1 Resuls in brief Resuls from he simulaions are given below, where Table 1 and 2 summarise he overall behaviour of he ess when he lag lengh of he es has been deermined using he Akaike and Schwarz crieria. Table 1 gives he average size over he parameer space considered for each sysem size. This should be compared o he nominal size of five percen and gives us an idea of he ess general size properies. However, since averages are used, we could be concerned abou ouliers ruining he resuls for ess ha perform well in general bu badly in a few cases. Therefore, Table 2 presens how ofen he ess reach an accepable size of five plus/minus wo and a half percen over he same parameer space and sysem size. Table 3 presens rejecion frequencies for he case when T = 200 and = 1. The repored values are rejecion frequencies of he null 9

10 hypohesis and should, jus like he values in Table 1, be compared o he nominal size of five percen. A complee overview of he resuls is given in Tables A1 o A4 in he appendix and is discussed in more deail below. Table 1. Fracion of rejecion of he null hypohesis when lag lengh in he ess was chosen using informaion crieria. Averages over all 27 combinaions of parameers for each sysem size. Sysem A S A S A S A AEG ( ) AEG ( ) W ( ) W ( ) J ( ) size J ( ) J ( ) J ( S ) All Sysem size J J ( S ) J J ( S ) All Sysem size refers o he number of variables in he daa generaing process, i.e. k+1. A and S in parenheses indicaes he usage of he Akaike and Schwarz crieria respecively. Superscrip means ha he Reinsel and Ahn finie sample correcion has been applied o he es saisic. Table 2. Number of imes when he empirical size was 5 ± 2.5 percen when lag lengh in he ess was chosen using informaion crieria. 27 combinaions of parameers for each sysem size. Sysem A S A S A S A AEG ( ) AEG ( ) W ( ) W ( ) J ( ) size J ( ) J ( ) J ( S ) 3 4/27 2/27 13/27 11/27 8/27 9/27 8/27 9/27 4 3/27 2/27 8/27 8/27 4/27 8/27 4/27 8/27 5 2/27 2/27 4/27 6/27 4/27 7/27 4/27 6/27 6 1/27 2/27 2/27 5/27 4/27 5/27 4/27 4/27 All 10/108 8/108 27/108 30/108 20/108 29/108 20/108 27/108 Sysem size J J ( S ) J J ( S ) 3 12/27 9/27 12/27 9/27 4 9/27 9/27 9/27 9/27 5 9/27 9/27 9/27 9/27 6 9/27 11/27 12/27 12/27 All 39/108 38/108 42/108 39/108 Sysem size refers o he number of variables in he daa generaing process, i.e. k+1. A and S in parenheses indicaes he usage of he Akaike and Schwarz crieria respecively. Superscrip means ha he Reinsel and Ahn finie sample correcion has been applied o he es saisic. The firs hing o noe from Table 1 is ha he average size disorion is large; he average size of he ess is beween and 0.333, which should be compared o he nominal size of five percen. The smalles average size disorions can be found when he Akaike informaion crierion is used o choose lag lengh for he Boswijk Wald es and he Johansen es adjused by he finie sample correcion of Reinsel and Ahn; 10

11 he Wald es has he bes resul wih an average size disorion one percenage poin smaller han he adjused es. I is, however, clear from Table 2 ha he adjused es ouperforms he Wald es, and all oher ess, in erms of how ofen he ess have an accepable size. The slighly worse performance in average size for he adjused es is found o mainly be due o some large size disorions when he variance of he error erm is high and he variance of he omied variable is low. I appears as if he Boswijk Wald es could be a compeiive alernaive in smaller sysems, bu ha he adjused Johansen es is more robus in general. This will sand as he general conclusion from he Mone Carlo sudy, bu le us now have a look a he deails of he sudy by invesigaing more closely how he differen ess respond o changes in parameers. Table 3. Fracion of rejecion of he null hypohesis when T = 200 and was chosen using informaion crieria Sysem A ηk = 1 and lag lengh in he ess AEG ( ) AEG ( S ) W W ( S ) J J ( S ) J ( S ) size Sysem size J J ( S ) J J ( S ) ηk Sysem size refers o he number of variables in he daa generaing process, i.e. k+1. A and S in parenheses indicaes he usage of he Akaike and Schwarz crieria respecively. Superscrip means ha he Reinsel and Ahn finie sample correcion has been applied o he es saisic. J 3.2 The AEG es Looking a he resuls in he appendix, i can be noed ha he AEG es has large size disorions, regardless of sysem size, when few lags are used. This is no compleely 11

12 unexpeced since he omission of one of he explanaory variables inroduces an ARIMA(0,1,1) srucure in he esimaed error erm. The size of he AEG es is monoonically decreasing wih respec o lag lengh, which in general means ha adding lags makes he es ge closer o he correc size of five percen. Furher i is found ha he size is almos in all cases monoonically decreasing wih η and increasing wih k. This makes sense since a smaller η, ceeris paribus, makes he MA-roo in he k error erm closer o minus uniy, as does a larger. The larges size disorions can, hence, as expeced be found when is large and η is small. In hose cases, he AEG k es even has he propery o spuriously rejec he null more ofen wih increasing sample size. Wih respec o sysem size, he AEG es shows no clear endency in size. When lag lengh is chosen using informaion crieria, he AEG es is increasing in size wih bu decreasing wih respec o η and he sample size. Unlike he case when k lag lengh was fixed, size is also found o be increasing wih sysem size. I can be noed ha he size disorions for he AEG es were fairly moderae given ha wo or hree lags are used. The raher depressing resuls using boh informaion crieria hough suggess ha we in pracice are likely o choose a lag lengh oo small o give he AEG es accepable size properies. This problem is no surprisingly worse when he more conservaive Schwarz crierion is used. 3.3 The Johansen and imum eigenvalue ess The wo Johansen ess behave in similar ways wih respec o parameers changed. Wihou finie sample correcion, i is almos always he case ha he size improves, i.e. decreases, wih he sample size. This comes as no surprise when asympoic criical values are used. Only in a few cases when is large and η is small do he Johansen k ess show he same behaviour as he AEG es o rejec he null more ofen wih increasing sample size. When few lags are used in he smalles sysems, size is fairly close o nominal in a few cases even when he sample size is smaller han 200 observaions he sample size ha oherwise seems required in order o make he Johansen ess perform reasonably well. Looking a he wo larges sysems, he resuls are less flaering; size disorions in he Johansen ess end o increase when he sysem 12

13 size increases, someimes o a muliple of he desired five percen. As can be seen in Table A1 o A4, size is monoonically increasing wih lag lengh, excep in he cases when = 2 and = 0. 5 or 1 when size decreases wih lag lengh. The ess show no sysemaic changes in size wih respec o eiher η or k. Swiching o informaion crieria o selec lag lengh, he behaviour of he size of he ess changes; however he general paern is he same regardless of wheher he Akaike or Schwarz crierion is used. Size is increasing wih respec o and decreasing wih respec o ηk ; wih respec o sample size he effec on es size is ambiguous. However, sysem size increases he size of he es and in oo many cases he size disorions mus be described as severe. Over all, boh he imum eigenvalue and he ess do no perform very well in erms of size and par of his is likely due o he asympoic criical values being inappropriae approximaions in he sample sizes considered here. The permanen overrejecion of he null of no coinegraion found here is in line wih he findings of Cheung and Lai (1993), as is he fairly general observaion in his sudy ha an increasing sysem size increases he size. The size properies generally improve markedly when he Reinsel and Ahn correcion is used o deal wih hese problems. When he finie sample correcion is applied, we find ha size is increasing wih and decreasing wih respec o ηk when he lag lengh is small. Furhermore, here is a endency for size o increase wih he sample size and decrease wih lag lengh, whereas here is no obvious effec wih respec o sysem size. Deermining he lag lengh using informaion crieria, boh ess respond similarly o changes in and ηk as when lag lengh is chosen arbirarily. In conras, increasing he sample size has a endency o decrease size in he small sysems and increase i when he sysem size is larger. Summarising he resuls for he Johansen ess, i is clear ha he finie sample correced es saisics reduce size disorions and are o be preferred. 13

14 3.4 The Boswijk Wald es The Boswijk Wald es shows very good resuls when hree lags are used he size disorions are hen very modes in mos cases, hough a few ouliers can be found when is large and η is small. When few lags are used, size in mos cases k decreases wih η and increases wih k. This paern does no, however, appear when hree lags are being used, regardless of error erm variances and sysem size. When lag lengh is deermined by he wo informaion crieria, he general paern seems o be ha size increases in and decreases wih η and sample size. However, apar k from fairly good properies in he small sysems, he size properies are no very impressive for he Boswijk Wald es anymore he empirical size is way above nominal. Similar o he resuls for he AEG es, i is obvious ha boh he Akaike and Schwarz informaion crieria choose a lag lengh oo small o make he es behave well in erms of size. 4. An empirical applicaion In his secion, he ess are applied o a model ha resembles he DGP considered in he Mone Carlo sudy. Herberson and Zoega (1999) sugges, using he naional-income accoun ideniy and he life-cycle heory of consumpion ogeher, ha he curren accoun should be a funcion of he age srucure. A naion largely a work should have curren accoun surpluses whereas a naion wih proporionaely more young and old people should have deficis. The inuiion is ha he young are saving for reiremen while he old are running down pas and fuure savings. Based on he above argumens, Herberson and Zoega iniially esimaed equaion (16) on a panel consising of 84 counries from 1960 o CA ( ) ε i = α + β 1 + (16) i D i 14

15 where CA and i Di are he curren accoun 4 and dependency raio 5 in counry i a ime. The purpose of he exercise was o see if here is any reason o expec demography o play a big role in he deerminaion of curren accoun surpluses. However, if he variables in equaion (16) are I(1) and no coinegraed, he regression is likely o yield spurious resuls. Hence, here is reason o be concerned abou he ime series properies of his regression, especially since equaion (16) looks raher incomplee in is specificaion. 6 I could be he case ha we have a misspecified sysem of he kind invesigaed in he previous secion in his paper; is likely o be exogenous and if an omied variable is exogenous as well and if all variables are I(1) hen he DGP is very similar o he one in he Mone Carlo sudy. D i In order o see how he differen coinegraion ess perform in an empirical siuaion, hey are applied o his model using Swedish daa. Being a small open economy, he effecs discussed by Herbersson and Zoega are fairly likely o be presen for Sweden. In erms of equaion (16), he quesion of ineres is hen wheher he Swedish CA and D are coinegraed, which of course also would imply ha CA and ( 1 D ) are coinegraed. Iniially, we have o invesigae wheher he CA and D are I(1) or I(0). If hey urn ou o be saionary in levels, he concep of coinegraion is irrelevan. Yearly daa from 1960 o 1990 on Swedish curren accoun and dependency raio was supplied by Cenral bank of Sweden and Saisics Sweden. The resuls from applying he ADF es o he wo series are repored in Table 4. Table 4. Resuls from uni roo ess ADF ( 0) ADF ( 1) ADF ( 2) ADF ( 3) Dependency raio ** Curren accoun Number in parenheses is lag lengh used in he esimaion, i.e. f. * significan a he 5% level The es resuls indicae ha a uni roo process generaed he curren accoun as he null hypohesis of a uni roo canno be rejeced for he series using any lag lengh. However, 4 The curren accoun is measured as a fracion of GDP 5 The dependency raio is defined as he fracion of he populaion ha is younger han 15 years of age or older han Herbersson and Zoega laer add anoher variable he governmen budge surplus o his equaion. I is of course sill worh paying aenion o he maer of coinegraion in his iniial sep. 15

16 i is no immediaely clear wheher he dependency raion is I(1) or I(0). The null hypohesis can be rejeced using one lagged difference, bu no when lag lengh is chosen using he Akaike or Schwarz informaion crieria. The conclusion drawn is herefore ha boh series conain uni roos. Turning o he quesion of coinegraion beween he wo variables, he resuls from applying he four coinegraion ess are given in Table 5. As is ofen he case when using real daa he resuls are ambiguous. Conradicing resuls should no be a surprise hough given he raher varying rejecion frequencies we found for he differen ess in he Mone Carlo sudy. Table 5. Resuls from ess of coinegraion beween CAB and dependency raio Lags AEG W J * * * J * * * J J * * * Lags refer o lag lengh used in he esimaion, i.e. f, p-1 and m. * significan a he 5% level The AEG es and he Boswijk Wald es do no find any suppor for coinegraion in he daa; he null hypohesis of no coinegraion canno be rejeced using any lag lengh. The wo Johansen ess on he oher hand rejec he null hypohesis for all lags excep zero when no finie sample correcion is applied. When he Reinsel and Ahn correcion is made o he es saisics, he null of no coinegraion is rejeced using one lag for he imum eigenvalue es and when using one and wo lags for he es. Leing informaion crieria decide he lag lengh even hough we have seen ha his sraegy need no produce he bes resuls i urns ou ha he Akaike crierion chooses wo lagged differences as opimal for he Johansen ess, whereas he Schwarz crierion chooses one. Hence, he Johansen ess unambiguously suppor coinegraion using he Schwarz crierion bu give a mixed resul when lag lengh is chosen according o he Akaike crierion. 16

17 Since we never know he rue daa generaing process in an empirical applicaion, i hen boils down o wheher he above resuls are due o 1): good size properies for he AEG and Boswijk Wald ess in combinaion wih size disorions for some versions of he Johansen ess or 2): low power for he AEG and Boswijk Wald ess in combinaion wih good power properies for he Johansen ess. In Mone Carlo sudies performed in his and oher papers, i has been shown ha especially he Boswijk Wald es has good size properies when he lag lengh is sufficienly large. Furhermore, he Boswijk Wald es also has power properies as good as, or beer han, he Johansen imum eigenvalue es as poined ou by Boswijk and Franses (1992) and Pesaveno (2000). This suppors he idea ha he rejecions for he Johansen ess are likely o be spurious and an oucome of size disorions, even hough he finie sample correced version of Johansen s es has been shown o be robus. This conclusion is suppored by he fac ha he esimaed coinegraing vecor from he Johansen es conradics heory he parameer on is highly significan, bu has he wrong sign. In conclusion, i seems doubful ha hese resuls should be inerpreed o be in favour of coinegraion beween CA and D D. Hence, one should hink a leas wice before running a regression like equaion (16) on he Swedish daa given he above resuls. 5. Conclusions When dealing wih ime series ha are I(1), he concep of coinegraion becomes crucial for he specificaion of he model. Using he bes available ess, one can reduce he probabiliy of esimaing economeric models ha are misspecified. Geing he long- and shor-run dynamics of a sysem righ will improve esimaion and, hence, our undersanding of economic relaionships. In his sudy, he small sample properies of four ess for coinegraion he AEG, Johansen s imum eigenvalue, Johansen s, and Boswijk Wald have been invesigaed in a Mone Carlo sudy. Misspecifying a sysem by omiing one of he explanaory variables is found o generae large size disorions of he ess in some cases. This is especially likely when he variance of he omied variable is small and he variance of he dependen variable is large. In his siuaion he likelihood of running a spurious regression when rying o esimae a coinegraing vecor using he firs sep of 17

18 he AEG es increases, or a vecor error correcion model may misakenly be employed when we in fac should esimae a VAR wihou any error correcion erms. The Johansen es adjused by he finie sample correcion of Reinsel and Ahn (1988) is found o have he mos robus performance when lag lengh in he es equaions is chosen according o radiional informaion crieria. Wihou he Reinsel and Ahn correcion, he wo Johansen ess perform worse regardless of specificaion and we can noe ha for sample sizes generally used by macroeconomiss, hey are likely o have considerable size disorions when a sysem has been misspecified. The AEG and, especially, he Boswijk Wald es perform raher well when he lag lengh is sufficien in he es equaions. However, boh he Akaike and Schwarz informaion crieria end o choose oo few lags, yielding fairly large size disorions for boh hese ess. Though he Wald es was found o be compeiive in smaller sysems, his mus be seen as a pracical limiaion of he AEG and he Wald es. 18

19 References Akaike, H. (1974), A New Look a he Saisical Model Idenificaion, IEEE Transacions on Auomaic Conrol 19, Banerjee, A., Dolado, J. J., Hendry, D. F., and Smih, G. W. (1986), Exploring Equilibrium Relaionships in Economerics hrough Saic Models: Some Mone Carlo Evidence, Oxford Bullein of Economics and Saisics 48, Bewley, R. and Yang, M. (1995), Tes for Coinegraion Based on Canonical Correlaion Analysis, Journal of he American Saisical Associaion 90, Bewley, R. and Yang, M. (1998), On he Size and Power of Sysem Tess for Coinegraion, The Review of Economics and Saisics 80, Boswijk, H. P. (1989), Esimaion and Tesing for Coinegraion wih Trended Variables: A Comparison of a Saic and a Dynamic Regression Procedure, Repor AE 12/89, Universiy of Amserdam. Boswijk, H. P. and Franses, P. H. (1992), Dynamic Specificaion and Coinegraion, Oxford Bullein of Economics and Saisics 54, Cheung, Y. and Lai, K. S. (1993), Finie-Sample Sizes of Johansen s Likelihood Raio Tess for Coinegraion, Oxford Bullein of Economics and Saisics 55, Engle, R. F. and Granger, C. W. J. (1987), Co-Inegraion and Error Correcion: Represenaion, Esimaion, and Tesing, Economerica 55, Granger, C. W. J. (1981), Some Properies of Time Series Daa and Their Use in Economeric Model Specificaion, Journal of Economerics 16, Granger, C. W. J. and Newbold, P. (1974), Spurious Regressions in Economerics, Journal of Economerics 2, Hansen, B. E. (1990), A Powerful, Simple Tes for Coinegraion Using Cochrane- Orcu, Unpublished manuscrip, Deparmen of Economics, Universiy of Rocheser. Hansen, B. E. (1992), Tess for Parameer Insabiliy in Regressions wih I(1) Processes, Journal of Business and Economic Saisics 10, Haug, A. A. (1996), Tess for Coinegraion A Mone Carlo Comparison, Journal of Economerics 71, Herbersson, T. T. and Zoega, G. (1999), Trade Surpluses and Life-Cycle Saving Behaviour, Economics Leers Johansen, S. (1988), Saisical Analysis of Coinegraion Vecors, Journal of Economic Dynamics and Conrol 12,

20 Johansen, S. (1991), Esimaion and Hypohesis Tesing of Coinegraion Vecors in Gaussian Vecor Auoregressive Models, Economerica 59, Johansen, S. and Juselius, K. (1990), Maximum Likelihood Esimaion and Inference on Coinegraion wih Applicaions o he Demand for Money, Oxford Bullein of Economics and Saisics 52, Johansen, S. (1995), Likelihood-Based Inference in Coinegraed Vecor Auoregressive Models. Oxford Universiy Press, New York. Kremers, J. J. M., Ericsson, N. R. and Dolado, J. J. (1992), The Power of Coinegraion Tess, Oxford Bullein of Economics and Saisics 54, Pesaveno, E. (2000), Analyical Evaluaion of he Power of Tess for he Absence of Coinegraion, Discussion Paper , Deparmen of Economics, Universiy of California, San Diego. Phillips, P. C. B. (1986), Undersanding Spurious Regressions in Economerics, Journal of Economerics 33, Phillips, P. C. B. (1991), Opimal Inference in Coinegraed Sysems, Economerica 59, Phillips, P. C. B. and Ouliaris, S. (1990), Asympoic Properies of Residual Based Tess for Coinegraion, Economerica 58, Reinsel, G. C. and Ahn, S. K. (1988), Asympoic Properies of he Likelihood Raio Tes for Coinegraion in he Nonsaionary Vecor AR Model, Technical Repor, Deparmen of Saisics, Universiy of Wisconsin-Madison. Said, S. E. and Dickey, D. A. (1984), Tesing for Uni Roos in Auoregressive Moving Average Models of Unknown Order, Biomerika 71, Sargan, J. D. and Bhargava, A. (1983), Tesing Residuals from Leas Squares Regressions from Being Generaed by he Gaussian Random Walk, Economerica 51, Schwarz, G. (1978), Esimaing he Dimension of a Model, Annals of Saisics 6, Sims, C. A. (1992), Inerpreing he Macroeconomic Time Series Facs. The Effec of Moneary Policy, European Economic Review 36, Sock, J. H. and Wason, M. W. (1988), Tesing for Common Trends, Journal of he American Saisical Associaion 83,

21 Appendix Table A1. Fracion of rejecion of he null hypohesis when here is one x-variable in esimaed equaion and wo in DGP, i.e. k+1 = T AEG ( ) AEG ( ) AEG ( ) AEG ( ) W ( ) W () W ( ) W ( 3) T J ( 0) J ( 1) J ( 2) J ( 3) J ( 0) J () 1 J ( 2) J ( 3)

22 T J ( 0) J ( 1) J ( 2) J ( 3) J ( 0) J () 1 J ( 2) J ( 3)

23 T AEG AEG ( S ) W W ( S ) J J ( S ) J ( S ) T J J ( S ) J J ( S ) J 23

24 Table A2. Fracion of rejecion of he null hypohesis when here are wo x-variables in esimaed equaion and hree in DGP, i.e. k+1 = 4 T AEG ( ) AEG ( ) AEG ( ) AEG ( ) W ( ) W () W ( 2) W ( 3) ηk T J ( 0) J ( 1) J ( 2) J ( 3) J ( 0) J () 1 J ( 2) J ( 3)

25 T J ( 0) J ( 1) J ( 2) J ( 3) J ( 0) J () 1 J ( 2) J ( 3)

26 T AEG AEG ( S ) W W ( S ) J J ( S ) J ( S ) T J J ( S ) J J ( S ) J 26