THE FORECASTING ABILITY OF A COINTEGRATED VAR DEMAND SYSTEM WITH ENDOGENOUS VS. EXOGENOUS EXPENDITURE VARIABLE

Transcription

1 WORKING PAPERS Invesgação - Trabalhos em curso - nº 109, Julho de 2001 THE FORECASTING ABILITY OF A COINTEGRATED VAR DEMAND SYSTEM WITH ENDOGENOUS VS. EXOGENOUS EXPENDITURE VARIABLE Margarda de Mello Kevn S. Nell FACULDADE DE ECONOMIA UNIVERSIDADE DO PORTO Faculdade de Economa do Poro - R. Dr. Robero Fras Poro - Porugal Tel Fax hp://

2 THE FORECASTING ABILITY OF A COINTEGRATED VAR DEMAND SYSTEM WITH ENDOGENOUS VS. EXOGENOUS EXPENDITURE VARIABLE: AN APPLICATION TO THE UK IMPORTS OF TOURISM FROM NEIGHBOURING COUNTRIES MARGARIDA DE MELLO Faculdade de Economa do Poro Rua Dr. Robero Fras Poro, Porugal emal: mmdm@clx.p KEVIN S. NELL Naonal Insue of Economc Polcy P. O. Box Braamfonen 2017, Johannesbourg, Souh Afrca ABSTRACT Ths paper uses Sms s VAR mehodology, as an alernave o Deaon and Muellbauer s AIDS approach, o esablsh he long-run relaonshps beween I(1) varables: oursm shares, oursm prces and UK oursm budge. The VAR deermnsc componens and ses of exogenous and endogenous varables are esablshed, and he Johansen s rank es s used o deermne he conegraed vecors n he sysem. The srucural form of he conegraed VAR s denfed and he long-run parameers are esmaed under several heorecal resrcons. The resrced conegraed VAR reveals self a heorecally conssen and sascally robus means o analyse he long-run demand behavour of UK ourss and an accurae forecaser of he desnaons shares.

3 1. INTRODUCTION The Almos Ideal Demand Sysem (AIDS) of Deaon and Muellbauer (1980a, 1980b) has been used, n oursm demand conexs, for esmang desnaons oursm shares and for verfyng consumers heory resrcons of homogeney and symmery (e.g. Papaheodorou, 1999; De Mello e al., 2001). The model s specfcaon ncludes an assumed endogenous-exogenous dvson of varables ha may be quesonable and he use of nonsaonary me seres. When dealng wh nonsaonary daa, he concep of conegraon s synonymous wh he concep of long-run equlbrum. Falure o esablsh conegraon ofen means he non-exsence of a seady sae relaonshp among he varables and s mporan o recognse he effec of un roos on he dsrbuon of esmaors (Harvey, 1990, p.83). Hence, he esmaon resuls obaned wh an AIDS model can be deemed spurous and he sascal nference nvald, f he assumpon of exogenous regressors does no hold and/or no conegraed relaonshp(s) exs. Consequenly, here seems o be a rsk nvolved n he esmaon of sysems wh nonsaonary daa whch regress endogenous varables on several assumed exogenous varables, whou sanconng her sascal valdy wh approprae esng and conegraon analyss. Gven ha he number of conegraed vecors s unknown, and gven he possbly of smulaneously deermned varables, emprcal analyss mus go one sep furher and specfy economerc models whch can be effcenly esmaed and valdly esed whn a sysem of equaons approach. The man goal of hs paper s o conrbue an emprcal bass for he valdaon or oherwse, of he esmaon and nference procedures mplemened wh an AIDS model for he UK oursm demand. We do so by usng Sms (1980) vecor auoregressve (VAR) approach o specfy he relaonshp beween desnaons oursm shares and her deermnans, Johansen s (1988) reduce rank es o esablsh he number of conegraed vecors n he sysem, he fndngs of Pesaran, Shn and Smh (2000) regardng he srucural analyss of conegraed VARs wh exogenous I(1) varables, and he procedures of Pesaran and Shn (1998) o exacly-denfy he longrun coeffcens of he conegraed vecors n accordance o he heorecal prncpals underlyng he AIDS approach. We provde emprcal evdence for sanconng he conegraed VAR and AIDS models as sascally robus and heorecally conssen means o produce vald and relable esmaes of he srucural parameers underlyng he relaonshps beween desnaons oursm shares, oursm prces and an orgn s 2

4 real per capa oursm budge. In addon, we confrm he compeence of he VAR model, boh n s general unresrced reduce-form and under he full se of heorecal resrcons, for provdng accurae forecass of he desnaons shares. The paper proceeds as follows. Secon 2, addresses he man feaures of he VAR mehodology, esablshes he order of negraon and he approprae lag-lengh of he varables, specfes he (unresrced) VAR model for he UK oursm demand and presens he forecasng resuls obaned wh hs specfcaon. Secon 3, apples he Johansen rank es o deermne he number of conegraed vecors and presens he conegraed srucural VAR esmaes under exacly- and over-denfyng resrcons. Secon 4, provdes he forecass obaned wh he conegraed VAR and compares hem wh hose obaned wh he unresrced VAR and AIDS model of De Mello e al (2001). Secon 5 concludes. 2. VAR MODELLING OF THE UK TOURISM DEMAND The problem of smulaneous bas s ofen presen n a srucural sysem because hs specfcaon expresses each endogenous varable as a separae funcon of oher endogenous varables, alongsde predeermned varables. Snce he endogenous varables are correlaed wh he error erms, he srucural coeffcens canno be conssenly esmaed by OLS. Ths problem can be removed f he srucural equaons are solved for he exsng endogenous varables, makng hem dependen solely on predeermned varables and sochasc dsurbances. Snce he former are assumed o be uncorrelaed wh he laer, OLS appled o he reduced-form equaons generaes conssen and asympocally effcen esmaes. Ye, he esmaes obaned wh hs procedure are hose of he reduced-form and no of he srucural-form coeffcens whch are ulmaely of neres. Snce he laer are combnaons of he former, he possbly exss ha he srucural coeffcens can be rereved from he reduced-form coeffcens. Wheher hs s he case, brngs abou he problem of denfcaon. Frequenly, he secre for denfcaon, s relaed o he use of zero resrcons. Ofen, however, models seem o be formulaed wh varables added o equaons and deleed from ohers merely o acheve denfcaon, and whou much economc usfcaon. Crcs o mul-equaon srucural modellng have been cenred on he role of zero resrcons and on he assumed exogenous/endogenous dvson of varables. Sms (1980), regards zero resrcons as ncredble and devses a new 3

5 approach for he specfcaon and esmaon of mul-equaon sysems, known as vecor auoregressve (VAR) mehodology, where all varables are endogenous and no zero resrcons are mposed. However, Sms VAR s ofen labelled an a-heorecal approach o long-run equlbrum analyss snce much of he long-run analyss whn hs purely-sascal approach s conduced whou provdng an explc accoun of he ype of equlbrum heory ha may underle, and emprcal applcaons of hs mehodology have focused on he sascal properes of he underlyng economc me seres, ofen a he expense of heorecal nsghs and economc reasonng (Pesaran, 1997, p. 178). Hence, he feaures ha make he VAR a flexble and smple ool, also mark an area of weaknesses: a reduced-form VAR s boh a ruly smulaneous sysem and a smple specfcaon snce all s varables are regarded as endogenous and requres lle more han he choce of approprae varables and a suable lag-lengh. Ye, unlke he radonal srucural sysems, an unresrced VAR does no use any a pror nformaon and, unless he underlyng srucural model can be denfed from he reduced-form, he nerpreaon of s esmaes s dffcul. If auocorrelaon exss n he error erms of a VAR, he predeermned rgh-hand sde varables can be correlaed wh he error erms leadng o nconssen esmaors. So, proper lag-lengh selecon s crucal n VAR modellng. However, he longer he laglengh, he faser degrees of freedom are eroded and, gven he lmed number of observaons generally avalable n mos emprcal analyss, he nroducon of several lags for each varable can be a problem. Sll, f an approprae lag-lengh can be esablshed, he error erms of each equaon are serally uncorrelaed and, as a VAR expresses he curren values of each endogenous varable as a funcon solely of predeermned varables and hese are no correlaed wh he error erms, each equaon can be esmaed by OLS provdng conssen and asympocally effcen esmaes. 1 A VAR can be vewed as a reduced-form sysem wh no exogenous varables specally adaped for forecasng. Even so, an unresrced VAR model may be overparameersed n he sense ha some lagged varables n he model could be properly deleed on he bass of sascal nsgnfcance. Ye, he advocaes of hs mehodology advse agans hs procedure argung ha he mposon of zero resrcons may suppress mporan nformaon, and ha he regressors n a VAR are lkely o be hghly 1 If he approprae lag-lengh s no he same for all varables, or f he equaons nclude dfferen regressors, Zellner s (1962) SURE mehod s more effcen han OLS for esmang a VAR. 4

6 collnear so ha he -ess on ndvdual coeffcens are no relable gudes for downszng he model. In pracce, however, s mpossble o avod he consderaon of pror resrcons. Sample sze consrans mean ha here s always a lm o he number of varables and lags ncluded. Even f he sample sze s no a problem, he possbly of gvng some srucure o a VAR and usng for economc analyss alongsde forecasng purposes, requres he mposon of resrcons. Well-founded heorecal consrans may help o ransform an unresrced VAR no a resrced model conssen wh even hghly dealed economc heores (Charemza and Deadman, 1997, p.157). The consderaon of such resrcons allows for denfcaon and economc nerpreaon of he srucural parameers n a way no possble wh he reduced-form. Furhermore, a pror nformaon concernng he parameers allows for esng resrcons whch can mprove he precson of esmaes and reduce he forecas error varance. Hence, even f forecasng s he man obecve, he down-szng of an over-parameersed VAR can help o mprove resuls. When emprcal analyss on he exsence of long-run relaonshps among more han wo non-saonary seres s conduced and here are doubs abou he exogenous naure of regressors, an approprae modellng sraegy consss of reang all varables as endogenous whn a reduced-form VAR framework. Then, usng one or more of he mehods avalable, ess for he exogeney of he se of varables n doub can be carred ou. Once he endogenous-exogenous dvson s esablshed, he Johansen approach can be used o es for he exsence of conegraed relaonshp(s). The number of conegraed vecors found esablshes he number of meanngful long-run relaons n he sysem. The esmaes of he long-run coeffcens can be assessed by mposng resrcons o exacly-denfy he underlyng srucural VAR. Once he srucural form s denfed, addonal resrcons makng he VAR compable wh specfc economc heores, can also be esed. We specfy a general unresrced VAR model of he UK oursm demand for France, Span and Porugal whch ncludes he varables n vecor z = [ WF,WS,WP,PF,PS,PP, E ]. Where WF, WS and WP represen, respecvely, he UK oursm expendure shares of France, Span and Porugal; PF, PS and PP represen oursm prces n France, Span and Porugal and E represens he UK real per capa oursm budge. All varables n z, and daa sources are descrbed n Appendx A, accordng o he defnons and daa sources used n De Mello e. al. (2001). 5

7 The specfcaon of a VAR model sars wh esablshng he order of negraon and he approprae lag-lengh of s varables. Hence, we sar by deermnng wheher he me seres n z are saonary and he approprae lag-lengh of he VAR. All esmaons and sascal ess mplemened below were compued usng Pesaran and Pesaran (1997) Mcrof Order of negraon of he varables ncluded n he VAR Table 1 shows he sascs and respecve crcal values a he 5% sgnfcance level of he Dckey-Fuller (1979, 1981) (DF) and Augmened Dckey-Fuller (ADF) un roo ess for he levels and frs dfferences of varables WF, WS, WP, PP, PS, PF and E. I also shows he Akake Informaon Creron (AIC) and Schwarz Bayesan Creron (SBC) for he lag-lengh selecon of he DF es equaons. Inser Table 1 here The ess ndcae ha all varables n levels are non-saonary and, excep for PP, all varables n frs dfferences are saonary. Ths means ha accordng o he DF and ADF ess, all varables, excep PP, can be consdered I(1) varables. Doubs abou he order of negraon of varables PP and PP can be cleared wh he Phllps-Peron (1988) es. The Phllps-Peron es s based on a smple DF regresson, such ha PP = + PP (1) The OLS esmaon resuls for equaon (1) are (-raos n brackes), PˆP = PP 1 (0.536) ( ) To carry ou he non-paramerc correcon of he sasc proposed by Phllps and Peron, we use he Whe and Newey-Wes covarance marx and compue he adused varances. The esmaes of regresson (1) usng he adused covarance marx are PˆP = (0.5112) ( ) PP 1 The adused rao for 1 coeffcen s now he vald sasc o compare wh he crcal value. As < we canno reec he hypohess of PP beng nonsaonary. Ths confrms he resul obaned wh he ADF es n Table 1 for he same 6

8 varable. However, he problem resdes n ha he ADF es ndcaes PP o be nonsaonary as well. So, we perform he Phllps-Peron es for PP, wh regresson: PP = ' +' PP (2) The OLS esmaon resuls for regresson (2) are PˆP = ( ) ( ) PP 1 The esmaon resuls for (2) usng he adused covarance marx are PˆP = ( ) ( ) PP 1 As > we canno reec he hypohess of PP beng saonary. Hence, he varable n levels PP s I(1) and we can consder all varables n levels as I(1). 3.2 Deermnaon of he order of he VAR Gven he number of varables and sample sze, he lag-lengh (p) for hs VAR model canno exceed wo. Gven hese lmaons, we used he AIC and SBC crera and he adused (for small samples) Lkelhood Rao (LR) es for selecng he order of he VAR wh he maxmum lag-lengh permed. The LR es reecs order zero bu canno reec a frs order VAR. The SBC creron clearly ndcaes he order of he VAR o be one. The AIC creron ndcaes p o be wo, bu by a very small margn. So, we selec a order one VAR. Ye, s always pruden o examne he resduals of he equaons n order o check for seral correlaon. Table 2 presens he LR sasc and he wo selecon crera for choosng he lag-lengh (p). Inser Table 2 here 3.3. The unresrced VAR specfcaon of he UK demand for oursm The reduced form of a frs order unresrced VAR of he UK oursm demand for France, Span and Porugal (denoed VAR I) can be wren as: 2 z * A0 + A1z * = + (3) 1 where z* = [WF, WS, PP, PS, PF, E ] 7

9 Snce he share varables observaons (WF, WS and WP ) sum up o uny, one of he share equaons s omed. We om he equaon for he share of Porugal (WP ). The esmaon resuls are nvaran whchever equaon s excluded and, by he addng-up propery, all coeffcen esmaes of he omed equaon can be rereved from he coeffcen esmaes of he oher wo. The sascal qualy of VAR I model s accessed by esmang (3) and compung relevan dagnosc sascs. Table 3 shows he esmaon resuls ( raos n brackes), he AIC and SBC crera and a se of sascs - adused R 2, F sasc and χ 2 sasc for dagnosc ess of seral correlaon, funconal form, error normaly and heeroscedascy (p values n brackes) - for all equaons ncluded n he basc unresrced VAR I srucure. Inser Table 3 here There s no sascal evdence of seral correlaon n VAR I equaons. Hence, he laglengh seleced seems o be adequae. However, he ess ndcae problems n he funconal form and error normaly for some equaons. We are neresed n he expendure share equaons and funconal form problems for hese, appear severe. In an AIDS sysem, he oursm shares of France (WF), Span (WS) and Porugal (WP) would be he only endogenous varables, and changes n hese varables would be explaned by a se of exogenous regressors whch ncludes oursm prces (PF, PS and PP) and he UK real per capa oursm expendure (E). Whn a VAR specfcaon, we queson he assumed exogeney of he prce varables. However, here seems o be no obvous heorecal or emprcal bass for challengng he mul-sage budgeng process underlyng he raonaly of an AIDS expendure share sysem, whch ses he varable UK real per capa expendure as an exogenous deermnan of he demand shares n (3). The reasons are as follows. A VAR model ses all s varables as endogenous. Hence, a b-dreconal cause-effec relaonshp (feedback) beween hem should exs. In a oursm demand conex, however, even f s reasonable o consder ha changes n he UK real per capa expendure affec he oursm shares of mporan UK holday desnaons such as 2 Ths basc specfcaon of he unresrced VAR I s laer subeced o modfcaons due o he nroducon of dummy varables and he exogeney assumpon concernng some regressors. 8

10 France, Span or Porugal, does no seem realsc o expec ha changes n hese shares affec he way n whch UK consumers allocae her budges. Emprcal evdence seems o suppor hs lne of reasonng. As can be nferred from he esmaon resuls of he equaon for E n Table 3, he 99% of hs varable s varaons explaned by he model le exclusvely on s own lagged value. No oher varable n ha equaon s ndvdually or only sascally sgnfcan. In fac, he F es for he on sgnfcance of all explanaory varables, excludng E -1, presens a value of 0.86 mplyng ha he hypohess of hese varables coeffcens beng zero canno be reeced. Furhermore, he esmaon resuls of he VAR equaons for WF and WS ndcae ha he lagged value of E does no affec sgnfcanly he curren values of hese oursm shares. To nvesgae furher he feaures of he lnk beween he expendure shares (W ) and he UK real per capa oursm budge (E ), we analyse he relaonshps beween he error erm of he condonal model for W and he sochasc dsurbance of he assumed daa generang process (d.g.p.) of E. Consder ha he h share equaon s W = α + α E + α W + u where = F, S, P (4) and assume ha E s a sochasc varable whch underlyng d.g.p. s E = E + ε ; 1 <1 and ε N(0, σ 2 ) (5) 1 1 If u and ε are no correlaed we can say ha EV(u, ε s )=0 for all, s (where EV sands for expeced value, no o be confused wh varable E). Then, s possble o rea E as f was fxed, ha s, E s ndependen of u such ha EV(E,u )=0. Hence, we can rea E as exogenous n erms of (4), wh he curren value of he UK real per capa expendure (E ) beng sad o Granger-cause W. Equaon (4) s a condonal model snce W s condonal on E, wh E beng deermned by he margnal model (5). 3 A varable canno be exogenous per se (Hendry, 1995, p.164). A varable can only be exogenous wh respec o a se of parameers of neres. Hence, f E s deemed o be exogenous wh respec o parameers α ( = 0, 1, 2) n (4), he margnal model (5) can be negleced and he condonal model (4) s complee and suffcen o susan vald 3 As noed n Harrs (1995), f (5) s reformulaed as E = E + W + ε, EV(E , u )=0 sll holds. However, as pas values of W now deermne E, E can only be consdered weakly exogenous n (4). The curren value E sll causes W bu no n he Granger sense, snce lags of W now deermne E. 9

11 nference. So, knowledge of he margnal model wll no sgnfcanly mprove he sascal or forecasng performance of he condonal model. Followng hs lne of reason, we run regresson (4) for he expendure shares of France, Span and Porugal and regresson (5) as a represenaon of he d.g.p. of E. We rereve he resdual seres of hese four regressons, namely u F, u S, u P sandng for he resdual seres of (4) for, respecvely, France, Span and Porugal, and ε sandng for he resduals of (5). We hen run regressons of he curren and lagged values (up o he ffh lag) of he resduals u ( = F, S, P) on he curren and lagged (up o he ffh lag) values of ε. The esmaon resuls of all 90 regressons ndcae no sgnfcan relaonshp lnkng he curren or lagged resduals of he condonal models o he curren and lagged resduals of he margnal model. These resuls can be vewed as an ndcaon ha knowledge of he margnal model does no mprove he sascal or forecasng performance of he condonal (on E -1 ) equaons for WF, WS, PP, PS and PF n he VAR and so E may be reaed as exogenous n he VAR gven by (3). The clam ha feedback effecs mgh be absen n he relaonshps beween varables UK real per capa expendure (E ) and expendure shares (W, = F, S), can be furher nvesgaed usng he causaly concep proposed by Granger (1969). A es for causaly s relaed o wheher he lags of a varable are sascally sgnfcan n he equaon of anoher varable. Usng Grffhs e al. s (1992, p. 695), defnon a varable y1 s sad o be Granger-caused by y2, f curren and pas nformaon on y2 help mprove forecass n y1. 4 Hence, f he lagged values of y2 do no mprove he forecass of y1,.e., f he lagged values of y2 are sascally nsgnfcan n he reduced form equaon for y1, hen y2 does no Granger-cause y1. A mulvarae generalsaon of he Granger-causaly es can be used o esablsh f one or more varables n a VAR should or should no negrae he group of endogenous varables. The use of he LR sasc for esng he null ha he coeffcens of a subse of varables n a VAR are zero, s called block Granger non-causaly es. Ths es provdes a sascal measure of he exen o whch lagged values of a se of varables (say E ), are mporan n predcng anoher se of varables (say W ), once lagged values of he laer (W -1 ) are ncluded n he model. The LR sasc value for block Granger non-causaly of varable E, esng he null ha he coeffcens of E -1 are zero 4 Granger s causaly concep does no mply a cause-effec relaon. Indeed, causaly has a meanng more on he lnes of o predc raher han o produce (Charemza and Deadman, 1997, p. 165). 10

12 n he block equaons for WF, WS, PP, PS and PF, s χ 2 (5)= As he 5% crcal value s , he null canno be reeced. Gven he above, we exclude E from he se of endogenous varables and re-formulae he VAR under hs assumpon. Ths new verson of he model s denoed VAR II. Table 4 shows he model s esmaes and he selecon crera and dagnosc sascs prevously used o access s qualy. The selecon crera and dagnosc ess ndcae VAR II o be sascally more robus han VAR I. However, problems wh he funconal form of he share equaons sll reman. Inser Table 4 here There s cause o beleve ha evens n he 1970s (change of polcal regmes n Porugal and Span and he ol crses) may have affeced he me pah of varables ncluded n he VAR. Addonally, he negraon process of Span and Porugal n he European Unon (EU) n 1986 s lkely o have affeced he UK oursm demand for s souhern neghbours. To accoun for he 1970s evens we add a dummy varable D1 akng he value of uny n he perod and zero oherwse. To accoun for he negraon of he wo Iberan counres n he EU, we use dummy varables D2 (akng he uny value n he perod and zero oherwse) and D3 (akng he uny value n he perod and zero oherwse) o spl he negraon process no wo sub-perods: he negraon perod ( ) and he pos-negraon perod ( ). These dummes are assumed o be exogenous. Ths hrd verson of he VAR s denoed VAR III. Table 5 shows he AIC and SBC crera and he same se of dagnosc sascs used before o access he qualy of he VAR III. Inser Table 5 here The resuls n Table 5 ndcae ha he sascal qualy of VAR III over-performs hose of VAR I and VAR II, parcularly wh respec o he expendure share equaons. Addonally, he LR es for block Granger non-causaly of varable E performed on VAR III specfcaon, confrms he resuls of he smlar es performed on VAR I. The LR es presens now he sasc value of χ 2 (5)=8.376, whch s well below he 5% crcal value (11.071), furher supporng he null ha he coeffcens of E -1 are sascally nsgnfcan n he block equaons of he VAR. The ess resuls 11

13 concernng he sgnfcance of he nercep, he on sgnfcance of he dummy varables, and he Granger block non-causaly of varable E -1 are shown n Table 6. The frs column presens he null hypohess for each es and shows he varables enerng he VAR under he unresrced hypohess (U), and he resrced null hypohess (R) (for smplcy, he me subscrps are omed). In each case, he se of endogenous varables s separaed from he se of deermnsc componens and exogenous varables by he symbol &. The second column presens he maxmum value of he lkelhood funcon (ML) for he unresrced and resrced alernaves. In he hrd column he LR sasc s compued. Under he null, LR s asympocally dsrbued as χ 2 wh degrees of freedom () equal o he number of resrcons. The null s reeced f LR sasc s larger han he relevan crcal value. Inser Table 6 here Alhough VAR III s sascally more robus han VAR I, he laer s a more general model, no beng resrced n any way, whle he former s a specfc model, beng a paral sysem condoned on exogenous varables. Hence, s neresng o use boh models for comparson purposes n accessng he feaures of he UK oursm demand for France, Span and Porugal, whn a reduced-form VAR specfcaon. However, a reduced-form VAR may no be he deal means o conduc economc analyss. Frs, because an a-heorecal model s unlke o produce esmaon resuls nerpreable whn he lms of economc heory. Second, because he economc nerpreaon of he srucural parameers s only possble f he underlyng srucural model s denfed from he reduced-form. Fnally, because he lag-srucure of he reduced-form s lkely o be over-parameersed causng mprecson n he coeffcens esmaes and obscurng he economc meanng of he long-run parameers. However, for forecasng purposes, a smple reduced-form VAR has s advanages. Hence, we use VAR I and VAR III reduced-forms for forecasng raher han for economc analyss Forecasng ably of he reduced-form VAR I and VAR III models To oban forecass for he UK oursm shares of France, Span and Porugal from VAR I and VAR III, we esmae hese models for he perod , usng he las four observaons for forecasng. Tables 7, 8 and 9 repor he acual and forecased values and he forecas errors for, respecvely, he shares of France, Span and Porugal, and a 12

14 number of summary sascs (mean absolue predcon errors MAE; mean squared predcon errors MSE; roo mean squared predcon errors RMSE) o evaluae he models forecasng accuracy. The forecas errors and all qualy crera ndcae VAR I o be a beer forecaser han VAR III. ***** Inser Tables 7, 8, and 9 here ***** The fac ha he varables ncluded n he VAR models are I(1) mples ha esmaon, sascal ess and forecasng procedures are srcly vald f conegrang relaonshp(s) exs lnkng he varables. Therefore, he nex sep for proceedng he emprcal analyss s o esablsh wheher conegraed vecors exs whn he VARs. To do so we use he Johansen s conegraon rank es. Besdes esablshng he procedure for conegraon rank es, he Johansen approach provdes a general framework for denfcaon, esmaon and hypohess esng n conegraed sysems (see Johansen, 1988, 1991, 1995 and 1996 and Johansen and Juselus, 1990 and 1992). Hence, gven a conegraed sysem, hs approach provdes a mehod for denfyng he srucural relaonshps underlyng he unresrced reducedform. However, he Johansen s emprcal process o exacly-denfy he long-run coeffcens may no always be adequae, parcularly n economc conexs where heory provdes srong, sensble and esable resrcons (Pesaran and Shn, 1998). In hese cases, he conegraed vecors mus be subec o exac- and over-denfyng resrcons suggesed by heory and oher relevan a pror nformaon, raher han beng subec o some normalsaon process ha does no consder he heorecal and emprcal framework whn whch he phenomenon evolves. As Harrs (1995, p.117) pons ou wha s becomng ncreasngly obvous s he need o ensure ha pror nformaon movaed by economc argumens forms he bass for mposng resrcons, and no he oher way around. Ths s parcularly mporan n a oursm conex nvolvng an AIDS sysem of equaons regressng (m-1) oursm shares on m oursm prces and an orgn s per capa oursm budge. In hs case, he number of long-run relaonshps predced by heory s (m 1),.e., he number of equaons n he sysem. Hence, f heory s rgh, he conegraon ess nvolvng a VAR of he UK oursm demand wh varables WF, WS, PP, PS, PF and E should ndcae ha he model s relevan long-run relaonshps are hose esablshed by he share equaons. 13

15 Furhermore, he denfcaon process of he srucural equaons should confrm ha her seady-sae form s ha of an AIDS model s equaons. Consequenly, n eher VAR I and VAR III, we expec o fnd exacly wo conegraed vecors and o denfy he srucural parameers wh resrcons ha mach hose of he normalsaon process used for denfyng he share equaons of an AIDS model. If hs s he case, hen we can subec he long-run relaonshps o furher hypohess esng, such as homogeney and symmery, and conrbue an emprcal bass for he valdaon of he prncpals of consumer heory whn an AIDS sysem framework usng a VAR specfcaon. 4. JOHANSEN S REDUCED RANK TEST Le z be a vecor of n poenally endogenous varables. Then, s possble o model z as an unresrced VAR nvolvng up o p-lags such ha z = A1z Apz p + u u IN(0, ) (6) where z s a (n x 1) vecor and each of he A s a (n x n) marx of parameers. The VAR model n (6) can be reformulaed as a vecor error-correcon (VEC) such ha: z = Γ z Γp 1 z p+ 1 + Πz p u (7) where Γ = I A... A ), = 1,...,p 1 and Π = I A... A ) = α ( 1 ( 1 p α represens he speed of adusmen marx and s he marx of long-run coeffcens such ha, z -p mplc n (7) represens up o (n 1) conegraon vecors n he mulvarae VAR. These conegrang vecors defne he relaonshps whch ensure ha he varables n z converge o her long-run soluons. If he varables n z are I(1) hen, z -p n (7) mus be saonary for u I(0) o be a whe nose. Ths happens when = α has reduced rank, ha s, when here are r (n 1) conegrang vecors n alongsde (n r) nonsaonary vecors. Hence, esng for conegraon amouns o deermnng he number of r lnearly ndependen columns n. The hypohess ha here are a mos r conegrang vecors (n r nonsaonary relaonshps) can be esed wh he egenvalue race sasc (λ race ) whch null s r=q (q=0, 1,, n 1) agans he alernave r q+1, and/or he maxmum egenvalue sasc (λ max ) whch null s ha here are r=q conegrang vecors agans he alernave r=q+1 exs. The resuls of hese ess are gven n Table 10. The frs column shows he 14

16 egenvalues assocaed wh each of he I(1) varables, ordered from hghes o lowes, necessary o compue λ max and λ race. The second column shows he varous hypohess o be esed, sarng wh no conegraon (r=0 or n r=6 n he case of VAR I, and r=0 or n r=5 n he case of VAR III) and followed by ncreasng numbers of conegraed vecors. The followng columns show he esmaed λ max and λ race and respecve 5% and 10% crcal values. The las column presens he SBC model selecon creron. Inser Table 10 here For VAR I, boh he λ max and λ race sascs gve evdence, a he 5% level, of wo conegraed vecors correspondng o he hgher egenvalues aached o he share equaons. Boh sascs assocaed wh he null of r=0 and r=1 reec hese hypohess (sasc value>crcal value) bu canno reec r=2 (sasc value<crcal value). The SBC creron furher suppors hese fndngs by selecng he model wh wo conegraed vecors. For VAR III, a he 5% level, λ max sasc suggess he exsence of wo conegrang vecors whle λ race sasc does no reec he hypohess of only one vecor. Ths dsagreemen s no uncommon, parcularly n cases of small samples and added dummy varables. However, we have enough evdence supporng he choce of r=2: we have λ max sasc clearly reecng he exsence of only one n favour of wo conegraed vecors; we have SBC creron selecng VAR III model wh wo conegrang vecors; we have heory suggesng he exsence of wo and no one longrun relaonshps whch s unmsakably suppored by boh es sascs and selecon creron n he more general VAR I model. Hence, gven ha evdence agans heory predcon seems weak, we proceed by seng r=2 for boh VAR I and VAR III. To denfy he srucural form of he wo conegraed vecors we follow Pesaran and Shn (1998) and use he heorecal exac-denfyng resrcons mplc n he share equaons of an AIDS model. Consder he followng noaon for marx of he wo conegrang vecors n VAR I wh varables WF WS PP PS PF E and nercep, ' I = and for marx of he wo conegrang vecors n VAR III wh varables WF, WS, PP, PS, PF, E, D1, D2, D3 and nercep 15

17 ' III = The heorecal resrcons whch exacly-denfy he conegraed vecors as he share equaons of an AIDS specfcaon n boh VAR I and VAR III models are gven by: H AIDS : = 1 = = 0 = 1 The coeffcens esmaes of he wo exacly-denfed conegraed vecors n VAR I and VAR III models are presened n Table 11 (asympoc raos n brackes). The hrd vecor corresponds o he share equaon for Porugal, and was rereved from he esmaes of he oher wo applyng he addng-up propery. Inser Table 11 here There s a sharp dfference beween he esmaes of VAR I and VAR III models, boh n magnude, sascal relevance and n her expeced sgns. For nsance, VAR I esmaes ndcae all coeffcens as sascally rrelevan a he 5% level n he share equaon for Porugal, and n he share equaons for France and Span only he prce of Span and he nercep as sgnfcan a he 5% level. In he equaon for Porugal, he own-prce and nercep esmaes have wrong sgns and mplausble magnudes and n he equaon for France, mplausble magnude s also he case for he nercep esmae. In conras, he esmaes of he conegraed VAR III model are overall sascally relevan, presen he expeced sgns and magnudes and gve plausble nformaon abou how he evens represened by he dummy varables affeced he UK oursm demand for he desnaons. Indeed, he coeffcens of D1 ndcae ha he ol crses and polcal changes n Porugal and Span, affeced sgnfcanly and negavely UK ourss preferences for hese desnaons, favourng France nsead n The coeffcens of D2 ndcae ha Span and Porugal s negraon process n he EU caused UK oursm flows o dver from France o he Iberan pennsula, alhough favourng more Span han Porugal. The D3 coeffcens, represenng he posnegraon perod, ndcae a recovery of he share for France a he expense of Span s whch seadly declnes (he openng of he channel unnel n 1994 may also have conrbue for hs resul). In he same perod, he share of Porugal shows an ncrease alhough no sascally sgnfcan. 16

18 We showed ha he nercep s a relevan deermnsc componen of he VAR specfcaons. We consdered ha me rends should no be ncluded as here s no sascal suppor for her presence. We found sascal suppor for no reecng he possbly of reang E as exogenous, and for ncludng varables D1, D2 and D3 as sascally relevan regressors. VAR III model negraes all hese feaures n s specfcaon whle VAR I consders only he former wo. Hence, he conssency and plausbly of resuls obaned wh VAR III n conras wh hose of VAR I, should be expeced. Hence, we consder VAR I no o be an approprae means o supply relable nformaon on he long-run demand behavour of UK ourss and proceed he analyss wh he conegraed VAR III model. In many cases, and parcularly n demand sysems, he focus of neres s on a se of hypohess relang o he long-run srucure of he model, whch s que ndependen of any shor-run dynamcs fed o he emprcal model (Chambers 1993, p.727). Lkewse, n he case of a VAR sysem for he UK oursm demand, our neres s focused on he srucural equaons upon whch parameer resrcons are o be mposed o es heorecal hypohess on he long-run equlbrum relaonshps. Theory suggess ha homogeney and symmery, reflecng he raonaly of consumers behavour, should hold n a sysem of demand equaons. In addon, and accordng o De Mello e al. s (2001) assumpon abou he compeve behavour of neghbourng desnaons, prce changes n France (Porugal) should no affec UK oursm demand for Porugal (France), whle prce changes n Span should affec sgnfcanly UK oursm demand for boh France and Porugal. To es hese hypohess we use he LR sasc on he conegraed VAR III model. Table 12 shows he ess resuls for he hypohess of null cross-prce effecs beween France and Porugal, homogeney and symmery and all hese hypohess smulaneously. The ess ndcae ha he se of hypohess canno be reeced. Hence, he conegraed VAR III comples wh he heorecal resrcons of homogeney and symmery and wh he asseron of null cross-prce effecs beween he share equaons for France and Porugal. Inser Table 12 here The coeffcens esmaes for he wo conegraed vecors of VAR III under he full se of resrcons are gven n Table 13 (asympoc raos n brackes). The resuls show he conegraed VAR III as a sascally robus, heorecally conssen and 17

19 emprcally plausble model, ndcang as an adequae bass for analysng he longrun behavour of he UK oursm demand for France Span and Porugal. Inser Table 13 here Even so, a more dealed nvesgaon requres he analyss of he relevan elascy values. We compue he expendure, and uncompensaed own- and cross-prce elasces, usng he long-run esmaes of he conegraed VAR III under he full se of resrcons and he formulae gven n Appendx A. Table 14 presens hese and he correspondng esmaes obaned wh he AIDS model of De Mello e al. (2001). 5 Inser Table 14 here The elasces esmaes of he VAR and AIDS models are smlar. The expendure elasces are close o uny for all desnaons n boh models and excep n he VAR equaon for Span, he own-prce elascy esmaes are close o 2. In he VAR equaon for Span hs esmae s roughly half of ha of s neghbours. The VAR cross-prce elasces esmaes gve he same ndcaons as hose of he AIDS model. For nsance, nsgnfcan cross-prce effecs beween he equaons for Porugal and France, ndcang ha he UK demand for Porugal (France) s no sensve o prce changes n France (Porugal), and sgnfcan cross-prce effecs beween Span and France and Span and Porugal, ndcang blaeral compeve behavour. These crossprce esmaes also mply ha he UK demand for Porugal or France s more sensve o prce-changes n Span han ha for Span s o prce changes n s neghbours. The resuls subsanae he mporance of he VAR approach and he Johansen s rank es n fndng sascal suppor for he exsence of he wo conegraed vecors whch endorse he share equaons of he AIDS sysem as he only meanngful long-run relaonshps. In addon, as showed by Pesaran and Shn (1998), also n he case of oursm share equaons he denfcaon of he long-run srucural model requres a pror nformaon provded by economc heory and knowledge of pragmac aspecs concernng he relaonshps beween he counres nvolved. Exogeney of regressors, marke condons, polcy regulaons, nsuonal aspecs, geographcal arbues, 5 We use De Mello e al. s (2001) esmaes for he second perod ( ), as hese correspond o more recen behavour of he UK oursm demand. 18

20 polcal upheavals and economc nsably, need o be appropraely modelled and esed whn a resrced VAR approach o oban heorecally conssen, emprcally plausble and sascally relable esmaes for he srucural parameers. As he conegraed VAR III model fully comples wh he heory predcons underlyng he AIDS approach, he smlary of her esmaes s no surprsng. Ths smlary gves furher suppor o he AIDS approach as a heorecal and emprcal robus means for economc analyss of long-run oursm demand. However, he analyss s no complee whou accessng he forecasng ably of he conegraed VAR III. 5. FORECASTING WITH A COINTEGRATED VAR SPECIFICATION For assessng he forecasng ably of conegraed VAR III under he full se of resrcons (denoed hereafer by over-var ) we esmae for he perod , leavng he las four observaons for forecasng purposes. To compare he forecasng accuracy of hs model wh ha of he unresrced VAR I of secon 2 (denoed hereafer pure-var ), and ha of he De Mello e al. s (2001) AIDS model, ables 15, 16 and 17 show he acual values, forecass and forecas errors of, respecvely, he oursm shares of France, Span and Porugal. 6 The correspondng summary sascs MAE, MSE and RMSE are repored n able 18. ************ Inser Tables 15, 16, 17, and 18 here ************ For an overvew of he hree models forecasng performance we dsplay Fgures 1, 2 and 3 showng a plo of he acual and forecased values of he desnaons oursm shares obaned wh he hree models. ******* Inser Fgures 1, 2, and 3 here ******* For he shares of France and Span, he over-var s he bes forecaser, followed by he AIDS model. For he share of Porugal, he pure-var s he bes forecaser followed by he over-var. For hs share, however, he dfferences beween he wo VAR models forecass are so small ha hey can be consdered equvalenly as excellen forecasers. 6 The forecass repored n De Mello e al. (2001) were compued for he las hree observaons ( ). The VAR forecass n hs sudy are compued for he las four observaons ( ). Hence, 19

21 Hence, we fnd VAR III specfcaon under he full se of heorecal resrcons as he bes forecasng devce. The forecasng performance of he AIDS model s also remarkable, parcularly n he cases of he share equaons for France and Span. Alhough he pure-var predcons are no as accurae as he oher models hey are, even so, farly precse. Ths brngs ou he unresrced VAR forecasng ably alongsde s remarkable quales of form smplcy, esmaon ease and assumpon mnmalsm. Indeed, f he man purpose of a research s o provde farly accurae forecass of expendure shares, he esmaon of a unresrced reduce-form VAR wh he approprae varables and lag-lengh, mgh be consdered he preferable approach, snce he qualy dfferences do no seem o usfy undergong he complexes of conegraon esng and denfcaon procedures of he srucural model. 6. CONCLUSION Economc models porrayng long-run relaonshps of me seres varables are a cenral aspec of heorecal and emprcal research. Par of he role of appled work s o esablsh approprae formal specfcaons whch can be consdered vald means o esmae he equlbrum pah of relevan varables and o es compeng heorecal hypohess underlyng hose specfcaons. However, esmaon of equaon sysems nvolvng long-run relaonshps of nonsaonary daa, dubous assumpons on he exogeney of regressors and he ncluson of zero resrcons wh poor economc bass, may lead o nvald nference and forecasng procedures. Ye, nference based on such sysems can be valdaed f one or more conegraed relaonshps exs and he exogeney assumpons and zero resrcons have heorecal and sascal suppor. Hence, ess mus be performed o deermne wheher a sysem s conegraed and s underlyng assumpons are sascally vald and, f so, proceed wh suable esmaon echnques whch can provde vald esmaes for s srucural parameers. Besdes esablshng a es procedure for deermnng he number of conegraed vecors, he Johansen approach provdes a general framework for denfcaon, esmaon and hypohess esng of he srucural model whn a VAR specfcaon. However, he Johansen s denfcaon process may no be approprae n economc conexs where heory provdes specfc rules o denfy he srucural parameers of o compare he forecasng ably of he VAR and AIDS models on an equal bass, we re-esmae he AIDS model of De Mello e al. for he perod , and oban forecass for he perod

22 he conegraed vecors. The AIDS model of he UK oursm demand for France, Span and Porugal, s a sysem of equaons whch ncludes nonsaonary I(1) seres and assumes exogeney for all s rgh-hand sde varables. The mplcaons of hese feaures can rsk he valdy of s esmaon resuls, sascal nference and forecass, f no conegrang relaonshps exs among he varables and/or he regressors exogeney assumpon does no hold. As an alernave o such an AIDS model, we specfed a reduced-form unresrced VAR sysem and used sascal ess o esablsh he lag-lengh, deermnsc componens and he endogenous/exogenous dvson of s varables. Once he approprae form of he VAR was n place, we used he Johansen rank es o deermne he number of conegraed vecors. Theory underlyng a sysem of equaons regressng wo desnaons oursm shares on oursm prces and an orgn s oursm budge, predcs he exsence of exacly wo long-run (conegraed) relaonshps. Therefore, n a VAR sysem wh he same varables, wo conegraed vecors should be accouned for. The conegraed rank es provded sascal evdence o suppor he heory predcons. The heorecal framework of an AIDS sysem of he UK oursm demand also esablshes he srucural form of he long-run relaonshps predcs. Hence, he srucural parameers of he conegraed vecors n he VAR should be exacly-denfed wh resrcons machng hose of he normalsaon process used o denfy he oursm share equaons of an AIDS sysem. Ths was done and he resulng srucural form of he conegraed VAR was hen subec o addonal resrcons such as homogeney, symmery and null cross-prce effecs beween he share equaons for Porugal and France. These resrcons could no be reeced. Consequenly, evdence was obaned on he capably of he conegraed VAR o comply wh heorecal predcons underlyng he raonaly of he UK oursm demand behavour and he desnaons compeve conduc. The esmaes of he srucural coeffcens of he conegraed VAR under he full se of resrcons were hen used o compue he expendure, own- and cross-prce elasces esmaes of he UK oursm demand. These esmaes, smlar o he correspondng ones obaned wh he AIDS model of De Mello e al. (2001), proved o be sascally relevan and emprcally plausble. Gven he heorecal conssency and sascal robusness of he conegraed VAR model under he full se of resrcons, s excellen predcve ably was expeced. Indeed, he qualy crera measurng he forecass accuracy of hs model s ndcae 21

23 as he bes forecasng model, when compared wh he reduce-form pure VAR and he AIDS models. Neverheless, hese same qualy crera sugges he reduced-form (unresrced) VAR o be a farly accurae forecasng devce. Ths fndng gves suppor o he clamed compeence of he VAR approach for forecasng purposes, and emphaszes he valuable quales of modellng smplcy and esmaon ease of a pure unresrced VAR for forecasng purposes. REFERENCES BUSINESS MONITOR MA6 ( ) Overseas Travel and Toursm, Governmen Sascal Servce, Cenral Sascal Offce, HMSO, London. CHAMBERS, M. J. (1993), Consumers demand n he long-run: some evdence from UK daa, Appled Economcs, 25, CHAREMZA, W. W. and DEADMAN D. F. (1997), New drecons n economerc pracce, 2 nd edon, Edward Elgar Publshng Lmed, Chelenham. DE MELLO, M., SINCLAIR, M. T. and PACK, A. (2001) A sysem of equaons model of UK oursm demand n neghbourng counres, Appled Economcs, forhcomng. DEATON, A. and MUELLBAUER, J. (1980a) Economcs and Consumer Behavour, Cambrdge Unversy Press, Cambrdge. DEATON, A. and MUELLBAUER, J. (1980b) An Almos Ideal Demand Sysem, The Amercan Economc Revew, 70, DICKEY, D. A. and W. A. FULLER (1979) Dsrbuons of he esmaors for auoregressve me seres wh a un roo, Journal of he Amercan Sascal Assocaon, 74, DICKEY, D. A. and W. A. FULLER (1981) Lkelhood rao sascs for auoregressve me seres wh a un roo, Economerca, 49, GRANGER, C. W. J. (1969) Invesgang causal relaons by economerc models and cross-specral mehods, Economerca, 37, GRIFFITHS, W. E., HILL R. C. and JUDGE G. G. (1993) Learnng and praccng economercs, John Wley & Sons, Inc., New York. HARRIS, R. I. D. (1995) Usng conegraon analyss n economerc modellng, Prence Hall, Hemel Hempsead. HARVEY, A. C. (1990) The economerc analyss of me seres, 2 nd edon, Phlp Allan, Hemel Hempsead. HENDRY, D. F. (1995) Dynamc Economercs, Oxford Unversy Press, Oxford. INTERNATIONAL MONETARY FUND ( ) Inernaonal Fnancal Sascs Yearbook, Inernaonal Moneary Fund, Washngon, D.C. 22

24 JOHANSEN, S. (1988) A sascal analyss of conegraon vecors. Journal of Economc Dynamcs and Conrol, 12, JOHANSEN, S. (1991) Esmaon and hypoheses esng of conegrang vecors n Gaussan vecor auoregressve models, Economerca, 59, JOHANSEN, S (1995) Idenfyng resrcons of lnear equaons wh applcaons o smulaneous equaons and conegraon, Journal of Economercs, 69, JOHANSEN, S (1996) Lkelhood-based nference n conegraon vecor auoregressve models, 2 nd edon, Oxford Unversy Press, Oxford. JOHANSEN, S. and JUSELIUS, K. (1990) Maxmum lkelhood esmaon and nference on conegraon wh applcaon o he demand for money. Oxford Bullen of Economcs and Sascs, 52, JOHANSEN, S. and JUSELIUS, K. (1992) Tesng srucural hypoheses n a mulvarae conegraon analyss of he PPP and UIP for he UK, Journal of Economercs, 53, PAPATHEODOROU, A. (1999) The demand for nernaonal oursm n he Mederranean regon, Appled Economcs, 31, PESARAN, M. H. (1997) The role of economc heory n modellng he long-run. Economc Journal, 107, PESARAN, M. H. and PESARAN, B. (1997) Workng wh mcrof 4.0: neracve economerc analyss, Oxford Unversy Press, New York. PESARAN, M. H. and SHIN, Y. (1998) Long-run srucural modellng, DAE Workng Papers Amalgamaed Seres no.9812, Deparmen of Appled Economcs, Unversy of Cambrdge. Revsed las n Aprl PESARAN, M. H., SHIN, Y. and SMITH, R. J. (2000) Srucural analyss of vecor error correcon models wh exogenous I(1) varables, Journal of Economercs, 97, PHILLIPS, P. C. B. and PERRON, P. (1988) Tesng for a un roo n me seres regresson, Bomerka, 75, SIMS, C. (1980) Macroeconomcs and realy. Economerca, 48, 1-48 Travel Trends: A repor on he 1994 (1995/6/7/8) nernaonal passenger survey, Governmen Sascal Servce, Cenral Sascal Offce, HMSO, London. ZELLNER, A. (1962) An effcen mehod of esmang seemngly unrelaed regressons and es for aggregaon bas, Journal of he Amercan Sascal Assocaon, 57,

25 Table 1: Un roo DF and ADF ess for he varables WF, WS, WP, PF, PS, PP and E Varable Tes Sasc AIC creron SBC creron Crcal value WF WF WS WS WP WP PF PF PS PS PP PP E E ADF(0) * 52.87* ADF(1) ADF(0) * 48.42* ADF(1) ADF(0) * 50.76* ADF(1) ADF(0) * 46.72* ADF(1) ADF(0) * 84.70* ADF(1) ADF(0) * 80.20* ADF(1) ADF(0) * 31.64* ADF(1) ADF(0) * 29.23* ADF(1) ADF(0) * 32.17* ADF(1) ADF(0) * 28.48* ADF(1) ADF(0) ADF(1) * 32.23* ADF(0) * 30.51* ADF(1) ADF(0) * 19.18* ADF(1) ADF(0) * 16.77* ADF(1) Table 2: AIC and SBC crera and adused LR es for selecng he order of he VAR Order (p) AIC SBC Adused LR es χ 2 (36) = (0.393) χ 2 (72) = (0.000) 24

26 Table 3. Esmaon resuls and sascal performance of he unresrced VAR I model REGRESSORS WF -1 (2.10) WS -1 (1.44) PP -1 (-1.14) PS -1 (4.85) PF -1 (-2.38) E -1 (-0.79) Inercep (-0.60) EQUATIONS WF WS PP PS PF E (-1.14) (-0.57) (1.56) (-4.45) (2.16) (0.00) (1.77) (-1.57) (-1.48) (7.05) (-0.49) (-1.29) (-1.23) (1.57) (-0.75) (-0.48) (1.10) (2.54) (-0.55) (-0.10) (0.57) (-1.32) (-0.78) (3.06) (-0.01) (0.17) (-1.35) (1.01) SLECTION CRITERIA AND DIAGNOSTIC STATISTICS (-0.15) (-0.61) (-1.62) (0.14) (1.09) (22.40) (0.59) AIC SBC Adused R F sasc Seral Correlaon 0.78(0.38) 2.48(0.12) 0.73(0.39) 0.19(0.67) 0.37(0.54) 0.58(0.45) Funconal Form 3.81(0.05) 8.66(0.00) 1.85(0.17) 7.39(0.01) 0.00(0.98) 0.18(0.67) Normaly 1.02(0.60) 1.56(0.46) 0.56(0.75) 4.82(0.09) 11.65(0.00) 1.66(0.44) Heeroscedascy 0.39(0.53) 1.19(0.28) 1.33(0.25) 0.28(0.60) 0.03(0.87) 0.21(0.65) Table 4. AIC and SBC selecon crera and dagnosc ess for he VAR II model SELECTION CRITERIA AND DIAGNOSTIC TESTS EQUATIONS WF WS PP PS PF AIC SBC Adused R F sasc Seral Correlaon 0.83(0.38) 2.16(0.14) 0.27(0.50) 0.46(0.67) 0.47(0.49) Funconal Form 3.28(0.07) 8.49(0.00) 0.70(0.40) 5.56(0.02) 0.02(0.90) Normaly 0.86(0.65) 1.55(0.46) 0.05(0.97) 3.04(0.22) 6.49(0.04) Heeroscedascy 0.42(0.52) 1.23(0.27) 0.84(0.36) 0.21(0.65) 0.00(0.95) 25