A Vecor Error Correcion Forecasing Model of he Greek Economy Thanassis Kazanas 1 kazanas@aueb.gr Absrac This paper discusses he specificaion of Vecor Error Correcion forecasing models ha are anchored by long-run equilibrium relaionships suggesed by economic heory. These relaions are idenified in, and are common o, a broad class of macroeconomic models. The models include four variables such as he HICP, he unemploymen rae, he real GDP, he GDP deflaor, he 10-years governmen bond, he curren accoun o GDP raio and he expors o GDP raio. We examine he esimaed model s sabiliy, and following he wo-sep approach, we assess he forecasing power of he esimaed VECM by performing dynamic forecass wihin and ou of sample. 1. Inroducion Numerous sudies of macroeconomic ime-series daa sugges a need for careful specificaion of he model s mulivariae sochasic srucure. Following he classic work of Nelson and Plosser (1982), many sudies have demonsraed ha macroeconomic ime series daa likely include componens generaed by permanen (or a leas highly persisen) shocks. Ye, economic heory suggess ha a leas some subses of economic variables do no drif hrough ime independenly of each oher; ulimaely, some combinaion of he variables in hese subses, perhaps nonlinear, revers o he mean of a sable sochasic process. Granger (1981) defined variables whose individual daa generaing processes are well-described as being driven by permanen shocks as inegraed of order 1, or I(1), and defined hose subses of variables for which here exis combinaions (linear or nonlinear) ha are well described as being driven by a daa generaing process subjec o only ransiory shocks as coinegraed. Many coinegraion sudies have shown ha some individually I(1) variables including real money balances, real income, inflaion, and nominal ineres raes may be combined in linear relaionships ha are saionary, or I(0). Evidence on he saionariy of linear money demand relaions has been presened by Hoffman and Rasche (1991), Johansen and Juselius (1990), Baba, Hendry, and Sarr (1992), Sock and Wason (1993), Hoffman and Rasche (1996a), Crowder, Hoffman and Rasche (1999) and Lucas (1994), among ohers. Evidence in 1 Acknowledgmens: The auhor would like o hank Efhymios Tsionas for helpful commens and suggesions. 1
favor of an equaion ha links he income velociy of money o nominal ineres raes, in several counries, is presened by Hoffman, Rasche and Tieslau (1995). Mishkin (1992), Crowder and Hoffman (1996) and Crowder, Hoffman and Rasche (1999) presen evidence of a Fisher equaion, and Campbell and Shiller (1987, 1988) have examined coinegraion among yields on asses wih differen erms o mauriy. Anderson, Hoffman and Rasche (2002) esimae a VECM model for he US ha includes six variables real GDP, he GDP deflaor, he CPI, M1, he federal funds rae, and he consanmauriy yield on 10-year Treasury securiies and four coinegraing vecors. Their forecass from he model for he 1990s compare favorably o alernaives, including hose made by governmen agencies and privae forecasers. Chrisofidis, Kourellos and Sylianou (2004) esimae a four variable VAR as well as a VECM model for he Cyprus economy using nominal gross domesic produc, oal liquidiy (M2), he average deposi rae, and he consumer price index. The VECM esimaion is exremely significan, since i no only provides useful informaion on he long run equilibrium relaionship of he variables bu, in addiion, is he basis for forecasing analysis. Our sudy describes an applicaion of VECM models o he forecasing of imporan Greek macroeconomic variables in he following quarers. We use quarerly daa for he HICP, he unemploymen rae, he real GDP, he GDP deflaor, he curren accoun o GDP raio, he expors o GDP raio and he 10-years governmen bond. An ou-of-sample assessmen shows ha he qualiy of he forecass supplied by his model is saisfacory. Our paper is organized as follows. Secion 2 describes he VECM models as well as he associaed esimaion and forecasing mehods. Secion 3 presens he daa used in our sudy and examines he forecasing performance of VECM models esed on heir sample base and on an ou-of-sample basis. 2. Vecor Auoregressive models and Coinegraion Analysis 2.1. Vecor Auoregressive models The Vecor Auoregressive model (VAR) was popularized by Sims (1980) as a model which disregards he heoreical resricions of simulaneous equaion, or srucural, models. The model is formed by using characerisics of our daa; herefore here are no resricions ha are based on economic heory. However, economic heory sill has an imporance for VAR modeling 2
when i comes o he selecion of variables. According o Sims here should no be any disincion beween endogenous and exogenous variables when here is rue simulaneiy among a se of variables. The VAR model can be seen as a generalizaion of he univariae auoregressive model and is used o capure he linear inerdependencies in muliple ime series. Is purpose is o describe he evoluion of a se of k endogenous variables based on heir own lags and he lags of he oher variables in he model. Regarding he assumpions of he VAR model, here are no many ha need o be considered. This is because he VAR model les he daa deermine he model and uses no or lile heoreical informaion abou he relaionships beween he variables. Excep for he assumpion of whie noise disurbance erms, i is beneficial o assume ha all he variables in he VAR model are saionary, o avoid spurious relaionships and oher undesirable effecs. If he variables are no saionary, hey have o be ransformed ino saionariy by aking differences. A sandard k variables VAR model of order p has he following form: p y A y BX u 0 i i i 1 where y k R is he k 1 vecor of he I(1) endogenous variables. X is a vecor of deerminisic variables which migh include a rend and dummies, 0 k R is a vecor of inerceps, A i is a k k coefficien marix, B is a coefficien marix, and u R k is a vecor of innovaions. The selecion of he final VAR for every combinaion of variables is based on he crierion of saisical adequacy. A model is said o be saisically adequae if all he underlying assumpions of he model are suppored by he daa. This is crucial because, if our model is saisically adequae, we are able o suppor saisically hypohesis esing, forecasing, causaliy ess, ec. More precisely, we may es for normaliy, for saic and dynamic heeroskedasiciy, for serial correlaion, for non lineariy, for omied variables, as well as sabiliy. An imporan issue in model specificaion is also model parameer sabiliy. Ofen srucural breaks characerize macroeconomic variables over a long period of ime. 3
2.2. Coinegraion Analysis and Vecor Error Correcion Model Economic heory ofen suggess ha cerain groups of economic variables should be linked by a long-run equilibrium relaionship. Alhough he variables may drif away from equilibrium for a while, economic forces may be expeced o ac so as o resore equilibrium. Variables which are I(1) end o diverge as n because heir uncondiional variances are proporional o he sample size. Thus i migh seem ha such variables could never be expeced o obey any sor of long-run equilibrium relaionship. Bu, in fac, i is possible for a group of variables o be I(1) and ye for cerain linear combinaions of hose variables o be I(0). If ha is he case, he variables are said o be coinegraed. If a group of variables is coinegraed, hey mus obey an equilibrium relaionship in he long run, alhough hey may diverge subsanially from equilibrium in he shor run. A vecor error correcion model (VECM) is a resriced VAR model in differences. The VECM specificaion resrics he long-run behavior of he endogenous variables o converge o heir long-run equilibrium relaionships, while allowing for shor-run dynamics (see, for example, Engle and Granger (1987). This is done by including an error correcion mechanism (ECM) in he model, which has proven o be very useful when i comes o modeling nonsaionary ime series. The VECM formulaion of he corresponding VAR represenaion can be wrien as: p 1 y y y BX u 0 i i 1 i 1 The y 1 is he error correcion erm and he k r marix Π shows how he sysem reacs o deviaions from he long-run equilibrium. The shor-run dynamics are ruled by i. When r is zero hen a process in differences is appropriae and when r k hen in levels. For 0 r k here exiss an ECM ha pushes back deviaions from he long-run equilibrium (characerized by he co-inegraing relaions). For a solid review of he VECM, see, for example, Johansen (1988, 1991, 1995). We may es for coinegraion in he conex of a sysem of equaions. Johansen and Juselius (1990, 1992) propose a es of his ype, which is based on canonical correlaions, using a Likelihood Raio Tes. The applicaion of his es requires he inclusion of exogenous 4
variables, e.g., an inercep and rend in he longrun relaionship and a linear rend in he shorrun relaionship. In addiion, Johansen, Mosconi and Nielsen (2000) as well as Hungnes (2005) consider he presence of dummies in he coinegraion relaionship when he variables are affeced by a number of breaks. Afer finding evidence supporing he exisence of a coinegraing relaionship among he examined variables, someone may esimae a VECM. As menioned before, a VEC Model is a resriced VAR which has coinegraion relaions buil ino he specificaion so ha i resrics he long-run behaviour of he endogenous variables o converge o heir coinegraing relaionships while allowing for shor-run adjusmen dynamics. The coinegraion erm is known as he correcion erm since he deviaion from long-run equilibrium is correced gradually hrough a series of parial shor-run adjusmens. In he conex of he VECM esimaion, Pairwise Granger Causaliy Tess and Impulse Response Funcion analysis can be used for economic policy evaluaion (see, e.g. Sims, 1980). The Impulse Response Funcion is he pah followed by y as i reurns o equilibrium when we shock he sysem by changing one of he innovaions ( zero. u ) for one period and hen reurning i o Anoher way of characerizing he dynamic behaviour of a VAR sysem is hrough Forecas Error Variance Decomposiion, which separaes he variaion in an endogenous variable ino he componen shocks o he VAR. If, for example, shocks o one variable fail o explain he forecas error variances of anoher variable (a all horizons), he second variable is said o be exogenous wih respec o he firs one. The oher exreme case is if he shocks o one variable explain all forecas variance of he second variable a all horizons, so ha he second variable is enirely endogenous wih respec o he firs. Since coinegraion is presen, i is exremely significan o model he shor-run adjusemen srucure, i.e he feedbacks o deviaions from he long run relaions, because i can reveal informaion on he underlying economic srucure. Modeling he feedback mechanisms in coinegraed VAR models is ypically done by esing he significance of he feedback coefficiens. These ess are called weak exogeneiy ess, because cerain ses of zero resricions imply long run weak exogeneiy wih respec o he coinegraing parameers. The concep of weak exogeneiy was defined by Engle, Hendry and Richard (1983) and is closely relaed o esing he feedback coefficiens. If all bu one variable in a sysem are weakly exogenous, hen 5
efficien inference abou he coinegraion parameers can be conduced in a single equaion framework. Choosing valid weak exogeneiy resricions is of major imporance, because policy implicaions are someimes based on he shor-run adjusmen srucure. According o Johansen (1995), here is a Likelihood Raio Tes ha may be used o es weak exogeneiy. The VECM presens no only he long-run relaionship of he variables, bu i has an addiional significan advanage: forecasing. According o Anderson, Hoffman and Rasche (2002) we may perform a wo-sage echnique, where we esimae an economic relaion using he echnique of a VECM and, on a second sage, we assess he qualiy of forecas oucome. Thus, in he conex of sochasic simulaion analysis we apply dynamic forecass (muli-sep forecass) using a large number of ieraions wihin and ou of he ime bounds of he observaions of he sample. Afer forecasing, we assess how far he esimaed model has approximaed he real-hisorical values. The closer he forecass are o he real values, he beer he forecasing power of he VECM considered. The algorihm used for he implemenaion of ieraions is he well-known Gauss-Seidel, which works by evaluaing each equaion in he order ha i appears in he model, and uses he new value of he lef-hand variable in an equaion as he value of ha variable when i appears in any laer equaion. 3. Empirical analysis 3.1. Daa Our daa se covers he period from he firs quarer of 2000 unil he firs quarer of 2017. All series were downloaded from Eurosa and OECD daabases. Some variables ha published monhly have been convered o quarerly frequency by aking he average of he corresponding quarer. Our daa se includes he real GDP, he unemploymen rae, he harmonized index of consumer prices, he curren accoun o GDP raio, he expors o GDP raio, he GDP deflaor, he 10-years governmen bond, he oil price and he real GDP of euro area. Appendix A provides variable descripions and sources. All he series, excep for he harmonized index of consumer prices, he curren accoun o GDP raio and he oil price, were seasonally adjused. So, using he TRAMO/SEATS filer we proceed o seasonal adjusmen of hese series. Table 1 presens briefly he descripive saisics for hose variables, while Figure 1, Figure 2 and Figure 3 presens he level, he level in logarihms and he firs difference graph respecively. 6
Table 1: Descripive Saisics Mean Median Maximum Minimum Sd. Dev. Real GDP 53,006.03 52,103.90 63,334.50 45,479.80 6,069.05 Real GDP EURO 2,346,579.00 2,389,139.00 2,547,553.00 2,099,481.00 117,035.30 Unemploymen rae 15.12 10.70 27.83 7.53 7.19 (%) HICP 90.74 94.08 103.70 70.12 10.87 Deflaor 91.23 95.05 101.82 74.30 8.52 Oil Prices 64.77 59.13 122.46 19.35 32.03 GB10Y (%) 7.62 5.47 25.40 3.41 4.96 Curren Accoun o -0.08-0.08 0.01-0.16 0.05 GDP (%) Expors o GDP (%) 24.49 23.17 34.35 18.33 4.70 Figure 1: level presenaion of he variables Y Y_EURO UN 65,000 2,600,000 30 60,000 55,000 50,000 2,500,000 2,400,000 2,300,000 2,200,000 2,100,000 25 20 15 10 45,000 2,000,000 5 HICP GDP Deflaor Oil price 110 105 140 100 100 95 120 100 90 90 80 80 85 60 70 80 75 40 20 60 70 0 Governmen bond 10y Curren accoun o GDP Expors o GDP 30.05 36 25.00 32 20 15 10 -.05 -.10 28 24 5 -.15 20 0 -.20 16 7
Figures 1 and 2 sugges ha mos series have a rend, whereas he presence of srucural breaks is also obvious. I is crucial o incorporae he srucural breaks using dummies in he VAR model, since hey affec heir shor run as well heir long-run relaionship. A firs glance, i seems ha he real GDP, he unemploymen rae, he real GDP of euro area, he en year governmen bond and he oil price have a srucural break in 2008. The harmonized index of consumer prices and he curren accoun o GDP raio have a srucural break in 2010. The influence of he srucural break is more obvious in Figure 3, where he series are presened in firs differences. Figure 2: log presenaion of he variables LOG(Y) LOG(Y_EURO) LOG(UN) 11.1 14.80 3.6 11.0 10.9 10.8 14.75 14.70 14.65 14.60 3.2 2.8 2.4 10.7 2000 2002 2004 2006 2008 2010 2012 2014 2016 14.55 2000 2002 2004 2006 2008 2010 2012 2014 2016 2.0 2000 2002 2004 2006 2008 2010 2012 2014 2016 4.7 LOG(HICP) 4.7 LOG(DEFL) 5.0 LOG(OILP) 4.6 4.5 4.4 4.3 4.6 4.5 4.4 4.5 4.0 3.5 3.0 4.2 2000 2002 2004 2006 2008 2010 2012 2014 2016 4.3 2000 2002 2004 2006 2008 2010 2012 2014 2016 2.5 2000 2002 2004 2006 2008 2010 2012 2014 2016 LOG(GB10Y) 3.5 3.0 2.5 2.0 1.5 1.0 2000 2002 2004 2006 2008 2010 2012 2014 2016 LOG(expors o GDP) 3.6 3.5 3.4 3.3 3.2 3.1 3.0 2.9 2000 2002 2004 2006 2008 2010 2012 2014 2016 8
Figure 3: firs difference presenaion of he variables D(LOG(Y)) D(LOG(Y_EURO)) D(LOG(UN)).04.02.15.02.01.10.00 -.02.00 -.01 -.02.05.00 -.04 -.03 -.05 -.06 -.04 -.10 D(LOG(HICP)) D(LOG(DEFL)) D(LOG(OILP)).020.04.4.015.03.2.010.02.0.005.01 -.2.000.00 -.4 -.005 -.01 -.6 -.010 -.02 -.8 D(LOG(GB10Y)) D(curren accoun o GDP) D(LOG(expors o GDP)).4.06.15.2.04.02.10.05.0.00.00 -.2 -.02 -.04 -.05 -.10 -.4 -.06 -.15 9
3.2. Esimaion of he model 3.2.1 Vecor Auoregressive Model resuls The esimaion of a VAR model requires esing he sabiliy of he series, beginning wih uni roo ess because, when he series under invesigaion are no sable, hen he esimaed resuls are no valid (spurious regression). Afer esing for he exisence of a uni roo in he series in he conex of exogenous as well as endogenous breaks, we find ha all variables have a uni roo. Table 2: VAR Lag Order Selecion Crieria Model 1 Endogenous variables: LOG(Y) LOG(HICP) LOG(UN) CAY Exogenous variables: C D(LOG(OILP)) D(LOG(Y_EURO)) @TREND Lag LogL LR FPE AIC SC HQ 0 470.033 NA 1.01E-11-13.97025-13.43501-13.75906 1 822.9946 619.0405 3.18E-16-24.3383-23.26783* -23.91593 2 855.2666 52.62809* 1.95e-16* -24.83897-23.23327-24.20542* 3 870.1084 22.37687 2.07E-16-24.80333-22.6624-23.9586 4 888.7753 25.84657 1.99E-16-24.88540* -22.20923-23.82948 Model 2 Endogenous variables: LOG(Y) LOG(P) LOG(GB10Y) LOG(UN) LOG(XY) Exogenous variables: C @TREND Lag LogL LR FPE AIC SC HQ 0 340.9016 NA 2.61E-11-10.18159-9.847067-10.0496 1 726.9436 688.9364 3.92E-16-21.29057-20.11975* -20.82861 2 763.0767 58.92474* 2.83e-16* -21.63313* -19.626-20.84119* 3 777.8706 21.84951 4.04E-16-21.3191-18.47567-20.19718 4 794.325 21.77048 5.67E-16-21.05615-17.37642-19.60426 * indicaes lag order seleced by he crierion LR: sequenial modified LR es saisic (each es a 5% level), FPE: Final predicion error AIC: Akaike informaion crierion, SC: Schwarz informaion crierion, HQ: Hannan-Quinn informaion crierion So, we examine he shor-run relaionship among he series, hrough he esimaion of alernaive VAR models over he whole sample period. Specifically, we esimae VAR models using wo ses of variables. Firs, we use as endogenous variables he real GDP, he HICP, he 10
unemploymen rae and he curren accoun o GDP raio. Moreover, we use he real GDP of Eurozone and he oil prices as exogenous variables. The endogenous variables are in logarihms excep for he curren accoun and he exogenous variables ha are in firs differences of heir logarihms. The specificaion of model 1 follows: 2 2 2 2 y y y y y y euro y y y y, i i p, i i u, i i c, i i oil ye i 1 i 1 i 1 i 1 y y p u cay oil y 2 2 2 2 p p p p p p euro p p p y, i i p, i i u, i i c, i i oil ye i 1 i 1 i 1 i 1 p y p u cay oil y 2 2 2 2 u u u u u u euro u u u y, i i p, i i u, i i c, i i oil ye i 1 i 1 i 1 i 1 u y p u cay oil y 2 2 2 2 ca ca ca ca ca ca euro ca ca ca y, i i p, i i u, i i c, i i oil ye i 1 i 1 i 1 i 1 cay y p u cay oil y In he second se, we use he real GDP, he GDP deflaor, he unemploymen rae, he en year governmen bond of Greece and he expors o GDP raio. All variables are in logarihms. So, model 2 akes he following form: 2 2 2 2 2 y y y y y y y y y, i i p, i i u, i i gb, i i ex, i i i 1 i 1 i 1 i 1 i 1 y y p u gb exy 2 2 2 2 2 p p p p p p p p y, i i p, i i u, i i gb, i i ex, i i i 1 i 1 i 1 i 1 i 1 p y p u gb exy 2 2 2 2 2 u u u u u u u u y, i i p, i i u, i i gb, i i ex, i i i 1 i 1 i 1 i 1 i 1 u y p u gb exy 2 2 2 2 2 gb gb gb gb gb gb gb gb y, i i p, i i u, i i gb, i i ex, i i i 1 i 1 i 1 i 1 i 1 gb y p u gb exy 2 2 2 2 2 ex ex ex ex ex ex ex ex y, i i p, i i u, i i gb, i i ex, i i i 1 i 1 i 1 i 1 i 1 exy y p u gb exy In order o es he saisical adequacy assumpion, for he wo ses of variables, we employ a series of misspecificaion ess which can be found in Table 2. In ligh of he ess underaken, he VAR model includes wo lags, a consan and a rend for boh se of variables. The corresponding esimaed VAR models are presened in ables 3.1 and 3.2. According o he esimaion resuls, i is obvious ha our variables are conneced wih a shor-run relaionship. Tables 3.1 and 3.2 sugges ha here is a srong posiive relaionship 11
beween variables and heir firs lagged value excep for he curren accoun o GDP raio in model 1. Table 3.1: Vecor Auoregression Esimaes of Model 1 LOG(Y) LOG(HICP) LOG(UN) CAY LOG(Y(-1)) 0.597656-0.009773-0.040752-0.075136 [ 4.85626] [-0.31023] [-0.12562] [-0.48135] LOG(Y(-2)) 0.337481 0.022408-0.459169 0.153816 [ 2.57271] [ 0.66731] [-1.32791] [ 0.92449] LOG(HICP(-1)) -1.29366 1.275148 1.164292-0.031698 [-2.97548] [ 11.4575] [ 1.01591] [-0.05748] LOG(HICP(-2)) 1.340385-0.353241-0.416357-0.315286 [ 3.14425] [-3.23706] [-0.37052] [-0.58312] LOG(UN(-1)) -0.159695 0.023144 1.393038 0.081642-0.05207-0.01333-0.13725-0.06604 [-3.06708] [ 1.73646] [ 10.1497] [ 1.23625] LOG(UN(-2)) 0.111661-0.017021-0.563362 0.03585 [ 2.42719] [-1.44536] [-4.64565] [ 0.61440] CAY(-1) 0.033919-0.051579-0.108204 0.038747 [ 0.33496] [-1.98980] [-0.40536] [ 0.30168] CAY(-2) 0.178496-0.037731 0.195319 0.178855 [ 1.73210] [-1.43033] [ 0.71902] [ 1.36838] C 0.434353 0.210157 1.817236 0.845671 [ 0.85204] [ 1.61048] [ 1.35234] [ 1.30793] D(LOG(OILP)) 0.012667 0.004772-0.055802-0.006534 [ 1.30652] [ 1.92298] [-2.18351] [-0.53132] D(LOG(Y_EURO)) 1.027502 0.123427-0.009205-0.887018 [ 3.39256] [ 1.59202] [-0.01153] [-2.30908] @TREND -0.000383 0.000348-0.00224 0.001045 [-0.85328] [ 3.02372] [-1.89142] [ 1.83373] R-squared 0.992122 0.999524 0.99658 0.926458 Adj. R-squared 0.990547 0.999429 0.995896 0.91175 12
Log likelihood 880.1303 AIC -24.83971 Schwarz crierion -23.26023 Table 3.2: Vecor Auoregression Esimaes of Model 2 LOG(Y) LOG(P) LOG(GB10Y) LOG(UN) LOG(XY) LOG(Y(-1)) 0.695352 0.002997 0.658509-0.333354 0.639059 [ 4.71577] [ 0.03190] [ 0.48390] [-0.94023] [ 1.05500] LOG(Y(-2)) 0.295002 0.153331-1.864019-0.189786-0.275223 [ 1.97384] [ 1.61005] [-1.35141] [-0.52812] [-0.44826] LOG(P(-1)) 0.362446 0.469051 2.038829-0.491674 0.115842 [ 1.77244] [ 3.59974] [ 1.08034] [-0.99996] [ 0.13790] LOG(P(-2)) -0.403208 0.262111-0.419569 1.073096-0.114289 [-2.17014] [ 2.21395] [-0.24469] [ 2.40202] [-0.14974] LOG(GB10Y(-1)) -0.015932-0.005716 1.252636 0.03378 0.087047 [-1.25092] [-0.70440] [ 10.6572] [ 1.10307] [ 1.66374] LOG(GB10Y(-2)) 0.004562 0.016296-0.522582-0.01774-0.062197 [ 0.34735] [ 1.94714] [-4.31117] [-0.56173] [-1.15271] LOG(UN(-1)) -0.131775 0.063861 1.31817 1.160082-0.146023 [-2.18068] [ 1.65852] [ 2.36365] [ 7.98417] [-0.58822] LOG(UN(-2)) 0.158972-0.063622-1.490322-0.302467 0.346632 [ 2.73128] [-1.71545] [-2.77445] [-2.16125] [ 1.44969] LOG(XY(-1)) -0.060376 0.000947 0.494215-0.027154 0.676482 [-1.88516] [ 0.04642] [ 1.67206] [-0.35262] [ 5.14167] LOG(XY(-2)) 0.02269 0.006926-0.252387-0.058995 0.032458 [ 0.72402] [ 0.34684] [-0.87264] [-0.78290] [ 0.25212] C 0.364585-0.560786 6.118339 3.712286-3.590339 [ 0.55854] [-1.34827] [ 1.01564] [ 2.36525] [-1.33891] @TREND -0.00016 0.000907-0.003985-0.000576-0.000336 [-0.38953] [ 3.45962] [-1.04981] [-0.58260] [-0.19888] R-squared 0.990602 0.994197 0.960266 0.996605 0.94304 Adj. R-squared 0.988723 0.993037 0.952319 0.995926 0.931648 Log likelihood 781.8695 13
AIC -21.54834 Schwarz crierion -19.57399 -saisics in [ ] 3.2.2 Granger Causaliy Analysis Our esimaion resuls provide evidence which suppors he exisence of a shor run relaionship among he variables. In order o verify his correlaion we perform Granger Causaliy Tess, which are presened in Tables 4.1 and 4.2 for each model correspondingly. Paricularly, we es he null hypohesis ha here is no Granger Causaliy relaionship in he sysem, for he above wo VAR models. For each equaion in he VAR models, he ables display (Wald) saisics for he join significance of each and of all oher lagged endogenous variables in ha equaion. Consequenly, he resuls obained from he VAR models, are confirmed as well in he Granger Causaliy analysis. Table 4.1: Pairwise Granger Causaliy Tess-Block Exogeneiy Wald Tess Dependen variable: LOG(Y) Excluded Chi-sq df Prob. LOG(HICP) 9.968789 2 0.0068 LOG(UN) 9.455642 2 0.0088 CAY 3.256026 2 0.1963 All 38.78238 6 0 Dependen variable: LOG(HICP) Excluded Chi-sq df Prob. LOG(Y) 0.671763 2 0.7147 LOG(UN) 3.015286 2 0.2214 CAY 6.621385 2 0.0365 All 23.48745 6 0.0006 Dependen variable: LOG(UN) Excluded Chi-sq df Prob. LOG(Y) 7.232862 2 0.0269 LOG(HICP) 7.188783 2 0.0275 CAY 0.630585 2 0.7296 All 13.1224 6 0.0411 Dependen variable: CAY Excluded Chi-sq df Prob. LOG(Y) 1.171322 2 0.5567 14
LOG(HICP) 6.671691 2 0.0356 LOG(UN) 10.25958 2 0.0059 All 28.33065 6 0.0001 Table 4.2: Pairwise Granger Causaliy Tess-Block Exogeneiy Wald Tess Dependen variable: LOG(Y) Excluded Chi-sq df Prob. LOG(P) 4.7685 2 0.0922 LOG(GB10Y) 2.66451 2 0.2639 LOG(UN) 8.967688 2 0.0113 LOG(XY) 4.79716 2 0.0908 All 39.62806 8 0 Dependen variable: LOG(P) Excluded Chi-sq df Prob. LOG(Y) 9.577332 2 0.0083 LOG(GB10Y) 5.654132 2 0.0592 LOG(UN) 2.955682 2 0.2281 LOG(XY) 0.354617 2 0.8375 All 33.45129 8 0.0001 Dependen variable: LOG(GB10Y) Excluded Chi-sq df Prob. LOG(Y) 3.396451 2 0.183 LOG(P) 3.188408 2 0.2031 LOG(UN) 8.325613 2 0.0156 LOG(XY) 3.194155 2 0.2025 All 16.66993 8 0.0337 Dependen variable: LOG(UN) Excluded Chi-sq df Prob. LOG(Y) 7.166373 2 0.0278 LOG(P) 10.74706 2 0.0046 LOG(GB10Y) 1.474215 2 0.4785 LOG(XY) 2.795834 2 0.2471 All 26.65596 8 0.0008 Dependen variable: LOG(XY) Excluded Chi-sq df Prob. LOG(Y) 1.82696 2 0.4011 LOG(P) 0.022648 2 0.9887 LOG(GB10Y) 2.838917 2 0.2418 LOG(UN) 7.838473 2 0.0199 15
All 14.68694 8 0.0655 3.2.3 Coinegraion Analysis Alhough he VAR resuls provide informaion abou he shor-run relaionship beween he macroeconomic variables, neverheless we do no know wha heir long-run behaviour is. The VECM no only gives an answer o he quesion of wheher he shor-run relaionship of he variables is persisen, bu also allows us o perform forecasing. The esimaion of he VECM requires firs o es for he exisence of coinegraion. We follow he Johansen and Juselius (1990, 1992) approach which is based on canonical correlaions. As we deermine ha he number of lags is wo in he above VAR models hen we should impose acually one lag in he VECM, in he coinegraion es. The resuls are presened in Tables 5.1 and 5.2 for each model respecively. Table 5.1: Johansen Coinegraion Tes for Model 1 Trend assumpion: Linear deerminisic rend (resriced) Series: LOG(Y) LOG(HICP) LOG(UN) CAY Exogenous series: D(LOG(OILP)) D(LOG(Y_EURO)) Warning: Criical values assume no exogenous series Lags inerval (in firs differences): 1 o 1 Unresriced Coinegraion Rank Tes (Trace) Hypohesized Trace 0.05 No. of CE(s) Eigenvalue Saisic Criical Value Prob.** None * 0.590825 118.229 63.8761 0 A mos 1 * 0.427183 58.35703 42.91525 0.0008 A mos 2 0.161393 21.02541 25.87211 0.1784 A mos 3 0.128726 9.232543 12.51798 0.1665 Trace es indicaes 2 coinegraing eqn(s) a he 0.05 level Unresriced Coinegraion Rank Tes (Maximum Eigenvalue) Hypohesized Max-Eigen 0.05 No. of CE(s) Eigenvalue Saisic Criical Value Prob.** None * 0.590825 59.872 32.11832 0 A mos 1 * 0.427183 37.33162 25.82321 0.001 A mos 2 0.161393 11.79287 19.38704 0.4347 A mos 3 0.128726 9.232543 12.51798 0.1665 16
Max-eigenvalue es indicaes 2 coinegraing eqn(s) a he 0.05 level * denoes rejecion of he hypohesis a he 0.05 level **MacKinnon-Haug-Michelis (1999) p-values Table 5.2: Johansen Coinegraion Tes for Model 2 Trend assumpion: Linear deerminisic rend (resriced) Series: LOG(Y) LOG(P) LOG(GB10Y) LOG(UN) LOG(XY) Lags inerval (in firs differences): 1 o 1 Unresriced Coinegraion Rank Tes (Trace) Hypohesized Trace 0.05 No. of CE(s) Eigenvalue Saisic Criical Value Prob.** None * 0.526899 134.9179 88.8038 0 A mos 1 * 0.376983 84.77207 63.8761 0.0003 A mos 2 * 0.293844 53.06888 42.91525 0.0036 A mos 3 * 0.26001 29.75826 25.87211 0.0156 A mos 4 0.133276 9.58336 12.51798 0.1474 Trace es indicaes 4 coinegraing eqn(s) a he 0.05 level Unresriced Coinegraion Rank Tes (Maximum Eigenvalue) Hypohesized Max-Eigen 0.05 No. of CE(s) Eigenvalue Saisic Criical Value Prob.** None * 0.526899 50.14586 38.33101 0.0015 A mos 1 0.376983 31.70319 32.11832 0.0561 A mos 2 0.293844 23.31062 25.82321 0.1037 A mos 3 * 0.26001 20.1749 19.38704 0.0384 A mos 4 0.133276 9.58336 12.51798 0.1474 Max-eigenvalue es indicaes 1 coinegraing eqn(s) a he 0.05 level * denoes rejecion of he hypohesis a he 0.05 level **MacKinnon-Haug-Michelis (1999) p-values Table 5.1 suggess ha, aking ino accoun he Trace Saisic and he Maximal Eigenvalue Saisic, we idenify he exisence of wo coinegraing relaionships in he fourvariable VAR wih wo exogenous variables a he 5%. Regarding Table 5.2, he Trace Saisic indicaes he exisence of four coinegraing relaionships while he Maximal Eigenvalue Saisic of one coinegraing equaion. Taking ino consideraion he Maximal Eigenvalue Saisic we proceed wih one coinegraing equaion a he 5% in he five variable VAR. As a resul, since boh models exhibi wo and one coinegraing relaionships beween he variables respecively, we move a sep furher for he esimaion of wo VEC models which 17
require no only he variables o be linked in he shor run, bu o be relaed in he long run via he exisence of coinegraion. 3.2.4 Vecor Error Correcion Esimaion In his secion we esimae a VECM model based on he four-variable VAR model wih wo exogenous variables in which we idenify wo coinegraing relaionships. The specificaion of he firs model follows: y c c c y c p c u c cay d d d y d p d u d cay 1 11 1 2 3 1 4 1 5 1 6 1 12 1 2 3 1 4 1 5 1 6 1 + y p u cay oil y euro y 11 1 12 1 13 1 14 1 15 16 p c c c y c p c u c cay d d d y d p d u d cay 2 21 1 2 3 1 4 1 5 1 6 1 22 1 2 3 1 4 1 5 1 6 1 + y p u cay oil y euro p 21 1 22 1 23 1 24 1 25 26 u c c c y c p c u c cay d d d y d p d u d cay 3 31 1 2 3 1 4 1 5 1 6 1 32 1 2 3 1 4 1 5 1 6 1 + y p u cay oil y + euro u 31 1 32 1 33 1 34 1 35 36 cay c c c y c p c u c cay d d d y d p d u d cay 4 41 1 2 3 1 4 1 5 1 6 1 42 1 2 3 1 4 1 5 1 6 1 + y p u cay oil y + euro ca 41 1 42 1 43 1 44 1 45 46 The VECM resuls are presened in Table 6.1. The wo coinegraed equaions summarize he long run behavior of he variables. The unemploymen rae is relaed negaively wih real GDP and HICP while he curren accoun o GDP raio is relaed posiively wih real GDP and negaively wih HICP. 18
Table 6.1: Vecor Error Correcion Esimaes of Model 1 Coinegraing Eq CoinEq1 CoinEq2 LOG(Y(-1)) 1 0 LOG(HICP(-1)) 0 1 LOG(UN(-1)) 0.746981 0.033553 [ 7.23459] [ 0.81463] CAY(-1) -1.564394 0.712404 [-2.04075] [ 2.32964] @TREND(00Q1) -0.00067-0.003704 [-0.49522] [-6.86692] C -9.48235-4.250204 Error Correcion: D(LOG(Y)) D(LOG(HICP)) D(LOG(UN)) D(CAY) CoinEq1-0.077591 0.014011-0.138376 0.210679 [-2.04041] [ 1.43840] [-1.33210] [ 4.37153] CoinEq2 0.027445-0.071297 0.459444-0.483209 [ 0.27927] [-2.83219] [ 1.71143] [-3.87973] D(LOG(Y(-1))) -0.299551-0.03099 0.176936-0.227474 [-2.40563] [-0.97157] [ 0.52017] [-1.44144] D(LOG(HICP(-1))) -1.600233 0.415285 1.773348 0.588114 [-4.19051] [ 4.24544] [ 1.70000] [ 1.21522] D(LOG(UN(-1))) -0.111685 0.016424 0.670524 0.007209 [-2.63532] [ 1.51287] [ 5.79192] [ 0.13421] D(CAY(-1)) -0.126406 0.025325-0.472863-0.235765 [-1.27743] [ 0.99908] [-1.74933] [-1.87998] C 0.006169 0.002647-0.002563 0.000856 [ 2.54542] [ 4.26319] [-0.38711] [ 0.27852] D(LOG(OILP)) 0.011489 0.0051-0.057949-0.008633 [ 1.17603] [ 2.03811] [-2.17147] [-0.69728] D(LOG(Y_EURO)) 1.189203 0.08566-1.003227-1.116934 [ 4.20869] [ 1.18348] [-1.29975] [-3.11908] R-squared 0.589426 0.760776 0.59974 0.503118 Adj. R-squared 0.532796 0.727779 0.544531 0.434583 Log likelihood 869.6176 AIC -24.5856 Schwarz crierion -23.07193 -saisics in [ ] 19
Then we esimae a VECM model based on he five-variable VAR model in which we idenify one coinegraing relaionship. The VECM for model 2 follows: y c c c y c p c u c gb c exy 1 1 1 2 3 1 4 1 5 1 6 1 7 1 + y p u gb exy y 11 1 12 1 13 1 14 1 15 1 p c c c y c p c u c gb c exy 2 2 1 2 3 1 4 1 5 1 6 1 7 1 + y p u gb exy p 21 1 22 1 23 1 24 1 25 1 u c c c y c p c u c gb c exy 3 3 1 2 3 1 4 1 5 1 6 1 7 1 + y p u gb exy u 31 1 32 1 33 1 34 1 35 1 gb c c c y c p c u c gb c exy 4 4 1 2 3 1 4 1 5 1 6 1 7 1 + y p u gb exy gb 41 1 42 1 43 1 44 1 45 1 exy c c c y c p c u c gb c exy 5 5 1 2 3 1 4 1 5 1 6 1 7 1 + y p u gb exy exy 51 1 52 1 53 1 54 1 55 1 The VECM resuls are presened in Table 6.2. The one coinegraed equaion indicaes ha he deflaor is relaed posiively wih real GDP while he unemploymen rae, he en-year governmen bond and he expors o GDP raio are relaed negaively wih real GDP. 20
Table 6.2: Vecor Error Correcion Esimaes of Model 2 Coinegraing Eq CoinEq1 LOG(Y(-1)) 1 LOG(P(-1)) -1.813251 [-8.22962] LOG(GB10Y(-1)) 0.016916 [ 0.64719] LOG(UN(-1)) 0.061825 [ 1.36445] LOG(XY(-1)) 0.099599 [ 1.24597] @TREND(00Q1) 0.005125 [ 3.62503] C -3.384602 Error Correcion D(LOG(Y)) D(LOG(P)) D(LOG(GB10Y)) D(LOG(UN)) D(LOG(XY)) CoinEq1 0.055303 0.134215 0.14339-0.147515-0.290079 [ 1.51454] [ 6.23298] [ 0.43124] [-1.64083] [-1.96217] D(LOG(Y(-1))) -0.143866-0.161535 1.113195 0.179324 1.041283 [-0.93961] [-1.78903] [ 0.79842] [ 0.47569] [ 1.67976] D(LOG(P(-1))) 0.421395-0.233928 1.673773-0.484957-0.017445 [ 2.26105] [-2.12845] [ 0.98625] [-1.05686] [-0.02312] D(LOG(GB10Y(-1))) -0.018479-0.008002 0.404505 0.052236 0.078965 [-1.39882] [-1.02714] [ 3.36261] [ 1.60601] [ 1.47640] D(LOG(UN(-1))) -0.121881 0.088687 0.911171 0.52099-0.008679 [-2.32354] [ 2.86705] [ 1.90758] [ 4.03401] [-0.04087] D(LOG(XY(-1))) -0.048782-0.00654 0.233285-0.015115-0.055453-0.03103-0.0183-0.28259-0.07641-0.12565 C [-1.57189] [-0.35734] [ 0.82551] [-0.19781] [-0.44134] -0.000323 0.003535-0.015058 0.006599 0.004364 R-squared 0.407418 0.422743 0.27521 0.530699 0.102839 Adj. R-squared 0.34816 0.365018 0.202731 0.483769 0.013123 Log likelihood 739.4835 AIC -20.85025 Schwarz crierion -19.50111 -saisics in [ ] 21
3.2.5 Variance Decomposiion Analysis Using he esimaed models, which provide informaion for he long-run relaionship of he variables, we perform Variance Decomposiion Analysis which is a way o characerize he dynamic behavior of he models. Table 7.1 suggess ha in he long run, he variaion of real GDP depends also on shocks o oher variables. Specifically, his percenage increases hrough ime and, in he las period, 45% of he oal change on he variance is due o he res variables. A similar siuaion holds for he res variables wih a noable impac on curren accoun o GDP raio. Table 7.1: Variance Decomposiion Analysis of Model 1 Period Variance Decomposiion of: LOG(Y) LOG(HICP) LOG(UN) CAY depending on: LOG(Y) LOG(HICP) LOG(UN) CAY 1 100 81.93349 82.14983 83.0403 2 86.45101 74.43673 78.9368 73.58469 3 78.27917 70.02948 75.0162 58.1004 4 71.10336 66.91025 71.41415 46.73156 5 66.66653 65.24427 68.77732 38.692 6 63.25093 64.49884 66.9229 33.12338 7 60.79371 64.28488 65.63066 29.10132 8 58.81965 64.27218 64.68719 25.9204 9 57.24809 64.2428 63.96879 23.39324 10 55.94111 64.02293 63.39974 21.32649 The dynamic behavior of he second model is similar o ha of he firs. More specifically, Table 7.2 indicaes ha he impac on variance decomposiion of he GDP deflaor from oher variables is very srong. Through ime, he influence increases and in he las period, 52% of he variaion of GDP deflaor is due o he oher variables. Regarding he unemploymen rae, he impac on is variaion from he res variables increases reaching a level of 39% in he las period. Finally, he variaion of he res hree variables, namely he real GDP, he en-year governmen bond and he expors o GDP raio, depends also on shocks o oher variables on average 20%-25% during he las period. 22
Consequenly, in he long run, he link beween he variables becomes more significan, since he variaion of a variable is due no only o own, bu o shocks from oher variables oo. Table 7.2: Variance Decomposiion Analysis of Model 2 Period Variance Decomposiion of: LOG(Y) LOG(P) LOG(GB10Y) LOG(UN) LOG(XY) depending on: LOG(Y) LOG(P) LOG(GB10Y) LOG(UN) LOG(XY) 1 100 86.58745 92.91475 80.99054 98.02334 2 93.21738 73.92174 90.94771 80.1267 93.34485 3 90.00836 70.40339 88.50224 76.73418 92.21756 4 87.9087 68.93714 86.56102 73.5492 91.91664 5 86.09093 67.29713 85.15449 70.83589 91.4441 6 84.65423 65.2412 84.15034 68.44148 90.87414 7 83.53245 62.48846 83.43355 66.30596 90.21641 8 82.62466 58.79223 82.91994 64.38511 89.45064 9 81.87602 54.18156 82.55039 62.64186 88.58277 10 81.2477 48.90437 82.28425 61.05046 87.62429 3.2.6 Forecasing Performance The VECMs are used o produce medium-erm forecass for main macroeconomic variables. According o he esimaed models, we make forecass for he endogenous variables for he nex wo years (eigh quarers). Regarding he firs model, we need o obain forecased values for he wo exogenous variables, namely he oil prices and he real GDP of Eurozone. For his reason, we examine alernaive univariae auoregressive models for each one of he wo variables and choose he model wih he minimum roo mean squared error. So, for oil price we esimae an AR(3) specificaion while for he real GDP of Eurozone an AR(2) model. Then, we may esimae heir eigh-quarer ahead forecass and use hem in order o esimae he forecased values of he endogenous variables. The esimaed forecass of he endogenous variables are presened in Table 8. This able displays he average of he growh rae of he seasonally adjused real GDP, he growh rae of he HICP, he growh rae of he GDP deflaor, he unemploymen rae, he curren accoun o GDP raio and he expors o GDP raio. All values are annually averages. In a second sage, following Anderson e al (2002), we assess he forecasing performance of he esimaed VECMs. We esimae each model during he sample period 2000:1 o 2014:4 and make forecass for he nex eigh quarers. Then we compare he forecased values 23
wih acual daa for he periods 2015:1 o 2016:4 and compue he corresponding RMSE crierion. These resuls are presened in he las column of Table 8. We may see ha model 2 performs beer in erms of real GDP. Table 8: Forecass Model 1 Variables 2017 2018 RMSE Real GDP seasonally adjused -0.6% -0.08% 641.21 HICP 1.5% 1.00% 2.86 Unemploymen rae 22.7% 23.1% 0.12 Curren accouno GDP raio -1.6% -1.2% 0.01 4. Conclusion Model 2 Variables 2017 2018 RMSE Real GDP seasonally adjused 0.61% 1.11% 649.15 10-year governmen bond 6.87% 6.51% 2.15 GDP deflaor 0.7% 1.87% 4.12 Unemploymen rae 22.65% 22.62% 0.05 Expors o GDP raio 32.14% 32.22% 0.04 Noe: RMSE sands for Mean Squared Error. This sudy has performed a forecasing exercise involving wo ime series daases for Greece. Due o he idenificaion of coinegraing relaionships in he variables, shor-erm forecass of GDP are esimaed using Johansen s VECM esimaion mehod using an informaion se ha proxies for he componens of expendiure based GDP wihin an open economy framework. For his purpose, he models are esimaed using quarerly daa on real GDP, he GDP price deflaor, HICP, unemploymen rae, 10yr governmen bond raes, expors o GDP raio and he curren accoun o GDP raio over he sample period 2000:1 o 2017:1. Then seven quarers ou of sample forecass are generaed under each model framework. Moreover, we assess he forecasing performance of he esimaed VECMs esimaing each model during he sample period 2000:1 o 2014:4, making forecass for he nex eigh quarers and comparing he forecased values wih acual daa. In addiion o he forecass, an effor is made o examine he relaionships among he variables. 24
Developing his research furher could ake ino accoun he fac ha he models presened here are linear by heir naure, and herefore fail o ake ino accoun nonlineariies in he daa. One of he responses o his problem wihin he lieraure has been he developmen of DSGE models, which are capable of handling boh srucural changes, as well as nonlineariies. The curren rend in forecasing is dominaed by he use of calibraed and esimaed versions of DSGE models ha have been shown o produce beer forecass relaive o radiional forecasing mehods in many cases (see, e.g, Zimmerman (2001)). Anoher poenial area o furher develop he work presened here, could be o pool ogeher he informaion se ino a panel of European counries. Wihin a panel VECM framework, he predicive abiliy of a candidae variable wihin he informaion se could be explored for he enire panel of counries. Analysis such as his may reveal poenial inerdependencies wihin he European group of counries. References Anderson, R.G., Hoffman, D.L., Rasche, R.H., (2002), A vecor-error correcion forecasing model of he US economy, Journal of Macroeconomics, 23, pp. 569-598. Baba, Y., D.F. Hendry and R.M. Sarr (1992), The Demand for M1 in he U.S.A, 1960-1988, Review of Economic Sudies, 59, pp. 25-61. Engle, R.F., Hendry, D.F., Richard, J.F., (1983), Exogeneiy, Economerica, 51, pp. 277-304. Engle, R.F., Granger, C.W.J., (1987), Coinegraion and error correcion: represenaion, esimaion and esing, Economerica, 55, pp. 277-304. Campbell, J.Y. and R.J. Shiller (1987), Coinegraion and Tess of Presen Value Models, Journal of Poliical Economy, 95, pp. 1062-88. Campbell, J.Y. and R.J. Shiller (1988), Inerpreing Coinegraed Models, Journal of Economic Dynamics and Conrol, 12, pp. 505-22. Chrisofides, L., Kourellos, A., Sylianou, I., (2006), A small macroeconomic model of he Cyprus economy, Economic Analysis Papers, No 02-06, Economics Research Cenre, Universiy of Cyprus. 25
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