Forecasting Performance of Alternative Error Correction Models

Transcription

1 MPRA Munich Personal RePEc Archive Forecasing Performance of Alernaive Error Correcion Models Javed Iqbal Karachi Universiy 19. March 2011 Online a hps://mpra.ub.uni-muenchen.de/29826/ MPRA Paper No , posed 30. March :55 UTC

2 Forecasing Performance of Alernaive Error Correcion Models By Javed Iqbal Deparmen of Saisics Karachi Universiy Karachi Pakisan Preliminary draf: No for publicaion I hank he paricipans of 5 h Mahemaics Colloquium, IoBM Karachi for commens suggesions. 1

3 Absrac Forecasing Performance of Alernaive Error Correcion Models I is well esablished ha regression analysis on non-saionary ime series daa may yield spurious resuls. An earlier response o his problem was o run regression wih firs difference of variables. Bu his ransformaion desroys any long-run informaion embodied in he levels of variables. According o Granger Represenaion Theorem (Engle and Granger, 1987) if variables are co-inegraed, here exis an error correcion mechanism which incorporaes long run informaion in modeling changes in variables. This mechanism employs an addiional lag value of he disequilibrium error as an addiional variable in modeling changes in variables. I has been argued ha ECM performs beer for long run forecas han a simple firs difference or level regression. This process conribues o he lieraure in wo imporan ways. Firsly empirical evidence does no exis on he relaive meris of ECM arrived a using alernaive coinegraion echniques. The hree popular co-inegraion procedures considered are he Engle-Granger (1987) wo sep procedure, he Johansen (1988) mulivariae sysem based echnique and he recenly developed Auoregressive Disribued Lag based echnique of Pesaran e al. (1996, 2001). Secondly, earlier sudies on he forecasing performance of he ECM employed macroeconomic daa on developed economies i.e. he US and he UK. By employing daa form he Asian counries and using absolue version of he purchasing power pariy and money demand funcion his paper compares forecas accuracy of he hree alernaive error correcion models in forecasing he nominal exchange rae and moneary aggregae (M2). 2

4 1. Inroducion: The Granger Represenaion Theorem (Engle and Granger, 1987) enables simulaneous modeling of firs difference and he levels of he variables using an error correcion mechanism which provides he framework for esimaion, forecasing and esing of coinegraed sysems. If X and Y are individually I(1) variables are co-inegraed wih coinegraion vecor 1,, ) he general form of he ECM can be expressed as ( 0 1 A ( L) Y B( L) X ( Y 1 0 1X 1) u (1) wih he lag polynomials A( L) 1 a... 2 p 1 L a2l a p L ; 2 B( L) b b L b L... b q L q where he lag operaor is defined as i LY Y i. In his model he coefficiens in he A(L) and B(L) represen he impac of shor changes while he long run effecs are given by he co-inegraion vecor 1,, ) and he conrols he speed of adjusmen shor ( 0 1 run changes owards long run pah. As co-inegraion and ECM provides a unified framework of molding boh long and shor run an ineresing quesion for researcher was wheher incorporaing he long-run resricion in an error correcion models yields superiors forecas in comparison wih pure firs difference models which do no impose co-inegraion resricions. On a heoreical ground co-inegraion is expeced o yield beer forecas as poined by Sock (1995, p-1) who assers ha If variables are co-inegraed, heir values are linked over he long run, and imposing his informaion can produce subsanial improvemens in forecas over long horizons. This asserion is based on heoreical resuls by Engle and Yoo (1986) 3

5 ha long horizon forecass from he co-inegraed sysems saisfy he co-inegraion relaionship exacly and ha he coinegraion combinaion of variables can be forecas wih finie long-horizon forecas error variance. A simulaion sudy by Engle and Yoo (1987) shows ha he wo sep EG ECM provide beer forecas compared o unresriced VAR paricularly a longer horizons while he similar simulaion sudy by Chambers(1993) furher corroboraed his resul using a nonlinear one-sep ECM. Using he same experimenal se up as Engle and Yoo, Clemens and Hendry (1995) find ha over-differencing he sysem resuls in inferior forecasing performance. In a simulaion sudy using a four-dimensional VAR(2) Reinsel and Ahn (1992) show ha forecas gains from co-inegraed sysem depends on proper specificaion of he number of uni roos and under specifying he number of uni roos resuls in poor performance for en o weny five seps ahead forecass whereas overspecificaion resuls in inferior shor-erm forecass Afer he pioneering wo-sep esimaor of he ECM parameers proposed by Engle and Granger (1987) several ECM echniques have been developed. The Engle-Granger echnique can idenify only a single equilibrium relaionship among he variables under sudy. Johansen (1988) proposed a frame work of esimaion and esing of vecor error correcion model (VECM) based on vecor auo regression (VAR) equaions. The VECM can be expressed as: Y Y p 1 jy j u (2) j 1 4

6 The is an m m marix conaining he long-run parameers. If here are r coinegraion vecors hen can be expressed as a produc of ow marices as ' wher boh and are m r marices. The marix conains he coefficiens of long-run relaionship and he conains he speed of adjusmen parameers which is also inerpreed as he weigh wih which each co-inegraion vecors appears in a given equaion. This approach can accommodae muliple equilibrium relaionships in he VECM. Boh of hese esimaion echniques assume ha he variables o be modeled are I(1). Recenly Pesaran, Shin and Smih (1996) and Pesarn (2001) proposed a echniques based on Auoregressive Disribued Lag (ARDL) model which allows boh I(0) and I(1) variables hus poenially avoids pre-es bias. In he lieraure some sudies have compared forecas abiliy of he ECM resuling from he Engle-Granger and he Johansen VECM echnique. However he lieraure does no provide empirical evidence regarding he forecas accuracy of he ARDL based ECM and is comparison wih EG and Johansen echniques. In addiion, mos of he empirical evidence employing real daa in forecas comparison comes from he developed economies. This sudy provides empirical evidence of forecasing performance of he ECM resuling from he hree echniques from Asian counries. 2. The Lieraure Hoffman and Rasche (1996) compared he forecas performance of a coinegraed sysem relaive o he forecas performance of a comparable VAR ha fails o recognize ha he sysem is characerized by coinegraion. They considered coinegraed sysem composing hree vecors, a money demand represenaion, a Fisher equaion, and a risk 5

7 premium capured by an ineres rae differenial. The daa were from he US economy. They found ha he advanage in imposing co-ingraiaion appears only a loner forecas horizon and his is also sensiive o he appropriae daa ransformaion. They considered 8 years ou-sample forecas horizon. Jansen and Wang (2006) invesigaed he forecasing performance of he Error Correcion model arising from he co-inegraing relaionship beween he equiy yield on he S&P 500 and he bond yield relaive o ha of univariae models. They found ha he Fed Model improves on he univariae model for longer-horizon forecass, and he nonlinear vecor error correcion model performs even beer han is linear version. The forecas horizon hey considered was 10 years. Wang and Bessler (2004) employed five US agriculure ime series. They used annual daa from 1867 o 1966 for model specificaion and daa for 1966 o 2000 were used for ou-of sample forecas evaluaion. Their resuls favored ECM for hree o four seps ahead forecass. However he differences in forecas obained from differen models were no saisically significan. Lin and Tsay (1996) considered boh simulaed and financial and macroeconomic real daa from he UK, Canada, Germany, France and Japan and ineres rae daa from he US and Taiwan. Their resuls are conradicory as he simulaed daa yield beer forecas fro he ECM whereas for real daa he performance of ECM is mixed. They aribue his conradicion o deficiency in forecas error measure which does no recognize ha forecas are ied ogeher in he long-run. This brief lieraure review indicaes ha a bes he resuls on relaive meri of imposing co-inegraion consrain are mixed. If here is some advanage on using ECM i occurs a 6

8 longer horizon. An imporan observaion from his review is ha very few sudies employ daa from he less developed economies such as Eas and Souh Asian economies. Also no sudy has ye considered forecasing performance of he newly developed ARDL based co-inegraion. I has been argued ha ARDL has imporan advanages over he Engle-Granger and Johansen approaches. Firsly i can be applied irrespecive of wheher underlying repressors are I(0) or I(1). Secondly in simulaion sudies i performs beer han EG and Johansen co-inegraion es in small samples. Thirdly appropriae modificaion of he orders of he ARDL model is sufficien o simulaneously correc for residual serial correlaion and he problem of endogenous variables. 3.The daa and he models The economic models we considered are he Purchasing Power Pariy and he money demand funcion. The absolue PPP saes ha exchange rae beween wo currencies adjus o remove any arbirage opporuniies (buy in a low price marke and sell wih a profi in a high price marke). If PPP holds hen in he long run exchange rae equals raio of prices in he wo economies. i.e. he inercep equals zero and slope equals 1 in he equaion: log( e) 0 1 log( CPI Raio) u (3) Secondly we considered demand of real money balances depends posiively on ransacion volume (oupu level) and negaively on cos of holding cash i.e. ineres rae i.e. log( M / CPI ) 0 1 log( Y ) 2 R u (4) 7

9 Thus he ask is o forecas exchange rae (local currency per dollar) and money sock (M2) from he alernaive ECM resuling from he hree co-inegraion echniques. The quarerly daa (1978Q1-2009Q4) of en Asian counries are employed namely 1. Korea 2. Singapore 3. Malaysia 4. Indonesia 5. Thailand 6. Philippines 7. Sri Lanka 8. India 9. Pakisan 10. Bangladesh. Ineres rae is measured by discoun rae, lending rae or money marke rae (whichever is available for full sample period) Oupu is measured by manufacuring producion index which indicae significan seasonaliy so quarerly dummies are added in esimaion. The daa comes mosly from IFS. Thai manufacuring producion index is obained from Bank of Thailand. The oupu daa for Sri Lanka are no available so money demand resuls are no presened for Sri Lanka. In he empirical analysis possess cerain challenges e.g. EG and Johansen require he preesing for uni roo in he variables and sricly speaking are valid if variables are I(1). However ARDL does no need such pre-esing. Uni roo ess on all he series were conduced using ADF, Phillips-Perron and KPSS mehods. In some cases EG, Johansen and ARDL co-inegraion ess could no uncover any co-inegraion however in mos cases he co-inegraion evidence comes from significance of Error Correcion erm. The analysis is conduced for all counries despie hese limiaions. We considered quarerly daa from 1978Q1 o 2009Q4. For model specificaion and esimaion we employ daa from 1978Q1 o 1994Q4 and he forecas evaluaion is conduced from he period 2005Q1 and 2009Q4. We employ Mean Absolue Percenage Error (MAPE) o evaluae he forecas accuracy. This measure eliminaes he effec of 8

10 scaling of variables so ha forecas error from counries is comparable MAPE is given by: 1 MAPE 100 H H 1 Y Yˆ Y Where Y and Yˆ represen acual and forecas respecively. 5. Resuls and Discussion The following ables (Table 1 and Table 2) presen he comparison of forecas accuracy based on MAPE. The bes ECM model in each case is highlighed. Generally he ARDL appears o yield lower forecas errors followed by Johansen echnique. This is he case for money sock forecas (Table 2) where poenially more han one co-inegraion vecors are possible. For Bangladesh he EG ECM yields he bes forecas for he wo variables. For Malaysia Johansen echnique appears o be superior. For India and Singapore he ARDL echnique resuls in he lowes forecas error. The resuls for oher counries are mixed for he wo variables. Table 1: MAPE of exchange rae forecas for five year forecas horizon Engle COUNTRIES Granger Johansen ARDL Bangladesh India Indonesia Korea Malaysia Pakisan Philippines Singapore Sri Lanka Thailand Noes: Schwarz crieria selecs lag 1 as opimal for Engle-Granger mehod for all en counries. Regression of ECM model wih his opimal lags indicae ha error correcion erm is insignifican only in Sri Lanka. For VECM esimaion using Johansen echnique opimal lags are obained by choosing lags based on AIC crieria and hen deermined using AIC hen insignifican lags were removed using join F-es. Same number of lags for each variable was employed in his case. Trace and Max ess did no provide evidence of coinegraion in some cases bu subsequen analysis by VECM models indicae ha loading coefficiens 9

11 was insignifican only in Indonesia. In oher cases loading coefficien was significan wih negaive sign in a leas one VECM equaion. Opimal lags using Schwarz crieria for ARDL is 1 for all counries. Wih opimal lags ECT erm is insignifican only in Indonesia. Table 2: MAPE of M2 forecas for five year forecas horizon Engle COUNTRIES Granger Johansen ARDL Bangladesh India Indonesia Korea Malaysia Pakisan Philippines Singapore Thailand Noes: Opimal lags for Engle-Granger es are 1 for all counies. In some counries Engle-Granger ADF es did no uncover co-inegraion bu subsequen in ECM model error correcion erm is insignifican only in Korea, Malaysia and Pakisan. Opimal lags for Johansen vary over differen counries using same lags for each variable. Trace and Max saisics do no indicae co-inegraion bu in VECM models he loading coefficiens was insignifican only in Indonesia. Opimal lags using Schwarz crieria using ARDL mehod are four for Korea, Philippines, Pakisan and Bangladesh; hree for Singapore and one for India, Malaysia, Thailand. Wih opimal lags error correcion erm is insignifican only in Malaysia and Pakisan Manufacuring producion for Sri Lanka is no available so money demand esimaion is no possible. 6. Conclusion I is well known ha regression analysis on non-saionary ime series daa may be spurious (non-sense) if he underlying variables are no co-inegraed. Error correcion models provide a convenien soluion for esimaion, esing and forecasing. However here are now differen co-inegraion esimaion and esing echniques have been developed. In his paper we have compared he forecas accuracy of hree popular error correcion models ha are derived from he Engle-Granger, Johansen and he ARDL echniques. The resuls indicae ha in general he ECM based on boh he ARDL and Johansen echniques ouperform he Engle-Granger echnique. The ARDL ECM resuls 10

12 in he bes performance in abou 48% of he cases whereas he Johansen s ECM yields he bes performance in abou 36% cases. The ARDL echnique appears o be superior even in cases where more han one co-inegraion relaionships are possible i.e. money demand model which involve hree variables in he sysem. The average MAPE for exchange rae forecas across en counries is 15%, 18.4% and 22.5% for he ARDL, Johansen and he EG echniques respecively. The average MAPE for M2 forecass are 7.3%, 9.9% and 13.6% for he ARDL, Johansen and he EG echniques respecively. Thus our analysis provides evidence in favor of he ARDL based ECM. Also i will be ineresing o compare forecas of ECM from alernaive echniques which do no impose coinegraion e.g. ARIMA and VAR echniques. This is reserved for fuure research. 11

13 References Chambers, M.J. (1993). A noe on forecasing in co-inegraed sysems. Compuers Mehod Applicaion s, 25, Clemens, M. P. and D. F. Hendry (1995), Forecasing in co-inegraed sysems. Journal of Applied Economerics10, Engle, R.F. and Granger, C.W.J. (1987).Co-inegraion and error correcion: Represenaion, esimaion and esing, Economerica 55, Engle, R.F. and Yoo, B.S. (1987). Forecasing and esing in co-inegraed sysems, he Journal of Economerics, 35, Haigh, M.S. (2000). Co-inegraion, unbiased expecaions and forecasing in he BIFFEX freigh fuures marke. Journal of Fuures Marke, 20, Hoffman, D.L and Rasche, R. H. (1996). Assessing Forecas Performance in a Coinegraed Sysem. Journal of Applied Economerics, Vol. 11, Jansen, D.W and Wang, Z. (2006). Evaluaing he Fed Model of Sock Price Valuaion: an Ou-of-Sample Forecasing Perspecive. Economeric Analysis of Financial and Economic Time Series/Par B Advances in Economerics, Volume 20, Johansens S. (1998). Saisical analysis of coinegraing vecors. Journal of Economics of Economics Dynamics and Conrol, 12,

14 Lin, J.L. and Tsay, R.S. (1996). Co-inegraion consrain and Forecasing: An empirical examinaion. Journal of Applied Economerics, Vol. 11, Pesaran, M. H., Shin, Y., and Smih, R. J. (1996) Bounds Tesing Approaches o he Analysis of Level Relaionships DEA working paper 9622, Deparmen of Applied Economics, Universiy of Cambridge. Pesaran, M. H., Shin, Y. and Smih, R. J. (2001) Bounds Tesing Approaches o he Analysis of Level Relaionships Journal of Applied Economerics 16, Sock, J.H.(1995). Poin forecass and predicion inervals for long-horizon forecas. Manuscrip, J.F.K. School of Governmen, Harvard Universiy. Wang, Z and Bessler, D.A. (2004). Forecasing performance of mulivariae ime series models wih full and reduced rank: An empirical examinaion. Inernaional Journal of Forecasing 20,