ECONOMETRIC MODELS AND THEIR ABILITY TO PREDICT GDP GROWTH OF THE CZECH REPUBLIC

ECONOMETRIC MODELS AND THEIR ABILITY TO PREDICT GDP GROWTH OF THE CZECH REPUBLIC [Ekoomerické modely a jejich schopos předvída růs HDP České republiky] Mila Bouda 1 1 Uiversiy of Ecoomics i Prague, Faculy of Iformaics ad Saisics, W. Churchill Sq. 4, 130 67 Prague, Czech Republic. Email:bouda.mil@sezam.cz Absrac: The paper deals wih forecasig he abiliy of he mos commo macroecoomic mehods. The mai goal is o predic he perceage GDP growh while usig may mehods ad a he ed assess he performace of hese mehods. The performace is measured by he Roo Mea Square Error saisics. Mehods used i his paper are: aive Auo Regressio wih oe lag, Vecor Auo Regressio wih wo lags, Bayesia Vecor Auo Regressio wih wo lags, Dyamic Sochasic Geeral Equilibrium (DSGE) ad a DSGE- VAR model which obais priors from a DSGE model ad he esimaes i like Vecor Auo Regressio. The ex coribuio of his paper is specificaio of a New Keyesia DSGE model ad codig i i Malab. Fial resuls summarize he performace of each mehod. O he oher had, srucural models as DSGE ad DSGE-VAR perform beer ha a bechmark. I case of a DSGE model i is maily caused by is srucural aure. This model also coais forward lookig variables which ake io accou he behavior of households ad firms which are basic corersoes of a NK DSGE model. The DSGE-VAR model performs beer ha a bechmark due o fac ha priors are ake from he DSGE model. I meas ha srucural iformaio ca be rasferred usig hese priors. Neverheless, accordig o he RMSE saisics he bes performig mehod is he DSGE model. Keywords: AR, VAR, BVAR, DSGE, DSGE-VAR. JEL classificaio: E12, E17 Doručeo redakci: 29.8.2013; Recezováo: 2.6.2014; 19.6.2014; Schváleo k publikováí: 23.9.2014 Iroducio Oe of he bigges problems of currely used macroecoomic models is he abiliy o predic srucural chages i he ecoomy. This issue is a very ho opic i he academic sphere as well as i he aioal baks. Currely, he ecoomerics allows a specificaio of various kids of models which ca be used for forecasig ad here is o geeral agreeme which mehod is he bes oe. The moivaio of his paper is o discuss all releva mehods ad assess heir predicio power. The mai goal of his paper is o predic he GDP growh of he Czech Republic while usig he followig mehods: aive Auo Regressio wih oe lag (AR(1)) which is used as a bechmark, Vecor Auo Regressio (VAR), Bayesia Vecor Auo Regressio (BVAR), Dyamic Sochasic Geeral Equilibrium (DSGE) ad Dyamic Sochasic Geeral Equilibrium Vecor Auo Regressio (DSGE-VAR). The firs meioed aive AR(1) is a bechmark which is a challege o overcome whils usig he oher mehods. The performace of hese esimaio mehods is measured by he Roo Mea Square Error (RMSE). A model wih a lower RMSE is geerally cosidered as a beer. The mai goal of he paper is o examie wheher he basic versios of curre ecoomeric ools are able o predic srucural chages i he ecoomy. Fially, he predicio abiliy of all mehods is assessed by he RMSE. Equally valuable is he specificaio ad codig of he New Keyesia DSGE model of he Czech Republic which is ake from Galí (2008). 5

1 Roo Mea Square Error (RMSE) This paper preses predicios of GDP usig he advaced ecoomeric mehods. Firs, i is ecessary o defie he measure which is used as a assessme of accuracy. For his purpose he Roo Mea Square Error (RMSE) is ake from Hušek (2007). RMSE is a frequely used measure of he differeces bewee values prediced by a model or a esimaor ad he values acually observed. The RMSE measure quaifies he effeciveess of each ecoomeric mehod ad i allows compariso of hese ecoomeric mehods. RMSE is defied as follows y yˆ 2 1 RMSE, (1) where is he umber of observaio, is he oal umber of observaios, y is he acual value ad ŷ is he fied value. 2 Daa The mehods use he various variables. The firs observed variable is Gross Domesic Produc (GDP) ad his variable is used i all models. The secod observed variable is iflaio. Time series are ake from he daabases of he Czech Naioal Bak called ARAD ad he Czech Saisical Office. Firs, oe has o rasform variables io he appropriae form. The GDP is seasoally adjused ad i cosa prices of year 2005. Nex, he GDP is rasformed io he year over year (YoY) perceage chages. Boh variables are i quarerly frequecies ad accordig o he KPSS es saioary. Figure 1 below shows he rasformed GDP ad iflaio. The ed of he ime series is highlighed because his ed will be esimaed by differe mehods ad hese values will be used as a bechmark. Figure 1: Boh observed variables Source: ow. May mehods have o fulfill he assumpio of saioariy. Therefore oe has o perform his saisical es. The saioariy may be verified by lookig a he daa or wih he exac saisical ess, see Greee (2012) or Arl ad Arlová (2009). The mehod used i his paper is a KPSS es, for deails see Kwiakowski (1992). The ull hypohesis says ha he ime series is saioary. This hypohesis is o rejeced (sigificace level is 5 %) for all of our ime series. Fially, we have 60 observaios ad he ime series are from Q1 1997 o Q4 2011. Daa from Q1 2012 o Q3 2012 are used for he evaluaio of forecased values. This period was chose because oe already has observed a ecoomic dowur i daa. If oe decided o choose he period from Q1 2008 o Q4 2009 he here is a problem wih daa because here is o ecoomic dowur i daa. I would be eve more challegig ha he 6

period from Q1 2012 o Q3 2012. I is impora o meio ha all models excep he AR(p) model use boh variables (GDP ad iflaio). AR(p) is uivariae echique ad for modelig is used oly GDP ime series. Each mehod uses differe legh of ime series because differe lags are applied i each model. Geerally, i ca be said ha his fac has egligible effec o he fial resuls. O he oher had, models wih lower lag are less complex bu heir advaage is ha hey may uilize more iformaio from ime series. I is impora o keep his iformaio i he model. 3 AR(p) The firs bechmark model is a aive AR(1). This model is furher used as a bechmark. The mai goal of all furher ecoomeric echiques is o overcome his AR(1). The esimaed model is expressed as GDP GDP (2) 1, where he GDP a ime is explaied oly by he GDP a ime 1. GDP is rasformed o he year over year perceage chages as is described i chaper 2. The esimaio of (2) is performed by a Ordiary Leas Squares (OLS) mehod. For deails abou AR(1) see Arl ad Arlová (2009). 4 VAR(p) The secod model used i his paper is a Vecor Auo Regressio (VAR) model which is a saisical model used o capure he liear ierdepedecies amog muliple ime series. Edogeous variables i he VAR are reaed symmerically i a srucural sese (alhough he esimaed quaiaive respose coefficies will o i geeral be he same); each variable has a equaio explaiig is evoluio based o is ow lags ad he lags of he oher model variables. VAR modelig does o require as much kowledge abou he forces ifluecig a variable as do srucural models wih simulaeous equaios: The oly prior kowledge required is a lis of variables which ca be hypohesized o affec each oher ier emporally. VAR models are esimaed usig he OLS mehod. For deails see Hušek (2007). I accordace wih he Schwarz Bayesia crierio (BIC), VAR(2) is esimaed which is cosruced as follows y 1y 1 2y 2 v, (3) where is a vecor of level cosas, y is a vecor of he observed curre ad lagged edogeous variables, i is a marix of he ukow parameers of edogeous variables ad is a vecor of ormally disribued error erms. Two edogeous variables are used for he predicio of GDP growh: he logarihmic differeces of GDP ad uemployme. As is meioed i chaper 2, boh edogeous variables fulfill he assumpio of saioariy. 5 BVAR(p) Cosider he followig VAR p model y y y x u (4) 11 p p, where 1 T is he ime idex, y is a colum vecor of edogeous variables, x is a u N 0, is he vecor of i.i.d. residuals, is colum vecor of exogeous variables, y y marix,,, 1 p are y y marices ad is y y marix. The marix form of he Bayesia form of our VAR model is Y X U. (5) u 7

I oher words i ca be wrie as 1 y1 y0 y1 p x 1 Y X. (6) p y T yt 1 yt p x T Before he Bayesia esimaio oe eeds o defie a prior disribuio over he parameers ad. Priors are made of hree compoes: diffuse prior, dummy observaios prior ad raiig sample prior. For more deails abou prior cosrucio see Doa (1984). For more deails abou he BVAR heory see Koop (2003). The fial esimaio is performed usig he Malab rouies developed by Sims (2003) ad he same daa as i he case of he VAR(2) model are used. 6 New Keyesia DSGE model The New Keyesia model (NKM) is formulaed accordig o Galí (2008) ad cosiss of hree ecoomic ages. Households purchase goods for cosumpio, hold moey ad bods, supply labor, ad maximize he expeced prese value of uiliy. Firms hire labor, produce ad sell differeiaed producs i moopolisically compeiive goods markes, ad maximize profis. The ceral bak corols he omial rae of ieres. Figure 2 shows he basic srucure ad dyamics of he NKM. The NKM cosiss of six geeral equilibrium equaios which are derived i appedix ad wo sochasic shocks defiiios. All equaios are log-liearized ad variables deoed by wave are gap variables. Le s sar wih he dyamic IS equaio 1 y i E 1 r E y 1, (7) where y is he oupu gap, i is he shor erm omial rae, E 1 is he expeced iflaio i he ex period, r is he aural rae of ieres, E y represes he expeced oupu gap i he ex period ad fially he parameer is he coefficie of risk aversio. Figure 2: The srucure of he New Keyesia model 1 Source: ow. 8

The secod equaio is called he New Keyesia Phillips curve (NKPC) E y (8) 1, where is he iflaio, is he household discou facor, ad i is he 1 oupu gap elasiciy of iflaio, is he coefficie of risk aversio, is he elasiciy of 1 1 1 1 labor supply, is he share of capial ad. 1 The hird equaio shows he evoluio of he aural rae of ieres r yae a 1, (9) where r is he aural rae of ieres, is he real ieres rae i he seady sae, 1 ya ad E a 1 is he expeced chage of echology progress i he 1 ex period. The fourh equaio is he ieres rae rule of he ceral bak, usually called a Taylor rule. For deails see Taylor (1993) i y, (10) y where i is he omial ieres rae, ad y is he sesiiviy of he ceral bak wih is a exogeous respec o iflaio ad oupu gap (boh are chose by he ceral bak), sochasic compoe wih a zero mea. The fifh equaio represes he producio fucio cosisig of echology ad labor y a 1, (11) where y is he oupu, a is he level of echology ad is he umber of worked hours. The sixh equaio is he ad-hoc moey demad m y i, (12) where m is he moey demad, is he elasiciy of he moey demad wih respec o he omial ieres rae. The las wo equaios represe he sochasic shocks. The firs oe is a echology shock which follows he AR(1) process a a a (13) 1, where he persisece of he echology shock 0;1 ad a is a zero mea whie oise process. The secod is a moeary policy shock which follows he AR(1) process (14) 1, where he persisece of he moeary policy shock 0;1. Posiive (egaive) realizaio of is ierpreed as a coracioary (expasioary) moeary policy shock, leadig o a rise (declie) i he omial ieres rae, give iflaio, ad he oupu gap. The NKM model coais he same observable variables as VAR(2) ad BVAR(2) models. I keeps cosisecy ad comparabiliy of hese models. The esimaio is performed usig he Dyare which is he mos powerful Malab package ha ca be used for he esimaio of geeral equilibrium models. The model is cosidered as a fully calibraed ad i meas ha all parameers i he model have fixed values or if hey are esimaed he here is some prior disribuio for hese parameers se. The priors of he parameers,,, y are ake from 9

he HUBERT model which is used by he Miisry of Fiace of he Czech Republic, see Šork e al. (2009). The res of parameers are calibraed accordig Galí e al. (2001). These were valid for he US ecoomy, hus I expec ha hese priors are i accordace wih he Czech ecoomy. For deailed calibraio see Table 1. Table 1: The calibraio of all srucural parameers Parameer Prior Descripio 0.5 share of capial 0.99 discou facor 1.5 elasiciy of subsiuio, m log 1, m 1.1 0.698 measure of price sickiess, 0 = prices are absoluely flexible 1 1 1 1 1 0.154 log real ieres rae i he seady sae, 0.0101 1 coefficie of risk aversio 1 elasiciy of labor supply 1.5 sesiiviy of he ceral bak wih respec o he iflaio 0.25 sesiiviy of he ceral bak wih respec o he oupu gap y a 0.975 persisece of he echology shock 0.5 persisece of he moeary policy shock 4 elasiciy of moey demad wih respec o he omial ieres rae Source: ow. The equilibrium of he NKM is characerized by he equaios (7) (14). These equaios are rewrie io he Dyare. I is a ope-source add-o which works i Malab or Ocave. For more deails see Adjemia (2012). I he Dyare a comprehesive package of Bayesia echiques is implemeed which are used for he esimae. These echiques are described i Koop (2003) or Hamilo (1994). The algorihm which is implemeed i he Dyare is described i Schorfheide (2000) ad if you wa o lear wih he Dyare package he read Griffoli (2010). The whole process of he esimaio of DSGE models is described by Villaverde (2009). The mai goal is o fid he poserior disribuio of all ukow parameers (codiioal o observed daa) ad i is performed by usig he Bayesia rule for he codiioal probabiliy. The poserior disribuio is obaied by he combiaio of a likelihood fucio ad prior disribuios of esimaed parameers. The likelihood fucio is esimaed by he Kalma filer. The poserior disribuio is very ofe a ukow disribuio ad hus i is ecessary o use a umerical echique o geerae he radom samples. The Dyare uses for his purpose he Meropolis-Hasigs algorihm. This algorihm allows he calculaio of he basic saisics ad momes. Table 2 coais parameer esimaes. Oe may see ha he prior ad he poserior values are similar. I meas ha he prior values were correcly calibraed. O he oher had here are very wide cofidece iervals. I is caused by he shor ime series. The Czech ecoomy offers oly 51 observaios ad i is o sufficie. If he resuls of he esimaio are isered io (10) we ca see ha he ceral bak is very sesiive o he level of iflaio ( ). O he oher had, he sesiiviy o he oupu gap ( y ) is o very suig. The elasiciy of labor supply ( ) is calibraed o oe ad he poserior mea is 0.79 ha meas ha he labor supply is o elasic i he Czech Republic. The share of capial is equal o 0.50 which meas 10

ha he share of labor is also 0.50. This kowledge ca be used for he fuure calibraio of he Cobb-Douglas producio fucio. Table 2: Esimaio resuls parameer prior poserior lower upper disribuio ps. dev. 0.50 0.4978 0.4177 0.5791 bea 0.05 1.00 0.7928 0.7140 0.8744 bea 0.05 1.50 1.4946 1.4144 1.5810 orm 0.05 0.25 0.2416 0.1695 0.3206 orm 0.05 y Source: ow. 7 DSGE-VAR This cocep is based o he recogiio of Sims (1980) ha here is a igh relaioship bewee dyamic equilibrium models ad VARs. Le us imagie he followig experime, where for he mome he vecor of he DSGE model parameers is fixed. Oe millio observaios are geeraed from he DSGE model. We geerae a sequece of shocks (moeary ad echology), feed hem hrough he DSGE model ad obai arificial daa. Nex, oe esimaes a VAR wih p lags o hese arificial daa. If he DSGE model is covariace saioary, he he esimaed VAR provides a approximaio o he DSGE model wih he propery ha is firs p auocovariaces of he DSGE model. By icludig more ad more lags oe ca mach more ad more auocovariaces ad icrease he accuracy of he VAR approximaio of he DSGE model. Now le us imagie ha he daa geeraio is repeaed by usig differe parameer values for he DSGE model. As log as he DSGE model parameer space is small compared o he VAR parameer space, a resricio fucio ca be raced ha maps he DSGE parameers io a VAR parameer subspace. Hece, esimaig a DSGE model is almos like esimaig a VAR wih cross-equaio resricio. Isead of dogmaically imposig he cross-coefficie resricios implied by he DSGE model o he VAR, oe will allow for deviaios. The overall magiude of hese deviaios is corolled by a hyperparameer,. Roughly speakig, if, he he resricios are sricly eforced, whereas if 0, he resricios are compleely igored i he esimaio of he VAR parameers. Compuaioal deails are described by Del Negro ad Schorfheide (2004). 8 Resuls Models which are specified i his paper are assessed by he RMSE which is defied by (1). This RMSE is able o assess he power of predicio. The lower value of he RMSE is beer fi o empirical daa. The summarizaio of he RMSE ad predicios of GDP are i Table 3. Table 3: Roo Mea Square Error (RMSE) of he esimaed models Model RMSE Q1 2012 Q2 2012 Q3 2012 AR(1) 2.2375 1.0 1.2 1.3 VAR(2) 2.4019 1.1 1.3 1.6 BVAR(2) 2.5249 1.3 1.4 1.6 DSGE 1.5041 0.6 0.4 0.4 DSGE-VAR(2) 2.0496 1.2 1.0 0.9 Source: ow. Figure 3 (lef graph) shows he ime series of he perceage YoY GDP growh which is a arge ad predicios (righ graph) obaied by mehods described i previous chapers. 11

As oe ca see i Figure 3 ad Table 3 he aive AR(1) model does o fi he observed GDP very well. O he oher had, his bechmark is o overcome by VAR(2) ad BVAR(2). This fac is caused by he mai characerisics of boh models. They obai iformaio oly from hisorical values of he observed variables. The mos appropriae ecoomeric ool is he DSGE model. This resul is maily caused by he specificaio of he model. The DSGE model cosiss of srucural equaios ad coais raioal expecaios. These properies improve forecas i periods wih sudde chages. The DSGE-VAR model is he secod bes performig mehod. I works like VAR(2) bu priors are obaied from he DSGE model. Figure 3: YoY growh of GDP (lef) ad predicios (righ) Source: ow. Coclusio The paper deals wih forecasig he abiliy of he mos commo macroecoomic mehods. The mai goal is o predic he perceage GDP growh while usig may mehods ad a he ed assess he performace of hese mehods. Models are esimaed i he raiig period from Q1 1997 o Q4 2011 ad validaed i he period from Q1 2012 o Q3 2012. The validaio is measured by he Roo Mea Square Error (RMSE) saisics. Mehods used i his paper are: Auo Regressio wih oe lag (AR(1)), Vecor Auo Regressio wih wo lags (VAR(2)), Bayesia Vecor Auo Regressio wih wo lags (BVAR(2)), Dyamic Sochasic Geeral Equilibrium (DSGE) ad Dyamic Sochasic Geeral Equilibrium Vecor Auo Regressio (DSGE-VAR) which obais priors from he DSGE model ad he esimaes i like VAR(2). Aoher coribuio of his paper is he specificaio of he New Keyesia (NK) DSGE model which is ake from Galí (2008) ad codig of his model i Malab. The NK model is specified as a closed ecoomy model which is a cerai simplificaio. The ope-ecoomy model is more complex ad geerally oe ca expec ha he performace of his model would be beer i case of he Czech Republic. Models specified i his paper are heoreically described ad for he iquisiive reader here are also may refereces o foreig ad domesic lieraure. Fial resuls are ierpreed as follows. The VAR(2) ad BVAR(2) models perform worse ha he bechmark AR(1) model which is caused by fac ha hese models obai oly hisorical iformaio ad his hisory is limiig hese models. O he oher had, srucural models as he DSGE ad DSGE-VAR perform beer ha a bechmark. I case of he DSGE model i is maily caused by is srucural specificaio. This model also coais forward lookig variables which ake io accou he aural behavior of households ad firms. These wo ages (households ad firms) are basic corersoes of he NK DSGE model. The DSGE-VAR model performs well due o fac ha priors are ake from he DSGE model. I meas ha srucural iformaio ca be rasferred by usig hese priors. 12

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