Forecast Intervals for Inflation in Romania

Transcription

1 Theoreical and Applied Economics Volume XVIII (011), No. 11(564), pp Forecas Inervals for Inflaion in Romania Mihaela BRATU Buchares Academy of Economic Sudies Absrac. In his paper I buil forecass inervals for he inflaion rae in Romania, using he quarerly prediced values provided by he Naional Bank of Romania for Firs, I used he hisorical errors mehod, which is he mos used mehod, especially by he cenral banks. Forecas inervals were buil considering ha he forecas error series is normally disribued of zero mean and sandard deviaion equal o he RMSE (roo mean squared error) corresponding o hisorical forecas errors. I inroduced as a measure of economic sae he indicaor relaive variance of he phenomenon a a specific ime in relaion wih he variance on he enire ime horizon. Then, I calculaed he relaive volailiy in order o know he change ha mus be brough o he roo mean squared error in order o ake ino accoun he sae of economy. Finally, I proposed a new way of building forecass inervals, when he dae series follows an auoregressive process of order 1. In his case he lengh of forecass inerval is smaller and I go a slighly higher relaive variance. I consider really necessary he building of forecass inervals, in order o have a measure of predicions uncerainy, which is quanified by he Naional Bank of Romania using he predicion inervals based on a simple mehodology. I calculaed he forecass inervals using MAE (mean absolue error), he indicaor chose by Naional Bank of Romania and he MSE (mean squared error) indicaor. Keywords: forecas inervals; hisorical forecass errors; roo mean squared error (RMSE); relaive variance; uncerainy. JEL Code: C3. REL Codes: 8F, 10B.

2 8 Mihaela Brau 1. Inroducion Economic forecass are used for a cerain purpose, mosly being relaed o he orienaion in aking economic decisions. However, hese forecass are affeced by uncerainy for which saisical measures of evaluaion are used. Public percepion ends o associae o macroeconomic forecass published by he governmen wih he precision domain, wih no prospec of uncerainy. Crozier shows ha he accompanying of forecass wih insrumens for measuring he uncerainy provides auonomy o public environmen involved in forecass developing. Simon shows ha he governmen uses various sraegies o minimize he uncerainy. Krause (00) demonsraes how risk managemen sraegies provide recommendaions on how o adap o changing economic condiions. Uncerainy is based essenially on associaing probabiliies o fuure evens verisimiliude. Sancu and Mihail (005) showed ha a he beginning of 50s he economic phenomena were analyzed from he deerminisic poin of view, bu he complexiy of economic behaviour made necessary he sochasic conceps in describing he evoluion of he economic processes and phenomena. Since predicing a variable by providing numerical values implies a high degree of uncerainy, he researchers have focused on he builing of inervals where he prediced value migh appear wih a cerain probabiliy. All he insiuions base heir forecass uncerainy on hisorical errors, bu even in his case Knüppel M. (009) poins ou ha he sudies based on his mehod of quanifing he uncerainy in lieraure are almos nonexisen, excep hose of Williams and Goodman. Chafield (1993) shows he necessiy o accompany he predicions by forecass inervals, which represen he uncerainy degree of variaion. The probabiliies of cerain evens can be given. Fair (000) emphasizes ha he possibiliy of an economic crisis should be specified wihin he forecas inerval. Afer a brief overview of he main achievemens in lieraure relaed o he consrucion of predicion inervals, I buil forecass inervals for quarerly inflaion rae prediced by he Naional Bank of Romania in using he hisorical errors mehod, aking hen ino accoun he sae of he economy. In addiion, given ha inflaion raes series follows an auoregressive process of order 1, I proposed a new mehod for building predicion inervals. Finally, I compared he quarerly forecasing inervals on horizon for inflaion in Romania using MAE wih hose using MSE as synheic indicaors of forecasing error.

3 Forecas Inervals for Inflaion in Romania 9. Forecas inervals The problem of building forecas inervals and he deerminaion of disribuions was approached quie lae in he lieraure, noable works in his area being wrien by Cogley, Adolfson, Clark and Jore, Giordani and Villiani. The resuls showed an imporan conclusion: in order o build a forecas inerval wih a cerain probabiliy, he model has o include variances deviaion in ime. Kjellberg and Villani (010) numbered he advanages and disadvanages of boh ypes of forecass, he ones based on models and hose buil by he expers. Forecas mehods based on models describe he complex relaionships using endogenous variables by is ransparence making easy he idenificaion of misakes ha generaed wrong predicions. The disadvanages are relaed o he difficuly of adaping he model o recen changes in he economy, as well as he oo simple form of he models. Chafield shows ha forecas inervals are ofen oo narrow no aking ino accoun he uncerainy relaed o model specificaion, problem ha is encounered also in he expers assessmen. Unlike he forecass based exclusive on models, exper assessmens modify immediaely o any change of informaion relaed o he prediced phenomenon. Disadvanages in expers assessmens are relaed jus o he low degree of ransparency, he difficuly of using many explanaory variables ouside an explici model. The way o build a forecass inerval is described by Granger, he rerospecive presenaion of he mehods being done by Chafield (1993). Chrisoffersen (1998) explains how o evaluae hese inervals while he mehods for measuring forecass densiy are inroduced only in 1999 by Diebold, who exends hem laer for bivariae daa. Wallis (003) is he firs one who proposes ess for forecass inervals, while Orok and Whieman (1997, 1998), Roberson (003) and Cogley (003) inroduce bayesian predicion inervals. Unlike oher mehods of building predicion inervals ha are specified in lieraure, he Bayesian ones also analyze he impac of esimaor error on inerval. Sock and Wason (1999, 003) specify he condiional disribuion funcion for k-seps-ahead forecass. Their approach is developed by Hansen (005), who buil asympoic forecass inervals o include he uncerainy deermined by he parameer esimaor. 3. Building predicion inervals based on hisorical forecas errors The building of inervals aking ino accoun he forecass accuracy is an effecive way o highligh he uncerainy ha accompanies any forecas made. In he following, I used hisorical forecas errors o deermine he forecas inerval for inflaion. I also used he projeced inflaion raes a he end of he year published by he Naional Bank of Romania for each quarer from 007 o 010. Forecas errors are calculaed as he difference beween expeced value and he regisered value. Forecas errors for each quarer are calculaed by RMSE.

4 30 Mihaela Brau Forecas inervals are buil considering ha he forecas error series is normally disribued of zero mean and sandard deviaion equal o he RMSE corresponding o hisorical forecas errors. For a probabiliy of (1-α), forecas inerval is calculaed: ( X ( k) zα / RMSE( k), X ( k) + zα / RMSE( k)), k = 1,..., K where: X ( k ) is puncual forecas for variable X + a ime ; z is he α/ quinile of sandardized normal disribuion. α / The able below displays he RMSE and lower and upper limis of he forecas inerval for inflaion prediced by he cenral bank wih a quarer before ( one-sep-ahead ). Table 1 The limis of he inflaion rae forecass inervals in Romania from 007 Q1 o 010 Q4 (based on hisorical forecass errors) Quarer RMSE Lower limi Upper limi 007 T T T T T T T T T T T T T T T T Source: calculaions made using daa from repors of inflaion of Naional Bank of Romania beween The forecas inervals based on RMSE are independen of he sae of he economy. Therefore, Blix and Sellin (1998) proposed he change of he mehod, so ha he inerval akes ino accoun of changes in he economy, muliplying RMSE by a facor of uncerainy subjecive chosen by he exper in forecasing. Anoher approach uses, for he series of observaions, a model in which ime varies. Theseries of quarerly inflaion raes follows an auoregressive AR process in which he series has a residual variance of sochasic ype. I is assumed he hypohesis ha errors are idenically disribued and follows a sandardized normal disribuion. Then, he regression model can be wrien: k

5 Forecas Inervals for Inflaion in Romania 31 ri = m + K k = 1 φ ( ri α e, k k m) + where α is he sandard deviaion of errors ln α = lnα 1 + ε, where ε follows a normal disribuion and lnα is a random walk We inroduce a new saisical measure called he relaive volailiy or relaive variance (variance of T momen in relaion wih he geomeric mean of variances corresponding o he inerval used o calculae RMSE), calculaed by αt he formula: β T = ; 1 1 and are he iniial momen and he final one 1 ˆ n n α = 1 of he period for which RMSE is calculaed, he ime of he inerval bounded of he wo momens is: n = 1 + 1, and αˆ T is a bayesian esimaion. The new inervals of variaion of forecas values will be calculaed as follows: X ( k) z α RMSE( k), X ( k) + z α RMSE( k)), k 1,..., K ( α / α / =. The relaive volailiy is 0,79. Relaive volailiy of Q4 of 010 was 1.79, which means ha a 6.1% decrease in he value of RMSE is necessary o ake ino accoun he changes in he economy. 4. The proposal of a new way o build forecas inervals for Romania Beween , inflaion raes calculaed a he end of he quarer may be represened by an AR process of order 1 (AR (1)). To deermine he inerval of variances of BNR predicions aking ino accoun he sae of he economy in each of he periods for which daa were recorded, he coefficien which muliplies RMSE is calculaed in differen way han ha recommended in he lieraure. Inflaion is modeled in as: r_inf = r_inf -1 + e. σ e For an AR process ( X = ϕ 1 X 1 + e ), he variance is: var( X ) =, 1+ ϕ1 where σ e AR process error variance. σ e 1.3 The variance of inflaion is: var( r _ inf ) = = = I inroduce as a measure of economic sae he indicaor δ relaivevariance of he phenomenon a a specific ime in relaion wih he variance on he enire [ et E( e )] ime horizon, which for T momen is calculaed as: δ T = = var( r _ inf)

6 3 Mihaela Brau Table The limis of he inflaion rae forecass inervals in Romania from 007 Q1 o 010 Q4 (based on own mehod) Quarer e [e E(e)] δ RMSE Lower limi Upper limi 007 T T T T T T T T T T T T T T T Source: calculaions made using daa from repors of inflaion of Naional Bank of Romania beween In his case, I obained a relaively large variance, which means ha i is necessary a decrease of RMSE value wih 66.1% if one akes ino he sae of he economy in he las quarer of The meodology used by naional bank of Romania and an alernaive o i Providing an evaluaion of uncerainy is relaed o he effeciveness wih which an insiuion fails o influence he economic aciviy. The mehodology used by BNR is a simple one, like measure of global medium uncerainy for he rae on inflaion based on is macroeconomic shor-erm forecas model is used he mean absolue error (MAE-mean absolue error). This synheic indicaor includes all effecs of unanicipaed pas shocks ha led he deviaion of he expeced values from he regisered ones. Based on his ype of error predicion, forecasing inervals are buil, BNR numbering several advanages of is mehodology: i considers all he previous shocks ha have affeced he rae of inflaion; i deermines a classificaion of he deviaions from he acual values in he hisory of projecions: deviaions ha deermined an overesimaion of he projeced inflaion and deviaions ha generaed an underesimaion; he mehodology excludes any arbirary assumpion abou he acion of individual risk facors;

7 Forecas Inervals for Inflaion in Romania 33 i allows he adjusmen of inervals of uncerainy, so ha hey reflec he assessmens of differen agens regarding he magniude of he fuure uncerainy in relaion wih he one of previous periods. Unlike he RMSE indicaor, he indicaor for forecasing error MAE is less sensiive o large predicion errors. If he daase is small MAE is recommended, bu he mos insiuions use RMSE as is uni of measuremen is he same as he one of he indicaor which is calculaed. RMSE is always a leas equal o he MAE. Equaliy occurs if he errors have he same magniude. The difference beween he MAE and he RMSE is higher, he greaer he variabiliy of he daa series. RMSE is affeced by generalized variance, he inerpolaion, he errors in he phase and by he presence of ouliers. I proposed a new measure for he predicion error by analogy wih he MSE (mean square error). This indicaor measures he difference beween he esimaor and he parameer value of his arge. In saisics, MSE is a funcion of risk, which signifies he expeced value of quadraic loss, ie i measures he mean square error. How MSE uni is he uni square of he indicaor, he square roo of MSE has a concree significance. By analogy wih he reasoning behind he calculaion of MSE, we consider he esimaor as he expeced value for he rae of inflaion (ri_p) and he parameer is acual value (ri_e) and i will resul: MSE(ri_p) = Var(ri_p) + [Bias ( ri _ p, ri _ e) ]. The resuled forecasing inervals are higher. Table 3 Quarerly forecasing inervals for inflaion in Romania on horizon (calculaed using he hisorical errors mehod, ex-pos echnical) li(mae) ls(mae) li(mse) ls(mse) 007T T T T T T T T T T T T T T T T Source: Processing based on daa from he BNR and he INS ( (li (MFA), li (MSE) and ls (MAE), ls (MSE), he lower respecively he upper forecasing inerval limi, inerval calculaed using MAE, respecively MSE).

8 34 Mihaela Brau I buil quarerly forecasing inervals on horizon for inflaion in Romania using MAE and MSE as synheic indicaors of forecasing error. I noiced ha he lengh of inervals based on MSE is lower, so he accuracy is higher. 6. Conclusions Based on daa of inflaion forecass provided quarerly by he Cenral Bank, forecas inervals were buil using he mehod of hisorical forecas errors. For Romania, when inflaion raes follows an AR (1), I have improved he echnique of building forecas inervals aking ino accoun he sae of he economy in each period for which daa were recorded. I recommend he use of inerval forecass by he Naional Bank of Romania based on RMSM or MSE, in order o have forecass accompanied by an objecive degree of uncerainy, fac ha improves he decisional process. References Blix, M., Sellin, P., Uncerainy Bands for Inflaion Forecass, Sveriges Riksbank Working Paper Series, 65, 1998 Chafield, C., Calculaing inerval forecass, Journal of Business and Economic Saisics, 11, 1993, pp Hansen, B., Inerval Forecass and Parameer Uncerainy, Journal of Economerics, 135, 005, pp Kjellberg, D., Villani, M., The Riksbank s communicaion of macroeconomic uncerainy, Economic Review, 1, 010, pp. -49 Knüppel, M., Schulefrankenfeld, G., How informaive are macroeconomic risk forecass? An examinaion of he Bank of England s inflaion forecass, Discussion Paper, Series 1, Economic Sudies, 1(14), 008, pp Krause, A., Coheren risk measuremen: an inroducion, Balance Shee, 10 (4), 00, pp Sancu, S., Mihail, N. (005). Decizii economice în condiţii de inceriudine, Ediura Economică, Bucureşi hp:// hp://