Syddansk Universitet. Forecasting Inflation Autoregressive Integrated Moving Average Model Iqbal, Muhammad; Naveed, Amjad

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1 Syddansk Universie Forecasing Inflaion Auoregressive Inegraed Moving Average Model Iqbal, Muhammad; Naveed, Amjad Published in: European Scienific Journal DOI: /esj.2016.v12n1p83 Publicaion dae: 2016 Documen version Publisher's PDF, also known as Version of record Documen license CC BY Ciaion for pulished version (APA): Iqbal, M., & Naveed, A. (2016). Forecasing Inflaion: Auoregressive Inegraed Moving Average Model. European Scienific Journal, 12(1), hps://doi.org/ /esj.2016.v12n1p83 General righs Copyrigh and moral righs for he publicaions made accessible in he public poral are reained by he auhors and/or oher copyrigh owners and i is a condiion of accessing publicaions ha users recognise and abide by he legal requiremens associaed wih hese righs. Users may download and prin one copy of any publicaion from he public poral for he purpose of privae sudy or research. You may no furher disribue he maerial or use i for any profi-making aciviy or commercial gain You may freely disribue he URL idenifying he publicaion in he public poral? Take down policy If you believe ha his documen breaches copyrigh please conac us providing deails, and we will remove access o he work immediaely and invesigae your claim. Download dae: 26. Dec. 2018

2 Forecasing Inflaion: Auoregressive Inegraed Moving Average Model Muhammad Iqbal, PhD fellow Deparmen of Economics, Universiy of Vienna, Ausria Amjad Naveed, Posdoc fellow Deparmen of Business and Economics, Universiy of Souhern Denmark, Denmark doi: /esj.2016.v12n1p83 URL:hp://dx.doi.org/ /esj.2016.v12n1p83 Absrac This sudy compares he forecasing performance of various Auoregressive inegraed moving average (ARIMA) models by using ime series daa. Primarily, The Box-Jenkins approach is considered here for forecasing. For empirical analysis, we used CPI as a proxy for inflaion and employed quarerly daa from 1970 o 2006 for Pakisan. The sudy classified wo imporan models for forecasing ou of many exising by aking ino accoun various iniial seps such as idenificaion, he order of inegraion and es for comparison. However, laer model 2 urn ou o be a beer model han model 1 afer considering forecased errors and he number of comparaive saisics. Keywords: Forecasing, Inflaion, Time series, ARIMA, Box-Jenkins approach Inroducion Persisen economic growh along wih low inflaion is he core aim of any macroeconomic policy. The inflaion rae is considered o be criical as an indicaor for a sable macroeconomic environmen. During he las hree decades inflaionary environmen has seen dramaic changes. For insance, from 1991 o 1994 inflaion rae in Pakisan ranges from 9.25 o 12.9 percen, whereas negaive or adequae growh in inernaional prices (in dollars) has been observed. The Sharp increase in prices was a resul of considerable depreciaion of exchange rae in Unconrolled moneary and fiscal policy of pressured reserves and he exchange rae. The rae of economic growh recovered srongly in he years 1991 and 1992, bu i was a shor-lived growh. In he 1993 growh rae ouched is lowes level 83

3 in las wo decades. In he nex wo year growh rae increased bu remain lower han he pas average. The rae of moneary growh raised o 12.6 percen in 1990 form ha of 4.6 percen in 1989 and since han is being in he range of 16 o 18 percen. High budge deficis in he preceding years cause he moneary expansion. Budge defici decreased o 5.8 percen in 1994 bu even han moneary growh was 16 percen. This growh could be credied o increase of foreign asses insead of domesic credi creaion. Thus, he reasons for he growh in money supply in 1994 were quie differen from he pas hree years. Inflaion rae remained beween 10 o 13 percen in Pakisan from 1991 o Pakisan. The perseverance of double digi inflaion and he large fiscal defici has been he main cause of macroeconomic shorcomings in he 1990s. Food and non-food inflaion added o he double digi inflaion, which were 11.6 and 10.3 percen respecively from 1991 o Inflaion averaged 3.5 percen during The main cause of his decline in he inflaion was low food inflaion. Furhermore, in he fiscal year 2003 he inflaion decline o 3.1 percen as compared o 3.5 percen during he fiscal year Food and non-food inflaion boh experienced a visible decrease. Afer ouching he lowes 1.4 percen in July 2003, CPI inflaion winessed a seep rise hrough mos of he fiscal year CPI inflaion was indeed influenced by inernaional prices. CPI non-food inflaion sars moving upward in March 2004 mainly due o house ren index (HRI). A number of sudies have been conduced o examine and evaluaing differen mehodologies o forecas he inflaion. One approach is relaed o Fama (1975) and exended by Fama and Gibbons (1982). This approach exracs from observed nominal ineres raes and marke expecaion of inflaion. They found ineres rae based models forecas beer resuls han he univariae ime series models. Meyler e al (1998) oulined Auoregressive inegraed moving average (ARIMA) models o forecasing inflaion in Ireland. Meyler e al (1998a) used Bayesian mehod o esimae vecor auoregressive (VAR) models o forecas inflaion in Ireland. Salam a.al (2007) employed ARIMA models on he Pakisan inflaion daa o find he bes model o forecas inflaion. Keeping in view he above menioned sudies and lieraure, presen sudy ries o use he approach used by Salam e al (2007) on an exend daa from he year 1970 o The sudy employed differen specificaion of ARIMA models o deermine a beer model of forecasing he inflaion. The ouline of sudy is as follow: secion 2 presens brief lieraure review. Secion 3 presens daa. The mehodology is discussed in secion 4 and secion 5 conains he resuls. Concluding remarks follow in secion 6. 84

4 Lieraure Review Quie a few sudies have examined he accuracy of differen inflaion forecasing models. Fama and Gibbons (1982) argue ha ineres rae model gives low error inflaion forecass han a univariae model. Meyler e al (1998) used ARIMA models o forecas he inflaion in Ireland. The sudy applied he Box-Jenkins approach and he objecive funcion mehods. The resuls implicae ARIMA forecas o be more reliable. Sekine (2001) calculaed he inflaion funcion and forecass one-year ahead inflaion for Japan. The sudy suggess markup relaionships, oupu gapes and excess money o be he deerminans of an equilibrium correcion model of inflaion. Callen and Chang (1999) argued ha he Reserve Bank of India has moved away from a broad money arge o muliple indicaors approach o conduc he moneary policy. The paper assesses which indicaors provide he mos useful informaion abou fuure inflaionary rends. I concludes ha he developmens in he moneary aggregaes remain an imporan indicaor of fuure inflaion. The concern wih inflaion is no only o mainain he macroeconomic sabiliy bu also because of he fac ha i his he poor paricularly hard. One may say ha he inflaion is he single bigges enemy of he poor. India has been reasonably successful in mainaining an accepable rae of he inflaion from he 1980s. Simone (2000) esimaes wo ime varying parameer models for inflaion for Chile. Box-Jenkins models ouperform he wo models for he shor erm ou of sample forecass. Dros e al (1997) say ha financial daa exhibi condiional heeroscedasiciy. GARCH ype models are ofen used o model his phenomenon. Sockon and Glassman (1987) find ha simple ARIMA model of inflaion should urn in such a respecable forecas performance relaive o he heoreically based specificaions. Salam e al (2007) have also recommended ARIMA models o forecas he inflaion using monhly CPI daa of Pakisan. Daa A long ime series daa is required for univariae ime series forecasing. I is usually recommended o have a leas 50 observaions. Using Box-Jenkins mehods can be problemaic wih oo few observaions. Consumer price index (CPI) is used as a proxy of inflaion in he presen sudy. The quarerly daa from 1970 o 2006 for Pakisan is aken from Inernaional Financial Saisics 2008 (IFS). We used daa from 1970 o 2004 for he esimaions and lef he las 8 values for he comparison of he forecas resuls. An imporan inflaion indicaor for Pakisan is CPI. I is calculaed by Federal Bureau of Saisics and published every monh. The CPI is an 85

5 esimaion of he price changes for a baske of goods (food, housing, educaion ec.). The goods are assigned weighs in order o ge a precise measure. CPI has been an imporan economic indicaor and is used by he cenral banks as he official measure of inflaion for evaluaing moneary policies. CPI is also used in he indexing of pension and superannuaion paymens. Many business conracs are revised o cope wih he changes in CPI. CPI is used for muliple purposes, some of hese are: Measure of changes in consumer prices. Compensaion index Cos of living index Indexaion of Governmen Reail rae deflaion Measure of changes in consumer prices. Naional accouning deflaion Mehodology The sudy applies Box-Jenkins (1976) approach for esimaion and forecasing. Auoregressive inegraed moving average (ARIMA) models are generalizaions of he simple auo regressive (AR) model ha use hree ools for modeling he serial correlaion in he disurbance: 1. The AR (1) model uses only he firs-order erm, bu in general, one may use addiional higher-order AR erms. Each AR erm corresponds o he use of a lagged value of he residual in he forecasing equaion for he uncondiional residual. An auoregressive model of order p, AR (p) has he form = α α α p p + ε 2. Each inegraion order corresponds o differencing he series being forecas. A firs-order inegraed componen means ha he forecasing model is designed for he firs difference of he original series. A secondorder componen corresponds o using second differences, and so on. = ε + θ1 ε 1 + θ2ε θqε q 3. A moving average model uses lagged values of he forecas error o improve he curren forecas. A firs-order moving average erm uses he mos recen forecas error; a second-order erm uses he forecas error from he wo mos recen periods, and so on. An MA (q) has he form: = ε + θ1 ε 1 + θ2ε θ qε q The auoregressive and moving average specificaions can be combined o form an ARMA (p,q) specificaion: = α α α p p + ε + θ1ε 1 + θ2ε θ qε q 86

6 Original CPI daa auocorrelaions and parial auocorrelaions are ploed for he idenificaion. Figure 1 shows he ACF and PACF of he original CPI series. A slowly decaying ACF suggess he non-saionariy of he series. In order o es he saionariy of he series, we used Dickey Fuller uni roo es (1979). The resuls are presen in able 1 and i also suggess he non-saionariy of he daa. Table 1. Uni Roo Tes -Saisic Prob.* Augmened Dickey-Fuller es saisic Tes criical values: 1% level % level % level *MacKinnon (1991) one-sided p- values Figure 1. Correlogram of Original CPI Series Box-Jenkins suggess differencing of he daa, based on he above resuls. A firs difference, he null hypohesis of uni roo is rejeced which implies daa is saionary a firs difference i.e. CPI daa is inegraed of order (1). In order o furher smooh he flucuaions, we used simple log of he price. Tesing suggess his variable is also non-saionary a level and saionary a firs difference. Resuls of uni roo es are presen in able 2. 87

7 Table 2. Uni Roo Tes (afer difference and log ransformaion) -Saisic Prob.* Augmened Dickey-Fuller es saisic Tes criical values: 1% level % level % level *MacKinnon (1991) one-sided p- values Afer deermining he order of inegraion o make he series saionary, nex sep is o deermine ARMA form of he model. Differen orders of AR and MA are used o deermine he suiable ARMA srucure. The crieria o selec he models are Akaike Informaion Crieria (AIC) and Schwarz Informaion Crieria (SIC). We performed he es ill AR (4) and MA (1). The resuls are presened in able 3 and 4. Boh AIC and SIC suggesed he model wih he following specificaion, Model 1 = α 0 + α1 4 + ε + θ1ε 1 The Box-Jenkins mehodology examines plos of sample ACF and PACF. By ploing he ACF and PACF of he above menioned model 1, we found ou some disracion a lag 10 (figure 2). We hen esablished anoher model wih he following specificaion. Model 2 = α0 + α1 4 + α ε + θ1ε 1 The ACF and PACF of model 2 are presened in figure 3. Table 3. Akaike Informaion Crieria (SIC) MA Order AR order AR order Table 4. Schwarz Informaion Crieria (SIC) MA Order

8 Figure 2. Correlogram of residuals Model 1 Figure 3. Correlogram of residuals Model 2 89

9 Resuls Having seleced wo ARIMA model specificaions we regressed hem o ge he forecass o compare he models. Graphical represenaions of he boh he models are presened in figure 4. Seemingly here is no much visible difference beween he wo forecass. However, in order o compare furher in deail, able 5 presens various comparaive saisics such as R 2, AIC, SIC, Roo Mean Squared Error, Mean Absolue Error, Theil Inequaliy Coefficien. The R-square is bi high for he model 2, suggesing i o be a beer model. According o AIC model 2 is beer han model 1. However, SIC suggess ha model 1 is slighly beer han model 2. Theil inequaliy coefficien is used o check he goodness of fi of he forecased models and he value ranges beween 0 and 1, where 0 indicaing a perfec fi. In our resuls according o Theil inequaliy coefficien model 2 is slighly beer han model 1. Figure 4. Forecass Model 1 Model Residual Acual Fied Residual Acual Fied Table 5. Comparaive saisics Model 1 Model 2 R-squared Adjused R-squared S.E. of regression Durbin-Wason sa Akaike info crierion Schwarz crierion Roo Mean Squared Error Mean Absolue Error Mean Abs. Percen Error Theil Inequaliy Coefficien Bias Proporion Variance Proporion Covariance Proporion

10 Table 6 shows he acual and forecased values of CPI o furher compare he performance of wo models. Based on he forecased errors of he models, model 2 looks beer han model 1. Table 6. Forecas comparison Quarer Acual CPI Forecas Model 1 Forecas Model 2 Forecas Error Model 1 Forecas Error Model Q Q Q Q Q Q Q Q Conclusion The sudy aemps o compare and selec an accurae model form various ARIMA models which possess high power of predicabiliy wih low error. The sudy adoped Box-Jenkins approach o forecasing. Various seps are aken in he process, including deermining he inegraion order, model idenificaion and hen comparison of he wo seleced models. The sudy is based on he quarerly CPI daa for Pakisan. Among he wo esimaed models, which were seleced from a number of model specificaions, model 2 has come ou o be slighly beer han model 1. For he purpose of comparison we used he graphical approach firs and hen we compared he saisics for he wo models (e.g. Theil inequaliy coefficien, AIC). A he end, he acual values of CPI for he quarers of 2005 and 2006 are also compared wih he forecased values. This also urns model 2 o be slighly beer. References: Brockwell, Peer J., and Richard A. Davis Inroducion o ime series and forecasing. 2nd ed. Springer exs in saisics. New York: Springer. Box, G. E. P. and G. M. Jenkins Time series analysis: forecasing and conrol, Holden-Day. Callen, Tim, and Dongkoo Chang Modeling and Forecasing Inflaion in India. IMF Working paper Sepember: SSRN: hp://ssrn.com/absrac= Dickey, David A., and Wayne A. Fuller Disribuion of he Esimaors for Auoregressive Time Series wih a Uni Roo. Journal of he American Saisical Associaion 77 (366):

11 Diebold, Francis Elemens of forecasing. 2nd ed. Cincinnai, Ohio: Souh-Wesern College Pub. Dros, Feike C., Chris A. Klaassen, and Bas J. M. Werker Adapive esimaion in ime series models. Analysis of Saisics 25 (2): Fama, Eugene F Shor-Term Ineres Raes as Predicors of Inflaion. American Economic Review 65 (3): Fama, Eugene F., and Michael R. Gibbons Inflaion, Real Reurns and Capial Invesmen. Journal of Moneary Economics 9 (3): MacKinnon, J. G. (1991), Criical values for coinegraion ess, Chaper 13 in Long-Run Economic Relaionships: Readings in Coinegraion, ed. R. F. Engle and C. W. J. Granger. Oxford, Oxford Universiy Press. Meyler, Aidan, Kenny Geoff, and Terry Quinn Forecasing Irish inflaion using ARIMA models. Cenral Bank and Financial Services Auhoriy of Ireland Technical Paper Series 3/RT/98: a. Bayesian VAR Models for Forecasing Irish Inflaion. Cenral Bank and Financial Services Auhoriy of Ireland Technical Paper Series 4/RT/98: Salam, Muhammad A., Shazia Salam, and Mee Feridun Modeling and Forecasing Pakisan s Inflaion by Using Time Series ARIMA Models. Economic Analysis Working Papers 6: Sekine, Toshiaka Modeling and Forecasing Inflaion in Japan. IMF Working paper June. Simone, Francisco N Forecasing Inflaion in Chile Using Sae- Space and Regime-Swiching Models. IMF Working paper Vol: SSRN: hp://ssrn.com/absrac= Sockon, David J., and James E. Glassman An Evaluaion of he Forecas Performance of Alernaive Models of Inflaion. The Review of Economics and Saisics 69: