FORECASTING IRISH INFLATION USING ARIMA MODELS

Transcription

1 TECHNICAL PAPER 3/RT/98 DECEMBER 998 FORECASTING IRISH INFLATION USING ARIMA MODELS BY AIDAN MEYLER *, GEOFF KENNY AND TERRY QUINN The views expressed in his paper are no necessarily held by he Cenral Bank of Ireland and are he personal responsibiliy of he auhors. The auhors would like o hank John Frain and paricipans a an inernal seminar in he Cenral Bank for commens on an earlier draf of his paper. All remaining errors and omissions are he auhors. Commens and criicisms are welcome. Economic Analysis, Research and Publicaions Deparmen, Cenral Bank of Ireland, PO Box 559, Dublin 2. * Auhor for correspondence: Aidan Meyler. enquiries@cenralbank.ie

2 ABSTRACT This paper oulines he pracical seps which need o be underaken o use auoregressive inegraed moving average (ARIMA) ime series models for forecasing Irish inflaion. A framework for ARIMA forecasing is drawn up. I considers wo alernaive approaches o he issue of idenifying ARIMA models - he Box Jenkins approach and he objecive penaly funcion mehods. The emphasis is on forecas performance which suggess more focus on minimising ou-of-sample forecas errors han on maximising in-sample goodness of fi. Thus, he approach followed is unashamedly one of model mining wih he aim of opimising forecas performance. Pracical issues in ARIMA ime series forecasing are illusraed wih reference o he harmonised index of consumer prices (HICP) and some of is major sub-componens.

3 . INTRODUCTION The primary focus of moneary policy, boh in Ireland and elsewhere, has radiionally been he mainenance of a low and sable rae of aggregae price inflaion as defined by commonly acceped measures such as he consumer price index. The underlying jusificaion for his objecive is he widespread consensus, suppored by numerous economic sudies, ha inflaion is cosly insofar as i undermines real, wealhenhancing, economic aciviy. From he beginning of 999, he Irish economy faces a new environmen in which moneary policy will be se by he Governing Council of he European Cenral Bank (ECB). The ECB is commied o a moneary policy which has he primary objecive of mainaining price sabiliy hroughou he eleven euro-area counries as a whole. 2 Regardless of he exac sraegy adoped by he ECB in he formulaion of moneary policy, i.e., argeing moneary aggregaes such as he broad money sock or direc inflaion argeing, he provision of opimal and imely inflaion forecass represens a key ingredien in designing moneary policies which are geared oward he achievemen of price sabiliy. While i could be argued ha Ireland s weigh in he overall euro-area price index is relaively small and, as such, Irish inflaion no longer warrans rigorous examinaion, i is imporan o noe ha Ireland has an inpu ino moneary policy decision making a he ECB ha is disproporionae o is economic size. However, a more compelling argumen for a coninued focus on forecasing Irish inflaion is he increased imporance of fiscal policy and wage bargaining negoiaions in he absence of independen moneary conrol. Arguably, he inflaion forecas should be given greaer weigh in fiscal policy and in wage negoiaions in Ireland han has been he case hereofore. Furhermore, given he possibiliy of susained Two recen examples are Feldsein (996) and Dosey and Ireland (996). Boh of hese sudies argue, in paricular, ha even low raes of inflaion of he order of 2% o 4% are highly cosly over he long-run he Governing Council of he ECB makes i clear ha i will base is decisions on moneary, economic and financial developmens in he euro area as a whole. The single moneary policy will adop a euro area-wide perspecive; i will no reac o specific regional or naional developmens (ECB, 998).

4 differences in inflaion raes across euro area currencies and he subsequen impac on compeiiveness, monioring and undersanding price developmens in individual economies will remain of significan imporance. In pas sudies of inflaion by he Cenral Bank of Ireland he emphasis has been on esing economic heory and on empirical analysis. Even hough some of hese sudies have been used as an inpu ino he forecasing process wihin he Bank, hey have no hereofore been subjec o rigorous forecas evaluaion echniques. This paper and Kenny e al (998) se ou o redress his deficiency and explicily use ime series echniques solely for forecasing purposes. There are a number of approaches available for forecasing economic ime series. One approach, which includes only he ime series being forecas, is known as univariae forecasing. Auoregressive inegraed moving average (ARIMA) modelling is a specific subse of univariae modelling, in which a ime series is expressed in erms of pas values of iself (he auoregressive componen) plus curren and lagged values of a whie noise error erm (he moving average componen). This paper focuses on ARIMA models. An alernaive approach is mulivariae ime series forecasing. Mulivariae models may consis of single equaion models wih exogenous explanaory variables or alernaively may include a srucural or non-srucural sysem of equaions. Parallel research is also being currenly underaken wihin he Cenral Bank of Ireland ino he use of Bayesian Vecor Auoregressive (BVAR) models for forecasing Irish inflaion (see Kenny e al, op. ci.). In pracice he formal economeric models oulined above are ofen supplemened by subjecive off-model inpus. Such informaion may include survey daa gahered from liaising wih reailers and manufacuring enerprises. Thus, inflaion forecasing is an ar raher han a hard science combining formal economeric echniques wih forecasers experience and experise. Pracical issues in relaion o ARIMA ime series forecasing are illusraed using he harmonised index of consumer prices (HICP) and some of is major sub-componens. The HICP was developed o allow comparison of inflaion raes across EU saes. 2

5 Prior o he developmen of he HICP, each sae consruced is own consumer price index (CPI), which could differ in how hey reaed cerain iems such as housing, healh, educaion and insurance. Previous work examining Irish inflaion has concenraed on he CPI. This paper offers an opporuniy o apply univariae echniques o he Irish HICP for he firs ime. 3 In his paper six ime series are examined. These allow us o consider many of he issues ha arise in ARIMA ime series forecasing. These series are he overall Harmonised Index of Consumer Prices (HICP) and he HICP broken down beween unprocessed foods (HICPA), processed food (HICPB), non-energy indusrial goods (HICPC), energy (HICPD), and services (HICPE). The emphasis is on forecas performance which suggess more focus on minimising ou-of-sample forecas errors han on maximising in-sample goodness of fi. Thus, he approach followed is unashamedly one of model mining wih he aim of opimising forecas performance. 4 The srucure of he paper is as follows: Secion 2 presens a brief inroducion o ARIMA modelling, oulining he main advanages and disadvanages. Secion 3 - he main secion of he paper - oulines a general framework for ARIMA forecasing, including a comparison of he radiional Box-Jenkins mehodology wih objecive penaly funcion mehods. A pracical applicaion of he framework is made wih reference o he HICP series and is major sub-componens. The discussion focuses on wo series - he overall HICP and he non-energy indusrial goods (HICPC) componen - as hese serve o highligh many of he issues encounered when using ARIMA models o forecas inflaion. Secion 4 briefly summarises a semi-auomaic algorihm developed in he preparaion of his paper. Secion 5 concludes and offers some observaions on he limiaions of ARIMA models. Appendix A provides a 3 For informaion on he consrucion of an hisorical series for he HICP in Ireland see Meyler e al (998). 4 Cecchei (995, pg. 99) finds ha, in his sudy, wheher a model fis well in-sample ells us virually nohing abou is ou-of-sample forecasing abiliy. However, in his paper, a posiive correlaion is generally found beween a model s in-sample explanaory power as ranked according o he penaly funcion crierion and is ou-of-sample forecas rank according o he sum of he average mean absolue error for each of he firs four seps ahead. 3

6 descripion of ARIMA models and some of heir heoreical properies. Appendices B and C presen resuls for he oher sub-componens of he HICP. 2. AN INTRODUCTION TO ARIMA MODELLING ARIMA mehods for forecasing ime series are essenially agnosic. Unlike oher mehods hey do no assume knowledge of any underlying economic model or srucural relaionships. I is assumed ha pas values of he series plus previous error erms conain informaion for he purposes of forecasing. The main advanage of ARIMA forecasing is ha i requires daa on he ime series in quesion only. Firs, his feaure is advanageous if one is forecasing a large number of ime series. Second, his avoids a problem ha occurs someimes wih mulivariae models. For example, consider a model including wages, prices and money. I is possible ha a consisen money series is only available for a shorer period of ime han he oher wo series, resricing he ime period over which he model can be esimaed. Third, wih mulivariae models, imeliness of daa can be a problem. If one consrucs a large srucural model conaining variables which are only published wih a long lag, such as wage daa, hen forecass using his model are condiional forecass based on forecass of he unavailable observaions, adding an addiional source of forecas uncerainy. Some disadvanages of ARIMA forecasing are ha: Some of he radiional model idenificaion echniques are subjecive and he reliabiliy of he chosen model can depend on he skill and experience of he forecaser (alhough his criicism ofen applies o oher modelling approaches as well). I is no embedded wihin any underlying heoreical model or srucural relaionships. The economic significance of he chosen model is herefore no clear. Furhermore, i is no possible o run policy simulaions wih ARIMA models, unlike wih srucural models. 5 5 For a discussion of his issue see Frain (pg. 2, 995). 4

7 ARIMA models are essenially backward looking. As such, hey are generally poor a predicing urning poins, unless he urning poin represens a reurn o a long-run equilibrium. However, ARIMA models have proven hemselves o be relaively robus especially when generaing shor-run inflaion forecass. ARIMA models frequenly ouperform more sophisicaed srucural models in erms of shor-run forecasing abiliy (see, for example, Sockon and Glassman (987) and Lierman (986)). Therefore, he ARIMA forecasing echnique oulined in his paper will no only provide a benchmark by which oher forecasing echniques may be appraised, bu will also provide an inpu ino forecasing in is own righ. Appendix A presens a descripion of ARIMA models and some of heir heoreical properies. A general noaion for a muliplicaive seasonal ARIMA models is ARIMA (p,d,q)(p,d,q), where p denoes he number of auoregressive erms, q denoes he number of moving average erms and d denoes he number of imes a series mus be differenced o induce saionariy. P denoes he number of seasonal auoregressive componens, Q denoes he number of seasonal moving average erms and D denoes he number of seasonal differences required o induce saionariy. This may be wrien as d D () φ( B) Φ( B) Y = θ( B) Θ( B) a s where, X = d sd Y is a saionary series, ( B) d = d D s D represens he number of regular differences and s = ( B ) represens he number of seasonal differences required o induce saionariy in Y, s is he seasonal span (hence for quarerly daa s = 4 and for monhly daa s = 2), 0 B is he backshif operaor (such ha B X ( ) = X, BX = X, B 2 X X =,..), 2 q θ B = + θ B+ θ B θ B is a q-order polynomial in he backshif operaor, 2 q 2 5

8 ( ) 2 p φ B = φ B φ B... φ B, 2 s 2s Φ( B) = Φ B Φ B.. Φ B s 2s s 2s Θ( B) = + Θ B + Θ B Θ B s 2s p Ps Qs Ps Qs, and, As shown in Appendix A any non-deerminisic saionary process can be approximaed by an ARMA process. The problem lies in ensuring he series is saionary and in deermining he order of p and q ha adequaely describes he ime series being examined. I is hese issues which are examined in he nex secion. 3. ARIMA FORECASTING IN PRACTICE This secion oulines a general ARIMA modelling and forecasing sraegy. Figure illusraes his process graphically. I is imporan o noe, however, ha his process is no a simple sequenial one, bu can involve ieraive loops depending on resuls obained a he diagnosic and forecasing sages. The firs sep is o collec and examine graphically and saisically he daa o be forecas. The second sep is o es wheher he daa are saionary or if differencing is required. Once he daa are rendered saionary one should seek o idenify and esimae he correc ARMA model. Two alernaive approaches o model idenificaion are considered - he Box-Jenkins mehodology and penaly funcion crieria. I is imporan ha any idenified model be subjec o a baery of diagnosic checks (usually based on checking he residuals) and sensiiviy analysis. For example, he esimaed parameers should be relaively robus wih respec o he ime frame chosen. Should he diagnosic checks indicae problems wih he idenified model one should reurn o he model idenificaion sage. Once a model or selecion of models has been chosen, he models should hen be used o forecas he ime series, preferably using ou-of-sample daa o evaluae he forecasing performance of he model. One common pifall of ARIMA modelling is o overfi he model a he idenificaion sage, which maximises he in-sample explanaory performance of he model bu may lead o poor ou-of-sample predicive power relaive o a more parsimonious model. Thus, if a model wih a large number 6

9 of AR and MA lags yields poor forecasing performance, i may be opimal o reurn o he model idenificaion sage and consider a more parsimonious model. FIGURE - ARIMA FORECASTING PROCEDURE Daa Collecion and Examinaion Deermine Saionariy of Time Series Model Idenificaion and Esimaion Diagnosic Checking Forecasing and Forecas Evaluaion 3.. STEP ONE - DATA COLLECTION AND EXAMINATION An economerician should always fall in love wih his/her daa A lenghy ime series of daa is required for univariae ime series forecasing. I is usually recommended ha a leas 50 observaions be available. Using eiher Box- Jenkins or objecive penaly funcion mehods can be problemaic if oo few observaions are available. Unforunaely, even if a long ime series is available, i is possible ha he series conains a srucural break which may necessiae only examining a sub-secion of he enire daa series, or alernaively using inervenion analysis or dummy variables. Thus, here may be some conflic beween he need for sufficien degrees of freedom for saisical robusness and having a shorer daa sample o avoid srucural breaks. 7

10 Graphically examining he daa is imporan. They should be examined in levels, logs, differences and seasonal differences. The series should be ploed agains ime o assess wheher any srucural breaks, ouliers or daa errors occur. If so one may need o consider use of inervenion or dummy variables. This sep may also reveal wheher here is a significan seasonal paern in he ime series. Consider, for example, a plo of he firs difference of he log of he HICP series for he period Q 976 o Q4 998 as shown in Figure 2. From his figure and Table i is eviden ha for he period 976 o 983, he mean rae of, and sandard deviaion of, inflaion was higher han for he period pos-983. Thus, i may be necessary o consider inclusion of an inervenion variable for he earlier period, or perhaps, o idenify and esimae he model for he laer period only. 6 FIGURE 2 - PLOT OF DLHICP 7 SERIES, 976Q-998Q4 8% 7% DLHICP 6% 5% 4% 3% 2% % 0% -% A more formal es (Perron, 989) for nonsaionariy, in he presence of srucural breaks, is considered below. 7 The following noaion is used in his paper. LHICP denoes he naural log of he HICP series. DLHICP denoes he LHICP series differenced once. DDLHICP denoes he LHICP series differenced wice. DsDLHICP denoes he seasonal difference of he DLHICP series. 8

11 TABLE - SUMMARY STATISTICS FOR DLHICP SERIES, 976Q-998Q4 Period Mean Sandard Deviaion 976Q-983Q Q-998Q Overall Period A plo of he firs difference of he log of he non-energy indusrial goods componen of he HICP (HICPC), given in Figure 3, also yields useful informaion. This series shows a decline in he mean rae of change similar o ha for he DLHICP series. Anoher noiceable feaure is he more violen oscillaion in recen periods. This appears o reflec a more pronounced seasonal paern wih deeper sales discouns and subsequen rebound in prices. The more pronounced seasonal sales paern in he HICPC index is considered furher below. FIGURE 3 - PLOT OF DLHICPC SERIES, 976Q-998Q4 5% 4% DLHICPC 3% 2% % 0% -% -2% Anoher way o examine he properies of a ime series is o plo is auocorrelogram. The auocorrelogram plos he auocorrelaion beween differing lag lenghs of he ime series. Ploing he auocorrelogram is a useful aid for deermining he saionariy of a ime series, and is also an imporan inpu ino Box-Jenkins model 9

12 idenificaion. The heoreical auocorrelogram for differen orders of AR, MA and ARMA models are oulined in secion dealing wih model idenificaion (Sep 3). The SACF may be consruced using equaion (A4). The maximum lag lengh considered is usually no more han n/4. Alhough sample auocorrelaions for lags in excess of wice he seasonal span (i.e., in excess of 8 for quarerly daa) should be reaed wih cauion. If a ime series is saionary hen is auocorrelogram should decay quie rapidly from is iniial value of uniy a zero lag. If he ime series is nonsaionary hen he auocorrelogram will only die ou gradually over ime. Figure 4 plos he auocorrelogram for he log of he HICP, he firs differences and second differences of he log and he seasonal difference of he firs difference (for he period 984Q - 998Q4). I would appear from Figure 4 ha he log of levels series is nonsaionary as he auocorrelaions decay slowly owards zero. A firs glance he firs difference series appears saionary alhough here seems o be some evidence of seasonal behaviour (he auocorrelaions a lags 4, 8 and 2 exhibi disincive behaviour and die ou quie slowly). The auocorrelogram of he second difference series is more volaile and may indicae over-differencing. The auocorrelaions of he seasonal differences of he firs difference series exhibi a quasi-sinusoidal decay paern, which may indicae he presence of complex roos. Based on a graphical examinaion of Figure 4, he firs difference of logs and he seasonal differences of he firs differences require more formal uni roo esing o deermine saionariy. 0

13 FIGURE 4 - AUTOCORRELOGRAM FOR LHICP, DLHICP, DDLHICP AND DSDLHICP LHICP DLHICP DDLHICP DsDLHICP Figure 5 displays he auocorrelograms for various ransformaions of he non-energy indusrial goods (HICPC) componen of he HICP. I is eviden ha he goods componen of he HICP exhibis a much sronger seasonal paern han he overall HICP. This paern is driven primarily by he Winer and Summer sales and he subsequen rebound. Based on Figure 5 here is a srong case for seasonally differencing he rae of inflaion in non-energy indusrial goods prices (i.e., seasonally differencing he firs difference of he log of he HICPC series).

14 FIGURE 5 - AUTOCORRELOGRAM FOR LHICPC, DLHICPC, DDLHICPC AND DSDLHICPC LHICP DLHICP DDLHICP DsDLHICP Alhough he auocorrelogram gives some indicaion as o wheher a series is saionary or nonsaionary, in more recen years a vas array of formal ess for saionariy wih known saisical properies have been developed STEP TWO - TESTING FOR STATIONARITY The ime series under consideraion mus be saionary before one can aemp o idenify a suiable ARMA model. A large lieraure has developed in recen years on he issue of esing ime series for saionariy and nonsaionariy (See, for example, Harris (995) and Banerjee e al (993)). For AR or ARMA models o be saionary i is necessary ha he modulus of he roos of he AR polynomial be greaer han uniy, and for he MA par o be inverible i is also necessary ha he roos of he MA polynomial lie ouside he uni circle. 2

15 The original Dickey-Fuller es considered he model X = ρx - + ε or alernaively X = (ρ-)x + ε. If he series conains a uni roo, hen ρ = (and ρ- = 0). The sandard -disribuion canno be used o es if ρ =, he Dickey-Fuller disribuion should be used insead. However, should ε be auocorrelaed, he Dickey-Fuller disribuion is no longer valid eiher. In his case, an alernaive model should be esimaed, where l lags of he firs difference of he series are added unil he series e displays no evidence of auocorrelaion. 8 (2) X = (ρ-)x + δ i X i l i= + e In his insance he Augmened Dickey Fuller es saisic should be used. Table 2 presens some summary resuls esing he HICP series and is major sub-componens for uni roos. The -adf saisic is he Augmened Dicky Fuller es saisic, under he null hypohesis ha ρ- = 0 (or equivalenly, ρ = ). The columns denoed lags indicae he number of lagged firs differences of he series ha were added o ensure whie noise error erms. The final column in each par of he able conains he 5 per cen and per cen criical values for he -adf saisic. For he log of he levels series, he series is esed for nonsaionariy around a consan and a rend. However, he differenced series are esed for nonsaionariy around a consan solely. The resuls from Table 2 indicae ha: none of he levels series, excep HICPA and HICPB, is saionary around a consan and rend; all of he series excep he HICPB and HICPC series are saionary if differenced once or more. 9 However, he -adf saisics on he HICP and HICPE series differenced once are relaively low, and may indicae he need o consider seasonal differencing. 0 This issue is considered furher below. 8 In pracice, he number of lag differenced erms o be added is deermined using model selecion crieria such as he AIC and BIC oulined below. This is usually sufficien o ensure a well-behaved error erm. 9 In general i is sufficien o es up o d = 2 and D =. 0 The conflicing resuls for he HICPB series and he resul for he HICPA series in levels indicae he low power of many uni roo ess. This is why i is necessary o use a number of alernaive ess o ensure consisency. 3

16 TABLE 2 - AUGMENTED DICKY-FULLER TESTS, 984Q - 998Q4 HICP HICPA HICPB 5% (%) -adf lags -adf lags -adf lags -adf * () Log of Level * ** (-4.) (2) s Diff. of () -3.8 ** ** (-3.5) (3) Seas. Diff. of (2) -5. ** 3-0. ** ** (-3.5) (4) 2nd Diff. of (2) -5.0 ** ** -9.2 ** -2.9 (-3.5) HICPC HICPD HICPE 5% (%) -adf lags -adf lags -adf lags -adf * () Log of Level (-4.) (2) s Diff. of () ** ** (-3.5) (3) Seas. Diff. of (2) -6.2 ** ** ** (-3.5) (4) 2nd Diff. of (2) -6.0 ** ** ** (-3.5) A number of less formal echniques exis for deermining saionariy of a ime series. As saed above examinaion of he auocorrelogram can be one useful indicaor. For example, consider a pure AR() process. The auocorrelaion a lag k of an AR() process is given by φ k. Thus if φ = he auocorrelogram does no decay over ime. In general, he sample auocorrelogram of a nonsaionary series will only decay very slowly owards zero. However, relying solely on he auocorrelogram o deermine he saionariy of a ime series ends o lead o over-differencing (Mills (990, pg. 2)). One way o check for over-differencing is o examine he variance of he process (Anderson, 976). In general he sample variance of a process will decrease unil he correc order of differencing is found, bu will increase hereafer if he process is over-differenced. Table 3 indicaes ha using log of levels is inappropriae for all he HICP series. The sample variances for he HICPC and HICPE series sugges seasonally differencing he inflaion rae could be necessary as he inflaion rae iself may be nonsaionary. For he oher series (HICP, HICPA, HICPB and HICPD), he sample variances indicae ha using he inflaion rae is sufficien o ensure saionariy, alhough he resuls are no always clear-cu. 4

17 TABLE 3 - SAMPLE VARIANCE, 984Q - 998Q4 HICP HICPA HICPB HICPC HICPD HICPE () Log of Level (2) s Diff. of () (3) Seas. Diff. of (2) (4) 2nd Diff. of (2) In summary, whils he adf es and analysis of he sample variance favour a single differencing of he overall HICP o induce saionariy, examinaion of he auocorrelogram suggess seasonal differencing. Furhermore, he variance of he seasonal difference of he firs difference of he LHICP series is only marginally higher han he LHICP series jus differenced once. In conras, he firs difference of he HICPC series fails he adf es a he 5 per cen confidence level. The seasonal difference of he firs difference of he HICPC series passes all he ess for saionariy. In general, when differen es saisics offer conflicing evidence, i is bes o bring forward boh alernaives o he forecasing sage, as he power of many ess for nonsaionariy can be quie low especially wih small sample sizes STEP THREE - MODEL IDENTIFICATION AND ESTIMATION Having deermined he correc order of differencing required o render he series saionary, he nex sep is o find an appropriae ARMA form o model he saionary series. There are wo main approaches o idenificaion of ARMA models in he lieraure. The radiional mehod uilises he Box-Jenkins procedure, in which an ieraive process of model idenificaion, model esimaion and model evaluaion is followed. The Box-Jenkins procedure is a quasi-formal approach wih model idenificaion relying on subjecive assessmen of plos of auocorrelograms and parial auocorrelograms of he series. Objecive measures of model suiabiliy, in paricular he penaly funcion crieria, have been used by some auhors insead of he radiional Box-Jenkins procedure. For a recen example of he use of objecive penaly funcion crieria see Gómez and Maravall (998). However, hese objecive measures are no wihou problems eiher. 5

18 Ouside of he Box-Jenkins and penaly funcion crierion mehods here are a number of alernaive idenificaion mehods proposed in he lieraure. These include, iner alia, he Corner mehod (Beguin e al, 980), he R and S Array mehod (Gray e al, 978), and canonical correlaion mehods (Tsay and Tiao, 985). These mehods are usually based on he properies of he auocorrelaion funcion and do no require esimaion of a range of models, which can be compuaionally expensive. This lack of compuaion is a significan advanage over he penaly funcion crierion oulined below. However, he problem wih mos auocorrelaion-based mehods is ha hey are no very useful for dealing wih seasonal daa. The seasonal naure of price daa makes hese alernaive mehods less aracive for he purposes of forecasing inflaion BOX-JENKINS METHODOLOGY The Box-Jenkins mehodology essenially involves examining plos of he sample auocorrelogram, parial auocorrelogram and inverse auocorrelogram and inferring from paerns observed in hese funcions he correc form of ARMA model o selec. The Box-Jenkins mehodology is no only abou model idenificaion bu is, in fac, an ieraive approach incorporaing model esimaion and diagnosic checking in addiion o model idenificaion. Theoreically Box-Jenkins model idenificaion is relaively easy if one has a pure AR or a pure MA process. However, in he case of mixed ARMA models (especially of high order) i can be difficul o inerpre sample ACFs and PACFs, and Box-Jenkins idenificaion becomes a highly subjecive exercise depending on he skill and experience of he forecaser. Random noise in ime series, especially price daa, makes Box-Jenkins model idenificaion even more problemaic. 6

19 Pure AR Process The auocorrelaions of a pure AR(p) process should decay gradually a increasing lag lengh. Hence, using an auocorrelogram i is no possible o differeniae beween a pure AR(3) model or a pure AR(4) model. However, he parial auocorrelaions of a pure AR(p) process do display disincive feaures. The parial auocorrelogram should die ou afer p lags. Thus, he parial auocorrelogram of a pure AR(3) process should die ou afer 3 lags, whereas ha of a pure AR(4) process would die ou afer 4 lags. Hence, for a pure AR(p) process he heoreical ACF and PACF are as follows: ACF(i) 0 i PACF(i) 0 i =,...,p PACF(i) = 0 i > p where i denoes he number of lags. Pure MA Process The behaviour of correlograms and parial auocorrelograms for pure MA(q) processes is he reverse of ha for pure AR processes. The auocorrelogram of a pure MA(q) process should die ou afer q lags. The parial auocorrelogram of a pure MA process, on he oher hand, only decays slowly over ime (similar o he behaviour of he auocorrelogram of a pure AR process). Thus, i should be impossible o disinguish beween he PACF of an MA(3) and MA(4) process, whereas he ACF of he MA(3) process should decay o zero afer 3 lags and he MA(4) process afer 4 lags. If he ARMA model φ( BX ) θ( ) = Ba is inverible, he inverse auocorrelogram of he series, X, is simply he auocorrelogram of he invered model (i.e., he dual of he original model) given by θ BX = φ Ba. See, Chafield (979) for a discussion of inverse auocorrelaion funcions. ( ) ( ) 7

20 Hence, for a pure MA(q) process he heoreical ACF and PACF are as follows: ACF(i) 0 i =,...,q ACF(i) = 0 i > q PACF(i) 0 i Thus if one has eiher a pure AR or MA process model idenificaion should be relaively sraighforward in heory. Furhermore he behaviour of he auocorrelogram and parial auocorrelogram can provide informaion on he AR and MA componens, in erms of sign or he exisence of complex roos. For example, he auocorrelaions of a pure AR() process wih a negaive roo should oscillae around zero and decay wih increases in lags, whereas he auocorrelaions of a pure AR() process wih a posiive roo should decay gradually and monoonically owards zero (assuming φ ). The auocorrelogram of an AR(p) process wih complex roos should exhibi a sinusoidal (or wave) paern. Mixed ARMA Processes Unforunaely, model idenificaion is grealy complicaed for mixed (i.e., ARMA) processes. The paerns of sample auocorrelaions and parial auocorrelaions of high order ARMA models are nooriously difficul o inerpre. Thus, model idenificaion using he Box-Jenkins procedures will be an ieraive process, wih Sep Four - diagnosic checking - deermining wheher alernaive models should be examined. See Box and Jenkins (976) for a deailed discussion of idenifying mixed ARMA process OBJECTIVE MODEL IDENTIFICATION Because of he highly subjecive naure of he Box-Jenkins mehodology, ime series analyss have sough alernaive objecive mehods for idenifying ARMA models. Penaly funcion saisics, such as Akaike Informaion Crierion [AIC] or Final Predicion Error [FPE] Crierion (Akaike, 974), Schwarz Crierion [SC] or Bayesian Informaion Crierion [BIC] (Schwarz, 978) and Hannan Quinn Crierion [HQC] 8

21 (Hannan, 980), have been used o assis ime series analyss in reconciling he need o minimise errors wih he conflicing desire for model parsimony. These saisics all ake he form minimising he sum of he residual sum of squares plus a penaly erm which incorporaes he number of esimaed parameer coefficiens o facor in model parsimony. These saisics ake he form rss k log log *, n n (3) BIC = + ( n) rss k log 2 * log( log )*, and n n (4) HQC = + ( n) rss k (5) AIC = log + * n 2 n where, k = number of coefficiens esimaed ( + p + q + P + Q) rss = residual sum of squares n = number of observaions. Assuming here is a rue ARMA model for he ime series, he BIC and HQC have he bes heoreical properies. The BIC is srongly consisen whereas AIC will usually resul in an overparameerised model; ha is a model wih oo many AR or MA erms (Mills 993, p.29). Indeed, i is easy o verify ha for n greaer han seven he BIC imposes a greaer penaly for addiional parameers han does he AIC. Gómez and Maravall (998, p.9) also favour he BIC over he AIC. Thus, in pracice, using he objecive model selecion crieria involves esimaing a range of models and he one wih he lowes informaion crierion is seleced. This can creae a number of difficulies. Firs, i can be compuaionally expensive using he penaly funcion crierion. Esimaing all possible models encompassed by a 9

22 (3,0,3)(2,0,2) model involves esimaing 44 differen models. Therefore he choice of maximum order is very imporan o avoid expensive compuaional requiremens. Unforunaely, here is no a priori informaion o assis in selecing he maximum order of he ARIMA model o esimaed. Moving from a maximum order of (3,0,3)(2,0,2) o (2,0,2)(,0,) reduces he number of models o be esimaed from 44 o 36. One useful rule of humb for deermining he maximum order is o selec a maximum for he regular erms of he seasonal span less one (i.e., hree for quarerly daa or eleven for monhly daa) and one for he seasonal erm. Thus for quarerly daa his would sugges esimaing an ARMA of maximum order (3,0,3)(,0,), which implies esimaing 64 differen models and calculaing 64 informaion crierion. 2 Second, he differen objecive model selecion crieria can sugges differen models. Tha is he ranking order based on he BIC will usually no be he same as under he AIC. Table 4 compares he op five ranking models under he BIC, HQC and AIC for he DLHICP series esimaed over he period 984Q - 998Q4. The op ranking model under he AIC only ranks sevenh using he BIC. Furhermore, he AIC generally favours a less parsimonious model han eiher he BIC or he HQC. TABLE 4 - COMPARISON OF RANKING BY CRITERION - DLHICP (984Q - 998Q4) BIC HQC AIC rank (0,0,0) x (,0,) (0,0,0) x (,0,) (3,0,0) x (,0,0) rank 2 (0,0,0) x (,0,0) (3,0,0) x (,0,0) (3,0,0) x (,0,) rank 3 (,0,0) x (,0,0) (,0,0) x (,0,0) (0,0,0) x (,0,) rank 4 (0,0,) x (,0,0) (,0,0) x (,0,) (,0,0) x (,0,) rank 5 (,0,0) x (,0,) (3,0,0) x (,0,) (0,0,) x (,0,) Third, even if one uilises only one measure (e.g., BIC), he difference beween he BIC saisic for differen models is someimes only marginal. Poski and Tremayne (987) sugges he idea of a model porfolio. This involves comparing alernaive 2 In addiion o he MA and AR dimensions, i is also necessary o deermine he correc level of differencing. For he analysis in his paper, all he series were also seasonally differenced in addiion o a single regular differencing and fied wih an ARMA model. 20

23 models o he bes model suggesed by he informaion crierion. Denoing he bes model as (p*,0,q*)(p*,0,q*), he following saisic is compued for each alernaive model { ( 0 0 ) (( 0 )( 0 ))} R= exp T BIC p*,, q* P*,, Q* BIC p,, q P,, Q 2. (6) ( )( ) Alhough his saisic has no formally defined criical values, i may be used o quanify he decisiveness wih which a paricular model can be rejeced compared o he bes model. Poski and Tremayne sugges ha a value of less han 0 as a suiable cu-off poin. This implies ha models wih an informaion crierion wihin 2log( 0) T of he bes model ener he model porfolio. Then, using he model porfolio approach, no only he bes model is used in he diagnosic and forecasing sages bu all models in he porfolio. Using he DLHICP series over he period 984Q-998Q4 (T = 60), he porfolio approach would consider hree exra models in addiion o he model which minimised he BIC saisic. The larges porfolio using any of he six series considered in his paper would conain five models. 3 Gómez and Maravall (998, p.2) sugges using balanced models where possible if wo models perform relaively similarly. In oher words, selec a model where p and q are relaively similar raher han choosing a model wih jus AR erms or jus MA erms. For example, if he informaion crieria sugges ha a (2,0,0) and a (,0,) model perform similarly hen Gómez and Maravall would sugges using he more balanced model (,0,). One benefi of choosing balanced models migh be ha i would be easier o idenify common facors in he AR and MA polynomials. If an ARIMA model has common facors i should be possible o represen he model in a more parsimonious manner by eliminaing he common facor from boh he AR and MA componens. 3 This was he DLHICPA series. Using he criical value above, only one model would ener he porfolio for he DLHICPB, DLHICPC and DLHICPE series. 2

24 A more general alernaive approach o he model porfolio approach oulined above is simply o selec he op five or en ranking models and carry hese forward o he diagnosic checking and forecasing sage. This is he approach adoped in he semiauomaic ARIMA model selecion algorihm oulined below. Table 5 below presens he op five ranking models as classified by he BIC for he DLHICP, DLHICPC and DsDLHICPC series. The models indicaed in bold ype are hose models which would ener he model porfolio using he crierion of 0 suggesed by Poski and Tremayne (987). 4 The resuls are shown for he regularly differenced (DLHICPC) and he regularly differenced plus seasonally differenced nonenergy indusrial goods series (DsDLHICPC) as he ess for nonsaionariy were indeerminae beween a single regular differencing and a seasonal differencing in addiion o he regular differencing. The BIC indicaes ha he regularly differenced series performs slighly beer han he seasonally differenced series, reflecing he ambiguiy over he correc order of differencing. TABLE 5 - TOP FIVE MODELS BASED ON BIC - DLHICP, DLHICPC AND DSDLHICPC (984Q - 998Q4) Rank DLHICP DLHICPC DLHICPC Model BIC Model BIC Model BIC (0,0,0) x (,0,) (0,0,0) x (,0,0) (0,0,0) x (,,) (0,0,0) x (,0,0) (0,0,0) x (,0,) (0,0,0) x (0,,) (,0,0) x (,0,0) (,0,0) x (,0,0) (0,0,0) x (0,,0) (0,0,) x (,0,0) (0,0,) x (,0,0) (0,0,0) x (,,0) (,0,0) x (,0,) (3,0,0) x (,0,0) (,0,0) x (,,) In summary, he main advanages and disadvanages of objecive penaly funcion crieria are as follows: Advanages of Objecive Penaly Funcion Crieria 22

25 Objecive measure wih no subjecive inerpreaion. Resuls are readily reproducible and verifiable. BIC and HQC are asympoically consisen. Disadvanages of Objecive Penaly Funcion Crieria Need o calculae a wide range of models. This can be compuaionally expensive. There are no heoreical guidelines for choosing he maximum order of ARIMA model o consider. Someimes here is lile o chose beween compeing models STEP FOUR - MODEL DIAGNOSTICS An economerician should never fall in love wih his/her model The fourh sep will be he formal assessmen of each of he ime series models. This will involve a rigorous assessmen of he diagnosic ess for each of he compeing models. As differen models may perform reasonably similarly, a number of alernaive formulaions may have o be reained a his sage o be furher assessed a he forecasing sage. There are a number of diagnosic ools available for ensuring a saisfacory model is arrived a. Ploing he residuals of he esimaed model is a useful diagnosic check. This should indicae any ouliers ha may affec parameer esimaes and also poin owards any possible auocorrelaion or heeroscedaciy problems. A second check of model suiabiliy is o plo he auocorrelogram of he residuals. If he model is correcly specified he residuals should be whie noise. Therefore, a plo of he auocorrelogram should immediaely die ou from one lag on. Any significan auocorrelaions may indicae ha he model is misspecified and may poin o he soluion. For example, if a (0,0,)(0,0,0) model of a quarerly ime series is esimaed, 4 Noe, however, ha an indicaion of he relaive sabiliy of he DLHICP series over he period (984Q - 998Q4) is ha fiing a sraigh line (i.e., a (0,0,0)(0,0,0) model) o he series ranks 2s ou of 64 models under he BIC (-0.327). 23

26 bu he auocorrelogram of he residuals indicaes a significan auocorrelaion a he fourh lag, his would indicae ha a (0,0,)(0,0,) model should be esimaed, as his migh remove he auocorrelaion a he fourh (seasonal) lag. Figure 6 plos he ACF of he residuals from he (0,0,0)(,0,) model of DLHICP fied over he period 984Q - 998Q4. In general, he auocorrelaions are no significanly differen from zero, however, he auocorrelaions a lags hree, five and eigh are marginally significan. FIGURE 6 - PLOT OF ACF OF RESIDUALS FROM (0,0,0)(,0,) MODEL OF DLHICP SERIES, 984Q-998Q ACF(resids) More formal es saisics exis which involve esing he residuals of he esimaed model. The Ljung-Box (978) Q saisic is he mos commonly used es saisic. The Q-saisic ess for auocorrelaion in he residuals where Q is defined as (7) ( ) ( 2) ( ) Qk = TT+ T i r i k i= 2 a χ k 2. Anoher essenial check is o es he robusness of a seleced model by esimaing i over a number of differen ime periods. If he parameer esimaes are no sable over 24

27 ime his indicaes ha furher consideraion will have o be given o he model. I may be ha he ime series conains a srucural break and ha, for he purposes of forecasing, only he period since he srucural break should be used when esimaing he model, as here can be a fundamenal conflic beween esimaing a model o maximise in-sample goodness-of-fi and opimising ou-of-sample forecas performance when he series conains a srucural break. A formal es for nonsaionariy in he presence of a srucural break may be carried ou using Perron s (989) augmened uni roo es. Examining he DLHICP daa using Perron s augmened uni roo es, allowing for a srucural break, indicaes ha he daa are saionary pos-983, wih he saisic on he srucural break being maximised around STEP FIVE - FORECASTING AND FORECAST EVALUATION If he univariae modelling procedure is being uilised for forecasing purposes hen his sep can also form an imporan par of he diagnosic checking. Using ARIMA models for forecasing is relaively sraighforward. For example, consider a non-seasonal (,0,) model. The esimaed model is given by (8) X = φ X + a + θ a Then he forecas value one period ahead condiional on all informaion up o ime,, is simply given by F (9) X+ = φx + θa as E (a +i ) equals zero i>0. Similarly, 25

28 F F F (0) X+ 2 = φ X+, as E (a + ) and E (a +2 ) equal zero and replace X + wih he value given in equaion (9). F () X = φ X and so on. F To assess he ou-of-sample forecasing abiliy of he model i is advisable o reain some observaions a he end of he sample period which are no used o esimae he model. One approach is o esimae he model recursively and forecas ahead a specific number of observaions. For example, consider a ime series wih daa available from 976Q o 998Q4 and we wish o forecas four seps ahead (i.e., 999Q-999Q4). We could iniially esimae he model over he period 976Q o 992Q4 and forecas four seps ahead. Then re-esimae he model over he period 976Q o 993Q and forecas four seps ahead. Repea his process unil he esimaion period leaves no ou of sample observaions available for forecas evaluaion (i.e., 976Q o 998Q4). Using he acual inflaion daa over he period 992Q o 998Q4, his allows us o calculae 24 one sep ahead forecas errors, 23 wo-sep ahead forecas errors,..., and 2 four-sep ahead forecas errors. These can be used o calculae saisics such as mean error (ME), mean absolue error (MAE), roo mean squared error (RMSE) and Theil s U. Denoing he forecas error as e = X value of he series and he forecas value), hen X (i.e., he difference beween he realised F (2) ME = F i= F e (3) MAE = F i= F e F 2 (4) RMSE = ( e ) F i= 26

29 (5) Theil s U = F F ( e ) i= 2 F F N ( e ) i= 2 = RMSE RMSE N where, F equals he number of ou-of-sample observaions reained for forecas evaluaion allowing for he forecas sep, and N denoes he naive model of no change in he modelled series from he las available observaion. One indicaion ha he model specificaion could be improved is if he ME for each of he five seps are eiher all posiive or all negaive. This would indicae ha he model is eiher forecasing oo low on average (if posiive) or oo high on average (if negaive). If he ME is of he same magniude as he MAE his would also indicae ha he model is forecasing consisenly eiher oo low (if he ME is posiive) or oo high (if he ME is negaive). The RMSE will always be a leas as large as he MAE. They will only be equal if all errors are exacly he same. Theil s U saisic calculaes he raio of he RMSE of he chosen model o he RMSE of he naive (i.e., assuming he value in he nex period is he same as he value in his period - no change in he dependen variable) forecasing model. 5 Thus, a value of one for he Theil saisic indicaes ha, on average, he RMSE of he chosen model is he same as he naive model. A Theil saisic in excess of one would lead one o reconsider he model as he simple naive model performs beer, on average. A Theil saisic less han one does no lead o auomaic accepance of he model, bu does indicae ha, on average, i performs beer han he naive model. The advanage of he Theil saisic is ha i is 5 The naive model for he one-sep ahead forecas assumes inflaion follows a random walk (i.e., Π = Π - + e ). 27

30 uniless as i compares he RMSE of he chosen model o ha of he naive forecas model. The ME, MAE and RMSE all vary depending on he dimension (or scale of measuremen) of he dependen variable. The Theil saisic also provides a quick comparison wih he no change model and, as such, is a measure for one-sep ahead forecass of he addiional forecasing informaion he model provides beyond a random walk model. An addiional es of he ARIMA model would be o compare is performance wih compeing models including alernaive ARIMA specificaions and mulivariae models. These ess should be carried ou on he range of models carried over from he diagnosic checking phase. Should some models forecas significanly worse han ohers his may be an indicaion of parameer insabiliy or uni roo problems if some of he facors of he AR or MA polynomials are close o or greaer han uniy. Table 6 below presens some forecas saisics for DLHICP series esimaed over he period 984Q o 998Q4. These saisics were calculaed by firs esimaing he model over he period 984Q o 992Q4 and forecasing four seps ahead. The model was hen recursively esimaed, sepping forward one quarer a a ime, and again forecas four seps ahead. As he sample period reaches 998Q, obviously one canno forecas four seps ahead. Hence he number of forecas observaions available declines wih each sep. The main poins o noe from Table 6 are as follows: The RMSE varies beween 0.43% and 0.38%. 6 This implies a 90 per cen confidence inerval of approximaely.4 per cen per quarer. Alhough his appears high relaive o he mean of he series, i compares favourably wih resuls repored by Cecchei (995) for he Unied Saes who calculaed a 90 per cen confidence inerval of approximaely.3 per cen for one sep ahead inflaion forecass by a commercial inflaion forecaser. These resuls also compare favourably wih Bayesian vecor auoregression (BVAR) resuls for forecasing a 6 The mean of he DLHICP series over he period 993 Q Q4 was 0.56 per cen. 28

31 shor horizons he Irish HICP as repored by Kenny e al (998). For longer horizons (i.e., in excess of 4 quarers) he BVAR model ouperforms he ARIMA model. The mean errors are significanly lower han he mean absolue error. This implies ha he forecass are neiher sysemaically over-forecasing or under-forecasing inflaion. The sign of he mean error varies by sep. Again his implies ha he forecass are neiher sysemaically over-forecasing or under-forecasing inflaion. The Theil saisics are consisenly below uniy, indicaing ha he seleced model ouperforms he simple naive model. However, noe ha a sraigh line model (i.e., (0,0,0)(0,0,0) model) also ouperforms he naive model. In fac he fied model (0,0,0)(,0,) is lile more han a sraigh line model wih allowance for a degree of seasonaliy. 7 Given he relaively sable paern of inflaion over he period, i is perhaps unsurprising ha his model ouperforms more elaborae models. I does indicae ha, over he period 993Q-998Q4, i would be difficul o improve on a very simple model of inflaion. Thus over a shor horizon forecas saisics from mulivariae models are unlikely o ouperform in any significan way he forecas saisics presened here for a relaively simple ARIMA model. TABLE 6 - FORECAST STATISTICS FOR (0,0,0)(,0,) MODEL OF DLHICP (993Q - 998Q4) Sep Mean Error Mean Abs. Error RMS Error Theil U N.Obs Avg Tables 7 and 8 presen forecas saisics using DLHICPC and DsDLHICPC over he same period. Examining he forecas saisics for DLHICPC firs, he impac of 29

32 seasonal sales is shown up in he Theil saisics for he second and fourh sep forecass, which are significanly higher han hose for oher seps. This is direcly as a resul of he volaile naure of he series, wih increases in one period generally being followed by decreases in he nex. Hence, ouperforming he naive forecas a odd seps would be expeced. However, a even seps (i.e., when Summer and Winer sales and here respecive rebounds coincide) he naive model will perform relaively well. Also noe ha he RMSE of he forecass is higher han for he overall HICP indicaing he greaer volailiy of he series. The RMSE vary beween 0.66% and 0.69% (compared o approximaely 0.42% for he overall HICP). There is lile difference beween Tables 7 and 8. The RMSE and Theil saisics are broadly similar, perhaps, reflecing he degree of ambiguiy presen in he ess for saionariy of he DLHICPC series. The resuls are obained for he DLHICPE (services) series are somewha similar o hose for he DLHICPC series. Based on hese resuls perhaps consideraion of Auoregressive Fracionally Inegraed Moving Average (ARFIMA) models for modelling he DLHICPC and DLHICPE series migh be jusified. TABLE 7 - FORECAST STATISTICS FOR (0,0,0)(,0,0) MODEL OF DLHICPC (993Q - 998Q4) Sep Mean Error Mean Abs. Error RMS Error Theil U N.Obs Avg The esimaed seasonal AR coefficien is 0.58 and he esimaed seasonal MA coefficien is