Forecasting with ARMA Models

Transcription

1 CS-BIGS 4(): CS-BIGS hp://wwwbenleyedu/csbigs/vol4-/jaggiapdf Forecasing wih ARMA Models Sanjiv Jaggia California Polyechnic Sae Universiy, San Luis Obispo, CA Increasing awareness and daa availabiliy has made saisical analysis an inegral par of managerial decision making Forecasing, in paricular, has become an essenial skill for all financial analyss, economiss, markeing and accouning professionals Arguably all business sudens should be educaed in forecasing echniques wih an emphasis on ARMA modeling While saisical packages such as Eviews have increasingly become user friendly, he discussion and he implemenaion of he imporan diagnosic analysis of ARMA models in popular undergraduae exbooks has no been oally saisfacory This paper succincly oulines his imporan diagnosic sep for ARMA modeling wih Eviews, using monhly daa on 3-monh governmen Treasury bills rae for illusraion The presenaion is accessible o readers wih an inermediae level of saisics Keywords: xxx Inroducion Forecasing is an imporan aspec of saisical analysis ha provides guidance for decisions in all areas of business I is imporan o be able o make sound forecass for variables such as sales, producion, invenory, ineres raes, exchange raes, real and financial asse prices for boh shor and long erm business planning In essence, he success of any business or governmen organizaion depends on he abiliy o accuraely forecas is revenue and expendiure Auo-regressive moving average (AR- MA) models provide a unifying framework for forecasing These models are aided by he abundance of high qualiy daa and easy esimaion and evaluaion by saisical packages There are numerous applicaions where researchers have employed basic as well as advanced ARMA models for forecasing While a seasoned researcher is well versed in hese models, he deails and easy o follow seps of AR- MA modeling, which may be invaluable o a business suden or a praciioner, are no available For insance, popular exbooks in economerics, including Using Economerics (0) by Sudenmund and Inroducory Economerics: A Modern Approach (009) by Jeffrey Wooldridge provide no formal discussion of ARMA models While specialized books such as Elemens of Forecasing (008) by Francis Diebold and Business Forecasing (009) by Hanke and Wichern discuss ARMA models, heir suggesed diagnosic analysis of he models has no been comprehensive For insance, hese books do no provide a unified reamen of in-sample and ou of sample analysis of compeing models This paper succincly oulines he imporan sep of diagnosic analysis of ARMA models wih Eviews, using monhly daa on 3-monh governmen Treasury bills rae for illusraion Advanced exbooks such as Enders (005) do he job properly

2 - 0 - Forecasing wih ARMA Models / Jaggia There are a number of useful issues ha sudens and praciioners mus undersand regarding he analysis of ARMA models Firs, an inuiive undersanding of saionariy is essenial since hese models can only be idenified for saionary series Informal graphs can be used o look for deparures from he consancy of he mean and he variance We generally difference he series o achieve consancy of he mean However, if boh he mean and he variance are ime-varying, we also need a dampening or a pre-differencing ransformaion This simple explanaion is lef ou in he exbooks by Diebold as well as Hanke and Wichern Since economic heory provides no guidance on ransformaions, he naural log or square roo pre-ransformaions may produce beer forecass Sudens and praciioners mus be advised o idenify several ARMA models wih various ransformaions and use model selecion crieria o pick he mos appropriae model Anoher issue ha has no been explicily saed is a consisen definiion of he residuals ha works for all models, wih or wihou pre-ransformaions Saisical packages usually sore he residuals of he esimaed models on he basis of pre-ransformaions Therefore, he model selecion measures ha emerge from such residuals are no comparable Finally, since forecasing is essenially an ou-of-sample problem, compeing models mus also be compared on he basis of heir performance ouside he sample period This imporan analysis of he ou of sample forecas errors based on saic and dynamic mehods has been lef ou in popular exbooks Model Idenificaion Le y, y,, yt represen a sample of T observaions of a variable of ineres y Consider a ime series { } y represening he monhly ineres rae of 3-monh Treasury bills for a sample period of January 984 o December 007 (see Table ) This applicaion is used hroughou he paper o moivae and implemen ARMA forecasing models Since saionariy is essenial for he idenificaion of an ARMA model, he firs sep is always o es for saionariy of he underlying series While many macroeconomic, business, and financial series are no saionary, he series can easily be made saionary by differencing wih or wihou pre-ransformaions Formally, { y } is said o be saionary if he mean, E( y ) = μ, he variance, Var( y) = E( y μ) = σ, and he covariance, Cov( y, y s) = E( y μ) ( y s μ) = γs, are all sable over ime For insance, for he series o be saionary, i mus no exhibi any sochasic rend (changing mean) or varying volailiy (changing variance) Whereas he sochasic rend is generally removed by firs differencing of he series, he varying volailiy can be reduced wih a naural log or square roo pre-differencing ransformaion As menioned earlier, he imporan discussion of pre-ransformaions is lef ou in he above exbooks alhough hey include he applicaions of growh raes based on log ransformaions A simple plo of he series is ofen used as an informal es for saionariy The series is considered saionary if he level and he spread do no exhibi any visual changes over ime The plo of he auocorrelaion funcion also serves as anoher visual es for saionariy 3 The series is considered saionary if he ACF cus off, or dies down, fairly quickly A plo of he monhly ineres rae series in Figure indicaes non-saionariy as i exhibis an obvious rend Figure 3-monh Treasury Bill Rae The differenced series specified in erms of a lag operaor L, y y = ( L) y, is ploed in Figure As is ofen he case wih graphical ess, i is no clear wheher or no he variance of he differenced series is ime invarian In such siuaions, i is recommended o also consider pre-ransformaions and use model selecion crieria o deermine he mos appropriae model for forecasing In he regression conex, i is analogous o spuriously comparing a linear wih a log-linear model on he basis of he compuer generaed R 3 In Eviews, choose 'Correlogram' from 'Series Saisics' o ge he auocorrelaion (ACF) and he parial auocorrelaion (PACF) funcions of he series

3 - - Forecasing wih ARMA Models / Jaggia Table 3-Monh Treasury Bill Rae Dae Tbill Dae Tbill Dae Tbill Dae Tbill Dae Tbill Dae Tbill Jan Jan Jan Jan Jan Jan Feb Feb-88 5 Feb Feb Feb Feb Mar Mar Mar Mar-9 49 Mar Mar Apr Apr Apr Apr Apr-00 5 Apr May May-88 May-9 33 May-9 50 May May-04 0 Jun Jun-88 4 Jun-9 3 Jun Jun Jun-04 7 Jul-84 0 Jul Jul-9 3 Jul-9 55 Jul Jul Aug Aug Aug-9 33 Aug Aug Aug Sep Sep Sep-9 9 Sep Sep Sep-04 5 Oc Oc Oc-9 8 Oc Oc-00 Oc-04 7 Nov-84 8 Nov Nov-9 33 Nov Nov-00 7 Nov Dec Dec Dec-9 3 Dec-9 49 Dec Dec-04 9 Jan Jan Jan Jan Jan-0 55 Jan Feb Feb Feb Feb Feb Feb Mar Mar Mar Mar Mar-0 44 Mar Apr Apr Apr Apr-97 5 Apr Apr May May May-93 9 May May-0 3 May Jun Jun Jun Jun Jun Jun Jul Jul Jul Jul Jul-0 35 Jul-05 3 Aug Aug Aug Aug Aug-0 33 Aug Sep Sep Sep Sep Sep-0 4 Sep Oc-85 7 Oc Oc Oc Oc-0 Oc Nov Nov Nov Nov Nov-0 87 Nov Dec Dec Dec Dec-97 5 Dec-0 9 Dec Jan Jan Jan Jan Jan-0 5 Jan-0 44 Feb-8 70 Feb Feb Feb Feb-0 73 Feb Mar-8 5 Mar Mar Mar Mar-0 79 Mar-0 45 Apr-8 0 Apr Apr Apr Apr-0 7 Apr-0 40 May-8 5 May May May May-0 73 May-0 47 Jun-8 Jun Jun Jun Jun-0 70 Jun Jul Jul-90 7 Jul Jul Jul-0 8 Jul Aug Aug Aug Aug Aug-0 Aug-0 49 Sep-8 5 Sep Sep-94 4 Sep-98 4 Sep-0 3 Sep-0 48 Oc-8 58 Oc Oc Oc Oc-0 58 Oc-0 49 Nov Nov Nov Nov Nov-0 3 Nov Dec Dec Dec Dec Dec-0 9 Dec Jan Jan-9 Jan Jan Jan-03 7 Jan Feb Feb Feb Feb Feb-03 7 Feb Mar Mar-9 59 Mar Mar Mar-03 3 Mar Apr Apr-9 55 Apr Apr Apr-03 3 Apr May-87 5 May-9 54 May May May May Jun Jun Jun Jun Jun Jun-07 4 Jul Jul Jul Jul Jul Jul Aug Aug Aug Aug Aug Aug Sep Sep-9 5 Sep Sep Sep Sep Oc-87 3 Oc Oc Oc Oc Oc Nov Nov-9 45 Nov Nov Nov Nov Dec Dec Dec Dec Dec Dec Source: Board of Governors of he Federal Reserve Sysem

4 - - Forecasing wih ARMA Models / Jaggia Figure Differenced 3-monh Treasury Bill Rae Figure 3 Differenced, in logs, 3-monh Treasury Bill Rae Figure 4 Differenced, in square roos, 3-monh Treasury Bill Raes Figures 3 and 4 conain he graphs of he series of differenced logs ( ln y ln y = ( L)ln y ) and differenced square roos ( y y = ( L) y ) respecively The ACF and he PACF of he hree ransformed series are presened in Table The simple plos as well as he ACF graphs sugges ha he ransformed series are all saionary 4 4 Dickey-Fuller ess for a uni roo, alhough no included in he paper, corroborae he findings of informal graphical ess Table Auocorrelaions and Parial Auocorrelaions of Transformed Series Lag ( L) Tbills ( L) ln( Tbills) ( L) Tbills AC PAC AC PAC AC PAC 0509* 0509* 050* 050* 050* 050* 000* * * * 0 034* 008* 045* 08* 4 09* * * * 00 04* * 05 08* 0 098* 05 07* 039* 7 0* * * * * * * Noes AC and PAC denoe auocorrelaions and parial auocorrelaions respecively and heir significance is compared wih ± / T =± 03 * denoes significance a he 5% level The general specificaion of an ARMA( p, q) model wih mean zero, where p and q represen he order of he auoregressive and moving average pars of he series, is given by: y = ϕy + ϕy + + ϕpy p + ε + θε + θε + + θqε q () p q p p = ( ϕ L ϕ L ϕ L )y = ( + θ L + θ L + + θ L ) ε where i=,,, p and j =,,, q < ϕi < and ϕ+ ϕ + + ϕp < θ j < and θ+ θ + + θp The error erm, ε is whie noise wih: E( ε ) = 0, Var( ε ) = σ, Cov( ε, ε ) = 0 s () s The sample auocorrelaions and parial auocorrelaions are compared wih heir heoreical counerpars o idenify he parameers p and q In general, an auoregressive (moving average) model is inferred if he sample auocorrelaions (parial auocorrelaions) decay oward zero and he sample auocorrelaions (parial auocorrelaions) cu off Le ρ s represen he auocorrelaion beween y and is s-period lagged value y s and le φ ss be he corresponding parial auocorrelaion In Table 3,

5 - 3 - Forecasing wih ARMA Models / Jaggia Table 3 Idenificaion of ARMA Models Model ACF PACF Whie Noise ρ s = 0 s φss 0 MA() Direc or oscillaory ρ 0, ρ s = 0 s > decay oward zero MA(q) Direc or oscillaory ρ 0,, ρq 0, ρs = 0 s > q decay oward zero AR() Direc or oscillaory decay oward zero φ 0, φss = 0 s > AR(p) we presen he crieria for choosing p and q based on he values of he sample auocorrelaions and parial auocorrelaions AR- MA(,) AR- MA(p,q) Direc or oscillaory decay oward zero Decay oward zero beginning from ρ Decay oward zero beginning from ρ q φ 0,, φpp 0, φ ss = 0 s > p Decay oward zero beginning from φ Decay oward zero beginning from φ pp In mos applicaions, he idenificaion may no lead o a single model and hence several models mus be examined wih and wihou pre-ransformaions As menioned earlier, since he underlying series is no saionary, ARMA( p, q) models are idenified wih y replaced wih ( L) y, ( L) ln y, or ( L) y We consider only hree models for illusraion Model is a consrained ARMA(,) model, idenified from he ACF and PACF graphs of a differenced series, ( Ly ) Models and 3 are boh consrained AR() models based on he ACF and PACF of differenced series wih a log, ( L) ln y, and a square roo, ( L) y, pre-ransformaion respecively The models, and heir corresponding esimaion commands in Eviews, are given below Model: [ ) AR ] ( ϕl ϕ L )( L) Tbill = ( + θ L) ε Eviews : d(tbill) MA( () Model 3 ( ϕl ϕl ϕl )( L)ln( Tbill ) = ε [ Eviews : d(log ( Tbill)) AR() AR3) AR() 3 Model3 (3) ] (4) [ R A ] (5) ( ϕl ϕl )( L) Tbill = ε Eviews : d( sqr(tbil l)) A () R() The above compeing models are esimaed wih he daa from 984:0-005: The las 4 observaions from 00 and 007 are lef ou o analyze ou of sample performance of hese models, which we will discuss laer In Table IV, we presen he esimaion resuls of he hree models All parameers of he models are saisically significan a a 5 percen level Table 4 Parameer Esimaes of Compeing ARMA Models Parameer Model Model Model 3 Esimaes 0348* 0435* 0438* ϕ (0080) (0055) (0054) 049* ϕ 3 (0057) 0749* 05* 083* ϕ (00589) (005) (00545) 093* θ (08) Noes Sandard errors are in parenheses below he parameer esimaes; * denoes significance a he 5% level; Esimaion period is 984:0 005: Model : ( ϕl ϕl )( L) Tbill = ( + θl) ε ; Model : 3 ( ϕl ϕ3l + ϕ L )( L) ln( Tbill ) = ε ; Model 3: ( ϕl ϕl )( L) Tbill = ε Apar from he saisical significance of he parameers, an imporan es is ha he residuals, e ˆ = y y, of he esimaed model are whie noise In oher words, for a model o be accepable, no discernable informaion should be lef back in he residuals A convenien es for whie noise is he Ljung-Box (978) es, also called he Q es, which is implemened as: k Q= n( n+ ) ˆ ρ () i i= ( n i) where ˆi ρ is he esimaed auocorrelaion of he residual a lag i and n 5 is he number of residuals used for he es Q has an asympoic chi-square disribuion wih degrees of freedom given by k-p-q The choice of k is somewha ad hoc, and i is common o perform his es for various reasonable values of k A model is accepable only when he null hypohesis ha he residuals are whie noise is no rejeced I is worh poining ou ha he Q es is implemened on he residuals and herefore he degrees of freedom of he es mus be adjused for he number of esimaed parameers, p+ q Eviews makes his adjusmen in he p-values of he Q es if i is performed as a residual es wihin esimaion I is common for sudens o ignore his imporan fac and misakenly perform he Q-es on he residuals ouside he model In fac, Diebold makes his misake and consequenly miscalculaes p-values hroughou he book 7 5 The number of residuals may be smaller han he original sample size due o differencing and he order of he auoregression For he Q-es of he residuals in Eviews ha is adjused for he degrees of freedom, choose Views afer esimaion and under Residuals Tess, compue Correlogram Q saisics 7 On page 03 of Chaper 0, "Puing i All Togeher", he repored p-values are spuriously large As a consequence his seleced model should have acually been discarded making he remaining discussion in his imporan chaper quie fuile

6 - 4 - Forecasing wih ARMA Models / Jaggia The op par of Table 5 presens he p-values of he Q es using k equal o,, and 8 Since he p-values are consisenly above he chosen significance level of 5 percen, he residuals from Models,, and 3 are found o be whie noise Models ha pass he whie noise es of he residuals are hen subjeced o a variey of model selecion crieria o selec he mos appropriae forecasing model Table 5 Model Selecion Crieria Applied o Compeing Models Model Selecion Crieria Model Model Model 3 p-value for he Q es (k=) 0493* 073* 039* p-value for he Q es (k=) 055* 0593* 0339* p-value for he Q es (k=8) 009* 085* 048* AIC of he residuals 00374@ SIC of he residuals @ RMSE of forecas Errors (Dynamic) @ RMSE of forecas Errors (Saic) @ 043 Noes * denoes ha he residuals are whie noise a he 5% denoes he bes model selecion measure AIC and SIC are based on he residuals from 984:08-005:; RMSE is based on he forecas errors from 00:0-007: Model : ( ϕl ϕl )( L) Tbill = ( + θl) ε ; 3 Model ( ϕl ϕ3l + ϕl )( L) ln( Tbill ) =ε ; Model 3: ( ϕl ϕl )( L) Tbill = ε Model Selecion Crieria Compeing models are compared on he basis of insample predicabiliy and ou-of-sample forecasing abiliy The comparison of in-sample predicabiliy of he esimaed model is based on he residuals Here is where he confusion lies The residuals repored by saisical packages are no comparable across models if hey are based on differen pre-ransformaions The defaul residuals compued by saisical packages are ( L)( y yˆ ) for Model, ( L)(ln y ln yˆ ) for Model and ( L) y yˆ for Model 3 In order o compue ( ) model selecion measures, i is imporan o use consisenly defined residuals, e = y yˆ, where yˆ can easily be obained for any model 8 This issue is no raised in he above menioned exbooks 8 In Eviews consisenly defined residuals, Resid, are easily compued by choosing Forecas from he menu and specifying he esimaion range wih Saic forecas Two commonly used model selecion crieria, based on he residuals, are he Akaike Informaion Crierion (AIC) and he Bayesian Informaion Crierion (BIC), which are derived as: 9 k e k/ n e AIC = exp ; SIC = n (7) n n n The AIC and SIC are based on he residual sum of squares and a penaly for he number of parameers used in he esimaed model They boh measure he informaion conen in he residuals; he smaller he value, he beer is he fi Since model selecion measures are used o compare models across ransformaions, one mus firs compue he residuals manually o derive AIC and SIC So far we have discussed model comparison on he basis of heir in-sample predicabiliy The AIC and SIC essenially compare compeing models on he basis of how well he esimaed models fi he sample daa While hese crieria impose a penaly for over-fiing, hey are imperfec and can be misleading Someimes a well fiing model may no perform so well in making ou-ofsample forecass Therefore, i is imporan o choose beween compeing models on he basis of boh in-sample predicabiliy as well as ou-of sample forecasing abiliy In order o analyze ou of sample performance of he models, we generally leave ou he las few observaions from esimaion If he sample consiss of T observaions, we esimae he model wih T and leave he remaining T for analyzing ou of sample performance, where T= T+ T 0 Alernaive models are compared on he basis of heir in-sample predicabiliy wih a shorened span of daa and also heir forecasing abiliy of he holdback period Ou-of-sample performance is compared on he basis of he forecas errors, f* = y * yˆ * for * = T +, T+,, T Dynamic as well as he saic forecass for T observaions, y ˆ *, are based on he parameer esimaes obained from he reduced sample of T observaions All dynamic forecass are condiional on 9 There are oher monoonic ransformaions, including he ones repored by Eviews ha can also be used for model comparison 0 The definiion of he holdback period is somewha ad hoc Furher, a large sample is needed o have reliable esimaes in he insample period and he forecas analysis in he ou-of-sample period The ideal mehod, hough ime consuming, is o make forecass on he basis of recursive esimaion In oher words, we firs esimae he model wih observaions o make a one-period ahead T

7 - 5 - Forecasing wih ARMA Models / Jaggia T he informaion a The saic mehod condiions on he acual value of he variable raher han is fied value o forecas beyond one period Therefore, while a firs period forecas beyond he esimaion sample is idenical, subsequen forecass can differ significanly beween he dynamic and saic mehods The saic and dynamic forecass are easily compued in Eviews by choosing Forecas and specifying he forecas range wih a saic or a dynamic mehod Wih boh ypes of forecass, he preferred model will have he smalles roo mean square error (RMSE) of he forecas errors where * RMSE( f ) = f (8) T Surprisingly hese simple model selecion crieria have no been implemened in sandard exbooks The lower porion of Table V conains he AIC and SIC of he consisenly defined residuals, and he RMSE of he forecas errors wih saic and dynamic mehods for each of he esimaed models, repored in Table IV As is ofen he case, he four model selecion crieria, namely AIC, SIC, RMSE (dynamic), and RMSE (saic), do no lead o a clear winner However, we choose Model 3 since i ouperforms he oher wo models in erms of SIC of he residuals and RMSE (dynamic) of he forecas errors Furhermore, Model 3 beas Model and Model individually in hree ou of four model selecion crieria Ineresingly, models idenified on he square roo preransformaion ouperforms all oher models As a final sep he chosen Model 3 is re-esimaed wih he enire daa from o make a forecas of 988 percen for January, 008 I is worh noing ha he above hree models used in his paper are for illusraion only There may be oher suiable models wih ransformaions including derending Since economic heory provides no guidance on ransformaions or ARMA parameers, sudens and praciioners should be encouraged o idenify all possible suiable models for comparison While he fuure is uncerain, praciioners mus use he available sample informaion o make he bes possible forecas her, while saisical packages such as Eviews have grealy simplified esimaion and analysis of he ARMA models, popular undergraduae exbooks do no provide a comprehensive reamen of he diagnosic analysis of hese models These books lack he emphasis on consisenly defined residuals o compare in-sample predicabiliy of compeing models wih and wihou preransformaions Furher, since forecasing is essenially an ou-of-sample problem, he imporan analysis of he ou of sample forecas errors based on saic and dynamic mehods has also been lef ou This paper succincly oulines his imporan diagnosic sep of ARMA modeling, using monhly daa on 3-monh Treasury bills rae for illusraion The easy o follow seps of ARMA modeling wih Eviews included in he paper can be invaluable o a business suden or a praciioner Correspondence: sjaggia@calpolyedu REFERENCES Diebold, F X 007 Elemens of Forecasing Thomson and Souh-Wesern Fourh Ediion Enders, W 005 Applied Economeric Time Series John Wiley & Sons Inc Second Ediion Hanke, JE and Wichern, DW 009 Business Forecasing Pearson Prenice Hall Ninh Ediion Sudenmund, A H 0 Using Economerics: A Pracical Guide Pearson Addison Wesley Sixh Ediion Wooldridge, JM 009 Inroducory Economerics: A Modern Approach Souh-Wesern Cengage Learning Fourh Ediion Conclusion Mos academics and praciioners are in favor of imparing modern forecasing ools o business and economics sudens Despie he obvious imporance, many universiies do no offer a specialized course in forecasing Fur- ŷ T + ŷ T ec + T forecas The model is hen re-esimaed wih + observaions o forecas