Chapter 3, Part IV: The Box-Jenkins Approach to Model Building
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1 Chaper 3, Par IV: The Box-Jenkins Approach o Model Building The ARMA models have been found o be quie useful for describing saionary nonseasonal ime series. A parial explanaion for his fac is provided by Wold s Theorem: "Any saionary series can be expressed as he sum of wo componens: a perfecly forecasable series and a moving average of possi- bly infinie order." In pracice, he only perfecly forecasable aspec of an economic series is he seasonal componen, if any. Thus, nonseasonal series can always be represened by an MA ( ) model, which in urn can usually be approximaed by an ARMA(p,q) model wih p +q small (i.e., wih a small number of parameers). Thus, he ARMA models can ypically provide an accurae ye parsimonious descripion of saionary nonseasonal series. In fac, mos economic series are nonsaionary and have a seasonal componen. This does no degrade he usefulness of ARMA models, however, since he raw daa may ypically be processed ( ofen by some form of differencing) o produce an approximaely saionary, nonseasonal series. This series may be forecas by fiing an appropriae ARMA model. Forecass of he original series may hen be obained by reversing he processing operaion. Specifically, he processing proceeds as follows. Seasonal componens may be removed by a ech- nique called "seasonal differencing", discussed in Chaper 4. Nonsaionariy can ofen be classified as a "rend in mean", or a "rend in variance". Trends in mean can usually be handled by ordinary differencing. An example is he series x = (a + b ) +ε. Trends in variance can ofen be convered ino rends in mean by aking logarihms, as wih he series x = exp (a + b ) exp (ε ). The rend in mean of log x can hen be removed by differencing. Since he echniques jus described are reasonably effecive, we can safely assume ha our daa (afer being suiably processed) forms a saionary nonseasonal ime series. Wha Is Model Building? So far, in our discussions of forecasing for saionary series, we have assumed ha he series acually obeys an ARMA(p,q) model, ha he model orders (i.e., p and q ) are known, and ha he corresponding parameer values are known as well. In pracice, we will simply have a series of daa
2 -2- values, and none of hese assumpions will be valid. Indeed, i is highly doubful ha our saionary series obeys an exac ARMA model. The main jusificaion for using such a model is no ha we believe i acually holds, bu insead ha we believe i can provide an accurae, parsimonious descripvalues for (p, q ), and how should we esimae he corresponding parameer values? Box and Jenkins ion of he daa, as discussed above. Sill, some imporan quesions remain: Wha are he appropriae refer o hese respecively as he ideni f icaion and esimaion sages of model building. We will describe how hese wo sages are implemened. Noe ha once a model has been idenified and is parameers esimaed, he resul is aken o be he rue model and forecass are obained accordingly. I is worh remembering, however, ha he fied model is almos cerainly no idenical o he rue model. This can resul in a ype of forecasing error (essenially ignored by mos auhors) which canno be easily gauged, and which can in fac be quie devasaing. As a minimum proecion agains such problems, we mus check ha he fied model is (or a leas seems o be) adequae. Such diagnosic check- ing is he final sage of he Box-Jenkins approach, and will be described. Model Idenificaion: The Correlogram and Parial Correlogram The class of ARMA models is quie large, and in pracice we mus decide which of hese models is mos appropriae for he daa a hand, x, x,...,x 1 2 n. The correlogram and parial correlogram are wo simple diagrams which can help us o make his decision (i.e., o "idenify he model"). We firs describe he correlogram, since i is concepually he simples. The heoreical correlogram is a plo of he heoreical auocorrelaions ρ = corr (x, x ) k k agains k. The sample correlogram is a plo agains k of he esimaed auocorrelaions n r = (x x)(x x) / (x x). k Σ =k + 1 k n Σ = 1 2 If he series were acually MA(q), is heoreical correlogram would "cu off" (i.e., ake he value zero) for k > q. Thus, we would expec ha he sample correlogram would have a similar (hough no idenical) shape o he heoreical correlogram, and would herefore say reasonably close o zero for
3 -3- k > q. Reversing his reasoning, we ge he following rule: If he correlogram seems o cu off for k > q, hen he appropriae model is MA(q). For AR(p) models, he auocorrelaions ρ k are approximaely (for large enough k ) ρ =A λ k k where λ < 1. Thus, for k large (say k p ), he correlogram would be expeced o decline seadily (if λ>0) or be bounded by a pair of declining curves (if λ<0). This paern of decline can ofen be dis- inguished from he "cuoff" described earlier, and should be aken as evidence ha he correc model is no MA. To acually idenify an AR model, however, we need a diagram which will have a more dis- incive shape when he series is acually AR. The parial correlogram is such a diagram. To define parial correlaions, suppose we fi an AR(k) model o our daa: x = â x + â x â x +ε. k1 1 k 2 2 kk k Then â is he esimae of he coefficien of x when a k h order AR is fied. Rewriing his as kk k x [â x â x ] = â x +ε, k1 1 k (k 1) (k 1) kk k w kk k e see ha â is a plausible esimae of he correlaion beween x and ha par of x which canno be forecas from x,...,x. â is called he parial correlaion beween x and x. I is he 1 (k 1) kk k esimaed correlaion beween x and x k afer he effecs of all inermediae x s on his correlaion are aken ou. Clearly, if he series is acually AR(p), hen he heoreical parial correlaions a kk will be zero for k > p. Thus, we can use he parial correlogram (i.e., a plo of he esimaed parial correlaion coefficiens) o idenify AR models: If he parial correlogram cus off for k > p, hen he appropriae model is AR(p). There is an ineresing dualiy (symmery) beween he properies of he correlogram and parial correlogram for pure AR and pure MA models. The behavior of a given diagram for a given model ype is he same as he behavior of he oher diagram for he oher model ype. We have already seen some evidence of his: The correlogram for an MA model and he parial correlogram for an AR model boh cu off. As we know, he correlogram for an AR model dies down (bu does no cu off). I can be shown ha he parial correlogram for an MA model dies down as well.
4 -4- A sill unanswered quesion is how we can idenify a mixed ARMA model. In his case, i can be shown ha he correlogram and parial correlogram boh die down (bu do no cu off). Thus, if boh diagrams die down, we can conclude ha he appropriae model is ARMA. Unforunaely, hough, he diagrams do no in his case help us o decide on he order (p, q ) of he mixed model. The following able summarizes he behavior of he diagrams. Behavior of Correlogram and Parial Correlogram for Various Models Correlogram Parial Correlogram AR Dies Down Cus Off MA Cus Off Dies Down ARMA Dies Down Dies Down Afer examining he correlogram and parial correlogram in he ligh of he above described pro- peries, we should be able o selec a few models which seem appropriae. (Unforunaely, he observed paerns are ofen no so clear as o unambiguously poin o a single model.) Anoher guiding principle in model idenificaion is ha of parsimony : The oal number of parameers in he model should be as small as possible (e.e., 3 or less, in he view of Box and Jenkins), subjec o he resricion ha he model provide an adequae descripion of he daa. If wo models appear o fi he daa equally well, he one wih he fewes parameers will always be preferred. Indeed, in his case he one wih he fewes parameers will almos cerainly produce he bes forecass. One reason is ha we can obain more precise (sable) parameer esimaes if he number of parameers is small. Besides faciliaing he idenificaion of models for saionary series, he correlogram can also diagnose nonsaionariy. If a series is nonsaionary (and needs o be differenced o produce a saionary series) hen he heoreical auocorrelaions will be nearly 1 for all k. Thus, if he esimaed correlogram fails o die down (or dies down very slowly), hen he series should be differenced. If he esimaed correlogram for he differenced series sill fails o die down, hen he series should be differenced once more. Noe, however, ha economic series ypically need o be differenced only once. If he series needs o be differenced d imes before an ARMA(p,q) model can be idenified, he original series is
5 -5- said o be an inegraed mixed auoregressive-moving average series, denoed ARIMA(p,d,q). The model idenificaion mehod jus described is he one advocaed by Box and Jenkins, and Granger (among ohers). Is usefulness has been amply demonsraed on acual daa, economic and oh- erwise. I is he mehod ha we will use in his course. The mehod does have some serious drawbacks, however: I is no enirely objecive, is implemenaion requires careful examinaion of he daa by a knowledgeable and experienced analys, and i may fail o unambiguously idenify a model. Since he publicaion of Box-Jenkins and Granger, several objecive mehods have been proposed and esed. These mehods auomaically selec a model wihou any inervenion from he user. Alhough here is no universal agreemen as he superioriy of he objecive mehods compared o he Box-Jenkins mehod, he poenial advanages of a high-qualiy auomaed mehod are quie srong. Sill, if an experienced analys is available, considerable insigh may be gained hrough examinaion of he correlo- gram and parial correlogram, even if an auomaed mehod is ulimaely used. We will discuss he new mehods more fully if ime permis. Esimaion In he las secion, we described ways of choosing an appropriae model. Sricly speaking, how- ever, "model idenificaion" consiss merely of selecing he form of he model, bu no he numerical values of is parameers. Suppose, for example, we have decided o fi an AR(1) model x = ax +ε. 1 Since he value of he parameer a is no known, i mus somehow be esimaed from he daa. Here, we describe mehods of esimaing he parameers of ARMA models. For pure AR models, here exis simple esimaion echniques, since here is a linear relaionship beween he auocorrelaions and he AR parameers. This relaionship can be invered, and hen he heoreical auocorrelaions can be replaced by heir esimaes, o yield esimaes of he AR parameers. In he AR(1) case, for example, we know ha ρ =a. Thus, we may esimae a by â = r. In general for he AR(p) model 1 1 x = ax + ax apx p +ε we obain a sysem of linear equaions called he Yule-Walker equaions by muliplying boh sides by
6 -6- x (k = 1,...,p), aking expecaions and hen normalizing. The k h equaion in he sysem is k ρ = a ρ + a ρ a ρ k 1 k 1 2 k 2 p k p. The esimaes â,...,â 1 k of he AR parameers are obained by solving his linear sysem, hereby obaining a formula for a,...,a in erms of ρ,...,ρ and hen replacing ρ,...,ρ by hei 1 p 1 p 1 p r esimaes r,...,r 1 p in his formula. This procedure is equivalen o solving he sysem r = â r + â r â r (k = 1,...,p k 1 k 1 2 k 2 p k p ) for â,...,â 1 p. The resuling values are called he Yule-Walker esimaes. I can be shown ha he Yule-Walker esimaed AR parameers always correspond o a saionary AR model. The siuaion for MA models is considerably more complicaed. The heoreical relaionship beween he parameers and auocorrelaions is no linear. For example, in he MA(1) x =ε +bε, we have ρ = b b 2 1 In his case, we ge a quadraic equaion for b, namely ρ b + ( 1)b +ρ =0, which has he wo solu ions b = 1± 1 4ρ. 2ρ I 1 can be shown ha ρ.5 for any MA(1) model, so he soluions will boh be real. The corresponding esimaes of b are bˆ = 1 2 1± 1 4r, 2r 1 and wo problems arise here. Firs, here is no guaranee ha 1 4r 1 2 > 0. Second, how do we decide which of he wo soluions o use? To answer his second quesion we mus define inveribiliy. An MA model is said o be inverible if i can be represened as (i.e., "invered o") a saionary infinie-order auoregression, AR ( ). Consider, for example, he MA(1) model x =ε +bε 1. If we consider his as a difference equaion for
7 -7- ε, we obain he soluion 2 k ε = x bx 1 + bx ( b ) x k If b > 1, an explosive series resuls and he curren ε canno be esimaed from pas x. Thus, o be useful for forecasing, he MA model mus be inverible. For he MA (q ) model, he inveribiliy condi- ion is ha he roo of larges magniude of he equaion z + bz b = 0 should have magniude less han one. q 1 q 1 q Reurning now o he issue of which soluion o choose for bˆ in he MA(1) case, i can be shown ha of he wo possible soluions, only one gives an inverible model. Esimaion for MA(q) models proceeds similarly. From he expressions for ρ,...,ρ, we obain a sysem of nonlinear equaion 1 q 1 q for he parameers b,...,b. This sysem will have many soluions, bu only one will give an inverible model. Compuer programs for fiing MA models will always choose his inverible model. Esimaion for mixed ARMA models proceeds by nonlinear mehods. The programs used will always choose a saionary, inverible model. The mehods jus described are hose given in Granger. All of hese exploi he he connecion beween he auocorrelaions and he parameers. In fac, here exis many oher esimaion echniques, including he very popular maximum likelihood mehod. This mehod assumes ha he innovaions are normally disribued, and hen explois his assumpion as fully as possible. Anoher popular mehod is leas squares, in which he sum of squared errors of he fied model (i.e., he sum of squares of he esimaed innovaions) is made as small as possible. Assuming normal innovaions he maximum likeli- hood and leas squares mehods are generally superior o he mehod described in Granger, paricularly when he model is near he nonsaionariy boundary (i.e., when he larges roo of he saionariy equa- ion has magniude close o 1). Diagnosic Checking Once a model has been idenified and esimaed, i is usually aken o he he rue model and forecass can be obained accordingly. As menioned earlier, i is virually cerain ha he esimaed model is no he rue model. To proec agains disasrous forecasing errors, he leas we can do is o
8 -8- check ha he fied model is a saisfacory one. This is done by he use of diagnosic checks.ifwe had a large amoun of daa, i would be feasible o break he daa ino wo pars, idenify and esimae he model on he firs par and check he qualiy of he forecass on he second par. This mehod, known as cross validaion, gives one of he few ways of obaining an hones esimae of forecasing error. Unforunaely, here is ypically no enough daa for cross-validaion o be used, so ha models are idenified, esimaed, and diagnosically checked on he same daa se. The mos commonly used mehod is o examine he correlogram of he residuals from he fied model o see if he residuals are a whie noise (as hey should be, if he model is correc). For example, he Box-Pierce es saisic is based on he sum of squares of he residual auocorrelaions. If his es saisic exceeds some criical value (found in a able), hen he model in quesion is declared o be inadequae. Unforunaely, his es is no very likely o flag inadequaely fiing models. Furhermore, even if a model is no found o be inadequae, he mehod provides no assessmen of he probable conribuion o forecas error due o he idenificaion and esimaion sages, and due o he difference beween he idenified and acual models.
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