Bayesian Approach for ARMA Process and Its Application

Transcription

1 Iteratioal Busiess Research October 008 Bayesia Aroach for ARMA Process ad Its Alicatio Chogju Fa Busiess School, Uiversity of Shaghai for Sciece ad Techology Shaghai 00093, Chia Sha Yao Busiess School, Uiversity of Shaghai for Sciece ad Techology Shaghai 00093, Chia The research is fiaced by the key research roject of Shaghai Muicial Educatio Commissio of Chia. No. 06ZZ34 (Sosorig iformatio) Abstract There have bee a lot of works relatig to time series aalysis. I this aer, the Bayesia aalysis method for ARMA model is discussed ad a alicatio examle is give. Firstly, the Bayesia theoretic results about AR model ad the determiatio aroach for model order are obtaied. The, the aroach are reseted for Bayesia aalysis of MA ad ARMA models. As its alicatio, the forecastig model for Shaghai real estate rice idex is set u ad the digital result shows well erformace. Keywords: Bayesia aalysis, ARMA, Ecoomic forecstig, Real estate rice idex. Itroductio The ast few decades have see the mai results of research o Bayesia time series aalysis method (Chogju Fa, 993). It has a short history. However, sice its secial sigificace of the method for fittig ecoomics data, research i this asect has see a vigorous develomet ad rogress very soo. The Bayesia aroach of ARMA model which is give by Broemelig ad Shaarawy (986) is articularly aroriate. With coditioal likelihood fuctio ad base o ormal-gamma rior distributio, they educed that both the osterior distributio of model arameters ad the oe-ste-ahead redictive distributio of AR model are t-distributio, ad the aly the results to MA ad ARMA model. Meawhile, they have show arithmetic to determiate the order umber of the models uder geeralized rior. I this aer, above methods are reorgaized, simlified ad exaded, articularly the roblem of the model order umber uder ormal - gamma rior distributio is discussed.. The results about AR model I this sectio, we cosider followig -th auto-regressive (AR) model Where, we assume that Y( t) is observatio at timet, ad Y ( t) Y ( t ) Y ( t ) e( t) (.) e(t), t (,, ) The ad are ukow arameter of the modelig. If we suose o observatio are available, deote Y () Y (), ad let S i 0,, are i.i.d. N(0, ). Let (Y(),,Y(i)) (Y(i ),, Y() ) S (-i) 49

2 Vol., No. 4 Iteratioal Busiess Research The the coditio desity of S (-) whe S is give is as follows: where f S S Yt Yt i ( ( ) / ;, ) ex{ ( ( ) i ( ) } i i ex{ ( S( ) X)( S( ) X )} (.) Y( ) Y() X Y ( ) Y ( ) We ca use this coditio likelihood fuctio istead of the exact likelihood fuctio whe is large relative to (Priestley M B., 98). I this sectio, the discussios are based o fuctio (.). I the model, we cosider the grou of cojugated rior distributio: ormal-gamma distributio. It ca be roved that the Bayesia aalysis method for AR model euates to the followig Bayesia multivariate regressive aalysis: S ( ) X E E ~ N(0, I) (.3) with the rior desity: ~ (, ) / (coditio rior desity of give ) ~ N mr, (,( ) ) The followig result ca be obtaied easily by the results of Bayesia multivariate regressive model: Theorem. Combiig (.) with (.4), It ca be obtaied the margial osterior desities of,, ad oe-ste-ahead redictive Y(+) are multivariate t-distributio, Gamma distributio ad uivariate t-distributio resectively, amely (.4) ~ t ( ˆ,, ) Q (.5) Where, ~ (, SSe) (.6) Y (+) ~ t Yˆ, D, ) (.7) ( ( ) X X X S ( ) (.8) SSe S S ( X S ) ( ) ( ) ( ) ( XX) ( XS( ) ) Q ( ) ( SSe)( X X ) L (Y,,Y ) () (-+) Yˆ ˆ ( ) L D L X X L SS e [ ( ) ]( ) ( ) (.9) 50

3 Iteratioal Busiess Research October 008 The followig four corollaries ca be obtaied easily from theorem, that is Corollary. With the suare loss, Bayes estimatio of is the weighted average of least suare estimator X X X S ad rior mea u, that is LS ( ) X X X X X X Proof: It ca be roved from euatios (.5) ad (.8) Corollary. With the suare loss, the Bayes estimatio of radom error variace, is LS ( ) ( SSe) (.0) Proof: It follows from (.6) that the margial osterior desities of is athwart Gamma distributio, the we ca obtai (.0) from the results about the exected value of athwart- Gamma distributio (See age 395 of Berger J. O., 980). Corollary 3. Oe-ste-ahead redictive Y (+) ca be divided to two arts as: ( ) ˆ (.) V L X X L where L( XX ) Lˆ is osterior variace of L. Proof: Combiig (.7) ad the results about variace of t distributio (See age 394 of Berger J. O., 980), we ca obtai the euatio V ( )[( ) ] D The we obtai this Corollary from (.9) ad (.0). Corollary 4. Whe suose =0, =0, =0, the euatios (.5), (.6) ad (.7) are the same with the results of geeralize rior desity (, ) i Jeffreys o-iformatio coditio. Euatio (.7) gives the distributio of oe-ste-ahead Y ( ). It ca be roved distributio of Y( is also uivariate ) t-distributio whe oe-ste-ahead Y( ) is give. But for the joit redictive distributio of Y ( ),, Y( ad sole k) multi-ste-ahead distributio Y ( k), we eed make use of digital algorithm. We will ot make further discussio i the aer. 3. The determiatio of model order umber Iferece o the order umber of the model, we eed use the followig results: Theorem. It follows with the result of Theorem, that () ˆ Q ( )~ t(0,, ) (3.) where () Piecemeal make the matrix above ˆ ˆ ˆ Q Q Q () () ˆ ˆ ˆ () () Q Q Q Q Q 5

4 Vol., No. 4 Iteratioal Busiess Research Where,, () ˆ are () order matrix, Q is order matrix resectively, ad. The the osterior distributio of whe () () 0 is give is as follows: 0~ t ( ˆ Q Q ˆ, () () () () ( ˆ Q ˆ ) () () ( Q Q Q Q ), ) (3.) Proof: It ca be obtaied from the roerty of multivariate t-distributio ad (.5). If the model order umber is, ot, the rior distributio (.4) ca be adjusted to exress whe 0 is give as follows: () / ~ N ( () (),( ) ) ~ (, () ( ) ()) (3.3) where u u u () () ad where u is (), is. Discussio about the order umber determiatio of model AR is as follows. It ca be show that there are two kids of recursio algorithms from Theorem about the osterior distributio of coefficiet arameter, which is <> We Suose that the real order umber of the model is ot bigger tha a iteger N. sice the osterior distributio of whe N is give, the we ca test by usig (3.). Suose H : 0 0, ad oosig H : 0, if accet H, it meas that the value of ca be desceded, so make 0 N, ad the adjust the rior distributio by (3.3). It ca be calculated recursively util the value of ca ot be desceded. <> Whe 0 is acceted, we ca use (3.) to take the ext test directly, the also by this way we ca determie the order umber fially. It ca be show the relatio betwee two kids of recursio algorithms by the followig Theorem 3. it is similar with the FPE rule of traditioal method. By usig the Corollary 3 to set u a ste of redictive error criterio to determiate order umber: Suose that N is bigger tha or eual to the model real order umber. Euatio (.) shows that we ca calculate the value of V whe =,, N, ad we also ca adjust to the rior distributio usig by (3.3). If the value of V is the smallest at ˆ, the we ca make ˆ as the estimator of. Theorem 3. If the order umber of AR model is, let Y( ),..., Y deote available observatios, with the rior ( ) distributio (3.3), the the osterior distributio of is (3.). () 4. The aalysis method of MA model Geerally -th movig average (MA) model ca be rereseted as Y() t et () et ( ) et ( ) (4.) 5

5 Iteratioal Busiess Research October 008 where we also assume that et (), t 0,,, areiid... N(0, ) corresodigly with likelihood fuctio L(, / S) ex Y( t) je( t j) i j (4.) If et () is give, the euatios (4.) ad (.) are the same i the form. The we kow all euitatio above is also suitable i the model. Relaced (,... ) by, by, S by S, Corresodigly we ca recast the model. For examle we ca obtai X e(0) e( ) e( ) e ( ) e ( ) e ( ) ad so o. Therefore, the key of aalysis method of MA model is how to assess the value ofet (). I Shaarawy ad Broemelig (984), the Box-Jekis aalysis method articularly aroriated. Estimatio ˆ ˆ * ( *,..., ) of ca be calculated firstly, the we ca get the estimatio by usig the followig formulas. et ˆ() Yt () et ( j) t,,..., et ˆ( ) 0 t0,,..., ˆ ˆ * j (4.3) j0 The i the cotext of the Bayesia aalysis aroach above we ca deduced the osterior distributios of ad Obviously, the aalysis aroach of MA model combied traditioal methods with Bayesia aroach ad the results are aroximately. A commo aroach assessmet for et () show by whe we see estimatio as observatios, There is, however, o how to get the estimatio of et () oly by Bayesia aalysis aroach is relatively little research ad usolved. 5. The aalysis method of ARMA model The aalysis method of ARMA model ARMA similar with the aalysis method of MA model.arma model ca be recast ito the followig Now firstly we also cosider about the likelihood fuctio of S Y() t Y( t ) Y( t ) e() t e( t) e( t) (5.) L(,, / S) ex Yt ( ) iyt ( i) jet ( j) i i j The difficulties are also how to getet (). Firstly we ca get ˆ ad ˆ * by Box-Jekis aalysis aroach, the from * followig euatio we ca estimate et ˆ( ): et ˆ( ) Yt ( ) ˆ ˆ ˆ * iyt ( i) * jet ( j) t,..., i j0 et ˆ( ) 0 t,,..., Therefore, It ca be obtaied the value by euatio (5.). Base o above euatio (5.), we ca obtai the osterior 53

6 Vol., No. 4 Iteratioal Busiess Research distributios of ad. Similarly Box-Jekis aalysis method hels to get the estimatio of et (). Ad we ca draw the coclusio from the discussio above that it is the oit to make well the model of MA. 6. A alicatio As we kow, ecoomical system is a comlex system ad i chage freuetly (Chogju Fa, 008). There is always ot eough history data for settig u the forecast model. Therefore, how to make use of Bayesia aroach to imrove the forecast recisio by the relevat data ad iformatio is a meaigful toic. Here we discuss the alicatio of Bayes aalysis aroach to the followig case. The geeral aalysis o real estate by usig rice idex has bee carried out for may years i Chia. The rice idex is comosed by the data from the market research. These data which kee track of the uotatios o the market all the times are come from differet estates ad become a dyamic grah used to observe the uotatio o the market. The uatitative research ad the recise descritio ad forecast o the orbit track of rice idex lay a major role i real estate research. This aer costructs the forecast modelig o Shaghai Comosite Idex of Chiese Restate from Jauary i 005 to May i 008. The data are deseasoized ad ormalized first. First, we get the rior distributio by usig Beijig Comosite Idex of Chiese Restate from October i 005 to March i 008, the set Shaghai model u. The calculatios: the order umber of the model is =3, ad we ca see Shaghai comosite idex of Chiese restate by Table as below. It is ca be see from table that error rate for oe ste forecastig of Aril, May ad Jue i the year 008 resectively is -0.49%, -0.34%, 0.09%. Better recisio ad well erformace ca be draw form results. (Isert Table here) 7. Coclusios This aer maily at studyig the Bayesia aroach for ARMA models ad illustratig the methods for forecastig Shaghai comosite idex of Chiese restate as a alicatio. Sice our isufficiet ecoomy data, it takes us difficulties to forecast just by settig u a model, however, i the view of a itroductio of a ew solutio thought which we ca use the rior distributed, it has commo sigificace. The alicatio results do show that the methods we roosed are effective. Ackowledgemets This research was suorted by the key research roject of Shaghai Muicial Educatio Commissio of Chia (No. 06ZZ34). Exresses the thaks. Refereces Broemelig, L. & Lad M. (984). O forecastig with uivariate autoregressive rocesses. Commuicatios i Statistics, 3. Broemelig, L. & Shaarawy. (986). A Bayesia aalysis of time series. Bayesia Iferece ad Decisio Techiues, Goel, P., Zeller, A. eds.. Berger J. O. (980). Statistical Decisio Theory. Sriger- Verlag, New York. Fa, Chogju. (993). The Bayesia methods i Time series aalysis. Joural of Xia statistical college,. Fa, Chogju & Hui Xu. (008). The Review o No-liear Aalysis ad Forecastig Methods of the Real Estate Market i Chia, Asia Social Sciece,. Hu, Zhag-mig. (007). Research o forecastig real estate rice idex based o eural etworks, Academic Joural of Zhogsha uiversity, 7 (): Priestley M B. (98). Sectral aalysis ad time series, Academic Press, Lodo. Shaarawy S. & Broemelig, L. (984). Bayesia ifereces ad forecasts with movig average rocesses. Commuicatios i Statistics, 3. Shaarawy S. & Broemelig, L. (985). Ifereces ad rodictio with ARMA rocesses. Commuicatios i Statistics, 4. Huimig Zhu, Yui Ha & Jicheg Zheg. (005). The Bayesia forecastig for AR () model based o ormal Gamma with cojugatio rior distributio, Statistics & decisio,. 54

7 Iteratioal Busiess Research October 008 Table. Predicted result of Shaghai comosite idex of Chiese restate Year/Moth Shaghai comosite idex of Chiese restate Oe ste forecastig Error rate for oe ste forecastig 08/ % 08/ % 08/ % 55