Advanced time-series analysis (University of Lund, Economic History Department)

Transcription

1 Advanced me-seres analss (Unvers of Lund, Economc Hsor Dearmen) 3 Jan-3 Februar and 6-3 March Lecure 4 Economerc echnues for saonar seres : Unvarae sochasc models wh Box- Jenns mehodolog, smle forecasng echnues. 4.a. Box-Jenns modelng mehod: buldng blocs Mos of he deas behnd he Box-Jenns mehod have mlcl been nroduced n he earler lecures. The dea s ha all covarance saonar me-seres can be modeled as he sum of hree es of me-seres: whe nose, auoregressve (AR), and movng average (MA). Ths echnue s ofen seen as a heorecal, b whch s mean ha unle srucural models, he deenden varable s no modeled b a se of exogenous regressors, bu b s earler value. Le us loo a hese hree fundamenal buldng blocs! Whe-nose (WN): Whe nose has he roeres of he deal dsurbance (or error) erm. I has zero mean, s homoscedasc and exhbs no seral correlaon whasoever. Tme seres ε s a whenose f: E( ), Var( ) for all, and Cov(, ) f. The whe nose s urel random, conans no useful nformaon whasoever. B above defnon, a whe nose s saonar. The correlogram of a random sandard normal varable (a whe-nose) Movng average (MA): A movng average rocess of order euals a (an oonal) consan, whe nose erm, and he lnear combnaon of he lagged errors from - o -., WN(, ) an MA() rocess. j j j

2 The execed value of an MA() rocess s: j j E E E, snce hs does no deend on, hs condon fulflls he j reuremens of saonar. The varance of an MA() rocess s: j j j j rocess has a fne varance. The frs-order auocovarance of an MA() rocess s, ha s, as long as s fne or j Cov(, ) E E E E and zero for hgher orders. The ACF s:,, The PACF can be obaned b nverng he MA() o an AR( ) rocess: L ( L) ( L) o or or Where he coeffcens are he resecve PACF coeffcens Correlogram of he MA() rocess:.8 for all j, hs j Auoregressve (AR) model: An auoregressve rocess of order euals a (an oonal) consan, whe nose erm, and he lnear combnaon of he lagged deenden varable from - o -. (, ) an AR() rocess. WN

3 The execed value of an AR() rocess s: E E( ) ( ) s fne. The varance of an AR() rocess s:, rovded, he execed value of an AR() rocess, ha s, as long as, hs rocess has a fne osve varance. The auocovarance of an AR() rocess s easl obaned b nverng o an nfne MA rocess: ( L ) 3 3 ( ) (...) L L L L Fron hs s obvous ha he dnamc mullers are for all lags. The covarance s: The ACF s: ACF The PACF s obvous:, f PACF( ), f Correlogram of he AR() rocess:.7 3

4 Hgher order AR or MA rocesses: No as f ou ever needed hs o do b hand, ou can guess how he ACF or PACF of a hgher order ssem would loo b facorzaon: L L hs s an AR() rocess, whch can be facorzed no: L L L L Where λ and λ are he characersc roos we alread now. L L L (...) ( L L...)( L L...) L L... So acuall wha ou have learned for frs order rocesses wll do for hgher order ssems. Correlogram of he AR() rocess:.5. Auoregressve Movng Average (ARMA) model: A combnaon of he AR and MA rocesses: j Usng lag oeraor:, (, ) hs s an ARMA(,) model. WN ( L L... L ) ( L L... L ) or ( L) ( L) or ( L) ( L) Boh he ACF and he PACF are domnaed b he exonenal deca. Correlogram of he ARMA(,) rocess:.5. 4

5 4.b. Esmaon The rocess s smle: f we have a covarance saonar me seres, wll wor fne. Se.: Chec f our seres are saonar wh a un roo es. If our seres are saonar, go ahead. If no, hen ae frs dfference and chec for saonar agan. The ARMA models can onl be fed on covarance saonar seres! If our varable was non-saonar, he resulng Box- Jenns e model s referred o as ARIMA(,d,) where d denoes he order of negraon for he varable. Se : You wll need o esmae an nal model: hs reures ha ou loo a he correlogram of he saonar varable. Neverheless, wh real daa ou wll rarel be able o mmedael fnd he rgh model. Possbl ou ae a good guess based on he correlogram and esmae a frs model. Se 3: You chec f he assumons of he model are fulflled: hese reure wo hngs: a. ha he resdual s a whe nose rocess. You should loo a he resdual correlogram and esecall he Q es. You should go on wh modelng unl no seral correlaon s o be found n our resdual. b. Several oher models can lead o serall uncorrelaed resduals, bu ou need o fnd a arsmonous model (ha f whou unnecessar exlanaor varables). Ths ma reure ha ou f several smlar models and choose he one wh he bes nformaon crera (Schwarz or Aae). Ideall ou have a small orfolo of models ha all lead o serall uncorrelaed resdual, bu one of hem wll have he bes f. Tha s he rgh model. An ARIMA(,,) model can be wren wh lag oeraors as follows: ( L) L L 5

6 4.c. Wh all hs? You ma as ourself wh such an n economc sense aheorecal mosel s useful o ou? Well, here are several cases when ou need o rel on a Box-Jenns mehod:. I s good for forecasng, acuall for shor-run forecasng s almos unbeaable even b comlex srucural models.. Undersandng hs mehod hels ou o learn a lo abou me-seres. 3. Somemes ou do no wan o exlan he execed value of he varable bu ou are more neresed n s varance (volal): now an ARMA s a good frs se. See: ARCH, GARCH 4. You can use for olc evaluaon (nervenon analss). Ths means ha ou model a rocess and use dumm varables o caure he effec of some change or even. 6