Stationarity and Error Correction

Transcription

1 Saioariy ad Error Correcio. Saioariy a. If a ie series of a rado variable Y has a fiie σ Y ad σ Y,Y-s or deeds oly o he lag legh s (s > ), bu o o, he series is saioary, or iegraed of order - I(). The rocess ha geeraes a saioary series is ie-ivaria - σ σ -s. b. If he series coais a ie red such as α, i) ake his o-saioary series saioary by firs differecig. Y Y α α β ( ) β. ii) If a series is oliear, firs differece he logliearized series. iii) If o-saioariy ses fro quadraic red, secod differece he series. Y ( Y Y ) ( Y Y ) Y Y Y c. For a regressio odel Y α β i X i Y ε, however, i is o always ecessary o ake all Y ad X i s saioary, for hey ay o be iegraed of he sae order. I suffices o coiegrae he, so ha E[ ε ε ] (whie-oise). This saisfies OLS assuio, e e. Now, ε is saioary because X i s ad Y are coiegraed.. Saoariy & Rado Walk a. Box-Pierce (or Poraeau) Q-saiic for Whie-oise Q k k ˆ ρk ~ χ w / df, where ˆ ρk k e H : No auocorrelaio H : Ay kid of auocorrelaio e e b. Lug-Box Q-Saisic For sall sale LJB Q* ( ) k ˆ ρ k k ~ χ w / df

2 c. Dickey-Fuller Ui Roo Tes for Rado Walk (Differece Saioary Nosaioariy) Y α ρy u w/ a deeriisic ie red or drif Y β ρy u H : ρ Rado Walk ρi Y β Y u Dickey-Fuller Geeralized H : ρ Rado Walk Y Y β ( ρ ) Y u, where ρ δ ad u θi ε θi β δ Y ε Augeed Dickey-Fuller (ADF U ) H : δ Rado Walk i β θ ε Augeed Dickey-Fuller Resriced (ADF R ) H : β δ or β, ρ. Rado Walk ( ESSR ESSU ) /( Q k) F, where Q k i his case. ESS /( k) Use Dickey-Fuller or McKio τ c ad F c. d. Egel-Grager Tes (Dickey-Fuller o Residuals) u u ρ ε H : ρ Rado Walk U u ζ i u γ γ δ u υ AEG H : δ Rado Walk However, esiaed u is based o he esiaed coiegraig araeer β. Therefore, DF ad ADF τ c ad F c are o quie aroriae. Therefore, fid criical values i he followig referces: i) Ecooerica vol , 5-7 ii) Joural of Ecooerics vol iii) Log-ru Ecooic Relaioshi: Readigs i coiegraio, Oxford Uiv. Press, 99, Chaer.

3 3. Coiegraio a. Coiegraio Defied If X ad Y are RW s, a liear cobiaio of X ad Y would also be a RW. Ye, a aricular liear cobiaio of he X αy ay be saioary. Thus, X ad Y ay be each I(), bu X αy ay be I(). The, X ad Y are coiegraed. E.g.) SR & LR i s, P X &P Y, where X&Y are close subsiues, &w i relaed arkes. b. Tesig for Coiegraio i) Coiegraio Regressio DW Tes Give Y α X u DW ( uˆ uˆ uˆ I geeral u will be I() if X&Y are I(). If so, DW will be close o, ad he wo series are o coiegraed. Thus, es H : DW lack of coiegraioif DW is sigificaly >, he he wo series ay robably be coiegraed. However, he sadard DW able is o alicable, because he H : DW i he coiegraio es, o H : DW. ii) Augeed Dickey-Fuller Tes o Error Ters Ru coiegraio regressio firs ad obai residual: uˆ The, esiae uˆ β i uˆ φ uˆ ε. ) Y αˆ βˆ X The es saisic is -saisic for φ, bu he usual -disribuio is o aroriae. Refer o Egel & Grager ADF -saisic. Criical Values for Tesig Coiegraio Level of sigificace (%) DW saisic ADF -saisic

4 c. Esiaio i he Presece of Coiegraio Corresodig o a Pair of coiegraed variables, he error correcio odel would be y β ε x ( y β x ) ε β x uˆ i) Esiae he coiegraig regressio y β x u ad obai residuals uˆ y β x. ii) Esiae he above y β x uˆ ε. iii) Drawback: A saic odel ca yield biased esiaes. 4. Error Correcio Mechais (Model) a. Error Correcio Defied If here exiss a LR equilibriu relaioshi bewee he wo ecooic variables behid he SR disequilibriu, a roorio of he disequilibriu i oe eriod is correced i he ex eriod hrough Error Correcio. For exale, if i holds i he LR Y KX l Y l K l X, i will be rue ha y k x ( y k x ) y x i he LR. However, i he SR y β u lagged aduse. x x αy The SR odel will be cosise wih he LR odel oly uder he followig codiios: [( α ) y * β ( β ) x*] [ y k x ], where y y - y*, x x - x* ad u i he LR. For his o hold, α β, y * k * x*, where β k *. α Le α γ β. α γ ad γ β. The, y β u or x ( γ β) x ( γ ) y y y u, or β ( x x ) γ ( x y ) y β u, x γ ( x y ) where γ x y ) caures he SR disequilibriu aduse. Tes γ for his. ( A ore geeral secificaio of ECM will be as follows: y β u x γ x γ y x π π y θ x θ y υ 4

5 b. Error Correcio Coad o SAS Error Correcio by SAS PROC AUTOREG is o squeeze ou whaever uexloied relaio is sill here bewee X i s ad Y, which is rereseed by he resece of auocorrelaio ad heeroskedasiciy. The saee HETERO ca be used wih PROC AUTOREG o es for he resece of heeroskedasiciy, he H of which is o heeroskedasiciy. c. Grager Causaliy Tes If X Grager-causes Y, bu Y does o, he as values of X should be able o hel redic fuure values of Y, bu o he oher way aroud. q iy i Y α β X u Uresriced Model Y α Y υ Resriced Model i i H : β X does o Grager-cause Y. H a q : β ( ESS Wald F-Saaisic: F ESS U R ESSU ) / q /( q) A reecio of H idicaes ha X Grager-causes Y. A well-kow exale of causliy bewee ieres rae ad oey suly is as follows: α ir i r β u i i α β r υ d. Wald (F) Saisic for Secificaio, Liear Resricio ad/or Orhogoaliy W [C(θ) - q]'{var[c(θ) - q]} - [C(θ) - q] ~ χ /df, where Var[C(θ) - q] CVar[θ]C' ad C C(θ)/ θ' (Check Gree o see if he es is ea for liear resricio or GMM.) All righs reserved by Dr. S.M. Jeff Hog. Do o quoe or coy wihou roer aribuio. 5