Chapter 5.docx

12 Pages
71 Views
Unlock Document

Department
Statistical Sciences
Course
STA457H1
Professor
Zhou Zhou
Semester
Winter

Description
Chapter 5 5.1 – Preliminary Estimation 5.1.1 – Yule­Walker Estimation AR(p) X tϕ X1 t−1…−ϕ X p t−p=Z (t) 2 Z tN 0,σ( ) Multiply  X t−1, Xt−2,…, X t−p  to  (¿)  and take expectation E [ Xt t−i]−ϕ E1X [ t−1 Xt−i]…−ϕ E X p [ t−pX t−i]E Z [ t t−i i=1,…,p γi−ϕ γ1 i−1−…−ϕ γ p i−p =0 γi=ϕ γ1 i−1+…ϕ γ p i−p ¿∗¿ ́ γp=Γ ϕp¿ p γ 1 ́p= ⋮ () γ p p Γ p γ[ i−ji,j=1 ¿∗¿ ́ Γ ́ ¿  implies that we can use the Method of Moments and replace  p  and  p  by  p  and  Γ p  and obtain the Yule­Walker estimation: ̂ ϕ pΓ γ p1́p Multiply  X t  to  (¿)  and take expectation 2 γ 0 −ϕ γ 1…1ϕ γ =E Z p =p Z ϕ[X t t] [ t( 1 t−1+…+ϕ X p t−p +Z t)] σ =γ 0 −ϕ γ ́T ́ p p Use the Method of Moments,  ̂ 2 ́T ̂ σ =γ 0 −ϕ γ ṕp Theorem: Simple Yule­Walker Estimation ́ −1 ̂ ϕ pR ̂p ρp σ =γ 0 1−( R ρ́ p̂ p1́ p) R p Here   is the autocorrelation matrix R = ̂ ρ p p [ i−ji, j=1 ρ 1 ρ p ⋮ () ρ p Proof: HW Proposition:  σ Pσ asn→∞ → Proof: not required Theorem: For large  n , ́ ́ 2 −1 √ ( ϕ pϕ D p)0,σ Γ( p ) → Remark: For AR(p) models, the Yule­Walker estimates are as efficient as the Maximum Likelihood Estimates. ́ −1 ϕp=Γ γp ́p Note: 1n−i ̂i= ∑ (X jX X)( j+1X́ ) nj=1 i=0,…, p Note: E X =0 [ i X ≈0 n−i n−i n−i n−i n−i ̂i≈1 ∑ X j j+1 1 ∑ X j( j+i−1 1…+X j+i−p pZ j+i= 1∑ Xj(́ j+i pZ i+j 1∑ X j ϕj+p 1 ∑ X j i+ j n j=1 n j=1 n j=1 nj=1 n j=1 n ̂ ̂ ́ 1 ́p≈Γ p +p ∑ X k k nk=p+1 X i−1 Xi= ⋮ (X) i−p n ϕ −ϕ ≈ Γ ̂−1 1 X Z p p p √ n k=p+1 k k Now 2 steps: (1) Show  1 n ∑ X k k N 0,( Γ 2 p) √n k=p+1 → (2) Show ΓpPΓ p → Then Slutsky’s Theorem implies  −1 1 n −1 2 −1 2 −1 −1 2 −1 Γ p ∑ X k DkΓ N p,σ ( Γ =N 0p) p Γ Γ ( p p p )N 0,( Γ p ) √ nk=p+1 → (2) By the weak law of large numbers, for each  n ̂ = 1 X X PE X X =γ k n j=k+1 j j−k→ [ j j−k] k (1) Apply the CLT of STA457 Y =X Z Note:  k k k  is a mean­zero stationary sequence Z E Z =0 2 Lemma: If  { i  are iid  [ i  and  Var [ ]i 1.96 √ 0.04/√100 ∴  Reject H0 5.1.3 – Innovations Algorithm MA(q) Theorem: The fitted innovations MA(m) is  X tZ +t Zm1 t−1+…+θ ̂mmZ t−m ZtWN 0,̂( vm) θ́ ̂ Here   m  and   m  are obtained by the innovations algorithm with ACVF replaced by sample ACVF 3 m m→∞ m →0 Choose   such that   with  n X MA(q) Note: Suppose  t θ θ ,…,θ One cannot just calculate  qi ’s as estimates of 1 q ̂ θqiθ i Right way: Select a relatively large   and find  θm1…,θ mm  and  vm 2 Set   θ1=θ m1,θ =θ qσ =mq ̂ ̂m 5.1.4 – Hannan­Rissanen Algorithm ARMA(p,q) Step 1:  A high order AR(m) model is fitted to data ( )  ̂ ̂ ̂ Then obtain the residuals  X tϕ Xm1 t−1…−ϕ mmX t−m ,  t=m+1,m+2,…,n Z ≈ Z Rationale:   t t Step 2: Once   Zt  are obtained Then fit the ordinary least squares n s(β)= ∑ ( −ϕ X
More Less

Related notes for STA457H1

Log In


OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.


Submit