{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

tutorial_8 - STA 4005 Tutorial 8 Method of Moment...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
STA 4005 Tutorial 8 Method of Moment Estimation (MME): For AR(p) process, the Yule-Walker equations are: ρ k = φ 1 ρ k - 1 + φ 2 ρ k - 2 + · · · + φ p ρ k - p for k 1 Replace ρ k by r k and solve ˆ φ 1 ˆ φ 2 . . . ˆ φ p = 1 r 1 r 2 . . . r p - 1 r 1 1 r 1 . . . r p - 2 . . . . . . . . . r p - 1 r p - 2 . . . . . . 1 - 1 r 1 r 2 . . . r p Example: From a series { Z t } of length 100, we have computed r 1 =0.7, r 2 =0.45, r 3 =0.3, ¯ Z = 2, and a sample variance 5. We consider an AR(2) process with a constant term θ 0 : Z t = θ o + φ 1 Z t - 1 + φ 2 Z t - 2 + a t where a t WN (0 , σ 2 ), estimate φ 1 , φ 2 , θ 0 and σ 2 by MME. Sol: Z t - μ = φ 1 ( Z t - 1 - μ ) + φ 2 ( Z t - 2 - μ ) + a t Z t = θ 0 + φ 1 Z t - 1 + φ 2 Z t - 2 + a t where θ 0 = μ (1 - φ 1 - φ 2 ). By MME, ˆ φ 1 ˆ φ 2 = 1 r 1 r 1 1 - 1 r 1 r 2 = r 1 (1 - r 2 ) 1 - r 2 1 r 2 - r 2 1 1 - r 2 1 = 0 . 75 - 0 . 08 ˆ σ 2 = S 2 (1 - ˆ φ 1 r 1 - ˆ φ 2 r 2 ) = 2 . 555 ˆ θ 0 = ¯ Z (1 - ˆ φ 1 - ˆ φ 2 ) = 0 . 66 Example: Consider the AR(2) process Z t = φZ t - 1 + φ 2 Z t - 2 + a t where a t WN (0 , σ 2 ) 1
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
a . For what value of φ it is stationary? b . After observing 200 samples, the sample moments were computed:
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern