12_06ans - STAT 420 Examples for Fall 2007 1 Consider the...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: STAT 420 Examples for 12/06/2007 Fall 2007 1. Consider the AR(2) process for which it is known that μ = 0, Y t = φ 1 Y t – 1 + φ 2 Y t – 2 + e t Based on a series of length N = 100, we observe y 99 = – 2, y 100 = 3. a) Suppose r 1 = – 0.50, r 2 = 0.55. Use the method of moments to estimate φ 1 and φ 2 . Yule-Walker equations for an AR(2) process: ρ 1 = φ 1 + φ 2 ρ 1 ρ 2 = φ 1 ρ 1 + φ 2 – 0.50 = φ 1 – 0.50 φ 2 × 2 – 1 = 2 φ 1 – φ 2 0.55 = – 0.50 φ 1 + φ 2 0.55 = – 0.50 φ 1 + φ 2 & – 0.45 = 1.5 φ 1 & 1 ˆ φ = – 0.30 & – 1 = 2 ( – 0.30 ) – φ 2 – 1 = – 0.60 – φ 2 & 2 ˆ φ = 0.40 b) Use your answers to part (a) to forecast y 101 , y 102 , and y 103 . Y N + 1 = μ + φ 1 ( Y N – μ ) + φ 2 ( Y N – 1 – μ ) + e N + 1 1 ˆ + N y = () 1 ˆ N y = E N ( Y N + 1 ) = μ + φ 1 ( y N – μ ) + φ 2 ( y N – 1 – μ ) Y N + 2 = μ + φ 1 ( Y N + 1 – μ ) + φ 2 ( Y N – μ ) + e N + 2 2 ˆ + N y = ( ) 2 ˆ N y = E N ( Y N + 2 ) = μ + φ 1 ( 1 ˆ + N y – μ ) + φ 2 ( y N – μ ) Y N + 3 = μ + φ 1 ( Y N + 2 – μ ) + φ 2 ( Y N + 1 – μ ) + e N + 3 3 ˆ + N y = ( ) 3 ˆ N y = E N ( Y N + 3 ) = μ + φ 1 ( 2 ˆ + N y – μ ) + φ 2 ( 1 ˆ + N y – μ ) … Y N +...
View Full Document

This note was uploaded on 04/29/2010 for the course STAT stat 420 taught by Professor Stepanov during the Spring '07 term at University of Illinois at Urbana–Champaign.

Page1 / 5

12_06ans - STAT 420 Examples for Fall 2007 1 Consider the...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online