# 420Pr8 - Practice Problems 1 Consider the MA(2 process for...

This preview shows pages 1–4. Sign up to view the full content.

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

View Full Document

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Practice Problems 1. Consider the MA(2) process for which it is known that μ = 0, Y t = e t – θ 1 e t – 1 – θ 2 e t – 2 where { e t } is zero-mean white noise ( i.i.d. N ( 0, 2 e σ ) ). a) Find the expression for Var ( Y t ) = Cov ( Y t , Y t ), Cov ( Y t , Y t + 1 ), and Cov ( Y t , Y t + 2 ), and Cov ( Y t , Y t + 3 ) in terms of θ 1 , θ 2 , and 2 e σ . b) Find the expression for ρ 1 , ρ 2 , and ρ 3 in terms of θ 1 and θ 2 . 2. 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 where { e t } is zero-mean white noise ( i.i.d. N ( 0, 2 e σ ) ). Find the expression for ρ 1 and ρ 2 in terms of φ 1 and φ 2 . 3. Consider the MA(2) process for which it is known that μ = 0, Y t = e t – θ 1 e t – 1 – θ 2 e t – 2 where { e t } is zero-mean white noise ( i.i.d. N ( 0, 2 e σ ) ). Based on a series of length N = 6, we observe y 1 y 2 y 3 y 4 y 5 y 6 4.4 – 2.0 – 6.3 4.1 5.6 – 6.1 a) Using e = 0, e – 1 = 0, calculate S ( θ 1 , θ 2 ) = & = N t t e 1 2 for θ 1 = 0.3, θ 2 = 0.4. b) For θ 1 = 0.3, θ 2 = 0.4, forecast y 7 , y 8 , y 9 , and y 10 . c)* For θ 1 = 0.3, θ 2 = 0.4, given 2 ˆ e σ = 16.3, calculate 95% probability limits for y 7 , y 8 , y 9 , and y 10 . 4. Consider the AR ( 2 ) processes Y & t – 0.3 Y & t – 1 – 0.1 Y & t – 2 = e t where { e t } is zero-mean white noise ( i.i.d. N ( 0, 2 e σ ) ), Y & t = Y t – μ . a) Based on a series of length N = 100, we observe …, y 98 = 152, y 99 = 156, y 100 = 147, y = 150. Forecast y 101 and y 102 . b) Use Yule-Walker equations to find ρ 1 and ρ 2 . c) Is this process stationary? 5. Determine whether the following processes are stationary. a) Y t – Y t – 1 = e t – 0.8 e t – 1 b) Y t – 0.39 Y t – 2 – 0.16 Y t – 4 = e t – 0.8 e t – 1 c) Y t – 0.7 Y t – 1 – 0.3 Y t – 2 = e t + 0.5 e t – 1 d) Y t – 0.9 Y t – 1 – 0.9 Y t – 2 = e t – 1.4 e t – 1 6. Consider the AR(2) process Y t = μ + φ 1 ( Y t – 1 – μ ) + φ 2 ( Y t – 2 – μ ) + e t Based on a series of length N = 60, we observe …, y 59 = 190, y 60 = 215, y = 200. a) Suppose r 1 = 0.40, r 2 = – 0.26. Use Yule-Walker equations to estimate φ 1 and φ 2 . b) If φ 1 and φ 2 are equal to your answers to part (a), is this process stationary? c) Use your answers to part (a) to forecast y 61 , y 62 , and y 63 . 7. The following sample ACF and PACF are from 3 simulated stationary time series....
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 / 11

420Pr8 - Practice Problems 1 Consider the MA(2 process for...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online