This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: STAT 443: Assignment 2 This assignment is to be handed in at the start of your lecture of Friday 2nd March . Please make sure that you write your name, student number and section number on your script. This material covers the slides on stationary processes and regression. 1 Stationary processes 1. Show the process { X t } defined by X t = Z t + C ( Z t 1 + Z t 2 + ) where C is a constant, Z t WN (0 , 2 ), is nonstationary. Also show that the series of first differences Y t = X t X t 1 is stationary and compute its acf. 2. Assume that { X t } is a stationary time series, with mean and ACVF ( h ). Let P n X n + h be the optimal (in terms of MSE) linear prediction of X n + h based on X 1 , ,X n . (a) Show explicitly how to construct P n X n + h in the case n = 2 and h = 2 (b) In the case n = 2 and h = 2, show that P n X n + h is an unbiased estimator i.e. E ( X n + h P n X n + h ) = 0 (c) In the case n = 2 and h = 2, show that P n X n + h is uncorrelated with X j i.e. E [( X n + h P n X n + h )( X j )] = 0 j = 1 , 2. 3. Consider the stationary solution to X t X t 1 = Z t + Z t 1 where Z t WN (0 , 2 ) (a) show by direct substitution that X t = Z t + ( + ) X j =1 j 1 Z t j is a solution. 1 (b) Show that when   < 1 lim N N X j =1  j 1  < . (c) Compute Cov ( X n ,X n +2 ). 4. (a) Find the ACVF of the time series Y t = Z t + 0 . 3 Z t 1 . 4 Z t 2 where Z t WN (0 , 1) (b) For this time series find P 1 Y 2 and its mean square error 5. Suppose that Z t is an iid sequence with Z t N (0 , 1). Consider the process defined by...
View Full
Document
 Winter '09
 YuliaGel
 Forecasting

Click to edit the document details