Winter 2016, PSTAT 174/274: Homework # 4 Assigned Thursday, January 28. Due in lecture on Thursday, February 4. All students in PSTAT 174/274 must answer questions 1 through 3 inclusive, and do the reading discussed in question 4. All PSTAT 274 students also must answer question 5. (Last Question is not required of PSTAT 174 students.) Clear, logical, and detailed working must be included to receive homework credit. 1. (a) Simulate a series of n = 500 Gaussian white noise observations as in Example 1.8 and compute the sample ACF, b(h), to lag 20. Compare the sample ACF you obtain to the actual ACF, (h). [Recall Example 1.19.] (b) Repeat part (a) using only n = 50. How does changing n aect the results?. 2. Identify the following models as ARMA(p; q) models (watch out for parameter redundancy), and determine whether they are causal and/or invertible, in case of either causality or invertibility and provide and plot the ...rst 25 values (by R) for the weights j and j . (a) Xt = 0:8Xt 1 0:15Xt 2 + Wt 0:3Wt 1 ; + Wt Wt 1 ; (b) Xt = Xt 1 0:5Xt 2 Bonus: provide a formula for the weights j and j . Compare your wheights (caculated by R) with the ones computed by R, (...rst 25). 1 3. Write each of the following models in terms of the backshift operator, and each corresponding polynomial (B) when writing each of these as Xt = (B)Zt . Label each answer with the corresponding letters. Which of these models are invertible? Answer yes/no, and explain brie y 2 why. Throughout the following, assume Zt W N (0; Z ). 1. Xt = Zt + :2Zt 1 = (B)Zt ; 2. Xt = Zt 2Zt 3. Xt = Zt 0:7Zt 1 + 0:3Zt 2 = (B)Zt ; 4. Xt = Zt 0:2Zt 1 0:7Zt 2 = (B)Zt ; 5. Xt = 1:2Xt 6. Xt = 1 1:2Xt 1 = (B)Zt ; + Zt , 1 + Zt , 2 4. (Nothing needs to be handed in for ??: question 5): Continue to review your introduction to probability and mathematical statistics material (at a level equivalent to PSTAT 120AB). It will help you later if you review these topics now. In particular: 1. Random variables, Independence, Expected value, Variance 2. Covariance, correlation 3. Test simple hypothesis 4. Multivariate Normal (Gaussian) distribution, more detailed when the dimension is 2. (indendence and uncorrelated) Moreover: a) Vectors in R3 , innerproduct, orthogonality, projection, matrixproduct, inverting matrices. b) Linear regression, linear model c) Geometric series d) Roots of a polynomial of order 2 Roots of a polynomial of order p. e) Complex numbers, unit circle 5. This problem also is required for students enrolled in PSTAT 274 ONLY Show that the autoregressive equations Xt = 'Xt 1 + Zt ; t = 0; 1; 2 : : : where fZt g W N (0; 2 ) and j'j = 1j have no stationary solution. HINT: Suppose there does exist a stationary solution fXt g and use the autoregressive equation to derive an expression for the variance of Xt 'n+1 Xt n 1 that contradicts the stationarity assumption. 3