Winter 2016, PSTAT 174/274: Homework # 6 Assigned Wednesday May 4. Due in lecture on Wendesday May 11. 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. Check the stationarity, causality, invertibility of the following model Xt = 5 + 0:75Xt 1 + Wt ; t = 0; 1; 2 : : : : Find the ...rst 4 - weights and the ...rst 4 - weights. (You may check your values by R). Calculate and sketch the ACF and PACF function for this model. Solution 2. (Book 3.11) Consider the MA(1) series Xt = Wt + #Wt 1 ; where Wt is white noise with variance 2 . (a) Derive the minimum mean-square error one-step forecast based on the in...nite past, and determine the mean-square error of this forecast. (b) (For 274 students only!) Let X n+1 be the truncated one-step-ahead forecast as given in (3.92). Show that E Xn+1 X n+1 2 = 2 1 + #2+2n Compare the result with (a), and indicate how well the in...nite approximation works in this case. . 1 3. The sunspot numbers {Xt , t =; 1; 2 : : : ; 100}, ...led as sunspots.dat, ...nd the mean b and sample autocovariances b (0), b (1), b (2), and b (3). Use these values to ...nd the Yule-Walker estimates of '1 , '2 , and 2 in the model Yt = '1 Yt 1 + '2 Yt 2 + Wt ; fWt g W N 0; 2 for the mean-corrected series Yt = Xt b, .t =; 1; 2 : : : ; 100. Assuming that the data really are a realization of an AR(2) process, ...nd 95% con...dence intervals for '1 , and '2 ; (see Property 3.8, p.Book p122.). Plot togehter and compare the theoretical (by the estimated parameters '1 , and '2 ) and estimated ACF and PACF, comment. 4. (Nothing needs to be handed in for question 4): 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. Conditional expectation 2. Complex numbers 3. Projection in Euclidean space say in R3 4. Book: Ch.: 3.5, 3.6 5. Inverse of a matrix, for instance inverse of a matrix by 2 2 and 3 3. 5. This problem also is required for students enrolled in PSTAT 274 ONLY For a causal AR(1) model, determine the general form of the m stept ahead forecast Xt+m and show E Xt+m t Xt+m 2 = '2m : 1 '2 21 Find the limit for this variance when m ! 1, comment. Solution 2